The feasibility of optical coherence elastography is examined and the steps necessary for registration of optical coherence tomography (OCT) images are developed. We address the drawbacks of conventional techniques for solving the "image registration problem and inverse elasticity problem (IEP)" and then propose a new scheme for simultaneous solution of both problems. We have previously exploited the kinematics of incompressible tissue as a side-constraint term in OCT registration, and we now assess how the integration of FEM-derived mechanics directly within the registration process, improves the overall quality. The new technique estimates fewer unknowns and still it generates more realistic strain/modulus maps with less sensitivity to local minima. It is composed of a series of computationally-efficient and robust algorithms to create elastograms and elastic modulus maps and it does not need to utilize a multi resolution grid. Finally, we apply our scheme to a porcine aorta and demonstrate its ability in recovering lipid pools both in the strain map and in the elastic modulus image.