The focus of this research is an extension of the concept of topological sensitivity to the sampling of heterogeneous viscoelastic solids and its applications to tissue elasticity imaging based on medical imaging data. Topological sensitivity, developed within the framework of qualitative inverse scattering solutions, is an effective non-iterative obstacle reconstruction method. However, its existing formulations are limited to "simple" background solids and remote sensing problems. In this study the concept of topological sensitivity is first extended, by way of dimensional analysis, to deal with preliminary reconstruction and characterization of defects in heterogeneous viscoelastic solids with piecewise-analytic constitutive and density properties. The main result is a sensitivity formula that takes a bi-linear form in terms of two visco-elastodynamic fields computed for the "defect-free" reference body, with the weighting coefficients depending on geometry of the vanishing defect, its material characteristics, and local properties of the reference solid. With further generalization and adaptation, the concept of topological sensitivity is then applied to tissue elasticity imaging using magnetic resonance imaging data and vibro-acoustography imaging data. In the former application, the formulation of topological sensitivity is generalized to allow averaged volumetric measurements and the nucleation of a dissimilar infinitesimal inclusion inside the measurement region. To overcome the difficulties associated with direct application to the entire organ, smaller cubic subdomains are taken as reference bodies whose background material properties required for topological sensitivity calculation, if unknown, can be estimated independently by a material sensitivity-based optimization approach. In the latter application, to cater to vibro-acoustography type of probing, the concept of topological sensitivity is adapted to transmission-mode inverse problems where the infinitesimal inclusion is created at the location of a point force and the asymptotic behavior of the perturbation field is fundamentally changed. The resulting sensitivity formula, expressed in terms of the adjoint field, features contractions of respective adjoint fields or their gradients with corresponding elasticity/inertial-contrast tensors. Prior knowledge about the viscoelastic behavior of the background material can be obtained from gradient-based minimization of a Bayesian cost function. The effectiveness of these developments is demonstrated through numerical examples and preliminary experimental results.