This paper presents new contributions to forward and inverse problems in 3D differential microwave imaging. This technique is well suited to monitor a pathological evolution, for instance the brain stroke of a patient. The forward problem is solved using a volume integral equation formulation, which enables to accurately compute the scattering by an heterogeneous medium such as the brain. In this contribution, several inversion regularization schemes are compared in order to retrieve the characteristics of the brain and more specifically to track the evolution of the pathology. These regularizations take advantage of the a priori sparsity knowledge inherent to the differential imaging scenario.