Automated design of pulse sequences for magnetic resonance fingerprinting using physics-inspired optimization

Abstract

Magnetic resonance fingerprinting (MRF) is a method to extract quantitative tissue properties such as [Formula: see text] and [Formula: see text] relaxation rates from arbitrary pulse sequences using conventional MRI hardware. MRF pulse sequences have thousands of tunable parameters, which can be chosen to maximize precision and minimize scan time. Here, we perform de novo automated design of MRF pulse sequences by applying physics-inspired optimization heuristics. Our experimental data suggest that systematic errors dominate over random errors in MRF scans under clinically relevant conditions of high undersampling. Thus, in contrast to prior optimization efforts, which focused on statistical error models, we use a cost function based on explicit first-principles simulation of systematic errors arising from Fourier undersampling and phase variation. The resulting pulse sequences display features qualitatively different from previously used MRF pulse sequences and achieve fourfold shorter scan time than prior human-designed sequences of equivalent precision in [Formula: see text] and [Formula: see text] Furthermore, the optimization algorithm has discovered the existence of MRF pulse sequences with intrinsic robustness against shading artifacts due to phase variation.

Keywords: magnetic resonance fingerprinting; magnetic resonance imaging; optimization; pulse sequence design.

Fig. 1.

Here, the overall structure of the pulse-sequence optimization process is illustrated. A physics-inspired optimization algorithm proposes one or more randomly generated initial pulse sequences, which are then given to a cost function, which returns a quality metric assessing their speed and accuracy. Based on this feedback, the optimization algorithm proposes updated sequences. The cycle of updating and reevaluation is repeated for a fixed number of iterations. The best sequence found during this process, as judged by the cost function, is produced as final output. Within the cost function, a full simulation of MRF process is performed. The discrepancy between the simulated ground truth $T_1$ and $T_2$ values in a brain slice and the corresponding values inferred by the standard MRF dictionary-matching procedure are used as a metric of accuracy.

Fig. 2.

$T_1$ (red) and $T_2$ (blue) map simulations of an optimized sequence (top two rows) and a standard human-designed sequence (1) (bottom two rows) incorporating phase variation. The error is modeled as a time-independent phase that varies quadratically along a chosen direction. Experimentally, one finds that this direction varies randomly from one scan to the next. In this figure, A includes no phase variation. B–E correspond to four example orientations for the phase variation. In vivo results for both sequences are shown in F.

Fig. 3.

The optimized sequences (Left column) display qualitatively different features than the standard human-designed sequence (Right column). In particular, the optimization algorithm consistently produces pulse sequences in which the TR duration is at its minimum allowed value for most of the TRs, but briefly spikes to much longer duration. The predicted magnitude of the magnetizations for white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) are shown for each sequence in units such that the initial inversion pulse achieves magnetization of magnitude 0.95. (The considerations behind modeling the initial magnetization as 0.95 are discussed in ref. .) Optimized sequence $o$ is chosen here as a representative example, about which further information is available in Fig. 5 and SI Appendix, Tables 1–4. The optimization that produced this sequence used $w_{\mathrm{C}\mathrm{S}\mathrm{F}}=0.024\hspace{0.17em}52$, $w_{\mathrm{W}\mathrm{M}}=1.000$, $w_2=12.02$, and $w_{\mathrm{m}\mathrm{a}\mathrm{g}}=0$. The pulse sequences are available for noncommercial research purposes from https://github.com/madan6711/Automatic-MRF-seq-design. A.U., arbitrary units; deg, degrees.

Fig. 4.

Comparison of standard optimization routines from SciPy (L-BFGS-B and SLSQP) against our simulated annealing implementation with nonisotropic moves (SA). The decrease in cost is plotted as a function of the number of queries made to the cost function. As the evaluation of the cost function is by far the most computationally intensive part of the algorithm, this is therefore a metric of the efficiency of the optimization method. The physics-inspired method’s performance varies depending on random seed. Here, the average performance across 200 trials is shown alongside the performance from the best of these trials. (As a meta-algorithm, one can run such trials in parallel and select the resulting sequence with lowest cost-function value.) For comparison, the cost-function value achieved by the standard sequence is shown as a dashed line.

Fig. 5.

Precision vs. duration tradeoff in T1 (Left panel) and T2 (Right panel) for optimized and unoptimized sequences. Here, the metrics of precision are the SDs in inferred $T_1$ and $T_2$ values, which we estimate from in vivo data using bootstrap statistics, as described in ref. . The standard sequence from (1) is truncated to TR counts from 480 to 3,000 in order to obtain scans of different durations, as illustrated by the gray tradeoff curve. The optimized data points are classified according to which terms were included in the cost function. Unsurprisingly, the sequences with best robustness against random error (err.) are obtained by heavily incentivizing large signal magnitude (i.e., magnetization) in the cost function. In vivo images corresponding the labeled data points (a–o) are shown in SI Appendix. Note that these bootstrap statistics are derived from in vivo experimental data, for which exact ground truth values of $T_1$ and $T_2$ are inaccessible. Thus, they can only assess scatter, and not bias.