Magnitude least squares optimization for parallel radio frequency excitation design demonstrated at 7 Tesla with eight channels

Abstract

Spatially tailored radio frequency (RF) excitations accelerated with parallel transmit systems provide the opportunity to create shaped volume excitations or mitigate inhomogeneous B 1 excitation profiles with clinically relevant pulse lengths. While such excitations are often designed as a least-squares optimized approximation to a target magnitude and phase profile, adherence to the target phase profile is usually not important as long as the excitation phase is slowly varying compared with the voxel dimension. In this work, we demonstrate a method for a magnitude least squares optimization of the target magnetization profile for multichannel parallel excitation to improve the magnitude profile and reduce the RF power at the cost of a less uniform phase profile. The method enables the designer to trade off the allowed spatial phase variation for the improvement in magnitude profile and reduction in RF power. We validate the method with simulation studies and demonstrate its performance in fourfold accelerated two-dimensional spiral ex- Parallel excitation offers a means of designing multidimensional radio frequency (RF) pulses using accelerated gradient trajectories resulting in a short pulse duration compared with single-channel excitation. Accelerations of four-to sixfold have been shown using an eight-channel transmit system (1), potentially enabling several important applications, including flexibly shaped excitation volumes, and mitigation of RF field inhomogeneity at high field. Various methods have been proposed for the design of such RF and gradient waveforms (2-5), primarily in the low flip domain (6), and successfully implemented on multichannel hardware (1,7).In this work, we propose an extension to the spatial domain parallel excitation pulse design method introduced by Grissom et al. (5), where we apply magnitude least squares optimization to improve excitation magnitude profile and reduce the required RF power at a cost of increased phase variations in the excitation pattern. However, for many excitation applications, such as when magnitude images are recorded, low-order spatial phase variations do not impose a significant penalty. In fact, they can potentially decrease the dynamic range requirements of the imaging (which can be extensive for high field threedimensional [3D] acquisitions) by reducing the amplitude at the center of k-space. We, therefore, developed a method for pulse calculation with an adjustable regularization parameter de-emphasizing the excitation phase profile and study the potential benefits in magnitude profile fidelity and SAR, which can accompany this relaxed constraint.The idea of permitting phase variation in the excitation profile has previously been exploited in several applications, including the design of quadratic-phase RF pulses (8,9), RF shimming (10 -13), and frequency-sweep pulses (14), with benefits such as improved magnitude transition bands for saturation pulses, homogeneity for RF shimming, and reduced RF peak power for frequency-sweep pulses. In this work,...

High‐resolution gradient‐recalled echo imaging at 9.4T using 16‐channel parallel transmit simultaneous multislice spokes excitations with slice‐by‐slice flip angle homogenization

“…Here, we briefly introduce the main mathematical formalism used in the spokes pulse design and the nomenclature that is used in Results and Discussion [for further details, see Grissom et al 15, Setsompop et al 22, and Sbrizzi et al 43]. Using small tip‐angle approximation 10, spokes pulses optimization can be represented as a regularized MLS minimization problem through the spatial domain method 15: $\mathbf{b}=\text{arg}\hspace{0.17em}\underset{\mathrm{boldb}}{\min }\left\{\Vert \left|\mathbf{A}\mathbf{b}\right|-{\left|\mathrm{boldm}\right|\Vert }_2^2+\lambda {\Vert \mathrm{boldb}\Vert }_2^2\hspace{0.17em}\right\},$where b is a column vector of length N Tx containing the complex RF scaling factors for each of the transmit channels, A is a N vox  ×  N Tx system matrix containing the complex transmit sensitivity from each of the N Tx transmit coils in each of the N vox region of interest voxels with the phase induced from the local B 0 offset and the spokes gradient blips, m is a column vector of length N vox representing the transverse magnetization target set in each of the N vox region of interest voxels, and $\lambda$ is the Tikhonov regularization parameter as a means to regularize the global RF power.…”

“…Using small tip‐angle approximation 10, spokes pulses optimization can be represented as a regularized MLS minimization problem through the spatial domain method 15: $\mathbf{b}=\text{arg}\hspace{0.17em}\underset{\mathrm{boldb}}{\min }\left\{\Vert \left|\mathbf{A}\mathbf{b}\right|-{\left|\mathrm{boldm}\right|\Vert }_2^2+\lambda {\Vert \mathrm{boldb}\Vert }_2^2\hspace{0.17em}\right\},$where b is a column vector of length N Tx containing the complex RF scaling factors for each of the transmit channels, A is a N vox  ×  N Tx system matrix containing the complex transmit sensitivity from each of the N Tx transmit coils in each of the N vox region of interest voxels with the phase induced from the local B 0 offset and the spokes gradient blips, m is a column vector of length N vox representing the transverse magnetization target set in each of the N vox region of interest voxels, and $\lambda$ is the Tikhonov regularization parameter as a means to regularize the global RF power. This problem can be solved efficiently by the multishift version of conjugate gradients for least‐squares (mCGLS) algorithm 43 together with the local variable exchange method 22. Using mCGLS, the solutions of a set of $\lambda$ can be obtained simultaneously 43.…”

“…Using mCGLS, the solutions of a set of $\lambda$ can be obtained simultaneously 43. MLS was chosen for this study instead of least‐squares because MLS has been shown to improve the achieved magnitude profile as well as lower the required RF power 22.…”

“…Static RF shim can be considered as the special case of one spoke. The phases and amplitudes required for the sinc subpulses can be found using magnitude least‐squares (MLS) optimization 22, while taking into account the subject‐specific B 0 field distribution and complex channel‐by‐channel ${\mathrm{normalB}}_1^{+}$ sensitivity maps with the formalism of the spatial domain method 15. For multislice acquisitions, slice‐ or slab‐specific spokes pulses have been shown to be beneficial for the overall achievable homogenization in the imaging region 11, 19, 23, 24, 25.…”

Fast design of local N‐gram‐specific absorption rate–optimized radiofrequency pulses for parallel transmit systems

“…In radiofrequency (RF) shimming, the interference of the B + 1 fields of the individual channels is optimized for excitation homogeneity (2)(3)(4)(5). Parallel transmit systems using spatially tailored RF pulses facilitate even more control of the local spin excitation within sufficiently short RF pulse durations and over a sufficient spectral range (6)(7)(8)(9). Experimental demonstrations of these techniques have been shown in recent years (10)(11)(12)(13).…”