Stanford provided all basic functions/M-files needed for Bloch simulation. The following MATLAB code snippet was made on top of these M-files to simulate the Hanh Echo:
T = 1000; % total duration df = 10; % Hz off-resonance. T1 = 600; % ms. T2 = 100; % ms. tau = 50; % ms. TE/2 TR = 500; % ms. alpha1 = pi/2; alpha2 = pi; % ===== Get the Propagation Matrix ====== [A,B] = freeprecess(tau,T1,T2,df); % ===== Simulate the Decay 90 - 180 ====== M = zeros(3,3); M(:,1)=[0;0;1]; % Starting magnetization. % Let M(:,1) be the magnetization just before the 90. % M(:,2) be just before the 180. % M(:,3) be at TE = 2*tau. Rflip = yrot(alpha1); Rrefoc = xrot(alpha2); M(:,2)=A*Rflip*M(:,1)+B; % just before - P2 M(:,3)=A*Rrefoc*M(:,2)+B; % Echo sig = abs(M(1,3)+1i*M(2,3)); %Echo Magnitude % ===== Simulate the Decay 90 - 90 ====== alpha1 = pi/2; alpha2 = pi/2; M1 = zeros(3,3); % Keep track of magnetization at all time points. M1(:,1)=[0;0;1]; % Starting magnetization. Rflip1 = yrot(alpha1); Rrefoc1 = xrot(alpha2); M1(:,2)=A*Rflip1*M1(:,1)+B; % before -P2 M1(:,3)=A*Rrefoc1*M1(:,2)+B; % Echo sig1 = abs(M1(1,3)+1i*M1(2,3)); disp(sig1/sig) % should be 0.5
According to some basic knowledges, a spin echo using 90-180 would produce the maximum echo amplitude. While the maximum SE(Hanh Echo) intensity of a 90°-90° pair is only half as large as that produced by a 90°-180° pair. This means sig1/sig should be 0.5 . But what I got is a number larger than 1. What's wrong with my simulation code? How to derive the following conclusion using Bloch Simulation?
In general, if the first RF-pulse has flip angle α1 and the second has flip angle α2, the maximum signal intensity of the Hahn echo will be smaller than the conventional (90°-180°) echo by a factor of (sin α1)•(sin² α2/2).