Abstract
We have recently demonstrated supervised deep learning methods for rapid generation of radiofrequency pulses in magnetic resonance imaging (https://doi.org/10.1002/mrm.27740, https://doi.org/10.1002/mrm.28667). Unlike the previous iterative optimization approaches, deep learning methods generate a pulse using a fixed number of floating-point operations - this is important in MRI, where patient-specific pulses preferably must be produced in real time. However, deep learning requires vast training libraries, which must be generated using the traditional methods, e.g., iterative quantum optimal control methods. Those methods are usually variations of gradient descent, and the calculation of the gradient of the performance metric with respect to the pulse waveform can be the most numerically intensive step. In this communication, we explore various ways in which the calculation of gradients in quantum optimal control theory may be accelerated. Four optimization avenues are explored: truncated commutator series expansions at zeroth and first order, a novel midpoint truncation scheme at first order, and the exact complex-step method. For the spin systems relevant to MRI, the first-order midpoint truncation is found to be sufficiently accurate, but also significantly faster than the machine precision gradient. This makes the generation of training databases for the machine learning methods considerably more realistic.
Highlights
Tailored radiofrequency (RF) pulses are used in advanced magnetic resonance imaging (MRI) applications: reduced field-of-view imaging[1,2,3], spectral-spatial selectivity[1,4], as well as imaging with inhomogeneous RF fields (B1+)[5], off-resonance effects[6–9] or gradient imperfections[10]
This DeepControl framework requires training libraries with hundreds of thousands of RF pulses generated, e.g., with optimal control, but a trained neural network predicts a pulse quickly, and with a hard guarantee on the wall clock time. These vast libraries require an improvement in our ability to run optimal control simulations and calls for optimization of the numerical efficiency of the gradient ascent pulse engineering (GRAPE) algorithm, in particular, in the part that deals with the gradient calculation
We analyze the trade-offs between CPU time and convergence rate for four different levels of approximation to the GRAPE gradient: standard zero and first order approximation, a novel midpoint first order approximation proposed here, and the exact, complex-step gradient
Summary
Tailored radiofrequency (RF) pulses are used in advanced magnetic resonance imaging (MRI) applications: reduced field-of-view imaging[1,2,3], spectral-spatial selectivity[1,4], as well as imaging with inhomogeneous RF fields (B1+)[5], off-resonance effects[6–9] or gradient imperfections[10]. We have recently proposed a neural network based method for generating RF pulses to alleviate the run time problem[22–24] This DeepControl framework requires training libraries with hundreds of thousands of RF pulses generated, e.g., with optimal control, but a trained neural network predicts a pulse quickly, and with a hard guarantee on the wall clock time. These vast libraries require an improvement in our ability to run optimal control simulations and calls for optimization of the numerical efficiency of the GRAPE algorithm, in particular, in the part that deals with the gradient calculation. We analyze the trade-offs between CPU time and convergence rate for four different levels of approximation to the GRAPE gradient: standard zero and first order approximation, a novel midpoint first order approximation proposed here, and the exact, complex-step gradient
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