On Variant Strategies to Solve the Magnitude Least Squares Optimization Problem in Parallel Transmission Pulse Design and Under Strict SAR and Power Constraints

Abstract

Parallel transmission is a very promising candidate technology to mitigate the inevitable radiofrequency (RF) field inhomogeneity in magnetic resonance imaging (MRI) at ultra-high field (UHF). For the first few years, pulse design utilizing this technique was expressed as a least squares problem with crude power regularizations aimed at controlling the specific absorption rate (SAR), hence the patient safety. This approach being suboptimal for many applications sensitive mostly to the magnitude of the spin excitation, and not its phase, the magnitude least squares (MLS) problem then was first formulated in 2007. Despite its importance and the availability of other powerful numerical optimization methods, the MLS problem yet has been faced almost exclusively by the pulse designer with the so-called variable exchange method. In this paper, we investigate various two-stage strategies consisting of different initializations and nonlinear programming approaches, and incorporate directly the strict SAR and hardware constraints. Several schemes such as sequential quadratic programming (SQP), interior point (I-P) methods, semidefinite programming (SDP) and magnitude squared least squares (MSLS) relaxations are studied both in the small and large tip angle regimes with RF and static field maps obtained in-vivo on a human brain at 7 Tesla. Convergence and robustness of the different approaches are analyzed, and recommendations to tackle this specific problem are finally given. Small tip angle and inversion pulses are returned in a few seconds and in under a minute respectively while respecting the constraints, allowing the use of the proposed approach in routine.

First and second order derivatives for optimizing parallel RF excitation waveforms

“…Thus, the same non-linear programming approaches as for solving the arbitrary flip angle problem are suitable for optimizing RF waveforms in the small tip angle regime, too. We are not aware of formulas for derivatives for the magnitude least squares objective function in the literature, but the derivatives of the magnitude-squared least squares objective function [33,23] are structurally similar to ours. The reason for also presenting derivatives of the small tip angle approximation is to compare their performance to the exact method, not only with respect to the achieved quality, but especially regarding computational effort and computation time.…”

“…Filter design methods [22] rely on the hard pulse approximation, but have not been extended to parallel RF waveform design yet. Other options are the use of numerical derivatives as in [23]. In contrast to these methods, our derivatives aim at the avoidance of approximations.…”

“…It requires a number of floating point operations, which is quadratic in the number of samples and channels and linear in the number of voxels. In the literature, second order derivatives are rarely deployed for optimizing RF waveforms, with the exception of [12,17,23], where quasi Newton approximations for the Hessian are investigated. In [35] second order derivatives are used in quadratic small tip angle approximation with quadratic constraints.…”

“…This solver gives us the option of processing exact Hessians or of using a limited memory version of the quasi Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) approximation [37]. The use of a BFGS approximation in the context of excitation waveform optimization has already been proposed and tested in [12,23].…”

“…We applied this interior point solution method to the examples of Grissom et al [20] and Hoyos-Idrobo et al [23]. The authors of [20] use a fast optimal control method in order to design selective 8 channel RF waveforms for flip angles of 90 and 180 degrees with underlying spiral and echo-planar gradient trajectories.…”