Multiple sclerosis is a neurodegenerative autoimmune disease of the central nervous system. It is characterized by the presence of evolving lesions and faster-than-normal brain atrophy. In clinical practice, magnetic resonance imaging (MRI) is preferred for diagnosis and monitoring of disease progression. However, visual analysis of longitudinal MRI scans is time-consuming and difficult. Assessing lesion evolution is subjective and prone to high intra- and inter-operator variability. These challenges highlight the need for automated tools to evaluate lesion progression. Conventional approaches for longitudinal brain MRI change detection involve sequentially processing different sources of changes. First, intensity discrepancies caused by radiofrequency bias are corrected. Then, registration is used as a pre-processing step before detecting relevant pathological changes. Deformable registration compensates for geometric distortions due to brain atrophy, but it can cause lesions of interest to appear deformed, reducing detection sensitivity. Rigid and affine registration are usually chosen as they preserve lesions, but anatomical distortions due to brain atrophy lead to false detections. The limitation of sequential approaches arises from the disjointed treatment of registration and change detection. In this article, we consider the interdependence of these processing steps by proposing a unified framework where these sub-problems are jointly solved. We formulate it as the minimization of a cost function that encompasses both registration and change detection. We experimentally demonstrate the benefits of this joint approach compared to its sequential counterpart on synthetic and real data.