Physics-informed neural network estimation of active material properties in time-dependent cardiac biomechanical models

TL;DR

The paper presents a physics-informed neural network framework to estimate time-dependent, spatially varying active contractility in cardiac tissue from limited displacement and strain data. By coupling two neural networks—one for displacement and one for the active stress field—and by enforcing the governing cardiac biomechanics via PDE residuals, the approach reconstructs active stress fields and detects fibrotic scars without requiring stress data. Key innovations include adaptive loss weighting, residual-based attention, exact Dirichlet boundary enforcement, regularisation tailored to boundary identifiability, and Fourier feature embeddings to capture high-frequency scar boundaries. The results show accurate displacement reconstruction and scar detection in both homogeneous and heterogeneous test cases, with robustness to noise and reduced computational cost due to decoupled space-time representations. The work holds potential clinical impact for non-invasive scar delineation and personalized cardiac therapy planning, while outlining pathways to extend to patient-specific geometries and realistic fiber architectures.

Abstract

Active stress models in cardiac biomechanics account for the mechanical deformation caused by muscle activity, thus providing a link between the electrophysiological and mechanical properties of the tissue. The accurate assessment of active stress parameters is fundamental for a precise understanding of myocardial function but remains difficult to achieve in a clinical setting, especially when only displacement and strain data from medical imaging modalities are available. This work investigates, through an in-silico study, the application of physics-informed neural networks (PINNs) for inferring active contractility parameters in time-dependent cardiac biomechanical models from these types of imaging data. In particular, by parametrising the sought state and parameter field with two neural networks, respectively, and formulating an energy minimisation problem to search for the optimal network parameters, we are able to reconstruct in various settings active stress fields in the presence of noise and with a high spatial resolution. To this end, we also advance the vanilla PINN learning algorithm with the use of adaptive weighting schemes, ad-hoc regularisation strategies, Fourier features, and suitable network architectures. In addition, we thoroughly analyse the influence of the loss weights in the reconstruction of active stress parameters. Finally, we apply the method to the characterisation of tissue inhomogeneities and detection of fibrotic scars in myocardial tissue. This approach opens a new pathway to significantly improve the diagnosis, treatment planning, and management of heart conditions associated with cardiac fibrosis.

Figures (17)

• Figure 1: Time-evolution of the Bestel-Clément-Sorine model with different values of the maximal active stiffness $\sigma_0>0$.
• Figure 2: Ground-truth FEM solution of the displacement field. The light-shaded cube represents the reference configuration, whereas the coloured object is in deformed configuration.
• Figure 3: Comparison of training and testing losses for displacement data and PDE discrepancy and the relative error on the parameter $S_a$. Left: Algorithm performance considering noiseless data. Right: Performance using data corrupted with noise corresponding to $LD = 0.05$. The solid line depicts the geometric mean over the seeds; the shaded region is the area spanned by the trajectories. The second dashed vertical line marks the selected end of training at 10k BFGS epochs, for which we report performance and associated errors. For completeness, we also display the algorithm's behavior up to 50k epochs.
• Figure 4: Analysis of apparent Pareto fronts (constant active stress parameter $S_a$, quasi-static approximation). The results show the different training trajectories $\mathbf{\mathcal{J}}_{\text{PDE}}$ and $\mathbf{\mathcal{J}}_{\text{BCN}}$ and the relative error in the parameter $S_a$, denoted by $\epsilon_{S_a; rel}$, with different weights for $\mathbf{\mathcal{J}}_{\text{OBS}}$, $\mathbf{\mathcal{J}}_{\text{PDE}}$, and $\mathbf{\mathcal{J}}_{\text{BCN}}$. For each weight combination, the sum $\Sigma=\lambda_{\text{OBS}}+\lambda_{\text{PDE}}+\lambda_{\text{BC}}$ is computed, and, based on this normalising factor, the fraction of each of the three weights $\lambda_i$, $i \in \{ \text{OBS}, \text{PDE}, \text{BC}\}$, is then given by $\lambda_i/\Sigma$. The colour encodes the fraction of the chosen PDE weight whereas the size of the end marker encodes the fraction of the BCN weight. Every trajectory represents an average over three different seeds. All trajectories start at the same configuration (indicated by a white square). The small coloured circle along the trajectory indicates the transition from Adam to BFGS optimisation. The endpoints are marked by different symbols according to the endvalue of the relative error $\epsilon_{S_a; rel}$: a star if $\epsilon_{S_a; rel} < 0.05$, a circle if $0.05 \leq \epsilon_{S_a; rel} < 0.1$, and a cross if $\epsilon_{S_a; rel}>0.1$. The noiseless case is shown in \ref{['fig:pareto:000']} whereas in \ref{['fig:pareto:005']} we consider $LD=0.05$ as a noise level.
• Figure 5: Time evolution of the PINN solution. Top row: ground-truth data corrupted with noise corresponding to $LD=0.05$ and then sliced along the $y$-axis. Middle row: PINN reconstruction of the displacement field. Bottom row: absolute error between the PINN reconstruction and the noise-free ground-truth solution.