The accumulation of soluble and insoluble amyloid-β (Aβ) in the brain indicates failure of elimination of Aβ from the brain with age and Alzheimer's disease (AD). There is a variety of mechanisms for elimination of Aβ from the brain. They include the action of microglia and enzymes together with receptor-mediated absorption of Aβ into the blood and periarterial lymphatic drainage of Aβ. Although the brain possesses no conventional lymphatics, experimental studies have shown that fluid and solutes, such as Aβ, are eliminated from the brain along 100 nm wide basement membranes in the walls of cerebral capillaries and arteries. This lymphatic drainage pathway is reflected in the deposition of Aβ in the walls of human arteries with age and AD as cerebral amyloid angiopathy (CAA). Initially, Aβ diffuses through the extracellular spaces of gray matter in the brain and then enters basement membranes in capillaries and arteries to flow out of the brain. Although diffusion through the extracellular spaces of the brain has been well characterized, the exact mechanism whereby perivascular elimination of Aβ occurs has not been resolved. Here we use a computational model to describe the process of periarterial drainage in the context of diffusion in the brain, demonstrating that periarterial drainage along basement membranes is very rapid compared with diffusion. Our results are a validation of experimental data and are significant in the context of failure of periarterial drainage as a mechanism underlying the pathogenesis of AD as well as complications associated with its immunotherapy.