Boundary homogenization and numerical modeling of solute transport across the blood-brain barrier

Effective clearance of amyloid-β (Aβ) 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 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 Aβ concentrations, we explore how realistic variations in parameter values drive changes in both steady-state Aβ concentration and transient dynamics. Simulations and analytical results reveal that Aβ 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 Aβ clearance during sequential sleep cycles and due to a pathological phenomenon, spreading depolarization. This work presents a biophysically consistent boundary condition for Aβ transport across the BBB, offering a powerful tool for studying brain waste clearance under both physiological and pathological conditions.