Topological sensitivity for vibro-acoustography applications

Abstract

To aid the material identification of tissue anomalies exposed by the vibro-acoustography approach to medical imaging that makes use of the acoustic radiation force, the focus of this study is an extension of the concept of topological sensitivity, originally developed for remote sensing applications, to transmission-mode inverse problems in elastodynamics where the physical excitation “taps” directly on an internal defect. The germane formula for the topological sensitivity is obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of an elastic inclusion, at the focal point of the acoustic radiation force, in an otherwise defect-free reference body. Due to the presence of a body force inside the nucleating defect, the underpinning limiting behavior of the elastodynamic perturbation is fundamentally different from that considered previously in the context of remote sensing, where the topological disturbance is by definition free of body forces. Under the premise of a ball-shaped nucleating inclusion, the formula for topological sensitivity is shown to be expressible in terms of the adjoint elastodynamic field and its gradients, contracted with sampling-point-independent tensors that are computable from two reference elastostatic solutions in R3. The proposed result is verified numerically by comparing the adjoint-field formula with its finite-difference approximation, computed for progressively smaller trial defects. On the basis of the new expression, a proposition is made wherein the variation of topological sensitivity versus the trial shear modulus of the nucleating defect at a fixed sampling, i.e. source point is capable of revealing (via the location of its infimum) whether the lesion, acted upon by a highly localized body force in the physical experiment, is “stiffer” or “softer” than the background solid. The utility of the idea is supported by a set of numerical results for a testing configuration that was motivated by a recent experimental study.