Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling

TL;DR

This work tackles the challenge of identifying constitutive models for tensional homeostasis in soft tissues under finite strain by embedding physics directly into neural networks. It extends inelastic Constitutive Artificial Neural Networks (iCANNs) with growth and homeostatic surfaces, using two feed-forward networks to discover the Helmholtz free energy and a convex pseudo potential, while enforcing thermodynamic consistency through a co-rotated intermediate framework and a Perzyna-type time evolution. Applied to stripe and cross tissue-equivalents, the approach learns interpretable material parameters and reproduces point-wise behavior, though structural simulations reveal limitations due to data sparsity, activation-function choices, and isotropy assumptions. The public code and data enable reproducibility, and the framework lays a path toward richer, uncertainty-aware discovery of tensional homeostasis mechanisms in biological tissues, with future work needed to extend to directional effects and more complex loading scenarios.

Abstract

Soft biological tissues exhibit a tendency to maintain a preferred state of tensile stress, known as tensional homeostasis, which is restored even after external mechanical stimuli. This macroscopic behavior can be described using the theory of kinematic growth, where the deformation gradient is multiplicatively decomposed into an elastic part and a part related to growth and remodeling. Recently, the concept of homeostatic surfaces was introduced to define the state of homeostasis and the evolution equations for inelastic deformations. However, identifying the optimal model and material parameters to accurately capture the macroscopic behavior of inelastic materials can only be accomplished with significant expertise, is often time-consuming, and prone to error, regardless of the specific inelastic phenomenon. To address this challenge, built-in physics machine learning algorithms offer significant potential. In this work, we extend our inelastic Constitutive Artificial Neural Networks (iCANNs) by incorporating kinematic growth and homeostatic surfaces to discover the scalar model equations, namely the Helmholtz free energy and the pseudo potential. The latter describes the state of homeostasis in a smeared sense. We evaluate the ability of the proposed network to learn from experimentally obtained tissue equivalent data at the material point level, assess its predictive accuracy beyond the training regime, and discuss its current limitations when applied at the structural level. Our source code, data, examples, and an implementation of the corresponding material subroutine are made accessible to the public at https://doi.org/10.5281/zenodo.13946282.

Figures (12)

• Figure 1: Schematic illustration of our feed-forward network for the Helmholtz free energy, $\psi$, embedded into the recurrent network architecture. The first layer computes the eigenvalues of $\bar{\bm{C}}_e$, subsequently the deformation is decomposed into isochoric and volumetric deformations. The last layer applies custom-designed activation functions including a first set of weights, $w_{\star,1}^\psi$. These functions are multiplied by a second set of weights, $w_{\star,2}^\psi$. Noteworthy, $\mathrm{det}\left(\bar{\bm{C}}_e\right)=\lambda_1\lambda_2\lambda_3$ holds.
• Figure 2: Schematic illustration of our feed-forward network for the pseudo potential, $\hat{\phi}$, embedded into the recurrent network architecture. The first layer computes the eigenvalues of $\bar{\bm{\Sigma}}$, subsequently the principal shear stresses are computed. The last layer applies custom-designed activation functions including a first set of weights, $w_{\star,3}^\phi$. These functions are multiplied by a second set of weights, $w_{\star}^\phi$.
• Figure 3: Loss during training of the iCANN for the stripe specimen (uniaxial stress state). Left: Training with $L_1$ regularization, cf. Table \ref{['tab:weights_stripe']}. Right: Training with $L_2$ regularization, cf. Table \ref{['tab:weights_stripe']}. The loss is plotted on a logarithmic scale. In both cases, $4,000$ epochs were used.
• Figure 4: Discovered model for the stripe specimen (uniaxial stress state). The experimental data is taken form eichinger2020. In both plots, the black curve represents the experimental mean from three specimens, while the shaded areas show the mean $\pm$ the standard error of the mean. The stress is plotted in terms of $\bm{S}$ in loading direction. Left: Training results for $L_1$ and $L_2$ regularization. In the experiment, the homeostatic force is reduced by $-10\%$ at $t=17$ [h]. Right: Testing results for $L_1$ and $L_2$ regularization, where the homeostatic force is increased by $+10\%$ at $t=17$ [h].
• Figure 5: Finite element discretization with $4,580$ elements for the stripe specimen. The thickness of the specimen is equal to $4\ [\text{mm}]$, while the cross section area is equal to $40\ \left[\text{mm}^2\right]$ in the center of the specimen. For discretization in thickness direction, $5$ elements were used across the thickness.

Theorems & Definitions (2)

• Remark 1
• Remark 2