Armin Rund^{1}, Christoph Stefan Aigner^{2}, Lena Nohava^{3}, Roberta Frass-Kriegl^{3}, Elmar Laistler^{3}, Karl Kunisch^{1,4}, and Rudolf Stollberger^{2}

^{1}Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria, ^{2}Institute of Medical Engineering, Graz University of Technology, Graz, Austria, ^{3}Division MR Physics, Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, Vienna, Austria, ^{4}Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria

### Synopsis

An optimal control framework for designing parallel
transmission RF pulses and gradient shapes is introduced. The optimal control
model includes technical constraints and a local SAR model based on the Q-matrix
formalism. Second-order optimization methods give RF pulses with enhanced
homogeneity of the excitation pattern and/or decreased local SAR. The optimized
results are tested in numerical experiments and validated with numerical
electromagnetic simulations.

### Purpose

The inclusion of B_{1}^{+} maps in the RF pulse design process for
parallel transmit (pTx^{1,2}) allows increasing the homogeneity of
multi-dimensional excitation patterns^{3,4}. Below, an optimal control method
is introduced for the joint design of multi-dimensional RF, and gradient shapes
with hard physical constraints for pTx with minimum local 10g-SAR using the Q-matrix
formalism^{5}.### Theory

Joint design of RF and gradient amplitudes
for fixed pulse duration is posed as an optimal control model for the
spin-domain Bloch equation with Cayley-Klein parameters ($$$\alpha,\beta$$$).
The slice profile accuracy for a 2D excitation is posed as inequality
constraints (in magnitude and phase). We control eight complex-valued RF pulses
$$$U_k(t)$$$ and two gradient components $$$(g_1(t),g_2(t))$$$
$$B_1=\sum_{k=1}^8 S_k(x)U_k(t), \qquad G_s=g_1(t)
x+g_2(t)y$$
using the RF coil sensitivities $$$S_k(x)$$$
of the RF channels. $$$(x,y)$$$ are the spatial coordinates. The task is to
minimize the time averaged maximum local 10g SAR modeled by the Q-matrix^{5}. Componentwise
amplitude constraints on RF and gradients are included by semismooth Newton
techniques. The more general bounds on the slew rate and the slice profile
accuracy are treated with a $$$L^p$$$ penalization^{6}. The optimization is done with
second-order methods using exact adjoint-based first derivatives and
quasi-Newton approximations for the second derivatives (BFGS) together with a
trust-region framework for robustness^{6}. ### Methods

The proposed
algorithm is implemented in MATLAB (The MathWorks, Inc, Natick, USA) and
applied to the design of pulses for a homogeneous two-dimensional 90° slice excitation
using a generic shielded 8-channel pTx loop head coil^{5} (Figure 1). We
used electromagnetic simulations (XFdtd 7.4, Remcom, State College, USA) and
circuit co-simulation (ADS, Keysight Technologies, Santa Clara, USA) to compute
the complex B_{1}^{+} maps in of a homogeneous spherical
phantom (d = 18cm, ε = 50.6, σ = 0.66S/m). The field of view was set to 220x220mm
with a 64x64 matrix resulting in 4096 spatial points. The circular
2-dimensional target pattern (ROI) was defined in the transverse plane in the
center of the sphere with a diameter of 180mm, with in-ROI and out-ROI
regions (Fig. 1, right). As the
initialization of the optimization, we used the spatial domain method^{3,4}
with excitation flip angle 90° (>83° in-ROI, <5° out-ROI), temporal discretization of 4µs, and
allowed error of 10% with a pulse duration of 12.4ms. For optimization, the
complex RF pulses were bound by a peak power of 1kW per channel and both
gradient channels limited to a peak amplitude of 40mT/m and a slew rate of
212T/m/s. For “low SAR”-optimization, the profile accuracy was constrained by
the mean squared error of the initialization (in-ROI magnitude and phase,
out-ROI magnitude). A factor six smaller in-ROI magnitude error was used for
“low error”-optimization. ### Results and Discussion

Figure 2 shows the RF and gradient shapes of
the initialization and after optimization. The initial spiral-out gradient
trajectory results in skewed power deposition at the center of k-space, thus
leading to high peak RF power. Both optimized results show significantly reduced
peak RF and a more uniform RF power deposition over time.
Figure 3 compares the peak RF power and
SAR of the investigated pulses. The “low error” optimum yields a higher profile
accuracy in-ROI, both in the root mean square deviation (RMSD) and in the infinity-norm
(higher minimum flip angle) and decreases peak RF power by 90%, and SAR by 68%.
The “low SAR” optimum holds the in-ROI RMSD of the initialization. The in-ROI
minimum flip angle is decreased by 1.6% which allows the optimization method to
reduce peak RF power by 97% and maximum SAR by 86%. Both optima exhibit the
same in-ROI phase RMSD as the initialization, but with smaller maximum
deviation.
Figure 4 shows the results of the Bloch
and SAR simulations. The first row shows the flip angle, the second row
contains the phase of the transversal magnetization ($$$2\alpha \beta^*$$$).
Excitation profiles are well defined and phase dispersion is slightly lower as
compared to the initial pulse. Rows 3-5 contain SAR maximum intensity
projections in all three directions. Notably, SAR values decrease globally
without significant changes in their spatial distribution.### Conclusion

Bloch and SAR simulations indicate the
feasibility of using the proposed optimal control based design method to
jointly compute complex RF pulses and gradient shapes for large flip angle pTx
excitation and its ability to reduce local SAR and peak RF power. ### Acknowledgements

Partly supported by Austrian Science Fund
(FWF) projects SFB F3209-18 (MOBIS) and P28059-N36 (pULSE), and by BioTechMed-Graz.### References

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