Soft Tissue Simulation and Force Estimation From Heterogeneous Structures Using Equivariant Graph Neural Networks.

Accurately simulating soft tissue deformation is crucial for surgical training, pre-operative planning, and real-time haptic feedback systems. While physics-based models such as the finite element method (FEM) provide high-fidelity results, they are often computationally expensive and require extensive preprocessing. We propose a graph neural network (GNN) architecture that predicts both tissue surface deformation and applied force from sparse point clouds. The model incorporates internal anatomical information through binary tissue profiles beneath each point and leverages E(n)-equivariant message passing to improve robustness. We collected experimental data that comprises a real silicone and bone-like phantom, and complemented it with synthetic simulations generated using FEM. Our model achieves a comparable performance to a baseline GNN on standard test cases and significantly outperforms it in rotated and cross-resolution scenarios, showing a strong generalization to unseen orientations and point densities. It also achieves a significant speed improvement, offering a solution for real-time applications. When fine-tuned on experimental data, the model maintains sub-millimeter deformation accuracy despite limited sample size and measurement noise. The results demonstrate that our approach offers an efficient, data-driven alternative to traditional simulations, capable of generalizing across anatomical configurations and supporting interactive surgical environments.