| Abstract: |
| Transport and clearance of amyloid-$\beta$ in the brain tissue involve coupled diffusion-reaction processes in the interstitial bulk and receptor-mediated interactions on cellular interfaces, leading naturally to a bulk-surface system. These mechanisms operate across strongly separated spatial scales: microscopic cellular geometries determine local transport behavior, whereas clinically relevant observations concern macroscopic tissue domains.
We formulate a microscale class I mixture model posed on a periodically perforated domain, incorporating bulk diffusion, nonlinear surface reactions, and interfacial exchange. Using two-scale convergence techniques, we rigorously derive an effective macroscopic model that retains explicit dependence on the underlying microstructure through effective transport and reaction coefficients. The analysis establishes a systematic link between microscopic structure and macroscopic transport laws, yielding a model suitable for large-scale simulation. |
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