Physics 12.0 of Magnetic Resonance
MRI produces high-resolution, high-contrast cross-sectional images throughout the head and body. Like ultrasound, it is non-invasive, limited mainly by power deposition. MRI is also limited by the fact that the signal is generated by the nuclei in the tissue and the only way to increase this signal is to increase the magnetic field, which is very expensive. Beyond imaging, MRI has functional aspects such as chemical species sensitivity, microscopic blood flow sensitivity that makes brain neuronal activity accessible and diffusional sensitivity to evaluate tissue microstructure. This section will cover the basics, without delving into imaging schemes. Later sections will cover imaging methods.
Microscopic Magnetization Macroscopic Magnetization Precession and Larmor Frequency Transverse and Longitudinal Magnetization RF Excitation Relaxation Bloch Equations Spin Echoes Contrast Mechanisms
Important Points from MRI Lecture 1
Electrons in atomic orbits and nucleons in nuclear shells have internal angular momentum, S, and orbital angular momentum, L, giving total angular momentum J=L+S. Electrons, protons and neutrons are Fermions with S=1/2 The total angular momentum, either electronic or nuclear is determined by the total angular momentum of unpaired electrons, neutrons or protons. Nuclei with Even #p-Even #n have J=0, Even #p-Odd #n nuclei or Odd #p-Even #n nuclei have J=1/2, 3/2, (half-integer), while Odd #p-Odd #n nuclei have J=1,2,(integer). Angular momentum of charged particles or internal charges (neutron) gives rise to magnetic properties such that electrons in atomic orbits or nuclei have magnetic moments (magnetization) m (or )=J for nuclei and m=-mBJ/ for electrons. is the gyromagnetic ratio in units of rad/sec-Tesla. For H1, =42.57MHz/T In general, a particle has a magnetic moment given by m=gmPJ where g is the g factor that depends on the particle and situation, mP is the Bohr magneton for the electron (mB=e/2me) and mP is the nuclear magneton for the nucleon (mN=e/2mN).
When placed in a magnetic field, B, magnetic moments of a spin=1/2 system can have two orientations (up or down) that have an energy difference of E=B. This is like E= where the frequency, = B
In a magnetic field nuclear magnetic moments precess at a Larmor frequency fL= B Since me~mN/1000, the resonant frequency associated with electrons is nearly 1000 times higher than for nuclei (ie, GHz vs MHz)
Important Points from MRI Lecture 1 (continued!)
A collection of nuclear magnetic moments in a field B0 are only weakly aligned with the field because of constant exchange of thermal energy from the surroundings. About 1 in 106 more moments are aligned with the field (lower energy state) than against it. This leads to a weak bulk magnetization, M, that is aligned with B0 because the phase angles of the individual nuclear moments (precessing) are randomized. For H1 the equilibrium magnetization depends on temperature, B0, proton density and . The electronic magnetic moments are responsible for the bulk magnetic properties of matter: diamagnetic, paramagnetic, ferromagnetic. These are summarized in the magnetic susceptibility constant,, of a material. Materials that have different values of will distort the magnetic field at their boundary according to: B= B0 The bulk magnetization, M, if tilted by an angle, , will precess due to the torque on it by the magnetic field. The precession is at the Larmor frequency, fL=B0. At 1T, for H1, fL=42.6MHz We defined the longitudinal and transverse magnetization We noted that a suitable coil placed around transverse magnetization would have a voltage induced by Faradays law of induction parametrized by the coil receptivity Br (field induced by a unit current) The induced signal depends on Mxy which depends on B02, the tilt angle and the sample volume The rotating frame was described to help understand how a relatively weak co-rotating (circularly polarized) B1 field (0.02 gauss) could tilt M away from equilibrium by = B1t, where t is the time that B1 is on.
dM
(T2 and related T2*) (T1)
dM
From integrating the Bloch equations
T1 Relaxation
After the RF is applied and the net magnetic moment is offset from the z axis (at value Mo), it will relax and align with the z axis again. If the flip angle is 90o then Mz is reduced to zero and the magnitude of the magnetic moment along the z axis at any time after the flip is given by Mz = Mo (1-e-t/T1) where T1 is a unique constant that characterizes the ability of the spin to interact with the lattice
What Affects T1 Relaxation Time?
After a flip, there are more spins in the higher energy state, N+, than the lower energy state, N-. The probability of a spontaneous transition is very small. All relaxation is through stimulated emission from the lattice (surroundings) For the N+ spins to relax, the surrounding lattice must provide a stimulation near the Larmor frequency to cause emission of energy from the nucleus.
What Affects T1 Relaxation Time?
For liquids and tissue the main effect is from molecular motions and the dipole-dipole interactions between nuclei in the sample. Each nucleus produces a magnetic field at each other nucleus, proportional to /r3 The B produced at the other proton varies in orientation and strength depending on the rotational speed of the molecule
B produced by other proton.
Correlation Time
Measure of the rotational speed of a molecule (in most cases, the molecule the atom hydrogen is bonded to) = 1/(frequency of rotation)
Both T1 and T2 relaxation are dependent of the correlation time related to temperature and molecular size
Frequency spectrum of molecular motions
Correlation Time
fundamental mechanism difference between T1 and T2 relaxation: T1-energy at fL T2-dephasing of Mxy
T 2*
* different isochromats dephase due to different local Larmor frequencies due to magnetic field inhomogeneities If no dephasing, then T2T1 (but T2 T1)
new symbol: T2* exp(-t/T2*)
Example: Consider 2 proton isochromats at different locations in a 1.5T magnet such that the B field differs by 20ppm. How long will it take before the isochromats are 1800 out of phase? Ans: B=20ppm of 1.5T so the Larmor frequency also differs by 20ppm. At 1.5T =42.57x106Hz/T x 1.5T x 20 x 10-6=1277.4 Hz. To be out of phase by cycle will take t=0.5cycle/(1277cycle/sec)=391microsec They dephase (lose coherence) rapidly. Thus to get a T2* of 391msec we would have to improve the homogeneity to 0.02ppm.
so, remarkably, we can refocus or recover some of our lost signal
slow
fast
T2 and
T2 describes the irreversible losses of the signal
T2* Relaxation
T2* can be broken up into two components: 1/T2* = 1/T2` + 1/T2 where T2` is the reversible decay and T2 is the irreversible decay. T2` is caused by inhomogenities in B0, which causes the change in frequency (1/T2= B0/2). T2 is caused mainly by dipole-dipole interactions, similar to those causing T1 relaxation. Note that this is a rate equation. If the rate (frequency) at which transverse magnetization disappears is R then we have:
R2*=R2+R2 or R2*=R2+ B0/2
T2 Irreversible Decay
Mainly due to dipole-dipole interactions with a small role played by the other interactions.
Can also be caused by diffusion between two different B0 inhomogeneities. Cannot be eliminated
Different for all materials and thus can be used to provide contrast in the image
Thus, although we would expect T2=T1, ie, Mx,y doesnt disappear until Mz is fully recovered, we actually have T1>T2>T2*
Dipole-Dipole Coupling
Strength of Interaction heavily dependent on distance, thus for pure water the strongest T2 effect on one H+ atom is the other H+ atom and the time they are in a particular orientation
Paramagnetic Effects
Paramagnetic atoms have unpaired electrons, which have a much higher magnetic moment than nuclei.
Thus, they are able to have a large dipole-dipole interaction with spinning nuclei and therefore increase T2 relaxation substantially.
Common T1 and T2 values
All values are for a B0 = 1.5T
Tissue/Fluid
Fat(adipose)
T1 (3.0T)
382
T1 (1.5T)
260
T2 (1.5T)
80
Liver
Kidneys Muscle White Matter Gray Matter CSF Whole Blood (deoxygenated) Whole Blood (oxygenated)
809
-898 832 1331 3700 ---
500
650 870 780 900 2400 1350 1350
40
60-75 45 90 100 160 50 200
Felix Bloch 1905-1983 Swiss born, naturalized US. 1952 Nobel prize with Purcell. Stanford, Harvard, Los Alamos, Radar, NMR
(we are using T2 here by assuming no inhomogeneity)
Example: Fun with the Bloch Eqns. Find the equations for the components of M in the xy plane and verify that the transverse relaxation after a pi/2 pulse (in x direction) satisfys the equations.
Mx T2 My T2 (M z T1 M0)
Mx d My dt Mz
M y Bz
M z By
M x Bz M z Bx M x B y M y Bx
after the pi/2 pulse we have Bx=By=0, Bz=B0 and the transverse Bloch eqns simplify to:
d Mx dt M y
M y B0 M x B0
Mx T2 My T2
we write the transverse magnetization as:
M xy
Mx My
M 0 sin e
j (2
0t
)
e
t / T2
the initial phase angle is /2 (y direction) since B1 is in the x direction
M 0 cos( 2 M 0 sin( 2
0t 0t
/ 2) e / 2) e
t / T2 t / T2
M 0 sin( 2 M 0 cos(2
0 t )e 0 t )e
t / T2 t / T2
to see if these satisfy the Bloch eqns we differentiate:
dM x dt
d dt 2
M 0 sin( 2
0 M 0 cos(2
0 t )e
t / T2
0 t )e
t / T2
1 M 0 sin( 2 T2
0 t )e
t / T2
B0 M y
1 Mx T2
and similarly for My
Spin Echo Parameters
TR is the time between 90 pulses is the time between the 90 pulse and the 180 pulse. It is also the time between the 180 and the echo Thus, TE, the time between the 90 pulse and the echo is equal to 2 Mxy = M(TR, T1) e-(TE/T2)
Note: We tend to use M0 and PD interchangeably, sorry
M z (Tr ) M xy (Te)
M 0 (1 e M 0e
Tr / T1
)
Te / T2 (*)
Proton Density
M z (Tr ) M xy (Te) M 0 (1 e M 0e
Tr / T1
)
Te / T2 (*)
Long TR with short TE provides proton density. Proton density Images normally have little contrast due to the uniformity of water Density in human tissue. Here good gray/white matter contrast exists.
from Dr. W. Block, U. Wisconsin, http://zoot.radiology.wisc.edu/~block/bme_530_lectures.html
T1-Weighting
Sagittal
Axial
Short echo time with moderate TR provides T1-weighting. Notice cerebral spinal fluid (CSF) is dark.
from Dr. W. Block, U. Wisconsin, http://zoot.radiology.wisc.edu/~block/bme_530_lectures.html
T2-Weighting
Long echo time with long TR provides T2-weighting. Notice bright CSF, tumor and grey/white matter contrast.
from Dr. W. Block, U. Wisconsin, http://zoot.radiology.wisc.edu/~block/bme_530_lectures.html
Multiple Echo Spin Echo
An echo is stored after each 180 spin reversal pulse. Each echo is used to build a different T2-weighted image. What would the pulse sequence look like?
from Dr. W. Block, U. Wisconsin, http://zoot.radiology.wisc.edu/~block/bme_530_lectures.html
Multiple Echo Example
from Dr. W. Block, U. Wisconsin, http://zoot.radiology.wisc.edu/~block/bme_530_lectures.html