Extended Phase Graphs (EPG) Purpose / Definition Propagation Gradients, Relaxation, RF Diffusion Examples 1
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; % 2
EPG Motivating Example: RF with Crushers G z 120xº RF G z M y M y M y = + M x M x M x 3
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 4
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; 78:397 407. Hennig J. Echoes How to Generate, Recognize, Use or Avoid Them in MR-Imaging Sequences, Part I. Concepts in Magnetic Resonance 1991; 3:125-143. 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; 51:68-80 Weigel M. Extended Phase Graphs: Dephasing, RF Pulses, and Echoes - Pure and Simple. JMRI 2014; 5
Basic Magnetization Vector Definitions Common representations for magnetization: y = + i y * = - i 2 4 M x M y Mz 3 5 = 2 4 0.5 0.5 0 0.5i 0.5i 0 0 0 1 3 5 2 4 M xy M xy M z 3 5 2 4 M xy M xy M z 3 5 = 2 4 1 i 0 1 i 0 0 0 1 3 5 2 4 M x M y Mz 3 5 6
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 7
EPG Basis: Graphical Fn and Zn are basis coefficients F+0... F-1 F-2... indicates direction) Z1 Z2 Mz Z0 8 Transverse basis are simple phase twists (sign of n Longitudinal basis are sinusoids F+2 F+1 ension Voxel Dim ension Voxel Dim ension Voxel Dim...
EPG Basis: Mathematically Transverse basis functions (F n ) are just phase twists: M xy (z) =F + 0 M xy (z) = Longitudinal basis functions (Z n ) are sinusoids: ( M z (z) =Real Z 0 +2 1X n= 1 NX n=1 F + n e 2 inz 1 + X F + n e 2 inz +(F n ) e 2 inz n=1 Z n e 2 inz ) F + 1 F - 1 Z 1 F n and Z n are the coefficients, but we sometimes use them to refer to the basis functions ( twists ) they multiply Although there are other basis definitions, this is consistent with that of Weigel et al. J Magn Reson 2010; 205:276-285 9
Magnetization to EPG Basis F + states: F + n F n = Z 1 0 M xy (z)e 2 inz dz F - states: F - n = F* -n F n = Z 1 0 M xy(z)e 2 inz dz Z states: Z n = Z 1 0 M z (z)e 2 inz dz Note redundancy F - n = (F + -n)* (Can use F + n and F - n for n>0, or just F + -n (all n) 10
Review Question What are the F + and F - states that represent this magnetization (entirely along M y )? Answers: A) F + 1=1, F - 1=1 z B) F + 1=1, F - 1=i C) F + 1=i, F - 1=i (z) = 0 (z) = 2cos(2πz) Recall: M xy (z) = 1X n= 1 F + n e 2 inz 11
Basis Functions in MR Sequences Key Point: Basis is easily propagated in MR sequences: Gradient (one cycle over voxel): Increase/decrease F + n or F- n state number (n), e.g. F+ n+1=f + n RF Pulse Mixes coefficients between Z n, F + n, and F- n [ or (F+ -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) 12
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 Rephasing for n<0 (or F - ) Generally can apply p cycles (increase n by p) F + 0 F - 2 F + n+1 = F+ n F + 1 F - 1 Z states are unaffected Assume the gradient such as a spoiler or crusher induces one twist cycle per voxel 13
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) Z 1 Z 1 Z 0 state also experiences recovery: Mz Z 0 Mz Z 0 Z 0 = M 0 (1-e -T/T1 )+ Z 0 e -T/T1 14
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 Longitudinal (Zn) states are also affected described shortly M x 15
EPG RF Rotations An RF pulse cannot change number of cycles Mixes F + n, F- n and Z n (details next ) F + 2 F - 2 Z2 16
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 Z n n 3 5 0 = 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 3 5 2 4 F + n F Z n n 3 5 Same R φ RF rotation as before, in [M xy,m xy *,M z ] T frame 17
Example: 180xº Refocusing Pulse Magnetization Starting as F + 1=1 18
Phase Graph States (Flow Chart) Gradient Transitions Transverse (y) Longitudinal (Mz) F + 0 F - 0 Z0 T1 Recovery F + 1 F + 2 F + N... F - 1 F - 2 F - N... RF Effects Z1 Z2 ZN... T2 Decay T1 Decay 19
Examples: For a 90x rotation: (Right-handed!) R x ( /2) = 2 4 0.5 0.5 i 0.5 0.5 i 0.5i +0.5i 0 3 5 For a 90y rotation: (Right-handed!) R y ( /2) = 2 4 0.5 0.5 1 0.5 0.5 1 0.5 0.5 0 3 5 2 4 F + n F Z n n 3 5 0 = 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 3 5 2 4 F + n F Z n n 3 5 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 exampleb2_23.m Crusher gradients cause twist cycle Relaxation between RF pulses (Here e -TE/T2 =0.81, e -TE/T1 =0.9) 180 x pulse rotation: Swap F n and F -n Invert Z n 2 4 F + n F Z n n 3 5 0 = 2 4 0 1 0 1 0 0 0 0 1 3 5 2 4 F + n F Z n n 3 5 21
Example 1: Ideal Spin Echo Train RF 90º 180 x º 180 x º Signal = 0.81 Signal = 0.66 G z Mz Z0 F + 0 F + 1 F - 1 F + 0 F + 1 F - 1 F + 0 22
Example 1: Summarized 23
Coherence Pathways: Spin Echo RF G z 90º 180 x º 180 x º 180 x º phase time F + 1 Z0 F - 1 F + 0 Z0 F + 1 Z0 F - 1 F + 0 Z0 F + 1 Z0 F - 1 F + 0 Z0 Transverse (F) Longitudinal (Z) Echo Points Diagram shows non-zero states and evolution of states Perfect 180º pulses keep spins in low-order states 24
Example 2: Non-180º Spin Echo Ideal spin echo train gives simple RF rotations Now assume refocusing flip angle of 130º Compare RF rotations: 180 x º 2 4 130 x º 0.18 0.82 0.77i 0.82 0.18 0.77i 0.38i 0.38i 0.64 3 5 Positive F + n states remain (from F+ n ) because magnetization is not perfectly reversed, generating higher order states Many more coherence pathways (see next slide ) 25
Coherences: Non-180º Spin Echo RF G z 90º 180º 180º 180º F + 2 phase time F + 1 Z0 F - 1 Transverse (F) Transverse, but no signal Longitudinal (Z) Echo Points F + 1 F + 0 Only F 0 produces a signal other F n states are perfectly dephased 26
Example 3: Stimulated Echo Sequence 60º 60º 60º 60º We will follow this sequence through time Show which states are populated at each point 27
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 Z0 Mz 28
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 Z0 F + 0 + Mz Z0 *Note that we show the voxel dimension along different axis for Z and F states 29
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 F + 0 F + 1 + Mz Z0 + Mz Z0 30
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 F + 1 F + 1 F - 1 Mz + F + 0 Z0 + + + + Mz Z0 Z1 31
Another Gradient Cycle The F - 1 state is refocused to F- 0 or F+ 0 The F + 0 and F+ 1 states become F+ 1 and F+ 2 The Z states are all unaffected The process continues! F + 1 F - 1 + + Z1 F + 2 F + 0 + + Z1 + F + 0 + Mz Z0 + F + 1 + Mz Z0 32
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) F + 2 Z2 phase F + 1 F + 0 F - 1 F - 2 Z1 Z0 time 33
Summary of Sequence Examples 90º and 180º RF pulses swap states Generally RF pulses mix states of order n Gradient pulses transition F + n to F+ n+p and 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 34
Matlab Formulations Single matrix called P or FZ (for example) Rows are F + n, F - n and Zn coefficients, Column each n RF, Relaxation are just matrix multiplications Gradients are shifts 35
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 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 36
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); (See epg_stim.m) % Z0=1 (Equilibrium) % RF pulse % Gradient/Relax % Signal from F0 37
Stimulated Echo Example (Cont) Calculate signal vs flip angle: fplot(epg_stim([x x x],[0,120]) 38
Diffusion Diffusion weighting can easily be applied Details in Weigel et al. J Magn Reson 2010; 205:276-285 Attenuation greater with n for both F n and Z n Requires physical 2π gradient twist k (m -1 ) If gradient is played, Δk=k, otherwise Δk=0 39
Summary 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) 40