While segmentation consists in partitioning a givenimage into meaningful constituents in order to identify relevantstructures such as homogeneous regions or edges, registration,given two images, aims at finding an optimal orientation-preserving one-to-one deformation aligning the structures visiblein an image into the corresponding ones in the other. Recently,intertwining both tasks into a single framework has proven toyield better results in terms of accuracy —in particular whenthe images exhibit weak boundary definition —and increase ofreliability of the encoded structure matching —since now, notonly based on intensity distribution comparison but also ongeometrical and topological features —. In line with this idea, wepropose going a step further by adding explicitly some dynamicsin the modelling,i.e., by making the minimization problemboth space and time-dependent so that the correlation betweenboth tasks is achieved through the process, connecting thus theproblem to an interpolation one. The shapes to be matched areviewed as Saint Venant-Kirchhoff materials, a special instanceof hyperelastic ones, and are implicitly modelled by level-setfunctions. These are evolved in order to minimize a functional in-cluding both a nonlinear-elasticity-based regularizer prescribingthe physical nature of the deformation and a term penalizingthe shape misalignment, thus promoting structure matchingrather than intensity pairing. Theoretical results emphasizingthe mathematical soundness of the model are provided, amongwhich the existence of minimizers and the existence of a weakviscosity solution to the related evolution problem. The model isthen applied to the longitudinal registration of hepatic dynamiccontrast-enhanced MRI sequences and shows good performance.This application has an important impact on the computer-aided follow-up of patients suffering from liver cancers.