Boundary Homogenization and Numerical Modeling of Solute Transport Across the Blood–Brain Barrier

Amyloid-β (Aβ) is a protein waste produced in the brain that, if not cleared, contributes to development of Alzheimer's disease. Aβ is cleared by crossing the blood-brain barrier (BBB) via receptor-mediated transport, but existing models oversimplify these receptor-driven dynamics. In this work, we present a novel homogenized boundary condition that models Aβ exchange between the brain and blood compartments via saturable, bidirectional transport mediated by finite-density receptors. Our formulation incorporates Michaelis–Menten kinetics, ensures mass conservation, and accounts for both the quantity and binding dynamics of receptors on either side of the BBB. We present a numerical model that couples this boundary formulation with idealized models of a capillary and adjacent brain tissue. The numerical model is verified against a steady-state analytical solution that we derive and then validate using experimental measurements of Aβ in mice. We find brain Aβ levels are highly sensitive to receptor number and blood flow, which both change with aging. We simulate more complex phenomena, including sleep and spreading depolarization. Ongoing efforts are focused on tuning and validating the model using in vivo experimental data to probe BBB properties that cannot be readily measured.