In this work we investigate a mathematical model to reconstruct the mechanical properties of an elastic medium, for the optical coherence elastography imaging modality. To this end, we start by considering a mathematical model for the mechanical deformation based on time-harmonic equations of linear elasticity. The mathematical model for solving this direct problem is the computational basis to address the inverse problem which consists of determining the set of parameters that characterize the mechanical properties of the medium knowing the displacement field for a given excitation. We formulate the inverse problem as PDE-constrained optimization problem, where the objective function measures the discrepancy between observations and predictions. We propose a derivative free trust-region method to solve this inverse problem and we report several computational results which illustrate its behavior in terms of accuracy and efficiency.