Principal-stretch-based constitutive neural networks autonomously discover a subclass of Ogden models for human brain tissue

Abstract

The soft tissue of the brain deforms in response to external stimuli, which can lead to traumatic brain injury. Constitutive models relate the stress in the brain to its deformation and accurate constitutive modeling is critical in finite element simulations to estimate injury risk. Traditionally, researchers first choose a constitutive model and then fit the model parameters to tension, compression, or shear experiments. In contrast, constitutive artificial neural networks enable automated model discovery without having to choose a specific model before learning the model parameters. Here we reverse engineer a constitutive artificial neural network that uses the principal stretches, raised to a wide range of exponential powers, as activation functions. Upon training, the network autonomously discovers a subclass of models with multiple Ogden terms that outperform popular constitutive models including the neo Hooke, Blatz Ko, and Mooney Rivlin models. While invariant-based networks fail to capture the pronounced tension–compression asymmetry of brain tissue, our principal-stretch-based network can simultaneously explain tension, compression, and shear data for the cortex, basal ganglia, corona radiata, and corpus callosum. Without fixing the number of terms a priori, our model self-selects the best subset of terms out of more than a million possible combinations, while simultaneously discovering the best model parameters and best experiment to train itself. Eliminating user-guided model selection has the potential to induce a paradigm shift in soft tissue modeling and democratize brain injury simulations. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.Statement of Significance: Understanding the constitutive response of the brain is critical to estimate brain injury risk, design protective devices, and predict surgical intervention. The current gold standard in constitutive modeling, first choosing a constitutive model and then fitting its parameters to data, is largely biased by user experience and personal preference. Constitutive artificial neural networks eliminate the need for user-guided model selection and enable automated model discovery. Here we reverse-engineer a constitutive artificial neural network with custom-designed activation functions from principal stretches raised to a wide range of exponential powers. When trained with data from human gray and white matter tissue, our network autonomously discovers a subclass of models with multiple Ogden terms that outperform popular constitutive models including the neo Hooke, Blatz Ko, and Mooney Rivlin models. While these classical invariant-based networks fail to capture the pronounced tension-compression asymmetry of brain tissue, our discovered principal-stretch-based models can simultaneously explain tension, compression, and shear data from the human cortex, basal ganglia, corona radiata, and corpus callosum. Without fixing the number of model terms a priori, our network self-selects the best subset of terms out of more than a million possible combinations, while simultaneously discovering the best model parameters, for example, shear moduli of 1.47kPa, 0.68kPa, 0.69kPa, and 0.29kPa for these four brain regions. Our findings are significant in that they eliminate user-guided model selection and have the potential to make brain modeling more accessible to a wide group of scientists with diverse training and backgrounds towards the ultimate goal to democratize human brain simulations.