bSSFP signal is dependent on isochromats' position along the z axis even after having applied RF.freq_offset

[danil1932]

What happened?

Hello,

I am simulating a bSSFP sequence used for Magnetic Resonance Fingerprinting. The issue I encounter occurs whenever I try to simulate the entire 5-slice acquisition, i.e. when I set a slice offset along z; naturally, I apply the corresponding frequency offset to the RF pulses (RF.freq_offset = Gz.amplitude * slice_position, with slice_position in meters)

The issue manifests itself as a frequency offset (df) of the bSSFP signal whenever the slice is not centered at z=0: df = (2 pi * RF_duration * Gz_amplitude * dz_offset) / (2 pi * TR). A simple example of this is shown in the Julia notebook ('DebuggingNotebook') attached and in the attached images: the signal for an isochromat at 0.01mm with RF centered at 0.01mm is different from the signal for an isochromat at 0mm with RF centered at 0mm. The signals at 0mm and 0.01 match almost perfectly for a spin at 0 Hz and a spin at 8.6Hz (calculated using the formula above for 1.6ms, 5000Hz/mm RF and TR=9ms; red and green lines on the 1st image). Even more curiously, the issue goes away when I remove the slice-select rephasers (which solves the problem, but only for a single spin whereas I'd like to simulate the slice profile).
More details on how the sequence was generated can be found in the 'SimpleSincTests' python notebook)

Any and all help would be greatly appreciated, please do let me know if there is some convention that I am not aware of. I have talked to a colleague of mine and he was just as confused as I was; we do not see the same frequency shift depending on the z position in our in vivo acquisitions which are well described by the signal simulated for z=0mm. Thank you in advance!

Best regards,
Dan

bSSFP_sequences.zip

DebuggingNotebook.ipynb

SimpleSincTests.ipynb

Environment

OS x86_64-linux-gnu
Julia 1.11.6
KomaMRIPlots 0.9.4
KomaMRIFiles 0.9.3
KomaMRI 0.9.2
KomaMRICore 0.9.3
KomaMRIBase 0.9.3

[cncastillo]

Hi. Thank you very much for this detailed report, this will be very useful for discovering the bug. Could you put in this issue these pulseq files to reproduce the problem?

seq0mm = read_seq(".../bSSFP_200Points_RF_freq_offset=0_NoRephasers.seq")
seq001mm = read_seq(".../bSSFP_200Points_RF_freq_offset=50Hz_NoRephasers.seq") 

I can test if the fix I have in mind solves the issue.

Cheers!

---

I will leave a few notes here for myself:

My intuition is that the problem comes from to the fact that we are assuming the phase_offset of the pulse to be the "initial phase" instead of the phase at the RF center. So right now we see something like this (the frequency offset adds an additional phase, so you get $\phi_1$ instead of $\phi_0$ at the RF center):

Related: #571. The most sane solution is probably to force the phase_offset to mean the phase at the RF center in the discretize step, that way we wouldn't need to compensate the phase added by the frequency offset. I believe this matches the scanner's behavior. In practice this means:

$$ B_{1}(t) = A(t) \exp(i (\phi(t) - \phi_{\Delta f})) $$

with

$$ \phi_{\Delta f} = -2\pi \int_{0}^{t_{\mathrm{center}}} \Delta f(t) \mathrm{d}t. $$

Here $A(t)$ is the amplitude modulation, $\phi(t)$ the phase modulation, and $\Delta f(t)$ the frequency modulation (constant in this case).

Even more curiously, the issue goes away when I remove the slice-select rephasers

The reason this solves the problem is most probably because it exactly does rewind the phase until the rf center.

[danil1932]

@cncastillo , thank you for taking the time to look into this! I added the archive with the four PyPulseq sequences to the original post.

Best regards,
Dan

[danil1932]

@cncastillo , I was wondering if you've been able to find out more in regard to this behavior? I tried setting RF.phase_offset for all RF pulses in the sequence according to the formula but it did not change the signal at 0.01 mm. I think the issue is more the gradual phase accumulation every cycle (which manifests itself as an apparent frequency offset I talk about in the original post) as opposed to a constant factor that needs to be substracted from every RF's phase. I may very well be mistaken but I think that a different reference point for the RF phase for a series of identical pulses would just result in the magnetization being flipped "the other way" (30° instead of 0° for example), making no difference when looking at the absolute value of the signal.

Best regards,
Dan

[cncastillo]

I actually I did look at it for 4-5 hours last week, but I haven't truly discovered the problem yet. The magnetization after the first pulse looks correct, after the second pulse there are significant magnitude differences. As you said, the phase shouldn't affect the magnitude, but the phase of the RF pulse does change the effective rotation vector,

$$ B=[B_{1,x},B_{1,y},\vec{x}\cdot\vec{G}-\Delta f_{\text{RF}} / (\gamma / 2\pi) ]. $$

I need to check if correcting it solves the problem. Also the gradient is very high (600 mT/m) in that sequence, so maybe there is an error that is negligible in other cases but it is not in this case.

Sorry for not being able to solve this one yet!

Small update:
Actually the first excitation is quite different already, so it might be related to the RF phase:

[danil1932]

Thank you for continuing to look into this, I really appreciate it!