Intravascular ultrasound (IVUS) is known to be the reference tool for preoperative vessel lesion assessments and for endovascular therapy planning. Nevertheless, IVUS echograms only provide subjective information about vessel wall lesions. Since changes in the vascular tissue stiffness are characteristic of vessel pathologies, catheter-based endovascular ultrasound elastography (EVE) has been proposed in the literature as a method for outlining the elastic properties of vessel walls. In this paper, the Lagrangian Speckle Model Estimator (LSME) is formulated for investigations in EVE, i.e., using a polar coordinate system. The method was implemented through an adapted version of the Levenberg-Marquardt minimization algorithm, using the optical flow equations to compute the Jacobbian matrix. The theoretical framework was validated with simulated ultrasound rf data of mechanically complex vessel wall pathologies. The results, corroborated with Ansys finite element software, demonstrated the potential of EVE to provide useful information about the heterogeneous nature of atherosclerotic plaques.