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Diffraction Single-slit Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2007 Ultrasound Lecture 2 Beamwidth TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 "z D Source:wikipedia ! In general Plane Wave Approximation Huygen s Principle $ 2"z ' 1 U(x 0 , y 0 ) = exp& j )F [ s(x, y)] k = x 0 ,k = y 0 , x y % #( z #z #z Example s(x, y) = rect(x /D)rect(y /D) 1 U(x 0 , y 0 ) = exp( jkz)D2 sinc ( Dk x ) sinc ( Dk y ) z $ x' $ 1 y' = exp( jkz)D2 sinc& D 0 ) sinc& Dk y 0 ) % #z ( % z #z ( Zeros occur at x 0 = n#z n#z and y 0 = D D r01 Wavenumber 2" k= # Beamwidth of the sinc function is TT Liu, BE280A, UCSD, Fall 2007 #z D TT Liu, BE280A, UCSD, Fall 2007 ! Oliquity Factor Anderson and Trahey 2000 ! ! 1 Small-Angle (paraxial) Approximation Fresnel Approximation r01 Approximates spherical wavefront with a parabolic phase pro le TT Liu, BE280A, UCSD, Fall 2007 Anderson and Trahey 2000 TT Liu, BE280A, UCSD, Fall 2007 Anderson and Trahey 2000 Fresnel Approximation 1 U(x 0 , y 0 ) = ## exp( jkr01 ) s( x1, y1 ) dx1dy1 j"r01 1 2 2 $ ## exp jk z 2 + ( x1 % x 0 ) + ( y1 % y 0 ) s( x1, y1 ) dx1dy1 j"z & jk exp( jkz) 2 2) $ ## exp( ( x % x ) + ( y1 % y 0 ) +s( x1, y1) dx1dy1 ' 2z 1 0 * j"z & & jk 2 )) exp( jkz) 2 = ( s(x 0 , y 0 ) ,,exp( ( x 0 + y 0 )++ ' 2z ** j"z ' U(x 0 , y 0 ) " = Fresnel Zone $$ $$ & jk exp( jkz) 2 2) exp( ( x % x 0 ) + ( y1 % y 0 ) +s( x1, y1) dx1dy1 ' 2z 1 * j#z & jk ) exp( jkz) 2 2 2 2 exp( ( x + y1 ) + ( x 0 + y 0 ) % 2( x1x 0 + y1y 0 ) +s( x1, y1)dx1dy1 ' 2z 1 * j#z ( ) (( )) ( ) ( ) = & jk 2 exp( jkz) 2) exp( ( x 0 + y 0 )+ , ' 2z * j#z & jk 2 ) & jk ) exp( jkz) $$ j#z exp( 2z ( x1 + y12 )+s( x1, y1) exp(% z ( x1x 0 + y1y 0 )+dx1dy1 ' * ' * - & jk 0 & jk 2 ) exp( jkz) 2) exp( ( x 0 + y 0 )+F /exp( ( x12 + y12 )+s( x1, y1 )2 ' 2z * . ' 2z * j#z 1 = ! Phase across transducer face TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 ! 2 Fraunhofer Condition Phase term due to position on transducer is Far- eld condition is k2 ( x1 + y12 ) << 1 2z k " z >> ( x12 + y12 ) = ( x12 + y12 ) 2 # ! k2 ( x1 + y12 ) 2z D12 " ! D1 ! D2 "z D2 2 D2 ! " "z D1 ! For a square DxD transducer, x12 + y12 = D2 /2 z >> ! "D 2 D2 $ 2# # ! ! TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 "z D =D z= D 2 / " ! Fresnel Zone U(x 0 , y 0 ) = Focusing in Fresnel Zone ) # jk , # jk 2 & exp( jkz) 2& exp% ( x 0 + y 0 )(F +exp% ( x12 + y12 )(s( x1, y1 ). $ 2z ' * $ 2z ' j"z - ! D c > c0 z0 # jk 2 & exp%" ( x1 + y12 )( $ 2z0 ' " jk 2 % exp$ ( x1 + y12 )' 2z0 # & TT Liu, BE280A, UCSD, Fall 2007 Anderson and Trahey 2000 TT Liu, BE280A, UCSD, Fall 2007 ! ! 3 Focusing in Fresnel Zone U(x 0 , y 0 ) = ) # jk , # jk 2 & exp( jkz) 2& exp% ( x 0 + y 0 )(F +exp% ( x12 + y12 )(s( x1, y1 ). $ 2z ' * $ 2z ' j"z - z0 Acoustic Lens D Make At the focal depth z =z0 ! U(x 0 , y 0 ) = # jk 2 & exp( jkz0 ) exp% ( x 0 + y 02 )(F [s( x1, y1)] j"z0 $ 2z0 ' "z0 = "F D c > c0 ! At the focal depth z =z0 U(x 0 , y 0 ) = = # jk 2 & * # jk & # jk & exp( jkz0 ) exp% ( x 0 + y 02 )(F ,exp% 2z ( x12 + y12 )( exp%) 2z ( x12 + y12 )(s( x1, y1)/ j"z0 $ 2z0 '+ $ 0 '$ ' . 0 # jk 2 & exp( jkz0 ) exp% ( x 0 + y 02 )(F [s( x1, y1)] j"z0 $ 2z0 ' Beamwidth at the focal depth is: "z0 D ! # jk 2 & s( x1, y1 ) = s0 ( x1, y1 ) exp%" ( x1 + y12 )( $ 2z0 ' TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, 2007 Fall ! ! ! Depth of Focus & jk 2 ) & jk ) When z " z 0 , the phase term is #$ = exp(% ( x1 + y12 )+ exp(% 2z ( x12 + y12 )+ * ' 2z0 *' and the lens is not perfectly focused. Consider variation in the x - direction. kx 2 & 1 1 ) #$ = (% + 2 ' z z0 * x 2 D2 = 2 8 2& ,D 1 1 ) If we want #$ = ( % + < 1 radian then 4- ' z z0 * For transducer of size D, 11 4#z %< . / < z z0 ,D2 D2 z 2 D2 0 The larger the D, the smaller the depth of focus. TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 ! 4 z0 Focusing with Phased Array D "z0 = "F D D ! TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 Anderson and Trahey 2000 Focusing and Steering Dynamic Focusing TT Liu, BE280A, UCSD, Fall 2007 Prince and Links 2005 TT Liu, BE280A, UCSD, Fall 2007 Prince and Links 2005 5 Doppler Effect "f = Doppler Effect 2vf 0 2vf 0 $ c #v c Example v = 50 cm/s c = 1500 m/s f0 = 5 MHz 2vf 0 = 3333 Hz c http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 ! Doppler CW Doppler http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 6 CW Doppler CW Doppler Resolution "f =1/Tobs -f0- f -f0 f0 f0+ f * Effective width = 1/Tobs "v = c"f c = 2 f 0 2Tobs f 0 Example Design goal : "v = 5 cm /s; f 0 = 5 MHz Tobs = c 1500m /s = = 3ms 2"vf 0 2(0.05m /s)(5 #10 6 ) Note that for a depth of 15 cm, it takes only 200 usec for echos to return. TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 ! PW Doppler PW Doppler http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 7 Aliasing Measure Doppler shifts at a specified range For unambiguous range, one pulse at a time. 2rmax (e.g. 200 usec for 15 cm depth) c TPR = To avoid aliasing require 1 > 2"f max TPR ! * - f f 1/TPR = - f f Aliased TT Liu, BE280A, UCSD, Fall 2007 Siemens Medical Systems; jnormal common carotid artery TT Liu, BE280A, UCSD, Fall 2007 - f f Aliasing Aliasing http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 8 Aliasing PW Doppler Velocity Resolution (same as with CW) 1 c Tobs > = "f 2"vf 0 Range Resolution cTpulse 2 We also need to make sure that particles remain within this region over the observation time Tobs "z cTpulse v max Tobs < "z # Tobs < = v max 2v max Want to interrogate velocities from a small region "z = http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htm chapter_01.htm TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007 ! PW Doppler Design Example R max = 6 cm " TPR = 2(0.06m) = 80 sec 1500m /s 1 4v f > 2#f max = max 0 TPR c c > v max " for f0 = 5MHz we find that v max < 93.75cm /s 4TPR f 0 If we choose #v = 1cm /s then Tobs = c = 15ms 2#v max f 0 Range resolution : #z > v max Tobs = 1.4cm Tpulse = 2#z = 18.8usec c TT Liu, BE280A, UCSD, Fall 2007 ! 9
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UCSD >> BE >> 280 (Fall, 2008)
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HMP 3 Health & Medical Professions Preparation Program GUIDELINES FOR A SUCCESSFUL MENTORSHIP Mentoring is a nurturing process that facilitates the growth and development of the mentee. The mentorship creates a relationship in which the mentor, an...
UCSD >> BE >> 280 (Fall, 2008)
BENG280A, Principles of Biomedical Imaging Fall Quarter 2005 INFORMATION SHEET Instructor: 09/22/05 HO #1 Thomas Liu, Department of Radiology Center for Functional Magnetic Resonance Imaging (fMRI), Room 1008 (858) 822-0542 , ttliu@ucsd.edu Eric Ro...
UCSD >> BE >> 280 (Fall, 2008)
Bioeng 280A: Principles of Biomedical Imaging Fall Quarter 2005 Preliminary Syllabus Week 1 Thursday Week 2 Tuesday Thursday Week 3 Tuesday Thursday Week 4 Tuesday Thursday Week 5 Tuesday Thursday Week 6 Tuesday Thursday Week 7 Tuesday Thursday Week ...
UCSD >> BE >> 280 (Fall, 2008)
BENG280A, Principles of Biomedical Imaging Fall Quarter 2005 HOMEWORK #1 Due by 4 p.m. Wednesday 10/5/04 09/27/04 Homework Policy: Turn in assignments to the TA. Late homeworks will be marked down 20% for every day that they are late. If you know t...
UCSD >> BE >> 280 (Fall, 2008)
BENG280A, Principles of Biomedical Imaging Fall Quarter 2005 HOMEWORK #2 Due by 4 p.m.,Wednesday 10/12/05 Readings: 1. Review last weeks reading as necessary. Read Chapter 2, sections 2.1-2.6. Problems: 1. 10/4/05 Draw a pulse sequence that achieve...
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