Inverse and optimum drive problems in ultrasonic imaging of soft tissue elasticity.

Abstract

Tissue elasticity may be used to distinguish normal and cancerous tissues. This new, emerging field of research produces a class of inverse problems that involve reconstructing the Young's modulus (elasticity) of tissue from mechanical strains induced inside the body part of interest by external forces. The resulting deformations (strains) are measured using ultrasonic imaging. The elastic equilibrium equations are then used to reconstruct the Young's modulus, with the measured strains as input data. This is non-trivial since the differential equations generally do not have an analytic solution. Several interesting and important questions arise in this problem. Two important problems dealt with here are: (1) the optimum drive problem (ODP) and (2) the inverse/reconstruction problem (IP). In the ODP, the question of optimum surface deformations is addressed, while measured strains are used to reconstruct the elasticity of the object in the IP, based on the solution to the ODP. Two methods are proposed to solve the IP. The first approach is a better conditioned problem, but requires precise boundary measurements which may not be available. This leads to a second approach, which yields a more ill-conditioned IP. Regularization techniques have to be used to solve the resulting system of equations. We use the generalized Tikhonov method to solve the IP. The practical issue of partial data availability is also addressed, and a new least-squares method is proposed to estimate the unavailable data from noisy measurements.