Computational Hemodynamics of the Cerebral Circulation: Multiscale modeling from the Circle of Willis to Cerebral Aneurysms

One-dimensional (1D) models are exploited for the representation of the complex system of cerebral arteries, featuring a peculiar structure called the circle of Willis. These models, based on the Euler equations, are unable to capture the local details of the blood flow but are suitable for the description of the pressure wave propagation in large vascular networks. The propagative phenomenon is driven by the mechanical interaction of the blood and the vessel wall, and is therefore affected by the mechanical features of the wall. Our 1D model takes into account the wall viscoelasticity, whose effects on the wave propagation phenomena are qualitatively studied in some numerical experiments representative of realistic conditions in the cardiovascular and cerebral arterial systems. The details of the blood flow can be studied by means of three-dimensional (3D) models, based on the Navier-Stokes equations for incompressible Newtonian fluids. These models can correctly describe blood flow patterns in medium and large arteries, and in particular allow the evaluation of the stress field in the fluid. Thus, it is possible to estimate the traction exerted by the blood flow on the vessel wall (wall shear stress, WSS). We also show that by exploiting the representation of the vascular tree in terms of centerlines, it is possible to easily identify regions of interest in the computational domain, in which to restrict the fluid dynamics analysis, and to study the correlation between fluid dynamics features and the location along the arterial tree. Cerebral aneurysms are a disease of the vascular wall causing a local dilation, which tends to grow and can rupture, leading to severe damage to the brain. The mechanisms of initiation, growth and rupture have not been completely explained yet, but the effects of blood flow on the vascular wall are generally accepted as risk factors. In the context of the Aneurisk project (www2.mate.polimi.it:9080/aneurisk) it was found that certain spatial patterns of radius and curvature are associated to the presence and to the position of an aneurysm in the cerebral vasculature. Starting from this observation, a classification strategy for vascular geometries has been devised. Blood flow has been simulated in patient-specific vascular geometries reconstructed in the context of the Aneurisk project, and an index of the mechanical load exerted by the blood on the vascular wall near the aneurysm has been defined. Moreover, we show that certain values of the mechanical load are associated to the presence and the location of an aneurysm in the cerebral circulation: adding this hemodynamic parameter in the classification technique improves its efficacy. The interaction between local and global phenomena is a typical feature of the circulatory system. It is believed to be crucial in the context of the cerebral circulation, since defects or diseases at the level of the circle of Willis can induce local flow conditions associated to the initiation of an aneurysm. Geometrical multiscale models are a promising tool for the modeling of this interaction. We present a geometrical multiscale model of the cerebral circulation, based on the coupling of a 1D representation of the circle of Willis and the 3D representation of a carotid artery (T. Passerini, M. de Luca, L. Formaggia, A. Quarteroni, and A. Veneziani, 2009). Moreover, we discuss a novel method to describe the interface between the two models. The number of potential applications of numerical models, due to their proven effectiveness in the study of vascular networks, calls for the design of efficient and robust software tools. The software specifically written in the context of this work for the simulation of the circulatory system is based on the C++ LifeV (www.lifev.org) library.