Physics-Informed Graph Neural Network Emulation for Rapid Cardiac Mechanics Modeling

Introduction: : Physics-based computational modeling of cardiac mechanics provides valuable insights into clinical diagnosis and treatment of heart diseases. However, conventional high-fidelity finite element method (FEM)-based simulations that include a detailed constitutive modeling of the passive mechanical behavior with local fiber structure, patient-specific geometry and active contractive behavior remain computationally prohibitive for clinical settings that require rapid evaluations. Artificial neural networks offer a promising alternative for accelerating these FEM-based simulations. Graph Neural Networks (GNNs) are adept at handling the irregular mesh geometries typical of patient-specific models, overcoming limitations of methods requiring structured data or low-rank approximations [1]. Extending from the existing physics-informed GNN (PI-GNN) framework [1] that predicts the passive mechanical behavior of cardiac only by minimizing the potential energy function of the myocardium, we develop a novel PI-GNN approach to rapidly predict the deformation of the heart by considering both passive and active cardiac mechanical behavior with detailed fiber structure. A key innovation is the integration of myocardial microstructure (fiber and sheet directions) into the graph topology and loss function, enhancing physical realism. Critically, this PI-GNN training paradigm does not require computationally expensive FEM simulation data, leveraging only the governing physical principles. We hypothesize that this approach yields fast, accurate, and physically plausible emulations of cardiac mechanical behavior, providing a foundation for future applications on developing novel diagnostic tools and treatments for heart diseases.

Materials and

Methods: : A subject-specific left ventricular (LV) geometry, reconstructed from 3D echocardiographic images [2], was discretized using a tetrahedral mesh comprising 1482 nodes and 7352 elements. Myocardial fiber and sheet orientations [3] were incorporated by encoding them as node and edge features within the graph representation. Diastolic filling boundary conditions were implemented via zero-displacement constraint at the LV base and an applied pressure load (1.06 kPa) on the endocardial surface. The passive mechanical behavior of the myocardium was modeled using an 8-parameter orthotropic Holzapfel-Ogden (HO) constitutive model [4], with material parameters (theta) sampled across their physiological ranges.

A GNN (K=5 message-passing steps, M=40 latent dimension, MLP hidden layers=[128, 128]), was implemented in JAX/Flax. This GNN processes the LV mesh graph, including integrated microstructure features, along with sampled HO parameters to predict nodal displacements. The GNN was trained for 15,000 epochs solely by minimizing a physics-based loss function (Figure 1). This loss comprised the total potential energy (calculated using predicted displacements, theta, the HO law, and microstructure) plus a Jacobian-based volumetric penalty. Numerical stability was improved using barrier transformation functions [1]. The Adam optimizer was employed (initial LR=5e-4, Batch Size=32). Crucially, no FEM simulation data was utilized for training. For validation, an independent test set (N=150) with ground truth displacements was generated using an FEM solver (FEniCS). Emulator performance was evaluated against FEM results using nodal displacement RMSE, relative errors (%) in potential energy, deformation gradient (Frobenius norm), and first invariant, plus computational inference speed.

Results, Conclusions, and Discussions::

Results: The PI-GNN, trained solely via potential energy minimization incorporating microstructure, demonstrated high fidelity on the unseen FEM test set (N=150). Visual comparison for the median error test case (Figure 2a-c) confirms strong qualitative agreement between the PI-GNN predicted L₂ displacement norm (||û||₂) and FEM ground truth (||u||₂), with nodal errors (||u-û||₂) predominantly below 0.02 cm. Quantitatively, the median absolute L₂ norm nodal displacement error was 9.98 × 10⁻³ cm (~2.6% relative to the median (||u||₂ of 0.382 cm). Median relative errors were notably low: 0.468% for potential energy, 0.812% for the deformation gradient, and 0.262% for the first invariant. Inference achieved a speedup exceeding 10,000x over FEM, enabling near real-time predictions. The training process was stable, with mean nodal displacement error converging below 0.01 cm over 15,000 epochs (Figure 2d).

Conclusions: This study validates that a PI-GNN, trained by minimizing potential energy derived from an orthotropic HO law with microstructure integration, accurately and efficiently emulates passive LV mechanics without using FEM simulation data for training. High fidelity (median displacement error of 9.98 × 10⁻³ cm and median relative errors below 0.82% for key mechanical quantities) and significant computational acceleration (>10,000x) were achieved, demonstrating the suitability of PI-GNNs for demanding biomechanical analyses, including inverse problems and potential clinical applications.
Discussions: The integration of myocardial microstructure (fiber and sheet directions) directly into the graph topology and physics-based loss significantly enhanced the model’s physical realism. While small residual errors remain in capturing full-field deformations compared to FEM, the overall energy-based performance (0.468% error) and strict adherence to physical constraints indicate strong potential for clinical and computational applications. Compared to data-driven methods, PI-GNNs provide robustness and generalizability without overfitting to training data, and the physical consistency (e.g., positive Jacobians) is critical when stress/strain or energy metrics are downstream targets. This work confirms the strong potential of simulation-free PI-GNN training for complex biomechanics. Future work will focus on incorporating active contraction models for full cardiac cycle emulation and validation on diverse patient-specific clinical datasets.

Acknowledgements and/or References (Optional)::

References:
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[3] L. Fan et al., “Comparison of Left Ventricular Function Derived from Subject-Specific Inverse Finite Element Modeling Based on 3D ECHO and Magnetic Resonance Images,” Bioengineering, vol. 11, no. 7, 2024, doi: 10.3390/bioengineering11070735.
[4] G. A. Holzapfel and R. W. Ogden, “Constitutive modelling of passive myocardium: A structurally based framework for material characterization,” Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., vol. 367, no. 1902, pp. 3445–3475, 2009, doi: 10.1098/rsta.2009.0091.