2D Spiral Excitation Pulses
2D spiral pulses can be designed for any flip angle using Fourier designs The flip angle is the Fourier transform of the k-space weighting. This was described in
“A k-space analysis of small-tip-angle excitation” J Pauly, D Nishimura, A Macovski Journal of Magnetic Resonance (1969) 81 (1), 43-56
“A linear class of large-tip-angle selective excitation pulses”, J Pauly, D Nishimura, A Macovski Journal of Magnetic Resonance (1969) 82 (3), 571-587
An example design is an eight turn pulse with 1 G/cm gradient, and 2 G/cm/ms slew rates (10 mT/m and 20 mT/m/s)
%use octave
try
cd rf_tools_octave
catch
try
cd ../rf_tools_octave
catch
try
cd ../rf_tools_octave
catch
cd ../../rf_tools_octave
end
end
end
pkg load signal
%use octave
[rf g] = dz2d(8,1,4,512,1,2);
re_rf = real(rf);
im_rf = imag(rf);
re_g = real(g);
im_g = imag(g);
Gradient duration is 8.205 ms
where the second argument is the spatial bandwidth in cycles/cm and the fourth argument is the space-bandwidth product (like TBW for 1D pulses).
The pulse ends up being 8.2 ms. We can plot the RF pulse and $G_x$ and $G_y$ gradients as
%use octave
t = [1:512]*8.205/512;
(View plot code)
Next we want to simulate the excitation profile. At this point it is scaled to 1 radian. To increase it to $\pi/2$
%use octave
rf = rf*pi/2;
The abr() simulator will take 2D pulses. First define spatial vectors
%use octave
x = [-32:32]/4;
y = [-32:32]/4;
mxy = ab2ex(abr(rf,g,x,y));
abs_mxy = abs(mxy);
re_mxy = real(mxy);
im_mxy = imag(mxy);
(View plot code)
This RF pulse is well refocused at the end of the pulse, as we can see by plotting $M_x$ and $-M_y$,
(View plot code)
(View plot code)
$M_x$ is only a couple of percent.
We can simulate over a broader spatial range to visualize the sidelobes which are inherent in this type of design
%use octave
x = [-32:32]/4;
y = [-32:32]/4;
x = 4*x;
y = 4*y;
mxy4 = ab2ex(abr(rf,g,x,y));
re_mxy4 = real(mxy4);
im_mxy4 = imag(mxy4);
(View plot code)
(View plot code)
Note that the main lobe is imaginary, the first sidelobe is real, and the second sidelobe is imaginary. Extra credit if you can explain this!
We can use the same pulse as a spin-echo pulse if we increase the flip angle to $\pi$, and make sure the gradients integrate to zero. Here we do this with a single sample, in fact you’d want to add an additional gradient lobe, or incorportate the area by adjusting one of the crusher gradients.
%use octave
x = [-32:32]/4;
y = [-32:32]/4;
rfse = rf*2;
rfse = [0; rf*2];
gse = [-sum(g); g];
mxyse = ab2se(abr(rfse,gse,x,y));
re_mxyse = real(mxyse);
(View plot code)
We’ve just plotted $M_y$ here, $M_x$ is again very small.
Things to try:
- Change the space-bandwidth product to 6 or 8
- Change the number of turns
- Do a two shot experiment, where g = -g for the second shot. Add Mxy’s.
- Use modern gradient numbers