Data-driven Modeling of the Mechanical Behavior of Anisotropic Soft Biological Tissue

Abstract

Constitutive models that describe the mechanical behavior of soft tissues have advanced greatly over the past few decades. These expert models are generalizable and require the calibration of a number of parameters to fit experimental data. However, inherent pitfalls stemming from the restriction to a specific functional form include poor fits to the data, non-uniqueness of fit, and high sensitivity to parameters. In this study we design and train fully connected neural networks as material models to replace or augment expert models. To guarantee objectivity, the neural network takes isochoric strain invariants as inputs, and outputs the value of strain energy and its derivatives with respect to the invariants. Convexity of the material model is enforced through the loss function. Direct prediction of the derivative functions -rather than just predicting the energy-serves two purposes: it provides flexibility during training, and it enables the calculation of the elasticity tensor through back-propagation. We showcase the ability of the neural network to learn the mechanical behavior of porcine and murine skin from biaxial test data. Crucially, we show that a multi-fidelity scheme which combines high fidelity experimental data with low fidelity analytical data yields the best performance. The neural network material model can then be interpreted as the best extension of an expert model: it learns the features that an expert has encoded in the analytical model while fitting the experimental data better. Finally, we implemented a general user material subroutine (UMAT) for the finite element software Abaqus and thereby make our advances available to the broader computational community. We expect that the methods and software generated in this work will broaden the use of data-driven constitutive models in biomedical applications.

FE${}^\textbf{ANN}$ $-$ An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

“…Probably, the most common technique is the application of artificial neural networks (ANNs), which have already been proposed in the early 90s by the pioneering work of Ghabussi et al [24]. In the last decades, ANNs have been intensively used for mechanical material modeling and simulations by means of the finite element method (FEM), e. g., in [25,26,27,28,29] among others.…”

Finite electro-elasticity with physics-augmented neural networks

“…Starting with the more conventional methods, by a special choice of input quantities, i.e., invariants [38,73], multiple constitutive requirements can be fulfilled at the same time. Using invariants as input quantities for feed-forward neural networks (FFNNs) [46] with scalar-valued output, highly flexible hyperelastic potentials [33] can be constructed [53,83]. However, FFNNs are in general not convex, and thus the models proposed in [53,83] do not fulfill the polyconvexity condition.…”

“…Using invariants as input quantities for feed-forward neural networks (FFNNs) [46] with scalar-valued output, highly flexible hyperelastic potentials [33] can be constructed [53,83]. However, FFNNs are in general not convex, and thus the models proposed in [53,83] do not fulfill the polyconvexity condition. For this, a special choice of network architecture is required.…”

Learning hyperelastic anisotropy from data via a tensor basis neural network

“…With this formulation they were able to connect the equivariant properties of the network to conservation laws. In a more specific setting, Tac et al [57] developed a neural network model of hyperelastic soft tissue material with two fiber families. The model employed a neural network for the isochoric component of the stress response with pre-selected invariant inputs.…”