**Unformatted text preview: **MAGNETIC RESONANCE IMAGING: PHYSICAL PRINCIPLES AND SEQUENCE DESIGN Professor of Radiology, Electrical Engineering and Biomedical Engineering Washington University, St. Louis, MO Professor of Physics Case Western Reserve University, Cleveland, OH E. Mark Haacke Institute Professor of Physics Case Western Reserve University, Cleveland, OH Robert W. Brown Michael R. Thompson
Picker International Highland Heights, OH Instructor, Department of Radiology Washington University, St. Louis, MO Ramesh Venkatesan This book is dedicated to our parents: Helena Doris Haacke Ewart Mortimer Haacke William James Brown Florence Elizabeth Brown Robert Thompson Mary Christina Thompson Ramasubramaniam Venkatesan Saroja Venkatesan v vi Foreword
Je rey L. Duerk I heard my rst lecture on an emerging eld in medical imaging known as Nuclear Magnetic Resonance Imaging in 1983 as an electrical engineering graduate student at The Ohio State University. I was captivated and soon moved to Cleveland, a city then considered by many to be a United States' center for the development of MR imaging and where both Picker International and Technicare were located a few miles apart. After studying many manuscripts, books and `primers,' I enrolled in a new Physics and Biomedical Engineering course at Case Western Reserve University denoted by PHYS/EBME 431: The Physics of Medical Imaging, taught by Prof. E. Mark Haacke. In large part, the present book has grown and evolved from the class notes and lectures from this course's o ering over the years. The power of Magnetic Resonance Imaging (MRI) in the diagnostic arena of patient care is unquestionable. A multitude of books exist to assist in the training/teaching of clinicians responsible for interpreting MR images. Since joining the faculty of Case Western Reserve University almost a decade ago, I have been asked by graduate students, new industry hires, and fellow professors (from both CWRU and institutions throughout the world), if there was a book I could recommend which would provide su cient depth in physics and MR imaging principles to serve as either a textbook or a complete tutorial for basic scientists. In my opinion, there were none which could provide the basic scientist with the tools to understand well the physics of MRI and also understand the engineering challenges necessary to develop the actual acquisitions (known as pulse sequences) which ultimately lead to the images. While there were no single sources available, I implored all to be patient. Today, I believe that patience has been rewarded as the book has arrived. While much has changed in the eld since my introduction to it in the early 1980's (e.g., the `N' in NMR Imaging and the company Technicare are both gone), the power of this book is that many central concepts in MRI are rather more permanent, and their coverage here is superb. The in uence by such notable predecessors as Abragam, Slichter, and Ernst is, at times, unmistakable. Mostly, the personal descriptive and analytical teaching style of Drs. Haacke, Brown, Thompson and Venkatesan builds an understanding of new concepts while clarifying old ones from the solid foundation provided earlier in the text. Another particular advantage of this book is that the notation is consistent, and located in a single reference the readers do not have to overcome notational di erences among our predecessors or di culties in separating fundamental concepts from advanced material. Importantly, virtually every homework problem in the text has been designed to emphasize a central concept crucial to MRI. When I page through the book, I am often able to nd the same derivations in the homework questions as in my log-books from the early part of my career in MRI. Insights from the authors are present throughout the text as well as within the problems they provide those vii viii less experienced with glimpses (which later become illuminating ashes (no pun intended)) into how MR physics and sequences work and how they can be taken advantage of in the application to new ideas. While I have been a co-instructor for EBME/PHYS 431 for a number of years and have used drafts of this book as the textbook, by now it bears little resemblance to the class notes of the initial o ering in 1985. For that matter, the eld of MRI has exploded with new techniques, new applications and far greater understanding and analysis of the innumerable aspects of the MRI hardware and software on image quality. I have bene ted from my long friendships with Drs. Haacke and Brown, and the more recent ones with Drs. Thompson and Venkatesan. If you were to walk into the CWRU MRI Laboratory today, you would nd no less than ve drafts of this book on the shelves. The greatest tribute to these authors in their e orts to compile an important comprehensive treatise on the physics of MRI and MR sequence design can be heard in our research group's discussions of new imaging techniques (and likely those in the future at other institutions world-wide) when someone beckons `Grab `Haacke and Brown'!' Je rey L. Duerk, Ph.D. Director, Physics Research Associate Professor-Departments of Radiology and Biomedical Engineering Case Western Reserve University Cleveland, OH January 20, 1999 ix Felix W. Wehrli Haacke et al.'s new book spans a signi cant portion of proton MRI concerned with the design of MRI pulse sequences and image phenomenology. The work, designed for the physicist and engineer, is organized in twenty-seven chapters. The rst eight chapters deal with the fundamentals of nuclear magnetic resonance, most of which is based on the classical Bloch formalism, except for a two chapter excursion into quantum mechanics. This portion of the book covers the basic NMR phenomena, and the concepts of signal detection and data acquisition. Chapters 9 and 10 introduce the spatial encoding principles, beginning with onedimensional Fourier imaging and its logical extension to a second and third spatial dimension. Chapter 11 treats continuous and discrete Fourier transforms, followed in Chapters 12 and 13 by sampling principles, ltering and a discussion of resolution. Chapter 14 may be regarded as the opening section of the book's second part exploring more advanced concepts, beginning with treatment of non-Cartesian imaging and reconstruction. In chapter 15, the properties of signal-to-noise are dealt with in detail including a discussion of the important scaling laws, followed, in chapter 16 by a return to a more advanced treatment of rf pulses, along with such concepts as spatially varying rf excitation and spin-tagging. Chapter 17 is dedicated to the various currently practiced methods for water-fat separation, and in chapters 18 and 19, the authors delve into the ever-growing area of fast imaging techniques. Chapter 18 is entirely dedicated to steady-state gradient-echo imaging methods to which the authors have themselves contributed a great deal since the inception of whole-body MRI. Chapters 19 and 20 address echo train methods focusing on EPI, T2 dephasing e ects and the resulting artifacts, ranging from intravoxel phase dispersion to spatial distortion. Chapter 21 is a brief introduction to the physics underlying di usion-weighted imaging and pertinent measurement techniques. Chapter 22 treats the quanti cation of the fundamental intrinsic parameters, spin density, T1 and T2 . Chapters 23 and 24 deal with the manifestions of motion and ow in terms of the resulting artifacts and their remedies, followed by a broad coverage of the major angiographic and ow quanti cation methods. The topic of chapter 25 is induced magnetism and its various manifestations, including a discussion of its most signi cant application | brain functional MRI exploiting the BOLD phenomenon. In chapter 26, the authors return to pulse sequence design, reviewing the design criteria for the most important pulse sequences and discussing potential artifacts. The nal chapter 27, at last, discusses hardware in terms of magnets, rf coils and gradients. This book is the result of a monumental ve-year e ort by Dr. Haacke and his coauthors to generate a high-level, comprehensive graduate and post-graduate level didactic text on the physics and engineering aspects of MRI. The work clearly targets the methodology of bulk proton imaging, deliberately ignoring chemical shift resolved imaging or treatment of biophysical aspects such as the mechanisms of relaxation in tissues. Understanding the book requires college-level vector calculus. However, many of the basic tools, such as Fourier transforms and the fundamentals of electromagnetism, are elaborated upon either in dedicated chapters or appendices. The problems interspersed in the text of all chapters are a major asset and will be appreciated by student and teacher alike. There is no doubt that the authors have succeeded in their e ort to create a textbook that nally lls a need which has persisted for years. Haacke et al.'s book is, in the reviewer's assessment, the most authoritative new text on the subject, likely to become an essential x tool for anyone actively working on MRI data acquisition and reconstruction techniques, but also for those with a desire to understand MR at a more than super cial level. The work is a rare synthesis of the authors' grasp of the subject, and their extensive practical experience, which they share with the reader through exceptional didactic skill. The book has few aws worth mention at all. First, not all chapters provide equal coverage of a targeted topic in that the book often emphasizes areas in which the authors have excelled themselves and thus are particularly experienced. Such a personal slant, of course, is very much in the nature of a treatise written by a single group of authors. On the other hand, the coherence in terms of depth of treatment, quality of illustrations and style, o ered here, is never achievable with edited books. A case in point of author-weighted subject treatment is fast imaging, which is heavy on steady-state imaging. The following chapter on echo-train imaging almost exclusively deals with EPI and only secondarily with RARE and its various embodiments. Likewise, di usion is treated only at its most fundamental level with little mention of anisotropic or restricted di usion, or di usion tensor imaging. Though the suggested reading list is helpful, a division into historic articles and those more easily accessible to the student would have been helpful since many of the historic papers cited would have to be retrieved from the library's storage rooms provided they are available at all. Finally, an introduction to the imaging hardware earlier (rather than as the last chapter) would help the novice bridging the gap between theory and instrumentation. None of the above, however, should detract from the book's high quality and practical usefulness. In summary, the authors need to be congratulated on a superb product a text vital to those concerned with MRI at a rigorous level. Magnetic Resonance Imaging: Physical Principles and Sequence Design is likely to become the daily companion of the MRI scientist and a reference standard for years to come. Felix W. Wehrli, Ph.D. Professor of Radiologic Science and Biophysics Editor-in-Chief, Magnetic Resonance in Medicine February 9, 1999 Preface
The principal motivation for this book is to create a self-contained text that could be used to teach the basics of magnetic resonance imaging to both graduate students and advanced undergraduate students. Although this is not a complete research treatise on MRI, it may also serve as a useful reference text for those experienced in the eld. Time and page limitations have made it impossible to include detailed discussions of exchange processes, rf penetration, k-t space, perfusion, and parametric reconstruction methods, to name a few important topics omitted. MR simulations, interactive MRI, and distance learning are other important issues that may be addressed in an expanded web-based companion volume in the future. We hope that the present text is a useful complement to the many technical details available on coil concepts in the MR technology book by Chen and Hoult and on di usion in the microscopic imaging book by Callaghan. To varying degrees, the chapters contain discussions of the technical details, homework problems, sequence concepts, and the resulting images. Key points are often highlighted by italicized text and single quotation marks usually signify the introduction of MR nomenclature or stylized language. Representative references appear at the end of each chapter, but only general review or introductory articles, or selected papers with which we are especially familiar, are referenced. It is beyond the scope of this book to make any attempt to present a complete bibliography. The rst fteen chapters of the text are introductory in nature and could perhaps serve as a one-semester course. After the brief preview given in chapter one, they wend their way from the basic dynamics of nuclear magnetic moments, to the concepts of imaging, and later to the e ects of reconstruction type, contrast and noise. The next eleven chapters represent the bulk of the imaging applications addressed they could either be covered in a second semester or the basic concepts of each could be interspersed with those of the earlier chapters comprising a faster paced single-semester course (which has been our tendency). The eleven chapters begin with brief excursions into the areas of rf pulse design and chemical shift imaging, and are followed by detailed discussions on fast imaging, magnetic eld inhomogeneity e ects, motion, ow, di usion, sequence design and artifacts. A uni ed discussion of the rf, gradient and main magnet coils is contained in the last chapter. Alternatively, we do nd appealing the assimilation of coil hardware issues with material in earlier chapters where appropriate. The appendices contain review material for basic electromagnetism and statistics as well as a list of acquisition parameters for the images in the book. xi xii Acknowledgments
Much like all major technologies, the development of magnetic resonance imaging has been a step-by-step process, building over many years on the ideas and experience of innumerable researchers in the eld. The development of this book has itself been based not only on many years of our teaching magnetic resonance imaging (we are well into our second decade), but also on the e orts of numerous colleagues and collaborators, both faculty and students. Sometimes, a particular discussion or imaging result or insight has been directly due to the e orts of a single M.S. or Ph.D. student. To all those who have directly or indirectly contributed, we are indebted. Speci cally, we thank the following people for reading di erent parts of the text and reviewing speci c chapters: Gabriele Beck, Andreas Brenner, James Brookeman, Mark Conradi, Lawrence Crooks, Thomas Dixon, Je Duerk, Jens Frahm, Jurgen Hennig, Christopher Hess, Steve Izen, Permi Jhooti, Stephan Kannengeiser, Peter Kingsley, Uwe Klose, Zhi-Pei Liang, Michal Lijowski, Robert Ogg, John Pauly, Jean Tkach, Yi Wang, Felix Wehrli, Robert Weisko , Eric Wong and Ian Young. We also owe our gratitude to the following people for assisting in either the collection or processing of the data for images shown in the text: Azim Celik, Xiaoping Ding, Karthikeyan Kuppusamy, Debiao Li, Weili Lin, Yi Wang and Yingjian Yu. Special thanks are also due to Peter Kingsley for reading many of the early versions of numerous chapters, to Marinus T. Vlaardingerbroek and Jacques den Boer for incorporating some of the nomenclature into their own book on MRI, and particularly to Norman Cheng for his participation in the entire process of scrutinizing the text, solving problems, and facilitating the nal editing of the book. The following students and fellow researchers need special mention for their feedback and corrections on several speci c aspects of di erent chapters: Hongyu An, Markus Barth, Hiro Fujita, Pilar Herrero, Frank Hoogenraad, Renate Jerecic, Shantanu Kaushikkar, Weigi Kong, Song Lai, Jingzhi Liu, Jurgen Reichenbach, Jacob Willig, Yingbiao Xu and Anne Marie Yunker. Finally, our thanks to Jan Lindley for her secretarial support during the last two years of this project. During our own tenure in this eld, we have personally bene ted from our involvement with both Siemens Medical Systems (Erlangen, Germany) and Picker International (Cleveland, Ohio). Many of the imaging methods have been developed thanks to a collaborative agreement with Siemens. All images presented in this text were acquired with a 1.5 T Siemens VISION scanner. We would like to thank the following people in Siemens for their support over the years: Richard Hausmann, Randall Kroeker, Gerhard Laub, Gerald Lenz, Wilfried Loe er, Hermann Requardt, Erich Reinhardt and Franz Schmitt. We would also like to thank Gordon DeMeester, Surya Mohapatra, Michael Morich and John Patrick at Picker for longstanding support and collaboration. xiii xiv Although we have played the role of teacher in giving this course, we have bene ted from the entire educational experience of preparing this book. We are grateful to the many students and colleagues who have taught us in this process over and above those mentioned so far. These include: Michael Martens, Todd Parrish, Cynthia Paschal, Labros Petropoulos, Shmaryu Shvartsman, Jean Tkach, Piotr Weilopolski and Fredy Zypman. The mistakes that remain are, of course, our responsibility alone. For these errors, we apologize in advance. We invite you to share your thoughts and to provide suggestions for improvements in the text. In this way, we can establish an updated list of corrections and additions, taking full advantage of the exciting new manner in which educational issues, open problems, and databases in MRI may now be addressed via the internet. On a more personal note, we are most happy to nally have this chance to thank our families and friends who have supported and sustained the writing e orts over the last ve years. Their patience and encouragement were crucial to the book's completion. Contents
1 Magnetic Resonance Imaging: A Preview
1.1 Magnetic Resonance Imaging: The Name . . . . . . . . . . . . . . . . . . . 1.2 The Origin of Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . 1.3 A Brief Overview of MRI Concepts . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Fundamental Interaction of a Proton Spin with the Magnetic Field 1.3.2 Equilibrium Alignment of Spin . . . . . . . . . . . . . . . . . . . . 1.3.3 Detecting the Magnetization of the System . . . . . . . . . . . . . . 1.3.4 Magnetic Resonance Spectroscopy . . . . . . . . . . . . . . . . . . . 1.3.5 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . 1.3.6 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.7 Resolution and Contrast . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8 Magnetic Field Strength . . . . . . . . . . . . . . . . . . . . . . . . 1.3.9 Key Developments in Magnetic Resonance . . . . . . . . . . . . . . 1.4 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Magnetic Moment in the Presence of a Magnetic Field 2.1.1 Torque on a Current Loop in a Magnetic Field . 2.1.2 Magnet Toy Model . . . . . . . . . . . . . . . . 2.2 Magnetic Moment with Spin: Equation of Motion . . . 2.2.1 Torque and Angular Momentum . . . . . . . . . 2.2.2 Angular Momentum of the Proton . . . . . . . . 2.2.3 Electrons and Other Elements . . . . . . . . . . 2.2.4 Equation of Motion . . . . . . . . . . . . . . . . 2.3 Precession Solution: Phase . . . . . . . . . . . . . . . . 2.3.1 Precession via the Gyroscope Analogy . . . . . 2.3.2 Geometrical Representation . . . . . . . . . . . 2.3.3 Cartesian Representation . . . . . . . . . . . . . 2.3.4 Matrix Representation . . . . . . . . . . . . . . 2.3.5 Complex Representations and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 . 3 . 3 . 4 . 5 . 7 . 7 . 8 . 9 . 10 . 10 . 13 . . . . . . . . . . . . . . 1 2 Classical Response of a Single Nucleus to a Magnetic Field 17 18 18 22 23 23 24 25 26 27 27 28 30 32 32 3 Rotating Reference Frames and Resonance 3.1 Rotating Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 The Rotating Frame for an RF Field . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 xv 35 xvi 3.2.2 Quadrature . . . . . . . . . . . . . . 3.3 Resonance Condition and the RF Pulse . . . 3.3.1 Flip-Angle Formula and Illustration . 3.3.2 RF Solutions . . . . . . . . . . . . . 3.3.3 Di erent Polarization Bases . . . . . 3.3.4 Laboratory Angle of Precession . . . 4.1 4.2 4.3 4.4 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 42 43 43 45 47 4 Magnetization, Relaxation and the Bloch Equation
Magnetization Vector . . . . . . . . . . . . . . Spin-Lattice Interaction and Regrowth Solution Spin-Spin Interaction and Transverse Decay . . Bloch Equation and Static-Field Solutions . . . The Combination of Static and RF Fields . . . ~ 4.5.1 Bloch Equation for Bext = B0z + B1 x0 . ^ ^ 4.5.2 Short-Lived RF Pulses . . . . . . . . . . 4.5.3 Long-Lived RF Pulses . . . . . . . . . . 51 51 52 56 58 60 60 61 62 66 70 71 72 74 75 78 5 The Quantum Mechanical Basis of Precession and Excitation
5.1 Discrete Angular Momentum and Energy . . . . . . . . 5.2 Quantum Operators and the Schrodinger Equation . . 5.2.1 Wave Functions . . . . . . . . . . . . . . . . . . 5.2.2 Momentum and Angular Momentum Operators 5.2.3 Spin Solutions for Constant Fields . . . . . . . . 5.3 Quantum Derivation of Precession . . . . . . . . . . . . 5.4 Quantum Derivation of RF Spin Tipping . . . . . . . . 65 6 The Quantum Mechanical Basis of Thermal Equilibrium and Longitudinal Relaxation 83 6.1 Boltzmann Equilibrium Values . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Quantum Basis of Longitudinal Relaxation . . . . . . . . . . . . . . . . . . . 87 6.3 The RF Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.1 Faraday Induction . . . . . . . . . . . . . . . . . . 7.2 The MRI Signal and the Principle of Reciprocity . 7.3 Signal from Precessing Magnetization . . . . . . . 7.3.1 General Expression . . . . . . . . . . . . . . 7.3.2 Spatial Independence . . . . . . . . . . . . . 7.3.3 Signal Demodulation . . . . . . . . . . . . . 7.3.4 Dependent Channels and Independent Coils 7.4 Dependence on System Parameters . . . . . . . . . 7.4.1 Homogeneous Limit . . . . . . . . . . . . . . 7.4.2 Relative Signal Strength . . . . . . . . . . . 7.4.3 Radiofrequency Field E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 97 99 99 101 102 105 105 106 107 108 7 Signal Detection Concepts 93 Contents 8.1 Free Induction Decay and T2 . . . . . . . . . . . . . . . . . . . . 8.1.1 FID Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Phase Behavior and Phase Conventions . . . . . . . . . . . 8.1.3 T2 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 The FID Sequence Diagram and Sampling . . . . . . . . . 8.2 The Spin Echo and T2 Measurements . . . . . . . . . . . . . . . . 8.2.1 The Spin Echo Method . . . . . . . . . . . . . . . . . . . . 8.2.2 Spin Echo Envelopes . . . . . . . . . . . . . . . . . . . . . 8.2.3 Limitations of the Spin Echo . . . . . . . . . . . . . . . . . 8.2.4 Spin Echo Sampling . . . . . . . . . . . . . . . . . . . . . 8.2.5 Multiple Spin Echo Experiments . . . . . . . . . . . . . . . 8.3 Repeated RF Pulse Structures . . . . . . . . . . . . . . . . . . . 8.3.1 The FID Signal from Repeated RF Pulse Structures . . . . 8.3.2 The Spin Echo Signal from Repeated RF Pulse Structures 8.4 Inversion Recovery and T1 Measurements . . . . . . . . . . . . . 8.4.1 T1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Repeated Inversion Recovery . . . . . . . . . . . . . . . . . 8.5 Spectroscopy and Chemical Shift . . . . . . . . . . . . . . . . . . 9.1 Signal and E ective Spin Density . . . . . . . . . . . . . . . 9.1.1 Complex Demodulated Signal . . . . . . . . . . . . . 9.1.2 Magnetization and E ective Spin Density . . . . . . . 9.2 Frequency Encoding and the Fourier Transform . . . . . . . 9.2.1 Frequency Encoding of the Spin Position . . . . . . . 9.2.2 The 1D Imaging Equation and the Fourier Transform 9.2.3 The Coverage of k-Space . . . . . . . . . . . . . . . . 9.2.4 Rect and Sinc Functions . . . . . . . . . . . . . . . . 9.3 Simple Two-Spin Example . . . . . . . . . . . . . . . . . . . 9.3.1 Dirac Delta Function . . . . . . . . . . . . . . . . . . 9.3.2 Imaging Sequence Diagrams Revisited . . . . . . . . 9.4 Gradient Echo and k-Space Diagrams . . . . . . . . . . . . . 9.4.1 The Gradient Echo . . . . . . . . . . . . . . . . . . . 9.4.2 General Spin Echo Imaging . . . . . . . . . . . . . . 9.4.3 Image Pro les . . . . . . . . . . . . . . . . . . . . . . 9.5 Gradient Directionality and Nonlinearity . . . . . . . . . . . 9.5.1 Frequency Encoding in an Arbitrary Direction . . . . 9.5.2 Nonlinear Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 112 112 113 115 117 118 118 121 122 123 123 124 125 127 129 130 133 134 140 140 141 142 142 143 144 144 145 148 149 149 150 154 157 159 159 162 8 Introductory Signal Acquisition Methods: Free Induction Decay, Spin Echoes, Inversion Recovery and Spectroscopy 111 9 One-Dimensional Fourier Imaging, k-Space and Gradient Echoes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10 Multi-Dimensional Fourier Imaging and Slice Excitation 10.1 Imaging in More Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.1.1 The Imaging Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.1.2 Single Excitation Traversal of k-Space . . . . . . . . . . . . . . . . . 169 165 xviii 10.2 10.3 10.4 10.5 10.1.3 Time Constraints and Collecting Data over Multiple Cycles . 10.1.4 Variations in k-Space Coverage . . . . . . . . . . . . . . . . Slice Selection with Boxcar Excitations . . . . . . . . . . . . . . . . 10.2.1 Slice Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Gradient Rephasing After Slice Selection . . . . . . . . . . . 10.2.3 Arbitrary Slice Orientation . . . . . . . . . . . . . . . . . . . 2D Imaging and k-Space . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Gradient Echo Example . . . . . . . . . . . . . . . . . . . . 10.3.2 Spin Echo Example . . . . . . . . . . . . . . . . . . . . . . . 3D Volume Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Short-TR 3D Gradient Echo Imaging . . . . . . . . . . . . . 10.4.2 Multi-Slice 2D Imaging . . . . . . . . . . . . . . . . . . . . . Chemical Shift Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 A 2D-Spatial 1D-Spectral Method . . . . . . . . . . . . . . . 10.5.2 A 3D-Spatial, 1D-Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 172 175 175 179 181 184 184 193 194 194 195 196 200 204 11 The Continuous and Discrete Fourier Transforms 11.1 The Continuous Fourier Transform . . . . . . . . . . . . . . . 11.2 Continuous Transform Properties and Phase Imaging . . . . . 11.2.1 Complexity of the Reconstructed Image . . . . . . . . . 11.2.2 The Shift Theorem . . . . . . . . . . . . . . . . . . . . 11.2.3 Phase Imaging and Phase Aliasing . . . . . . . . . . . 11.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Convolution Theorem . . . . . . . . . . . . . . . . . . . 11.2.6 Convolution Associativity . . . . . . . . . . . . . . . . 11.2.7 Derivative Theorem . . . . . . . . . . . . . . . . . . . . 11.2.8 Fourier Transform Symmetries . . . . . . . . . . . . . . 11.2.9 Summary of Continuous Fourier Transform Properties . 11.3 Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . 11.3.1 Heaviside Function . . . . . . . . . . . . . . . . . . . . 11.3.2 Lorentzian Form . . . . . . . . . . . . . . . . . . . . . 11.3.3 The Sampling Function . . . . . . . . . . . . . . . . . . 11.4 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . 11.5 Discrete Transform Properties . . . . . . . . . . . . . . . . . . 11.5.1 The Discrete Convolution Theorem . . . . . . . . . . . 11.5.2 Summary of Discrete Fourier Transform Properties . . 207 208 209 211 211 212 215 215 218 218 220 221 221 221 223 223 224 226 227 228 232 232 234 239 239 241 244 12 Sampling and Aliasing in Image Reconstruction 12.1 In nite Sampling, Aliasing and the Nyquist Criterion . . . . . . . . . . . . 12.1.1 In nite Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Nyquist Sampling Criterion . . . . . . . . . . . . . . . . . . . . . . 12.2 Finite Sampling, Image Reconstruction and the Discrete Fourier Transform 12.2.1 Finite Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Reconstructed Spin Density . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Discrete and Truncated Sampling of ^(x): Resolution . . . . . . . . 231 Contents 12.2.4 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 12.2.5 Practical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 RF Coils, Noise and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 RF Field-of-View Considerations . . . . . . . . . . . . . . . . . . . 12.3.2 Analog Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Avoiding Aliasing in 3D Imaging . . . . . . . . . . . . . . . . . . . 12.4 Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Aliasing from Interleaved Sampling . . . . . . . . . . . . . . . . . . 12.4.2 Aliasing from Digital-to-Analog Error in the Gradient Speci cation . . . . . . . . . xix 245 247 248 248 248 253 253 253 261 13 Filtering and Resolution in Fourier Transform Image Reconstruction 13.1 Review of Fourier Transform Image Reconstruction . . . . . . . . . . . . . . 13.1.1 Fourier Encoding and Fourier Inversion . . . . . . . . . . . . . . . . 13.1.2 In nite Sampling and Fourier Series . . . . . . . . . . . . . . . . . . 13.1.3 Limited-Fourier Imaging and Aliasing . . . . . . . . . . . . . . . . . 13.1.4 Signal Series and Spatial Resolution . . . . . . . . . . . . . . . . . . 13.2 Filters and Point Spread Functions . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Point Spread Due to Truncation . . . . . . . . . . . . . . . . . . . . 13.2.2 Point Spread for Truncated and Sampled Data . . . . . . . . . . . . 13.2.3 Point Spread for Additional Filters . . . . . . . . . . . . . . . . . . . 13.3 Gibbs Ringing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Gibbs Overshoot and Undershoot . . . . . . . . . . . . . . . . . . . . 13.3.2 Gibbs Oscillation Frequency . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Reducing Gibbs Ringing by Filtering . . . . . . . . . . . . . . . . . . 13.4 Spatial Resolution in MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Resolution after Additional Filtering of the Data . . . . . . . . . . . 13.4.2 Other Measures of Resolution . . . . . . . . . . . . . . . . . . . . . . 13.5 Filtering Due to T2 and T2 Decay . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Gradient Echo Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Spin Echo Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Zero Filled Interpolation, Sub-Voxel Fourier Transform Shift Concepts and Point Spread Function E ects . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.1 Zero Padding and the Fast Fourier Transform . . . . . . . . . . . . . 13.6.2 Equivalence of Zero Filled Signal and the Sub-Voxel Shifted Signal . 13.6.3 Point Spread E ects on the Image Signal Based on the Object Position Relative to the Reconstructed Voxels . . . . . . . . . . . . . . . . . . 13.7 Partial Fourier Imaging and Reconstruction . . . . . . . . . . . . . . . . . . 13.7.1 Forcing Conjugate Symmetry on Complex Objects . . . . . . . . . . . 13.7.2 Iterative Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.3 Some Implementation Issues . . . . . . . . . . . . . . . . . . . . . . 13.8 Digital Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 266 267 267 267 268 269 269 270 271 272 272 274 275 277 282 283 285 286 287 289 289 290 291 292 295 296 298 299 xx Contents 14.1 Radial k-Space Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Coverage of k-Space at Di erent Angles . . . . . . . . . . . . . . . . 14.1.2 Two Radial Fourier Transform Examples . . . . . . . . . . . . . . . . 14.1.3 Inversion for Image Reconstruction . . . . . . . . . . . . . . . . . . . 14.2 Sampling Radial k-Space and Nyquist Limits . . . . . . . . . . . . . . . . . . 14.3 Projections and the Radon Transform . . . . . . . . . . . . . . . . . . . . . . 14.4 Methods of Projection Reconstruction with Radial Coverage . . . . . . . . . 14.4.1 X-Ray Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Back-Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Projection Slice Theorem and the Fourier Reconstruction Method . . 14.4.4 Filtered Back-Projection Method . . . . . . . . . . . . . . . . . . . . 14.4.5 Reconstruction of MR Images from Radial Data . . . . . . . . . . . . 14.5 Three-Dimensional Radial k-Space Coverage . . . . . . . . . . . . . . . . . . 14.6 Radial Coverage Versus Cartesian k-Space Coverage . . . . . . . . . . . . . . 14.6.1 Image Distortion Due to O -Resonance E ects: Cartesian Coverage Versus Radial Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 E ects of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Cartesian Sampling of Radially Collected Data . . . . . . . . . . . . . 14 Projection Reconstruction of Images 303 304 305 306 307 308 314 315 316 317 319 320 322 323 326 327 329 329 15 Signal, Contrast and Noise 15.1 Signal and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 15.1.1 The Voxel Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 15.1.2 The Noise in MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 15.1.3 Dependence of the Noise on Imaging Parameters . . . . . . . . . . . . 334 15.1.4 Improving SNR by Averaging over Multiple Acquisitions . . . . . . . 337 15.1.5 Measurement of 0 and Estimation of SNR . . . . . . . . . . . . . . . 340 15.2 SNR Dependence on Imaging Parameters . . . . . . . . . . . . . . . . . . . . 340 15.2.1 Generalized Dependence of SNR in 3D Imaging on Imaging Parameters340 15.2.2 SNR Dependence on Read Direction Parameters . . . . . . . . . . . . 341 15.2.3 SNR Dependence on Phase Encoding Parameters . . . . . . . . . . . 346 15.2.4 SNR in 2D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.2.5 Imaging E ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 15.3 Contrast, Contrast-to-Noise and Visibility . . . . . . . . . . . . . . . . . . . 349 15.3.1 Contrast and Contrast-to-Noise Ratio . . . . . . . . . . . . . . . . . . 349 15.3.2 Object Visibility and the Rose Criterion . . . . . . . . . . . . . . . . 350 15.4 Contrast Mechanisms in MRI and Contrast Maximization . . . . . . . . . . . 352 15.4.1 Three Important Types of Contrast . . . . . . . . . . . . . . . . . . . 352 15.4.2 Spin Density Weighting . . . . . . . . . . . . . . . . . . . . . . . . . 352 15.4.3 T1-Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 15.4.4 T2 -Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 15.4.5 Summary of Contrast Results . . . . . . . . . . . . . . . . . . . . . . 362 15.4.6 A Special Case: T1 -Weighting and Tissue Nulling with Inversion Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 15.5 Contrast Enhancement with T1 -Shortening Agents . . . . . . . . . . . . . . . 367 331 Contents 15.6 Partial Volume E ects, CNR and Resolution . . . . 15.7 SNR in Magnitude and Phase Images . . . . . . . . 15.7.1 Magnitude Image SNR . . . . . . . . . . . . 15.7.2 Phase Image SNR . . . . . . . . . . . . . . . 15.8 SNR as a Function of Field Strength . . . . . . . . 15.8.1 Frequency Dependence of the Noise in MRI 15.8.2 SNR Dependence on Field Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 370 374 374 375 376 377 378 16 A Closer Look at Radiofrequency Pulses 16.1 Relating RF Fields and Measured Spin Density . . . . . . . . . . . . . 16.1.1 RF Pulse Shapes and Apodization . . . . . . . . . . . . . . . . 16.1.2 Numerical Solutions of the Bloch Equations . . . . . . . . . . . 16.2 Implementing Slice Selection . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Calibrating the RF Field . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Checking the RF Pro le . . . . . . . . . . . . . . . . . . . . . . 16.4 Low Flip Angle Excitation and Rephasing Gradients . . . . . . . . . . 16.5 Spatially Varying RF Excitation . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Two-Dimensional `Beam' Excitation . . . . . . . . . . . . . . . 16.5.2 Time Varying Gradients and Slice Selection . . . . . . . . . . . 16.5.3 3D Spatially Selective Excitations in the Low Flip Angle Limit . 16.6 RF Pulse Characteristics: Flip Angle and RF Power . . . . . . . . . . . 16.6.1 Analysis of Slice Selection Parameters . . . . . . . . . . . . . . . 16.7 Spin Tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.1 Tagging with Gradients Applied Between RF Pulses . . . . . . . 16.7.2 Multiple RF and Gradient Pulses for Tagging . . . . . . . . . . 16.7.3 Summary of Tagging Applications . . . . . . . . . . . . . . . . . 17.1 The E ect of Chemical Shift in Imaging . . . . . . . . . . 17.1.1 Fat Shift Artifact . . . . . . . . . . . . . . . . . . . 17.2 Selective Excitation and Tissue Nulling . . . . . . . . . . . 17.2.1 Selective Excitation and Saturation . . . . . . . . . 17.2.2 Tissue Nulling with Inversion Recovery . . . . . . . 17.3 Multiple Point Water/Fat Separation Methods . . . . . . . 17.3.1 Gradient Echo Sequence for Water/Fat Separation . 17.3.2 Single-Echo Separation . . . . . . . . . . . . . . . . 17.3.3 Spin Echo Approach . . . . . . . . . . . . . . . . . 17.3.4 Two-Point Separation . . . . . . . . . . . . . . . . 17.3.5 Three-Point Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 382 385 387 388 392 393 397 399 399 402 404 406 407 411 411 415 416 421 422 428 428 429 431 431 436 439 440 444 452 454 460 463 17 Water/Fat Separation Techniques 421 18 Fast Imaging in the Steady State 18.1 Short-TR, Spoiled, Gradient Echo Imaging . . . . . . . . . . . . 18.1.1 Expression for the Steady-State Incoherent (SSI) Signal . 18.1.2 Approach to Incoherent Steady-State . . . . . . . . . . . 18.1.3 Generating a Constant Transverse Magnetization . . . . 451 xxii 18.1.4 Nonideal Slice Pro le E ects on the SSI Signal . . . . . . . . . . . 18.2 Short-TR, Coherent, Gradient Echo Imaging . . . . . . . . . . . . . . . . 18.2.1 Steady-State Free Precession: The Equilibrium Signal . . . . . . . 18.2.2 Approach to Coherent Steady-State . . . . . . . . . . . . . . . . . 18.2.3 Utility of SSC Imaging . . . . . . . . . . . . . . . . . . . . . . . . 18.3 SSFP Signal Formation Mechanisms . . . . . . . . . . . . . . . . . . . . 18.3.1 Magnetization Rotation E ects of an Arbitrary Flip Angle Pulse . 18.3.2 Multi-Pulse Experiments and Echoes . . . . . . . . . . . . . . . . 18.4 Understanding Spoiling Mechanisms . . . . . . . . . . . . . . . . . . . . . 18.4.1 General Principles of Spoiling . . . . . . . . . . . . . . . . . . . . 18.4.2 A Detailed Discussion of Spoiling . . . . . . . . . . . . . . . . . . 18.4.3 Practical Implementation of Spoiling . . . . . . . . . . . . . . . . 18.4.4 RF Spoiled SSI Sequence Implementation . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . 466 467 472 477 480 482 482 485 500 500 501 506 510 19 Segmented k-Space and Echo Planar Imaging 19.1 Reducing Scan Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1 Reducing TR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Reducing the Number of Phase/Partition Encoding Steps . . . . . . . 19.1.3 Fixing the Number of Acquisitions . . . . . . . . . . . . . . . . . . . 19.1.4 Partial Fourier Data Acquisition . . . . . . . . . . . . . . . . . . . . . 19.2 Segmented k-Space: Phase Encoding Multiple k-Space Lines per RF Excitation for Gradient Echo Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Conventional Multiple Echo Acquisition . . . . . . . . . . . . . . . . 19.2.2 Phase Encoding Between Gradient Echoes . . . . . . . . . . . . . . . 19.3 Echo Planar Imaging (EPI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 An In-Depth Analysis of the EPI Imaging Parameters . . . . . . . . . 19.3.2 Signal-to-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Alternate Forms of Conventional EPI . . . . . . . . . . . . . . . . . . . . . . 19.4.1 Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.2 Segmented EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.3 Angled k-Space EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.4 Segmented EPI with Oscillating Gradients . . . . . . . . . . . . . . . 19.4.5 Trapezoidal Versus Oscillating Waveforms . . . . . . . . . . . . . . . 19.5 Artifacts and Phase Correction . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.1 Phase Errors and Their Correction . . . . . . . . . . . . . . . . . . . 19.5.2 Chemical Shift and Geometric Distortion . . . . . . . . . . . . . . . . 19.5.3 Geometric Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.4 T2 -Filter E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.5 Ghosting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Spiral Forms of EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.1 Square-Spiral EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.2 Spiral EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7 An Overview of EPI Properties . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.1 Speed of EPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.2 Contrast Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 514 514 514 516 516 516 517 520 524 528 530 533 533 534 537 542 545 546 546 547 549 551 552 552 552 556 560 560 562 Contents xxiii 19.7.3 Field-of-View and Resolution in the Phase Encoding Direction . . . . 562 19.7.4 EPI Safety Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 19.8 Phase Encoding Between Spin Echoes and Segmented Acquisition . . . . . . 563 20 Magnetic Field Inhomogeneity E ects and T2 Dephasing 20.1 Image Distortion Due to Field E ects . . . . . . . . . . . . . . . . . . . . . . 570 20.1.1 Distortion Due to Background Gradients Parallel to the Read Direction570 20.1.2 Distortion Due to Gradient Perpendicular to the Read Direction . . . 575 20.1.3 Slice Select Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . 578 20.2 Echo Shifting Due to Field Inhomogeneities in Gradient Echo Imaging . . . . 580 20.2.1 Echo Shift in Terms of Number of Sampled Points . . . . . . . . . . . 583 20.2.2 Echo Shift Due to Background Phase/Partition Encoding Gradients . 585 20.2.3 Echo Shift Due to Background Gradients Parallel to the Slice Select Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 20.2.4 Echo Shift Due to Background Gradients Orthogonal to the Slice Select Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 20.3 Methods for Minimizing Distortion and Echo Shifting Artifacts . . . . . . . . 587 20.3.1 Distortion Versus Dephasing . . . . . . . . . . . . . . . . . . . . . . . 587 20.3.2 High Resolution and Phase Dispersion . . . . . . . . . . . . . . . . . 588 20.3.3 2D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 20.3.4 2D Imaging with Variable Rephasing Gradients . . . . . . . . . . . . 593 20.3.5 3D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 20.3.6 Phase Encoded 2D and 3D Imaging with Single-Point Sampling: A Limited Version of CSI . . . . . . . . . . . . . . . . . . . . . . . . . . 600 20.3.7 Spectrally Resolved 2D and 3D Imaging . . . . . . . . . . . . . . . . 600 20.3.8 Understanding the Recovered Signal with Spectral Collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 20.4 Empirical T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 20.4.1 Arbitrariness of T2 Modeling of Gradient Echo Signal Envelopes . . . 602 20.4.2 The Spin Echo Signal Envelope and the Magnetic Field Density of States604 20.4.3 Decaying Signal Envelopes and Integrated Signal Conservation . . . . 605 20.4.4 Obtaining a Lorentzian Density of States: A Simple Argument . . . . 609 20.4.5 Predicting the E ects of Arbitrary Field Inhomogeneities on the Image 609 20.5 Predicting T2 for Random Susceptibility Producing Structures . . . . . . . . 611 20.6 Correcting Geometric Distortion . . . . . . . . . . . . . . . . . . . . . . . . . 614 21.1 Simple Model for Intrinsic T2 . . . . . . . . . . . . . 21.1.1 Gaussian Behavior for Random Spin Systems 21.1.2 Brownian Motion and T2 Signal Loss . . . . . 21.2 Simple Model for Di usion . . . . . . . . . . . . . . . 21.3 Carr-Purcell Mechanism . . . . . . . . . . . . . . . . 21.4 Meiboom-Gill Improvement . . . . . . . . . . . . . . 21.5 The Bloch-Torrey Equation . . . . . . . . . . . . . . 21.5.1 The Gradient Echo Case for a Bipolar Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 21 Random Walks, Relaxation and Di usion 619 620 620 621 622 624 627 627 628 xxiv Contents 21.5.2 The Spin Echo Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 21.5.3 Velocity Compensated Di usion Weighted Sequences . . . . . . . . . 631 21.6 Some Practical Examples of Di usion Imaging . . . . . . . . . . . . . . . . . 631 22 Spin Density, T1 and T2 Quanti cation Methods in MR Imaging 22.1 Simplistic Estimates of 0 , T1 and T2 . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Spin Density Measurement . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 T1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3 T2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Estimating T1 and T2 from Signal Ratio Measurements . . . . . . . . . . . . 22.2.1 T1 Estimation from a Signal Ratio Measurement . . . . . . . . . . . . 22.2.2 T2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Estimating T1 and T2 from Multiple Signal Measurements . . . . . . . . . . . 22.3.1 Parameter Estimation from Multiple Signal Measurements . . . . . . 22.3.2 T1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.3 T2 and T2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Other Methods for Spin Density and T1 Estimation . . . . . . . . . . . . . . 22.4.1 The Look-Locker Method . . . . . . . . . . . . . . . . . . . . . . . . 22.4.2 T1 Estimation from SSI Measurements at Multiple Flip Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Practical Issues Related to T1 and T2 Measurements . . . . . . . . . . . . . . 22.5.1 Inaccuracies Due to Nonideal Slice Pro le . . . . . . . . . . . . . . . 22.5.2 Other Sources of Inaccuracies in Relaxation Time and Spin Density Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.3 Advanced Sequence Design for Relaxation Time and Spin Density Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.4 Choice of Number of Signal Measurement Points . . . . . . . . . . . . 22.6 Calibration Materials for Relaxation Time Measurements . . . . . . . . . . . 23.1 E ects on Spin Phase from Motion along the Read Direction . . . . . . . . . 23.1.1 Spin Phase Due to Constant Velocity Flow or Motion in the Read Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1.2 E ects of Constant Velocity Flow on the Image . . . . . . . . . . . . 23.2 Velocity Compensation along the Read and Slice Select Directions . . . . . . 23.2.1 Velocity Compensation Concepts . . . . . . . . . . . . . . . . . . . . 23.2.2 Velocity Compensation along the Slice Select Direction . . . . . . . . 23.3 Ghosting Due to Periodic Motion . . . . . . . . . . . . . . . . . . . . . . . . 23.3.1 Ghosting Due to Periodic Flow . . . . . . . . . . . . . . . . . . . . . 23.3.2 Sinusoidal Translational Motion . . . . . . . . . . . . . . . . . . . . . 23.3.3 Examples of Ghosting from Pulsatile Flow . . . . . . . . . . . . . . . 23.4 Velocity Compensation along Phase Encoding Directions . . . . . . . . . . . 23.4.1 E ects of Constant Velocity Flow in the Phase Encoding Direction: The Misregistration Artifact . . . . . . . . . . . . . . . . . . . . . . . 23.4.2 Phase Variation View of the Shift Artifact . . . . . . . . . . . . . . . 637 639 639 639 640 640 641 646 648 648 648 650 650 650 654 657 657 661 663 664 665 23 Motion Artifacts and Flow Compensation 669 670 670 673 676 676 682 682 682 684 688 690 690 690 Contents xxv 23.4.3 Velocity Compensating Phase Encoding Gradients . . . . . . . . . . . 694 23.5 Maximum Intensity Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 697 24 MR Angiography and Flow Quanti cation 24.1 In ow or Time-of-Flight (TOF) E ects . . . . . . . . . . . . . . . . . . . 24.1.1 Critical Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.2 Approach to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 24.1.3 2D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.4 3D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.5 Understanding In ow E ects for Small Velocities . . . . . . . . . 24.2 TOF Contrast, Contrast Agents and Spin Density/T2 -Weighting . . . . . 24.2.1 Contrast Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.2 Suppressing Signal from In owing Blood Using an Inversion Pulse 24.2.3 Suppressing Signal from In owing Blood Using a Saturation Pulse 24.3 Phase Contrast and Velocity Quanti cation . . . . . . . . . . . . . . . . 24.3.1 Phase Subtraction and Complex Division for Measuring Velocity . 24.3.2 Four-Point Velocity Vector Extraction . . . . . . . . . . . . . . . 24.4 Flow Quanti cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.1 Cardiac Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 704 704 705 707 709 713 713 714 720 721 725 727 731 734 735 742 742 743 744 746 746 747 749 749 751 753 755 757 757 762 765 765 766 767 769 769 771 25 Magnetic Properties of Tissues: Theory and Measurement 25.1 Paramagnetism, Diamagnetism and Ferromagnetism . . . . . . . . . . . . . . 25.1.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.2 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ 25.2 Permeability and Susceptibility: The H Field . . . . . . . . . . . . . . . . . ~ 25.2.1 Permeability and the H Field . . . . . . . . . . . . . . . . . . . . . . 25.2.2 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Objects in External Fields: The Lorentz Sphere . . . . . . . . . . . . . . . . 25.3.1 Spherical Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.2 In nite Cylindrical Body . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.3 Local Field Cancellation via Molecular Demagnetization . . . . . . . 25.3.4 Sphere and Cylinder Examples Revisited: The Physical Internal Fields 25.4 Susceptibility Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.1 Phase Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.2 Magnitude Measurements . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Brain Functional MRI and the BOLD Phenomenon . . . . . . . . . . . . . . 25.5.1 Estimation of Oxygenation Levels . . . . . . . . . . . . . . . . . . . . 25.5.2 Deoxyhemoglobin Concentration and Flow . . . . . . . . . . . . . . . 25.5.3 Functional MR Imaging (fMRI): An Example . . . . . . . . . . . . . 25.6 Signal Behavior in the Presence of Deoxygenated Blood . . . . . . . . . . . . 25.6.1 The MR Properties of Blood . . . . . . . . . . . . . . . . . . . . . . . 25.6.2 Two-Compartment Partial Volume E ects on Signal Loss . . . . . . . 741 xxvi Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Sequence Design, Artifacts and Nomenclature 26.1 Sequence Design and Imaging Parameters . . . . . . . . . . . . . . . . 26.1.1 Slice Select Gradient . . . . . . . . . . . . . . . . . . . . . . . . 26.1.2 Phase Encoding Gradient . . . . . . . . . . . . . . . . . . . . . 26.1.3 Read Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1.4 Data Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Early Spin Echo Imaging Sequences . . . . . . . . . . . . . . . . . . . . 26.2.1 Single and Multi-Echo Spin Echo Sequences . . . . . . . . . . . 26.2.2 Inversion Recovery . . . . . . . . . . . . . . . . . . . . . . . . . 26.2.3 Spin Echo with Phase Encoding Between Echoes . . . . . . . . 26.3 Fast Short TR Imaging Sequences . . . . . . . . . . . . . . . . . . . . . 26.3.1 Steady-State Incoherent: Gradient Echo . . . . . . . . . . . . . 26.3.2 Steady-State Incoherent: Spin Echo . . . . . . . . . . . . . . . . 26.3.3 Steady-State Coherent Imaging . . . . . . . . . . . . . . . . . . 26.3.4 Pulse Train Methods . . . . . . . . . . . . . . . . . . . . . . . . 26.3.5 Magnetization Prepared Sequences . . . . . . . . . . . . . . . . 26.4 Imaging Tricks and Image Artifacts . . . . . . . . . . . . . . . . . . . . 26.4.1 Readout Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 26.4.2 Dealing with System Instabilities . . . . . . . . . . . . . . . . . 26.4.3 DC and Line Artifacts . . . . . . . . . . . . . . . . . . . . . . . 26.4.4 Noise Spikes and Constant-Frequency Noise . . . . . . . . . . . 26.5 Sequence Adjectives and Nomenclature . . . . . . . . . . . . . . . . . . 26.5.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.2 Some Other Descriptive Adjectives and Some Speci c Examples 27.1 The Circular Loop as an Example . . . . . . . . . . . 27.1.1 Quality of Field . . . . . . . . . . . . . . . . . 27.2 The Main Magnet Coil . . . . . . . . . . . . . . . . . 27.2.1 Classic Designs . . . . . . . . . . . . . . . . . 27.2.2 Desirable Properties of Main Magnets . . . . . 27.2.3 Shielding . . . . . . . . . . . . . . . . . . . . . 27.2.4 Shimming . . . . . . . . . . . . . . . . . . . . 27.3 Linearly Varying Field Gradients . . . . . . . . . . . 27.3.1 Classic Designs . . . . . . . . . . . . . . . . . 27.3.2 Calculating Linearly Varying Fields . . . . . . 27.3.3 Desirable Properties of Linear Gradient Coils 27.3.4 Eddy Currents and dB=dt . . . . . . . . . . . 27.3.5 Active Shielding . . . . . . . . . . . . . . . . . 27.3.6 `Linearly Varying' Magnetic Fields . . . . . . 27.4 RF Transmit and Receive Coils . . . . . . . . . . . . 27.4.1 Transmit Coils . . . . . . . . . . . . . . . . . 27.4.2 Receive Coils or RF Probes . . . . . . . . . . 27.4.3 Classic Designs . . . . . . . . . . . . . . . . . 27.4.4 RF Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 782 782 785 786 787 787 787 790 791 793 793 795 795 798 799 800 801 803 806 813 816 816 820 828 829 831 831 835 840 841 842 842 844 845 846 848 848 850 851 852 853 858 27 Introduction to MRI Coils and Magnets 827 Contents xxvii 27.4.5 Power Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858 A Electromagnetic Principles: A Brief Overview
A.1 A.2 A.3 A.4 A.5 A.6 Maxwell's Equations . . . . . . . . . Faraday's Law of Induction . . . . . Electromagnetic Forces . . . . . . . . Dipoles in an Electromagnetic Field . Formulae for Electromagnetic Energy Static Magnetic Field Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 864 864 865 866 867 867 869 870 871 871 871 872 872 874 875 876 877 878 878 B Statistics B.1 Accuracy Versus Precision . . . . . . . . . . . . B.1.1 Mean and Standard Deviation . . . . . . B.2 The Gaussian Probability Distribution . . . . . B.2.1 Probability Distribution . . . . . . . . . B.2.2 z-Score . . . . . . . . . . . . . . . . . . . B.2.3 Quoting Errors and Con dence Intervals B.3 Type I and Type II Errors . . . . . . . . . . . . B.4 Sum over Several Random Variables . . . . . . . B.4.1 Multiple Noise Sources . . . . . . . . . . B.5 Rayleigh Distribution . . . . . . . . . . . . . . . B.6 Experimental Validation of Noise Distributions . B.6.1 Histogram Analysis . . . . . . . . . . . . B.6.2 Mean and Standard Deviation . . . . . . 869 C Imaging Parameters to Accompany Figures Index 883 893 xxviii Contents Chapter 1 Magnetic Resonance Imaging: A Preview
Chapter Contents
1.1 1.2 1.3 1.4
Magnetic Resonance Imaging: The Name The Origin of Magnetic Resonance Imaging A Brief Overview of MRI Concepts Suggested Reading Introduction
The primary purpose of this chapter is to provide a succinct overview of the basic principles involved in the process of using nuclear magnetic resonance for imaging. This overview is in the form of a list of results, without derivation, in order to provide a story line and goals for the reader to follow in the detailed treatment of this material in the coming chapters. The chapter begins with an explanation of the name, `magnetic resonance imaging,' or MRI, followed by a short and inadequate history1 of some of the developments that led to the discovery of key imaging concepts. The third, and largest, section is a preview of the subsequent twenty-six chapters. Finally, some relevant reference texts and articles are listed in the suggested reading section. 1.1 Magnetic Resonance Imaging: The Name
Magnetic resonance imaging is a relatively new discipline in the realm of applied sciences. A main thrust has come from the imaging of soft tissues in the human body and metabolic processes therein, such that it occupies a strong position in biomedical science applications.
A great deal of the history of nuclear magnetic resonance and MRI is presented in volume one of the Encyclopedia of Nuclear Magnetic Resonance which is listed in the historical/review references.
1 1 2 Chapter 1. Magnetic Resonance Imaging: A Preview MRI is a powerful imaging modality because of its exibility and sensitivity to a broad range of tissue properties. One of the original reasons for the excitement about MRI was, and continues to be, its relative safety, where the `noninvasive' nature of the magnetic elds employed make it possible to diagnose conditions of people of almost any age. Today MRI also o ers great promise in understanding much more about the human body, both its form and its function. MRI stems from the application of nuclear magnetic resonance (NMR) to radiological imaging. The adjective `magnetic' refers to the use of an assortment of magnetic elds and `resonance' refers to the need to match the (radio)frequency of an oscillating magnetic eld to the `precessional' frequency of the spin of some nucleus (hence the `nuclear') in a tissue molecule. It might be more accurate to refer to this eld as NMRI rather than MRI, but there is widespread concern over any phrase containing the word `nuclear.' Although the nuclear component simply refers to a benign role of the `spin' of the nucleus in the process, the word has been suppressed and the public and the profession have embraced the MRI acronym. 1.2 The Origin of Magnetic Resonance Imaging
To describe the history of any technological advance in a given eld is a very di cult and sensitive issue to o er a brief and incomplete account is fraught with peril. Still, the beginning student may be aided and inspired by even a short historical discussion. It may be said that MRI had its beginnings in 1973 with the seminal papers by Lauterbur and Mans eld. It was already well known that the intrinsic angular momentum (or `spin') of a hydrogen nucleus (the proton) in a magnetic eld precesses about that eld at the `Larmor frequency' which, in turn, depends linearly on the magnitude of the eld itself. Their idea was very simple. If a spatially varying magnetic eld is introduced across the object, the Larmor frequencies are also spatially varying. They proposed and showed that the di erent frequency components of the signal could be separated to give spatial information about the object. This key point of spatially encoding the data opened the door to MR imaging. Others also recognized the importance of this area, with early attention brought to tumor detection by Damadian. Something may be learned here by the beginning student. Often the basic step toward new developments, which may become quite complicated as a whole, is a simple connecting idea. The concept of using a magnetic eld gradient was one such `aha' that captured the essence of MRI as it is practiced today, much like the coupling of the nuclear spin of the proton to the magnetic eld was the key to the early experiments by both Bloch's group and Purcell's group in their pioneering work in NMR. The concept of nuclear magnetic resonance had its underpinnings with the discovery of the spin nature of the proton. Leaning on the work of Stern and Gerlach from the early 1920's, Rabi and coworkers pursued the spin of the proton and its interaction with a magnetic eld in the 1930's. With this foundation in hand in 1946, Bloch and Purcell extended these early quantum mechanical concepts to a measurement of an e ect of the precession of the spins around a magnetic eld. Not only did these gentlemen successfully measure a precessional signal from a water sample and a para n sample, respectively, but they 1.3. A Brief Overview of MRI Concepts 3 explained precociously many of the experimental and theoretical details that we continue to draw from still today. For this work, they shared the Nobel prize in physics in 1952. 1.3 A Brief Overview of MRI Concepts
In this section, we attempt to introduce and list for the reader the basic elements that make MRI possible. The derivations, explanations and related details of these results are covered principally in Chs. 2-15. Although the later chapters, Chs. 16-27, are focused on advanced applications and concepts, additional related results are presented there as well. 1.3.1 Fundamental Interaction of a Proton Spin with the Magnetic Field We have alluded to the idea that MRI is based on the interaction of a nuclear spin with an ~ external magnetic eld, B0 . The dominant nucleus in MRI is the proton in hydrogen and its interaction with the external eld results in the precession of the proton spin about the eld direction (see Fig. 1.1). Imaging of humans rests on the ability to manipulate, with a combination of magnetic elds, and then detect, the bulk precession of the hydrogen spins in water, fat and other organic molecules. about another xed axis caused by the application of a torque in the direction of the precession. The interaction of the proton's spin with the magnetic eld produces the torque, causing it to ~ ~ precess about B0 as the xed axis. When looking down from above the vector B0 , the precession of the magnetic moment vector ~ , which is proportional to the spin vector, is clockwise. For the customary counterclockwise de nition of polar angles, the di erential d shown is negative. Fig. 1.1: By de nition, precession is the circular motion of the axis of rotation of a spinning body The basic motion of the proton spin may be understood by imagining it as a spinning gyroscope that is also electrically charged. It thus possesses an e ective loop of electric 4 Chapter 1. Magnetic Resonance Imaging: A Preview current around the same axis about which it is spinning. This e ective current loop is capable of interacting with external magnetic elds as well as producing its own magnetic eld. We describe the strength with which the loop interacts with an external eld, as well as the strength with which the loop produces its own eld, in terms of the same `magnetic dipole moment' vector ~ . The direction of this vector is nothing other than the spin axis itself and, like a compass needle, the magnetic moment vector will tend to align itself along ~ any external static magnetic eld, B0. This is like the initial tendency of a gyroscope to fall in the direction of gravity. However, the tendencies are complicated by the spin in exactly the same way as is the gyroscopic motion. Instead of `falling' along the eld direction, the magnetic moment vector, like the spinning gyroscope, will precess around the eld direction. In Ch. 2, we nd that the precession angular frequency for the proton magnetic moment vector (and for the spin axis as well) is given by ~ ! ~ 0 = B0 (1.1) where is a constant called the gyromagnetic ratio. In water, the hydrogen proton has a value of roughly 2.68 108 rad/s/Tesla (so that { =(2 ) is 42.6 MHz/Tesla). For a 2 T eld, for example, the spins precess at a radiofrequency of 85.2 MHz, just below the FM range for radio broadcasting. (We use SI units throughout the text.) This precession frequency is referred to as the Larmor frequency and (1.1) is referred to as the Larmor equation. 1.3.2 Equilibrium Alignment of Spin
The magnetic moment vector for a typical proton is prevented from relaxing fully to an alignment along the external magnetic eld because of thermal energy associated with the absolute temperature T . From the discussions in Chs. 5 and 6, we can compare the magnetic eld interaction with the average thermal energy kT , where k is the Boltzmann's constant. At human body temperatures, the thermal energy is millions of times larger than the quantum energy di erence for parallel alignment (lower energy) versus anti-parallel alignment (higher energy). For a proton with only two quantum spin states, these are the only two possible alignments. Signi cantly, the frequency in the quantum energy di erence, h!0, is nothing other than the Larmor precession frequency (1.1) where h h=(2 ) in terms of Planck's quantum constant h. The extreme smallness of the quantum spin energy compared with the thermal energy means that the fraction h!0=(kT ) 1. In that case, the Boltzmann probability discussion of Ch. 6 demonstrates why the number of spins parallel to the magnetic eld exceeding the number anti-parallel to that eld, the `spin excess,' is also very small. Speci cally, the spin excess is suppressed by a factor involving that fraction: spin excess ' N h!0 (1.2) 2kT where N is the total number of spins present in the sample. In the rst problem below, it is found that the spin excess is only one in a million spins even for a magnetic eld strength as large as 0.3 T. 1.3. A Brief Overview of MRI Concepts 5 Problem 1.1
Using h = 1.05 10;34 Joule s, k = 1.38 10;23 Joule/K and T = 300 K, nd the spin excess as a fraction of N at 0.3 Tesla. Since the spin excess is millions of times smaller than the total number of proton spins, it might be guessed that no signi cant signal would be detected at room temperature. However, there are Avogadro numbers of protons in a few grams of tissue. Consider the average magnetic dipole density, or `longitudinal equilibrium magnetization' M0 for the component of the magnetic moment vector along the external eld direction. For a sample with 0 de ned as the number of protons per unit volume (or the `spin density'), the longitudinal equilibrium magnetization is found in Ch. 6 to be given by the proton magnetic moment component h=2 multiplied by the relative spin excess (1.2) times the spin density. Noting (1.1), it is thus given by 2 h2 (1.3) M0 = 0 kT B0 4 This equilibrium value, while limited by the spin excess, leads to measurable NMR e ects to be described next. 1.3.3 Detecting the Magnetization of the System Even in a macroscopic body, a bulk nonvanishing spin excess is not enough to guarantee a detectable signal. In a classical picture (which is derived from the quantum underpinnings in Ch. 5), the magnetization vector (the magnetic moment vector density due to the spin population) must be tipped away from the external eld direction in order to set it into precession. The magnetic eld produced by the aggregate proton spins will precess along with the magnetization yielding a changing ux in any nearby (`receiver') coil (Ch. 7). To accomplish this, as discussed in Chs. 3 and 4, the magnetization can be rotated away from its alignment along the B0 axis (i.e., from its longitudinal direction) by applying a radiofrequency (rf) magnetic eld for a short time (an rf `pulse'). This rf pulse is produced from another nearby `transmit' coil (which may be the same as the receiver coil, provided its radiofrequency is tuned to the Larmor frequency (see Fig. 1.2). This is the resonance condition in MRI described earlier and it ensures that the precessing spin gets a continuously synchronized push (rotation) away from the longitudinal direction (the z-axis, say). ~ ~ Suppose that M has been rotated by an rf pulse to a direction orthogonal to B0 = B0 z. (The rf pulse that tips all the original longitudinal magnetization, or z-magnetization, ^ through an angle of 90 into the transverse, or x-y, plane is called a =2-pulse), The resulting `transverse magnetization' has magnitude M0 and begins to precess clockwise in the x-y plane. Its rectangular components have sinusoidal time dependence with frequency given by the Larmor frequency. As de ned in Ch. 4, the complex magnetization is M+ (t) Mx (t) + iMy (t) = M0 e;i!0 t+i 0 (1.4) 6 Chapter 1. Magnetic Resonance Imaging: A Preview in terms of the magnitude and the (polar angle) phase. This shows an important connection between the time-dependence of the complex phase and the rotation of the magnetization. The phase angle gives the direction in the x-y plane of the two-dimensional transverse magnetization vector. The initial phase 0 is determined by the choice of rotational axis for the initial rotation into the transverse plane. For the example in Fig. 1.2, 0 = =2. Fig. 1.2: Illustration from Ch. 3 of the e ect of an rf pulse on an individual magnetic moment ~ . ~ (a) In a frame rotating about B0 (which is along z , say) at the Larmor frequency (with coordinates ^ 0 , y 0 and z 0 = z ), there is no observed precession about B . Upon application of an rf magnetic ~0 x 0 , the magnetic moment is rotated about x0 at a rate corresponding to eld pulse applied along x ^ ^ the frequency !1 = B1 determined by the amplitude of the rf eld, B1 . A =2 ip relative to its starting position along z 0 is achieved in a time rf provided that !1 rf = =2. (b) The behavior ^
of the same magnetic moment rotation is observed to be more complicated in the xed laboratory frame. This picture has been constructed for the case !1 = 0:06 !0 . In actual MR applications, the frequency !1 would be much smaller in relation to !0 , but then the spiraling would be too dense to illustrate. The signal analyzed in Ch. 7 corresponds to the voltage induced in a receive coil from the time-varying magnetic ux that, in turn, is produced by a rotating magnetization. The inductive coupling of the receive coil to the magnetization may be described, according to a reciprocity principle, as equivalent to a constant ux, produced by a unit current owing around the receive coil, that penetrates the precessing magnetization of the sample. The voltage, or electromotive force (emf ), induced in the receive coil is given by I d ~ ~ emf = ; dt (M Brf )d3r (1.5) ~ where Brf is the static eld produced by the receive coil per unit current. Ignoring any spatial variations and noting that the time derivative of the phase in the transverse magnetization (1.4) dominates the time derivative in (1.5), the emf is proportional to !0 M0. (In other words, the dominant time dependence is due to precession.) From (1.3) and (1.1), it follows that the signal from an MR experiment will depend on the square of the static magnetic eld B0 3 2 (1.6) signal / B0 0 T 1.3. A Brief Overview of MRI Concepts 7 The interest in higher elds stems from the growth of the signal with eld strength we return later in the preview to address the eld dependence of the signal-to-noise ratio. 1.3.4 Magnetic Resonance Spectroscopy Hydrogen protons in di erent molecules are immersed in slightly di erent magnetic environments, even in the presence of identical external magnetic elds. That is, di erent chemical compounds have slightly di erent local magnetic elds which means that the local Larmor frequency is `chemically shifted' to di erent values !0 (j ) depending upon the molecular species type j . Chemical shift imaging is discussed in Chs. 8, 10 and 17. Chemical shift imaging may be considered as adding another dimension, corresponding to the frequency range of di erent species. Each species will have its own contribution to the total signal. For instance, the transverse magnetization (1.4) gives a signal signal / X
j M0 (j )!0(j )e;i!0 (j)t+i (j) (1.7) Here, the time derivative of the phase in (1.4) again yielded the factors !0(j ) in the leading terms. The goal in magnetic eld spectroscopy is to nd the relative (spectral) amplitudes of the di erent frequency components, M0 (j ), in (1.7), whose spin densities 0 (i) would enter via (1.3). This may be analyzed utilizing a Fourier transform to map the time domain back into a frequency domain. The goal in chemical shift imaging is the spatial disentangling of the signals of di erent tissue (spectral) components, such as water and fat. 1.3.5 Magnetic Resonance Imaging The goal of imaging is to correlate a series of signal measurements with the spatial locations of the various sources. When all protons are represented by just one chemical species such as water, then the above spectroscopic analysis simply gives the total signal from all spins regardless of their spatial location in the static magnetic eld, as long as that eld is uniform. We now utilize the fact that the addition of a spatially changing magnetic eld across the sample produces a signal with spatially varying frequency components according to !(x) = B (x) (1.8) where x denotes the spatial coordinate along the direction of the gradient of the eld. This means that the spectral components now represent spatial information and, in turn, leads to the possibility that the signal could be `inverted' and the physical object could be reconstructed (Chs. 9 and 14) or `imaged.' The inversion of the signal is greatly facilitated through a connection to Fourier transforms (Chs. 9, 10 and 11). By constructing an additional coil (a linear gradient coil) that ~ changes the original eld B0 linearly in some direction, the phase in (1.4) becomes linear in the coordinates of that direction, so that the mapping back and forth between signal space and the image position space may be carried out with a Fourier transform. With more gradient coils, data reconstruction by inverse Fourier transformation can be carried in more 8 Chapter 1. Magnetic Resonance Imaging: A Preview spatial dimensions. Two- and three-dimensional `imaging' in MRI is elegantly realized with this powerful mathematical tool (Ch. 10). In particular, the application of a nite bandwidth rf excitation centered at the Larmor frequency of the combined static eld plus a gradient eld leads to the excitation of a layer, or slice, of spins orthogonal to that gradient with slice thickness TH , say (see Fig. 1.3). By employing di erent con gurations of gradient coils, the choice of gradient direction is completely exible, a powerful procedure allowing slices to be acquired in any orientation. No physical rotation of the sample is required. Fig. 1.3: The precession frequency (f = !=(2 )) in the laboratory frame is a function of position along the slice select axis. The original static eld B0 has been supplemented with a eld with constant gradient Gz in the z -direction. The central frequency and spectral bandwidth of the rf pulse ( f BWrf , the shaded horizontal strip) are such that the slice of thickness z TH (the
shaded vertical strip) is uniformly `excited' (i.e., all spins in the slice have the resonance condition satis ed). The fact that the slice is o set from the origin in the z direction by z0 implies that the center frequency of the rf pulse must be o set from the static Larmor frequency f0 = {B0 by {Gz z0 as has been shown along the frequency axis. An important factor in the strength of the signal has been omitted in the above discussions and must be considered. It is the `spin-lattice' decay or relaxation of the signal due to the interactions of the spins with their surroundings. After the magnetization has been rotated into the transverse plane, it will tend to grow back along the direction of the static eld ~ B0 , chosen here to be z . This rate of regrowth can be characterized by a time constant T1 ^ called the longitudinal relaxation time and arises from the interaction between the spins and the atomic neighborhood. The magnetization time evolution is described by the solutions given in Ch. 4 for the famous Bloch equations which incorporate both relaxation and precession e ects. For an initial situation where Mz (0) = 0 (for example, the condition achieved following the application of a =2-pulse), the subsequent regrowth of Mz is given by Mz (t) = M0 (1 ; e;t=T1 ) (1.9) 1.3.6 Relaxation Times 1.3. A Brief Overview of MRI Concepts 9 If the data are sampled following the application of another rf pulse at a time short compared to T1 , the longitudinal magnetization Mz ( ) is suppressed according to (1.9). Therefore, any transverse magnetization obtained by an rf rotation of Mz ( ) into the transverse plane will also be suppressed. With the recognition of another relaxation e ect, a more realistic assessment of the MRI signal may be achieved. The `dephasing' of clusters of spins represents a `spin-spin' decay of the transverse magnetization before data sampling can occur. Consider an experiment (Ch. 8) where a =2 rf pulse is applied at interval time TR where any previous transverse magnetization has decayed away due to the spin-spin e ect and only the longitudinal magnetization corresponding to (1.9) remains to be rotated into the transverse plane. If the signal data are instantaneous sampled at a time TE (`echo time,' see below for an explanation of this nomenclature) following the rf pulse, the signal is proportional to the magnitude of the transverse magnetization given by M?(TE ) = M0 (1 ; e;TR=T1 )e;TE =T2 (1.10) In (1.10), the e;TE =T2 is the spin-spin decay factor characterized by the time constant T2 it is caused by the decorrelation between (dephasing of) the di erent spins. Their phases disperse due to variations in the local precessional frequencies. In general, signals would su er additional suppression due to dephasing from external eld inhomogeneities (T2 would be replaced by a smaller relaxation time T2 < T2). But a `rephasing' or `echoing' of this source of dispersion has been assumed in (1.10) such that the additional suppression has been avoided. This can be achieved by an additional rf pulse application, where the basic idea is to ip all the spins 180 in the transverse plane. The dephasing is reversed and the refocusing of any external eld dispersion occurs at the echo time TE . The three tissue parameters (the spin density and the two relaxation times, T1 and T2 ) play principal roles throughout the book. Their speci c measurements using MR techniques are the subject of Ch. 22. 1.3.7 Resolution and Contrast An interesting aspect of MRI is the fact that resolution (the size of the spatial features that can be distinguished) does not depend upon the wavelength of the input rf eld. Radiofrequencies generally have wavelengths on the order of meters, yet resolution in an MR image is on the order of millimeters. In fact, the inherent resolution in MR is a function of the way the signal and noise are sampled and ltered (see Chs. 12, 13 and 15) and it is ultimately limited only by the di usion (Ch. 21) of the protons through the tissue and the local magnetic eld nonuniformities around the proton. The success of MRI goes beyond resolution and is understood by recognizing its large number of useful variables. MRI can be used to di erentiate between materials because of its sensitivity to proton densities, relaxation times, temperature, proton motion, the chemical shift in the Larmor frequencies, and tissue heterogeneity, as examples. This large set of variables permits images to be generated with di erent levels of contrast based upon the desired usage. Therefore, MRI is more versatile than those imaging techniques restricted to only one type of contrast. 10 Chapter 1. Magnetic Resonance Imaging: A Preview Contrast-to-noise and contrast mechanisms are rst described in Ch. 15 but important aspects of image contrast already can be understood from (1.10). Examining the behavior of the exponentials, we see that for long TR (relative to T1 ) and short TE (relative to T2 ), the image will be sensitive only to the tissue spin density (Fig. 1.4a). For TE ' T2 and long TR , the image is weighted by both spin density and T2 (Fig. 1.4b) and the contrast between tissues with di erent T2 is enhanced. Often the spin density-weighted images and T2 -weighted images exhibit similar contrast features, the latter enhancing the former. Lastly, for TR T1, and short TE , the image is weighted by both spin density and T1 (Fig. 1.4c). The interest in higher elds stems from the fact that the signal-to-noise ratio (SNR is the subject of Ch. 15) increases with eld strength. While machines providing lower elds (less than 0.5 T) are less expensive, they produce lower SNR, as compared to mid elds (0.5 T to 1.0 T) and high elds (higher than 1.0 T). The signal (1.6) exhibits quadratic growth with B0 but this is partially o set by the fact that the noise has linear B0 dependence at high elds. In the range from 0.5 T to 4.0 T, the implied linear growth of SNR with eld strength has been experimentally validated in human experiments. There also has been concern about rf heating and nonuniform rf elds, where wavelengths nally play a role, as higher elds are considered. (Table 1.1 shows a comparison of free-space wavelengths for the di erent frequency ranges of familiar electromagnetic wave categories.) However, we have noted that the eld strength of 1.0 T corresponds to 42.6 MHz, and this implies a rather long seven-meter free-space wavelength for imaging protons. Hence the rf eld wavelengths for higher magnetic elds fall below 1 m only above 7.0 T. On the other hand, in humans, the relative electrical permittivity r is about 50 near 1 T to 2 T, and the interior wavelengths are therefore reduced by a factor of 1=p r inside the body. This reduces the e ective wavelength to about 1 m at 1 T for in vivo imaging and rf eld nonuniformity must be considered at higher B0 values. More detailed rf pulse considerations are the subject of Ch. 16. The design issues for the coils producing the static eld, the gradient elds and the rf elds are discussed in Ch. 27 where SNR is also revisited in terms of the high eld dependence. 1.3.8 Magnetic Field Strength Problem 1.2
Find the frequency and free-space wavelength associated with the rf eld required for proton magnetic resonance at each of the di erent B0 values of a) 0.04 T, b) 0.2 T, c) 1.5 T and d) 8 T. 1.3.9 Key Developments in Magnetic Resonance An important mechanism in MRI is the `echoing' capability discussed earlier for the recovery of some of the signal lost to transverse relaxation. (We should also mention `gradient 1.3. A Brief Overview of MRI Concepts 11 (a) (b) (c)
Fig. 1.4: Images of the human head with di erent forms of contrast: (a) a spin density-weighted image, (b) a T2 -weighted image and (c) a T1 -weighted image. These di erent acquisitions can be
seen to create di erent contrasts between white matter, gray matter and cerebrospinal uid. They all reveal excellent anatomic detail. 12 Category Subcategory Chapter 1. Magnetic Resonance Imaging: A Preview Frequency (MHz) Field strength (T) Wavelength (m) 0.03-0.3 0.3-3 0.54-1.6 3-30 30-300 54-216 300-3 103 3 103-3 104 104-3 105 7 10;4-7 10;3 7 10;3-0.07 0.013-0.038 0.07-0.7 0.7-7 1.27-5.07 7-70 70-700 233-7 103 104-103 103-102 555-188 102-10 10-1 5.55-1.39 1-0.1 0.1-0.01 0.3-10;3 radio waves LF (long wave) MF (medium wave) AM radio (MF) HF (short wave) VHF (short wave) FM radio (VHF) UHF SHF microwaves Table 1.1: Range of radio and microwave frequencies. The letters F, L, M, H, V, U and S refer to
frequency, low, medium, high, very, ultra and super, respectively. Associated free-space wavelengths and NMR eld strengths for protons are given here. echoes' where dephasing brought about by the external gradient eld itself is countered by reversing the gradient direction during its application.) When Hahn happened on the `spin echo' concept, it was through an application of multiple rf pulses. This concept made it possible to collect data in what would otherwise be considered poor experimental conditions (inhomogeneous magnetic elds) where little signal would remain. What he had found initially to be a spurious signal later became a workhorse in NMR and in clinical MRI where disease states are often clearly diagnosed in T2-weighted images (see Fig. 1.4b). In a somewhat related manner, what presents itself as an unexpected and even bothersome image inaccuracy, or an `artifact,' may end up opening new doors and even a new direction of research. Two fairly recent examples of this include MR angiography (the study of blood vessels using MRI) and MR brain functional imaging. The rst example came about because of a thrust to eliminate ow and motional blurring. In so doing, early researchers came up with the concept of ow compensation. Eliminating the artifacts in this way led to the very pleasant surprise of enhanced images of blood vessels Thus began the sub eld of MR angiography. The phenomena of blood ow and heart and lung motion and the methods for obtaining their images are detailed in Chs. 23 and 24. The second example occurred when the loss of signal due to the susceptibility of a contrast agent was so large it led to a dramatic loss of signal on its rst pass through the brain. It was soon realized that veins also had a di erent susceptibility than the rest of the brain and, hence, veins could be recognized by their phase alone or the signal loss they caused at long echo times. When blood ow changes, so does the blood's deoxyhemoglobin concentration and, therefore, the blood's susceptibility and its e ect on the signal also changes. Evidently, blood ow changes occur when the brain activates, which, in turn, leads to signal changes. 1.4. Suggested Reading 13 Today, we can measure these changes in a matter of seconds, i.e., in a sense, we can detect the brain function. This is the basis of MR brain functional imaging known as fMRI and has led to a major focus in the elds of neuroscience and neuroradiology. It is considered in Ch. 25. Besides artifacts, there were other reasons to improve the MR methodology. Spin echo scans took a long time to acquire and could not o er dynamic imaging to study, for example, the beating heart. To speed up data acquisition, researchers pushed toward fast `gradient echo' imaging (where no extra refocusing rf pulses are applied) which reduced scan times from minutes to seconds (see Ch. 18) and to `echo planar' methods which could acquire data in a fraction of a second (see Ch. 19). These methods had direct implications on MR angiography and functional brain imaging as well as cardiac MRI. However, they still su er from inhomogeneous elds which cause signal loss and image distortion (see Ch. 20). These are less of a problem today because of improvements in main magnet design (see Ch. 27), but some remnant inhomogeneity e ects remain caused by the local elds in the body itself (see Ch. 25). Even these remnants are being addressed today with better sequence designs (see Ch. 26) and improvements in our understanding of the principles of magnetic resonance imaging. We turn now to the details and the exposition of these and other key concepts in MRI in the coming chapters. It is hoped that the following discussions will help elucidate those remarks that were inadequately explained in this brief overview. 1.4 Suggested Reading
Certain references that o er a good background for this opening chapter are given next. More general references are listed later. F. Bloch. Nuclear induction. Phys. Rev., 70: 460, 1946. R. Damadian. Tumor detection by nuclear magnetic resonance. Science, 171: 1151, 1971. W. Gerlach and O. Stern. Ueber die Richtungsquantelung im Magnetfeld. Ann. Phys., 74: 673, 1924. E. L. Hahn. Spin Echoes. Phys. Rev., 80: 580, 1950. P. C. Lauterbur. Image formation by induced local interactions: Examples employing NMR. Nature, 242: 190, 1973. P. Mans eld and P. K. Grannell. NMR `di raction' in solids? J. Phys. C: Solid State Phys., 6: L422, 1973. E. M. Purcell, H. C. Torrey and and R. V. Pound. Resonance absorption by nuclear magnetic moments in a solid. Phys. Rev., 69: 37, 1946. I. I. Rabi, J. R. Zacharias, S. Millman and P. Kusch. A new method of measuring nuclear magnetic moments. Phys. Rev., 53: 318, 1938. 14 Chapter 1. Magnetic Resonance Imaging: A Preview There are a number of texts and articles on magnetic resonance. We have grouped these according to (1) primary technical references, which contain a great deal of relevant information to this text, (2) secondary references, both technical and clinical which also contain much useful information, (3) tertiary references, which will broaden the reader's perspective on MR in general and (4) some interesting review articles and texts to add overview and historical perspectives. Primary Technical References
A. Abragam. The Principles of Nuclear Magnetism. Oxford, 1961. P. T. Callaghan. Principles of Nuclear Magnetic Resonance Microscopy. Oxford, New York, 1991. C.-N. Chen and D. I. Hoult. Biomedical Magnetic Resonance Technology. Bristol, Philadelphia, 1989. P. Mans eld and P. G. Morris. NMR Imaging in Biomedicine. Supplement 2, Advances in Magnetic Resonance, Ed. J. S. Waugh, Academic Press, New York, 1982. P. G. Morris. Nuclear Magnetic Resonance Imaging in Medicine and Biology. Oxford, New York, 1986. A. M. Parikh. Magnetic Resonance Imaging Techniques. Elsevier, New York, 1992. M. T. Vlaardingerbroek and J. A. den Boer. Magnetic Resonance Imaging: Theory and Practice. Springer-Verlag, Berlin, 1996. Secondary Technical and Clinical References
E. R. Andrew. Nuclear Magnetic Resonance. Cambridge at the University Press, New York, 1955. R. R. Edelman, J. R. Hesselink and M. B. Zlatkin. MRI: Clinical Magnetic Resonance Imaging. 2nd ed., Saunders, Philadelphia, 1996. D. G. Gadian. Nuclear Magnetic Resonance and Its Applications to Living Systems, 2nd ed., Oxford, University Press, 1995. J. E. Potchen, E. M. Haacke, J. E. Siebert and A. Gottschalk. Magnetic Resonance Angigraphy. Concepts & Applications. Mosby, St. Louis, 1993. R. T. Schumacher. Introduction to Magnetic Resonance. W. A. Benjamin, Inc., New York, 1970. C. P. Slichter. Principles of Magnetic Resonance. Springer Verlag, III ed., New York, 1990. D. D. Stark and W. G. Bradley, Jr. Magnetic Resonance Imaging. Mosby Year Book, II ed., St. Louis, 1992. 1.4. Suggested Reading 15 F. W. Wehrli. Fast-scan Magnetic Resonance: Principles and Applications. Raven Press, New York, 1990. Tertiary References A. Bax. Two-dimensional Nuclear Magnetic Resonance in Liquids. Delft University Press, Delft, Holland, 1982. P. T. Beall, S. R. Amtey and S. R. Kasturi. NMR Data Handbook for Biomedical Applications. Pergamon Press, New York, 1984. J. D. de Certaines, W. M. M. J. Bovee and F. Podo, eds.. Magnetic Resonance Spectroscopy in Biology and Medicine. Functional and Pathological Tissue Characterization. Pergamon Press, Oxford, 1992. R. R. Ernst, G. Bodenhausen and A. Wokaun. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford, New York, 1987. W. W. Paudler. Nuclear Magnetic Resonance: General Concepts and Applications. Wiley, New York, 1987. C. P. Poole. Theory of Magnetic Resonance. Wiley, New York, 1987. N. Salibi and M. A. Brown. Clinical MR Spectroscopy: First Principles. Wiley-Liss, New York, 1998. D. M. S. Bagguley, ed.. Pulsed Magnetic Resonance: NMR, ESR, and Optics. A Recognition of E. L. Hahn. Oxford Science, Oxford, 1992. B. Blumich and W. Kuhn, eds.. Magnetic Resonance Microscopy. Methods and Applications in Materials Science, Agriculture and Biomedicine. VCH, Weinheim, 1992. G. M. Grant and R. K. Harris, eds.. Encyclopedia of Nuclear Magnetic Resonance, Volume 1, Historical Perspectives. John Wiley and Sons, Chichester, England, 1996. W. S. Hinshaw and A. H. Lent. An introduction to NMR imaging: From the Bloch equation to the imaging equation. Proc. IEEE, 71: 338, 1983. I. L. Pykett, J. H. Newhouse, F. S. Buonanno, T. J. Brady, M. R. Goldman, J. P. Kistler and G. M. Pohost. Principles of nuclear magnetic resonance imaging. Radiology, 143: 157, 1982. Historical/Review References 16 Chapter 1. Magnetic Resonance Imaging: A Preview Chapter 2 Classical Response of a Single Nucleus to a Magnetic Field
Chapter Contents
2.1 Magnetic Moment in the Presence of a Magnetic Field 2.2 Magnetic Moment with Spin: Equation of Motion 2.3 Precession Solution: Phase Summary: The concept of a magnetic moment is described. The classical equations of motion for the magnetic moment in the presence of an external eld are developed. Solutions for static elds are found and their connection to gyroscopic precession is made. Introduction
Magnetic resonance imaging (MRI) works because we can observe the way the protons in the human body respond to external magnetic elds. The MRI experiment is really a combination of a two-step process where, in the rst stage, the proton `spin' orientation is manipulated by an assortment of applied magnetic elds. In the second stage, changes in orientation can be measured through the interaction of the proton's magnetic eld with a coil detector. Although each proton eld is minuscule, a signi cant signal can be measured resulting from the sum of all elds of all a ected protons of the body. In this chapter, and the next, the focus is on the basic element of the rst stage, a single proton's response to an external eld, ignoring the interactions of each proton with its surroundings. These important interactions are deferred to the fourth chapter, where we shall include their e ects in the equations of motion. Extensive use is made of a classical picture where the proton is viewed as a tiny spinning charge with an attendant circulating electric current. Indeed, much of MRI theory can be understood classically, although in Ch. 5 we will show how to derive the principal results of this chapter in the more fundamental quantum mechanical framework. 17 18 Chapter 2. Classical Nucleus in Magnetic Field 2.1 Magnetic Moment in the Presence of a Magnetic Field Fig. 2.1: Circular current loop depicted in two di erent orientations relative to a uniform magnetic eld. The forces on representative di erential current segments are shown (one d~ is explicitly shown `
in each case). The rst (a) shows the current plane perpendicular to the eld where there is no net twist (torque) the second (b) shows the current plane at an arbitrary angle to the eld where there is a nonzero torque. 2.1.1 Torque on a Current Loop in a Magnetic Field We begin with a study of magnetic forces on current-carrying conductors, in order to introduce the interaction of a proton with an external magnetic eld. A circular loop of current I and area A is pictured with two di erent orientations in Fig. 2.1. If an external magnetic ~ eld B is turned on, the loop will feel a di erential force on each of its di erential segments d~ given by the basic Lorentz force law (see Appendix A) ` ~ dF = Id~ B ` ~ (2.1) This cross product implies that the di erential force is perpendicular to the plane de ned by two vectors: the current segment and the magnetic eld evaluated at that segment. It is in the direction that a right-hand screw, perpendicular to the plane, advances when rotated from the current segment vector to the magnetic eld vector. The total force on the circular loop, and indeed on any closed loop, due to a uniform (constant over space) external magnetic eld is zero. As an example, the symmetrical vector sum of the di erential forces on either current loop in Fig. 2.1 clearly vanishes for any loop orientation. To prove it in general, the result Hfollows from an integral over (2.1), where I ~ and B can be taken outside the integral, since d~ = 0 for an integration around any closed ` 2.1. Magnetic Moment in the Presence of a Magnetic Field 19 path. Now zero total force means zero change in the total momentum ~, from Newton's law, p ~ p F = d~ (2.2) dt Therefore, a current loop initially at rest in a spatially constant magnetic eld stays at rest. (For the present, we also assume the eld is constant in time.) But this is not the whole story. A current loop can be rotated by the eld, depending on the loop's orientation. Rotations can arise from forces applied o center, even when the vector sum of all forces ~ cancels out. The vector quantity used to describe the rotation of an object is torque (N ).1 If the sum of the di erential torque contributions, ~ r ~ dN = ~ dF (2.3) is nonzero, the current loop can be expected to rotate, instantaneously, at least, about the ~ axis along dN . In (2.3), ~ is the position vector from, say, the center of the loop to the point r of application of the di erential force. Problem 2.1
IfH the total force on a current loop (or any system, for that matter) is zero ~ ~ Hr ~ ( dF = 0), show that the total torque N = ~ dF is independent of the choice of origin. Hint: Change to a primed coordinate system where ~ = r0 + ~0, with r ~ r an arbitrary shift ~0. r In the special case where the plane of the circular loop is perpendicular to a constant ~ magnetic eld, Fig. 2.1a, each ~ is parallel to its dF so each di erential torque is zero. But r in the general case, Fig. 2.1b where the plane is at some arbitrary angle to that eld, it is evident that there is a net torque rotating the loop back into the perpendicular plane. We come back to what the proton actually does in response to a similar torque, after a discussion of a basic torque formula. The formula for the net torque on any current distribution, which is exact in a constant magnetic eld, is given in terms of the `magnetic dipole moment' or simply `magnetic moment' ~ : ~ ~ N =~ B (2.4) with ~ itself to be discussed shortly. This cross product can be taken with respect to any point, but because the net force is zero, the net torque vector is independent of the origin chosen (see the previous problem). In place of a general argument for (2.4),2 we present a speci c calculation of the torque on a circular loop. This serves as an example of how a loop magnetic moment arises in electromagnetic formulas, and the result can be used to verify the form (2.4). But rst a formula for the magnetic moment of a current loop is needed.
1 2 Discussions of torque can be found in most introductory physics textbooks. See Appendix A for further remarks and references. 20 Chapter 2. Classical Nucleus in Magnetic Field To prescribe the magnetic moment for a planar loop, imagine a right-hand screw piercing the interior of the loop, and perpendicular to the plane of the loop. De ne a unit vector n ^ to point along the direction the screw advances, if the screw is rotated in the same sense as the current ows. The magnetic moment vector for planar loops is given by ~ = IA n ^ (2.5) where A is the area of the loop interior. A sample planar moment is illustrated in Fig. 2.2, along with an alternative description of the right-hand rule. Fig. 2.2: A loop with current I lying in a plane. The perpendicular unit vector n points up from ^ the side where the area A is on our left if we were to walk along the current path in the direction of current ow. The magnetic moment for the loop is IAn. ^ The torque due to a constant magnetic eld is to be calculated for a circular loop with radius R and current I centered in the x-y plane. Let the eld lie in the y-z plane and have magnitude B . The di erential torque on d~ can be written quite generally as ` ~ r ~ ` r dN = ~ (Id~ B ) = Id~ (B ~) ; I B (d~ ~) ` ~ ` ~ r (2.6) Use has been made of the double cross-product formula3 after combining (2.1) and (2.3). The torque can be calculated with respect to the origin (we have seen that torque on a closed current loop is independent of this choice). The magnetic eld and the cylindrical unit vectors shown in Fig. 2.3a are ~ B = B (cos z + sin y) ^ ^ ^ = cos x + sin y ^ ^ ^ = ; sin x + cos y ^ ^ (2.7) where the angles are also illustrated in that gure. With d~ = Rd ^ and ~ = R ^, the second ` r scalar product in (2.6) is zero, and a reduction of the rst scalar product using (2.7) yields ~ dN = IBR2 sin sin ^ d (2.8) An integration of (2.8) over the polar angle , with ^ from (2.7), gives Rthe total torque. R There is no net y-component because 02 d sin cos = 0. The integral 02 d sin2 = 3 The `BAC-CAB' rule is A ~ (B C ) = B (A C ) ; C (A B ). ~ ~ ~ ~ ~ ~ ~ ~ 2.1. Magnetic Moment in the Presence of a Magnetic Field is needed4 for the calculation of the net x-component, leading to 21 ~ N = ;I R2 B sin x ^ (2.9) ~ Equation (2.9) is exactly ~ B in view of the fact that the magnetic dipole moment for the circular loop of Fig. 2.3a is ~ = I R2 z ^
(2.10) The reader is invited to investigate, in similar fashion, both the force and the torque for the example of a rectangular loop in Prob. 2.2. Exact for constant elds, the torque formula (2.4) is also very accurate for small loops in a spatially varying eld. The only requirement is that the loop scale (say, its diameter D) must be much less than the typical distances over which the eld changes. (For example, j B j ' [email protected][email protected] << jB j.) Corrections would arise from, for example, `higher moments' such as electric quadrupole moments. In the case of a proton, however, the electric quadrupole moment, and all other higher moments, are zero. Fig. 2.3: Circular (a) and rectangular (b) current loops lying in the x-y plane and a magneticcharge pair along the z -axis (c), all experiencing a constant magnetic eld. In each case, the eld lies in the z -y plane.
Such integrals lead to a rule of thumb: The average value of sin2 or cos2 over any multiple of =2 is 1=2.
4 22 Chapter 2. Classical Nucleus in Magnetic Field Problem 2.2
Consider the constant magnetic eld in the z-y plane at an angle with the z-axis as shown in Fig. 2.3b. A current I ows in a rectangular loop with sides a and b lying in the x-y plane. a) Show that the di erential forces on the four respective current legs have the forms IB (; cos y + sin z ) jdxj, ;IB (sin z ; cos y) jdxj, IB cos x jdyj, ^ ^ ^ ^ ^ and ;IB cos x jdyj. ^ b) Show that, after integration, the total force on the loop is zero. c) Show that the di erential torques on the four respective current legs relative to the center of the loop have the forms xx; a y] IB (; cos y +sin z ) jdxj], ^ ^ ^ ^ b^ 2 b^ xx + a y] ;IB (sin z ; cos y) jdxj], 2 x + yy] IB cos x jdyj], and ; 2 x + ^ 2^ ^ ^ ^ ^ yy] ;IB cos x jdyj]. ^ ^ d) Show that the integration of the results in (c) is simpli ed by the vanishing of certain integrands odd in x or y, leading to a net torque on the loop given by ;IBab sin x. Again, the double cross product formula could have been ^ used to bypass the force calculation, in achieving this result. e) Find the magnetic moment vector for this loop from (2.5) and show that the total torque found in (d) agrees with the formula (2.4). 2.1.2 Magnet Toy Model A magnetic dipole moment can be viewed as a pair of magnetic charges with equal magnitudes and opposite signs, in analogy with the way electric charges lead to electric dipole moments. The word `dipole' has its origin in this model, which is an acceptable alternative to the current loop picture as long as we use it to investigate only the eld outside the moment structure. The eld inside for the charge pair is di erent than that for the current loop. It should also be noted that magnetic monopoles have yet to be observed in nature. The magnetic moment in Fig. 2.3c resembles a bar magnet with the positive and negative magnetic charges playing the role of the north and south poles, respectively. The analogy to an electric dipole moment (Appendix A) implies that the magnetic moment vector is ~ m = nqm d ^ (2.11) where d is the distance between the magnetic charges, and n points along the line directed ^ from the negative charge ;qm to the positive charge qm . We wish to check that the net torque on the pair of magnetic charges has the expected form (2.4). The force on a magnetic charge qm due to a magnetic eld is like that on an electric charge due to an electric eld. It is ~ ~ Fm = qm B (2.12) 2.2. Magnetic Moment with Spin: Equation of Motion 23 Since the total force is zero, for convenience we can choose the torque reference point to lie on the negative charge (the origin in Fig. 2.3c). With respect to that point, the net torque is that due to the force on the charge at z = d, ~ ~ Nm = (^d) (qmB ) z (2.13) Expression (2.13) obviously agrees with (2.4) and implies a rotation of the magnetic-charge dipole toward the eld direction. The torque equation (2.4) is the starting point for future classical discussions of the magnetic-moment behavior in the presence of a magnetic eld. It applies as well to elds that vary in space and time. 2.2 Magnetic Moment with Spin: Equation of Motion
The lesson so far is that a magnetic moment, such as that corresponding to a current loop or a bar magnet, will try to line up along the direction of an external magnetic eld. This is much as a falling pendulum tries to align itself with the direction of gravity.5 If the moment is associated with an angular momentum (a `spinning'), then the motion is changed. To see what the new motion is, we introduce in this section the general di erential equation for angular momentum in the presence of external torque and the atomic relation between intrinsic angular momenta and magnetic moments. The resulting di erential equation is solved for a special case in the next section. ~ Nonzero total torque on a system implies that the system's total angular momentum J must change according to ~ ~ dJ = N (2.14) dt This equation, discussed in most introductory mechanics textbooks, can be derived, as a problem, for a single point mass. 2.2.1 Torque and Angular Momentum Problem 2.3
Consider a point mass m moving at velocity ~ (t) with position ~(t) de ned by v r ~ some origin. Its angular momentum relative to that origin is therefore J = ~ p with p = m~ . Derive (2.14) by taking the time derivative of this angular r ~ ~ v momentum. Note (2.2) and (2.3). The generality of (2.14) follows by considering a system as a limit of many point particles. The total angular momentum is the corresponding limit of ~ Xr p (2.15) J = ~i ~i
i
5 We return to a related gravitational analogy in Sec. 2.3. 24 with respect to some origin. Chapter 2. Classical Nucleus in Magnetic Field We next formulate the connection between the proton intrinsic angular momentum (or what is often referred to as its `spin') and its moment. The connections for other nuclear particles are also of interest. The proton spin can be thought of as leading to a circulating electric current, and, hence, an associated magnetic moment. The direct relationship between the magnetic moment and the spin angular momentum vector is found from experiment, ~ ~= J (2.16) The proportionality constant in (2.16) is called the gyromagnetic (or magnetogyric) ratio and depends on the particle or nucleus. For the proton, it is found to be6 = 2:675 108 rad=s=T (2.17) or, what may be referred to as `gamma-bar,' where T is the Tesla unit of magnetic eld and is equal to 10,000 Gauss (G). Of all the numbers in MR, { is probably the one most often used in back-of-the-envelope calculations. From (2.16) we are justi ed, in any discussion, to refer either to spin, or to the magnetic dipole moment, since they track each other. It is useful to compare the experimental values for the gyromagnetic ratios with a formula for a simply structured system. Consider a point particle with charge q, mass m, and speed v going in a circle of radius r. A calculation carried out in the following problem yields the result for the gyromagnetic ratio of a point particle (2.19) (point charge in circular motion) = 2q m This is not an accurate formula for the nuclear particles of interest, but it does help us understand the di erences due to mass. The simple result (2.19) also arises as a coe cient in basic magnetic moment quantities. In Ch. 5, we will introduce the quantum unit of angular momentum h h=2 (also referred to as `h-bar') where h is Planck's constant. For the charge e and mass me of the electron, the basic magnetic moment unit is the Bohr magneton eh = 9.27 10;24 A m2 (2.20) B 2me For the same charge but with a proton mass, the nuclear magneton is eh = 5.05 10;27 A m2 (2.21) n 2mp
6 2.2.2 Angular Momentum of the Proton { = 2 = 42:58 MHz=T (2.18) To this accuracy, the measured value for a proton bound in H2 O is the same as that for a free proton. 2.2. Magnetic Moment with Spin: Equation of Motion 25 Problem 2.4
a) Show that the angular momentum ~ p of the circulating particle with r ~ respect to the center is mrvn where n is a unit vector perpendicular to the ^ ^ plane of the circle. Here, n points in a direction given by the right-hand ^ rule applied to the particle's motion. b) Show that the magnetic moment associated with the motion of the point charge is qvr=2 and thus that the gyromagnetic ratio is given by (2.19). c) Evaluate numerically the gyromagnetic ratio (2.19), choosing the same mass (1:67 10;27 kg) and charge (1:60 10;19 C) as for a proton. The di erence between your answer and (2.17) is due to the more complicated motion of the proton constituents, the `quarks.' For related reasons, a neutron has a nonvanishing magnetic moment despite its zero overall charge. 2.2.3 Electrons and Other Elements From the mass dependence of the previous exercise (see Prob. 2.4), it is not surprising that the factors can vary from one particle to another, if only because their masses may di er. Indeed, the electron factor is expected to be much larger than that for the proton in view of the inverse mass dependence. The di erence between the observed ratio j ej = 658
p (2.22) and the measured mass ratio mp=me = 1836 (the electron mass is 9:11 10;31 kg) is due to the di erence in the structure of the two particles. The electron has no apparent size, while the proton has a size on the order of 1 fermi (10;15 m) and is a complex composite of quarks. They do, however, have exactly the same spin. Why do we not use electron imaging? The principal reason is the striking di erence in the frequency with which a magnetic moment precesses about a static magnetic eld, and which is discussed in the next section. The precession frequency is proportional to the gyromagnetic ratio and, with the di erence shown in (2.22), it is much larger for the electron. In the standard MRI experiment, an additional, oscillating magnetic eld is required,7 whose frequency is matched with the precession frequency. For static elds in the Tesla range, a radiofrequency eld in the microwave spectrum is thus needed for electron experiments. However, too much energy would be deposited in human bodies, if electron spins were `excited' by these rf elds. Other problems associated with eld inhomogeneities and signal-to-noise (subjects of importance in much of the remainder of this book), imply that reducing the frequency by reducing the static eld strength is not a readily available alternative.
7 The additional eld is detailed in the next chapter. 26 Chapter 2. Classical Nucleus in Magnetic Field For nuclei, the rst requirement is nonzero intrinsic angular momentum (total `spin'). It might be guessed, incorrectly, that magnetic moments of heavier nuclei would be rather smaller, roughly by the inverse ratio of their total masses to the proton mass, than that for a proton. In reality, they usually are not very much smaller, nor are they ever very much larger. Only the `outer shell' nucleons contribute to the total angular momentum of heavier nuclei the total nuclear mass is not relevant to the determination of the factor. In general, protons and, separately, neutrons pair up as much as possible inside of a nucleus, with their spins and orbital motions canceling. Consider the di erent nuclear cases. Each `even-even' nucleus (even numbers of protons and even numbers of neutrons) has zero total angular momentum, and, hence, zero magnetic moment. For this reason, we cannot image the 16 O and 12 C in our bodies with the MR techniques under discussion. The magnetic moment of an even-odd nucleus can be approximately understood in terms of the single unpaired nucleon, but admixtures of states di ering in the con gurations of the other nucleons must sometimes be taken into account. The unpaired proton and neutron in odd-odd nuclei are not in the same orbital state and do not conspire to give zero spin, in general. For example, nitrogen has twice the spin of hydrogen, and a nonvanishing magnetic moment. The factors in (2.16) for nuclei with nonzero angular momentum are consequently within an order of magnitude or so of that for the proton. The relation to the nuclear magnetic moment does involve the nuclear spin IN = IN (2.23) The gyromagnetic ratios are determined by measurements, and their values are often rather smaller than that for the proton. Smaller values for are not the only reason, however, that imaging of elements other than hydrogen is di cult in the human body. The problem is usually one of low concentration. Still, sodium (23 Na) and phosphorus (31 P) are of imaging interest in view of their nonvanishing magnetic moments (their spins are `3/2' and `1/2' in quantum units h in terms of which the proton has spin `1/2,' see Ch. 5). Their relative factors, spin, and concentration are listed together with other nuclei of interest in Table 2.1. 2.2.4 Equation of Motion
Using both the relation (2.16) between the spin and the magnetic moment and the expression ~ (2.4) for torque on a magnetic moment due to an external magnetic eld B , we nd that (2.14) reduces to d~ = ~ B ~ (2.24) dt This fundamental equation of motion is at the heart of the rotations and precessions that we shall frequently discuss. It is a simple version of the Bloch equation to be presented in Ch. 4. Important corrections arise from the interactions of spins with their surroundings, processes which are referred to as `relaxation' phenomena. 2.3. Precession Solution: Phase Nucleus hydrogen 1H sodium 23 Na phosphorus 31 P oxygen 17 O uorine 19 F Spin Magnetic moment 1/2 2.793 3/2 2.216 1/2 1.131 5/2 -1.893 1/2 2.627 { Abundance in human body 42.58 88 M 11.27 80 mM 17.25 75 mM -5.77 16 mM 40.08 4 M 27 Table 2.1: List of selected nuclear species with their spins (in units of h where the proton has spin 1=2), their associated magnetic moments in units of a nuclear magneton n (2.21), gyromagnetic ratios { (in units of MHz/T), and their relative body abundances (1 M = 1 molar = 1 mole/liter). For comparison, the hydrogen (1 H) molarity of water is 110 M, and brain gray matter, for example, has a water content of 80% leading to an abundance of 88 M. The quoted body abundances will vary from tissue to tissue. Certain common elements are omitted, such as 12 C and 16 O, because their nuclear spins (and hence their nuclear magnetic moments) are zero. A negative sign for the moment and gyromagnetic ratio refers to the fact that the magnetic moment is anti-parallel to the angular momentum vector. 2.3 Precession Solution: Phase
The di erential equation (2.24) for a static eld is readily solved, and the corresponding precessional motion is important to the MR application. Di erent approaches to this solution are described below. A comparison with a well-known gravitational analogy is brie y discussed before the magnetic moment case is detailed. 2.3.1 Precession via the Gyroscope Analogy
If instead of a pendulum we consider an object spinning along its axis, a gyroscope or spinning top, experience tells us that it does not fall but rather precesses around the gravity axis. The precession in Fig. 2.4a is the result of a gravitational torque perpendicular to the spin axis. Exactly analogous precession takes place for a spinning magnetic moment experiencing torque from a constant external magnetic eld. However, the torque,8 and hence the precession, are in the opposite direction. Compare the gure to Fig. 2.5. There is a direct correspondence between the equations for a magnetic moment and a spinning top immersed in constant vertical magnetic and gravitational elds, respectively. It is easy to show that the force on a rigid body like a top due to a uniform gravitational pull on all of its parts is equivalent to a total force mg applied at its center of mass (c.m.). The resultant torque with respect to the bottom support pivot is ~c:m: (;mgz ) where r ^ ~c:m: is the position of the c.m. relative to the pivot. The torque is in the direction of r increasing azimuthal angle in Fig. 2.4b, and, as it will be proven when the corresponding magnetic-moment case is solved later, the top precesses along a cone at constant .
The pendulum tends to fall down toward the gravity direction, while the magnetic moment, in the absence of the spin e ect, tends to swing up toward the magnetic eld direction.
8 28 Chapter 2. Classical Nucleus in Magnetic Field (a) (b) a constant gravitational eld. (b) The corresponding angular momentum diagram showing how the gravitational torque leads to precession. The precession is in the opposite sense to that for a magnetic moment immersed in a magnetic eld pointing in the positive z -direction (see Fig. 2.5). Fig. 2.4: (a) A symmetrical spinning top with spin angular velocity ~ and mass m precessing in The torque expressions B~ z and ;mg~c:m: z map into one another for ~ parallel ^ r ^ to ~c:m:. Furthermore, the angular momentum for a symmetrical rigid body is I ~ with I the r moment of inertia and ~ the angular velocity vector pointing along the spin axis. This, too, ~ ~ is parallel to the angular momentum associated with ~ . The equation dJ=dt = N looks the same for both. More general top trajectories, including nutations which follow an initial downward `shove,' have counterparts (and some di erences) in the magnetic moment case. This is especially true when time-dependent magnetic elds are considered in the next chapter. 2.3.2 Geometrical Representation
Some general remarks can be made about the motion predicted by the `magnetic torque' equation (2.24). When the time rate of change of a vector is proportional to a cross product involving that vector, we immediately see its magnitude = j~ j is unchanged. Problem 2.5
Demonstrate that (2.24) implies d =dt = 0. Hint: Form a scalar (dot) product of both sides of (2.24) with ~ . 2.3. Precession Solution: Phase 29 Fig. 2.5: Clockwise precession of a proton's spin about a magnetic eld. As shown, the di erential d is negative. The magnitude may be xed but the direction is changing. The instantaneous change in the magnetic moment direction is equivalent to an instantaneous left-handed rotation about ~ B , the other vector in the cross product. To see the rotation and get the instantaneous rotation frequency, consider Fig. 2.5. The di erential change in the moment in time dt is ~ ~ d~ = ~ Bdt, which is perpendicular to the plane de ned by ~ and B . This pushes the ~ tip ~ (when viewing from `above' with B pointing at the viewer) on a clockwise precession ~ around a circular path. The tip would stay on that same circle if B were constant in time. ~ If d is the angle subtended by d~ , and is the angle between ~ and B , the geometry of Fig. 2.5 indicates that jd~ j = sin jd j (2.25) On the other hand, ~ jd~ j = j~ B jdt = B sin dt (2.26) ~ A comparison gives B jdtj = jd j with B jBj, giving the well-known Larmor precession formula, ! d = B (2.27) dt ~ along an instantaneous axis de ned by a left-handed screw rotation about B . That is, d = ;! (2.28) dt so that the angular velocity vector is9 ! = ;!z ~ ^ (2.29)
We shall use the convention that angular frequencies are positive. The rotation sense can be indicated by specifying the angular velocity vector, about whose direction the rotation is right-handed. Therefore, the rotation indicated by (2.28) is left-handed with respect to the positive z -axis.
9 30 = ;!0 t + Chapter 2. Classical Nucleus in Magnetic Field ~ If the eld is along the z-axis and constant in time, B = B0z , the solution of (2.28) is ^
0 (constant eld case) (2.30) where 0 is the initial angle. Again, notice the minus sign ( is the usual azimuthal angle de ned in right-handed fashion around the z-axis) (2.30) shows constant left-handed precession around the eld direction. From now on, we de ne the Larmor frequency for the constant eld case to be !L(constant eld) !0 B0 (2.31) Fig. 2.6: The vector ~ (t) is obtained by the clockwise rotation of ~ (0) through an angle = !0t about the z -axis ( = ; + 0 ). The Cartesian coordinates of ~ (t), shown in (a), can be found by rotating the individual components of ~ (0), as shown in (b). Note that 0 x (0), 0 y (0). x y The z -component of the vector is not changed only the transverse components are shown in the
B0 z . ^ gures. The clockwise rotation corresponds to magnetic-moment precession about the static eld 2.3.3 Cartesian Representation What about a formula for ~ (t) in terms of Cartesian axes? As an alternative to working with (2.30), it is possible to use the rotation picture developed above and trigonometry to ~ derive the answer for B = B0z . Note rst that the z-component of the magnetic moment is ^ unchanged during the rotation. In Fig. 2.6, the new x and y components resulting from the separate rotations of the two vectors x(0)^ and y (0)^ are shown. Adding these, we can x y obtain formulas for the total x and y components of ~ . The combined answer in terms of 2.3. Precession Solution: Phase their initial values has the rotation form 31 ~ (t) = x(t)^ + y (t)^ + z (t)^ x y z
with
x(t) y (t) z (t) (2.32) = = = x (0) cos !0 t + y (0) sin !0 t y (0) cos !0 t ; x(0) sin !0 t z (0) (2.33) Problem 2.6
It will be useful in later discussions to have the answer (2.33) rederived as a solution to the di erential equation (2.24). ~ a) For B = B0 z , show that the vector di erential equation (2.24) decomposes ^ into the three Cartesian equations d x = y B0 = !0 y dt d y = ; B = ;! x 0 0 x dt d z = 0 (2.34) dt b) By taking additional derivatives, show that the rst two equations in (2.34) can be decoupled to give d2 x = ;!2 0 dt2 d2 y = ;!2 0 dt2
x y (2.35) These decoupled second-order di erential equations have familiar solutions of the general form C1 cos !0t + C2 sin !0t. c) By putting the general solutions back into the rst-order di erential equations (2.34), and by assuming the initial conditions used previously, show that you recover (2.33). 32 Chapter 2. Classical Nucleus in Magnetic Field 2.3.4 Matrix Representation It is useful to unite (2.32) and (2.33) in matrix notation. ~ (t) = Rz (!0t)~ (0) (2.36) Here we mix notation a bit, and view vectors as column matrices. For example, 0 1 x(t) ~ (t) = B y (t) C (2.37) @ A (t) z Rz ( ) in (2.36) corresponds to a clockwise rotation of vectors through an angle about the z axis and can be represented by the matrix 0 1 cos sin 0 Rz ( ) = B ; sin cos 0 C (2.38) @ A 0 0 1 This is yet another mathematical representation of the physical picture of the magnetic moment precessing at a constant angular frequency around a constant magnetic eld. Problem 2.7
Show that, if (2.38) is substituted into (2.36), then (2.32) and (2.33) are recovered, assuming = !0t. 2.3.5 Complex Representations and Phase The fact that the principal action of the proton spin in a constant magnetic eld is a rotation in the two-dimensional transverse plane suggests that a complex number representation will be useful. The two degrees of freedom, x and y , can be given in terms of the real and imaginary parts of (2.39) + (t) = x(t) + i y (t) The addition of the rst di erential equation plus i times the second in (2.34) yields d + = ;i! (2.40) 0 + dt The solution of (2.40) is perhaps the easiest of all, ;i! t (2.41) +(t) = +(0)e 0 In (2.41), e;i!0t represents a clockwise rotation in the complex plane, or equivalently, about z , consistent with the precession picture drawn earlier. It is seen that phase directly relates ^ to position and is of utmost importance in the description of spin motion. 2.3. Precession Solution: Phase 33 Problem 2.8
Show that the real and imaginary parts of (2.41) agree with the rst two solutions in (2.33). In view of the importance of phase, it is useful to introduce a standard phase notation. In general, the complex number +(t) can be written in terms of its magnitude and phase
+ (t) = j +(t)je i (t) (2.42) (2.43) (2.44) The solution (2.41) has constant magnitude j +(t)j = j +(0)j
Thus that solution can be re-expressed as
i (t) + (t) = j +(0)je 0 where the phase is (2.45) The adaptation of the static- eld phase (2.45) to include the e ects of other elds is a primary subject in Ch. 20. 0 (t) = ;!0 t + 0 (0) 34 Chapter 2. Classical Nucleus in Magnetic Field Suggested Reading
For a review of some of the basic physics concepts, the following three texts are useful: H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, 1980. D. Halliday, R. Resnick and K. S. Krane. Physics. John Wiley and Sons, New York, 1992. J. D. Jackson. Classical Electrodynamics. John Wiley and Sons, New York, 1975. For a more comprehensive list of active NMR elements (i.e., those with nonzero spin), see, for example: P. T. Beall, S. R. Amtey and S. R. Kasturi. NMR Data Handbook for Biomedical Applications. Pergamon Press, New York, 1984. Chapter 3 Rotating Reference Frames and Resonance
3.1 Rotating Reference Frames 3.2 The Rotating Frame for an RF Field 3.3 Resonance Condition and the RF Pulse Summary: The combined e ect of two perpendicular elds, one a static eld and the other Chapter Contents a much smaller rf eld, is considered. The motion of a magnetic moment immersed in these elds is analyzed by the use of a rotating reference frame de ned in terms of the frequency of the harmonic eld. It is shown that the magnetic moment is most e ciently rotated away from the static- eld direction when the frequency of the rf eld matches the Larmor precessional frequency, a resonance condition. Introduction
We have seen that, at any given point in time, the interaction of a classical magnetic moment with an external magnetic eld is equivalent to an instantaneous rotation of the moment about the eld. For a static eld, the rotation is a constant precession about the xed eld axis. In the present chapter, we wish to consider the e ect of adding other magnetic elds. We are most interested in the combination of a radiofrequency (rf) eld perpendicular to a much larger constant eld. The interest in the additional eld stems from a new application of the lesson from the last chapter. The act of turning on a perpendicular eld for some period of time should tip any magnetic moment, initially aligned along the original static eld, away from that direction. Such rotations leave the classical moment precessing at an angle1 around the
The more fundamental quantum picture allows only two spin states for any measurements along the static direction, either parallel or anti-parallel. The use of a classical picture of a moment or spin precessing at some angle is still appropriate in MR, as explained in Ch. 5.
1 35 36 Chapter 3. Rotating Reference Frames and Resonance original static eld. Even very weak rf elds can be used, as we shall see, to rotate an initially aligned spin into the plane transverse to the static eld, or, for that matter, to any angle away from its initial alignment. It is to be shown in this chapter that the key is to tune the radiofrequency to the Larmor frequency. The rotation will be most e ective when this `resonance condition' is satis ed.2 The rf eld described above is the so-called `transmit' eld it is produced by a coil system separate from the static eld source. The precessing (or `excited') protons produce their own rf eld at the Larmor frequency. The rf eld produced by the protons can be detected by the transmit system, or by an altogether di erent rf coil system. See Chs. 7 and 27. 3.1 Rotating Reference Frames
Suppose we are making magnetic moment measurements in a reference frame rotating at the Larmor precession frequency. Such frames rotate clockwise around the z axis as seen from above the origin (z > 0) in a laboratory frame with a constant magnetic eld pointing in the positive z direction. See Fig. 3.1. From our rotating perspective, the spin axis is not moving at all. This reference frame has proven to be very useful in describing MRI experiments, and a mathematical framework is given for it below. Fig. 3.1: The sense of the rotation of a primed reference frame in which a magnetic moment is at rest. The primed frame rotates clockwise around the z = z 0 axis for a static magnetic eld pointing in the +^ direction according to a laboratory observer positioned above (z > 0) the x-y plane. The z rotation frequency is the Larmor precession frequency (2.31). In Fig. 3.2, a laboratory ( xed) reference frame, denoted by the unprimed Cartesian coordinates (x y z) and their associated unit vectors, is compared to another frame denoted
Adding a static perpendicular eld, on the other hand, to the original static eld would not yield the same kind of rotation. According to the solutions worked out in Ch. 2, it instead leads to a precession about the new, resultant eld direction.
2 3.1. Rotating Reference Frames 37 Fig. 3.2: The primed reference frame rotating according to a general angular velocity ~ relative to the unprimed reference frame. The rotation is counterclockwise around ~ . by primed quantities (x0 y0 z0 ), which is rotating about an arbitrary axis with respect to the xed frame. The instantaneous rotation of this frame is de ned by a rotational angular velocity vector ~ . The direction of this vector is the axis around which the primed frame is being rotated and its magnitude is the angular speed of rotation in radians per second. The ~ angular velocity may be changing with time, ~ = ~ (t). We go on to consider some vector C xed (at rest) in the rotating frame, whence it must be rotating in the laboratory. ~ Any vector C instantaneously rotated by ~ has its time rate of change with respect to the laboratory frame coordinates given by ~ dC = ~ C ~ dt (3.1) ~ One way to see (3.1) is to rewrite C in terms of its components parallel and perpendicular to ~ ~ ~ ~ C (t) = Ck + C? (3.2)
The parallel component is unchanged ~ dCk = 0 (3.3) The perpendicular component has a di erential change whose magnitude can be calculated just as the di erential arc length d was in Fig. 2.5. This step is left as an exercise for the reader. The combination of (3.2), (3.3), and the next equation, (3.4), yields (3.1). 38 Chapter 3. Rotating Reference Frames and Resonance Problem 3.1
Reversing the argument given in the discussion of Sec. 2.3 and de ning the unit ~ vector n parallel to ~ C , show that ^ ~ ~ ^ ~ dC? = dtjC?jn = ~ Cdt (3.4) ~ ~ Now replace C by a more general vector function V not necessarily at rest in the rotating ~ reference frame. In general, V is time dependent in both frames, and its respective Cartesian components in the two frames remain di erent because of the rotation. In the unprimed (inertial) frame where the unit vectors are xed in time: ~ V (t) = Vx(t)^ + Vy (t)^ + Vz (t)^ x y z (3.5) ~ In the primed frame, de ne the components of V with respect to the primed unit vectors, ~ ~ V 0(t) = V (t) = Vx (t)^0 (t) + Vy (t)^0(t) + Vz (t)^0 (t) x y z (3.6)
0 0 0 ~ ~ There is no need to distinguish between V 0 and V since they refer to the same vector. The primed unit vectors are rotating relative to the unprimed, so they are time dependent. ~ Either (3.5) or (3.6) can be used to represent V (t), and to compute its time derivative. The two ways must give the same answer, i.e., dVx x + dVy y + dVz z dt ^ dt ^ dt ^ ^0 ^0 ^0 = dVx x0 (t) + dVy y0(t) + dVz z0 (t) + Vx dx (t) + Vy dy (t) + Vz dz (t) (3.7) ^ ^ ^ dt dt dt dt dt dt ~ The unit vectors x0, y0, z0 are examples of the C vectors discussed above. For instance, ^ ^ ^ dx0 = ~ x0 ^ ^ (3.8) dt The time derivative of (3.5) thus can be written in convective form as
0 0 0 0 0 0 0 10 ~ ~ dV = @ dV A + ~ V ~ dt dt (3.9) where the primed derivative is de ned as 0 10 ~ @ dV A dt dVx (t) x0 (t) + dVy (t) y0(t) + dVz (t) z0 (t) dt ^ dt ^ dt ^
0 0 0 (3.10) This derivative represents the rate of change of the vector quantity with respect to the rotating reference frame. 3.2. The Rotating Frame for an RF Field 39 ~ Consider ~ as the vector quantity V (t) in (3.9) ! d~ = d~ 0 + ~ ~ dt dt (3.11) (3.12) (3.13) d~ = ~ B ~ dt From a comparison of (3.11) and (3.12), we can bank on the primed rate to be ! d~ 0 = ~ B ~ eff dt
where the `e ective magnetic eld' in the rotating frame is On the other hand, (2.24) is ~ ~ ~ Beff = B + (3.14) The e ective magnetic eld is useful in the determination of the precessional motion, and a key concept in the magnetic moment analysis in general. In the primed (rotating) frame, ~ is observed to have motion equivalent to the instantaneous rotation due to a `total' magnetic eld given by (3.14). This e ective eld is the superposition of the external magnetic eld plus a ctitious magnetic eld whose magnitude is j~ j= and whose direction is the same as ~ that of the vector ~ . The rotational motion around Beff is the by now familiar clockwise or left-handed precession (for an instant, at least) looking backwards along its direction. The freedom to choose the primed frame leads to a rapid rederivation of the solution ~ found for the constant magnetic eld problem. Take ~ = = ;B . Then (d~ =dt)0 = 0 from ~ (3.13), so ~ is constant in the primed frame. For the example where B = B0 z, it means ^ ~ rotates in the unprimed frame at a xed angle with respect to the z-axis, and at xed angular velocity ~ = ; B0 z (that is, left-handedly or clockwise around z). This implies the ^ solution (2.41) or (2.45). ~ Consider a proton spin aligned along B0 z . An rf magnetic eld B1 is added to `tip' the ^ 3 According to (3.13), B must have components in the x or y direction ~1 spin away from z . ^ (i.e., `transverse' or `perpendicular' components) in order to rotate ~ away from the z axis. The Larmor frequency, to which the applied transmit eld's frequency should be matched, generally lies in the rf range for conventional MRI (see Ch. 2). Recall that this eld is referred to as the transmit rf eld.
Throughout the book, this will often be referred to as a spin ip, in view of the common usage of phrases like ` ip angle.' The terminology is motivated by the quantum picture (see Ch. 5) where, as we have stated in an earlier footnote, the proton has only `spin-up' or `spin-down.' Nevertheless, a classical picture of continuous spin rotations remains adequate for the present discussion.
3 3.2 The Rotating Frame for an RF Field 40 Chapter 3. Rotating Reference Frames and Resonance 3.2.1 Polarization In the present context, `polarization' refers to the direction of the magnetic eld,4 so a sinusoidal eld linearly polarized along the x-axis is ~ lin 1 B1 = blin cos !t x ^ (3.15) There is additional time dependence in the amplitude blin representing the need to turn the 1 eld on and o . For the present, however, we ignore such time dependence, since, in practice, blin varies only over times much larger than the rf laboratory period 2 =!. 1 The usefulness of a rotating reference frame in the analysis of the static eld is incentive to consider such frames for the combined static plus rf elds. The motivation can be made more compelling by the following picture: Imagine a spin precessing at a small angle around the static eld direction. To tip this spin down to a larger angle, an additional eld should be synchronized to push the spin down at a given position every time the spin comes back around in its precession to this same position. (This is much like timing the pushing of a child on a swing it serves as a useful analog here and for the resonance condition discussed in the next section.) In the reference frame language, it is necessary to have some transverse component of the B1 eld at rest (constant), or nearly so, in the rotating reference frame. Let us in fact de ne the rotating reference frame using the laboratory frequency ! of the rf eld, instead of !0.5 Take the primed frame to be that frame undergoing negative (clockwise) rotation, with respect to the laboratory frame, around the z axis with angular frequency !. The corresponding angular velocity is ~ = ;! z ^ (3.16) The unit vector x0 rotates clockwise, with time, in the negative direction. ^ x0 = x cos !t ; y sin !t = Rz (!t)^ ^ ^ ^ x
assuming x0 (0) = x(0). Similarly, ^ ^ (3.17) (3.18) y0 = x sin !t + y cos !t = Rz (!t)^ ^ ^ ^ y ~ The B1 eld can be expressed in terms of the primed basis. The inversion of (3.17) and (3.18) is x = x0 cos !t + y0 sin !t = Rz (;!t)^0 ^ ^ ^ x (3.19) y = ;x0 sin !t + y0 cos !t = Rz (;!t)^0 ^ ^ ^ y (3.20) equivalent, as noted, to the rotation of the primed unit vectors by Rz (;!t) in the positive -direction. From double-angle trigonometric formulas, (3.15) and (3.19) lead to ~ lin 1 1 ^ B1 = 2 blin x0(1 + cos 2!t) + y0 sin 2!t] ^ (3.21)
In most electromagnetic discussions, polarization refers to the electric eld direction. This is more natural. The Larmor frequency changes from particle species to species, and it varies with the magnetic eld strength. The frequency !, on the other hand, is under experimental control.
4 5 3.2. The Rotating Frame for an RF Field 41 Equation (3.21) displays a constant term plus two oscillating terms whose e ects average to zero over times that are half multiples of the rf period or, more relevantly, for all times large compared to the rf period. The de nition for an average over a time interval T is given by 1 Z T dtf (t) < f (t) > T (3.22) 0 With the details left as a problem for the reader, the primed-frame time averaged value of (3.21) is 1 ^ ~ lin < B1 >primed= 2 blinx0 (3.23) 1 for constant blin . 1 Problem 3.2
Show that the average (3.22) over time interval T for f (t) = sin (n!t) or f (t) = cos (n!t) for any positive integer n is zero when a) T = 2m =! for any positive integer m (or T = m =! for even n) or when the following limit is taken b) T >> 2 =! The reduced amplitude 1 blin in (3.23) is noteworthy. It implies that only half of the 2 1 original linearly polarized eld (3.15) amplitude is available in the rotating reference frame for tipping the spin. Also, (3.23) is still valid for a time dependent blin (t), provided it changes 1 only over time scales much larger than the rf period 2 =!. It has been noted earlier that such scales are consistent with MR applications. In that case, blin(t) can be considered to 1 be a constant in the integration for time averaging. 3.2.2 Quadrature We can construct an e cient rf eld where the original laboratory amplitude is the same as the amplitude in the rotating reference frame. A `left-circularly polarized' rf eld that is constant in the rotating frame de ned by (3.16) is obtained by a superposition. Adding two linearly polarized rf elds with the same frequency and peak amplitude, but perpendicular to each other and 90 out of phase (with respect to time dependence), gives ~ cir B1 = B1 (^ cos !t ; y sin !t) x ^ (3.24) The combination of unit vectors in (3.24) is nothing other than x0 , so that equivalently ^ ~ cir B1 = B1 x0 ^ (3.25) It is obvious that the eld (3.25) is `at rest' in the frame (3.16). Importantly, the (generally time dependent) amplitude B1(t) is the full rotating frame amplitude available for spin 42 Chapter 3. Rotating Reference Frames and Resonance ipping. In the discussion of Ch. 27, it will be seen that less power is required from the rf coil ampli er to produce a given spin ip with left-circularly polarized elds than with linearly polarized elds. This advantage is related to the factor of one-half found in (3.23). The 90 phase di erence between the x and y coordinates of the laboratory eld (3.24) has led to calling (3.25) a `quadrature' eld. Because of the power advantage, and other factors concerning signal-to-noise and the need for rf spatial homogeneity, such rotating elds are commonly used in MR. 3.3 Resonance Condition and the RF Pulse
The equation of motion (3.13) in the rotating reference frame for a spin immersed in the ~ combination of the constant eld B0 = B0 z and the left-circularly polarized eld (3.25) is ^ d~ dt !0 z0 (!0 ; !) + x0 !1] ^ ^ ~ = ~ Beff
= ~ (3.26) setting z0 = z for the rotation (3.16). Appearing in (3.26) are the Larmor frequency !0 ^ ^ B0 , the rf laboratory frequency !, and the spin-precession frequency !1 generated by the circularly polarized rf eld (3.24), ! 1 B1 (3.27) The e ective magnetic eld is ~ Beff z0 (!0 ; !) + x0!1 ] = ^ ^ (3.28) Equation (3.26) in the primed coordinates is an important result. In general, it states that there is left-handed (clockwise) precession in the primed frame around the axis de ned ~ by Beff . Leaving a more general precession discussion to Sec. 3.3.4, we focus now on a special rotating reference frame, where the applied rf frequency ! matches the Larmor frequency !0. The rst term is then eliminated in (3.28), leading to the cornerstone equation of motion ! d~ 0 = ! ~ x0 ^ 1 dt (when ! = !0) (3.29) There is then only a precession about the x0 axis with the precessional frequency !1 given ^ in (3.29). The stipulation for a given static uniform eld B0 is that ! = !0 (on-resonance condition) (3.30) the origin of the `resonance' reference in the acronyms NMR and MRI. Under this condition, the B1 eld is maximally synchronized to tip the spin around the x0 -axis. 3.3. Resonance Condition and the RF Pulse 43 3.3.1 Flip-Angle Formula and Illustration A B1 eld applied on-resonance for a nite time is called an `rf pulse.' Suppose the rf eld is turned on (quickly) to a constant value B1 x0 for a time interval and then it is just as ^ rapidly turned o . From the precession lessons of Ch. 2 (or from the explicit solutions of the next subsection), (3.29) implies that the spin rotates through the angle = B1 (3.31) around x0 . For example, the size of B1 required for a 90 ip angle over 1:0 ms is 5:9 T ^ (0:059G) for protons. Despite the simplicity, this computation is given as a problem, in view of its importance to MR analysis. Problem 3.3
In many MR experiments it is necessary to ip a spin, which is initially along the z axis, into the x0-y0 plane by using an appropriate rf pulse. This is referred to as a `90 ' or ` =2' pulse. If the desired rf pulse time interval is 1.0 ms, what B1 magnitude in T (1T=10 000G) is required for a) a proton spin? (answer given above) b) an electron spin? (See (2.22).) c) How many Larmor precession cycles take place in the laboratory frame, for B0 = 1:0T, during the =2 ip of the proton spin in (a)? See Fig. 3.3. An illustration of various spin trajectories is helpful in highlighting the e ectiveness of being on-resonance for tipping spins. Trajectories in the primed and unprimed frames for both o -resonance and on-resonance conditions are shown in Fig. 3.3. The pictures are generated from the solutions of Sec. 2.3 which are easily adapted to a constant B1x0 in ^ the rotating frame. (See the following subsection for both constant and time dependent adaptations.) The pure on-resonance rotation about the x0 axis is observed in Fig. 3.3a. The corresponding spiraling down found in the laboratory is shown in Fig. 3.3b. The o resonance motion in the primed frame (Fig. 3.3c) shows the expected precession about the total eld. In Fig. 3.3d, the laboratory picture is that of a superposition of this precession on top of the rotation of the primed frame. On-resonance, even a weak rf eld would readily rotate the spin down into the transverse plane. By contrast, the farther o -resonance a spin becomes, the closer the e ective eld is to the static eld, and the less the spin is tipped from the vertical. 3.3.2 RF Solutions The fact that the magnetic resonance e ect is more easily understood in the corresponding rotating reference frame is also re ected by the manner in which an analytical solution is 44 Chapter 3. Rotating Reference Frames and Resonance Fig. 3.3: An on-resonance =2 spin ip as viewed in the primed (a) and unprimed (b) frames for ! = !0 and !1 = 0:06 !0 . An o -resonance trajectory as viewed in the primed (c) and unprimed (d) frames corresponds to the o set value, ! = 0:85 !0 , with !1 = 0:06 !0 . In MR applications, the frequency !1 would be much smaller in relation to the rf frequency, but the spiraling would then
be much too dense to illustrate. See Prob. 3.3. found for the motion of the magnetic moment. In the rotating frame, let the rf eld be constant along x0, ^ ~ B1 = B1 x0 ^ (3.32) The total e ective eld (3.28) on-resonance is thus given by (3.32). The magnetic moment vector motion is found by transcribing the solution (2.33) according to the substitutions z ! x0, y ! z0 , x ! y0
x (t) y (t) z (t)
0 0 0 = x (0) = y (0) cos 1(t) + z (0) sin 1(t) = ; y (0) sin 1(t) + z (0) cos 1 (t)
0 0 0 0 0 (3.33) with (3.34) in terms of the precession frequency, !1 B1 , and the rotating frame components of the initial vector value, ~ (0). The solution may be rewritten as a rotation matrix operation. 1 (t) = !1 t ~ (t) = Rx ( 1(t))~ (0)
0 (3.35) using 0 1 0 Rx ( ) = B 0 cos @
0 0 ; sin 0 sin C A cos 1
(3.36) 3.3. Resonance Condition and the RF Pulse 45 The general time dependent case (with arbitrary initial time t0 ) is solved by the substitution 1 ! 1 (t) and ~ (0) ! ~ (t0 ) where
1 (t) = Z t1
t0 dt0!1(t0 ) (3.37) in which !1 (t) = B1(t) (3.38) This is required for rf pulses of nite time duration and more general temporal envelopes. A sequence of such rf pulses can be modeled by products of rotations. For example, the (clockwise) rotation through an angle 1 around x0 (due to the application of (3.32) for a ^ nite time interval) followed by the rotation through 2 around y0 (applying B1y (t)^0 for ^ y some later, non-overlapping, time interval) yields ~ (t) = Ry ( 2 )Rx ( 1 )~ (t0) (3.39) with 0 1 cos 0 ; sin Ry ( ) = B 0 1 0 C (3.40) @ A sin 0 cos
0 0 0 0 Problem 3.4
A static eld points uniformly along the positive z-axis. A classical spinning particle, with positive gyromagnetic ratio and xed magnetic moment magnitude , has its spin initially in the direction of the static eld. A circularly polarized rf eld points along the y0 axis with time-dependent amplitude B1y ^ applied on-resonance starting at t=0. a) Give expressions analogous to (3.33) for all three magnetic-moment vector components in the rotating (prime) reference frame for time t > 0. Your answer will be in terms of a de nite integral. b) Show that the equation of motion (2.24) is satis ed by your answers in (a) ~ for B ! B1y y0. ^ c) Find the generalization of (2.35) needed for this time dependent case.
0 0 3.3.3 Di erent Polarization Bases In going from a linearly polarized rf eld to a circularly polarized rf eld, it is natural to replace the 2D Cartesian basis (^ and y, say) by a complete basis made up of the left-circular x ^ (or left-circulating) unit vector xleft = x cos !t ; y sin !t = x0 ^ ^ ^ ^ (3.41) 46 Chapter 3. Rotating Reference Frames and Resonance and the right-circular (right-circulating) unit vector xright = x cos !t + y sin !t ^ ^ ^ (3.42) ~ left ^ It has been seen that the rf eld B1 / xleft is maximally e ective in tipping the spin ~ right ^ around the x0-axis. A right-circularly polarized eld B1 / xright is completely ine ective, according to the problem which follows. The right-circular eld would be appropriate, on the other hand, for a reference frame rotating in right-handed fashion (counterclockwise) with frequency ! about the z-axis. Problem 3.5
Show that xright = x0 cos 2!t + y0 sin 2!t ^ ^ ^ (3.43) using steps like those used in deriving (3.21). Its time average is clearly zero. Another basis of much importance to MR is the complex representation of vectors in the transverse plane (cf. Sec. 2.3.4). Recall that in this basis the real part is the x-component and the imaginary part is the y-component. In terms of a given amplitude B1, left-handed and right-handed rf elds can be written, respectively, as
left B L B1 1 right B R B1 1 where the clockwise (left-handed or `negative') component is L B1 = B1 e;i!t (3.44) (3.45) (3.46) (3.47) and the counterclockwise (right-handed or `positive') component is
R B1 = B1ei!t Only the former component (3.46) is e ective, on-resonance, in tipping spins. The linear (along x) rf eld (3.15) can be rewritten as
lin L R B1 = B1 + B1 = 2B1 cos !t (3.48) (3.49) provided the identi cation is made that 1 B1 = 2 blin 1 It is veri ed that only half of the original amplitude is present in the left-handed component in (3.48). A linear eld must have a peak amplitude of 2B1 in order to have a weighting of B1 in its left-handed component. 3.3. Resonance Condition and the RF Pulse 47 3.3.4 Laboratory Angle of Precession A return to the general picture in the rotating reference frame, on- or o -resonance, can be made through the e ective eld (3.14). (Recall that ! is the rf frequency and !0 = B0 and they are not necessarily equal in this discussion.) For the combined static and rf elds, ~ B = (B ; ! )^ + B x0 z ^ (3.50)
eff
0 1 ~ ~ rewriting (3.28). If the angle between Beff and B0 is de ned to be (Fig. 3.4), q q 2 2 !eff = Beff = (B0 ; != )2 + B1 = (!0 ; !)2 + !1 (3.53) The equation of motion (3.26) dictates that, for constant !0 and !1, the magnetic moment ~ precesses around Beff in the rotating reference frame with frequency !eff , as exempli ed in Fig. 3.4 (see also Fig. 3.3). This precession incorporates the primary lessons of both the present and previous chapters. where ; 0 cos = B0B != = !! ; ! eff eff sin = !1 !eff (3.51) (3.52) Fig. 3.4: The precession of a magnetic moment around the e ective magnetic eld in the rotating
reference frame. The angle between the spin magnetic moment ~ and the static eld direction z shown ^ in Fig. 3.4 is the subject of the next problem. There the request is to show that cos (t) = cos2 + cos !eff t sin2 = 1 ; 2 sin2 sin2 !eff t (3.54) 2 48 Chapter 3. Rotating Reference Frames and Resonance The expression (3.54) yields the behavior of the z-component of the magnetic moment as a function of time. Problem 3.6 Let (t) be the polar angle of ~ (t) as shown in Fig. 3.4. Show that cos (t), measuring the amount of magnetic moment, or spin, left along z , is given in ^ terms of and !eff by (3.54). Hint: Decompose ~ (t) into components parallel ~ and perpendicular to Beff . Then project these components onto the z-axis. 3.3. Resonance Condition and the RF Pulse 49 Suggested Reading
A review of the relevant general concepts of rotation and precession in classical mechanics and NMR can be found in the following two texts: H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, 1980. C. P. Slichter. Principles of Magnetic Resonance. Springer Verlag, III ed., New York, 1990. The next article presents an outstanding discussion of NMR in rotating reference frames: I. I. Rabi, N. F. Ramsey and J. Schwinger. Rotating coordinates in magnetic resonance problems. Rev. of Modern Phys., 26: 167, 1954. 50 Chapter 3. Rotating Reference Frames and Resonance Chapter 4 Magnetization, Relaxation and the Bloch Equation
4.1 Magnetization Vector 4.2 Spin-Lattice Interaction and Regrowth Solution 4.3 Spin-Spin Interaction and Transverse Decay 4.4 Bloch Equation and Static-Field Solutions 4.5 The Combination of Static and RF Fields Summary: The interactions of spins with their surroundings are modeled in the presence Chapter Contents of external eld e ects by the phenomenological Bloch equation. The relaxation decay times T1 , T2 , T20 and T2 are introduced. Solutions of the Bloch equation are given for constant and harmonic elds. Introduction
Thus far, the response of an isolated proton's spin in an external magnetic eld has been modeled by the classical equations of motion of a single magnetic moment. The interactions of the proton spin with its neighboring atoms lead to important modi cations to this behavior. The local elds change the spin precession frequency, and the proton can exchange spin energy with the surroundings. In this chapter, we model these e ects, as guided by experiment, after introducing the average magnetic dipole moment density (`magnetization'). 4.1 Magnetization Vector
For images of a macroscopic body, we focus on protons, introducing their local magnetic ~ r moment per unit volume, or magnetization, as M (~ t). Consider a volume element (`voxel') 51 52 Chapter 4. Magnetization, Relaxation and the Bloch Equation with volume V small enough that external elds are to a good approximation constant over V ,1 but big enough to contain a large number of protons. The magnetization is ~ 1 X ~i M=V (4.1) protons
in V The set of spins in V is called a spin `isochromat,' which can be de ned to be an ensemble or domain of spins with the same phase. With the neglect of the proton interactions with their environment, a sum over the equations of motion for the individual spins (2.24) yields 1 X d~ i = X ~ B ~ (4.2) V i dt V i i ext or ~ dM = M B ~ ~ ext (non-interacting protons) (4.3) dt It is most advantageous to analyze the magnetization, and its di erential equation, in terms of parallel and perpendicular components de ned relative to the static main magnet ~ eld, Bext = B0z . The parallel, or `longitudinal' component of the magnetization is ^ Mk = Mz
The transverse components are (4.4) ~ M? = Mxx + My y ^ ^ (4.5) The corresponding components of the cross product in (4.3) lead to decoupled equations dMz = 0 (non-interacting protons) (4.6) dt and ~ dM? = M B ~ ? ~ ext (non-interacting protons) (4.7) dt The modeling of the proton interactions with its neighborhood leads to additional terms in (4.6) and (4.7) which depend on decay parameters, and these parameters are di erent in the two equations. This di erence is related to the fact that, in contrast to a given magnetic moment, the magnitude of the macroscopic magnetization is not xed, since it is ~ the vector sum of (many) proton spins. The components of M parallel and perpendicular to the external eld `relax' di erently in the approach to their equilibrium values. 4.2 Spin-Lattice Interaction and Regrowth Solution
Equation (4.6) is certainly wrong for interacting protons, insomuch as their moments try to align with the external eld through the exchange of energy with the surroundings. To understand the origin of the missing term, an energy argument is helpful.
1
1 The external elds are assumed to vary spatially only over scales much larger than V 3 . 4.2. Spin-Lattice Interaction and Regrowth Solution 53 The classical formula for the potential energy associated with a magnetic moment immersed in a magnetic eld is (Appendix A) ~ U = ;~ B (4.8) This implies that the moment will tend to line up parallel to the eld in order to reach its minimum energy state, if energy can be transferred away. Since the protons are considered to be in thermal contact with the lattice of nearby atoms, the thermal motion present in the lattice can account for any change in a given proton spin energy (4.8). In the quantum language developed in the next chapter, a spin can exchange a quantum of energy with the lattice. The magnetization version of (4.8) is the potential energy density ~ ~ UM = ;M B = ;Mk B0 (4.9) involving only the longitudinal component of the magnetization. Although the transverse components can be ignored in discussing the energy, it follows that, as the longitudinal magnetization returns to its equilibrium value M0 , the transverse magnetization must vanish. (In fact, the transverse magnetization can vanish more quickly due to `dephasing,' see the next section.) The equilibrium value relevant to room temperatures obeys Curie's law in its dependence on the absolute temperature T and the external eld, M0 = C B0 (4.10) T The constant C is derived in Ch. 6 for protons as well as for other particles with di erent spins. It is helpful to preview some of the discussion in Ch. 6 with respect to (4.10). In the applications to MRI, (4.10) is very small compared to the maximum possible magnetization (which would be the product of the spin density times the individual spin magnetic moment). Since the proton spin energy (4.8) is tiny compared with the thermal energy scale kT (k is the Boltzmann's constant and T is in Kelvin) at room temperature, there is only a minuscule energy advantage for a spin moment to be aligned with the magnetic eld. In consequence, only a very small fraction (about ve in one million for a eld strength of 1.5 T) of parallel spins exceed anti-parallel spins for eld strengths of interest. Fortunately, Avogadro's number is so large, that, for example, on the order of 1018 excess proton spins are aligned along a 0.5 T eld in one mole of water (see Ch. 6). Hence the magnetization M0 is still big enough to be measured. Suppose the equilibrium magnetization of a body is disturbed (by, say, the temporary application of an rf pulse) from its equilibrium value. As a result of the continued presence of the static eld, the magnetization returns to its equilibrium magnetization vector M0 z . ^ In the remainder of this section, the relaxation of the longitudinal component to M0 is discussed, and, in the next section, the relaxation of the transverse components to zero is described. Introduction of T1 A constant interaction growth rate from the proton interactions with the lattice (see Ch. 6), implies that the rate of change of the longitudinal magnetization, dMz (t)=dt, is proportional 54 Chapter 4. Magnetization, Relaxation and the Bloch Equation to the di erence M0 ; Mz . The proportionality constant is empirically determined, and represents the inverse of the time scale of the growth rate. Equation (4.6) is replaced by2 dMz = 1 (M ; M ) ~ (Bext k z ) ^ (4.11) z dt T1 0 where T1 is the experimental `spin-lattice relaxation time.' The relaxation parameter T1 ranges from tens to thousands of milliseconds for protons in human tissue over the B0 eld strengths of interest (0.01 T and higher.) Typical values for various tissues are shown in Table 4.1. Tissue T1 (ms) T2 (ms) gray matter (GM) 950 100 white matter (WM) 600 80 muscle 900 50 cerebrospinal uid (CSF) 4500 2200 fat 250 60 blood 1200 100-2003 Table 4.1: Representative values of relaxation parameters T1 (see Sec. 4.2) and T2 (see Sec. 4.3), in milliseconds, for hydrogen components of di erent human body tissues at B0 = 1:5 T and 37 C
(human body temperature). These are only approximate values see Ch. 22. Problem 4.1
Derive (4.12) by solving the rst-order di erential equation (4.11). Hint: One method is to use an integrating factor. Another is simply to put the magnetization and time variables on the opposite sides of the equation and integrate. The solution of (4.11) can be found, for example, by the procedures outlined in Prob. 4.1. After the application of an rf pulse, the longitudinal magnetization displays an exponential form showing the evolution from the initial value, Mz (0), to the equilibrium value, M0 : ~ Mz (t) = Mz (0)e;t=T1 + M0 (1 ; e;t=T1 ) (Bext k z ) ^ (4.12) ~ We reiterate that this solution corresponds to the situation where B = B0z and M0 is the ^ 4 The solution for an arbitrary starting point will be of value in multiple equilibrium value.
2 The absence of an explicit eld dependence in the solution should not be misunderstood. It is only because of the external eld that the longitudinal magnetization evolves to M0 . 3 The higher value pertains to arterial blood and the lower value to venous blood. 4 If the external eld were not uniform, the solution would only refer to a given point in space. The dependence on ~ has been suppressed. r 4.2. Spin-Lattice Interaction and Regrowth Solution rf pulse experiments: 55 Mz (t) = Mz (t0)e;(t;t0 )=T1 + M0 (1 ; e;(t;t0 )=T1 ) ~ (Bext k z ) ^ (4.13) Problem 4.2
The key equation (4.12) can be used to investigate general questions. If unmagnetized material is placed in a region with a nite static eld at t = 0 (Mz (0) = 0): a) Find the time it takes, in units of T1, for the longitudinal magnetization to reach 90% of M0 . b) Find an approximate formula for Mz (t) of this material in the limit that t << T1 . Use this formula to nd the initial (t = 0) slope of Mz (t), and compare the answer to the general formula indicated in Fig. 4.1, when Mz (0) = 0.
Formula (4.12) is the key to understanding the regrowth, after an initial disturbance, of longitudinal magnetization. Throughout the discussion in later chapters of this book, it is often necessary to determine how much longitudinal magnetization is available to be rotated back into the transverse plane by a given sequence of rf pulses. An illustration of the exponential regrowth for a given initial value is presented in Fig. 4.1a. The time scale for regrowth is seen to be determined by T1 . Fig. 4.1: (a) The regrowth of the longitudinal component of magnetization from the initial value Mz (0) to the equilibrium value M0 . (b) The decay of the magnitude of the transverse magnetization
from an initial value. 56 Chapter 4. Magnetization, Relaxation and the Bloch Equation 4.3 Spin-Spin Interaction and Transverse Decay
An important mechanism for the decay of the transverse magnetization is as follows. Spins experience local elds which are combinations of the applied eld and the elds of their neighbors. Since variations in the local elds lead to di erent local precessional frequencies, the individual spins tend to fan out in time, as shown in Fig. 4.2, reducing the net magnetization vector. The `fanning out' is usually referred to as `dephasing.' The total transverse magnetization is the vector (or complex) sum of all the individual transverse components. Fig. 4.2: The upper sequence shows a 90 tip of a set of spins (isochromats) into the transverse plane such that they all lie along the y-axis (laboratory frame) at some instant in time, as shown in the middle gure. Precession of the individual spins in the x-y plane immediately follows (the
recovery of longitudinal magnetization is ignored since the focus is on transverse magnetization dephasing e ects). The lower sequence shows the same process in terms of the net transverse magnetization which decreases in magnitude during the precession because of the fanning out of the spins. Introduction of T2
The characterization of the overall rate of reduction in transverse magnetization brings forth another experimental parameter, the `spin-spin' relaxation time T2. The di erential equation (4.7) is changed by the addition of a decay rate term ~ d M? = M B ; 1 M ~ ? ~ ext ~ dt T2 ? (4.14) 4.3. Spin-Spin Interaction and Transverse Decay 57 ~ The additional term leads to exponential decay of any initial value for M?. This is most easily seen in the rotating reference frame, where the di erential equation has a standard decay-rate form5 0 1 ~ 0 dM? A = ; 1 M ~ @ (rotating frame) (4.15) dt T2 ? with the solution ~ ~ M?(t) = M?(0)e;t=T2 (rotating frame) (4.16) ~ Equation (4.16) describes the exponential decay of the magnitude M? jM?j of the transverse magnetization in either the laboratory or the rotating reference frame. A sample curve for the decay is displayed in Fig. 4.1b. Owing to the fact that the `spin-spin' interactions include the collective dephasing e ect, where no energy is lost, as well as the same spin-lattice couplings giving rise to T1 e ects, (4.14) corresponds to a higher relaxation rate than (4.11). De ne the relaxation rates by R1 1=T1 and R2 1=T2 (4.17) Then R2 > R1 or T2 < T1 (4.18) The relaxation parameter T2 is on the order of tens of milliseconds for protons in most human tissue (see Table 4.1 for a variety of T2 tissue values). It is approximately constant over the B0 range of interest for a given tissue. The values of T2 are much shorter for solids (on the order of microseconds) and much longer for liquids (on the order of seconds).
In practice, there is an additional dephasing of the magnetization introduced by external eld ~ inhomogeneities. This reduction in an initial value of M? can sometimes be characterized by a separate decay time T20 .6 The total relaxation rate, de ned as R2 , is the sum of the internal and external relaxation rates 0 R2 = R2 + R2 (4.19) In terms of an overall relaxation time T2 1=R2 , 1 = 1 + 1 (4.20) T2 T2 T20 As we shall see later, the loss of transverse magnetization due to T20 is `recoverable.' To the extent that T20 e ects dominate the `fan out' shown in Fig. 4.2, an additional pulse can be designed so as to lead to a rephasing of the spins, a reversal of the dephasing caused by the external eld inhomogeneities. It is possible to recover their initial phase relationship ~ corresponding to the initial value of M?. Referred to as `creating an echo,' this process will be described in detail in Ch. 8. The intrinsic T2 losses are not recoverable they are related to local, random, time-dependent eld variations.
5
0 Introduction of T2 and T20 ~ ~ Recall from Ch. 3 that there is no need to distinguish between M and M . There is no guarantee that local eld inhomogeneities lead to an exponential signal decay, but they are assumed to do so in this discussion.
6 58 Chapter 4. Magnetization, Relaxation and the Bloch Equation 4.4 Bloch Equation and Static-Field Solutions
The di erential equations (4.11) and (4.14) for magnetization in the presence of a magnetic eld and with relaxation terms can be combined into one vector equation, ~ dM = M B + 1 (M ; M )^ ; 1 M ~ ~ ext ~ (4.21) z z dt T1 0 T2 ? This empirical vector equation is referred to as the Bloch equation. The relaxation terms describe the return to equilibrium, but only for a eld pointing along the z-axis. The quantum mechanical underpinnings of the Bloch equation are described in Chs. 5 and 6. ~ Let us solve the Bloch equation for the constant eld case, Bext = B0 z. A calculation of ^ the components of the cross product in (4.21) produces the three component equations dMz = M0 ; Mz (4.22) dt T1 dMx = ! M ; Mx (4.23) 0 y dt T2 dMy = ;! M ; My (4.24) 0 x dt T2 The rst equation is the same as (4.11) whose solution is (4.12). For the last two equations, the relaxation terms can be easily eliminated by the change of variables, Mx = mxe;t=T2 and My = my e;t=T2 (i.e., by the introduction of integrating factors). The resulting di erential equations for mx and my have exactly the form of the equations found, and solved, for x and y in Ch. 2.7 In terms of the original variables, the complete set of solutions is therefore Mx(t) = e;t=T2 (Mx(0) cos !0 t + My (0) sin !0t) (4.25) ;t=T2 My (t) = e (My (0) cos !0t ; Mx(0) sin !0t) (4.26) ;t=T1 ;t=T1 Mz (t) = Mz (0)e + M0 (1 ; e ) (4.27) The equilibrium or steady-state solution can be found from the asymptotic limit t ! 1 of (4.25)-(4.27). In that limit, all the exponentials vanish implying the steady-state solution Mx (1) = My (1) = 0 Mz (1) = M0 (4.28) Problem 4.3
A direct derivation of the steady-state solution, when it exists, of a system of di erential equations can often be found by the following procedure. Assuming that the system evolves to constant values for large times, all time derivatives can be set to zero. The problem reduces to a system that can often be solved analytically. Use this procedure to nd the steady-state solution directly from (4.22), (4.23) and (4.24), verifying (4.28).
7 See, in particular, Prob. 2.6. 4.4. Bloch Equation and Static-Field Solutions 59 The general time-dependent solution for the transverse components, (4.25)-(4.26), is seen to have sinusoidal terms modi ed by a decay factor. The sinusoidal terms correspond to the precessional motion discussed in Ch. 2, and the damping factor comes from the transverse ~ relaxation e ect. The magnitude jM j is not xed: The longitudinal component relaxes from its initial value to the equilibrium value M0 the transverse component rotates clockwise and it decreases in magnitude. Recall that the transverse decay time T2 is in general di erent from (smaller than) the longitudinal decay time T1 . An example of the resulting left-handed `corkscrew' trajectory for an initial magnetization lying in the transverse plane is illustrated in Fig. 4.3. Fig. 4.3: The trajectory of the tip of the magnetization vector showing the combined regrowth of the longitudinal magnetization and decay of the transverse components. The initial value was along the y axis and the reference frame is the laboratory. Phase Description
The general solutions could also be simpli ed8 in their description by employing the complex representation of Sec. 2.3.5 M+ (t) Mx(t) + iMy (t) (4.29) With the details left to the next problem, the solution for a static eld in this representation is M+ (t) = e;i!0 t;t=T2 M+(0)
8 (4.30) The solutions also may be described by the rotation matrices introduced in Ch. 2. 60 Chapter 4. Magnetization, Relaxation and the Bloch Equation Problem 4.4
a) Find the di erential equation for M+ (t) analogous to (2.40) and show that its solution is (4.30). b) Show that (4.30) is equivalent to (4.25)-(4.26). c) Repeat (a) for M; Mx ; iMy = ;i(My + iMx ) (4.31) As a follow-up on the remarks of Sec. 2.3.5 about the phase of a single moment, the phase of the complex representation of the magnetization in solutions like (4.30) plays a key role in characterizing imaging signals. The generalization of (2.41) is M+(t) = jM+(t)jei (t) = M?(t)ei (t) (4.32) noting that the magnitude of the complex magnetization (4.29) is the same as the magnitude M? of the transverse vector (4.5). For the static- eld solution, M?(t) = e;t=T2 M?(0) (4.33) and (t) = ;!0 t + (0) (4.34) In addition, the phase is often given with reference to rotating frames where, for the staticeld case in the Larmor rotating frame, it becomes a constant, = (0). 4.5 The Combination of Static and RF Fields ~ An rf eld needs to be added to the static eld in order to tip M from its equilibrium direction. The precession of the resulting transverse component of magnetization produces its own (rotating) eld, which can be detected with a nearby coil (see Ch. 7). The analysis of the resulting motion is done expeditiously, as we have shown in Ch. 3, in rotating reference frames.
~ 4.5.1 Bloch Equation for Bext = B0z + B1x0 ^ ^ ~ Following Sec. 3.3.3, we add a left-circularly polarized rf eld B1 which is at rest in the 0 rotating frame and parallel to x ( see Ch. 3). The total external eld is ^ ~ Bext = B0z + B1 x0 ^ ^ (4.35) The e ective eld in that frame is ~ Beff = (B0 ; ! )^ + B1x0 z ^ (4.36) 4.5. The Combination of Static and RF Fields 61 ~ To nd the Bloch component equations for (4.36), the rf eld B1 is assumed to have a 9 It is supposed that there still exists a z -component magnitude much smaller than B0. equilibrium value M0 , and decay constants T1 and T2 . The reader is asked to show that the component Bloch equations in the primed coordinates take the form ! dMz 0 = ;! M + M0 ; Mz 1 y dt T1 !0 dMx = ! M ; Mx y dt T2 !0 dMy = ; ! M + ! M ; My x 1 z dt T2
0 0 0 0 0 0 (4.37) (4.38)
0 (4.39) with ! !0 ; ! (4.40) We are reminded that !0 is the Larmor frequency, !1 is the spin frequency due to the rf eld, and ! is the rf laboratory frequency of oscillation. The ! terms in the above equations are `o -resonance' contributions. They may represent deviations from ideal conditions due to static eld impurities or variations in the applied rf frequency. Problem 4.5
Taking note of (3.26), demonstrate that in the primed basis (4.21) reduces to (4.37)-(4.39). The above di erential equations contain frequency terms, which combine to produce an instantaneous rotation about an e ective eld, and decay constant terms. On-resonance ~ (! = !0) precession (with frequency !1 = B1 ) around B1 of the components transverse to 0 x is evidently superimposed on the relaxation decay in (4.37) and (4.39). In the original ^ unprimed frame, this is a nutation superimposed on the decay. 4.5.2 Short-Lived RF Pulses The rf pulses in most MR measurements are designed to have a very small time duration, rf . Furthermore, the typical values for !1 are much greater than the decay rates 1=T1 and 1=T2. The solution of the di erential equations (4.37)-(4.39) can therefore be carried out in two steps, each of which is already familiar to us. In the rst step, the relaxation terms can be ignored relative to the frequency terms. ~ Upon the replacement M ! ~ , the resulting equations are identical to (3.26), and the simple solutions of (3.26) can be employed in the present case. In the second step, the rf pulse is considered to be turned o (!1 = 0). Now the equations are just what was solved in the previous section, aside from a transformation to the rf rotating
9 For many imaging applications, it is also turned o a large fraction of the time. 62 Chapter 4. Magnetization, Relaxation and the Bloch Equation reference frame. Upon the replacement !0 ! ! in (4.25)-(4.27), the motion in the rotating frame is described by Mx (t) = e;t=T2 (Mx (0) cos !t + My (0) sin !t) My (t) = e;t=T2 (My (0) cos !t ; Mx (0) sin !t) Mz (t) = Mz (0)e;t=T1 + M0 (1 ; e;t=T1 )
0 0 0 0 0 0 (4.41) (4.42) (4.43) 4.5.3 Long-Lived RF Pulses
The rf eld is kept on for a relatively long time in some applications. The sample is then said to be `saturated' and the long-term behavior of the magnetization can be described by steady-state solutions. Steady State
To nd the steady-state solutions, all time derivatives are set equal to zero in (4.37)-(4.39). We shall rst derive them in the limit, !1 = 0. In this limit, the steady-state solution is easily seen to be
ss Mzss = M0 Mx = Myss = 0
0 0 (4.44)
0 0 The above zeroth-order (in !1 ) solution implies that as !1 ! 0 the transverse magnetization components must vanish. Hence, for small but nonzero !1 , Mx and My must be 2 O(!1) (i.e., at most rst-order in !1).10 Therefore, from (4.37) we nd M0 ; Mz = O(!1 ). Then, correct in rst-order !1, the steady-state solution must satisfy Myss
0 ss Mx
0 Mzss = M0 1 ss ! ; T Mx = 0 2 1 M ss = M ! !+T y 0 1 2
0 0 (4.45) (4.46) (4.47) The solutions of (4.46) and (4.47) are T ss Mx = M0 1 + ( !! 2T )2 !1T2 2 1 Myss = M0 1 + ( ! T )2 !1T2
0 0 (4.48) (4.49) 2 2 correct to O(!1 ). The general-order solutions are treated in a problem. O(xn ) means that the terms vanish at least as fast as xn as x ! 0. In the Taylor series for a function f (x), f (x) = f (0) + xf (0) + x2 f (0)=2 + : : :, the rst term is O(1), the second is O(x), the third is O(x2 ),
10
0 00 and so forth. 4.5. The Combination of Static and RF Fields 63 Problem 4.6
Return to the Bloch equations (4.37)-(4.39) and solve them in the steady state for arbitrary B1 , obtaining ss Mx = M0 ! T2 !1T2 (4.50) D 1 Myss = M0 D !1T2 (4.51) 2 (4.52) Mzss = M0 1 + ( D! T2 ) with
0 0 2 D = 1 + ( ! T2 )2 + !1 T1T2 (4.53) Notice that these reduce to (4.48), (4.49), and (4.45), respectively, for small ss !1. In particular, show that Mx and Myss are O(!1 ) and that M0 ; Mzss = 2 O(!1 ), consistent with the previous discussion. Also, show that the steady-state magnetization develops a phase shift in the x-y plane whose magnitude is
0 0 j j = cot;1( ! T2 ) (4.54) 64 Chapter 4. Magnetization, Relaxation and the Bloch Equation Suggested Reading
The basic concepts of NMR appear in the following early papers: F. A. Bloch. Nuclear induction. Phys. Rev., 70: 460, 1946. F. A. Bloch, W. W. Hansen and M. Packard. Nuclear induction. Phys. Rev., 69: 127, 1946. F. A. Bloch, W. W. Hansen and M. Packard. The nuclear induction experiment. Phys. Rev., 70: 474, 1946. E. M. Purcell, H. C. Torrey and R. V. Pound. Resonance absorption by nuclear magnetic moments in a solid. Phys. Rev., 69: 37, 1946. Chapter 5 The Quantum Mechanical Basis of Precession and Excitation
Chapter Contents
5.1 5.2 5.3 5.4
Discrete Angular Momentum and Energy Quantum Operators and the Schrodinger Equation Quantum Derivation of Precession Quantum Derivation of RF Spin Tipping Summary: The history of the quantization of spin is brie y reviewed. A derivation of precession in a constant eld is presented in the context of quantum mechanics. A quantum mechanical derivation of the action of an rf eld is also derived. This chapter and the following one (Ch. 6) could be omitted in a rst MRI study. Introduction
In quantum mechanics, the wave nature of matter associated with probability amplitudes leads to discrete values for energy, momentum, and angular momentum, just as the waves on violin strings lead to discrete values of frequency, or harmonics. The primary implication for the spin vector of a proton is that only two discrete (quantized) values are found for any measurement of a given spin component. The objective in the present chapter is to show that this is consistent with the classical picture of spins precessing about a magnetic eld. We begin with a little of the history of the measurements of quantum spin interactions with magnetic elds. The quantum mechanical framework for calculations of these interactions, in which the Schrodinger equation plays the central role, is introduced. The use of the classical precession picture for static elds, and of the rf induced rotation of magnetic moments in the rotating reference frame, is justi ed by speci c quantum analysis. In each case, we choose to make an explicit matrix element derivation of the corresponding 65 66 Chapter 5. QM Introduction to MR Fig. 5.1: The historical Stern-Gerlach experiment: Using a vertical (z-axis) magnetic eld gradient from an electromagnet to split a beam of silver atoms in the direction of the gradient. Two views of the experiment are shown. In (a), the direction of the beam is out of the page. In (b), a side view of the beam is shown. MRI phenomenon, rather than one based on quantum operator techniques. The latter1 are important, especially in NMR studies, but perhaps what is presented here is more quickly assimilated for the issues directly at hand. The quantum basis for the thermal interactions of spins and the spin relaxation phenomena is the subject of the next chapter. 5.1 Discrete Angular Momentum and Energy
Discreteness of the energy levels of the proton's magnetic moment interaction with a magnetic eld is related to the discreteness of the proton's intrinsic angular momentum, or spin. In fact, this is the historical path along which scientists came to the conclusion that spin was quantized. In the early 1920's, Stern and Gerlach experimented with a beam of neutral silver atoms passing through a perpendicular (vertical) magnetic eld gradient (see Fig. 5.1). Even with zero electric charge, a magnetic force is exerted on any atom that possesses a nonzero magnetic moment, as is the case for a silver atom, in a spatially varying magnetic eld. To see this, an expression for the force can be found from the gradient of the magnetic ~ potential energy U in (4.8). For a eld B , ~ ~ ~ ~ F = ;rU = r(~ B ) (5.1) The eld produced by the magnet in the gure has y- and z-components, and both are spatially varying. But, averaged over time, the y-component of the magnetic moment (and its spatial gradient) in the central region is zero. This follows from the expected classical ~ precession2 about the z-axis. Therefore, ~ B may be replaced by z Bz in (5.1) and, since
1 2 Operator techniques are described in the references. The quantum mechanical justi cation of the precession picture is the subject of Secs. 5.3 and 5.4. 5.1. Discrete Angular Momentum and Energy 67 Fig. 5.2: Two magnetic monopoles, of equal and opposite charge, lying a distance d apart in the ~ y-z plane, and experiencing a magnetic eld B (y z ) whose components also lie in that plane. The unit vector n is directed from the negative charge toward the positive charge. ^ ~ is independent of position, the z-component of (5.1) becomes (5.2) Fz = z @Bz z Gz @z The notation Gz for the z-derivative of the vertical component of the eld will be useful in later imaging discussions about ` eld gradients.' Problem 5.1
The concept of a magnetic charge has been introduced in Ch. 2. Two oppositely charged magnetic monopoles, as shown in Fig. 5.2, have a magnetic moment given by ~ = qm dn, and the force on a magnetic charge qm due to a magnetic ^ ~ ~ ~ eld B is given by F = qmB . (These results are analogous to the electric dipole moment and the force on an electric charge due to an electric eld. See Appendix A.) Suppose that the z-component of the eld depends only on z, so that its Taylor series is (5.3) Bz (z) = Bz (0) + z dBz (0) + ::: dz In the limit of small separation d (point dipole limit), show that the z-component z of the force on the two monopoles is qmd cos dBdz(0) and compare this answer to (5.2). The experiment of Stern and Gerlach had the two necessary ingredients entering into (5.2), gradients and moments. They used electromagnets with vertical pole faces where the 68 Chapter 5. QM Introduction to MR z-component of the eld increases as the upper pole is approached (Fig. 5.1a). The angular momentum of an unpaired electron in a silver atom gives rise to an atomic magnetic moment3 whose component parallel to the eld gradient would lead to a de ection according to (5.2). The relation analogous to (2.16) is ~ ~ e = e Je (5.4) ~ in terms of the spin angular momentum Je associated with the electron. The classical conclusion from (5.2) and (5.4) is that the beam should be de ected through a spread of angles, due to a continuous spread of magnetic moment values, or more fundamentally, a continuous spread of angular momentum values. Working at the time when many profound quantum discoveries were being made, Stern and Gerlach had the speci c goal of searching for quantization e ects, and they found them. Instead of a smear of de ection angles, their measurements showed that the beam split vertically into two beams, corresponding to two discrete values for the z-component of the angular momentum of the electron. Although it was not known at that time, we now understand the angular momentum of the unpaired silver electron to be entirely due to its spin there is no orbital angular momentum for this electron state. The spin component in (5.2) must be quantized. With the motivation given above, let us describe, as an experimental fact, the quantization of angular momentum for the general case. (It is to be kept in mind, however, that the theoretical underpinnings discussed in the next section can be used to derive these results.) ~ A measurement of, say, the z component of any atomic or nuclear angular momentum J leads to integer (or half-integer) multiples of h, which is Planck's constant divided by 2 , h
The measured values are where the 2j + 1 values of mj are h 2 (5.5) (5.6) (5.7) Jz = mj h mj = ;j ;j + 1 ::: j ; 1 j and j is a positive integer or half-integer whose relation to the total angular momentum is discussed below. The mj are sometimes called magnetic quantum numbers because of their role in magnetic eld experiments like that of Stern and Gerlach. The quantum number j is associated with the magnitude of the total angular momentum ~ according to J J 2 = j (j + 1)h2 (5.8) where
3 1 j = 0 2 1 3 ::: 2 (5.9) In this case, the nuclear magnetic moment is negligible the reader is referred to the discussion in Ch. 2. 5.1. Discrete Angular Momentum and Energy 69 Problem 5.2
A theoretical argument can be made to derive (5.8) from (5.7). Consider making many measurements of Jz2 for some isolated system (or over a large ensemble of identical systems) which has a xed magnitude J . With no outside interaction, each mj value is equally probable. Furthermore, the average values of all components-squared must come out the same for this isotropic situation. (Separate measurements of the components are contemplated, since sharp, simultaneous measurements of more than one component cannot be made. See the next two sections for a further discussion of these quantum e ects.) It is thus assumed that 2 < Jx >=< Jy2 >=< Jz2 > (5.10) Starting with the relation between the magnitude and the components, J 2 = 2 Jx + Jy2 + Jz2 , nd (5.8) from an average value of Jz2 . Hint:
j X
;j m2 = j (2j + 1)(j + 1) j 3 (5.11) ~ The total angular momentum for atomic and nuclear systems has a contribution L from ~ orbital motion and a contribution S from the intrinsic spin, ~ ~ ~ J =L+S (5.12)
Each of these vectors has analogous quantum numbers. For the orbital angular momentum ~ L, the number corresponding to j is called l and is an integer only. Its magnitude satis es L2 = l(l + 1)h2 where l = 0 1 2 ::: (5.13) There are 2l + 1 observed values of ml such that Lz = mlh for any experiment set up to ~ measure the z component of orbital momentum. The spin angular momentum vector S has a quantum number s that can take on both integer and half-integer values. The same form holds for its magnitude that 1 S 2 = s(s + 1)h2 where s = 0 2 1 3 ::: (5.14) 2 with 2s +1 values of ms pertaining to the z-component Sz = msh of the spin vector. Finally, the range of j values for a given set of l and s is found from j = jl ; sj jl ; sj + 1 ::: l + s ; 1 l + s (5.15) The two de ection angles observed in the Stern-Gerlach experiment determine the spin quantum number s for the electron. From the above discussion, and for no orbital angular 1 momentum, it must be that s = 2 and ms = 1 . The electron has `spin one-half.' At the 2 70 Chapter 5. QM Introduction to MR time of the experiment, by the way, it was not known whether an electron had any spin at all. We see that the existence of spin itself is inferred from such measurements. Return now to the proton. Experiments in the years following the work of Stern and Gerlach showed the proton to have spin one-half as well. Thus the magnetic moment (2.16) is discretized and (4.8) leads to discrete energy values. ~ E = ;~ B = ; z Bz = ; mshBz (5.16) 1 parallel to eld ms = 2 ) spin anti;parallel to eld (5.17) spin The two proton energy levels predicted by (5.16) are exhibited in Fig. 5.3. This is an example
of the general Zeeman e ect where atomic or nuclear magnetic moments in the presence of an external magnetic eld lead to splittings in the atomic or nuclear energy levels. with4 Problem 5.3
In atomic physics, the magnetic moment associated with an intrinsic spin angular ~ momentum S is written in terms of the Lande g-factor, ~ ~= S (5.18) where with the `magneton' factor =g
M M (5.19) = e (5.20) 2M In (5.20), e is the magnitude of the particle charge (1:60 10;19 Coulomb for electrons and protons) and M is the particle mass (9:11 10;31 kg for the electron and 1:67 10;27 kg for the proton, leading to the Bohr magneton and the nuclear magneton factor, respectively see Ch. 2). Calculate the gyromagnetic factors (5.19) for the electron and proton, respectively, given the experimental g-factors, ge = 2:01 and gp = 5:58. Compare your answers with those in Sec. 2.2. 5.2 Quantum Operators and the Schrodinger Equation
The challenge that arises in view of the quantum nature of the proton spin degree of freedom is to justify the use of classical precession for the proton spin motion. In this section, the tools needed for a quantum analysis are introduced after a simple, but important, connection
The two ms values are also referred to as `spin-up' and `spin-down,' respectively, stemming from a reference to the z axis.
4 5.2. Quantum Operators and the Schrodinger Equation 71 Fig. 5.3: The Zeeman energy levels for a spin one-half system and a positive gyromagnetic ratio. The spin is parallel to the external eld B0 z in the lower energy state. The wavy vertical line ^
represents a transition from the higher to the lower state by photon emission. between precession and quantum energy di erences is made. The precession justi cation is presented in Sec. 5.3. The magnitude of the energy absorbed or released by the proton spin system, upon a transition (up or down) between the higher and the lower energy states, is found from (5.16) to be 1 1 (5.21) E = E ms = ; 2 ; E ms = + 1 = 1 hB0 ; ; 2 hB0 = h!0 2 2 ~ for a constant eld B = B0z . The frequency in (5.21) associated with the emission or ^ absorption of the quantum of energy (the photon) is nothing other than the Larmor precession frequency !0 = B0 (5.22) It is observed that the frequency of the rf photon that is emitted during such transitions (Fig. 5.3), or that can be used to stimulate such transitions, is the same as the classical precession frequency of the proton magnetic moment. In Larmor precession, however, the magnetic potential energy is constant. To uncover the full quantum description of precession, and to understand better the identity between the frequency associated with the transition from one state to the next and the Larmor frequency, let us develop predictions for the measurement of the magnetic moment. Quantum mechanics is the description of physical states and transitions between those states by wave functions (or wave amplitudes). These wave functions are complex numbers, now a necessity and not merely a convenience, and the square of their modulus (absolute value) is a probability density. For example, a particle wave function (~ t) implies r 5.2.1 Wave Functions Z ignoring spin for the moment. V j (~ t)j2dV = probability of nding the particle in volume V at time t r (5.23) 72 Chapter 5. QM Introduction to MR subject to boundary and initial conditions. The `operator' H (the energy or Hamiltonian operator) is typically a combination of functions and derivatives it operates on . The method of separation of variables, = (~)f (t), can be used to solve (5.24) if H has no r explicit time dependence. In that case, a solution is = (~)e; h Et r provided the time-independent Schrodinger Equation is satis ed,
i It is the wave nature of that leads to discrete energy levels. In de ning the framework of quantum mechanics, the wave functions are assumed to satisfy the Schrodinger di erential wave equation H = ih @ (5.24) @t (5.25) (5.26) H =E According to (5.26), the Hamiltonian operates on to give the energy E of `stationary states,' i.e., those states described by the single term (5.25). For single-particle systems with interactions describable by force potentials 5 where p is the particle momentum and m its mass. The operator r2 = @ 2 [email protected] + @ 2 [email protected] + ~ 2 [email protected] 2 arises from the kinetic energy term and U is the potential energy term. @ The operator representation of the kinetic energy in the second step of (5.27) follows from the momentum operator6 ~ p = ;ihr ~ (5.28) @ Components of (5.28), such as px = ;ih @x , satisfy the commutation relations7 p2 ~ h2 H = 2m + U = ;m r2 + U 2 (5.27) 5.2.2 Momentum and Angular Momentum Operators px x] = py y] = pz z] = ;ih px y] = px z] = py x] = py z] = pz x] = pz y] = 0
with (5.29) (5.30) A B ] AB ; BA 5 These are `non-relativistic' particles, meaning the speeds are much less than the speed of light. The deeply bound electrons in heavy elements have important relativistic corrections. 6 For example, the action of this operator on the `plane-wave' state ei~ ~ yields the momentum `eigenvalue' kr ~ . A particle parameter has thereby been connected with a wave parameter (the wave vector ~ ) p = hk ~ k consistent with the de Broglie relation p = h= where k = 2 = with wavelength . 7 To verify these, distribute the action of the derivative in, for example, the quantity p x]f (x) for an x arbitrary function f (x). 5.2. Quantum Operators and the Schrodinger Equation 73 Commutation relations for spin operators are of special interest. The quantum algebra for angular momentum components, in general, is the set Jx Jy ] = ihJz Jy Jz ] = ihJx Jz Jx] = ihJy (5.31) ~ which also holds for the components of the orbital angular momentum L and intrinsic spin ~ S , separately. In fact, it is straightforward, though tedious, to use (5.29) to check that the components of the operator representation ~ r ~ L = ~ p = ;ih~ r r ~ (5.32) satisfy (5.31) with Jx Jy Jz replaced by Lx Ly Lz , respectively. We shall represent spin angular momenta with nite matrices, instead of derivatives. (Because of the di erence in the commutation relations, matrix representations of linear momenta and position need to be in nite-dimensional.) In particular, the matrix representation for the spin one-half vector is ~ S = h~ (5.33) 2 where the three linearly independent `Pauli spin matrices' are ! ! ! 0 1 0 ;i 1 0 (5.34) x= 1 0 y= i 0 z = 0 ;1 It is left for Prob. 5.4 to show that this spin vector has the correct magnitude and that its components satisfy the correct algebra. The two spin states of a spin-1/2 system can be represented by two-component column vectors. The two-by-two matrices of (5.34) correspond to operators on the spin states. Problem 5.4 Consider the spin one-half vector given by (5.33) in terms of (5.34). 1 a) Show that its square satis es (5.14) for s = 2 . Note that the right-hand-side of (5.14), in ! case, is a matrix proportional to the 2 2 identity matrix this 1 0 . I= 0 1 b) Show that the components satisfy the same algebra as in (5.31), but with ~ ~ J replaced by S . Explain why the algebra itself is a restriction on the magnitude. For the record, it is noted that the well-known Heisenberg uncertainty principle corresponds to the fact that certain operators do not commute. The non-commutativity of px, x, for example, is intimately related to the impossibility of measuring them at the same time. Similarly, the commutation relations (5.31) imply that no two (or more) components of angular momentum can be simultaneously measured. In the limit h ! 0, all commutators are zero and the classical determinism is recovered. The reader is referred to standard textbooks on quantum mechanics for elaboration of these remarks. 74 Chapter 5. QM Introduction to MR Consider the magnetic-moment interaction (only) of a particle, with a gyromagnetic ratio ~ , in a constant external (non-operator) magnetic eld B = B0z . The general solution will ^ be described rst, and its derivation for an example is detailed afterwards. The potential energy is ~ U = ;~ B = ; Jz B0 (5.35) in terms of the operator Jz . The use of a (square) matrix representation for Jz implies that the wave function in (5.26) on which it operates is a column matrix. For states with xed angular momentum j , the column entries j mj refer to di erent values mj of the angular momentum component Jz as previewed in (5.7). A general solution of the linear Schrodinger di erential equation for a given j , but possibly di erent energies, is the superposition of terms of the form (5.25). If only the mj dependence of the energies is shown, the superposition is (~ t) = r
+j X 5.2.3 Spin Solutions for Constant Fields mj =;j Cmj r ; Em t j mj (~)e h j i (5.36) The coe cients Cmj are complex numbers to be determined by initial conditions. With the neglect of the kinetic energy contribution, the energies are the eigenvalues of (5.35) Emj = ; mj hB0 (5.37) to be compared to (5.16). A concrete example of (5.36) is derived below for the spin one-half case. Consider the speci c case of a proton at rest and immersed in the constant external eld. Let us show how the solution of the Schrodinger equation for zero kinetic energy leads to the quantum numbers and matrix elements introduced above, for a spin one-half particle ~ ~ with no orbital motion. Using the potential energy (5.35), the equality J = S , and the representation (5.33), the Hamiltonian in (5.27) becomes
1 h! H = ; B0 Sz = ; 20 0 + 10 2 h!0 ! (5.38) The Pauli representation for z has led to a Hamiltonian that is already diagonalized. The two solutions8 to (5.26) are thus easily found by inspection to be
+
; 1 +1=2 = 0 ! 0 ;1=2 = 1 =E where ! spin parallel (`spin up') spin anti-parallel (`spin down') 1 E = 2 h!0 (5.39) (5.40) (5.41) such that H
8 There are two linearly independent column matrices. 5.3. Quantum Derivation of Precession The solutions
+ 75 are indeed `eigenfunctions' of Sz as well as of H 1 Sz = 2 h (5.42) 1 with the expected `eigenvalues' ms = 2 . We have found that the simplest matrix representation satisfying the spin commutator algebra leads to Schrodinger solutions with the eigenvalues for the energy and the z-component of the angular momentum expected for a spin one-half particle. In simpli ed subscript form, the version of (5.36) for the proton is and
; (t) = X 1 m= 2 Cm m e; hi Em t (5.43) It is noticed that the absence of any ~ dependence originates from the neglect of the proton's r orbital and translational motion. 5.3 Quantum Derivation of Precession
What does quantum mechanics predict for a measurement of the magnetic moment as a function of time, ~ (t), for a proton at rest in a constant eld? In the quantum framework, this is de ned by an average value9 (or `expectation value') for the general state described by (5.43), < j~ j > ~ dV = y~ V XX y ~ = V Cm Cm m S m e hi (Em ;Em)t
y Z m m 0 0 0 (5.44) 0 where, as before, the subscripts on the magnetic quantum numbers are dropped (e.g., ms 1 ). The Hermitian adjoint of a matrix (such as y) is a combined transpose and m= 2 ~ complex conjugate operation. The positioning of ~ = S between y (a row matrix) and (a column matrix) re ects the operator nature of the spin angular momentum vector (and hence of the magnetic moment). The factor V arises from the trivial integration over the volume containing the proton we recall is independent of ~. r The normalization condition on the total probability implies a condition on the complex coe cients in (5.43). The proton must be somewhere in V , i.e., < j > Z y dV = y V =V V XX
m m
0 Cm Cm
0 i (E ;Em )t y m me h m
0 0 =1 (5.45) (5.46) This leads to (see Prob. 5.5)
9 X
m jCmj2 = 1 For a simple, classical probability density distribution ( ), the average value of a quantity is found R by evaluating the integral < > = dV ( ). This is generalized in quantum mechanics, as shown, to a density matrix involving the wave functions. 76 Chapter 5. QM Introduction to MR Problem 5.5
We want to provide the derivation of (5.46). From the explicit solutions (5.39) and (5.40), show that y (5.47) m m = mm where the Kronecker delta ( m = m0 = 1 (5.48) mm 0 m 6= m0
0 0 0 Thus show that (5.45) leads to (5.46).
y ~ The key ingredient in the reduction of (5.44) is m S m . For example, (5.39), (5.40), and y (5.34) can be used to show that m y m is zero if m = m0 , ;i if m0 = 1=2 and m = ;1=2, or +i if m0 = ;1=2 and m = 1=2. We can summarize all results as a vector equation y ^ ^ ^ (5.49) m ~ m = x m ;m + 2miy m ;m + 2mz m m
0 0 0 0 0 0 Problem 5.6
Complete the derivation of (5.49). With (5.21), (5.33), and (5.34), the y-component of (5.44), for example, reduces to the real part of a complex number 1 < y > < j y (t)j > = 2 h V ;iC+ C;e;i!0t + iC;C+e+i!0 t] = hV Re iC; C+e+i!0t ] (5.50) in which C+ C+1=2, C; C;1=2. If a polar coordinate representation is used, C+ = a+ei + C; = a;ei (5.51) then (5.46) requires V (a2 + a2 ) = 1. This is satis ed by a+ p1V cos , a; p1V sin . + ; Hence (5.50) becomes < y > = h sin cos sin ( ; ; + ; !0 t). The calculation of the other two components (see the following problem) with =2, ; ; + 0 leads to < x > = 2h sin cos ( 0 ; !0t) (5.52) (5.53) < y > = 2h sin sin ( 0 ; !0t) < z > = 2h cos (5.54)
; 5.3. Quantum Derivation of Precession 77 The set of expectation values for the magnetic moment components represents a vector of magnitude h=2 precessing clockwise about the z-axis at a xed polar angle . The initial azimuthal angle is 0. The motion is illustrated in Fig. 5.4. Problem 5.7
a) Derive the other two components (5.52) and (5.54). b) What is the expectation value of + = x + i y ? We observe that the quantum mechanical state consisting of an arbitrary superposition of a parallel spin state and an anti-parallel spin state leads to an expectation value which is precisely the clockwise precession predicted classically for a proton.10 The relative weighting (a+=a;) determines the polar angle and the relative phase ( ;; + ) xes the initial azimuthal angle. Fig. 5.4: The precession of the quantum expectation value of the magnetic moment operator in the presence of a constant external eld pointed in the z -direction. Quantum mechanics shows itself only through the h factor coming from the spin dependence of the moment magnitude h=2. The proton spin is fundamentally a quantum property.
10 78 Chapter 5. QM Introduction to MR 5.4 Quantum Derivation of RF Spin Tipping
Attention is now turned to a magnetic moment immersed in the combined static and leftcircularly polarized rf eld (3.41) ~ B (t) = B0 z + B1(^ cos !t ; y sin !t) ^ x ^ (5.55) We should like to justify the classical picture where, in the rotating reference frame de ned by the rf eld, and on-resonance, the motion is simply a rotation of the magnetic moment about the x0 -axis (the rf eld direction). The calculations proceed in a manner similar to those in the preceding section. The Hamiltonian for a proton at rest in the eld (5.55) is ~ H (t) = ;~ B (t) = ; 2h ( z B0 + ( x cos !t ; y sin !t)B1 ) (5.56) recalling the Pauli spin vector de nition of (5.33). The time dependence of the Hamiltonian will have to be considered, but it is still possible to use the method of separation of variables in the solution of the Schrodinger equation for the present case. It is useful to write (5.24) in terms of the matrix elements of H (t). From (5.34), the static- eld term is ! h B = ; h!0 1 0 (5.57) ;2 z 0 2 0 ;1 and the component of ~ along x0 is ^
x
0 = x cos !t ; y sin !t = e;i!t 0 ei!t 0 ! (5.58) ! @ = ;h !0 !1 ei!t ih @t (5.59) 2 !1e;i!t ;!0 with the usual identi cations for the frequencies, !0 B0 and !1 B1. Note that, as in the previous section, the h factor can be canceled out, a rst indication that classical behavior will be recovered. A simple set of di erential equations can be found by a change of variables. Consider the following (completely general) form for the wave function 0 i!0 t=2 ! 0 i!0 t=2 + 0 (t) e;i!0 t=2 = 1 (t)e = 1 (t) + e (5.60) ; 2 0 ;i!0 t=2 2 (t)e The superposition of the `complete set' of column matrices, + and ; , leads to di erential 0 0 equations, with no !0 terms, for the new wave variables, 1 (t) and 2 (t).11 After a cancellai!0 t=2 factors, and the assumption that the tion of those terms by the time derivatives of the e
In fact, the change of variables (5.60), as the primes suggest, is the wave function transformation from the laboratory to the Larmor rotating frame.
11 The Schrodinger equation becomes 5.4. Quantum Derivation of RF Spin Tipping 79 where the ci are, in general, complex constants. Also, from (5.61) and (5.62), c3 = ;ic2 (5.66) c4 = ic1 (5.67) With the wave function known, the expectation values of the di erent components of the magnetic moment vector can be computed. The expectation values of interest are h z i, and the rotating transverse components, h x i and h y i, since the classical behavior predicts precession about the x0 -axis. The rst calculation (cf. (5.44)) yields 1 0 0 (5.68) h z i h j z j i = 2 h V j 1 j2 ; j 2j2 To determine h x i and h y i, the combinations of Pauli spin matrices that arise are x given in (5.58), and ! 0 ;ei!t (5.69) y = x sin !t + y cos !t = i e;i!t 0 The remaining 2 2 matrix elements can be reduced to 0 0 h x i = hV Ref( 1) 2g (5.70) 0 0 h y i = hV Imf( 1 ) 2 g (5.71)
0 0 0 0 0 0 0 0 rf eld is on-resonance (i.e., ! = !0 ), the result is the coupled system for the column-matrix components 0 d 1 = i! 0 (5.61) dt0 2 1 2 d 2 = i! 0 (5.62) dt 2 1 1 Because of the on-resonance condition, the coupled equations have no explicit time dependence in their coe cient. Taking a second time-derivative of (5.61)-(5.62), a technique which was used to decouple a similar set of di erential equations in Sec. 2.3.3 (Prob. 2.6), gives 0 d2 1 2 1 2 0 (5.63) 2 = ; 4 !1 1 2 dt Equations (5.63) have the general solutions !1t !1t 0 (5.64) 1 (t) = c1 cos 2 + c2 sin 2 !1t !1t 0 (5.65) 2 (t) = c3 cos 2 + c4 sin 2 Problem 5.8
Provide the details for the calculations leading from (5.64) and (5.65) to (5.68), (5.70) and (5.71). 80 Chapter 5. QM Introduction to MR The next step is to write the magnetic moment expectation values in terms of the explicit solutions, (5.64) and (5.65). Two computations provide the necessary ingredients
0 0 j 1 j2 ; j 2j2 = (jc1j2 ; jc2j2) cos !1t + 2Refc1 c2g sin !1t 1 0 0 ( 1 ) 2 = i 2 (jc1j2 ; jc2j2) sin !1t ; iRefc1c2g cos !1t + Imfa bg (5.72) (5.73) (5.74) The condition (5.45) on the wave function normalization is now
0 0 h j i = V j 1j2 + j 2 j2 = V jc1j2 + jc2j2 = 1 The general solution of the constraint (5.74) is c1 p1V cos 2 e;i 1 and c2 p1V sin 2 e;i 2 . The combination (5.72), (5.73), and the constraint solutions transforms (5.70), (5.71) and (5.68) into h x (t)i = 2h sin cos (5.75) (5.76) h y (t)i = 2h cos sin !1t + sin cos !1t sin ] h z (t)i = 2h cos cos !1t ; sin sin !1t sin ] (5.77) with 1 ; 2 ; =2. The initial values of the expectation components are h x (0)i = 2h sin cos (5.78) (5.79) h y (0)i = 2h sin sin h z (0)i = 2h cos (5.80) where it is noted that h x (t)i is independent of time. Equations (5.78)-(5.80) describe an arbitrary initial orientation for a vector of magnitude 2h . In spherical coordinates, the initial direction is seen to have a polar angle of and an azimuthal angle of . The expectation-value components (5.75)-(5.77) can be rewritten in terms of the initial components (5.78)-(5.80),
0 0 0 0 0 h x (t)i = h x (0)i h y (t)i = h y (0)i cos !1t + h z (0)i sin !1t h z (t)i = ;h y (0)i sin !1t + h z (0)i cos !1t
0 0 0 0 0 (5.81) (5.82) (5.83) Equations (5.82)-(5.83) represent a vector, of xed magnitude, which is precessing clockwise about the x0-axis, in the rotating reference frame, with a precession frequency !1 as predicted by classical theory. Arbitrary initial conditions are obviously accommodated. 5.4. Quantum Derivation of RF Spin Tipping 81 Suggested Reading
Excellent reviews of the basic quantum aspects required to understand the NMR signal appear in the following three references: A. Abragam. The Principles of Nuclear Magnetism. Clarendon Press, New York, 1961. I. I. Rabi, N. F. Ramsey and J. Schwinger. Rotating coordinates in magnetic resonance problems. Rev. of Modern Phys., 26: 167, 1954. C. P. Slichter. Principles of Magnetic Resonance. Springer-Verlag, III ed., New York, 1990. 82 Chapter 5. QM Introduction to MR Chapter 6 The Quantum Mechanical Basis of Thermal Equilibrium and Longitudinal Relaxation
Chapter Contents
6.1 Boltzmann Equilibrium Values 6.2 Quantum Basis of Longitudinal Relaxation 6.3 The RF Field Summary: The equilibrium magnetization M0 is derived for a system of particles with spin at temperature T in the presence of a static external eld. The thermal quantum basis of the Bloch equations for combined static and rf elds is brie y discussed. Introduction
In the quantum description of precession in both the laboratory and rotating reference frames, given in Ch. 5, the relaxation mechanisms were ignored. We turn here to the quantum description of the interactions that lead to both spin-lattice and spin-spin relaxation. The thermal equilibrium value of the magnetization is determined, in the presence of relaxation, for a constant external magnetic eld. This can be referred to as the Boltzmann equilibrium value it applies when the time between experiments is long compared to the relaxation time scale. The relaxation terms used in the previous chapters can themselves be derived in a quantum framework. The quantum basis for these terms and the driving terms used to model an rf magnetic eld is very brie y touched upon. Transient and steady-state solutions of simple versions of the Bloch equations are studied. 83 84 Chapter 6. QM Basis of Relaxation 6.1 Boltzmann Equilibrium Values
The equilibrium value M0 was required in the solution (4.12) of the Bloch equation for the longitudinal magnetization, Mz (t). This value, arising in the relaxation limit Mz (1) = M0 , represents the trade-o between the tendency of a spin system to align itself with the external eld (the lowest energy state), and its ability to gain energy from thermal contact. Interactions beyond those with the external elds are at the heart of these exchanges. To set the stage for a discussion of the exchange interactions, consider two systems in external magnetic elds, with gravity ignored. First, the classical motion of a point particle with charge q and velocity ~ is governed by the Lorentz magnetic force law (see Appendix v A) ~ v ~ F = q~ B (6.1) ~ in the presence of an external magnetic eld B . The particle traces out helical paths along the eld lines as time goes on. Although the Lorentz force is observed to be perpendicular to the motion, so that no work is performed on the particle, radiation and other dissipative interactions will eventually leave the particle at rest. The second system is a compass needle initially swinging back and forth in response to torque from an external eld (see Ch. 2). It will slow down due to friction, with smaller and smaller oscillations around the position of zero torque. The point particle stops, and the bar magnet ultimately settles down along the eld direction, because energy is given o to the surroundings. On the other hand, if the particle and the bar magnet were in thermal contact with other material, and at (absolute) temperature T , they would retain kinetic energy on the order of kT , where k is Boltzmann's constant. The exchange interactions will leave these systems (as well as a system of spins) in equilibrium somewhere above the (ground) state of lowest energy, depending on how large kT is. The probability of nding a system with energy , while in contact with a much larger system (the `reservoir' depicted in Fig. 6.1) at temperature T , is equal to the normalized Boltzmann factor,1 e; =kT P( ) = Z (6.2) The normalization divisor is the partition function, the sum over all weighting factors, X Z = e; =kT (6.3) A system of interest is a spin in thermal contact with the rest of a set of N spins and with the background lattice all at temperature T . The number N is taken to be very large along with the size of the lattice. To nd the thermal equilibrium value of Mz , consider the calculation of the z-component of the average total magnetic moment for N spins distributed over all possible magnetic spin states, neglecting translational motion. This brings together the thermal interactions and the quantum basis of the magnetization. The quantization axis is chosen along the external eld direction (the z-axis, as usual) and the case of a general spin s, with magnetic number ms m, is analyzed. The thermal average of the magnetization is M0 =
1 0 s X m=;s P ( (m)) z (m) (6.4) See any introductory thermal physics text for a discussion of the Boltzmann factor. 6.1. Boltzmann Equilibrium Values 85 where 0 = N=V is the density of spins per unit volume in the homogeneous isochromat of volume V .2 From (5.16), = ;mh!0 (6.5) (6.6) z=m h The explicit expression for the equilibrium magnetization is N h Ps =;s memu m M0 = V Ps emu (6.7) m=;s with (6.8) u h!0 kT Fig. 6.1: A small system in thermal equilibrium with a reservoir at temperature T . The total
energy
0 is conserved between the two systems. Problem 6.1
Find the value of (6.7) in the limits a) T ! 1. Your limit should be consistent with the expectation of the spins being equally distributed over all 2s + 1 states. b) T ! 0. Your limit should be consistent with the spins all dropping to the bottom (ground) state. Hint: The maximum magnetization is: M0max = 0 hs corresponding to m = mmax = s for every spin. There is no longer any thermal interaction.
In this chapter, 0 describes the true density of spins per unit volume. However, in later chapters, other factors will be absorbed into 0 , and it will be used to describe the sensitivity of the MRI experiment to a group of spins. The term `spin density' is still employed in these instances, but the reader is cautioned that it is not always a measure of the number of spins per unit volume.
2 86 Chapter 6. QM Basis of Relaxation 0 !1 h!0 + O @ h!0 2 A + ::: = 1+ kT kT ' 1 + 6:6 10;6B0 (6.9) For the nite m values of interest, emu is also very close to unity, leading to a simpli ed limit for (6.7). For arbitrary spins, it is to be shown in Prob. 6.2 that 1) 2 2 M0 ' 0 s(s +kT h B0 (h!0 kT ) (6.10) 3 This form of M0 will be used in Ch. 7 for de ning the relative signal strengths of di erent elements in an MR experiment.
exp h!0 kT The expression (6.7) can be simpli ed in MR, because the nuclear magnetic energies are so much smaller than room-temperature thermal energies. Having referred, in Ch. 4, to the extremely small `excess spin' in the direction of the magnetic eld, we can now explain that it comes from the Boltzmann factor. For human body temperature (310 K), and protons, the basic exponent unit in (6.7)-(6.8) has the numerical value u ' 6:6 10;6B0 for B0 in Tesla. Hence the basic exponential is unity to within ten parts per million for eld strengths in that range, ! Problem 6.2
Derive (6.10) as the leading term in a 1=T expansion of (6.7) using emu ' 1+ mu. s X Hint: Since m = 0, the rst-order term, such as that in (6.9), must be kept. ;s Now see Prob. 5.2. The limit (6.10) constitutes a quantum derivation of the experimental Curie's law for magnetization, which states that the magnetization should be proportional to 1=T , but with the bonus of determining the coe cient. Its vanishing in the limit of T ! 1 should agree with the reader's result for part (a) in Prob. 6.1. For a proton (s = 1=2), (6.10) becomes (proton h!0 kT ) (6.11) kT B0 Finally, let us revisit these results in terms of the proton spin excess, de ned by the difference between the number of spins parallel (N (")) and anti-parallel (N (#)) to the external eld:
0 1 M0 ' 4 2 h2 N N (") ; N (#) = N (P+ ; P;) ' Nu 2 (6.12) 6.2. Quantum Basis of Longitudinal Relaxation
1 where the Boltzmann probability (6.2) for the two spin- 2 states (m = 1 2 87 or =
1 h! ) 2 0 is u=2 P = eu=2e+ e;u=2 (6.13) Problem 6.3
A homogeneous sample of protons with gyromagnetic ratio is immersed in a uniform static eld B0 and at arbitrary absolute temperature T . Find an expression in terms of the hyperbolic tangent for the average value (thermal equilibrium) of the z-component of the magnetic moment vector for one of these protons. Show that this gives the expected T ! 0 and T ! 1 results. Problem 6.4
a) Derive the approximation in (6.12). b) Show that (6.12) implies an excess of 5 out of every 106 spins at 1.5 T. 6.2 Quantum Basis of Longitudinal Relaxation
The next task is to understand, by an example, how the Bloch relaxation terms themselves can be derived within the quantum approach. The local interactions leading to relaxation are treated as perturbations of the spin-system external magnetic- eld quantum states. The example to be considered is the z-component of the Bloch equations for a constant external eld. Suppose there are N protons at rest and subjected to the constant magnetic eld B0z , ^ with no other external elds. (We return later to the description of the rf eld interactions in this framework.) Each can be in either of the two spin states, m = 1=2. The lower energy state pertains to m = +1=2 where the magnetic moment and the eld are parallel. None of the protons could make a change of state without additional forces, reminiscent of the earlier remarks about point charges and compass needles. Turning on the interactions with the neighborhood atomic and nuclear lattice allows state transitions. These small disturbances can be analyzed in `time-dependent perturbation theory.3 ' The lowest-order transition probability rate to go from state i to state f (or a set of nal states ff g) due to a small perturbation potential V , as it occurs in the Schrodinger
For a detailed discussion of time-dependent perturbation theory, see any introductory quantum mechanics text.
3 88 equation, is Chapter 6. QM Basis of Relaxation Wfi = 2h j< f jV ji >j2 (Ef ) (6.14) The importance of this formula in quantum calculations has led to it being called the `golden rule number two.' The occurrence of the square of the modulus of the rst-order transition amplitude in (6.14) is the familiar connection between wave amplitude and probability. The notation for the o -diagonal transition matrix element < f jV ji > follows that in Ch. 5. The density (Ef ) of nal states per unit energy reduces for one nal state to a Dirac delta function (see Ch. 10 for the de nition of this function) expressing the conservation of energy
(Ef ) = (Ef ; Ei ) (6.15) The derivation of (6.14) can be found in the references. Labels are needed for the transition probabilities and the number of states. Any transition down in energy from m = ;1=2 to m = +1=2 for one proton must be accompanied by a transition upwards in energy from some lower lattice state l to a higher lattice state h (l; ! h+). The energy jumps, up and down, must be equal, by the law of conservation of energy. The lattice states must include other nucleon spin states. For the transition up in energy for a proton, the picture is reversed (h+ ! l;). See Fig. 6.2. We denote the respective probability rates (6.14) for these two processes by Wh+ l; and Wl; h+. (Strictly speaking, these are the rates per proton per lattice site.) Finally, if N is the initial number of spins with m = 1=2, respectively, and if nh (nl ) is the number of initial lattice states with the higher (lower) energy, the rate of change of the +1=2 spin-state number is dN+ = W N n ; W N n (6.16) h+ l; ; l l; h+ + h dt The (large) numbers of lattice states, nh and nl , are independent of changes in N+, and are related to each other by the Boltzmann factor ratio, nh = e; hB0 =kT (6.17) nl which is just the ratio of the average values for nh and nl . The probabilities in (6.16) are equal in lowest-order perturbation theory, Wh+ l; = Wl; h+ W (6.18) This follows from the `detailed balance' relation Wfi = Wif which holds for any Hermitian4 potential V . In that case, < f jV ji >=< ijV jf > in the formula (6.14), using (6.15) for a single state f . Also, in terms of the spin excess pointing along the eld, N N+ ; N; (6.19)
and the ( xed) total number of spin states N = N+ + N;
4 (6.20)
y Recall from Sec. 5.3 that Hermiticity of a matrix M implies that M M T = M. 6.2. Quantum Basis of Longitudinal Relaxation 89 is explained in the text. Fig. 6.2: The response of the lattice to transitions between the two proton spin states. The labeling the spin-state numbers can be rewritten N =N 2 N
From (6.16), (6.18) and (6.21), (6.21) d( N ) = WN (n ; n ) ; W N (n + n ) l h l h dt In the relaxed or equilibrium steady state, d( N )=dt = 0, leading to
( N )0 = (nl ; nh) N (nl + nh) (equilibrium) (6.22) (6.23) The zero subscript refers to the equilibrium value, achieved in the long-time limit. Problem 6.5
Show that (6.23) and (6.17) yield a) the common ratio for the equilibrium values N; N+ ! = nh = e; hB0 =kT nl 0 (6.24) b) a high temperature limit consistent with (6.11). 90 Chapter 6. QM Basis of Relaxation With the coe cient of N on the right-hand-side of (6.22) identi ed as the spin-relaxation decay rate, 1 W (n + n ) (6.25) l h T then
1 d( N ) = ( N )0 ; N (6.26) dt T1 1 Finally, multiply (6.26) by the z-component of the magnetic moment, 2 h, and average over the volume, as in (4.1), to get Mz for a proton. The result is dMz = M0 ; Mz (6.27) dt T1 This is the same form as that in (4.11). 6.3 The RF Field
The external eld cross products in the Bloch di erential equations, which drive the solutions, are the remaining terms to be justi ed. For the constant external eld, the quantum ~ ~ derivation of the expression M B0 follows from a volume average of the magnetic moment equations of motion found by the methods of Ch. 4. The time-dependent rf eld cannot be found quite so straightforwardly. However, we can analyze e ects of rf elds with amplitudes that are small compared to the constant eld, using the same perturbation theory used in the discussion of relaxation. Consider N protons at rest and immersed in the constant magnetic eld B0 z , but with ^ no lattice interactions. Turning on the rf eld can cause transitions between the two spin states for each proton. The fact that the rf eld is perpendicular to the z-axis produces complications in the quantum calculations. The amplitude in the transition rate (6.14) ~ involves a perturbation potential ;~ B1, where spin-vector components transverse to the quantization axis arise. Spin operators x and y , which were discussed in Ch. 5, lead to o -diagonal matrix elements, and hence to cross terms in the square of the probability amplitudes. The di erent magnetization components become mixed together in the Bloch equations, as a result of these complications. If the vector nature of both the rf eld and the magnetization is ignored, the origin of an external eld driving term is easy to understand. For the remainder of this argument, we shall make that simpli cation. For such rf elds, de ne the probability rate for a proton to drop to m = +1=2 from m = ;1=2 as w+;. The probability rate for the proton to go up to m = ;1=2 from m = +1=2 is w;+. It is important to note that energy exchange takes place through the interaction with the external eld no lattice neighborhood interaction is needed for the exchange, so that interaction is also ignored. The `detailed balance' relation is w+; = w;+ w (6.28) so dN+ = N w ; N w = (N ; N )w (6.29) ; +; + ;+ ; + dt 6.3. The RF Field 91 Here, Boltzmann factors and lattice population numbers have not entered into the analysis. Using (6.21) for N , we obtain d( N ) = ;2w N (6.30) dt for constant N . Problem 6.6 Solve the di erential equation (6.30) in terms of the initial condition N (0). Conclude from the solution that any initial spin excess disappears for constant w. Within the approximations of the model, the driving term, due to the external rf eld, can be joined to the relaxation rate (6.26) d( N ) = ( N )0 ; N ; 2w N (6.31) dt T1 Equation (6.31) is an uncoupled version of the z-component Bloch equation (4.21). To derive the correctly coupled form of the equations (4.22)-(4.24), a more detailed examination of the quantum transition amplitudes is required. It can be shown, however, that the general relaxation-plus-driving-term structure arises in much the same way as in the above. And the solutions of the diagonal version (6.31) are also instructive. For example, the steady-state, or equilibrium, solution is N )0 (6.32) Neq = 1(+ 2wT
1 92 Chapter 6. QM Basis of Relaxation Suggested Reading
The fundamental concepts of the quantum processes are given in the following four references: N. Bloembergen, E. M. Purcell and R. V. Pound. Relaxation e ects in nuclear magnetic resonance experiments. Phys. Rev., 73: 679, 1948. E. M. Purcell, H. C. Torrey and R. V. Pound. Resonance absorption by nuclear magnetic moments in a solid. Phys. Rev., 69: 37, 1946. I. I. Rabi, N. F. Ramsey and J. Schwinger. Rotating coordinates in magnetic resonance problems. Rev. of Modern Phys., 26: 167, 1954. C. P. Slichter. Principles of Magnetic Resonance. Springer Verlag, III ed., New York, 1990. Chapter 7 Signal Detection Concepts
Chapter Contents
7.1 7.2 7.3 7.4
Faraday Induction The MRI Signal and the Principle of Reciprocity Signal from Precessing Magnetization Dependence on System Parameters Summary: The physical principles of MR signal detection are derived from Faraday's law of electromagnetic induction. The principle of reciprocity is used to obtain an expression for the MR signal in terms of the sample magnetization and the eld of the detector coils. Both real valued and complex valued demodulated signal formulas are presented. The strength of the MR signal as a function of the static and rf elds, the gyromagnetic ratio, the natural abundance, and other parameters is discussed. Introduction
In the previous chapters, the central theme has been to detail the changes in the magnetization of a sample that are produced by external magnetic elds. One such case included rotating the magnetization away from its alignment along a static eld by the application of a perturbing, transient rf eld perpendicular to the static eld direction. The rotation is actually a spiraling-downward motion due to the continuing precession about the static eld direction. Once the magnetization has a transverse component, the detection of its precession about B0 can be considered. It is the spin's own magnetic eld lines that are swept along with the precession which forms the key to observing the magnetization. An electromotive force (emf ) would be created in any coil through which the spin's magnetic ux sweeps, a consequence of Faraday's law. The time-dependent form of this current carries the information that is eventually transformed into an image of the sample. Let us look more closely at the signal generated by the precessing magnetization. Consider a single magnetic moment as a bar magnet precessing about an axis perpendicular to 93 94 Chapter 7. Signal Detection Concepts its length. The magnetic ux of a rotating bar magnet is rotated as well, and an emf may be induced in nearby coil elements. We refer the reader to Fig. 7.1. On a macroscopic scale, this system is recognized as an electrical generator. The electrical power in use today is generally produced by forcibly spinning giant electromagnets near conducting coils. The MR signal is detected using the same principles: an rf coil is placed near a body which contains a large number of tiny rotating nuclear magnetic moments. Both the transmit and receive coils are designed for optimal performance at the same Larmor frequency so they may e ciently tip the magnetization and detect the resulting signal, respectively. Indeed, a single coil can be used for both purposes. A great deal of interesting engineering goes into the detection of the MR signal. In this chapter, the basic physical principles involved will be discussed, and the engineering issues will be considered in later chapters dealing with coil design. Below, the relationship between the strength of the MR signal and the magnetization of the sample is quanti ed, starting with coil emf considerations. The di erent variables upon which the signal depends are exhibited, and the signal processing issues, such as demodulation, are explored. In the next chapter, simple but important signal detection experiments are introduced. 7.1 Faraday Induction
The emf induced in a coil by a change in its magnetic ux environment can be calculated by Faraday's law of induction (see Appendix A): where is the ux through the coil,1 = emf = ; d dt (7.1) ~ ~ B dS (7.2) coil area The ux can be thought of as proportional to the number of eld lines penetrating the e ective area presented by the loops making up the coil. The number of ux lines is a convenient picture, and is arbitrarily normalized. Equation (7.2) is, by contrast, well-de ned. The currents induced in a conducting loop by a changing ux produce a eld which opposes the changes induced by the external eld.2 In the application of (7.1) to MRI, a study of elementary examples with wire loops is helpful. The rst example is a xed coil in a time-dependent, xed-axis magnetic eld. Consider a harmonic (i.e., sinusoidal in time) magnetic eld with angular frequency ! making an angle to the normal of the plane of the square coil shown in Fig. 7.2a. The magnetic eld is given by ~ B (t) = B (sin y + cos z ) sin !t ^ ^
1 Z (7.3) ~ The vector dS has magnitude dS and is normal to the di erential area in the direction chosen for the de nition of positive ux. 2 The result that induced currents produce elds that oppose externally induced ux changes is referred to as Lenz's law. 7.1. Faraday Induction 95 macroscopic bar magnet is shown in (a). Part (b) of the gure models the magnet as the combined e ect of many individual spins (small dark arrows). The depiction of individual spins in (b) is only symbolic since an average sample contains trillions of trillions of nuclei. According to the discussion in Sec. 7.1, the ux through the coil is minimized when the bar length is in the plane of the coil (c) and maximized when it is perpendicular to that plane (d). In order for the emf to be nonzero, the ux must change in time and some coil element must have a component of its direction that is perpendicular to the applied ux. Fig. 7.1: This gure illustrates the ux coupling of a magnet with a loop of wire. A single rotating 96 Chapter 7. Signal Detection Concepts for constant B > 0. The eld (7.3) is taken to be uniform over the space of the coil, at any ~ instant in time.3 The coil is chosen to lie in the x-y plane so that dS = dxdyz. From (7.1) ^ and (7.2), the emf generated in the coil by the time-varying magnetic eld is d Z L=2 dx Z L=2 dyz B (t) emf = ; dt ^ ~ ;L=2 ;L=2 = ;L2 B! cos cos !t (7.4) At t = 0, for example, the emf is negative, producing a clockwise (negative ^-direction) current in the loop. This can be veri ed by using the right-hand rule to see in which direction the current would have to ow to produce a eld which opposes the change in ~ the applied eld B (t). The appearance of the ! factor in (7.4), which results from the time derivative, is an example of the signi cant fact that the emf increases with higher frequency. The next example is a rotating coil in a constant eld, which is the subject of Prob. 7.1. This case has an induced emf identical to that which would arise if, instead, the coil were stationary and the eld rotated, a situation more closely related to the MRI experiment. In a standard MRI experiment, the eld associated with a precessing magnetization sweeps past xed receiving coils. Problem 7.2 addresses a simple situation of this kind. Fig. 7.2: Three related examples of loops of wires experiencing changing ux as a function of time: (a) A square coil is stationary, and the applied eld oscillates along a xed axis. (b) A circular coil rotates in a static eld. (c) A stationary coil is immersed in a rotating, xed-magnitude eld (an example of special relevance to MR measurements).
3 A magnetic eld that varies in time must also vary in space, according to Maxwell's equations (see Appendix A). However, such spatial variations may be ignored for distances small compared to the wavelength, = 2 c=!. At 1.0 T the free-space wavelength for electromagnetic radiation is about 7 m. 7.2. The MRI Signal and the Principle of Reciprocity 97 Problem 7.1
a) A vertical circular loop of wire with radius a rotates about the z-axis at constant angular frequency, as shown in Fig. 7.2b, in a uniform magnetic eld. If the loop lies initially in the y-z plane, the normal to the plane of the loop changes with time according to n(t) = x cos !t + y sin !t. The magnetic ^ ^ ^ ~ = B x. Find the emf induced in this coil. eld is given by B ^ ~ b) Consider another orientation of the loop, assuming that B remains the same, ~ i.e., B = B x. What emf will be induced in the loop if it initially lies ^ in the x-y plane, and is rotated about the x-axis? In this case, n(t) = ^ y sin !t + z cos !t. ^ ^ Problem 7.2
Consider the situation in Fig. 7.2c where the coil lies in the x-y plane and a spatially independent eld rotates about the x-axis: ~ B (t) = ;B sin !tz + B cos !ty ^ ^ Show that the emf induced in the coil is L2 B! cos !t. (Note that such elds can be produced by, for example, `birdcage' coils, provided the rf wavelengths are not too small. See Ch. 27.) 7.2 The MRI Signal and the Principle of Reciprocity
Equation (7.2) can be converted into a form which is more useful for MRI, where the roles of the magnetization source and the detection coil are reversed. Switching their roles in an equation for the signal is an example of the principle of reciprocity which will be described later on in this section. There is a magnetic eld associated with the magnetization of a sample arising from the e ective current density ~ r ~ ~ r JM (~ t) = r M (~ t) (7.5) ~ ~ ^ A current density J implies jJ j charge per unit time per unit area in the direction of J . The curl operation in (7.5) computes the net `circulation' of the magnetization. To get a feel for this, magnetization can be approximated as a density distribution of many small current loops. The net currents associated with these loops correspond to the e ective current 98 Chapter 7. Signal Detection Concepts density (7.5). Some background for the magnetization current and the vector calculus to follow is found in Appendix A. The vector potential at position ~ stemming from a source current, such as (7.5), is r Z ~ r0 ~ (~) = 0 d3r0 J (~ )0 Ar 4 (7.6) j~ ; ~ j r r where the time dependence has been suppressed and the `gauge choice' is left understood. The e ects due to the time delay between the source and the measurement of the eld are ignored. The magnetic eld is calculated from ~ ~ ~ B=r A (7.7) Using (7.7) and Stokes' theorem,4 it is possible to write the ux (7.2) through a coil in terms of the vector potential = Z area ~ ~ B dS = Z area ~ ~ ~ (r A) dS = d~ A l ~ I (7.8) A detailed derivation demonstrating that the ux M through a coil due to a magnetization source can be related to a ux due to the coil that goes through the magnetization is given next. The use of (7.5), (7.6) and (7.8), an integration by parts (where a surface ~ ~ ~ ~ ~ ~ term can be ignored for nite sources), and the vector identity, A (B C ) = ;(A C ) B , respectively, give
M 3 2 Z ~ ~ r r0 M (~ 0 ) 5 = d~ 4 4 0 d3r0 j~ ; ~ 0 j l r r " ! # Z I 1 0 3 r 0 d~ 0 0 ~ ~ r = 4 d l ;r j~ ; ~ 0j M (~ ) r0 r 2 Z I d~ 13 0 ~ r ~ = d3r0M (~ 0) 4r0 @ j~ ;l~ 0j A5 4 r r I (7.9) The version of (7.6) for current loops, now evaluated at position ~ 0, r I Id~ l ~ (~ 0) = 0 Ar 4 j~ ; ~ 0j r r (7.10) ~ shows that the curl of the line integral over the current path in (7.9) is actually Breceive, the magnetic eld per unit current that would be produced by the coil at the point ~ 0, r 0 I 1 d~ A l ~ ~r ~ Breceive(~ 0 ) = B (~ 0)=I = r0 @ 4 0 j~ ; ~ 0j r r r (7.11) Stokes' theorem states that the surface integral of the scalar product between the curl of an arbitrary vector function ~ and R normal to the surface is equal to the line integral of ~ around the closed path a the a ~ ~ a H l a bounding the surface: dS (r ~ ) = d~ ~
4 7.3. Signal from Precessing Magnetization Finally, the ux can be written as Z M (t) = 99 ~ d3rBreceive(~) M (~ t) r ~ r (7.12) sample where the time dependence of the magnetization is now made explicit. The fact that the ux ~ in (7.12) depends upon Breceive, the `receive' eld5 produced by the detection coil at all points where the magnetization is nonzero, is an example of the principle of reciprocity. The original expression as a surface integration over the detection coil area has been replaced by a volume integration over the region of nonzero magnetization. That is, the ux through the detection coil due to the magnetization can be found instead by calculating the ux that would emanate from the detection coil, per unit current, through the (rotating) magnetization. The emf induced in the coil is expressed as d emf = ; dt M (t) dZ ~ r ~ = ; dt d3rM (~ t) Breceive(~) r (7.13) sample
Equation (7.13) is a key formula for understanding the factors which a ect signal amplitude. transmit is implicit in the The dependence of the emf on the excitation or transmit eld B1 ~ dependence of (7.13) on the magnetization M . Problem 7.3
Consider two xed loops of wire, each carrying the same current I (t). Show that the emf induced in coil 1 by coil 2 is identical to the emf induced in coil 2 by coil 1, an example of a reciprocity relation. Hint: Consider (7.8) and the vector potential for current loops (7.10). 7.3 Signal from Precessing Magnetization
The fundamental signal in an MR experiment comes from the detection of the emf predicted by (7.13) for precessing magnetization. In this section, the prediction is analyzed further, in terms of variables involving the static and rf elds, and the properties of the sample of spins. The experiment described below from a given sample is generally referred to as a `free induction decay.' This experiment is discussed in more detail in the next chapter. 7.3.1 General Expression
5 It is assumed that the sample is immersed in a static, uniform eld B0 z and has been ^ `excited' by some rf pulse so that there exists, at time t, transverse components, Mx My of
Recall that the term `magnetic eld per unit current' refers to magnetic eld divided by a constant ~ ~ current, B. In later discussions, the notation B will be used to describe this quantity. 100 Chapter 7. Signal Detection Concepts magnetization, in addition to a longitudinal component, Mz . The signal measured by some system of electronics is proportional6 to (7.13), which can be written out in terms of the individual components as Z h i receive r receive r receive r signal / ; d d3r Bx (~)Mx (~ t) + By (~)My (~ t) + Bz (~)Mz (~ t) (7.14) r r r dt ~ r When the known solutions for M (~ t) are inserted into the integrand, the evaluation of the time-derivative, discussed below, shows that the longitudinal magnetization can be neglected, even when there is a nonzero z-component for the receive-coil eld. The static- eld solutions (4.25)-(4.27) hold for each ~. The longitudinal generalization is r r r Mz (~ t) = e;t=T1 (~) Mz (~ 0) + (1 ; e;t=T1 (~) )M0 r r (7.15) It is especially useful to employ the complex representation for the transverse components, not just in taking the time derivative, but for a variety of phase discussions to follow in subsequent chapters. Noting (4.32) through (4.34), r M+ (~ t) = e;t=T2 (~) e;i!0t M+(~ 0) r r ;t=T2 (~) ;i!0 t+i 0 (~) re r M (~ 0) = e (7.16) ? r The phase 0 and magnitude M?(~ 0) are determined by the initial rf pulse conditions. The r rectangular components are found through the real and imaginary parts, Mx = ReM+ My = ImM+ (7.17) The time derivative in (7.14) can be taken inside the integrand (and inside the Re and Im operations when (7.16) and (7.17) are substituted into (7.14)) to operate directly on (7.15) and (7.16). For static elds at the Tesla level and protons, the Larmor frequency !0 is at least four orders-of-magnitude larger than typical values of 1=T1 and 1=T2. Hence the derivative of the e;t=T1 and e;t=T2 factors most certainly can be neglected,7 compared with the derivative of the e;i!0 t factor. With that approximation understood,
r receive r receive r signal / !0 d3re;t=T2 (~) Bx (~) Re iM+ (~ 0)e;i!0t + By (~) Im iM+(~ 0)e;i!0t r r r receive r / !0 d3re;t=T2 (~) M?(~ 0) Bx (~) sin (!0t ; 0(~)) r r receive r +By (~) cos (!0t ; 0(~))] r Z Z h i (7.18) The leading term in the time-derivative calculation indicates that the rapid Larmor oscillations of the transverse magnetization induce the dominant signal in the receive coil. It q 2 also explains why the transverse magnetization M+, or its magnitude M? = Mx + My2 , is referenced, heuristically, as the `signal.8'
The proportionality factor depends on ampli er gain and other factors, as determined by the detection scheme. 7 There are certain cases, such as in solids, where T is on the order of a microsecond or less, and such 2 approximations may no longer be valid. 8 The complete Larmor-frequency dependence of the signal, including implicit ! dependence in M , is 0 + discussed in the next section.
6 7.3. Signal from Precessing Magnetization 101 The expression for the signal can be further simpli ed. The receive eld laboratory components may be written quite generally in terms of the magnitude B? and angle B in the parametrization receive B cos receive B sin Bx By (7.19) ? B ? B With the possible position dependence made explicit and the trigonometric identity sin(a + b) = sin a cos b + cos a sin b,
r signal / !0 d3re;t=T2 (~)M?(~ 0)B?(~) sin (!0t + B (~) ; 0(~)) r r r r Z (7.20) The expression (7.20) for the signal is easily modi ed for the more general situation. The replacement T2 ! T2 must be made in the presence of external eld inhomogeneities, although this distinction is ignored in the current chapter. A time-independent (or timeaveraged) variation in the z-component of the local magnetic eld, other than those already taken into account through T20 , may arise, for example, from the gradient elds used in imaging as detailed in later chapters. These eld variations change the precession frequency according to !(~) = !0 + !(~) r r (7.21) The correction !(~) has been ignored in the outside factor !0 in (7.18), but it cannot r be omitted in the phase of the sinusoid.9 Finally, in the case where additional elds have time dependence (e.g., when they are turned on and o ), the phase ;i!(~)t is replaced by r R t dt0!(~ t0), see (3.37). ;i 0 r 7.3.2 Spatial Independence
The limit where all spatial dependence can be neglected, applies, for example, to studies of small homogeneous samples. Consider all quantities inside the integral (7.20) to be independent of ~. If the sample volume is Vs, then r signal / !0Vse;t=T2 M?B? sin (!0t +
B ; 0) (space-independent limit) (7.22) For the precession frequency to be constant everywhere, the static eld B0 must also be uniform throughout space. The constant phases in (7.22) are especially important in the comparison of signals from di erent sources where, for example, it is necessary to see whether cancellations occur. Practice in their computation is the subject of the next problem.
A simple, but relevant example is cos(106 + ). Variations of in the interval (0 2 ), though minuscule with respect to one million, can change the cosine value over the full range (;1 1).
9 102 Chapter 7. Signal Detection Concepts Problem 7.4
transmit eld lying along x0 in Consider a =2-pulse produced by a left-circular B1 ^ the rotating frame. It results in the rotation of the equilibrium magnetization of a sample into the +^0 direction at t = 0 (at which time y = y0). (The magnetization y ^ ^ would subsequently precess such that it remains pointed along the +^0 direction y in the rotating frame.) a) Find the phase angle 0 in (7.22) corresponding to this initial condition. transmit b) Find 0 for the imaging scenario where the =2-pulse is produced by B1 along ;y0 (so that the magnetization is rotated into +^ at t = 0 and remains ^ x 0 pointed along the +^ direction in the rotating frame thereafter). x transmit along +^0. c) Repeat (b) for B1 y transmit along ;x0 . d) Repeat (b) for B1 ^ The rapid oscillations at the frequency !0 in the above signal expressions are removed, in practice, by an electronic step of `demodulation,' which is tantamount to viewing the signal from a rotating reference frame. Demodulation corresponds to the multiplication of the signal by a sinusoid or cosinusoid with a frequency at or near !0 . Strictly speaking, the transmit or `irradiation' rf frequency !rf is this frequency, but in most experiments !rf is chosen equal to !0. The space-independent formula (7.22) is used below to illustrate a demodulated signal. We take the reference signal, which is separately generated for the demodulation, to have frequency = !0 + ! where ! is referred to as the `o set' frequency from the Larmor frequency !0. Both sine and cosine multiplication are considered, corresponding, respectively, to data storage in two `channels,' a `real' array of numbers and an `imaginary' array of numbers. The origin of the terminology is made evident in the following. Consider rst the multiplication of (7.22) by sin (!0 + !)t.10 With the suppression of the 1 remaining factors, another trigonometric identity sin a sin b = 2 (cos (a ; b) ; cos (a + b)) leads to demodulated signal / reference signal induced emf / sin (!0 + !)t sin (!0t + ) 1 / 2 (cos ( ! t ; ) ; cos ((2!0 + !)t + )) (7.23)
10 7.3.3 Signal Demodulation Real Channel An analog-multiplier electronic circuit is used to combine the signal with the reference sinusoid. 7.3. Signal from Precessing Magnetization 103 with B ; 0 . A split into two frequencies is exhibited. Low pass ltering11 applied to the demodulated signal (Fig. 7.3) eliminates the high frequency component in (7.23), 1 demodulated and ltered signal / 2 cos ( ! t ; ) = 1 Re(ei ! t;i ) 2 (real channel) (7.24) Together with the T2 envelope factor, this yields the time dependence of the so-called `real' channel of the signal. Imaginary Channel
The imaginary channel may be de ned by multiplying the signal by the demodulation factor ; cos (!0 + !)t (the negative rather than positive cosinusoid has been chosen as a convention) and carrying out steps analogous to the above analysis. The result is demodulated and ltered signal / 1 sin ( ! t ; ) = 1 Im(ei ! t;i ) 2 2 (imaginary channel) (7.25) Problem 7.5
Derive the time-dependence (7.25) for the imaginary channel of the demodulated and ltered signal. Laboratory Versus Demodulated Signals
The frequencies of the original and demodulated signals are compared in Fig. 7.3. The gure is applicable for either the real or imaginary channel, showing how the original signal at the Larmor frequency is demodulated into two components, one close to zero frequency and the other nearly twice the Larmor frequency, for small o set !. See (7.23), for example. The ltered signal, (7.24) or (7.25), is left with only the small frequency component. In summary, the signal, as measured in the laboratory, oscillates rapidly near the Larmor frequency. The demodulated12 signal, essentially the signal measured in the rotating frame de ned by the reference frequency = !0 + !, is free of the rapid Larmor oscillation. It oscillates at the o set frequency !.
In the past, this has been done with analog lters such as the Butterworth lter which have slow fallo away from zero frequency. Today, however, digital lters which have much improved step-like edges are used to perform low pass ltering. 12 From now on, demodulation will usually refer to the combined demodulation and ltering operation.
11 104 Chapter 7. Signal Detection Concepts Fig. 7.3: The shifts in frequency for signals from a uniform sample due to demodulation and ltering, with small o set !. The thick bars represent the signal at the indicated frequencies. Problem 7.6
Consider the signal resulting from two spin isochromats with identical spin densities but di erent frequencies of precession !a = !0 + ! and !b = !0 ; !. The total signal for this experiment is just the linear addition of the signal from each isochromat. Find the demodulated signal (with zero o set, ! = 0) from the twospin system and compare it to the demodulated signal (with o set) represented by (7.24) or (7.25). Note: It will be evident in the solution that the signal from two spin isochromats with slightly di erent frequencies (a di erence represented by a small !) exhibits beats. See the discussion on beating in Ch. 8. Complex Signal
The real and imaginary forms of the two channel signals, (7.24) or (7.25), suggest the de nition of a complex quantity that is quite useful. Consider the complex demodulated signal, s(t), de ned as s(t) sre(t) + isim (t) (7.26) where sre and sim refer to the real and imaginary channel demodulated signals. In place of (7.22), we have s(t) / !0Vse;t=T2 M?B?ei ( ;!0 )t+ 0 ; ] (space-independent limit) (7.27)
B 7.4. Dependence on System Parameters 105 in terms of the reference signal frequency . The real (imaginary) part of this expression yields the real (imaginary) channels. More generally, (7.20) is replaced by
r s(t) / !0 d3re;t=T2 (~) M?(~ 0)B?(~)ei(( r r Z ;!0 )t+ 0 (~); r B (~)) r (7.28) Alternate forms with a change in the overall sign on the phase are possible. The integrand can also be written in complex notation. From (7.16) and the related de nition based upon (7.19) receive receive B+ Bx + iBy = B?ei (7.29) we obtain
B Z s(t) / !0 d3rM+(~ t)B+(~) r r (7.30) This expression can itself be generalized for the more detailed experiments involving additional rf pulses and gradient elds. 7.3.4 Dependent Channels and Independent Coils
The above derivation of the real and imaginary channel demodulated signals was performed ~ for arbitrary receiver coil eld directions. Whether Breceive pointed in the laboratory xdirection ( B = 0) or along the y-direction ( B = =2), or any other (spatially independent) direction in the x-y plane, the complex signal remained proportional to ei ! t . Only the phase B changes in (7.27). In this sense, one coil is as good as another, and two (or more) coils do not yield any more information. The signal from two uncoupled coils (coils that have small ux `linkage') can be used, however, to obtain an improvement in the ratio of signal to noise. The noise in one of these uncoupled coils is independent of the noise in the other whereas, for a single coil, the noise in one channel is the same as that in the other (i.e., the noise is correlated between the channels). These issues are discussed in more detail in Ch. 15. A single quadrature, or circularly polarized (CP), receive coil can also be used to increase the signal.13 This corresponds to demodulation with a sinusoid in the x-direction, say, and a cosinusoid in the y-direction. The resulting signal is the same for both: (7.24), with B = 0, is the same as (7.25), with B = =2. To the extent that the `cross-channel' noise is uncorrelated, the CP coil will also increase the signal-to-noise ratio. 7.4 Dependence on System Parameters
Let us investigate what (7.22) tells us about the variables on which the signal depends. Despite its lack of spatial information, this simple approximation is quite useful in understanding various imaging issues.
13 Such coils are discussed further in Ch. 27. 106 Chapter 7. Signal Detection Concepts 7.4.1 Homogeneous Limit Suppose that the equilibrium magnetization M0 z is independent of position (i.e., the sample ^ is homogeneous), and it is uniformly rotated (i.e., the rf eld is homogeneous) into the transverse plane with an on-resonance =2-pulse. Further, suppose that relaxation e ects may be neglected, and that the static eld is also perfectly homogeneous. The proportionality (7.27) can be turned into an equation for the signal amplitude, if the electronic ampli cation factors are ignored,14 jsj = !0M0 B?Vs (7.31) Recall that M0 is proportional to B0 from (6.11). Therefore, in the small sample limit, 2 2 (7.31) predicts a growth of B0 (or, equivalently, !0 from !0 = B0) in the signal as a function of eld. A general analysis, including space dependence and di erences in the sampled nuclei, is complicated, and electronic and sample noise must also be considered. For example, at high elds, noise also increases linearly with B0 so that, by the above estimate, the more pertinent quantity, the signal-to-noise ratio, is proportional to B0 .15 It is the increase of signal-to-noise ratio with frequency that accounts for the interest in higher- eld imaging in MRI. Indeed, elds in excess of 10 T are already in use for NMR microscopy experiments. Higher elds have additional attraction for spectroscopists since the chemical shift dispersion increases linearly with eld strength. Problem 7.7
The following problem will be of recurring interest when signal-to-noise issues are discussed. Consider a cubical volume of a uniform water sample, 10 cm on a side, in a constant magnetic eld of 1 T, and at a temperature of 300 K. Assume the approximation given by (7.31). a) Estimate the emf induced in a coil which has a constant magnetic eld per unit current over the sample of 1 Gauss/A. Assume a proton NMR experiment was performed. b) What is the percentage increase or decrease in the signal from this sample if the length of the cubical side is increased to 20 cm? c) What new eld strength is required in order that the original sample gives the same signal strength as that found in part (b)? d) What new temperature is required for the original sample to give the same signal strength as that found in part (b)? There are a number of technical di culties associated with higher eld imaging, however, and whether there is an optimal eld strength for MRI remains to be seen. (See Ch. 15 for
Even in the presence of T2 decay and without demodulation, the expression (7.31) still represents the peak signal obtained at t = 0 for a homogeneous sample. 15 More general expressions for signal-to-noise in MR experiments are discussed in Ch. 15.
14 7.4. Dependence on System Parameters 107 some additional insight into this question.) Optimal eld strength is also likely to depend upon the application. The signal dependence on Vs is crucial to very high resolution imaging, since the signal in each voxel is limited by its volume and the available magnetization for the given eld and temperature conditions. In general, as resolution is increased, the available signal decreases. This is easy to understand by considering the volume integral over the entire sample as a summation over smaller volumes (voxels, say) within the sample. The smaller the individual volume, the less signal it will produce. In order to obtain enough signal from the tiny volumes ( 1000 m3 for m = one micron) sought after by microscopists, elds of approximately 7 T and higher are usually employed. 7.4.2 Relative Signal Strength What we call the relative signal strength R of an MR experiment for a particular nuclear species can also be analyzed through (7.31).16 The rst step is to nd the dependence of the equilibrium-magnetization formula (6.10) on the gyromagnetic ratio i and spin si of a speci c nucleus i: M0i / aisi(si + 1) i2 (7.32) where ai is the natural abundance (the fractional occurrence of a given stable isotope relative to all stable isotopes). The relative signal strength Ri of a given nuclear species i can be de ned from the species-dependent factors in (7.31) combined with (7.32). Noting !0 / , we nd
(7.33) The weighting ri is the relative abundance in the human body of the given element referenced to some nucleus. In this text, signal strength is quoted relative to 1H in gray matter. Using instead of !0 means that only for the same static eld can a comparison of two di erent elements be made through (7.33). The parameters for several other elements relative to 1H are presented in Table 7.1. Its sizable gyromagnetic ratio and large fractional presence explain why 1 H is the subject of choice for imaging of humans by nuclear magnetic resonance. Ri j ij3riai si(si + 1) Problem 7.8 Assume that 1H has a signal strength normalized to unity for an experiment performed on a human subject. Using (7.33) and the data from Table 7.1, verify the calculated values given in that table for the relative signal strengths Ri=R(1 H) of the following nuclear species: (i) 23Na (ii) 17 O (iii) 31P. The more common de nition for the sensitivity of an MR experiment is based on the signal-to-noise ratio of a given chemical species relative to a xed standard species, usually 1 H, which is normalized to have a sensitivity of unity.
16 108 Nucleus i
1 H, 23 Na, 31 P, Chapter 7. Signal Detection Concepts {i (MHz/T) 42.5764 11.2686 17.2510 -5.7741 40.0765 ri
1.0 9.1 10;4 8.5 10;4 0.5 4:5 10;8 ai
1.0 1.0 1.0 3.7 10;4 1.0 si Ri =R(1H) at 1 T
1 2 3 2 1 2 5 2 1 2 gray matter average tissue average tissue gray matter average tissue 1.0 8.5 10;5 5.7 10;5 5:4 10;6 3.8 10;8 17 O, 19 F, Table 7.1: Table of {, natural abundance a, fraction of all isotopes r relative to 1 H in gray matter, spin s in units of h (see Ch. 5), and calculated relative sensitivity of other elements of interest. The values presented refer to a xed eld strength. Note that a negative value of { means that precession occurs in the counterclockwise direction. The product of ai ri and the molarity 88 M for 1 H in gray matter yields the relative body abundances of Table 2.1 . 7.4.3 Radiofrequency Field E ects
Up to now, we have assumed that the transmit and receive rf coils produce uniform elds over the imaging volume. If this is not the case, the image intensity will vary as a function of position, even for a uniform sample. The image will appear to be brighter or darker in transmit (tips spins) or Breceive regions where either one or both of the rf elds, referred to by B1 (measures signal), vary. The changes in image intensity as a function of eld depend on ip angle and which eld is being discussed. Assuming a uniform rotation by the transmit coil, the image will be darker in regions where Breceive is smaller, and brighter where Breceive is larger. The e ect of the transmit eld is more complex. Consider the case of a =2 ip angle. Any eld which is greater or less than ideal (i.e., than that strength resulting in exactly a 90o rotation), results in reduced transverse magnetization and, therefore, a decrease in available signal. This statement is valid for a single pulse experiment only. However, if a ip angle of less than =2 is desired, the picture is more complicated, and will be discussed in Ch. 8. These e ects can lead to quadratic inhomogeneity e ects if the same coil is used for transmit and tipping the spins and detecting the signal, or if similar eld pro les exist in B1 Breceive. Transmit and receive coils are carefully chosen in each imaging situation to maximize image quality, but it is still necessary to be aware of coil e ects when viewing an image. More of these considerations will be discussed in later chapters where speci c applications and design issues are addressed. 7.4. Dependence on System Parameters 109 Problem 7.9
Investigate the e ect of using di erent combinations of rf systems to tip the spins and measure the resulting signal. a) Assume two identical spins are excited by a uniform eld, such that they both receive a perfect =2-rotation into the transverse plane. What will be the relative di erence in the signal between the two spins if Breceive(spin 1) = C , a constant, and Breceive(spin 2) = 1=2 C ? b) Assume now that the two spins from (a) are excited by di erent transtransmit (spin 1) = C 0 , another conmit elds. The rst spin experiences B1 stant, and receives a =2 ip. The second spin only experiences a eld of transmit (spin 2) = 1=2 C 0 , and therefore is not rotated by the same angle. B1 If Breceive is identical for the two spins, what is the relative di erence in their measured signal? c) It appears from the preceding problems that if only one of the transmit and receive elds is inhomogeneous, but not both, then it might be best to transmit with an inhomogeneous rf eld, and receive with a homogeneous eld. However, this is not how experiments are normally performed. Explain why this might be the case. Hint: Recall that the received signal depends upon B = B=I and preview Ch. 15. 110 Chapter 7. Signal Detection Concepts Suggested Reading
A good introduction to basic NMR signal detection is given in: W. S. Hinshaw and A. H. Lent. An introduction to NMR imaging: From the Bloch equation to the imaging equation. Proc. IEEE, 71: 338, 1983. The basic interaction between the nuclear magnetic moment and the rf coil as well as the reciprocity concept are introduced in this widely quoted paper: D. I. Hoult and R. E. Richards. The signal-to-noise ratio of the nuclear magnetic resonance experiment. J. Magn. Reson., 24: 71, 1976. A complete description of magnet moments appears in the following text: J.D. Jackson. Classical Electrodynamics. John Wiley and Sons, New York, 1975. A list of other NMR active elements (i.e., those with nonzero spin) can be found in: P. T. Beall, S. R. Amtey and S. R. Kasturi. NMR Data Handbook for Biomedical Applications. Pergamon Press, New York, 1984. Chapter 8 Introductory Signal Acquisition Methods: Free Induction Decay, Spin Echoes, Inversion Recovery and Spectroscopy
Chapter Contents
8.1 8.2 8.3 8.4 8.5
Free Induction Decay and T2 The Spin Echo and T2 Measurements Repeated RF Pulse Structures Inversion Recovery and T1 Measurements Spectroscopy and Chemical Shift Summary: Basic MR experiments are described, including the detection of a global signal
from all the excited spins in a sample, the measurement of eld inhomogeneities through T2 decay, the reduction of such eld e ects through the spin echo method, and variations on that method. The signals for commonly repeated sequences, such as the free induction decay and spin echo, are analyzed. A discussion of chemical shifts is based on di erences in the Larmor frequency for the same nucleus in di erent chemical structures. Introduction
The signal received by the rf detector coil is determined not just by the properties of the body but also by a rich set of magnetic eld possibilities. In this chapter, we begin the study of the signal by considering simple, though important, rf eld choices. The rst is a single rf excitation pulse applied uniformly to the sample, producing a signal associated with the collective Larmor precession of all excited spins. The second is a pair of rf pulses 111 112 Chapter 8. Introductory Signal Acquisition Methods the excitation pulse followed by another pulse whose purpose is to help recover, via an `echo,' some of the signal lost to T20 relaxation. The third is one of repeating, in various combinations, a given set of rf pulses. The fourth corresponds to an important method called `inversion recovery,' where the spins are inverted prior to creating the transverse magnetization. Inversion recovery is useful in highlighting di erences in T1 behavior, and in `nulling' the signal from tissues. Four sections are devoted to these choices and a rstpass approach for measuring T2 and T1 is also presented. A more in-depth study of these parameters using modern imaging sequences appears in Ch. 22. The `sequence diagram' is also introduced, which represents all rf pulses and other relevant quantities ( eld gradients will be included in Ch. 9) that are applied during the entire experiment in an easy-to-understand format. As more complex sequence diagrams are introduced in future chapters, it will be seen that a single experiment may involve changing certain parameters and repeating the modi ed rf structure or, in the case of `multiple data acquisitions,' simply repeating a xed rf structure. Spectroscopy is brie y introduced in the last section. The signal from components with di erent Larmor frequencies is collected to obtain information about the constituents of the sample. 8.1 Free Induction Decay and T2
The simplest MRI experiment involves detecting a global signal from a sample. Consider a =2-pulse applied uniformly to proton spins in a static magnetic eld associated, say, with any hydrogen atoms present (a macroscopic set). The pulse rotates the longitudinal magnetization (the excess spins) into the transverse plane after which the tipped spins freely and collectively precess. As discussed in Ch. 7, the total time-varying coherent magnetic eld derived from the sum over all precessing proton spin elds would induce a small emf in any rf coil properly oriented to detect the corresponding ux changes. This experiment is called a free induction decay (FID). The signal expressions from the previous chapter can be applied directly to its analysis. An FID is performed routinely on MRI machines to tune rf coils and optimize system response. It can be used to locate the resonance peak for water and determine the rf amplitude and duration necessary to produce a maximum signal. 8.1.1 FID Signal The groundwork for the theoretical expression for the FID signal has been laid in Ch. 7. From (7.28), the complex form for a demodulated signal due to an rf spin ip at t = 0 is
r s(t) / !0 d3 re;t=T2 (~) B?(~)M?(~ 0)ei(( r r Z ;!(~))t+ 0 (~); r r B (~)) r (8.1) with demodulation reference frequency and eld angle B . Over the whole sample, there is, in general, spatial dependence in T2 and the initial magnetization (even with a perfect =2-pulse, the equilibrium value M0 depends on the local spin density). The possibility that the precession frequency can change with position has also been included, ! = !(~). r 8.1. Free Induction Decay and T2 113 Consider the following simple FID experiment. Assume that, in fact, the precession frequency is constant (! = !0) over all space, and that the initial magnetization phase, the direction of the receive eld, and the decay constant are also space-independent. The associated factors can be taken outside of the integral in (8.1) s(t) / !0e;t=T2 ei(( ;!0 )t+ 0 ; B ) Z d3rB?(~)M?(~ 0) r r (8.2) Nonuniformities in the receive coil, and especially in the sample, are still taken into account. The complex exponential gives way to a cosine or a sine depending on whether the real or the imaginary channel signal is detected. The expression (8.2) can be used to understand several examples of signal time dependence. In the laboratory frame, where no demodulation is applied ( = 0), rapid oscillations at frequency ! are damped by the relaxation factor, as shown in Fig. 8.1a. Demodulation onresonance ( = !0 ) eliminates the sinusoid, with only the T2 envelope remaining (Fig. 8.1b). A demodulation slightly o -resonance ( ; ! = !) leaves a low frequency component in Fig. 8.1c. Lastly, the superposition of demodulated signals (at some central value for ) from several spin populations, each with a di erent Larmor frequency, leads to a sum of terms (8.2), yielding a signal that decays faster than a single demodulated signal which only has T2 damping (Fig. 8.1d). The dephasing among the populations causes the reduced signal, resembling a T20 e ect. The transmit rf eld pulses shown in Fig. 8.1 show rapid oscillations in the laboratory and none in the rotating frame. Indeed, the left-circular B1 eld, given by (3.24), is sinusoidal in the x y coordinates, and it is constant in the x0 y0 coordinates. The remaining gures of this chapter show the Larmor-frame rf pulses. It is important to note that narrow `boxcar' pulses (or `hard pulses') correspond to larger frequency spreads than what is generally desired. In fact, time pro les such as `sinc' functions (Ch. 9) are required in imaging applications, where rf pulses with su ciently limited frequency spread are utilized.1 The more relevant rf pulse time graphs are discussed and illustrated in Ch. 10. 8.1.2 Phase Behavior and Phase Conventions It is recalled that the contribution to the phase of (8.1) due to the magnetization (or spin isochromat) at ~ is r (~ t) = ;!(~)t + 0(~) = ; Bz (~)t + 0 (~) r r r r r (8.3) Starting at the initial value 0 at t = 0, the phase is seen to accumulate an additional amount, at time t, due to the precession about the z-component of the eld. From the discussion in Sec. 7.3.1, where it was assumed that the transverse magnetization is created by an initial rf pulse, the angle 0, the phase of the initial magnetization M+(~ 0), is determined r by the direction of the rf pulse. See Prob. 7.4. Initial phases, such as those in (8.3), often can (and will) be rede ned or ignored in subsequent equations. It is possible to choose the origin along any axis of convenience. Although polar coordinates in the x-y plane are usually de ned such that = 0 along
1 Nevertheless, narrow boxcar pulses, or hard pulses, will be the staple for the remainder of this and the following chapters. 114 Chapter 8. Introductory Signal Acquisition Methods Fig. 8.1: (a) The FID signal in the laboratory frame for all spins precessing at the same Larmor frequency. The laboratory rf transmit eld oscillates at that frequency. (b) The same experiment but as measured in the Larmor rotating frame (i.e., demodulated). The rotating frame rf eld is at `rest.' (c) The demodulated FID signal when the demodulation is not exactly at the Larmor frequency. (d) The total demodulated FID signal from several isochromats, each with slightly di erent Larmor frequencies, exhibiting a decay, with slow oscillations, that is faster than T2 decay alone, due to dephasing. The demodulation in (d) is determined by the average (fast) frequency. Note that the laboratory signal in (a) is only suggestive since the oscillations seen in practice are too rapid to display. The T20 e ects have not been included in any of the curves note that the di erences in case (d) could be considered, alternatively, to be due to eld inhomogeneities. 8.1. Free Induction Decay and T2 115 the positive x-axis, it is frequently useful to choose the zero point to be along the initial magnetization. For example, the choice for the zero angle would be along the y axis for a ^ =2-pulse ( eld) direction along x. At t = 0, x and y coincide with their respective primed, ^ ^ ^ rotating counterparts, x0 and y0. However, the counterclockwise rotation for the de nition ^ ^ of increasing will be maintained. 8.1.3 It has been discussed previously that the dephasing of the magnetization caused by eld inhomogeneities produces additional suppression of the signal in (8.1). The damping of the magnitude of the transverse magnetization was described in Ch. 4 by the replacement of T2 by T2 M?(~ t) = M?(~ 0)e;t=T2 r r (8.4) Although qualitatively the same shape as shown in Fig. 8.1b, the new envelope factor typically decays much faster in time. The decay time T2 represents a combination of external eld induced (T20 ) and thermodynamic (T2 ) e ects, 1 = 1 + 1 (8.5) T2 T2 T20 The quantity T20 is both machine and sample dependent.2 The characteristic which most distinguishes it from T2 , besides the fact that it can be much smaller, is that the T20 signal loss is recoverable (reversible). The details behind the reversibility are presented in Sec. 8.2. The T20 dependence is used in FID measurements to determine whether the homogeneity of the main magnet is adequate. A uniform sample is placed in the magnet, and the FID is then observed. If the decay is too rapid (Fig. 8.1d), poor magnet homogeneity is indicated, and corrections to the main eld are made in an attempt to recover the pristine signal. The reader is referred to Ch. 27, where static- eld coil construction issues are discussed. Although T2 can be obtained by tting the data to the exponential form in (8.4), separating T2 and T20 is more di cult. A method for measuring T2 is presented in this chapter after an introduction of the spin echo. T2 Decay T20 Estimates The origin of the T20 part of the decay is the collective e ect of the dephasing of the isochromats of spins. This can be described by a dephasing limit (i.e., t ! 1) X i (~ t) e r dephasing 0 (8.6)
sample
; ! A demonstration of the vanishing of the sum in (8.6) for randomly distributed phases is contained in Sec. 21.1, while a related integral version is found in Sec. 10.2. The sum corresponds to a discrete version of the demodulated signal (8.1) where the remaining eld
In particular, T2 is a local quantity, and images are frequently found to be of poorer quality away from the main-coil center. The static eld variations across a unit voxel are smaller near the center where the eld uniformity is usually best.
2
0 116 Chapter 8. Introductory Signal Acquisition Methods
0 and magnetization factors have been suppressed. It has been noted in Ch. 4, that only under certain circumstances would the sum be proportional to the exponential form e;t=T2 (and consistent with the T20 dependence in (8.4)). A simple phase argument can be applied to estimate the size of the decay time T20 . This argument carries the additional bene t of introducing a way to connect other eld variations, such as external linear elds and internal susceptibility e ects (Ch. 25), to phase changes. Consider the phase (8.3) for a spin isochromat at position ~ where there is a time-independent r spatial variation B (~) in the z-component of the total magnetic eld. With the neglect of r the initial phase, the laboratory accumulated phase is (~ t) = ; (B0 + B (~)) t r r (8.7) The reader is asked to carry out a numerical estimate of T20 , using (8.7), in the next problem. Problem 8.1
It is not necessary to consider how the sum of phases in (8.6) leads to an exponential decay, in order to nd an estimate of the value of T20 and to understand how it is a ected by eld strength and homogeneity. The following modest calculation will su ce. Suppose two proton spins are positioned at ~1 and ~2 , respectively, where B (~1 ) = r r r 6 ) B , and B (~ ) = ; ppm, with respect + ppm (parts per million) ( =10 r2 0 to an arbitrary B0. Find a formula for the time it takes the two spins to become radians out of phase with each other, so that the net magnetization of the pair is zero. Discuss the dependence of this time on and B0 and nd a numerical estimate of T20 , as the time of the rst zero crossing of the signal for = 1:0 and B0 = 1.5 T. This problem is also representative of two spins with symmetrically chemically shifted frequencies relative to some base-line frequency. The concept of chemical shift is introduced later in this chapter. Although simple phase arguments are most useful in understanding various e ects, care must be taken not to double count. For example, external eld inhomogeneities are already accounted for when the exponential factor e;t=T2 is included, as in the formula for the magnitude of the magnetization, (8.4). The phase of the complex magnetization M+ is rede ned, when that factor is present, so as not to include the e ects of those inhomogeneities. A similar situation occurs for the intrinsic T2 damping local eld e ects are already included, at least in part, by the factor e;t=T2 . An experimental method for removing the dephasing represented by the e;t=T2 factor is described in the next section. The following problem is also on phase dispersion, where an approximate value of a eld gradient is to be calculated.
0 0 8.1. Free Induction Decay and T2 117 Problem 8.2
The parameter T20 is associated with the (relatively smooth) variation in the zcomponent of the external eld. An estimate of the average gradient in this component can be found from a given phase variation. If the z-component changes from B0 + B (~1) to B0 + B (~2 ), then the average gradient of that component r r between the two points ~1 and ~2 can be de ned as r r r2 ; r G j B (~~) ; ~ B (~1 )j (8.8) jr2 r1j Suppose two protons are situated at these points. If j~2 ; ~1 j is 2 mm, and if r r there is no initial phase di erence between their spins, nd the value of G leading to a 2 di erence in phase, after a time 5 ms, for the two proton spins. The average gradient found in this problem is not dissimilar to that present in the body, at the interface between tissue and air, as caused by magnetic susceptibility di erences, for static elds at the Tesla level. That is, even for a perfectly homogeneous static eld, there are inhomogeneities due to di erences in local eld shielding (see Ch. 20). Fig. 8.2: Sequence diagram for a repeated FID experiment. Repetition of the rf pulse and sampling, with repeat time TR , is indicated by the dotted line and arrow in this gure, but the fact that the
process, or `cycle,' is usually repeated is left understood in most diagrams. The ADC line represents the activity of the analog-to-digital converter, which is the device used to sample the signal over time Ts . Figure 8.2 is a sequence diagram representing one cycle of a simple FID experiment in which the repetition is indicated explicitly by a dotted line. This chapter notwithstanding, the dotted line and the repeat time TR are customarily left out of the sequence diagrams, and the repetition is implied by other features, such as broken lines or `gradient tables' (see Ch. 10). 8.1.4 The FID Sequence Diagram and Sampling 118 Chapter 8. Introductory Signal Acquisition Methods The continuous emf signal is measured by the analog-to-digital converter (ADC) as shown in Fig. 8.2. The measurement is carried out at a nite set of discrete steps in time. If N points are collected at uniform t intervals, the total sampling time is Ts = (N ; 1) t (8.9) Such sampling represents an approximation of the signal. The number and frequency of such steps needed for an accurate representation is discussed in Ch 12. 8.2 The Spin Echo and T2 Measurements
In the circumstance where the external eld is not particularly uniform, the time constant T20 can often be su ciently small that 1=T20 dominates 1=T2, and a severe extrinsic signal loss may result. Fortunately, this e ect can be reversed by a well-known rf pulse sequence called the `spin echo method.' 8.2.1 The Spin Echo Method The spin echo sequence is based on the application of two rf pulses: a =2-pulse followed by a -pulse (or `refocusing' pulse). The plot showing the recovery of T20 signal loss resulting from this sequence is shown in Fig. 8.3. To understand the signal plot, we shall analyze three steps. In the rst step, it is su ciently general to assume that the magnetization of a sample is tipped by the rst pulse immediately into the transverse plane. Suppose this happens instantaneously at t = 0 such that, initially, the (excess) spins point along the y0 axis. (Recall ^ that (the eld of) such a pulse lies along x0 , where the pulse is assumed to be applied in the ^ rotating frame.) The spins at di erent positions ~ begin to dephase, relative to each other, r as they experience di erent eld strengths, each of which, in general, is not exactly equal to B0 . The accumulated phase of a spin at ~ in the rotating frame is found by subtracting the r Larmor term in (8.7). The phase in this frame relative to the y0-axis is (~ t) = ; r B (~) t r for 0 < t < (8.10) where 0 = 0 in (8.3). The entire second step, for present purposes, is considered to be instantaneous. Another rf pulse with twice the amplitude, but otherwise identical to the rst, is then applied. The second pulse is along y0, rotating the spins about the y0-axis through the angle , at time . ^ The spins which had previously accumulated extra positive phase now have, at the instant after the -pulse,3 the negative of that phase, and vice versa,4 = ; (~ ;) r = B (~) r (8.11) 3 We shall use the notation f (t ) to refer to the limits lim f (t j j). 0 4 The action of the -pulse, ! ; , would be changed to ! ; , for example, if the = 0 line were (~ r
+)
! 0 rede ned to lie along the x -axis. 8.2. The Spin Echo and T2 Measurements 119 Fig. 8.3: A =2-pulse is applied along the positive x0-axis at t = 0, and a -pulse is applied along the positive y0 -axis at t = to invert the phase that the spins have accumulated. The spins then `rephase,' producing what is called an `echo.' Notice that the signal strength is still limited by the T2 envelope at the echo time TE = 2 , as measured from the center of the rf pulse. The corresponding exponentials for the decay envelopes are found in (8.18). Here, and in later gures, the -pulse is drawn at twice the height, for the same width, as the =2-pulse, corresponding to the need for twice the B1 amplitude to get twice the angle of spin rotation. The echo shows a positive local maximum, instead of a negative local minimum, since the magnetization is rephased along +^0 . The subscripts on the rf pulse brackets denote the axis along which the rf eld is applied. y In the third step, the spins continue, after time , to accumulate phase according to (8.10) (~ t) = (~ + ) ; B (~)(t ; ) r r r = ; B (~)(t ; 2 ) r = ; B (~)(t ; TE ) r t> with the echo time de ned by (8.12) TE 2 (8.13) Since the rate at which phase is accumulated by each spin is unchanged, all of the spins will return to = 0 at the same time, the echo time. Equation (8.12) shows that the
accumulated phase of all spins experiencing a time-independent eld variation will return to zero at t = TE , regardless of the value of B and the position ~. Therefore, it is obvious that r 120 Chapter 8. Introductory Signal Acquisition Methods all of the spins will be realigned at the same time, and the realignment of the spins is called a spin echo.5 This particular rf pulse combination produces a positive echo along y 0 (i.e., ^ the echo refocuses all spins along the positive y0 direction). Figure 8.4 presents a simpli ed picture of the behavior of the individual spins during the experiment. Fig. 8.4: A simulation of an ensemble of spins in the rotating reference frame during a spin echo experiment. A =2-pulse rotates the spins, around the x0 -axis, into the transverse plane where they begin to precess. The spins accumulate extra phase, until this accumulation is inverted by the -pulse. The spins continue to collect extra phase at the same rate and, at a later time, all spins return to the positive y0 -axis together, forming an echo. The echo amplitude is still reduced, however, by the intrinsic T2 decay. 5 This type of echo is also called a Hahn or rf echo. 8.2. The Spin Echo and T2 Measurements 121 Problem 8.3
It is also possible to generate a spin echo with the same rotational axis for the =2 and -pulses. Consider, for example, the case where the =2 and -pulses are both applied along the x0 -axis (so the spins are always rotated clockwise about the x0 -axis). Write the equations for the phase during the experiment in analogy ~ with (8.11)-(8.12), using the convention that the phase is zero when M is parallel 0 to y . Show that an echo is produced along the direction opposite to that which ^ the magnetization had immediately after the =2 excitation (i.e., a negative-echo along y0). Signal plots would therefore display a `negative' peak at the echo. ^ 8.2.2 Spin Echo Envelopes The next question is about the explicit form of the decay-and-growth echo exponentials. The envelope of M?(~ t) for the spin echo experiment can be found by addressing the time r SE dependence in the relaxation rate R2 for the spin echo experiment. This is determined by the time rate of change in the rotating reference frame (see Ch. 4) dM? dt !0 relaxation SE = ;R2 M? (8.14) where the primed derivative is de ned in (3.10). All other terms in the Bloch equation for the transverse magnetization are neglected. It has been noted in an earlier discussion that the microscopic justi cation for the rate equation (8.14), which is based on an analysis of the dephasing cancellations, is found in Ch. 20. SE Before the -pulse and after the echo, R2 is the usual sum of extrinsic and intrinsic terms. Between the -pulse and the echo, the extrinsic rate is negative, because the phase di erences due to eld inhomogeneities are being refocused. This corresponds to the change in sign implied by (8.12) for d =dt.6 We have 8 0 > R2 + R2 < SE (t) = ;R0 + R R2 > R20 + R2 : 2 2 0<t< < t < 2 = TE t > 2 = TE (8.15) to be compared with the simple FID rate
0 R2 = R2 + R2 (8.16) SE Within each interval, R2 is constant, and the solution to (8.14) has the usual exponential form 2 (each interval) (8.17) M?(t) = M?(t0)e;(t;t0 )RSE
6 If there is exponential decay during dephasing then there is exponential growth during rephasing. 122 Chapter 8. Introductory Signal Acquisition Methods Although any t0 within the given interval can be taken as an initial time, the intention is to use (8.17) to evolve the magnetization from the beginning of each interval (t0 = 0 TE respectively). Matching the exponential solutions of the separate di erential equations at the two boundaries of the three intervals leads to the spin echo envelope (magnitude) for the whole experiment. Leaving the details to a problem, we nd 8 ;t=T2 >e 0<t< < ;t=T2 ;(T ;t)=T E 2 M?(t) = M?(0) > e e <t<2 =T : e;t=T2 e;(t;TE )=T2 = e;t=T2 eTE =T2 t > 2 = TE E
0 0 0 (8.18) in terms of the decay times7 and the initial magnetization M?(0) at t = 0. The spatial dependence has been suppressed everywhere. The envelope shown in Fig. 8.3 corresponds to (8.18). Problem 8.4
Derive (8.18) from the solutions to the di erential equations in each of the three intervals. The corresponding changes in the relaxation rate are given by (8.15). The initial value is M?(0) at t = 0, and M?(t) is continuous over time. 8.2.3 Limitations of the Spin Echo The signal peaks at the echo, as seen in Fig. 8.3, but it recovers only to the level dictated by the T2 envelope. The spin echo does not reduce the e ect of T2 decay on the signal. The reason lies in the rapid time uctuations in the intrinsic local elds. The inhomogeneities in these internal elds do not stay xed in time after the -pulse, the rates at which phase is accumulated change with time, and, in general, no refocusing is possible. Fortunately, the irreversibility of the intrinsic T2 loss is not a severe limitation for liquids (T2 is often three orders of magnitude shorter for solids). The time interval over which data are collected often can be made short compared with T2 . Examples of other time-dependent spin e ects include motion (macroscopic or microscopic) where the spins move from place to place during the experiment, such that they experience di erent elds at di erent locations. The spin echo will not refocus these types of inhomogeneities.8 Although the spin echo presents a way of reversing the e ect of T20 , the recovery is complete only when (t) backtracks to zero for all of the spins. If the spins are not at this point, then the data will be tainted by T20 decay, despite the e ort to refocus the spins. For instance, in Fig. 8.3, only the data point collected at t = TE is completely refocused. The
Recall from Ch. 4 that R2 1=T2, R2 1=T2 , and R2 1=T2. The e ects that coherent motion can have on MRI will be discussed in the chapter on ow (Ch. 24). Inferior rf pulses and coils can also a ect the phase of the signal.
7
0 0 8 8.2. The Spin Echo and T2 Measurements 123 problem is acute when B0 is rather inhomogeneous, since a very short T20 implies a rapid drop-o in either direction away from the echo. 8.2.4 Spin Echo Sampling The measurement of the signal raises the question of how the data are to be collected. We have noted that data will be sampled at multiple time points (i.e., a time series). The acquisition of single data points9 at two di erent echo times can be used to determine the spin-spin decay constant, if its spatial dependence is ignored. From the combination of (8.18) and (8.2), the demodulated signal at the spin echo, t = TE , is s(TE ) / !0 e ;TE =T2 Z d3rB?(~)M?(~ 0) r r (8.19) where the initial phases have been ignored. Varying the time of the -pulse in a second 0 experiment leads to a di erent echo time (TE > TE , say) where Z 0 s(TE ) / !0 e;TE =T2 d3rB?(~)M?(~ 0) r r
0 (8.20) From these two data points, an estimate for T2 can be obtained from the ratio of the two signals, (8.19) and (8.20). The explicit and implicit proportionality constants can be taken to be the same, and the ratio can be solved to nd 0 E T T2 = ln (sTT ;=sET 0 )) (8.21) ( E) ( E Multiple points can be sampled near a single echo for more information. A su ciently large time series may be used to spectrally resolve the signal, a subject of importance in `chemical shift' imaging. See Sec. 8.5 and Ch. 10. Rather than repeating an experiment with a di erent echo time to measure T2 , it is common to collect data for more than one echo of the original rf excitation. This is accomplished by applying multiple -pulses after a single =2-pulse (see Fig. 8.5). For de niteness, it may be assumed that these are all along the y0 axis, so that the repeated refocusing is always along ^ that same axis this is not a general rule. For uniform spacing, the nth -pulse is at the time (2n ; 1) . The signal at each echo can be measured and plotted, as a function of time or n, on a semi-log scale where the echoes occur at tn = 2n = nTE . A t to a straight line will reveal the relaxation rate. The spin echo experiments to measure T2 are conceptually simple but turn out to be fraught with di culties. In an introductory discussion such as given here, the -pulses are assumed to be perfect: The ip angle rotations are taken to be instantaneous and exactly =2 or over the whole sample. Also, the spatial di usion of the spins is ignored. Di usion is a
The need for a small frequency spread or `bandwidth,' provided by a low pass lter with a narrow window to suppress high frequency noise, is investigated in Ch. 12.
9 8.2.5 Multiple Spin Echo Experiments 124 Chapter 8. Introductory Signal Acquisition Methods Fig. 8.5: A signal arising from multiple spin echoes generated by regularly repeated pulses. problem because eld inhomogeneities lead to changes in the Larmor frequency when spins meander into di erent spatial regions. These and other factors lead to accumulated errors in the determination of relaxation parameters. We shall return to such issues in Chs. 16, 21 and 22, where realistic rf pulses and methods to overcome a variety of di culties are discussed. 8.3 Repeated RF Pulse Structures
A major issue in imaging, and indeed in NMR experiments in general, is the requirement of an adequate signal, relative to the noise present. It may be necessary to repeat the experiment a number of times, averaging the measurements for a nal result. This yields a general improvement in the ratio of signal to noise, in terms of the number of repetitions, as will be discussed in Ch. 15. For an experiment with repeated sequences, however, the available transverse magnetization (and hence the signal) will depend upon the repetition time TR , which determines the amount of regrowth of longitudinal magnetization Mz (t).10 Recall that the transverse magnetization is established initially by the tipping of the longitudinal magnetization into the transverse plane. Fig. 8.6 illustrates the magnetization history of a repeated FID example. An important issue, consequently, is the choice of TR . Repeating the rf pulse structure (as it is de ned in the sequence diagram) after too short an interval will not leave time for Mz (t) to relax to the maximum longitudinal equilibrium magnetization, M0 . (For very short TR , the signal becomes so small, it is referred to as being `saturated.' See Ch. 6.) Waiting a longer time to repeat the data acquisition step is ine cient. A compromise must be reached.
10 See Ch. 18 for a more detailed discussion of signal behavior for arbitrary TR and ip angle. 8.3. Repeated RF Pulse Structures 125 Fig. 8.6: The behavior of the transverse and longitudinal magnetization for a regularly repeated FID experiment, with xed repeat time TR . In this section, formulas for the longitudinal and transverse magnetization will be derived for experiments in which FID and spin echo sequences are repeated. It is kept in mind that the signal is directly related to an integration over the transverse magnetization in expressions such as (8.2). The integrals and spatial dependence will be left understood in what follows, phases are neglected, and the ~ dependence will be suppressed in the equations. Since the r signal is directly proportional to the transverse magnetization, the magnitude M? will be studied as a function of time. The important variables are the repeat time TR and, where applicable, the -pulse time . We also introduce the nomenclature of repeated rf pulse structures and cycles within a loop structure of sequence diagrams. The detailed question of nding the optimal value of TR is considered in Ch. 15. There it will be seen that the choice depends upon the desired `contrast' in the image. 8.3.1 The FID Signal from Repeated RF Pulse Structures
In the canonical FID experiment, a single =2-pulse is applied and the signal can be measured immediately. In this section, an expression for the FID signal as a function of time for regularly repeated =2-pulses (i.e., constant TR ) will be found. The simplifying assumption, 126 Chapter 8. Introductory Signal Acquisition Methods TR T2 , is used, implying that the transverse magnetization has decayed completely by the end of any given repetition (prior to the next =2-pulse). Assume that the longitudinal magnetization available for the rst sequence is M0 . Since the interest here is to perform an e cient experiment, the repeat time will not be too large: TR < T1 . In this case, only the rst rf pulse may see the maximum longitudinal magnetization M0 . From the second sequence onwards, the longitudinal magnetization will recover only as far as TR allows. As we will see, the signal behavior after the rst cycle will be similar from one repetition of the rf pulse structure to the next, so a straightforward averaging will improve the signal-to-noise ratio.11 The signal from the rst cycle is usually not used, since the initial magnetization may be di erent from the equilibrium value. De ne t = 0 at the center of the rst =2-pulse. The pulse is assumed to be su ciently narrow that its action is instantaneously to tip the magnetization from the z-axis into the transverse plane Mz (0+) = 0 M?(0+) = Mz (0;) = M0
(8.22) (8.23) Following the pulse, the time-evolution of the transverse and longitudinal magnetization are described by the Bloch solutions (Ch. 4) in terms of the initial values (8.22) and (8.23) Mz (t) = Mz (0+)e;t=T1 + M0 (1 ; e;t=T1 ) = M0 (1 ; e;t=T1 ) M?(t) = M?(0+)e;t=T2 = M0e;t=T2 (8.24) (8.25) for 0 < t < TR . After the rst cycle, (8.22) and (8.23) are replaced by the action of the second pulse at t = TR
+ Mz (TR ) = 0 + ; M?(TR ) = Mz (TR ) = M0(1 ; e;TR=T1 ) (8.26) (8.27) ; in which (8.24) was used to evaluate Mz (TR ). (Remember it is assumed that there is no ; transverse magnetization left at t = TR to get tipped into the z-direction.) The Bloch solutions for the second interval are combined with (8.26) and (8.27) to give + Mz (t) = Mz (TR )e;(t;TR )=T1 + M0 1 ; e;(t;TR )=T1 = M0 1 ; e;(t;TR )=T1 (8.28) + M?(t) = M?(TR )e;(t;TR )=T2 = M0 1 ; e;TR =T1 e;(t;TR )=T2 (8.29) for TR < t < 2TR. The pattern is now clear. For the nth cycle, the longitudinal magnetization is zero at the beginning and evolves to a common value at the end
; Mz (t = nTR ) = M0 (1 ; e;TR=T1 )
11 (8.30) This is only true when =2-pulses are employed. For the more general case of variable ip angles, see Ch. 18. 8.3. Repeated RF Pulse Structures 127 independent of n. Therefore, the same longitudinal value, for n 2, is tipped into the transverse plane at the start of each cycle, to initialize the evolution of the transverse magnetization. The `signal' as a function of time in the nth cycle is universally12 M?(tn) = M0 (1 ; e;TR =T1 )e;tn =T2 (n 2) (8.31) for 0 < tn < TR where the time during the cycle tn is de ned in terms of the total experimental time t by t tn + (n ; 1)TR (8.32) The signal for the repeated FID experiment13 is thus reduced by the factor 1 ; e;TR =T1 at each instant in time, relative to that for a single FID run. It is essentially this factor that is plotted in Fig. 8.7, which shows the maximum value of the signal, in each interval, for the repeated FID measurement. This corresponds to (8.31) at tn = 0 as a function of TR . Fig. 8.7: The demodulated signal, for a repeated FID experiment, as a function of TR . The signal is assumed to be measured immediately following the =2-pulse, in a given cycle. Problem 8.5
Experiments with =2-pulses and short TR can be expected to have reduced signal. That is, in the limit that TR becomes much less than T1 (but still much ; larger than T2 ), show that Mz (nTR ) is proportional to TR =T1 . Consider a spin echo experiment with regularly repeated rf pulse structures under the same simplifying assumption that TR T2 . The sequence is diagramed in Fig. 8.8. Again, the
n.
12 13 8.3.2 The Spin Echo Signal from Repeated RF Pulse Structures
0 We might very well use t , in place of tn , to describe the cycle time in view of the lack of dependence on For short TR , this is sometimes called a `partial saturation' experiment. 128 Chapter 8. Introductory Signal Acquisition Methods Fig. 8.8: The sequence diagram for a repeated spin echo experiment. A long sampling time is not
required when only one data point is measured. corresponding FID expression (8.30). e ect of the =2-pulse for the rst cycle is given by Mz (0+) = 0 (8.33) + ) = M (0; ) = M M?(0 (8.34) z 0 The Bloch solution (4.12) for the longitudinal magnetization can be applied to the two intervals before and after the -pulse at t = . Noting (8.34), ( + ;t=T1 ;t=T1 = 0 (1 ;t=T1 Mz (t) = Mz (0+)e ;(t; )+ 1M0 (1 ; e ; e;)(t; )M1 ) ; e ) 0 < t < T (8.35) =T + M0 (1 =T Mz ( )e <t< R From the action of the -pulse and the rst solution in (8.35), Mz changes sign so that Mz ( +) = ;Mz ( ;) = ;M0 (1 ; e; =T1 ) (8.36) The second solution in (8.35) thus becomes Mz (t) = M0(1 ; 2e;(t; )=T1 + e;t=T1 ) < t < TR (8.37) The transverse magnetization is given by (8.18) with M?(0+) = M0. At the echo, with all T20 e ects rephased, it is M?(TE ) = M0 e;TE =T2 (8.38) ; The longitudinal magnetization at the end of the rst cycle, with the value Mz (TR ), gets tipped into the transverse plane by the next =2-pulse. The transverse magnetization evolves with this initial value to its magnitude at the echo in the second cycle ; M?(TR + TE ) = Mz (TR )e;TE =T2 (8.39) where, from (8.37), ; Mz (TR ) = M0 (1 ; 2e;(TR ; )=T1 + e;TR=T1 ) (8.40) In practice, << TR , and independent of the size of T1 , (8.40) is well approximated by14 ; TR (8.41) Mz (TR ) ' M0 (1 ; e;TR =T1 ) 14 This answer could have been written down immediately. In the limit that ! 0, (8.37) reduces to the 8.4. Inversion Recovery and T1 Measurements 129 The longitudinal magnetization at the end of the second cycle has an expression identical to (8.37), because its initial value was again zero. (As in the FID case, the limit TR >> T2 implies that M?(TR ) = 0 so that the excitation pulse will not tip any longitudinal magnetization back along the z-axis, i.e., Mz (0+) = 0.) n For each ensuing cycle, the answers repeat. The longitudinal magnetization evolves to the approximate answer ; Mz (nTR ) ' M0 (1 ; e;TR =T1 ) n 1 (8.42) The transverse magnetization at an arbitrary echo is M?(2 + nTR ) ' M0 (1 ; e;TR =T1 )e;2 =T2 n 1 (8.43) recalling TE = 2 . For other times in each cycle, formulas for the signal can be adapted from (8.18). In conclusion, the measured signal at any echo has the same TR =T1 dependence, for << TR , as the repeated FID experiment, but without the severe T2 suppression. As expected, the -pulse continues to serve to refocus static eld inhomogeneities, so that the magnetization at the center of the echo is only a function of T2 . The variability of the signal as a function of sequence timing (through the parameters TR or ) is quite evident in (8.43). This variability is of great utility to MRI in de ning image contrast as will be seen throughout the remainder of the text. Multiple Spin Echo Sequence Diagram Consider a pulse structure in which a single =2-pulse is followed by multiple -pulses. After each -pulse, a new echo can be measured. The signal and sequence diagram are exempli ed by Figs. 8.5 and 8.9, corresponding to four and three -pulses, respectively. A related problem for two -pulses appears next. Problem 8.6
A repeated `double spin echo' sequence is a pulse structure with a single =2pulse followed by two -pulses. The rst -pulse follows the =2-pulse by a time interval and the two -pulses are separated from each other by 2 . a) Draw the sequence diagram for this experiment. b) Derive the signal expression for the two echoes. 8.4 Inversion Recovery and T1 Measurements
The FID and spin echo experiments are useful for determining the T2 properties of a sample. As single experiments, however, they are not sensitive to T1 . As repeated experiments, on 130 Chapter 8. Introductory Signal Acquisition Methods Fig. 8.9: Sequence diagram for a three-echo example of the multi-echo spin echo experiment. the other hand, their signals do depend on T1 , and the spin-lattice parameters could, in principle, be determined by performing a number of repeated sequences with di erent values of TR . Such experiments, however, take a relatively long time. There is another experiment, called `inversion recovery,' that is sensitive to T1 . Although this method can be employed to get an accurate value of T1, the determination likewise cannot be made from a single experiment. Inversion recovery is similar to an FID experiment, but with an additional -pulse employed to invert the magnetization and de ned to occur at a time interval TI before the =2-pulse. The sequence diagram is found in Fig. 8.10a. The regrowth of the longitudinal component of the magnetization between these pulses leads to a strong signal dependence on TI . 8.4.1 In the measurement of T1 through inversion recovery, it is necessary to keep track of the longitudinal component of the magnetization during the period between the - and =2pulses. The longitudinal magnetization right after the initial -pulse is the negative of the equilibrium value T1 Measurement Mz (0+) = ;M0 Mz (t) = ;M0 e;t=T1 + M0 (1 ; e;t=T1 ) = M0(1 ; 2e;t=T1 ) (8.44) The time t = TI has been de ned as the time of the =2-pulse. The magnetization regrows to its equilibrium value in the interval between the pulses according to (4.12) 0 < t < TI (8.45) After the longitudinal magnetization is tipped into the transverse plane to provide the initial signal, the magnitude of the transverse magnetization evolves as M?(t) = M0 (1 ; 2e;TI =T1 ) e;(t;TI )=T2 t > TI (8.46) As advertised, the magnitude of the measured signal is modulated by the T1 -dependent factor 1 ; 2e;TI =T1 . This can be compared with the factor j1 ; e;TR=T1 j from (8.43) for the repeated spin echo experiment. The factor of 2 occurring in (8.46) makes it possible to nd 8.4. Inversion Recovery and T1 Measurements 131 Fig. 8.10: (a) The inversion recovery experiment consists of a combination of two rf pulses. The rst pulse inverts the longitudinal magnetization. The second pulse tips the longitudinal magnetization into the transverse plane so an FID signal may be measured. (b) The spin echo inversion recovery experiment consists of three rf pulses. The rst pulse inverts the longitudinal magnetization, the second pulse tips the longitudinal magnetization into the transverse plane, and the third pulse inverts both the longitudinal and transverse magnetization leading to an echo. This experiment and a repeated version are the subjects of Prob. 8.7 and Prob. 8.9, respectively. 132 Chapter 8. Introductory Signal Acquisition Methods a nite value of TI where the signal is zero. Therefore, an accurate way to determine T1 is to use the fact that the signal vanishes when TI null = T1 ln 2 (8.47) This implies that the signal is a sensitive function of TI , and, hence, of T1, as illustrated in Fig. 8.11. To nd T1, TI is varied until a zero in the signal is located. Signal zeros can be accurately determined, so that T1 , at least for a uniform sample, can be precisely measured using this method. Fig. 8.11: The magnitude of the signal for an inversion recovery as a function of TI , the time between rf pulses. Notice that the signal vanishes at TI = T1 ln 2 which corresponds to the point where the longitudinal magnetization has regrown from ;M0 to zero. There is additional utility associated with the zero in the signal. If a sample is being studied which contains two substances or tissues with di erent values of T1, then an experiment can be performed with TI chosen such that the signal from one of the tissues is zero. That component is said to be `nulled.' In this way, di erent components of a sample may be studied selectively. Problem 8.7
For a single repetition of the cycle of the spin echo inversion recovery sequence in Fig. 8.10b, show that M?(TE + TI ) is given by M?(TE + TI ) = M0 1 ; 2e;TI =T1 e;TE =T2 (8.48) 8.4. Inversion Recovery and T1 Measurements 133 8.4.2 Repeated Inversion Recovery Fig. 8.12: A repeated inversion recovery experiment. Following Sec. 8.3, a natural extension is to consider a repeated sequence of the inversion recovery rf structure. The repeated inversion recovery experiment is diagramed in Fig. 8.12. The e ect of the rst -pulse is to invert the equilibrium magnetization, but no transverse magnetization is created, i.e., Mz (0+) = ;Mz (0;) = ;M0 M?(0+) = 0 ( rst cycle) (8.49) (8.50) An increasingly familiar calculation leads to the magnetization (the longitudinal component and the magnitude of the transverse component) during the period between the -pulse and the =2-pulse as given by Mz (t) = ;Mz (0;)e;t=T1 + M0 (1 ; e;t=T1 ) = M0(1 ; 2e;t=T1 ) M?(t) = 0 0 < t < TI (8.51) The =2-pulse converts the longitudinal magnetization into transverse magnetization whose precession can be detected as the MR signal Mz (TI+) = 0 M?(TI+) = ;Mz (0;)e;TI =T1 + M0(1 ; e;TI =T1 ) = jM0 (1 ; 2e;TI =T1 )j Mz (t) = M0 1 ; e;(t;TI )=T1 M?(t) = ;Mz (0;)e;TI =T1 + M0(1 ; e;TI =T1 ) e;(t;TI )=T2 = M0 (1 ; 2e;TI =T1 ) e;(t;TI )=T2 TI < t < TR (8.52) (8.53) The longitudinal magnetization and the magnitude of the transverse magnetization for the time interval between the =2-pulse and the next -pulse are found in the usual way to be (8.54) 134 Chapter 8. Introductory Signal Acquisition Methods
; Mz (TR ) = M0 1 ; e;(TR;TI )=T1 ; The longitudinal magnetization at t = TR , (8.55) corresponds to the regrowth from t = TI , the time the =2-pulse tipped all of the magnetization into the transverse plane, to the end of the repetition. Since Mz (TI+) = 0 for all repetitions of the sequence, Mz (TR ) will also be the same in every cycle. The signal during the nth cycle of the repeated experiment is simple to nd, if it is ; assumed that TR >> T2 . This condition implies M?(tn = TR ) = 0 at the end of each cycle. Substituting the expression (8.55) for Mz (0;) generalizes (8.54) to M?(tn ) = ;M0 (1 ; e;(TR ;TI )=T1 )e;TI =T1 + M0 (1 ; e;TI =T1 ) e;(tn ;TI )=T2 = M0 1 + e;TR =T1 ; 2e;TI =T1 e;(tn ;TI )=T2 TI < t n < T R
in each cycle. (8.56) Problem 8.8
As a generalization of (8.47), show that the value of TI for which the signal from a repeated inversion recovery experiment is zero is TI = T1 ln 1 + e2 TR=T1 (8.57) ; Problem 8.9
Derive the following formula, for the magnitude of the transverse magnetization, for any repetition of a =2 spin echo inversion recovery experiment, Fig. 8.10b: M?(TE + TI ) = M0 1 + 2e;(TR;TE =2)=T1 ; e;TR=T1 ; 2e;TI =T1 e;TE =T2 (8.58) 8.5 Spectroscopy and Chemical Shift
The FID experiment can be used to determine the presence of a particular nuclear species. Consider a sample immersed in a perfectly homogeneous static eld long enough to reach equilibrium magnetization. The Larmor frequency for the sample depends on the species !0i = iB0 (8.59) 8.5. Spectroscopy and Chemical Shift 135 where i denotes the nucleus of interest. A narrow-band rf pulse can be applied at the Larmor frequency of a particular nuclear species, such that other species with di erent frequencies are left unexcited. The abundance of a given species may be found by calibrating the signal intensity against samples with known properties. The frequencies which give rise to an FID signal help to determine the composition of a sample based upon knowledge of i for various nuclei. In this experiment, only the presence of the desired element is determined, not its spatial distribution. A more practical approach involves applying a broad-band transmit rf pulse, containing a wide spectrum of frequencies, and obtaining a more complicated FID signal. This signal contains information about the set of all nuclear species in the sample whose frequencies lie within the pulse spectrum. The bandwidth of the analog receive lter must also be wide enough to accommodate the range of excited frequencies. A decomposition of the signal using Fourier analysis can lead to a resolution of which species are present, and of their relative population. Such experiments are sensitive to the molecular environment in which the nucleus nds itself. There are small, but measurable, di erences in the degree of magnetic shielding due to the local environment, implying slightly di erent Larmor frequencies. These frequency changes can be described in terms of `chemical shifts.' The chemical shift is an assumed linear response of the electronic structure to the external eld, and it is de ned by Bshifted (j ) = (1 ; j )B0 (8.60) where j refers to the chemical compound. A positive (negative) chemical shift implies shielding (anti-shielding). Details about the role of chemical shifts in imaging are presented in Ch. 17 and other chapters dealing with sequence design. As an example, hydrogen nuclei in water are not the only source of signal for 1H MR. They can also be found in many organic compounds, where there are di erences in their frequencies. Body fat (lipids) have a mean chemical shift of 3.35 ppm. This corresponds, for instance, to a Larmor frequency shift of 215 Hz at 1.5 T relative to water. In general, an MR experiment designed to determine the abundance, but not the spatial distribution, of di erent molecules or nuclei in a sample, is referred to as an MR spectroscopic experiment. Hybrid experiments that provide information about the spatial distribution and abundance of di erent chemical species are referred to as chemical shift imaging experiments (see Chs. 10 and 17). The primary focus of this text, however, is the spatial imaging of a single chemical species or MRI, and not MR spectroscopy (MRS). Sampling and Spectroscopy Experiments
Consider a population of di erent nuclear species. Let the j th species contain Nj hydrogen nuclei with chemical shift j . The phase of each nucleus as a function of time is
j (t) = ; B0 (1 ; j )t (8.61) 136 Chapter 8. Introductory Signal Acquisition Methods assuming a homogeneous eld B0 . If all other spatial dependence is ignored as well,15 the demodulated signal from this set of spins is given by s(t) / X
j Nj ei j B0 t (8.62) Relaxation damping has also been neglected. The spatial integral (8.1) is replaced by a sum over the individual spins, and Nj is related to M? according to the discussion in Ch. 4. A spectral analysis is made of the signal in order to extract estimates of the relative spin densities corresponding to each Nj . The data can be collected using the FID structure shown in Fig. 8.1 or the spin echo structure of Fig. 8.3. In both cases, multiple data points are needed to resolve the contributions from the di erent species, as discussed in Sec. 8.2.4. For a continuous set of Nj , the sum is replaced by an integral over a continuous distribution in , and the signal becomes Z s(t) / d N ( )ei B0 t (8.63) where, as a portent of things to come, we observe that the signal (8.63) is a Fourier transform. It is through inverse transforms (or inverse Fourier series) that the signal can be processed in order to determine the original species pro le, and the next chapter presents the transform connection as a central theme. 15 Such assumptions are quite reasonable for the small samples used in NMR spectroscopy. 8.5. Spectroscopy and Chemical Shift 137 Suggested Reading
The fundamental concept of an echo is introduced in: E. L. Hahn. Spin Echoes. Phys. Rev., 80: 580, 1950. Basic Fourier transform concepts as applied to NMR are presented in: C. T. Farrar and E. D. Becker. Pulse and Fourier Transform NMR. Theory and Methods. Academic Press, New York, 1971. 138 Chapter 8. Introductory Signal Acquisition Methods Chapter 9 One-Dimensional Fourier Imaging, k-Space and Gradient Echoes
Chapter Contents
9.1 9.2 9.3 9.4 9.5
Signal and E ective Spin Density Frequency Encoding and the Fourier Transform Simple Two-Spin Example Gradient Echo and k-Space Diagrams Gradient Directionality and Nonlinearity Summary: The basic tenets are established for imaging with linear gradient elds. The MR signal is connected to the Fourier transform of the spin density, and the image reconstruction is described in terms of the inverse Fourier transform. The role of the gradient structure is studied with respect to phase refocusing, k-space coverage, and alternative gradient directions. Sequence diagrams are generalized to include gradient eld information. Introduction
In the previous chapter, a discussion of the global magnetic resonance signal from a sample was presented. The goal of MR imaging, however, is not only to establish the presence of di erent nuclei, but also to determine the spatial distribution of a given species within the sample. Since identical nuclei precess at di erent rates in locations where the magnetic eld has changed, the local spatial distribution of the spins can be determined from the frequency content of the resulting MR signal, provided a well-de ned spatial eld variation is superimposed on the homogeneous static eld. To impose spatial dependence, the uniform static eld can be augmented by a smaller, linearly varying magnetic eld. In this chapter, an analysis of the signals resulting from a `linear gradient eld' is given. For clarity, we can restrict ourselves to one dimension 139 140 Chapter 9. One-Dimensional Fourier Imaging in learning the basic concepts and the principal role of the gradient elds. The fact that the signal is e ectively a Fourier transform of the spin density is already evident in this case. A simple two-spin model and other examples serve to highlight some signal processing mathematics, such as that involving the Dirac delta function. The 2D and 3D imaging problems are considered in the next chapter. The coverage of k-space is pivotal to reconstructing an image of the sample by inverseFourier-transform techniques. The relationship between the choice of gradient elds and the way in which k-space is covered is introduced. The standard sequence diagram is expanded to include the gradient structures. The implications of arbitrary gradient directions and more general gradient behavior are discussed at the end of the chapter. 9.1 Signal and E ective Spin Density
The imaging of a body commonly refers to the determination of the amplitude (magnitude) of the spin distribution, or spin density, rather than the magnetization. Groundwork on the relationship between the signal and the spin density is laid in this section. The de nition of what is meant by a spin density, in three dimensions as well as in one dimension, is addressed. While it is convenient to ignore relaxation e ects in an introductory discussion of imaging, the results are not entirely academic. They are valid for data sampling taking place over times small compared to T2 . 9.1.1 Complex Demodulated Signal Let us de ne s(t) as the complex demodulated signal given by an adaptation of (7.28). To turn the proportionality into an equation, is introduced as a constant which includes the gain factors from the electronic detection system. The transmitting and receiving rf coils are considered to be su ciently uniform, so that the initial magnetization phase 0, the receive eld directional phase B , and the receive eld amplitude B? are all independent of position. In view of the focus on position dependence, all constant phases can be set equal to zero or absorbed into . The signal1 is generalized to include a space- and time-dependent precession frequency !(~ t). With the neglect of relaxation e ects r Z s(t) = !0 B? d3rM?(~ 0)ei( r Zt
0 t+ (~ t)) r (9.1) where is the reference (demodulation) frequency (with no o set, = !0). The angle is the accumulated phase (3.37) with the counterclockwise positive sign convention (~ t) = ; r dt0!(~ t0) r (9.2) where ! = !0 only if there is a uniform static eld. In the discussions of this chapter, the addition of a gradient eld is assumed to be the primary reason that the precession frequency ! is a function of both position and time. When the gradient eld is not present, the static eld is assumed to be strictly homogeneous (and !0 strictly constant).
1 Hereafter, the fact that the signal is demodulated and complex is left understood. 9.1. Signal and E ective Spin Density 141 9.1.2 Magnetization and E ective Spin Density With the neglect of relaxation e ects, and the assumption of, say, a perfect =2-pulse applied uniformly over the sample, the initial transverse magnetization is simply the equilibrium magnetization M0 . From (6.11), the equilibrium magnetization can be expressed in terms of the proton spin density. The resulting expression involving the temperature T and static eld B0 (the gradient eld is not yet turned on) is 2 2 M?(~ 0) = M0 (~) = 4 0 (~) kT B0 r r 1 r h (9.3) No restriction to one dimension has yet been made the quantity 0 (~) is the number of r proton spins per unit volume in three dimensions. From the combination of (9.1) and (9.3), Z r s(t) = d3 r (~)ei( t+ (~ t)) r (9.4) where the e ective spin density2 (~) has been introduced r
2 2 r h (9.5) (~) !0 B?M0 (~) = 4 !0 B? 0 (~) kT B0 r r 1 If the receive eld magnitude B?, for example, is not homogeneous, its spatial dependence can be included in (9.5). The relaxation factors may also be incorporated in the de nition. The present interest is the case where the phase factor in (9.4) depends only on one dimension, which can be de ned as the z coordinate. The signal can then be written as s(t) = dz (z)ei(
in terms of the e ective 1D spin density3 (z) Z t+ (z t)) (9.6) (9.7) ZZ dxdy (~) r All integration limits are determined by the region of nonzero spin density. The `linear density' (z) is thus obtained by projection of the other two dimensions onto the z-axis. The integration over the other two coordinates leaves a one-dimensional problem. A caveat is that the same symbol is used for both the three-dimensional and onedimensional e ective densities. Also, it is observed that they are not equal to, but rather, only proportional to, the actual number of proton spins per unit volume or length, respectively, with a proportionality coe cient including temperature, frequency, electronics, eld and other factors. Nevertheless, we shall, for convenience, call them `spin densities.' Neglecting relaxation e ects causes little error in a `single-acquisition' experiment, as alluded to earlier, if the total sampling time Ts is much less than T2 . (In (9.6), 0 t Ts.)
When the constant phases 0 and are retained, the e ective spin density is complex. Di erences in the precession frequencies of the various nuclear species excited by the given rf pulse are ignored.
2 3
B 142 Chapter 9. One-Dimensional Fourier Imaging For multiple acquisitions, the examples in Ch. 8 have shown that there are both T1 and T2 dependencies in spin echo experiments. Equation (9.6) remains valid for such repeated experiments, provided the repetition time is large enough, i.e., TR T1, and the echo time is small enough, i.e., TE T2. Under these conditions, e;TR =T1 is negligible, and e;TE =T2 is close to unity. When relaxation e ects cannot be ignored, (z) should be replaced by (z T1 T2 ), and in the general case, (~) by (~ T1 T2 ). r r 9.2 Frequency Encoding and the Fourier Transform
The object of imaging is to determine the spin density (z) of a sample from the measurement of the signal as a function of time. The rst step is to connect the spin precession to its position, and the second is to recognize that this connection implies that the signal is a well-known linear integral transform of the spin density. 9.2.1 Frequency Encoding of the Spin Position The Larmor frequency of a spin will be linearly proportional to its position z if the static eld is augmented with a linearly varying eld.4 The maximum of the linearly changing eld at any point in the system is usually considerably smaller in magnitude than the static eld.5 The magnetization is considered to have been tipped, initially, into the transverse plane, so that it is already precessing freely with angular frequency !0, before the gradient eld is brought into play. (Throughout the discussion of this chapter, it is assumed that the rf pulses and the gradient elds are not applied at the same time. In the next chapter, important techniques arising from their simultaneous application are considered.) If a (spatially) linearly varying eld is added to the static eld, then the z-component of the eld is Bz (z t) = B0 + zG(t) (9.8) The quantity G is the (spatially) constant gradient in the z-direction, Gz @Bz [email protected] (9.9) and for now G is set equal to Gz . The time dependence of G(t) reminds us that the gradient may undergo quite general modi cations during the course of an experiment. De ne the variation in the angular frequency of the spins by !(z t) !0 + !G(z t)
4 (9.10) ignored (see Ch. 27), although their e ects in certain `phase sensitive' methods are not always negligible. 5 In MRI experiments, G is on the order of tens of mT/m while the spatial extent is on the order of a few tens of cm. As a result, the gradient creates frequencies on the order of a few dozen to a few hundred kHz at most. The signal frequency components induced in the receiving coil have di erences in the audiofrequency range forming the envelope for the extremely fast oscillations at the Larmor frequency (on the order of MHz) in the radiofrequency range. This situation is analogous to the waves transmitted by FM radio stations, with the Larmor frequency playing the role of the carrier frequency. ~ ~ ~ ~ r B = 0 and r B = 0. If these components are very small compared to B0 they may, in general, be The `linear eld gradient' must have other components in order to satisfy the basic Maxwell's equations, 9.2. Frequency Encoding and the Fourier Transform For (9.8), the deviation from the Larmor frequency is linear in both z and G 143 (9.11) !G(z t) = zG(t) The use of a gradient to establish a relation, such as (9.11), between the position of spins along some direction and their precessional rates is referred to as frequency encoding along that direction.6 The object of the standard MR imaging of humans is to extract information about the hydrogen distribution of the sample, in this case by looking at the signal due to frequency encoded proton spins. The accumulated phase, up to time t, due to the applied gradient is
G (z t) = ; Zt = ; z 0Z dt0!G(z t0 )
t
0 (9.12) (9.13) dt0G(t0) where it is recalled that the gradient is assumed to be applied only after the initial rf excitation at t = 0. 9.2.2 The 1D Imaging Equation and the Fourier Transform
The signal (9.6), with demodulating frequency given by = !0 and precession frequency (9.10), is (9.14) Z s(t) = dz (z)ei G (z t) where the phase, after demodulation, is determined by the gradient eld. Equation (9.14) is applicable to measurements with gradient elds that have arbitrary z dependence it is often referred to as the 1D imaging equation. It can be generalized by replacing G by the angle due to all gradient and rf eld variations. Remember that any initial (spatially constant) phase is ignored. The explicit z-dependence in the phase (9.12) for the linear eld leads to s(k) = dz (z)e;i2 k(t) = { Z kz (9.15) (9.16) where the time dependence resides implicitly in the spatial frequency k = k(t) with Zt
0 dt0 G(t0) This expression shows that, when linear gradients are implemented, the signal s(k) is the Fourier transform of the spin density of the sample. The spin density is said to be Fourier encoded along z by the linear gradient. The fact that the signal and the density are related by such a well studied linear transform is a boon to MRI. A detailed treatment of the Fourier transform and some of its properties is
It is this simple but critical realization that is the cornerstone of MR imaging and led to the seminal paper by Lauterbur (see the list of references at the end of this chapter).
6 144 Chapter 9. One-Dimensional Fourier Imaging found in Ch. 11. The foremost property of the Fourier transform is its well-de ned inverse. Given s(k) for all k, the spin density of the sample can be found by taking the inverse Fourier transform of the signal Z (z) = dk s(k)e+i2 kz (9.17) The signal s(k) and the image (z) are a `Fourier transform pair.' It is important to understand that any direction could have been chosen for this one-dimensional example. In future chapters, the `frequency encoding' or `read direction' is chosen as the x-direction, rather than the z-direction. 9.2.3 The Coverage of k-Space It is most useful to formulate imaging arguments in terms of image (z) space and data (k) space, in view of the Fourier transform connection. The inverse transform implies that the spin density can be reconstructed from the signal, if the latter is collected over a su ciently large set of k values. The integration in (9.17) requires `good coverage' of k-space. The dependence on time and the applied gradient amplitude is the key to covering a large enough range in k. In the case where the applied gradient is constant over the whole time interval (0 t), (9.16) reduces to k = {Gt (9.18) Therefore, to collect a uniform distribution of points in k-space, it is only necessary to sample the signal at a constant rate in the presence of a constant gradient. Sampling both negative and positive values for k can be achieved by changing the sign of the gradient. Speci c procedures for such coverage are detailed in Sec. 9.4. If the signal could be measured continuously over a very long time and an accurate integration carried out in (9.17), then a faithful picture of the spin density would be found. However, several factors prevent collection of continuous data over all k-space. First, the signal must be sampled in a nite amount of time and second, relaxation e ects wipe out the signal within a nite period. In actuality, the data collected will be a truncated and discretized version of s(k), and it is necessary to carefully study the discrete Fourier transform to understand the impact these two modi cations have upon the resulting image (see Chs. 11 and 12) relative to the continuous analytic form given in (9.17). 9.2.4 Rect and Sinc Functions There are two important functions in upcoming discussions of signal analysis, the `boxcar' or rect function 8 > 0 z < ;1=2 < rect(z) > 1 ;1=2 z 1=2 (9.19) : 0 z > 1=2 and the sinc function sinc(z) = sin z z (9.20) 9.3. Simple Two-Spin Example 145 The functions rect(z) and sinc(k) are a Fourier transform pair, a fact highlighted in the rst problem to follow. Following the previous chapter, time pro les of the rf pulses continue to be shown as rect functions (recall that narrow rect functions have been referred to as hard pulses). It was also noted there that, in general, rect functions are more appropriately used for the frequency pro les, rather than the time pro les, of the rf elds. It is now seen that this means realistic rf time pro les should be modeled by sinc functions. Such pro les are implemented in Ch. 10. Problem 9.1
Consider a boxcar spin-density distribution with width z0 , centered at z = 0, and given by (z) = 0 rect(z=z0 ). Find the signal s(k) for this spin density from (9.15). The answer will involve the sinc function, sinc( kz0 ). Then check, using integral tables for example, that the answer gives back the correct spin density through the inverse transform (9.17). (See also Prob. 9.3.) Problem 9.2
Assume a typical clinical static eld of B0 = 1:5 T and a linear gradient eld with G @Bz [email protected] = 10 mT/m. The volume of the sample to be imaged (the `imaging volume') is 50 cm DSV (`diameter spherical volume'). a) Compare the maximum change in the eld introduced by the gradient with that introduced by eld inhomogeneities of 10 ppm. b) Compare the maximum change in frequency induced by the gradient with the Larmor frequency produced by the main eld. c) Explain (qualitatively) how eld inhomogeneities might a ect the image. 9.3 Simple Two-Spin Example
Consider a pair of classical spins7 (a `dumbbell' of two point spins) lying along the z-axis, at z = z0 . Although individual classical spins are not experimentally relevant, the simplicity of the model is useful in demonstrating the 1D imaging method. Suppose the spins have reached equilibrium alignment along the static eld (Fig. 9.1a). If an rf pulse is applied to tip the spins into the transverse plane, a single-frequency signal results in the absence of
It is perhaps more appropriate to think of these as two spin isochromats. In any case, we wish to consider continuous changes in the spin direction we avoid referring to (quantum mechanical) proton spins for this reason.
7 146 Chapter 9. One-Dimensional Fourier Imaging a gradient. If the signal shown in Fig. 9.1b were demodulated at the Larmor frequency, it would be constant. If the experiment is run again in the presence of the eld gradient (Fig. 9.1c), during the time interval (t1 t2 ), the precessional rates of the two spins will di er slightly from the Larmor frequency. For G > 0 and gyromagnetic ratio , the spin at z0 rotates clockwise and the spin at ;z0 rotates counterclockwise at the same rate, in the rotating reference frame. Therefore, while the constant gradient G is applied, with t1 < t < t2, the spin at z0 has rotated through an angle, (z0 t) = ; Gz0 (t ; t1 ), and the spin at ;z0 through an angle, (;z0 t) = Gz0 (t ; t1). The signal in this case can be directly calculated since the integrals over the spin density reduce to the sum of the contributions from each spin. Recall the analogous sum in (8.62) and let t1 = 0 for convenience. The net signal is s(t) = s0e;i Gz0 t + s0ei Gz0 t = 2s0 cos ( Gtz0) 0 < t < t2 (9.21) or s(k) = 2s0 cos (2 kz0 ) 0 < k < k2 {Gt2 (9.22) with k from (9.18) and s0 the (common) magnitude of the signal associated with each spin. Equation (9.21) exhibits the expected beat envelope from two identical sources at di erent frequencies. (The superposition of the envelope on the fast carrier frequency corresponds to the laboratory signal shown in Fig. 9.1c.) Given this signal, and knowledge of the applied gradient, the distribution of the spins follows. In this case, the beat frequency {Gz0 implies the distance between the spins is 2z0. Albeit a very simple example, this demonstrates how the frequencies imposed by gradients yield information about the spatial distribution of the spins in a sample. It is useful to verify that the signal leads directly to (z) through the inverse Fourier transform. By changing the sign on G, (9.22) may be considered to be valid for all k a later discussion is directed to techniques for acquiring data at both positive and negative k values. From (9.17) Z1 (z) = dk 2s0 cos (2 kz0)ei2 kz = s0 dk ei2 k(z+z0 ) + ei2 k(z;z0) ;1 = s0 (z + z0 ) + (z ; z0)] (9.23) In the last step, it is recognized that the integrals over the exponentials are Dirac delta functions. For now, it need only be noted that the delta function (z ; z0 ) is, qualitatively, a spike at z = z0 with, in the limit, an in nite height, zero width, and unit area. In the next section, the Dirac delta function is discussed in more detail. Also, in view of its importance in signal processing, the representation Z1 (z) = dkei2 kz (9.24) is derived in the same section. Thus, the spin density (9.23) is exactly what is expected, for two point spins at z0 .
;1 ;1 Z1 9.3. Simple Two-Spin Example 147 a `spin dumbbell' with no MRI signal from the connecting rod) are to be determined by imaging. The z -component of the laboratory magnetic eld is plotted to the left of the dumbbells. The rotating frame orientation of the spin in each dumbbell is displayed at the right of the diagram. The sequence diagrams underneath represent the rotating frame input rf (assumed to be along x0 ^ and producing a =2 rotation), the applied gradient, and the laboratory received signal (with T2 decay neglected), before demodulation, as a function of time. Steps (a), (b), and (c) are described in the text. Note that Gz has the constant value G in part (c) between the times t1 and t2 . Fig. 9.1: Example of a 1D MRI experiment. The positions of two point spins (depicted as circles on 148 Chapter 9. One-Dimensional Fourier Imaging 9.3.1 Dirac Delta Function
The Dirac delta function8 is the zero width limit of a sharply peaked distribution with xed unit area (z ; a) = 0 if z 6= a and (9.25) Z z2
z1 dz (z ; a) = 1 a 2 (z1 z2 ) 0 a 62 (z1 z2 ) ( (9.26) The delta function is a lter or sifting function since it has the property of being able to select a particular value of a function according to Z1
;1 dz (z ; a)f (z) = f (a) (9.27) Let us derive, as a limit, the representation of the delta function in (9.24). De ne I (z K ) dkei2 kz = sin (2 Kz) z = 2K sinc(2 Kz)
;K ZK (9.28) The integrated result has been written in terms of the sinc function (9.20). Next, the large K limits of (9.28) and its integral need to be investigated. Note that, for z = 0, lim I (0 K ) = 1, since, as a limit, sinc(0) = 1. For z 6= 0, I (z K ) oscillates increasingly K !1 rapidly as K grows. As K approaches in nity, the function oscillates so rapidly that any function f (z) it multiplies will be averaged to zero everywhere except near z = 0, where it will sift out the value f (0). Finally, an integration over any interval including the origin gives unity for K ! 1. The remaining details of the arguments showing that
K !1 lim I (z K ) = (z) (9.29) are the subject of the following problem.
The Dirac delta function is an example of a generalized function which, despite its singular properties, can be manipulated according to the usual rules of calculus.
8 9.4. Gradient Echo and k-Space Diagrams 149 Problem 9.3
a) In order to obtain a qualitative understanding of how the function I (z K ) = 2K sinc(2 Kz) approaches a delta function, plot it as a function of z around the origin for the following values of K : i) K = 1= ii) K = 10= iii) K = 100= It will be noticed that, as K grows larger, the rst zero crossing occurs closer and closer to zero, the function oscillates faster, and its `energy' is more and more concentrated at the origin. b) Show that for any positive pair, a > 0 b > 0
K !1 lim Zb ;a 2K sinc(2 Kz) dz = 1 given the integral value Z1
;1 dw sin w = w (9.30) 9.3.2 Imaging Sequence Diagrams Revisited
Sequence diagrams have been introduced in the previous chapter. In Fig. 9.1, the sequence diagrams have been generalized to include gradient structure. More examples appear in subseqent gures of this chapter, and gradients for additional dimensions are added to the diagrams of the next chapter in the discussion of multi-dimensional imaging. The diagrams remain easy to understand. Recall that time is displayed on the horizontal axis. The vertical axis indicates whether the given quantity is active or not, and the activation refers, for example, to the B1 magnetic eld on the rf axis or Gz = dBz =dz on the gradient axis. 9.4 Gradient Echo and k-Space Diagrams
Suppose the two spins in the dumbbell example are replaced by a `cylinder' of an arbitrary zdistribution of spins (Fig. 9.2). The positions of the spins are frequency encoded by an applied gradient, and Fourier techniques are available, as discussed, to image the spin distribution. Figure 9.2 carries us through an MR experiment where the equilibrium magnetization M0 is attained (see Fig. 9.2a) and acted upon by a 90 rf pulse along x0 to create a transverse ^ magnetization also of amplitude M0 over the whole sample. The decaying signal in Fig. 9.2b represents the FID in the laboratory frame. Applying a gradient along z causes a more ^ rapid decay exhibiting beat frequencies (Fig. 9.2c). The transverse components of spins at di erent z-locations are shown projected onto the x-y plane to illustrate the dephasing of 150 Chapter 9. One-Dimensional Fourier Imaging the signal. By reversing the gradient polarity, an echo can be formed at t0 = 0 (Fig. 9.2d), according to the following explanation. Signal measurements over a su cient range of k are needed to reasonably reconstruct the spin density. The principal technique for obtaining negative, as well as positive, k values is the `gradient echo' method. A single, constant gradient (such as in Fig. 9.2c) is restricted in the k interval it generates, while a combination of gradients (such as in Fig. 9.2d) can be used to expand the k range from negative to positive k values. The time at which the data corresponding to the k = 0 point (the echo) is measured can be moved, for example, to the center of a particular gradient `lobe.' In particular, combinations of gradient lobes permit the recovery of a signal loss due to the presence of the gradients themselves. The net signal from a set of spins will begin to disappear, after a gradient eld is applied, for the same reason as that underlying T20 decay.9 Since the applied gradient is a eld inhomogeneity, the spins dephase, as illustrated by the `fanout' around = 0 in Fig. 9.2c for constant gradients. (For a given sign on the gradient, both positive and negative phases are found for any distribution of spins straddling the origin, z = 0.) A series of gradient pulses can be used to form an echo similar to the spin, or rf, echo described in Ch. 8. A simple example of a gradient echo is developed below. It should be noted that the loss of signal due to the presence of the gradient is expected from the property of the Fourier transform. Roughly speaking, the Fourier transform is a (continuous) sum of di erent phases which interfere with each other. The longer the cylinder of spins, the faster the Fourier transform drops o with k (i.e., with time in the imaging experiment). It is observed in Fig. 9.2c that the signal is (half of) a sinc function with a fallo expected from the dephasing arguments. (Rephasing can be achieved, as discussed below, such that the full sinc function signal is produced (Fig. 9.2d).) In essence, the details of the dephasing determine the Fourier transform, and this dephasing represents the encoded information content. The associated coverage of k values can be described by a k-space diagram. The k-space diagram for the simple boxcar gradient in Fig. 9.1c is shown in Fig. 9.3. For a constant gradient, the k value is proportional to the imaging time according to (9.18). In this case, only half of k-space is covered for a gradient with a given sign. Other examples are exhibited below for gradient echo experiments. The e ect on the signal due to the gradient manipulation is studied through the expression for the signal strength, (9.14). Recall that the relation between the phase and an arbitrary linear gradient in the z-direction is given by (9.13). 9.4.1 The Gradient Echo
The gradient dephasing of spins in an object can be understood using phase arguments similar to those employed in Ch. 8 for the rf echo. Let us analyze the speci c gradient sequence included in Fig. 9.2d. A constant negative gradient (Gz = ;G for G > 0) is present in the time interval (t1 t2). From (9.13), the phase accumulation due to the gradients for a spin at z, and at a time t,
The total signal decay in the presence of an applied gradient is sometimes characterized by the relaxation time, T2 , analogous to T2 for static- eld inhomogeneities.
9 9.4. Gradient Echo and k-Space Diagrams 151 Fig. 9.2: A consecutive set of time events in a 1D MRI experiment. This illustration follows the description of Fig. 9.1, except the two point spins are replaced by a `cylinder' containing an arbitrary distribution of spins. Parts (a) and (b) are the same as in the previous gure, and (c) and (d) are described in the text. The magnetic eld plotted at the left of the cylinder in (c) refers to the eld in the time interval (t1 t2 ), and, in (d), to the interval (t3 t4 ). For comparison, the spin isochromats are pictured lying in a single plane, despite their di erent z coordinates. Note that Gz = ;G in part (c) and has both a negative and a positive lobe of strength G in part (d). 152 Chapter 9. One-Dimensional Fourier Imaging Fig. 9.3: The k-space coverage for an FID imaging experiment. The time t00 = (t ; t1 ) is de ned from the beginning of the gradient. The total sampling time is Ts . Notice that only half of k-space is covered by this experiment. There is no progression through k-space until time t1 in Fig. 9.1c which is equivalent to t00 = 0. during the application of the rst gradient lobe is
G (z t) = + Gz(t ; t1 ) t1 < t < t2 (9.31) The sign is consistent with the counterclockwise precession expected for a negative gradient and positive z-coordinate. The second gradient lobe is positive (Gz = G in terms of the same parameter G) during the time interval (t3 t4 ). The phase behavior relative to the y0-axis at any time in this interval is t3 < t < t4 (9.32) G (z t) = + Gz (t2 ; t1 ) ; Gz (t ; t3 ) Notice that there is no phase change during the period when the applied gradient is zero. The gradient echo is now evident. The phase (9.32) returns to zero (Fig. 9.2d) at t = t3 + t2 ; t1 TE (9.33) for all z. The echo corresponds to that time during the second gradient lobe where the evolved area under the second lobe just cancels the area of the rst lobe. The cancellation suggests the general condition for a gradient echo, Z G(t)dt = 0 (9.34) i.e., the zeroth moment of G(t) vanishes. The gradient echo analysis in imaging revolves around this criterion. Returning to the example, the second gradient time interval can be chosen, as in Fig. 9.2d, such that the echo occurs at its center that is, when (t4 ; t3 )=2 = t2 ; t1 . It is useful to de ne a new time coordinate with its origin at the echo t0 t ; t3 ; (t2 ; t1) = t ; TE
Thus, (9.32) can be rewritten
G (z (9.35) (9.36) t) = ; Gzt0 ; (t4 ; t3 )=2 < t0 < (t4 ; t3 )=2 with an obvious zero at t0 = 0. It is during the time interval indicated in (9.36) that the data are usually sampled. 9.4. Gradient Echo and k-Space Diagrams 153 The signal (9.14) can be written as a function of t0 over the region in which Gz = G. During the time the second gradient is turned on, the signal is Z s(t ) = dz (z)e;i Gzt Z = dz (z)e;i2 k(t )z
0
0 0 ; (t4 ; t3 )=2 < t0 < (t4 ; t3)=2 (9.37) where k = {Gt0 from (9.16). With the time-shift to t0, the signal (9.37) is expressed in a form convenient to analyze the data as referenced from the center of the second lobe. (The second lobe is called the `rephasing gradient lobe' of the read gradient, while the rst lobe is referred to as the `dephasing lobe' for that gradient.) Such time-shifts are regularly used in MR analysis. The above experiment thus leads to a range of negative and positive k-space points for measurements made symmetrically about the echo. The negative gradient lobe has been utilized to create an echo in the middle of the positive rephasing gradient where k = 0. A k-space diagram for the gradient echo experiment is shown in Fig. 9.4a. The signal in k-space is a simple transcription of (9.37), Z s(k) = dz (z)e;i2 kz ; kmax < k < kmax (9.38) where kmax = {G(t4 ; t3)=2. Image reconstruction is also possible with just positive k-space values (see Ch. 13 for a further discussion of FID imaging with partial Fourier reconstruction). In this case, the imaging experiment is performed with a single, positive, read gradient lobe. A k-space diagram for the FID imaging experiment has already appeared in Fig. 9.3. The limits on k-space sampling are 0 < k < kmax , where kmax = {GTs, for a constant gradient G > 0 and sampling time Ts. Problem 9.4
Spins with gyromagnetic ratio are uniformly distributed with linear spin density 0 along the z -axis from ;z0 to z0 in a 1D imaging experiment. Suppose they are excited at t = 0 by an rf pulse such that the signal at that instant would be given by Z z0 s(t = 0) = dz 0 = 2z0 0 (9.39) A negative constant gradient eld ;G is immediately applied at t = 0+ and ipped to the positive constant gradient eld +G at time t = T . Find an expression for the signal for t > T and show that it exhibits a gradient echo at time t = 2T .
;z0 154 Chapter 9. One-Dimensional Fourier Imaging Fig. 9.4: The k-space coverage for the basic gradient echo experiment and the two variants of the spin echo experiment presented in the text. Diagram (a) applies to the basic gradient echo experiment and to the spin echo variant where all frequency encoding occurs after the -pulse (see Fig. 9.6). Diagram (b) shows the spin echo variant where both gradient lobes are positive (see Fig. 9.5). In all cases, kmax = {G(t4 ; t3 )=2. The dashed line represents the action of the -pulse which changes k to ;k. 9.4.2 General Spin Echo Imaging
In Ch. 8, the spin echo experiment was introduced as a method to refocus the phase accumulation due to static- eld inhomogeneities. While the gradient echo refocuses the phase induced by the application of gradients, it does not refocus the dephasing due to static- eld inhomogeneities. In many cases, T2 decay can cause a signi cant reduction in signal before the gradient echo is performed. Under these circumstances, it may be bene cial to perform a spin echo in concert with the gradient echo, to refocus the static eld inhomogeneities and, thereby, increase the signal at the gradient echo. In general, the use of both a gradient echo and a spin echo during an imaging experiment is referred to simply as a spin echo experiment. It is necessary, however, to distinguish an imaging spin echo experiment from the global spin echo method described in Ch. 8.10 If imaging is being discussed, then the spin echo sequence is a combination of multiple rf pulses and a set of applied gradients,11 but, in a global signal detection experiment, it may only refer to the rf pulses.
An alternate description is to refer to spin echoes induced by rf pulses as rf echoes or Hahn echoes. Both rf echoes and gradient echoes are, after all, spin rephasing, or spin echoes. 11 The time variations of gradients will be referred to as `gradient pulses.'
10 9.4. Gradient Echo and k-Space Diagrams 155 Two Variants of the 1D Spin Echo Imaging Experiment
There are two primary methods in which the spin echo may be incorporated with imaging. The rst is to apply a -pulse in between two gradient pulses, where both gradient pulses have the same polarity. See Fig. 9.5 for the timing and gradient parameters. In the presence of magnetic eld inhomogeneities, the phase behavior for a spin isochromat at the position z during the rst gradient pulse is (z t) = ; B (z)t ; Gz(t ; t1 ) t1 < t < t2 (9.40) for the initial =2-pulse ( eld) applied along x0 and de ned with respect to the y0-axis so ^ that (z 0) = 0 (see Sec. 8.2 ). In (9.40), the z-component of the static eld is B0 + B (z), and the constant gradient G is taken to be positive. For the -pulse applied along y0 at ^ t = , all of the phase, including that due to the gradients, is inverted, ! ; .12 During the application of the second gradient lobe, the phase evolves according to (z t) = B (z) + Gz(t2 ; t1) ; B (z)(t ; ) ; Gz(t ; t3) t3 < t < t4 (9.41) If, as indicated in the gure, the time of the spin echo, t = 2 , coincides with the time of the gradient echo, TE t3 + (t2 ; t1 ), then the expression (9.41) for the phase vanishes at that common point. With this constraint on , the total phase induced by the applied-gradient and static- eld inhomogeneities has been refocused. The key to the arrangement is that the -pulse e ectively changes the sign on the rst gradient, so that the two gradient lobes can produce the same cancellation seen in (9.32). The associated k-space coverage is displayed in Fig. 9.4b. A second, essentially equivalent method of collecting spin echo data in conjunction with a gradient echo uses a `bipolar' gradient structure. Two gradient lobes of opposite polarity follow the -pulse (Fig. 9.6). The evolution of the phase is worked out in the following problem, with the result that the phase expression pertaining to the time interval of the second gradient lobe shows the same zero as in (9.41). With the times ti rede ned by Fig. 9.6, it is clear that (z TE ) = 0, if 2 = t3 + (t2 ; t1). The rf echo and gradient echo mechanisms work independently in the second method and, as before, they combine to give maximal signal at TE . Therefore, the k-space diagram is the same as that for the regular gradient echo sequence, which was shown in Fig. 9.4a and similar forms of (9.36) to (9.38) are valid for the spin echo as well. This exercise demonstrates that several di erent data acquisition schemes can lead to the same k-space coverage.
The conclusions in these discussions do not depend on the choices of the rf pulse eld directions and the = 0 axis. For other choices, constant phases may arise that are nonzero, but that are the same for all isochromats.
12 156 Chapter 9. One-Dimensional Fourier Imaging Fig. 9.5: Sequence diagram for a spin echo 1D imaging experiment with a -pulse between two gradient lobes of the same polarity. The directions along which the rf elds point are indicated by the subscripts and correspond to the choices in the text, but the echo is independent of these choices. Fig. 9.6: Sequence diagram for a spin echo 1D imaging experiment with a -pulse preceding two gradient lobes of opposite polarity. The subscripts on the rf ip angles again denote the direction ~ of the applied magnetic eld B1 . 9.4. Gradient Echo and k-Space Diagrams 157 Problem 9.5
a) In terms of the timing and gradient parameters shown in Fig. 9.6, show that the phase during the rst (negative) gradient lobe is (z t) = Gz(t ; t1 ) ; B (z)(t ; 2 ) t1 < t < t2 (9.42) t3 < t < t4 (9.43) b) Show that the phase during the second (positive) gradient lobe is (z t) = Gz(t2 ; t1 ) ; which is identical to (9.41). B (z)(t ; 2 ) ; Gz(t ; t3 ) In both sequences, the common denominator is the occurrence of the gradient echo and spin echo at the same time. Although more advanced imaging discussions are necessary to fully understand di erences between these two sequences, it is possible to give one reason for favoring the second method. The position encoding in the second example takes place at a time closer to the sampling of the data and, as may be anticipated, motion e ects will be minimized (see Ch. 23). 9.4.3 Image Pro les MR images, in general, are shown in a grayscale format where brightness (whiteness) indicates higher spin density. Most images are displayed such that only the highest spin density in an object appears white and areas with lower values of ^ appear in correspondingly darker shades of gray. Zero spin density appears black. Figure 9.7 demonstrates how a 1D projection of an object is related to both an e ective spin density and an image. In Fig. 9.7a, a pencil is shown where the spin density in the eraser end is constant and only half of that of the constant spin density found in the remainder of the pencil. The pencil also tapers linearly to zero at the nib. Figure 9.7b is a plot of the 1D (projected) physical spin density in the pencil as a function of z. The linear gradient eld shown in Fig. 9.7c could be used to obtain the k-space data necessary to reconstruct a 1D image. In order to view the reconstructed 1D `line' image, the density ^(z) is shown as a row of pixels with nite width in Fig. 9.7d. Notice that in this 1D imaging discussion, where all spins at a given frequency have been projected onto the same z-position, relaxation e ects have been neglected. The image pro le of the pencil in Fig. 9.7b is an example of a conventional graph of e ective spin density along the imaging axis. In general, this may be generated along an arbitrary line in higher dimensions, and it is useful for the determination of speci c properties of the reconstructed 3D spin density which are di cult to otherwise discern. For instance, consider the linear decrease in signal at the end of the pencil. This feature is immediately evident in the pro le, but the linearity may not be readily apparent in the 1D image (Fig. 9.7d). 158 Chapter 9. One-Dimensional Fourier Imaging Fig. 9.7: The 1D imaging example of a pencil (where the density di erences due to the lead and lead tip are ignored). The physical object being imaged is shown in (a), and the physical 1D projected spin density, or image pro le, is plotted in (b). A 1D MRI of the pencil can be carried out using the magnetic eld with the linear gradient plotted in (c). In (d), the resulting image is shown as a line of pixels z , where the thick black border is added for visual contrast. The projection onto 1D at z0 , for example, is carried out by three integrations: Two integrations are over the transverse dimensions of the pencil, and the third integration is over z . 9.5. Gradient Directionality and Nonlinearity 159 Problem 9.6
In this problem, we study the e ects of relaxation on the signal for the pencil described above and in Fig. 9.7. Assume that 0 = 1 and that relaxation e ects during data sampling can be neglected, i.e., Ts T2 . A simple gradient echo experiment (Fig. 9.2d) with a =2-pulse is considered. a) Assume that all data are collected in one excitation so that the longitudinal relaxation of the sample is not a factor. For T2 eraser = 20 ms, T2 pencil = 50 ms, and TE = 30 ms, plot the reconstructed image pro le (where only the relaxation e ects are taken into account). b) Instead, assume that the experiment must be repeated many times to acquire an adequate amount of data. In this case, the signal will be a ected by both longitudinal and transverse relaxation e ects. Assume that TE ' 0, T1 eraser = 400 ms, T1 pencil = 3000 ms, and TR = 500 ms. Plot the reconstructed image pro le in this case. See Sec. 8.3. This problem is an introductory example of how imaging parameters can a ect the resulting reconstructed pro le. Contrast mechanisms based on di erences in relaxation times are developed further in Ch. 15. A practical illustration of the concepts of the previous discussion is found by performing a gradient echo experiment. A spatially uniform cuboidal gel (a `phantom') is imaged along its longest dimension. The other two dimensions are much smaller in comparison. The resultant 1D image is shown in Fig. 9.8b, where the oscillations at the edge in the object pro le are an example of Gibbs ringing (see Ch. 12). The concave dip in the image pro le is presumably due to an rf nonuniformity caused by the gel electrical conductivity (see Ch. 27). 9.5 Gradient Directionality and Nonlinearity
9.5.1 Frequency Encoding in an Arbitrary Direction
Although the z-direction was chosen for the 1D spatial encoding of spins, there is no restriction on the direction along which the gradient can be applied. A constant gradient ~ vector G may be used to de ne an arbitrary coordinate direction, with respect to which ~ the z-component of the associated applied magnetic eld B g is linearly changing.13 The superscript g is used to remind the reader that an applied gradient is being considered. The
13 ~ The fact that G is constant in space means that Bz grows linearly in that direction. 160 Chapter 9. One-Dimensional Fourier Imaging Fig. 9.8: A 1D imaging experiment on a thin cuboidal object of uniform spin density using a gradient echo sequence. (a) Expected physical 1D pro le. (b) Measured pro le at two di erent TE values (the solid line is obtained at TE = 6 ms and the dashed line is obtained at TE = 20 ms). As expected, there is a drop in signal at the longer TE value. (c) A 1D image where the line of pixels is constructed as in Fig. 9.7d. The thickness of the line is exaggerated for visualization purposes. Due to the choice of relationship between image brightness and signal strength, the pro le variations seen in (b) are not visible in the image. 9.5. Gradient Directionality and Nonlinearity gradient vector is de ned by14 ~ G(t) 161 g ^ g ^ g = x @ Bz + y @ Bz + z @ Bz ^ @x @y @z Gx(t)^ + Gy (t)^ + Gz (t)^ x y z (9.44) The general linear gradient is a superposition of three linear gradients, Gx Gy Gz along the three orthogonal directions. Each gradient corresponds to a magnetic eld whose zcomponent varies linearly along the given direction and this same z-component augments the static eld. The x-gradient, y-gradient, and z-gradient z-components of the magnetic eld vary linearly in x, y, and z, respectively. ~ A gradient G with a xed direction15 corresponds to one coordinate along one direction. The z-component of the total (static plus linear) magnetic eld can be written as ~ r Bz ( t) = B0 + G(t) ~ = B0 + G(t) (9.45) ~ r rBzg (~) where the magnitude of the gradient vector is G, and the variable is de ned as the position ~ along the direction of G, ^ r G ~ (9.46) The application of the eld (9.45) leads to a formula for the (spatially varying) precessional frequency, ~ r !( t) = !0 + G(t) ~ = !0 + G(t) (9.47) as a generalization of (9.10). The associated phase accumulation is
G( t) = ; ~ r Zt
0 ~ dt G(t0 ) = ;
0 Zt
0 dt0G(t0) (9.48) (9.49) Equation (9.16) is now extended to ~ (t) = { k s(t) = Zt
0 ~ G(t0 )dt0
= d ( )e;i2 The combination of these results gives the more general 1D imaging equation Z d3 r (~)e r ;i2 ~ (t) ~ k r Z k(t) (9.50) (9.51) (9.52) A fuller reference to the k-space directions is obtained by rewriting (9.50) as Z kr ~ ) = d3r (~)e;i2 ~ ~ s(k r
() The 1D spin density along is
14 ZZ d d (~) r We remind the reader that the unprimed coordinates are assigned to the laboratory, rather than the rotating, reference frame. 15 This implies that the individual components of G have exactly the same time dependence. Their ratios ~ are (any) constants. 162 Chapter 9. One-Dimensional Fourier Imaging ~ The variables and are coordinates for a plane perpendicular to the gradient G direction. The resultant signal (9.50) remains a Fourier integral and one-dimensional imaging can ~ ~ thus be conducted in the direction of G. The image along the direction of G, as before, can be reconstructed using an inverse Fourier transform. The directionality of the gradient eld and, hence, of the image can be varied by changing the relative amplitudes of the three components Gx, Gy and Gz more details concerning the relationship of the components to the net direction are found in Sec. 10.1. (This is also a topic of discussion in Ch. 14 where radial k-space coverage is considered.) In fact, now the reader begins to see some of the exibility inherent in MRI, that, without physically rotating the sample, the viewing angle or imaging direction can be modi ed by manipulation of the imaging gradients. 9.5.2 Nonlinear Gradients It is possible to alter the uniform static eld with something other than a linear gradient eld. As long as there are no points where Bz has the same value, i.e., as long as it is singlevalued, the Fourier transform of the signal can be constructed. The Fourier transform, and knowledge of the magnetic eld, can then be combined to relate the frequency and position information, and to create a faithful image. It is actually impossible to generate a perfectly linear gradient in free space and, in some imaging circumstances, it is necessary to account for the distortion associated with the gradients after the Fourier transform is performed. Still, gradient coil designers strive for linearity over a limited imaging volume, and consider linearity, speed, and image quality in their speci cations. In most whole-body MRI machines, the maximum gradient deviation remains within 5% of the desired value (see Ch. 20 for speci c image distortion discussions). If an application presented itself where extremely fast data acquisition were required, and unlimited computing power were available, poorer gradients might be a practical alternative. 9.5. Gradient Directionality and Nonlinearity 163 Suggested Reading
The following two references discuss basic k-space concepts: S. Ljunggren. A simple graphical representation of Fourier-based imaging methods. J. Magn. Reson., 54: 338, 1983. D. B. Twieg. The k-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods. Med. Phys., 10: 610, 1983. Elementary aspects of sampling and Nyquist limits are presented in: O. E. Brigham. The Fast Fourier Transform. Prentice-Hall, Englewood Cli s, New Jersey, 1974. The basic pioneering concepts of magnetic resonance imaging are introduced in: P. C. Lauterbur. Image formation by induced local interactions. Examples employing magnetic resonance. Nature, 243: 190, 1973. P. Mans eld and P. K. Grannell. NMR `di raction' in solids? J. Phys. C: Solid State Phys., 6: L422, 1973. 164 Chapter 9. One-Dimensional Fourier Imaging Chapter 10 Multi-Dimensional Fourier Imaging and Slice Excitation
Chapter Contents
10.1 10.2 10.3 10.4 10.5
Imaging in More Dimensions Slice Selection with Boxcar Excitations 2D Imaging and k-Space 3D Volume Imaging Chemical Shift Imaging Summary: Imaging in more than one dimension is introduced. The notion of k-space is extended to multiple dimensions. Generic 2D and 3D imaging sequences are introduced. Excitation of a speci c slice is described as a technique for the creation of 2D images. The signals contributed by di erent chemical species during the same experiment are studied as imaging in an additional dimension. Introduction
The previous chapter focused on the 1D version of an MR imaging experiment. It was discovered there that the signal from the excited spins with a transverse component precessing in a linearly changing magnetic eld is a Fourier transform of the e ective spin density along the linear gradient direction ( , say). The transform variable of the data is the spatial frequency k , which may be varied by virtue of its dependence on time and the gradient strength. The signal can be inverse Fourier transformed in the direction to pro le the e ective spin density. In the rst section of this chapter, the generalization of 1D Fourier encoding to two and three dimensions is introduced. A component of the spatial frequency vector ~ is associated k with each direction, and with a gradient-vector component for that direction. The resulting 165 166 Chapter 10. Multi-Dimensional Imaging expression for the signal is a two- or three-dimensional Fourier transform of the e ective spin density. The technique of covering k-space by varying gradient amplitudes and the notion of `phase encoding' are introduced. Rather than collecting data over the entire third dimension, a `slice select' technique is introduced in Sec. 10.2 whereby the signal is limited by an rf excitation with a su ciently narrow bandwidth to that from a thin slice perpendicular to the third direction. The two `in-plane' dimensions of the slice are those referred to in `2D Fourier imaging' in Sec. 10.3. Two common techniques for working in three dimensions, multi-slice 2D imaging and 3D imaging, are introduced in the fourth section. The multi-slices are obtained by a series of rf pulses with di erent center frequencies. The 3D imaging is the partitioning of a single slab by the use of a phase encoding gradient. A central consideration is the time taken to collect data. The last section examines the di erence in signals from protons in di erent molecular environments due to chemical shifts. The chemical shift is an additional degree of freedom and the distribution of the signal in this variable can be couched as a dependence on an additional imaging dimension. 10.1 Imaging in More Dimensions
Imaging in 1D was predicated on separating out the unique information associated with each spatial position by frequency encoding the data. A new construct, k-space, could be used to express the data in terms of the Fourier transform of the spin density. Di erent points in k-space can then be collected by sampling the data during the ADC on-time, Ts, as shown in Fig. 10.1. This would produce a k-space representation of s(kx) as a set of points sampled along the continuous line in Fig. 10.1b. Similarly, enough information must be collected in the y and z directions to make it possible to extract (~) in three dimensions. In the r following discussion, we rst assert that the same Fourier transform representation of the data can be found in 3D and that the gradients in y and z can be used to extend k-space to 3D as well. Afterward, we shall demonstrate how to physically accomplish this, despite the loss of signal due to relaxation. 10.1.1 The Imaging Equation
The 3D Representation
Consider the extension of the one-dimensional imaging equation (9.15) to all three spatial dimensions.1 The signal from a single rf excitation of the whole sample in the presence of a set of three orthogonal gradients may be written as the 3D Fourier transform2 Z kr s(~ ) = d3r (~)e;i2 ~ ~ k r (10.1) or ZZZ s(kx ky kz ) = dx dy dz (x y z)e;i2 (kxx+ky y+kz z) = F (x y z)] (10.2)
For the moment, we defer the question concerning the way the image is to be viewed or presented. Although they are important, relaxation factors continue to be omitted in the introductory imaging equations.
1 2 10.1. Imaging in More Dimensions 167 Fig. 10.1: (a) A 1D imaging protocol for a gradient echo sequence structure with sampling restricted to the ADC period Ts between t3 and t4 . (b) The associated sampled k-space coverage running from kmin to kmax . Although jkmin j ' kmax , the fact that they need not be the same is illustrated
in this diagram and is a subject of interest in later chapters. This is called the 3D imaging equation. The three implicitly time-dependent components of ~ are related to the respective gradient-component integrals (cf. 9.16) k Zt Zt Zt 0 0 0 0 kx(t) = { Gx(t )dt ky (t) = { Gy (t )dt kz (t) = { Gz (t0 )dt0 (10.3) where the integrations run from the onset of the gradient to time t. The challenge is to manipulate the application of the gradients in such a way that 3D k-space can be sampled su ciently. This is required in order that the reconstructed image ^(~), which is the inverse r ~ ) (a density distribution derived from a set of Fourier transform of the measured data, sm (k discrete data, see Ch. 12), Z ^(~) = d3ksm (~ )ei2 ~ ~ r k kr (10.4) 168 Chapter 10. Multi-Dimensional Imaging be an accurate estimate of the physical density (~). In the discussions to follow, this will r be accomplished by independently varying Gx, Gy and Gz to sweep over the 3D k-space. The 2D Representation
For a planar object, or for a thin slice through a 3D object, where 2D information alone is required, only kx and ky need to be sampled. The sequence diagram in Fig. 10.2 employs just one rf pulse to cover 2D k-space with a set of discretely sampled parallel lines as shown in Fig. 10.3. Each set of points along a given ky line represents the acquisition of `phase encoded' 1D data, since the phase of each point along y is encoded as Gy y y. This y-dependent phase contribution is unchanged during data sampling along the kx -axis.3 We return to a detailed discussion of Figs. 10.2 and 10.3 in the next subsection. Fig. 10.2: Sequence diagram for coverage of 2D k-space with a single rf excitation. To continue echoing the data, the read gradient is reversed after each application of Gy . The `phase encoding' steps Gy are applied when the read gradient is o . Consider the imaging of a planar portion of a three-dimensional object with physical density (~). As an example of the phase encoding described above, all information along y r at a given x is projected onto x for the line ky = 0, i.e., Z s(kx 0) = d3 r (~)e;i2 kxx r (10.5) An inverse Fourier transformation of this line of data leads to the 1D projection reconstructed image according to the following calculation Z Z Z i2 kxx = d3 r0 (r0 ) dk e;i2 kx (x;x ) ~ ^1D (x) = dkx s(kx 0)e x ZZZ ZZ = dx0dy0dz0 (x0 y0 z0 ) (x ; x0 ) = dydz (x y z) (10.6)
0 The use of the phrase `phase encoding' can be misleading, since all gradients or eld inhomogeneities add phase to the magnetization, but it is universally used when the encoding gradient is varied in stepwise fashion.
3 10.1. Imaging in More Dimensions 169 Fig. 10.3: The k-space coverage for a 2D example. Each dot represents a sampled point. Lines
of connected dots are shown to be along the read direction, referring to data collected during the same read period. The arrows indicate the chronological order of data acquisition. The generalization of the encoding to arbitrary ky is s(kx ky ) = ZZZ dx dy dz (x y z)e;i2 (kx x+ky y) (10.7) After a calculation similar to (10.6), the result of inverse Fourier transformation with respect to both kx and ky is the 2D image projected along z ^ ^(x y) = dkxs(kx ky Z )ei2 (kxx+ky y) = dz (x y z) Z (10.8) 10.1.2 Single Excitation Traversal of k-Space
1D Coverage
In the 1D example of the gradient echo sequence, the path through k-space was seen to start at zero and then move to the left as the negative gradient was applied from t1 to t2 (Fig. 10.1). Just after it is turned o , a large negative k value has been reached. When the gradient is switched on again, this time to a positive value, the direction of coverage in k-space is reversed, and the data are sampled between t3 and t4 . The `dephasing gradient' is seen to be a gradient which determines the starting point of the sampling in k-space, but data usually are not taken over this lobe. 2D Coverage A standard coverage of 2D k-space corresponds to the series of parallel lines in a plane. The rst two gradients along x and y in Fig. 10.2 move the position in k-space from the ^ ^ origin to the bottom left corner in Fig. 10.3. This traversal of 2D k-space is obtained by alternately turning on the Gx gradient and then the Gy gradient as illustrated in Fig. 10.2. The bottom line in k-space is obtained by the usual read gradient structure during sampling of the data. Applying a positive Gy with just enough amplitude to carry ky up one line 170 Chapter 10. Multi-Dimensional Imaging moves the position in k-space to the right side of the second line from the bottom. To bring it back across kx requires applying a negative lobe of the same time duration as the positive rephasing gradient applied for the bottom line. Repeating the same steps, and each time reversing the polarity of the x-gradient, will carry the position up through the (kx,ky ) plane. The vertical spacing of the set of horizontal read gradient lines is determined by the step size in ky which is in turn determined by Gy (see below). The data are not collected during the short vertical steps drawn in Fig. 10.3. We return to a detailed analysis of the signal in 2D imaging in Sec. 10.3. 3D Coverage and Data Collection The above coverage of 2D k-space can be generalized to 3D by considering a series of planes leading to a three-dimensional set of parallel lines. The set can be obtained in 3D k-space by discrete sampling, or phase encoding, in both of the directions ky and kz .4 The imaging of three dimensions by the use of two phase encoding directions perpendicular to the read direction is called `3D imaging,' to contrast it with `multi-slice 2D imaging.' The di erence between the two approaches has to do with how the rf excitation is carried out. In the latter, a series of rf pulses de nes the series of slices lling out the third dimension. In the former, an rf pulse may be used to excite a `thicker' slice which is then phase encoded (or, as it is often called, `partition encoded') in the slice select gradient direction a series of such rf pulses is needed to run through all the partition encoding and phase encoding gradient values. (See Sec. 10.4 for further details about both approaches.) How do we change the gradients to achieve the discretized coverage in 3D imaging? Let the x-gradient de ne the read direction of the set of lines. The read sampling along the line may be carried out, as before, with measurements at nite time steps t during the continuous application of a gradient Gx. The associated step in the kx direction is kx = { Gx t (10.9) The orthogonal gradients, Gy and Gz , are turned o during the read sampling, in order to keep each line parallel to the x-axis. Before the read data are taken, the (ky kz ) position of each line is determined by applying the orthogonal gradients for, say, times y and z . After a given line has been sampled, an adjacent parallel line is approached by turning the orthogonal gradients back on, for the same times y or z with the same amplitudes, either Gy or Gz . The corresponding shifts in k-space are ky = { Gy y kz = { Gz z (10.10) Once again, to continue echoing, the kx = 0 point is traversed alternately either left-to-right or right-to-left while the read gradient oscillates from a positive to a negative value from one line to the next. The phases associated with the orthogonal, or phase encoding, gradients, Gy and Gz , are indeed xed along each line during the taking of data.
4 The relationships of both the spacings and the limitations on the number of steps to the degradation of the image quality are detailed in Chs. 12 and 13. 10.1. Imaging in More Dimensions 171 Fig. 10.4: A sequence diagram for imaging an entire object in 3D. The role played by each gradient in the experiment is described in the gure. The location of a sampled point in each direction of k- space is proportional to the area under the corresponding applied gradients between the rf excitation pulse and the sampled point (see Sec. 10.3). The ellipses denote the systematic repetition of the cycle of rf and variable gradient pulse amplitudes shown above required to cover k-space. In later chapters, we discuss the manner in which the transverse magnetization is suppressed or `spoiled' just before the start of each new cycle. 10.1.3 Time Constraints and Collecting Data over Multiple Cycles It is rare to collect all k-space points following a single rf excitation. Only a small number of lines of k-space can be collected after each rf excitation before the signal is lost due to T2 or T2 decay. In conventional imaging, often only one line of k-space data is collected following an rf excitation (the process is illustrated in Fig. 10.4). After a new rf pulse, the phase encoding gradients are incrementally increased/decreased with step sizes Gy and Gz appearing in (10.10), followed by the acquisition of another line of k-space data. This process is repeated every TR until all of the necessary k-space data are acquired. In terms 172 Chapter 10. Multi-Dimensional Imaging of a sequence diagram, the set of phase encoding gradient values corresponding to their incrementation as carried out over the multiple TR intervals is represented as a `table' (such as shown in Fig. 10.5c), with an arrow pointing in the direction of incrementation, which also indicates the direction of k-space traversal in the phase encoding direction. In this case, as suggested in Figs. 10.4 and 10.5, all kx lines are collected from left-to-right in contrast to the example in Fig. 10.3. The possible loss of signal due to transverse relaxation e ects is best discussed in terms of the `total acquisition time' for an MR experiment. Consider rst a 3D k-space data collection and let Ny and Nz denote the phase encoding steps for the two directions perpendicular to the read axis. It takes Ny Nz di erent repetitions (di erent lines) of the experiment for each unique pair of phase encoding gradient values (each line). The total time for a 3D imaging method is given by Tacq = Ny Nz TR (10.11) For a 2D imaging experiment, the total acquisition time is simply Tacq = Ny TR (10.12) See Prob. 10.1 for an example calculation of the total acquisition time for a 3D experiment. Given the large amount of data which needs to be calculated, it is understood why k-space coverage, at least for simpler MR systems, is often limited to one line at a time.5 For both 3D and 2D experiments, when data is collected Nacq times to improve signal-to-noise, the total MR imaging time is increased by a factor Nacq . Problem 10.1
Consider a uniform cubical sample of water. a) If the entire sample is excited, how long would it take to collect 64 64 lines (64 phase encoding points along each of the transverse axes, i.e., those axes perpendicular to the original `read' line of data), if TR = 20 ms? This is the procedure described as 3D volume imaging in Sec. 10.5. b) Based on part (a) and the values of T2 in Table 4.1, explain why k-space data is often collected only one line at a time (but see Ch. 19). 10.1.4 Variations in k-Space Coverage
5 Recall that phase encoding is the process of stepping to a speci c location in k-space in a plane de ned by two orthogonal dimensions before a line of data is acquired along the
When signal-to-noise and gradient capabilities are su cient, either multiple gradient echo or short-TR sequences can be used to shorten acquisition times. These concepts are discussed in detail in Chs. 18 and 19. 10.1. Imaging in More Dimensions 173 Fig. 10.5: (a) A typical frequency encoding read gradient structure where discrete time samples lead to discrete 1D k-space coverage along a read direction. (b) The coverage along a given phase encoding direction is carried out with one k-space step for each rf pulse. Di erent k-space samples are obtained by varying the amplitude of the gradient for each pulse repetition. The dashed line indicates that this `cycle' of the sequence structure is repeated. (c) A simpler representation of the entire sequence structure indicated in (b). The series of gradient amplitudes, and the direction of changing amplitude, are now represented by the stepped column or `phase encoding table' and its arrow. It is noted that either frequency encoding or phase encoding could be used for coverage of any k-space line, see Sec. 10.1.4. 174 Chapter 10. Multi-Dimensional Imaging read direction perpendicular to that plane. Whatever the scheme, the basic requirement of sampling the data is to visit each lattice point of the k-space volume-of-interest. The plane or set of lines in 2D or 3D imaging could be replaced, in principle, by an arbitrary trajectory through the k-space region of interest as long as the same points are sampled. A rich variety of rf and gradient combinations can be used to e ect the same coverage of k-space, as will be discovered throughout the text. In fact, there are alternatives for the coverage of a single line itself. Instead of the series of time steps t made during the application of a constant gradient G0 (Fig. 10.5a), the gradient strength may be stepped by an amount Gx to generate the same kx (Fig. 10.5b). The time the variable gradient is applied may be xed at x and a gradient table. This is exactly how the phase encoding shifts (10.10) are generated, and it would have been possible to introduce phase encoding by rst showing this as an alternative to frequency encoding of the line (see Prob. 10.2). A new rf excitation is required for each new gradient strength, however, and the time taken is increased thereby.6 Problem 10.2
We have introduced the concept of phase encoding as a separate means to image the spatial distribution of (~) orthogonal to the read gradient. However, it can r also be used to replace the read gradient. To accomplish this, assume that the 1D experiment is repeated Nx times with a repeat time TR . Use only the dephasing lobe of the read gradient, of length , (the readout portion is suppressed) and assume that only one point of the data is acquired at the end of this dephasing gradient. Assume that the amplitude of the gradient is changed from rf pulse to rf pulse by an amount Gx and sampling takes place at the same point in time after the gradient is applied. Only a single point would be sampled in this scenario. a) Describe how the amplitude of the dephasing lobe must be varied from one rf pulse to the next to get the equivalent coverage of k-space to that in Fig. 10.5a where kx = {Gx t
b) Speci cally show that (10.13) (10.14) Gx = Gx t
if the sampling in k-space is identical for each approach. 6 A third alternative is to increase the time x with Gx xed. The acquisition time is increased even more. 10.2. Slice Selection with Boxcar Excitations 175 10.2 Slice Selection with Boxcar Excitations
In general, magnetic resonance images are produced by exciting a single thin slice of the body by using a combination of gradient elds and `spatially selective' rf pulses. The slices are thinner for the 2D imaging methods. Radiofrequency pulses are spatially selective by having a nite region of support in the frequency content of the transmitted pulse, i.e., by having a nite bandwidth. The presence beforehand of a linear gradient means that only the slice corresponding to the region of support in the Larmor frequency domain is excited. The one-to-one correspondence of a given distance along the gradient direction to a particular Larmor frequency leads to the possibility of tuning the rf pulse frequency to excite a slice at a desired spatial location. 10.2.1 Slice Selection Let us de ne the `slice select axis' as the direction perpendicular to the plane of the desired slice. Accordingly, the gradient along this axis is de ned as the slice select gradient. Choosing the z-axis as the slice select axis leads to a transverse slice of the body. If the y-axis is chosen, the slice is referred to as coronal. Finally, if the x-axis is chosen, the slice is said to be sagittal. The nomenclature is exhibited in Fig. 10.6, and summarized in Table 10.1. The presence of a slice select gradient causes the frequency of precession to be a linear function of position along the corresponding slice select axis. The usual convention is to choose the z-direction as this axis (corresponding to transverse slices). The frequency at position z (in cycles per second) is f (z) = f0 + {Gz z (10.15) where f0 = {B0 is the Larmor precession frequency at z = 0. This linear relation between f (z) and z is illustrated in Fig. 10.7. The speci c goal is to excite uniformly a slice such that all spins in the slice have identical phase and ip angle after slice selection. To excite an in nitesimal slice through z0 , the rf pulse must be tuned to the frequency f (z0 ) given by (10.15). Since the frequency spread of a realistic rf pulse is bounded (i.e., bandlimited), a region of nite thickness along the z direction within the object would have its spins tipped, while spins outside this region would ideally remain aligned with B0 . To excite a slice of nite thickness extending from z0 ; z=2 to z0 + z=2, the rf pulse should have a frequency pro le, in the rotating frame, which is unity over the range f of frequencies from ( {Gz z0 ; {Gz z=2) to ( {Gz z0 + {Gz z=2) and zero outside (Fig. 10.8). The bandwidth BWrf of the rf pulse, i.e., the width f of its region-of-support in the frequency domain, is given by BWrf f (10.16) = ( {Gz z0 + {Gz z=2) ; ( {Gz z0 ; {Gz z=2) (10.17) = {Gz z (10.18) In summary, thanks to the presence of the gradient Gz , there is a range of frequencies which can be excited to create transverse magnetisation in a slice with thickness z orthogonal to 176 Chapter 10. Multi-Dimensional Imaging the z-axis. This range of frequencies of bandwidth BWrf is e ected by the electronics driving the transmit coil. It is convenient to introduce a notation for the slice thickness7 z TH (10.19) As a result, the slice thickness TH is a function of the bandwidth BWrf of the rf pulse and the applied gradient: TH = BWrf (10.20) {Gz Note that the various quantities de ned in the above discussion are illustrated in Figs. 10.7 and 10.8. In order to get a uniform ip angle across the slice, the analytic form of the rf excitation pro le, as a function of frequency, must be proportional to a boxcar function, rect(f= f ) of bandwidth f . This implies that the temporal envelope of the rf pulse B1 (t), which is the inverse Fourier transform of the frequency pro le (see Ch. 16), is a sinc function. Using the results of Prob. 9.1 in the time-frequency domain B1(t) / sinc( f t) (10.21) The sinc envelope corresponds to the amplitude modulation of the rf oscillations in the laboratory frame.8 The center frequency of the excited bandwidth is {Gz z0 in the Larmor rotating frame (Fig. 10.8) and f0 + {Gz z0 in the laboratory frame (Fig. 10.7). The former is in the audiofrequency (kHz) range while the latter frequency is in the radiofrequency (MHz) range in MRI applications. (Compare this comment to footnote 5 in Sec. 9.2.) It is observed that the rst zero crossing of the expression (10.21) occurs at a time t1 = 1= f . (The zeros of sinc(x) sin x=x are x = n for all nonzero integers n.) While the sinc function is ultimately damped to zero only at in nity, a realistic rf pulse is necessarily a time-truncated version of the sinc function, being on only for time rf , say. The number nzc of sinc zero crossings in this time is an important quantity in the design of an rf pulse. The total number of crossings is given by9 nzc = rf =t1 ] = f rf ] (10.22) where the square brackets ] are used to specify the operation `largest integer less than or equal to.' The larger the number of zero crossings, the closer the excitation pro le approaches the ideal boxcar. More discussion on the design of rf pulses is found in Ch. 16.
In the case of 2D imaging, the thickness of the imaging slice z and the excited slice TH are equivalent. However, in the case of 3D imaging, the thickness of the entire excited region TH is divided up into a series of e ective 2D images of thickness z . See Secs. 10.1 and 10.4. 8 The temporal envelopes in the sequence diagrams are shown henceforth as sinc functions they are sometimes called `soft pulses,' in contrast to the `hard pulses' described in earlier chapters. 9 To guarantee an even number of zero crossings, set n = 2 zc rf =2t1 ].
7 10.2. Slice Selection with Boxcar Excitations Applied slice select gradient Name Slice plane orientation Gx sagittal parallel to y-z plane Gy coronal parallel to x-z plane Gz transverse parallel to x-y plane 177 plane produced by it. The names have an anatomical basis. It is common in MRI to de ne these planes in terms of the magnet axes. Table 10.1: The relationship between the slice select gradient used and the orientation of the slice Problem 10.3
Suppose a slice select gradient Gz = 20 mT/m is used to excite a slice which is 10 mm thick. a) What is the required rf bandwidth? b) If Gz is reduced to 5 mT/m, but rf is unchanged, by what factor is the total number of zero crossings increased or decreased? Fig. 10.6: The orientation of the orthogonal slice planes correlated with the standard axes of a whole-body MRI magnet system. The subject is usually placed head rst into the magnet and carried in by a sliding gantry on the patient table. The feet point along the +^, left shoulder along z +^, and nose along +^ when the person is supine (lying on his/her back). x y 178 Chapter 10. Multi-Dimensional Imaging Fig. 10.7: The precession frequency (f ) in the laboratory frame as a function of position along the slice select axis when a gradient Gz is applied along the z -direction. The frequency bandwidth f BWrf (the shaded horizontal strip) is such that the slice region of thickness z (the shaded vertical strip) is symmetrically excited. Since the slice is o set from the origin in the z direction by z0 , the center frequency of the rf pulse is o set from the static Larmor frequency {B0 by {Gz z0
as indicated. Fig. 10.8: Excited magnetization (normalized) boxcar frequency pro le in the Larmor rotating
frame. Its temporal pro le is that of an ideal (in nitely long) sinc pulse. 10.2. Slice Selection with Boxcar Excitations 179 10.2.2 Gradient Rephasing After Slice Selection Any attempt to uniformly excite a slice faces the same signal-loss mechanism discussed in Sec. 9.4 concerning the gradient echo method. The slice select gradient induces dephasing in the slice select direction across the nite slice thickness. Moreover, the defocusing begins to occur during the excitation, since the slice select gradient is assumed to be turned on already by the time the rf pulse is initiated. (In comparison, all sequences treated in Ch. 9 had gradients that were not applied during the rf pulse.) As in the gradient echo 1D imaging case, when an echo is sought, the signal can be recovered by adding a rephasing gradient. To set the stage for a good rst approximation of the structure needed for the follow-up gradient, consider the slice to be excited instantaneously at t = 0 (see Fig. 10.9) in the midst of a constant gradient strength, Gz = Gss. The transverse magnetization at position z along the excited slice gains phase (z t) at the time t in the Larmor rotating frame, given by (9.13) (z t) = ; Gsszt (10.23) Since the rf pulse excites magnetization only across the slice of thickness z, the signal at time t follows the form (9.14) involving the (e ectively) one-dimensional spin density (z) de ned earlier in (9.7). The spin density can, in addition, be taken as constant over thin slices (but not the phase which is a function of z) yielding s(t) ' (z0 ) Z z0 + 2z
z0 ; 2
z ei (z t) dz (10.24) As time progresses, spins at di erent z positions accumulate di erent amounts of phase. The resulting decrease in the signal stems from the vanishing of the integral Z z0+ 2z
z0 ; 2
z ei (z t) dz dephasing
; ! 0 (10.25) which is an example of the general dephasing result (8.6). The dephasing occurs along the gradient direction (i.e., perpendicular to the slice) and the signal, in general, decreases with an increase in the slice thickness and/or gradient strength. Fig. 10.9: A slice select gradient to excite a slice parallel to the x-y plane. 180 Chapter 10. Multi-Dimensional Imaging The procedure to correct for this phase accumulation is to have the gradient reversed after the rf pulse is turned o , such that the spins realign in the transverse plane at the end of the reversed gradient lobe (see Fig. 10.9). (The simple relationship between these two gradient lobes is discussed below.) As a result, the excited spins in the slice are all at zero phase after the application of this rephasing lobe. The slice is now ready to be imaged. In the aforementioned approximation, where it is assumed that the spins are tipped instantaneously into the transverse plane at the center of the rf pulse (at t = 0), the dephasing takes place only during the second half of the slice select gradient applied during the rf pulse. The requirement on the rephasing-gradient structure is easy to nd. Recall from (9.34) that the total area under the gradients, from the time t = 0, will evolve to zero at the instant of refocusing. For a symmetric slice select gradient with jGz j = jGrephase j R dtG rephase R dtG = total area under Grephase = 50% total area under Gss ss (10.26) including the possibility of arbitrary time dependence for each gradient lobe. The signs of Gss and Grephase are assumed to be xed and opposite to each other. If, for example, they are both boxcar pulses, and are related by Grephase = ;2Gss, then the rephase gradient would have to be applied for only one quarter of the total length of time as the slice select gradient. See the forthcoming problem and gures for more examples. We shall return later in this book to the fact that the 50% result of (10.26) is only valid for small ip angles. In a more accurate calculation, the constraint on the rephasing gradient depends upon the details of the rf pulse. For the example of a slice selective =2-pulse with a sinc envelope, the ideal rephasing gradient should have an area which equals 52% of the area of the original slice select gradient. This is discussed further in Ch. 16. In practice, the amplitude of the rephasing gradient is varied until the maximum signal is obtained. Problem 10.4
Consider the excitation of a slice of thickness z=4 mm, by a bandlimited rf pulse on for a time duration rf = 4 ms. Assume that the ip angles of the spins are dominated by the exact (time) center of the rf pulse. If the applied gradient strength is given as 10 mT/m: a) Find the di erence in accumulated phase between two spins on opposite sides of the slice (i.e., at z = z0 z=2) at the end of the rf pulse. b) Show that, in general, = 0 for all spins at the end of the rephase gradient lobe, under the constraint (10.26). c) How long must the rephasing gradient be applied, for complete refocusing, if Grephase = ;Gss and the two gradients are boxcar pulses? What is the duration of the total slice select process, including the time that both Gss and Grephase are applied? 10.2. Slice Selection with Boxcar Excitations 181 Fig. 10.10: Slice selection at an arbitrary direction by a gradient vector perpendicular to the slice selection axis. Here the arbitrary gradient vector requires all three gradients Gx , Gy and Gz to be
used. The spherical angles of the gradient vector are the polar angle and the azimuthal angle . 10.2.3 Arbitrary Slice Orientation
An advantage of MRI is that an arbitrary slice direction can be selected without using moving parts or repositioning the body. The expression for the precessional frequency of spins in the presence of an arbitrary gradient vector was obtained in (9.50). Slice excitation, as shown in Fig. 10.7, is a frequency encoding process. When an arbitrary gradient vector is used in combination with a sinc pulse in the time domain, a slice perpendicular to that gradient vector is selected (see Fig. 10.10). The creation of an arbitrary gradient vector requires all three Cartesian gradients to be used simultaneously, and has been discussed already in Sec. 9.5.1. Their time dependence must be the same, to within an overall positive or negative proportionality constant. A sequence diagram corresponding to excitation of an arbitrary slice is presented in Fig. 10.11. The di erent weightings of the Cartesian gradients can be transcribed into spherical ~ ~ angles. If G = (G0x G0y G0z ), the slice select axis de ned by G is at the angle relative to the z-axis given by q 2 G0x + G2y 0 ;1 = tan (10.27) G0z Its projection into the x-y plane makes an angle relative to the x-axis where = tan;1 G0y G0x The angles are shown in Fig. 10.10. (10.28) 182 Chapter 10. Multi-Dimensional Imaging excite an arbitrary slice. The ratios of the di erent components are xed during the time the three gradients are applied. Fig. 10.11: Sequence diagram showing gradient waveforms (with rephasing lobes) required to It must also be remembered that the read and phase encoding directions are required to be orthogonal to the slice select direction. The Cartesian components for the associated ~ ~ ~ ~ gradients must therefore satisfy the orthogonality conditions GR Gss = GPE Gss = 0: The slice excited along an arbitrary direction is thus described by the same number of angles as that used for a rigid body (for example, the Euler angles). If the slice requires only one angle to x its orientation (say, one rotation from a standard z-slice con guration), it is called an `oblique slice.' If two are required, it is called `double oblique.' Images for conventional slice orientations and an oblique-slice orientation are shown in Fig. 10.12. Problem 10.5
Assume that the slice select gradient amplitudes are changed from those shown in Fig. 10.11 to be G0x = ;G0y = 2G0z . What are the angles and of the resulting slice? Draw the excited plane (use Fig. 10.10 as a guide). 10.2. Slice Selection with Boxcar Excitations 183 (a) (b) (c) (d) Fig. 10.12: Images of the human brain, obtained with a 2D gradient echo imaging sequence, in the three standard orientations along with an image for an oblique orientation. (a) Transverse image, (b) sagittal image, (c) coronal image and (d) an oblique plane obtained by tilting slightly (by 9 ) the slice select plane for a coronal image toward the transverse plane (by applying a small Gz component in addition to Gy for the slice select gradient). 184 Chapter 10. Multi-Dimensional Imaging 10.3 2D Imaging and k-Space
Two classes of typical 2D imaging sequences are described. The rst involves the =2-pulse gradient echo sequence, which is one of the simplest to analyze. Second, attention is turned to spin echo experiments. Coverage of k-space for both cases is given. In these examples, the convention will be to take the two imaging dimensions in the x-y plane, implying that the slice select gradient is along the z-axis. The phase encoding and read gradients are along the y-axis and x-axis, respectively. 10.3.1 Gradient Echo Example Figure 10.13 depicts a conventional 2D gradient echo experiment. A constant z-gradient is applied during the rf pulse for a total time rf , and then reversed immediately following the pulse. To rephase the spins in the slice, the requirement from (10.26) is that, if the amplitude Grephase of the follow-up lobe equals ;Gss, it must be on for a time rf =2. As required, the area under the rephase lobe then counterbalances half of the area under the slice select lobe10 (these areas are denoted by -B=2 and B , respectively, in the notation of the gure). At the end of the rephase lobe of the slice select gradient, all the transverse magnetization components within the slice are in phase, with a common accumulated phase value = 0. The signal from the whole slice, as de ned by a demodulated version of (9.4), is the zero phase slice integration Slice Selecting s( rf ) = ZZ dxdy "Z z0 + z 2
z0 ; 2z (x y z)dz # (10.29) For the given slice in z, the purpose of the next steps is to probe the two-dimensional dependence of the e ective 3D spin density (x y z) via gradients applied along the x- and y-directions. Phase Encoding the Data After the slice select gradient is turned o , it is followed by a phase encoding gradient (the second gradient labeled Gy PE in Fig. 10.13). Assume for the moment that no other gradient is applied during this period. The magnetization accumulates a y-dependent phase when a gradient is kept on in the y direction for a time PE = y , say. For a boxcar gradient Gy ( Gy PE ), a simple y-dependent adaptation of the phase (10.23) leads to a signal immediately following the phase encoding (see Fig. 10.13) given by s( rf + y ) =
10 Z "Z "Z z0+ 2z
z0 ; 2z dx (only Gy is applied during y ) (x y z) dz e # ;i2 {Gy y y dy # (10.30) Hereafter, the total gradient structure including the rephasing lobe is referred to as the slice select gradient. In this case, it is Gz (t) for all t. 10.3. 2D Imaging and k-Space 185 Fig. 10.13: Sequence diagram for a 2D gradient echo imaging sequence. The rf pulse is a sinc function in time, corresponding to a boxcar frequency spectrum. The design of the rephasing lobe of the slice select z -gradient is described in Sec. 10.2.2. The phase encoding y-gradient is pictured as a series of horizontal lines to denote that it is being stepped regularly through increasing values during di erent repetition periods. The x-gradient has the read gradient echo structure explained in Sec. 9.4. The symbols A B , and B=2 refer to areas under the gradient lobes, and important timings and gradient strengths are also marked. Note that in this 2D sequence structure no gradients overlap each other in time, in contradistinction to the structure shown in Fig. 10.14. 186 Chapter 10. Multi-Dimensional Imaging The encoded phase11 can be written in terms of the k-space spatial frequency associated with the y direction ky (Gy ) = {Gy y (10.31) The independent variable Gy will be varied in steplike fashion (step sizes GPE ), ultimately to gather information about the y dependence of the spin density. This will be a repeated experiment (see Figs. 10.5b and 10.5c) where y stays the same in every cycle. We return later to the manner in which Gy and, hence, ky , are stepped in value. To establish a ky value before the data are collected, the phase encoding gradient waveform should appear before the read lobe (i.e., the gradient lobe(s) applied during data sampling) of the third, or read, gradient Gx. It can, however, be superimposed on the dephasing lobe of the read gradient and/or on the rephasing lobe of the slice select gradient see the later discussion on the gradient superposition principle. Reading the Data
The third linear magnetic eld to be applied corresponds to the read gradient echo structure already detailed in the previous chapter. The gradient Gx has a negative dephasing lobe followed by the read (or rephasing) lobe (a previous example is found in Fig. 9.6 for the `constant gradient' or boxcar case), during which time the signal is measured. The position along the x-axis is encoded according to the 1D Fourier discussion in Ch. 9 that axis determines the so-called `frequency encoding' direction. In terms of the time-shifted variable t0 t ; TE , the phase in (9.48) can be augmented by the x-axis version of (10.30), to give an expression for the signal. (The magnetic elds, and hence the phases, can be superposed: see the discussion below.) For the boxcar gradient and timing parameters de ned in Fig. 10.13, s(t Gy ) =
0 Z "Z "Z z0+ 2z
z0 ; 2
z (x y z) dz e # ;i2 {Gy y y dy # e;i2 {Gxt x dx
0 ;Ts=2 < t0 < Ts=2 (10.32) where z0 refers to the center of the excited slice. The spatial frequency12 associated with the x-direction is kx(t0 ) = {Gxt0 (10.33) A change of independent variables in (10.32) leads to13 s(kx ky ) =
11 Z Z "Z z0+ 2z
z0 ; 2
z (x y z)dz e;i2 # (kx x+ky y) dxdy (10.34) Following earlier notation, the gradient Gy and the k-space variable ky used to describe the phase encoding direction are more generally referred to as GPE and kPE , respectively. 12 Also following earlier notation, the read gradient G and its k -space variable k are more generally x x referred to as GR and kR , respectively. 13 As before, the symbol s for signal is used in both cases for convenience. This use reminds us once again about the notational trap that (10.34) is not obtained from (10.32) by a strict substitution, for example, of kx for t. 10.3. 2D Imaging and k-Space 187 The integrated z-dependence is often suppressed, giving for a slice centered at z0 ZZ s(kx ky ) = (x y z0)e;i2 (kx x+ky y)dxdy (10.35) Of course, the integral over z in (10.34) reminds us that the 2D image contains a sum or projection over all information in the slice. It is evident in (10.32) that the signal obtained by measuring the received emf over a period of time Ts, in the presence of a gradient echo structure in the read direction after it is phase encoded by a xed value of Gy , gives a line in the k-space set of 2D Fourier transform values of the e ective 2D spin density for the selected slice. The phase encoding line in kspace is itself sampled at discrete points during the time interval Ts. Thus, stepped changes in Gy produce a series of sampled lines (see Fig. 10.15a), in a coverage of the 2D k-space pertaining to the 2D Fourier transform. The z-,y-,x-axes of Fig. 10.6 are the conventional choices for the slice select, phase encoding and read directions, respectively, for 2D imaging. Experiments may be performed with di erent axes for the di erent gradients. It is important to identify the gradient axes in the interpretation of the images produced, because artifacts14 are often formed in the image which are unique to each axis. For example, several artifacts associated with physiological motion are more pronounced along the phase encoding axis. The appearance and location of these artifacts can be altered by using a di erent choice of encoding axes (see Ch. 23). Problem 10.6
a) In both the slice select and read directions, it is often desirable to keep the gradients used prior to sampling the data as short as possible to reduce the e ects of motion. Neglecting the phase encoding time, how can this be accomplished while satisfying the conditions on the area of the slice select rephasing and the read dephasing gradients if Gss, GR, rf and Ts are xed for other reasons? Remember that you are constrained by a maximum possible gradient strength in each direction. b) Draw a sequence diagram showing the shortest possible echo time, assuming that the slice select and read gradients used are 0:1Gmax, where Gmax is the maximum available gradient strength for either gradient. Label the magnitudes and duration of all gradients used in terms of rf , Ts and Gmax . Superposition of Phase E ects
It was noted earlier that the phase encoding gradient and the dephasing lobe of the read gradient in the 2D gradient echo sequence shown in Fig. 10.13 can be applied simultaneously.
An artifact is de ned as a false feature in the image. It is created by some imperfect process in the data collection or in the inverse Fourier transform. See Chs. 11 and 12.
14 188 Chapter 10. Multi-Dimensional Imaging The rephasing lobe of the slice select gradient can also be applied at the same time. In examples of sequences to follow this discussion, all these three lobes are generally shown to coincide in time with each other (see, for example, Fig. 10.14 where all these three lobes are all turned on simultaneously.) How is it that we can have these gradients on simultaneously before the data are read out, given that they act as a single e ective gradient vector if applied during slice selection or during data sampling? The answer is that the phase accumulation for any superposition of contributions to the z-component of the magnetic eld, referring back to the general precession solutions of Chs. 2 and 3, is just the sum of the phases for each contribution. Any additional eld just adds a term to the exponent in solutions such as those in (2.41) including the generalizations to time dependent elds and the associated phase integrals in (3.37). This superposition principle for the phase has already been used, in e ect, in the discussions for the construction of arbitrary gradient directions in Secs. 9.5 and 10.2.3. The point now is that the gradient components must have di erent time dependence as well. In the typical 2D sequence, such as in Fig. 10.14, the phase along each direction serves a di erent purpose. The terms along x and y encode the data to extract information in the ^ ^ excited plane, while the gradient lobe in the z direction serves the purpose of completing the ^ slice selection process. The processes just described are independent of the occurrences in the other directions. The only exceptions to this rule of independence of action are during the time periods of slice selection and the data measurement (where there is rf excitation or signal detection). In these two cases, the presence of additional gradients would lead to a rotation of the encoded spatial axis (just as the slice select gradient axis is changed by adding other gradient components). k-Space Coverage A su ciently large set of kx and ky values is required in order to invert the 2D Fourier transform (10.32) and accurately reconstruct the image of the spin density. A measure of how large this set must be is the subject of Ch. 12. Presently, the discussion is focused on the procedure for covering a given set of k-space data. The k-space coverages associated with the sequences shown in Figs. 10.13 and 10.14 are pictured in Fig. 10.15. The initial phase, before the application of the phase encoding and read gradients, corresponds to the 2D origin, kx = ky = 0 for every repeat cycle. (Any initial phase from the slice select gradient is rephased to zero by its rephasing lobe.) The combination of the rst (dephasing) lobe of the read gradient and a particular step of the phase encoding gradient lobe is applied to locate the point (kR min ky (Gy )) in k-space, where ky (Gy ) is given in (10.31) and kR min = ; {GxTs=2 (10.36) This is represented by the long dark arrows which are right-angled in Fig. 10.15a where the read gradient dephasing lobe follows the phase encoding lobe and which are diagonal in Fig. 10.15b when these lobes overlap. The second, positive part of the read gradient lobe moves the signal through a set of (Nx, say) steps while data are collected (the series of short sequential horizontal arrows in Fig. 10.15). Multiple lines in the phase encoding gradient indicate the stepped changes (Ny in all, say) for each repetition of the experiment. Each time this gradient is altered, a new value of ky is chosen (the long-dashed dark arrows), and 10.3. 2D Imaging and k-Space 189 Fig. 10.14: A 2D gradient echo sequence that is more typical than that of Fig. 10.13. The rephase lobe of the slice select gradient, the dephase lobe of the read gradient and the phase encoding gradient table are all switched on at the same time. This shortens the echo time, a desirable feature in most applications. 190 Chapter 10. Multi-Dimensional Imaging a new set of data is sampled along kx. The complete path that a sequence takes through k-space is often referred to as its k-space trajectory. The sizes of the vertical and horizontal steps taken in k-space are considered next. Fig. 10.15: Traversal of k-space for typical gradient echo experiments. The signal may be considered to be initialized at kx = ky = 0 with any accumulated phase due to the slice select gradient reset to zero by its rephasing lobe. (a) For the sequence in Fig. 10.13, the phase encoding gradient and the dephasing lobe of the read gradient are applied at di erent times. After each new rf pulse, the k-space trajectory would rst move vertically along the ky direction (dark solid arrow and dark small dashed arrows) to the current phase encoded spatial frequency over the period of time the phase encoding gradient lobe is applied. This is then followed by a horizontal traversal to kR min while the read dephasing lobe is on (lightened arrows). (b) For the sequence shown in Fig. 10.14, the combined e ect of the phase encoding lobe, at a given (stepped) Gy value, and the dephasing read gradient lobe is to move the signal to a chosen k-space value moving along a diagonal trajectory where both kx and ky change simultaneously as a function of time (long bold arrow). The read gradient is then reversed to bring the spins back through a complete set of kx values (short bold arrows) leaving ky unaltered during data sampling. The experiment is repeated by returning to a new starting ky value (long-dashed dark arrows) to obtain a complete set of data (lightened arrows). The e ects of the slice select rephasing gradient (the third dimension) are not included in the 2D k-space description. Phase Encoding Order The k-space coverage for the 2D imaging sequences has been found in the above discussion to be represented by Ny horizontal lines of 2D k-space each with Nx sampled points. Practically, there is a desire to have Ny an even number in `fast Fourier transform' analysis has been noted (i.e., Ny 2ny ), so that the lines are typically indexed in a minimally asymmetric manner 10.3. 2D Imaging and k-Space 191 from m = ;ny to m = (ny ; 1) (see Chs. 11 and 12). The value of Gy is incremented by a xed amount Gy at each step, such that Gy min = ;ny Gy and Gy max = (ny ; 1) Gy .15 At the step corresponding to index m, the y-gradient is Gy (m) = m Gy . This particular stepping order is referred to as a sequential acquisition or `sequential ordering' of the phase encoding steps. Recall that the arrow on the sequence representation, or gradient table, of the phase encoding, such as in Fig. 10.13, indicates the direction of the steps taken in the phase encoding k-space direction. An arrow pointing upward represents k-space coverage from ky min = {Gy min y kPE min to ky max = {Gy max y kPE max while a downwardpointing arrow would have indicated the opposite coverage. Although sequential ordering is the most commonly used ordering of phase encoding steps, there are two other ordering schemes that are used occasionally (see Chs. 18 and 19 for discussions of the circumstances under which they are advantageous). The rst is called a `centric reordering' scheme where the kPE = 0 line is collected rst, then followed by kPE = ; ky , kPE = ky , kPE = ;2 ky , kPE = 2 ky , and so on outward. An expression for ky is found below. The other scheme is a reordering such that the lines in k-space are collected starting from the outermost lines, alternating between positive and negative values, and moving inward until the kPE = 0 line is collected last. This is referred to as `reverse centric' coverage and leads to what can be thought of as a high pass lter since it enhances the larger spatial frequencies. These three phase encoding ordering schemes are illustrated in Fig. 10.16. kx-ky Coverage and Time In the archetypal sequential acquisition schemes shown in Fig. 10.14, the neighboring lines of data,16 each of which are at a constant kPE , are separated by
(10.37) { GPE PE The position of the k-space samples in the read direction, on the other hand, is determined by the sampling rate rather than a change in gradient strength. If the data are sampled in time steps t, then the associated k-space sampling interval in the read direction is
y ky kPE = { Gy (10.38) {GR t The signal is measured over Nx NR steps for a given value of m in the gradient table. Each new rf excitation is performed a time TR after the previous excitation, until k-space has been adequately sampled along both directions. How sparse this sampling can be (that is, how large t and GPE can be) is dictated by the Nyquist criterion, which is discussed in detail in Ch. 12. How far out k-space is sampled is de ned by kR min, kR max, kPE min and kPE max, and these boundaries determine the spatial resolution (see Chs. 12 and 13). Although a vertical pair of neighboring sampled points may be separated by an identical distance in k-space as for a neighboring horizontal pair ( kx = ky ), they may be very much separated in their times of acquisition. In the present case, nearest neighbors on the
Thus, for large ny , k-space coverage is approximately symmetric about Gy = 0. We refer to di erent phase encoding `lines' as the data collected for a given ky value. These are also referred to as phase encoding `views.'
15 16 kx kR = {Gx t 192 Chapter 10. Multi-Dimensional Imaging x-axis are separated by t, but adjacent points along the y-axis will be separated by a time interval of TR per acquisition. Fig. 10.16: Di erent ordering schemes for the phase encoding gradient table. (a) A sequential ordering scheme showing the acquisition of a linearly increasing phase encoding line number as a function of rf pulse number. (b) A centric reordering scheme where the rst gradient amplitude is zero, the second G, the third ; G, the fourth 2 G, and so on. (c) A reverse-centric reordering scheme where the gradient settings begin at the maximum magnitudes and work inward. The arrows shown indicate the time direction of the collection of the phase encoding steps. 10.3. 2D Imaging and k-Space 193 Alternate Phase Encoding Scheme
In the sequential phase encoding scheme discussed above, the phase encoding gradient is increased through a table of values starting from a negative maximum to a positive maximum by steps of Gy . It has been assumed that a pulsed gradient waveform with a variable amplitude Gy is switched on for a time y , so that ky (Gy ) = {Gy y . The same phase encoding could have been done by using a constant gradient value G0 (say) while varying y the time ty for each given phase encoding step. Then ky (ty ) = {G0 ty . While of historical y interest,17 this scheme leads to longer and longer echo times and hence an increase in the minimum-achievable TR value. However, for systems which do not have variable gradient strengths, this is an alternative method for phase encoding, and o ers the same k-space coverage. Fig. 10.17: Sequence diagram for a 2D spin echo imaging sequence. The arrow pointing downwards in the gradient table, indicating that the phase encoding gradient is stepped sequentially from its most positive value to its most negative value, has the same e ect as an upward arrow in a sequence without a -pulse. There is no rephasing gradient required for the -pulse as a symmetric slice select gradient is self refocusing. 10.3.2 Spin Echo Example The sequence diagram for a conventional 2D spin echo imaging experiment is shown in Fig. 10.17. As discussed in Sec. 9.4.2, there are two options for doing the 1D spin echo imaging experiment based on the placement of the readout gradient dephasing lobe. There are also two options for the phase encoding gradient in the 2D spin echo imaging sequence, corresponding to placing it either before or after the -pulse. The phase encoding gradient is placed before the -pulse in the example in the gure. While it can be shown that the full
17 See the reference to Kumar, Welti, and Ernst in the suggested reading. 194 Chapter 10. Multi-Dimensional Imaging 2D k-space is covered in both cases, there is a di erence in how this space is covered. This is the subject of Prob. 10.7. The practical implications of these two phase encoding choices are discussed in Chs. 23 and 26. Problem 10.7 Assume that the phase is de ned such that a -pulse reverses the accumulated phase of all spins: ! ; . In terms of k-space, the -pulse is equivalent to a re ection through the origin: ~ ! ;~ . k k a) Show that the k-space diagram for the spin echo experiment shown in Fig. 10.17 is identical to that in Fig. 10.15b which corresponds to the gradient echo sequence shown in Fig. 10.14. b) Find the k-space diagram for the spin echo experiment shown in Fig. 10.17 if the phase encoding gradient table is applied after the -pulse, but before the read gradient. (The dephasing read gradient lobe is unchanged.) 10.4 3D Volume Imaging
More often than not, information is required over a three-dimensional volume for imaging. In Sec. 10.1, we have introduced the two approaches to be described here. One involves 3D methods with short TR values (see Ch. 18 for a discussion of fast imaging). The other is the multi-slice 2D method where the 2D imaging techniques of the previous section are generalized to a series of slices for the coverage of a 3D volume. 10.4.1 Short-TR 3D Gradient Echo Imaging
The conventional 3D imaging sequence is created from a 2D imaging sequence by adding a gradient table in order to phase encode along the slice select direction as well. An example is shown for a gradient echo sequence in Fig. 10.18. As mentioned, the process of phase encoding the slice select direction is often called partition encoding. The excited slice is usually called a slab (with thickness TH ) and the reconstructed multiple slices (with thickness z = TH=Nz according to the Nyquist discussions in Chs. 12 and 13) are known as partitions. Each new rf pulse excites the same slab, but is followed by a di erent phase encoding gradient setting. What are the possible advantages to collecting data with a 3D volume method, in comparison with multi-slice imaging? First, the ability to change the number of the Nz phase encoding steps over the slab means that there is control over the smallness of the partition thickness z = TH=Nz without any limitation on the rf amplitude or duration. Second, consecutive slices can be adjacent, in contrast to 2D imaging where the rf pulse leaks into the neighboring slices and there is always a slice gap in practice. Third, larger rf bandwidths can be used for thicker slabs in 3D imaging, and the rf pulse time rf can be shortened, making 10.4. 3D Volume Imaging 195 it possible to reduce TE . Fourth, both short TE and the high spatial resolution achievable in the partition encoding direction help reduce signal loss due to T2 dephasing (see Ch. 20 for a detailed discussion). Fifth, the signal-to-noise ratio can be enhanced even for thin slices because of the parameters available in 3D imaging (see Ch. 15 for the dependence of SNR on TR and Nz ), but achieving this may come at the expense of increased imaging time. Fig. 10.18: A 3D gradient echo imaging sequence where the two directions orthogonal to the read
direction are phase encoded. 10.4.2 Multi-Slice 2D Imaging
The multi-slice 2D approach for the coverage of a 3D volume-of-interest (VOI) is accomplished by the application of a number of rf pulses within a single repeat time. Each rf pulse is centered at a di erent frequency and excites a di erent slice. In an original single rf pulse short-TE , intermediate-TR sequence, the additional rf pulses can be applied during the idle time between the end of data sampling and the rest of the cycle time TR to excite other slices in the VOI. 196 Chapter 10. Multi-Dimensional Imaging Because of imperfect rf pro les, however, the immediate neighborhood of an excited slice is also partly excited. Hence, this region cannot be included in the following slice, if the spins do not have time to recover toward equilibrium. In multi-slice methods, it is common either to leave a gap between slices so that contiguous coverage is not obtained, or to excite all the odd-numbered slices rst and then all the even-numbered slices afterwards (the slices are said to be `interleaved' in the odd-even procedure), in order to avoid the `slice crosstalk' e ect (the alteration of signal in one slice caused by overlapping rf pulses from an adjacent slice). Figure 10.19 exhibits the sequence diagram and frequency encoding graph for leaving gaps between slices. In this gure, !d represents the frequency associated with a slice gap (d) such that !d < !slice. Speci cally, d = !d= Gz = !slice ; 2 BW G
z (10.39) Although the multi-slice 2D method appears to be rather time consuming (ine cient) for large TR , scans with longer repeat times can be made more e cient through the usage of multiple gradient echoes and spin echoes (see Ch. 19). Very long values of TR with a reduced number of repetitions and reduced scan time are possible see the discussion of segmented k-space coverage in Ch. 26. For comparison, a set of images obtained with a 2D spin echo sequence and a set of images with a 3D short-TR gradient echo sequence are shown in Fig. 10.20. Although comparable volume coverage was obtained (with the 2D method) for one image orientation in about one third the time of the 3D imaging experiment, the total acquisition time for obtaining 2D images in all three orientations would be about the same as the 3D imaging time. While all three slice orientations can be reconstructed from a single 3D data set, the resulting image quality may su er if the resolution in each direction is not the same. The image quality is reduced by the interpolation required to view the 3D data from di erent orientations. 10.5 Chemical Shift Imaging
In previous discussions in this chapter, the chemical shift e ect has been ignored. The MRI signal for living tissue is largely made up of 1H in water, and the e ects on the image from other hydrogen nuclei are typically small. An important exception to this assumption, however, is fat (CH2 based tissue elements), which is chemically shifted from H2O and can have a signi cant signal associated with it. The encoding of measurable contributions from di erent species such as water, fat, choline, creatine, n-acetyl aspartate (NAA) and lactate, leads to an additional dimension for imaging.18
Studies in the biomedical sciences are addressed to other chemical species contributing to the 1 H spectrum in humans. For example, one of the species seen in a hydrogen spectrum is a lactate spectral peak an elevated lactate content in some location in the human body points to an ine cient and typically anomalous anaerobic glucose metabolism for energy production as opposed to the more e cient and typically normal aerobic glucose metabolism.
18 10.5. Chemical Shift Imaging 197 (a) (b)
Fig. 10.19: (a) A multi-slice imaging experiment where a number of slices are collected during each TR by stepping through a series rf pulse frequencies. (b) The eld-frequency plot where the
center frequency of the rf pulse is varied from one slice to the next in the presence of the slice select gradient. To insert a gap between two adjacent slices, the quantity !d = !slice ; 2 BW must be positive, in order to avoid `interslice crosstalk.' The frequency bandwidth is BW for all rf pulses. 198 Chapter 10. Multi-Dimensional Imaging (a) (b) (c) (d) 10.5. Chemical Shift Imaging 199 (e) (f) (g) (h) Fig. 10.20: Images acquired using a 2D spin echo sequence in (a) the transverse plane, (c) the sagittal plane, (e) the coronal plane, and (g) a plane obtained such that the selected slice is slightly rotated from the coronal plane into the transverse plane (see Fig. 10.12). Images of the same slices (b), (d), (f) and (h) reconstructed from a 3D whole-head imaging experiment with a short-TR , 3D gradient echo sequence. Image (b) is slightly o set from (a) due to subject motion. Note the reasonably similar tissue di erentiation in both sets of images. The signal from the top of the head is reduced in the spin echo case relative to the gradient echo because of a drop-o in rf excitation sensitivity. This e ect is discussed later in Ch. 18. 200 Chapter 10. Multi-Dimensional Imaging Chemical shift imaging refers to the process of selectively imaging (or obtaining the spatial distribution of) identical nuclei that experience di erent levels of magnetic shielding due to their chemical environments. These shielding e ects cause chemical shifts which were brie y introduced in Sec. 8.5. There are a number of methodologies for performing a spectrally selective MRI sequence, including `selective excitation,' `selective saturation,' and using the chemical shift to generate additional frequency or phase encoding of the spins. The last method is detailed below in terms of an added imaging dimension. Other methods for chemical shift imaging are discussed in Ch. 17. We noted in Sec. 8.5 that di erences in magnetic shielding cause nuclei in various chemical compounds to precess at slightly di erent rates. It is possible to use this frequency variation in a manner similar to that employed by frequency encoding gradients. From (8.60), the shift in the frequency due to the chemical shift is ! ( ) = ; B0 = ; !0 (10.40) Hence, a chemical shift frequency encoding `axis' can be de ned in terms of = ; !=!0, which is typically quoted in ppm. This creates another e ective imaging axis,19 leading to, for example, a 4D space (or 3D) made up of three spatial dimensions x, y, z, (or two spatial dimensions x, y) and a spectral dimension . In the corresponding expression for the signal, the chemical shift phase e ects are to be added to those due to a general set of applied gradients leading altogether to a 4D Fourier transform for 3D spatial volume imaging, for example. Assuming a perfectly uniform B0 , the generalization of (9.15) is s(~ t) = k Z d3r d (~ )e;i2 r (~ ~; f0 t) kr (10.41) in terms of the Larmor frequency f0 . The signal is now expressed in terms of an additional Fourier transform over the population of chemically shifted species evaluated at the point in the dimensionless space given by f0 t. In order to image along the `chemical shift' dimension, some modi cations are made in the previous sequences. In typical chemical shift imaging, data are read along the time axis which means that the transverse magnetization is allowed to evolve over time under the in uence of the chemical shifts while data is measured. That is, each spatial axis is treated as a phase encoding axis with gradients used to locate a position in 2D or 3D k-space before data are taken. A line of chemical shift data is collected at a speci c point in k-space during each repetition of the experiment. A sequence that could be utilized for the phase encoding of a 2D k-space in the chemical shift data collection is shown in Fig. 10.21. Note that no imaging gradients are applied during the time data would be measured around the echo. 10.5.1 A 2D-Spatial 1D-Spectral Method
19 Consider how a set of 2D images of di erent chemical species might be found from the 2D chemical shift imaging (2D CSI) experiment of Fig. 10.21. A simple illustration of the associated 2D k-space data from such an experiment is given in Fig. 10.22, while the
It is evident, however, in (10.40) that is a dimensionless quantity. 10.5. Chemical Shift Imaging 201 Fig. 10.21: A 2D spin echo chemical shift imaging sequence for the acquisition of a chemical shift
spectrum. corresponding 2D-spatial, 1D-spectral representation might look like that in Fig. 10.23. The set of images for the di erent chemical species generated is suggested by the stack of cartoons in Fig. 10.24. In more detail and following the previous discussions, a thin slice of the body is chosen with a combination of rf pulses and gradients, leading to an e ective 2D spin density. For this chemical shift experiment, each point in 2D k-space is encoded with two phase encoding tables, one in x and one in y. After spatial phase encoding, the data are sampled at n points each separated in time by t to obtain information along the chemical shift axis. A 3D Fourier inverse transform of the collected data leads to an image whose third axis contains information about di erent chemical species. By `slicing' the created 3D image set, it is possible to produce a set of images, each corresponding to a particular value of (Fig. 10.24). In this manner, it is possible to obtain separate images of water, fat, lactate, and so forth, showing their respective spatial distributions in the body. The presence of eld inhomogeneities complicates the process, and methods to deal with this in the simpli ed twospecies case of extracting water and fat images are discussed in Ch. 17. Generalizations to the multi-species case are also considered in that chapter. As in the previous formula development, the 2D CSI signal can be conveniently written as a function of t0 , the time de ned about the echo. It is given by s(kx ky t ) = d
0 Z Z dx dy (x y )e;i2 (kx x+ky y; f0 t )
0 (10.42) A 3D inverse Fourier transformation of the signal thus yields an image ^(x y ) over a slice de ned by slice selection in the z-direction. The similarity between this experiment and the 202 Chapter 10. Multi-Dimensional Imaging Fig. 10.22: The real part of the signal centered around the spin echo shown as a function of time for each sampled k-space point for a boxcar chemical shift distribution (i.e., a sample where the di erent spectral components are present in equal amounts). The frequency content of the signal at each point contains chemical shift information, as well as the spatial information encoded by the gradients. The latter is represented by the variation (reduction) in the spin echo signal observed with an increase in distance from the origin of k-space. Fig. 10.23: A simple illustration of an example of a chemical shift spectrum collected at every point in 2D x-space. Note that the spatially varying spectral distribution is illustrated by the varying
shape and strength of peaks in the spectra shown for each voxel. 10.5. Chemical Shift Imaging 203 Fig. 10.24: The layers of images obtained with the chemical shift techniques described in
Sec. 10.5.1. The image is sliced along the chemical shift axis to provide images of the di erent chemically shifted nuclei in the sample. For example, one can think of creating a water image, a lactate image, an NAA (n-acetyl aspartate) image and a fat image. standard spatial 3D imaging is re ected by the formula for the total 2D CSI acquisition time which is the same as (10.11) with Nz replaced by Nx
Tacq = NxNy TR (10.43) Problem 10.8
a) Show that the information collected at each time point prior to Fourier transformation of the signal along t0 in (10.42) already provides a useful 2D image at t0 = 0. b) What changes in the image as t0 deviates from zero? The 2D CSI experiment could also be performed using an FID rather than a spin echo to collect the data. Such an idea is attractive at rst sight since it is possible to begin collecting data immediately after the phase encoding gradients, leading to shortened TR values and reduced imaging time. As mentioned in Ch. 8, an FID signal contains an adequate amount of information to obtain spectral information. There are a number of technical problems, however, which are primarily associated with the di culty in collecting data about the t = 0 point. The absence of this data can lead to baseline or low frequency signal loss errors in the reconstructed images. These associated problems are eliminated in the spin echo case, 204 Chapter 10. Multi-Dimensional Imaging where the chemical species come together in phase at the echo, de ning the total integrated (over all chemical species) spin density information correctly for each voxel. Problem 10.9
Given the following imaging parameters, determine Tacq in each case. a) Find Tacq for a 2D spatial imaging experiment with Ny = 256 and TR = 1000 ms. b) Find Tacq for a 2D spatial 1D spectral CSI experiment with Nx = Ny = 256 and TR = 1000 ms. c) Find Tacq for a 3D spatial imaging experiment with Nx = Ny = 256, Nz = 128, and TR = 1000 ms. d) Find Tacq for a 4D imaging experiment with Nx = Ny = Nz = 16 and TR = 1000 ms. e) Discuss the implications of your results to the imaging of humans. Assume it is di cult for an average patient to stay in the imaging environment (lying still inside the bore of a magnet) for more than 30 minutes without becoming uncomfortable. Usually 10 minutes is assumed to be an upper limit for a single MRI scan. Assume also that you are imaging the entire human head which requires about 19.2 cm (left-to-right) 25.6 cm (head-to-foot with oversampling) 22.4 cm (front-to-back) of total spatial coverage. Given that the spatial resolution, as de ned later in Ch. 13, is the ratio of FOV to number of encoded points, discuss the trade-o in spatial resolution versus imaging time in going from part (c) to part (d). It is apparent that 4D imaging is also possible, but with a signi cantly longer imaging time. All three spatial directions can be phase encoded before data are read along the -direction. The 3D CSI signal for data acquisition about the center of a spin echo in terms of t0 for each read period is 10.5.2 A 3D-Spatial, 1D-Spectral Method
Z Z s(kx ky kz t0 ) = d d3r (x y z )e;i2 (kx x+ky y+kz z ; f0 t )
0 (10.44) The total acquisition time when all three axes are phase encoded is given by Tacq = NxNy Nz TR (10.45) The extra factor of Nz in comparison with (10.43) makes 4D imaging impractical for human imaging in most situations. It is, nevertheless, a powerful method for imaging and spectroscopy of inanimate samples and sedated animals at high elds where times on the order 10.5. Chemical Shift Imaging 205 of hours are taken to perform the experiments to obtain high SNR. On humans, it would be necessary to sacri ce spatial resolution or spatial coverage for reduced imaging time and reasonable SNR, when all four dimensions need to be encoded. 206 Chapter 10. Multi-Dimensional Imaging Suggested Reading
The current approach to phase encoding appears in: J. M. Hutchison, R. J. Sutherland and J. R. Mallard. NMR imaging: image recovery under magnetic elds with large nonuniformities. J. Phys. E.: Scient. Instrum., 11: 217, 1978. The `alternate phase encoding scheme' for Fourier imaging was introduced in the following paper: A. Kumar, D. Welti and R. R. Ernst. NMR Fourier Zeugmatography. J. Magn. Reson., 18: 69, 1975. The concepts of k-space are introduced in: S. Ljunggren. A simple graphical representation of Fourier-based imaging methods. J. Magn. Reson., 54: 338, 1983. D. B. Twieg. The k-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods. Med. Phys., 10: 610, 1983. An introduction to chemical shift imaging can be found in: T. R. Brown, B. M. Kincaid and K. Ugurbil. NMR chemical shift imaging in three dimensions. Proc. Natl. Acad. of Sciences, 79: 3523, 1982. R. E. Sepponen, J. T. Sipponen and J. I. Tanttu. A method for chemical shift imaging: demonstration of bone marrow involvement with proton chemical shift imaging. J. Comput. Assist. Tomogr., 8: 858, 1984. Chapter 11 The Continuous and Discrete Fourier Transforms
Chapter Contents
11.1 The Continuous Fourier Transform 11.2 Continuous Transform Properties and Phase Imaging 11.3 Fourier Transform Pairs 11.4 The Discrete Fourier Transform 11.5 Discrete Transform Properties Summary: Continuous and discrete Fourier transforms pertinent to MRI are detailed. A set of tables describing various Fourier transform pairs and general properties is given. Features of imaging that are directly related to the properties are described and analyzed. Phase imaging is introduced. Although the discussions in this chapter are addressed to 1D transforms, the results are easily generalized to higher dimensions. Introduction
An important connection between the MR signal and the continuous Fourier transform has been demonstrated in the previous two chapters. In the absence of relaxation e ects, the signal s(~ ) was shown to be the Fourier transform of the spin density (~), so an inverse k r Fourier transform performed on the data leads to a reconstructed spatial image ^(~). The r caret sign over the indicates that this represents an estimate of the e ective spin density. To better understand this transformation and its numerical approximations, basic properties of the continuous and discrete Fourier transforms are studied in the present chapter. Descriptions of certain experimental imaging results in terms of the transform properties and theorems are included, particularly with respect to artifacts in the data. Certain artifacts are connected with the fact that ^ may be complex, and its phase leads to the concept of `phase imaging.' 207 208 Chapter 11. The Continuous and Discrete Fourier Transforms The de nitions and properties of the continuous transform are contained in the rst and second sections, respectively. Properties refer to theorems, symmetries, and convolution, and the experimental relevance of some of these properties is also a subject of the second section. Among the Fourier transform pairs (functions and their transforms) of importance to MRI, and discussed in the third section, is the sampling function and its transform, both of which are `comb' functions. The de nitions and properties of the discrete transform round out the chapter in the last two sections. 11.1 The Continuous Fourier Transform
The primary function of the Fourier transform is to map from position space to its `conjugate' space, and vice versa through the inverse transform. The ability to represent a function in the two domains is important in many applications. In the case of MRI, the one-dimensional imaging problem leads to the Fourier integral s(k) = Z1 ;1 dx (x)e;i2 kx (11.1) as shown in (9.15). The spin density in x-space (position space) is transformed into its associated k-space (spatial-frequency space) counterpart, which is directly related to the signal. In the following, several basic Fourier transform concepts and properties are presented, with only a little attention paid to general theory. The emphasis will be on useful formulas for the evaluation of (11.1). Although only one-dimensional transforms are considered in what follows, 2D and 3D transforms are straightforward generalizations. In a more general notation,1 the 1D transform can be written2 H (k) F (h(x)) =
along with its inverse
;1 Z1 ;1 dx h(x)e;i2 kx (11.2) (11.3) h(x) F (H (k)) = Z1
;1 dk H (k)e+i2 kx The consistency of the pair of equations (11.2) and (11.3) can be shown using the Dirac delta function (or impulse function) introduced in Ch. 9. The delta functions in either space (x-space and k-space) have the following representations (cf. (9.24)) (k ; k0) = (x ; x0 ) =
1 Z1 Z ;1 1 ;1 dx e;i2 dk e+i2 (k;k0 )x (11.4) (11.5) k(x;x0 ) The inclusion of the 2 factors in the exponentials, a convention already employed in the imaging p equations, eliminates the need for 1=2 (or 1= 2 ) normalization coe cients outside of one or the other (or both) Fourier integrals. 2 A lower-case letter is generally used to denote a function in one domain, and its transform is denoted by the corresponding upper-case letter. This convention has not been followed in the connection between (x) and s(k). 11.2. Continuous Transform Properties and Phase Imaging Insertion of (11.2) into (11.3) yields3 209 h(x) = Z1 Z;1 1 dx0 h(x0 ) Z1
;1 dk e;i2 k(x ;x)
0 = dx0 h(x0 ) (x ; x0 ) ;1 = h(x) (11.6) The inverse Fourier transform can now be put to work. If the signal s(k) were known for all k, (11.1) could be inverted to give (x) = Z1 ;1 dk s(k)ei2 kx (11.7) In fact, as has been emphasized in various portions of the past three chapters, it has been emphasized that the signal can only be nitely sampled, corresponding to the so-called measured signal distribution sm(k).4 The (continuous) inverse Fourier transform of sm yields the reconstructed image Z1 ^(x) = dk sm (k)ei2 kx (11.8) The di erences between ^(x) and (x) comprise an important topic recurring throughout this book and especially in Chs. 12-15.
;1 Problem 11.1
a) Write the Cartesian 2D forms of (11.2) and (11.3). b) If s(kx ky ) = 0 AB sinc( kxA) sinc( ky B ), describe the object (x y) that produced this signal. (See Ch. 9 or Table 11.3.) An illustration of this 2D Fourier pair is in Figs. 11.1a and 11.1b. 11.2 Continuous Transform Properties and Phase Imaging
It is necessary to study the general properties of the continuous Fourier transform in order to understand the di erences between the reconstructed MR image ^(x) and the physical spin density (x). Besides the discrete nature of the sampling, di erences between (x) and ^(x) can be the result of errors in data acquisition and signal processing. The term artifact
It is assumed throughout that for the physical applications considered the mathematical steps employed, e.g., changing order of integration, are valid. In particular, the steps taken are justi ed even for generalized functions, such as the Dirac delta function, which are neither continuous, di erentiable, nor square integrable. 4 While we often refer to this as simply the `measured signal,' it is seen in Ch. 12 that this is in fact a sum of delta functions.
3 210 Chapter 11. The Continuous and Discrete Fourier Transforms (a) (b) (c) (d) Fig. 11.1: Examples of the relationship between the magnitude of the signal s(k) and the magnitude of the reconstructed spin density ^(x). An image (a) of a rectangular object is shown alongside its k-space representation (b). The k-space data are more spread out along the direction parallel to the shorter side of the rectangle, re ecting the increase in higher spatial frequency components expected for smaller objects see Prob. 11.1. The image and the data shown in (c) and (d), respectively, are for a transverse slice through a human head. The fact that the k-space data has the shape of a ball is a result of the roundness of the head and the shape is not very di erent from circular-phantom data (see Ch. 14). 11.2. Continuous Transform Properties and Phase Imaging 211 introduced in Ch. 10 is generally used to describe any of the di erences between ^(x) and (x). Speci c di erences can often be directly related to a particular feature of the Fourier transform. For instance, ^(x) often appears complex because of experimental conditions (see below). Some Fourier transform properties of particular interest to MRI are discussed in the remainder of this section. A summary of general Fourier transform properties can be found in the tables. Although (x) is strictly a real quantity, ^(x) need not be real. The simplest example that demonstrates this is the presence of a global (constant) phase shift 0 leading to a modi ed signal s(k) = ei 0 s(k) ~ (11.9) where the `true' s(k) is de ned as the Fourier transform of (x). This can arise from the real and imaginary channels being switched or from incorrect demodulation and leads to ^(x) = ei 0 (x) (11.10) The real part of ^(x) will not give the physical image, unless the presence of 0 is understood and a correction made. One solution is to take the magnitude of ^(x) to get (x) independent of 0 , a common practice owing to the many sources of global phase errors creeping into the data. This `magnitude' image is the image most commonly used. The local phase error is the subject of the next subsection. 11.2.1 Complexity of the Reconstructed Image 11.2.2 The Shift Theorem The `time shifting' relation (or Fourier transform shift theorem) found in Table 11.1 tells of the consequences of shifts in the echo relative to its expected location or of incorrectly referenced signal demodulation (i.e., demodulation at the wrong frequency). This will now be seen to lead to a complex reconstructed image ^(x). Consider a signal which is shifted in k-space by k0 so that sm (k) ! sm(k ; k0). In particular, the center, or echo, of the signal is moved to k = k0 (the shift of the signal to the left in Fig. 11.2 corresponds to k0 < 0). The e ect of this shift on a reconstructed image ^(x) is found by taking the inverse Fourier transform of sm (k ; k0) ^(x) = F ;1 (sm (k ; k0 )) Z1 = dk sm (k ; k0)ei2 kx = dk0 sm(k0 )ei2 k x ;1 = ei2 k0x ^expected(x) (11.11) Equation (11.11) implies that a shift of the origin in k-space creates an additional phase (which is linear in that shift) in the reconstructed image ^(x). Since the magnitude is not altered, (x) is faithfully reconstructed by a magnitude operation (i.e., j ^j = in this case). Local phase errors are removed in the same manner as global ones.
0 ei2 k0x ;1 Z1 212 Chapter 11. The Continuous and Discrete Fourier Transforms The other case to be considered is the introduction of a linear phase shift in the kspace data. The following problem demonstrates that a linear phase shift of the signal, sm (k) ! sm (k)e;i2 kx0 , leads to a spatial shift of x0 in the position of the reconstructed image. Problem 11.2
Show that h(x ; x0 ) = F ;1(H (k)e;i2 kx0 ). Explain how this result leads to a spatial shift in the reconstructed image when the signal is incorrectly demodulated, so that sm(t) ! sm (t)e;i !t . To what demodulation frequency does this correspond? Fig. 11.2: Example of a read gradient structure leading to a shift of the echo in the sampling window. The area under the gradient between t1 and t2 is A and the area under the read gradient 0 itself is 2B . The echo is shifted from TE to TE < TE if A is chosen to be less than B . This results in a shift of k0 < 0 in the read direction k-space variable. 11.2.3 Phase Imaging and Phase Aliasing This is an appropriate place to introduce `phase imaging,' following the establishment of both image reconstruction and the shift theorem. While it is a magnitude image that is typically presented in the 2D plots viewed, additional information remains in the phase of the complex reconstructed image ^. It is useful to plot 2D phase images where each pixel intensity is proportional to the calculated phase value for the complex local signal (or `voxel signal'). We shall see that the phase artifact due to an uncentered gradient echo is best visualized in such a phase image. 11.2. Continuous Transform Properties and Phase Imaging The phase image (x y) is obtained from (x y) Arg ^(x y) = tan
;1 213 Im ^(x y) Re ^(x y) ! (11.12) The inverse-tangent function shown is calculated to nd the 2D intensity plots. Since the inverse-tangent is periodic over the interval ; ), the phase image can only be mapped into this interval, although the true phase value might take on any real value. The end result of this is a form of phase aliasing5 where spins with phase values di ering by multiples of 2 would have the same intensity.6 If this occurs predominately in one of the two dimensions, it shows up as bands, or `zebra stripe' artifacts, in (x y) intensity plots, and it is illustrated in Figs. 11.3c and 11.3d. One cause of the zebra stripes is an `asymmetry' in the echo position. The gradientinduced echo corresponds to the point k = 0 (k kx kR) where the readout gradient phase is zero for all spins. The assumption is made that the echo occurs at the center of the sampling window. However, when the gradient echo is not centered in this window, either by design or by an error, the measured signal (sa (k), say) with k = 0 still de ned at the center is not the true s(k). It is instead sa (k) = s(k ; k0 ), in terms of the shift k0 in the k-space origin along the read direction. According to the shift theorem, this changes the phase in the reconstructed x-space image by the amount (x) = 2 k0 x (11.13) In a plot over a given range of x-values, the phase shift (11.13) will lead to a phase image with more and more aliasing bands, the larger the value of k0 . Measurements, such as the widths of the horizontal bands in Fig. 11.3 (where the x-axis is vertical), can be used to nd the spatial frequency k0 associated with the zebra stripe artifact. With this analysis, adjustments can be made to recenter the echo. Problem 11.3
Explain how you could use the result (11.13) of the Fourier shift theorem analysis to nd the time of the center of the echo given a zebra stripe of width x and a readout gradient strength G. In general, the time point where the echo occurs may have spatial dependence on y as well as x. Local eld inhomogeneities lead to such dependence for local phases in ^(x y) and can cause serious artifacts, but by studying the phase variations, eld inhomogeneities can be probed. The deviations from strictly parallel bands seen in Figs. 11.3c and 11.3d are due to the presence of local eld errors.
This is also referred to as phase `wrapping,' `foldover,' or `wraparound.' The usefulness of knowing the true phase value such as in MR velocity quanti cation (Ch. 24) and in computing the actual echo shift in the case considered next, prompts the need for `phase unwrapping' algorithms.
5 6 214 Chapter 11. The Continuous and Discrete Fourier Transforms (a) (b) (c) (d) Fig. 11.3: (a) A 2D gradient echo magnitude image of the head. (b) The corresponding phase image of the head when the echo is centered in the read data acquisition window. (c) A phase image taken from the same data that produced (b) except that the echo is shifted `one sample point' away from the center of the sampling window. (d) A phase image where the echo is shifted ve points away from the center of the sampling window. The restricted variation of the phase from ; to creates the banding referred to as the `zebra stripe' artifact. The phase shift occurs along the read (vertical) direction, and the bands are perpendicular to this direction. This is an example of how knowledge of the theory behind an artifact gives information about the sequence acquisition. 11.2. Continuous Transform Properties and Phase Imaging 215 11.2.4 Duality
The shift theorem is also evident in the transform of a shifted delta function, (x) ! (x;x0 ). With H (k) found from (11.2), the transform pair exhibit the expected phase shift h(x) = (x ; x0 ) H (k) = e;i2 k x0 (11.14) Alternatively, the shifted delta function is seen to give rise to a periodic transform with period 1=x0 (i.e., invariance under k ! k + 1=x0 ). This is relevant to sampling, as will be seen in Ch. 12. Another Fourier pair is given by H (;x) = ei2 k0 x h(k) = (k ; k0) (11.15) which is a simple illustration of the general duality property of the Fourier transform. This F is the property that the replacement of x ! k and k ! ;x in a transform pair h(x) , H (k), F yields a new transform pair H (;x) , h(k). Knowledge of one Fourier transform pair leads to another pair without additional calculations. 11.2.5 Convolution Theorem
Many signal modi cations leading to variations in ^(x) can be described in terms of functions or ` lters' that multiply the unperturbed signal (see Chs. 12 and 13). As an example, consider T2 decay during data acquisition. The measured signal can be modeled as the product of s(k) in the absence of relaxation and an exponential decaying function. The (inverse) Fourier transform of products of such functions is needed for the reconstructed spin density. The convolution theorem provides a method for understanding the Fourier transform of the product of two or more functions in terms of their individual transforms. The Fourier transform of the product of two functions is the convolution of the Fourier transforms of each function F (g(x) h(x)) = G(k) H (k) (11.16) where the convolution operator is de ned by G(k) H (k) Z +1
;1 dk0 G(k0)H (k ; k0) (11.17) The proof of the convolution theorem is left to Prob. 11.4. 216 Chapter 11. The Continuous and Discrete Fourier Transforms Problem 11.4
a) Prove the convolution theorem given in (11.16) and Table 11.1, F (g(x)h(x)) = G(k) H (k) Z1 ;1 dk0 G(k0)H (k ; k0 ) Hint: Replace g(x) and h(x) by their Fourier transform representations inside the transform of the product. b) Show that Parseval's theorem of Table 11.1 follows directly from the convolution theorem upon the replacement of h(x) by g (x). The integral in (11.17) shows that nding the convolution of two functions at the position x involves re ecting one function through the origin, displacing it by x, and nding the area of the product of the two functions. A graphical representation is useful for a qualitative, and sometimes quantitative, understanding of convolution. As an example, consider the convolution of a right-triangular (ramp) function with itself. The ramp function is de ned by h(x) = x 0 ( 0<x<1 otherwise (11.18) The convolution is described graphically in Fig. 11.4. The value of the convolution is displayed at the right for ve di erent x values. Consistent with the ve cases shown, a direct integration leads to 8 1 3 > <1 6x h(x) h(x) = > 6 (2 ; x)(x2 + 2x ; 2) : 0 0<x<1 1 x<2 0 otherwise (11.19) The details are the subject of Prob. 11.5, after which appears a similar, but simpler, example. Problem 11.5
Derive (11.19) by direct integration. 11.2. Continuous Transform Properties and Phase Imaging 217 Fig. 11.4: A combined graphical and numerical example in convolution. The convolution of a ramp function with itself involves an integration over the product (which is not shown) of the mirror-re ected ramp functions h(x0 ) and h(x ; x0 ) both of which are plotted for ve di erent x values. See (11.18), (11.19) and Prob. 11.5. 218 Chapter 11. The Continuous and Discrete Fourier Transforms Problem 11.6
a) Show that the (isosceles) triangle function (x) de ned as (x) = 1 ; jxj ;1 x 1 0 elsewhere ( ) (11.20) can be obtained as the convolution of two rect functions with both unit width and unit height (x) = rect(x) rect(x) (11.21) as shown in Fig. 11.5. Do this either graphically or by direct integration. b) Derive the Fourier transform of (x) by direct integration of (11.20). c) Rederive the same Fourier transform using the convolution theorem. 11.2.6 Convolution Associativity The convolution of three functions arises in the analysis of the e ect of nite sampling of k-space data on the reconstructed image in Ch. 12. In that analysis, the rst convolution includes the sampling e ect, while a second convolution is required to include the nite length of data collection. It does not matter in which order the convolutions are done, since the convolution operation is associative. Problem 11.7
Prove the associativity: a(x) (b(x) c(x)) = (a(x) b(x)) c(x). 11.2.7 Derivative Theorem It is possible to derive important information about the boundaries in an object by analyzing the derivative of the image. The derivative theorem states that if f (x) and F (k) are Fourier transform pairs, then F (f 0(x)) = i2 kF (k) (11.22) (See the following problem.) As a result, a `derivative image' ^0(x) can be found directly by taking the inverse Fourier transform of the product of s(k) with i2 k. An example is shown in Fig. 11.6 which also illustrates the fact that derivative images magnify the e ects of noise 11.2. Continuous Transform Properties and Phase Imaging 219 Fig. 11.5: A shortcut to the Fourier transform of the product of two sinc functions through the
convolution of two boxcars. 220 Chapter 11. The Continuous and Discrete Fourier Transforms
m (k), (k). Upon multiplication by k, the measured signal, sm (k) + values, where noise is relatively more important. is enhanced at large k (a) (b) Fig. 11.6: A 2D magnitude image (a) and an image of the 1D-derivative (b) for a `resolution phantom.' The 1D derivative is taken in the vertical direction, so that the derivative image shows enhanced upper and lower edges (those edges with components perpendicular to the vertical direction). Notice that noise is also enhanced in the derivative image. Problem 11.8
Derive the Fourier derivative theorem (see Table 11.1). 11.2.8 Fourier Transform Symmetries Odd and even symmetries play a signi cant role in the Fourier transform description of imaging and in certain image reconstruction methods. As an example, consider a real function h(x) = h (x), of particular interest since the spin density is just such a function. The real and imaginary parts of its transform are even (symmetric) and odd (anti-symmetric), respectively, under k ! ;k, Re H (k)] = Re H (;k)] and Im H (k)] = ;Im H (;k)] (see Table 11.2). In order to see this, the transform (11.2) can be rewritten in terms of cosine and sine transforms Z1 dx h(x) cos 2 kx ; i sin 2 kx] (11.23) H (k ) =
;1 11.3. Fourier Transform Pairs 221 With h(x) real, the real (imaginary) part of H (k) is given by the cosinusoid (sinusoid) term in (11.23), which is even (odd) in k. The above example implies that the Fourier transform of a purely real function possesses complex conjugate symmetry: H (k) = H (;k). When (x) is real, the signal s(k) has this symmetry, a fact which is exploited in `partial Fourier imaging' (see Ch. 13). The procedure is to cover only half of k-space, with the other half is generated by assuming conjugate symmetry. The following problem requires similar analysis for a purely imaginary h(x). Problem 11.9
Find the even/odd symmetries for the real and imaginary parts of the transform of a purely imaginary function h(x). Compare your answers with those given in Table 11.2. 11.2.9 Summary of Continuous Fourier Transform Properties The properties of the Fourier transform and their relevance to MRI discussed in this section have been summarized in Table 11.1. Some additional properties of interest are included in that table. The compilation should prove useful in certain Fourier applications in the remainder of the text, and the reader may wish to make note of its location. Another table noted in the previous problem catalogs the odd/even, real/imaginary, and continuous/discrete properties of Fourier transform pairs. Table 11.2 is of particular value in MRI where knowledge of the sample can be used to predict properties of the measured signal. In particular, the fact that the object is real is exploited in half Fourier or partial Fourier imaging, as mentioned earlier in this section. A detailed discussion of partial Fourier imaging appears in Ch. 13. 11.3 Fourier Transform Pairs
There are several Fourier transform pairs which are ubiquitous in discussions of MRI. The derivations of some of the more involved Fourier transforms are presented below. Table 11.3 contains ve transform pairs of importance and appears at the end of the section. 11.3.1 Heaviside Function As an example of a Fourier transform important to MRI, consider the Heaviside function (k), 8 >1 k>0 < k=0 (k) = > 1=2 (11.24) :0 k<0 The inverse Fourier transform of (k) is given in terms of a Dirac delta function. To derive this distribution, insert an exponential e;2 jkj where the limit ! 0+ is to be taken after 222 Property linearity duality space scaling Chapter 11. The Continuous and Discrete Fourier Transforms Mathematical expression F (h1(x) + h2 (x)) = F (h1(x)) + F (h2(x)) F ( h(x)) = F (h(x))
F if h(x) , H (k) F then H (;x) , h(k) F (h( x)) = j 1 j H
R k kx0 kx kx space shifting F (h(x ; x0)) = H (k)e;i2
1 alternate form h (x) = ;1 dk H (k)e;i2 = F ;1(H (;k)) even h(x) odd h(x) convolution derivative Parseval R R0 H (k) = R01 dx h(x)e;i2 kx + ;1 dx h(x)e;i2 = 01 dx h(x) e;i2 kx + ei2 kx R = 2 01 dx h(x) cos 2 kx R H (k) = 01 dx h(x) e;i2 kx ; ei2 kx R = ;2i 01 dx h(x) sin 2 kx F (g(x)h(x)) = G1k) H (k) R(
=
;1 dk0 G(k0)H (k ; k0 ) F ( dh ) = i2 kH (k) dx R 1 dx jh(x)j2 = R 1 dk jH (k)j2 ;1 ;1 Table 11.1: Summary of the mathematical properties of the continuous Fourier transform. H (k) Real part Imaginary part real even odd imaginary odd even
plications. h(x) Table 11.2: Symmetry properties of the continuous Fourier transform important for imaging ap- 11.3. Fourier Transform Pairs the other operations are completed. The transform now becomes 223 where h (x) has been used to denote the Fourier transform of (k) to avoid con ict with the use of as an angle. The identi cation of the -function and the principal value operation7 P is based on the identities (x) = lim+ 1 2 + x2 (11.26) !0 1 x P ( x ) = lim+ 2 + x2 (11.27) !0 h (x) = F ( (k)) = lim+ dk !0 ;1 1 lim 1 = 1 lim = 2 !0+ ; ix 2 !0+ = 1 (x) + i P ( 1 ) 2 2 x
;1 Z1 (k)ei2 kxe;2 jkj = dk e;2 x 2 + x2 + i 2 + x2
lim !0+
0 Z1 k( ;ix) (11.25) 11.3.2 Lorentzian Form
Problem 11.10 The Fourier transform of exponentials representing signal decay around echoes yields a `Lorentzian' function, an example of which is described in the following problem. Spectroscopists study `metabolites' by consideration of the area under the frequency or spectral response of the MR signal. Assume that the e ective signal decay in the time domain is of the form h(t) = e;jtj=T2 . Note that f and t are the Fourier conjugate variables in this problem, instead of x and k. a) Derive H (f ) and compare your answer to that in Table 11.3. b) Using the form found in part (a), nd the total area under the curve H (f ). How does it depend upon T2? c) Alternatively, can you nd the area under H (f ) from h(t)? d) In practice, the true t = 0 point is not sampled. Why does this imply that the area must be found in the frequency domain? 11.3.3 The Sampling Function
x = 0 singularity of the function 1=x.
7 The modeling of uniform sampling is discussed in the next chapter. Discrete sampling is represented by the product of the continuous signal with a sampling function, and the
The principal value instruction requires that the region (- ) be excluded in the integration over the 224 Chapter 11. The Continuous and Discrete Fourier Transforms convolution theorem can be used to nd the Fourier transform of the product. However, before convolution can be applied, the Fourier transform of the sampling function must be known. The sampling function, de ned as u(k), is a sum of delta functions such that its product with the continuous signal is zero except at the points where the data were sampled, as desired. It is 1 X u(k) = k (k ; p k ) (11.28) By substituting (11.28) into (11.3), an intermediate step in nding U (x), the inverse Fourier transform of u(k), is given by
p=;1 U (x) = F (u(k)) = k 1 X Z1 p=;1
1 X ;1 dk (k ; p
1 X k)ei2 kx = k 1 X p=;1 ei2 p kx (11.29) Using the Fourier series identity (see the following problem)
n=;1 ei2 na = m=;1 (a ; m) (11.30) (11.31) gives8 U (x) = 1 X Therefore, the Fourier transform U (x) is also a sampling function. This property plays an important role in establishing the periodicity of the reconstructed spin density image ^(x). q=;1 (x ; q= k) Problem 11.11
a) Prove the identity given in (11.30). (Hint: Consider integrations over small intervals that either include or exclude the region where the argument of one of the -functions vanishes.) b) Show that the Fourier transform of U (x) collapses back to (11.28). 11.4 The Discrete Fourier Transform
The discrete Fourier transform (DFT) can be introduced independently as a mathematical tool which maps a function de ned at a nite set of uniformly spaced points into a like number of uniformly sampled points in conjugate space. A discrete inverse Fourier transform (DIFT) can also be de ned, leading to a discrete Fourier transform pair. The discrete 8 Note the delta function identity (ax) = (x)=jaj. 11.4. The Discrete Fourier Transform rect function Gaussian sampling (or `comb') function Lorentzian form Heaviside step function
1 X 225 rect
x W x2
F , F , W sinc( Wk) e; k
1 X e;
n=;1 k2 F (x ; n= k) , p=;1 (k ; p k) 2 1+4 2 x2 F , F , e;jkj
1 2 1 (k) ; 2i P ( k ) (x) Table 11.3: Fourier transform pairs which are relevant to a large number of MRI discussions found throughout the text. Note that the P in the expression of the Heaviside function refers to the
principal value of the integral. Fourier transform represents an exact transform, and it can be viewed as a special case of the continuous Fourier transform.9 As used in MRI, the discrete Fourier transform is an approximation of the continuous Fourier transform. Although the signal in MRI is continuous over all k-space, uniform sampling over a nite amount of time leads to a measured signal that is best described as a nite set of uniformly spaced measurements approximating the continuous signal. The measured signal describes a function which may be transformed using the discrete Fourier transform. Chapters 12 and 13 contain a detailed analysis of the e ects of approximating the continuous signal. The discrete Fourier transform as a special case of the continuous Fourier transform is developed in anticipation of these discussions in Ch. 12. It is also useful to present discrete Fourier transform properties similar to those laid out in Sec. 11.1 for the continuous Fourier transform. Multi-dimensional generalizations are not di cult to make for the various properties discussed. A notation is used where h and H are replaced by g and G, respectively, to highlight the di erence between the continuous pairs and the discrete pairs. With a length scale L, the discrete Fourier transform is de ned by p G L D(g) = g qL e;i2 pq=2n 2n q=;n p = ;n ;n + 1 ::: 0 ::: n ; 2 n ; 1 n;1 X (11.32) which can be written in terms of more familiar x-space and k-space notation if k = 1=L
It may also be viewed as a truncated version of an in nite Fourier series, the latter the periodic limit of a Fourier transform.
9 226 and L = 2n x Chapter 11. The Continuous and Discrete Fourier Transforms G(p k) = n;1 X q=;n g(q x)e;i2 pq x k (11.33) This notation will be further developed, and explained in Ch. 12. The corresponding de nition for the discrete inverse Fourier transform is n;1 X p g qL D;1(G) = 21 G L ei2 pq=2n 2n n p=;n q = ;n ;n + 1 ::: 0 ::: n ; 2 n ; 1 (11.34) The coe cient 1=2n is the normalization factor that could be inserted into one or the other of the summations. Again, (11.34) can be written in terms of k-space and x-space,
n;1 X g(q x) = 21 G (p k) ei2 n p=;n pq=2n (11.35) The demonstration that (11.32) and (11.34) constitute a discrete Fourier pair rests on the identity n;1 X i2 pk=N ;i2 rk=N e e = N pr (11.36) with
k=;n N = 2n (11.37) The Kronecker delta function pr was de ned in Ch. 5 as ( 1 for p = r (11.38) pr = 0 otherwise Using (11.34), the object is recovered perfectly by using the discrete inverse Fourier transform only if it is actually a set of points of amplitude g qL . 2n Problem 11.12
a) Prove the identity (11.36). b) Substitute (11.32) into (11.34) to show that the discrete Fourier transform in fact gives rise to a transform pair. 11.5 Discrete Transform Properties
Mirroring those of the continuous transform, some discrete transform properties are shown in Table 11.4, where g(q) and G(p) are shorthand, in the remainder of this section, for the 11.5. Discrete Transform Properties 227 respective quantities (11.34) and (11.32). Since many properties of the two transforms are similar, and have already been outlined for the continuous transform, fewer are presented here. The proofs of several properties are left to a problem. This section serves more as an illustration of how to manipulate the nite sums than a further demonstration of transform properties. The development of these properties parallels the steps required for the derivation of the properties in Table 11.1. 11.5.1 The Discrete Convolution Theorem
The convolution theorem states that g1 (q) g2(q) , G1(p) G2 (p)
where discrete convolution is de ned by10
n;1 X G1 (p) G2(p) = 21 G1(r)G2 (p ; r) n r=;n D (11.39) (11.40) To prove the theorem, substitute the discrete Fourier transform de nitions of G1 and G2 on the right-hand-side of (11.40) and use (11.36) to show
n;1 n;1 n;1 X X X g1(q)e;i2 rq=2n g2(q0)e;i2 G1(p) G2 (p) = 21 n r=;n q=;n q =;n n;1 n;1 X X = 1 g (q)g (q0)2n q q e;i2 pq =2n 2n q=;n q =;n 1 2 n; 1 X = g1(q)g2(q)e;i2 pq=2n
0 0 0 0 (p;r)q =2n
0 = q=;n D (g1(q) g2 (q)) (11.41) Problem 11.13
Assuming periodicity of g(q): a) Prove the space shifting property of Table 11.4. b) Prove Parseval's theorem as shown in Table 11.4. Arguments that lie outside the range (;n n ; 1) must be mapped back into the range by the `mod 2n' operation.
10 228 Chapter 11. The Continuous and Discrete Fourier Transforms Problem 11.14
Find the even/odd symmetries for the real and imaginary parts of the transform G(p) of a purely real function g(q). Compare your result with the entry in Table 11.2. 11.5.2 Summary of Discrete Fourier Transform Properties As was the case with the continuous Fourier transform, only a selection of transform properties have been discussed in detail. Table 11.4 includes additional properties of the discrete Fourier transform. Description of property convolution linearity symmetry space shifting alternate form Parseval's Theorem Mathematical expression D(g1(q) g2 (q)) = G1 (p) G2(p) = 21n
g(q) ! G(p) n;1 X r=;n G1(r)G2 (p ; r) D(g1(q) + g2(q)) = D(g1(q)) + D(g2(q))
1 N G(p) ! g (;q ) 1 0 n;1 X g(q) = @ 21n G (p)e;i2 pq=2nA
p=;n n;1 X q=;n D(g(q ; j )) = G(p)e;i2 pj=2n jg(q)j2 = 21n n;1 X p=;n jG(p)j2 Table 11.4: Properties of D, the discrete Fourier transform. 11.5. Discrete Transform Properties 229 Suggested Reading
Some notation and several conventions have been borrowed for this chapter from: O. E. Brigham. The Fast Fourier Transform. Prentice-Hall, Englewood Cli s, New Jersey, 1974. Another excellent text from which to study the Fourier transform is: R. N. Bracewell. The Fourier Transform and Its Applications. McGraw Hill, New York, 1986. 230 Chapter 11. The Continuous and Discrete Fourier Transforms Chapter 12 Sampling and Aliasing in Image Reconstruction
Chapter Contents
12.1 12.2 12.3 12.4
In nite Sampling, Aliasing and the Nyquist Criterion Finite Sampling, Image Reconstruction and the Discrete Fourier Transform RF Coils, Noise and Filtering Nonuniform Sampling Summary: Signal sampling and the Nyquist sampling criterion are presented. Image re- construction and the role of the discrete Fourier transform are discussed, along with a rst introduction to resolution. Descriptions of aliasing artifacts, analog ltering, nonuniform sampling and other practical imaging considerations are made. Introduction
This chapter addresses some of the e ects that data collection methods have on the image. It is recalled that the signal is proportional to the emf induced in the receive coil by the rotating magnetization. This emf gives rise to a continuous or analog signal which is detected, sampled, and stored as nite, digitized data. Measurements of the MRI signal necessarily involve truncations and discrete sampling of k-space. It is critical to acquire data in such a way as to generate a faithful representation of the spin density after a Fourier transform is performed on the signal. In the rst section, discretization of in nite data is shown to lead to the Nyquist sampling rule for minimizing certain aliasing image errors (reconstruction artifacts). Truncation of the data is discussed in the second section, along with the truncated and discretized reconstructed image. The conditions under which the sampled signal and the sampled image are connected by the discrete version of the Fourier transform are also described. The relationship of aliasing to rf coil properties, noise, and analog lters is considered in Sec. 12.3. Several consequences of 231 232 Chapter 12. Sampling and Aliasing in Image Reconstruction inadequate or incorrect sampling, especially with respect to nonuniformities in k-space, are covered in the last section. 12.1 In nite Sampling, Aliasing and the Nyquist Criterion
We consider in this section the mathematical modeling of in nite sampling, using the sampling function introduced in Ch. 11. The errors associated with a small sampling rate are considered. Consider the discretization of the measurement, but without any limitation on the (in nite) number of discrete steps. Although it is not possible to collect an in nite set of data, it is instructive to understand rst the consequences of such `in nite samplings.' A subsequent step will be to consider a nite sample size. The MR signal is generally collected over a set of uniformly spaced points in k-space. For a constant read gradient GR, such sampling is achieved along the read direction by taking data at uniform intervals t in time, with the by now familiar relationship between the k-space step kR k and t k = {GR t (12.1) The measured signal is s(p k) at the step denoted by the integer p. The set of all (positive and negative) integers corresponds to the in nite sampling limit. Consider the multiplication of the continuous signal by a `comb' or `sampling' function. Recall that the sampling function is the in nite sum of evenly spaced Dirac delta functions introduced in Ch. 11 comb function(k) sampling function(k) u(k) = k
1 X 12.1.1 In nite Sampling p=;1 (k ; p k ) (12.2) with a generic constant spacing k. The multiplication yields the signal distribution1 corresponding to in nite sampling: s1(k) s(k) u(k) 1 X = k s(p k) (k ; p k)
p=;1 (12.3) The coe cients of the delta functions are the aforementioned sampled signals s(p k). The sampling function, a signal example, their respective Fourier transforms, and the signal distribution are all shown in Fig. 12.1.
This distribution is an in nite sum of delta functions, but it has the same units as a signal. It is a `signal density' multiplied by k.
1 12.1. In nite Sampling, Aliasing and the Nyquist Criterion 233 Fig. 12.1: The Fourier transform pairs for a signal example and the sampling function are illustrated in (a) and (b). The functions sinc2 (k) and u(k), and their respective (inverse) transforms, are the
subjects of problems in Ch. 11. The product of the sampling function in (a) with the signal in (b) yields the sampled signal, and its inverse transform, in (c). The Fourier transform of the sampled signal, which may also be computed by the indicated convolution of the individual inverse transforms, is the rst topic in the present chapter. Aliasing, the second topic, is avoided in this gure by the choice of a su ciently small k-space step k and, hence, a su ciently large ` eld-ofview' Lx 1= k. 234 Chapter 12. Sampling and Aliasing in Image Reconstruction The inverse transform of s1(k) leads to an approximation, or reconstructed image, of the physical density (x). With a portion of the notation already introduced in (11.8), the reconstructed image for in nite sampling is found to be ^1(x) = dk s1(k)ei2 kx 3 Z1 2 X 1 = dk 4 k s(p k) (k ; p k)5 ei2
;1 ;1 Z1 kx = k X
1 p=;1 p=;1 s(p k)ei2 p kx (12.4) This expression is an in nite Fourier series and represents a histogram approximation of the continuous inverse transform (11.7), i.e., of (x). In the event that (x) vanishes outside of a nite interval, the periodic Fourier series yields an in nite set of exact copies of the physical spin density, provided that the period associated with the copies is larger than the interval. A period that is too small gives rise to an `aliasing' problem discussed in the next subsection. Before addressing the important question of the relation of the period to k, we show how convolution of delta functions is a simple way to examine the multiple copies. The product of s(k) and u(k) in the de nition of the sampled signal (12.2) implies that ^1(x) can be described equivalently by the convolution of (x) and U (x) (the latter is the inverse Fourier transform of u(k) discussed in Ch. 11): ^1(x) = (x) U (x) The function U (x) is again a comb, as shown in Sec. 11.3.3 , (12.5) (12.6) U (x) = 1 X q=;1 (x ; q= k) The convolution of any function f (x) with a delta function (x ; x0 ) is easily seen to be Z f (x) (x ; x0 ) = dx0 f (x0) (x ; x0 ; x0) = f (x ; x0 ) (12.7) Thus the convolution of the in nite sum (12.6) with any function gives an in nite series, each term of which is a copy of the function displaced from the next by the interval 1= k. That is, (12.5) becomes 1 X ^1(x) = (x ; q= k) (12.8) An in nite series of copies of the triangle function is illustrated in Fig. 12.2, which focuses on the convolution indicated in Fig. 12.1.
q=;1 12.1.2 Nyquist Sampling Criterion From the above discussion and the gures, spatial periodicity is a prominent feature of ^1(x). It is observed from the expression (12.4) that the periodicity corresponds to the fact that 12.1. In nite Sampling, Aliasing and the Nyquist Criterion 235 Fig. 12.2: The image ^1(x) = (x) U (x) for in nite sampling of (the Fourier transform of) a triangle function (see Fig. 12.1). Notice that the physical spin density is now repeated over all space, with periodicity ^1 (x) = ^1 (x + L), where L = 1= k (Sec. 12.2.3). The spacing k is small enough that L exceeds the object (triangle) size and aliasing is avoided. 236 Chapter 12. Sampling and Aliasing in Image Reconstruction ei2 p k x is unchanged, for any p, if x ! x + 1= k. This replication concept is evident from (12.8) and demonstrates that ^1(x) is translationally invariant. That is,
^1(x) = ^1(x + 1= k) (12.9) Thus the periodicity of ^1(x) is given by the reciprocal of the spacing of the delta functions in the Fourier transform of the sampling function, 1= k L FOV (12.10) The uniform spacing between data points k is 1=L where L is the spatial interval over which the reconstructed image repeats itself. The interval L is called the eld-of-view (FOV). To construct an image, one of these copies is chosen. Although the copies of (x) are adequately separated in the gures shown (Fig. 12.2), this need not be the case. If the images overlap, then there will be signi cant di erences between (x) and what is displayed as an image. This type of artifact is generally referred to as `aliasing.2' It is possible to nd a general requirement, or criterion, for the sampling rate to eliminate this artifact. The criterion can be introduced by considering what happens if L is too small. The example at the top of Fig. 12.3 depicts a case where the spatial period L of the images in x-space is greater than the length (call it A) of the object being imaged. However, if L < A, then those parts of the image corresponding to the pixels near the edges of the object, which have to be assigned somewhere in the image, will be mapped back inside of L. The object is `aliased' due to the overlap of the repeated images, with the left edge mapped into the right side of the interval L, and vice versa for the right edge. See the example at the bottom of Fig. 12.3. An image illustration of such aliasing appears in Fig. 12.4. To avoid this type of image error, data must be sampled such that the inverse of the sampling step in k-space is larger than the size A of the object to be imaged. In one dimension, this means that the FOV must be larger than the object size: 1 L > A or k < A (12.11) Equation (12.11) is referred to as the Nyquist sampling criterion. The above discussion carries the hidden assumption that the signals from all spins in the object are detected. The ability to suppress aliasing by reducing the region to which the receive coil is sensitive is considered in Sec. 12.3.
Besides the names of phase wrapping, foldover, and wraparound indicated in Sec. 12.4.1 aliasing is also called `ghosting.'
2 12.1. In nite Sampling, Aliasing and the Nyquist Criterion 237 Fig. 12.3: The in nite sampling image ^1(x) is reconstructed for a ramp-like spin density. The Nyquist relation is satis ed in the top gure where the sample size is less than the FOV (A < L) while it is not satis ed in the bottom gure where A > L. Aliasing occurs for the latter, where copies of the object overlap and image distortion results, assuming the receive coil detects the signal from the entire object. The left (right) piece of the ramp is mapped inside the right (left) border of the L interval. Fig. 12.4: An aliased image of the head and neck. Notice that the base of the neck is seen to appear at the top of the head (see arrows). In this example, the FOV parallel to the head-foot (cranial-caudal) axis was chosen to be too short, resulting in the aliasing. That is, the rf coil picked up signal from a region larger than the FOV. 238 Chapter 12. Sampling and Aliasing in Image Reconstruction Problem 12.1
a) Redraw Fig. 12.3 for a boxcar spin density function. b) Redraw Fig. 12.3 for the triangular spin density function of Figs. 12.1 and 12.2. c) In imaging the human torso, including the shoulders and arms, it is typical to acquire a 2D coronal image (see Table 10.1). Discuss what aliasing might look like for an FOV along the shoulder-to-shoulder axis that only covers the width of the chest but not the arms. Nyquist in the Read Direction In the case where sampled data are separated by a time interval t and a gradient GR (t) is applied along the read direction, (12.11) can be rewritten as Z t+ t 1 1 kR { dt0 GR (t0) = L < A (12.12) t R R For the special case of a boxcar gradient, the left-hand-side of (12.12) is kR = {GR t (rectangular gradient lobe) (12.13) where the dimension of the sample in the read direction is AR . The combination of (12.12) and (12.13) is often taken as an equality, (12.14) {GR tLR = 1 which can be used to nd one parameter when the others are xed. Although it is easier to consider the Nyquist relation in terms of x- and k-space, for practical reasons it is necessary to understand the rate at which data must be sampled in time. The de nition of the associated Nyquist sampling frequency during the application of a constant read gradient GR leads to fR BWR 1 = {GR LR > {GR AR (12.15) tR Problem 12.2
Determine the Nyquist frequency and the maximum allowable value kmax , before aliasing occurs, of a 1D water sample of 0.5 m in length. Assume that the applied gradient strength is either a) GR = 10 mT/m, b) GR = 25 mT/m, or c) GR = 50 mT/m. 12.2. Finite Sampling, Image Reconstruction and the DFT 239 Nyquist in the Phase Encoding Direction
The Nyquist relation applies to all three orthogonal directions. In the phase encoding direction (or partition encoding direction in 3D imaging) the Nyquist criterion speci es that kPE { Z t+ PE
t dt0 GPE (t0 ) = L1 < A1 PE PE
(rectangular gradient lobe) (12.16) Once again, for the special case of a boxcar gradient, the left-hand-side of (12.16) is kPE = { GPE PE (12.17) where PE is the duration of the phase encoding gradient pulse. The dimension of the sample along the direction under consideration is APE . In this case, the strength of the phase encoding gradient is being varied, but the duration of the gradient is xed. 12.2 Finite Sampling, Image Reconstruction and the Discrete Fourier Transform
The treatment of the MRI signal is not complete until the limits are imposed on the time during which measurements are taken. It is the reconstructed image derived from the corresponding truncated data set that is de ned by the function ^(x). For in nite sampling, this has the previous function ^1(x) as its limit. 12.2.1 Finite Sampling
The data truncation, or `windowing,' is modeled mathematically by multiplying the sampled data by the rect function introduced in Ch. 9. The boundaries of the rect function must be de ned carefully, so that the product of the sampling and the window functions reduces to the standard MRI sampling convention previewed in Sec. 10.3, and reintroduced in the last chapter through the discrete Fourier transform. The details are left to the problem given below and the result is
1 2 u(k) rect k +W k ! = k n;1 X p=;n (k ; p k) (12.18) where the total number of points is N = 2n and W 2n k = N k (12.19) An integration over (12.18) yields k for each sampled point, which may be interpreted as 1 2 k on each side of the point. Thus 2n k, and not (2n ; 1) k , is the total interval covered in k-space, encompassing 2n total points. 240 Chapter 12. Sampling and Aliasing in Image Reconstruction Problem 12.3
Derive (12.18). That is, show that the summation limits on the right-hand side follow from (12.2) and the argument in the rect function given on the left-hand side. The nal expression for the signal distribution corresponding to nite sampling, or `measured' signal distribution, sm (k), is the product of three functions (signal, sampling, and window) given by ! k+1 k 2 sm (k) = s(k) u(k) rect W = k n;1 X p=;n s(p k) (k ; p k) (12.20) The result is the expected discrete and nite sum of delta function terms whose coe cients are again the sampled signals s(p k). We consider the inverse Fourier transform of this signal in the next subsection. Standard MRI Sampling
It has been noted that the distribution of points in (12.20) follows the standard convention where k-space is sampled slightly asymmetrically with an even number of points N = 2n spread uniformly over the region ;n k (n ; 1) k]. If n >> 1, then the sampling is symmetric, to good numerical approximation. The reason for this convention is that an even number of points (usually 2p points, for some p) is sampled in the implementation of a `fast Fourier transform.' The FFT is an e cient computer algorithm for calculating the discrete Fourier transform of an object. Consider what happens to the window function in (12.18) if it instead had been de ned symmetrically with respect to the origin. With all points sampled at intervals of k, 2n + 1 points would then have fallen inside the sampling window (just barely: two would lie on its edges). By contrast, the shift of (1=2) k to the left in the rect argument leads to only 2n sampled points, each separated by k and covering the range of integers -n,n-1]. See the examples in Fig. 12.5. The Nyquist theorem ensures adequate sampling of the data, but does not directly address what happens when the data are not sampled at the assumed points. For example, shifts in the sampled points are common in MRI, either due to experimental design or physical errors. The subject of the problem in the next subsection is to show that a uniform k-space shift in the data leads to a phase shift in the image that varies linearly as a function of position. The phase shift is anticipated from the continuous Fourier transform analysis in Sec. 11.2.2, but there are other errors present in the truncated and sampled data owing to the k-space shift, which can be analyzed with the discrete Fourier transform. An important signal loss arises if the shift is great enough to leave the signal peak at k = 0 close to the edge or even 12.2. Finite Sampling, Image Reconstruction and the DFT 241 Fig. 12.5: Simple examples of (a) `truly' symmetric sampling of k-space and (b) standard sampling as de ned in MRI, both with an even number of points. In (a), both the sampling function u(k)
and the window function have been displaced from their standard-MRI-sampling de nitions by the amount k=2 to the right. In (b), both are in standard position. out of the sampling window. Nevertheless, the phase shift artifact is the principal di culty, leading to artifacts in Re ^(x). The procedure described in Ch. 11 involving the magnitude j ^(x)j is a fortiori a method of choice in displaying images. 12.2.2 Reconstructed Spin Density
The reconstructed spin density for nite sampling is called ^(x) and it is de ned by the inverse Fourier transform of (12.20) ^(x) dk sm(k)ei2 kx ;1 Z 1 n;1 X = k dk s(p k) (k ; p k)ei2
= Z1 kx k n;1 X ;1 p=;n p=;n s(p k)ei2 p kx (12.21) It is apparent that the periodicity (12.9) survives in (12.21) ^(x) = ^(x + 1= k) so the Nyquist criterion applies to the truncated data as well. (12.22) 242 Chapter 12. Sampling and Aliasing in Image Reconstruction Fig. 12.6: The convolution of the unperturbed spin density with the Fourier transforms of the sampling and rect functions. The blurring and creation of multiple images shown here are present in all MRI images, even with L chosen to be larger than the object size. A single image is chosen from the set and the blurring is minimized if a wide rect function is employed (so that the sinc function is approximately a delta function). The gure is not accurately drawn in that the sinc function should be much narrower, and there should be rounding at the peaks and dipping below zero at the minima for ^(x). Also, the phase accompanying asymmetric sampling has been ignored. 12.2. Finite Sampling, Image Reconstruction and the DFT 243 Problem 12.4
Consider a shift K in the k-space data that can be acceptably approximated by K = r k for some integer r. Assume that the entire echo signal remains within the sampling window. Justify the use of the discrete Fourier transform shift theorem in Table 11.4 for this situation and show that the theorem leads to a spatially varying linear phase in the reconstructed image (12.21). As a special case, explain how the results for truly symmetric sampling (for example, Fig. 12.5a) can be used to derive the results expected for `MRI-symmetric' sampling (for example, Fig. 12.5b). Discuss the complications that arise for a shift that is not an integer multiple of k. The fact that ^(x) is not expected to represent exactly the physical spin density (x), in view of the limited data set, is made evident in (12.21). The nite sampling has led to a reconstructed spin density that is an approximation of the result (12.4) for in nite sampling, which, in turn, is a Fourier series representation of the physical spin density. Recall that the latter leads to exact copies of a localized (x) for su ciently small values of k. The fact that the former is a discrete Fourier transform that may give an accurate representation of the physical spin density for a given `resolution' is the subject we shall develop in the discussion to follow. The relationship between n and resolution will be part of that discussion. It is recalled that di erences between ^(x) and (x) are de ned as artifacts (Ch. 11). An artifact arising from the truncation is `blurring' and it, along with other features, can be illustrated as follows. Since sm is the product shown in (12.20), the Fourier transform in (12.21) may be reconsidered as a convolution of the three functions ^(x) = (x) U (x) W sinc( Wx) e;i x k (12.23) where the Fourier transform of the rect function in (12.23) is found by using the shift theorem on the transform of rect(k=W ) (see Ch. 11). The result of the convolution of the rst two functions, obtained in (12.4) and shown in the top portion of Fig. 12.6, is itself to be convolved with the sinc function (the next portion of the gure), the inverse Fourier transform of the rect function.3 The reconstructed image ^(x) obtained after the modi cations of sampling and truncation is shown at the bottom of the gure. Notice that the convolution of the in nitely sampled signal with the sinc function blurs the bandlimited images into each other since the sinc function is not bandlimited. All images include this e ect, but it is generally negligible for a wide rect function (long boxcar), except when sharp boundaries are present in ^(x) (see Ch. 13). A long boxcar gives rise to a narrowly peaked sinc function, approaching a delta function in the limit of in nite boxcar sampling. In this regard, and for a connection to resolution, see Prob. 12.7.
It is remembered from Ch. 11 that the convolution operation is associative so that the order does not matter here.
3 244 Chapter 12. Sampling and Aliasing in Image Reconstruction 12.2.3 Discrete and Truncated Sampling of ^(x): Resolution The measurement of x-space is itself discretized and truncated. The spatial range, or spatial period, over which the spin density is imaged has already been de ned as L. The truncation is for the obvious reason that there is no new information to be gained from the repetitions of the image. The discretization is over spatial steps (with uniform step size x) since there is a lower limit to the spatial information, or `resolution,' available in the reconstructed spin density (see Prob. 12.7). A fundamental reason for the lower limit is the fact that truncation of k-space data leads to the x-space blurring described previously. As in the k-space discussion, nite sampling can be modeled as the product of ^(x), a sampling function, and a rect function. The sampling function resembles (12.6), but with steps x it is given by 1 X ~ U (x) = (x ; q x) (12.24) For the moment, we consider the number of sampling points to be equal to n0 , rather than n and we perform a calculation like that in (12.20). The complete expression of the `measured' reconstructed spin density distribution for 2n0 sampled points becomes ! 1 x+ 2 x ~ ^m (x) = ^(x) U (x) rect L =
q=;1 x n ;1 X
0 with the relation q=;n 0 ^(q x) (x ; q x) (12.25) L = 2n0 x (12.26) analogous to that for W in (12.19). The resolution x = L=(2n0) is thus xed by n0 and L, and is commonly referred to as the image voxel (or pixel) size. We observe that the coe cients of the delta functions (and x) are the sampled reconstructed densities ^(q x) in parallel with the way that the sampled signals s(p k) appeared in (12.20). Recall that the inverse Fourier transform of the signal distribution sm (k) gave the reconstructed spin density function ^(x). Similarly, the Fourier transform of the reconstructed spin density distribution ^(x) is a signal function, which we shall call s(k), given ^ by Z s(k) = dx ^m (x)e;i2 kx ^ n ;1 X = x ^(q x)e;i2 kq x (12.27)
0 It is seen that, for large n0 and small x, (12.27) approaches the continuous Fourier transform of ^(x), and ^(x) approaches the continuous inverse transform of s(k). ^ But for large n and small k, we have already seen in (12.21) that ^(x) approaches the continuous inverse transform of s(k). The question addressed in the next subsection is about the conditions under which s(k) may be identi ed with s(k). This will lead to the result ^ that the discrete set of signals s(p k) may be mapped into the discrete set of reconstructed spin densities ^(q x) by the discrete inverse Fourier transform. q=;n 0 12.2. Finite Sampling, Image Reconstruction and the DFT 245 12.2.4 Discrete Fourier Transform The condition that the two sets s(p k) and ^(q x) form a discrete Fourier transform pair is simply that the number of steps in both domains (k-space and x-space) must be equal n = n0 (12.28) It is reasonable that the same size of data sets is needed to transform back and forth between domains. Also, see the argument in the next problem given below. To reach this result, we begin by showing that s(r k) reduces to s(r k) under (12.28) ^ for integer r. Replace ^(q x) in (12.27) by the sum (12.21) of terms s(p k) s(r k) = ^
= x k x k n;1 n ;1 X X
0 p=;n q=;n n;1 n ;1 X X
0 0 s(p k)ei2 s(p k)ei2 p k q x e;i2 r k q x
(p;r) k q x p=;n q=;n (12.29) 0 If the condition n = n0 is imposed, then (12.10) and (12.26) combine to give 1 k x = L 2L = 21 n n and (12.29) reduces to
n;1 n;1 X X s(r=L) = 21 ^ s(p=L)ei n p=;n q=;n q(p;r)=n (12.30) (12.31) (12.32) (12.33) Using the identity (11.36), n;1 X q=;n ei q(p;r)=n = 2n pr it is found that s(r=L) is point by point equal to the signal where it is originally sampled: ^ s(r=L) = s(r=L) ^ It follows that the two discrete and truncated (or bandlimited) functions, the sampled signal and the sampled reconstructed spin density, are related by s(p=L) =
^(qL=2n) = and de ning x k n;1 X q=;n n;1 X p=;n ^(qL=2n)e;i pq=n s(p=L)ei pq=n (12.34) (12.35) ^MRI (qL=2n) = ^(qL=2n) x 246 then Chapter 12. Sampling and Aliasing in Image Reconstruction s(p=L) = n;1 X ^MRI (qL=2n) = 2n s(p=L)ei p=;n q=;n X 1 n;1 ^MRI (qL=2n)e;i
pq=n pq=n (12.36) This is precisely the discrete Fourier transform pair discussed in Ch. 11. It is also easy to show that the Nyquist symmetry continues to hold for this pair. Further, it demonstrates that ^MRI (x), the discrete Fourier transform of s(p=L), which is what is displayed in an MRI image, is proportional to the physical spin density times the volume of the voxel. However, for the sake of brevity, and where no ambiguity occurs, we continue to use ^(~), the e ective r spin density, to discuss the MR image. Problem 12.5
Argue that we must have n = n0 by rst showing that s(k) in (12.27) is periodic ^ with period 1= x. Then relate this period to L, on one hand, and to the total k-space interval 2n k, on the other hand. Problem 12.6
Nyquist symmetry: Show that the expression for ^(q x) in (12.34) is invariant under x ! x + L, i.e., under q ! q + L= x. Problem 12.7
For convenience, consider a symmetric window, rect(k=W ), in this problem. Show that the product of W with the rst zero crossing of the Fourier transform of this window is constant, and nd that constant. Compare this result to (12.30), noting that the rst zero crossing distance represents the accuracy of measuring distances in x-space, the `spatial resolution' discussed above and in Ch. 13. This result is reminiscent of the Heisenberg uncertainty principle in quantum mechanics (Ch. 5). Given that W controls how small a window in k-space is examined, what does this `uncertainty' result imply about blurring in MRI? 12.2. Finite Sampling, Image Reconstruction and the DFT 247 12.2.5 Practical Parameters
The following problem provides a summary for sampling parameters. The role of the Nyquist criterion is highlighted in the calculations for both 2D and 3D imaging. Problem 12.8
Assume the following 2D imaging parameters: Lx = Ly = 256 mm Nx = Ny = 256 TH = 5 mm TR = 600 ms. Assume that x, y, and z are the read, ^ ^ ^ phase encoding, and slice select directions, respectively. Also suppose that Ts = 5.12 ms and PE = 2.56 ms and that the rf excitation bandwidth BWrf is 2 kHz. See Ch. 10 for the sequence diagrams. a) Find the readout bandwidth, BWR , the Nyquist sampling interval tR in the read direction, the Nyquist sampling interval kx in the kx direction, and the strength of the read gradient (Gx) used. b) What is the Nyquist sampling interval ky in the ky direction? What is the gradient step size Gy in the phase encoding table? What is the strength of the maximum value Gy max = Ny Gy =2 of the phase encoding gradient? c) What is the slice select gradient Gss used? d) What is the total imaging time Tacq ? e) What happens to Gx, Gy max and Gss when Nx and Ny are doubled at the same time that TH is halved while all other quantities are unchanged? f) How does Gx change when Lx is halved while all other parameters are held xed? How does it change if, instead, Ts is changed to 2.56 ms while Lx is unchanged from 256 mm and Nx is xed at 256? g) Suppose the imaging is performed as a 3D imaging experiment with Lz TH = 32 mm and Nz = 16, with all other parameters the same as in the original problem statement (the partition encoding and phase encoding gradient times are the same: z = PE = 2.56 ms). i) What is Gss? (The z-axis is now the `slab' selection axis and the partition encoding gradient may be referred to as either Gss or Gz .) ii) What is Gz and what is Gz max? iii) What is the total imaging time if (i) TR were 600 ms (ii) TR were 60 ms? How does Tacq in either case compare with the imaging time in the 2D imaging experiment? iv) How does Gss compare between the 2D and 3D imaging experiments? 248 Chapter 12. Sampling and Aliasing in Image Reconstruction 12.3 RF Coils, Noise and Filtering
The choice of FOV rests on a number of additional factors in imaging, besides the size of the object. We consider in this section the e ects of variations in the spatial regions that are excited by the rf transmission coils, or that are receive-coil-sensitive, or that are directly ltered by electronic means. An illustration of these considerations may be made involving the rf receive coil. If that coil is sensitive to MR signals or noise outside of the frequency range determined by GR and LR in the read direction (BWread = {GRLR ), then there will be aliasing in the reconstructed image according to the Nyquist relation. The e ective size of the spin system may be larger than the body size AR. 12.3.1 RF Field-of-View Considerations In order to avoid aliasing and still image e ciently, LR L must be chosen carefully. A simple set of 1D imaging situations is presented in Fig. 12.7 to illustrate di erent factors that determine a proper choice of L. In Fig. 12.7a, L is determined by the actual physical extent A of (x) since the entire object is excited by the rf pulse. The region of receive coil sensitivity matches the excitation range. In Fig. 12.7b, L has been chosen to be less than the physical extent of (x) because the region of rf excitation is less than the length of the sample. Since there will be no signal from regions where the spins have not been tipped into the transverse plane there is no reason for L to be larger than the rf excited region. (See later for complications due to noise stemming from those regions.) In Fig. 12.7c, the receive coil is only sensitive to a region smaller than A. There will be no signal detected outside of the receive coil sensitivity, and therefore L does not need to extend beyond this range. 12.3.2 Analog Filtering The subject of noise in the MR signal is an important one and Ch. 15 is devoted to its detailed discussion. As a preliminary topic, we describe here some aspects of the ltering of noise, as they relate to the FOV. The total measured signal can be modeled by the sum of a signal s(k) and a noise function (k). In view of the results of the previous section, the total signal is considered to be the nite and discretized set of measurements related to the nite and discretized set of reconstructed spin density data through the discrete Fourier transform. Since the transform is linear, the reconstructed image is also the summation of the MR image and a noise image D s(k) + (k) , ^(x) + ^(x) (12.37) where ^(x) D;1 ( (k)). The noise in MR is usually assumed to be white in nature, i.e., power is uniformly distributed as a function of frequency. This means that a reconstructed image will have noise aliased into the image from frequencies outside of the region-of-interest (ROI). In the read direction, however, it is possible to limit the frequency content of the measured signal to a narrow frequency band by ltering the data. In general, modi cations of the signal in MRI are referred to as lters. In the next chapter, lters corresponding to multiplication 12.3. RF Coils, Noise and Filtering 249 varied. As described in the text, the FOV is chosen in each case such that no aliasing appears in the reconstructed image. Fig. 12.7: Three 1D imaging conditions where the parameters a ecting the choice of L have been 250 Chapter 12. Sampling and Aliasing in Image Reconstruction of the k-space data will be discussed at length. The speci c electronics of frequency lters relevant to the present topic is beyond the scope of this text, but the lter e ects on the reconstructed image can be examined. Since the data received in the read direction by the electronics are continuously acquired, they can be analog ltered before sampling. Analog ltering is equivalent to multiplying the signal as a function of frequency by a lter function H (f ) as shown in Fig. 12.8. The linear relationship between frequency and position in an MRI experiment implies the lter can also be thought of as modifying the x-space image. Phase encoding, however, is performed before data are read, so analog ltering cannot be carried out along the phase encoding direction. Fig. 12.8: Example of a low pass frequency response (top gure) for an electronics analog lter. Frequency encoding along the read direction leads directly to its corresponding spatial response (bottom gure). The negative frequencies are a result of the demodulation of the Larmor frequency in the signal. The transition frequencies, fflat and fcutoff , are described in the text, and correspond to the spatial positions, xflat and xcutoff , respectively. There are two primary reasons to perform analog ltering of the MRI data. The rst, as mentioned, is to limit the e ects of noise from outside of the desired bandwidth. The second includes cases where the body excited in the read direction is larger than the ROI of the 12.3. RF Coils, Noise and Filtering 251 imaging experiment. For example, measurements for a transverse image of the human chest for the study of the heart (the ROI) may include the signal from the body over the whole arm-to-arm range (the read direction). A lter can be used to limit the frequency content of the signal in order to get an image of the heart alone. In the ideal case, the analog lter behaves like a rect function allowing all frequencies within its bandwidth and rejecting all frequencies outside its bandwidth. However, lters with such sharp frequency transitions are unrealistic. The typical realization has a smooth roll-o from `pass-band' to `reject-band' (e.g., Fig. 12.8), the roll-o rate being determined by the type of the lter. In this case, it is possible to identify a at pass-band, or low pass, region jf j < fflat , in terms of a maximum at frequency fflat . The reject band region may be de ned by jf j > fcutoff , introducing an outer edge cuto frequency fcutoff . The transition roll-o region lies between the at and cuto frequencies. In Fig. 12.9, three possible implementations of analog ltering are illustrated. Figure 12.9a shows the case where the ltered frequency band closely matches the size of the object being imaged (2xflat AR ).4 In this case, the FOV in the read direction LR can be chosen equal to 2xcutoff , so that there is no noise outside of the imaging region that leaks back inside. Therefore, the ltered reconstructed image, de ned by ^f , is given by ^f (x) = H (x) (^(x) + ^(x)) (12.38) It is necessary to be careful in matching the lter with the object size since, if AR > 2xflat , the image will lose intensity near the edges of the object being imaged. Figure 12.9b shows the case where the lter extends beyond the physical body and includes some frequencies which are not of interest. Since LR is less than 2xcutoff , any data obtained outside of this region will be aliased back into the image. The noise between the lter cuto s and the boundaries of the FOV on the left (right) will be mapped in reverse order inside and to the right (left) of the FOV interval. If the lter range is not so large that multiple aliasing has to be considered, a formula for the ltered reconstructed spin density may be given by 8 > H (x) (^(x) + ^(x)) + H (x ; LR )^(x ; LR ) 0 x LR =2 > | {z } > > < aliased noise ^f (x) = > (12.39) > H (x) (^(x) + ^(x)) + H (x + LR )^(x + LR ) ;LR =2 < x < 0 > | {z } > : aliased noise The fact that (x) = 0 for jxj > LR =2 (i.e., AR < LR ) means there is aliasing only of noise and not of the MR spin signal. The noise pro le does not have uniform variance, due to the aliasing e ect, and is a function of the width of the lter. As an example, if L is only slightly less than 2xcutoff , the noise will appear slightly larger at the outer edges of the image than in the center. Finally, in Fig. 12.9c the case is shown where 2xcutoff < AR . In this case, part of the object will be truncated by the lter, and the subinterval in x that is reconstructed may be found from (12.38). This method can be used to select a small volume or avoid aliasing of a large object.
4 The encoded positions xflat and xcutoff are illustrated in Fig. 12.8. 252 Chapter 12. Sampling and Aliasing in Image Reconstruction Fig. 12.9: In (a), the analog lter H (x) has been chosen to closely match the object being imaged, minimizing the e ect of noise (x) outside of the body. In (b), the applied analog lter extends beyond the boundaries of the desired object, and thereby includes additional noise contributions which will be aliased back into a reconstructed image. In (c), the analog lter is used to isolate a region within the body by ltering out the contribution from those areas not being imaged. 12.4. Nonuniform Sampling 253 In the above cases, the range of the lter was varied for a xed FOV and body size. In fact, noise and signal aliasing both can be eliminated for the read direction if LR is made larger than either the lter range or AR . This will not increase sampling time, alter the resolution x, nor change the SNR of the experiment (see Ch. 15). The only cost is in the increased storage and processing time needed for the additional data. Sampling extra data outside of the ROI in order to ensure that aliasing is avoided is generally referred to as oversampling. 12.3.3 Avoiding Aliasing in 3D Imaging In 3D imaging, Lz is often chosen to be equal to TH . However, this choice implies that the signal contributions of spins that are excited outside the slice or slab pro le (Fig. 12.10a) due to rf leakage will be aliased into the imaging region as shown in Fig. 12.10b. Images with and without this aliasing are displayed in Fig. 12.11. It is clear that the artifact seriously degrades the quality of the resulting images. Aliasing can be avoided in this situation by choosing Lz to be greater than TH (Fig. 12.10c). Unfortunately, since z = Lz =(2nz ), an increase in Lz forces the partition number nz to be larger for the same resolution, z, and, accordingly, data acquisition time also increases. 12.4 Nonuniform Sampling
Along any line of k-space data there exist factors such as eddy currents, analog-to-digital conversion errors, and timing errors that may lead to nonuniform spacing of the data points. These types of error may occur along read lines, or along phase/partition encoding lines perpendicular to a given read position. The nonuniform sampling is shown to lead to aliasing in the images. In the situation where k-space sampling of a line is not uniform, it is generally possible to model the data as being made up of several uniformly sampled data sets that are shifted in k-space from each other. To investigate the presence of aliasing, Fourier transforms are taken along the read lines or the phase encoding lines. In the following subsection, a prototypical 1D situation where two uniformly spaced data sets are combined to generate an image is studied. Afterwards, a brief introduction to analog-to-digital errors is made in the second subsection. 12.4.1 Aliasing from Interleaved Sampling Suppose the reciprocal of the minimum required FOV for a given 1D object is k = 1=L, and the data sets containing the odd points,5 so(k), and even points, se(k), are collected along a line such that the separation between two adjacent points in either data set is 2 k. A complete set obtained by interleaving the odd and even points symmetrically would have the uniform k-space step size ( k) required to avoid aliasing.
More generally, we might refer to odd lines and even lines in 2D imaging as those lines in k-space with odd index or even index where the line running through the origin is even. These are the perpendicular phase encoding lines referred to just previously.
5 254 Chapter 12. Sampling and Aliasing in Image Reconstruction Fig. 12.10: The rf excitation pro le of a typical slice select pulse in a 3D imaging experiment is shown in (a) along with the assumed slice or slab thickness TH . If a 3D experiment is done and L is chosen to be equal to TH , then a signi cant amount of aliasing from the tail of the excitation will be aliased into the slice as shown in (b). Recall that the upper tail in (a) aliases into the lower portion in (b), as indicated by the long arrow, and vice versa. If L is chosen larger, as shown in (c), then there will be no aliasing, but the number of partitions will have to be increased to maintain the same resolution z . Also, the imaging amplitude drops to zero outside the slice. Correctly Interleaved Data
The sampling function u(k) can be written as the sum of two functions ue(k) and uo(k) u(k) = ue(k) + uo(k)
which are de ned as (12.40) (12.41) (12.42) ue(k) = uo(k) = k k 1 X m=;1
1 X (k ; 2m k) (k ; (2m + 1) k) m=;1 12.4. Nonuniform Sampling 255 (a) (b) (c) (d) Fig. 12.11: Four partitions from a single 3D experiment where images (a) and (b) show two una- liased images and (c) and (d) exhibit aliasing. The head image (a) and neck image (b) correspond to partitions closer to the middle of the total slab, while the head image in (c) is from the top of the slab and shows aliasing of the neck into the brain (white arrows). The neck image in (d) shows signi cant aliasing from the brain. The fact that the aliased information from the brain is as bright as the neck tissue in (d) is due to the inhomogeneity of the rf coil used for both receive and transmit (see Ch. 16) resulting in a reduced ip angle in the neck region. 256 Chapter 12. Sampling and Aliasing in Image Reconstruction That is, ue(k) samples the even index k-space points whereas uo(k) samples the odd points. Multiplication in k-space by these comb functions leads to measured data sets se(2m k) and so((2m + 1) k), respectively.6 The reconstructed image for in nite sampling is a convolution of the physical spin density and U (x), the inverse transform of the sampling function. The latter is found to be U (x) = Ue (x) + Uo(x)
where the identities in Sec. 11.2.9 can be employed to obtain
1 X Ue (x) = F ;1fue(k)g = k (2 k x ; q) q=;1 1 1 X x ; qL = 2 2 q=;1 (12.43) (12.44) and, with further assistance from the shift theorem (11.11), Uo(x) X = 1 2 q=;1
1 F ;1fuo(k)g = F ;1fue(k ; k)g 1 1 ei2 kx X x ; q L = 2 2
q=;1 x ; q L eiq 2
1 X (12.45) The simple result (12.7) for any convolution involving a delta function leads to the respective series for the convolution of (x) with Ue(x) and Uo (x): ^e(x) and ^o (x) (x) Ue(x) = 1 2 q=;1
1 1 X (x) Uo (x) = 2 x ; qL 2 x ; q L eiq 2 (12.46) (12.47) The expressions (12.46) and (12.47) are the ingredients for the following image formula ^1(x) = F ;1fs1(k)g = F ;1fse(k)g + F ;1fso(k)g = ^e (x) + ^o (x) (12.48) q=;1 Since the odd terms cancel between the two series, the standard result for in nite sampling is recovered, 1 X ^1(x) = (x ; rL) (12.49)
r=;1 In nite sampling is considered for simplicity but without losing too much generality. As shown earlier, the nite data collection window leads to another convolution in the image domain and blurring occurs. E ects on the image reconstruction of discrete sampling can be considered independent of these windowing e ects since the convolution operator is associative as shown in Ch. 11.
6 12.4. Nonuniform Sampling 257 which exhibits an in nite number of Nyquist copies. In the interval x = ; L to x = L , the series (12.49) reduces to one term, ^1 (x) = (x) 2 2 under the assumption that the spin density vanishes outside that interval, and there is no aliasing. There is aliasing, however, in the individual odd and even series where the range of nonzero values for (x) implies that three terms q = ;1, 0 and +1 contribute to the series (12.46) and (12.47). Also, these contributions can be better circumscribed by splitting the physical spin density into its left-side and right-side values de ned by (x) = l (x) + r (x) where and
l (x) r (x) (12.50) (12.51) (
0 (x) ;L=2 < x 0 otherwise (x) 0 < x < L=2 (12.52) 0 otherwise The left/right decomposition is particularly useful for describing aliasing features inside the FOV and leads to ^e(x) = 1 x + L + (x) + x ; L 2 2 2 1 (x) + 1 L + x; L = 2 l 2 r x+ 2 2 ( ; L=2 < x < L=2 (12.53) ^o(x) = 1 ; x + L + (x) ; x ; L 2 2 2 L + x; L 1 (x) ; 1 ; L=2 < x < L=2 (12.54) = 2 l 2 r x+ 2 2 The extra terms in each equation are left-hand and right-hand functions representing aliasing that cancels if the even and odd sets are combined according to (12.48). The aliasing e ect in images reconstructed with just the even or odd points alone can be investigated through (12.53) and (12.54). Consider the simple boxcar spin density illustrated in Fig. 12.12. The reader is asked to verify the plots shown for ^e(x) and ^o (x) in the gure, which exhibit aliasing features adjacent to the object (the object is only one-third of the FOV). Their sum, however, is correctly interleaved and gives the expected result (x). Problem 12.9
Employ (12.53) and (12.54) to derive the plots showing aliased features for ^e(x) and ^o(x) for the boxcar object in Fig. 12.12. 258 Chapter 12. Sampling and Aliasing in Image Reconstruction Fig. 12.12: Reconstructed images for in nite sampling of a unit boxcar object centered in the FOV with width equal to L=3. Aliasing in the uniform 1D k-space sampling case occurs whenever the odd points or even points are missing along a line in k-space. There is no aliasing for the given FOV, however, provided that the point separation is as shown for u(k), and, indeed, interleaving the even and odd series leads to ^1 (x) = (x). The vertical arrows represent the relative strengths
of the delta functions and there is an in nite number of sampling points in each row of vertical arrows. 12.4. Nonuniform Sampling 259 Incorrectly Interleaved Data
In the incorrectly interleaved case, the separation between the interleaved points is not equal to k. Consider the case where ue(k) is still given by (12.41), but the odd sampling function is shifted slightly in k-space and is modeled by uo i(k) = k 1 X where ;1 < < 1 ( = 0 represents perfect sampling for the odd points). The subscript i is present to remind the reader that this is the `incorrectly interleaved case' for the odd-point sampling function. From the shift theorem for the Fourier transform of (12.55), the x-space sampling function is found to be 1 1 X x ; q L ei q(1; ) Uo i(x) = 2 (12.56) 2 q=;1 Steps taken in the previous, correctly interleaved, case can be repeated such that ^o i (x) is found to be 1 ^o i(x) = 2 x + L e;i (1; ) + (x) + x ; L ei (1; ) 2 2 1 (x) + 1 L e;i (1; ) + x ; L ei (1; ) = x+ 2 l 2 2 r 2 (12.57) The aliasing terms exhibit additional phase shifts beyond those found previously, and now would not cancel their counterparts in (12.53). Let ^i(x) represent the incorrectly interleaved reconstructed image. The combination of (12.53) and (12.57) gives i (1; ) ;i (1; ) L ^i (x) = (x) + 1 + e 2 x + L + 1 + e2 r l x; 2 2 (12.58) It is observed that, for 6= 0, the aliased information is not cancelled by the addition of the two data sets. Furthermore, neither adding the magnitudes of the separate series nor taking the magnitude of the sum (12.58) eliminates aliasing. m=;1 (k ; (2m + 1 ; ) k) (12.55) ;L=2 < x < L=2 ;L=2 < x < L=2 Problem 12.10
Redo Fig. 12.12 for both the real and imaginary parts of ^i(x), ^e(x) and ^o i(x) utilizing (12.58) and = 0:5. Comment on the amplitude of the ghosts relative to the original object. The object is the same boxcar distribution as in Prob. 12.9. 260 Chapter 12. Sampling and Aliasing in Image Reconstruction Another advantage of describing these images in terms of their right and left values is that the nal image can be written in terms of matrix equations for the left and right sides of ^e and ^o i. This treatment gives a more manageable analytical approach to an arbitrary number of interleaves, and it also lends itself more readily to numerical computer work. Consider a matrix representation for (12.58). Noting that l (x ; L=2) produces right-side values, the right-hand-side equation, in particular, can be written as ^r (x) = ^re (x) + ^ro i (x) (12.59) The even and odd components of the right half of the reconstructed image can be written in matrix notation as 1 ^re (x) = 1 1 ^ro i (x) 2 1 ei (1; ! !
) l x; r (x) !
L 2 0<x< L 2 (12.60) The right and left parts of (x) may now be found by inverting the matrix transformation in (12.60) yielding
r (x) L l x; 2 ! = (ei (1; ) ; 1) 2 ei ;1 (1; ) ;1
1 ! ^re (x) ^ro i (x) ! 0 < x < L (12.61) 2 By translating l (x ; L=2) to l (x), the original physical spin density is recovered. When is known, two images, e(x) and o i(x), are reconstructed. This is accomplished by taking the full data set s(k) and replacing all so i(k) points with zero to get the se(k) data set and replacing all se(k) points with zero to obtain the so i(k) data set, respectively. The resulting two images, (both of them aliased in this case) are then processed using (12.61) to extract an unaliased image. Problem 12.11
Using (12.61) and = 0, show that r (x) and l (x ; L=2) turn out to give the correct right and left sides of ^(x), respectively. Hint: see (12.53) and (12.54). Aliasing in the incorrectly interleaved case occurs simply because there are two di erent k-space step sizes. If this is extended to many di erent step sizes, the above methods can be generalized to multiple k-space interleaves. In certain sequences, the most e ciently implemented versions cover k-space in such a way that nonuniform samples in k-space are obtained, and this nonuniformity, together with any resulting aliasing, can be analyzed using multiple interleave formalisms. The following problem deals with such a generalization of the interleaved k-space sampling problem. 12.4. Nonuniform Sampling 261 Problem 12.12
a) Consider a four-point interleaved problem (i.e., the case where every fourth point is collected in one experiment, and the entire k-space coverage is obtained after four experiments) where the four sampling functions are de ned as 1 X uj (k) = k (k ; (4m + j ) k) (12.62) and j = 0 1 2 3. Generalize the analysis carried out in the previous discussion to this case. Note that in this case, the FOV must be divided into quarters for a similar analysis to be performed. Verify the following matrix equation in this case
m=;1 (12.63) b) Rewrite the 4 4 matrix in (a) for the case where sampling functions 1 2 3 are shifted in k-space by 1 k, 2 k and 3 k, respectively. c) Qualitatively describe the e ect on the image when 1 , 2 and 3 are all nonzero in terms of the aliasing and the noise present in the reconstructed image. 0 ^ (x) 1 0 1 B ^0 (x) C 1 B 1 B 1 C= B B ^ (x) C 4 B 1 @ 2 A @ ^3 (x) 1 1 i ;1 ;i 1 ;1 1 ;1 1 ;i ;1 i 10 1 (x) C B (x ; L=4) C CB C B (x ; L=2) C C [email protected] A (x ; 3L=4) L <x< L 4 2 12.4.2 Aliasing from Digital-to-Analog Error in the Gradient Speci cation
The gradient step, Gy G for phase encoding required to give a large eld-of-view can be su ciently small that round-o errors result. Consider an example where G = 12:5 GDAC min where GDAC min is the smallest possible step size that the MR gradient ampli er can produce. The phase encoding table would lead to the series 12.5, 25, 37.5, 50, ... (in units of GDAC min), but, since only integer units are possible in the digitalto-analog conversion (DAC), the series 13, 25, 38, 50, ... is realized instead. In terms of gradient steps, the series would be characterized as G0 , 2 G, 2 G+ G0 , 4 G, etc., where G0 = 13 GDAC min. This is an example of the even and odd interleaving problem where, ignoring the change in units, the series corresponds to k = 12:5 kDAC min and k = 0:5 kDAC min in the earlier notation. Aliasing would arise from the indicated shift for nonzero values. 262 Chapter 12. Sampling and Aliasing in Image Reconstruction Problem 12.13
Assume that the round o error for a system is such that GDAC min
Answer the following questions. Gy ! = 15 4 a) What is for the point where Gy = Gy in this case? b) With what frequency will points be sampled at the correct location in kspace given the above parameters? The nonuniform sampling that occurs in this case varies periodically throughout k-space. c) Based upon the sampling variations with what period do you expect to see the object repeated in the image? 12.4. Nonuniform Sampling 263 Suggested Reading
Sampling principles are addressed in detail in: R. N. Bracewell. The Fourier Transform and its Applications. McGraw Hill, New York, 1986. O. E. Brigham. The Fast Fourier Transform. Prentice-Hall, Englewood Cli s, New Jersey, 1974. A. V. Oppenheim and R. W. Schafer. Discrete-time Signal Processing. Prentice-Hall, Englewood Cli s, New Jersey, 1989. A. Papoulis. The Fourier Integral and its Applications. McGraw-Hill, New York, 1987. 264 Chapter 12. Sampling and Aliasing in Image Reconstruction Chapter 13 Filtering and Resolution in Fourier Transform Image Reconstruction
Chapter Contents
13.1 13.2 13.3 13.4 13.5 13.6
Review of Fourier Transform Image Reconstruction Filters and Point Spread Functions Gibbs Ringing Spatial Resolution in MRI Filtering Due to T2 and T2 Decay Zero Filled Interpolation, Sub-Voxel Fourier Transform Shift Concepts and Point Spread Function E ects 13.7 Partial Fourier Imaging and Reconstruction 13.8 Digital Truncation Summary: The point spread function introduced by reconstruction from limited Fourier
data is established. The Gibbs ringing associated with truncated data is considered along with its reduction by image ltering. The spatial resolution of an image reconstruction method is formally de ned and discussed. Partial Fourier data collection and image reconstruction of asymmetric k-space data are presented. Digital data truncation e ects and methods to overcome them are also reviewed. Introduction
Studying discrete Fourier transforms is only the initial step in developing an accurate MR image reconstruction. The e ects of Fourier inversion from sampled and limited data (` ltered' data) are quantitatively modeled in terms of the `point spread' or `blur' function (Sec. 13.2). 265 266 Chapter 13. Filtering and Resolution The point spread function arises from the convolution of the Fourier transforms of the signal and the lters. One of the artifacts that arise from the presence of an oscillatory point spread function is Gibbs ringing which occurs at all step-like discontinuities in the object (Sec. 13.3). Spatial ltering is used to reduce Gibbs ringing at the expense of blurring the reconstructed image. Spatial resolution of a given reconstruction method is also formally de ned (Sec. 13.4). From this de nition, it is shown that, when transverse relaxation e ects during sampling can be neglected, the spatial resolution is described by the pixel size. When spatial ltering is used to reduce Gibbs ringing, the e ective spatial resolution is degraded. Similarly, transverse decay e ects during data sampling are shown to worsen the spatial resolution. Using the formalism of specifying the blur by a simpler de nition of the full width at half maximum of the point spread function leads to certain interesting results. For a spin echo sequence, two blurring functions exist: one corresponding to a k-space asymmetric T2 decay lter, and the other corresponding to a k-space symmetric T20 decay lter whose blurring e ects are found to be insigni cant even for data sampling times Ts which are comparable to T20 and T2 (Sec. 13.5). For a gradient echo sequence, it is found that FID sampling and image reconstruction by k-space data complex conjugate symmetrization (partial Fourier imaging) lead to an image with better spatial resolution than image reconstruction from a symmetrically sampled k-space data set (Sec. 13.5). The demodulated, ampli ed and sampled coil-detected emf is rst discretized or digitized by an analog-to-digital converter (ADC) before being stored in numeric form on a computer for image reconstruction. These stored data represent the collected k-space data. Some important practical ADC issues related to the selection of ampli er gain are considered, and the e ects of data truncation, when the gain is incorrectly set high enough that some of the k-space data of small objects are lost, are discussed. A few simple methods for removing this `ADC noise' are introduced. The limitations of a low resolution ADC are brie y discussed in relation to the acquisition of 3D data. In the following two sections, two important properties of discrete Fourier transform image reconstruction are discussed: Gibbs ringing, an end result of the collection of only a nite amount of discrete k-space data, and the spatial resolution obtained when the discrete Fourier transform is used for reconstruction. The e ects of Gibbs ringing may be reduced by ` ltering' the k-space data before reconstruction. This is done, however, at the expense of spatial resolution. 13.1 Review of Fourier Transform Image Reconstruction
In the previous chapter, the MR imaging experiment has been introduced as one where the data are discretely collected over a nite region of k-space, and an image is subsequently obtained by a discrete inverse Fourier transform of that data. Chapter 12 contained, in particular, an explanation of the necessary density of k-space sampling so that the image can be reconstructed without aliasing. All these discussions describe the simplest data collection strategies for a given object. In this section, the progress made so far on the image 13.1. Review of Fourier Transform Image Reconstruction reconstruction problem is reviewed and summarized. 267 13.1.1 Fourier Encoding and Fourier Inversion In Chs. 9 and 10, it was shown that the measured MR data are equivalent to a k-space representation of the excited magnetization of the sample. That is, 3D spatial encoding through magnetic eld gradients leads to a signal which is a Fourier integral of the e ective spin density ZZZ s(kx ky kz ) = dx dy dz (x y z)e;i2 (kx x+ky y+kz z) (13.1) Such developments are referred to as Fourier encoding methods. The reconstruction method for nding (x y z) through the inverse Fourier integral (x y z) = ZZZ dkx dky dkz s(kx ky kz )ei2 (kx x+ky y+kz z ) (13.2) is referred to as Fourier inversion. 13.1.2 In nite Sampling and Fourier Series Data measurement, as described in Ch. 12, is done by sampling the Fourier encoded data rather than by continuous monitoring, leading to discrete k-space coverage. When in nite k-space sampling is carried out, an inverse Fourier series replaces the continuous inverse Fourier transform. The 1D series1 has been de ned previously as ^1(x) and given by (12.4) ^1(x) = k
1 X p=;1 s(p k)ei2 p kx (13.3) The theory of in nite Fourier series can be applied here. For example, the reconstruction ^1(x) converges, in the absence of aliasing, to (x) in the mean squared sense when the 1 object (x) has no step discontinuities, and to 2 (x+ ) + (x; ) when a step discontinuity 0 0 occurs at x = x0 . 13.1.3 Limited-Fourier Imaging and Aliasing Time limitations restrict the number of data samples collected per readout (Nx), as well as the number of phase encoding (Ny ) and partition encoding (Nz ) steps. With only partial coverage of k-space, the inversion problem is called a limited-Fourier inversion problem. The 1D reconstructed image, de ned as ^(x), is the truncated version (12.20) ^(x) = k
1 n;1 X p=;n s(p k)ei2 p kx (13.4) As in previous chapters, often we restrict ourselves to one dimension. Thanks to the separability of the discrete Fourier transform, the image reconstruction in the other two dimensions can be assumed to be carried out independently and in a similar fashion. 268 Chapter 13. Filtering and Resolution Now it is the theory of discrete Fourier transforms that is relevant. The transform sum is over N = 2n data points in k-space, with individual widths k for each of the N points. The total width W of k-space coverage is W = N k = 2n k = kmax ; kmin + k (13.5) We recall that (13.4) follows (13.3) in exhibiting the periodicity x ! x +1= k. Therefore, to avoid aliasing, the intervals k must satisfy the Nyquist criterion k 1 L 1 A (13.6) That is, the FOV, L, must be greater than the physical size A of the object (more generally, the region to which the experiment is sensitive). 13.1.4 Signal Series and Spatial Resolution
The other member of the discrete transform pair (12.34) is the nite series for the signal over truncated and discretized spatial positions
n;1 X q=;n s(k) = ^(q x)e;i2 kq x (13.7) The evaluations of ^(x) are in terms of the spatial step size x, L 1 x = N = N1 k = W (13.8) Equation (13.8) provides a starting point in the discussion of `spatial resolution,' or the size of the smallest feature that can be measured for a given object. A familiar aspect of Fourier analysis is the need for large values of k in order to describe smaller and smaller spatial features this is re ected in the combination of (13.5) and (13.8). Given the linear dependence of k on the applied gradient, it follows that an increase in gradient strength implies a smaller value for x. The e ects of the size of W on image quality are considered in Sec. 13.4. There are other important factors in the determination of the best possible spatial resolution, such as the signal-to-noise ratio, which is the subject of Ch. 15. The ltering of data is fundamentally connected to spatial resolution and is introduced in the next section. These issues notwithstanding, a naive lower limit on x is instructive, especially in comparison with the more realistic estimates addressed later in the chapter. This is considered in the following problem. 13.2. Filters and Point Spread Functions 269 Problem 13.1
What is the best resolution x that can be expected from proton imaging for a read gradient of 20 mT/m, without regard to signal-to-noise? Assume that the sampling time is limited by the intrinsic spin-spin relaxation time, taking a representative value for T2 of 50 ms. Investigate the change in your answer for the largest read gradient strength in commercial use. 13.2 Filters and Point Spread Functions
In this section, a framework for quantitatively understanding the e ects of sampling and truncation in terms of ` lters' and their associated Fourier transforms is developed. A kspace lter2 is any function H (k) that multiplies the k-space MRI data. The inverse Fourier transform of a k-space lter is de ned to be the associated point spread function h(x). The reconstructed image ^(x) may then be found by convolution of h(x) with (x). The convolution operation is a convenient procedure for the investigation of any `blurring' that occurs due to the lter, as noted in the last chapter. The imaging artifacts created by di erent lters, such as truncation and sampling, are considered below. An easy way to understand the e ect of a lter is to look at the simple example of (x) = (x). In this case, ^(x) will be equal to the Fourier transform of the lter (hence the name, `point spread function'), since convolution of a function with (x) returns the function.3 The reconstructed image will now have nite values at locations other than the origin it has been smeared or `blurred' by the e ect of the lter and the reconstructed image is not equivalent to the physical spin density. This example gives insight into how a lter will a ect an arbitrary image, since any object spin density can be described as a sum of individual spins or delta functions. As an image of a delta function produced by the lter, the point spread function is also called the `impulse response function.' 13.2.1 Point Spread Due to Truncation Truncation of data is equivalent to saying that s(k) has been ltered by the rect function. Suppose that the k-space data are collected continuously, but only from ;n k to (n ; 1) k, the limits in (13.4). Then the collected data are said to be truncated, or windowed, by the lter Hw (k) given by the rect function in (12.18)
1 k+2 k Hw (k) rect W
2 ! (13.9) For consistent notation, an x-space lter, such as that already introduced in Sec. 12.3, is referred to as H (x). The caveat again arises that H (k) and H (x) are generic labels and there is no implied relation between them. 3 From Sec. 12.1.1, we recall that the convolution of f (x) with (x ; x ) is equal to f (x ; x ). 0 0 270 The collected data are computed by Chapter 13. Filtering and Resolution sw (k) = s(k)Hw (k) (13.10) where s(k) is the untruncated continuous Fourier transform of (x). The reconstructed image from this data is ^w (x) = F ;1 s(k)Hw (k)] = (x) hw (x) (13.11) where hw (x) is de ned to be the inverse Fourier transform, or the point spread function, of Hw (k). From (12.23), hw (x) = W sinc( Wx)e;i k x (13.12) The blurring due to the convolution of the spin density with (13.12) has been described earlier in Sec. 12.2. In the present language, the function hw (x) spreads the information from each point and maps spin information from a range of physical locations into each position x. Attention is now directed to include the e ects of discrete sampling.
The modeling of both windowing and discrete sampling is through the product (12.18) of a rect function with the sampling function, ! 1 1 n;1 X k+2 k X Hws(k) k rect (k ; p k) = k (k ; p k) (13.13) W p=;1 p=;n The measured data are the nite set of s(p k) arising as coe cients in the distribution sm (k) given in (12.20)
n;1 X p=;n 13.2.2 Point Spread for Truncated and Sampled Data sm(k) sws(k) = s(k)Hws(k) = k s(p k) (k ; p k) (13.14) From (13.13) and (13.14), the Fourier inversion yields the image (12.21) ^(x) ^ws(x) = Z1 ;1 dk sm (k)ei2 kx = k n;1 X p=;n s(p k)ei2 p kx (13.15) The original notation had no subscript for the nitely sampled reconstructed spin density. Alternatively, ^(x) can be expressed as the convolution ^(x) = (x) hws(x)
n;1 X p=;n (13.16) The inverse Fourier transform hws(x), or the point spread function in the nite-Fourier inversion case, is found simply by replacing s(k) by unity in (13.15). The result is hws(x) = k ei2 p kx (13.17) 13.2. Filters and Point Spread Functions 271 The explicit summation of (13.17) is given as a problem, and the following simpli ed expression is obtained sinc( hws(x) = W sinc( Wx)) e;i k x (13.18) kx Problem 13.2
a) Derive the identity
n;1 X p=;n xp = x;n 11; xx ; 2n (13.19) b) Derive (13.18) from (13.17). c) Show that the phase factor e;i x k is eliminated by symmetrizing the sum from ;n to n, rather than from ;n to n ; 1. As in the continuous case, the point spread function hws(x) creates a blurred version of (x) to yield ^(x). The primary di erence between hw (x) and hws(x) is that the latter also leads to aliasing, or overlapping, of the images corresponding to the blurring of the spatially repeated images into one another. However, while hw (x) only includes the e ect of truncation, it accounts for most of the blur in an image. 13.2.3 Point Spread for Additional Filters Other factors that modify the MRI signal may also be modeled as lters. They include the relaxation exponentials, involving T2 or T2 , which change the magnitude of the signal. Furthermore, it is sometimes useful to modify the data after collection (i.e., `post-process' the data) in order to modify the `blurring' caused by hws(k), minimize other artifacts, or enhance certain image features. This is often done by applying another lter to the data. The lters in this text will be denoted by Hfilter (k) where the generic subscript filter will be the name of the speci c lter under discussion. The same subscript will be added to the reconstruction spin density or image, so that, for several lters, we would write ^filter 1 filter 2 . The standard windowing and sampling lters will be left understood, consistent with the previous usage in representing the signal (13.14) and the reconstructed image (13.15). The signal obtained by the addition of one lter to the standard windowing and sampling can be written in a variety of ways sm filter (k) = sm(k) Hfilter (k) = s(k) Hws(k) Hfilter (k) ^ s(k) Hws filter (k) (13.20) where Hws filter (k) is shorthand for all three lters applied to the data. As a result, the reconstructed image can be written in terms of various convolutions 272 Chapter 13. Filtering and Resolution ~filter (x) = ^(x) hfilter (x) = (x) hws(x) hfilter (x) (x) hws filter (x) (13.21) Throughout the remainder of this text, lters will be considered in terms of their own point spread functions hfilter (x), or of the combined point spread function hws filter (x) for all lters. Examples of such k-space lters are treated throughout the remainder of this chapter. 13.3 Gibbs Ringing
It is well-known that Gibbs ringing accompanies Fourier series representations of functions with step discontinuities. The presence of any bright objects or sharp transitions from one tissue to another can lead to the Gibbs e ect as an artifact in MRI. This is particularly serious if it mimics certain disease states. In the present section, Gibbs ringing is introduced, illustrations are provided, and the reduction of the Gibbs e ect by ltering is discussed. In the numerical evaluation of a Fourier series, and especially in the nite series de ned in the discrete Fourier transform, Gibbs ringing arises as an oscillating overshoot and undershoot in the immediate neighborhood of a step discontinuity. The overshoot (above the top of the step) and undershoot (below the bottom of the step) have the property that, as the total number N of terms in the series tends to in nity, both the peak overshoot and the peak undershoot approach a limiting value of approximately 9% of the step height. Using the reconstructed spin density as an example, and assuming that a step discontinuity occurs at x = x0 , lim ^(x0 ) ; (x0 ) ' 0:09 (x+) ; (x; ) (13.22) 0 0 N !1 13.3.1 Gibbs Overshoot and Undershoot where x0 lim x0 . A derivation of the `9%' result employing the convolution (13.16) !0+ is presented next. This derivation also is useful for showing how other properties of Gibbs ringing may be analyzed with the convolution representation. An alternative demonstration using Fourier series is in the form of a problem given later. Suppose we have a function f (x) which is continuous at all points except at x = 0, where it has a step discontinuity. With a step amplitude of jf (0+) ; f (0;)j at that discontinuity, f (x) can be represented by the sum fc(x), and f (x) = fc(x) + (f (0+) ; f (0;)) (x) (13.23) where fc(x) is a continuous function and (x) is the by-now familiar Heaviside step-function. Since the Gibbs e ect originates from windowing the data, a window lter is assumed to be applied to the k-space representation of f (x). For convenience, consider a symmetric window with width W sym Hw = rect(k=W ) (13.24) 13.3. Gibbs Ringing which implies that the reconstructed image is given by the convolution f^(x) = f (x) hsym(x) w where the symmetric point spread function is (13.12) without the phase factor 273 (13.25) (13.26) hsym (x) = W sinc( Wx) w The next step is to compare the values of the reconstructed function at the two di erent positions, x = 0 and x = x, in the limit x ! 0 or W = 1= x ! 1. The identity (9.29) gives lim hsym (x) = (x) (13.27) W !1 w Equations (13.23), (13.27), and (12.7) combine to give lim f^(x) = fc(x) + (f (0+) ; f (0;)) Wlim W !1 !1 Z1
0 hsym(x ; x0 )dx0 w
(x ; x0) (13.28) Before taking the limit, the integral in (13.28) can be reduced to Z1 hsym (x ; x0 )dx0 0 w hsym (X )dX w ;1 Z Wx sin y = 1 y dy ;1 = 1 + 1 Si( Wx) 2
= Zx where X (13.29) where Si(x) is the `Sine integral' de ned as Z x sin y dy 0 y and Si(;1) = ; =2.4 With Si(0) = 0 and Si( ) ' 1:8519,
Si(x)
lim f^(0) = f (0;) + 1 (f (0+) ; f (0;)) W !1 2 1 (f (0;) + f (0+)) = 2 (13.30) (13.31) and (recall W = 1= x) 1 lim f^( x) ' fc( x) + (f (0+) ; f (0;)) 2 + 1 + 0:09 W !1 2 +) ; f (0;)) ' f ( x) + 0:09(f (0 (13.32) noting f ( x) = fc( x) + (f (0+) ; f (0;)). In similar fashion, it can be shown that lim f^(; x) = f (; x) ; 0:09(f (0+) ; f (0;)). W !1
4 See also (9.30). 274 Chapter 13. Filtering and Resolution 13.3.2 Gibbs Oscillation Frequency
The above demonstration makes it clear that, if N is large and N ! 2N , the peak-to-peak di erence in the ringing is invariant. However, the peaks of the overshoot and undershoot move half the distance closer to the discontinuity than before. For an in nite Fourier series, the oscillations are thereby squeezed into an in nitesimal physical region. For a nite Fourier series (such as in the discrete Fourier transform), the oscillations have a nite width. In particular, an invariance property of Gibbs ringing can be seen from the expression (13.18) for hws(x). For a xed FOV and N ! 2N , we have W ! 2W and x = 1=W ! x=2. More generally, the product Wx = Wq x = q depends only on the pixel number q. Thus the oscillations in the sinc( Wx) factor in the numerator5 of (13.18), which are responsible for the Gibbs e ect, scale with the pixel size. The lter is e ectively invariant in that it is a function of pixel number only. This property of invariance for Gibbs ringing is also illustrated in the problem given below. Both properties are shown in Fig. 13.1. Problem 13.3
Consider the function f (x) = rect(x=A) with its two discontinuities at x = ;A=2 and x = A=2. a) Find a cosine Fourier series for this function and investigate the Gibbs ringing at one of the discontinuities by numerical evaluation of the rst N terms. Find the 9% overshoot and undershoot peaks, and show that the distance of these peaks from the discontinuity is proportional to 1=N for large N . b) Also comment on the possible increase in the e ective amplitude of the Gibbs ringing when these two edges are at some separation which leads to a coherent addition of the ringing. Problem 13.4
Gibbs ringing can occur from a discontinuity in magnitude or phase. Investigate the e ects of nite sampling for the same object f (x) used in the previous problem but with a phase (x) = 0 for x 0 and (x) = =2 for x > 0. Find the complex Fourier transform of f (x) and then use a DIFT to estimate the image. Assume L = 2A = 256 mm and N = 256. The argument, or spatial frequency, of the sinc function in the denominator is reduced by a factor of 1=N , in comparison with that of the numerator.
5 13.3. Gibbs Ringing 275 Fig. 13.1: Gibbs oscillations resulting from the convolution of a step discontinuity with the point spread function associated with nite and discrete sampling in k-space. For large N , the amplitude
of the truncation artifact remains invariant as a function of pixel number. The oscillations, however, increase in spatial frequency, with a scale set by the resolution. 13.3.3 Reducing Gibbs Ringing by Filtering
Gibbs ringing can seriously degrade an image and its interpretation. Increasing N for a xed FOV alleviates the problem in view of the fact that the spatial distance over which the ringing propagates is reduced. Hence, running a scan twice with two di erent values of N can help determine whether or not an object is a Gibbs artifact (see Fig. 13.1). There may be insu cient time, however, to double the number of scans whenever artifacts appear. A di erent remedy is the use of an additional lter to smooth out the image. retrospectively evaluating, or postprocessing, the data. If the data set is multiplied by a function which vanishes, along with its rst derivative, at k = kmax , it is said to have been apodized. Again, for convenience, k-space is ltered symmetrically, with W = 2kmax . For 276 example, if s(k) is multiplied by Chapter 13. Filtering and Resolution ! 1 + cos 2Wk k HHanning (k) = = cos2 W (13.33) 2 it has been apodized with a Hanning lter. If k ! kmax , then 2 k=W ! , and it is 0 veri ed that HHanning (k) ! 0, and HHanning (k) ! 0, The Fourier transform of HHanning (k) easily follows from its decomposition into exponentials, 1 hHanning (x) = 2 (x) + 1 ( (x ; x) + (x + x)) (13.34) 4 The corresponding ltered reconstructed spin density is also simple to determine ^Hanning (x) = hHanning (x) ^(x) = 1 ^(x ; x) + 1 ^(x) + 1 ^(x + x) (13.35) 4 2 4 It is seen that this k-space lter corresponds to an `averaging' in the image domain.
Since images are generally presented as pixel maps, the images and the lter point spread functions may be speci ed discretely and indexed by integers. In this representation, a 2D image, for example, is a 2D matrix of numbers. A 1D lter such as h(x) is represented by a 1D vector which is usually referred to as convolution mask. The physical dimensions are added separately as scales. The Hanning result (13.35) may be rewritten as X ^Hanning (q x) = mHanning r (q)^(r x) r 1 1 (13.36) = 4 ^ ((q ; 1) x) + 2 ^(q x) + 1 ^ ((q + 1) x) 4 where the Hanning three-point mask has been de ned by 8 1=4 r = q ; 1 > > 1=2 r = q < mHanning r (q) = > 1=4 r = q + 1 (13.37) > : 0 otherwise The Hanning mask acts only on the pixel q under consideration and its two immediate neighbors. Since masks are speci ed by a vector of length equaling the number of a ected neighborhood pixels, hHanning (x) is said to be represented by a vector of length 3. The number 3 for the Hanning mask vector length is related to the requirement that the function and its rst derivative vanish at the maximum k values. It leads to smoothing of oscillations due to sharp features in the data. In the convolution picture, Gibbs ringing is understood by the oscillations from one pixel to the next produced, in symmetric windowing, by the function hsym (x) = W sinc( Wx) (13.38) w Convolution Masks 13.4. Spatial Resolution in MRI 277 The averaging over three neighboring pixels implied by (13.36) reduces much of the ringing, as illustrated in Fig. 13.2. Fig. 13.2: The model object is a rect function. The reconstructed image is obtained by windowing
and sampling (DFT image reconstruction). The ltered image is found by apodization with the Hanning lter. The Gibbs ringing is dramatically reduced after ltering but at the expense of a loss of resolution. 13.4 Spatial Resolution in MRI
The spatial resolution of an imaging method refers to the smallest resolvable distance between two di erent objects, or two di erent features of the same object. The de nition for this distance is subject, however, to the choice of object, the limitations on measurements, psychological matters and more. It is recalled that the point spread function can be used to quantify the spread or blur due to the reconstruction method. Indeed, the point spread function for a perfect experiment with in nite data could be described by a delta function. 278 Chapter 13. Filtering and Resolution The convolution with a delta function yields a perfect reconstruction6 with vanishing voxel size. In a real experiment, a nonzero spatial resolution may be de ned in terms of a point spread function. A rst de nition of spatial resolution may be chosen to be the area under h(x), for an arbitrary imaging method with a point spread function h(x), divided by the value of the lter at the origin: spatial resolution of lter xfilter `blur' of lter = h (0) filter = Hfilter (0) h (0)
filter 1 Z1 ;1 dx hfilter (x)
(13.39) This gives an e ective measure of the spread or width of the point spread function.7 For a boxcar function, for example, whose height is hfilter (0), xfilter is its width. In general, it may be assumed that the lter has a global maximum at the origin, H (k) H (0). In those cases where distributions, such as delta functions, are used, the spatial resolution may need to be de ned as a limit. Problem 13.5 Find the spatial resolution xGaussian associated with a Gaussian lter h(x) = e;ax2
for a real constant a > 0. In the discrete Fourier transform reconstruction case, hws(x) is periodic with period L and the de nition (13.39) would lead to an in nite width. Consider, instead, the spatial resolution of the windowed and sampled MR image to be de ned in terms of the area within the FOV Z L=2 xMRI filter MR spatial resolution with lter = h 1 (0) dx hws filter (x) ;L=2 ws filter (13.40) This de nition is used to estimate the smallest separation between two objects which can be determined with a standard MRI reconstruction with additional lters. The integration in (13.40) is readily carried out, in the absence of an additional lter, utilizing the representation (13.17) and giving
The reader is warned about ambiguities in the language used to describe resolution. Better or improved or `higher' resolution means smaller or lower or decreased resolvable distances. Worse or `lower' resolution means larger or increased distances. 7 As indicated in the equation, the spatial resolution x filter is also called the `blur' of the given set of lters.
6 13.4. Spatial Resolution in MRI 279 window and sampling resolution with no additional ltering n;1 Z L=2 X = Pn;11 dxei2 p kx 1 p=;n ;L=2 p =;n X 1 n;1 L = 2n p0 p=;n 1 = 2n1 k = W = x = the Fourier pixel size (13.41) According to the de nition (13.40), the smallest distance that can be resolved with a discrete Fourier transform reconstruction is exactly the same as the `Fourier pixel size' x already introduced in the previous chapter and in Sec. 13.1.3.
0 xMRI Spatial Resolution and k-Space Coverage
The broader the coverage in k-space, the better the MR spatial resolution. This well-known statement follows from (13.41) and brings up the relationship of k-space coverage to the reconstruction of objects with structures over di erent scales. Consider 2D images with sampling windows of variable sizes Nx Ny and xed L. Increases in Nx and Ny lead to decreases in x and y, respectively. The reduced voxel size (in this case, a pixel size) should lead to improved or higher resolution. An example is presented in Fig. 13.3 involving a `resolution phantom,' where a hierarchy of di erent sized holes are studied under changes in the voxel size. The results of reconstructing images from windows of size 64 64, 128 128, 256 256 and 512 512 with a xed FOV of 256 mm are shown. The coverage of a larger window in k-space improves the de nition of the small holes or `resolution elements.' In particular, the Gibbs ringing pattern becomes less noticeable as the voxel size decreases. An additional e ect to be discussed in Ch. 15 is p that a halving of voxel size in each direction leads to a 2 loss in signal-to-noise ratio. An interesting enhancement of the Gibbs artifact occurs when two or more edges from a series of objects in a row come close enough together that the ringing from each edge coherently combines with that from all the other edges. In this instance, the ringing extends signi cantly past the edge of the outermost object (see arrows in Figs. 13.3a through 13.3c). The row where this occurs shifts as a function of resolution. Information Content in k-Space
The improvement in spatial resolution brought about by the inclusion of larger k-space values brings up a more general point. Pictures in k-space show the spatial frequency components necessary for the Fourier representation of the physical spin density function. Low spatial frequency components are needed for spatially slowly changing parts of the object, while the high spatial frequency components must be included to represent small structures whose size is on the order of the voxel size. Small features of particular interest in MRI are tissue boundaries. 280 Chapter 13. Filtering and Resolution (a) (b) (c) (d) Fig. 13.3: Resolution phantom images reconstructed from ltering a 512 512 matrix size acqui- sition in a xed FOV of 256 mm. The reconstructions are from the reduction of the full window down to a (a) 64 64 window, (b) 128 128 window, (c) 256 256 window, and also the full (d) 512 512 window. The improved resolution and reduction in Gibbs ringing e ects are evident for each increase in the window size, which implies a decrease in the pixel size x y. 13.4. Spatial Resolution in MRI 281 The relationship of regions in k-space to di erent scales may be better understood by considering two extreme limits: a 1D object lling the entire FOV and a 1D object whose size equals x. For a constant pro le, the rst object maps to an `impulse' at k = 0 see the following problem. More generally, data points in the neighborhood containing the k = 0 point are needed to represent such slowly varying structures in the object. An example of the information content in the `low frequency components' was demonstrated in Fig. 13.3a, where the 512-point data set was truncated down to 64 points (e ectively, a low pass lter was applied in each direction). Evidently, almost all information about the large structures is already contained in the central 1/64th fraction of the 512 512 data set. (a) (b) Fig. 13.4: A result of high pass ltering, in k-space, of the same imaging object as in Fig. (13.3). (a) The removal of a central portion of k-space signal data (a center square that is 1/64th of the 512 512 window has been replaced by zeros). (b) The corresponding reconstructed image is left with only the edges (i.e., the rapid spatial variations) and the small resolution elements highlighted. Although the image is similar to the 1D derivative image in Fig. 11.6, it is more akin to a 2D derivative image since a 2D k-space lter was used. The small resolution elements, on the other hand, are indistinguishable in Fig. 13.3a because of severe blurring, in contrast to the high resolution image of Fig. 13.3d. An object with constant pro le whose size equals x has a sinc function for its k-space representation, whose rst zero crossing occurs only at 2kmax this is also expanded upon in the next problem. While this means that all collected k-space points contain some information about such small objects, the high spatial frequency components are especially necessary. The image in Fig. 13.4b has been obtained by setting the central part of k-space to zero (the 282 Chapter 13. Filtering and Resolution high pass ltering depicted in Fig. 13.4a), the complement to data used for reconstructing the image shown in Fig. 13.3a. The edges of the hierarchy of small holes and of the disk itself are the only features remaining in the image. Problem 13.6
In this problem, some very familiar calculations are revisited in the context of the information content of k-space. It is helpful to recall that sampling and ltering in k-space correspond to processing the analog signal obtained from the continuous Fourier transform of the spin density. To reconstruct the image from the sampled signal data, a discrete Fourier transform is analyzed. a) Find the analog signal for a constant spin density lling the FOV, where (x) = 0 rect(x=L). Show that sampling this signal at intervals of p k ( k is determined by the Nyquist criterion) for any integer p gives zero except at p = 0. b) Find the sinc function that represents the analog signal for a constant spin density occupying only one voxel, (x) = 0 rect(x= x). Discuss the implications of the rst zero crossing of the sinc function. How many pixels wide would the object have to be so that the rst zero crossing of the sinc function is at the edge of k-space? The addition of another lter to the windowing and sampling of data implies the multiplication of the signal by both Hws(k) and Hfilter (k) in k-space. From (13.15), it is easy to see that the corresponding point spread function hws filter (x) may be expressed by 13.4.1 Resolution after Additional Filtering of the Data
n;1 X p=;n hws filter (x) = k xMRI filter Hfilter (p k)ei2 p kx (13.42) Starting with (13.40) and (13.42), it is also simple to follow the steps in (13.41) to obtain MR "spatial resolution with lter # 1 Hfilter (0) = k Pn;1 (13.43) 0 p =;n Hfilter (p k) It may be assumed that the global maximum of the k-space lters occurs at k = 0, i.e., Hfilter (0) Hfilter (k) for all k since these lters generally are required to be low pass lters. The result (13.43) then leads to 1 xMRI filter 2n1 k W = x (13.44)
0 13.4. Spatial Resolution in MRI 283 In consequence, the resolution with additional ltering is worse than the resolution from direct discrete Fourier transform reconstruction alone. 13.4.2 Other Measures of Resolution There is a separate approach to de ning xfilter 1 filter 2 that does not require a calculation such as that in (13.40). Instead, it is based on the intrinsic blurring of a point isochromat created by the lter. Imagine two point objects a ected by a lter H (k) with a point spread function h(x). Consider the signal due to the two objects, A at x = 0 and B at x = xB , as shown in Fig. 13.5. Since the image at xB will be a ected by A because of the lter blur, the contribution at the point xB due to A is ^due to A(xB ) = a (x) h(x)jx=xB filter = a h(xB ) (13.45) The contribution at xB due to object B itself is ^due to B (xB ) = b (x ; xB ) h(x)jx=xB filter = b h(0) (13.46) Thus, the nal reconstructed image at xB is given by ^two spins(xB ) = b h(0) + a h(xB ) (13.47) filter Fig. 13.5: The spin density for 2 point objects A and B is given by (x) = a (x)+ b (x ; xB ). The reconstructed spin density for object A is also shown (dashed line), corresponding to a truncation kspace lter, and given by the point spread function times the magnitude a. In contrast to Fig. 13.6,
spatial resolution can be de ned in terms of the zero crossings of the point spread function. Equation (13.47) demonstrates the obscuring of the point object at B due to the blur from A. For illustration purposes, consider the lter to be a window function. It is assumed that object B can be resolved in the presence of A when the distance between the two objects is greater than or equal to the distance of the rst zero crossing of the lter. With this de nition, the rst zero crossing of (13.12), or (13.18), gives xalternative = 1=W , which is equivalent to the earlier result for xMRI in (13.41). 284 Chapter 13. Filtering and Resolution Full Width Half Maximum and Full Width Tenth Maximum Approximations
The de nition of spatial resolution in terms of zero crossings is not appropriate for a positive de nite, or negative de nite, point spread function, such as that shown in Fig. 13.6. An alternate criterion based on the lter pro le is often used.8 The xfilter may be de ned as the `width' of the lter, for a symmetric lter with a maximum at its center. The `width' may, in turn, be de ned as twice the distance from the center to the point where the lter has fallen to a fraction of its maximum. Therefore, xfilter is found by solving the equation given by h( xfilter =2) = h(0) (13.48) The ambiguity in determining whether two objects can be resolved is re ected in the choice of . A well-known convention is to employ the full width half maximum for the width, xfilter FWHMfilter (13.49) corresponding to = 1=2. A more demanding criterion follows from the use of the full width tenth maximum, xfilter FWTMfilter (13.50) for which = 1=10. Problem 13.7
a) Find the xGaussian using both the FWHM and the FWTM de nitions for the Gaussian lter of Prob. 13.5, and compare your answers to the result for that problem. b) Repeat the two determinations in (a) but for the window lter whose point spread function is hfilter (x) = W sinc( Wx). Compare your answer to the de nition based on the rst zero crossing result. A question arises about how to nd the total resolution for two or more lters applied to the same data. If two lters of the same shape are multiplied in the k-space domain, it is common to add their individual resolution, or blur, together via xfilter 1 filter 2 xfilter 1 + xfilter 2 (13.51) This is an identity if the FWHM rule is used to de ne x and both lters have exponential form in the k-space domain (see the following problem). If one or another does not have exponential form, the total blur may have to be calculated for the combined ltering, including the e ects of nite sampling. While it is tempting to write xMRI filter xMRI + xfilter ,
In general, a statistical analysis is necessary to determine the probability that two objects may be distinguished given any resulting overlap. Nevertheless, the expressions given in the text for xMRI filter work well for the lters commonly applied in MRI.
8 13.5. Filtering Due to T2 and T2 Decay 285 Fig. 13.6: Example of several impulse objects, and their convolution with a positive de nite point
spread function. The spatial resolution may be de ned in terms of the `width' of the point spread function. the MRI window and sampling lter are not exponential in form and, in general, this approximation is not valid. Some examples in which the approximation can be compared to the exact total resolution are found in the next section. Problem 13.8 Prove that (13.51) is an identity for exponential forms, by adding the FWHM for the Fourier transforms of e;ajtj and e;bjtj and comparing the result to the FWHM of e;(a+b)jtj . These forms are pertinent to SE imaging see the following section. 13.5 Filtering Due to T2 and T2 Decay
Sampling and windowing along with the procedures chosen to enhance image clarity are part of the data processing and have been described as applied lters. Other lters are inherent to the MR experiment and must be considered in any case. Signal variations as a function of time due to T2 or T2 decay during data sampling in the read direction can be modeled in terms of intrinsic lters. The form is determined from the exponential expressions found earlier for transverse relaxation. In particular, such innate ltering imposes limits on the best spatial resolution which can be achieved in an experiment. Even if in nite time were available for sampling, and continuous data could be acquired, the signal would still be ltered by the transverse decay envelope leading to a limited resolution. 286 Chapter 13. Filtering and Resolution The previous results, for which the Fourier pixel size was the lower limit on the spatial resolution, depended, for gradient echo sequences, on the implicit assumption that the sampling time Ts is short compared to T2 . In the case when Ts is comparable to T2 , the exponential decay of the signal during sampling causes a signi cant ltering action on s(k), given by the familiar form e;t=T2 where t = 0 is chosen to be the time center of the rf pulse. Since k = {Gt0 with t0 = t ; TE , the conversion to a lter in k-space is found from t = k + TE (13.52) {G To learn the essential e ects, consider a simple model of symmetric windowing, with no sampling, along with the T2 decay. The corresponding k-space lter is ! k ;k= {GT2 ;TE =T2 Hw(sym) T2 (k) = e e rect W (13.53) Since this lter, though real, is not symmetric about k = 0, the material in Ch. 11 suggests that its Fourier inversion should lead to a complex lter point spread function. The details of the inverse transform are left to the next problem and do indeed give the complex form Ts =(2T2 );i x= x ; e;Ts =(2T2 )+i x= x hw(sym) T2 (x) = e;TE =T2 e (13.54) Ts x=T2 ; i2 x where (13.55) Ts = W = G1 x {G { 13.5.1 Gradient Echo Sequence Problem 13.9
Derive (13.54) from the inverse Fourier transform of (13.53). The T2 Filter E ect on Resolution The spatial resolution (13.39) is appropriate for the combined lter (13.53) in the absence of sampling (and its Nyquist periodicity). It is found immediately that H (0) xw(sym) T2 = h w(sym) T2 (0) w(sym) T2 x = (13.56) sinch( 2Ts2 ) T introducing a hyperbolic version of the sinc function x ;x sinch(x) sinh(x) = e ;xe x 2 (13.57) 13.5. Filtering Due to T2 and T2 Decay Since sinch(x) 1 ; 1 x2 for small x, 6 287 xw(sym) T2 0 !1 1 Ts 2 A x @1 + 6 2T2 (13.58) for small Ts=(2T2 ). Even for Ts = T2 , the deviation from unity of the coe cient in (13.58) is only 4%. Therefore, even if the sampling time is approximately equal to T2 , the resolution of the experiment will be equal to the ideal case to within a few percent. The FWHM Approximation
It is of interest to examine the magnitude of hw T2 (x) and to look at the full-width at half maximum (FWHM). This gives an estimate of the spatial extent of the blur caused by the lter. The FWHM is a good rst-order guess of the spatial resolution when the integral in (13.39) is di cult to evaluate. If e;Ts=2T2 1, then
Ts =(2T2 ) hw T2 (x) ' (T T2 2e + (2 xT )2 jTs x + i2 xT2 j s x) 2 (13.59) and this function has a FWHM of T FWHMT2 = 3 T s 2 p ! x (13.60) This FWHM represents an additional blur over and above that caused by the usual sampling window. Problem 13.10 a) Derive (13.60) from (13.59). b) If T2 = 20 ms, N = 256, G = 5 mT/m, and L = 256 mm, then nd t and FWHMT2 for the gradient echo experiment shown in Fig. 10.14. 13.5.2 Spin Echo Sequence The ltering e ects due to transverse relaxation on the spin echo experiment are best described in terms of two separate lters, one representing the e ect of T2, and the other the e ect of T20 . This analysis follows that in Ch. 8 and Fig. 8.3, where it was seen that there are two parts to the envelope describing the spin echo signal evolution during data sampling. First, there is a symmetric T20 decay envelope centered about the instant of the spin echo (and, hence, around the k-space origin). Second, there is the asymmetric T2 decay envelope 288 Chapter 13. Filtering and Resolution which goes as e;TE =T2 e;k= {GT2 . This lter therefore behaves very similar to HT2 (k) with T2 replaced by T2 , HSE (k) = HSE T2 (k)HSE T2 (k) = e;TE =T2 e;k= {GT2 e;jkj= {GT2
0 0 0 (13.61) An e ective analysis of HSE T2 (k) was already performed for the gradient echo experiment with T2 replaced by T2 . The e ect of HSE T2 (k) will be investigated here in order to nd a nal result for the ltering e ect of transverse relaxation on the spin echo experiment. The nite windowed form is given by Hw SE T2 (k) = e;jkj=( {GT2 ) rect k +W k=2
0 0 ! (13.62) for the symmetrized data set. For the analog case, the k=2 term in the argument of the rect function is not present. This analog lter function is both real and symmetric and leads to the point spread function 0 hSE T2 (x) = 1 + 4 22{GT22T 02 x2 (13.63) {2G 2 which is also real and symmetric. The FWHM of hSE T2 (x) is given by
0 0 FWHMSE T2 = 1=( {GT20 ) ! x Ts = T20
0 (13.64) In e ect, the total additional loss of resolution (increase in xMRI filter ) due to signal decay during sampling in a spin echo experiment can be approximated by the sum of the two blurring functions, i.e., FWHMSE = FWHMSE T2 + FWHMSE T2 p ! 3 x Ts + x Ts = T2 T20
0 (13.65) Quantitative Values of Filtering Due to Transverse Relaxation for the Spin Echo Experiment
It is now reasonable to answer the question as to what limiting restriction should be placed on the duration of Ts such that signi cant loss of spatial resolution does not occur due to transverse relaxation. For example, in the spin echo case even for Ts = 2T2 , the image blur is small (and not signi cant enough) due to the T20 component of T2 . For the T2 decay part, Ts = T2 has a similar e ect. Such limiting values can be calculated based on subjective limits set by the user of the MR system on how much blurring is considered signi cant. In any case, it is always good to keep Ts as short as possible. What limits how short Ts can be? Ts is rst limited by the available maximum gradient strength for a given x. Further, 13.6. Zero Padding 289 it will be shown in Ch. 15 that the SNR/pixel in the reconstructed image is proportional to p Ts, so SNR is yet another criterion which limits Ts. Field inhomogeneities of the system can also dictate upper limits on Ts through T20 . Problem 13.11
It is useful to have an expression which determines the blur that is introduced by transverse relaxation during sampling. Find the spin echo parallel of (13.58) which is the ratio of the ideal MR resolution to the resolution achieved in the presence of transverse relaxation. What is the percentage deviation between the ideal resolution and that found when Ts = T2 = 2T20 ? 13.6 Zero Filled Interpolation, Sub-Voxel Fourier Transform Shift Concepts and Point Spread Function E ects
A well-known and often-used method of interpolation when incomplete Fourier data are available is to ll out the k-space data with additional zeros such that the pixel size after image reconstruction meets the interpolation desired. In this section, the equivalence of zero padded interpolation and sub-voxel shift using the shift theorem is shown. A discussion on the e ects of the point spread function on the interpolated values shows that up to a 36% signal loss can be obtained if a point like object is not centered within a reconstructed voxel. Sub-voxel shifts done by using the shift theorem are shown to recover this signal loss. Despite the apparent high resolution in terms of image voxel size in the image display, the spatial resolution does not change it is still determined by the Fourier pixel size since the point spread function does not change its form. In the `symmetric' N -sample case, the discrete Fourier transform reconstruction gives ^N DFT(q x) = k
N=2;1 X p=;N=2 13.6.1 Zero Padding and the Fast Fourier Transform
s(p k)ei2
p q=N (13.66) When N is a power of 2 and Nimage equals N , a very e cient version of the discrete Fourier transform called the fast Fourier transform (FFT) can be used for image reconstruction. When N is not a power of 2, the k-space data are often `zero padded' to the nearest power of 2, say Nimage, and an Nimage-point image is reconstructed using the FFT. In the FFT reconstruction, the pixel size xpixel is given by pixel size (13.67) xpixel = N L = N 1 k image image 290 Chapter 13. Filtering and Resolution In the symmetrically collected data set where Nimage = N , 1 x = N1 k = W Fourier transform pixel size (13.68) 1 When x equals W the reciprocal of the extent of k-space coverage, the pixel size is sometimes also referred to as the Fourier pixel size. This is used to distinguish this particular pixel size from the case where the k-space data are zero padded to some other size and the image is reconstructed at a pixel size smaller than the Fourier pixel size. Zero padding adds no new information at the level of x=2 but it does provide an interpolated image. where ;N q (N ; 1). Two separate images can also be constructed from the zero lled data set by constructing images at the even and odd data points separately. Consider only the even data points q = 2r where ;N=2 r (N=2 ; 1). The image reconstructed from these points is N=2;1 X s(p k)ei p2r=N = 1 ^m (r x) (13.71) ^m zero even(r x) = 21 N p=;N=2 2 = signal from the even-indexed voxels of the zero lled image and the signal from the odd-indexed zero lled image voxels is X 1 N=2;1 s(p k)ei (2r+1)p=N ^m zero odd((2r + 1) x=2) = 2N p=;N=2 X 1 N=2;1 hs(p k)ei p=N i ei2 rp=N (13.72) = 2N p=;N=2 Suppose that N k-space data points are collected symmetrically. Filling this data set symmetrically to 2N points and reconstructing the image at 2N points allows the image to be reconstructed at a voxel size of 2x while the spatial resolution of the image still remains x, i.e., the image is interpolated to obtain half-voxel shifted signal values. Suppose sm zero(k) represents the zero lled data set. Then 8 < s(p k) ; N p N ; 1 2 2 sm zero(p k) = : (13.69) 0 ;N p ; N ; 1 and N p (N ; 1) 2 2 ;1 If D2N represents the 2N -point discrete inverse Fourier transform operator, then ^m zero(q x=2) = ;1 ; D2N fsm zero(p k)g, and ^(q x) = DN1fsm(k)g N ;1 X 2 qp ^m zero(q x=2) = 21 sm zero(p k)ei 2N N p=;N N=2;1 X i qp=N = 21 (13.70) N p=;N=2 s(p k)e 13.6.2 Equivalence of Zero Filled Signal and the Sub-Voxel Shifted Signal 13.6. Zero Padding 291 If the half-voxel shifted image obtained using the shift theorem is ^shift(q x) whose N point discrete Fourier transform is equal to sshift(p k) = ei
then ^shift(q x) is given by p=N s(p k)
p=N ei2 qp=N (13.73) X 1 N=2;1 s(p k)ei ^shift(q x) = N
p=;N=2 (13.74) Comparing (13.72) and (13.74), it can be seen that the image computed at the interpolated voxels from the zero lled signal is the same as that from the sub-voxel shifted image voxels obtained using the shift theorem. 13.6.3 Point Spread E ects on the Image Signal Based on the Object Position Relative to the Reconstructed Voxels In the section on the point spread function corresponding to discrete Fourier transform image reconstruction from truncated data, it was noted that when ^, the image, is reconstructed at a continuous-valued position variable, say x, the reconstructed image is the convolution of the original object with the point spread function hws(x). The sub-voxel shifted image function ^shift(q x) is obtained by sampling this convolved image function at discrete points x = q x. There is an interesting property of the shift theorem that is derived from this convolution with hws(x). For example, consider a boxcar object of unit signal and in nitesimal width w x centered about x = xs with jxs j x=2. The resultant image function is ^bc(q x) = dx0 (x0 )hws(q x ; x0 ) Z (13.75) where the subscript `bc' indicates `boxcar object.' The signal obtained at q = 0, the voxel nearest to the center of the object is ^bc(0) = dx0 hws(;x0 ) xs ;w=2 ' whws(;xs ) Z xs+w=2 (13.76) The approximation in (13.76) is valid since w is an in nitesimal width, and hws(x) is a smoothly varying function about x = xs. If the image is reconstructed on a grid which is shifted by half a pixel by using the shift theorem, the reconstructed signal would have been ^s (0) ' whws(xs ; x=2) bc (13.77) where the superscript s is used to indicate the shifted reconstruction. Comparing the reconstructed signals in (13.76) and (13.77), the signal is reduced in the unshifted case by the ratio R = ^s (0)= ^bc(0) (13.78) bc 292 which, for s = x=2, gives Chapter 13. Filtering and Resolution R = jhws(; x=2)j=jhws(0)j (13.79) In the limit of k ! 0, jhws(x)j ! W sinc( Wx). Hence, jhws(; x=2)j = jhws( x=2)j ! sinc( =2) = 2W= = 0:64W , i.e., a 36% reduced signal is obtained when the object is centered half a voxel away from the voxel center. The shift theorem allows us to overcome this worst-case 36% underestimation of the signal by centering the object within a reconstructed voxel. However, recovering the `missing' 36% from a shifted object does not imply that the image is resolved any more accurately, i.e., resolution is still limited to x and an object which is only half a pixel in size with amplitude 2a will show itself as an object a pixel in size with an amplitude a. This property of the Fourier transform shift theorem, used to correct the reconstructed signal for the signal loss in the worst-case scenario where an object is shared equally by two neighboring voxels, is itself something which can be taken advantage of in a number of practical situations. For example, zero lled interpolation has been in use for a long time in NMR spectroscopy for improved spectral peak estimation. With improved peak estimation comes improved estimates of the spectral position. Extending the same ideas to imaging, it is increasingly being used for improved object signal estimation by overcoming the partial volume averaging9 e ect due to sharing of the object by two neighboring voxels. 13.7 Partial Fourier Imaging and Reconstruction
In certain cases, k-space data are collected asymmetrically, i.e., the k-space origin is not centered within the acquisition window. In some applications where minimizing imaging time is of the essence, k-space is covered asymmetrically in the phase encoding direction. Here, the number of phase encoding lines required by Nyquist is collected only in one half of k-space, while the other half (usually the negative half of k-space) is covered only partially. Since phase encoding lines are usually separated in time by TR , collecting fewer phase encoding lines implies a shortening of the total imaging time. The amount of time saved is determined by the `degree of asymmetry.' In other applications such as ow imaging and in imaging areas with large local eld inhomogeneities, short gradient echo times are preferred. One of the most popular means to achieve this is to obtain asymmetric echoes in the readout direction. These methods of asymmetric k-space coverage are generically classi ed as partial Fourier imaging methods. Any Fourier imaging done such that k-space is covered over the region ;n; k n+ k], where n; n+, is classi ed as a partial Fourier imaging scheme. The degree of asymmetry `asym' is de ned as ;n asym = n+ + n; (13.80) n
; In the partial Fourier imaging scheme just formalized, n; can take on values in the range 0 n+]. As a result, asym can take on all values between 1 (completely asymmetric case, i.e., when only the positive half of k-space is covered) and 0 (completely symmetric case, i.e., symmetric k-space coverage). For example, consider the read gradient in Fig. 13.7. It
9 + Partial volume averaging is de ned and discussed in Ch. 15. 13.7. Partial Fourier Imaging and Reconstruction 293 has n; points before the echo and n+ ; 1 points after. The total read sampling time is N t where N is de ned to be N = n; + n+ (13.81) The eld echo time FE is referred to as the time from the beginning of the application of gradients in the read direction until the echo occurs in the sampling window. From Fig. 13.7, the minimum eld echo time is given by FE = 3 rt + 2n; t (13.82) This is the rst and only gradient eld echo along the read direction in this sequence.10 The echo time is then TE = rf =2 + 3 rt + 2n; t (13.83) Clearly, as n; approaches zero, TE can be reduced to its minimum value. Ideally, the image is reconstructed from the asymmetric echo data by nding s(;k) via the complex conjugate symmetry relation s(;k) = s (k) (13.84) when (x) is a real function. Now, recreating a new data set with 2n+ points symmetrically distributed about the echo is straightforward. If n+ is not a multiple of 2, the data can be zero lled to accomodate the FFT algorithm. Another way to view totally asymmetric data (i.e., n; = 0) is to look at the case when the data s(k) is multiplied by a lter equivalent to the Heaviside step function (k). = 1 (x) + i P 1 (13.85) 2 2 x (The details of this Fourier transform can be found in Ch. 11.) Application of this lter, leads to the following reconstructed spin density ^H (x) = (x) h (x) = 1 (x) + 1 (x) i P 1 (13.86) 2 2 x This shows that when (x) is real, it can be found after this lter is applied by taking 2Re(^H (x)). In practice, because of eld inhomogeneities and ow, (x) is e ectively complex, (x) = real (x) + i imag: (x). This means that Re(^H (x)) will no longer represent (x) since the complex terms in (x) and F (x)] will mix components. More complicated schemes need to be designed to deal with this problem in that case.
The reader is reminded that eld echoes, gradient echoes, or gradient eld echoes all refer to times, after the initiation of the rst read lobe, when the area under the read gradient goes to zero. The phase, due to the read gradients only, is zero for all stationary spins at these times. These echoes should not be confused with spin echoes which are rf echoes that occur when the phase due to static eld inhomogeneities for stationary spins is everywhere returned to zero.
10 h (x) = F ;1( (k)) = Z1
0 dk ei2 kx 294 Chapter 13. Filtering and Resolution (a) (b)
Fig. 13.7: (a) Readout gradient waveform and timings for the asymmetric sampling scheme, and (b) the corresponding k-space coverage. The shaded regions A and B have been shown explicitly
to clarify the location of the gradient echo. 13.7. Partial Fourier Imaging and Reconstruction 295 The above discussion of the Heaviside function did not take into account sampling or truncation. If partial Fourier reconstruction is considered from a discrete perspective, the lter H (k) (equivalent to the Heaviside unit step function (k) in the continuous case) becomes 8 > 0 k<0 <1 H + (k) = > 2 k = 0 (13.87) : 1 k>0 1 The factor of 2 at the origin can be understood by assuming that symmetric sampling should be recovered by the addition of two heaviside functions. This is the case, if H (k) is de ned to be 8 > 1 k<0 <1 H (k) = > 2 k = 0 (13.88) : 0 k>0
; ; One approach to reconstructing a real image from asymmetric data is to force the data to obey complex conjugate symmetry. However, for a complex object (x ) =
re (x) + i im (x) 13.7.1 Forcing Conjugate Symmetry on Complex Objects (13.89) where re(x) and im (x) are real functions, this will result in an image which does not represent the actual object. Due to the linearity of the Fourier transform, it is possible to view this object as giving rise to two separate signals sre(k) F re(x)] sim (k) F i im (x)] (13.90) Just as the signal sre(k) from the real part of (x) obeys (13.84) (complex conjugate symmetry), the signal from im (x) obeys the relation sim (k) = ;sim (;k) (13.91) Suppose the k-space data are acquired only for k > 0. Complex conjugate symmetry is then used to create the uncollected data and form a double-sided data set ( 0 m k) s(k) = ss ((;k)) k < 0 ^ (13.92) (m k
Rewriting sm (k) and (sm (;k)) in terms of their contributions from the real and imaginary parts of the object and applying complex conjugate symmetry to the real and imaginary parts yields s(k) = ^ sre(k) + sim(k) k 0 sre(k) ; sim (k) k < 0 ( )! ;1 k < 0 = sre(k) + sim (k) 1 k 0 ( (13.93) 296 Chapter 13. Filtering and Resolution Taking the Fourier transform of this merged data set (^(k)) gives s ( )! ;1 k < 0 ;1 ^(x) = re(x) + F sim(k) (13.94) 1 k 0 1 = re(x) + (i im (x)) i P x (13.95) = re(x) ; im (x) 1 P 1 x which is clearly not equal to (x) unless im(x) is zero.11 Therefore, if an object is complex and a partial Fourier reconstruction method assuming a real object is applied, the resulting image will not be an accurate representation of (x). Collecting only part of the negative half of k-space allows for an iterative, constrained image reconstruction scheme. Two powerful constraints that can be used in the reconstruction are: (i) the k-space data that were collected represent the best knowledge about the object new k-space data will therefore be created only for the (n+ ; n;) uncollected points, and (ii) the low spatial frequency phase image that can be reconstructed with the rst 2n; points covering ;n; k n; ; 1 represents our best knowledge about the phase of the complex spatial domain data. The new 2n+-point image will then have its phase constrained to that of the 2n; original low k-space data zero lled to 2n+ points. It is also assumed that for an n-dimensional k-space data set, the remaining (n-1) dimensions have already been reconstructed by inverse Fourier transformation. The iterative reconstruction algorithm is detailed in a stepwise fashion next. Step 1: Create a truncated and zero padded (to 2n+ points) data set s (p k) from ~ the measured data set sm(p k) as follows: 8 >0 ;n+ p ;n; ; 1 < s (p k) = > sm (p k) ;n; p n; ; 1 ~ (13.96) :0 n; p n+ ; 1
Step 2: Step 3: 13.7.2 Iterative Reconstruction This data set is used to obtain a low frequency phase estimate ^(q x): h i ^(q x) arg D;1 (~ (p k)) s (13.97) Initialize an iteration counter j to zero. An initial 2n+-point image is obtained from a zero padded version of the measured data set, say, s0 (p k): ( 1 s0 (p k) = sm(p k) ;n; p n+ ;; 1 (13.98) 0 ;n+ p ;n; The reconstructed image, D;1 (s0 (p k)), is denoted by ^0(q x). The function in (13.94) which is -1 for k < 0 and +1 for k 0 is sometimes referred to as the `signum function.' This function Sgn(k) is given by Sgn(k) 2 (k) ; 1. Hence, the resulting inverse Fourier ; transform is ix .
11 13.7. Partial Fourier Imaging and Reconstruction The actual iterative part of the algorithm starts from the next step.
Step 4: 297 An intermediate (j + 1)st iterated complex image is obtained from ~j+1(q x) = j ^j (q x)j ei ^(q
x) (13.99) Step 5: This intermediate complex image is then Fourier transformed to create an intermediate k-space data set sj+1(p k) given by ~ sj+1(p k) D (~j+1(q x)) ~
Step 6: (13.100) sj+1(p k) has nonzero data for ;n+ p ;n+ ; 1 unlike the measured ~ data sm (p k). The complex data covering the range ;n+ k to ;n; k are substituted into a 2n+-point data area containing the measured k-space data for other values of p to give a (j + 1)st iterate k-space data set ( 1 sj+1(p k) = sm(p(p k) ) ;n; p n+ ;; 1 (13.101) sj+1 k ;n+ p ;n; ~
This is inverse Fourier transformed to create the (j + 1)st complex image iterate: ^j+1(q x) = D;1 (sj+1(p k)) (13.102) If j ^j+1(q x) ; ^j (q x)j is `su ciently small' (as determined by a predesigned su cient convergence criterion), then go to step 9 otherwise, go to step 10. Output ^j+1(q x) as the nal reconstructed image and end the algorithm. Increment iteration counter j by 1 and return to step 4. Step 7: Step 8: Step 9: Step 10: In the above algorithm, it is important that the replacement data sj+1(p k) for each iter~ ation are consistent with the collected data sm(p k) around the merging point. Otherwise truncation artifacts due to the k-space discontinuity can be found in the reconstructed image. It is therefore useful to merge the two data sets with a u-point Hanning lter using an averaging procedure during the last iteration to avoid these artifacts. That is, the last k-space iterate is given by 8 s (p k) ;n; + u p n+ ; 1 > m > s (p k) < ~jn ;n+ p ;n; ; 1 +1 sj+1(p k) = > 1 sm (p k) h1 + cos + (p+n ) i + u >2 h (p+n ) io ;n : p ;n; + u ; 1 sj+1(p k) 1 + cos u ~ ;
; ; (13.103) 298 Chapter 13. Filtering and Resolution Problem 13.12
In Ch. 9, a short discussion on k-space sampling during gradient ramp-up was presented. This technique nds practical use in short-TE asymmetric echo imaging to obtain shorter TE 's. For each sequence, some minimum number of points must be collected before the echo for a stable partial Fourier reconstruction to be performed. In certain ultrashort imaging applications, even the time required to collect as few as 32 points before the echo is considered to be too long. In such applications, sampling during gradient ramp-up provides the required shortening of TE . Partial Fourier image reconstruction is then done on these data to obtain improved image information. How many extra k-space points would be obtained if the ramp-up and ramp-down times of the dephasing and rephasing lobes are both 1 ms, the read gradient strength is 25 mT/m and t is a) 20 s or b) 4 s? With this particular design, the number of points recovered from the ramp-up to a constant read gradient will be the same as which occurs during the usual uniform sampling before the echo. Those points sampled during the ramp-up will need to be nonuniform in time to create uniformly sampled points in k-space. 13.7.3 Some Implementation Issues
The complete discussion of the algorithm was built on a 1D basis. Consider the general asymmetrically collected 3D k-space data problem. Let the data collected be represented by s(kx ky kz ). What kind of complex conjugate symmetry does this data set possess when (x y z) is real? To answer this question, write out the expression for the complex conjugate of s(kx ky kz ) in terms of a Fourier transform: s (kx ky kz ) = ZZZ ZZZ dx dy dz (x y z)e;i2 (kx x+ky y+kz z) = dx dy dz (x y z)ei2 = s(;kx ;ky ;kz ) (kx x+ky y+kz z ) (13.104) In short, for multi-dimensional k-space data, complex conjugate symmetry holds for points which are re ections of each other about the k-space origin. How does this important property manifest itself from a practical point of view? For example, in the 2D case, it is tempting to ask whether collecting only a single quadrant of k-space would be enough to reconstruct the image using complex conjugate symmetrization. The answer is simply no. Suppose that points were collected such that they covered only the rst quadrant of k-space. The conjugate symmetry property (13.104) implies that only k-space samples which lie in the third quadrant can be obtained the second and fourth quadrants are not lled at all. 13.8. Digital Truncation 299 When Is the Complex Conjugate Step Performed for Multi-Dimensional Data
Data can only be collected asymmetrically in one dimension such as the read direction to reduce TE , or one of the phase encoding directions to reduce the total acquisition time. The property (13.104) also has another implication from an e cient algorithmic point of view. There are two ways asymmetrically collected multi-dimensional k-space data can be reconstructed: one is to store the complete 2D k-space data into an array and complex conjugate symmetrize the data using (13.104). The second method is rst to inverse Fourier transform all other directions in which data were collected symmetrically, and then apply a 1D complex conjugate symmetrization to the 1D k-space data. Suppose data were collected symmetrically along ky and asymmetrically along kx to obtain a k-space data set s(kx ky ). Fourier inverting along y yields a function with a single spatial dimension and a single k-space dimension, say ^y (kx y): Z ^y (kx y) = dky s(kx ky )ei2 ky y (13.105) Now, ^y (kx y) satis es (13.84), i.e., ^y (;kx y) = ^y (kx y)] (13.106) using which the uncollected data points can be lled for Fourier inversion. Obviously, the second method is more e ciently implemented from a memory size requirement point of view since all operations are separable in the di erent directions. Faster Convergence of the Iterative Method
In practice, the imaged object may have signi cant imaginary parts because of rapidly varying background eld inhomogeneities. As seen from (13.95), it is then possible to have signi cant reduction in the image amplitude by forcing complex conjugate symmetry on the data. This is the reason why the all-positive zero padded reconstruction magnitude image, j ^0(q x)j, was used as a starting constraint in the iterative reconstruction method. However, this image still su ers from the complex blurring, introduced by the multiplication by the Heaviside step function in k-space, which can often be signi cant, and can lead to a rather slow convergence of the iterative process. For small phase errors, j ^0(q x)j in step 3 can be replaced by the absolute value of the image created after forcing complex conjugate symmetry on the original data. 13.8 Digital Truncation
When the data are sampled, they are converted from an analog (or continuous) signal into a digital signal (i.e., a combination of bits) via an ADC (analog-to-digital converter). Today, a 16-bit ADC is often used. How does an ADC operate? An ADC has a peak input voltage dynamic range speci cation (from say ;V to +V volts) over which it works. A 16-bit ADC, for example, would take any input voltage Vin and allocate it to a certain 16-bit output stream. This bit stream is based on which of 216 voltage levels Vin is allocated to. Since the voltage range between ;V and +V has to be allocated to a total of 216 voltage levels, 300 Chapter 13. Filtering and Resolution 2V the voltage di erence separating two levels is 216 = 2;15V . For a commercially available ADC, V is on the order of a few tens of volts. Therefore, the detected and demodulated emf corresponding to the NMR signal (which is of the order of a fraction of a mV) has to be ampli ed prior to its input to the ADC. Suppose the ampli cation is xed based on the signal from a homogeneous object of spin density 0 which lls the entire eld-of-view of interest using the entire dynamic range of the ADC. The k-space signal for such an object is s(kx ky ) = 00 LxLy kx = ky = 0 elsewhere
In terms of the reconstructed voxel size and number of collected k-space samples, ( (13.107) s(kx ky ) = NxNy 0 ( 2 = N 0 0 ( 0 2 x y kx = ky = 0 elsewhere kx = ky = 0 elsewhere (13.108) if Nx = Ny = N and x = y = . Suppose now that the ampli cation is such that 0 2 , the signal per voxel, corresponds to the voltage di erence between two adjacent voltage levels. Then, for N = 256 = 28, the ADC limit will have been reached. For an image of an object twice the size of the rst object, i.e., for N = 512, N 2 = 218 and the last two bits will be lost from the 18-bit data that this input would require. For the homogeneous object, this loss does not matter since there is only one nonzero k-space sample. However, suppose another object occupying exactly one voxel and with spin density 0 is superimposed exactly at the center of the homogeneous object. The signal from this object, say s(kx ky ), is: s(kx ky ) = 0 2 (13.109) This adds a constant voltage level corresponding to the unit voltage level which is usually represented by a bit 1 as the last bit, and zeros at all other bit positions. Unfortunately, since the last 2 bits are lost, i.e., all signal values below 4 units will be suppressed to level 0, this single-voxel object will not be seen at all! This problem of loss of small object visibility is especially exacerbated in 3D imaging where Nz voxels, each with unit thickness, are reconstructed (Nz can be 64 or 128 in practice). If the same ampli er gain is used, any object or set of objects whose signal sums to Nz 0 2 z will have its signal contribution suppressed in the digitized k-space signal. Most information about these objects will be lost, and they will appear as noise in the image. This noise is usually referred to as `digitization or discretization noise.' One possible solution to avoid this problem is to collect the data twice, once with central k-space and once with the remaining part of k-space so that neither has peak amplitudes which exceed 216. An alternate method would be to apply an automatic gain control (AGC) to the data to enhance signal at the edge of k-space so that the overall dynamic range is reduced. Then the data could be rescaled digitally to the values that would have been measured had an AGC lter not been applied. If the central k-space data have a gain of 96 dB, and the discrete inverse Fourier transform is taken by zero lling the missing (but to be collected) data then the central point s(0) with 13.8. Digital Truncation 301 value N 2 0 x y transforms to 0 in the image domain for all x y. Note that the image is at, and has no need for great dynamic range at all. The image will be limited in resolution by the nite window W1 (see Fig. 13.8). After the DFT, the data from the high k-space region (W2 ; W1 ) will be rescaled appropriately and added to improve the resolution of the image. Problem 13.13
Consider a 3D object that lls the FOV and has unit spin density for each voxel. When a 3D data set is collected with Nx = Ny = 128 and Nz = 32, the maximum signal at kx = ky = kz = 0 for a centered echo will be 219 units. Generally, the ADC is only 16 bits (i.e., it will recognize 216 individual voltage levels). a) If the system rounds o to zero for any number less than unity, explain how having insu cient ADC range acts as a lter on the data. b) What e ect does this have on resolution in this problem and in general? Assume that the underlying object lls the FOV.
1 Imagine next a narrow object, of width 4 x and amplitude 1 , superimposed on the original object. The signal s(k) from this little object will be spread out over all k-space with roughly a constant amplitude. c) Show for 1 < 32 that the signal from this object outside ~ = 0 will be k either 0 or 1 when the signal from the entire object is normalized to 216. Fig. 13.8: Scheme for overcoming the data truncation artifact by acquiring the central k-space rst and acquiring the higher k-space a second time so that both scans are automatically gain-controlled. 302 Chapter 13. Filtering and Resolution Suggested Reading
The following articles present methods for constrained partial Fourier image reconstruction: E. M. Haacke, E. D. Lindskog and W. Lin. A fast, iterative partial Fourier technique capable of local phase recovery. J. Magn. Reson., 92: 126, 1991. Z.-P. Liang, F. E. Boada, R. T. Constable, E. M. Haacke, P. C. Lauterbur and M. R. Smith. Constrained reconstruction methods in MR imaging. Reviews of Magn. Reson. Med., 4: 67, 1992. P. Margosian, F. Schmitt and D. E. Purdy. Faster MR imaging: Imaging with half the data. Health Care Instr., 1: 195, 1986. The following articles present the ltering e ects of T2 and T2 decay during data sampling: E. M. Haacke. The e ects of nite sampling in spin echo or eld-echo magnetic resonance imaging. Magn. Reson. Med., 4: 407, 1987. J. P. Mugler III and J. R. Brookeman. The optimum data sampling period for maximum signal-to-noise ratio in MR imaging. Reviews in Magn. Reson. Med., 3: 761, 1993. The next text is an excellent reference on how to design and implement digital lters: R. W. Hamming. Digital Filters. III edition. Prentice-Hall, Englewood Cli s, New Jersey, 1989. General integral formulae and the concept of principal value can be found in the next two references, respectively: M. Abramowitz and I. A. Stegun, eds.. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55, U.S. Govt. Printing O ce, Washington, D.C., 1964. R. V. Churchill and J. W. Brown. Complex Variables and Applications. McGraw Hill, New York, 1990. Chapter 14 Projection Reconstruction of Images
Chapter Contents
14.1 Radial k-Space Coverage 14.2 Sampling Radial k-Space and Nyquist Limits 14.3 Projections and the Radon Transform 14.4 Methods of Projection Reconstruction with Radial Coverage 14.5 Three-Dimensional Radial k-Space Coverage 14.6 Radial Coverage Versus Cartesian k-Space Coverage Summary: Radial methods of k-space coverage are detailed as alternatives to Cartesian
Fourier k-space coverage. Reconstruction methods for such data are also discussed. The relationship between di erent reconstruction approaches is addressed. Certain advantages and disadvantages of the radial k-space coverage method are presented. Introduction
The traversal of k-space can be carried out in a variety of ways, most of which have real physical analogs in terms of sequence design. In this chapter, radial coverage of k-space is considered, a method of historical consequence and of contemporary interest for specialized applications. The method is commonly known as projection reconstruction.1 There is a broad experience that exists in this approach to image reconstruction in the world of xray tomography where there are entire books devoted to the subject. The advantages and di culties in projection reconstruction are the present subject, but it is to be kept in mind that a variety of other ways to cover k-space are addressed in later chapters.
It might be argued that this term should be reserved for the process of creating the image from the radially collected data. However, in keeping with the spirit of the common usage of the term in the MR literature, we reserve the usage of projection reconstruction to mean both the radial k-space coverage method and the reconstruction method.
1 303 304 Chapter 14. Projection Reconstruction of Images The choice of reconstruction method is critical, in general, and is especially true in image reconstruction from radially collected data because of the many available methods for their reconstruction. Artifacts associated with one method may be eliminated by re-sorting to other methods. Although new artifacts necessarily replace the old when this change is made, the new artifacts may be more acceptable. It is such practical issues which ultimately decide which reconstruction method and what type of k-space coverage are used in a given application. Projection reconstruction techniques are used on data collected with the aforementioned radial k-space coverage, whereas Fourier reconstruction methods are used to transform data obtained using rectangular k-space coverage. The theory behind various reconstruction methods for radially acquired data and some discussion on the implementation issues are presented in this chapter. The chapter begins with a rewriting of the measured signal in terms of polar coordinates which are better suited for the description of radially covered k-space data. This is followed by a discussion on how the gradients have to be varied so that radial k-space coverage can be obtained. Image reconstruction in polar coordinates is then brie y described to set the stage for later discussions. 14.1 Radial k-Space Coverage
To set the stage for a di erent approach to taking k-space data, the standard coverage described in Ch. 10 should be reviewed. The archetypical 2D imaging method encodes in the kx and ky directions by reading out a line of data with both read gradient Gx and phase encoding gradient Gy xed. The latter is changed in amplitude at each subsequent rf pulse structure. The corresponding signal at time t is s(t y Gx Gy ) = Z z dz ZZ dxdy (x y z)e;i2 {(Gxxt+Gy y y ) (14.1) for a slice of thickness z perpendicular to the z-axis. The phase encoding has taken place during time y . The relaxation e ects and overall constants are absorbed into the spin density. With the z integration over the slice thickness included in a further rede nition of the spin density, a more compact formula for the signal can be written as s(t y Gx Gy ) = ZZ dxdy (x y)e;i2 {(Gxxt+Gy y y ) (14.2) The measurements implied by (14.2) are a horizontal line by line accumulation of the points in the (kx ky ) plane where, for pulsed gradients, the coordinates kx(t) and ky ( y ) are determined by kx = {Gxt and ky = {Gy y (14.3) leading to the more conventional form s(kx ky ) =
where ~ = (kx ky ) and ~ = (x y). k r ZZ kr dxdy (x y)e;i2 ~ ~ (14.4) 14.1. Radial k-Space Coverage 305 14.1.1 Coverage of k-Space at Di erent Angles Rather than varying just the phase encoding gradient, suppose that the two gradients Gx and Gy are turned on at the same time. Then the net gradient vector points (and the data are frequency encoded) along an arbitrary angle with respect to the x-axis (the reader is referred back to Ch. 9 to the discussion about readout in an arbitrary direction).2 The overall gradient strength G and angle are given by q G G = G2 + G2 and = tan;1 Gy (14.5) x y x such that Gx = G cos and Gy = G sin (14.6) Under these circumstances, the signal can be written3 ZZ s(t G ) = dxdy (x y)e;i2 {G(x cos +y sin )t (14.7) Following Ch. 9, the connection to ~ -space is made through the de nition of the k-space k vector along the frequency encoding direction as ~ {Gt k ~ (14.8) ~ where G = (Gx Gy ). So equations analogous to (14.5) and (14.6) hold in terms of k and the same angle , where ! q2 2 ;1 ky (14.9) k = kx + ky and = tan k x such that kx = k cos and ky = k sin (14.10) Substituting ~ = (kx ky ) from (14.10) and ~ = (x y) into (14.4), k r s(k ) = ZZ dxdy (x y)e;i2 k(x cos +y sin ) (14.11) Finally, the polar variables k spur a corresponding change from rectangular to polar coordinates in (x y). Letting x = r cos and y = r sin (14.12) the signal becomes s(k ) =
2 ZZ rdrd (r )e;i2 kr cos ( ; ) (14.13) The angles and are displayed in Fig. 14.1.
Recall that a given gradient direction means that the z -component of the magnetic eld changes linearly along that direction. 3 Throughout this chapter, and elsewhere in the book, it will be most convenient if the same symbol can be used for a function, even though the variables have been changed. For example, we may refer to f (x y) or f (r ) depending on whether Cartesian or polar coordinates are used. We may also use the same symbol for more variables, i.e., f (x y z ). 306 Chapter 14. Projection Reconstruction of Images Fig. 14.1: The familiar polar and Cartesian relationships are shown for (r ) and (k ). 14.1.2 Two Radial Fourier Transform Examples
(r) = N0 (r ; r0) Consider two examples of spin densities independent of the polar angle . First, a spin density representing an in nitesimally thin ring at r = r0 is (14.14) where N0 is the number of spins at any given angle. The double integration (14.13) is reduced to a form independent of , as expected, owing to the periodicity in the integration4 sring (k) = N0 r0 d e;i2 kr0 cos ( ; ) 0 = 2 N0r0 J0(2 kr0) Z2 (14.15) The transform has been related to the zeroth order Bessel function J0. The general expression for the nth -order Bessel function is in Z 2 d e;iu cos +in Jn(u) 2 (14.16) 0 These functions serve as a set of radial basis functions analogous to the trigonometric Fourier series. The second example is that of a disk of spins, (r ) = in which the Heaviside or (u)
4 0 (r0 ; r) (14.17) (14.18) step function is utilized ( 1 if u > 0 0 if u < 0 A shift in the integration variable may be used to show that integrals such as independent of for any function f . R 2 d f (cos( ; 0 )) are 14.1. Radial k-Space Coverage 307 d With the identity uJ0(u) = du (uJ1(u)), the Fourier transform (14.13) for the disk is found to be Z r0 sdisk (k) = 2 0 dr rJ0(2 kr) 0 = 0 r0J1(2 kr0 )=k (14.19) (14.20) In Fig. 14.2, a plot of (14.20) is compared with the product of sinc functions found in the 2D Cartesian transform of a box shaped spin density. The shapes in k-space are similar. Problem 14.1
Compare the Taylor series expansions up to and including fourth-order in k for sinc(k A) and J1( kd)=( kd=2) where A is the width of the rect function and d is the diameter of the disk. Using these series, or some other computation, compare the two functions through a plot on the same graph for A = d. Fig. 14.2: Comparison between (a), the Fourier transform of a disk of diameter d of uniform spin density, and (b), the Fourier transform of a square of side d with uniform spin density. 14.1.3 Inversion for Image Reconstruction
(x y) = Su cient k-space data will allow a determination of the image through the inverse Fourier transform of (14.4) or (14.13). In Cartesian coordinates the image is found from ZZ dkxdky s(kx ky )ei2 (kx x+ky y) (14.21) 308 and in polar coordinates from (r ) = Chapter 14. Projection Reconstruction of Images ZZ kdkd s(k )ei2 kr cos ( ; ) (14.22) The degree to which the image is faithfully reconstructed depends on the completeness of k-space coverage. The next question is the manner in which the array of vector values of ~ needed for the k above inverse transforms is measured. Di erent points along a line in ~ -space collinear with a k ~ given G can be sampled over time, or by changing the magnitude of the gradient. To sample lines at di erent angles, and sweep out a 2D area in ~ -space, the gradient vector direction k can be changed systematically. After a discussion of the number of data points required, various formulas for analyzing data gathered in radial sweeps of ~ -space are detailed in the k rest of the chapter. Data can also be sampled in k-space and then interpolated to a Cartesian coverage to allow the use of (14.21). Similarly, (r ) is usually interpolated to a Cartesian grid to display the image. 14.2 Sampling Radial k-Space and Nyquist Limits
Radial Coverage with an Echo
Assume for a given readout that it is possible by gradient reversal or by the appropriate spin echo sequence to cover both positive and negative halves of each line covered in k-space for a xed (see, for example, Fig. 14.3a). Then the interval can be limited to 180 . To maintain coverage of the complete kx-ky plane, k can be rede ned as plus or minus the magnitude of ~ with the limits, k ;kmax k kmax (14.23) where kmax = {GTs=2. The angular coverage is also limited to collect a nite number of angles. For N uniformly spaced readouts (N di erent gradient directions, Fig. 14.3b), N = (14.24) Problem 14.2
For the convention using both positive and negative values of k, show that the polar variable relations (14.10) are replaced by
2 2 a) k = sgn(ky ) kx + ky ky b) = tan;1( kx ) mod q 14.2. Sampling Radial k-Space and Nyquist Limits 309 Radial Coverage with FIDs
Although there is an inherent advantage to using an echo to collect a frequency encoded symmetric data set, the original bene t gained in radial coverage (i.e., not having to wait for the phase encoding before readout) is lost. Most of the current applications of radial k-space coverage involve imaging of short T2 or T2 tissues. Sampling of the k = 0 point as quickly as possible is achieved only in the limit of sampling an FID. In this case, each readout covers the range 0 k kmax, and the gradient angle must now be varied over the range 0 < 2 to cover a circular region of radius kmax in k-space. The sequence diagram for such an acquisition method is shown in Fig. 14.3c. In any FID sampling scheme, the true k = 0 point, i.e., the time point where all isochromats are in phase, is not adequately sampled because of the nite width of the rf pulse. In those cases, the measured k-space function is actually a shifted version of s(k ), (s(k + ), say). This k-space shift manifests itself as a spatially linearly varying phase when the measured data are inverse Fourier transformed and image reconstruction then creates artifacts due to phase cancellation. An estimate of this linearly varying phase is therefore required before image reconstruction can be successfully performed on a newly created data set corrected for this shift. Even if the phase shift is successfully used to estimate the k-space shift , it is still di cult to extrapolate the collected data to estimate s(0 ) especially in an imaging experiment performed over a large sample because of the presence of B0 inhomogeneities. However, as seen later (in Sec. 14.4.4), the signal at the k = 0 sample is not required for image reconstruction as it is forced to be zero in the reconstruction method anyway (see the description of ` ltered back projection' and the `M- lter' in Sec 14.4.4). Another major disadvantage with FID sampling is that if there are any remnant eddy currents (discussed in Ch. 27) immediately after the ramp-up of the read gradients, these might cause missampling of k-space, making the prediction of the s(0 ) point di cult in practice. As a result of the di culties with the FID method, the sampling scheme is often a compromise between the ease of reconstruction of the symmetric radial coverage and the short intervals of the rst readout sample of the FID radial coverage method (see Fig. 14.3c). This compromise is achieved by using highly asymmetric spin echoes or gradient echoes. The collection of a few negative k-space points in an asymmetric data collection scheme also allows image reconstruction with 0 and the use of partial Fourier image reconstruction (see Ch. 13). Nyquist Limits in Radial Sampling
The Nyquist limitations on the step sizes in k-space need to be adapted to the polar coordinates. The di erences compared to Cartesian step sizes change the nature of aliasing in important ways. The step sizes in the radial direction and in the direction, respectively, kr and k , are illustrated in Fig. 14.4. The key is that the latter depends on the distance from the origin k =k kmax (14.25) 310 Chapter 14. Projection Reconstruction of Images 14.2. Sampling Radial k-Space and Nyquist Limits 311 Fig. 14.3: Sequence diagrams and associated k-space coverage for radial coverage. Spin echo sampling is illustrated in (a) and (b). The frequency encoding gradients, one of which steps through a cosinusoidal table, and the other which steps through a sinusoidal table, are varied such that lies in the range 0 . FID sampling is illustrated in (c) and (d), where the frequency encoding gradients vary such that lies in the range 0 2 . This variant has found increasing usage in imaging short T2 or short T2 tissues. A partial Fourier method necessitates the use of asymmetric radial sampling as shown in (e) and (f). To prevent aliasing, the largest angular step for xed must follow the Nyquist criterion (Ch. 12) kmax 1=L (14.26) for the eld-of-view L. Hence the minimum number of angular views required to avoid aliasing is n = = kmax L (14.27) The radial step must similarly satisfy kr = 1=L (14.28) From (14.8), the time steps along the gradient direction are thereby limited to t = kr =( {G) 1=( {GL) (14.29) The minimum total number of samples required in radial sampling is nr = 2kmax = kr 2kmax L (14.30) This is the same condition as that for the read direction in the Cartesian k-space coverage method. Therefore the total number of points required to obtain an unaliased image 312 Chapter 14. Projection Reconstruction of Images with equal spatial resolution in both directions, i.e., with equal radial and angular spatial resolution when an echo is collected, is nr n 2 (kmaxL)2 (14.31) Fig. 14.4: Radial k-space coverage involves obtaining samples in k-space that fall on concentric circles. The separation between any two adjacent circles de nes the separation kr in the radial sampling direction, whereas, the angular separation between any two successive angular lines in k-space (such as between the two example lines shown in the gure) de nes . The two quantities kr and are constrained by the Nyquist sampling criterion. Comparison of Projection Reconstruction and 2D k-Space
The number (14.31) can be compared to the number of points required in Cartesian coverage. The number in either the x or y directions is the same as (14.30), so the total for the standard 14.2. Sampling Radial k-Space and Nyquist Limits two-dimensional imaging method is5 313 (14.32) nxny 4(kmaxL)2 The simple fact that 2 > 4 means there are more sampling points in the circle of diameter L (pertinent to the radial coverage) than there are in the (larger) square of side L (pertinent to Cartesian coverage). In order to get adequate sampling at large k, there is oversampling (i.e., sampling more densely than Nyquist requires) of the smaller k values where k is reduced proportionate to the k value. The total acquisition time for the projection reconstruction method is thus =2 times longer, and the extra points collected result from the increased scan time. Looking ahead to Ch. 15, it is noted that this translates into better signal to noise for the `radial readout,' but the price paid is the longer time taken for the measurement. Furthermore, this potential increase in SNR is realized only for a subregion of sampled points. The illustration that the density of coverage in k-space varies inversely as the distance from the origin in k-space is presented in Fig. 14.4. We are reminded that the density of coverage in Cartesian methods is uniform. This varying density of coverage (leading to nonuniform coverage) usually has implications that relate to aliasing artifacts (as discussed brie y in Ch. 12). On the other hand, because the denser sampling of small k-space values implies relatively more information about the large structures of the image, the corresponding image reconstruction creates an image with a higher signal to noise ratio at the large distance scales in contrast with images reconstructed from Cartesian data. The high density in the center of k-space can lead to coherent or colored (in comparison with incoherent or white) noise in the object which can create a streaking of noise across pixels (see Ch. 15). Problem 14.3
a) For a circular eld-of-view of diameter L = 256 mm and kmax = 1 mm;1, nd the minimum number of angular views (n ) that would be required to avoid aliasing in (i) an FID radial sampling scheme, and (ii) a spin echo sampling scheme. b) Similarly, nd the minimum number of views (nr ) required along the radial k-space variable direction. c) If TR = 600 ms, what is the minimum total imaging time for (i) the FID radial sampling scheme, and (ii) the spin echo sampling scheme? It is interesting to note that the spatial resolution in the x or y direction is limited in the 2D approach, since each gradient has an upper limit, which in turn restricts k-space coverage. But for , the resolution in the direction, no such limit exists. Therefore, by
We consider the same maximum for k in both coverages in order to have the same spatial resolution in the radial direction, r = 1=(2kmax), as in either x or y direction.
5 314 Chapter 14. Projection Reconstruction of Images an increase in angular acquisition, an object, such as a wheel with many spokes, can be resolved to the degree of accuracy desired. 14.3 Projections and the Radon Transform
A rotated coordinate system can be exploited for the case of a xed gradient direction. It is observed that the phase in (14.11) depends only on the combination x0 = x cos + y sin (14.33) This is expected since the rotation through an angle to a new coordinate system leads to ~ r a frequency encoding axis x0 parallel to the gradient direction (G ~ = Gx0) (see Ch. 9). It is natural to go to an integration over the primed coordinates since all spins along a given line perpendicular to the x0 axis have constant phase. A change of integration variables in (14.11) from (x y) to (x0 y0), for example, yields6 s(k ) =
where (x0 y0) can be written (x0 y0) ZZ x kk ^
0 dx0 dy0 (x0 y0)e;i2 ~ kx 0 (14.34) (14.35) ZZ dxdy (x y) (x0 ; x cos ; y sin ) (y0 + x sin ; y cos ) The integration restrictions indicated imply that the Cartesian coordinates x0 y0 must be chosen such that the x0 -axis is at an angle with respect to the original x-axis. That is, x0 is given by (14.33) and y0 by y0 = ;x sin + y cos (14.36) 0 0 The axes x y are a (counterclockwise) rotated version of the x y axes. The fact that the phase in (14.34) is independent of y0 focuses attention onto the integral P (x ) =
0 Z In the above notation, the projection P (x0 ) represents an integration along a line, at an angle of =2 + with respect to the x-axis, passing through the point x0 . This line integral follows the Cartesian ray perpendicular to the gradient direction, and is sometimes referred to as a `ray sum,' a nomenclature growing out of the x-ray analog discussed below. The ray for x0 = x00 and a given is shown in Fig. 14.5. The reference to a `ray sum' is appropriate in view of the discrete steps taken in measurements and in numerical integration over that discrete data set. The (discrete) set of all ray sums at a given is the full `projection' for a speci ed gradient direction. The continuous set of projections at all possible angles of an object (x y) comprise the set of Radon transforms . The Radon transform is de ned as (x ) R
0
6
0 0 x 0 dy0 (x0 y0) (14.37) Z1Z1 ;1 ;1 dxdy (x y) (x0 ; x cos ; y sin ) 0 (14.38) dx dy for the same reason a rotated pixel is unchanged in area or shape. The Jacobian for the transformation represented by (14.33) and (14.36) is unity. More simply, dxdy = 14.4. Methods of Projection Reconstruction 315 Fig. 14.5: Notation and framework for de ning the projection of an object (shown here as a shaded surface) for rays (such as the one shown at a normal distance of x00 ) making some arbitrary angle with respect to the x-axis. which is nothing more than what is shown pictorially in Fig. 14.6. Hence, a set of `projections' as de ned above and collected at some nite set of angles, is a discrete approximation to the Radon transform of (x y). Problem 14.4
Compute the Radon transform of the following circular disk of uniform spin density: x2 + y2 2 (x y) = 00 elsewhere r0 ( 14.4 Methods of Projection Reconstruction with Radial Coverage
There are several ways of extracting an image from the data obtained by the projection approach. We consider three methods of projection reconstruction, the rst a trivial and coarse approximation of the object, and the other two the most commonly used methods, after a brief connection to the reconstruction technique used in x-ray tomography is made. 316 Chapter 14. Projection Reconstruction of Images Fig. 14.6: A complete sampled set of sums for a set of parallel rays de ned by the gradient direction.
A continuous version of this so called `parallel ray projection' for a given object measured over all possible angles is the Radon transform of the object. 14.4.1 X-Ray Analog
The gathering of data over rays projected through objects has its historical precedence in x-ray tomography. An x-ray beam is attenuated as it passes through the body to be imaged. With an initial rate I0 of photons per unit time (at the same energy), the attenuation through a homogeneous thickness L is given by the familiar exponential formula, I = I0 e; L (14.39) where is an absorption constant that depends on the electron density of the material. For a heterogeneous body with a spatially varying attenuation constant (x y) and a beam direction of =2 + with respect to the x-axis, the primed coordinates in the previous projection discussion can also be employed in the more general attenuation formula, giving I (x0 ) = I0(x0 )e; R L dy (x y )
0 0 0 (14.40) The tomographic ray sum is de ned by the exponent in (14.40) Px;ray (x0 ) = ln (I0(x0 )=I (x0)) = Z L dy0 (x0 y0) (14.41) This measures the shading found in x-ray lm and represents the total (i.e., integrated) absorption along the ray de ned by the polar coordinate pair (x0 ). Comparison of (14.37) with (14.41) shows a close connection between x-ray imaging and the radial readout measurement in MRI. 14.4. Methods of Projection Reconstruction 317 14.4.2 Back-Projection Method The line integral in (14.37) or (14.41), associated with a ray such as that shown in Fig. 14.5, constitutes a signal whose components come from somewhere along the line of pixels orthogonal to the x0 -axis at some xed x0 value. A rst approximation to image inversion from this data is to assume the signal has had equal contributions from each pixel in the line a graphical representation of this assumption is to shade each pixel with its share of the total weight corresponding to the value of the original line integral. (Thus, in rst approximation, the share is 1=nr of the integral value.) Each daughter signal (the line integral) is apportioned back equally on each parent pixel (in the reconstructed image). The image obtained is the net shading achieved when all rays are added together. This process of image formation is called `back-projection.' In Fig. 14.7, a set of four rays is illustrated along with the corresponding signal shading. The example of back-projection shows only a rough similarity to the original object. Although a larger number of rays can improve the result, residual `starring' remains at the edges, and corrections must be applied. A ltered version of the method is discussed after the next subsection. There is also some residual nonzero signal (bleeding out of the signal) in the reconstructed image where originally the spin density was zero. Fig. 14.7: The back-projection method is the simplest rst approximation to the image in recon- struction from projections. Note the `starring' artifact is obtained even for the simple case of a disk of uniform spin density. This bleeding out (or smearing) artifact can be demonstrated easily with the simple example of an object with a point spin density. Suppose that the spin density model is such that it has unit spin density at the point x = x0 (illustrated as point P in Fig. 14.8a). Then (x y) = (x ; x0 ) (y) (14.42) and the Radon transform (or projection) of evaluated at a distance x0 away from the origin 318 along a direction making an angle
0 Chapter 14. Projection Reconstruction of Images is given as (14.43) (x0 0 ) = (x0 ; x0 cos 0 ) Back-projection of some discrete set of projections representing samples of the continuous Radon transform of the object requires that each given projection be taken and reprojected into the image matrix and the intensities at a given pixel from all reprojected rays be summed to give the nal image R(x y). Since each projection at angle has unit amplitude at x0 = x cos , and is zero everywhere else (from (14.43)), each reprojection will create a line of unit intensity, each of which passes the point P in the reconstructed image, while one and only one reprojected nonzero ray passes through every other point (see Fig. 14.8b). In other words, the reconstructed image has unit intensity everywhere, and an intensity n at P . The nonzero background intensity represents the starring or blurring e ect, and the relative intensity of the artifact is reduced as the number of angular samples n increases. Fig. 14.8: (a) Pictorial representation for a particular projection computed such that the point P is mapped to the point where x00 = x0 cos 0 on the projection. See Fig. 14.5 for details. (b)
The reprojection of the collected projection data creates lines of unit intensity, all of which pass through P , but only one of which passes through any other point. The nal back-projected image has a value n at point P and a value of unity everywhere else. In summary, in the case where is covered continuously over 0 to , the back-projected image R(x y), represented by B , is given by R (x y ) B = Z 0 d P (x cos + y sin ) (14.44) The formula for the back-projected image that allows for rays that are not necessarily distributed continuously, but covers 0 through with the most general case of unequal angular intervals of i is X R(x y) = P (x cos i + y sin i ) i (14.45)
i 14.4. Methods of Projection Reconstruction 319 Aside from units, R(x y) re ects the reconstructed spin density corresponding to the starred construction described earlier. For a given x y, the expression (14.33) changes from angle to angle, and is generalized to give x0i = x cos i + y sin i (14.46) The inherent di culty in using the back-projection method to recover an acceptable approximation of the object spin density, even in the ideal case, has been made evident. It is usually discussed as a method of projection reconstruction, however, since it is a useful tool for laying the groundwork for other reconstruction methods to follow. In another approach to image reconstruction from projections, we return to the relation of the measured signal to the two-dimensional Fourier transform of the spin density in k-space. 14.4.3 Projection Slice Theorem and the Fourier Reconstruction Method
Projection Slice Theorem
In terms of the line projection (14.37), (14.34) becomes Z s(k ) = dx0P (x0 )e;i2 kx 0 (14.47) The angle is xed in this integration and, hence, it is seen that s(k ) is the Fourier transform, with respect to x0 , of the line projection P , with a xed spectator parameter. This is the projection slice or central slice theorem which says that the two-dimensional Fourier transform s(~ ) of the spin density is the one-dimensional Fourier transform of the k line projection integral. The one-dimensional transform is with respect to the coordinate along the direction of ~ . The interrelationship among , and s, as represented by the k projection slice theorem, is illustrated in Fig. 14.9. Fig. 14.9: Interrelationships among the original spin density , its Radon transform , and its 2D Fourier transform s. In the gure, F represents the 2D Fourier transform operator, and Fk
represents the 1D radial Fourier transform operating on each projection. 320 Chapter 14. Projection Reconstruction of Images Fourier Reconstruction
In review of the standard 2D Fourier MR imaging method, position (with respect to the two orthogonal directions within the slice) is encoded into the phase of the collected data at di erent instants of time. Recall that one direction involves `phase encoding' and the other, `frequency encoding.' Because the encodings are independent of each other, the Fourier inversion operator is separable in the two directions, and it can be implemented as two 1D inversions, one following the other. However, in the radial sampling method, both spatial variables are encoded at the same instant of time. Therefore, a true 2D inversion would be needed if the reconstruction were performed directly on the collected data, for which certain fast Fourier implementation algorithms do not apply. Fortunately, we can make use of the projection slice theorem in a conversion from radial k-space data to an interpolated rectangular grid. The 2D Fourier inversion can then be employed and fast Fourier transform numerical techniques can be utilized in such a `Fourier reconstruction method.' Problems occur near the edge of k-space in accurately doing this (because of the reduced density of k-space samples there), leading to aliasing of edges (represented mainly by the high spatial frequency data) unless oversampling is done in the direction to improve interpolation. 14.4.4 Filtered Back-Projection Method
R(x y) = There is a way to nd the two-dimensional Fourier transform of (x y) directly from the back-projection formulas. Consider (14.44), which is the integral limit of (14.45), Z remembering that there is additional dependence in x0 through (14.33). The projection slice theorem (14.47) implies that the line projection P (x0 ) is given by a 1D inverse Fourier transform of s(k ) along the k direction. Substitution of P (x0 ) in (14.48) yields 0 d P (x0 ) (14.48) R(x y) = Z Note that the k-integration in (14.49) includes negative values. We can reduce the k range to the positive axis, returning to a magnitude de nition for k by doubling the integration range, Z2 Z1 kr R(x y) = d dk s(k )ei2 ~ ~ (14.50) 0 0 where we have also backtracked to the original form of the exponent in (14.50) involving the two-dimensional vectors ~ and ~. k r It is observed in (14.50) that, if the integrand is multiplied and divided by k, the full polar-variable measure of kdkd = d2k appears. Thus 1=k times s(k ) must be the Fourier transform of R. We have the Fourier pair in 2D Z s(k ) ~ r R(x y) = d2 k k ei2 k ~ = F ;1(s=k) (14.51) Z s(k ) = d2rR(x y)e;i2 ~ ~ = F (R) kr (14.52) k 0 d Z1 ;1 dk s(k )ei2 kx 0 (14.49) 14.4. Methods of Projection Reconstruction 321 The procedure can now be made clear. The Fourier transform s(k ) of an object is found by rst determining the Fourier transform of R(x y) and then multiplying it by k (or jkj, if we go back to including negative values for k). In detail, rewriting the spin density as the inverse Fourier transform of s(k ) but integrated over the ranges suitable for projection variables and expressing the exponent ~ ~ = kx0 (from (14.22)) yields k r (x y ) = Z The integrand now suggests the use of a ` ltered' projection that is the inverse Fourier transform of jkjs(k ) instead of s(k ), as it naturally arises in (14.53), by 0 d Z1 jkjdk s(k )ei2 kx ;1 0 (14.53) P lter(x0 ) = Z1 jkjdk s(kx ky )ei2 kx ;1 0 (14.54) Finally, the image is obtained from the inverse Fourier transform of the ltered k-space data. Multiplication by jkj is both a cure and disease for this image reconstruction problem. Note that this `M- lter' (so-called because of its `M'-like shape in k-space see Fig. 14.10) is a high pass lter since it enhances the high spatial frequencies while eliminating the zero spatial frequency. Remember the comment in the back-projection section that the simple reprojection of the collected projections creates blurring and the starring artifact. This high pass lter serves to remove the blurring, and an acceptable image is obtained using the ltered back-projection method (see Fig. 14.11 for a comparison between images reconstructed by back-projection and ltered back-projection). Although the object is now ideally reconstructed, the noise at the higher k-space samples gets exaggerated by the linear lter jkj. As a result, the reconstructed image has noise which becomes correlated even though the noise in the collected data is white and uncorrelated. Fig. 14.10: The M- lter k-space response. In practice, it is common to smooth the M- lter transition to zero at kmax to avoid truncation artifacts (ringing) at tissue boundaries by applying
a Hanning lter following the M- lter. 322 Chapter 14. Projection Reconstruction of Images Fig. 14.11: Comparison between (a) back-projection and (b) ltered back-projection reconstruction
from projections of a disk of uniform spin density. Note that the ltering has eliminated the star artifact seen in the back-projection. 14.4.5 Reconstruction of MR Images from Radial Data
Note that the data collected in the projection reconstruction scheme are the 1D Fourier transforms of projections of the spin density function along di erent directions. Therefore, methods such as ltered back-projection have the added advantage of needing less reconstruction time because the time required for the 1D Fourier transforms is saved. Therefore, the ltered back-projection method applicable to MR data is to apply the M- lter to s(k ) by multiplying it by jkj. Then take the inverse Fourier transform and back-project to create the image. In operator form, this translates into ^(x y) = f BFk;1]jkjs(k )g(x y) (14.55) Similarly, the Fourier regridding and reconstruction of MR images is made easier by the data already being in k-space. The only required operation on the collected data is to regrid the radially sampled data to Cartesian coordinates so that a 2D fast Fourier transformation would yield the nal image. One remark about a practical issue is worth mention here. If the projections obtained by inverse Fourier transforming each line of collected data at di erent angles have the origin mapped to di erent points, the reconstructed image will be badly blurred. One way to remedy this is to note that the center of mass of each projection is supposed to be spatially invariant. Using this fact, each projection can be centered correctly by estimating the center of mass of each projection and forcing them all to be at some arbitrary point. 14.5. Three-Dimensional Radial k-Space Coverage 323 Problem 14.5
In this problem, we will derive the form of another image reconstruction method which is equivalent to ltered back-projection, and is commonly known as the `convolution back-projection' reconstruction method. a) Starting with the nal expression for the ` ltered projection' P lter(x0 ) in (14.54) and the convolution theorem, show that P lter(x0 ) = F ;1s(k )] F ;1jkj] (14.56) b) Show that the inverse 1D Fourier transform of jkj evaluated at r is ;1=(2 2r2). Hence, write an equivalent expression for the ` ltered projection' as a convolution integral. c) Write the equivalent set of operations for convolution back-projection from radial MR data based on your result in part (b). 14.5 Three-Dimensional Radial k-Space Coverage
The signal measured from an experiment with an arbitrary frequency encoding gradient is the Fourier transform of the line integral or ray projection of the spin density distribution of the object along the gradient direction only when a slice selective pulse is used. When such a selective pulse is not applied, and instead a nonselective pulse is applied, the measured signal is now a planar projection or planar integration, the plane again being perpendicular to the gradient direction de ned by the polar angle and azimuthal angle . Suppose we consider a gradient applied at an angle relative to the z-axis and angle relative to the x-axis. Then, for points P de ned by the Cartesian coordinates (x y z) lying along some arbitrary plane perpendicular to some point R de ning the projection plane and at a distance r away from the origin, de ned by the spherical polar coordinates (r ) (see Fig. 14.12), the vectors ;! ;! OP and OR are de ned respectively, as: OP = x^ + y^ + z^ x y z ;! OR = r sin cos x + r sin sin y + r cos z ^ ^ ^
;! ;! (14.57) (14.58) Hence, the normal vector PR drawn from P to the line OR de ning the frequency encoding axis is given by RP = OP ; OR = (x ; r sin cos )^ + (y ; r sin sin )^ + (z ; r cos )^ x y z ;! ;! ;! (14.59) 324
;! ;! Chapter 14. Projection Reconstruction of Images
;! ;! Since OR and RP are perpendicular to each other, OR PR = 0, and that de nes the equation of the projection plane, i.e., (x ; r sin cos )(r sin cos ) + (y ; r sin sin )(r sin sin ) + (z ; r cos )(r cos ) = 0 (14.60) which, on simpli cation yields the equation of the projection plane as: r = x sin cos + y sin sin + z cos (projection plane equation) (14.61) Fig. 14.12: Projection plane;! nition. Here the projection plane at R(r ) is de ned to be de ;! perpendicular to ;! vector OR. One such point on the projection plane is P , with PR being the perpendicular to OR. By collecting a set of these planar projections for di erent values of and covering a sphere for obtaining isotropic spatial resolution, the entire spherical volume can be reconstructed into voxels (or volume elements) that are represented by a uniform spin density estimate (x y z). This 3D spin density distribution can be written in terms of its 3D Fourier transform, s(kx ky kz ) as: (x y z) = ZZZ dkxdky dkz s(kx ky kz )ei2 (kx x+ky y+kz z ) (14.62) which is the same representation as the reconstructed image from the signal measured one line at a time in k-space by `reading out' after the signal is phase encoded in the other two directions by constant phase encoding and partition encoding gradients. Repeating such an experiment with di erent values of the phase encoding and partition encoding gradients allows for the complete coverage of 3D k-space. This same k-space inversion method is valid for data encoded radially over a sphere such that one radial line is collected in k-space making angles and with respect to the z and x-axes respectively, and parallel to the read 14.5. Three-Dimensional Radial k-Space Coverage 325 gradient direction during a single readout. Repeated measurements with read gradients at di erent and values such that they cover the required extent in k-space within a spherical volume allow inversion by converting (14.62) into its polar form, giving (x y z ) (~) = r Z2
0 d Z 0 d Z1
0 dk k2 sin s(k kr )ei2 ~ ~ (14.63) where ~ has spherical polar coordinates of (k ). k Image reconstruction from 3D radial k-space data can be carried out similar to reconstruction from 2D data, using either the 3D version of the projection slice theorem or a new ltered back-projection method. To mathematically state the 3D projection slice theorem, we again de ne the planar projection along the ( ) direction, P (r),7 and its 1D radial Fourier transform, Fr fP (r)g or F (k), say. P (r) =
and ZZZ dxdydz (x y z) (r ; x sin cos ; y sin sin ; z cos ) F (k) Fr fP (r)g(k) s(k (14.64) (14.65) (14.66) Z1
0 drP (r)e;i2 kr Given these notations, the projection slice theorem states that ) = F (k) In other words, each measured line F (k), the Fourier transform of the planar projection P (r), equals that line from the 3D k-space data s(k ) along the same orientation as the read gradient. Therefore, either one of the two expressions (14.62) or (14.63) can be used for computing the required 3D spin density distribution (x y z). The main di erences between using the radial representation and Cartesian representation is that there is no fast Fourier transform method to implement (14.63) as in the 2D case, whereas, once the Cartesian representation is obtained approximately by interpolating s(k ) into a Cartesian coordinate system, the 3D inversion can be achieved separably in the three directions using FFTs. To derive the 3D ltered back-projection method, we rst note that P (r) can be written in terms of F (k) as P (r) =
= Z1
1 Z0 dkF (k)ei2 dks(k kr kr ;1 )ei2 (14.67) Using a previous notation (14.54), let us de ne a ` ltered projection:' P
7 lt (r) Z1
0 dkk2 sin s(k )ei2 kr (14.68) The set of all planar projections, de ned over all possible (r object. ), is the 3D Radon transform of the 326 Then, from (14.63) and (14.64) (~) = (x y z) = r Chapter 14. Projection Reconstruction of Images Z2 Z
0 If the 3D back-projection operator B3 acting on some set of planar projections P lt(r) is de ned as the operation 0 d d P lt(x sin cos + y sin sin + z cos ) (14.69) B3 P lt(r)
then R(x y z) Z Z2
0 0 d d sin P lt(r)jr=x sin cos +y sin sin +z cos (14.70) (x y z) = B3P lt(r) (x y z) (14.71) i.e., the 3D spin density distribution is obtained as the 3D back-projection of the inverse Fourier transform of the collected k-space data s(k ) that is ` ltered' by the k-space function k2 sin . Only the continuous case in k-space has been considered up to now. In reality, only samples are collected in k-space. Nyquist limits similar to the 2D case can be derived for the 3D case too. To determine how the gradient direction is stepped, the usual method is to assume that isotropic spatial resolution is required. This means that the solid angle8 sin d d formed by the elemental surface area strip on the unit sphere by some azimuthal angular increment d from some initial angle and by some polar angle increment d from some initial angle , is the same for all ( ) pairs (see Fig. 14.13). Since the solid angle is sin d d , this means that if is incremented in uniform intervals of , is incremented as sin . For isotropic resolution, = . Hence, the discrete ltered 3D back-projection is obtained as X X lt (x y z ) = 2 P m n (x sin m cos n + y sin m sin n + z cos m ) (14.72) for such a k-space coverage.
m n h i 14.6 Radial Coverage Versus Cartesian k-Space Coverage
In the Cartesian k-space sampling scheme, a nite amount of time is spent after the rf pulse for phase encoding to encode the directions other than the frequency encoding direction. In radial coverage, this nite time is saved, as the readout can be started as soon as the slice select rephase lobe is completed. This saving is very useful in imaging certain short T2 tissues, and in microscopic imaging where susceptibility e ects cause the T2 to be extremely short. Of course, with a nonselective rf pulse (rf pulse with a sharp boxcar rf envelope) with a frequency response broad enough to excite all spins in the object lying within the region of sensitivity of the transmit coil, even the nite time required for the rephase lobe of the slice select gradient is not required. Then, an FID sampling is approximately achieved.
Remember, solid angles are de ned as elemental surface areas on the unit sphere. For example, the solid angle subtended by the entire spherical surface is 4 steradians, since the total surface area of the unit sphere is in fact 4 .
8 14.6. Radial Coverage Versus Cartesian k-Space Coverage 327 Fig. 14.13: Illustration of the solid angle de ning the elemental area sin d d on the unit circle. Note that for isotropic resolution in the nal 3D reconstruction, this solid angle formed by elemental changes d and d in the azimuthal and polar angles remains constant for all and . One di culty with radial k-space coverage is the radial blurring of point objects that occurs in the presence of eld inhomogeneities. On the other hand, in Fourier imaging, this resonance o set e ect shifts the location of the a ected spins along the read direction only. As a result, in both cases, the point spread function changes from point to point as a function of the eld inhomogeneity relative to the read gradient strength. 14.6.1 Image Distortion Due to O -Resonance E ects: Cartesian Coverage Versus Radial Sampling E ects of B0 Inhomogeneities in Cartesian Imaging
In this and the next subsection, the e ects of eld inhomogeneities on the imaging point spread function are considered (no discrete approximations are considered, and the data and image functions are (continuous) Fourier transform pairs this implies that the point spread function in the absence of any inhomogeneity is a Dirac delta function in either case). To this e ect, the image for data collected when the object is a point (i.e., an impulse at some arbitrary position) gives the point spread function. Let us consider the object to be a unit impulse at r0 = (x0 y0), i.e., ~ (x y) = (x ; x0 ) (y ; y0) (14.73) Suppose that the eld is inhomogeneous, and the inhomogeneity at (x0 y0) is B (x0 y0). Therefore, the measured signal as a function of time is s(t) = e;i2 { B(x0 y0 )t e;i2 ( {GRx0(t;TE )+ky y0 ) (14.74) 328 Chapter 14. Projection Reconstruction of Images Since kx equals {GRt0 where t0 = (t ; TE ), {t0 in the above expression can be replaced by kx=GR, and the signal can be written as a function of kx as
B(x0 y0 ) s(k) = e;i2 kx GR e;i2 { B(x0 y0 )TE e;i2 (kxx0 +ky y0) (14.75) Inverse Fourier transforming and neglecting the constant phase factor rst term and recognizing that the third term is in fact the Fourier transform of (x y) (from (11.5)) gives the generalized point spread function in the presence of an inhomogeneity at position (x y) as ! B (x0 y0) (y ; y ) h B c(x y) = x ; x0 ; G (14.76) 0 R As before, this new point spread function only convolves with any other point spread function, whether it is due to signal decay during sampling or to the discrete Fourier transformation. Since this is a delta function, it obviously does not change the shape of the the imaging point spread function. The only e ect is that the coordinate position in the read direction is translated by an amount x0 + B (x0 y0)=GR in the presence of a continuously-varying eld inhomogeneity function, this leads to a distortion of the object. This distortion e ect is the subject of detailed discussion in Ch. 20. E ects of B0 Inhomogeneities in Radial Imaging
An analysis along lines similar to the one above is carried out for radial imaging here. The Radon transform of ~(x y) (de ned in (14.38)) yields the projection P (x0 ) = ZZ dxdy (x ; x0) (y ; y0) (x0 ; x cos ; y sin ) (14.77) Similar to (14.47), the collected k-space data in the presence of B (x0 y0) are then given by Z ZZ s(k ) = dx0e;i2 kx dxdy (x ; x0 ) (y ; y0) (x0 ; x cos ; y sin ) e;i2 { B(x0 y0)t (14.78) 0 0 In the above equation, the radial k-space variable k isq by k {Gt , and t (t + TE ). given Note that G is independent of , and is given by G = G2 + G2 with Gx and Gy varying as x y G cos and G sin for a given projection angle . Upon simpli cation, (14.78) yields s(k ) = e;i2 k(x0 cos +y0 sin ) e;i2 k B(x0 y0)=G e;i2 { B(x0 y0)TE (14.79) Let us use the ltered back-projection method as a tool for illustration here. The reconstructed image evaluated at ~ = (x y) (neglecting the constant phase term e;i2 { B(x0 y0)TE ) r is given by
0 ^(x y) = Z2
0 d Z1
0 dk kei2 k(x cos +y sin ) e;i2 k(x0 cos +y0 sin ) e;i2 k B(x0 y0 )=G (14.80) To determine the relevant point spread function due to the presence of the eld inhomogeneity, (14.80) is rewritten using the convolution theorem (see Ch. 11, for example) as ^(x y) = F2;1 e;i2 k(x0 cos +y0 sin ) F2;1 e;i2 k B(x0 y0)=G (14.81) D D 14.6. Radial Coverage Versus Cartesian k-Space Coverage 329 From (11.5), the rst term is a delta function positioned at x = x0 y = y0, the same as the object's spin density. The second term, the point spread function (hradial(x y)), is clearly B not a delta function, unless B (x0 y0) is zero. The 2D inverse Fourier transform needed to evaluate hradial(x y) can be performed in B radial k-space, where each point is de ned by the coordinates (k ). Let us de ne a r0 as B (x0 y0) G for convenience. Now, performing the integration over rst gives the result Z1 hradial(r ) = 2 dk kei2 r0 k J0(2 kr) B r0
0 (14.82) (14.83) which is independent of the polar angle in the spatial coordinate system used to describe the reconstruction point (x y). This means that, instead of the local, spatial shifting of the point as occurs in Fourier imaging, the e ect in radial k-space coverage is to create a radial smear. Frequency shifts due to chemical shifts also manifest themselves in a similar fashion. 14.6.2 E ects of Motion One of the nice features of projection reconstruction on the other hand is the e ect of object motion during data acquisition on the reconstructed image. In Fourier imaging, periodic motion causes periodic ghosts whose number within the FOV, and distance from each other are determined by the period of the motion relative to TR . In any case, the rst ghost, which is closest to the object, is the most signi cant amongst these. In the worst-case scenario, this ghost might lie very close to the object-of-interest, leading sometimes to false diagnosis. On the other hand, periodic motion causes radial streaking in radial imaging. However, by nature, the streaks have very low intensity very close to the moving object, and the rst signi cant streak occurs farther away from it. As a result, object visibility is not hindered by the presence of any signi cant nearby ghosts with projection reconstruction. However, there is still a smearing or blurring of the image due to the motion. 14.6.3 Cartesian Sampling of Radially Collected Data It is not uncommon to resample radial data onto a Cartesian grid. This has the advantage of allowing a straightforward 2D Fourier transform reconstruction to be performed to create the image. The noise spatial frequency is white when a 2DFT is used compared to a varied spatial frequency response for projection reconstruction. The latter has heavily ltered low spatial frequency noise. The e ects of motion during data collection still cause a blur rather than ghosting despite the fact that a 2DFT was performed. The regridding does lead to aliasing of the image due to insu ciently accurate data interpolation. This can be partially overcome by oversampling the radial data but requires increased data acquisition time. 330 Chapter 14. Projection Reconstruction of Images Suggested Reading
Two excellent texts that cover the basic concepts and reconstruction of projection data are: S. R. Deans. The Radon Transform and Some of its Applications. John Wiley and Sons, New York, 1983. G. T. Herman. Image Reconstruction from Projections: The Fundamentals of Computerized Tomography. Academic Press, New York, 1980. General sampling concepts for di erent k-space schemes are covered in: D. B. Twieg. The k-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods. Med. Phys., 10: 610, 1983. The point spread function resulting from polar sampling is given in the following two papers: M. L. Lauzon and B. K. Rutt. E ects of polar sampling in k-space. Magn. Reson. Med., 36: 940, 1996. K. Sche er and J. Hennig. Reduced circular eld-of-view imaging. Magn. Reson. Med., 40: 474, 1998. Chapter 15 Signal, Contrast and Noise
Chapter Contents
15.1 Signal and Noise 15.2 SNR Dependence on Imaging Parameters 15.3 Contrast, Contrast-to-Noise and Visibility 15.4 Contrast Mechanisms in MRI and Contrast Maximization 15.5 Contrast Enhancement with T1 -Shortening Agents 15.6 Partial Volume E ects, CNR and Resolution 15.7 SNR in Magnitude and Phase Images 15.8 SNR as a Function of Field Strength Summary: Quantitative methods for understanding the e ects of noise in an image are introduced. The important imaging parameter, the signal-to-noise ratio (SNR), is studied in detail. The e ects of imaging parameters such as spatial resolution, Ts, and the number of measured data samples Nx, Ny and Nz on SNR are discussed. Contrast-to-noise ratio (CNR) and visibility concepts are developed as measures of useful information in an image. The issue of SNR as a function of eld strength is brie y considered. Introduction
All physical measurements include either random or systematic noise, which can seriously a ect the accuracy or interpretation of a measurement. The degree to which noise a ects a measurement is generally characterized by the signal-to-noise ratio (SNR). In MRI, the SNR is a key parameter for determining the e ectiveness of any given imaging experiment. If the SNR is not high enough, it becomes impossible to di erentiate tissues from one another or the background. Since the signal has already been discussed at some length in previous chapters, a study of the properties of the noise, and development of expressions for SNR 331 332 Chapter 15. Signal, Contrast and Noise which depend on the imaging parameters are made the major focus of this chapter.1 SNR as a function of resolution, readout bandwidth (BWread), and imaging time is developed. p In MRI, every factor of 2 improvement in SNR allows a doubling of resolution in one direction and has an order of magnitude psychological e ect on the quality of the image. It is, therefore, necessary to strive in all possible ways to optimize SNR for a xed spatial resolution. In Ch. 13, the issue of spatial resolution was studied, and it was determined that MRI can localize signals to very small regions. Localization of the signal is only half of the battle in an MRI experiment. If the signals from di erent tissues are all the same, a uniform image would be obtained giving no useful anatomic or diagnostic information. The real objective is to localize the signal and be able to di erentiate diseased from healthy tissue and one tissue type from another. This implies that there must be a signal di erence, referred to as contrast, between di erent tissues. Combining SNR with contrast leads to the quantity contrast-to-noise ratio (CNR) which is the real `measure' of the usefulness of an experiment. The second half of this chapter deals with the study of CNR in MRI. It will be found that there are many variables which may be employed to generate contrast in the MRI experiment, and several speci c forms of MRI contrast are discussed in depth. 15.1 Signal and Noise
The ability to ascertain whether some object is within a voxel or not depends critically on whether the signal for the object in that voxel can be distinguished from noise. It is the manifestation of the noise in the image, not the noise in the raw data, that is critical here. In this section, the translation from one domain to the other is considered. 15.1.1 The Voxel Signal The general k-space signal encoding scheme leads to a signal expression (10.2) s(~ ) = k s(kx ky kz ) = Z V kr d3r (~)e;i2 ~ ~ r (15.1)
(kx x+ky y+kz z ) When the signal expression in (15.1) is written for the 3D Fourier imaging case, it becomes ZZZ dxdydz (x y z)e;i2 (15.2) As in earlier chapters, the data are assumed to be sampled at a set of points in 3D k-space. These points are separated by kx, ky and kz , respectively, in the three orthogonal k-space directions, and cover the range de ned by the set of integers p0 q0 and r0 2 (;Nx Nx ; 1) (;Ny Ny ; 1) (;Nz Nz ; 1): The image is reconstructed by applying a discrete inverse Fourier transform to the set of data represented by sm(p0 kx q0 ky r0 kz ). Likewise, the image is represented by
Imaging parameters are those variables which are free to be chosen in an experiment, such as TR , TE , readout bandwidth, etc., and not the physical constants associated with the sample such as , T1 or T2.
1 15.1. Signal and Noise 333 ^m (p x q y r z) where p, q and r typically cover the same range as p0, q0 and r0. The signal at a given point in the image is pp qq rr X 0 1 ^m (p x q y r z) = N N N s(p kx q0 ky r0 kz )ei2 Nx + Ny + Nz (15.3) x y zp q r The e ective spin-density ^m (p x q y r z ) is also known as the `voxel signal' S (~) since it r is the signal which will be represented in the volume element x y z at position (p x, q y , r z) in the reconstructed image.2 Recall that ^m (p x q y r z) often not only represents the transverse magnetization within the voxel volume but also contains a handful of scaling factors and relaxation parameters.3 In Secs. 15.1-15.3, it is assumed that voxels are small volumes with uniform spin densities ( (x) = a constant). Therefore, all arguments in these sections neglect the possibility of a voxel containing more than one tissue type. Section 15.4 introduces the subject of voxels which are not homogeneous. From Ch. 13, it must also be remembered that the presence of the nite-width point spread function of the discrete Fourier transform (13.17) and the nite blur due to T2 decay during sampling (13.53) cause the presence of a nonzero contribution from magnetization outside the voxel boundaries to overlap or contribute to the voxel of interest. From the discussion on spatial resolution in Ch. 13, the voxel size also represents the area under the point spread function normalized to the zero value of the point spread function. As a result, whenever the spatial resolution is improved by reducing x, y or z, the total contribution to the voxel signal is reduced proportionately (see, for example, (12.25)) for example, if x is halved, ^m (p x q y r z) is also halved. Therefore, in general ^m (p x q y r z) / x y z voxel volume Vvoxel (15.4) The signal limit of (7.20) shows that the peak signal for a homogeneous sample (at t = 0 in an FID experiment, or at t = TE for a 90 gradient echo or spin echo imaging experiment) when transverse signal decay and ampli cation factors are neglected is given by (7.22) signal for homogeneous object = !0M0 B?Vsample (15.5) where B? is the transverse eld amplitude produced by the receive coil when a unit current is passed through it. Since the voxel is assumed to be a small homogeneous volume element in these discussions, (6.10) can be used to rewrite (15.5) for proton imaging as S ^m (p x q y r z) 1 3 2 h2 2 2 / !0 3kT B0 B?(p x q y r z) x y z 3 h2 / 4kT B02 B?(p x q y r z) Vvoxel (15.6)
0 0 0 0 0 0 The reader may recall from Ch. 11 that the Fourier transform of a function represented by a lowercase letter is represented by an uppercase letter. Here lowercase s is used to represent k-space data, s(~ ), and an k uppercase S is used for the image S (~), usually when the image voxel signal is being discussed. r 3 The use of e ective spin density is no more appropriate than using transverse magnetization or number of spins in a voxel. It has been used in this way because all voxels in a conventional Fourier imaging approach have the same dimensions, and hence, the actual spin density times this volume has the same relative behavior from voxel to voxel as what we have called, for simplicity, e ective spin density.
2 334 Chapter 15. Signal, Contrast and Noise It is important to note the di erent dependencies in (15.6). Of interest is the direct depen~ ~ dence on the voxel volume, on M B (which is proportional to 0B?) which is discussed in 2 , the topic of discussion of a later section in this chapter, as well Secs. 7.3 and 7.4, and on B0 as the inverse dependence on the sample temperature (which is discussed in Ch. 6). One of the primary goals of a well-conducted MRI experiment is to obtain enough voxel signal relative to noise (as measured by the ratio of the voxel signal to the noise standard deviation, or signal-to-noise ratio SNR) to observe tissues of interest. Generally, the noise voltage derives from random uctuations in the receive coil electronics and the sample. Even though there are other sources of signal uctuations such as digitization noise and pseudorandom ghosting due to moving spins, these sources are minimized in an ideal experiment. The variance4 of the uctuating noise voltage is presented here, without proof,5 to be 2 var(emfnoise ) thermal / (emfnoise ; emfnoise )2 = 4kT R BW (15.7) where the horizontal bar over a value implies an average value, R is the e ective resistance of the coil loaded by the body, and BW is the bandwidth of the noise-voltage detecting system (in NMR, both the signal and the noise are detected by the receive coil and the bandwidth of reception is determined by the cuto frequency of the analog low pass lter, BWread ). The proportionality to BW is the principal feature of (15.7), inasmuch as the temperature and resistance of the coils and bodies are not variable. The random thermal uctuations in the measured signal (as represented in (15.7)) are called `white' uctuations because they are characterized by equal expected noise power components at all frequencies within the readout bandwidth.6 The noise `variance' of the body and coil together is the sum of variances since these statistical processes are independent, leading to 2 2 k 2 k 2 k k (15.8) thermal (~ ) = body (~ ) + coil (~ ) + electronics(~ ) 2 for all k-space values. For all further subsections, the shorthand notation m is typically 2 used in place of thermal where the subscript `m' connotes `measured.' It is seen that an e ective or total resistance can be inserted into (15.7) representing the sum of the individual components Reff = Rbody + Rcoil + Relectronics (15.9) 15.1.2 The Noise in MRI 15.1.3 Dependence of the Noise on Imaging Parameters
4 Imaging parameters refer to those factors which can be chosen in an experiment and are not intrinsic properties of the sample. Imaging parameters include factors such as TE , TR , If the probability density function of a random variable x is f (x) so that dx f (x) = 1, the mean of a R function g(x) denoted by g(x) is given by g(x) = dx g(x)f (x). The variance is de ned to be var(g(x)) = R dxf (x)(g(x) ; g(x) )2 . 5 For a nice derivation of the Nyquist theorem, see, for example, Kittel in the suggested reading. This expression is generally valid up to the order of 1014 Hz for protons. 6 Since white noise has zero mean, the standard deviation is also the square root of the mean of the noise emf squared, i.e., the root mean squared (or rms) value. R 15.1. Signal and Noise 335 x, y and z, etc., for example, which can be changed in the experiment. The number of spins in the sample, its temperature, and relaxation times are not usually considered imaging parameters. In this section, the e ects of relaxation parameters on the experiment are neglected. Just as the voxel signal depends on the voxel volume which is an imaging parameter, the variance of the voxel signal due to noise is also found to depend on the choice of imaging parameters through the discrete inverse Fourier transform. As before, the measured k-space signal can be thought of as the sum of the true k-space signal s(k) with white noise (k) added to it to give the noisy measured signal sm (k): sm (k) = s(k) + (k) (15.10) For this characterization, the noise autocorrelation function is de ned as R ( ) (kp) (kq )j (kp;kq ) (15.11) and the Fourier transform of R ( ), r (f ), is known as the spectral density. R ( ) is given 2 by an impulse of strength m , i.e., 2 R( )= m ( ) (15.12) and the spectral density is seen to be white since Z r (f ) d R ( )e;i2 f 2 = m (15.13) The above expression is valid for continuous k-space data measurement. In the discrete case, 2 (kp) (kq ) = m pq (15.14) where now kp p k and kq = q k and the Dirac delta is replaced by a Kronecker delta. As before, this implies that any two white noise samples are uncorrelated and the expected 2 noise power is m . White noise is also typically characterized as being Gaussian (or normal) distributed 2 with zero mean and variance m . This distribution is denoted by N (0 m). Two Gaussian distributed random variables which are uncorrelated are also independent. This information makes it possible to evaluate the statistical properties of the noise in the image domain. Using the de nition of the discrete inverse Fourier transform acting on (k), the white noise transforms to 1 X (p0 k)ei2 p kp x (15.15) (p x) = N p in the image domain. Recall that the k-space variable (p0 k) is assumed to have a distribution of the form N (0 m). Taking the expectation of both sides, we get 1 X (p0 k)ei2 p kp x (p x) = N p = 0 (15.16)
0 0 0 0 336 Chapter 15. Signal, Contrast and Noise
p x(p k;q k)
0 0 Taking the variance of both sides yields a result independent of p 1 X X (p0 k) (q0 k)ei2 2 var( (p x)) 0 (p x) = N 2 p q =
2 m N2 2 m XX
p q
0 0 0 0 pqe
0 0 i2 p x(p k;q k)
0 0 = (15.17) N where the independence of the random variables (p0 k) and (q0 k) as de ned in (15.14) is 2 2 used. Note that m is the measured variance of any point in k-space, while 0 will be used to indicate the noise variance in the image domain. Hence, the variance measured in any voxel in image space is N times smaller than in the detected signal and is the same for all voxels. It is also common to quote the noise as the standard deviation 0 (p x). Inp conclusion, from (15.17), as N increases to aN (a > 1), 0(p x) decreases by the factor 1= a. Problem 15.1
Show that ^(0) is nothing more than the average of all k-space data. Hence, rederive (15.17) for the voxel at p = 0. Problem 15.2
In Ch. 13, the partial Fourier imaging problem was presented and the special form of its reconstruction was discussed in detail. a) Suppose only half the number of k-space lines are collected in the phase encoding direction (^) to save imaging time (i.e., only one half of k-space is y covered). How does this a ect the noise in the reconstructed image? b) Generalize this to the case where n; points are collected in the negative ky direction and n+ points are collected in the positive de nite ky direction (i.e., ky 0). The image can now be rewritten, due to the linearity of the Fourier transform, as ^m (p x) = ^m 0 (p x) + (p x) (15.18) 2 where (p x) has mean 0 and variance 0 (p x) independent of p. In (15.18), ^m 0 represents the pristine image without any noise. As a reminder, the generalization of the expression (15.17) to two dimensions gives
2 0 (p x)j2D = N m N 2 x y (15.19) 15.1. Signal and Noise and to three dimensions gives
2 0 (p 2 337 x)j3D = N Nm N x y z (15.20) Repeating an entire imaging experiment Nacq times7 and averaging the signal over these Nacq measurements to improve the SNR is common practice. The MRI system typically adds the signals directly to one another, and does not store them separately, saving a great deal of data space. The averaged k-space sample sm av (k) of sm(k) is: 15.1.4 Improving SNR by Averaging over Multiple Acquisitions X 1 Nacq s (k) sm av (k) = N mi acq i=1
This implies that (15.21) X 1 Nacq s (k) = 1 (N s(k)) = s(k) sm av (k) = N mi Nacq acq acq i=1 (15.22) The noise from each of the Nacq acquisitions is assumed to be statistically independent from 2 one acquisition to the next. As a result, the noise variance m from each measurement adds in quadrature to the total noise variance of the averaged signal sm av (k), i.e.,
2 m av Nacq X var(sm i(k)) var(sm av (k)) = N1 2 acq i=1 2 = m (k) Nacq m av (k) = (15.23) (15.24) Therefore, qm (k) Nacq The SNR of the k-space signal becomes q k) SNR(k) = sm av (k) = Nacq s((k) m av (k) m (15.25) i.e., the SNR improves as the square root of the number of acquisitions if the noise is uncorrelated from one experiment to the next. However, other sources of systematic noise from 2 2 the MR experiment can lead to m being greater than thermal and these sources will not be reduced the same way by averaging (see Appendix B).
These repetitions are sometimes done in consecutive TR intervals (with the phase encoding gradient(s) at the same value) for minimizing k-space data inconsistencies. In practice, the repetition rate for the acquisition loop is determined by the intended application of the sequence.
7 338 Chapter 15. Signal, Contrast and Noise The noise for a given voxel has already been shown to be proportional to m . Hence, the q same Nacq dependence carries over into the expression for the SNR/voxel, i.e., SNR=voxel(p x q y r z) / as well. q Nacq
0 (15.26) Phase Encoding Order When Multiple Acquisitions Are Averaged
Figure 15.1 shows one particular repetition structure for a sequence performing Nacq acquisitions before the phase encoding gradient amplitude is augmented. In other applications, this might not be the method of choice for averaging over multiple acquisitions. For example, it is typical in cardiac imaging to actually have the loop structure inside out from the case shown in the gure, i.e., a new acquisition is started after all phase encoding steps have been collected, and the images are averaged nally. Fig. 15.1: Sequence diagram explicitly demonstrating multiple acquisitions for a 2DFT acquisition method. The phase encoding gradient table is shown in a di erent form from the usual to remind the reader that the phase encoding gradient is held at one constant value in the acquisition loop. The solid line is supposed to indicate this particular gradient value. The dashed line shows the limit to which the phase encoding gradient is increased at the end of the imaging experiment. 15.1. Signal and Noise 339 Averaging Unequal Voxel Signals from a Set of Multiple Coils
In MRI, it is becoming increasingly common to use multiple smaller coils to enhance the SNR. Smaller coils pick up less noise than larger coils since they magnetically couple to a smaller volume of the sample. The signal picked up by each coil is used to create individual images which are then combined in some way to create a nal image. Unfortunately, these smaller coils, usually lying on the surface of the sample (surface coils), have progressively rapidly weakening B1 receive elds as a function of distance away from the coil surface. Therefore, the intensity of images reconstructed from these coils also varies spatially (see Sec. 7.4). In order to construct a nal image with reasonable homogeneity of signal and optimal SNR, the images generated by the individual coils must be combined. Problem 15.3
Suppose that a two-coil system is used such that the voxel signal picked up by coil 1 (S1) at a point ~ a equals a, and that picked up by coil 2 (S2) at the same x point equals a with 1. The standard deviation of the noise associated with either coil is assumed to be the same (say 0 ). Suppose that the two images are added together such that the voxel signal (^m (~ a )) from coil 2 is multiplied by x where is also 1. a) Write an expression for the SNR/voxel after addition of the two weighted voxel signals as a function of given that the noise distributions in the two coils are statistically independent and are identically equal to each other. b) Maximize the SNR/voxel as a function of and show that the SNR is maximized when equals . c) Maximize the SNR/voxel when 2 = 0 1 , where 0 is a real constant. From the above problem, the optimal linear combination of the two signals is not direct addition as a rst guess might suggest. Instead, it is given by the sum of the squares of the signals picked up by the two coils (the case where equals ) normalized to the larger of the two signals. What is the logic behind such a combination? If the two signals are equal, direct addition yields the optimal SNR. However, when one of the two signals is smaller than the other, weighting the smaller measurement by the square of the ratio of the smaller to larger signal produces the optimal SNR. For example, by the time the second signal is one- fth of the rst, there is no need to keep its signal since noise dominates in this case. In this squared sum scheme, the second signal contributes only 4% to the signal, preventing this domination by noise. Recall that the noise picked up by the coil manifests itself as noise with equal variance throughout the image so that, as B? falls o , and the voxel signal decreases while the noise does not. 340 Chapter 15. Signal, Contrast and Noise
0 15.1.5 Measurement of As obtained in (15.19) and (15.20), the standard deviation of the signal in any voxel due to white noise added to the k-space data is independent of voxel number, i.e., the image white noise standard deviation is independent of position in the image. Henceforth, 0 and not 0 (p x) is used to denote the voxel signal standard deviation. In a homogeneous region of a tissue of interest in the image, the average voxel signal over that region is a good estimate of the tissue voxel signal S . In the absence of any systematic variations such as Gibbs ringing, when the SNR in that region is high enough, it is shown later in Sec. 15.7 that the standard deviation in this region is also a very good estimate of 0 . A better way to measure 0 is to measure the average value or standard deviation of a region-of-interest outside the object where there is no signal (see Fig. 15.2). As shown in Appendix B, the voxel signal here is `Rayleigh distributed' in the magnitude image. The mean and standard deviation of this random variable are related to the standard deviation of the underlying Gaussian distributed white noise as 1:253 0 and 0:665 0 , respectively. Obtaining 0 from either measurement in the background noise gives a more accurate estimate than trying to measure the standard deviation in the image itself (unless the image is perfectly uniform in the region being evaluated). The ratio of the average signal to the estimated value of 0 then gives an SNR estimate. and Estimation of SNR 15.2 SNR Dependence on Imaging Parameters
In this section, several imaging parameter dependencies of the SNR on a voxel basis are summarized in di erent forms. The dependence of SNR/voxel on imaging parameters such as the number of acquisitions Nacq , the number of k-space samples Nx Ny and Nz , the readout bandwidth dependence and voxel dimensions x y and z (or TH in 2D) are discussed in detail. The dependence of SNR on spatial resolution is given utmost importance, and some compromises which have to be kept in mind while improving spatial resolution are presented. 15.2.1 Generalized Dependence of SNR in 3D Imaging on Imaging Parameters
q x y z Nacq SNR=voxel / q BWread
Nx Ny Nz The SNR dependence on imaging parameters is complicated by the noise behavior. Noise depends on many di erent imaging parameters. From (15.6), (15.7), and (15.19) (15.27) Substituting t = BW1read gives SNR=voxel / x y z Nacq NxNy Nz t Since Ts = Nx t, substituting this into (15.28) yields SNR=voxel / x y z Nacq Ny Nz Ts q (15.28) (15.29) q 15.2. SNR Dependence on Imaging Parameters 341 Equations (15.27)-(15.29) can be re-written in a number of ways, depending on the parameters which are to be considered. It is necessary to keep in mind that although any parameter in (15.27)-(15.29) may be varied without altering the validity of the relations, the following relations hold (a) Lx = Nx x (b) Ly = Ny y (c) Lz = Nz z (d) Ts = Nx t (15.30) 1 = G L (f) (e) BWread = t { x x BW=voxel = BWread =Nx These interrelations exist implicitly in each of the above expressions for the SNR. Therefore, whenever a parameter in a given expression for SNR is varied, the resultant e ects on the rest of the parameters must be checked. Also, it is obvious that through these relations a number of other expressions for the SNR may be developed to highlight the e ects of varying a certain subset of these parameters. Often an expression for SNR will be accompanied by a condition that some quantity be kept constant which limits how certain parameters may be varied. Finally, there are other dependencies of the signal related to data acquisition and tissue properties, which will be dealt with in Sec. 15.4. To calculate what happens in the case of a xed variable, an implicit form can be substituted into (15.29). For example, since x = Lx=Nx, if x is xed, Lx and Nx cannot vary arbitrarily and (15.28) or (15.29) are better written as N SNR=voxel / Lx y z NyNz t x s x=Lx =Nx =constant (15.31) Other such specialized proportionalities can be derived. An increase in the spatial resolution by a factor of 2 in both in-plane directions leads to a factor of 2 loss in SNR when the increased spatial resolution is achieved by maintaining the read gradient xed while Ts is doubled (from (15.29)). Such an example is shown in Fig. 15.2, from which the mean and standard deviation estimates were obtained. Taking the ratio of the mean of the local background to the noise standard deviation in the two cases demonstrates the consistency of the measured SNR with that expected from (15.29). Doing this in a region where the pro le is at yields an SNR/voxel of 187.5 for Fig. 15.2a and 91.0 for Fig. 15.2b. From (15.29), with all other parameters maintained constant, the SNR dependence on read direction parameters can be reduced to SNR=voxeljread / x Ts 15.2.2 SNR Dependence on Read Direction Parameters
q (15.32) Although (15.32) depends only upon two parameters, dependencies on Lx , BW , etc. are implicit in this expression (see (15.29) and (15.30)). Due to the importance of understanding the e ects of altering read direction parameters, several example situations are shown in Table 15.1. Case 1 is chosen as a reference and given an SNR of unity. Also, note that in order to simplify the treatment, all of the situations in Table 15.1 are restricted to doubling 342 Chapter 15. Signal, Contrast and Noise (a) (b) (c) (d) Fig. 15.2: Two images collected with identical TR , TE and Gx. However, Nx, Ny and Ts in (b) are two times the same values in (a), leading to an improvement in spatial resolution. This increase in resolution leads to a reduction of the SNR by a factor of 2. Noise standard deviation inside the object is estimated by taking 1/1.253 ' 0.8 times the mean measured in a region outside the phantom where there is no signal (see Appendix B). This noise-only region has to lie reasonably away from the edge of the object, and there should be no artifacts nearby. The pro les in (c) and (d) illustrate the larger variation in noise for (a) versus (b), respectively. They also show the higher resolution associated with (d), where ve distinct dips (each corresponding to one resolution element) are visible versus only three in (c). The pro les were taken through the row cutting through the last row of smallest resolution elements. 15.2. SNR Dependence on Imaging Parameters 343 or halving the parameters involved. There are also practical aspects to these choices, which will be discussed below. From (15.32), it appears that it might be possible to obtain very high resolution without reducing SNR by shrinking x while increasing Ts. Realistically, however, there is a limit to how long Ts can be before the signal is seriously degraded by T2 e ects. Oversampling to Avoid Aliasing
Although the topic of aliasing has been dealt with in Ch. 12, it is useful here to revisit it in terms of how changing the FOV a ects the SNR in a given experiment. In Table 15.1, cases 2 and 3 demonstrate that altering the FOV does not alter the SNR of an experiment as long as Ts and x are unchanged. There are two important implications of this result. First, it is possible to avoid aliasing in a given experiment by doubling the FOV in the read direction. This is accomplished by collecting twice as many points without varying the read gradient Gx or Ts (referred to as oversampling (case 3)), which neither degrades nor improves the SNR of the experiment. Therefore, in many practical imaging situations, oversampling is used to double the FOV in the read direction and reduce aliasing artifacts without sacri cing SNR or lengthening Ts or the acquisition time. Consider acquiring a transverse image of the human chest, for example, where the left to right dimension of the experiment is very large and is most likely to be chosen as the read direction. Oversampling in the read direction is then used to guarantee that the FOV extends past the patient's arms without increasing imaging time, or degrading SNR. Alternatively, for a small object, where aliasing is not a problem and data storage space is a premium, choosing a smaller FOV and collecting fewer data points (case 2) does not reduce SNR. Degrading Resolution to Increase SNR
In case 4, x is increased by a factor of 2, the number of data p points collected is halved and Ts is also halved. As seen from Table 15.1, SNR is increased by 2. If lower resolution can be tolerated, this increase in SNR could lead to better tissue recognition, and the reduction in Ts could be bene cial in reducing the chemical shift artifact (discussed in detail in Ch. 17), in overcoming static eld inhomogeneity e ects (since BW /voxel is increased) and in reducing relaxation e ects during sampling. In case 5, resolution is reduced by a factor of 2, and SNR is doubled because this increase in resolution is achieved while Ts is held constant. A doubling of SNR is a signi cant SNR improvement. Note that this e ect is achieved by reducing the read gradient by a factor of 2. In most cases, reducing the gradient is not a problem, but it must be kept in mind that if the applied gradient strength reduces to a level comparable to those produced by local eld inhomogeneities, then severe image distortion will result (as detailed in Ch. 20). This trick of using a xed Ts with the read gradient halved to acquire low resolution images and obtain a factor of 2 improvement in SNR is commonly used in clinical applications. 344 Chapter 15. Signal, Contrast and Noise Case x Nx Lx Gx t Ts SNR Reference case 1 x0 N0 L0 G0 t0 Ts 0 1 Data reduction and oversampling 2 3 x0 x0 N0 =2 L0 =2
2N0 2L0 G0 G0 2 t0 Ts 0 Ts 0 1 1 t0 =2 Degrading spatial resolution 4 5 2 x0 N0 =2 2 x0 N0 =2 L0 L0 G0 t0 Ts 0=2 Ts 0 p 2 G0=2 2 t0 2 Improving spatial resolution 6 7 8 9 x0 =2 x0 =2 N0 N0 L0 =2 L0 L0 G0
2G0 2 t0 2Ts 0 1= 2 p L0 =2 2G0 G0 t0 t0 =2 t0 Ts 0 Ts 0 1=2 1=2 x0 =2 2N0 x0 =2 2N0 2Ts 0 1= 2 p Table 15.1: Table showing the SNR/voxel when the voxel size is changed in the read direction under di erent conditions. The SNR/voxel is given relative to that of case 1 and can be seen to correlate exactly with x(Ts )1=2 or equivalently, ( x=Gx )1=2 . Note that Ts and Gx vary similarly from baseline conditions in the di erent cases. 15.2. SNR Dependence on Imaging Parameters 345 Improving Resolution in the Read Direction
In both cases 6 and 7, x is halved by reducing the FOV by a factor of 2 without changing the number of sampled points. This should be done only if the FOV in case 1 is greater than or equal to twice the width of the object in the read direction so that aliasing will be avoided. In case 6, Ts is doubled by doubling t. This method leads to a reduction of SNR of only p 2 since BW /voxel is again halved. Realistically, Ts must be short enough in case 1 so that when it is doubled, it is still much shorter than T2 . In case 7, Lx is halved by doubling the read gradient Gx. This approach requires that enough gradient power be available to double G0. Unfortunately, SNR is reduced by a factor of 2, making this a very ine cient approach. In case 8, x is halved by doubling Nx and keeping Lx and Ts xed which requires doubling Gx and halving t0 . This also leads to a halving of the SNR. In case 9, x is halved by doubling Nx while keeping both Lx p Gx xed. This requires doubling the total sampling and time, but is accompanied by only a 2 reduction in SNR. This is probably the optimal way to double the resolution of the experiment as long as the increased Ts is not comparable to T2 , and storing the extra data is not a problem, although it is similar to case 6 which requires less storage space but a smaller object. Problem 15.4
A better understanding and feel for the SNR variation in the read direction can be obtained by relating the SNR to a combination of x and the BW /voxel. a) Show that SNR/voxel / x= BW=voxel. b) When x is halved with a corresponding doubling of Gx (as in cases 7 and 8 in Table 15.1), how does BW /voxel vary in these two cases? What happens to the voxel signal? What happens to the SNR/voxel? Does the FOV change? c) When x is halved by increasing Nx without a corresponding doubling of Gx (as in case 9 in Table 15.1), how does BW /voxel vary? What happens to the voxel signal and the SNR/voxel? d) Find two other ways to double x of the experiment depicted in Table 15.1. (Hint: See cases 6 through 9.) e) Given the following parameters, nd x0 and the proportional SNR/voxel as given in (15.32). FOVread = 256 mm, Nx = 256, Gx = 2.5 mT/m, Gmax = 15 mT/m, with the T2 of the tissue of interest being 20 ms. Assume that you would like to reduce x0 in this case to x0 =2. Discuss how you might best accomplish this in terms of getting optimal SNR/voxel. For deriving this result, assume that Ts has to be less than or equal to T2 . Since Ts equals 1=( {Gx x), the proportionality in (15.32) can be rewritten as q 346 SNR=voxel / Chapter 15. Signal, Contrast and Noise s (15.33) Either expression contains no hidden dependencies and requires none of the imaging parameters to be xed, i.e., they can be applied to any situation for computing the relative SNR p change. According to this expression, the SNR/voxel decreases only by a factor of 2 for every halving of x as long as the read gradient is xed. SNR however reduces by a factor of 2 if this improvement in spatial resolution is accompanied by a doubling of Gx. Note consistency with these observations for cases 6 through 9 in Table 15.1. The only way to improve the SNR when x is made smaller is to increase Ts accordingly. Unfortunately, this is possible only up to a point, either before relaxation e ects begin to reduce the k-space signal or before Ts starts to limit the minimum TR value. The features in cases 6 through 9 are also summarized in Fig. 15.3. / x Gx q x Ts e ects on SNR. This gure summarizes cases 6 through 9 in Table 7.1, showing also what parameters were changed in comparison with case 1 to attain high resolution. For Ts xed, either approach yields a loss of 2 in SNR while for Ts doubled, only a loss of square root of 2 in SNR occurs. Fig. 15.3: Di erent ways of achieving improved spatial resolution in the read direction, and their 15.2.3 SNR Dependence on Phase Encoding Parameters Pulling out just the SNR dependence on parameters which change only the image characteristics in the in-plane and through-plane phase encoding directions (^ and z , respectively) y ^ gives q SNR=voxel / y z Ny Nz (15.34) 15.2. SNR Dependence on Imaging Parameters 347 (recall Ly = Ny y and Lz = Nz z). There are only two alternate methods for improving spatial resolution in the phase encoding directions: rst, decreasing y or z by increasing Ny or Nz while keeping Ly or Lz xed (case 3 in Table 15.2), or second, decreasing y or z by decreasing Ly or Lz while keeping Ny or Nz xed (case 2 in Table 15.2). Consider halving y using either of these methods. In the rst method, the SNR dep creases only by a factor of 2, whereas in the second method, the SNR decreases by a factor of 2. What are the advantages or disadvantages of either method? The rst method, requiring doubling Ny , takes twice as long to complete compared to the second method. If the SNR is good enough and aliasing is avoided, the second method is twice as time-e cient as p the rst while having 2 worse SNR in comparison with the rst. Of course, in the same imaging time as required by the rst method, two acquisitions can be performed with the p second method to reclaim the factor of 2 SNR loss, while maintaining the advantage of requiring less image storage space. These two cases are highlighted in Table 15.2. Case y Ny Ly Gy max TT SNR Reference case 1 y0 Ny 0 Ly 0 Gy 0 TT0 1 Improved spatial resolution 2 3 y0 =2 Ny 0 Ly 0/2 2Gy 0 y0 =2 2Ny 0 Ly 0 TT0 1/2 2Gy 0 2TT0 1/ 2 p Table 15.2: Two di erent ways to improve spatial resolution in the phase encoding direction, and
their e ects on the SNR and total imaging time. 15.2.4 SNR in 2D Imaging The SNR expression in (15.29) can be rewritten for a 2D imaging experiment. The voxel now has dimensions of x y TH . Also, Nz is replaced by unity. Therefore q (SNR=voxel)j2D / x yTH Ny Ts (15.35) In other words, a 2D imaging experiment performed with exactly the same imaging paramep ters (including TR, TE and ip angle) with TH = z has Nz worse SNR in comparison with 348 Chapter 15. Signal, Contrast and Noise the 3D imaging experiment. However, the 2D imaging experiment requires an imaging time which is Nz times shorter than the 3D experiment. However, it is typically possible to collect the data for only one slice per TR when the TR value in the 2D imaging experiment equals the practically sensible choice of short TR in the 3D imaging experiment. To obtain the same spatial coverage in the slice select direction requires Nz imaging experiments, increasing the total imaging time by Nz in the 2D imaging case. If the same imaging time is used for both the 2D and 3D imaging experiments, the same volume of p coverage and the same contrast as the 3D can be achieved in 2D imaging albeit only with Nz less SNR. Or a single slice with the same SNR as the 3D experiment (obtained by imaging with Nacq2D = Nz ) can be obtained. The most e cient means of collecting 2D data and covering the same region-ofinterest as in the 3D imaging case is to use TR2D = Nz TR3D and use a multi-slice acquisition8 (although this guarantees neither that the 2D SNR is as good as the 3D SNR as discussed below nor that the contrast is comparable with the 3D imaging method). 15.2.5 Imaging E ciency In clinical applications, patient comfort and patient throughput are important day-to-day issues. The longer the imaging time per patient, the more patient discomfort and the less throughput. In Fourier imaging, the total imaging time for each imaging experiment is dependent directly on the number of sequence cycle repetitions. Consider the 3D imaging case, where the total imaging time is given by TT = Nacq Ny Nz TR (15.36) The general wisdom is that, since (15.37) (SNR=voxel)3D / y z Ny Nz with Ly and Lz constant, if either Ny or Nz is halved and consequently y or z is doubled, p respectively, not only is the SNR improved by a factor of 2, but the imaging time is also halved. As a consequence, if the same imaging time as that for the better resolution scan p is used, the SNR can be improved further by an additional factor of 2 by averaging 2 acquisitions at lower resolution. Thereby, the SNR-per-unit time is doubled in comparison with the high resolution image. A time-normalized SNR is therefore considered a measure of imaging e ciency. The normalization to time is done not on a per unit-time basis, but p on a per square root of time basis, since, for a xed TR , SNR is proportional to TT = q Nacq Ny Nz TR . Therefore, for a xed TR , the imaging e ciency is de ned as q q x y z Nacq Ny Nz Ts q Nacq Ny Nz q / x y z Ts (15.38) In other words, an image with better spatial resolution, such as with y or z halved, is considered only half as SNR-e cient as an image acquired with voxel size y or z,
(SNRpvoxel)3D / = TT
8 See Ch. 10 for more details. 15.3. Contrast, Contrast-to-Noise and Visibility 349 respectively. Again, this measure does not include the role of voxel size relative to object size. The main conclusion is that high resolution can be achieved e ciently only in the read direction, and even that is limited by the need to keep Ts on the order of T2 or less. 15.3 Contrast, Contrast-to-Noise and Visibility
Even the highest signal-to-noise ratio does not guarantee a useful image. An important aim of imaging for diagnostic purposes is to be able to distinguish between diseased and neighboring normal tissues. If the imaging method used does not have a signal-manipulating mechanism which produces di erent signals for the two tissues, distinguishing the two tissues is not possible. MRI is blessed with an abundance of signal-manipulating mechanisms, as the signal is dependent on a wide variety of tissue parameters. The problem of distinguishing a given (diseased) structure from surrounding (normal) tissue in the presence of added white noise falls under the broad category of the `signal detection' problem, and requires an understanding of the importance of Contrast-to-Noise Ratio (CNR). 15.3.1 Contrast and Contrast-to-Noise Ratio
The common way to look at this problem is to examine the absolute di erence in the signal between the two tissues of interest. If these tissues are labeled A and B , their signal di erence9 is de ned as the `contrast:' CAB SA ; SB (15.39) where SA and SB are the voxel signals from tissues A and B , respectively. Although the inherent contrast may be large enough to detect a change, if the noise is too large, the signal di erence would not be visible to the eye or to a simple signal di erence threshold algorithm. The more appropriate measure is the ratio of the contrast to the noise standard deviation10 known as the contrast-to-noise ratio CNR: CNRAB CAB = SA ; SB = SNR ; SNR A B
0 0 (15.40) The utility of this de nition is best illustrated with a simple statistics discussion. For Gaussian distributed white noise, the probability that two tissues are di erent if CNRAB equals p p 2 2 is 95% and if CNRAB equals 3 2 is 99%. Ideally we would like to design the MR experiment to have su cient spatial resolution to resolve the two tissues of interest (such as gray matter and white matter) and to have a high enough CNR that they can be distinguished from each other.
It is also common to refer to contrast as CAB =SA or CAB =SB . We de ne these as relative contrast ratios. 10 Since no physical subtraction is performed in the image, the standard deviation used is that common to p both tissues A and B, i.e., 0 . If a contrast image were created, then 0 would be replaced by 2 0 .
9 350 Chapter 15. Signal, Contrast and Noise 15.3.2 Object Visibility and the Rose Criterion
eff If multiple independent signal measurements Nacq are performed, an average of these signal measurements implies that the e ective noise standard deviation becomes =q Nacq 0 (15.41) In an image where tissue A occupies nvoxel voxels, each of which has signal SA with independent additive white noise with standard deviation 0 , a similar voxel-averaging scenario can be incorporated into the detection criterion (with Nacq replaced by nvoxel ) via a new measure referred to as the `object visibility' or VAB CAB = CAB pnvoxel = CNRAB pnvoxel
eff
0 (15.42) Again, the noise in each voxel of the image, 0, is assumed to be the same for both tissues. The Rose Criterion
The visibility threshold can be determined empirically. Based on experiments with human observers detecting circular objects shone on a television screen, Rose found that random uctuations in the photon ux forming the object confused the observer, and a minimum `SNR' was required for con dent object detection. Depending on the observer's expertise and what is being observed, a detection SNR threshold varying between 3 and 5 was found to be required for object recognition. This requirement is known as the `Rose Criterion' in the diagnostic imaging literature. A visibility threshold of about 4 can be reinterpreted for the tissue discrimination problem as follows: a Gaussian model for the additive white noise is found to be a rather good approximation for the thermal noise. With this assumption made, the image signal from tissues A and B can be modeled by two Gaussian distributions centered at SA and SB , respectively, with the same standard deviation 0. Now, if (SA ; SB ) = 4 0 , as it would be for single-voxel tissues A and B , the two Gaussian distributions are separated by 4 0 (see Fig. 15.4). Since a distance 2 0 to one side of the mean covers about 97.5% of the area under a Gaussian, it is reasonable to expect the human observer to choose a threshold which is 2 0 away from either SA or SB . Then, the probability of the observer incorrectly classifying a voxel as belonging to one or the other tissue is 0.025, i.e., this is a 1-in-40 chance occurrence. That is, the probability of detection of the tissue is very high: for a multi-voxel object when V equals 4, only 1 in 40 voxels in the object will be classi ed incorrectly by the observer as a background voxel, and vice versa for a single-voxel object, there is only a 1-in-40 chance of not detecting it.11 This description is essentially a rule of thumb, as the perception of the object is likely to be much more complicated. p 11 As for footnote 10, visibility has been de ned relative to 2 0 . If the latter were chosen, the 0 , not p above argument is still valid but the quoted visibility in the case being considered would be 2 2 rather than
4. 15.3. Contrast, Contrast-to-Noise and Visibility 351 Fig. 15.4: Image signal distribution of two tissues A and B with added Gaussian distributed white noise. When (SA ; SB ) equals 4 0 , the probability of an error in the detection of tissue A is about
0.025 when a threshold of 2 0 is used. Likewise, the probability of mis-classifying a noise point as an object is also 0.025. Problem 15.5
a) Show for cases 6 and 9 of Table 15.1 that V remains constant for a multivoxel object. b) It has been suggested that the Great Wall of China can be seen as the only man-made structure visible from outer space, yet it is only a few meters wide. Postulate why this might be possible based upon a visibility argument. The e ect of object size, 0 and contrast level on object visibility can be visually demonstrated to be determined by the quantity V as it was de ned in (15.42) by observing the images in Fig. 15.5. The model image with no noise added is shown in Fig. 15.5a. The simulated objects are disk-like objects of linearly decreasing radius (one voxel radius to ve voxel radius) going from top to bottom and linearly increasing signal values going from left to right (doubling from column 1 to column 5 actual values can be computed from SNR values quoted in the caption for Fig. 15.5) imaged in a zero signal background. The model image is assumed to be all real, whereas the noise is assumed to have uncorrelated and equal expected power in both the real and imaginary image channels. The noisy images were created by adding the real channel noise to the model and taking the magnitude of the two image channels after this addition. As described in the earlier discussion, the smaller disks become undistinguishable from noise at the lower SNR levels while the larger disks are detectable even at degraded SNR levels. Also, the higher the true contrast of the object, the higher the SNR degradation must be before such an object becomes undetectable. It is easy to note 352 Chapter 15. Signal, Contrast and Noise that at a given SNR level (images at di erent SNR levels are shown in Figs. 15.5b-15.5d), an imaginary diagonal line can be drawn separating the barely detectable objects from the undetectable objects and the clearly detectable objects. It is found that the objects along such a diagonal line have a constant value of the product of the signal with the square root of the number of voxels occupied by the object. This shows that the threshold of visibility is determined by the quantity V de ned in (15.42). 15.4 Contrast Mechanisms in MRI and Contrast Maximization
As mentioned earlier, MRI has the exibility to manipulate the tissue signal in many ways, leading to numerous contrast mechanisms. The exibility arises from the MR signal dependence on many imaging parameters and tissue parameters. The most basic contrast generating mechanisms are based on spin density, and T1 and T2 di erences between tissues. Others are ow, magnetic susceptibility di erences, magnetization transfer contrast, tissue saturation methods, contrast enhancing agents and di usion, all of which are discussed in one place or another in later chapters of this book. In this section, the focus is on three basic forms of contrast: spin-density weighted contrast, T1-weighted contrast, and T2 (or T2 )-weighted contrast. A 90 gradient echo experiment is used as an example of how to obtain di erent forms of contrast. These results are identical to those that would be found for a 90 spin echo experiment under the assumption that TE TR and T2 is replaced by T2 . Di erent expressions and relations must be derived for other imaging techniques. 15.4.1 Three Important Types of Contrast Although each type of contrast is designed to enhance di erences in one of the speci ed parameters ( 0 , T1 or T2 ), the signal is a function of all three variables, and each must be kept in mind when determining overall image contrast. As an illustrative example, the contrast between tissues A and B for a 900 ip angle gradient echo experiment (see Fig. 15.6) is CAB = SA(TE ) ; SB (TE ) = ;T =T ;T =T ;T =T ;T =T 0 A (1 ; e R 1 A )e E 2 A ; 0 B (1 ; e R 1 B )e E 2 B (15.43) Note that the signal is assumed to be determined by the tissue signal solution from the Bloch equation at the echo time TE . Remember, this time corresponds to the k = 0 sample in the read direction. CAB can then be maximized with respect to either TR or TE . 15.4.2 Spin Density Weighting In order to get contrast based primarily on 0 , the T1 and T2 dependence of the gradient echo tissue signals must be minimized. When the argument of an exponential is small, an appropriate approximation to e;x is (1 ; x) which is better written as 1 ; O(x). If the exponent is large and negative, e;x can be approximated by zero. It is seen that in order 15.4. Contrast Mechanisms in MRI and Contrast Maximization 353 (a) (b) (c) (d) of CNR, and how their detection for a given CNR depends on the object size. For the cases shown, CNR is the same as SNR since each feature is being compared to the background noise. (a) Model of circles with no background noise (SNR = 1) (b) SNR = 4 (c) SNR = 2 and (d) SNR = 1. (The SNR values quoted for these images are measured relative to the objects with lowest signal (column 1). As mentioned in the text, SNR doubles for the rightmost column in comparison with the leftmost column in each image.) Fig. 15.5: Images designed to show how visibility of large and small objects changes as a function 354 Chapter 15. Signal, Contrast and Noise to maintain adequate signal and get contrast based primarily upon spin-density, appropiate choices of TE and TR would be TE T2A B ) e;TE =T2 ! 1 TR T1A B ) e;TR =T1 ! 0 and expression (15.43) for the contrast between tissues becomes, CAB = ( 0 A ; 0 B ) ! ! TE + TE ;TR =T1 A ;TR =T1 B ; 0A e +T +T 0B e 2A 2B + higher order and cross terms (15.44) (15.45) ' 0A ; 0B (15.46) Fig. 15.6: A 2D gradient echo sequence diagram. 15.4. Contrast Mechanisms in MRI and Contrast Maximization 355 In this approximation, the contrast does not depend upon TR or TE , and need not be extremized relative to either TR or TE . This gives a general rule for spin density weighting: keep TR much longer than the longest T1 component keep TE much shorter than the shortest of T2A B the gradient echo image contrast is then primarily determined by spin density di erences. Similar rules will be formed for T1 -weighting and T2 -weighting based on the practical limits imposed on TE and TR . These limits are summarized in Table 15.3. The actual error in this approximation of spin density weighting of the signal depends on the coe cients for TE =T2 and TR =T1, the former vanishing only linearly in TE =T2 . In practice, this means that most imaging experiments still have an error of a few percent even if TE is as low as a few milliseconds since some typical T2 values are on the order of tens of milliseconds. Problem 15.6
Estimating water content of one tissue relative to another from their contrast differences for a supposedly spin density weighted sequence may require including rst order e ects of nonzero TE and nite TR . Consider a gradient echo experiment with TR = 5 sec, TE = 5 ms and a =2-pulse. Assume that 0 , T1 and T2 for two adjacent tissues are 1.0, 2 sec, 20 ms and 0.8, 1 sec, 50 ms, respectively. What are the e ects of nonzero TE and nite TR on the contrast in this supposedly spin density weighted sequence? Problem 15.7
a) Show that the expression for contrast for a 90 repeated spin echo experiment is given by CAB = SA ; SB = 0 A(1 ; 2e;(TR ; )=T1 A + e;TR =T1 A )e;TE =T2 A ; 0 B (1 ; 2e;(TR ; )=T1 B + e;TR=T1 B )e;TE =T2 B (15.47) ;TR =T1 A ;TE =T2 A ;TR =T1 B ;TE =T2 B ' 0 A(1 ; e )e ; 0 B (1 ; e )e for TR TE = 2 (15.48)
where is the time between the 90 and 180 rf pulses. b) Under what conditions would you expect to be able to obtain a more accurate spin density-weighted image with a spin echo instead of a gradient echo experiment? Practically, the minimum TE is limited by the available gradient strength. In fact, this limitation made the imaging of rigid solids impossible for many years because their T2 values 356 Chapter 15. Signal, Contrast and Noise are on the order of a few hundred microseconds to several milliseconds, and no hardware was available which could form an echo before the signal was gone. However, with modern hardware and modern imaging techniques, solids imaging is now viable. TE is also limited by the highest acceptable readout bandwidth/voxel (or lowest possible Ts) for SNR and object visibility reasons. The maximum value of TR , on the other hand, is constrained by imaging time and imaging e ciency reasons. Therefore, true spin density weighting using a 90 gradient echo sequence is practically achievable only for tissues with long enough T2 's and short enough T1 's which allow TE and TR choices which satisfy the constraints imposed by the gradient strength limitation, the SNR and imaging time. Good spin density-weighted contrast is available for most purposes without requiring a zero TE , or in nite TR . Normal soft tissue T1 values are quite di erent from one another. For this reason, T1 -weighted contrast o ers a very powerful method for delineation of di erent tissues. To obtain spin density weighting, TE and TR were chosen to reduce the e ect of T1 and T2. For T1 and T2 weighting, only the e ects of T2 and T1 di erences, respectively, can be minimized. The e ects of spin density di erences cannot be neglected. For T1 weighting, T2 e ects have to be minimized. Using the gradient echo example as before, the choice of a very short TE again reduces any T2 (or T2) contrast, i.e., TE is chosen such that TE T2A B ) e;TE =T2 ! 1 (15.49) and the expression for the contrast is now CAB = SA(TE ) ; SB (TE ) ' 0 A(1 ; e;TR=T1 A ) ; 0 B (1 ; e;TR =T1 B ) TE T2A B = ( 0 A ; 0 B ) ; ( 0 A e;TR=T1 A ; 0 B e;TR=T1 B ) (15.50) Since there is no transverse relaxation dependence in the above expression, this expression is equally valid for a spin echo sequence as well. It is typical that T1 and T2 correlate with spin density, i.e., a tissue with higher spin density usually has longer T1 and T2 values, and tissues with lower spin density usually have shorter T1 and T2 values. As a result, while T1 weighting depicts tissues with longer T1 values with low signal and short-T1 tissues with higher signal, the spin density contrast counteracts this e ect. Hence, a unique choice of TR which maximizes the T1 -weighted contrast should exist. To optimize the T1 -weighted contrast, (15.50) is extremized with respect to TR. Di erentiating CAB with respect to TR and setting it equal to zero leads to the relation: ;T =T ;TR =T1 B 0 Ae R 1 A = 0 B eT (15.51) T1 A 1B Solving for TR from (15.51) gives the optimal TR : 15.4.3 T1-Weighting TRopt = ln T1 B 1 T1 B 0B ; ln T01 A A 1 ; T1 A (15.52) 15.4. Contrast Mechanisms in MRI and Contrast Maximization 357 Some a priori knowledge of tissue properties is clearly very useful. When several tissues are present, it may be di cult to choose a single TR which optimizes all contrast and two scans with two di erent TR values would be required. In principle, T1's of all tissues could then be found (see Ch. 22). This optimal value of TR can also be obtained graphically by plotting the expression for CAB from (15.50) as a function of TR . Consider one such plot shown in Fig. 15.7a. At long TR values, all tissues will have relaxed completely, and only spin density contrast is obtained, i.e., the contrast curve approaches a constant value asymptotically. At low values of TR such that TR T1 (this de nes the T1-weighted contrast regime), where the signal is inversely proportional to T1 , the tissue with lower T1 has a higher signal. In the case of gray matter and white matter, since white matter has the lower T1, it has a higher signal at short TR values. However, since white matter also has a smaller spin density than gray matter, once TR becomes comparable to T1 , gray matter starts growing towards a higher value, crossing the white matter curve towards its higher spin density value. The crossover point represents a `null point' between gray matter and white matter where there is no contrast. In between a TR value of 0 where the contrast is zero and the null point, there must be a maximum, and this represents the TR value which gives the optimal T1-weighted gray matter-white matter contrast. Two other examples, GM/CSF (Fig. 15.7b) and GM/lesion12 ( 0 = 0.8, T1 = 1.5 sec Fig. 15.7c) are also shown for comparison. Again, the previous observations are obeyed in both cases, and the range of TR choices for the spin density weighting or T1 weighting regimes can be determined from these plots. Problem 15.8
a) Show that for a 90 ip angle, short-TE gradient echo sequence, there is a choice of TR where gray matter (gray matter spin density relative to water = 0.8 T1 = 950 ms at 1.5 T) and white matter (white matter spin density relative to water = 0.65 T1 = 600 ms at 1.5 T) are iso-intense (i.e., they have the same signal). This represents a `crossover point' on the contrast curve. What TR does this crossover occur at? b) Explain why there is such a crossover point (a plot of the two signals as a function of TR would be helpful in making your argument). c) From (15.52), nd the TR which optimizes the gray matter/white matter contrast. The reader will nd that these values are not perfect in real studies since the rf slice pro le is not a boxcar function and all of the spins in the slice are not tipped by =2. When this occurs, the tissues behave as if they have shorter than expected T1 .
A `lesion' is used to indicate abnormal tissue which contains T1 and T2 values larger than those of normal gray matter.
12 358 Chapter 15. Signal, Contrast and Noise Fig. 15.7: CAB as a function of TR for (a) GM/WM (b) GM/CSF (c) GM/lesion in the case of a T1 -weighted 2D or 3D imaging experiment assuming ideal rf pulses. As demarcated in (a), two regions can be identi ed for each plot: one where the contrast is T1 weighted (region A in (a) for example), and another where the contrast is spin density-weighted (region B in (a) for example). The gure shows that a unique TR value which optimizes either contrast can be identi ed for each tissue pair of interest (which varies according to the intended application). The tissue parameters 0 and T1 used to simulate these gures came from Table 4.1 in Ch. 4. The lesion parameters were chosen to be 0 = 0:8 and T1 = 1:5 sec. 15.4. Contrast Mechanisms in MRI and Contrast Maximization 359 The presence of a null point in the contrast curves was already noted. Its determination for the particular case of GM/WM contrast is the subject of the Prob. 15.8. Optimal TR for Tissues with Similar Spin Densities and Fractionally Di erent T1 The optimal value for TR obtained in (15.52) represents the most general case of two tissues A and B which have di erent spin densities. In the early stages of the formation of certain diseased states, it is not uncommon to nd the diseased tissue with a spin density which is very comparable to its normal neighbor, i.e.,
0A ' 0B 0 (15.53) and a T1 which is fractionally di erent. That is, T1 B = T1 A(1 + ) with
The expression for the contrast is !0 (15.54) CAB = ! TR ' 0e T1 A which is again a function of TR . Maximizing with respect to TR then yields
;TR =T1 A ' ;T =T 0e R 1 A ;T =T 0e R 1 A ; e;TR=((1+ )T1 A ) e TR =T1 A ; 1 since e;TR=(1+ )T1 A ' e;(1; )TR =T1 A
(15.55) (15.56) TRopt = T1 A i.e., for two tissues with comparable spin density and slightly di erent T1 values, the optimal TR to choose is the T1 value of the shorter T1 tissue. Problem 15.9
Derive an expression similar to (15.56) for a general value of . How does this value of TRopt normalized to T1 A compare with the choice of a TR of an average value of T1 A and T1 B normalized to T1 A? What is the range of values up to which the average value serves as a reasonable approximation to the optimal TR ? The fact that the contrast is optimized by a TR value comparable to the average of the T1 values of the two tissues is used as a general rule of thumb for choosing TR for tissues with comparable spin density values. 15.4.4 The third basic contrast generating mechanism is based on di erences in the transverse decay characteristics. Most disease states are characterized by an elevated T2 . Since the T2 values T2 -Weighting 360 Chapter 15. Signal, Contrast and Noise are only on the order of tens of milliseconds whereas T1 values are typically on the order of a second, a small increase in T2 corresponds to a larger percentage increase than the same increase in T1 . As a result, T2 is found to be a sensitive indicator of disease. T2 weighting can be obtained by using spin echo sequences. T2 weighting also plays a useful role when local magnetic eld susceptibility di erences between tissues are present. If eld changes occur su ciently rapidly across a voxel, additional signal loss will occur when gradient echo sequences are used. For this reason, T2 -weighted images are used to study brain activity in brain functional imaging studies (as discussed in Ch. 25). To avoid contributions from T1 confounding the contrast, TR is chosen such that13 TR T1A B ) e;TR=T1 ! 0 (15.57) in which case the gradient echo contrast is given by CAB = ;T =T 0 Ae E 2 A ; ;T =T 0 Be E 2 B (15.58) Figure 15.8b shows a plot of TE versus contrast for (15.58) using gray matter ( 0 = 0.8 and T2 = 0.1 sec) and CSF ( 0 = 1.0 and T2 = 2 sec) as tissues A and B . Since GM has a T2 value which is much shorter than that of CSF, the optimal TE value is expected to be long compared to the T2 value of gray matter and short compared to the T2 value of CSF. On the other hand, when gray and white matter signals are considered as functions of TE at long TR values (Fig. 15.8a), WM ( 0 = 0.65 and T2 = 0.08 sec) always has a signal that is lower than that of GM. So, the optimal GM-WM contrast is produced by a very short TE , in the spin density-weighted regime. A similar contrast curve can be generated for any two tissues of interest, whose relative spin density and T2 values are known. For example, the case of GM/lesion contrast is considered in Fig. 15.8c. The lesion is assumed to have a 0 of 0.8 and a T2 of 350 ms. Since the contrast expression (15.58) contains only TE dependence, optimal contrast is obtained by extremizing CAB relative to TE . The TE at which the contrast is optimized is TEopt = ln T2 B 1 T2 B 0B ; ln T02 A A 1 ; T2 A (15.59) As previously discussed, a similar expression is obtained for a spin echo experiment in terms of TE with T2 replaced by T2 . Expressions for the optimal TE in the special case of only a fractional di erence in T2 also gives results identical in form to the TR choice for optimal T1 -weighted contrast: that is, a TE approximately equal to T2 A should be chosen.
It is also possible to obtain spin density or T2 -weighting with shorter TR but then the rf pulse angle must be reduced to a value much less than =2. A discussion of this class of experiments is reserved for the fast imaging discussion in Ch. 18.
13 15.4. Contrast Mechanisms in MRI and Contrast Maximization 361 Fig. 15.8: CAB as a function of TE for (a) GM/WM (b) CSF/GM and (c) lesion/GM for a T2 weighted scan. CAB is given in units relative to a maximal value of unity. An optimal TE value
can also be obtained for each pair of tissues from such plots. 362 Chapter 15. Signal, Contrast and Noise Problem 15.10
a) Suppose that the relative spin density ( 0 ), T1 and T2 values of a certain tissue (gray matter, say) are 0.8, 1 s and 100 ms, respectively. When disease sets in, suppose the local water content increases from 0 to 10% in each voxel. This two compartment model implies that the fractional volume content of water in a voxel increases from 0 to 0.1 and that of the tissue decreases from 1.0 to 0.9. (The displaced tissue is simply pushed into an adjacent voxel.) If the T1 and T2 of water are 4 s and 2 s, respectively, and the R1 and R2 values of healthy tissue and water average according to volume fraction of occupation of the voxel, what are the percentage increases in the e ective T1 and T2 values in the diseased tissue? b) Hence, which of the two mechanisms, T1 - or T2-weighting, is more sensitive to small changes in local water content (compare the percentage change from values in normal tissues between T1 and T2)? c) Suppose that there is a 2% increase in the tissue volume due to elevated local water content. What is the percentage increase in the spin density 0 (remember that the relative spin density of water is 1.0)? d) What does T2 of this tissue change to if its intrinsic T2 value is 100 ms? Find the optimal TE for distinguishing this `diseased tissue' from its normal neighbor neglecting the change in spin density calculated in part (c). Hence, show that the optimal TE in the case of two tissues with comparable spin densities but di erent T2 values such that T2B = T2A (1 + ) is comparable to the average of the two T2 values. The conclusion from the above problem is that T2-weighting might be the intrinsic contrast mechanism of choice for distinguishing diseased states from normal tissue. In fact, it therefore comes as no surprise that T2 -weighted spin echo sequences are used for a variety of clinical applications. 15.4.5 Summary of Contrast Results
The general appearance of spin density, T1 - and T2-weighted images of the brain is depicted in the images shown in Fig. 15.9. An optimal and yet practicable set of imaging parameters was used to obtain these images. Clearly, the T1-weighted imaging method seems to be the most e cient in achieving the contrast required. An intermediate TR value of about 600 ms gives optimal GM-WM contrast and shows good di erentiation between these two structures and CSF and fat. Fat, with the lowest T1 value amongst these four tissues, is shown as the brightest structure. On the other hand, CSF is shown with almost no noticeable signal because of its long T1 . White matter is the structure which is shown as the bright structure in 15.4. Contrast Mechanisms in MRI and Contrast Maximization Type of contrast TR TE spin density as long as possible as short as possible T1 -weighted on the order of the T1 values as short as possible T2 -weighted as long as possible on the order of the T2 values Table 15.3: General set of rules for generating tissue contrast. 363 the brain, while gray matter is shown with a lower gray level, all consistent with T1 weighting. On the other hand, when it comes to spin density or T2 weighting, it is impractical to design the sequence with a TR which is on the order of a few T1's of CSF. Therefore, it is typical to choose a TR of about twice the T1 of gray matter. At this TR , CSF is almost iso-intense with white matter, and gray matter is found to have the highest signal. As described here, the spin density and T2 -weighted images shown in Figs. 15.9b and 15.9c were obtained using a TR of 2 sec. On the T2-weighted image, typically obtained using a longer TE at the same TR as the spin density-weighted image, CSF has the highest signal while white matter has the lowest signal. A set of general rules for choosing TE and TR for a =2 gradient echo (or spin echo) experiment are outlined in Table 15.3. Problem 15.11
Tissues A and B are found to have the following properties when imaged at a certain eld strength Quantity A B T1 600 ms 300 ms intrinsic T2 ( B0 = 0) 80 ms 60 ms normalized 0 1.0 0.8 Assume that 0 A= 0 is 40:1 and, in order for two tissues to be reliably di erentiated, that the CNR must be greater than or equal to 4. Consider the gradient echo example presented earlier in this section. It may be necessary to numerically generate some plots for various sections of this problem. a) If TE is kept as short as possible (say TE = 20 ms) to minimize T2 contrast, what is the minimum value of TR that may be used to obtain a T1 -weighted image with adequate CNR such that A and B are reliably di erentiated? b) If TR is 2000 ms, nd the maximum value of TE that may be used to obtain a T2-weighted image with adequate CNR. 364 Chapter 15. Signal, Contrast and Noise 15.4.6 A Special Case: T1-Weighting and Tissue Nulling with Inversion Recovery
Another commonly used mechanism for generating T1 contrast is to use an inversion recovery (IR) sequence. Image contrast with the inversion recovery sequence can be adjusted with both TR and TI . There is interest in nding out the most time-e cient way to obtain T1 weighting at some value (not necessarily the optimal value) of TR or TI . This is obtained by looking at the di erential change in signal relative to a di erential increase in TR or TI . For the gradient echo sequence, @CAB = @TR 0A TR e;TR =T1 A ; T1 A 0B TR e;TR =T1 B T1 B (15.60) whereas, for the inversion-recovery sequence (whose signal expression is given by (8.46)) @CAB = 2 @TI TI e;TI =T1 A ; 0A T1 A TI e;TI =T1 B 0B T1 B ! for TR T1 (15.61) i.e., for the same increase in TR or TI , the inversion-recovery signal has twice the signal change as the spin echo signal change. If two tissues with only fractional T1 di erences are considered, this directly correlates to a doubling of the contrast in the inversion-recovery sequence. This was the reason for the early popularity of the inversion-recovery sequence for obtaining T1 weighting. The inversion process o ers the ability to null a speci c tissue by an appropriate choice of TI . Some examples of this feature unique to the inversion recovery sequence are shown in Fig. 15.10. Here, four di erent choices of TI can be used to null fat, white matter, gray matter and cerebrospinal uid, respectively. The rst three are used in T1 -weighted IR imaging methods and the fourth in a T2 -weighted sequence to better di erentiate small lesions otherwise obscured by CSF. The rst is useful to image near the orbits where fat can obscure the optic nerve, for example. The third shows a more conventional T1 -weighted image. The fourth nulls CSF and any other water-like components but leaves good signal from pathological states with only slightly elevated water content relative to normal tissue. Problem 15.12
Consider the implementation of an inversion-recovery sequence with TI chosen to null the signal (implies TI null = T1 ln 2 in the limit that TR >> T1 ) for either fat or water at 1.5 T. Find TI null for water and fat whose T1 values are T1 w = 4 s and T1 f = 250 ms respectively. 15.4. Contrast Mechanisms in MRI and Contrast Maximization 365 (a) (b) (c)
Fig. 15.9: Di erent forms of contrast generated by varying the imaging parameters with a spin echo sequence. (a) Spin density-weighted image. (b) T2 -weighted image. (c) T1 -weighted image. Although (a) is supposed to indicate spin density, it fails to do so for CSF since T1 of CSF is too long (4.5 sec) relative to the TR (2.5 sec). Gray/white matter contrast is good. In (b), the globus
pallidus (arrow) appears quite dark (since the iron in it causes a di usion weighted signal loss see Ch. 21). As expected, CSF is bright because of its long T2 (2 sec). Lastly, in (c), gray/white matter contrast is reversed and CSF is heavily suppressed. The dark CSF regions seem to visually enhance the overall contrast in the image. 366 Chapter 15. Signal, Contrast and Noise (a) (b) (c) (d) Fig. 15.10: Di erent tissues nulled with an inversion recovery sequence. (a) A short TI is chosen to null fat. (b) An intermediate TI is chosen to null WM. (c) A higher TI nulls GM. (d) A very long TI nulls CSF, which is typically used in a T2 -weighted sequence to null the otherwise very high
signal producing CSF. 15.5. Contrast Enhancement with T1 -Shortening Agents 367 15.5 Contrast Enhancement with T1-Shortening Agents
In fast imaging, where TR is much less than T1 , the signal for a 90 ip angle is proportional to TR =T1 and, in this regime, tissues with shorter T1 have a higher signal than tissues with longer T1 and the contrast is predominantly T1 -weighted. Certain external agents can be introduced into speci c targeted tissues where these agents act to reduce the T1 of that tissue. Figure 15.11 demonstrates the T1 shortening e ect of one of these agents. Suppose that the targeted tissue (in the case shown in Fig. 15.11, the targeted tissue is blood) has similar NMR properties as its neighboring tissue causing an inability to di erentiate the two using any of the three contrast mechanisms discussed in Sec. 15.4 but contains a di erent signal response to the contrast agent. By delivering the T1 -shortening agent to the tissue of interest, the targeted signal is increased while the signal from the background remains the same when a T1 -weighted sequence is used. This increases the contrast between the two tissues and, for this reason, such agents are commonly referred to as `contrast agents.' In general, the increase in relaxation rate after T1 shortening is found to be directly proportional to the concentration C of the contrast agent delivered to the tissue. If T1 0 is the intrinsic T1 of the tissue and T1 (C ) is its shortened value, it is found that 1 R1 (C ) T (C ) = T1 + 1C 1 10 R1 0 + 1C (15.62) where the constant of proportionality 1 is called the longitudinal relaxivity (T1 relaxivity) with units of (mmol/l);1sec;1 ,14 a property speci c to the composition of the contrast agent. Figure 15.12a demonstrates the e ect of a contrast agent in vivo on the T1 of blood for which T1 0 ' 1200 ms at 1.5 T. In addition to shortening the T1 , these contrast agents also tend to shorten the T2 of the tissue in a similar fashion to the T1 shortening, i.e., 1 R2(C ) T (C ) = T1 + 2C 2 20 = R2 0 + 2 C (15.63) where 2 is the transverse relaxivity of the contrast agent. Figure 15.12b shows the e ect of contrast agent dosage on the T2 of blood (which has an intrinsic T2 ranging from 100 ms for venous blood to 200 ms for arterial blood). This concomitant decrease in T2 tends to partly counterbalance the e ects of shortened T1. For many T1 -shortening contrast agents, the transverse and longitudinal relaxation rates are comparable in magnitude ( 1 ' 2 ). Since R2 0 > R1 0, a given increase in the concentration leads to a larger magnitude change in T1 and, hence, in the signal due to this shortened T1 e ect than the e ect caused by the T2 reduction. It is only when 2 C becomes comparable to R2 0 that the signal loss due to the T2 shortening becomes signi cant, and starts to overwhelm the enhanced T1 -weighted contrast. In fact, this crossover point between signal loss due to T2 shortening and signal increase due to T1 shortening de nes an optimal contrast agent dosage.
14 Recall that 1 mmol/l is also written as 1 mM (millimolar). 368 Chapter 15. Signal, Contrast and Noise (a) (b) (c)
Fig. 15.11: T1 reduction e ects of a contrast agent. (a) A single slice from a 3D T1-weighted data
set before contrast agent was injected. (b) Image of the same slice after an intravenous injection of a T1 shortening contrast agent. Since the T1 of blood is shortened, the blood vessels light up, and are seen as bright spots in the image. Also, there is a loss of contrast between WM/GM because of the larger blood volume content of GM (hence, its signal increases more than that of WM, reducing contrast). (c) Subtraction of the rst image from the second image depicts this signal increase predominantly in the blood vessels, but also from the blood-containing tissue itself. Only those voxels with a positive subtraction are shown, voxels with a negative subtraction are set to zero, leading to the isolated black dots in this image. 15.5. Contrast Enhancement with T1 -Shortening Agents 369 concentration delivered to the tissue of interest. The contrast agent is supposed to have longitudinal and transverse relaxivities of 5 /mM/sec and 6 /mM/sec, respectively. Notice that the relative change in T1 for a given concentration is much larger than that for T2 . (c) Blood signal in relative units as a function of contrast agent concentration plotted for a spin echo acquisition with 600 ms TR and 20 ms TE . (T1 and T2 of blood were taken to be 1200 ms and 150 ms, respectively.) Note that, as the concentration increases, the T2 reduction e ect takes over and causes a reduction of blood signal. In essence, an optimal concentration which maximizes the signal exists. Fig. 15.12: T1 and T2 are plotted in (a) and (b), respectively, as functions of contrast agent 370 Chapter 15. Signal, Contrast and Noise Problem 15.13
If a dose of 0.1 mmole/kg is given to a 50 kg person with 5 liters of blood in his/her body, what is the expected T1 of the blood when the contrast agent is well-mixed in the blood? Assume the T1-relaxivity of the contrast agent is 5/mM/sec and T1 of blood is 1200 ms. In practice, some of the agent is taken up by other organs in the body and this `extravasation' can lead to the e ective blood volume (or e ective distributed volume of the contrast agent) being 10 liters for purposes of this calculation. How does this a ect the above estimate for T1? A major clinical application of T1 shortening contrast agents at present is intended for the improved detection of small lesions. Typically, these lesions have a fractionally increased water content leading to best depiction of these lesions in a T2-weighted image before contrast agent is injected. Despite being the most sensitive contrast mechanism, it is not possible for the T2 -weighted image to depict very small lesions which are averaged with neighboring tissue of comparable NMR tissue properties. The use of an intravenous injection of a T1 shortening agent is indicated for such patients. Most lesions are found to have a rich blood supply. Therefore, the lesions have contrast agent delivered locally, leading to lesion signal enhancement and improved lesion detection. 15.6 Partial Volume E ects, CNR and Resolution In the earlier discussion on object visibility, it was seen that V is invariant as a function of voxel size for multi-voxel objects under the assumption that the voxel signal came from a homogeneous chunk of tissue within the voxel. What then is the advantage of using a smaller voxel size if reducing the voxel size serves only to worsen the SNR while only maintaining object visibility? In this section, resolution e ects on object visibility are discussed in detail, leading to a theoretical understanding of the e ects of spatial resolution on the diagnostic value of an image. If tissue A is smaller than a voxel, it shares the voxel with the background tissue, and tissue A is said to be `partial volumed.' Such a partial volume model is shown in Fig. 15.13. If the point spread e ects of the image reconstruction are neglected, a simple model for the combined signal from the voxel is the summed fractional signals of tissues A and B . Let SA and SB represent the signal for tissues A and B , respectively, when they occupy an entire voxel of size x (i.e., SA is the signal from tissue A if equals 1 in Fig. 15.13a). In the general case, if is the fraction of voxel q occupied by tissue A, the signal for voxel q when = 1 (Fig. 15.13a) is the sum of the voxel signal contributions ^A q and ^B q from A and B , respectively, i.e., ^voxel q = ^A q + ^B q (15.64) Now ^A q SA and (15.65) 15.6. Partial Volume E ects, CNR and Resolution ^B q and hence from (15.64) The image contrast is then (1 ; )SB 371 (15.66) (15.67) = (SA ; SB ) (15.68) ^voxel q = SA + (1 ; )SB Cq q+1 = voxel with A and B homogeneous voxel with B |^voxel } {z q ; |^voxel q+1 {z } The conclusion is that if a certain tissue is partial volumed, its contrast is reduced according to the fraction of the voxel it is occupying. Fig. 15.13: Model of a 1D voxel q which is occupied by two di erent materials or tissues. Voxels (q ; 1) and (q + 1) contain only background tissue with signal magnitude SB in (a) where the voxel size is x, while the central voxel q contains a combination of tissues A and B , with tissue A of size x ( < 1) occupying a fraction of the voxel. In (b), the voxel size is reduced to x ( < 1) and tissue A now occupies the fraction = of the central voxel while the signal from voxels (q ; 1) and (q + 1) are also reduced to SB . In the case where the voxel size is reduced, i.e., < 1, tissue A occupies the fraction of the entire voxel volume and B occupies the remaining 1 ; fraction of the voxel. Also, the signal from voxels q ; 1 and q + 1 is now SB . In a similar fashion, the signal from A, when it occupies a complete voxel of size x is SA, leading to ^q A ! ( SA) (15.69) 372 and Chapter 15. Signal, Contrast and Noise ^q B ! ( ; ) ( SB ) = ( ; )SB (15.70) Hence the contrast is Cq q+1j <1 = ^voxel q ; ^voxel q 1 = ( SA + ( ; )SB ) ; SB = (SA ; SB ) (15.71) i.e., the contrast does not change. Two cases can be considered for increasing the spatial resolution (say, making ! =2): case 1, doubling Ts or the number of phase encoding steps so that the SNR reduces by only p 2 and case 2, halving the FOV keeping N xed, leading to a loss of 2 in SNR. As the voxel p size is decreased by a factor of 2, the CNR (and visibility) increases by 2 in case 1 and stays the same in case 2. If the SNR is good enough, this result indicates it is possible that a small partial volumed object will become visible as x is reduced. In summary, the utility of high resolution lies in its ability to make partial volumed objects which are smaller than the voxel size visible, even though the SNR decreases as x is reduced. Although at rst sight it seems like tissue A has been conveniently chosen to be at the center of the voxel in (a) for this improved visibility result, even in the more general case of an arbitrarily positioned A, it can always be moved into the center of the voxel by using the Fourier transform shift theorem (see Ch. 11 for details) for which case this improved visibility result holds. Problem 15.14
Consider an object of size x=4 (say a small blood vessel) and centered in a 1D voxel of size x as shown in Fig. 15.14a. Suppose (SA ; SB )= 0 is 2.0 when the voxel size is x. Now suppose the voxel size is halved (case 1) to x=2 (see Fig. 15.14b) and then halved again (case 2) to x=4 (see Fig. 15.14c). a) If Sq = is 8.0 when the voxel size is x, what is the SNR at the two smaller voxel sizes shown in Figs. 15.14b and 15.14c? Assume that SNRq is reduced p by 2 each time the resolution is halved. The common misconception is that because the SNR is reduced, the diagnostic value of the image is lost. Evidently, object visibility, rather than SNR, is the more appropriate index of clinical utility. b) What happens to V at the two smaller voxel sizes in (i) case 1, and (ii) case 2? Is the object visible at any of the three voxel sizes (i) in case 1, and (ii) case 2? Assume that the visibility threshold is 3. The important point is that if there is an object of interest which is invisible at a given voxel size because of partial volume e ects, it may be possible to make it visible by reducing the voxel size. Other additional options include using a contrast agent to increase signal or averaging over multiple acquisitions to reduce the noise. 15.6. Partial Volume E ects, CNR and Resolution 373 Fig. 15.14: (a) A 1D voxel of dimension x contains a square object x=4 in size. The voxel size is reduced to (b) x=2 and (c) x=4 to demonstrate the potential e ects of partial voluming. For case 1, the actual CNR between tissues from (a) to (c) increases by a factor of 2 (as discussed in Sec. 15.6). This can also be understood by considering the change in voxel signal for A when B is zero. For part (a), the signal in voxelpq will appear as SA =4 and remain at this value for parts (b) and (c) while the noise decreases by 2 at each step. 374 Chapter 15. Signal, Contrast and Noise 15.7 SNR in Magnitude and Phase Images
Phase o ers some fascinating ways to enhance information about certain features in MR images. When the eld is perfect and no motion is present, the expected phase is zero. White noise will also have a magnitude and phase. We consider here the contribution of noise to the magnitude and phase images (j ^j and ^) in the case where the SNR is much greater than unity. 15.7.1 Magnitude Image SNR
The complex signal can be written as or as ^ = R + iI (15.72) ^ = m ei (15.73) Let m in the absence of noise be a. If the noise-free phase is ^ then the real and imaginary parts, R and I , respectively, of ^ are R = a cos ^ + 1 (15.74) I = a sin ^ + 2 (15.75) where R and I are the real and imaginary parts of the reconstructed complex voxel signal, respectively, and 1 and 2 are noise samples from a Gaussian distribution with mean zero and variance 2. The mean values of R and I , R and I , respectively, are R = a cos ^ (15.76) I = a sin ^ (15.77) Their variances are both equal to 2. The magnitude
m m of the voxel signal is found from With the approximation made for signal is, as expected, given by = (R2 + I 2 )1=2 2 2 = (a2 + 2 2a cos ^ + 2 1a sin ^ + 1 + 2 )1=2 ' a 1 + a2 cos ^ + a1 sin ^
m m (15.78) in (15.78), the mean of the magnitude of the voxel =a (15.79) To obtain the variance of the magnitude, the noise is taken to be uncorrelated between the real and imaginary channels. Then, from (15.78), var( m ) = var( 2 ) cos2 ^ + var( 1) sin2 ^ = 2
2 mag (15.80) 15.7. SNR in Magnitude and Phase Images 375 i.e., if the SNR of the magnitude of the voxel signal is much larger than unity, the mean of the magnitude is the true magnitude value, and its variance is the same as the variance in either the real or imaginary part of the voxel signal. Problem 15.15
An alternate approach can be taken to obtain (15.79) and (15.80) for the case of large SNR. In this limit, the added white noise can be approximated as producing a small error in the measurement of the true magnitude ( m ). 1) The measured voxel signal magnitude m can then be expanded in a Taylor series around the value m by viewing m as a function of R and I , the real and imaginary parts of the voxel signal, respectively. With the SNR large, the approximation of its Taylor series up to the linear term is good enough. Using this approximation, write an expression for the error in the measured magnitude, m m ; m , in terms of the errors in R and I . 2) What is the mean value of m ? Assume that I = R = 0. 3) What is the error variance ( m ; m )2 ? Do you obtain the same result as in (15.80)? The phase of the signal can be found from tan ^ = I=R (15.81) Using the approach of di erentials on (15.81) to determine the error as in Prob. 15.15 gives ^ sec2 ^ = I ; R I2 (15.82) R R Hence, ^ = 0 and the variance of the phase is given by the mean squared error: ! I 2 I 2 + R2 cos4 ^ 2 = ^ R I 2 ! R2 2 = 2 cos4 ^ Rm 4 = and the standard deviation of the phase phase ' ( ^2)1=2 = = m . This is sometimes written as (15.84) phase = 1=SNRmag (units are radians)
2 2 m 15.7.2 Phase Image SNR (15.83) 376 or in degrees as
phase Chapter 15. Signal, Contrast and Noise 180 = (SNR ) (15.85) mag where SNRmag is the SNR of the voxel signal magnitude. This tells us that at voxels where the SNR of the magnitude is very high, there is a relatively low error in phase measurements. On the other hand, the error is very large if the voxel magnitude has very low SNR. One of the interesting features of the phase is that if there are artifacts which are of very low magnitude in comparison with the object signal (such as ghosts), the magnitude image does not show these features well. However, these artifacts still have some nonzero phase and even the smallest magnitude e ects are highlighted. Such an example is shown in Fig. 15.15. (a) (b) Fig. 15.15: Low amplitude features such as ghosts are typically highlighted in the phase image. Although a shifted ghost is present in the phase encoding direction, (a) it does not show on the magnitude image, yet, it is well depicted in (b) the phase image. 15.8 SNR as a Function of Field Strength
Finding the optimal eld strength for imaging has always been complicated and controversial since many factors a ect the outcome. Perhaps the three most important are spin density, T1 and eld inhomogeneity, all of which lead to modi cations of SNR and CNR as a function of eld strength. In this section, we will give a prediction of the signal as a function of eld strength based on (15.9) and (15.87) and a practical estimate of the ratio of the signi cance of the two terms in (15.87) at one eld strength (the two terms being formed by bulking together the 15.8. SNR as a Function of Field Strength 377 coil and electronics resistances as the sum of electronic resistance Rcoil , Relectronics and the magnetically induced losses due to the presence of the body, Rbody ). 15.8.1 Frequency Dependence of the Noise in MRI It was observed in Ch. 7 that the induced emf is proportional to !0M0 . Since M0 / !0 too, the induced emf increases as the square of the eld strength. In a circuit with a frequency-independent resistance, the rms noise voltage is also independent of the frequency (from (15.7)), depending only on the bandwidth of data collection. Unfortunately in MR, the e ective resistance `seen' by the signal-receiving electronics is found to depend on the operating frequency. This dependence causes the rms noise voltage to be frequency-dependent. The resistance of the coil and the electronics can be bulked together as one component contributing to the noise, while the sample resistance corresponds to the other major contributor to the noise voltage. At low frequencies, the coil and electronics resistance dominate over the sample resistance at room temperatures, whereas at high frequencies, the sample resistance of human tissues dominates the coil and electronics resistance. Let !0 mid be some frequency where this transition of noise source dominance begins to occur: ( Rcoil (!0) + Relectronics(!0) !0 !0 mid Reff (!0) (15.86) R (! ) ! !
sample 0
0 0 mid It has been found that the combined coil and electronics resistances have a square root 2 dependence on !0 , whereas, the sample resistance has an !0 dependence, i.e., (p Reff (!0 ) / !!0 !0 2 0 !0
1=4 / !0 !0 thermal ! ! !0 mid !0 mid (15.87) and, from (15.7) !0 mid (15.88) !0 mid 0 0 2 since thermal / Reff . As a result, when the experiment is designed such that the noise is white noise dominated, the SNR as a function of frequency !0 is such that ( 2 1=4 7=4 =!0 SNR(!0) / !0!2=! = !0 !0 !0 mid (15.89) =! ! !
0 0 0 0 0 mid ( i.e., at low eld strengths, a 7/4 power law improvement in SNR with eld strength occurs, whereas at high eld strengths, only a linear increase in SNR is obtained. The practical implications of this frequency dependence will be discussed in more detail in Sec. 15.8.2. 2 It su ces to say here that the SNR increases not as !0 as it would have if thermal were 7=4 frequency-independent. It increases only as !0 until some transition frequency, after which it increases further slowly for !0 !0 mid . It is also worth reminding the reader here that at a given frequency, the rms noise voltage depends on the measurement bandwidth BWread. 378 Chapter 15. Signal, Contrast and Noise We start with a low eld system (B0 low = 0:065 T), where one expects the noise to be dominated by the electronics. In this case, using a high temperature superconducting surface coil, it has been shown that the noise is reduced by a factor of 2 compared to a conventional coil at room temperature. If one assumes that the remaining noise is from magnetic losses, then it is possible to calculate the ratio of Rbody to Relectronics and predict the SNR increase from 0.065 T to a high eld such as 1.5 T, for example. Using the above information in the generalized expression for SNR given the eld dependence of Reff in (15.87), 2 SNR(!) = (!2 + !p!)1=2 (15.90) a 3=2 implies a = 3!low where here !low = = B0 low and ! = B0 (for simplicity the subscript 0 used in the previous subsection is dropped from ! at this point). The constant contains the system scale factor and all other terms on which signal depends. Using the fact that the noise is reduced by a factor of 2, as mentioned above, yields 15.8.2 SNR Dependence on Field Strength 2 6 SNR(!) = SNR(!low ) 6 6 4 !low ! 2 ! !low 2 +3 q !low ! 3 7 7 1=2 7 5 (15.91) Fig. 15.16 demonstrates that above 0.5 T, the SNR is linear in ! to within 6.3% of the predicted value and, below 0.04 T, it is proportional to !7=4 within 37%. Fig. 15.16: Plot of SNR as a function of eld strength. (a) Overall dependence shown over a range
of 0.001 T to 5.0 T. (b) Comparison of the dependence with the 7/4 power law (shown as the curve with the short dashed line) and linear behavior (long-dashed line) over a range of 0.001 T to 0.1 T. Implications for Low Field Strength Imaging The possibility of imaging at low elds appears rather dismal from these predictions assuming that the imaging parameters at 1.5 T are ideal. However, at lower elds both T1 and 15.8. SNR as a Function of Field Strength susceptibility e ects are reduced. For example, it has been empirically shown that 379 T1 (!) = 1=(A= ! + B ) p (15.92) This inverse square root dependence of the longitudinal relaxation rate on the eld strength was observed by Escanye in the article cited in the suggested reading found at the end of the chapter. For the tissues evaluated, B was found to be very close to zero, i.e., the relaxation rate was found to vary inversely as the eld strength. Recall that for spin echo or gradient p echo p imaging for short TR , the signal is proportional to TR =T1 so a ! decrease in T1 leads to a ! increase in signal. Problem 15.16
Another estimate for the eld dependence of tissue T1 is T1 (!) = a!b (15.93) This power law dependence of T1 on eld strength was proposed by Bottomley in the article cited at the end of this chapter. For gray matter, the values of a and b were found to be 0.00362 and 0.3082, respectively for white matter, the values are 0.00152 and 0.3477, respectively. For (15.93), plot T1 (!) from 1 MHz to 200 MHz for GM and WM. Note that in comparison with the square root increase in T1 with eld strength described in the text, this predicts approximately a cube root increase in T1 with eld strength. What a ects do you expect to see for T1 weighted scans as the eld changes from 1.5 T to 3.0 T? Also at lower elds, as long as Ts=2 is much shorter than TE at 1.5 T, BWread for a xed x can be reduced proportional to eld strength thanks to lower susceptibilities at lower elds and hence reduced signal dephasing and image distortion artifacts. Therefore, down to p some eld strength where the limiting case is reached, SNR goes up by another factor of !. Lo and behold, dependence on ! from the previous SNR argument, under some circumstances, is exactly balanced by the opposite dependence on ! from the two arguments just described. Problem 15.17
At high magnetic elds, R2 of blood becomes a function of eld strength. Assume that R2 varies directly with magnetic eld. Show that the SNR would vary as pB e; TE B0 for a xed echo time if the readout bandwidth is also increased 0 to avoid further geometric distortion as B0 increases. Comment on an optimal choice of B0 under these conditions. 380 Chapter 15. Signal, Contrast and Noise Suggested Reading
Basic concepts of noise are well described in the following two books: C. Kittel. Thermal Physics. John Wiley and Sons, Inc., New York, 1969. H. L. Van Trees. Detection, Estimation, and Modulation Theory. Part I. John Wiley and Sons, Inc., New York, 1968. The concepts of visibility and detectibililty are introduced in: A. Rose. Vision: Human and Electronic. Plenum Press, New York, 1985. A measure of imaging e ciency is proposed in: W. A. Edelstein, G. H. Glover, C. J. Hardy and R. W. Redington. The intrinsic signal-to-noise ratio in NMR imaging, Magn. Reson. Med., 3: 604, 1986. Issues of contrast and visibility in MR imaging are covered in the following two papers: R. T. Constable and R. M. Henkelman. Contrast, resolution, and detectability in MR imaging, J. Comput. Assist. Tomogr., 15: 297, 1991. R. Venkatesan and E. M. Haacke. Role of high resolution in magnetic resonance (MR) imaging: Applications to MR angiography, intracranial T1-weighted imaging, and image interpolation, Int. J. Imaging Sys. Technol., 8: 529, 1997. The square root eld strength dependence of T1 was proposed in: J. M. Escanye, D. Canet and J. Robert. Frequency dependence of water proton longitudinal nuclear magnetic relaxation times in mouse tissues at 20 C, Biochim. Biophys. Acta, 721: 305, 1982. The cube root eld strength dependence suggested in Prob. 15.16 was observed in: P. A. Bottomley, C. J. Hardy, R. E. Argersinger and G. Allen-Moore. A review of 1 H nuclear magnetic resonance relaxation in pathology: Are T1 and T2 diagnostic? Med. Phys., 14: 1, 1987. Chapter 16 A Closer Look at Radiofrequency Pulses
16.1 Relating RF Fields and Measured Spin Density 16.2 Implementing Slice Selection 16.3 Calibrating the RF Field 16.4 Low Flip Angle Excitation and Rephasing Gradients 16.5 Spatially Varying RF Excitation 16.6 RF Pulse Characteristics: Flip Angle and RF Power 16.7 Spin Tagging Summary: The measured spin density is related to the rf elds in terms of the receive Chapter Contents coil eld, transmit coil eld, and the slice selection process. The slice select pro le as a Fourier transform of the time-dependent B1 eld is expanded upon. A practical example of slice select sequence design is given which takes into account nite gradient rise times and spoiling. Methods for calibrating the rf transmit amplitude and slice select properties are discussed. Solutions of the Bloch equations for low ip angles are presented which can be used to select an arbitrary 3D region. Spin tagging is introduced in the nal section. Introduction
The image, or measured spin density in an MRI experiment, is not necessarily equivalent to the physical spin density of the object being imaged, even when relaxation and Fourier transform e ects are considered. The image is actually a picture of the signal received by the rf receive coil or rf probe. It is, in fact, proportional to the product of the receive coil's eld and the transverse magnetization (which itself depends on the transmit coil's eld). In 381 382 Chapter 16. A Closer Look at Radiofrequency Pulses this chapter, a study is made of how the applied gradients, transmit rf eld, and receive rf eld relate to the measured spin density. In Ch. 10, the notion of using nite bandwidth rf pulses in combination with applied gradients to excite a particular region of spins was introduced. It was assumed there that the rf eld was perfectly uniform and that a rectangular spectral excitation could be achieved. The Fourier transform dictates that a rectangular excitation in the frequency domain requires a sinc pro le in the time domain extending over all time. In practice, the rf pulse can only last a few milliseconds and, therefore, a perfectly rectangular excitation pro le cannot be achieved. A general introduction relating the applied rf eld to the measured spin density will be presented rst, and then several sections will explain speci c experimental scenarios for rf calibration, speci c pulses, and rf power deposition. (a) (b) Fig. 16.1: A uniform spin density object is placed in two di erent coils. (a) The rst image is acquired with a birdcage coil (see Ch. 27) used for both transmission of the rf pulse and reception of the signal. B1 (~) for this birdcage coil is approximately uniform over the object, and a fairly r uniform image results. (b) The same birdcage coil as in (a) was used to tip the spins, but a surface receive leads to a change in intensity coil was used to receive the signal. The spatial dependence of B1 of the image as a function of position across the image. 16.1 Relating RF Fields and Measured Spin Density
The focus of this section is to understand how variations in the rf eld will cause variations in image intensity that are independent of the physical spin density. In Ch. 7, it was shown 16.1. Relating RF Fields and Measured Spin Density 383 that the signal measured in an MRI experiment is proportional to the emf picked up by the rf receive coil which, in turn, is proportional to the scalar product of the transverse magnetization and the receive rf eld (see (7.20)). The signal received from a di erential volume element d3 r is ds(~) / M?(~)B1 ? (~)d3r r r receive r (16.1) where the time dependence of the signal has been ignored. After Fourier transforming a frequency encoded signal with respect to time, an image is produced which is proportional to the same terms ^(~) / M?(~)B1 ? (~) r r receive r (16.2) The reconstructed image is said to represent the physical spin density, although it is seen that other quantities are involved in this expression. Remember that M?(~) is usually generated r from the creation of transverse magnetization during the slice select process. To nd the relationship between the measured spin density and the physical spin density, it is necessary transmit r to determine how M?(~) is related to B1 ? (~ t) and the applied slice select gradients. r The magnitude of the perpendicular component of the magnetization, neglecting relaxation and assuming the equilibrium magnetization M0 is available, is M?(~) = M0(~) sin (~) r r r (16.3) The spatial e ects of the transmit rf eld and applied gradients are re ected in (~). It is r generally assumed in imaging that (~) is constant over the excited region, but this is the r case only if the rf eld is perfectly uniform and perfect slice selection occurs. However, neither of these conditions exists in reality (see Fig. 16.1). Slice Selection and the Spatial Dependence of B1
In the following example, the interaction between the spatial dependence of the transmit rf eld and slice selection is considered. In a manner similar to frequency encoding of the spins, the rf and the slice select gradient work in tandem to excite spins as a function of frequency (or spatial position). In the presence of an applied gradient eld, the Larmor frequency in the rotating reference frame is a function of position: ~r r f (~ t) = {G(~ t) ~ r (16.4) Assuming that this discussion is limited to selecting a transverse slice, only a frequency change along z should be imposed giving f (z t) = {Gz (t)z
1? 1? (16.5) The frequency content of a time-dependent rf pulse is found by taking its Fourier transform Z transmit (~ f (z )) = 1 1 dtB transmit (~ t)e;i2 f (z)t B r r (16.6)
rf
;1 transmit r The normalization is chosen such that B1 ? (~ f ) evaluated at f = 0 can also be viewed transmit r as the average value of B1 ? (~ t). 384 Chapter 16. A Closer Look at Radiofrequency Pulses Assuming that Gz is constant in time, the usual variable k = {Gz t may be used to change from t-space and f -space to k-space and z-space:
transmit r B1 ? (~ z) = G1 { z Z1 rf ;1 transmit r dkB1 ? (~ k)e;i2 kz (16.7) The (~ z) argument is meant to denote that z-dependence arises from the spatial dependence r of both the rf eld and the slice select eld gradient. Finally, an approximation is used to relate the B1 eld to : it is assumed that the rf acts as if it is on-resonance at each frequency in the rf pulse of bandwidth BW during the rf pulse time rf , leading to
transmit r (~ z) = B1 ? (~ z) r rf (16.8) In order to nd exactly how the spins behave as a function of the applied rf, the Bloch equations must be solved (see Ch. 3 and Sec. 16.4). Equations (16.3) and (16.8) show that ^(~ z) is proportional to the sine of the Fourier r transform of the time-dependent rf pulse. It is also proportional to the receive coil eld yielding ^(~ z) / M0(~)B1 ? (~) sin ( (~ z)) r r receive r r (16.9) Since the rf pulse is not in nite in time, but truncated, the selectively excited spins will not transmit r represent a perfect rectangle along the slice select direction. Also, since B1 ? (~) is not uniform over all space, there will be some variation of ip angle in the plane of the slice. Further, if the B0 eld or slice select gradient varies as a function of position, the linear relation between z and f will be altered, causing excited slices to be `distorted' by the eld inhomogeneity (see Ch. 20). Problem 16.1
If the rf eld is nonuniform across a slice, the resulting signal in the image will be modi ed because the excited transverse magnetization is modi ed. Consider a typical slice select rf/gradient combination where a sinc pulse is used to create a rectangular slice pro le. The following assumptions are made: The rf pulse is su ciently long that the frequency response of the pulse is a perfect rect function. A uniform sample is being imaged (M0(~) = M0 ). r The slice is bounded by z1 and z2 with z1 < z2 , (z1) = =2. transmit transmit transmit B1 ? (z) varies linearly such that B1 ? (z1) = 2B1 ? (z2). receive r B1 (~) is constant over the region of interest. Plot ^(z) at a xed (x y) coordinate across the slice. 16.1. Relating RF Fields and Measured Spin Density 385 16.1.1 RF Pulse Shapes and Apodization All rf pulses are modulated in time, since they are at the very least nite in their time duration, which implies they also have a modi ed frequency distribution. This situation can be modeled mathematically by multiplying the ideal eld B1 ideal (t) by a rect function which e ectively represents the act of time truncation ! t B (t) = B (t) rect (16.10)
1 1 ideal rf Truncation leads to a frequency response (spatial response for the slice selection process) which is the convolution of the ideal response with a sinc function B1(f ) = B1 ideal (f ) sinc ( f rf ) (16.11) As the duration of the rf pulse increases, the function rf sinc( f rf ) approaches a -function (see Ch. 9 Prob. 9.3), and B1 (f ) approaches B1 ideal (f ) but truncation artifacts will still exist, i.e., Gibbs ringing will occur near sharp boundaries. In practice, an additional apodizing function is used to bring B1 ideal (t) smoothly to zero and reduce truncation e ects. Therefore, a more general expression for the time dependence of the eld is ! t (16.12) B (t) = B (t) a(t) rect
1 1 ideal where a(t) is an apodizing function and the frequency response of the pulse becomes rf B1(f ) = B1 ideal (f ) A(f ) sinc ( f rf ) (16.13) The convolution theorem can be used to investigate the actual frequency content of the pulse. An example of the e ects of truncation and apodization on an ideal eld pro le as a function of time and frequency are shown in Fig. 16.2. In Fig. 16.2a, a truncated sinc function is shown, and its associated ip angle pro le, (z), is shown in Fig. 16.2b. Figure 16.2c shows the same sinc function after apodization with a Hanning lter (see Ch. 13). The excitation pro le shown in Fig. 16.2d demonstrates the smoothing e ect which apodization introduces. It is important to reduce the ripples extending past the slice of interest as they either contribute signal from outside the slice of interest for 2D imaging or alias into the images in 3D imaging (see Ch. 13). Filtering the time domain input limits the spatial extent of the excited spins and the excitation pro le, but increases the FWHM relative to the original pro le width. Hard and Soft Pulses
A terminology has developed di erentiating between pulses which are spectrally selective and those which excite a large range of frequencies. Short duration, high amplitude pulses generally truncated with a rect function are referred to as `hard' pulses, since they are approximately an impulse in the time domain. However, their short duration in the time domain leads to a broad excitation pro le in the frequency domain. For this reason, hard pulses are also referred to as nonselective pulses. They are generally used for saturation in 386 Chapter 16. A Closer Look at Radiofrequency Pulses (a) (b) (c) (d) Fig. 16.2: (a) A sinc pulse truncated by a rect lter and (b) its corresponding ip angle pro le, which is contaminated by Gibbs ringing. (c) A smoothly truncated or apodized sinc function (a Hanning lter has been used), and its resulting response (d) is spatially smoother than that shown in (b). The di erences between (b) and (d) demonstrate how smooth truncation of an rf pulse with an apodizing function can lead to a reduction of Gibbs ringing. The ltered pro le is also broader than the original by the fraction as shown in (d). 16.1. Relating RF Fields and Measured Spin Density 387 the absence of gradients, or, for spectroscopy where the goal is to excite a large number of frequencies. Pulses which are lower in amplitude, and longer in duration may be used to select a narrow range of frequencies. Pulses which t this description are often referred to as `soft' pulses. The rf pulses employed during slice selection are examples of soft pulses.
x y -Pulses Often, the direction along which B1 points in the rotating reference frame during the application of the rf pulse is assumed to be irrelevant. However, there are applications where the relative phase of the applied B1 eld (i.e., the direction of the applied rf eld) is important to the result. In these cases, a convention must be used to describe the axis along which B1 is applied. The convention used in this text is to subscript the ip angle with the axis along which the pulse has been applied. Whenever x or y appear in a sequence diagram the reader should pay special attention to how the phase of the rf pulse a ects the result. If is given by itself, it is usually safe to assume that the rf pulse may be applied along any axis. 16.1.2 Numerical Solutions of the Bloch Equations
The arguments presented so far assume that the spins behave in a simple linear fashion so transmit doubles so does . In practice, this approximation only holds that, for example, if B1 true in the low ip angle regime. To nd out what happens for larger ip angles, the Bloch equations must be solved. In the examples to follow, a numerical method has been used to solve the Bloch equations during a slice select pulse at several ip angles. It is assumed that transmit is spatially homogeneous for these simulations and applied along the x-axis. The B1 ^ solutions obtained also include the e ects of dephasing, which occur while the slice select gradient is applied, and rephasing must be performed after the rf pulse is turned o to get an accurate description of the magnetization as it appears just prior to frequency encoding. Figure 16.3 shows the magnetization as a function of z for a low ip angle pulse which has been rephased with a gradient area equal to half of the slice select lobe. This result shows that the low ip angle approximation is accurate, since the 50% refocusing has eliminated any dephasing (My is zero), and the only remnant errors in the pro le are due to truncation of the rf pulse. Figure 16.4 shows a =2-pulse excitation pro le, after refocusing with an area of 52% which leads to the least amount of Mx for this ip angle. The presence of a nonzero Mx component introduces a z-dependent phase behavior. The choice of 51.75% refocusing leads to a at, essentially zero, Mx pro le. The need for a non 50% refocusing pulse is indicative of the nonlinearities of the Bloch equations. In order to determine the result of a large ip angle pulse, a numerical method should be employed. This simulation was also used to demonstrate the behavior of the transverse magnetization as a function of time during the rf pulse for spins on-resonance and o -resonance. The solid line in Fig. 16.5 shows M?(t) during an applied sinc pulse for an on-resonance spin. The dashed line in Fig. 16.5 shows M?(t) during the same pulse for an o -resonance spin that is at the edge of the slice (whose resonance o set is f = BW=2, where BW is 388 Chapter 16. A Closer Look at Radiofrequency Pulses the rf excitation bandwidth). Similarly, an o -resonance spin that has a frequency o set of f = BW is also shown for comparison. Fig. 16.3: Mx(z) (dashed line) and My (z) (solid line) are shown for a 5 sinc pulse, as found by numerically integrating the Bloch equations. This result validates the low ip angle approximation that a 50% slice select refocusing gradient should give a maximal response for My (z ), with a negligible contribution from Mx (z ). 16.2 Implementing Slice Selection
Instantaneously pulsed, square-gradient waveforms are not achievable in practice. In this section, symmetrical trapezoidal gradient waveforms are considered as models of realistic gradient pro les with nonzero ramp-up and ramp-down times. The sequence diagram for slice selection is shown in Fig. 16.6. It is assumed that all transverse magnetization is zero prior to the start of the rf excitation, but as a design measure, Gss is frequently applied prior to starting the rf pulse to dephase any remnant transverse magnetization as well as to stabilize the gradient. The gradient area between t0 and t2 (A + B in Fig. 16.6) is adjusted to dephase all spins across the selected slice (cf. (10.25)) Z z0+ TH 2
z0 ; TH 2 dz ei (z) = 0 (16.14) 16.2. Implementing Slice Selection 389 Fig. 16.4: Mx(z) and My (z) are shown for a 90 sinc pulse, as found by numerically integrating the Bloch equations. The area of the refocusing gradient pulse used in (a) is 50% and in (b) is 52%. The Mx (z ) component can be made at if the refocusing is chosen to be 51.75%. 390 Chapter 16. A Closer Look at Radiofrequency Pulses Fig. 16.5: M?(t) during an applied sinc rf pulse for on-resonance and o -resonance spins. Here, f is the frequency o set in Hz and BWrf is the rf excitation bandwidth. Fig. 16.6: Sequence diagram for the trapezoidal-gradient slice selection. 16.2. Implementing Slice Selection 391 where TH is the slice thickness, and it is assumed that the spin density perpendicular to the plane all have the same initial phase. The phase (z) is that accumulated by a spin at position z during the time between t0 and t2, and is linear in z, (z) = ; z Z t2
t0 dt Gss(t) (16.15) The dephasing constraint (16.14) will be satis ed if the di erence (z0 +TH=2); (z0;TH=2) is a nonzero integer multiple of 2 . The minimum area to accomplish this is thereby found to satisfy (A + B )TH = 2 (16.16) (see Ch. 10). Typically, since G0z is xed by the slice selection process, (16.15) is used to compute t2 ; t0 (see Prob. 16.2). The rf pulse is assumed to be a sinc pulse, and starts at t = t2 and ends at t = t4 . After the rf pulse is turned o , the positive lobe is ramped down to zero and a negative lobe (also of trapezoidal shape) is applied to rephase the spins. The cancellation of gradient area to rephase the spins requires D+E =F +G+H (16.17) in terms of the areas in Fig. 16.6. This equation is based upon the assumption that all spins are tipped at the exact time center of the rf pulse, and dephase during the remainder of the slice select gradient pulse, which is accurate for the low ip angle approximation (see Sec. 16.4 for further details). If the rephasing lobe is symmetric and has the same magnitude (G0z ) for its plateau amplitude, (16.17) restricts the timings of Fig. 16.6 according to (t4 ; t3) + 1 (t5 ; t4) = t8 ; t6 (16.18) 2 since (t9 ; t8 ) = (t7 ; t6 ). Note also that the rf pulse has been assumed to be symmetric about its center, which occurs at time t3 , so the gradient area from t2 to t3 is equal to the area between t3 and t4 . Problem 16.2
Given t0 = 0.0 ms, t1 = 0.6 ms, TH = 2 mm, and G0z = 23.5 mT/m, nd the minimum value of t2 such that any remnant transverse magnetization is dephased, along the slice select direction, by an integer multiple of 2 . Consider a speci c imaging example. Take rf = t4 ; t2 = 5:12 ms and let t6 = t5 (the negative lobe immediately follows the positive lobe). Assume a slice thickness TH = 2 mm, G0z = 23:5 mT/m, and a rise time of 0:6 ms. The bandwidth of the rf pulse is then BWrf = { TH G0z = 2 kHz (16.19) 392 Chapter 16. A Closer Look at Radiofrequency Pulses The rst zero crossing of the sinc pulse used to generate the slice pro le in this case occurs 1 at a time tzc = BWrf from the center of the rf pulse, or 0:5 ms. This gives the number of zero crossings in rf to be nzc = 10 which is usually considered good enough for obtaining well-de ned slice pro les, especially after apodizing the rf pulse. Problem 16.3
Given the slice select parameters G0z = 23:5 mT/m, TH = 2 mm, and 5:12 ms, answer the following questions.
rf = a) If G0z is reduced to 5 mT/m and BWrf to 400 Hz, what is the e ect on the rf pulse shape? Discuss the e ect in terms of the time of occurrence of the rst zero crossing of the sinc function and the number of zero crossings within the rf excitation time, rf . b) Assuming the same number of zero crossings are desired in the rf pulse used in (a) as occur during the given rf pulse, what would be the duration of the pulse in (a) if TH remains at 2 mm? Is your rf reasonable for a slice select rf pulse? 16.3 Calibrating the RF Field
When a body is loaded into the rf coil, the amplitude of the rf pulse must be varied until the ip angle reaches a speci c value, usually chosen to be 90 or 180 (see Fig. 16.7). Ideally, the rf transmit and receive coils have uniform spatial response. By exciting the whole body and varying the amplitude of the B1 -pulse, the system can be calibrated as just described. For example, when the rf pulse creates an angle which goes through 90 , the transverse magnetization (and, hence, the measured signal) reaches a maximum when goes through 180 , the transverse magnetization passes through zero (Fig. 16.7). For the signal maximum to occur at = 90 and the correct calibration to be done, the repeat time of the experiment must be large compared to the longitudinal relaxation times of the sample TR T1 (see Ch. 18 for more details on the e ects of non =2-pulses when TR is on the order of, or less than, T1). An initial approach to calibrate the rf amplitude might be to nd the maximum of the signal and associate this with a 90 -pulse. The problem with this approach is choosing a maximum value for the transverse magnetization (M?) in the presence of noise. Fitting M?( ) to sin also helps to reduce the errors due to noise. It is much easier to nd the signal zero near 180 where the slope of M?( ) is the largest (see Fig. 16.7). The value for B1 is then chosen for a given relative to this value. Actually, since the rf transmitter has a nonlinear response a calibration curve must be made to obtain accurate values of as a function of the input power to the system. 16.3. Calibrating the RF Field 393 (a) (b) Fig. 16.7: (a) M? as a function of with associated error bars during measurement of this function. The dashed line in (a) represents the measurements, and the solid line represents the least square t to the data. Error bars in the plot represent 2 noise standard deviations. (b) Plot of the ideal current increase (solid line) and the current amplitude (dashed line) required because of transmitter nonlinearities to generate the correct -pulse as a function of . For transmit coils which are inhomogeneous, the choice of which current amplitude or B1 amplitude applied by the system gives a zero crossing will change as the object size and spin transmit is homogeneous only over roughly the central half density change. For example, if B1 of its volume then for a small object in the center of the coil, B1 associated with the 180 pulse (referred to as B1 180 ) will be correct everywhere. If the object is large compared to the uniform region of the coil, then B1 = B1 180 will not necessarily generate a zero crossing. This is due to the fact that for spins not at the coil center, B1 at those positions will create an angle greater than (or less than) 180 even when B1 at the center equals B1 180 . Since the pick-up coil receives an integrated signal, it will take a eld at the center smaller than (or greater than) B1 180 to generate a 180 -pulse in the outer regions of the phantom. The measured value of 180 will occur when the volume integral of the transverse magnetization is zero. 16.3.1 Checking the RF Pro le
It is also necessary to calibrate or measure the e ective slice created by the rf transmit pulse, slice select gradient, and rephasing gradient. The slice pro le should be calibrated to quote the correct slice thickness. Also, when using fast imaging methods or 3D imaging, the shape of the pro le can profoundly a ect the results. As discussed earlier, there are many reasons why the rf pro le does not match the desired result of a perfect rectangular pro le. Fortunately, the rf slice select pro le can be measured by an experiment where the read direction is placed along the slice select axis. 394 Chapter 16. A Closer Look at Radiofrequency Pulses Fig. 16.8: A sequence design for measuring the rf pro le along the slice select direction. The slice select gradient is replaced using a gradient of amplitude G1z . This gradient is kept constant and
used to dephase the spins after which the spins are rephased by switching the polarity as shown in the gure. Fig. 16.9: An alternate sequence design for measuring the slice select rf pro le. A read gradient along z can be placed as shown in Fig. 16.8 or Fig. 16.9 to generate an echo along the z-direction. Notice that the rephasing lobe may be incorporated into the echo generating gradients, since the amount of dephasing during the slice select gradient is assumed to be known. Therefore, (16.20) TH = G1 T { 1z s and since G1z can be made large and Ts long, it is possible to acquire many points through the slice. Recall the convention that TH describes the thickness of the excited slice, and z refers to the imaging resolution parameter. This approach can be used for a single acquisition 16.3. Calibrating the RF Field 395 which then projects the slice pro le over all (x y), it can be run with phase encoding along y at which point the signal is projected only over x, or it can be collected in a 3D mode with phase encoding along both x and y. Problem 16.4
Assume that it is desirable in some situations to measure the rf slice select pro le by doubling the time duration of the rephase gradient lobe as shown in Fig. 16.9. a) Find the resolution z that can be achieved in this experiment. b) Relate z to the slice thickness, and the number of zero crossings in the rf pulse, assuming that a sinc pulse is used to excite the rectangular slice. c) Given that an average rf pulse is 3 ms in duration, and a slice thickness of 5 mm may be excited with G1 = 5 mT/m, nd the number of zero crossings in the rf pulse, and determine z. Does this resolution appear to be adequate relative to the slice thickness? d) Assume that an additional positive gradient lobe of area C is added to an rf slice select gradient after the rf pulse is turned o , and then a negative gradient lobe is used to create the echo. Draw a sequence diagram for this experiment, and label the areas of each gradient lobe. Is this design a more time-e cient way to generate an echo than the design shown in Fig. 16.8. Note that the gradient strength during the additional positive lobe need not equal that during the application of the rf. De ning Slice Thickness
The term slice thickness is used to de ne the thickness of the region along the slice select direction which contributes signal to an MR image. In 3D imaging, it describes the length of the region which must be partition encoded to avoid aliasing. Due to the nite time duration of the applied rf pulses, it is not possible to achieve a perfectly rectangular slice excitation, instead the region of spins a ected by the rf pulse goes smoothly to zero. As a result, it is necessary to de ne some criterion for quoting the slice thickness based upon the region where appreciable signal is contributed to the image. In general, the FWHM (Full Width Half Maximum) of the measured slice select pro le is used to de ne slice thickness as shown in Fig. 16.10. Similar to the use of the FWHM in de ning resolution, it is necessary to keep in mind that this de nition is somewhat arbitrary. For a slice whose pro le approaches the ideal rectangular case, this de nition is reasonable since very little transverse magnetization is created beyond the FWHM (see Fig. 16.10) and, therefore, the de nition is accurate. If, instead, a Gaussian excitation is chosen, then there may be an appreciable amount of signal created outside of the FWHM of the image pro le and, in this case, the FWTM (Full Width Tenth Maximum) might be a better choice for slice thickness. It is important to remember 396 Chapter 16. A Closer Look at Radiofrequency Pulses that the thickness of the image pro le is being discussed here. The FWHM of the applied rf pulse does not necessarily match the FWHM of the slice measured in the image (see Secs. 16.1 and 18.1.4). It is the image slice pro le that is of interest. Consider two cases where this might be important, 3D imaging and 2D multi-slice imaging. If FOVz is chosen to be TH in a 3D experiment then there may be a signi cant amount of aliasing in the image from the spins excited between TH=2 and TH 0=2 (see Sec. 12.3.3). In a 2D multi-slice experiment, if TH is equal to the separation between slices, then there will be `crosstalk' between slices. Crosstalk refers to the situation where the excitation of one slice, signi cantly a ects the signal from a neighboring slice. To avoid excessive crosstalk in 2D multi-slice imaging, the distance between slice centers should be set to TH 0 outside of which there is essentially no slice excitation. Fig. 16.10: Illustration of the FWHM of a theoretical slice pro le generated from an apodized sinc pulse. Although, TH = FWHM is often chosen as the e ective slice thickness, clearly information
from outside this region is excited as well. Usually, excitation is well-behaved and no further excitation takes place outside the boundaries demarcated by TH 0 =2. Problem 16.5
a) Using a graphical example, demonstrate that the FWHM of ^(z) is equal to or greater than the FWHM of the rf transmit eld B1 (~ z). Consider a r triangular position dependence for B1 (~ z) as an example. r b) Under what conditions would you expect the FWHM of B1 (~ z) to be equivr alent to that for ^(z)? 16.4. Low Flip Angle Excitation and Rephasing Gradients 397 16.4 Low Flip Angle Excitation and Rephasing Gradients
A more versatile solution to the behavior of the magnetization in the presence of an applied rf eld and gradients can be obtained for the case of low ip angle rf pulses. As a demonstration of the utility of this method, it is used to demonstrate spins dephasing across a slice during slice selection and to show that the area under the rephase pulse must be equal to half of the area under the gradient while the rf eld is on. An analysis of the Bloch equations for small tip angles lends insight into the refocusing problem and how oscillating gradients may be used to excite localized regions in 3D. The magnetization is analyzed in the rotating reference frame on-resonance for all spins. It is assumed that all o -resonance spins end up with zero transverse magnetization (see Fig. 16.5 and the associated discussion in Sec. 16.1.2). The equation of motion in the presence of gradients and an rf eld over times small compared to the relaxation times is ~ dM = M B + (G ~)^ ~ ~1 ~ r z (16.21) dt ~ If the components of the rf eld lie in the x-y plane, i.e., B1 = (B1x B1y 0), (16.21) can be rewritten as 0 1 1 0 10 ~ r dMx=dt 0 G ~ ;B1y Mx B dMy =dt C = B ;G ~ 0 B C B My C (16.22) @ A @ ~ r A 1x A @ dMz =dt Mz B1y ;B1x 0 If only small tip angles are considered such that cos ' 1 and Mz ' M0 , then the transverse and longitudinal equations can be decoupled using arguments similar to those presented in Ch. 4. The resulting equations for Mx and My can be combined by choosing M+ Mx + iMy
and (16.23) B1+ B1x + iB1y (16.24) then the equation of motion reduces to ~ r dM+ =dt = ;i (G ~)M+ + i B1+ M0 (16.25) ~ where the explicit spatial dependence of G and B1 are left as understood and Mz has been replaced by M0 in (16.25). Dephasing across a Slice
A general solution can be used to illustrate the dephasing of the spins across the slice. Assume initial conditions of Mx(; rf =2) = My (; rf =2) = 0, Mz0 = M0 , and that B1(t) is nonzero over the interval (; rf =2 rf =2). Using, for example, the integrating factor technique, M+(~ r rf ) = i M0 Z rf =2
; rf =2 k r dt B1(t)e;i2 ~ (t) ~ (16.26) 398 where Chapter 16. A Closer Look at Radiofrequency Pulses ~ (t) = { k Z
t rf =2 ~ ds G(s) (16.27) (16.28)
~r dt B1(t)ei G ~t ~ ~ For G(s) = G, a constant,
and ~ (t) = {( rf =2 ; t)G ~ k M+(~ r ~r ;i G ~ rf =2 rf =2) = i M0 e Z rf =2 = i ; rf =2 ~ ~ rf =2 ;1 M0 e;i G r F (B1 (t)) (16.29) where B1(t) is taken to be zero outside the rf window. Fig. 16.11: The rf pulse and slice select gradient waveforms (assumed to be along z). Refocusing ^ occurs thanks to the e ect of the negative lobe with amplitude Grp. Since B1 (t) is a real and symmetric function, its Fourier transform is also real and sym~ r rf metric. Therefore, the explicit phase term e;i G ~ 2 in (16.29) represents the dephasing across the slice during slice selection. This indicates a need for a `rephasing gradient' of amplitude Grp after the rf pulse so that the phase it generates cancels the phase term ~ r2 ; G ~ rf = ; G0 z z rf in the above equation. Referring to Fig. 16.11, the phase rp 2 generated at the end of the rephase gradient lobe of amplitude Grp must be
rp = Grpz ~ ^ r
rp rp = Grpz rp (16.30) (16.31) such that Grp satis es the equation Grp = G0 z rf =2 i.e., a 50% refocusing pulse should work. If the area Grp rp satis es (16.31), it brings all spins back into phase along the slice select direction, and shows if Grp = G0 z then rp = rf =2 will 16.5. Spatially Varying RF Excitation 399 lead to refocusing. This discussion expands upon the earlier discussion in Ch. 10 where the simple example of an instantaneous rf pulse at t = rf =2 was used to demonstrate that the spins saw only G0 z for rf =2. In that case, Grp was required to refocus exactly one-half of the area under G0 z . Intuitively, the factor of one-half makes sense if one views the rf pulse as instantaneously exciting the spins at t = 0. Problem 16.6
If a -pulse is being applied during a constant gradient, no refocusing is required. Describe why -pulses need no rephasing lobes. 16.5 Spatially Varying RF Excitation
It is possible to generalize the treatment of the previous section to excite an arbitrary 3D region of space. The mathematical details of these excitations, however, can be somewhat laborious. Therefore, an example of how conventional slice select pulses may be used to excite a rectangular region of space is introduced rst. Also, as a further introduction to the concepts of slice pro le excitations, the 1D example used for selecting a slice will be reintroduced. The notion of varying the time dependence of the gradients while the B1 eld is being applied is considered. Finally, an introduction to the full 3D excitation will be given, where B1 and all three gradients are applied in a time dependent fashion to achieve the nal spatial excitation pro le. The reader is reminded that although usually a gradient along one of the Cartesian axes is used for de ning the slice select direction, in general multiple gradients can be applied to generate oblique slices as discussed in Ch. 10. 16.5.1 Two-Dimensional `Beam' Excitation It is possible to excite a `beam' of spins spatially with a spin echo, instead of a slice. This is useful to avoid aliasing when a small FOV in a large object is desired. A sequence diagram for implementing a `beam' excitation is shown in Fig. 16.12. The rst rf pulse and slice select gradient excite a slice in the conventional manner. During the -pulse, a gradient lobe is placed perpendicular to the original slice select axis. Therefore, the -pulse rotates spins in a slice perpendicular to the original slice. Only the rectangular set of spins which have experienced both pulses, as shown in Fig. 16.12b will form a spin echo. The slice thickness for the /2 and -pulses in each direction, respectively, determine the corresponding smaller than usual elds-of-view in the phase encoding or partition encoding directions (see Fig. 16.12). In general, the slice select gradient associated with the -pulse is placed along the phase encoding direction to reduce the total imaging time (TT = Ly TR = y) while maintaining or improving resolution. The positioning and shape of the read gradient lobes is also important when employing this method. The slice selective -pulse acts on all spins in the x-z plane, and there will be some of these spins that undergo only a =2-rotation (see Ch. 18), and contribute to the MRI signal. Although the -pulse may be very well designed, all spins 400 Chapter 16. A Closer Look at Radiofrequency Pulses experiencing a =2-pulse in the x-z plane will contribute to the MRI signal at TE . In order to minimize their e ect on the image, additional gradients are added to the dephase and rephase gradient lobes (shown as shaded areas in Fig. 16.12). The gradient echo still occurs at TE where the gradient areas are balanced, but the additional area before data collection dephases spins tipped from the z-axis into the transverse plane by the -pulse. The additional gradient should be chosen to su ciently dephase this magnetization before data are collected in a manner similar to that described in Sec. 16.2 for eliminating unwanted magnetization prior to slice selection. The di erence is that, in this case, the goal is to ensure that the unwanted magnetization is dephased across a pixel width along the read direction at the echo, and not along the slice select direction. Problem 16.7
In certain applications, imaging of ow, for example (see Ch. 23), it is useful to have a slice excitation pro le that increases linearly in the direction perpendicular to the plane of the slice (a ramp pro le). The speci c utility of this pulse will be described in Ch. 23. Consider the following representation of the variable spatial excitation desired for B1(~ f ): r 1 z z (16.32) (z) = 0 rect TH + 1 z TH rect TH where 0 is the nal desired ip angle at the center of the excited region and 1 is the desired ip angle at the edge of the slice. Show that the inverse Fourier transform of this pro le leads to the input rf eld as a function of time given by B1 (t) = TH 0 sinc where 0 = " 1 0;i 2 0 (cos 0 ; sinc 0) TH {Gsst # (16.33) This input rf and ideal spatial response are shown in Fig. 16.13. It is seen from (16.33) that phase and amplitude control of the rf is necessary to generate a linearly varying slice pro le. Other con gurations of the gradients are possible as long as they take into account the spins excited by the -pulse and minimize their e ect on the image. This approach is even more valuable when 3D imaging must be done, but a limited eld-of-view is desired. Problem 16.8
Draw a sequence diagram designed to excite a cubical volume within a larger physical body by adding an additional -pulse and gradient to Fig. 16.12. 16.5. Spatially Varying RF Excitation 401 beam of spins is excited. The number of phase encoding lines needed for the same resolution can be reduced by exciting smaller elds-of-view in two directions. This is achieved for a spin echo sequence, for example, by exciting a plane in one direction during the 2 -pulse and by exciting a plane perpendicular to the rst plane during the -pulse. In this example, slice selection takes place along z for the 2 -pulse and along y for the -pulse. The dephasing and rephasing portions of the read gradient are usually lengthened (see shaded area along Gx R) to allow `crushing' or dephasing of any transverse magnetization which may be created by the -pulse. Fig. 16.12: Sequence diagram (a) and schematic of the resulting excitation (b) when a rectangular 402 Chapter 16. A Closer Look at Radiofrequency Pulses Problem 16.9
Due to the nonlinearities of the Bloch equations, a -pulse also creates a nite amount of transverse magnetization (see Ch. 18). For phase encoding prior to the -pulse, this unwanted transverse component always leaks into the sampling window with the same intensity. What is the e ect on a conventional 2D spin echo sequence of this extraneous signal? This artifact is often called the `zipper artifact.' Hint: Recall that the Fourier transform of a constant function is a -function. (a)
which produce the slice select pro le in (a). (b) Fig. 16.13: (a) (z) for a linearly varying rf pulse slice select experiment. (b) B1x (t) and B1y (t) 16.5.2 Time Varying Gradients and Slice Selection
It can be seen from the previous arguments that the spatial region of excitation will be altered if the applied gradient varies during the slice select process. Low ip angle imaging is of particular interest in fast imaging applications (see Ch. 18) and, in this case, reducing the slice select time is often desirable. In these cases, it would be optimal to design the rf and gradient pulse so that the rf is on throughout the application of the gradient, including ramp times for example. Alternately, with extremely powerful gradients, a trapezoidal gradient 16.5. Spatially Varying RF Excitation 403 may be exchanged for a triangular waveform. Therefore, it is desirable to understand how to deal with a time-varying gradient pulse. Before embarking on a theoretical description, a simple conceptual explanation of the problem is useful. Varying the time dependence of the gradients will not change the fact that M+(~) is proportional to the Fourier transform of B1 (~ ). However, when G(t) is a constant, r k B1 (t)dt and B1(k)dk can be related by a simple change of variables since dk=dt = {G0 z . When Gss varies as a function of time, relating B1(t)dt to B1 (k)dk is not as easy, although the basic concept is identical. The key is understanding that the quantity of interest is the amount of B1 deposited in a given region of k-space per unit time given by B1(k(t))=jdk(t)=dtj whether the gradients are time-dependent or not. This quantity, generally referred to as the `B1 weighting of k-space,' gives a representation of B1 (t) in k-space. The following arguments will be restricted to 1D since the interest is in selecting a slice. Consider the general solution for the magnetization in 1D as a generalization of (16.26) M+ ( z rf =2) = i M0 Z
; rf =2 rf =2 dt B1(t)ei2 k(t)z (16.34) and k(t) 6= constant in this case because the applied gradient will vary as a function of time. Using the appropriate Jacobian, this expression can be rewritten in terms of k(t) to recapture the Fourier transform relationship between the spatial excitation and the input B1 (t) M+ (z rf =2) = i M0 Z ( rf =2) Z k( rf =2) rf
(; =2) dt B1(t)ei2 dk w(k)ei2 k(t)z kz = i M0 where k(; rf =2) (16.35) 1 w(k(t)) = j BG((k((tt)) j { k )) (16.36) The slice select pro le is now the inverse Fourier transform of this quantity which is, as discussed earlier, the k-space weighting of B1. As long as w(k(t)) is not a multiple valued function, evaluation of (16.35) should be fairly straightforward. This corresponds to using a single positive or negative lobe for slice selection. The following problem gives an example of how this formalism can be used, and should further clarify why it is a useful tool for slice selection. 404 Chapter 16. A Closer Look at Radiofrequency Pulses Problem 16.10
Assume that in order to minimize eddy currents, and maximize the switching speed of the gradient, a cosine shape might be chosen for the slice select gradient lobe ( rf rf t Gss(t) = G0 cos rf ; 2 < t < 2 (16.37) 0 all other times Assume that a Gaussian slice select pro le is desired, i.e., M+(z) / e; B1 (t) / 1 z2 (16.38)
t= rf ))2 = 1 Show that the B1(t) which generates this M+(z) is
rf cos ( t= rf )e;( {G0 rf sin ( (16.39) (Hint: Take the limits in (16.35) to be ;1 to 1.) 16.5.3 3D Spatially Selective Excitations in the Low Flip Angle Limit
The previous subsections described how time dependent gradients and rf pulses can be used to obtain di erent spatial excitation pro les in 1D. A natural extension of these concepts involves applying all three gradients, and obtaining arbitrary volume excitation pro les. Motivation for these methods has already appeared in this chapter. Figure 16.12 demonstrates how the combination of a =2 and a -pulse can be used to excite a rectangular ~ region. In fact, this set of pulses has built into it variable B1 (t) and G(t) and helps the ~ reader get a avor of how rf and G(t) can be varied to achieve variable spatial excitation. If the times separating the pulses and gradients shown in Fig. 16.12 are reduced to zero, a single composite pulse of gradients and rf designed to excite a volume emerges. Using this example as motivation, the low ip angle solution (16.26) will be developed for selectively exciting 3D volumes. ~ As mentioned, if G(t) is not a constant while the rf is applied, then the convenient Fourier transform relationship between the excited region, and B1 (t) is lost. However, after some analysis, a Fourier transform relationship can be found between the B1 weighting of k-space as determined by the gradient waveforms and the spatially excited magnetization. The key di erence between the slice select case, and the 2D and 3D cases, is that higher dimensional spaces cannot be covered completely, and a path through k-space must be de ned. The path must cover a su cient region of k-space with an adequate density. Since the Fourier transform is used, the criteria which denote adequate coverage of the path are given by the 16.5. Spatially Varying RF Excitation same rules used for sampling k-space in imaging. Begin by rewriting (16.26) as 405 M+ (~ r rf ) = i M0 Z rf =2 Equation (16.40) can be written as an integral over a 3D -function which describes the path taken through k-space Z rf =2 Z M+(~ rf ) = i M0 r dt B1(t) d3k 3 (~ (t) ; ~ )ei2 ~ ~ k k kr (16.41) and upon interchange of integration
; ; rf =2 k r dt B1(t)ei2 ~ (t) ~ (16.40) rf =2 M+ (~ r rf ) = i M0 Z Z d3k Z
; rf =2 rf =2 dt B1(t) 3 (~ (t) ; ~ ) k k ! kr ei2 ~ ~ M+ (~ rf ) = i M0 d3k b1 (~ )ei2 ~ ~ r k kr where M+(~ rf ) is now the Fourier transform of the quantity r Z rf =2 b1 (~ ) = k dt B1(t) 3(~ (t) ; ~ ) k k
; (16.42) (16.43) where b1 (~ ) describes the B1 weighting of k-space. Therefore, although the desired form k of M+(~) may be known, and an inverse Fourier transform can be used to nd b1(k), the r speci c forms of B1 (t) and the k-space trajectory must still be determined. In order to make the k-space weighting more obvious, it is useful to multiply and divide k( ~ b1 (k) by the k-space velocity generated by the gradients, d~dtt) = {G(t), so that Z rf =2 B1(t) ~ b1 (~ ) = k dt ~ 3 (~ (t) ; ~ ) dk(t) k k dt ; rf =2 j {G(t)j Z rf =2 ~ 1 dt 3(~ (t) ; ~ ) dk(t) k k dt = B~(k) j {G(k)j ; rf =2 = w(~ )P (~ ) k k (16.44) where the rst factor explicitly demonstrates the weighting, or amount of B1 associated with a given k-space di erential B 1 (16.45) w(~ (t)) = B~(t) = ~ 1(t) k j {G(t)j jdk(t)=dtj The integrand describes the path through k-space Z k ~ ) = dt d~ (t) 3 (~ ; ~ (t)) P (k k k (16.46) dt As long as the path adequately samples k-space, it is possible to set the desired k-space weighting equal to (16.45). These fundamental relations are often used by forcing a given coverage of k-space, a spiral coverage, for example, and a given desired slice pro le in 2D or 3D, such as a Gaussian. Similar to the 1D slice select example, (16.44) is used to nd w(~ ) k from B1 (t) via (16.45). rf =2 406 Chapter 16. A Closer Look at Radiofrequency Pulses 16.6 RF Pulse Characteristics: Flip Angle and RF Power
The previous discussions on rf pulse design for slice selection or saturation have been concerned only with the optimal preparation of the magnetization for imaging. However, when dealing with rf pulses, there are several other practical aspects of the rf pulse design that should be understood. First of all, it is useful to understand how much power must be supplied by the rf ampli ers in order to achieve a given ip angle in a given time for a speci ed set of slice parameters. A closely related issue of special interest for human imaging is the fact that some of the applied rf power will be deposited in the MRI sample as heat. In human imaging, strict guidelines are set for the SAR (Speci c Absorption Rate) of rf power by the body. Therefore, it is important to understand the power and energy characteristics associated with rf pulses. In this section, symmetric pulses applied during constant gradients are again considered, and the energy and maximum ip angle associated with several potential slice select pulses are found. The power deposited in the body will also brie y be examined here. It will be found that the expressions for the energy associated with a pulse are easier to understand in the spatial domain, rather than the time domain, but since they are proportional to the integral over the square of the eld, they are equivalent. The total energy associated with an rf pulse can be found using a standard electromagnetic formula. A derivation of this formula, however, requires knowledge of Maxwell's equations and general electromagnetic theory which are beyond the scope of this text, so it is presented without proof, and the reader is directed to any college level electromagnetic text for an in-depth derivation. The total energy associated with the rf magnetic eld is the time integral over the entire pulse of the power delivered to the system as given by ! Z rf dt Z d3rB (~ t)B (~ t) W ( rf ) = 2 0 (16.47) 1 r 1 r 0 The spatial dependence of B1 is the same for each pulse, and, therefore, the spatial integral in (16.47) is only a scale factor, since it is identical for every rf pulse. The pertinent information for the MRI experiment is contained in the term Wp = dt B1(t)B1 (t) Z (16.48) where the spatial dependence is neglected because the eld is assumed to be uniform over the region of interest. This term will be referred to as `Parseval Energy' here because Parseval's theorem (Ch. 11) asserts the equivalence of the time integrated `power' in both the time and frequency domains.1 From Parseval's theorem, it can also be expressed in the frequency domain as Z Wp = df B1(f )B1 (f ) (16.49) where B1(f ) = F (B1(t)). The complex representation of the spatial magnetic eld in the two-dimensional transverse plane (Ch. 2) is employed.
Notice that, as written, this `energy' does not have the correct units. It is not a real energy in the physical sense, but represents the relevant terms from the complete energy expression and is easier to deal with logistically.
1 16.6. RF Pulse Characteristics: Flip Angle and RF Power 407 Another parameter which is important to the de nition of an rf pulse is the angle through which it rotates the magnetization. Here, the ip angle at the center of the BW or position center of the selected slice is used to characterize the ip angle associated with the applied pulse. This reference tip angle 0 now requires an integral for its de nition = dt B1(t) = {B1(f )jf =0 Recall that B1 (f ) has units of eld times time.
0 Z (16.50) 16.6.1 Analysis of Slice Selection Parameters It is useful to look at how the slice select gradient, ip angle, pulse shape, Parseval energy, and applied B1 eld are related during the slice select phase of the MRI sequence. In this discussion, an example where the slice thickness and ip angle are kept constant will be considered, and the e ects of varying other parameters will be studied. Several di erent excitation pro les will be considered here (see Fig. 16.14). One aspect of the rf pulses which is di cult to determine analytically is the e ect of truncating waveforms that are ideally in nite in time. Although these lter e ects have been discussed at length, quantifying their e ects is di cult. In general, the longer a waveform is, the closer it comes to producing the desired pro le. Therefore, in what follows, it is assumed that a characteristic time can be de ned for each pulse, and the time duration of the rf pulse is some multiple of this time so that the resulting slice pro le is a reasonable approximation of the ideal case. Sinc Pulse
Consider generating a rectangular slice pro le. This implies that a sinc pulse will be applied in the time domain. B1(t) is then B (t) = B sinc t (16.51)
1 1 sinc where sinc pulse is zc from Sec. 16.2. The thickness of the excited slice associated with this sinc (16.52) TH = BWrf = G 1 {G0 z { 0 z sinc The characteristic ip angle and Parseval energy for this pulse must also be found. The integrals (16.48) and (16.50) can be computed, for example, by complex contour integration.2 The Parseval energy is equal to Z 1 sin2( t ) 2 sinc dt = B 2 (16.53) Wp = B1 2 t2 1 sinc :
;1 2 sinc Examples of contour integration can be found in any complex analysis text (see the Suggested Reading list, for example).
2 408 Chapter 16. A Closer Look at Radiofrequency Pulses (a) (b) (c) (d) Fig. 16.14: The graphs in (a) and (b) show a Gaussian pulse and its corresponding excitation pro le. A sinc pulse appears in (c) and its corresponding excitation pro le appears in (d). Alternatively, (d) represents a rect pulse, and (c) represents its excitation pro le. 16.6. RF Pulse Characteristics: Flip Angle and RF Power 409 where it is assumed that the pulse duration is su ciently long that it can be approximated by the in nite time result. Equation (16.53) can be veri ed from the frequency domain perspective by a simple integration of (16.49)
2 Wp = B1 2 sinc BWrf 2 = B1 sinc (16.54) The central ip angle associated with this pulse is
0 = B1 sinc (16.55) Recall that the Fourier transform of the above sinc pulse is the boxcar function, B1 (f ) = B1 sincrect(f sinc), corresponding to the spread (bandwidth) of frequencies, BWrf = 1= sinc. So all the frequencies between ;1=2 sinc and 1=2 sinc are tipped by the angle 0 in (16.55). The B1 and time dependence of Parseval energy and ip angle will be the same for all pulses, but the factors which multiply this dependence also determine the e ciency of a pulse for giving a certain ip angle within a desired amount of time with minimal energy. Knowing these parameters by themselves is useful for determining the energy used by the pulse, which is proportional to the energy deposited in the body, relative to the ip angle generated. However, it is also important to understand what happens when these parameters are varied to see what exibility exists in slice select design (assuming TH and 0 remain xed). In order to minimize TE , for example, it might be desirable for a given experiment to minimize the time duration of the rf pulse. This will be accomplished if rf is xed to be a multiple of sinc and sinc is reduced (i.e., the bandwidth of the rf pulse is increased). In this case, if (16.56) sinc ! sinc and
0 and TH are kept constant, then Gss ! Gss B1 ! B1 Wp ! Wp (16.57) If equals 1=2, for example, twice the slice select gradient, the B1 amplitude, and the SAR value must all be physically available for the pulse to be used. Another crucial point is that four times as much rf power must be available from the rf ampli er, since power = energy / Wp time
rf (16.58) In general, there is an increase in power from the pulse of 1= 2. For this reason, very short time duration, larger ip angle pulses may be impossible to use because they require too much rf ampli er power. Of course, it is also possible to look at situations where other parameters are varied or xed the above is only an introduction to the topic. 410 Chapter 16. A Closer Look at Radiofrequency Pulses Problem 16.11
Using the given parameters, answer the questions listed below. Assume that a sinc rf pulse excites a BWrf = 1 kHz for a rectangular slice pro le. In order to get a smooth slice pro le with little ringing, assume that rf = 6 sinc. a) If TH = 3 mm, what value of G0 z is required? b) By what factors are G0 z , B1 and Wp changed if TH is reduced to 1 mm or increased to 30 mm, and the ip angle is kept the same? Simple Harmonic Pulse
Another example is the nite-time, rect pulse. Consider a sinusoidal pulse on-resonance with !0 and applied for a time rect with constant amplitude. In the rotating frame, (16.59) B (t) = B rect t
1 1 rect The integrations in (16.48) and (16.50) are just the reverse of the previous example, and give
2 Wp = B1 rect rect (16.60) (16.61) and
0 = B1 Gaussian Pulse
The third example is a Gaussian pulse
2 B1 (t) = B1 e;t2 =(2 gauss ) This pulse decays to e;1 of its maximum (at t = 0) by the time t = 2 and peak angle are now
2 Wp = B1 p (16.62)
gauss . The energy (16.63) Z1 ;1 2 dt e gauss = ; t2 p 2 gauss B1 and = 2 B1 gauss (16.64) respectively. Proof of these formulae are left as exercises for the reader. The results derived for the three pulse envelopes are summarized in Table (16.1).
0 p 16.7. Spin Tagging Pulse envelope, B1 (t) Tip angle Parseval energy t 2 B1 rect rect B1 rect B1 rect t 2 B1 sinc sinc p B1 sinc p B1 2 sinc 2 B1 e;t2 =2 gauss 2 B1 gauss B1 gauss Table 16.1: Summary of pulse envelope properties 411 Problem 16.12
The slice thickness associated with most pulses is given by the FWHM of their Fourier transform response. a) Find the thickness of a slice selected with the Gaussian pulse from (16.62), assuming that the slice thickness is de ned by the FWHM of the pro le of the excitation. b) Assuming that 0 and TH are xed, what would be the e ect on B1, Gss, Wp for the Gaussian pulse, if gauss ! gauss? Assume that the duration of the pulse is a xed multiple of gauss. Problem 16.13
Show by direct calculation that the Parseval energy in both domains is the same for the Gaussian pulse. 16.7 Spin Tagging
Di erent combinations of rf pulse and gradient shapes can be used to excite spins with speci c frequencies or to track the motion of a particular group of spins. These rf pulse and gradient combinations are generally referred to as spin tagging methods. In this section two special cases are examined, both of which accomplish similar goals. 16.7.1 Tagging with Gradients Applied Between RF Pulses The application of a series of rf pulses interspersed with gradients has an intriguing e ect on the signal (see Fig. 16.15). The simplest example to consider is the case of excitation by 412 Chapter 16. A Closer Look at Radiofrequency Pulses two rf pulses, where both pulses are the same except one rotates the spins about x0 and the ^ 0 other about -^ (i.e., the rf pulses are applied 180 out-of-phase with respect to each other). x For an ideal system, all spins will end up back along z . If a gradient is on between the two ^ pulses, the phase of the spins will spread out as a function of position along the direction of the applied gradient (see Fig. 16.15b). Those spins that end up along x0 , for example, will ^ not be a ected by the second =2-pulse (see Fig. 16.15c). If all of the spins that remain in the transverse plane are then dephased with additional `spoiling' gradients (see Fig. 16.15d), only those spins that were returned to the longitudinal direction by the second rf pulse will be a ected by the slice selective excitation pulse used to create the imaging plane. This results in a spatial modulation of spin density along the direction the spins were dephased in the image (see Fig. 16.15e). The exact behavior of the spin system under the rf pulse sequence ( =2)x ; ( =2)-x is considered next.3 From this point forward, the prime will be dropped and it will assumed that all calculations are done in the rotating reference frame. (a) (b) (c) (d) (e) Fig. 16.15: Tagging and slice select design for a spatially modulated spin tagging imaging sequence. The diagrams at the bottom of the gure show the behavior of the magnetization in the transverse plane. (a) shows all spins along x tipped into the transverse plane with a nonselective =2-pulse along x. (b) shows the dephasing of the spins after the application of an additional spatial tagging ^ x-gradient. (c) shows how only some of the spins are a ected by the next rf pulse along -^, as a x result of the tagging. (d) demonstrates that the spoiling gradients (those highlighted by shading) dephase all of the transverse magnetization. (e) shows that the nal excitation slice selective pulse only tips those spins that were returned to the z -axis into the transverse plane, resulting in spatially modulated tagging along the x-direction. Subsequent to this, the normal phase encoding and read gradients would appear. If the applied gradient is along the x-direction and of amplitude Gx, then the phase
transmit is directed along. axis tells which axis the B1
3 Recall the notation employed to describe these rf pulses is ( )axis where describes the ip angle, and 16.7. Spin Tagging developed over a time
x 413 is (x x) = Gx xx (16.65) Everywhere where (x x ) is a multiple of 2 (i.e., at the time of the second pulse, the spins are back where they began), the signal will be rotated back along the z direction. In general, after both pulses have been applied, the remaining transverse magnetization along the x-axis will be M?(x) = jM0 sin ( Gxx x )j (16.66) This transverse magnetization can be spoiled with additional gradient pulses, leaving the longitudinal magnetization Mz (x) = M0 cos ( Gxx x) (16.67) If an image is acquired immediately after this sequence of events, a series of dark bands (regions of zero or low signal) will appear in the image. The spacing of the dark bands occurs over a distance of 1=(2 {Gx x). The larger Gx or x , the smaller the spacing. From an imaging perspective, the location of the signal zeroes (dark bands) is a type of resolution in the image. If the desired spacing of the zeroes is m pixels, then the Nyquist relation 1 m x=2 G { x
x (16.68) becomes the appropriate constraint. The size of the gradient required to obtain this spacing relative to the read gradient can be found by using the Nyquist criterion for x and yields Gx = 21 m Ts G r
x (16.69) The thickness of the bright region of the tag xw can be estimated as the FWHM of the longitudinal magnetization in (16.67) as Gxxw x = 46
or xw = 3 2 Gx x (16.70) There will be true signal zeros in the tags only if another rf pulse is applied immediately after the -x-pulse (see Fig. 16.15). Clearly, the longer the time interval between the ( =2)-x-pulse and the -pulse, the more the spins which were left in the transverse plane after the second pulse have recovered along z and the more nonzero the magnetization in the dark regions becomes. 414 Chapter 16. A Closer Look at Radiofrequency Pulses Problem 16.14
a) If the full width at one tenth maximum is used to nd xw in (16.70) instead of the FWHM, what is the expression for xw ? b) Plot the spin density as a function of position. Discuss why the tags are not sharply de ned. c) Why is it that the tags begin to vanish if the time of application of the -pulse after the preparatory tagging pulse becomes greater than the T1 of the tissue? An example application to a phantom, with spatial modulation applied along two directions, is shown in Fig. 16.16a. The dark bands are angled here (see Prob. 16.16) and are well-de ned relative to the excited signal. (a) (b) Fig. 16.16: In (a), an image of a phantom with tagging created by applying gradients between rf
pulses is shown, and in (b) a similar image of the same phantom is shown, but now collected using multiple rf pulses for tagging. These images demonstrate that either approach can be used to tag a group of spins. 16.7. Spin Tagging 415 Problem 16.15
Draw a sequence diagram that could produce Fig. 16.16a. Hint: Notice the angle the tags make with the x- and y-axes. As has been shown rf and gradient pulses, in combination, can be used for more than just exciting a slab for 2D or 3D imaging. In this subsection, a di erent approach is taken to prepare a set of parallel tags across an object. In fact, this simple concept rst espoused for spectroscopy purposes (see the reference for Ernst and Anderson in the suggested reading) also contains the seeds for a number of rapid imaging methods which use the echo train following a xed series of rf pulses (see Ch. 18). The desired excitation is shown in the upper left corner of Fig. 16.17. It is possible to describe this excitation as the convolution of a thin rect function with a comb function ! x U x M?(x) = M0 rect x (16.71) xtags w where xw is the thickness of the tags, and xtags is the distance between their centers. In the frequency domain, assuming that f = {Gxx, these equations can be rewritten as ! ! {Gxx U {Gxx M?(f ) = M0 rect G x x {! w { !Gxxtags = M0 rect ff U f f (16.72)
w tags 16.7.2 Multiple RF and Gradient Pulses for Tagging where fw = {Gxxw and ftags = {Gxxtags . The equivalent convolution is shown in the upper right hand side of Fig. 16.17. As described throughout the text, for a constant gradient on during the rf excitations, the frequency dependence and time dependence of the pulse are related by the Fourier transform. Therefore, using Fourier transform relations, the desired rf input necessary to produce such an excitation can be found. This result is shown in the bottom of Fig. 16.17 and is given by B1 (k) = B1 u (k xtags ) sinc (k xw )
where k = {Gxt, and, therefore, in terms of time and gradient variables (16.73) B1 (t) = B1u ( {Gxtxtags ) sinc ( {Gxtxw ) (16.74) It is seen that the desired input is the product of a comb function and a sinc function. Again, simple Fourier transform concepts, particularly the convolution theorem, have been applied to understand what appears to be a complex mathematical problem. Of course, the rf and 416 Chapter 16. A Closer Look at Radiofrequency Pulses gradient pulse duration cannot last for an in nite amount of time. Assume that m hard rf pulses (the short duration delta-like pulses in (16.74)) are applied, then the total gradient and rf pulse duration time is Tpulse duration Tpd = m G 1 (16.75) { xxtags Therefore, the time or k-space domain is further ltered by the function rect(t=Tpd) rect(k=( {GxTpd)) whose Fourier transform is g(x) = sinc ( {GxTpd x) (16.76) To nd the blurring this implies for the tags, the convolution of g(x) with rect(x=xw ) must be performed. Again, after blurring, the FWHM of the tags can be used to approximate their width. Of course, the longer the duration of the gradient and rf pulse combination, the closer g(x) will come to approximating a delta function, and the less blurring will be found. An example of tagging, on a phantom, produced using this method is shown in Fig. 16.16b. 16.7.3 Summary of Tagging Applications Tagging of this type in imaging is generally used to study the motion of myocardium, CSF, or blood ow, for example. An in vivo example of the use of rf tags is shown in Fig. 16.18. In general, the tagged planes are perpendicular to the imaging plane, and the slice select excitation will alter the magnetization created by the tags, and the speci c modi cation depends upon the imaging application. An important issue is to make sure that the slice excitation does not alter the magnetization in such a way as to create or destroy magnetization preferentially in the bright or dark region, or the tags will be lost or distorted. Again, if the data are not interrogated immediately after these pulses, Mz will grow back to M0 and contrast will be lost. Problem 16.16 If a set of tags with xw = 5 mm, and xtags = 10 mm is desired and Gmax = 25 mT/m, answer the following questions. a) Using the given parameters, nd the time between the multiple rf hard pulses. b) For m = 10, nd Tpd. c) Find the FWHM of g(x) for this pulse, and compare it to xw . Based upon your answer would you expect that m = 10 is a reasonable number to use? d) Draw a sequence diagram showing the preparatory tagging pulse, the appropriate imaging gradients and where the data are sampled. 16.7. Spin Tagging 417 Fig. 16.17: A set of thin excited stripes (a) are desired in an image, to give a bright/dark pattern across the eld-of-view. The associated set of rect functions can be described as the convolution of a narrow rect function with a comb function (b). This implies that the input rf (c) should be the product of a comb function and a very broad sinc function (d). Practically, the short hard pulses represented by the arrows in part (c) are of nite duration, but we ignore that for this presentation. 418 Chapter 16. A Closer Look at Radiofrequency Pulses (a) (b) (c) (d) myocardium via the distortion of the tags using gradient encoding. As time evolves going from (a) to (d), the tags begin to lose their contrast because of longitudinal magnetization recovery. Fig. 16.18: Four frames at di erent points in the cardiac cycle showing the movement of the 16.7. Spin Tagging 419 Suggested Reading
A theory of small tip-angle approximation of viewing slice selection in k-space was developed in this article: J. Pauly, D. Nishimura, and A. Macovski. A k-space analysis of small tip angle excitation, J. Magn. Reson., 81: 43, 1989. The following book is an excellent reference on Complex Analysis. The topic of contour integration is nicely developed in this text: R. V. Churchill and J. W. Brown. Complex Variables and Applications. McGraw Hill, New York, NY, 1990. The next three articles are useful introductory references on the applications of the two approaches to spatial tagging in MRI: L. Axel and L. Dougherty. MR imaging of motion with spatial modulation of magnetization, Radiology, 171: 841, 1989. T. J. Mosher and M. B. Smith. A DANTE tagging sequence for the evaluation of translational sample motion, Magn. Reson. Med., 15: 334, 1990. E. A. Zerhouni, D. M. Parish, W. J. Rogers, A. Yang, and E. P. Shapiro. Human heart: tagging with MR imaging - a method for noninvasive assessment of myocardial motion, Radiology, 169: 59, 1988. The basic concepts of `tagging' were actually rst evident in the frequency localization described in: R. R. Ernst and W. A. Anderson. Application of Fourier transform spectroscopy to magnetic resonance. Rev. Sci. Instrum., 37: 93, 1966. 420 Chapter 16. A Closer Look at Radiofrequency Pulses Chapter 17 Water/Fat Separation Techniques
Chapter Contents
17.1 The E ect of Chemical Shift in Imaging 17.2 Selective Excitation and Tissue Nulling 17.3 Multiple Point Water/Fat Separation Methods Summary: The basic concepts of separating water and fat, or any two spectrally di erent
components, are discussed. The concept of inversion recovery is reintroduced for water/fat imaging. A description is given for the preferential nulling of speci c tissues with spectrally selective rf pulses. A class of chemical shift imaging methods to separate two spectral components is presented. Introduction
There are many parts of the body that will bene t from good water/fat separation including the optic nerve, bone marrow, the breast, the heart and knee. Further, any area being imaged by a surface coil (see Ch. 27) will have enhanced signal for tissue near the coil such as the surface layer of fat. Separating water and fat leads to two separate images with improved contrast and a reduction of artifacts caused by the interference of fat and water. Quanti cation of how much fat or water (i.e., their relative spin densities) is in a given voxel for a given tissue can also be of clinical value. This chapter addresses water/fat nulling techniques and the simplest two-point approach to water/fat separation, followed by a more complex three-point separation method to extract local eld inhomogeneity e ects. These latter methods represent a marriage of imaging and spectroscopic techniques. 17.1 The E ect of Chemical Shift in Imaging
Even for a perfectly uniform external static eld, local elds vary at the molecular level. For example, the protons in water (H2O) see a di erent eld from those in a lipid-based 421 422 Chapter 17. Water/Fat Separation Techniques or fatty compound (which contain CH2 and CH3 ). The former represents the `water' signal (from water-bearing material or tissue) while the latter represents the `fat' signal. The fat is shifted to a lower frequency (see Fig. 17.1) so that the di erence between their precession frequencies ffw is given by ffw ff ; fw = ; fw {B0 (17.1) where the su x w stands for water, f for fat, fw (a positive quantity in this case) is the chemical shift between water and fat expressed as a fraction of the eld B0, and ffw refers to the frequency shift of fat relative to water.1 The subscripts w and f denote water and fat in (17.1), but this equation can describe the chemical shift for any two compounds containing the same MR nucleus and the methods outlined in this chapter will work equally well. Most fat in the human body has fw = 3.35 ppm (= 3:35 10;6), which leads to a frequency shift of 214 Hz at 1.5 T (see Fig. 17.1a). For the remainder of this chapter, the focus will be on imaging objects containing only water and fat. The spin density of water 0 w can be greater than the spin density of fat 0 f in healthy tissue. However, the voxel signal from fat for a short-TR experiment can be signi cantly larger than the signal from water since fat has a small T1 relative to most other tissues. Consider a fast, short-TR , T1 -weighted experiment where TR = 40 ms (such as in Fig. 17.1b) and the ip angle is set to 90 . The signal from water will be suppressed by TR =T1w , and from fat by TR =T1f , i.e., the water/fat image intensity ratio is reduced by T1f =T1w . Since T1f is so short, this ratio can be 1/3 to 1/5, leading to a signi cant suppression of the water signal relative to fat (see, for example, Fig. 17.1b). This problem of high signal from fat is further exacerbated when a surface coil is used for signal reception because of the proximity of fat to the coil (see Ch. 27). 17.1.1 Fat Shift Artifact
The fat misregistration will manifest along a frequency encoding direction (read or slice select) since fat is shifted in frequency relative to water.2 For example, in the rotating frame for water, the precession frequency of fatty tissue in the presence of a read gradient GRx is ^ ff (x) = {GR x + ffw (x) (17.2) This shift in frequency from the expected value creates a problem in the resulting image if the frequency spread per voxel (or bandwidth per voxel, fvoxel ) is not much greater than ffw . In the read and slice select directions, where a spin's position is frequency encoded, this frequency di erence causes a `spatial misregistration' of spins belonging to fatty tissue.
Note that this conventional choice of de ning chemical shift implies that a positive chemical shift is associated with a lower frequency for fat relative to water. It is typical in NMR spectroscopy to quote all 1 H resonance peaks relative to tetra-methyl silane or TMS. TMS is chosen because it is highly diamagnetic, and also has a 13 C peak. All biologically common substances, including water, are slightly paramagnetic relative to TMS and, therefore, all other spectral peaks are higher in frequency than this. For example, water is approximately 4.69 ppm higher in frequency than TMS. 2 The situation is more complicated than this when echo planar imaging is used (see Ch. 19).
1 17.1. The E ect of Chemical Shift in Imaging 423 For example, when the voxel size x = 1 mm and GR = 10 mT/m, fvoxel = 426 Hz, recalling that fvoxel = {GR x (17.3) The ratio of the chemical shift frequency to the frequency content per voxel gives the fractional number of voxels the fat is shifted as Fig. 17.1: Water and fat frequency components obtained after Fourier transformation of an FID from the marrow of a volunteer. (a) when TR = 1 and (b) when TR = 40 ms. Water in tissue has both a longer T2 and a narrower spectral peak than fat. Contrary to spectroscopic convention,
frequency is shown here increasing from left to right. 424 Chapter 17. Water/Fat Separation Techniques (17.4) (17.5) Nshift = fffw voxel Hence, the actual physical distance the fat is shifted in the image is xshift = Nshift x Consider the example where both water and fat are both present in a voxel. Further assume they appear only at the center of the voxel. When the read gradient is su ciently large so that Nshift < 0:5, then all of the fat signal will remain within the same voxel as the water (Fig. 17.2a). For lower gradients, Nshift increases and the fat signal begins invading the neighboring voxels (Figs. 17.2b and 17.2c). Equation (17.5) tells us that the lower the bandwidth per voxel, the worse the spatial misregistration artifact, and the higher the bandwidth per voxel, the less severe the fat shift.3 In this example then, only when the bandwidth/voxel is greater than 2 ffw = 428 Hz does the fat from the center of the voxel get mapped within the right voxel, as shown in Fig. 17.2a for the case when the BW /voxel is 500 Hz. Figure 17.3 demonstrates how the spin density is altered for the the example where fat and water are uniformly distributed throughout several voxels, and the fat is shifted by 3 voxels relative to water. These idealistic viewpoints must be tempered by the knowledge that each voxel is blurred by the point spread function, and the mix of fat and water in each voxel is unknown. Problem 17.1 Assume that a voxel centered at x0 with width x contains both water and fat uniformly distributed throughout the voxel. a) How large must the read gradient GR be so that 80% of the fat lies within the same voxel as the water when x = 1 mm? Assume that all the fat sits at one frequency with fw = 3:35 ppm and B0 = 1:0 T. b) In what direction along x is fat shifted? As an example of the shift artifact, consider a sagittal slice through the leg. The direction of the fat shift will depend on the sign of the read gradient. Figure 17.4a shows an image obtained using a positive read gradient and Fig. 17.4b shows an image acquired with a negative read gradient. The regions where the fat overlaps water are very bright. Regions where fat has shifted away from the tissue now have a black border or interface (illustrated by the arrows in Figs. 17.4c and 17.4d). For example, the regions that contain blood vessels now look like black holes in the shifted fat because their signal did not get displaced. Other artifacts exist due to the changing phase of the fat signal relative to water, but this will be covered in Sec. 17.3. To minimize the fat shift artifact it is necessary to use a high BW which
3 For a detailed discussion of this and other such eld inhomogeneity e ects, see Ch. 20. 17.1. The E ect of Chemical Shift in Imaging 425 that a set of samples (acting like point objects) containing a mix of water and fat are positioned at the center of each voxel. The frequency di erence between the water and fat components creates a spatial shift of the fat component relative to the water component in the image. Two adjacent voxels are shown, each of which contains water and fat. The voxels are large in size relative to the FWHM of the water or fat spectra. All blurring due to nite sampling is ignored in this example. Assuming that a 1.5 T eld is being used for imaging, the following cases are shown: (a) when the BW /voxel is 500 Hz, (b) if the BW /voxel is changed to 214 Hz, exactly the frequency di erence between water and fat, the fat signal is shifted by exactly one voxel, (c) a BW /voxel of 172 Hz. Fig. 17.2: A 1D example of the shift of fat relative to water in an image is shown. It is assumed 426 Chapter 17. Water/Fat Separation Techniques leads to reduced SNR. When a higher SNR is desired and a lower BW is used, methods to eliminate the fat signal (and, hence, fat shift artifacts) are often sought. (a) (b) Fig. 17.3: The top of (a) shows a 1D physical spin density where fat is surrounded by some water- bearing tissue. The top of (b) shows where the fat and water will be spatially encoded as a function of their frequency. The fat is assumed to shift three voxels to the left. The lower portion of each gure represents the amplitude which will be seen in the image. The fat is assumed to have a lower signal than the water. (b) demonstrates the e ect on the resulting image. The x-axis is marked in intervals of the voxel size x. Problem 17.2
Sketch a gure similar to Fig. 17.3, but in the slice select direction showing which regions of water and fat are excited when a conventional rf slice selective excitation is applied. Assume that no rf pulses have been applied prior to the excitation pulse, and that fat and water are uniformly mixed within the region being excited. Label the distance between the center of the excited fat and water regions in terms of the slice select gradient, and chemical shift between fat and water. 17.1. The E ect of Chemical Shift in Imaging 427 (a) (b) (c) (d) Fig. 17.4: Sagittal images across the human knee obtained with a gradient echo sequence obtained at an echo time where water and fat are in-phase (see for de nition, Sec. 17.3.1). In (a), the image was obtained with a positive read gradient and in (b), with a negative read gradient. Notice that the fat has shifted in the opposite direction in the read direction (up/down direction in the image) between the two cases, which is clearly appreciable in the zoomed areas around the knee joint of (a) and (b), shown in (c) and (d), respectively. The read gradient used here gives a BW /voxel of 325 Hz leading to a 0.68 voxel shift. 428 Chapter 17. Water/Fat Separation Techniques 17.2 Selective Excitation and Tissue Nulling
A number of unique approaches to eliminate signal from either water or fat have been developed. In this section, the concepts of selectively saturating a tissue or selectively nulling a tissue are introduced. A third approach to simultaneously extract information about both tissues is presented in the following section. Each method has its own advantages and disadvantages. 17.2.1 Selective Excitation and Saturation
Fat and water have di erent Larmor frequencies, therefore, in a perfectly homogeneous eld, a su ciently narrow band rf pulse can be used to tip either species into the transverse plane. If such a pulse is used to excite fat, for example, then a conventional slice select rf/gradient pulse combination applied shortly thereafter will only tip the water's magnetization into the transverse plane (Fig. 17.5). Since the fat has just been excited, its longitudinal magnetization has not had time to regrow and there is no fat component to tip into the transverse plane. Hence, the resulting signal measurement should be primarily from water. It is said that the rst rf pulse has saturated the signal from fat (see also Ch. 16), and this pulse is referred to as a fat saturation pulse. Of course, to image just fat, the roles of water and fat can be reversed. Fig. 17.5: The rf pulse sequence used for imaging one species (say, water). A spectrally selective, fat excitation pulse (90 ) is followed immediately (after all transverse magnetization is spoiled) by a spatially-selective pulse to excite water and at the same time saturate fat. The fat signal is dephased prior to water excitation by the spoiling lobe (shown as the shaded region in the gure). Depending on the time between the =2 and -pulses, the =2-pulse is usually made somewhat larger in ip angle angle to account for fat regrowth at the -pulse. The di culty of this method is that poor eld homogeneity will cause the fat selective pulse to excite water instead if { B = ffw . In that case, there will be regions where the water rather than the fat will be saturated. Although static magnetic elds are more homogeneous today, alternate approaches to separate water and fat that succeed independent of the presence of B0 are desirable. 17.2. Selective Excitation and Tissue Nulling 429 17.2.2 Tissue Nulling with Inversion Recovery
One method of eliminating signal from fat is to take advantage of the di erences between T1 tissue relaxation times by using an inversion recovery sequence. The e ectiveness of this approach will not depend upon eld homogeneity. By inverting the fat and water longitudinal magnetization and then collecting the data at the zero crossing of the fat signal (Fig. 17.6), the fat signal will be nulled independent of static eld variations. (Although, the inversion pulse does have some dependence on the local eld.) The water component of most tissues has a long T1 relative to fat so that for short inversion time TI most of the tissue signal remains large and negative. This inversion recovery sequence (see Ch. 8 and 26) needs a long TR to allow all tissues to return to M0 before the next inversion pulse so that the acquisition time is rather long. It is also ine cient because fewer slices are acquired than with a conventional spin echo scan. Nevertheless, with the development of echo planar imaging where the entire brain can be covered in a few seconds, inversion recovery applications become more feasible (see Ch. 19). A comparison between fat saturation and fat nulling using an IR pulse is shown in Fig. 17.7. The former fails behind the knee where the local eld inhomogeneity causes a signi cant shift in background frequency, while the latter is independent of background eld. Fig. 17.6: Inversion recovery signal behavior and the choice of the recovery period to null a speci c tissue. If data can be collected at the zero crossing (null point) of fat TI = tnull f or water TI = tnull w that particular tissue can be suppressed from the image. In this gure, fat and muscle (with essentially no fat component) signals are plotted for an IR sequence using a 1 s TR . From Ch. 8, the null point for tissues in the nite TR case is given by TI null = T1 ln (1+e 2 R =T1 ) . For TR T much greater than T1 , TI null is usually 150 to 170 ms for fat at 1.5 T.
; 430 Chapter 17. Water/Fat Separation Techniques (a) (b) (c) (d) Fig. 17.7: (a) Fat nulled image obtained using an inversion time (TI ) of 150 ms. (b) Water-nulled image obtained using a TI of 300 ms. Both IR images were acquired with a TR of 1 sec. As seen from the curves plotted in Fig. 17.6, tnull w for water in the muscle is about 300 ms. This is what is observed in (b). In (c) and (d), a frequency selective saturation pulse was used. (c) Fat saturated image, and (d) water-saturated image. In (c) it is apparent that the static eld is not uniform behind the knee (arrow) as some of the fat has not been fully saturated (compare with image (a) see also the phase in Fig. 17.15b in the same region). 17.3. Multiple Point Water/Fat Separation Methods 431 Problem 17.3
The above discussion suggests the possibility of nulling two tissues if water, fat, and silicon are present, for example. Discuss how you might null water and fat to image silicon using both inversion recovery and saturation techniques. Assume that the chemical shift of silicon relative to water is 4.69 ppm (i.e., it is shifted in the direction of fat, just further along the frequency axis) and its T1 is the same as the water containing tissue but is much longer than that of fat. 17.3 Multiple Point Water/Fat Separation Methods
Chemical shift imaging as described in Ch. 10 is a long spectral phase encoding procedure designed for the situation where no, or little, a priori knowledge exists about the spectrum. When only two hydrogen carrying compounds with known chemical shifts are present, the imaging scheme to acquire the requisite information to separate two spectral components is much simpli ed. For the water/fat separation techniques of this section, each voxel is modeled using a simple partial volume model (see Ch. 15 for further details) containing some unknown fractional contributions to the voxel signal from water and fat. The fact that the fat signal in a given voxel may have come from a di erent spatial location becomes irrelevant in this section, since the goal is to separate the water and fat signals. The water/fat separation process determines the relative signal contribution from water and fat, respectively, to each voxel. In this section, modi cations of the sequence design are used to collect two or three images with di erent phase information that can be manipulated to nd separate water and fat images. 17.3.1 Gradient Echo Sequence for Water/Fat Separation
For a simple gradient echo sequence, the complex image as a function of time in the presence of water and fat will behave as ^(TE ) = ^w (TE ) + ^f (TE ) (17.6) where ^(TE ) represents the complex image obtained at some chosen TE value, which contains unknown contributions from water and fat. It is assumed that the demodulation frequency is set to the water resonance frequency. In the rst few introductory subsections (Sec. 17.3.1 to 17.3.4), it is also assumed that the static eld is homogeneous throughout the imaged volume. Using these assumptions it is useful to look at the phase of fat and water during 432
w (t0 ) Chapter 17. Water/Fat Separation Techniques = = the sampling period4 to understand the e ect of the frequency shift for a speci c echo time frequency encoding ;2 {Gxxt0 ;2{zkxx | } (17.7) = ;2 { (Gxx + Bwf ) t0 + ;2 { Bwf TE = ;2 kx (x + Bwf =Gx) + ; !wf TE} (17.8) | {z } | {z frequency encoding additional phase where the di erence between the frequency encoding terms in (17.7) and (17.8) leads to the frequency encoded fat shift discussed in the previous section. In this section, methods to use the additional phase associated with fat to separate the fat and water signals will be presented. Notice that even if the phase information can be used to separate the fat and water signals the fat image will still be spatially shifted along the frequency encoding direction.5 Equations (17.7) and (17.8) imply that ^w is real, while ^f is complex, having a TE dependent phase. The complex voxel signal (17.6) can be rewritten as
f (t0 ) and ^(TE ) = ^w m + ^f m e;i !fw TE (17.9) The subscript m is used to denote the magnitude of the image. In this discussion of water/fat separation, the key issue is the di erence in phase generated by the frequency di erence between water and fat as a function of TE . By acquiring images at di erent values of TE , the phase information can be used to separate the fat and water signals. Equation (17.9) leads to a beat frequency pattern (see Fig. 17.8, for example) where the water and fat spins are out-of-phase (`opposed phase') at a time when !fw TE op(n) = (2n + 1)
and in-phase at a time when (17.10) !fw TE in(n) = 2n (17.11) where n is an integer `op' is used as a short-form for opposed phase, and `in' for in-phase. The n = 0 in-phase condition is achieved with an FID sampling method or, in a symmetric spin echo imaging case, where both water and fat are in-phase at the same time as the occurence of the k = 0 point. For later use, de ne !fw TE 90(n) = 2n + =2
(17.12) For the water/fat case, TE op(0) = 2 1fw ' 2:335 ms and TE in(1) ' 4:67 ms at 1.5 T. Di erent f fat components in the body may have di erent chemical shifts care should be taken to determine the chemical shift for the region-of-interest. 4 Recall that t = t ; T where t = 0 at the center of the rf pulse. E
0 Chapter 20 contains a more in-depth treatment of the relationship between phase variations during sampling, and associated image e ects.
5 17.3. Multiple Point Water/Fat Separation Methods 433 Problem 17.4
a) Rewrite (17.9) including T2 e ects. b) Replot Fig. 17.8 for the main static eld equal to 0.5 T and 0.1 T assuming T2 has not changed. c) How do the curves in (b) scale as a function of eld strength? d) Replot Fig. 17.8 when T2w = 100 ms and T2f = 60 ms. e) Plot ^(TE ) as a function of = !fw TE for w = f . Assume the same values of T2 given in part (d), and B0 = 0.5 T. Problem 17.5
For a gradient echo sequence, (17.11) is valid only for n 0. Show that no such restriction exists for a spin echo sequence, but that some negative values of n are also accessible to experiment. Fig. 17.8: The beating envelope of the combined water and fat signals as a function of time with f = 0:2 w . The T2 of water and fat are assumed equal to 80 ms. The markings on the time scale
are of special importance in that they represent the times at which water and fat seeing the same eld are in-phase at 1.445 T. Note that T1 e ects are neglected, i.e., an in nite TR is assumed. From (17.9), the phase of the fat signal contribution as a function of TE plays a very important role in determining the summed complex voxel signal. The phase of the voxel 434 Chapter 17. Water/Fat Separation Techniques signal, on the other hand, additionally depends on the relative signal contributions from the two species. Fig. 17.8 is an illustration of the time progression of the combined water/fat signal. The fat signal contribution to the echo is seen to pass through a progression of phase values relative to water as echo time increases. As seen earlier, the phase plays a key role in water/fat separation. What role does the TE -dependent phase of the fat play in a magnitude image? First, we formally de ne the partial volume model for the voxel signal. If f is the relative strength of the fat signal contribution to the voxel signal in magnitude,6 i.e.,
f ^f m ^f m + ^w m
!fw TE ) (17.13) then (17.9) can be rewritten as ^(TE ) = (^w m + ^f m )((1 ; f ) + f e;i In the case when !fw TE equals , the voxel signal is ^(TE ) = (^w m + ^f m )(1 ; 2 f ) out-of-phase case (17.15) (17.14) Clearly, when a fraction of the voxel signal comes from fat, and fat is out-of-phase relative to water, the fat fraction f exactly cancels o an equal fraction ( f ) of water, leading to a signal-contributing fraction of (1 ; 2 f ), as demonstrated by (17.15). From (17.9), the magnitude of the voxel signal as a function of echo time TE is j ^(TE )j = (^w m + ^f m cos !fw TE )2 + ^2 m sin2 !fw TE f
= ^2 m + ^2 m + 2^w m ^f m cos !fw TE w f
1=2 1= 2 (17.16) where the T2 decay has been ignored. Fig. 17.9 shows the spin behavior at three di erent phase angles for water and fat spins. The main theme of this illustration is that water and fat are two di erent phasors, and their sum is determined by a phasor sum, not a simple scalar sum. It also reminds us that (17.16) is nothing but the magnitude of the vector sum of two vectors which are separated by an angular di erence of !fw TE . This magnitude signal is plotted as a function of the volume fraction of fat in Fig. 17.10. When both water and fat occupy the same voxel, and fat is out of phase, signal cancellation will occur. Speci cally, when f = 0.5, total signal cancellation occurs. This almost always happens when the fat shifts more than a voxel away from its actual physical location in an out-of-phase image, since there is likely to be water at the voxel where the fat ends up.
Here, f is de ned as the relative strength of the fat signal contribution to the voxel signal in the spirit that, although it appears to be the case, f is not the fractional voxel spin density it is only the fractional signal contribution, determined also by the T1 and T2 weighting which are, in turn, in uenced by the choice of TR and TE .
6 17.3. Multiple Point Water/Fat Separation Methods 435 w
(a) f w
(b) f w f
(c) Fig. 17.9: Phasor diagrams representing water and fat spins when they are (a) in-phase, (b) 90 out-of-phase, and (c) 180 out-of-phase at TE in(n), TE 90 (n) and TE op(n), respectively. Fig. 17.10: Plot of the magnitude of the signal from a voxel containing both water and fat as a function of the fraction of fat signal for an opposed-phase image (solid line). There is total cancellation of signal when f = 0:5. This lack of signal shows as a black line artifact at fat/water interfaces in opposed phase images and is observed in a number of the images in this chapter. Shown in dotted lines (for comparison purposes) are the voxel signal magnitudes of voxels containing only water or only fat, occupying volume fractions (1 ; f ) and f , respectively, in these voxels. Problem 17.6
Show that (17.16) becomes a) j ^(TE )j = (^w m + ^f m ) for !fw TE = 0, and b) j ^(TE )j = j ^w m ; ^f m j for !fw TE = . 436 Chapter 17. Water/Fat Separation Techniques 17.3.2 Single-Echo Separation For a homogeneous eld, two pieces of information, the real and the imaginary part, can be obtained from a complex data set. Assume spins start in-phase at TE = 0 (Fig. 17.9) and end up 90 out-of-phase at TE 90(0) = 1.168 ms at 1.5 T (from (17.12)). Collecting an image at TE 90(0) gives ^(TE 90(0)) = ^w m + i ^f m (17.17) So, the real part of the voxel signal represents the water fraction, while the imaginary part of the voxel signal represents that of fat. Since sampling does take a nite amount of time, it is not usually possible to obtain a TE so short that TE = TE 90(0) when a high resolution image is also desired. The next available echo times where the water signal is real and the fat signal is imaginary occur at the 270 point before the rst in-phase point (TE = 3.503 ms at 1.5 T), or the rst 90 point after the rst in-phase point (TE = TE 90(1) = 5.838 ms). For inhomogeneous elds and imaging at 1.5 T with a TE = 4.67 ms = TE in(1), the in-phase complex image is given by ^in(n = 1) = (^w m + ^f m )e;i = (^w m + ^f m )e;i
BTE in (1)
B Complex In-Phase and 90 Out-of-Phase Method (in-phase) (17.18) 5 while the 90 out-of-phase complex image, obtained at an echo time of TE 90(1) = 4 TE in(1) is given by ^90 (n = 1) = (^w m + ^f me;i(5=4)TE in (1) = (^w m + i ^f m )e;i(5=4) B !fw )e;i(5=4) BTE in (1) (90 out-of-phase) (17.19) where B = BTE in(1) is the phase gained due to the presence of magnetic eld inhomogeneities for the in-phase image. The (5=4) !f TE in(1) term comes from the fact that the second echo is acquired at the (2 + =2) relative phase value between water and fat which is at the time 5=4TE in(1) of the in-phase image. The phase shift B can be found from the 5 rst image. This can then be used to remove the ; 4 B phase term from the second image and then the real and imaginary channels used to obtain water and fat images separate from each other. The di culty here occurs when the phase due to eld inhomogeneities aliases (because phase is computed only over the range ; ], any value of phase above wraps back to a negative value and it is not possible to correctly predict the correction - 5 B ). Another 4 problem can be the presence of a TE -independent phase shift ei 0 where 0 is positiondependent (this can happen due to rf penetration and the nite, nonzero conductivity of the tissue). These di culties and other practical issues arising with this method are illustrated in the examples shown in Figs. 17.11 and 17.12. 17.3. Multiple Point Water/Fat Separation Methods 437 (a) (b) (c) (d) Fig. 17.11: Complex images used for the complex in-phase and 90 out-of-phase method. (a) In-phase magnitude image, (b) in-phase phase image, (c) 90 out-of-phase magnitude image, and (d) 90 out-of-phase phase image. The read direction is cranio-caudal (the vertical direction). The phase of water and fat are not equal in part (b) because of rf penetration e ects this di erence is referred to as 0 in the text. 438 Chapter 17. Water/Fat Separation Techniques (a) (b) (c) (d) Fig. 17.12: Steps involved in extracting water and fat images from the 90 method are: unwrapping the in-phase phase image and using that to correct the 90 out-of-phase phase image for B e ects,
and then displaying the real part as the water image and imaginary part as the fat image. (a) Partially unwrapped in-phase phase image. (b) Attempted correction of 90 out-of-phase phase image. (c) Water image. (d) Fat image. The incorrect separation at the top part of the leg occurs because of a large phase wrap (caused by the rapid eld variations away from the center of the magnet) that cannot be corrected using only a single phase image for unwrapping. 17.3. Multiple Point Water/Fat Separation Methods 439 Problem 17.7
As discussed, fat appearing in one voxel may actually come from another location. Show for a gradient echo experiment that the complete expression for the spin density is (~) = ^w (~)e;i r r with
B(~)TE r + ^f (~f )e;i ( r B; f B0 )TE (17.20) ~f = ~ ; fw B0 x r r G ^ (17.21) R The shift from ~ to ~f is exactly the same e ect that typically causes the spatial r r shifting of the fat observed in Fig. 17.4 as the BW /voxel changes. The high values of GR generally used limit spatial shifts to a few voxels so that the assumption made by the three-point method in the next section that both water and fat experience the same 0 and B is valid. Fig. 17.13: The upper rf trace (a) represents conventional rf pulse timings relative to GR , while
the lower rf trace (b) shows a shifted -pulse timing. Shifting the -pulse by an appropriate time without changing the timing of the read gradient can be used to produce easily the rst =2 or out-of-phase point at the time 2 . This time can be made arbitrarily short if eld inhomogeneities are large. 17.3.3 Spin Echo Approach One problem with the gradient echo approach is that the image can be acquired only for n > 0 (see (17.10) and (17.11)), whether an in-phase image, a 90 out-of-phase image, or an opposed-phase image is required. This causes a problem when either a single-point (see 440 Chapter 17. Water/Fat Separation Techniques Sec. 17.3.2) or two-point (see Sec. 17.3.4) water/fat separation is performed in the presence of large local eld inhomogeneities. This problem can be overcome when a spin echo sequence is used and the -pulse is shifted toward the =2-pulse by a time (Fig. 17.13). In the presence of a eld inhomogeneity B , the rf echo is shifted by t = 2 (see Prob. 17.8) relative to the instant when the k = 0 sample occurs. This di erence in times between the rf echo7 and the gradient echo acts like a gradient echo TE since fat magnetization builds up phase relative to water starting from this time-point. Setting 2 = 1.168 ms produces the rst 90 out-of-phase water/fat image. Setting 2 = 2.336 ms produces the rst opposed-phase image. Problem 17.8
For the asymmetric spin echo sequence (acquired by shifting the -pulse earlier in time by seconds in a sequence designed to acquire an image with a symmetric spin echo) in Fig. 17.13, show that the phase during the ADC data collection window is given for water and fat, respectively, by
w (t ) 0 f (t )
0 = ; = ; B (t0 + 2 ) ; GR xt0 B (t0 + 2 ) + fw B0 (t0 + 2 ) ; GR xt0 where t0 is the time from the center of the readout where k = 0 is sampled. Clearly, when t0 = 0, (t0) equals ;2 B for water, and ;2 ( B ; fw B0) for fat, assuming the same eld inhomogeneities are experienced by both the water and fat fractions. Although the upcoming subsections focus on the gradient echo sequence, this spin echo approach can always be used instead. An obvious approach to separating water and fat is to use just two complex gradient echo images one an in-phase image (obtained at a TE = TE in(n)), and another, an opposed-phase image (TE = TE op(n)). The complex voxel signals in these two cases in the absence of any eld inhomogeneities are: ^in (n) = ^w m + ^f m (17.22) ^op(n) = ^w m ; ^f m (17.23) Hence, a rst attempt to obtain water and fat images would be to use the simple algorithm 1 ^w m = 2 (^in + ^op ) (17.24) (17.25) ^f m = 1 (^in ; ^op) 2
The rf echo is said to occur at that time-point when all static eld inhomogeneities, including chemical shifts, are refocused.
7 17.3.4 Two-Point Separation 17.3. Multiple Point Water/Fat Separation Methods 441 Examples of in-phase and opposed-phase magnitude images and their corresponding phase images are shown in Fig. 17.14. T2 e ects have been ignored to this point. Since ^in(n) and ^op(n) are acquired at di erent echo times, (17.22) and (17.23) are not strictly valid, especially for tissues with short T2 values such as bone marrow. To reduce possible errors due to T2 e ects, a third image can be collected to help obtain an in-phase image with approximately the same T2 weighting as the opposed-phase image. A third image ^in (n + 1) is collected and averaged with the rst in-phase image (two in-phase images surrounding the out-of-phase image in the sequence of three images) to create a new ^in, giving both in-phase and opposed-phase images for the calculations in (17.24) and (17.25) with roughly the same T2 weighting (see Fig. 17.15). Problem 17.9
Show that when (TE in(n + 1) ; TE in(n))=T2 << 1 for both fat and water, the last statement in the text is valid. In other words, if ^in (n) is replaced by 1 2 (^in (n) + ^in (n + 1)) in (17.24) and (17.25), errors due to T2 signal changes are removed. Problem 17.10
Assuming perfect magnetic eld conditions, if two gradient echo images are collected at di erent echo times, the resultant magnitude images can be reworked to give a water image and a fat image. a) To eliminate the e ects of any overall phase or systematic errors, magnitude images are often used. Suppose two magnitude images are collected at TE in(n) and TE op(n). Write expressions for j ^inj and j ^opj in terms of w and f . b) As a naive approach, it is quite possible to attempt to separate water and fat from the magnitude images using the same sum and di erence approach applied to a pair of in-phase and opposed-phase images. What do the sum ^sum and the di erence j ^inj + j ^opj ^diff j ^inj ; j ^opj of these two magnitude images yield? Speci cally, show that when it is not known whether ^f m or ^w m is the larger of the two, there is an ambiguity as to which image, ^sum or ^diff represents fat or water. 442 Chapter 17. Water/Fat Separation Techniques (a) (b) (c) (d) Fig. 17.14: Gradient echo images collected for complex two-point water/fat separation at 1.5 T. These two images were collected at TE values of 4.5 ms and 6.75 ms, respectively. (a) In-phase magnitude image and (b) in-phase phase image. (c) Opposed-phase magnitude image, and (d) opposed-phase phase image. Note the cancellation artifacts (arrows) at the fat/water boundaries in the magnitude image shown in (c) (see the discussion in Sec. 17.3.1 concerning the e ect illustrated in Fig. 17.4). 17.3. Multiple Point Water/Fat Separation Methods 443 (a) (b) (c) (d) Fig. 17.15: Use of a second in-phase image to overcome T2 -weighting di erences between the in- phase and opposed-phase images of Fig. 17.14. (a) Second in-phase magnitude image obtained at a TE of 9 ms, and (b) corresponding phase image. (c) The averaged magnitude image obtained from the two in-phase images. (d) The opposed-phase magnitude image, shown again for comparison purposes. Images (c) and (d) can be used to obtain separate water and fat images using the twopoint complex separation method. The in-phase phase image (b) shows some rapid and signi cant variation behind the knee and in the upper and lower regions of the FOV. The former is caused by the knee itself, the latter can be corrected for, if the background static eld variation is known. 444 Chapter 17. Water/Fat Separation Techniques 17.3.5 Three-Point Separation In practice, the static eld is not perfectly homogeneous, and this can lead to a reversal of water/fat roles again as in the magnitude approach, even for the complex method. Assuming that both water and fat are reconstructed within a voxel and experience the same 0 and B , then ^ = (^w + ^f e;i !fw TE )e;i BTE e;i 0 (17.26) where 0 is the contribution from an arbitrary global phase shift a ecting all voxels and can also be caused by a spatially-varying rf penetration or tissue conductivity. Acquiring two in-phase scans (^in(n) ^in(n + 1)) and one opposed-phase scan (^op(n)) makes it possible to nd ^w ^f B0 and 0. Recall that ^(x) is complex and can be written as ^(x) = j ^(x)jei
(x) (17.27) To avoid the fat's confounding e ect on phase, B is determined via the two in-phase images
in (n + 1) ; in (n) = ;( BTE in(n + 1) + 0 ) ; ( BTE in(n) + 0) = ; B (TE in(n + 1) ; TE in(n)) (17.28)
0 and = ; in(n) + BTE in(n) (17.29) The next step is to correct the out-of-phase complex image for 0 and B e ects by multiplying ^op by ei 0 +i BTE op(n) using 0 and B from (17.28) and (17.29), respectively, and also averaging the magnitude images ^in(n) and ^in(n +1) to get the same e ective T2 decay as ^op 1 ^0in = 2 (j ^in(n)j + j ^in(n + 1)j) (17.30) ^0op = ^op ei 0+i B0 TE op(n) (17.31) 0 from In principle, the resulting image ^0op is a real image (however, a magnitude operation still causes a loss of sign, so complex notation should be maintained) and nally 1 ^w = 2 (^0in + ^0op) (17.32) ^f = 1 (^0in ; ^0op) (17.33) 2 The same data used in Fig. 17.14 through Fig. 17.16 are used in this example. The necessary phase images for processing the data are shown in Figs. 17.17a, 17.17b, 17.17c and the resulting 0 and B images in Fig. 17.17d and Fig. 17.18a, respectively. The corrected opposed phase image is shown in Fig. 17.18b. Using (17.32) and (17.33), the nal water and fat images, corrected for eld inhomogeneity e ects are obtained in Figs. 17.18c and 17.18d. As expected, the mixture of water and fat signal behind the knee evident in Fig. 17.16c is gone in Fig. 17.18c. 17.3. Multiple Point Water/Fat Separation Methods 445 (a) (b) (c) (d) (d) fat image obtained with correction. Artifacts (see arrow) still exist behind the knee caused by the presence of eld inhomogeneities (methods of dealing with these are discussed in Sec. 17.3.5). Fig. 17.16: Two-point water/fat separated images with and without correction for di erences in T2 weighting. (a) Water image and (b) fat image obtained without correction. (c) Water image and 446 Chapter 17. Water/Fat Separation Techniques (a) (b) (c) (d) Fig. 17.17: Di erent steps involved in the three-point water/fat separation method using gradient echo images. Here, the separation of the di erent components of the TE -independent phase di erence 0 is detailed. First, the global, spatially linearly-varying phase due to non-centering of the echo in either direction is determined. Following the removal of this global phase shift, the spatially-variant but still TE -invariant 0 phase variation can be determined. Parts (a), (b) and (c) show the echo shifted phase images for the rst in-phase image, the opposed-phase image and the second in-phase image, respectively. Part (d) shows the extracted 0 contribution. Evidently, water and fat have consistently di erent values of 0 , probably caused by their di erences in conductivity. 17.3. Multiple Point Water/Fat Separation Methods 447 (a) (b) (c) (d) nation of B , and (b) the correction of the opposed-phase phase image. (c) The water image, and (d) fat image obtained by applying the two-point complex separation method to ^0in and ^0op . Local eld corrections have successfully removed the artifact behind the knee. Fig. 17.18: Remaining steps in the three-point water/fat separation method involve: (a) determi- 448 Chapter 17. Water/Fat Separation Techniques Problem 17.11
You are interested in quantifying the amount of water and fat in a given voxel, perhaps in bone marrow for example. You incorrectly judge the rst in-phase time to be 4.5 ms instead of 4.67 ms, and all your other times are now appropriate multiples of 4.5 ms rather than 4.67 ms. (Unfortunately, fat in di erent parts of the body has a variable chemical shift making quanti cation with a given set of three echoes tenuous unless the chemical shift is known.) Assume B0 = 1.5 T. Take the fat signal fraction to be 80% and water signal fraction to be 20% in a given voxel. a) Calculate the chemical shift corresponding to an in-phase echo time of 4.5 ms. b) Neglecting eld inhomogeneity e ects and T2 di erences between the tissues, nd ^w using 4.5 ms as the in-phase image and 6.75 ms as the opposed-phase image. c) How big is this error relative to the expected value of 80%? 17.3. Multiple Point Water/Fat Separation Methods 449 Suggested Reading
Chemical shift imaging was independently introduced in the following three papers: T. R. Brown, B. M. Kincaid and K. Ugurbil. NMR chemical shift imaging in three dimensions. Proc. Natl. Acad. Sci., 79: 3523, 1982. I. L. Pykett and B. R. Rosen. Nuclear magnetic resonance: in vivo proton chemical shift imaging. Radiology, 149: 197, 1983. R. E. Sepponen, J. T. Sipponen and J. I. Tanttu. A method for chemical shift imaging: demonstration of bone marrow involvement with proton chemical shift imaging. J. Comput. Assist. Tomogr., 8: 58, 1984. An early two-point chemical shift water/fat separation concept was presented in: W. T. Dixon. Simple proton spectroscopic imaging. Radiology, 153: 189, 1984. A review of fat saturation and chemical shift e ects appear in: E. M. Haacke, J. L. Patrick, G. W. Lenz and T. Parrish. The separation of water and lipid components in the presence of eld inhomogeneities. Rev. Magn. Reson. Med., 1: 123, 1986. 450 Chapter 17. Water/Fat Separation Techniques Chapter 18 Fast Imaging in the Steady State
Chapter Contents
18.1 18.2 18.3 18.4
Short-TR , Spoiled, Gradient Echo Imaging Short-TR , Coherent, Gradient Echo Imaging SSFP Signal Formation Mechanisms Understanding Spoiling Mechanisms Summary: This chapter contains an in-depth discussion of the signal behavior when TR becomes on the order of T1 and/or T2 . The concepts of incoherent and coherent steady-state signals are discussed, the former leading to a spin density-weighted or T1-weighted image and the latter a (T1 =T2)-weighted image. Radiofrequency (rf) spoiling is introduced to show how the steady-state transverse magnetization can be made zero prior to each new rf pulse even if TR is less than T2. Introduction
Fast imaging is perhaps one of the most interesting areas of MRI. Here, and in the next chapter, the foundation for certain basic approaches is laid out, including both short-TR imaging and echo planar imaging methods. It is intriguing that the limits of continuous rf scanning (which was how the experiments in the early days of NMR were done) are being approached again as repetition times (TR values) become shorter and shorter. In this chapter, the focus is on understanding the build-up of the magnetization to steady-state and the practical implementation of the simplest forms of imaging in the steady-state. Although it is possible to implement short-TR spin echo or arbitrary multiple pulse steady-state imaging sequences as well, this chapter focuses on the steady-state built up from conventional, single echo, gradient echo methods. When a spin system is repeatedly disturbed by a fast repetition of rf pulses and the transverse magnetization available just before each rf pulse is purposefully made zero, the transverse magnetization after each new rf pulse approaches a steady-state value which is 451 452 Chapter 18. Fast Imaging in the Steady State smaller than the thermal equilibrium value. The spin system takes a nite number of pulses before this steady-state is reached in a time that depends on both the T1 of the tissue and the ip angle of the rf pulse. Sequences utilizing a steady-state approach can be broadly classi ed as steady-state coherent (SSC) and steady-state incoherent (SSI) sequences. The main di erence between the two lies in whether or not the transverse magnetization is allowed to go naturally to steady-state between successive rf pulses. As the nomenclature suggests, the SSI sequences are based on the elimination, or the `spoiling,' of any remnant transverse magnetization prior to the occurrence of each new rf pulse. On the other hand, in SSC sequences, both transverse and longitudinal magnetization components at the end of a repetition period contribute to the signal in the next cycle, leading to a di erent magnetization response. The rst two sections in the chapter are dedicated to detailed derivations of expressions for the steady-state magnetization components and their approach to steady-state equilibrium. The nal two sections focus on the formation of echoes with multiple rf pulses and their elimination. These discussions lead to insights on coherent steady-state signal formation mechanisms and the understanding of the e ectiveness of practical implementations of spoiling. 18.1 Short-TR, Spoiled, Gradient Echo Imaging
Before an expression for the signal in the spoiled steady-state is obtained, it is worthwhile to see how the signal reaches a steady-state from its initial thermal equilibrium value. For a spin system initially at thermal equilibrium, the longitudinal magnetization is M0 . Let this spin system be acted on by a series of identical rf pulses of ip angle . After the rst rf pulse, the longitudinal magnetization is given by Mz (0+) = M0 cos
and the transverse magnetization by (18.1) (18.2) M?(0+) = M0 sin Between the rst and second rf pulses, the longitudinal magnetization grows from the initial value Mz (0+) toward M0 according to the Bloch equations (4.21). The longitudinal component has not necessarily reached its thermal equilibrium value by the time of the second rf pulse. At the same time, suppose that the transverse magnetization has been destroyed by some means, a discussion of which is presented in a later section on spoiling mechanisms. ; Mz (TR ) is again transformed into longitudinal and transverse components by the second rf pulse, and this process continues. After a su cient number of cycles, the process reaches a steady-state de ned by a recurrence of the same magnetization values and identical behavior in each rf cycle. This process is illustrated for one cycle in the steady-state for an arbitrary -pulse in Fig. 18.1a. In this steady-state limit, the magnetization is periodic with period TR . The steady-state values are reached when the loss in longitudinal magnetization due to its tipping by the rf pulse is exactly counterbalanced by its growth due to T1 recovery during the inter-pulse period. In 18.1. Short-TR, Spoiled, Gradient Echo Imaging 453 general, there is an initial transient behavior before the magnetization settles into this periodicity. Notice from Fig. 18.1b that the magnetization reaches steady-state after the second pulse when = =2. In general, the number of pulses required to reach steady-state is a function of the ip angle and TR . The process of achieving steady-state and the dependence on the number of pulses required to reach steady-state are described in detail in the next two subsections. (a) (b)
Fig. 18.1: Evolution of the magnetization components at successive rf pulses (a) for an arbitrary ip angle and one steady-state rf cycle, and (b) for = =2 and the rst two rf pulses. The evolution ; in (a) begins at time t = nTR where only the steady-state magnetization Mze is available. The application of a rf pulse at t = nTR creates a new transverse magnetization component Mze sin while leaving behind longitudinal magnetization Mze cos . The di erence between the longitudinal magnetization before and after the rf pulse is represented by Mze . Between the nth and the (n + 1)st pulse, the longitudinal magnetization starts regrowing towards M0 by T1 relaxation and the transverse component starts decaying towards zero due to T2 relaxation. This is depicted by the components Mz (tn + ) (lengthened) and M? (tn + ) (shortened in length). The evolution in (b) shows that, when = =2, it takes only two rf pulses for the magnetization to reach steady- state. This occurs because each 90 -pulse tips all longitudinal magnetization, and it recovers to the same value M0 (1 ; e;TR =T1 ) during each subsequent period. For the rst rf pulse alone, the initial magnetization is M0 hence it takes two rf pulses to reach steady-state. 454 Chapter 18. Fast Imaging in the Steady State 18.1.1 Expression for the Steady-State Incoherent (SSI) Signal We begin with the simplest concepts that describe short-TR steady-state incoherent imaging.1 ; Consider a gradient echo sequence where TR T2 . This is the situation where M?(nTR ) = 0, where t = nTR is the instant of occurrence of the (n + 1)st rf pulse. This is a naturally spoiled sequence where all transverse magnetization is essentially zero before the next rf pulse. We wish to nd a solution to the steady-state longitudinal magnetization for a given isochromat, for given values of T1 T2 TR and ip angle .2 The following analysis is valid for an isochromat of spins where there are no T2 e ects. The macroscopic spin phase e ects will be considered after the equilibrium conditions are derived. The transverse magnetization decays during each evolution period according to (4.33) M?(tn) = M?(0+)e;tn =T2 0 < tn < TR (18.3) where M?(0+) is a shorthand notation for M?(tn = 0+). The total time from the rst pulse is t, and the relative time within each cycle is de ned as tn t ; nTR (18.4) The regrowth of longitudinal magnetization during this period is Mz (tn) = M0 (1 ; e;tn =T1 ) + Mz (0;)e;tn =T1 (18.5) where Mz (0;) is shorthand notation for Mz (tn = 0;). Equations (18.3) and (18.5) can be rewritten by reintroducing the total time: ; + M?((n + 1)TR ) = M?(nTR )E2 ; = Mz (nTR ) sin E2 (18.6) which goes to zero as E2 goes to zero and, under these circumstances, ; ; Mz ((n + 1)TR ) = Mz (nTR ) cos E1 + M0 (1 ; E1 ) (18.7) where E1 and E2 are de ned by E1 e;TR=T1 (18.8) and E2 e;TR=T2 (18.9) for convenience. The attainment of the steady-state by the time of the N th pulse implies that the value of Mz just prior to each subsequent rf pulse is unchanged from cycle to cycle. De ne the steady-state equilibrium value of Mz to be Mze so that ; Mz (mTR ) = Mze m N (18.10)
These imaging methods are often referred to as FLASH (Fast Low Angle SHot) or spoiled GRASS (Gradient Refocused Acquisition in the Steady-State). We have been reticent to use acronyms in this text but these two are in common usage. In the nomenclature introduced here, they are referred to as short-TR SSI gradient echo methods. 2 When the isochromat is replaced by a voxel, T must be replaced by T for gradient echo imaging. 2 2
1 18.1. Short-TR, Spoiled, Gradient Echo Imaging From (18.7) and (18.10), Mze must satisfy 455 Mze = MzeE1 cos + M0 (1 ; E1)
This yields the steady-state or equilibrium value (18.11) M0 E Mze = (1 ;(1 ;cos1 )) E
1 (18.12) Plugging in this result into (18.3) gives ; M?( tn) = Mze sin e;tn =T2 = M0 sinE(1cos E)1) e;tn =T2 (1 ; 1 0 < tn < TR (18.13) under steady-state equilibrium. The only changes in (18.13) needed to represent the steady-state signal from a voxel containing several isochromats are the replacements of M0 by 0 (the voxel spin density) and T2 by T2 . Hence, ; ^( TE ) = 0 sin (1 (1 E E1 ) ) e;TE =T2 (18.14) ; 1 cos Figure 18.2 shows the general behavior of ^ as a function of for both white matter (WM) and gray matter (GM).3 The maximum signal occurs for an intermediate angle (less than =2) de ned as the Ernst angle E . The Ernst angle is the subject of Prob. 18.2 and is given in (18.15). Note that (18.14) vanishes at = 0 due to the sin factor (the E1 dependence cancels out in this limit). But the E1 dependence in the numerator is not canceled for larger angles, and yields an approximate suppression factor TR =T1 for small TR . In fact, when = 90 , the signal is directly proportional to TR =T1, as illustrated in Fig. 18.2b. Problem 18.1
a) Plot 0(M?( tn = 0)=M0) versus for 0 , with TR = 40 ms, for CSF (cerebrospinal uid), WM, GM and fatty tissue. Estimates for the associated relative spin density, T1 and T2 values at 1.5 T are given in Table 18.1. b) Show that the crossover point for GM/WM signal plots occurs for ' 17 when TE = 0. The gray matter of the cortex of the brain contains the cell bodies of the neurons. The white matter includes the axons which serve as the communication routes from these neurons to other neurons. The neurons are the source of electrical impulses controlling brain and bodily function.
3 456 Chapter 18. Fast Imaging in the Steady State Fig. 18.2: Comparison of the SSI signal behavior of gray matter (GM) and white matter (WM) for the representative values of TR = 40 ms and TE = 0 ms. (a) Small ip angle signal behavior.
The angles E GM and E WM denote the Ernst angles of GM and WM, respectively, and c denotes the ip angle at which the two curves cross each other (i.e., the crossover point). (b) SSI signal behavior for the range 0 . 18.1. Short-TR, Spoiled, Gradient Echo Imaging Tissue CSF WM GM fatty tissue
1.5 T. Here,
0 457 T1 (ms) T2 (ms) 1.0 4500 2200 0.65 600 80 0.8 950 100 0.9 250 60
0 Table 18.1: Estimated NMR properties of cerebrospinal uid (CSF), WM, GM and fatty tissue at
is the spin density relative to CSF. Problem 18.2
a) Show that the maximum SSI signal occurs at the Ernst angle = cos;1 E1 b) For TR
E (18.15) (18.16) T1 , show that
E in radians, and T ' 2T R
1 s v s u (1 ; E ) u TR 1 t M?( E ) = M0 (1 + E ) ' M0 2T 1 1
1 M?( E ) ' 2 M0
E (18.17) (18.18) Hence, show that where
E is in radians. An interesting feature of the SSI sequence is that it is possible to obtain spin-density weighting even for small TR values (TR T1). In the low ip angle, short-TR limit, the SSI signal will be seen to be linearly proportional to 0 and , and independent of T1 and T2 . The linear relationship to ip angle is shown in the plots of Fig. 18.2a. Further, note that the slope is lower for WM which has a lower relative spin density as shown in Table 18.1. At low ip angles, the expression for ^ can be approximated by ^( ) ' 1 ; E1 1 ;
0 (1 ; E1 )
2 2 = 1+ 0 1 E1 2 2 (1;E1 ) << E (18.19) 458 If, in addition, TR Chapter 18. Fast Imaging in the Steady State T1 , the spin density (18.19) can be further approximated by
^( ) ' 1+
0 T1 2TR 2 ' 1+ 0
E 2 (18.20) The last approximation comes from the short-TR result for E obtained in Prob. 18.2(b). It is observed in (18.20), that the low ip angle linear approximation is valid to within roughly 1% as long as is less than a third of E or so since the nonlinear corrections are second-order in = E . Other features of practical interest are the zero crossing of the SSI signal for all tissues when the ip angle equals 180 and the achievement of T1 -weighted contrast (suppression of long T1 tissues) past the Ernst angle. This latter feature is seen in Fig. 18.2b where WM has a higher signal than GM (GM has a longer T1 value see Table 18.1) for ip angles greater than E WM. The contrast stays approximately constant past the Ernst angle, a feature which is utilized to obtain T1-weighted contrast at ip angles much lower than 90 . The important features of SSI signal behavior are demonstrated in the images shown in Fig. 18.3. For TR = 25 ms, E CSF ' 6 , E GM ' 13 , E WM ' 16 and E fat ' 25 . With these conditions, it is expected that CSF will have the highest signal (highest 0) in a = 2 image, while WM will have the lowest signal as seen in Fig. 18.3a. WM and GM have a signal crossover for = 10 , yielding no WM/GM contrast while CSF is T1 -weighted, leading to a heavy suppression of its signal (Fig. 18.3b). For = 20 , good T1 -weighted contrast is obtained between GM and WM. CSF is suppressed further and fat now has the highest signal (Fig. 18.3c), all the necessary characteristics of a T1 -weighted image as discussed in Ch. 15. These ndings are further illustrated in the plots of the voxel signal as a function of ip angle for these four tissues in Fig. 18.3d. Matrix Formulation of Signal Behavior
It is useful to repeat the steady-state analysis of the magnetization in terms of matrices. Let 0 1 Mx(nTR ) ~ M (nTR ) = B My (nTR ) C @ A Mz (nTR ) (18.21) where the time argument is the total time t. Reverting to relative time t0 = t ; nTR with respect to the nth rf pulse, the Bloch equations with relaxation terms included, (4.22) to (4.24), have the solution ~ ~ Mn(t0 ) = D(t0)Rx( )Mn (0;) + M0 (1 ; E1)^ z
for an on-resonance isochromat. In (18.22),
0 (18.22) 0 ;t =T2 1 e 0 0 C D(t0) B 0 e;t =T2 0 @ A ;t =T1 0 0 e
0 0 (18.23) 18.1. Short-TR, Spoiled, Gradient Echo Imaging 459 (a) (b) (c) (d) Fig. 18.3: Images obtained with a 3D SSI sequence at di erent ip angles using a 20 ms TR . (a) Image for = 2 is spin density-weighted: CSF is the brightest, GM is intermediate and WM has the lowest signal. (b) Image for = 10 shows the e ect due to the crossover between the GM and WM signal curves note the absence of GM-WM contrast. (c) Image for = 20 is T1 -weighted the appearance is reversed from that of the spin density-weighted image. (d) The measured image signal ^ in arbitrary units as a function of ip angle from this data set. 460 and Chapter 18. Fast Imaging in the Steady State 0 sin C (18.24) A 0 ; sin cos It is recalled from Ch. 2 that Rx( ) implies a clockwise rotation through an angle about the x-axis. At equilibrium, ~ ~ ; ~ Mn+1 (0;) Mn (TR ) = Mn(0;) (18.25) The use of (18.25) in (18.22) gives ~ Mn (0;) = (I ; D(TR )Rx( ));1M0 (1 ; E1 )^ z (18.26) 0 1 0 Rx( ) B 0 cos @ 1 where I is the identity matrix. In the instance when E2 ! 0 (when TR T2 ), (18.12) is recovered. If E2 6= 0, the magnetization behavior is a rather complicated function of T1 and T2 . An expression for the signal in the coherent steady-state case will be obtained in due course. Incoherent steady-state methods commonly use TR values which are short enough that natural spoiling does not occur. These implementations use transverse magnetization spoiling mechanisms which require the knowledge of steady-state free precession. A detailed discussion of these modern implementations is contained in the nal section of this chapter. For the time-being, it is assumed that spoiling is achieved prior to each new rf pulse even in the short-TR case without asking how it is done. 18.1.2 Approach to Incoherent Steady-State Although equilibrium solutions have been found, the general transient behavior of the magnetization is yet to be evaluated. Some tissues will reach a steady-state equilibrium more quickly than others, depending on their T1 values. The mathematical description of this approach to steady-state is the goal of this subsection. If the steady-state is reached before data acquisition is started, then there will be no ltering e ects in the phase encoding direction. Otherwise, the signal changes from cycle to cycle, and the image quality can su er. This blurring is reminiscent of the ltering e ects in the read direction due to T2 decay, as discussed in Ch. 13. This is described at the end of the discussion in this subsection. Let us evaluate how many pulses it takes for a tissue magnetization to approach some fraction of its steady-state value. For convenience, the short-hand notation of Mz (n) is used to denote Mz (nTR ). From (18.7), which required that the transverse magnetization just prior to any rf pulse is zero, the longitudinal magnetization at the end of the (n ; 1)st repetition period is given by Mz;(n) = Mz; (n ; 1) cos E1 + M0(1 ; E1 )
A rst recursion of this equation in terms of Mz;(n ; 2) is n 1 (18.27) Mz; (n) = Mz;(n ; 2)(cos E1 )2 + M0(1 ; E1 ) cos E1 + M0 (1 ; E1) n 2 (18.28) 18.1. Short-TR, Spoiled, Gradient Echo Imaging Continuing this iteration yields 461 Mz (n ) =
; "n;1 X
l=0 (cos E1 )l (1 ; E1 )M0 # + M0 (cos E1 )n n 1 ; (cos n = M0 (1 ; E1) (1 1 ; cos E1 ) ) + M0(cos E1)n n 1 (18.29) E1 This expression is most easily evaluated at = E . Using cos E = E1 from (18.15), (18.29) becomes 2n 2 Mz; (n E ) = M0 (11; E1 ) + M0 E1 n n 1 (18.30) + E1 In the new notation, the steady-state value is ; (18.31) nlim Mz (n E ) = Mze ( E ) = M0 =(1 + E1 ) !1
The relative error in estimating Mz; (n) by Mz; (1) at the (n + 1)st pulse for the Ernst angle is given by Mz; (n E ) ; Mze( E ) = E 2n+1 (18.32) 1 Mze( E ) Hence, the number of pulses n required to reduce (18.32) to a given is 1 (18.33) n = ; 2T1 ln ; 2 TR where the square brackets denote the next largest integer of the argument. For the case when TR = 40 ms, the approximate values of n for = 0:01 and = 0:1 for di erent tissues are tabulated in Table 18.2. The TR = 400 ms case is tabulated in Table 18.3. Problem 18.3
Show that n TR for = 0:1 is on the order of the T1 of the tissue. Assume that = E . Qualitatively argue that this is a reasonable answer. See Fig. 18.4. CSF GM WM Fat = 0:01 250 54 36 16 = 0:1 125 27 18 8 Table 18.2: The pulse number n for two values and di erent tissues at their respective Ernst angles at 1.5 T when TR = 40 ms. See Fig. 18.4a. The situation in the general case is described from (18.29): Mz; (n ) = Mze( ) (1 ; (E1 cos )n) + M0 (E1 cos )n = (M0 ; Mze( ))(E1 cos )n + Mze( ) (18.34) 462 Chapter 18. Fast Imaging in the Steady State CSF GM WM Fat = 0:01 26 6 4 2 = 0:1 13 3 2 1 Table 18.3: As in Table 18.2, but for TR = 400 ms. See Fig. 18.4b. Equation (18.34) describes an exponential decay of Mz; (n) from M0 to Mze with a rate determined by the quantity E1 cos . This means that the approach to steady state is faster if either E1 or cos or their product is small. So, shorter T1 tissues reach steady-state faster as a function of rf pulse number similarly, all tissues reach steady-state faster for longer TR values and as increases in the range 0 90 . The plots for magnetization as a function of pulse number n for = 10 are shown for two di erent TR values in Fig. 18.4. This gure demonstrates that the steady-state equilibrium longitudinal magnetization can be small compared to the thermal equilibrium value of M0 . Fig. 18.4: Plot of Mz;(n) as a function of rf pulse number n for GM, WM, CSF and fatty tissue at a common ip angle = 10 for (a) TR = 40 ms (b) and TR = 400 ms. Note the di erence between
these data, and Tables 18.2 and 18.3 where the individual Ernst angles are used. Even though Mze( E ) from tissues with long T1 values may be small, it is possible to get a rather large signal during the smaller n, pre-steady-state cycles, despite the ltering e ect. For example, CSF could appear falsely bright in a T1-weighted image ( > E ) if n for CSF is greater than the number of phase encoding lines in a 2D scan. Note the small rf pulse number regime in Figs. 18.4a and 18.4b where the CSF signal is much greater than its equilibrium value. To obtain the expected contrast, it is to have reached steady-state before the imaging experiment is carried out. Only after n cycles, for 1, can true T1 -weighted 18.1. Short-TR, Spoiled, Gradient Echo Imaging 463 data be obtained without ltering e ects. This can signi cantly lengthen the imaging time for a short-TR 2D scan, but it is less of a problem in a short-TR 3D scan. If the k-space data are collected sequentially from ;kPE to +kPE , a relatively large transient signal does not a ect the image very much since the ltering occurs only at high k-space values where the absolute signal is small. In fact, this high pass ltering enhances the edge information on the image and could be quite advantageous. Further, by the time the origin of k-space is reached, the signal will generally have reached steady-state and the image contrast is not reversed as in the case brie y discussed in the previous paragraph. 18.1.3 Generating a Constant Transverse Magnetization
Maximizing signal for a xed TR by choosing a speci c ip angle (the Ernst angle) is important in fast imaging. However, this may not be the most e cient method to obtain high signal if only a limited number of rf pulses is employed to complete the entire imaging experiment. For short-TR 2D scans, it is possible to eliminate the ltering e ect caused by the variations in magnetization at each new rf pulse during the approach to steady-state equilibrium. Suppose that the transverse magnetization excited by each rf pulse is forced to be equal to that created by the previous rf pulse. That is, the second pulse is designed to return the transverse magnetization to M0 sin 0 , its value after the rst pulse. The third pulse is designed to obtain the same transverse magnetization value and so on. By making
+ M? (n + 1 + n+1 ) = M? (n n ) (18.35) for any n, the ltering e ect can be successfully eliminated. Equivalently, the ip angle n is modi ed from one pulse to the next such that the longitudinal magnetization Mz; (n n) satis es the condition Mz;(n + 1) sin n+1 = Mz; (n) sin n (18.36) This works as long as (18.36) does not require n+1 to be larger than =2. This results from the fact that if the longitudinal magnetization recovered in a given cycle is less than the desired transverse magnetization, even the maximum rotation of =2 is insu cient to recover the correct transverse magnetization. As a result, n = =2 is often used as the terminating condition for this method of producing constant magnetization. 464 Chapter 18. Fast Imaging in the Steady State Problem 18.4
From (18.36), show that the longitudinal magnetization just after the nth pulse is M0 sin 0 = tan n where 0 is the starting angle. a) Using this result, show that sin sin E1 tan 0 + (1 ; E1) = sin 0 n;1 n Given E1 and 0 , this equation can be used to generate fashion. b) Replicate the result in Fig. 18.5. (18.37)
n in an iterative The method of changing the ip angle as a function of rf pulse number is typically used in extremely short-TR 2D SSI imaging applications. A predetermined terminal ip angle is required to occur after a target number of rf pulses p. From the previous problem, it is seen that there is no analytic closed form for n as a function of n. The values for f ng are evaluated numerically the results of one such simulation are shown in Fig. 18.5. It is possible to make some general remarks concerning the behavior of n in the above method. Once the spin system is disturbed by the rst rf pulse, there is less longitudinal magnetization available for the following rf pulses since TR is extremely short (see Fig. 18.4). The ip angle must therefore increase with rf pulse number to create a constant transverse magnetization. If 0 is too small, the longitudinal magnetization is barely disturbed and almost full magnetization is available for the next pulse(s), and n increases very slowly with n. If the terminal ip angle is 90 , as it is in most 2D imaging applications, this target is reached only after a very large number of pulses. If it is desired that the total number of pulses not be too large, then the value of 0 should not be too small. The increase in ip angle with n, which is slow if 0 is small and accelerates as n increases, is apparent in Fig. 18.5. The gure shows the case where p is chosen to be 30, corresponding to 0 approximately 13 for a tissue with T1 = 950 ms and TR = 10 ms. The advantage of forcing steady-state using variable ip angles is illustrated in Fig. 18.6. An image obtained with xed ip angle is compared to an image obtained with a ip angle varied to excite constant magnetization at each new rf pulse. The former exhibits more blurring and some ghosting in the phase encoding direction. As discussed before, the blurring in the xed ip angle case (Fig. 18.6a) is a result of the exponential approach to steady-state as a function of rf pulse number of the excited magnetization, which is then phase encoded. The ghosting is a result of some periodic amplitude inconsistencies created along the phase encoding direction as the k-space data were acquired eight lines at a time with 15 pulses following which the longitudinal magnetization was allowed to relax towards M0 for roughly a second. (For a detailed discussion of this `segmentated' approach to imaging and reasons for this ghosting, see Ch. 19.) 18.1. Short-TR, Spoiled, Gradient Echo Imaging 465 Fig. 18.5: Use of a variable ip angle which increases as a function of rf pulse number in order to force a constant transverse magnetization from one rf pulse to the next. The plot was obtained for a terminal pulse number p = 30, with T1 = 950 ms and TR = 10 ms. (a) (b) Fig. 18.6: A comparison of the use of variable versus xed ip angle in short-TR 2D imaging. (a) An image obtained with xed ip angle appears more blurred in the phase encoding direction than (b) an image obtained according to the method described in Sec. 18.1.3. The blurring occurs in (a) due to the fact that the signal varies with rf pulse number before steady-state has been achieved. 466 Chapter 18. Fast Imaging in the Steady State 18.1.4 Nonideal Slice Pro le E ects on the SSI Signal It follows from the discussion in the last two subsections that care has to be taken in 2D SSI imaging to ensure that steady-state has been reached. Otherwise, image quality su ers. Two-dimensional SSI imaging also su ers from the e ects of signal integration over nonideal rf slice pro les (which lead to spatially varying ip angles along the slice select direction). A discussion of these e ects is the focus of this subsection. An e ective spin density for a single 2D slice, with z as the slice select direction, is given by an integration over the rf slice pro le, ^2D = Z1 ;1 dz ^( (z)) (18.38) The ip angle varies over the slice owing to the deviations from a rect function in the slice excitation pro le.4 The trapezoidal function shown in Fig. 18.7a may be used to model an rf pulse pro le. In region I of the gure, the ip angle is linear in z, z (z) = 0 ((B + B )) for ; B z ;A (18.39) ;A with 0 the maximum ip angle. With a constant pro le ( = 0 ) in region II and a linearly decreasing angle in region III, the integration in (18.38) gives Z B 0 (1 ; E1 ) sin (z) ^2D trap( 0 ) = 1 ; E1 cos (z) dz ;B ( ) 2A sin 0 + 2(B ; A) Z 0 sin d = 0 (1 ; E1 ) 1 ; E cos (18.40) 0 1 ; E1 cos 1 0 0 ( !) A sin 0 + (B ; A) ln 1 ; E1 cos 0 (18.41) = 2 0(1 ; E1) 1 ; E cos 1 ; E1 1 0 0 E1 For comparison, an ideal rect function excitation (constant ip angle 0) over a slice of thickness 2B yields5 ; E sin ^2D rect( 0 ) = 2B 0 (1 ; E 1 )cos 0 (18.42) 1
1 0 Plots of both ^2D rect( 0 ) and ^2D trap ( 0 ) as functions of 0 are presented for a trapezoidal slice pro le (Fig. 18.7a) are shown in Figs. 18.7c and 18.7d. They contrast ideal SSI behavior with that resulting from an integration over a more realistic rf pro le. The shape of the integrated voxel signal as a function of is similar to that expected from (18.42), but with an e ectively shorter T1.6
These variations in ip angle should not be confused with the in-plane and through-plane ip angle variations due to transmit coil inhomogeneities. 5 This result is obtained assuming that 0 is the relative spin density value obtained for a unit slice thickness. 6 The results quoted for T values at 1.5 T in Table 18.1 for GM and WM are based on 2D measurements 1 and may be signi cantly lower than their actual values. Nevertheless, they do correctly predict the signal behavior for these scans using similar nonideal rf pro les (see Ch. 22 for more details).
4 18.2. Short-TR, Coherent, Gradient Echo Imaging 467 Another important point revealed in Fig. 18.7b is how the actual slice thickness should be quoted in terms of the FWHM of the magnetization pro le for 0 values greater than the Ernst angle. For 0 > E , the magnetization pro le ^2D trap (z) is bimodal in appearance with two local maxima occurring at the two values of z where (z) equals E . Clearly, when the slice thickness is quoted at a given ip angle, it should not be quoted in terms of the FWHM of the slice pro le itself it should be quoted in terms of the FWHM of the signal pro le. For example, FWHM2 as measured for 0 greater than the Ernst angle is larger than FWHM1 , that measured at 0 < E (Fig. 18.7b). Problem 18.5
Reproduce Fig. 18.7c and add curves for white matter using the values of 0 and T1 from Table 18.1. Hence conclude that the contrast between GM and WM as a function of ip angle has changed in comparison with the ideal results illustrated in Fig. 18.2. The plots in Fig. 18.7 and the results in Prob. 18.5 demonstrate that the contrast can be signi cantly a ected by integration across the slice pro le. The main aspect to be noted is the attening of the signal response as a function of 0 as seen in Figs. 18.7c and 18.7d for gray matter, leading to a reduction of contrast between GM and WM, for example. On the other hand, in 3D imaging, where individual 3D partitions are thin relative to changes in rf pro le (or spatial ip angle variations), the e ective spin density is well approximated by ^2D rect( 0 ). This e ect is demonstrated in Fig. 18.8, where the GM/WM contrast seen in a single 3D partition from a T1-weighted acquisition is reduced in the corresponding 2D image obtained for the same ip angle. 18.2 Short-TR, Coherent, Gradient Echo Imaging
When TR is on the order of, or less than, T2 and no attempt is made to spoil the transverse magnetization, there will be a coherent build-up of signal towards a steady-state value which is a mixture of longitudinal and transverse components present at the initial state. This dependence on the initial magnetization is illustrated in Figs. 18.9 and 18.10 by assuming two di erent initial conditions. In Fig. 18.9, it is assumed that equal magnetization exists along both y and z axes, while in Fig. 18.10 it is assumed that equal magnetization components exist along all three axes. From Figs. 18.9 and 18.10, it is also seen that the steady-state magnetization components depend on the eld seen by the isochromat of interest. That is, two isochromats seeing different static elds will reach di erent steady-state values after free precession. In an imaging experiment, di erent gradients are used from one rf pulse to the next. These cause additional magnetic eld inhomogeneities during the transverse magnetization evolution period, which can also a ect the steady-state signal. As before, we consider an isochromat of spins where there are no T2 e ects. 468 Chapter 18. Fast Imaging in the Steady State Fig. 18.7: (a) Trapezoidal rf slice pro le. (b) SSI signal as a function of position z along the slice select direction in the presence of a trapezoidal rf pro le with A = B=2 = 0:5 for gray matter. The signal response is shown for TR = 40 ms for values of 0 = 10 (less than the Ernst angle solid line
curve) and 90 (greater than the Ernst angle dashed line curve). (c) ^2D trap ( 0 ) and ^2D rect( 0 ) as functions of 0 for the same values of A and B as in (b). (d) Same as (c), except now A equals zero, so the pro le is triangular rather than trapezoidal. 18.2. Short-TR, Coherent, Gradient Echo Imaging 469 (a) (b) Fig. 18.8: Reduced T1 -weighted contrast in 2D imaging due to slice pro le integration. Although both (a) the 2D image and (b) the 3D image shown were obtained with the same ip angle of 20 , there is reduced T1 -weighted GM-WM contrast in the 2D image. The signal loss in the region of the nasal sinuses (in the top half of the image) in the 2D image is due to T2 dephasing over the 4 mm thick 2D slice. This is not evident in the 3D case which is a sum over four 1 mm thick magnitude images (the reasons for the absence of signal loss in the 3D case are discussed in detail in Ch. 20). 470 Chapter 18. Fast Imaging in the Steady State (a) (b)
Fig. 18.9: Demonstration of coherent steady-state formation. (a) Sequence diagram showing an
rf pulse repetition example which leads to a steady-state. (b) The case when equal magnetization is available along the y and z axes at the beginning of a free precession period, and a 45 -pulse is applied along the x axis. The e ective z magnetization after the rf pulse is zero. Assume that the magnetization considered here is from a resonant isochromat of spins. Their precession angle in the rotating frame is zero, leaving them all along y throughout the evolution period. It is now easy to ^ visualize that, depending on the TR value, the y magnetization decays due to T2 decay while the z magnetization recovers towards M0 and, at the end of the evolution period, the magnetization state returns to the value it had just before the previous rf pulse, leading to a steady-state. 18.2. Short-TR, Coherent, Gradient Echo Imaging 471 (a) (b)
Fig. 18.10: (a) Sequence diagram showing a second rf pulse repetition example which leads to a di erent steady-state. (b) Steady-state formation with equal x, y and z components just before the rst rf pulse. The presence of a B0 inhomogeneity is assumed to cause a =2 precession of transverse magnetization during the evolution period. Following the second rf pulse, the third free precession period leads to the initial condition (at t = 0; ), thereby creating the required periodicity + to reach steady state. This is indicated by the arrow going from the state at t = 2TR to the state at t = 0; . It is also assumed that TR is so short that all relaxation e ects can be neglected. 472 Chapter 18. Fast Imaging in the Steady State Problem 18.6
a) Starting with the same initial magnetization vector as in Fig. 18.9, show the build-up to steady-state when the precession angle per TR is . b) Starting from thermal equilibrium conditions (i.e., Mz (0;) = M0 , Mx(0;) = My (0;) = 0), show that steady-state is reached after the rst free precession period for a 45x- 90x, 90-x]repeat sequence when the precession angle of the transverse magnetization per TR is zero. Assume that TR is short enough to also ignore all relaxation e ects. From Figs. 18.9, 18.10 and Prob. 18.6, it is seen that the steady-state magnetization depends on the ip angle, the initial magnetization components and TR . Further, in Figs. 18.9 and 18.10, the events that were followed carefully were the initial condition existing just before an rf pulse, the immediate e ect of the rf pulse, and the nal condition created at the end of the free precession period without paying much attention to the details of how the nal condition was achieved. It seems that the steady-state magnetization is determined by the total precession angle of the transverse magnetization components during each TR , i.e., the steady-state magnetization depends on the quantity
total (TR ) = BTR + ~ r Z TR
0 ~ G(t)dt (18.43) where the two terms are the static and gradient eld inhomogeneity-induced resonance o set angles, respectively. The quantity total is referred to as the resonance o set angle. To let all isochromats contained in the imaged volume reach steady-state free precession (SSFP) equilibrium depending only on the `free' precession angle (the precession angle with no gradients on in this case, the background eld inhomogeneities, B , are the source of resonance o set), all the gradients used must have zero zeroth moment over each repetition period, i.e., they should all be `balanced' over each repetition period. Such a balanced gradient coherent steady-state sequence implementation is shown in Fig. 18.11.7 For the rest of the discussion in this section, it is assumed that all imaging gradients used in the sequence are balanced, thereby the resonance o set angle depends only on the static eld inhomogeneities. For this reason, the subscript total is dropped from total for the remaining part of this section. In this section, the equilibrium solution will be found under a variety of conditions which a ect the phase of the transverse magnetization. Let (t) = ! t with ! = !0 ; ! = B .
It is common to use the term `SSFP sequence' in the MR imaging literature to loosely mean a balanced gradient coherent steady-state sequence. The terms SSC sequence and SSFP sequence are interchangeably used in this sense in the text too.
7 18.2.1 Steady-State Free Precession: The Equilibrium Signal 18.2. Short-TR, Coherent, Gradient Echo Imaging 473 Fig. 18.11: A typical SSC imaging sequence implementation. Note the balancing of all imaging gradients to let all isochromats evolve under equivalent free precession conditions over each repetition period una ected by a particular phase encoding or read gradient structure. Here, ! is referred to as the resonance o set frequency, and (TR ), the resonance o set (angle). For nTR t (n + 1)TR , t0 = t ; nTR , and the argument n representing the time nTR ,
+ Mx(n t0 ) = Mx (n) cos (t0) + My+ (n) sin (t0)]e;t =T2 + My (n t0 ) = ;Mx (n) sin (t0) + My+ (n) cos (t0)]e;t =T2 Mz (n t0 ) = Mz+ (n)e;t =T1 + M0(1 ; e;t =T1 )
0 0 0 0 (18.44) (18.45) (18.46) For a clockwise rotation about x, the matrix representation for (18.44)-(18.46) is found using ^ 0 0 B 1 cos Rx( ) = @ 0
0 0 ; sin
0 0 0 sin C A cos 0 0 1 (18.47) with 0 ;t =T2 e t0 e;t =T2 sin 0 B ;e;t =T2cos ((t)0) e;t =T2 cos (t 0) D(t0 ) = @ sin (t )
0 With these de nitions, and 0 0 e ;t0 =T1 1 C A (18.48) (18.49) (18.50) ~ ~ M + (n) = Rx( )M ; (n) ~ ~ M ; (n + 1) = D(TR)M + (n) + M0 (1 ; E1)^ z ~ ~ Now, M ; (n + 1) can be set equal to M ; (n) to nd the steady-state value. 474 Chapter 18. Fast Imaging in the Steady State An alternate approach is to use the equation (18.49) and the identity ~ ~ M ; (n) = D(TR )M +(n ; 1) + M0 (1 ; E1)^ z (18.51) ~ ~ and then set M + (n) = M + (n ; 1) to solve for the steady-state value for M + . The solutions are quite simple and compact in matrix form. From (18.49) and (18.50), as n approaches 1, ~ M ; (1) = (I ; D(TR )Rx( ));1M0 (1 ; E1 )^ z
while, from (18.49) and (18.51), (18.52) (18.53) ~ M + (1) = Rx( )(I ; Rx( )D(TR ));1M0 (1 ; E1)^ z ~ The tedious part is to nd the vector components of M from the compact results of (18.52) ~ or (18.53). Writing out the individual components of the magnetization vectors M ; (1) and ~ + (1): M
; Mx (1) = M0 (1 ; E1 ) E2 sind sin My; (1) = M0 (1 ; E1 ) E2 sin (cos ; E2 ) d Mz; (1) = M0 (1 ; E1 ) (1 ; E2 cos ) ; E2 cos (cos ; E2 )] d (18.54) (18.55) (18.56) (18.57) (18.58) (18.59) and
+ ; Mx (1) = Mx (1) My+ (1) = M0 (1 ; E1 ) sin (1 ;dE2 cos ) (1 Mz+ (1) = M0 (1 ; E1 ) E2 (E2 ; cos ) + d ; E2 cos ) cos ] where E2 e;TR =T2 and d = (1 ; E1 cos )(1 ; E2 cos ) ; E2 (E1 ; cos )(E2 ; cos )
(18.60) Problem 18.7
From (18.52), derive (18.54) through (18.56). From your results, verify (18.57) through (18.59) using (18.49). 18.2. Short-TR, Coherent, Gradient Echo Imaging 475 Problem 18.8
a) Show that for = E , the Ernst angle, My+ (1) collapses to the expression in (18.13) evaluated at t = 0. b) Show that for such that cos = E2, My+(1) again collapses to the expression in (18.13) evaluated at t = 0. + + c) What is Mx (1) in parts (a)R and (b)? Is M+ (1) in part (a) equal to R M + d j. The steady-state ~+ that in part (b)? Show that j ; M+ d j = j ; y obtained by integrating over the entire range of values is also known as the `resonance o set averaged steady-state.' It is a useful extension of the SSFP limit, whereby a -independent signal which equals the SSI signal value is obtained under two conditions: either the ip angle must equal the Ernst angle, or must equal cos;1 E2 . It is integration across which incorporates variations in the local magnetic eld. Clearly, the SSFP signal is a complicated function of , E1 , E2 and . The dependence on and are summarized for di erent tissues in Fig. 18.12 for a choice of TR T2 of all the tissues. These plots show that for small ip angles (for example, see curves for = 10 ), a large signal response is achieved only when is close to 0 or 360 . On the other hand, for large ip angles (for example, see curves for = 70 ), a uniformly high signal is obtained for a range of values centered around = 180 . Since the static eld varies as a function of position, varies as a function of position within the image and, typically, this leads to signal variation for SSC images of homogeneous objects. However, B (~) does not usually vary over a single voxel. As a result, each voxel r can be considered an isochromat whose signal expression is de ned based on (18.54)-(18.59). Practically, if there is a signi cant eld variation across the object, this will lead to a changing (~) and the dependence of the magnetization components on the resonance o set angle (see r Fig. 18.12) means that uniformity will be degraded in the reconstructed image. The measured signal in a voxel for a given echo time will be found by integrating M+ (1)e;TE =T2 over the physical distribution of values present in that voxel (see Sec. 20.4). Up to this point, all eld dependence is carried by and, therefore, E2 should not be replaced by E2 even though this is a gradient echo experiment. The above mentioned integration is what may eventually lead to e;TE =T2 being replaced by e;TE =T2 in the voxel. However, if remains close to 180 , this is not problematic as long as the variation in across the object is small. This is typically achieved by shimming the eld well, and keeping TR as short as possible. On the other hand, if (~) ranges in the neighborhood of 0 or 360 , r the signal variation is drastically enhanced. The question arises at this point as to how (~) can be biased around to obtain a r homogeneous image. An additional means to change the resonance o set angle other than using an unbalanced gradient is to make the rf phase a function of rf pulse number, typically 476 Chapter 18. Fast Imaging in the Steady State Fig. 18.12: Plot of the magnitude of the steady-state voxel signal, ^+ , as a function of 1 di erent tissues for = 70 and = 10 . A TR of 10 ms and a TE of 0 ms were assumed. for 18.2. Short-TR, Coherent, Gradient Echo Imaging 477 by incrementing the phase linearly.8 The linear rf phase increment is represented by the quantity rf . Although the steady-state magnetization in the presence of a linear rf phase increment does not conform to the solutions of (18.54)-(18.59), in the general case, it can be shown that the steady-state magnetization satisfy these equations when rf equals 180 , with e ectively replaced by ( + ). Now, signal uniformity can be preserved for large ip angles by keeping BTR small. These e ects and the contrasting signal behavior as functions of for small ip angle versus large ip angle excitation are summarized in the imaging results obtained on a homogeneous phantom shown in Fig. 18.13. As seen from Fig. 18.12, these images show peak signal on the low ip angle images (c and d) only where the signal in the corresponding rf alternated large ip angle image (a and b, respectively) is at its lowest. Similarly, note the transformation of low signal areas into peak signal areas between the non-alternated (a and c) and alternated rf (b and d, respectively) image acquisitions. Problem 18.9
a) Show that when an rf pulse with a phase (clockwise angle with respect ~ ~ to x) and ip angle is applied, M + = Rx( )Rz (; )M ; . Hint: First, ^ ; ~ write the transverse components of M in terms of components along and perpendicular to the rf axis. b) Rewrite the steady-state vector equations (18.49) and (18.50) for the rf alternated case, i.e., for rf = 180 . c) Hence show that the steady-state magnetization components satisfy (18.54)(18.59) with ! + . 18.2.2 Approach to Coherent Steady-State As in the case of incoherent steady-state, the magnetization takes a nite time to reach steady-state in the coherent or unspoiled case as well. For two arbitrary initial magnetization vector conditions, this was qualitatively shown to vary depending on the resonance o set angle and ip angle in Figs. 18.9 and 18.10. A further look at the steady-state magnetization components de ned in (18.57)-(18.56) tells us that the approach to steady-state also depends on T1, T2 and TR . The complicated dependence of the SSC signal on these quantities makes an analytical writing of the approach to steady-state along lines similar to the SSI case very complicated. For this reason, an understanding of the dependence on these parameters can be obtained only by Bloch equation simulation for a xed set of values of , TR and for a given tissue of interest. A plot showing the multi-parameter dependence of the approach to steady-state for di erent tissues (i.e., di erent E1 and E2 values) is given in Fig. 18.14. The
In any multi-pulse experiment such as in imaging, it is assumed that the demodulator and the rf transmitter are in phase for each readout despite the linear rf phase increment. This requires the demodulator phase to be incremented the same way as the rf pulse.
8 478 Chapter 18. Fast Imaging in the Steady State (a) (b) (c) (d) Fig. 18.13: Images of a homogeneous phantom obtained using an SSFP sequence. A TR of 7 ms was used to image the phantom which had a T1 of 350 ms and T2 of 200 ms. Images (a) and (b) were obtained using non-alternating and alternating rf pulses, respectively, at a ip angle of 90 . Images (c) and (d) were obtained with a ip angle of 2 also with non-alternating and alternating rf pulses, respectively. The background eld homogeneity was changed by adding a gradient o set of 0.004 mT/m along both the read and phase encoding directions, resulting in a linearly varying resonance o set along a 45 angle as displayed by the valleys and peaks in the signal in these images. 18.2. Short-TR, Coherent, Gradient Echo Imaging 479 Fig. 18.14: Approach to steady-state of My+ and Mz+ for three di erent tissues at a TR of 20 ms for = =2 and = 180 . (a) My+ as a function of rf pulse number n. (b) Mz+ as a function of rf pulse + number n. Mx is not shown for this case since the above plots were generated by assuming thermal equilibrium initial conditions and, for this particular initial condition, Mx is zero throughout the experiment. The additional chemical shift-induced resonance o set angle for fat is assumed to be 8 , so that the e ective (total) resonance o set angle for fat in the plot is 9 . 480 Chapter 18. Fast Imaging in the Steady State approach to steady-state is also changed by changing the ip angle or . As in the SSI case, the di erent magnetization components reach steady-state in a time which is on the order of T1 . 18.2.3 Utility of SSC Imaging Steady-state coherent imaging is of practical interest despite its complicated signal behavior. To appreciate this, it is instructive to examine the limit TR T2 T1 with set to either + zero or . In either case, Mx = 0. When = , My+(1) = (1 ; E M0 (1); E1 )(sin ; cos ) 1 cos ; E2 E1 M0 sin ' T1 (18.61) + 1 ; cos T1 ; 1 T2 T2 as both E1 and E2 can be approximated with 1 ; TR and 1 ; TR , respectively. Therefore, T1 T2 the contrast obtained with an SSFP sequence is essentially T2 =T1-weighted, which is yet another form of contrast available in MRI. The optimal signal as a function of ip angle for a xed can be shown to occur at the ip angle opt such that 1 (18.62) cos opt = 1E+ + E2 (cos ; E2))==(1 ; E2 cos )) E1 E2 (cos ; E2 (1 ; E2 cos In the case when ' , i.e., the alternating rf case for the short TR limit, the above expression approximates to (see Prob. 18.10) cos opt ' T1 =T2 ; 1 (18.63) T1 =T2 + 1 with the peak signal being proportional to (18.64) My+ (1)j = opt ' 1 M0 T2 2 T1 It is seen from (18.64) that tissues whose T2 -to-T1 ratios are high have high signal. For example, a tissue such as CSF, whose T1-to-T2 ratio is roughly 2 (see Table 18.1), opt equals 70 . At this ip angle, the peak signal for = is found from (18.64) to be roughly 0:35 0 (see also Fig. 18.12b). That is, a signal which equals 35% of the highest signal that can ever be attained for CSF (remember that, in the in nite TR limit, the signal is given by 0 see Chs. 8 and 15) is obtained at the shortest possible TR . Another interesting phenomenon which is noted in Fig. 18.12a is that CSF has the same peak signal response (roughly 0.35 0) even in the low ip angle regime. This leads to a very interesting high signal regime: if the background static eld can be designed such that all isochromats experience a B which leads to the unique value of where this peak value occurs, a large signal is obtained at short TR values with small ip angle excitation. s 18.2. Short-TR, Coherent, Gradient Echo Imaging 481 Problem 18.10
Using (18.61), show that (18.63) holds exactly when = . Both opt and My+ ( opt ) are determined by the ratio (T1 =T2) when ' . Both values are independent of TR , and this is of utmost interest. (a) (b) (c) (d) Fig. 18.15: Images obtained with a fully-balanced 3D SSC imaging sequence for = 35 and TR = 7.3 ms: (a) non-alternating rf (i.e., rf = 0) (b) alternating rf (i.e., rf = ). (c) A T1 -weighted image and (d) a T2 -weighted image at the same slice location are shown for comparison. The T2 - weighted image in (d) is a thicker slice (5 mm thick in comparison with 2 mm thick slices for the others). Hence, the slight di erence in anatomical depiction from the other three images. It is seen from (18.64) that SSC images are T2 =T1-weighted. This is demonstrated in images of the brain shown in Fig. 18.15. Since CSF has the highest T2-to-T1 ratio (see Table 18.1), it has the highest signal. On the other hand, GM and WM appear about the same because of their comparable T2 -to-T1 ratios. This e ect is further demonstrated by 482 Chapter 18. Fast Imaging in the Steady State comparing this di erent contrast mechanism with the more common T1 and T2 -weighted contrast (see Figs. 18.15c and 18.15d). Another important point is the complementarity of images acquired with alternating rf and non-alternating rf (Figs. 18.15a and 18.15b). It is seen that if the local B value causes the signal to be low in one image, the same inhomogeneity causes the signal to be high in the other image (see the structure indicated by the lled arrow and the blood vessel indicated by the open arrow, for example). Evidently, a homogeneous image can be obtained by choosing the higher of the two voxel signal values on a pixel-by-pixel basis. Equation (18.64) shows that SSC imaging represents the most e cient way to collect data for a given tissue using as short a TR as possible. This TR independence of the SSC signal leads to a TR independent SNR as long as TR can be reduced without a concomitant decrease in Ts and hence the noise BW /voxel. In comparison, the signal at the optimal p ip angle for SSI imaging (i.e., = E ) is proportional to TR for short TR (see part (b) of Prob. 18.2). The TR independence of the SSC signal has been untapped until now for several reasons. One relates to available gradient strengths and the other to the background eld inhomogeneity. For example, if TR = 10 ms (which is easily achievable on commercial scanners for 1 mm3 voxels in 3D imaging), only a B of about 0.75 ppm is required to create a resonance o set range of at 1.5 T. As discussed earlier, this leads to image signal inhomogeneity and also a TR dependence of the signal. The shorter TR can made, the less of a problem B will be. Overall, as magnet homogeneity improves, with better shimming and faster gradient switching capability, there will be an increasing number of applications of extremely short-TR SSC imaging. 18.3 SSFP Signal Formation Mechanisms
In this section, a purely classical treatment of SSFP signal formation principles is developed using the concept of spin-coherence pathways. It is important to state that, since the presentation is classical, the usage of the word `spin' in spin phase pathway is, for the purposes of this section, synonymous with `isochromat magnetization,' i.e., the treatment is of a spin population, not a single spin. It is worthwhile to note that a parallel and more complete quantum mechanical treatment exists for the probabilities of evolution of a spin's phase after being acted upon by a multitude of rf pulses. This treatment requires the usage of spin operator arithmetic which is not treated in this text. It is in keeping with the spirit of oneto-one equivalence of the two theories that, in the treatment given here, spin phase pathway `probabilities' are replaced with `observable signal amplitudes' and/or `fraction of isochromat population,' which is a measure of the population probability along a phase pathway. 18.3.1 Magnetization Rotation E ects of an Arbitrary Flip Angle Pulse
First, the e ect of an arbitrary ip angle rf pulse on any prior magnetization is considered. For this purpose, only the magnetization just following the rf pulse is of interest. From the Bloch equations, the magnetization response to an on-resonance -pulse applied along the 18.3. SSFP Signal Formation Mechanisms 483 (18.65) x-direction in the rotating frame is described by ~ ~ M + = Rx( )M ; if relaxation e ects can be neglected during the rf pulse (i.e., the rf pulse is assumed to be instantaneous). Expanding (18.65) into its component equations gives
+ Mx = Mx = Mx
; ; cos2 ! 2 + sin2 !! 2
; My+ = My; cos + Mz; sin = My; cos2 2 ; sin2 2 Mz+
= ;My sin + Mz cos = ;My sin + Mz
; ; ; ! !!
2 (18.66) cos2 ! + Mz; sin ; sin2 !!
2 (18.67) (18.68) Using the notation M+ = Mx + iMy for the complex transverse magnetization established earlier in Ch. 4 (Equation (4.29)), (18.67) and (18.68) can be combined to give
+ ; M+ = M+ cos2 2 + M+ sin2 2 + iMz; sin ! ! i + = M ; cos2 ; 2 ; Mz ; Mz sin 2 ; 2 ((M+ ); ; M+ ) sin z 2 ! ; ! (18.69) (18.70) In the above expressions, the isochromat populations M+ and M+ are counter-rotating partners. Since only the y-component of the transverse magnetization is rotated by the -pulse applied along x, these two counter-rotating isochromat populations are used to represent ^ My during free precession periods. Switching back to the notation of M+ and M; of Ch. 4 (Equations (4.29) and (4.31)) instead of the cumbersome notation of M+ and M+ , (18.69) and (18.70) can be rewritten as
+ M+ Mz+ 2 ! + iMz sin i ; ; = Mz; cos2 2 ; Mz; sin2 2 ; 2 (M; ; M+ ) sin = M+
; ; ; ; cos2 2! + M ! sin2 ! (18.71) (18.72) ; One needs to stop and consider what isochromat population M; represents. One way to think of it is as a ` ctitious, non-observable' representation of the tipped transverse magnetization which precesses in a direction opposite to the natural direction (the natural direction is clockwise in a positive eld o set from resonance). In this framework, the y-component of the transverse magnetization is made up of an observable and a non-observable component during the free precession period. Additionally, (18.71) shows that any arbitrary ip angle rf ; pulse ( 6= 0) is capable of converting the non-observable magnetization (M; ) into observable 2 ( =2) of it into M following the rf pulse (second term in magnetization by converting sin + (18.71)). In e ect, there is a change in the sense of precession of this magnetization, exactly Two Views of Thinking about the Population Represented by M; 484 Chapter 18. Fast Imaging in the Steady State the same e ect as a 180 -pulse would have had on observable transverse magnetization. It now seems necessary to include a third equation representing the fate of the non-observable population created by the rf pulse (obtained by complex conjugate of (18.71)):
; (18.73) 2 + M+ 2 ; i(Mz ) sin In other words, there are two parallel existences for the isochromat population, one as an observable sub-population and another as a non-observable sub-population, the rf pulse acting to mix the two populations and later leading to echoes. To state this behavior concisely, any rf pulse automatically creates both fresh observable (the third component of (18.73)) and non-observable transverse magnetization (third component of the complex conjugate version of (18.71)) from z-magnetization components. On prevailing transverse magnetization, the rf pulse partly converts from observable to non-observable magnetization and vice versa as well as partly preserving the population type. In a second view, which is purely classical with no quantum mechanical undertones, the ; explanation is based on the use of just (18.69) and (18.70). Here, the population M+ is interpreted as an isochromat population which precesses normally during the free precession period and whose phase is inverted instantaneously by the rf pulse. In this viewpoint, the rf pulse achieves an instantaneous complex conjugate of a sub-population of the prevalent transverse magnetization. This is the interpretation used in the remainder of this section. M+
; =M ; ; cos2 ! ; sin2 ! Equation (18.69) reveals that the fraction cos2 ( =2) of the population of spins originally in the transverse plane maintain their precession phase as it was before the rf pulse. From (18.70), the same fraction of the population of spins originally in the longitudinal direction stay there. Since this isochromat sub-population remains una ected by the rf pulse, it appears as though it has been e ectively acted upon by a 0 -pulse. The fraction sin2 ( 2 ) of the spin population originally in the transverse plane stay there after having their precession phase instantaneously inverted at the time of occurrence of the rf pulse. A similar fraction of any prior longitudinal magnetization is also inverted by the rf pulse. This part of the spin population appears to have been acted upon by an ideal 180 pulse. Finally, sin of the original y-magnetization is converted into negative z-magnetization (see (18.68)) and the same fraction of the original z-magnetization is converted into magnetization in the y-direction after the pulse. This spin-population has e ectively been acted upon by a 90 pulse. This discussion demonstrates that a single rf pulse appears to act as a weighted combination of three components: a 0 component of amplitude cos2 ( =2), a 90 component of amplitude sin and a 180 component of amplitude sin2 ( =2). Further, the y-magnetization that is converted into negative z-magnetization by the 90 component of the rf pulse consists of two sub-populations: half of this population maintains its phase prior to the rf pulse (fourth term in (18.70)) and the other half has its phase inverted prior to the end of the rf pulse (third term in (18.70)). Since all phase history is remembered by spins, these two sub-populations will reappear with opposite phases when they are reconverted to transverse magnetization by a later rf pulse. Interpretation of (18.69) and (18.70) 18.3. SSFP Signal Formation Mechanisms 485 We summarize next the e ects of the 0 -pulse, the 90 -pulse and the 180 -pulse on the precession phase of the magnetization after the rf pulse. The 0 -pulse does nothing to the magnetization phase development of isochromats in the transverse plane continues in the same sense as before the pulse and longitudinal magnetization stays in that direction recovering towards M0 . The 180 -pulse, on the other hand, causes phase rewinding following the rf pulse by inverting the accumulated phase of transverse magnetization prior to the pulse. This leads to a refocusing of all static magnetic eld inhomogeneities, and spin echoes are produced. In similar fashion, it also inverts any previously present longitudinal magnetization. The 90 -pulse creates transverse magnetization out of longitudinal magnetization and original transverse magnetization turns into longitudinal magnetization. The transverse magnetization converted to new longitudinal magnetization after the pulse remembers its phase history, i.e., its total accumulated phase up to that time-point this process is usually termed `phase storage.' It is important to note that (18.70) states that the 90 component converts half the free-precessed transverse magnetization directly into longitudinal magnetization whereas another half has its phase inverted before this phase is stored. These observations are summarized in Table 18.4. Flip E ective E ect on angle amplitude prior transverse magnetization prior longitudinal magnetization 2 ( =2) stays as transverse magnetization stays as longitudinal magnetization 0 cos continues to evolve later continues regrowth towards M0 90 sin tipped to longitudinal axis storage of accumulated phase and inverted phase in equal 1 numbers ( 2 sin ) during the following free precession period phase inverted rephased later tipping of fresh and stored magnetization to transverse plane magnetization inversion 180 sin2 ( =2) Table 18.4: Summary of the e ect of an arbitrary ip angle rf pulse. As a result of any arbitrary rf pulse acting as a combination of 0 -, 90 - and 180 -pulse components as discussed in Sec. 18.3.1, any repetitive sequence of rf pulses acting on a spin population leads to various combinations of echoes. In this section, the discussion focuses on the development of echoes in arbitrarily time-spaced multi-pulse experiments leading eventually to an interpretation of the SSFP signal as a combination of a multitude of echoes. This understanding is achieved by looking at the limiting case of an experiment containing an in nite number of periodically repeated rf pulses acting on some initial magnetization. The discussion progresses from a presentation of the formation of echoes in two- and threepulse experiments to arbitrarily spaced n-pulse experiments which lead to the SSFP limit. 18.3.2 Multi-Pulse Experiments and Echoes 486 Chapter 18. Fast Imaging in the Steady State Certain terminologies and pictorial representations which aid in the determination of echo locations and echo amplitudes are developed from the results of Sec. 18.3.1. Echoes from 90 - -90 Pulse Sequences: Hahn Echoes
In Ch. 8, the formation of a spin echo following a 90 -180 rf pulse pair was described. In this section, we will see how a partial spin echo is formed even when a 90 -90 rf pulse pair is applied (Fig. 18.16). Fig. 18.16: A 90 - -90 pulse sequence. Let us start with all magnetization along the z-direction (Fig. 18.17a). Assume that both 90 pulses are applied along the x-direction as shown in Fig. 18.16. The second pulse occurs a time after the rst pulse (Fig. 18.16). The rst pulse tips all magnetization down to the transverse plane (Fig. 18.17b). At time after the rst pulse, it is assumed that the isochromats are dephased and are uniformly distributed over the unit circle (Fig. 18.17c) because they experience a linear gradient as shown in Fig. 18.17a. The second 90 -pulse (occurring at t = ) rotates these isochromats spread over a unit circle lying in the x-y plane into the x-z plane, with the x-components preserved and the y-components converted to negative z-components (Fig. 18.17d). Assume that almost no longitudinal recovery occurs in the time . Starting at time t = +, the isochromats 2, 3, 4, 6, 7 and 8 with nonzero x-components just after the second 90 -pulse (t = + ) start precessing again, precessing by the same angle in the x-y plane as they did in the free precession period following the rst pulse. As a result, at time after the second pulse, all these isochromats end up on one side of the y-axis (Fig. 18.17e). This leads to a partial refocusing of magnetization along the negative y-axis, and is called the `Hahn echo.' Of course, since two of the eight isochromats (1 and 5) have been converted to longitudinal magnetization, the spin echo obtained has a smaller magnitude than the 90 -180 spin echo.9 It is for this reason that it is referred to as a partial echo.
For those interested in the complete spin picture including the z -components when the partial echo occurs, imagine a gure eight wrapped around a unit sphere's surface with the point of intersection of the two loops of the gure eight positioned at the point (0,-1,0) (the contour de ned by the equation = ). This contour de nes the locus of vectors originally uniformly distributed in the x-y plane.
9 18.3. SSFP Signal Formation Mechanisms 487 Fig. 18.17: Illustration of the refocusing of magnetization after two 90 pulses. (a) Magnetic eld seen by eight isochromats in the rotating frame. Isochromat 5 is on-resonance. (b) The rst 90 -pulse tips all z -magnetization into the y-axis. (c) Free precession in the presence of a gradient as shown in (a) occurs for the next seconds a set of of eight isochromats with phases uniformly spread over the unit circle at time after the rst pulse is considered. This is caused by the dephasing action of the eld gradient shown in (a). Remember that our convention is that the isochromats precess clockwise in the presence of a positive eld o set in the rotating frame. This makes the isochromats attain the positions shown in (c). (d) The second 90 -pulse rotates the eight isochromats by 90 with respect to the x-axis and puts them in the x-z plane. (e) Isochromats with nonzero x-components after the second pulse precess by the same precession angle as they did in the rst free precession period. Hence, seconds later, all of these spins end up with negative y-components this creates the Hahn echo. 488 Chapter 18. Fast Imaging in the Steady State Problem 18.11
Redraw with accompanying detailed captions Fig. 18.17 for the case when the magnetic eld shown in part (a) has a sign change. This problem is given as an exercise to ensure that the reader appreciates the di culties involved in arriving at the nal result using the vector model which uses a spin phasor pictorial representation. On the other hand, the quantitative reduction of the echo amplitude is easily obtained by looking at the population of spins in the transverse plane after the rst 90 -pulse which were acted upon by the 180 -like component of the second 90 -pulse. For a general -degree pulse, this population is given by sin2 ( =2) for = 90 , the echo amplitude is therefore half the magnetization that was tipped into the transverse plane by the rst 90 -pulse (refer to (18.69)), a result which is required to be shown pictorially in an involved fashion in Prob. 18.11. Problem 18.12 Based on the discussion about the e ects of arbitrary ip angle rf pulses on the magnetization, it is easy to visualize the formation of a weak echo in a sequence based on the generalization of the ip angles of the two rf pulses in the Hahn echo sequence to be arbitrary ip angles 1 and 2. a) What is the amplitude of this echo for a single pulse pair experiment? b) For 1 = 10 and 2 = 180 , what is the amplitude of this echo relative to a `full' spin echo? c) How does the echo amplitude change when 2 is changed to 60 ? How does this change when 1 = 60 and 2 = 180 ? This part of the problem is useful in redesigning short-TR spin echo sequences for improved T1 -weighted contrast. Problem 18.13
A problem closely related to Prob. 18.12 leads to a very interesting and important result which has some practical implications for spin echo imaging. Here we consider a conventional spin echo sequence using a ( =2)- pulse pair. Even in well-designed rf transmit coils, it is very di cult to create a homogeneous B1 eld over a large volume comparable to the coil's dimensions. 18.3. SSFP Signal Formation Mechanisms In such a case, in areas within the coil excitation volume where there are B1 eld inhomogeneities, the =2-pulse acts like a -pulse ( arbitrary) and the -pulse acts like a pulse with ip angle 2 . What is the relative amplitude of the spin echo signal obtained in such a region in comparison with a region where the B1 eld is correct so that a full echo is obtained? The computation of this relative amplitude would not have been as straightforward as it turns out here if one were to attempt this calculation without using the formulas and interpretation derived in Sec. 18.3.1. This reduced signal (assuming ) exacerbates the B1 inhomogeneity and is a hindrance when the goal is to obtain a uniform signal response over the volume of the coil. One such example is illustrated in Fig. 18.18. 489 (a) (b) Fig. 18.18: Signal drop-o in spin echo imaging due to rf eld inhomogneity. (a) Image of a uniform object when the object sees a homogeneous rf transmit eld. (b) Image of the same object when it is imaged in a coil whose volume of transmit eld homogeneity is smaller than the object dimension in the up-down direction. This shows the signal drop-o due to an inhomogeneous rf transmit eld. As expected, the signal drops o as sin3 . The Three-Pulse Experiment, Phase Memory and Stimulated Echoes
The previous discussion focused on the mechanism of formation of an echo following two =2-pulses (or any two arbitrary ip angle rf pulses). The addition of more =2-pulses to this pulse sequence will lead to multiple echoes. For example, a total of ve echoes can be observed with three 90 pulses (see Fig. 18.19). To understand the formation of these ve 490 Chapter 18. Fast Imaging in the Steady State Fig. 18.19: Pulse sequence for a generalized three-pulse experiment. echoes, it is necessary to discuss phase memory, spin phase pathways and coherence levels in detail. These and two examples for the three pulse experiment are addressed below. Phase memory is the phenomenon by which isochromats with a history of having been in the transverse plane during previous free precession periods remember their phase accumulated in the past. For example, any transverse magnetization which was recently converted to longitudinal magnetization remembers its earlier accumulated phase value with the same or opposite sign with equal probability. This phase is stored by the isochromats, and the stored phase will reappear when a second rf pulse reconverts them back to transverse magnetization. In the previous discussion, even while following just eight isochromats, it was already becoming di cult to keep track of each isochromat at the end of two pulses and a free-precession period following it. A new pictorial representation which is more intuitively appealing and less complicated is therefore developed to understand echo formation in multi-pulse experiments. We are interested in two aspects: locating the occurrence of echoes, and determining their amplitudes. Echo locations in time can be determined by following the phase of each transverse component and keeping track of the phase memory of any longitudinal component. The phase evolution of a given component between rf pulses is represented by so-called spin phase pathways which plot a non-dimensional quantity which re ects the phase development of several di erent isochromats as a function of time. Echo amplitudes can be determined by keeping track of the fraction of the magnetization based on (18.69) and (18.70) for the pathway leading to the echo of interest. To create the multitude of spin phase pathways, an initial transverse magnetization is allowed to evolve in phase along each possible phase evolution path by the action of rf pulses. This leads to what are called `extended spin phase diagrams.' To remain exact, the time evolution of amplitudes along each pathway between rf pulses should be relaxation-weighted as well. First, longitudinal magnetization pathways with phase memory have a T1 -dependent exponential decay during their stay along the longitudinal axis. i) phase memory ii) extended spin phase diagrams, spin phase pathways and spin con gurations 18.3. SSFP Signal Formation Mechanisms 491 Second, fresh longitudinal magnetization will grow exponentially towards the thermal equilibrium value. Third, T2 decay of transverse magnetization occurs in addition to the free precession which occurs in the period between rf pulses. These relaxation properties can be used to one's advantage in devising imaging schemes with special weighting mechanisms. Some terminology and descriptions about extended spin phase diagrams need to be established. First, the `spin phase diagram' is a tree of the evolving phase of a starting spin-population as a function of time as it is acted upon by multiple rf pulses each branch of this tree represents a `spin con guration' and each sub-pathway represents one particular fraction of the starting spin population, and is called a `spin phase pathway.' Second, each rf pulse is a branching point for a spin con guration there are either three or four branches (three branches originate from each longitudinal con guration and four branches originate from each transverse con guration) corresponding to new con gurations created by the 0 -like, the 90 -like and the 180 -like components of the pulse (see Fig. 18.20). Each branch has an associated weight that speci es the fraction of the magnetization which goes o into a given branching con guration. Third, an echo is said to occur along a spin pathway whenever there is a zero crossing by a particular con guration along that pathway. The vertex diagrams of Fig. 18.20 need to be understood rst. As mentioned in the previous paragraph, each rf pulse acts as a branching point for a pathway. Let us rst look at the four possible branches emanating from transverse con gurations (Fig. 18.20a). Suppose this con guration has dephased only during the free-precession period prior to the rf pulse. This isochromat population is split into four possible sub-populations: one which continues to dephase (by the 0 component solid line in Fig. 18.20a), one whose accumulated phase is inverted by the rf pulse and will form an echo (by the 180 component dashed line in Fig. 18.20a), one which is converted to Mz which stores the phase accumulated prior to the rf pulse (by the 90 component dot-dashed line in Fig. 18.20a) and one which is converted to Mz where the prior phase accumulation is inverted prior to storage (by the 90 component dotted line in Fig. 18.20a). Associated fractions of the population along each branch are given in the gure. Three branches are similarly created from each longitudinal con guration, one acted on by the 0 component which continues to stay along z (solid line in Fig. 18.20b), a ^ second by the 90 component which is converted to transverse magnetization (dashed line in Fig. 18.20b) and a third by the 180 component which puts it along -^ (dotted line in z Fig. 18.20b). Since two sub-populations coincide along z, it is convenient to take their sum ^ (yielding a fraction cos ) and consider only two branches emanating from any longitudinal con guration prior to an rf pulse. iii) coherence levels One of the interesting notions which results from the vertex diagrams of Fig. 18.20 is that of a coherence level. To introduce the concept of a coherence level, let us consider a periodically repeated rf pulse experiment. Suppose we are interested in the magnetization at the end of the (N ; 1)st repetition period. Let a single 90 pulse be placed arbitrarily at the end of one of the repetition periods in this cycle and a single 180 -pulse be placed at the end of the N th repetition period. The rest of the rf pulses are 0 rf pulses. Let the 90 -pulse occur after m repetition periods, where 0 m (N ; 1). Now, consider the 180 -pulse at the end of the N th repetition period. Prior to the -pulse, the transverse magnetization would have gone through n dephase periods where n ranges from N to 1 corresponding to m = 0 to 492 Chapter 18. Fast Imaging in the Steady State Fig. 18.20: Vertex diagram at an arbitrary rf pulse for (a) a transverse con guration, and (b) a longitudinal con guration. The corresponding fractions of the population prior to the pulse which belong to a given branch are also marked in the vicinity of each newly created con guration. Here, n represents the number of dephasing cycles a given con guration has undergone and is referred to as the `coherence level.' The numbers along the pathways represent the amplitude reductions. 18.3. SSFP Signal Formation Mechanisms 493 m = (N ; 1), respectively. The -pulse creates rephasing of the transverse magnetization, whereby the n = 0 in-phase condition is reached only after (N ; m ; 1) further repetition periods following the -pulse.10 It must be remembered that each repetition period following the -pulse adds unity to the number of dephase periods a spin con guration has seen. Clearly, n must be negative immediately after the -pulse for the n = 0 condition to be reached later, the value of n being given by (N ; m) and the possible values ranging from ;N to -1. From the previous discussion, it is seen that the number n qualitatively describes the condition of the isochromat population: `dephased' (n > 0), `rephasing' (n < 0) or `fullyrephased' (n = 0). This description of an entire sub-population of isochromats is achieved without necessarily delving into the details of how each individual isochromat's phase has evolved (which depends on B and TR ). Because of the quantized integer nature of this qualitative description of the phase evolution of the isochromats, the quantum number n is called a `coherence level' representing isochromats experiencing di erent elds, but all of which can be coherently moved to another level by a new rf pulse. Since the coherence level represents the condition of the isochromats on a time scale which is quantized in terms of an arti cially-introduced, xed repetition period and the spin phase continues to evolve linearly with time, the magnetization state at any intermediate time-point is described by a linear increase from one coherence level to the next (upper) coherence level during each free precession period. Let this B -independent quantity representing the coherence level occupied by a particular con guration viewed as a continuous function of time be denoted by q(t). To get the actual phase (t) of some isochromat experiencing a static eld inhomogeneity of B , the resonance o set angle per repetition period, BTR , is multiplied by q(t), that is,
(t) = BTR q(t) (18.74) iv) two speci c examples of echo amplitude computation for the three-pulse experiment Using an extended spin phase diagram, the ve echo forming pathways for the three-pulse sequence shown in Fig. 18.19 can be obtained. The extended spin phase diagram for this sequence is shown in Fig. 18.21. The times of occurrence of the ve echoes are easily determined from this diagram. This diagram also provides the amplitudes of the individual echoes as described in the next few paragraphs. Of all the ve echoes, one of them, (echo number 3 in Fig. 18.19) is the most interesting. Unlike the rest, all of which echo through the refocusing action of the 180 -pulse component of the -pulse while being in the transverse plane, this echo is formed from the refocusing action of the 90 -like component of the -pulse acting on a fraction of longitudinal magnetization which has stored phase history. This echo is called a `stimulated echo,' and requires the concept of phase memory to describe its formation. Echo amplitudes are easily obtained. Indexed transverse magnetization vectors M are used to represent di erent con gurations. An index 0 indicates phase storage (occupying a
10 Here n is used to symbolize the number of dephase periods experienced by a spin con guration and the adjective `in-phase' symbolizes that each isochromat represented by a con guration has a phase which equals its starting phase value when it was rst converted to transverse magnetization. 494 Chapter 18. Fast Imaging in the Steady State Fig. 18.21: Extended spin phase diagram as a function of time for an arbitrary-angle three-pulse experiment. The abscissa is time and the ordinate is phase. Note that the spin populations under consideration along any diagonal pathway have phase changes, whereas those forming the horizontal pathways have no phase changes during the free precession period. It should be noted that the con gurations labeled M0 , M0 0 and M0 0 0 are the least interesting as these represent the spin population growing towards thermal equilibrium and have experienced no prior dephasing. The three pathways M1 ;1 1 , M0 0 1 and M1 0 1 are not shown since they do not contribute to any further echoes after 3 in this three pulse sequence. 18.3. SSFP Signal Formation Mechanisms 495 nonzero, positive coherence level) or regrowing longitudinal magnetization (occupying the zero coherence level) an index 0 indicates storage of inverted phase (occupying a nonzero, negative coherence level) an index 1 indicates continued dephasing of transverse magnetization and an index -1 indicates rephasing because of phase inversion. As a spin population experiences multiple pulses, more indices are added to each con guration in chronological order (see, for example, Fig. 18.21). For example, the pathway M0 1 ;1 represents a spin con guration which stayed in the longitudinal plane after the rst rf pulse, then got tipped to the transverse plane by the second rf pulse, dephases in the period between pulses 2 and 3 (coherence level 1), is inverted instantaneously by the second rf pulse (instantaneous jump to coherence level -1) and then rephases during the second free precession period to form an echo (echo SE4 in Fig. 18.21) 2 seconds following the third pulse (coherence level 0). See Fig. 18.22a for calculation of the echo amplitude. Starting with unit longitudinal magnetization, cos 1 stays as longitudinal magnetization, sin 2 of that fraction is converted to transverse magnetization and nally sin2 ( 3 =2) of this fraction forms the echo of interest. The echo amplitude is therefore given by cos 1 sin 2 sin2 ( 3 =2).11 Similarly, M1 0 1 represents a spin pathway with an isochromat sub-population which was converted from longitudinal to transverse magnetization by the rst pulse, gained phase between pulses 1 and 2 (coherence level 1), got stored as longitudinal magnetization at the second pulse with its phase inverted (coherence level -1), and got reconverted to transverse magnetization which rewinds its earlier phase after the third pulse and echoes (coherence level 0). This sub-population would therefore echo following the third pulse after a time equal to the period between the rst and the second pulse equaling the dephasing time duration. Refer to Fig. 18.22b for computation of this echo's amplitude. Starting with unit magnetization before pulse 1, sin 1 of that population gets converted to transverse 1 magnetization (from (18.69)) 2 sin 2 of this population then gets stored as longitudinal magnetization with its phase inverted and sin 3 of this population gets converted into transverse magnetization, which rewinds to form the stimulated echo. The echo amplitude is therefore given by 1 sin 1 sin 2 sin 3 . 2 Problem 18.14
Show that when relaxation e ects are included in the calculation of the relative stimulated echo amplitude for the M1 0 1 con guration (echo number 3 in 1 Fig. 18.19) is 2 sin 1 sin 2 sin 3 e;2 1 =T2 e; 2 =T1 where 1 and 2 are de ned in Fig. 18.19. How does this expression change when the signal is the resultant of a few di erent isochromats, and T2 e ects have to be considered? (Hint: Remember that there is T1 decay of the stored magnetization during the period 2 .) Relaxation e ects are neglected here, but the results with relaxation e ects included for all ve echoes are given in Table 18.5.
11 496 Chapter 18. Fast Imaging in the Steady State Fig. 18.22: Two particular pathways from Fig. 18.21 with the fraction of spin population associated for each of the free precession periods indicated. (a) Pathway (M0 1 ;1 ) involved in the formation of the echo SE4. (b) Pathway (M1 0 1 ) involved in the formation of the stimulated echo (SE3). Echo Time of echo Amplitude SE1 TE 1 = 2 1 sin 1 sin2 22 e;TE 1=T2 SE2 TE 2 = 2 2 sin 1 sin2 22 sin2 23 e;TE 2=T2 1 sin sin sin e; 2 =T1 e;2 1 =T2 SE3 TE 3 = 2 1 + 2 1 2 3 2 ; 1 =T1 SE4 TE 4 = 1 + 2 2 1 ; (1 ; cos 1 )e ] sin 2 sin2 23 e;2 2 =T2 SE5 TE 5 = 2( 1 + 2 ) sin 1 cos2 22 sin2 23 e;TE 5=T2 Table 18.5: Amplitudes and times of occurrence of the ve echoes in the three-pulse arbitrary ip angle experiment. 18.3. SSFP Signal Formation Mechanisms Echo Description Pathway forming echo SE1 Hahn partial echo M1 ;1 SE2 virtual echo M1 ;1 ;1 SE3 stimulated echo M1 0 1 SE4 echo from pulse 2 M0 1 ;1 SE5 echo from pulse 1 M1 1 ;1 497 Table 18.6: Descriptions of, and pathways forming, the ve echoes in the general three-pulse
experiment. Problem 18.15
Based on the pathways given in Table 18.6, verify the amplitudes of the ve echoes SE1, SE2, SE3, SE4 and SE5 as given in Table 18.5. How do these respective amplitudes change if T2 dependence is considered? Discuss why SE2 is referred to as a virtual echo and does not exist for certain pulse timings. (Hint: Consider the case when 2 is less than 1 .) Maximum Number of Echoes in a Stopped-Pulse Experiment
The spin phase diagram can easily be extended to arbitrary multi-pulse sequences. A stopped-pulse experiment is one where a set of rf pulses spaced arbitrarily from each other is stopped abruptly after a number of rf pulses. The two-pulse and three-pulse experiments discussed before are just speci c examples of stopped-pulse experiments. Once the rf pulses are stopped, no phase inversion of transverse con gurations can occur nevertheless, the phase of any transverse con gurations already created continues to evolve. As a result of this phase evolution, those pathways occupying negative coherence levels following the last pulse will echo at later time instants. The maximum number of echoes that can be obtained at the end of a train of n pulses needs to be determined. For this purpose, the maximum possible number of transverse con gurations following the nth pulse (Tn) needs to be determined rst. Tn can be determined iteratively using mathematical induction. It must be remembered that Tn has contributions from the longitudinal con gurations present prior to the nth pulse (Zn;1, say). Iterative, inter-dependent expressions will next be obtained for Tn and Zn. i) iterative determination of Tn There are Tn;1 transverse con gurations created by the 0 -like component and the 180 -like component of the nth pulse acting on the previous transverse magnetization. A single new transverse con guration is created from fresh longitudinal magnetization by the (n ; 1)st pulse. Finally, Zn;1 transverse con gurations are created from longitudinal magnetization 498 Chapter 18. Fast Imaging in the Steady State occupying nonzero coherence levels. (Zn;1 is de ned as the number of longitudinal con gurations occupying a nonzero coherence level following the (n ; 1)st pulse.) Therefore, Tn = 2Tn;1 + Zn;1 + 1
Similarly, (18.75) (18.76) 3n;1 as seen in Zn = 2Tn;1 + Zn;1 With T0 = Z0 = 0, applying the above two formulas iteratively gives Tn Prob. 18.16. Problem 18.16 In this problem, we derive the result that Tn = 3n;1. a) First, show from (18.75) and (18.76) that Tn satis es the recursive equation Tn = 3Tn;1. b) Hence, obtain the result that Tn = 3n;1 by applying the initial condition. Except for the one newly formed transverse con guration, the remaining con gurations are equally distributed amongst positive and negative coherence levels after the nth pulse. Amongst these con gurations, only those occupying negative coherence levels will echo at a later time. This implies that the maximum number of echoes En which occur after the nth pulse equals n;1 En = (3 2 ; 1) (18.77) which equals the number of negative coherence level transverse con gurations. This maximum number can be achieved only if the timings between the n pulses are designed carefully so that none of these echoes overlap in time. In essence, rf pulse timing is a crucial determinant of whether the number of echoes increases only linearly with the number of pulses (as occurs when the inter rf pulse period is a constant), or reaches the maximum limit of (18.77). For example, in a three-pulse stopped-pulse experiment, En E3 = 4. Remember that En represents only the number of echoes which occur at the end of the rf pulse train, and the additional multiple echoes which occur in between adjacent rf pulses are not counted. These echoes are considered to be `virtual' echoes, as they are not utilized in such stopped-pulse experiments. It is worthwhile to note that the En echoes are formed by pathways which go through both possible echo formation mechanisms, either due to the refocusing e ect of a given rf pulse (the 180 component) acting on dephased transverse con gurations, or due to the stimulated echo formation mechanism. The utility of a stopped-pulse approach is that, if these echoes are well-separated, a complete image may be obtained by phase encoding each echo di erently to cover the k-space required for image reconstruction. For example, a 7-pulse experiment leads to a maximum of ii) maximum number of echoes 18.3. SSFP Signal Formation Mechanisms
(36 ;1) 2 499 echoes this provides the number of echoes needed to obtain 365 phase encoding steps, provided that there is enough separation between successive echoes to allow the application of the phase encoding gradient and to allow coverage of the required k-space sampling window in the read direction. Problem 18.17 a) What is the number of echoes for an n-pulse stopped pulse experiment when the spacing between the rf pulses is equal? b) Many other echoes (as given by En) are formed. Where do they appear? c) Can each echo therefore be considered the sum of several echoes? The answers to the questions in this problem provide the basis of the explanation behind the formation of the coherent steady-state signal. SSFP Signal as a Sum of Multiple Echoes
The motivation for the detailed discussion of multi-pulse experiments was to lead the reader to understand the formation of multiple echoes in arbitrary multi-pulse experiments. The ultimate aim is to understand the constituent parts of the coherent steady-state signal which is obtained in the in nite pulse number limit. The SSFP experiment is actually a rather easy-to-visualize version of the arbitrary multipulse experiment. Here, the rf pulse separation is xed (to be TR ) so that, in steady-state, multiple overlapping echoes occur in a periodic fashion, concurrent with each new rf pulse. As we have seen, half of the transverse spin con gurations following an rf pulse will form echoes some time later. The free precession period of any con guration is an integer multiple of TR . Therefore, multiple spin phase pathways created by previous rf pulses and occupying coherence level -1 following the previous pulse converge to form echoes at the instant of occurrence of each new rf pulse. When the spin system reaches steady-state, each of these echoes, aptly named `rf echoes,' are formed by an in nite sum of echoes created from an in nitely large collection of pathways, each pathway having di erent relaxation weighting factors and, hence, di erent amplitudes. After the rf pulse ends, yet another new free precession period begins, all possible con gurations are created and an FID is produced. In summary, the SSFP signal is composed of an rf echo which coincides with each new rf pulse's occurrence and an FID during the following free precession period. A plot of the SSFP signal from one rf pulse to the next will therefore consist of a decaying envelope (the FID) just after the rf pulse which rises to a maximum at the next rf pulse (the rf echo). This is illustrated in Fig. 18.23. 500 Chapter 18. Fast Imaging in the Steady State Fig. 18.23: The signal collected after one rf pulse up to the next following the establishment of coherent steady-state. The FID was collected on a water/fat phantom, which is the reason for the periodic modulation of the real and imaginary parts of the FID. Note the initial decaying part, the FID, and the increasing envelope towards the end of the free precession period to form the rf echo. 18.4 Understanding Spoiling Mechanisms
A detailed discussion of spoiling using mechanisms other than the natural T2 signal loss in cases where TR is comparable to T2 has been postponed till now since understanding the process requires some of the concepts discussed in the previous section. This section discusses, from basic principles, the means by which the signal in a short-TR experiment can be made to approach the SSI limit using external spoiling mechanisms. 18.4.1 General Principles of Spoiling When TR is on the order of or less than T2 , a coherent steady-state magnetization with nonzero transverse components occurs prior to each rf pulse as long as the resonance o set angle for any isochromat is independent of rf pulse number. As discussed at the end of Sec. 18.3, the characteristic of this SSFP buildup is that several pathways add together coherently at each new rf pulse to create a refocusing or `echo' at the instant of each new rf pulse.12 Of course, there are several other pathways with transverse con gurations which do not contribute to the echo, and still contribute to the signal following the current rf pulse. However, these con gurations are vulnerable to dephasing due to T20 mechanisms and the presence of gradient waveforms with nonzero moments that are applied during the current free precession period. To create SSI equilibrium means that all transverse magnetization con gurations within each voxel must be forced to have zero contribution to the signal by the end of each free precession period.
The echoing pathways are comprised of only those isochromats with transverse con gurations in coherence level -1 or longitudinal con gurations in coherence level 0 immediately after the previous rf pulse.
12 18.4. Understanding Spoiling Mechanisms 501 It is easy to achieve dephasing just before each new rf pulse for the con gurations not ~ forming echoes. The presence of some xed gradient waveform G(t) between each pair of adjacent rf pulses which satis es Z Z 3 r TR dt~ G(t) = 2 d r ~ (18.78) for each repetition cycle ensures that
voxel
0 =0 (18.79) i.e., all these con gurations within a voxel are completely dephased. On the other hand, this ~ rf pulse number independent gradient waveform G(t) behaves like a static eld inhomogeneity, and its e ect is `refocused' for all echo forming pathways. As a result, their coherent buildup towards rf echo formation is undisturbed. To eliminate the contribution of the various echoing pathways to the voxel signal, their phases need to be scrambled in such a manner that they add together in a destructive fashion and do not, in e ect, contribute to the signal following the rf pulse. This phase scrambling is achieved by making the resonance o set angle a function of rf pulse number. There are two ways of achieving this: one is to use an rf pulse number dependent rf phase, and the second is to use variable gradients whose strengths vary as a function of rf pulse number, i.e., by using unrefocused gradient tables between rf pulses. The rst is known as `rf spoiling' and the second as `gradient spoiling.'
voxel Z r T ~ d3re;i ~ 0 R dtG(t) R Problem 18.18
~ Consider a voxel where an extraneous gradient G0 is present in addition to the ~ . Let jG0j jGj. Discuss why the use of G such that it ~ dephasing gradient G distributes isochromats within a voxel over a large integer multiple of 2 in phase is recommended in this case. 18.4.2 A Detailed Discussion of Spoiling In an imaging experiment, the aim of spoiling is to eliminate the contribution of any prior transverse magnetization to each voxel's signal following a new rf pulse. It should be remembered that in the absence of any gradients, the magnetization in each voxel can be considered to be formed by one set of isochromats because B (~) is usually a slowly varying function r of ~. In the ensuing discussion, it is critical to realize that we need to be considering a set r of isochromats within an imaging voxel which see a variation in resonance o set angle from ; to . This variation is necessary to force the vector sum of the transverse magnetization components generated by certain pathways to zero. We will see that spoiling is achieved by a combination of a dephasing gradient and an rf pulse number dependent addition to the free precession resonance o set. The former serves the purpose of dephasing the non-echo forming con gurations, and the latter serves the purpose of creating a pathway-dependent phase at a given rf pulse such that all echo forming pathways vectorially add to zero just before each new rf pulse, achieving the conditions necessary for SSI steady-state formation. 502 Chapter 18. Fast Imaging in the Steady State Spin Phase Diagram Explanation with Examples
The e ects of dephasing gradients and a pulse number dependent additional phase on di erent pathways are best illustrated with spin phase diagrams. (In Fig. 18.24 through 18.27, the ordinate represents the phase of the spin isochromats.) Let us consider the three pathways shown in Fig. 18.24 with transverse con gurations prior to the fth rf pulse. In the freely precessing case, two of these pathways (pathways 1 and 2 in Fig. 18.24) end up with zero phase (creating a partial echo) while the third pathway (pathway 3 in Fig. 18.24) ends up with a nonzero phase. In general, this leads to a complex voxel signal after rf pulse number 5. Fig. 18.24: Di erent spin pathways add together to create the SSFP signal and the rf echo.
the resonance o set angle. is Gradient E ects
With a constant gradient introduced between each adjacent pair of rf pulses, several isochromats with position-dependent precession frequencies are formed within the voxel. The population amongst these isochromats which evolve along phase pathways 1 and 2 still rephase fully just before the fth rf pulse and form an rf echo (see Fig. 18.25a). However, the isochromats evolving along pathway 3 develop di erent phase values and are dephased by the presence of the gradient (see Fig. 18.25b). The amount of dephasing and hence these isochromats' contribution to the signal following the fth rf pulse depends both on the strength and duration of the gradient. With this in mind, we ignore pathway 3 for the next discussion assuming that it contributes very little or no signal at the end. E ects of RF Pulse Number Dependent Phase
With the inclusion of an additional phase to each pathway which is a function of the rf pulse number n, say '(n), the pathways 1 and 2 do not add together in phase anymore (see Fig. 18.26). However, the e ects of any magnetic eld inhomogeneity which is independent of the repetition period number are still refocused in these pathways. This includes the presence of the constant gradient summoned in Fig. 18.25 to dephase the non-echo forming 18.4. Understanding Spoiling Mechanisms 503 (a) (b)
of SSFP equilibrium. Again, echoing pathways come together in phase at di erent rf pulses while non-echoing pathways become dephased. (b) The presence of a gradient of xed strength occurring every repetition period creates many isochromats within a voxel. The spin population within each isochromat (de ned by all spins corresponding to a given position ~) which evolve along a non-echo r pathway such as pathway 3 gain di erent phase values. This leads to their complete dephasing when the phase of the di erent isochromats ranges from ; to . The end result is that these pathways do not contribute to the FID following each new rf pulse. Fig. 18.25: (a) Presence of a constant gradient between each rf pulse does not damage the formation 504 Chapter 18. Fast Imaging in the Steady State pathway 3. Because of their di erent phases, the combined contribution to the signal of pathways 1 and 2 following rf pulse 5 is reduced. Fig. 18.26: Introduction of extraneous phase which is a function of the rf pulse number makes the phase of a given con guration depend on the pathway it goes through to reach the rf pulse. This prevents constructive interference of di erent con gurations and promotes destructive interference of di erent con gurations just before the next rf pulse. Generalization and Extension to Multiple Spin Phase Pathways
In all the scenarios discussed before, only two broad classes of generalized spin phase pathways were considered: the echo forming pathways and the non-echo forming pathways. Let fpij g be the indices representing the state of the spin population forming pathway i's conguration during the j th free precession period (for example, see Fig. 18.27). For example, since pathway 1 corresponds to M1 1 ;1 1 (using the notation of Fig. 18.21), p11 = 1, p12 = 1, p13 = ;1 and p14 = 1. Assume that B does not change within the voxel, i.e., the voxel can be considered an isochromat. Each isochromat's phase is therefore incremented in any evolution period by BTR as long as pij does not equal 0. At the rf pulse, the phase accumulated prior to the rf pulse is either maintained (for the cases pij = 0 or 1) or is inverted (pij = -1 or 0 ). Therefore, the phase of isochromats forming pathway i just before the (n +1)st pulse satis es the relation: 8 > pin i(n) + < (n + 1) = > i(n) i : ; i (n) l pin 6= 0 pin = 0 pin = 0 (18.80) 18.4. Understanding Spoiling Mechanisms 505 Fig. 18.27: Illustration of the use of spin phase pathway indices using the example of pathway 2. Hence, i equals 2, and j varies for each TR . where l is an integer ranging from ;(n ; 2) to n.13 Amongst the multitude of possible pathways, only those with transverse con gurations satisfying i(n) = 0 (i.e., pathways with same number of periods of precession before and after phase inversion) form the so-called `rf echo' at the nth rf pulse. The non-echo pathways, representing isochromats whose phase values span the whole range of ;(n ; 2) to n , have a partially reduced contribution to the signal following the nth rf pulse. The reason that their contribution is only partially reduced is because the fractions of the population contributing to each pathway (and the relative amplitude of a given pathway's contribution to the signal) are still di erent from one another. Luckily, there are some special cases where the combined signal essentially sums to zero. With a constant gradient turned on for some xed time in every repetition period, the phase accumulated up to pulse n additionally becomes a function of position ~. However, it r is still independent of n. That is, the accumulated phase still obeys (18.80) with i(n + 1) replaced by the phaseRaccumulated in the presence of gradients, grad i(~ n + 1), and by r TR dtG(t).14 The e ect on echo forming pathways remains the same. ~ (~) r BTR + ~ 0 r These pathways continue to form echoes which add together coherently. The gradient causes isochromats from di erent locations within each voxel to go to a di erent SSFP equilibrium magnetization because of the spatially varying resonance o set. When the gradient is such that it causes a 2 phase shift across a voxel, the voxel signal is a resonance o set averaged steady-state signal. One distinction is that, unlike SSFP, the non-echoing pathways are dephased, and they do not contribute to the signal, unlike in the free precession case. With an rf pulse number dependent extraneous phase '(~ n) added, the total accumur lated phase total i(~ n + 1) in the most general case is given by r r total i (~ n + 1) = r grad i (~ n + 1) + 'i(~ n + 1) r (18.81) pulse, where a phase inversion occurs. The isochromats precess for another TR , picking up an extra . On the other hand, l = n corresponds to the complete continuous phase evolution case. 14 Here, ~ is the position inside a voxel and B is assumed to be constant inside a voxel and, consequently, r contains no ~ dependence. r where 'i(~ n + 1) is the phase accumulated by pathway i due to this extraneous phase. r As reasoned before, the non-echo pathways are ignored for this discussion too. For all the isochromats evolving along the di erent echo forming pathways, the contribution from the 13 l = ;(n ; 2) corresponds to that pathway where there is a continuous evolution of phase up to the nth 506 Chapter 18. Fast Imaging in the Steady State rst term in the above sum goes to zero. The second term creates a pathway-speci c phase accumulation for the originally echo forming pathways such that there are equal number of positive and negative rf phase terms added to obtain this value. Hence 'i(~ n) = r
= r total i (~ K X
k=1 n) for all echoing pathways K X '(~ n ; mk ) ; '(~ n ; ml ) r r
l=1 (18.82) where fmk g and fml g represent those rf pulse numbers relative to pulse number n at which there is either continued phase evolution, or phase inversion, respectively. The number 2K ( n ; 1) represents the total number of free precession periods for which the spin population along pathway i stays in the transverse plane. If the fraction of the spin population (or their corresponding amplitudes) forming each of these pathways is the same, the simple requirement to spoil the signal would be to let '(~ n) span the entire range of phase values ranging from ; to so that they will be r dephased. Their relative amplitudes not being the same, however, means that the form for '(~ n) needs to be obtained numerically. r As brie y mentioned before, there are two ways of adding an rf pulse number dependent phase to the transverse magnetization: variable rf phase, and varying gradients. In the rst case, there is no spatial dependence on ', and is replaced by 'rf (n). The use of a variable gradient table for spoiling, on the other hand, creates a position dependence of '. This means that spin populations evolving along the same pathway but at di erent positions in the image will be spoiled to di erent degrees, creating signal inhomogeneities. This aspect becomes more apparent after a discussion of rf spoiling, since the gradient spoiling method creates a spatially-varying continuum of rf pulse number dependent phase values while the rf spoiling method creates a spatially constant phase. Hence, gradient spoiling is understood by a simple extension of the rf spoiling method. 18.4.3 Practical Implementation of Spoiling RF Spoiling
All that needs to be determined here is a simple function for a variable phase 'rf (n) which, in the steady-state, will create an e ective signal which equals the SSI signal. This method of spoiling by varying the phase of the rf pulse as a function of rf pulse number is called `rf spoiling.' One observation is crucial to understanding the form of the function 'rf (n). At steadystate, the same combinations of di erent echo forming pathways with the same amplitude add together at each new rf pulse. Therefore, the phase of each such echo forming pathway fig with rf spoiling should be independent of rf pulse number for the attainment of steadystate by the magnetization, i.e., X
k 'rf (n ; mk ) ; X
l 'rf (n ; ml ) independent of n (18.83) 18.4. Understanding Spoiling Mechanisms 507 The condition (18.83) is satis ed only by a linearly varying function of rf pulse number. Hence 'rf (n) must be of the form 'rf (n) n 1 + 0 (18.84) 0 can be made equal to zero without any loss of generality because a constant rf phase only adds a constant phase to the reconstructed image. Determination of 1 is done numerically, as brie y mentioned before. This numerical procedure is described next. Fig. 18.28: Plot of rf spoiled signal as a function of 1 for gray matter at a TR of 15ms. The ideal SSI equilibrium signal for each tissue is plotted as a dotted line to indicate the equality of the two signals when 1 is either 117 or 123 . A similar equality of the rf spoiled signal and ideal SSI signal is found to occur for other tissues too. The time evolution of a set of isochromats with resonance o set values covering the entire set of phase values from - to + is simulated using the Bloch equations with relaxation terms included. These isochromats are supposed to represent the entire set of spins within a voxel ~ which experience a constant gradient G during each precession period, creating the required resonance o set distribution within the voxel. To each isochromat's position-dependent 508 Chapter 18. Fast Imaging in the Steady State resonance o set, a position-independent but linearly increasing rf pulse number-dependent phase n 1 is added during each precession period. The steady-state signal is estimated as the magnitude of the complex sum of the transverse components of these isochromats after an evolution time of about 3T1 (by which time it is assumed that the spins have reached steady-state as described in Sec. 18.1.2). It is found that a value of 1 of 117 or 123 makes the steady-state rf spoiled signal equal to the ideal SSI equilibrium value for all values of E1 E2 and of practical interest. One such example steady-state signal as a function of 1 is plotted for gray matter (T1 = 950 ms and T2 = 100 ms) for a TR of 15 ms (which is much shorter than the T2 's of di erent tissues in the brain) in Fig. 18.28. This plot shows that the use of an 1 value of either 117 or 123 is successful in achieving the SSI signal at steady-state. The same rf spoiled signal for 1 = 117 is plotted as a function of ip angle in Fig. 18.29, and compared with the ideally expected SSI equilibrium signal as a function of ip angle at a TR of 25 ms. The close matching of the two curves shows the success of rf spoiling even at extremely short TR s when 1 = 117 . A similar result is obtained for 1 = 123 . Fig. 18.29: Comparison of rf spoiled signal (solid line) as a function of ip angle for gray matter with the ideal SSI signal (dashed line) at a TR of 25ms, and 1 = 117 . Note the close matching of the two curves for all ip angles indicating the success of rf spoiling. Similar results are obtained for other tissues as well. Gradient Spoiling
The presence of a linearly increasing unbalanced gradient table in each precession period also satis es the condition (18.84) at each position along the direction of application of the gradient table. However, the phase increment, 1 , takes on all possible values as a function of position. As seen in Fig. 18.28, there are multiple 1 values where the steady-state signal di ers from the ideal SSI signal signi cantly. At positions which experience this 1 value, the signal is overwhelmingly large and di erent from the signal at other positions. That is, the gradient-spoiled signal will be spatially varying. This e ect is observed when images of homogeneous objects are obtained using large ip angles, where the coherent steady-state 18.4. Understanding Spoiling Mechanisms 509 signal can be quite large in comparison with the SSI signal when certain special conditions are met. Problem 18.19 The spatial inhomogeneity of a gradient-spoiled image can be taken advantage to measure 1 . This problem introduces this application of a gradient-spoiled sequence structure. a) As seen from Fig. 18.28, the spoiled signal is a maximum when 1 equals zero. For a 2D imaging sequence with an unbalanced phase encoding gradient (with the component of the sequence labeled (B) removed from the sequence shown in Fig. 18.30) and no rf phase increment, is there a position along the phase encoding direction which satis es this? If so, which position? b) Show that the position where this signal maximum occurs is shifted in the phase encoding direction when a constant rf phase increment 1 is introduced additionally. Relate this shift to 1 . c) How much is this position shift in number of voxels for 1 = 117 ? Fig. 18.30: An rf spoiled implementation of an SSI imaging sequence. Note that the areas under
the read gradient during the second half of sampling and the rest of the read gradient waveform (shown with denser shading) are the same. 510 Chapter 18. Fast Imaging in the Steady State 18.4.4 RF Spoiled SSI Sequence Implementation This subsection serves as a summarizing point for explaining the special features of a typical implementation of an rf spoiled SSI sequence shown in Fig. 18.30. The salient features of interest are labeled and highlighted in Fig. 18.30. As shown in the gure, a dephasing gradient (labeled (A)) is applied in every repetition period and the rf phase (labeled (C)) in the rotating frame is incremented by 117 or 123 . Further, all phase encoding gradients are refocused (the refocusing lobes being labeled (B)). Feature (B) ensures that no spatial inhomogeneity of signal as in a gradient-spoiled experiment is seen. Feature (A) dephases all the non-echoing pathways during each precession period. For example, when used in the read direction, this dephasing gradient lobe Gdephase (t) must satisfy the rst moment condition Z TR (Gdephase (t) + GR(t))dt mGTs (18.85) 0 where m is any positive integer and G is the frequency encoding read gradient amplitude. 18.4. Understanding Spoiling Mechanisms 511 Suggested Reading
The following article covering both the coherent and incoherent steady-states, gives expressions for ip angles which maximize the signal (hence the name `Ernst angle'): R. R. Ernst and W. A. Anderson. Application of Fourier transform spectroscopy to magnetic resonance. Rev. Sci. Instrum., 37: 93, 1966. Using just classical Bloch equation solutions, the following paper rst described the action of any arbitrary rf pulse as a composite of 0 , 90 and 180 pulses in its action on magnetization: D. E. Woessner. E ects of di usion in nuclear magnetic resonance spin echo experiments. J. Chem. Phys., 34: 2057, 1961. Spin-phase diagrams to locate echoes and determine their amplitudes when a set of rf pulses act on a population of spins are described in: R. Kaiser, E. Bartholdi and R. R. Ernst. Di usion and eld-gradient e ects in NMR Fourier spectroscopy. J. Chem. Phys., 60: 2966, 1974. A method for phase encoding rf echoes resulting from a stopped-pulse experiment to obtain images was rst described in: J. Hennig and M. Hodapp. Burst imaging. Magma, 1: 39, 1993. A speci c solution to the rf spoiling problem is given in: Y. Zur, M. L. Wood and L. J. Neuringer. Spoiling of transverse magnetization in steady-state sequences. Magn. Reson. Med., 21: 251, 1991. The next three articles listed here are modern reviews of the eld of steady-state, fast imaging methods: E. M. Haacke, P. A. Weilopolski and J. A. Tkach. A comprehensive technical review of short TR, fast magnetic resonance imaging techniques. Rev. Magn. Reson. Med., 3: 53, 1991. J. Hennig. Echoes - How to generate, recognize, use or avoid them in MR imaging sequences. Part I: Fundamental and not so fundamental properties of spin echoes. Concepts in Magn. Reson., 3: 125, 1991. J. Hennig. Part II: Echoes in imaging sequences. Concepts in Magn. Reson., 3: 179, 1991. The following four papers proposed di erent short TE , fast imaging methods: G. M. Bydder and I. R. Young. Clinical use of the partial and saturation recoverey sequences in MR imaging. J. Comput. Assist. Tomogr., 9: 1020, 1985. 512 Chapter 18. Fast Imaging in the Steady State A. Haase, J. Frahm, D. Matthei, W. Hannicke and K.-D. Merboldt. FLASH imaging: Rapid imaging using low ip angle pulses. J. Magn. Reson., 67: 256, 1986. A. Oppelt, R. Graumann, H. Barfuss, H. Fischer, W. Hertl and W. Schajor. FISP: A new fast MRI sequence. Electromedica, 3: 15, 1986. P. van der Muelen, J. P. Croen and J. J. M. Cuppen. Very fast MR imaging by eld echoes and small angle excitation. Magn. Reson. Imag., 3: 297, 1985. Chapter 19 Segmented k-Space and Echo Planar Imaging
Chapter Contents
Echo Imaging 19.1 Reducing Scan Times 19.2 Segmented k-Space: Phase Encoding Multiple k-Space Lines per RF Excitation for Gradient 19.3 19.4 19.5 19.6 19.7 19.8
Echo Planar Imaging (EPI) Alternate Forms of Conventional EPI Artifacts and Phase Correction Spiral Forms of EPI An Overview of EPI Properties Phase Encoding Between Spin Echoes and Segmented Acquisition Summary: Methods of phase encoding between gradient echoes to reduce the number of rf pulses required to create a 2D image are developed. This concept is extended to allow a single shot acquisition referred to as echo planar imaging or EPI. Both conventional and spiral EPI methods are introduced. These concepts are extended to phase encoding between -pulses to allow for rapid T2 weighted spin echo imaging. Introduction
This chapter introduces the reader to alternate forms of rapid imaging, as compared to the short-TR methods discussed in the previous chapter. The concept of k-space segmentation is discussed, and two examples of methods where multiple k-space lines per rf excitation are phase encoded between gradient echoes or spin echoes are given. Di erent methods of 513 514 Chapter 19. Segmented k-Space and Echo Planar Imaging echo planar imaging (EPI) are introduced, including conventional EPI, spiral EPI and squarespiral EPI. Technical limitations of each method are reviewed. The thrust for such single shot acquisitions has been driven by cardiac imaging, functional imaging and di usion weighted imaging applications. These studies require rapid temporal acquisition over 3D volumes. One form of EPI that exists in the above-mentioned list includes collecting multiple lines of data by phase encoding between spin echoes to allow for rapid T2 -weighted data acquisition. Since T2 -weighted imaging has high clinical utility, reduction of the usually long scan times is critical. This is achieved using phase encoding between spin echoes, di erent variants of which are discussed in the last section of this chapter. 19.1 Reducing Scan Times
As an introduction to reducing imaging time, a review of what leads to long or short scan times is useful. The total scan or acquisition time TT in MRI is determined by the repeat time TR , the number of phase encoding steps Ny , the number of partition encoding steps Nz , and the number of acquisitions Nacq , giving TT = Nacq Nz Ny TR (19.1) How can scan time be reduced? In the previous chapter it was demonstrated that TR could be reduced and yet T1 -weighting or T1=T2 -weighting could be maintained when large ip angles ( > E ) are used. In the following discussion, the role of each term in (19.1) in reducing TT and their resulting e ects on contrast will be considered. One way to reduce TT is to shorten the repetition time. Bear in mind, however, as TR ! TR with < 1, that there is a concomitant p change in signal-to-noise such that, at best, (at the Ernst angle de ned p Ch. 18), SNR ! SNR. There is no escaping the limit imposed by in nature that SNR / time. For a simple spin echo scan, where the rst pulse is a =2-pulse, SNR ! SNR since the signal is directly proportional to TR =T1 when TR T1. Of course, as TR changes, contrast also changes, and if it changes for the worse, the short TR scan may not be useful. Another way to reduce TT is to cut back on the number of phase encoding or partition encoding steps. This approach is useful because contrast is unchanged, although partial volume e ects will become worse (see Ch. 15). There are two ways to accomplish this. First, assume Ly and Lz remain xed. Then, for Ny ! Ny with < 1, the acquisition time is reduced by a factor , but the resolution becomes a factor worse (i.e., y ! y= ). See Figs. 19.1a and 19.1b. On the other hand, SNR is increased by a factor p1 . (Recall that the signal goes up by 1= , but the noise standard deviation goes up by only p1 .) If resolution is too important to be sacri ced, then this is a poor strategy for reducing scan time. 19.1.1 Reducing TR 19.1.2 Reducing the Number of Phase/Partition Encoding Steps 19.1. Reducing Scan Times 515 Fig. 19.1: (a) Conventional 2D k-space coverage with x = Lx=Nx and y = Ly =Ny . (b) Reducing Ny by a factor of two (from 2ny to ny ) by sampling only the central part of k-space increases the pixel size y by a factor of two (ky max is now cut in half). (c) Reducing acquisition time while maintaining the same resolution. In this case, ky max is not changed but ky has become 2 ky (Ly has become Ly =2), ny has become ny =2, so that y remains invariant. (d) Example of k-space
coverage for a partial Fourier method. In the case shown, standard MRI sampling is taking place in the read direction, but less data is collected in the phase encoding direction to reduce imaging time. The k-space data are collected from ;n; ky to (n+ ; 1) ky where n; is usually chosen y y y to be small relative to n+ . The missing negative ky -space data are reconstructed using the partial y Fourier reconstruction method discussed in Ch. 13. 516 Chapter 19. Segmented k-Space and Echo Planar Imaging The second method maintains resolution by reducing Ny and Ly by the same factor of , p thereby keeping y xed (see Fig. 19.1c). Now TT ! TT but SNR is reduced to SNR. This loss in SNR is unavoidable when smaller elds-of-view are used. In order to avoid aliasing, the object size Ay in the y-direction must be larger than Ly . A third method to reduce scan time is to keep Nacq = 1, which is feasible if there is su cient SNR. Using a small eld-of-view has been seen to be very ine cient, causing a reduction by p at least in SNR when TT ! TT . By using better rf coil designs such as smaller coils or a set of multiple coils,1 SNR can be increased (see Ch. 27). Better SNR also allows a shorter TR to be used. Hence, even rf coil design can be considered part of a faster imaging strategy. If SNR is not a limiting factor, the proposed short-TR methods of the previous chapter, or those described in this chapter, will be more practical. 19.1.3 Fixing the Number of Acquisitions 19.1.4 Partial Fourier Data Acquisition A fourth approach is to force Nacq to be e ectively less than unity. This is possible, thanks to the complex conjugate property of the Fourier transform of real objects and the associated reconstruction method called partial Fourier reconstruction (see Ch. 13). In this method, only Ny (0:5 1:0) lines are collected which reduces TT to TT . Imaging schemes that employ other methods for speeding up the process can be further accelerated using this capability. The reduction is by at most a factor of 2, the maximum referring to the ability to relate one half of 2D k-space to the other half. Also, due to the reduction in the number of data points collected for similar imaging parameters, the signal-to-noise ratio of the partial p Fourier data is reduced by . Problem 19.1
Of the methods discussed so far to reduce imaging time, which ones leave the contrast unchanged? Is CNR also kept constant in those cases where the contrast is unchanged? 19.2 Segmented k-Space: Phase Encoding Multiple kSpace Lines per RF Excitation for Gradient Echo Imaging
Section 19.1 constitutes a review of methods employed in previous chapters. The main focus of this chapter involves the concept of collecting multiple lines of k-space data after
A set of multiple rf coils used to collect data simultaneously in the same experiment is often referred to as a phased array coil.
1 19.2. Multiple Gradient Echoes 517 a single rf excitation. In this section, an example is studied where two lines of k-space are acquired after one rf pulse. The approach of collecting more than one line of k-space after a single echo is a form of `segmented k-space coverage.' However, a more general de nition of segmented k-space coverage would be the collection of multiple k-space lines in a set, each set being collected in a given order, but not constituting a complete coverage of k-space until all of the sets are merged together. It is then possible to collect one line of k-space for each rf pulse and yet still create a segmented k-space coverage. For example, if the even lines ;ny ;(ny ; 2) ;(ny ; 4) ::: are collected using single echo sequences during one cardiac cycle and odd lines ;(ny ; 1) ;(ny ; 3) ;(ny ; 5) ::: during the next, then this is referred to as a two-segment coverage of k-space relative to the cardiac cycle. In a sense, the conventional 2D Fourier transform coverage of k-space requiring Ny phase encoding steps is an Ny -segment coverage. However, since it is in such common use, this adjective is never used in practice. Fig. 19.2: The read and phase encoding gradient waveforms with an arbitrary ip angle for (a) a simple double gradient echo sequence, and (b) a segmented k-space double gradient echo sequence. (a) (b) Note that a small phase encoding gradient blip waveform has been added between the two gradient echoes in (b). This encodes the next line in 2D k-space for the second echo. The stepped phase encoding table covers all even k-space lines (assuming ny is a multiple of 2) in the rst echo while the odd lines are generated after the short phase encoding gradient (shown as a triangular waveform) and are collected in the second echo. This constitutes a simple form of segmented k-space coverage. 19.2.1 Conventional Multiple Echo Acquisition
2 The double gradient echo2 is a good introduction to collecting more than one line of k-space data after a single rf excitation. The double gradient echo experiment can be used in two In review, recall that a gradient echo occurs at any time after the application of the rst read gradient lobe where the accumulated phase due to the read gradient is zero for all stationary spins. Gradient echoes 518 Chapter 19. Segmented k-Space and Echo Planar Imaging ways. One is to collect two images with di erent contrast levels by collecting the same line of k-space with di erent values of TE . The second is to employ phase encoding between the two gradient echoes, thereby collecting multiple lines of k-space data to create a single image in half of the total imaging time. The sequence diagrams for the two cases are shown in Fig. 19.2 and the k-space coverages appear in Fig. 19.3. Assume that the double gradient echo is to be used to collect two images. The singleimage case is discussed in the next subsection. In the double-image case, the k-space lines covered for each echo per rf excitation will be the same but, as shown in Fig. 19.3a, the direction of traversal through k-space in the read direction is opposite for the two echoes. The reason it may be desirable to collect two echoes in this way is to get images with di erent T2 or ow contrast (see Sec. 24.3 on phase contrast imaging in Ch. 24) in the same time. Consider, for example, an experiment with a relatively long TR . This experiment could be used to generate a spin density weighted image if TE << T2 , or a T2 weighted image if TE ' T2 average . However, using the double gradient echo, and making proper choices of TE 1 and TE 2 both images can be acquired in half of the time than if the scan were run twice with a di erent TE each time. Problem 19.2
a) Draw a sequence diagram for an experiment where two images with di erent contrast are collected (no phase encoding between the echoes) following a single =2 excitation pulse. Design the sequence such that the rst echo provides data for a T2 -weighted image, and the second echo provides data for a T2-weighted image. b) Assuming an experiment is performed with TR >> T1 , modify the sequence diagram in part (a) to employ three echoes. Design the timings of the experiment such that one of the resulting images is spin density weighted, one is T2 weighted, and one is T2 weighted. c) Show that data acquired with the simple double gradient echo acquisition illustrated in the right-hand-side of Fig. 19.3a leads to a spatially reversed second image unless the data are reordered (i.e., put in the order shown in the left-hand side of Fig. 19.3a). More gradient echoes may be employed after the single rf pulse their number is limited only by TR, the available gradient strengths and rise times, and the T2 of the object. Such procedures are referred to as multi-echo image acquisitions.
are also referred to as eld echoes or gradient eld echoes in the MRI literature. A gradient echo should not be confused with a spin echo which is an rf echo that occurs when all of the phase accumulated by stationary spins due to static eld variations is refocused to zero. This double gradient echo scan is also referred to as a two echo, multi-gradient echo scan. 19.2. Multiple Gradient Echoes 519 Fig. 19.3: Part (a) depicts the 2D k-space coverage of the double gradient echo sequence when it is used to collect two images and the same line of k-space is covered twice after each rf excitation. The left-hand side of (a) shows the k-space coverage for the image collected from the rst echo, and the right-hand side of (a) shows the k-space coverage for the image collected from the second echo. Shown in (b) is the temporal coverage of k-space when additional phase encoding is applied between the two echoes. The upward arrow in (b) indicates the movement in k-space due to the application
of the triangular gradient blip in the phase encoding direction. The dashed lines represent the reversed k-space order in which the data are temporally collected. Notice that only half as many rf excitations are needed to cover the same region of k-space and that the return pathway in k-space from the second gradient echo is running from right to left. 520 Chapter 19. Segmented k-Space and Echo Planar Imaging 19.2.2 Phase Encoding Between Gradient Echoes
The previous application of the conventional double gradient echo experiment did not allow the collection of a single image in less time. Although several images could be collected in the same time as one, the time needed to collect one image did not change. The primary goal of this chapter is to show how to collect a single image in a much shorter period of time than that given by (19.1). This may be accomplished by acquiring more than one line of k-space after a single rf excitation. In other words, a full 2D, and possibly a 3D region of k-space may be acquired after each rf excitation, instead of just a line. Phase encoding can be introduced between the two gradient echoes so two lines of k-space are acquired in the double gradient echo experiment. The k-space structure is shown in Fig. 19.3b. By phase encoding between two echoes, a factor of two in time can be saved over that of normal k-space coverage. The phase encoding is accomplished by applying a short gradient pulse or `blip' of the phase encoding gradient between the sampling times (the triangular waveform added to the phase encoding gradient waveform in Fig. 19.2b). In terms of k-space coverage, every odd line from the second echo is interleaved between every even line from the rst to create a hybrid or segmented k-space. Factors of two can be quite signi cant in terms of time saved, especially for rather long sequences. Reordering the Data
In previous experiments dealing with single echoes, the direction of traversal in k-space was identical for each repetition of a cycle. However, when multiple gradient echoes are collected after a single excitation, this is not always the case. An examination of the data collected during the double gradient echo experiment is a useful introduction to how data must often be rearranged after collection before being used to reconstruct an image (see Fig.