An Elastic Registration Algorithm based on Strain Energy Minimization and its Application to Prostate MR images

B. Zhang, R. P. Gullapalli Magnetic Resonance Research Center, Dept of Radiology, University of Maryland School of Medicine, Baltimore, MD, United States Background Magnetic resonance imaging using the endorectal coil (erMRI) has become the clinical standard in the diagnosis of prostate cancer as it provides excellent quality high resolution images. However the images and spectra obtained using the endorectal coil experience certain amount of displacement from their original position and the prostate is imaged in this displaced, distorted position. Such displacement and distortion during diagnosis leads to uncertainties in the localization of prostate cancer during therapeutic intervention as in the case of radiation therapy. Image registration is a necessary exercise to transform the diagnostic images to their undistorted state. Rigid body registration is inadequate since the prostate encounters non-rigid elastic deformation. Elastic registration is of particular interest in erMRI since it takes into account the physical process that prostate has experienced during the medical imaging procedure. Traditional elastic image registration schemes derive forces from image data using some similarity measure and then deform the source image into target. [1,2] Instead of computing the forces and solving the Navier-Lame equation for deformation, the proposed registration algorithm in this work models the image as a dynamic system in equilibrium and derives the deformation using the principle of strain energy minimization. Method Strain Energy Minimization According to the principles of dynamics, the potential energy function has a stationary value if the system is conservative and is in equilibrium. Especially, if the system is stable, then the potential energy function is minimized. [3] In prostate erMRI, the prostate is in equilibrium before and after the insertion of endorectal coil, hence the above theorem can be applied to derive the underlying deformation within the prostate. Treating the prostate as an incompressible elastic body, the potential energy function is purely the strain energy U. It is defined as