Estimating uncertainty in inverse elasticity with application to quantitative elastography

Abstract

We consider an inverse elasticity problem motivated by medical ultrasound imaging: Given a displacement field measured in a 2D domain, determine the modulus distribution in that domain. An iterative approach to solve the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field and any assumed a priori knowledge of the modulus distribution. All such inverse problem solutions are subject to uncertainties in the data, however, which results in uncertainties in the predictions. For diagnostic purposes, it is desirable to know the confidence intervals within which the stiffness at a point might reside. The focus of this presentation is the computation of said confidence intervals. We discuss the formulation of the problem within a Bayesian context. We derive a formal solution for the a posteriori probability distribution of the modulus. We prove bounds on uncertainty in terms of the data at the continuous level and discuss the computational solution of the problem at the discrete level.