  Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with
high risk of rupture. An aneurysmal rupture could cause permanent disability
and even death. Finding and treating aneurysms before their rupture is very
difficult since symptoms can be easily attributed mistakenly to other common
brain diseases. Mathematical models could highlight possible mechanisms of
aneurismal development and suggest specialized biomarkers for aneurysms.
Existing mathematical models of intracranial aneurysms focus on mechanical
interactions between blood flow and arteries. However, these models cannot
apply to microaneurysms since the anatomy and physiology at the length scale of
cerebral microcirculation are different. In this paper we propose a mechanism
for the formation of microaneurysms that involves the chemo-mechanical coupling
of blood and endothelial and neuroglial cells. We model the blood as a
non-local non-Newtonian incompressible fluid and solve analytically the
Poiseuille flow of such a fluid through an axi-symmetric circular rigid and
impermeable pipe in the presence of wall slip. The spatial derivatives of the
proposed generalization of the rate of deformation tensor are expressed using
Caputo fractional derivatives. The wall slip is represented by the classic
Navier law and a generalization of this law involving fractional derivatives.
Numerical simulations suggest that hypertension could contribute to
microaneurysmal formation.
