Real-time nonlinear inversion of magnetic resonance elastography with operator learning

TL;DR

This paper addresses the slow computation of nonlinear inversion in brain magnetic resonance elastography by introducing oNLI, a neural operator that learns a resolution-invariant mapping from complex curl displacement fields to the complex shear modulus $\mu$. It implements a Fourier neural operator to realize real-time inversion and further enhances spatial fidelity with a SPADE variant that injects anatomical priors from $6$-region brain segmentations. In a 10-fold cross-validated retrospective study of 61 healthy subjects, oNLI markedly improves predictive accuracy over a 3D U-Net baseline and achieves up to a $30{,}000\times$ speedup over conventional nonlinear inversion while maintaining fine-grained spatial detail, with $r$ values reaching the upper end of the range across major regions. These findings suggest real-time, high-fidelity MRE inversion is feasible in clinical workflows, potentially enabling rapid biomarker development for aging and disease.

Abstract

$\textbf{Purpose:}$ To develop and evaluate an operator learning framework for nonlinear inversion (NLI) of brain magnetic resonance elastography (MRE) data, which enables real-time inversion of elastograms with comparable spatial accuracy to NLI. $\textbf{Materials and Methods:}$ In this retrospective study, 3D MRE data from 61 individuals (mean age, 37.4 years; 34 female) were used for development of the framework. A predictive deep operator learning framework (oNLI) was trained using 10-fold cross-validation, with the complex curl of the measured displacement field as inputs and NLI-derived reference elastograms as outputs. A structural prior mechanism, analogous to Soft Prior Regularization in the MRE literature, was incorporated to improve spatial accuracy. Subject-level evaluation metrics included Pearson's correlation coefficient, absolute relative error, and structural similarity index measure between predicted and reference elastograms across brain regions of different sizes to understand accuracy. Statistical analyses included paired t-tests comparing the proposed oNLI variants to the convolutional neural network baselines. $\textbf{Results:}$ Whole brain absolute percent error was 8.4 $\pm$ 0.5 ($μ'$) and 10.0 $\pm$ 0.7 ($μ''$) for oNLI and 15.8 $\pm$ 0.8 ($μ'$) and 26.1 $\pm$ 1.1 ($μ''$) for CNNs. Additionally, oNLI outperformed convolutional architectures as per Pearson's correlation coefficient, $r$, in the whole brain and across all subregions for both the storage modulus and loss modulus (p < 0.05). $\textbf{Conclusion:}$ The oNLI framework enables real-time MRE inversion (30,000x speedup), outperforming CNN-based approaches and maintaining the fine-grained spatial accuracy achievable with NLI in the brain.

Figures (6)

• Figure 1: a) oNLI overview: Complex 3D displacement maps, $\mathbf{u}(\mathbf{x})=(u_x, u_y, u_z)\in\mathbb{C}^3$ for $x$ on the imaging domain $x\in\Omega\subset\mathbb{R}^3$, are input to the neural operator, their curl is computed, lifted to a higher dimension by linear mapping $\mathscr{P}$, processed by $T$ Operator layers, and projected to the complex shear modulus, $\mu(\mathbf{x})\in\mathbb{C}$, by the linear map $\mathscr{Q}$. b) Diagrams of operator layers used in this work to instantiate oNLI. Top: Diagram of Fourier Layer introduced in li2021fourierneuraloperatorparametric, where the operator kernel is parametrized in the Fourier domain. Bottom: Diagram of the SPADE layer used in the SPADE-oNLI variant. Anatomical T1 scans are processed using SynthSeg billot2023synthseg to obtain anatomical segmentations, which are passed through convolutional layers to extract spatially varying statistics, $\gamma$ and $\beta$. These statistics are then used to modulate the affine parameters of instance normalization.
• Figure 2: Histogram showing the age distribution of the subjects used for training and validation, including male (blue) and female (orange) partitions for each bin.
• Figure 3: T1-weighted MRI (left) and corresponding 6-region segmentation mask (right) generated using SynthSeg. Regions include: background (black), cortical gray matter (red), white matter (blue), subcortical gray matter (green), brainstem/cerebellum (purple), and cerebrospinal fluid (CSF; orange).
• Figure 4: Ground truth vs. predicted mean storage and loss moduli ($\mu'$ and $\mu"$, respectively) across validation subjects ($n=56$) at three actuation frequencies (30, 50, and 70 Hz). Pearson's r with respect to a linear regression fit is reported for each comparison. Across the regions of interest (cerebral cortex, white matter, thalamus, and hippocampus), SPADE-oNLI performs the best with respect to Pearson's r.
• Figure 5: Absolute percent error of the storage and loss moduli ($\mu'$ and $\mu"$, respectively) as evaluated against ground truth mean values for the cerebral cortex, white matter, thalamus and hippocampus. oNLI shows a significant error reduction from U-Net across all regions and both moduli except for the storage modulus in the thalamus, and SPADE-oNLI shows a significant error reduction from U-Net across all cases. ns: p $\geq$ 0.5; *: p $<$ 0.05; **: p $<$ 0.01; ***: p $<$ 0.001; ****; p $<$ 0.0001.