Diffusion Magnetic Resonance Imaging (dMRI) is a promising tool to obtain useful infor- mation on microscopic structure and has been extensively applied to biological tissues. We establish a new macroscopic model from homogenization theory to obtain the aggregate dMRI signal measured in practice in the case of intermediate water exchange across cellular membranes. Based on a particular scaling of the permeability condition modeling cellular membranes, this model accurately reproduces the memory effects observed in practice. Explicit formulae given by homogenization for the coeffcients of this model emphasize their link to the relevant physiological quantities, and the inverse problem of retrieving these coefficients from a realistic set of measurements is considered.