Synopsis

Keywords: Parallel Transmit & Multiband, RF Pulse Design & Fields, Python, PyTorch, Optimization

Traditional pTx pulse design focuses on optimizing RF pulses with a fixed or parametrized gradient waveform. Utilizing fast GPUs for autodifferentiation, the optimization process becomes sufficiently efficient that RF pulses and the underlying gradient waveforms can be freely optimized concurrently within short time. This allows for the time-efficient creation of pTx pulses without prior knowledge about suitable gradient trajectories. Still, restrictions e.g. due to hardware limitations may be included into the optimization process.
We provide a pulse design toolbox and demonstrate its ability to generate universal small-FA excitation pulses as well as large-FA pulses.

Introduction

State-of-the-art pTx applications optimize individual RF waveforms for multiple transmit channels, while applying gradient waveforms at the same time. Often this waveform is either predefined or derived from earlier optimization steps, in which a parameterized waveform was generated^1,2. This had to be done to keep computational complexity manageable, in order to reduce calculation times.In more recent work gradient waveforms have been optimized either as a set of splines or completely freely together with a single RF waveform^3,4.
Modern optimization methods and ever-increasing computational power make free optimization of both parallel transmit pulses and gradient waveforms possible within reasonable time. Here we present a toolbox which uses PyTorch⁵, which was originally designed for machine learning, for simultaneous optimization of RF and gradient waveforms. PyTorch's autodifferentiation capabilities allow for automatic creation of jacobian matrices for functions with arbitrary complexity, which improves the performance of many optimization algorithms, thus reducing the required time for optimization.We present MPRAGE and GRE images acquired with a subject-specific tailored pulse (TP) which was optimized within approximately one minute. We also present universal pulses (UP) that were generated using this toolbox.

Methods

Software for pTx pulse optimization was developed in Python 3.9 using PyTorch 1.13.0. Local SAR supervision was performed using a Virtual Observation Points (VOP) model with 208 VOPs, which was specifically generated for this RF coil. All measurements were performed on a 9.4T whole-body MRI scanner (Magnetom 9.4T Plus, Siemens Healthineers, Germany) with 16 independent transmit channels, using a custom built head coil (16 transmit, 31 receive elements)⁶. Data was acquired from 14 healthy adults in accordance with the local ethics committee. $$$B_1^+$$$ and $$$B_0$$$ maps were acquired using 3D satTFL⁷ (TR=2.44 ms, TE=0.75 ms, GRAPPA 2x2, FA=2°/70°, matrix: 64x64x64, nominal resolution 3.5 mm isotropic).
A full Bloch simulation taking $$$B_1^+$$$ and $$$B_0$$$ maps into account was implemented for arbitrary RF and gradient pulses. Additionally, the small tip angle approximation (STA) was implemented allowing for faster simulation of small tip angle RF pulses by using the spatial domain method⁸. Optimization of RF and gradient waveforms was performed using the AdamW algorithm. As it employs a gradient descent based method, the optimization greatly benefits from PyTorch's autodifferentiation capabilities. The learning rate was set to $$$4\cdot10^{-4}$$$ and the optimizer was reinitialized after every set of 1000 iterations, using the former set's best performing pulse as start value. The cost function used root mean square error (RMSE) and quadratic hinge functions of the system/SAR limits as metric. Optimization was run on a computer equipped with a NVIDIA RTX A6000 GPU (48GiB VRAM).
A pTx excitation pulse (FA=10°, SAR limit 2.3 W/kg, TR=25.5 ms, Pulse duration 520 µs, raster time 10 µs) was designed as UP on a database of 9 subjects and also as TP for 5 additional subjects. Optimization of all three gradient axes and 16 complex RF waveforms was run on the STA model for 2,500 iterations (TP) resp. 7,500 iterations (UP). The UP was then refined with 2500 iterations of the Bloch optimizer.The pulses were used in two imaging sequences, where they were scaled down to 9° (MPRAGE) resp. 5° (GRE) and compared to a CP mode pulse as defined by the MR scanner.
Furthermore, a pTx pulse with nominal flip angle of 90° was designed. This was done by optimizing a 9° pulse using 3500 iterations of the STA simulation. It was then scaled up by a factor of 10 and refined with 1000 iterations of the Bloch simulation. This pulse was designed without SAR constraint since it was only played out once, as preparation pulse for a satTFL sequence.

Results

Calculation of the UP took 20.5 minutes for 9 subjects. When simulating the UP on the 5 subjects which were not part of the UP design database, the NRMSE to the target FA was (12.3±1.7)%.The tailored pulses for the same 5 subjects took approx. 67 seconds each to optimize and resulted in a NRMSE of (8.2±1.0)%. Figure 4 compares the FA distribution for all three pulses in one subject. It clearly shows that the CP-mode pulse not only has the widest FA distribution, but also misses the mean target FA of 10° by circa 40%.
As expected the tailored pulse performs best, with a narrow FA distribution.
The large FA pulse took 3.5 minutes to optimize. In simulation it achieved an RMSE of 7.1%. The measured FA map is shown in Figure 5. It had an RMSE of 8.35%, compared to the CP-mode's RMSE of 39.58%. The mean FA was measured at 88.6°(TP) resp. 52.2°(CP).

Conclusion

We demonstrated that the new approach is faster, easier to implement and has more degrees of freedom than previous approaches. Tailored RF pulses for common imaging sequences can be designed rapidly. We designed a universal pulse which was successfully applied and showed that the simulation gives accurate results in the low- and high FA regime.

Acknowledgements

Funding by the Deutsche Forschungsgemeinschaft (DFG –­German Research Foundation) under the Reinhart Koselleck Programme (DFG SCHE 658/12) and by the European Research Council (ERC Advanced Grant No 834940, SpreadMRI) is gratefully acknowledged.