Abstract
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 aneurysmal 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 be applied 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.
Highlights
Cerebral aneurysms are abnormal swellings of the vasculature with high rupture risk
Most cerebral aneurysms develop near bifurcations of the large arteries [1], microaneurysms can form on arterioles located either in the retinas of patients with diabetes [2,3] or deeper inside the brain tissue [4,5]
Since the application we are interested in is cerebral blood flow and in our mathematical model, we did not incorporate yet possible neuroglial effects on the vasculature, we show results for flow through an artery and an arteriole
Summary
Cerebral aneurysms are abnormal swellings of the vasculature with high rupture risk. most cerebral aneurysms develop near bifurcations of the large arteries [1], microaneurysms can form on arterioles located either in the retinas of patients with diabetes [2,3] or deeper inside the brain tissue [4,5]. A mathematical model capable of predicting the formation, growth and risk of rupture of a microaneurysm needs to incorporate the well-known non-Newtonian behavior of blood when flowing through smaller vessels [21,27,28], the deformability of the vascular wall [29] and relevant mechanotransduction processes taking place among red blood cells, endothelial cells and astrocytes [23,25]. We would like to point out that the model presented in this paper is a generalization of the classical continuum mechanics that provides a more flexible mathematical framework for the inclusion of entangled chemo-mechanical processes taking place at various length scales inside living systems (such as blood flow in vivo) characterized by actively interacting components and numerous physical phenomena. The paper ends with a section of conclusions and future work
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