Bayesian inverse problems for soft tissue mechanics

Over the last few decades, a large body of literature has focussed on mathematically modelling the mechanical behaviour of biological soft tissues within a continuum mechanics framework. Particular attention has focused on deriving constitutive equations within the context of large strain (nonlinear) elasticity and viscoelasticity. Many of the most widely used models take a phenomenological approach (for example, by proposing a strain energy which is an exponential function of the strain invariants, with free parameters that can be fitted to experimental data); however, this approach cannot be used to predict the effects of microstructural changes on macroscale mechanics. An alternative approach explicitly models the microstructure of the soft tissue, with measurable parameters to describe the geometry of the collagen fibre network that makes up the tissue, along with simple (linear) constitutive equations for the collagen fibres themselves. These linear constitutive models fit experimental data on individual collagen fibres well; however, due to the difficulty in precisely measuring the geometries of, and forces acting on, fibres that can be as small as tens of nanometres in diameter, the reported values of the constitutive parameters (e.g. the collagen Young's modulus) vary by orders of magnitude. This uncertainty in the values of these parameters makes it difficult to make quantitative predictions with deterministic, microstructural models; therefore, an alternative approach is needed which accounts for the uncertainty.

In this project, we will invoke the Bayesian framework, which, by incorporating the models of soft tissues with prior beliefs and macroscale experimental data, will give us a posterior probability distribution of the parameters conditioned on the observations. These distributions contain not only information about the likely values of the parameters, but also allow us to quantify and assess the uncertainty inherent in the estimates that arise from them. In practice this can be achieved by implementing Markov chain Monte Carlo (MCMC) methods, which through construction of an ergodic Markov chain with stationary distribution equal to the posterior distribution, allows us to sample from the distribution in order to characterise it. This might be done through the implementation of existing MCMC methods, or through the design of new methodologies which are able to efficiently target the types of distributions which arise from these inverse problems.