| Summary: | In this thesis we study an inverse problem arising in Magnetic Resonance Elastography (MRE) which is a noninvasive method for quantifying soft tissue stiffness. We develop a Bayesian formulation of this problem which involves inferring (heterogeneous) elastic properties in the time-harmonic purely elastic or viscoelastic wave equation. We apply modern Ensemble Kalman Inversion (EKI) algorithms which are derivative-free and provide robust approximations of the Bayesian posterior in a computationally tractable manner. Moreover, we show how parametrisations of EKI can be used to design effective inversions of properties with complex geometries relevant to the detection of diseased tissue via MRE.
In in-silico experiments, we showcase under the viscoelastic and purely elastic modelling assumptions that EKI can provide accurate estimates of the unknown local tissue stiffness and we also discuss limitations of both models. In particular we test EKI using the viscoelastic model in virtual experiments with complex geometries and unknown elastic properties that occur, for example, in brain MRE. We demonstrate how our algorithms are able to successfully discover cancer tissue and provide confidence intervals for the estimates and predictions of tissue stiffness, which can be diagnostically valuable for physicians. Furthermore, we analyse the influence of the prior, the amount of noise in the data and the ensemble size on posterior estimates provided by EKI and discuss the design of informative priors for EKI.
|
|---|