Traumatic brain injuries (TBI) lead to approximately 53,000 fatalities annually in the US alone. Finite element (FE) computational models of the head and brain are extensively used to study the effects of impact loading and develop protective gear. An FE model, however, is only as good as the underlying constitutive equation describing the material behavior of the various constituents, especially the soft matter. Obtaining constitutive equation parameters is an exercise involving inverse problems and optimization. We consider the Bayesian framework for inverse problems and pose the problem on an expanded stochastic space to solve for the distribution of random variables instead of the deterministic inverse methods. This is a natural choice given the large amount  of uncertainty  inherent to soft tissue mechanical testing.

We propose a novel holistic Bayesian framework for parameter estimation and model selection based on the parallel nested Monte Carlo sampling algorithm MULTINEST, wherein we consider four different factors t reliably choose a parsimonious model from the candidate set of models. These are the qualitative fit of the model to the experimental data, evidence values, maximum likelihood values, and the landscape of the likelihood function.

Optimal experimental design for linear viscoelastic models is considered, where in a novel analytical expression for the Fisher information based optimality criteria is developed that could be used to find optimal experimental configurations without complex numerical evaluations and compared well with the results from numerical Fisher information and the Kullback-Leibler divergence.

We use the ideas learned from the model selection framework and the Fisher information to build stochastic hyper-viscoelastic constitutive models for human brain and use it to study kinematic injury criteria for traumatic brain injury. The injury criteria are obtained by propagation the uncertainty in the material models to the kinematic response of human brain (using finite element model of human brain). Given the complexity of the model, we build surrogate models for kinematic response in the Bayesian framework using a novel, fast nearest neighbor Gaussian process (NNGP) method inspired from spatial statistics for efficient future use.