Forward and inverse problems in elasticity imaging of soft tissues

In elasticity imaging, a surface deformation is applied to an object using small pistons, and the resulting induced strains in the interior of the object are measured using ultrasonic imaging. Two important problems are considered: (1) the forward problem of determining the strains induced by a known deformation of an object with known elasticity; and (2) the inverse problem of reconstructing elasticity from measured strains and the equations of equilibrium. The method of finite differences is used to solve the forward problem for a given piston configuration; some nontrivial issues arise in determining boundary conditions. The finite difference equations are then rearranged into a linear system of equations which formulates the inverse problem; this system can be solved for the unknown elasticities. This formulation of the inverse problem is completely consistent with the forward problem; this is useful for iterative methods in which the deformation is adaptively changed. A comparison between simulated and actual measured results demonstrate the feasibility of the proposed procedure