Diseases like cancer or arteriosclerosis often cause changes of tissue stiffness on the micrometer scale. Elastography is a common technique for medical diagnostics developed to detect these changes. We consider a complex problem of estimating both the internal displacement field and the material parameters of an object which is being subjected to a deformation. In particular, we present our recently developed elastographic optical flow method (EOFM) for motion detection from optical coherence tomography images. This method takes into account experimental constraints, such as appropriate boundary conditions, the use of speckle information, as well as the inclusion of structural information derived from knowledge of the background material. Furthermore, we present numerical results based on both simulated and experimental data from an elastography experiment and discuss the material parameter estimation from these data.