Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods

Abstract

A variable echo-number method for estimating R 2 in MRI-based polymer gel dosimetry Med. Phys. 38, 975 (2011) The use of increasingly strong magnetic fields in magnetic resonance imaging (MRI) improves sensitivity, susceptibility contrast, and spatial or spectral resolution for functional and localized spectroscopic imaging applications. However, along with these benefits come the challenges of increasing static field (B 0 ) and rf field (B 1 ) inhomogeneities induced by radial field susceptibility differences and poorer dielectric properties of objects in the scanner. Increasing fields also impose the need for rf irradiation at higher frequencies which may lead to elevated patient energy absorption, eventually posing a safety risk. These reasons have motivated the use of multidimensional rf pulses and parallel rf transmission, and their combination with tailoring of rf pulses for fast and low-power rf performance. For the latter application, analytical and approximate solutions are well-established in linear regimes, however, with increasing nonlinearities and constraints on the rf pulses, numerical iterative methods become attractive. Among such procedures, optimal control methods have recently demonstrated great potential. Here, we present a Krotov-based optimal control approach which as compared to earlier approaches provides very fast, monotonic convergence even without educated initial guesses. This is essential for in vivo MRI applications. The method is compared to a second-order gradient ascent method relying on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method, and a hybrid scheme Krotov-BFGS is also introduced in this study. These optimal control approaches are demonstrated by the design of a 2D spatial selective rf pulse exciting the letters "JCP" in a water phantom.

Training Schrödinger’s cat: quantum optimal control

“…Motivated by magnetic resonance imaging applications, optimal control pulses for MRI have been developed [88,[324][325][326]330,[397][398][399][400]. Applications include improved spatially selective excitation schemes [106,401,402], pulses with minimal radio-frequency (RF) power and pulses that counteract RF inhomogeneity in parallel transmission at ultra-high field [330,398].…”

First and second order derivatives for optimizing parallel RF excitation waveforms

“…; Cg minimizing the value F mls , which is calculated through Eqs. (13), (14), (15), (9), (11), (12), (16), (17), (18), and (19). Frequently, additional restrictions are imposed on the variables, in order to comply with hardware limitations and patient safety regulations.…”

“…(10) of [12], represents an approximation. Similarly, optimization procedures derived from optimal control methods [18,16,17] are usually based on single time point derivatives and thus incur certain inaccuracies when they are applied to entire sample time intervals. For example the transition from the single time point t derivative in Eq.…”

“…A well-developed approach for parallel RF waveform optimization is via optimal control theory [13][14][15][16][17][18]. In this approach, the RF pulse design is cast as a variational problem, for which optimality conditions in the form of differential equations can be derived.…”

“…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.…”