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Cerebral aneurysms and microaneurysms are abnormal vascular dilatations withhigh risk of rupture. An aneurysmal rupture could cause permanent disabilityand even death. Finding and treating aneurysms before their rupture is verydifficult since symptoms can be easily attributed mistakenly to other commonbrain diseases. Mathematical models could highlight possible mechanisms ofaneurismal development and suggest specialized biomarkers for aneurysms.Existing mathematical models of intracranial aneurysms focus on mechanicalinteractions between blood flow and arteries. However, these models cannotapply to microaneurysms since the anatomy and physiology at the length scale ofcerebral microcirculation are different. In this paper we propose a mechanismfor the formation of microaneurysms that involves the chemo-mechanical couplingof blood and endothelial and neuroglial cells. We model the blood as anon-local non-Newtonian incompressible fluid and solve analytically thePoiseuille flow of such a fluid through an axi-symmetric circular rigid andimpermeable pipe in the presence of wall slip. The spatial derivatives of theproposed generalization of the rate of deformation tensor are expressed usingCaputo fractional derivatives. The wall slip is represented by the classicNavier law and a generalization of this law involving fractional derivatives.Numerical simulations suggest that hypertension could contribute tomicroaneurysmal formation.

Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms