In this study, we address the inverse problem of recovering the Lamé parameters ( $\lambda , \mu $ ) and the density $\rho $ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$ . This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters ( $\lambda , \mu $ ) are known and we look for the inverse problem of recovering the density $\rho $ . In this context, we derive a constructive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $\lambda $ , $\mu $ , and $\rho $ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $\lambda $ , $\mu $ , and $\rho $ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.
