A growing body of evidence suggests that cerebrospinal fluid circulates through the brain to sweep away high-molecular-weight solutes. Multiple studies demonstrate that flow through this pathway, often referred to as the glymphatic system, is most active during sleep. We numerically model the clearance of amyloid-$\beta$ (a high-molecular-weight protein connected to Alzheimer's disease) from the brain interstitium by combined diffusion and glymphatic advection. We first compare the clearance for a range of different flow conditions and quantify the relation between the clearance rates and Péclet number Pe. We then simulate protein buildup using a reaction-advection-diffusion equation based on the Smoluchowski aggregation scheme and quantify the buildup for different Pe. We find that for flows with $\text{Pe}\gtrsim 1$, the rate of accumulation of heavy aggregates decreases exponentially with Pe. We finally explore the effect of the sleep-wake cycle by incorporating a variation in the flow speed motivated by experimental measurements. We find that periods of sleep lead to better clearance of intermediate protein aggregates and deter the buildup of large aggregates in the brain. In a conservative estimate, for $\text{Pe}\approx 1$, we find a $32\%$ reduction in the buildup rate of heavier protein aggregates compared to purely diffusive clearance.

• Received 1 September 2021
• Revised 21 January 2022
• Accepted 6 February 2022

DOI:https://doi.org/10.1103/PhysRevE.105.024405