Assessing the vulnerability of atherosclerotic plaques requires an accurate knowledge of the mechanical properties of the plaque constituents. It is possible to measure displacements in vivo inside a plaque using magnetic resonance imaging. An important issue is to solve the inverse problem that consists in estimating the elastic properties inside the plaque from measured displacements. This study focuses on the identifiability of elastic parameters e.g. on the compromise between identification time and identification accuracy. An idealised plane strain. Finite Element (FE) model is used. The effects of the FE mesh, of the a priori assumptions about the constituents, of the measurement resolution and of the data noise are numerically investigated.