Computational Hemodynamics of Cerebral Vasculature

POLITECNICO DI MILANO Dipartimento di Matematica F. Brioschi Ph. D. course in Mathematical Engineering XXI cycle Computational hemodynamics of the cerebral circulation: multiscale modeling from the circle of Willis to cerebral aneurysms Ph. D. candidate: Tiziano PASSERINI Milano, 2009 Supervisor: Prof. Alessandro VENEZIANI To Lucia Contents Abstract 1 1 Introduction 3 1.1 Anatomy and physiology of the cerebral circulation . 3 1.1.1 The circle of Willis . 4 1.2 Morphology and fluid dynamics of cerebral aneurysms . 6 1.2.1 The role of hemodynamics . 10 1.3 Modeling the cerebral circulation . 12 1.3.1 The circle of Willis . 13 1.3.2 Cerebral aneurysms . 14 2 One-dimensional models for blood flow problems 15 2.1 Wave propagation phenomena in the cardiovascular system . 15 2.1.1 Modeling the vascular wall . 16 2.2 Formulation of the model . 17 2.2.1 A viscoelastic structural model for the vessel wall . 19 2.2.2 The linearized model . 21 2.3 Networks of 1D models . 27 2.4 Numerical discretization . 28 2.4.1 Numerical solution of the viscoelastic wall model . 29 2.5 Results and discussion . 30 2.5.1 Validation of the numerical model versus an analytical solution . 30 2.5.2 Wave propagation in a single 1-D vessel: a Gaussian pulse wave 32 2.5.3 Wave propagation in a single 1-D vessel: a sinusoidal wave . 34 2.5.4 A 1D model network: the circle of Willis . 35 3 Three-dimensional models for blood flow problems 39 3.1 Blood flow features in arteries . 39 3.2 Geometry and Flow . 40 3.2.1 Reynolds number . 41 3.2.2 Dean number . 41 3.2.3 Womersley number and Reduced Velocity . 43 3.3 The Navier-Stokes equations . 44 3.3.1 Formulation . 44 3.3.2 Numerical discretization . 48 3.4 Wall shear stress in the Navier-Stokes problem . 50 i Contents 3.4.1 Approximation for the velocity gradient . 50 3.4.2 Oscillatory Shear Index . 52 3.5 Working on regions of interest . 53 3.5.1 Decomposition of bifurcation branches . 53 3.5.2 Relating surface points to centerlines . 54 4 An application of three-dimensional modeling 59 4.1 Cerebral hemodynamics . 59 4.2 The Aneurisk project . 60 4.3 Hemodynamic features of the Internal Carotid Artery . 62 4.3.1 Discussion . 71 4.3.2 Wall shear stress as a classification parameter . 74 5 A geometrical multiscale model of the cerebral circulation 77 5.1 The compliant vessel problem . 77 5.2 Matching conditions in 3D rigid/1D multiscale models . 78 5.2.1 Numerical algorithm . 79 5.2.2 Matching conditions including compliance . 80 5.2.3 Parameters estimation . 85 5.2.4 Results . 86 5.3 A 1D-3D-1D coupling . 88 5.3.1 Results . 89 5.4 The 3D carotid model and the multiscale coupling . 91 5.4.1 Remarks and perspectives . 93 6 Computational tools 94 6.1 An introductory note on C++ . 94 6.2 LifeV: a C++ finite element library . 95 6.2.1 Code features . 96 6.3 Implementation of networks of 1D models . 97 6.3.1 Building the graph . 100 6.3.2 Interface conditions . 102 6.3.3 A simple example . 106 7 Conclusions 109 Acknowledgements 111 ii Abstract In this work we address the mathematical and numerical modeling of cerebral circulation. In particular, one-dimensional (1D) models are exploited for the representation of the complex system of cerebral arteries, featuring a peculiar structure called 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. This 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. Chap. 2 deals with 1D models taking into account the wall viscoelasticity. In particular, the derivation of the nonlinear model is presented in Sec. 2.2, while a linearized set of equations is presented in Sec. 2.2.2. An analytical solution is found for the latter formulation and is used to validate the adopted numerical scheme (Sec. 2.4 and Sec. 2.5). Finally, the effect of wall viscoelasticity on the wave propagation phenomena is 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 intro- duced in Sec. 3.3. 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, defined in Sec. 3.4). Moreover, 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: this approach is presented in Sec. 3.5. 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, as discussed in Sec. 1.2. In the context of Aneurisk project, an extensive statistical investigation has been conducted on the geometrical features of the internal carotid artery, finding that certain spatial patterns of radius and curvature are associated to the presence and to the position of an aneurysm in the cerebral vasculature (Sec. 4.2). Starting from this observation, a classification strategy for vascular geometries has been devised. In the present work, blood flow has been simulated in the 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. Finally, it has been shown that certain values of the mechanical load are associated to the presence and the location of an aneurysm in the cerebral circulation. Adding this 1 Contents hemodynamic parameter in the classification technique improves its efficacy (Sec. 4.3). 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 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 promis- ing tool for the modeling of this interaction. They are based on the coupling of reduced models taking into account the dynamics of the vascular network and detailed models describing the local blood features. In Sec. 5.4 a geometrical multiscale model of the cerebral circulation is presented, based on the coupling of a 1D representation of the circle of Willis and the 3D representation of a carotid artery. A novel method to describe the interface between the two models is discussed in Sec. 5.2. The number of potential applications of reduced models, due to their proven effec- tiveness in the study of vascular networks, calls for the design of efficient and robust software tools. In Chap. 6 we address this issue, by presenting some excerpts of the software specifically written in the context of this work for the simulation of the circulatory system (Sec. 6.3). 2 1 Introduction In this Chapter we discuss the motivation of this work, assessing the problems of interest. A description of the cerebral circulatory system and a review on the state of the art knowledge on cerebral aneurysms are presented in Sec. 1.1 and Sec. 1.2, respectively. Most of the material here presented is taken from the work by Khurana & Spetzler [65]. More details and additional references to the medical literature for these topics can be found therein. The modeling of cerebral circulation, with specific attention to the blood flow problems related to the development of vascular diseases, can enhance the comprehension of the pathology mechanisms and therefore help in devising treatment procedures. On the other hand, the complexity of the physical systems at hand calls for the definition of effective modeling strategies, balancing the need for a detailed description of the physical phenomena and the computational cost. These issues, together with a description of the original contribution of this work in the presented framework, are discussed in Sec. 1.3. 1.1 Anatomy and physiology of the cerebral circulation Cerebral vasculature is a complex structure, ensuring the adequate perfusion to all the brain districts [39]. Cerebral blood vessels are responsible for feeding the brain with oxygen and nutrients (brain arteries) and for the draining of metabolic waste products from the brain (brain veins). To illustrate the typical features of a cerebral artery, we refer for the sake of clarity to the schematic representation of its cross section, depicted in Fig. 1.1. The intima of brain arteries (the innermost part of the wall) is composed of a single layer of endothe- lial cells (represented as light blue cells in the figure), resting on a protein-rich layer called the basal lamina (inner part of the black circle). The outer part of the black circle represents the elastic lamina, whose main component is elastin protein, while smooth muscle cells (large red cells) form the media.

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