Atrial constitutive neural networks

TL;DR

This work tackles the challenge of accurately characterizing passive atrial mechanics by introducing constitutive neural networks that automatically identify a microstructure-informed free energy model from biaxial tensile data. The approach enforces thermodynamic consistency, incompressibility, and polyconvexity while learning a transversely isotropic constitutive law \\psi(I_1, I_2, I_{4,11}, I_{4,22}, I_{5,11}, I_{5,22}) using a three-layer neural network and automatic differentiation. Results on left and right atrial tissue from a single patient reveal a robust four-term model structure: an isotropic linear plus exponential quadratic term driven by \\ I_2^{3/2} and two fifth-invariant terms capturing anisotropy from orthogonal collagen fiber families; comparable models emerge for both atria with high predictive accuracy (average \\overline{R^2} ≈ 0.993–0.994). The findings offer a data-driven, physically constrained pathway to accurate atrial FE simulations, enabling improved planning of interventions and design of tissue-engineered constructs in a precision-mmedicine context.

Abstract

This work presents a novel approach for characterizing the mechanical behavior of atrial tissue using constitutive neural networks. Based on experimental biaxial tensile test data of healthy human atria, we automatically discover the most appropriate constitutive material model, thereby overcoming the limitations of traditional, pre-defined models. This approach offers a new perspective on modeling atrial mechanics and is a significant step towards improved simulation and prediction of cardiac health.

Figures (3)

• Figure 1: Constitutive neural network. Transversely isotropic, perfectly incompressible, constitutive neural network with three hidden layers to approximate the free-energy function $\psi(I_1, I_2, I_{4,11}, I_{4,22}, I_{5,11}, I_{5,22})$ as a function of the deformation gradient $\bm{F}$ using sixteen terms. The zeroth layer computes the no growth corrected deformation invariants from the network input. The first layer generates powers $(\circ)$ and $(\circ)^2$ of the zeroth and the second layer applies the identity $(\circ)$ and exponential function $(\rm{exp}(\circ))$ to these powers.
• Figure 2: Discovered constitutive model for left atrial tissue. Piola stresses $P$ as functions of stretches $\lambda$ of the constitutive neural network from Fig. \ref{['fig01']}, trained with all 1:0.5, 1:0.75, 1:1, 0.75:1, and 0.5:1 (left-to-right columns) $t_2$:$t_1$ tension ratios experiments on the anterior left atrial tissue sample of patient A4 simultaneously. Each individual stretch-stress curve's $R^2$ goodness of fit with respect to the original experimental data is shown in the top left corner.
• Figure 3: Discovered constitutive model for right atrial tissue. Piola stresses $P$ as functions of stretches $\lambda$ of the constitutive neural network from Fig. \ref{['fig01']}, trained with all 1:0.5, 1:0.75, 1:1, 0.75:1, and 0.5:1 (left-to-right columns) $t_2$:$t_1$ tension ratios experiments on the right atrial tissue sample of patient A4 simultaneously. Each individual stretch-stress curve's $R^2$ goodness of fit with respect to the original experimental data is shown in the top left corner.