A Generative Model of Hyperelastic Strain Energy Density Functions for Real-Time Simulation of Brain Tissue Deformation

Abstract

Purpose.  The goal of this paper is to build a simulation environment that allows for the prediction of patient-specific tissue response by drawing samples from a generative model with a probability distribution. We propose a Gaussian Process (GP) regression approach to learn distributions over strain energy density functions including elastography, linear and hyperelastic models reported in the literature. Methods.  We gather a total of 73 models characterising elastic properties of brain white matter, grey matter and abnormalities and express them as strain energy density functions. A multi-output GP is used to quantify means and confidence intervals across each anatomical region and model. We sample the GP distribution and use nonlinear optimisation to fit a Neo-Hookean meta-model to guarantee stable strain energy functions. We validate the Neo-Hookean meta-model by fitting known strain energy density functions from the literature and report optimisation cost. We also validate the ability of the GP to approximate elastic properties of tissue given a reference deformed state using simulation. Results.  The GP was able to capture confidence intervals of varying strain ranges; the GP parameters and optimisation costs indicated a higher variability of hyperelastic models compared to elastography and linear models. Although one term is insufficient to fully capture hyperelastic models with higher number of terms, the resulting meta model is stable for real-time simulation within a wider range of stretches captured during mechanical characterisation of soft tissue. We demonstrated that our approach was able to approximate known elastic properties of tissue with a root-mean-squared error of 0.6 mm of node displacements when drawing six samples from a distribution of hyperelastic white matter. Conclusion.  In this initial proof-of-concept, we demonstrated a GP-based approach to estimate the elastic behaviour of brain tissue through simulation by sampling a generative model comprising elastic models found in the literature.