A generic physics-informed neural network-based constitutive model for soft biological tissues

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Abstract

Constitutive modeling is a cornerstone for stress analysis of mechanical behaviors of biological soft tissues. Recently, it has been shown that machine learning (ML) techniques, trained by supervised learning, are powerful in building a direct linkage between input and output, which can be the strain and stress relation in constitutive modeling. In this study, we developed a novel generic physics-informed neural network material (NNMat) model which employs a hierarchical learning strategy by following the steps: (1) establishing constitutive laws to describe general characteristic behaviors of a class of materials; (2) determining constitutive parameters for an individual subject. A novel neural network structure was proposed which has two sets of parameters: (1) a class parameter set for characterizing the general elastic properties; and (2) a subject parameter set (three parameters) for describing individual material response. The trained NNMat model may be directly adopted for a different subject without re-training the class parameters, and only the subject parameters are considered as constitutive parameters. Skip connections are utilized in the neural network to facilitate hierarchical learning. A convexity constraint was imposed to the NNMat model to ensure that the constitutive model is physically relevant. The NNMat model was trained, cross-validated and tested using biaxial testing data of 63 ascending thoracic aortic aneurysm tissue samples, which was compared to expert-constructed models (Holzapfel-Gasser-Ogden, Gasser–Ogden–Holzapfel, and four-fiber families) using the same fitting and testing procedure. Our results demonstrated that the NNMat model has a significantly better performance in both fitting (R2 value of 0.9632 vs 0.9019, p=0.0053) and testing (R2 value of 0.9471 vs 0.8556, p=0.0203) than the Holzapfel–Gasser–Ogden model. The proposed NNMat model provides a convenient and general methodology for constitutive modeling.

Introduction

Constitutive modeling is a cornerstone for stress analysis of mechanical behaviors of biological soft tissues [1], [2], [3].Among the three key components required to solve a continuum biomechanics problem, i.e., the geometry (the domain of interest), the constitutive relations (how the material responds to applied loads under conditions of interest), and the applied loads (or associated boundary conditions), the identification of a robust constitutive model is probably the most challenging one to obtain and the key to success in this approach [4].

Currently, the approach to identify a robust constitutive model follows the DEICE procedure [4]: (1) Delineation of general characteristic behaviors, (2) Establishment of an appropriate theoretical framework, (3) Identification of specific functional forms of the constitutive relation, (4) Calculation of the values of the material parameters, and (5) Evaluation of the predictive capability of the final constitutive relation. In this approach, a domain expert, (i.e., a biomechanicist with years of advanced training), plays a central role in the first 3 steps. A classic example is how Dr. Y. C. Fung discovered the famous Green-strain based, exponential form of the strain–energy function for soft tissues, iconized now as the Fung-elastic model [5], [6]. Briefly, Fung showed that preconditioned soft tissue can be considered pseudo-elastic [5], [6], and the slope of load–deflection curve is proportional to the load in uniaxial elongation tests of rabbit mesentery [7]. Consequently, an exponential function was used to account for the nonlinearity of the stress–strain curve for soft tissues. Indeed, the Green-strain based orthotropic form of the strain–energy function constructed by Fung provides excellent fitting capability with experimental data. To study biaxial mechanical properties of myocardial tissues, Humphrey et al. [8] performed constant invariant biaxial experiments, in which each of the strain invariant was independently varied, to infer specific functional forms of strain invariant-based constitutive equations. Based on the experimental observations, a polynomial form of the strain–energy function was devised [8]. To formulate a microstructurally-motivated constitutive model, Holzapfel et al. [9] modeled the arterial tissue as bi-layer fiber-reinforced composite, in which the contributions of a ground matrix and collagen fibers can be modeled separately in a strain–energy function.

Constitutive models [6], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] constructed by biomechanics experts have been widely adopted to model mechanical behaviors of soft tissues. By following the 3rd and 4th steps of the DEICE procedure, the specific formulations of these models usually contain several constitutive parameters that can be adjusted to describe constitutive behaviors of an individual subject (e.g. a tissue sample); therefore, the expert-constructed models can be used to describe constitutive behavior of a new subject (within the same class of materials) without deriving new constitutive equations. In addition, these expert-constructed constitutive models demonstrate excellent in-sample descriptive/fitting capability (e.g., R2 value is high when fitting to mechanical testing data). However, their out-of-sample predictive capability may be limited when new data (i.e., data that are not used in the fitting) is employed to assess their performance [19].

Recently, machine learning (ML) techniques, especially deep neural networks have led to revolutionary breakthrough in many applications [20], [21], [22], [23], [24], [25], [26], [27], [28], including recent works [29], [30], [31], [32], [33], [34] in the field of biomechanics. Since ML techniques are capable of automatically discovering and capturing complex multi-dimensional input–output dependencies without the need of manually deriving specific functional forms, we hypothesize that a generic ML-based constitutive model can be developed and can have a similar, if not better, performance compared to the expert-constructed constitutive models.

Based on the universal function approximation theorems, a neural network with adequate capacities can approximate any continuous function with a small error [35], [36], [37], [38]. Traditional feedforward fully-connected neural networks (FFNN) have been used to model the strain (input) and stress (output) relations [39], [40], [41]. However, such FFNN-based model uses all of its parameters (a.k.a. weights and biases) to construct the constitutive relation for an individual subject, which does not strictly follow the 3rd and 4th steps in the DEICE procedure; therefore, it often contains hundreds to thousands of constitutive parameters. Compared to an expert-constructed model, a FFNN-based constitutive model has three major disadvantages: (1) a large number of constitutive parameters with no physical meanings, in contrast to only a few constitutive parameters in an expert-constructed model. (2) An expert-constructed model can not only delineate and capture the general mechanical behaviors of a class of materials, but also can accurately model an individual subject (e.g. individual material responses) by fine tuning the constitutive parameters. A FFNN-based model, however, cannot capture general characteristic behaviors of a class of materials, i.e., it cannot utilize data from multiple subjects (e.g. tissue samples from many patients) for better modeling of an individual subject (e.g. a tissue sample from a single patient). The model parameters of FFNN-based models for different subjects are completely independent to each other. (3) a FFNN-based model cannot guarantee its convexity, which is important for ensuring the model is physically meaningful with unambiguous mechanical behaviors [9].

In this study, we developed a novel neural network-based material model (NNMat) which employs a physics constraint and a hierarchical learning strategy (Fig. 1): (1) establishing constitutive laws to describe general characteristic behaviors of a class of materials; (2) determining constitutive parameters for an individual subject. These two steps are equivalent to 3rd and 4th steps of the DEICE procedure. The neural network structure consists of two parameter sets corresponding to the two steps: (1) a “class” parameter set for characterizing the general elastic properties of the class of materials; and (2) a “subject” parameter set with three parameters for modeling individual material response. Skip connections are utilized in the neural network structure to facilitate hierarchical learning. Hence, the class parameters can function as the expert-constructed constitutive equations, and the NNMat model has only three constitutive parameters. The trained NNMat model may be directly adopted for a different subject without re-training the class parameters. The predictive capability of the proposed NNMat model is compared with the expert-constructed constitutive models (Holzapfel–Gasser–Ogden [9], Gasser–Ogden–Holzapfel [11], and four-fiber families [18]).

Section snippets

Constitutive modeling of soft biological tissues

Soft biological tissues comprise bundles of collagen fibers embedded in a ground matrix and can be regarded as fiber-reinforced composites. Constitutive modeling of the hyperelastic tissues is often achieved by specifying the strain energy density W as a function of deformation gradient W(F), where F represents the deformation gradient tensor. Microstructurally-motivated constitutive models have become increasingly utilized for soft tissues, in which the contributions of the matrix and collagen

Cross validation and testing

In this study, stress–strain data (63 patients) was split into two sets: a training and validation set (57 patients) and a testing set (6 patients). In the training mode, parameters in both the class set and subject set are updated. While in the fitting mode, the class parameters are fixed and only the three subject parameters are adjusted to optimal for an individual subject. For the testing/validation mode, all the parameters are fixed. The network structure and hyperparameters, e.g., α and β

Cross validation

Using the training and validation set of 57 patients, grid search was performed to select the weights α and β. To reduce computational cost, the number of training epochs was set to be 1000 for the grid search. It is convenient to examine the convexity of a strain energy density function with respect to two in-plane components of the Green strain E11 and E22 while the shear component E12 is set to zero [47]. To examine its convexity, using the trained and fitted NNMat model, the second

Discussion

In this study, a novel generic physics-informed machine learning model was proposed for constitutive modeling of soft biological tissues. The proposed NNMat model utilizes a hierarchical learning strategy: it can learn from data of multiple subjects to improve its prediction for individuals. The structure of the NNMat model consists of a class parameter set for characterizing hyperelastic properties of a class of materials and a subject parameter set (three parameters) for fitting mechanical

Conclusions

In this study, a physics-informed machine learning model was proposed for constitutive modeling of soft biological tissues. A neural network material model (NNMat) with novel structure and hierarchical learning strategy is proposed. The NNMat model consists of two parameter sets: the class parameter set for characterizing the general elastic properties of a class of materials and the subject parameter set with three parameters for individual material response. Skip connections are utilized in

Declaration of Competing Interest

Dr. Wei Sun is a co-founder and serves as the Chief Scientific Advisor of Dura Biotech. He has received compensation and owns equity in the company. The other authors declare no competing interests.

Acknowledgments

This study is supported in part by the grants of AHA 18TPA34230083 and NIH HL142036. Minliang Liu is supported by an AHA predoctoral fellowship 19PRE34430060.

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