Automated model discovery for human brain using Constitutive Artificial Neural Networks

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The brain is our softest and most vulnerable organ, and understanding its physics is a challenging but significant task. Throughout the past decade, numerous competing models have emerged to characterize its response to mechanical loading. However, selecting the best constitutive model remains a heuristic process that strongly depends on user experience and personal preference. Here we challenge the conventional wisdom to first select a constitutive model and then fit its parameters to data. Instead, we propose a new strategy that simultaneously discovers both model and parameters. We integrate more than a century of knowledge in thermodynamics and state-of-the-art machine learning to build a Constitutive Artificial Neural Network that enables automated model discovery. Our design paradigm is to reverse engineer the network from a set of functional building blocks that are, by design, a generalization of popular constitutive models, including the neo Hookean, Blatz Ko, Mooney Rivlin, Demiray, Gent, and Holzapfel models. By constraining input, output, activation functions, and architecture, our network a priori satisfies thermodynamic consistency, objectivity, symmetry, and polyconvexity. We demonstrate that–out of more than 4000 models–our network autonomously discovers the model and parameters that best characterize the behavior of human gray and white matter under tension, compression, and shear. Importantly, our network weights translate naturally into physically meaningful parameters, such as shear moduli of 1.82kPa, 0.88kPa, 0.94kPa, and 0.54kPa for the cortex, basal ganglia, corona radiata, and corpus callosum. Our results suggest that Constitutive Artificial Neural Networks have the potential to induce a paradigm shift in soft tissue modeling, from user-defined model selection to automated model discovery. Our source code, data, and examples are available at Statement of significance: Human brain is ultrasoft, difficult to test, and challenging to model. Numerous competing constitutive models exist, but selecting the best model remains a matter of personal preference. Here we automate the process of model selection. We formulate the problem of autonomous model discovery as a neural network and capitalize on the powerful optimizers in deep learning. However, rather than using a conventional neural network, we reverse engineer our own Constitutive Artificial Neural Network from a set of modular building blocks, which we rationalize from common constitutive models. When trained with tension, compression, and shear experiments of gray and white matter, our network simultaneously discovers both model and parameters that describes the data better than any existing invariant-based model. Our network could induce a paradigm shift from user-defined model selection to automated model discovery.




Linka, K., St. Pierre, S. R., & Kuhl, E. (2023). Automated model discovery for human brain using Constitutive Artificial Neural Networks. Acta Biomaterialia, 160, 134–151.

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