Brain activity produces waste proteins such as amyloid-β, whose accumulation is an early hallmark of Alzheimer’s disease. Amyloid-β  dynamics are governed by cell-scale processes, namely production at neurons, diffusion and aggregation in the extracellular space, and receptor-mediated transport across the blood–brain barrier. The resulting accumulation, however, plays out at the tissue scale.

In this talk, we introduce a multi-species reaction–diffusion model in a periodically perforated domain, which is coupled to nonlinear receptor  dynamics on capillary surfaces. Since the receptors do not diffuse, they  lack spatial regularity, which makes the analysis more delicate. We  prove well-posedness with uniform L∞ bounds and, using two-scale  convergence, derive an effective macroscopic model. Its diffusion tensor  comes from a cell problem, surface processes become volume sources, and  the receptors remain a cell-scale variable. Numerical experiments show  that, at fixed porosity, the shape and curvature of cells alone can change the effective transport by tens of percent.