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

Effective clearance of amyloid- $\beta (A\beta$ ) from the brain is essential for preventing neurodegenerative diseases such as Alzheimer's. A significant portion of this clearance occurs through the blood-brain barrier (BBB) via receptor-mediated transport. However, current models fail to capture the complex kinetics and spatial heterogeneity of receptors at the BBB. In this study, we derive a novel boundary condition that accounts for finite receptor kinetics, receptor density, and bidirectional transport across the BBB. Specifically, we develop a nonlinear homogenized boundary condition that ensures mass conservation and incorporates receptor-mediated Michaelis-Menten kinetics. We then implement this boundary condition in a cylindrical geometry representing a capillary surrounded by brain tissue. After verifying that the model matches an analytical steady state solution that we derive and that it yields realistic blood $A\beta$ concentrations, we explore how realistic variations in parameter values drive changes in both steady state $A\beta$ concentration and transient dynamics. Simulations and analytical results reveal that $A\beta$ concentrations in the brain are sensitive to receptor number ratios, while concentrations in the blood are primarily affected by the blood clearance rate. Additionally, we use the model to investigate $A\beta$ clearance during sequential sleep cycles and due to a pathological phenomenon, spreading depolarization. This work presents the first biophysically consistent boundary condition for $A\beta$ transport across the BBB, offering a powerful tool for studying brain waste clearance under both physiological and pathological conditions.