Source-linked AI summary
Generalizing Soft Tissue Deformation and Force Prediction Across Material Stiffness and Geometry
Madina Kojanazarova, Sidaty El Hadramy, Philippe C. Cattin
TL;DR
The paper addresses limited evidence for constitutive-model selection across stiffness levels and weak surrogate generalization across stiffnesses and geometries. It calibrates SOFA hyperelastic models with silicone-beam experiments, then trains a softness-conditioned equivariant GNN on calibrated simulations. The model achieves sub-millimeter mean deformation accuracy at close to real-time inference speed, while force prediction varies with calibration consistency and constitutive model.
Problem
Existing approaches lack systematic guidance for constitutive-model selection across stiffness levels and do not generally generalize across varying stiffnesses and unseen geometries.
Method
The paper calibrates SOFA hyperelastic models using gravity-loaded silicone beams, generates multi-stiffness interaction data, and trains a softness-conditioned equivariant GNN surrogate.
Results
Sub-millimeter mean deformation accuracy is achieved with 0.010±0.004 s inference time, while force errors depend on constitutive model and softness level.
Takeaways & Limitations
Carefully calibrated FEM data can support fast surrogate prediction across three softness levels and unseen inclusion geometries.
Abstract
from arXiv · showhide
Accurate soft tissue simulation is essential for surgical training, pre-operative planning, and haptic feedback systems. While learning-based surrogate models trained on data using the finite element method (FEM) offer a promising path to real-time inference, their reliability depends on well-calibrated constitutive models. Existing approaches neither provide systematic guidance on model selection across stiffness levels, nor generalize across different tissue stiffnesses or geometries. We perform a comprehensive calibration of hyperelastic constitutive models in the SOFA Framework using gravity-loaded silicone beams with different stiffnesses. Using calibrated simulations as training data, we use a softness conditioned equivariant graph neural network, enabling deformation and force prediction across multiple tissue types and unseen geometries. Our model achieves sub-millimeter mean deformation accuracy at 0.010s inference time, while showing that force prediction quality is directly tied to upstream calibration consistency.
1 Introduction
The paper develops a unified pipeline that calibrates FEM material models and trains a softness-conditioned GNN to predict soft-tissue deformation and force across stiffnesses and unseen geometries. It targets the real-time limitations and generalization gaps of existing simulation surrogates.
- FEM can accurately model soft-tissue deformation, but solving large nonlinear systems at every time step typically prevents real-time use.
- Fast surrogate models approximate offline FEM responses at inference time, while GNNs suit irregular meshes and patient-specific geometries.
- Existing surrogates generally do not simultaneously generalize across unseen geometries and varying material stiffnesses.
- Constitutive-model selection and calibration are critical because parameter overfitting can corrupt FEM training data and the resulting surrogate.
- The proposed pipeline calibrates SOFA hyperelastic models using silicone beams, generates poking-interaction data across stiffnesses and embedded geometries, and trains a softness-conditioned equivariant GNN.
2 Background
The background presents hyperelastic constitutive models as strain-energy formulations with different microstructural assumptions and accuracy–stability trade-offs. It also introduces equivariant GNNs as suitable surrogates for unstructured soft-tissue data.
- Hyperelastic soft-tissue models represent nonlinear stress–strain behavior under large deformation through a strain-energy density function Ψ.
- Hyperelastic formulations differ in their assumptions about material microstructure and their trade-offs between accuracy and numerical stability.
- Saint Venant-Kirchhoff is computationally efficient but can produce non-physical, non-monotonic stresses under large compression.
- Neo-Hookean is a simple two-parameter hyperelastic model that captures qualitative rubber-like nonlinearity but not pronounced large-stretch strain stiffening.
- Stable Neo-Hookean modifies the volumetric term to maintain a positive semi-definite Hessian during inversion or extreme compression.
- Mooney-Rivlin adds a second invariant term for wider-range stress–strain fitting, whereas Ogden uses principal stretches for greater large-strain flexibility at higher computational cost.
- Arruda-Boyce models limiting polymer-chain extensibility and the resulting strain stiffening, which simpler models cannot capture.
- Equivariant GNNs use local message passing on unstructured data and generalize across rotations and translations without explicit augmentation.
3 Methods
The methods calibrate hyperelastic models against gravity-loaded silicone beams, then generate poking simulations across stiffness levels and unseen rigid-inclusion geometries for GNN training. The pipeline encodes material stiffness continuously and trains on deformation and force outputs from FEM simulations.
- Physical Setup: Silicone beams used softener ratios S0, S0.5, and S1 to represent three stiffness levels, with A:B fixed at 1:1.The mixtures correspond to A:B:S ratios of 1:1:0, 1:1:0.5, and 1:1:1.
- Physical Setup: Beams were tested at 50 mm and 100 mm as cantilevers and at 150 mm in a clamped-clamped configuration under gravity.Deformation was recorded after stabilization, using traced profiles to reconstruct the reference deformed geometry.
- Biomechanical Model & Parameter Optimization: All applicable SOFA hyperelastic models were parameter-optimized by matching simulated and reference beam deformations across softness levels and beam lengths.Simulations ran to equilibrium using an RMS nodal-displacement criterion, and calibration results reported surface distance, HD95, Dice, volume difference, and simulation time.
- Biomechanical Model & Parameter Optimization: Ogden and Mooney-Rivlin were selected for data generation because they achieved the most reliable fits across all three softness levels.The resulting datasets used both constitutive models across the three stiffness conditions.
- Data Generation: Training geometries comprised eleven randomly positioned rigid-inclusion shapes, including cubes, spheres, and cylinders, embedded in a 100 × 100 × 32 mm3 soft-tissue volume.Each shape was poked at fixed locations shared across softness levels and at independently sampled locations to vary contact configurations.
- Data Generation: The dataset contained 1320 pokings, with 14305 Ogden snapshots and 13672 Mooney-Rivlin snapshots, recording surface-node positions and total reaction force.Near-static frames were removed using a 1 mm contact-tip displacement threshold.
- Training Details: Surface point clouds used 32 × 32 sampled points, while material stiffness was represented continuously from 0 to 1 and assigned by softener ratio.S0, S0.5, and S1 mapped to stiffness values 0.8, 0.4, and 0.0; models minimized deformation-distance and force-prediction losses.
4 Results & Discussion
The softness-conditioned GNN predicts deformation consistently across constitutive-model datasets, while force accuracy varies with softness, model choice, and training-data consistency. Inference remains close to real time.
- Deformation accuracy: 0.095±0.155 mm and 0.114±0.198 mm mean Euclidean deformation errors were below the 0.156±0.298 mm single-material baseline for Mooney-Rivlin and Ogden, respectively.The results indicate that multi-softness training with continuous stiffness encoding did not degrade deformation accuracy relative to the cited baseline.
- Deformation accuracy: 2−2.5% mean relative tip error remained consistent across softness levels for both constitutive models.This further supports reliable deformation prediction across the tested stiffness range.
- Force prediction: 0.258 N to 0.537 N force errors were low and consistent across softness levels for Mooney-Rivlin, whereas Ogden reached 2.269±6.358 N for the stiffest mixture S0.Force errors therefore depended strongly on constitutive model and softness level, with high-variance Ogden targets making force regression harder.
- Effect of constitutive model: Mooney-Rivlin produced more consistent predictions because its force distributions were more uniform, while Ogden covered richer force responses at the cost of higher variance.The authors associate the difference with training-data characteristics rather than superior physical accuracy of Mooney-Rivlin.
- Inference speed: 0.010±0.004 s per inference, or 100 Hz, provided close to real-time predictions for interactive surgical simulation.The reported speed applies across both datasets.
5 Conclusions
The paper combines calibrated SOFA hyperelastic simulations with a softness-conditioned GNN to generalize soft-tissue prediction across stiffness levels and unseen geometries. It maintains sub-millimeter mean deformation accuracy at close to real-time speed, while force quality remains tied to calibration consistency.
- The proposed pipeline combines systematic calibration of SOFA hyperelastic constitutive models with a softness-conditioned graph neural network surrogate.
- Ogden and Mooney-Rivlin were identified as the most reliable models across the tested stiffness levels, although current constitutive models have limitations for very compliant materials.
- The stiffness-conditioned GNN achieved accurate deformation and force prediction across three trained softness levels and unseen inclusion geometries, with sub-millimeter mean deformation accuracy.
- The findings support carefully calibrated FEM simulations as training data for fast, generalizable surrogates in surgical simulation.
- The reported 0.010 s inference speed demonstrates practical viability for interactive surgical training systems with haptic feedback.