Convolutional Deep Operator Networks for Learning Nonlinear Focused Ultrasound Wave Propagation in Heterogeneous Spinal Cord Anatomy

TL;DR

The paper tackles the challenge of predicting nonlinear focused ultrasound pressure fields in patient-specific spinal cords, where geometric complexity and tissue heterogeneity impede real-time planning. It introduces a convolutional DeepONet that learns the PDE solution operator from simulations, enabling rapid, real-time predictions across varying anatomies and transducer placements. The approach achieves about a 2% relative $L_2$ loss on unseen anatomies and delivers inference in ~0.05 s for eight source locations, dramatically faster than the high-fidelity solver (~76 minutes). This surrogate enables efficient intraoperative parameter sweeps for precise, individualized transducer positioning, with potential extensions to segmentation and direct use of raw ultrasound data.

Abstract

Focused ultrasound (FUS) therapy is a promising tool for optimally targeted treatment of spinal cord injuries (SCI), offering submillimeter precision to enhance blood flow at injury sites while minimizing impact on surrounding tissues. However, its efficacy is highly sensitive to the placement of the ultrasound source, as the spinal cord's complex geometry and acoustic heterogeneity distort and attenuate the FUS signal. Current approaches rely on computer simulations to solve the governing wave propagation equations and compute patient-specific pressure maps using ultrasound images of the spinal cord anatomy. While accurate, these high-fidelity simulations are computationally intensive, taking up to hours to complete parameter sweeps, which is impractical for real-time surgical decision-making. To address this bottleneck, we propose a convolutional deep operator network (DeepONet) to rapidly predict FUS pressure fields in patient spinal cords. Unlike conventional neural networks, DeepONets are well equipped to approximate the solution operator of the parametric partial differential equations (PDEs) that govern the behavior of FUS waves with varying initial and boundary conditions (i.e., new transducer locations or spinal cord geometries) without requiring extensive simulations. Trained on simulated pressure maps across diverse patient anatomies, this surrogate model achieves real-time predictions with only a 2% loss on the test set, significantly accelerating the modeling of nonlinear physical systems in heterogeneous domains. By facilitating rapid parameter sweeps in surgical settings, this work provides a crucial step toward precise and individualized solutions in neurosurgical treatments.

Figures (5)

• Figure 1: Example ultrasound spinal cord images in the dataset along with their segmented masks and visualizations of corresponding pressure maps at various source locations.
• Figure 2: Visualization of proposed model architecture. This model consists of two branch nets for encoding the input function space (spinal cord image and transducer location) and a trunk net for encoding the domain of the output functions (discrete coordinates at which they are evaluated). The outputs of these deep neural networks are then merged with a dot product to approximate the true solution generator $\mathcal{G}$ with the neural operator $\mathcal{G}_{\theta}$. The loss is minimized to obtain the optimal set of parameters ($\theta$) for the solution operator during the training process.
• Figure 3: Visualization of results from the proposed neural operator model, including the ultrasound image of the injured spinal cord, the corresponding segmented mask used as input for the neural operator, high-fidelity ground truth pressure maps at source locations 4 and 8 obtained from k-Wave, the model’s predicted pressure maps for those locations, and the difference between the predictions and the ground truth.
• Figure 4: The loss for the training and validation set recorded at each training epoch for the proposed neural operator model.
• Figure 5: Visualization of the predicted pressure maps from the baseline comparison models for source location 4, including results from a convolutional neural network (CNN) trained only on source location 4, and fully convolutional networks (FCNs) trained on source location 4 and all source locations.