Computational Hemodynamics of Cerebral Circulation Multiscale Modelling From the Circle of Willis to...

7/28/2019 Computational Hemodynamics of Cerebral Circulation Multiscale Modelling From the Circle of Willis to Cerebral An

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POLITECNICO DI MILANO

Dipartimento di Matematica F. Brioschi

Ph. D. course in Mathematical EngineeringXXI cycle

Computational hemodynamics

of the cerebral circulation:

multiscale modeling from the circle of Willis

to cerebral aneurysms

Ph. D. candidate: Tiziano PASSERINIMat. D02436

MSc Degree in Biomedical EngineeringPolitecnico di Milano

Supervisor: Prof. Alessandro VENEZIANI

Tutor: Prof. Alessandro VENEZIANI

Coordinator: Prof. Paolo BISCARI

Milano, 2009


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Contents

Abstract 1

1 Introduction 3

1.1 Anatomy and physiology of the cerebral circulation . . . . . . . . . . . . 31.1.1 The circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Morphology and fluid dynamics of cerebral aneurysms . . . . . . . . . . 61.2.1 The role of hemodynamics . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Modeling the cerebral circulation . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 The circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Cerebral aneurysms . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 One-dimensional models for blood flow problems 15

2.1 Wave propagation phenomena in the cardiovascular system . . . . . . . 152.1.1 Modeling the vascular wall . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 A viscoelastic structural model for the vessel wall . . . . . . . . . 192.2.2 The linearized model . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Networks of 1D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Numerical discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Numerical solution of the viscoelastic wall model . . . . . . . . . 292.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.1 Validation of the numerical model versus an analytical solution . 302.5.2 Wave propagation in a single 1-D vessel: a Gaussian pulse wave 322.5.3 Wave propagation in a single 1-D vessel: a sinusoidal wave . . . 342.5.4 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 . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Geometry and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 Dean number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.3 Womersley number and Reduced Velocity . . . . . . . . . . . . . 43

3.3 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 Numerical discretization . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Wall shear stress in the Navier-Stokes problem . . . . . . . . . . . . . . . 50

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3.4.1 Approximation for the velocity gradient . . . . . . . . . . . . . . . 503.4.2 Oscillatory Shear Index . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Working on regions of interest . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.1 Decomposition of bifurcation branches . . . . . . . . . . . . . . . 53

3.5.2 Relating surface points to centerlines . . . . . . . . . . . . . . . . 544 An application of three-dimensional modeling 59

4.1 Cerebral hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 The Aneurisk project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Hemodynamic features of the Internal Carotid Artery . . . . . . . . . . . 62

4.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.2 Wall shear stress as a classification parameter . . . . . . . . . . . . 74

5 A geometrical multiscale model of the cerebral circulation 77

5.1 The compliant vessel problem . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Matching conditions in 3D rigid/1D multiscale models . . . . . . . . . . 78

5.2.1 Numerical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2.2 Matching conditions including compliance . . . . . . . . . . . . . 805.2.3 Parameters estimation . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 A 1D-3D-1D coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 The 3D carotid model and the multiscale coupling . . . . . . . . . . . . . 915.4.1 Remarks and perspectives . . . . . . . . . . . . . . . . . . . . . . . 93

6 Computational tools 94

6.1 An introductory note on C++ . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 LifeV: a C++ finite element library . . . . . . . . . . . . . . . . . . . . . . 956.2.1 Code features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3 Implementation of networks of 1D models . . . . . . . . . . . . . . . . . 976.3.1 Building the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.3 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Conclusions 109

Acknowledgements 111

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Abstract

In this work we address the mathematical and numerical modeling of cerebral circu-lation. In particular, one-dimensional (1D) models are exploited for the representationof the complex system of cerebral arteries, featuring a peculiar structure called circleof Willis. These models, based on the Euler equations, are unable to capture the lo-cal details of the blood flow but are suitable for the description of the pressure wavepropagation in large vascular networks. This phenomenon is driven by the mechanicalinteraction of the blood and the vessel wall, and is therefore affected by the mechanicalfeatures of the wall. Chap. 2 deals with 1D models taking into account the wall vis-

coelasticity. 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 isfound 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 propaga-tion phenomena is studied in some numerical experiments representative of realisticconditions in the cardiovascular and cerebral arterial systems.

The details of the blood flowcan be studied by means of three-dimensional (3D) mod-els, 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 mediumand 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 thevascular tree in terms of centerlines, it is possible to easily identify regions of inter-est in the computational domain, in which to restrict the fluid dynamics analysis: thisapproach is presented in Sec. 3.5.

Cerebral aneurysms are a disease of the vascular wall causing a local dilation, whichtends to grow and can rupture, leading to severe damage to the brain. The mechanismsof initiation, growth and rupture have not been completely explained yet, but the effectsof blood flow on the vascular wall are generally accepted as risk factors, as discussedin 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 thatcertain 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 thisobservation, a classification strategy for vascular geometries has been devised. In thepresent work, blood flow has been simulated in the patient-specific vascular geometriesreconstructed in the context of the Aneurisk project, and an index of the mechanicalload 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 tothe presence and the location of an aneurysm in the cerebral circulation. Adding this

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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 cir-

culatory system. It is believed to be crucial in the context of cerebral circulation, sincedefects or diseases at the level of the circle of Willis can induce local flow conditions as-

sociated 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 reducedmodels taking into account the dynamics of the vascular network and detailed mod-els describing the local blood features. In Sec. 5.4 a geometrical multiscale model ofthe cerebral circulation is presented, based on the coupling of a 1D representation ofthe circle of Willis and the 3D representation of a carotid artery. A novel method todescribe 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 robustsoftware tools. In Chap. 6 we address this issue, by presenting some excerpts of thesoftware specifically written in the context of this work for the simulation of the circu-

latory system (Sec. 6.3).

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1 Introduction

In this Chapter we discuss the motivation of this work, assessing the problems of inter-est. A description of the cerebral circulatory system and a review on the state of the artknowledge 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 befound therein.

The modeling of cerebral circulation, with specific attention to the blood flow prob-

lems related to the development of vascular diseases, can enhance the comprehensionof the pathology mechanisms and therefore help in devising treatment procedures. Onthe other hand, the complexity of the physical systems at hand calls for the definition ofeffective modeling strategies, balancing the need for a detailed description of the phys-ical phenomena and the computational cost. These issues, together with a descriptionof the original contribution of this work in the presented framework, are discussed inSec. 1.3.

1.1 Anatomy and physiology of the cerebral circulation

Cerebral vasculature is a complex structure, ensuring the adequate perfusion to all thebrain districts [39]. Cerebral blood vessels are responsible for feeding the brain withoxygen and nutrients (brain arteries) and for the draining of metabolic waste productsfrom the brain (brain veins).

To illustrate the typical features of a cerebral artery, we refer for the sake of clarityto 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 ofendothe-lial cells (represented as light blue cells in the figure), resting on a protein-rich layercalled 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 smoothmuscle cells (large red cells) form the media. Fibroblasts (thin green cells) and nervefibers (orange fibers) are located in the adventitia (the outermost layer of the wall) andare respectively responsible for the production ofcollagen fibers and for the innerva-tion of smooth muscle cells. The astrocytes, one of which is shown in the figure asa dark blue cell, are present only at the level of the smallest brain vessels (the braincapillaries) and provide biochemical support to the endothelial cells [65].

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Figure 1.1: Cross-section of a brain artery, showing the layers and components of thewall. The innermost part is a hollow space (the lumen) containing serum and

blood cells. The cells here illustrated are not to scale for the vessels in andaround the circle of Willis. from http://www.brain-aneurysm.com/

1.1.1 The circle of Willis

Four main arteries enter from the neck under the surface of the brain. The two internalcarotid arteries enter at the front, while the two vertebral arteries enter at the back. Allthe four of this trunks end in a ring of arteries known as the circle of Willis (see Fig. 1.2,

left). This is the main collateral pathway of the cerebral circulation (see Fig. 1.2, right),made of the right and left posterior cerebral arteries (rPCA and lPCA), the right andleft posterior communicating arteries (rPCoA and lPCoA), the right and left anteriorcerebral arteries (rACA and lACA) and the anterior communicating artery (ACoA).The two internal carotid arteries (rICA and lICA) feed the anterior circulation, delivering

blood in the anterior part of the brain, while the two vertebral arteries (rVA and lVA)join into the basilar artery (BA), feeding the posterior circulation which delivers blood inthe posterior region of the brain.

All the arteries forming the circle lie on the surface of the brain in the so-called sub-arachnoid space. From these vessels depart smaller arterial branches such as the perforat-ing arteries, which supply the deep structures of the brain, and thepial arteries. The latter

course over the brain surface (cortex) and into the brain valleys (sulci), originating per-forating arterioles feeding the deeper cerebral tissue. The arterioles end in capillaries,which drain first into venules and then into larger veins. A high-volume, low-pressurevenous system (the dural venous sinuses) collects blood and empties into the jugularveins in the neck, eventually closing the circuit into the right atrium of the heart.

The complex structure of the circle of Willis has two advantages. On the one hand itcan supply blood to the brain even when one or more vessels are occluded or missing.

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Figure 1.2: Representation of the circle of Willis. Left: overview of the undersurfaceof the brain. Right: the arteries composing the ring. from http://www.wikipedia.org

It is well known in fact that in almost 50% of the population one of the branches ofthe circle is absent or partially developed [74], but this finding is regarded as a normalvariation of brain vessels anatomy. On the other hand, the circle protects the brain fromdisuniform or excess supply of blood, distributing it uniformly.

The study of blood flows in normal cerebral arteries and the circle of Willis is es-sential for better understanding the hemodynamics environment in which pathologiessuch as aneurysms develop, and is relevant in clinical practice for many intracranial orextracranial procedures like the endoarterectomy, the carotid stenting or the compres-sion carotid test (see e.g. [60]).

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Figure 1.3: A saccular brain aneurysm (A) arising from the wall of a brain artery(ba). Black arrows indicate the aneurysm neck. from http://www.brain-aneurysm.com/

1.2 Morphology and fluid dynamics of cerebral aneurysms

An aneurysm (named after the greek worda n e r i s m a

, meaning widening), is a sac-likestructure which forms where the blood vessel wall weakens, ballooning outwards (see

Fig. 1.3). The most common type of cerebral aneurysm is the saccular or berry aneurysm,similar to a sack sticking from the side of a blood vessel wall. It is usually characterisedby a neck region (indicated in Fig. 1.3 by black arrows), and tends to grow and rupture.Less frequently, fusiform cerebral aneurysms are found: they look like vessels expandedin all directions, do not feature a neck region and they seldom rupture. Furthermore,they are typically associated to fatty plaque or atherosclerosis in the artery or with aninjury or break in the arterial wall. From now on, we will focus our attention on berryaneurysms, due to their greater clinical relevance.

Classification

Aneurysms can be classified according to their size, as shown in the following table:

Diameter Class

< 10 mm Small11 - 15 mm Large20 - 24 mm Near-giant

> 25 mm Giant

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Small and large aneurysms behave actually in similar ways in that they tend to growand rupture, while most of the near-giant and giant aneurysm cause symptoms by com-pressing or irritating the surrounding brain structures. However, a threshold value forthe diameter has not been precisely defined, and this explains the uncertain classifica-

tion of aneurysms with diameter comprised between 16 and 19 mm.

Location

Most brain aneurysms form on the arteries of the circle of Willis or from their mainbranches. Moreover, most tend to occur in the anterior circulation, preferentially inregions where arteries branch. Indeed brain blood vessels could be naturally weaker insuch locations, which are also preferential sites for fatty plaques deposition [65].

An extensive statistical investigation of the location of cerebral aneurysms has beenone of the goals of the Aneurisk research project which motivated the present work. Wewill discuss this point more thoroughly in Chap. 4.

Risk factors

Aneurysms may be congenital, but most of them are nowadays thought to be acquired.The main risk factors for aneurysm formation are listed in the following table:

The main risk factors for aneurysm formation

HypertensionPrevious aneurysmFamily history of brain aneurysmConnective tissue disorderOlder than 40 yearsFemaleBlood vessel injury or dissection

Some inherited genetic defects may predispose to the forming of aneurysms and becompounded by added insults due for instance to smoking or hypertension.

The hemodynamic factor is considered most relevant in the initiation of aneurysms.This topic will be dicussed later on in this Chapter and will be further expanded inChap. 4. Indeed, the Aneurisk project proposed an integrated analysis of the morpho-logical and fluid dynamics features of pathologic vessels, with the aim of defining aclassification of vascular geometries based on the probability of developing an aneu-rysm in specific locations [119].

Symptoms

Most aneurysms are silent, and are discovered at the time of rupture. The typical symp-tom associated to this event is a sudden, extremely severe headache. In the minority ofcases, the aneurysm may be found because of symptoms caused by the mass effect,in other words the compression or irritation of surrounding brain structures due to the

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Magnetic resonance techniques (MRI, MRA) are less invasive than cerebral angiog-raphy, but have a limitation in that they cannot detect the smallest aneurysms as wellas cerebral angiography can.

A new technique which is recently gaining popularity is CTA: it is based on a com-

bination of computed tomography (CT) scanning and angiography. More precisely, anintravenous dye is injected into the patient during CT scanning. The resulting tech-nique is quicker, cheaper and less invasive than the traditional cerebral angiographyand able to produce high-resolution, color and 3D images.

Ultrasound techniques and common radiography have no role in the detection ofaneurysms [65].

Treatment

If an aneurysm is detected but has not ruptured, the choice between immediate treat-ment or observation is controversial. The latter implies that the patients need to un-dergo repeated scans to determine if the aneurysm is enlarging, therefore facing therisk of excessive postponement of the treatment and, depending on the imaging tech-nique, the exposition to multiple invasive procedures. The former exposes patients to

perioperatory risks associated to the chosen procedures.The general criterium associating a risk of rupture to aneurysms based on their size is

not practically accepted, since it is believed that each brain aneurysm should be evalu-ated on an individual basis, with consideration of patients age and medical conditions(in particular the history of previous SAHs), the aneurysm site, size and shape [146].

The first option for the treatment is open surgery, which is usually recommended

as early as possible after a rupture. Most of the different types of open surgery arebased on the insertion of metallic clips across the neck of the aneurysm (direct clipping)or across the arteries feeding or draining the sac, in order to exclude it from the bloodpathway or to make it clot off and eventually shrink. Another therapeutic choice, lesscertain than the clipping, is the surgical reconstruction of the aneurysmal part of thewall.

On the other hand, endovascular intervention requires the insertion of a catheter,typically into the femoral artery, which is navigated through the aorta and up into the

brain to the region of the aneurysm. Then platinum microcoils or a glue or other com-posite materials can be placed in the lumen of the aneurysm in order to slow the flowof blood. Alternatively, a balloon can be placed in the parent artery feeding the aneu-

rysm, or a stent can be inserted across the aneurysmal portion of the artery to cut off itsblood supply. Even combinations of the presented procedures can be performed. In allcases, open surgery is not needed, the effectiveness of the treatment can be compara-

ble to that of surgery especially in small aneurysms and sometimes aneurysms whichwould be difficultly reached by open surgery can be treated endovascularly. However,aneurysms treated by coiling may persist or reoccur, thus needing to be treated again(by recoiling or open surgery) [84].

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1.2.1 The role of hemodynamics

It is accepted in the literature that hemodynamics plays a major role in the process ofaneurysm formation, progression and rupture. This introduction briefly summarizesthe state of the art knowledge on the topic, following the excellent review recently pro-posed by Sforza et al. [124].

Arteries feature an adaptive response to blood flow and in particular to wall shearstress (WSS, see Chap. 3). A chronic increase of the WSS, due to increased blood flow,causes a reaction by endothelial cells and smooth muscle cells, which leads to vesselenlargement in order to reduce WSS to physiological values [38, 76]. However, thiskind ofstructural remodeling can be potentially destructive, when triggered by locallyincreased WSS: in this situation, a damage to the arterial wall and a subsequent focalenlargement may take place [124].

On the other hand, endothelial cells can sense WSS and consequently adapt their spa-tial organization: uniform shear stress fields cause the cells to be stretched and aligned

in the direction of the flow, while irregular shapes and orientation are assumed underthe action of low and oscillatory wall shear stress. The latter situation promotes intimalwall thickening and potentially atherogenesis [31, 43, 50, 68], however in the particularcase of cerebral aneurysms could be a protective factor against wall weakening andrupture [124].

Many clinical and experimental observations support the theory of a relation be-tween cerebral aneurysm initiation and the effects of high-flow hemodynamic forceson the arterial wall. Studies pointed out the association of cerebral aneurysms with ar-terial anatomic variations and pathological conditions such as hypoplasia or occlusionof a segment of the circle of Willis [64, 81, 117]. High-flow arteriovenous malforma-tions inducing a local increase of blood flow in the cerebral circulation [96] can promote

the disease. Furthermore, aneurysms usually localize in sites of flow separation andelevated WSS such as bifurcations. These conditions were found to be associated inanimal models to fragmentation of the internal elastic lamina of blood vessels [130],alterations in the endothelial phenotype or endothelial damage [129]. Moreover, ex-perimental cerebral aneurysms can be created in rats and primates through systemichypertension and increased blood flow [58,66,67,90].

Aneurysm growth is nowadays understood as a passive yield to blood pressure.While the aneurysm diameter increases, the wall progressively heals and thickens.Hystological evidences and direct measurements on cadaveric and surgical specimensshow that the aneurysmal wall is mostly composed by collagen and that it can toleratestresses in the range of those imposed in vivo by the mean blood pressure. The rupture

of an aneurysm is thought to be the result of a process of weakening of the wall, whosemechanisms have not been explained yet. In particular, it is not clear if either low orhigh shear stresses have to be considered the main responsibles.

According to the high-flow theory, the process of wall remodeling and potential de-generation is induced by elevated WSS [91]. More precisely, the arterial wall can weakenunder the action of abnormal shear stress fields, due to biochemical processes leadingultimately to apoptosis of the smooth muscle cells and loss of arterial tone [51]. There-

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fore, the prevalence of blood pressure forces over internal wall stress forces may causea local dilation, which then grows under the action of non physiological blood shearstresses. The wall stiffens, because of stretching of elastin and collagen fibers in the me-dial and adventitial layers. Eventually an equilibrium can be reached, in which elastin

and collagen are constantly under a non physiologically large mechanical load: in thissituation wall remodeling may take place.

Low blood flows in the aneurysm can cause blood stagnation in the dome, and this isbelieved to be the major responsible for wall damage in the low-flow theory. Stagnationpromotes the aggregation of red cells, the accumulation and the adhesion of plateletsand leukocytes along the intimal surface [57]. This may be a cause of inflammation, dueto the infiltration of white blood cells and fibrin in the intimal layer [29]. The wall tissuethen degenerates and becomes unable to support blood pressure with physiologicaltensile forces. In this situation the aneurysmal wall progressively thins and may finallyrupture.

As previously discussed, a strong correlation between the size of aneurysms andtheir rate of rupture has been documented in literature. This led to the definition of aclinical measure termed aspect ratio (defined as the depth of the aneurysm divided bythe neck width): it has been found that an aspect ratio bigger than 1.6 is correlated to arisk of rupture [139]. On the other hand, it is known that flow velocities in aneurysmsdepend inversely on the volume [72,98,131] and that shear stresses in the sac are usu-ally significantly lower than in the parent artery, in particular for bigger aneurysms.These evidences support the theory of a decisive role oflow shear stress in the rupturemechanism.

However, recent patient-specific modeling based on computational fluid dynamics (CFD)showed that areas of elevated shear stress are commonly found in the body and domeof aneurysms, even if the spatial average WSS is still lower than in the parent artery.Thus, the size and position of the flow impingement region, and therefore the pres-ence ofhigh shear stresses on the wall may represent other risk factors for aneurysmrupture [22]. Moreover, narrow necks in large aneurysms geometrically induce concen-trated inflow jets and localized impact zones: the correlation between big aspect ratioand rupture rate may then be explained also by the high stress theory.

During its growth, an aneurysm moves in the peri-aneurysmal environment (PAE),coming in contact with structures such as bone, brain tissues, nerves and dura mater.

A clinical evidence of this phenomenon comes from symptoms related to the pressureexerted by the aneurysm on the surroundings, such as bone erosion, obstructive hydro-cephalus and cranial nerve palsy [63,105]. The effect of PAE on the aneurysm evolutionis not well known. The contact with external structures can be protective for the an-eurysm in that it can locally decrease stresses [122]. However, complex interactionswith the PAE can cause non uniformly distributed or unbalanced contact, with eitherprotective or detrimental effect on the evolution of the aneurysm [116].

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1.3 Modeling the cerebral circulation

The complexity of the vascular system demands for the set up of convenient mathemat-ical and numerical models. Computational hemodynamics is basically based on three

classes of models, featuring a different level of detail in the space dependence.Fully three-dimensional models (3D, see Chap. 3) are based on the incompressible Na-vier-Stokes equations possibly coupled to appropriate models that describe the bloodrheology and the deformation of the vascular tissue. These models are well suitedfor investigating the effects of the geometry on the blood flow and the possible phys-iopathological impact of hemodynamics. Unfortunately, the high computational costsrestrict their use to contiguous vascular districts only on a space scale of few centime-ters or fractions of meter at most (see e.g. [8], [56], [107]).

By exploiting the cylindrical geometry of vessels, it is possible to resort to one dimen-sional models (1D), reducing the space dependence to the vessel axial coordinate only(see Chap. 2). These models are basically given by the well known Euler equations

and provide an optimal tool for the analysis of wave propagation phenomena in thevascular system. They are convenient when the interest is on obtaining pressure dy-namics in a large part of the vascular tree with reasonably low computational costs(see [47,89,97]). However, the space dependence still retained in these models inhibitstheir use for the whole circulatory system. In fact, it would be unfeasible to follow thegeometrical details of the whole network of capillaries, smaller arteries and veins.

A compartmental representation of the vascular system leads to a further simplifi-cation in mathematical modeling, based on the analogy between hydraulic networks andelectrical circuits. The fundamental ingredient of these lumped parameter models (0D) arethe Kirchhoff laws, which lead to systems of differential-algebraic equations. Thesemodels can provide a representation of a large part or even the whole circulatory sys-

tem, since they get rid of the explicit space dependence. They can include the presenceof the heart, the venous system, and self-regulating and metabolic dynamics, in a sim-ple way and with low computational costs (see e. g. [89,97]).

All these models have peculiar mathematical features. They are able to capture dif-ferent aspects of the circulatory system that are however coupled together in reality.In fact, the intrinsic robustness of the vascular system, still able to provide blood todistricts affected by a vascular occlusion thanks to the development of compensatorydynamics, strongly relies on this coupling of different space scales. Feedback mecha-nisms essential to the correct functioning of the vascular system work over the spacescale of the entire network, even if they are activated by local phenomena such as anocclusion or the local demand of more oxygen by an organ. This is particularly evident

in the cerebral vasculature, as mentioned earlier in this Chapter.To devise numerical models able to cope with coupled dynamics ranging on differ-ent space scales a geometrical multiscale approach has been proposed in [47]. Followingthis approach, the three different classes of models are mathematically coupled in aunique numerical model. Despite the intuitiveness of this approach, many difficultiesarise when trying to mix numerically the different features of mathematical models,which are self-consistent and however not intended to work together. Some of these

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difficulties have been extensively discussed recently in [112].

1.3.1 The circle of Willis

Several studies have been carried out for devising a quantitative analysis of the bloodflow in the circle of Willis. After the first works based on hydraulic or electric analogmodels [11,26,41,88,115], most of the research has been based on modeling the circle ofWillis as a set of 1D Euler problems (see Chap. 2) representing each branch of the circle,with an appropriate modeling of the bifurcations [2,32,61,62,77,78,143]. More recently,metabolic models have been added to simulate cerebral auto-regulation, which is afeedback mechanism driving an appropriate blood supply into the circle on the basis ofoxygen demand by the brain [3]. Furthermore, a complete 3D image based numericalmodel of the circle of Willis has been presented in [20]. This model, however, requiresmedical data that are currently beyond the usual availability in common practice, and

is computationally intensive compared with the 1D counterpart.In the present work, the modeling of the circle of Willis is addressed from several dif-

ferent viewpoints. The features of the arterial ringper se are discussed in Chap. 2, wherea one-dimensional model (previously published by Alastruey et al. [2]) is studied withparticular attention to the problem of correctly modeling the mechanical behaviour ofthe arterial wall. Its viscoelastic features affect indeed the time and space pattern ofpressure waves propagating in the cerebral circulatory system, as can be seen by com-paring the results obtained with a viscoelastic model for the wall to those obtained byusing a linear elastic model (see Sec. 2.5.4). The computational study here presentedis carried out with a software tool specifically written and based on the C++ finite ele-ment library LifeV (see Sec. 6.2). The cerebral circulation is represented as a networkof interacting vessels, each one described by a 1D model. The design of algorithms anddata structures for the implementation of this approach is presented in Chap. 6.

The arteries of the circle of Willis can suffer from pathologies such as cerebral an-eurysms, associated to local damages of the vascular wall or induced by geometricalfeatures of the vessels which need to be studied in detail. Reduced models (such as1D models) are not suitable for this task: on the other hand, a full 3D modeling of alarge and complex system of arteries can be unaffordable, both because of high compu-tational costs and because of the lack of medical data to completely set up the problem.In Chap. 5 we present a geometrical multiscale model for the cerebral circulation, cou-pling a detailed 3D model of a carotid bifurcation together with a reduced 1D model of

the circle of Willis. The different models entail different assumptions on the mechan-ical behaviour of the vascular wall: its compliance is the driving mechanism for thepropagation of pressure and flow rate waves, and is differently modeled at differentgeometric scales. Proper matching conditions have been devised to retrieve the correctdescription of the dynamics of the coupled system (see Sec. 5.2).

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1 Introduction

1.3.2 Cerebral aneurysms

In the last years, the study of the blood flow dynamics of cerebral aneurysms has beencarried out with different tools. Experimental and clinical studies, focused on idealizedaneurysm geometries or on surgically created aneurysms on animals, were able to showthe complexity of intra-cerebral hemodynamics [121]: however, they did not explain therelation between hemodynamics and clinical events. The same limitation holds for invitro studies, which on the other hand can give a very detailed description of the flowmechanics inside idealized geometries [73]: the main drawback for this approach is theunfeasibility of patient-specific analyses.

Computational models have been extensively and successfully used due to their ca-pabilities in circumvent some limits of the other approaches. In particular, thanks torecent advances in medical imaging tools, it is relatively easy to obtain accurate patient-specific geometrical models of cerebral circulation. The blood motion inside arteriesand aneurysms can be then simulated by means of CFD techniques [21, 59, 133] or ex-perimental studies based on realistic anatomical models reconstructed from images us-ing rapid prototyping techniques [136]. The limitation of these approaches is mainlytheir validation, since the in vivo correct estimation of blood flow patterns is still anopen problem within nowadays imaging technology. However, employing virtual orsimulated angiography, it has been shown that CFD models are able to reproduce theflow patterns observed in vivo during angiographic examinations [23,44].

In the context of the Aneurisk project (see Chap. 4) a study of the internal carotidartery as a preferential site for aneurysms formation has been proposed. More precisely,starting from patient-specific geometrical modeling based on medical images [103], theparent arteries have been classified on the basis of their morphological features [120].These features have been found to be significantly correlated to the presence and thelocation of aneurysms. A CFD analysis on the same dataset shows that a similar corre-lation holds with hemodynamics features of the parent artery (see Sec. 4.3). More thanthat, we show that by considering fluid dynamics parameters together with geometri-cal parameters for the description of the considered cerebral vessels, the classificationcan be enhanced. It is indeed our belief that an integrated approach, starting from themedical image and systematically collecting different sources of information for thecharacterization of the physical system at hand, can lead to a greater insight in the un-derstanding of the pathology development.

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2 One-dimensional models for blood flowproblems

Reduced models for blood flow problems prove to be effective in capturing the mainfeatures of the wave propagation phenomena in the human cardiovascular system [19,45, 126]. In particular, one-dimensional models based on the Euler equations offer areliable description of the mechanics of blood-vessel interaction under the assumptionof cylindrical arteries, the direction of the cylinder axis being the main direction offlow considered in the model. This approximation easily applies to large parts of the

circulatory system, whenever we are not interested in the detailed description of flowfeatures in complex vascular geometries such as bifurcations, stenoses, aneurysms [46,125].

In this Chapter we present a quick review of 1D models for blood flow problemsand their application. We start by recalling the well known Euler equations (Sec. 2.2),focusing on different models for the vessel mechanics and in particular on a simple wayto take into account the viscoelastic features of the vascular wall (Sec. 2.2.1).

Under proper assumptions, an analytical solution for a linearized version of the Eulerequations can be obtained. Its derivation and the validation of the numerical discretiza-tion used to solve the equations are presented in Sec. 2.2.2 and Sec. 2.5.1 respectively.The fully non linear problem is solved in some test cases (Sec. 2.5), showing the ability

of the model at hand to capture the main features of the studied problems.In the spirit of 1D representation, the circulatory system as a whole can be seen asa network of interconnected vessels. By this representation we build one-dimensionalmodels of large regions of the circulatory system (Sec. 2.3), each vessel being described

by Euler equations. The application of this approach to the study of cerebral circulationis discussed in Sec. 2.5.4.

2.1 Wave propagation phenomena in the cardiovascular

system

The circulatory system is responsible for the distribution of blood flow through thehuman body. Blood is pumped by the heart into the network ofarteries, reaches thecapillaries where most of the biochemical phenomena associated to the tissue nutritiontake place, and is finally collected by the network ofveins bringing it back to the heart(see Fig. 2.1).

We can divide each cardiac cycle in an early phase (systole), associated to the ejectionof blood from the hearts ventricles, and a late phase (diastole), in which blood motion

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2 One-dimensional models for blood flow problems

Figure 2.1: Schematic representation of the human cardiovascular system. In each car-diac cycle, blood flows from the heart towards the peripheral circulation(arterioles, capillaries) and is collected back to the heart from the veins. fromhttp://www.williamsclass.com/

is driven by the compliance of the vascular wall. In systole, the contraction of the heartinduces a pressure wave which travels along the arterial tree causing the dilation of

the vessels. In diastole, arteries deflate and push blood towards the capillaries and thevenous compartment, featuring the so-called reservoir effect [1].The study of the time and space pattern of pressure and flow rate waves propagat-

ing in the circulatory system can help in understanding the correlation between localpathologies and systemic features. An interesting case in this respect is the effect ofarterial remodeling and stiffening due to aging or diseases (such as atherosclerosis);this is a documented cause of increased systolic pressure due to pressure wave reflec-tions, and it is associated to overload to the left ventricle (the so-called hemodynamicoverload [75]). This condition can determine left ventricular hypertrophy and alteredcoronary perfusion, with consequent heart damage.

2.1.1 Modeling the vascular wall

The interaction between blood and the vascular wall plays a fundamental role in thefunctionality of cardiovascular system. Indeed, the mechanical properties of the walldetermine the wave propagation, and this suggests that pathologies which affect thewall may be associated to non physiological pressure waveforms. Besides giving a

better insight on the behaviour of the wall under the effect of stresses exerted by the

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blood flow, an accurate mechanical modeling of the vessels could in principle allow thedetection of vascular diseases from information on the pulse propagation patterns ofpressure and flow rate in the circulatory system [30].

The mechanical modeling of blood vessels requires the definition of a constitutive

law describing the relationship between stress and strain fields in the vessel structure.The latter being a complex layered tissue, its mechanical characterization is still an openproblem. Many different constitutive models have been proposed in the literature: ves-sel wall can be treated either as a homogeneous material or described by a heterogenousmodel taking into account the micro-structure (cells, fibers and their mechanical inter-action) [144]. Hereafter we will focus our attention on homogeneous models, since inthe spirit of 1D representation the local detail of the physical phenomena at hand can

be foresaken.In the simplest approach, the wall can be treated as a linearly elastic membrane

[36,45,125]. This leads to a reliable description of the main features of the wave propa-gation, both in physiological and pathological situations [3]. Still, an oversimplified lin-

ear mechanical model for the vessel wall structure is not able to reproduce its viscoelas-tic behaviour, which is observed in vivo. Several different approaches have been pro-posed to address the modeling of viscoelastic features of vessel vascular wall [144]. Ar-mentano et al. showed that even a simple Kelvin-Voigt type model can be used to obtaina good agreement between in vivo measured data and numerical experiments [9,28]. Asimilar approach was followed by Canic et al. [19], who exploited a linearly viscoelasticcylindrical Koiter shell model for the arterial wall, based again on a Kelvin-Voigt typedescription of the structure viscoelastic features. A slightly more complex model wasemployed by Bessems et al. [17], who described the wall of large arteries with the stan-dard linear solid approximation. This same approximation was employed by Olufsen etal. [36], who also noted that the strain relaxation, which is not modeled by the simpler

Kelvin-Voigt model, can be relevant in the study of large arteries [140].In the following we extend the analysis on a previously published model for blood

flow in viscoelastic vessels [46], with the aim of validating the numerical scheme thereproposed against an analytical solution for a linearized version of the equations. More-over we highlight the viscoelastic features in the arterial wall dynamics, which are notcaptured from linearly elastic structural models, as we show in some test cases. Finallywe use the presented model to devise a one-dimensional description of the cerebralcirculation, based on the work by Alastruey et al. [3].

2.2 Formulation of the model

Let us consider a one-dimensional domain R representing the cylindrical vesseldepicted in Fig. 2.2 and let I R be a time interval. Given S(x, t) a cross sectionlocated along the vessel at axial coordinate x, considered at time t, A(x, t) is the areaofS, P(x, t) is the mean pressure on Sand Q(x, t) is the fluid velocity flux through S.For all x and for all t I we can express the fluid mass conservation principle(2.1a) and the fluid momentum conservation principle (2.1b) by means of the Euler

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Figure 2.2: A cylindrical compliant vessel. The shaded plane highlights a section Sataxial coordinate x and at time t.

equations [40]:A

t+

Q

x= 0 (2.1a)

Q

t+

x

Q2

A

+

A

P

x+ KR

Q

A= 0 (2.1b)

In (2.1), is the so-called momentum-flux correction (or Coriolis) coefficient, is the fluidmass density and KR is a strictly positive quantity which represents the viscous resis-tance of the flow per unit length of tube.

The closure of the previous system of two equations in the three unknowns A, Pand Q can be recovered by introducing a relation linking the pressure P to the area A

(see [49]), thus taking into account the vessel wall mechanics. Let us denote by Pext thepressure external to the vessel: the wall mechanics can therefore be given in terms of afunction establishing the dependence of the transmural pressure P(x, t) Pext on thevessel kinematics (in turn driven by the blood flow):

P(x, t) Pext = (A(x, t); x, t). (2.2)

We may define in a simple yet rather general way, as a function ofA (together withits derivatives) and of a set of parameters which may depend on x, t or A. Under thehypothesis that the pressure depends on A, on the reference cross-sectional area A0 andon parameters = (0, 1, . . . , p) describing the mechanical properties of the wall, a

possible choice for is:

(A(x, t); A0(x), 0(x), 1(x)) = 0(x)

A

A0(x)

1(x) 1

. (2.3)

For the ease of the notation, we will hereby refer to A0 and , noting that in generalthey are to be considered as functions of the axial coordinate x.

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In the previous, 0 is an elastic coefficient, while 1 > 0 is normally obtained byfitting the stress-strain response curves obtained by experiments. Whenever 1 = 12and 0 = 1A0 =

1A0

h0E12 , (2.3) is equivalent to the following relation:

(A(x, t); A0, ) = A A0A0

, (2.4)

which is derived from the linear elastic law for the wall mechanics of a cylindricalvessel and where E(x) is the Youngs modulus, h0 the wall thickness and the Poissonratio [49].

The adoption of a linearly elastic model for the vessel wall mechanics is convenientsince it simplifies the derivation of the equations, and is still able to capture the mainfeatures of the wave propagation phenomena in vascular system [16,46,82]. However,more accurate and complex mechanical models can be exploited, accounting for thevessel wall inelastic behaviour which is verified in vivo [144]: in the following we willdiscuss this aspect more thoroughly.

Remark Set U = [A Q]T. We derive a conservative form of system (2.1) [45]:U

t+

F(U)

x+ B(U) = 0 , (2.5)

where

F(U) =

Q

Q2

A+ C1

, B(U) =

0

KRQ

A+

A

x C1

x

.

We denote by C1 the following quantity

C1 =

A

A0

c21d , c1 =

A

A

,

where c1 is referred to as the celerity of the propagation of waves along the tube andA0 indicates a reference value for A, here taken equal to the cross-sectional area in anunloaded configuration.

2.2.1 A viscoelastic structural model for the vessel wall

As we already pointed out in Chap. 1, vascular wall is a complex biological tissue,formed by different materials organized in an anisotropic structure [53]. The interplayof the different anatomical components determines its mechanical behaviour [9,10,144].

A simple model, derived from the Navier equation for linearly elastic membranes,was proposed in [113]. It is referred to as the generalized rod model, since it takes intoaccount inner longitudinal actions, in a way similar to what is done in the classicalvibrating rod equation:

P Pext = a + b 2

t2+ c

4

x4 d

2

x2+ g

t

(2.6)

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where

= R R0 =

A A0

is the wall radial displacement, a,b, c,

d are positive coefficients and g

t

is a generic

function of the time derivative of the displacement. According to (2.6), the transmuralpressure on the vascular wall is balanced by five terms, describing different mechanicalfeatures of the structure.

The elastic response of the material is represented by the first term a, while theinertial effects are described by the term involving the second order time derivative ofthe wall displacement, where b = wh is the product of the wall mass density and thewall thickness. Resistance to bendings is expressed in this model by the term involvinga fourth order space derivative, while resistance to traction is taken into account by theterm involving the second order space derivative.

One of the most interesting mechanical features of the vascular wall is its viscoelastic

nature. Arteries exhibit creep, stress relaxation and hysteresis in the stress-strain re-lation. Equation (2.6) accounts for viscoelastic effects, by describing them with term

g

t

. Following formal mathematical arguments, Quarteroni et al [113] proposed

the following formulation

g

t

= e

3

t2x,

involving a third order mixed derivative of, which shows good agreement with ex-perimental results [113]. For the sake of simplicity, we will consider hereafter a simplerterm, based on the Voigt viscoelastic model [52]. Moreover, we will neglect for the sakeof simplicity the other non elastic effects, setting b = c = d = 0.

This leads to the following differential equation linking the transmural pressure tothe wall radial displacement :

P Pext = a + t

, (2.7)

where is the so-called viscoelastic modulus.Recalling that =

AA0

, and noting that typically in hemodynamic problems the

range of variation of cross-sectional area A is small, we approximate

t

=1

2AA

t 1

2A0 A

t

.

Moreover, the elastic response term can be recast in form (2.4), by setting a =

A0

.

The wall mechanics model may now be rewritten in terms ofA including viscoelas-ticity as follows:

P Pext = (A(x, t); A0, a) + At

(2.8)

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with = 2A0

. Therefore, assuming Pext independent ofx,

P

x=

A

A

x+

A0

dA0dx

+

a

da

dx+

2A

xt,

and we note that the second order mixed derivative of A can be recast into a secondorder derivative ofQ by exploiting the mass conservation equation (2.1a).

Substitution of the previous in (2.1b) gives:

Q

t+

x

Q2

A

+

A

A

A

x

2Q

x2+ KR

Q

A+

A

A0

dA0dx

+A

a

da

dx= 0 .

With respect to the conservative form (2.5), we set

F =

F1F2

, F1 = Q , F2 =

Q2

A+ C1

and, analogously,

B =

B1B2

, B1 = 0 , B2 = KR

Q

A+

A

A0

dA0dx

+A

a

da

dx C1

x

so that system (2.1) can be rewritten as follows:

A

t+

Q

x= 0 (2.9a)

Q

t+

F2x

A

2Q

x2+ B2 = 0 (2.9b)

Clearly, the introduction of the viscoelastic term makes this system of equations nolonger hyperbolic. However, we may assume that the elastic response function playsa leading role in determining the wall mechanics. On the basis of this assumption, anoperator splitting approach can be devised [45]. More details on this technique will bepresented later on in Sec. 2.4.1.

2.2.2 The linearized model

A linearized version of equations (2.1) can be derived in the following way. First, we

neglect the nonlinear term, therefore

x Q2

A = 0; moreover, we linearize the coef-

ficients with respect to A, setting A(x, t) A0. The resulting linear system of first orderpartial differential equations reads:

A

t+

Q

x= 0 (2.10a)

Q

t+

A0

P

x+

KRA0

Q = 0 . (2.10b)

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We will consider relation (2.8) linking the pressure to the cross-sectional area, and as-sume for the sake of simplicity = 0, therefore

P

x

=

A

A

x

+

A0

dA0

dx

+

a

da

dx

.

Now (2.10b) becomes

Q

t+

A0

A

A

x+

KRA0

Q +A0

A0

dA0dx

+A0

a

da

dx= 0 ,

and system (2.10) may be written in non conservative form as follows:

U

t+ HL

U

x+ SL = 0 , (2.11)

where

HL =

0 1A0

A0

, SL =

0KR

A0Q +

A0

A0

dA0dx

+A0

a

da

dx

.

A conservative form reads:

U

t+

FLx

+ BL = 0 , (2.12)

where

FL =

QCL

, BL = SL 0CL

x

,

and CL =AA0

c2Ld.System (2.11) is said to be strictly hyperbolic if H is similar to a diagonal matrix

and its eigenvalues are real and distinct. In particular, the eigenvalues of matrix HL

are 1,2 = cL, cL =

A0

Abeing the wave celerity in the linearized problem. A

necessary and sufficient condition for the eigenvalues to be real and distinct is

A> 0,

which is satisfied being typically A > 0 in blood flow problems and

A=

a

2

A.

Linearization of the previous relation yields

A a

2

A0= a .

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In the following we set cL =

A0

a.

We now denote by L, R the matrices whose rows (columns) are the left (right) eigen-vectors ofHL, respectively:

L =

lT1lT2

, R =

r1 r2

,

with the additional (non restrictive) hypothesis LR = I. Then

LHLR = = diag(1, 2) .

and the following equivalent form for system (2.11) is obtained:

LU

t+ L

U

x+ LSL = 0 . (2.13)

If there exist two quantities W1, W2 such that

W

U= L , W = [W1, W2]

T ,

then we can rewrite system (2.13) in diagonal form:

W

t+

W

x+ GL = 0 , (2.14)

where

GL = LSL

W

A0

dA0

dx

W

a

da

dx

.

The values W1, W2 are the so-called Riemann invariants for the hyperbolic system athand.

Left eigenvectors l1,2 read

l1,2 =

cL1

where = (A, Q) is an arbitrary, positive smooth function of its arguments. Therefore

W1A

= cL ,W1Q

= ,W2A

= cL , W2Q

= .

We now impose the integrability of the two differential forms W1 and W2 by choosing such that2Wi

AQ=

2WiQA

, i = 1, 2 ,

which yields

cL Q

=

A

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thus we may simply choose = 1.We now find (see e. g. [49] for a detailed presentation of the procedure) that the

linearized characteristic variables are given by integration of the resulting differentialform:

W1,2 = cLA + Q .We choose (A0, 0) as the zero state in the (A, Q) plane, in which the characteristic vari-ables are zero, and find after integration:

W1,2 = Q cL(A A0) .

Adding viscoelasticity

Lets now consider the viscoelastic term > 0 in (2.8): this yields

P

x= a

A

x+

2A

xt+

A0

dA0

dx+

a

da

dx,

and with arguments similar to those leading to system (2.9) we obtain

A

t+

Q

x= 0 (2.15a)

Q

t+

FL2x

A0

2Q

x2+ BL2 = 0 , (2.15b)

with

FL = FL1FL2 , FL1 = Q , FL2 = CL

and

BL =

BL1BL2

, BL1 = 0 , BL2 =

KRA0

Q +A0

A0

dA0dx

+A0

a

da

dx CL

x.

An analytical solution

The linearized equations (2.15) describe the propagation of area and flow rate waves inthe space-time domain. We may look for solutions in the form ofharmonic waves:

A(x, t) = A(k)expi(k)t kx (2.16a)

Q(x, t) = Q(k)exp

i

(k)t kx . (2.16b)In the previous, k is the wave number, defined as the number of complete oscillationsin the range x [0, 2]; is the (angular) frequency and A(k), Q(k) represent the waveamplitudes in x = 0, t = 0. In general, ,k, A(k), Q(k) C, however it is understoodthat we will be interested in the real part of the solution.

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Substituting (2.16) in the linearized Euler equations yields:iA ikQ

exp

i

(k)t kx = 0 (2.17a)iC1kA + i + k

2C2 + C3Q expi(k)t kx = 0 . (2.17b)For the sake of simplicity, in the previous we assume that all the parameters are constantwith respect to x and set

C1 =A0a

, C2 =

A0

, C3 =

KRA0

. (2.18)

Moreover, we omit to indicate explicitly the dependency ofA and Q on k.The problem of finding solutions to system (2.17) for each t and x is recast into the

existence of non trivial solutions to the following linear system:

iA

ikQ = 0 (2.19a)

(iC1k)A + (i + k2C2 + C3)Q = 0 , (2.19b)which yields the following condition:

(iC3 ) + k2(C1 + iC2) = 0 . (2.20)We can now study the dispersion relation (k), linking the angular frequency to the

wave number: solutions to system (2.17) are travelling waves with angular frequency

1,2(k) =i

C2k2 + C3

(C2k2 + C3)2 + 4C1k22

.

For the problem at hand, the phase velocity cp = (k)/k depends on the wave number,so that the solution to system (2.17) will be affected by wave dispersion: in other words,this means that waves with different wave length propagate with different speed, given

by

cp(k) =i (C2k + C3/k)

(C2k + C3/k)2 + 4C1

2.

We can conversely express the wave number k as a function of the frequency :

k() =

( iC3)C1 + iC2

, (2.21)

and we note that k is in general a complex number even when is real. It can betherefore written as

k = (k) + i(k) . (2.22)We remark that, due to the first equation of system (2.17), the solution is such that

i(k)A = ikQ ,

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which implies that, if R, then A or Q (or both) are complex numbers. We maychoose A R, therefore (recalling (2.22)):

A(x, t) = A(k)exp(k)x expi(k)t (k)xQ(x, t) =

Q(k) + iQ(k) exp(k)x expi(k)t (k)x .

In particular, we are interested in the real part of the solution, which reads

A(x, t) = A(k)exp(k)x cos(k)t (k)x (2.24a)Q(x, t) = exp(k)x(Q)cost (k)x (Q)sint (k)x . (2.24b)It follows from the previous that the imaginary part of the wave number is associated

to an exponential factor modulating the wave amplitudes. When (k) < 0, this is adamping factor corresponding to an exponential decay with variable x.

We now consider the contribution of the three different terms C1, C2 and C3 to thedefinition ofk. When C1 = 0 and C2 = C3 = 0,

k() =

2

C1,

where we choose the positive root. To force C3 = 0 means that we are considering aninviscid fluid, for which the friction term KR vanishes. On the other hand, C2 is equalto 0 when we neglect the viscoelasticity of the blood vessel wall (see (2.18) and (2.7)).In this case, k is a real number and the solutions we find are a set of travelling waves ofthe form (2.16).

IfC1, C3 = 0, the definition ofk reads

k() =

( iC3)C1

,

and k has an imaginary part. This case corresponds to the problem of a viscous fluidflowing inside a linearly elastic vessel. We can reformulate the previous definition asfollows:

k2() =2

C1 i C3

C1,

which in polar notation reads

k2() = r exp

i( + 2n)

, n N ,

r =

2

C1

2+

C3

C1

2,

= arctan

C3

.

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Therefore

k() =

r exp

i

2+ n

, n N ,

and we choose one of the two roots, such that (k) > 0. We remark that, since (k2) > 0and (k2) < 0, /2 < < 0: this implies

2+ n , n N [/4, 0]

or

2+ n , n N [3/4 , ] ,

the latter being excluded by the request (k) > 0. This ensures that (k) < 0 and there-fore the term C3 = 0 is responsible for an exponential damping of the signal amplitude.

In the general case C1

= 0, C2

= 0, C3

= 0, recalling (2.21),

k2() =( iC3)(C1 iC2)

C21 + C22

2=

2(C1 C2C3)C21 + C2

i (C1C3 + C22)

C21 + C2.

With a similar procedure to that applied to the previous case it can be seen that, underthe hypothesis C1 > C2C3, corresponding to assume

A0

2> KR ,

i. e. a Young modulus large enough, both the fluid viscosity (term C3) and the vis-coelasticity of the wall (term C2) cause an exponential decay of the travelling waves

amplitudes.

2.3 Networks of 1D models

Once having set up the model for a single tube, we can move towards the study ofnet-works: from a mathematical point of view, this means to find suitable interface condi-tions for connected tubes. Following [49], we adopt a domain decomposition approach,and request that the solutions in interfacing domains are such that the conservation ofcertain physical quantities is ensured at the interface.

Lets consider as an example the two tubes depicted in Fig. 2.3. We take as a referencesystem the simmetry axis x, which is the same for both vessels, and impose that themass flow and the total pressure are conserved across the interface:

Q1 = Q2 t > 0, at x = Pt,1 = Pt,2

(2.25)

where Pt = P+ 12(QA)

2 stands for the total pressure and the subscripts 1, 2 indicate thatthe subscripted quantity is relative to tube 1 or 2, respectively.

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x =

x

x = L

1 2

x = 0

Figure 2.3: Two connected tubes 1 and 2: the interface is located at coordinate x = .

Formaggia et al. [45] proved that this set of interface conditions guarantees an energyinequality for the coupled problem. On the other hand, they noted that typically in

blood flow problems the value of the pressure P is much greater than the kinetic energy12(

QA )

2, therefore in practice the continuity of pressure can be prescribed at the interface

without encountering stability problems.Other possible interface conditions can be designed to take explicitly into account

the fact that the total pressure decreases as a function of the flow rate, along the flowdirection, in correspondence with the interface . The second of (2.25) may then then

be written as follows [45]:

Pt,2 = Pt,1 sign(Q)f(Q) t > 0, at x =

f being a positive, monotone function satisfying f(0) = 0 and referred to as dissipationfunction. However, appropriate formulations for f are usually not available, therefore a

typical choice is f 0 corresponding to the continuity of the total pressure.

2.4 Numerical discretization

Let us refer for the sake of simplicity to system 2.5. Following [45], we adopt a dis-cretization based on a second order Taylor-Galerkin scheme which can be seen as a gen-eralization of the classical Lax-Wendroff scheme for systems of conservation laws [71].Given Un the approximation of the solution U(tn) at time tn, Un+1 is obtained by solv-

ing the following system:

Un+1 = Un t x

Fn t

2HnSn

+

t2

2

SnU

Fn

x+

x

Hn

Fn

x

t

Sn +t

2SnUS

n

, n = 0, 1, . . . , (2.26)

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where SnU =S

U(Un) and Hn, Sn and Fn are defined in a similar way. The Galerkin

finite element method is applied to (2.26), yielding

U

n+1

h , h

= (Un

h, h) + t

FLW(Un

h),

hx

+

t2

2

SU(Un

h)

F(Unh)

x , h

t2

2

H(Unh)

F

x(Unh),

hx

t (SLW(Unh), h) , h V0h. (2.27)

In the previous equation we used the notation FLW = F(Uh) + t2 FU(Uh)S(Uh) andSLW = S(Uh) +

t2 SU(Uh)S(Uh). All the details on the derivation of the scheme can

be found in [45]. In all the simulations presented in Sec. 2.5, we adopt a linear approxi-mation of the solution, based on P1 finite elements.

2.4.1 Numerical solution of the viscoelastic wall model

The addition of a viscoelastic term to the constitutive law for the vessel wall (see (2.8))and the adoption of the operator splitting approach previously mentioned yield thefollowing equivalent form of system (2.9) [45]:

A

t+

Q

x= 0 (2.28a)

Qet

+ F2(A, Q)

x= B2(A, Q) (2.28b)

Qvt

A

2Q

x2= 0 , (2.28c)

where Q is decomposed into two contributionsQ = Qe + Qv ,

due to the elastic and viscoelastic behaviour of the wall mechanics, respectively. Equa-tions (2.28a), (2.28b) compose a hyperbolic system involving the time derivative ofQe,while (2.28c) is a parabolic equation of variable Qv.

On each time interval [tn, tn+1], n 0, the first two equations in (2.28) are solved bythe Taylor-Galerkin scheme previously presented. The explicit time advancing schemegives An+1 and Qn+1e as functions ofA

n and Qn. The third equation is used to correctthe flow rate, and is solved by adopting an implicit Euler time advancing scheme andthe following finite element formulation [45]: given An+1h and (Qe)

n+1h , find (Qv)h

V0h

such that 1

An+1h(Qv)

n+1h , h

+ t

Qn+1h

x,

hx

= 0, h V0h ,

where we exploit the simplifying assumption that homogeneous Dirichlet boundaryconditions are imposed to the correction term Qv. This corresponds to correcting the

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flow rate only inside the computational domain, and not on the boundary. Moreover,this allows an easy treatment of branchings or anastomoses of vessels: the correctionterm vanishing on the interface between different models, there is no need of decou-pling the term in the different segments.

Now, knowing that Qn+1

= Qn+1e + Q

n+1v , we can write

1

An+1h(Qv)

n+1h , h

+ t

(Qv)

n+1h

x,

hx

=

t

(Qe)

n+1h

x,

hx

, h V0h . (2.29)

2.5 Results and discussion

This section presents a set of numerical experiments designed to test the reliability ofthe model. Particular attention is devoted to the effects of the wall viscoelasticity on thepropagation of waves in blood vessels.

We start by using analytical solutions (2.24) as a benchmark case, for validating thecode. Then we discuss the effect of dissipative terms associated with the fluid viscosityand the wall viscoelasticity. More precisely, following the arguments exploited for thelinearized model, we analyse the role of viscous dissipations in the non linear model(2.1). Furthermore, a simple numerical experiment shows that the model at hand canreproduce the hysteresis in the P(A) relation, which is a typical viscoelastic feature of

blood vessels in vivo.Finally, a model for the circle of Willis is presented, based on the published work by

Alastruey et al. [3] and modified by including a description of the viscoelastic effects inthe vessel wall mechanics. A comparison of the results obtained by the two models isdrawn at a qualitative level.

2.5.1 Validation of the numerical model versus an analytical solution

We simulate the propagation of a cosinusoidal flow rate wave in a cylindrical vessel.The solution we are looking for is of the form (2.24), which is the real part of solution(2.16), when we assume , A R. We will assume = 2 s1, and obtain the corre-sponding solution exploiting the dispersion relation (2.21).

The geometrical and physical features of the simulated vessel are summarized inTab. 2.1. The tube at hand is very long, in order to clearly show the damping effectof the wall viscoelasticity and blood viscosity on the wave amplitude, discussed inSec. 2.2.2. The wall mechanical parameters are in the range of physiological values forlarge arteries (see e. g. [52]). In particular, the value for the viscoelastic modulus = 6 104 dyn s cm3 is taken from [19] and corresponds to the estimated viscous modulusof a human femoral artery.

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Name Symbol Value Measurement unitlength L 1000 cmradius R0 1 cmthickness h 0.15 cm

mass density 1.05 g cm3

Poisson modulus 0.5 -Youngs modulus E 4 106 dyn cm2viscoelastic modulus [28] 6 104 dyn s cm3friction parameter KR 2.633 P

Table 2.1: Geometrical and mechanical parameters for the simulated vessel.

Based on these values, we find

C1 = 3.8095 105 dyn cm g1 , C2 = 2.8571 104 dyn cm s g1 , C3 = 0.83811 g cm2 s1

and k(2) = (k) + i(k) = 0.0093286 i0.0027481 .The initial conditions for the problem are

A(x, 0) = A0 =

Q(x, 0) = exp(k)x(Q)cos(k)x (Q)sin(k)x .

Recalling the first equation in system 2.19 and knowing that A = A(0, 0) = , we findthat

Q =

kA = (Q) + i(Q) = 619.76 + i 182.58 .

The boundary conditions on the left and right boundaries prescribe two periodic flowrates:

Q(xb, t) = exp(k)xb(Q)cost(k)xb(Q)sint(k)xb , xb = 0, 10 m .

The wave propagation is simulated on the time interval t [0, 1] seconds, with atime step ofdt = 104 s. The mesh size is dx = 0.5 cm. Since we are considering aviscoelastic model for the arterial wall and blood is considered as a Newtonian fluid,the wave amplitude is damped by an exponential factor (see Sec. 2.2.2). This effect isclearly visible in Fig. 2.4, where red lines represent the damped amplitude of the wave:

Qdamp(x, t) =

Q(k)

exp

(k)x

. (2.30)

We remark that the operator splitting approach presented in Sec. 2.4.1 is able to re-cover a good approximation of the analytical solution (see Fig. 2.5). We estimated theapproximation error as follows

Qdx QexactL2(0,T;L2())QexactL2(0,T;L2())

= 1.2122 104 ,

where we indicate by Qexact the analytical solution and by Qdx the numerical solution.

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0 2 4 6 8 102000

1500

1000

500

0

500

1000

1500

2000

x

Q

t = 0

t = 0.4

t = 0.8

Figure 2.4: Solution of the linearized model with the presented numerical setup. Y-axis:flow rate (in cm3/s); X-axis: tube axial coordinate (in m). The plot shows

snapshots of the travelling wave at different time. The superimposed redlines represent the damping term (2.30) associated to blood viscosity andwall viscoelasticity.

0 2 4 6 8 101000

500

0

500

1000

1500

2000

x

Q

analytical solution

computed solution

Figure 2.5: Solution of the linearized model with the presented numerical setup. Y-axis:flow rate (in cm3/s); X-axis: tube axial coordinate (in m). The plot shows thesuperposition of the analytical solution (blue) and the numerical solution(red) on the whole domain, at t = 0.1s.

2.5.2 Wave propagation in a single 1-D vessel: a Gaussian pulse wave

This numerical experiment describes the propagation along a vessel of a narrow, Gaus-sian shaped wave, a continuous approximation to a unit pulse (t) located at t = t0(i. e. (t0) = 1 and (t) = 0 for t = 0). The unit pulse waveform was used in [145] totrack the multiple transmissions and reflections in the arterial system, while its Gaus-

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sian approximation was used in [4] for the study of the effects of outflow boundaryconditions in 1D blood flow simulations. Here we aim to evaluate the dissipative ef-fects of wall viscoelasticity and blood viscosity on the travelling wave amplitude.

The simulated vessel has the same characteristics as the vessel described in the pre-

vious section (see Tab. 2.1), but is described this time by the non linear model (2.1). Theboundary condition prescribed on the left boundary is a flow rate of the form:

Q(xl, t) = exp

t t0

,

with = 0.01 and t0 = 0.05 s. Absorbing boundary conditions are prescribed on theright boundary [49]. The wave propagation is simulated on the time interval t [0, 1]seconds, with a time step ofdt = 104 s. The mesh size is dx = 0.5 cm.

The results of the numerical experiment of propagation are shown in Fig. 2.6. When

considering an inviscid flow inside an elastic shell, we can see that the shape of thewave travelling along the vessel is not altered (red line). The addition of fluid viscosityto the model attenuates the wave amplitude, in a similar way with respect to what can

be seen in the linearized model (green line).If we model blood as a viscous fluid and the wall as a series of viscoelastic rings (see

eq. (2.7)), we observe that the wave amplitude is extremely reduced (blue line). Again,this is qualitatively in accordance with what has already been said about the linearproblem.

0 0.2 0.4 0.6 0.8 10.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Q

Figure 2.6: Propagation of a gaussian flow rate wave in a 10m long vessel (here only thetract x [0, 1]m is represented) . Viscoelasticity of the wall and blood viscos-ity attenuate the amplitude of the travelling wave. Red: elastic wall, inviscidfluid; Green: elastic wall, viscous fluid; Blue: viscoelastic wall, viscous fluid.X-axis: position along the tube (m); Y-axis: flow rate (cm3/s)

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2.5.3 Wave propagation in a single 1-D vessel: a sinusoidal wave

This numerical experiment describes the propagation of a half-sinusoidal input wavealong a vessel. The inflow condition mimics a realistic cardiac output, while a lumpedparameter model of the peripheral circulation is coupled to the outflow of the vessel.More precisely, the resistance R and the compliance C of vessels peripheral to the con-sidered 1D domain are simulated by a three-element windkessel model, as proposed

by Alastruey et al. [4]. They showed that by coupling this model to a 1D representationof the aorta, some features of in vivo aortic measurements can be reproduced, such asthe pressure dicrotic notch and the exponential diastolic decay of the pressure.

We reproduce here the same experiment, by considering a 40 cm long vessel, withthe same other geometric and mechanical features as the vessel considered in previoussections (see Tab. 2.1), and described again by the non linear model (2.1). Moreover,R = 1.89 103 dyn s cm5 , C = 6.3 104 cm5 GPa1. The initial conditions for thevessel are A = A0 = and Q = 0. The flow rate boundary condition prescribed on theleft boundary is a periodic function of time, with period T = 1s:

Q(t) =

310 sin(2T t) 0 t < s

0 s t < Twith s = 0.3 s. The simulation was run with dx = 0.5 cm and dt = 104 s.

(a) Flow rate waveform. The thin line represents theinflow waveform.

(b) Pressure waveform

Figure 2.7: Flow rate and pressure waveforms, once a quasi-steady state is reached, inthe middle of a 1D vessel coupled with a 0D outflow model. The wall vis-coelasticity affects the wave propagation.

The present model is able to capture a slightly increased wave speed associated toviscoelastic phenomena (Fig. 2.7). Moreover, Fig. 2.8 (left) shows that the contributionof the viscoelastic term to the overall pressure (see (2.8)) is not negligible.

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(a) Comparison between elastic and viscoelas-tic contribution to the overall pressure wave.

(b) A-P curve for the sin wave propagation nu-merical experiment.

Figure 2.8: Propagation of a half-sinusoidal flow rate wave in a 40 cm long vessel.

The A-P curve (Fig. 2.8, (b)) shows hysteresis, which is a typical behaviour of a vis-coelastic vascular wall. In particular, it can be seen at a qualitative level that the dias-tolic phase, in which P is proportional to A, is clearly distinct from the systolic phasein which pressure and area waves show a significant phase shift. A more quantitativeanalysis of these results, based possibly on the comparison with clinically measureddata or with other available models in the literature, is required to assess the accuracyof the model in the description of the physical phenomena at hand.

2.5.4 A 1D model network: the circle of Willis

The proposed simulation is based on the set-up presented by Alastruey et al. [3]. Thecircle of Willis is immersed in a larger network of 1D models describing the main ar-teries bringing blood to the brain (see Fig. 2.9), and the inflow boundary condition forthe whole network is provided by the heart. Peripheral circulation is accounted for bya three elements Windkessel model coupled to each outflow of the network [2].

In our model the network is represented by an oriented graph (see Chap. 6). Theedges of the graph correspond to the vessels, while the nodes are the junctions. Eachedge is described by system (2.1) where appropriate initial conditions are assumed.The junctions are modelled by prescribing balance equations for the mass and the totalpressure Pt = P + 1/2(

QA)

2 (see [45,82] and Sec. 2.3).The flow rate boundary condition prescribed on the left boundary of the vessel rep-

resenting the aortic arch is a periodic function of time, with period T = 1s [3]:

Q(t) =

485 sin(2T t) 0 t < s

0 s t < T

with s = 0.3 s. The simulation was run with dx = 0.5 cm and dt = 104 s.

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Figure 2.9: 1D model of the circle of Willis: embedding into a larger arterial network(from [3], courtesy of Dr. J. Alastruey). The name and the characteristics of thenumbered vessels in figure are reported in Fig. 2.10.

In Fig. 2.11 we illustrate a snapshot of the solution in the brachial artery, comparingthe pressure waveforms obtained by the elastic and viscoelastic models for the vesselwall. The viscoelastic modulus = 104 dyn s cm3 is taken equal in all the vessels (itis in the range of values proposed in [19] for the femoral arteries, and is assumed here

as a reference value for medium-size vessels). It can be seen that the wave propagationspeed is increased in the viscoelastic wall, as we alr

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