RF Pulse Design. Multi-dimensional Excitation I. M229 Advanced Topics in MRI Kyung Sung, Ph.D Class Business

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1 RF Pulse Design Multi-dimensional Excitation I M229 Advanced Topics in MRI Kyung Sung, Ph.D Class Business Office hours - Instructors: Fri 10-12pm TAs: Xinran Zhong and Zhaohuan Zhang (time: TBD) - s beforehand would be helpful Homework 1 will be out on 4/12 (due on 4/26) Papers and Slides

2 <latexit sha1_base64="kxyuryaalgfopqia+zphcykspiw=">aaacc3icdva9swnben2l3/erammzjaixcxcialoqg0ufjazy8djbtjile3vn7pwqqnob/4qnhsk2/ge7/42bgmhpbzm83pthd16ysghqdd+cznz8wuls8kp2dw19yzo3td0wcao51hksy90mmqepfnrroirmooffoyslchay8s+uqrsrqxooe2hhrkdev3cgvgpyer/7giz6cq6olxqg7mwtgnhfh1m635n0ifdws+wya0f/e6/ktlegm5wfuve/e/m0aovcmmnanptge8q0ci5hnpvtawnja9adlqwkrwdao+kty7pnlq7txtqwqjpvv26mwgtmmartzmswb356e/evr5vi97g9eipjert/ekibsooxnqrdo0idrzm0hhet7f8p7zpnonr4sjaez0vp/6rxuplcknd+wkhuz3esk12sj0xiksnsiafkjnqjjzfkjjyqr+fwuxeenoep0ywz29kh3+c8vapl3jpc</latexit> <latexit sha1_base64="kxyuryaalgfopqia+zphcykspiw=">aaacc3icdva9swnben2l3/erammzjaixcxcialoqg0ufjazy8djbtjile3vn7pwqqnob/4qnhsk2/ge7/42bgmhpbzm83pthd16ysghqdd+cznz8wuls8kp2dw19yzo3td0wcao51hksy90mmqepfnrroirmooffoyslchay8s+uqrsrqxooe2hhrkdev3cgvgpyer/7giz6cq6olxqg7mwtgnhfh1m635n0ifdws+wya0f/e6/ktlegm5wfuve/e/m0aovcmmnanptge8q0ci5hnpvtawnja9adlqwkrwdao+kty7pnlq7txtqwqjpvv26mwgtmmartzmswb356e/evr5vi97g9eipjert/ekibsooxnqrdo0idrzm0hhet7f8p7zpnonr4sjaez0vp/6rxuplcknd+wkhuz3esk12sj0xiksnsiafkjnqjjzfkjjyqr+fwuxeenoep0ywz29kh3+c8vapl3jpc</latexit> <latexit sha1_base64="kxyuryaalgfopqia+zphcykspiw=">aaacc3icdva9swnben2l3/erammzjaixcxcialoqg0ufjazy8djbtjile3vn7pwqqnob/4qnhsk2/ge7/42bgmhpbzm83pthd16ysghqdd+cznz8wuls8kp2dw19yzo3td0wcao51hksy90mmqepfnrroirmooffoyslchay8s+uqrsrqxooe2hhrkdev3cgvgpyer/7giz6cq6olxqg7mwtgnhfh1m635n0ifdws+wya0f/e6/ktlegm5wfuve/e/m0aovcmmnanptge8q0ci5hnpvtawnja9adlqwkrwdao+kty7pnlq7txtqwqjpvv26mwgtmmartzmswb356e/evr5vi97g9eipjert/ekibsooxnqrdo0idrzm0hhet7f8p7zpnonr4sjaez0vp/6rxuplcknd+wkhuz3esk12sj0xiksnsiafkjnqjjzfkjjyqr+fwuxeenoep0ywz29kh3+c8vapl3jpc</latexit> <latexit sha1_base64="kxyuryaalgfopqia+zphcykspiw=">aaacc3icdva9swnben2l3/erammzjaixcxcialoqg0ufjazy8djbtjile3vn7pwqqnob/4qnhsk2/ge7/42bgmhpbzm83pthd16ysghqdd+cznz8wuls8kp2dw19yzo3td0wcao51hksy90mmqepfnrroirmooffoyslchay8s+uqrsrqxooe2hhrkdev3cgvgpyer/7giz6cq6olxqg7mwtgnhfh1m635n0ifdws+wya0f/e6/ktlegm5wfuve/e/m0aovcmmnanptge8q0ci5hnpvtawnja9adlqwkrwdao+kty7pnlq7txtqwqjpvv26mwgtmmartzmswb356e/evr5vi97g9eipjert/ekibsooxnqrdo0idrzm0hhet7f8p7zpnonr4sjaez0vp/6rxuplcknd+wkhuz3esk12sj0xiksnsiafkjnqjjzfkjjyqr+fwuxeenoep0ywz29kh3+c8vapl3jpc</latexit> Today s Topics Recap of adiabatic pulses Small tip approximation Excitation k-space interpretation Design of 2D excitation pulses - Spiral pulse design Recap of Adiabatic Pulses - A special class of RF pulses that can achieve uniform flip angle - Flip angle is independent of the applied B1 field 6= Z T 0 B 1 ( )d - Slice profile of an adiabatic pulse is obtained using Bloch simulations - Does not follow the small tip approximation

3 Adiabatic Pulses Non-adiabatic Pulses / Amplitude and frequency modulation Long duration (8-12 ms) High B1 amplitude (>12 µt) Generally NOT multipurpose (inversion pulses cannot be used for refocusing, etc.) Amplitude modulation Short duration (0.3-1 ms) Low B1 amplitude Generally multi-purpose (inversion pulses can be used for refocusing, etc.) Adiabatic Pulses Frequency modulated pulses: envelop frequency sweep d ~ M dt = ~ M ~ B eff where

4 Adiabatic Inversion AFP A(t) ω 1 (t) t ω 0 t Adiabatic Excitation: BIR Pulses BIR: B1 Insensitive Rotation Most popular: BIR-4 Pulse

5 Adiabatic Excitation: BIR4 Pulse a b c d Bloch Simulator -

6

7 Small Tip Approximation Bloch Equation (at on-resonance) d ~ M rot dt = ~ M rot ~ B eff where When we simplify the cross product, d ~ M dt = 0!(z) 0!(z) 0! 1 (t) 0! 1 (t) 0 1 A

8 Small Tip Approximation d ~ M dt = 0!(z) 0!(z) 0! 1 (t) 0! 1 (t) 0 1 A M z M 0 small tip-angle approximation sin θ θ cos θ 1 M z M 0 constant First order linear differential equation. Easily solved. Solving a first order linear differential equation: M r (,z)=im 0 e i!(z) /2 FT 1D {! 1 (t + 2 )} f= ( /2 )G z z (See the note for complete derivation)

9 To the board... Small Tip Approximation Fourier transform - For small tip angles, the slice or frequency profile is well approximated by the Fourier transform of B1(t) - The approximation works surprisingly well even for flip angles up to 90

10 Shaped Pulses Pauly, J. J. Magn. Reson (1989) small-angle approximation still works reasonably well for flip angles that aren t necessarily small Multi-Dimensional Excitation RF Pulses 1D vs. N-D RF pulses Small tip angle approximation revisited Excitation k-space interpretation 1D examples in excitation k-space Excitation k-space integrals 2D excitation pulse design steps 2D spiral pulse design example EPI pulse design, spectral-spatial pulses (next lecture)

11 What is Multi-Dimensional Excitation? Multi-dimensional excitation occurs when using multi-dimensional RF pulses in MRI/NMR, i.e. 2D or 3D RF pulses 1D vs. N-D RF Pulses z z 2D/N-D Pulse Design Requires: - Specific B1 waveform - Specific gradient waveforms Selective along z only y Selective along z y x x Selective along y 1D pulses are selective along 1 dimension, typically the slice select dimension 2D pulses are selective along 2 dimensions So, a 2D pulse would select a long cylinder instead of a slice The shape of the cross section depends on the 2D RF pulse

12 Excitation k-space Interpretation Small Tip Approximation

13 Small Tip Approximation Let us define: One-Dimensional Example B1(t) T Gz(t) k(t,t0) t1 t2 t3 t4 t5 t6 t7 Consider the value of k at s = t 1, t 2, t 7

14 One-Dimensional Example This gives magnetization at t = t0, the end of the pulse Looks like you scan across k-space, then return to origin Evolution of Magnetization During Pulse RF pulse goes in at DC (kz = 0) Gradients move previously applied weighting around Think of the RF as writing an analog waveform in k-space Same idea applies to reception

15 Other 1D Examples B1(t) Gz(t) k(t,t0) t0 Other 1D Examples B1(t) Gz(t) k(t,t0) t0

16 Other 1D Examples B1(t) Gz(t) k(t,t0) t0 Multiple Excitations Most acquisition methods require several repetitions to make an image - e.g., 128 phase encodes Data is combined to reconstruct an image Same idea works for excitation!

17 Simple 1D Example Sum the data from two acquisitions Same profile as slice selective pulse, but zero echo time Small Tip Approximation Solution as k-space Integral = Substituting and changing the order of integration:

18 where The magnetization is the inverse transform of p(k) We want this to be a unit delta, multiplied by a weighting function Small Tip Approximation Solution as k-space Integral Multiply and divide by k (s,t) : W(k(s,t)) Unit Delta If we assume W(k) is single-valued

19 Small Tip Approximation Solution as k-space Integral k-space weighting k-space sampling Small Tip Approximation Solution as k-space Integral So, the inverse Fourier transform of the k-space weighting will give us the excitation profile! k-space weighting

20 <latexit sha1_base64="rb2+fkgjh3kocij4dhy0xck8cpu=">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</latexit> <latexit sha1_base64="rb2+fkgjh3kocij4dhy0xck8cpu=">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</latexit> <latexit sha1_base64="rb2+fkgjh3kocij4dhy0xck8cpu=">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</latexit> <latexit sha1_base64="rb2+fkgjh3kocij4dhy0xck8cpu=">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</latexit> Design of 2D Excitation Pulses 2D Pulse Design 1. Choose a k-space trajectory 2. Choose a weighting function W ( ~ k) <latexit sha1_base64="mym179xgu2ktppiozatvbhaf2zi=">aaab8xicdvdjsgnbek1xjxglevtsgir4ctmiag5blx4jmawtifr0apimpt1dd08gdpkllx4u8erfepnv7cyc64ocx3tvvnulesg1cd13z2l5zxvtpber39za3tkt7o03djwqhnuwi1i1aqprcil1w43avqkqrohazjc8mvrnesrny3lrxgn6ee1lhnjgjzxumqxocfk2njx0c0w3xkm4fuq38crudevyonytvhv6musjliyjqnxbcxpjz1qzzgro8p1uy0lzkpaxbamkewo/m108icdw6zewvrakitp160rgi63huwa7i2og+qc3ff/y2qkjl/ymyyq1knl8uzgkymiyfz/0uejmxngsyhs3txi2oioyy0pk2xa+pyx/k8zp2xpl3s1zsxq5icmhh3aejfdghkpwdtwoawmj9/ait452hpxn52xeuuqszg7gg5zxd2gvklk=</latexit> <latexit sha1_base64="mym179xgu2ktppiozatvbhaf2zi=">aaab8xicdvdjsgnbek1xjxglevtsgir4ctmiag5blx4jmawtifr0apimpt1dd08gdpkllx4u8erfepnv7cyc64ocx3tvvnulesg1cd13z2l5zxvtpber39za3tkt7o03djwqhnuwi1i1aqprcil1w43avqkqrohazjc8mvrnesrny3lrxgn6ee1lhnjgjzxumqxocfk2njx0c0w3xkm4fuq38crudevyonytvhv6musjliyjqnxbcxpjz1qzzgro8p1uy0lzkpaxbamkewo/m108icdw6zewvrakitp160rgi63huwa7i2og+qc3ff/y2qkjl/ymyyq1knl8uzgkymiyfz/0uejmxngsyhs3txi2oioyy0pk2xa+pyx/k8zp2xpl3s1zsxq5icmhh3aejfdghkpwdtwoawmj9/ait452hpxn52xeuuqszg7gg5zxd2gvklk=</latexit> <latexit sha1_base64="mym179xgu2ktppiozatvbhaf2zi=">aaab8xicdvdjsgnbek1xjxglevtsgir4ctmiag5blx4jmawtifr0apimpt1dd08gdpkllx4u8erfepnv7cyc64ocx3tvvnulesg1cd13z2l5zxvtpber39za3tkt7o03djwqhnuwi1i1aqprcil1w43avqkqrohazjc8mvrnesrny3lrxgn6ee1lhnjgjzxumqxocfk2njx0c0w3xkm4fuq38crudevyonytvhv6musjliyjqnxbcxpjz1qzzgro8p1uy0lzkpaxbamkewo/m108icdw6zewvrakitp160rgi63huwa7i2og+qc3ff/y2qkjl/ymyyq1knl8uzgkymiyfz/0uejmxngsyhs3txi2oioyy0pk2xa+pyx/k8zp2xpl3s1zsxq5icmhh3aejfdghkpwdtwoawmj9/ait452hpxn52xeuuqszg7gg5zxd2gvklk=</latexit> <latexit sha1_base64="mym179xgu2ktppiozatvbhaf2zi=">aaab8xicdvdjsgnbek1xjxglevtsgir4ctmiag5blx4jmawtifr0apimpt1dd08gdpkllx4u8erfepnv7cyc64ocx3tvvnulesg1cd13z2l5zxvtpber39za3tkt7o03djwqhnuwi1i1aqprcil1w43avqkqrohazjc8mvrnesrny3lrxgn6ee1lhnjgjzxumqxocfk2njx0c0w3xkm4fuq38crudevyonytvhv6musjliyjqnxbcxpjz1qzzgro8p1uy0lzkpaxbamkewo/m108icdw6zewvrakitp160rgi63huwa7i2og+qc3ff/y2qkjl/ymyyq1knl8uzgkymiyfz/0uejmxngsyhs3txi2oioyy0pk2xa+pyx/k8zp2xpl3s1zsxq5icmhh3aejfdghkpwdtwoawmj9/ait452hpxn52xeuuqszg7gg5zxd2gvklk=</latexit> 3. Design the RF pulse B 1 (s) = 1 W ( ~ k) k 0 (s, t)

21 1. Choose a k-space trajectory Select a k-space trajectory that uniformly covers k-space - k-space extent ( k max, k max ) spatial resolution - sampling density ( k) spatial FOV 1. Choose a k-space trajectory Two most common choices: Spiral is common for pencil beams EPI is common for spectral-spatial pulses

22 2. Choose a weighting function An excitation profile is the inverse Fourier transform of the weighting function If you know what excitation profile you want, its Fourier transform will be the weighting function Localized excitation low-pass k-space weighting 2. Choose a weighting function k-space weighting Excitation profile

23 3. Design the RF pulse B 1 needs to be scaled for flip angle 2D Spiral Pulse Design Two major choices: - Resolution r Smallest volume / minimum transition width - Field-of-View (FOV) Distance to center of first sidelobe

24 k-space Trajectory k y Δk = 2k max 2N k max k max k x Impulse Response N Turn Spiral FOV = 2NΔr Δr = 1 2k max 2D Spiral Pulse Design Spiral Gradient Design - Constant angular rate spiral - Constant slew rate spiral

25

26 2D Spiral Pulse Design Truncated Jinc Weighting Minimum transition width, but residual ripples 2D Spiral Pulse Design Windowed Jinc Weighting Window Function Doubled transition width, but smoother response

27 2D Spiral Pulse Design Calculation of the RF pulse - given W(k) and k(s,t) - needs to be scaled for flip angle - k (s,t) is an estimate of the density compensation function d(t)

28 Conclusions - N-D RF pulses are selective in N-dimensions - The small tip approximation can be extended to describe the excitation k-space - The small tip approximation solution can be used to show that the excitation profile of an N-D pulse is given by the inverse Fourier transform of the excitation k-space weighting Conclusions - An N-D RF pulse can be designed by: Choosing a k-space trajectory Choosing a k-space weighting function Then calculating the B 1 (t) and G(t) functions

29 Next Class N-D pulses with EPI trajectory Spatial-spectral pulses Matlab demo of N-D pulses Thank You! - Further reading Read Spatial-Spectral Pulses p Acknowledgments John Pauly s EE469b (RF Pulse Design for MRI) Shams Rashid, Ph.D. Kyung Sung, PhD ksung@mednet.ucla.edu