Appendix A: Bloch Simulator
A numerical Bloch simulator was implemented, simulating one single voxel during the QRAPMASTER acquisition. The voxel contained 40 spinisochromats implemented as column vectors, each containing the magnetisation in the x, y and z direction, . The spinisochromats were aligned at equidistant positions, d, in the through-plane direction of a simulated 5-mm slice. The evolution of the magnetisation vector was calculated in time steps of Δt = 1 ns according to:
[3]
Where was the rotation around the z-axis caused by the gradient field strength S(t) at position d and time point t. The rotation angle of was calculated as:
[4]
Where d = 0 for the centre of the slice and was the gyro-magnetic ratio (1H). represented the rotation around the x- or y-axis imposed by the amplitude of the radio frequent RFpulse at each time point t. The rotation angle of was calculated as:
[5]
where (t) was the amplitude of the RFpulse at time point t. described the relaxation behaviour of the spins, with the longitudinal relaxation rate R1 and the transverse relaxation rate R2 as:
[6]
The start value of the magnetisation for each spinisochromat at time t = 0 was set to .
The RF pulses, gradients and timings of the QRAPMSTER sequence were implemented as a script for the Bloch simulation. For each combination of saturation delay, TDi, and each echo time, TEj, the vector sum of all the spinisochromats was recorded, forming a matrix, E, with the magnetisation states, :
[7]
To simulate the presence of noise in the sequence, a matrix, N, containing random white noise in the interval [-1, 1] was scaled with a factor SNR, to reflect the SNR ratio of the measurement, and this was added to E:
[8]
The simulator was implemented in MATLAB R2008a (MathWorks, 2008).