Small Tip Angle Approximation
Based on the results in page 41 of the book "Handbook of MRI Pulse Sequences" by Bernstein et al.
50In this example, we will showcase a common approximation in MRI, the small tip angle approximation. For this, we will simulate a slice profile for spins with positions $z\in[-2,\,2]\,\mathrm{cm}$ and with a gradient $G_{z}$ so their frequencies are mapped to $f\in[-5,\,5]\,\mathrm{kHz}$. To start, we define an RF pulse with a flip angle of 30 deg and pulse duration of $T_{\mathrm{rf}}=3.2\,\mathrm{ms}$.
B1 = 4.92e-6
Trf = 3.2e-3
zmax = 2e-2
fmax = 5e3
z = range(-zmax, zmax, 400)
Gz = fmax / (ฮณ * zmax)-5000.0:25.06265664160401:5000.0The designed RF pulse is presented in the figure below, where the additional gradient refocuses the spins' phase after the excitation.
seq = PulseDesigner.RF_sinc(B1, Trf, sys; G=[0;0;Gz], TBP=8)
p2 = plot_seq(seq; max_rf_samples=Inf, slider=false)"../assets/42-seq.html"Now we will perform the simulation using the function simulate_slice_profile. Note that we modified ฮt_rf in sim_params to match the resolution of the waveform.
sim_params = Dict{String, Any}("ฮt_rf" => Trf / length(seq.RF.A[1]))
M = simulate_slice_profile(seq; z, sim_params)"../assets/4b-profile.html"This produces the following slice profile:
As you can see, for a flip angle of 30 deg, the slice profile is very close to the small tip angle approximation (the Fourier transform of $B_{1}(t)$).
But what will happen if we use a flip angle of 120 deg instead?
ฮฑ_desired = 120 + 0im # The multiplication of a complex number scales the RF pulse of a Sequence
ฮฑ = get_flip_angles(seq)[1] # Previous FA approx 30 deg
seq = (ฮฑ_desired / ฮฑ) * seq # Scaling the pulse to have a flip angle of 120
M = simulate_slice_profile(seq; z, sim_params)"../assets/4a-profile.html"For this case, the small tip angle approximation breaks ๐ข, thus, the reason for its name!
This basic sinc pulse is not designed to be $B_{1}$-insensitive. Some adiabatic RF pulses have been proposed to achieve this. Watch out for a future example showing these adiabatic RF pulses ๐.
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