Sherry Huang^{1}, Darryl C. Jacob^{2,3}, Michael Beverland^{3}, Stephen Jordan^{3}, Helmut G. Katzgraber^{3}, Matthias Troyer^{3}, Rasim Boyacioglu^{4}, Yun Jiang^{4}, Dan Ma^{4}, Mark A. Griswold^{4}, Julie Love^{3}, and Debra F. McGivney^{4}

RF pulse design is crucial in creating the desired magnetization profile which is the basis of Magnetic Resonance Imaging. There are various methods to generate the RF pulse and gradient waveforms based on Fourier relationships, filter design, or optimizations. These methods rely on assumptions and approximations due to computational power constraints. Here we present preliminary results of using quantum inspired algorithms for Bloch simulation and RF pulse design optimization.

Theory: Here we used a relatively straightforward cost function that minimized the difference between the target magnetization profile and experimental profile plus regularization factors.

$$Cost Function= argmin ||u(x)-f(b(t))||^2 +a||g(b(t))||^2$$

Where $$$u(x)$$$ is the target excitation pattern, $$$f(b(t))$$$ is the output of a test excitation pattern, $$$b(t)$$$ includes both $$$B_1(t)$$$ and $$$G(t)$$$, and $$$g(b(t))$$$ is a function of the input waveform, which depends on the physical limits and safety concerns of the system. Because QIO methods have flexibility in the cost function design, in comparison to traditional approaches, the regularization term could include conditions such as SAR limits, maximum gradient slew rate and amplitude, maximum B1 slew rate and amplitude, the nonlinearity of RF Power Amplifiers, and maximum RF duration, to name a few. The quantum-inspired optimizer then samples combinations of RF pulses and gradient to converge at a solution.

Experiments: We first replicated the Bloch simulation on a quantum-inspired
simulator based on a custom implementation of the fourth order Runge-Kutta
(RK4) method^{8}. It was validated against the same simulation using
MATLAB^{9}. The capability of the quantum-inspired optimizer was then
tested to optimize an RF pulse given the magnetization profile (figure 2a) and the
gradient waveform (figure 1a). The desired magnetization profile was specified
as:

$$ M(x)=[M_x,M_y,M_z] \begin{cases} [0,0,1]& \text{if } x\, is \,outside\, the\, slice \\ [0, sin(\alpha), cos(\alpha)]& \text{else} \end{cases}$$

where $$$\alpha$$$ was the flip angle. In this experiment, the regularization factor was weighted on the RF pulse’s power, with a constant multiplier of 0.1. 2000 points were simulated. In this example, the optimization algorithm was based on Microsoft’s quantum-inspired implementation of Simulated Annealing (SA). We optimized in the time domain, which resulted in jagged output; therefore, we also explored the Gaussian Wavelets domain, as well as Sinc Parameterization to reduce cost in timing and constraining pulse smoothness.

- Pauly, J., Nishimura, D. & Macovski, A. A k-space analysis of small-tip-angle excitation. Journal of Magnetic Resonance 213, 544–557 (2011).
- Conolly, S., Nishimura, D. & Macovski, A. Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem. IEEE Transactions on Medical Imaging 5, 106–115 (1986).
- Pauly, J., Roux, P. L., Nishimura, D. & Macovski, A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm [NMR imaging]. IEEE Transactions on Medical Imaging 10, 53–65 (1991).
- Conolly, S., Nishimura, D., Macovski, A. & Glover, G. Variable-rate selective excitation. Journal of Magnetic Resonance (1969) 78, 440–458 (1988).
- Lee, D. et al. VERSE-Guided Numerical RF Pulse Design: A Fast Method for Peak RF Power Control. Magn Reson Med 67, 353–362 (2012).
- Anand, C. K., Stoyan, S. J. & Terlaky, T. The gVERSE RF Pulse: An Optimal Approach to MRI Pulse Design. in Modeling, Simulation and Optimization of Complex Processes 25–48 (Springer, Berlin, Heidelberg, 2008).
- Max-SAT 2016 - Eleventh Max-SAT Evaluation. Available at: http://maxsat.ia.udl.cat/results-incomplete/. (Accessed: 5th November 2018).
- Press, W. H. & Vetterling, W. T. Numerical recipes: the art of scientific computing. (Cambridge Univ. Pr., 1988).
- Hargreaves, B. Bloch Equation Simulation

Figure 1: a) Bloch simulation in MATLAB with 238 time
points. B) Bloch simulation on a quantum simulator with 238 time points. The
magnetization profiles are the same. This will serve as the $$$f(b(t))$$$ function in optimizations below.

Figure 2: a) This shows $$$u(x)$$$ in the optimizer which is the ideal top hat
magnetization profile. b) Result from the optimizer (shown in blue) in
comparison to simulation in figure 1b. as the reference (shown in green). The
gradient waveform is the same as figure 1b. Without any other constraints, the
optimizer output shows a very similar pulse shape to a Sinc pulse, and the
output magnetization profile is comparable to the given profile, with a better
performance in side lobes and roll off. The jaggedness in the RF pulse waveform
can be ameliorated with physical hardware constraints.

Figure 3: Gaussian
wavelet domain optimization result. The magnitude of the RF pulse at each time
point can be treated as the sum of many independent, normally distributed,
random variables. Therefore, the waveform has been constrained in the form of
Gaussian wavelets. The output pulse becomes much smoother than direct time
domain optimization. There are less ripples in the slice profile direction than
regular Sinc pulse. (The pulse is inverted due to a sign difference in MATLAB
and the optimizer).

Figure 4: Sinc parametrization optimization result.
Sinc parametrization uses the height and width of Sinc as optimization
variables. Sign inversion is also present in this simulation. This shows that the
optimization result could converge to a theoretical pulse shape.