Extended Phase Graphs (EPG)

Transcription

1 Extended Phase Graphs (EPG) Purpose / Definition Propagation Gradients, Relaxation, RF Diffusion Examples 133

2 EPG Motivating Example: RF with Crushers RF G z Crushers are used to suppress spins that do not experience a 180º rotation Dephasing/Rephasing work if 180º is perfect Bloch simulation: Must simulate positions across a voxel, then sum all spins: >>w = 360*gamma*G*T*z >>M = zrot(w)*xrot(alpha)*zrot(w)*m; % 134

3 EPG Motivating Example: RF with Crushers G z 120xº RF G z M y M y M y = + M x M x M x 135

4 EPG Motivating Example: RF with Crushers RF G z Brute-force simulation works, but little intuition Quantized gradients produce dephased cycles Decompose elliptical distributions to +/- circular 136

5 Extended Phase Graphs: Purpose One cycle of phase twist from a spoiler gradient (for example) Hennig J. Multiecho imaging sequences with low refocusing flip angles. J Magn Reson 1988; 78: Hennig J. Echoes How to Generate, Recognize, Use or Avoid Them in MR-Imaging Sequences, Part I. Concepts in Magnetic Resonance 1991; 3: Hennig J. et al. Calculation of flip angles or echo trains with predefined amplitudes with the extended phase graph (EPG)- algorithm: principles and applications to hyperecho and TRAPS sequences. MRM 2004; 51:

6 Magnetization Vector Definitions Common representations for magnetization: 2 4 M x M y Mz 3 5 = i 0.5i M xy M xy M z M xy M xy M z 3 5 = i 0 1 i M x M y Mz

7 The EPG Basis Consider spins in a voxel: z is the location (0 to 1) across the voxel M xy (z) and M xy *(z) are the transverse magnetization M z (z) is the longitudinal magnetization Represent the (huge) number of spins using a compact basis set of F n and Z n coefficients Motivation: Propagation of the basis functions through RF, gradients, relaxation is fairly simple 139

8 Fn and Zn are basis coefficients EPG Basis: Graphical F0... F-1 F-2... indicates direction) Z1 Z2 Mz 140 Transverse basis are simple phase twists (sign of n Longitudinal basis are sinusoids F2 F1 sion en Voxel Dim ension Voxel Dim Voxel n Dimensio...

9 EPG Basis: Mathematically Transverse basis functions (F n ) are just phase twists: M xy (z) = X 1 n= N Longitudinal basis functions (Z n ) are sinusoids: M z (z) =Real F ne 2 inz + ( Z 0 +2 NX n=1 NX F n e 2 inz n=0 Z n e 2 inz ) F n and Z n are the basis coefficients, but we sometimes use them to refer to the basis functions they multiply F1 F-1 Z1 Although there are other basis definitions, this is consistent with that of Weigel et al. J Magn Reson 2010; 205:

10 Magnetization to EPG Basis Positive F states (n>0): Negative F states (n<0): Z states (n>=0): F n = F n = F-n Z n = Z 1 0 Z 1 0 Z 1 0 M xy (z)e 2 inz dz y(z)e 2 inz dz M z (z)e 2 inz dz 142

11 Review Question What are the F states that represent this magnetization (entirely along M y )? (z) = 0 Answers: (z) = 2cos(2πz) A) F1=1, F-1=1 z B) F1=1, F-1=i C) F1=i, F-1=i Recall: 143

12 Basis Functions in MR Sequences Key Point: Basis is easily propagated in MR sequences: Gradient (one cycle over voxel): Increase/decrease F n state number (n) or Fn+1=Fn RF Pulse Mixes coefficients between F n, F -n and Z n Relaxation T 2 decay attenuates Fn coefficients T 1 recovery attenuates Z n coefficients, and enhances Z 0 Diffusion Increasing attenuation with n (described later) 144

13 EPG Propagation: Gradient Gradients induce one cycle (2π) of phase across a voxel Magnetization in F n moves to F n+1 Dephasing for n>=0 F0 F n+1 = F n F1 Rephasing for n<0 Generally can apply p cycles (increase n by p) F-2 F-1 Z states are unaffected Assume the gradient such as a spoiler or crusher induces one twist cycle per voxel 145

14 EPG Relaxation over Period T Transverse states: F-2 F-2 F n = F n e -T/T2 Z n states attenuated: Z n = Z n e -T/T1 (n>0) Z1 Z1 Z 0 state also experiences recovery: Mz Mz Z 0 = M 0 (1-e -T/T1 )+ Z 0 e -T/T1 146

15 EPG RF: Transverse Effects First consider transverse spins after one cycle of dephasing: After a 60 x º tip, the transverse distribution is elliptical, but can be decomposed into opposite circular twists (F 1 and F -1 ) F 1 M y M x Transverse View M y M y M y M z = + M x M y 3D view M x Transverse View M x F 1 F -1 M x Longitudinal (Zn) states are also affected described shortly 147

16 EPG RF Rotations An RF pulse cannot change number of cycles Mixes F n, F -n and Z n (details next ) F2 F-2 Z2 148

17 EPG RF Rotations Dephased magnetization generally has an elliptical distribution, even after additional nutations (flips) Can decompose into a sum of opposite circular twists (previous slide) and cosines along M z Can derive (trigonometrically) for flip angle α about a transverse axis with angle φ from M x : 2 4 F n F n Z n = 2 4 cos 2 ( /2) e 2i sin 2 ( /2) ie i sin e 2i sin 2 ( /2) cos 2 ( /2) ie i sin i/2e i sin i/2e i sin cos F n F n Z n 3 5 Same Rφ RF rotation as before, in [y,y,mz] T frame 149

18 Phase Graph States (Flow Chart) Gradient Transitions Transverse (y) Longitudinal (Mz) F0 F0 * T1 Recovery F1 F2 FN... F-1 F-2 F-N... RF Effects Z1 Z2 ZN... T2 Decay T1 Decay 150

19 Examples: For a 90x rotation: (Right-handed!) R x ( /2) = i i 0.5i +0.5i For a 90y rotation: (Right-handed!) R y ( /2) = F n F n Z n = 2 4 cos 2 ( /2) e 2i sin 2 ( /2) ie i sin e 2i sin 2 ( /2) cos 2 ( /2) ie i sin i/2e i sin i/2e i sin cos F n F n Z n

20 Example 1: Ideal Spin Echo Train RF 90 y º 180 x º 180 x º G z 90 excitation transfers Z 0 to F 0 Crusher gradients cause twist cycle Relaxation between RF pulses (Here e -TE/T2 =0.81) 180 x pulse rotation: Swap F n and F -n Invert Z n 2 4 F n F n Z n = F n F n Z n

21 Example 1: Ideal Spin Echo Train RF 90º 180 x º 180 x º Signal = 0.81 Signal = 0.66 G z F0 F1 F-1 F0 F1 F-1 F0 Mz 153

22 Coherence Pathways: Spin Echo RF G z 90º 180 x º 180 x º 180 x º F1 F1 F1 phase time F-1 F0 F-1 F0 F-1 F0 Transverse (F) Longitudinal (Z) Echo Points Diagram shows non-zero states and evolution of states Perfect 180º pulses keep spins in low-order states 154

23 Example 2: Non-180º Spin Echo Ideal spin echo train gives simple RF rotations Now assume refocusing flip angles of 130º Compare RF rotations: 180 x º x º i i 0.38i 0.38i Positive F n states remain because magnetization is not perfectly reversed, generating higher order states Many more coherence pathways (see next slide ) 155

24 Coherences: Non-180º Spin Echo RF G z 90º 180º 180º 180º F2 F1 F1 phase time F-1 F0 Transverse (F) Transverse, but no signal Longitudinal (Z) Echo Points Only F 0 produces a signal other F n states are perfectly dephased 156

25 Example 3: Stimulated Echo Sequence 60º 60º 60º 60º We will follow this sequence through time Show which states are populated at each point 157

26 Example 3: Stimulated Echo Sequence 60º 60º 60º 60º Prior to the first pulse, all spins lie along M z This is Z 0 =1 Mz 158

27 Stimulated Echo: Excitation: F 0 60º 60º 60º 60º After a 60º pulse, transverse spins are aligned along M y (F 0 =sin60º) One half (cos60º) of the magnetization is still represented by Z 0 Mz F0 + Mz *Note that we show the voxel dimension along different axis for Z and F states 159

28 One Gradient Cycle 60º 60º 60º 60º The gradient twists the spins represented by F 0 We call this state F 1, where the 1 indicates one cycle of phase (F 1 =0.86) The spins represented by Z 0 are unaffected F0 F1 + Mz + Mz 160

29 Another Excitation (60º) 60º 60º 60º 60º The Z 0 magnetization is again split to F 0 and Z 0 The F 1 magnetization is split three ways, to F 1, F -1 (reverse twisted) and Z 1 F1 Mz + F0 F1 + + F-1 Z1 + + Mz 161

30 Another Gradient Cycle The F-1 state is refocused to F0 The F0 and F1 states become F1 and F2 The Z states are all unaffected F-1 Z F0 + sion en Voxel Dim + Voxe ion l Dimens F0 Z1 + sion en Voxel Dim Mz + F2 F1 + + Mz F1 The process continues! sion en Voxel Dim

31 Example 3: Coherence Pathways The stimulated echo sequence coherence diagram is shown below Compare with F and Z states on prior slide (location of arrow) RF / Gradients 60º 60º 60º 60º Transverse (F) Longitudinal (Z) F2 F1 Z2 Z1 phase F0 F-1 time F-2 163

32 Summary of Sequence Examples 90º and 180º RF pulses swap states Generally RF pulses mix states of order n Gradient pulses transition all F n to F n+p Coherence diagrams show progression through F and/or Z states to echo formation Signal calculation examples actually quantify the population of each state 164

33 Matlab Formulations Single matrix called P or FZ (for example) Rows are Fn, F-n and Zn coefficients, Column each n RF, Relaxation are just matrix multiplications Gradients are shifts 165

34 Matlab Formulations EPG simulations can be easily built-up using modular functions: (bmr.stanford.edu/epg) Transition functions: epg_rf.m Applies RF to Q matrix epg_grad.m Applies gradient to Q matrix epg_grelax.m Gradient, relaxation and diffusion epg_gdiff.m Include diffusion effects Transformation to/from (M x, M y, M z ): epg_spins2fz.m Convert M vectors to F,Z state matrix Q epg_fz2spins.m Convert F,Z state matrix Q to M vectors 166

35 Stimulated-Echo Example RF G z?? Simulate 3 Steps: RF, gradient and relaxation Sample Matlab code: function [S,Q] = epg_stim(flips) Q = [0 0 1]'; for k=1:length(flips) Q = epg_rf(q,flips(k)*pi/180,pi/2); Q = epg_grelax(q,1,.2,0,1,0,1); end; S = Q(1,1); % =1 (Equilibrium) % RF pulse % Gradient/Relax % Signal from F0 167

36 Stimulated Echo Example (Cont) Calculate signal vs flip angle: fplot(epg_stim([x x x],[0,120]) 168

37 EPG Summary / Other Directions F n, Z n basis represents many spins in a voxel MR operations on F n, Z n states are simple Coherence diagrams show which states are non-zero Signal at any time is F 0. Other states are dephased Matrix formulation allows easy Matlab simulations: Construct sequence of RF, gradient, relaxation, diffusion Transient and steady state signals by looping Can simulate multiple gradient directions Can also simulate multiple diffusion directions (Weigel 2010) 169