Uncertainty in inverse elasticity problems

The non-invasive differential diagnosis of breast masses through ultrasound imaging motivates the following class of elastic inverse problems: Given one or more measurements of the displacement field within an elastic material, determine the material property distribution within the material. This thesis is focused on uncertainty quantification in inverse problem solutions, with application to inverse problems in linear and nonlinear elasticity. We consider the inverse nonlinear elasticity problem in the context of Bayesian statistics. We show the well-known result that computing the Maximum A Posteriori (MAP) estimate is consistent with previous optimization formulations of the inverse elasticity problem. We show further that certainty in this estimate may be quantified using concepts from information theory, specifically, information gain as measured by the Kullback-Leibler (K-L) divergence and mutual information. A particular challenge in this context is the computational expense associated with computing these quantities. A key contribution of this work is a novel approach that exploits the mathematical structure of the inverse problem and properties of conjugate gradient method to make these calculations feasible. A focus of this work is estimating the spatial distribution of the elastic nonlinearity of a material. Measurement sensitivity to the nonlinearity is much higher for large (finite) strains than for smaller strains, and so large strains tend to be used for such measurements. Measurements of larger deformations, however, tend to show greater levels of noise. A key finding of this work is that, when identifying nonlinear elastic properties, information gain can be used to characterize a trade-off between larger strains with higher noise levels and smaller strains with lower noise levels. These results can be used to inform experimental design. An approach often used to estimate both linear and nonlinear elastic property distributions is to do so sequentially: Use a small strain deformation to estimate the linear properties, and a large strain deformation to estimate the nonlinearity. A key finding of this work is that accurate characterization of the joint posterior probability distribution over both linear and nonlinear elastic parameters requires that the estimates be performed jointly rather than sequentially. All the methods described above are demonstrated in applications to problems in elasticity for both simulated data as well as clinically measured data (obtained in vivo). In the context of the clinical data, we evaluate repeatability of measurements and parameter reconstructions in a clinical setting