computational hemodynamics of cerebral vasculature

POLITECNICO DI MILANODipartimento di Matematica F. Brioschi

Ph. D. course in Mathematical EngineeringXXI cycle

Computational hemodynamics of thecerebral circulation: multiscale modeling

from the circle of Willis to cerebralaneurysms

Ph. D. candidate:Tiziano PASSERINI

Milano, 2009

Supervisor: Prof. Alessandro VENEZIANI

To Lucia

Contents

Abstract 1

1 Introduction 31.1 Anatomy and physiology of the cerebral circulation . . . . . . . . . . . . 3

1.1.1 The circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Morphology and fluid dynamics of cerebral aneurysms . . . . . . . . . . 6

1.2.1 The role of hemodynamics . . . . . . . . . . . . . . . . . . . . . . 101.3 Modeling the cerebral circulation . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 The circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Cerebral aneurysms . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 One-dimensional models for blood flow problems 152.1 Wave propagation phenomena in the cardiovascular system . . . . . . . 15

2.1.1 Modeling the vascular wall . . . . . . . . . . . . . . . . . . . . . . 162.2 Formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 A viscoelastic structural model for the vessel wall . . . . . . . . . 192.2.2 The linearized model . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Networks of 1D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Numerical discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 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 393.1 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 . . . . . . . . . . . . . . . 533.5.2 Relating surface points to centerlines . . . . . . . . . . . . . . . . 54

4 An application of three-dimensional modeling 594.1 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 775.1 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 946.1 An introductory note on C++ . . . . . . . . . . . . . . . . . . . . . . . . . 946.2 LifeV: a C++ finite element library . . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 Code features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3 Implementation of networks of 1D models . . . . . . . . . . . . . . . . . 97

6.3.1 Building the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.3 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Conclusions 109

Acknowledgements 111

ii

Abstract

In this work we address the mathematical and numerical modeling of cerebral 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 flow can 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 hasbeen conducted on the geometrical features of the internal carotid artery, finding thatcertain spatial patterns of radius and curvature are associated to the presence and tothe 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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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 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 ofbrain 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 layercalled the basal lamina (inner part of the black circle). The outer part of the black circlerepresents 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 of collagen 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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1 Introduction

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 andblood 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, deliveringblood 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 the pial arteries. The lattercourse 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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http://www.brain-aneurysm.com/

1 Introduction

Vertebralartery

Basilarartery

Pontinearteries

Anteriorspinalartery

Posteriorinferiorcerebellarartery

Anteriorinferiorcerebellarartery

Superiorcerebellarartery

Posteriorcerebralartery

Anteriorcerebralartery

Internalcarotidartery

Posteriorcommunicating

artery

Middlecerebralartery

Anteriorcommunicating

artery

Anteriorchoroidalartery

Ophthalmicartery

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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http://www.wikipedia.org

http://www.wikipedia.org

1 Introduction

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 word aneÔrisma, meaning widening), is a sac-likestructure which forms where the blood vessel wall weakens, ballooning outwards (seeFig. 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 Small

11 - 15 mm Large20 - 24 mm Near-giant> 25 mm Giant

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1 Introduction

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 formationHypertensionPrevious 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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1 Introduction

aneurysm large size. In this case, symptoms include chronic headaches, nausea, loss offunctions in the nerve bundles in the brain causing disturbs such as double vision. Inother cases, aneurysms may be found simply by chance.

Rupture

The rupture of an aneurysm is an event which can cause a stroke, the so-called brainattack, with effects comparable to those of the blockage of a blood vessel. Some an-eurysms, prior to the rupture, tear a little and release a small amount of blood: thisevent is referred to as a warning leak. The bleed occurring after the rupture is known assubaracnoid hemorrage (SAH).

In case the flow slows down in the aneurysm lumen, a thrombus may form, eitherbefore or after the rupture. Thrombosis can stop the bleeding after a rupture, but mayalso cause additional stresses to be exerted on the wall, by transmitting the blood pul-sation through the mass of the clot. Moreover, the thrombus may host small channelsof blood (recanalization), which can be associated to the growth, rupture and reruptureof the aneurysm.

On the other hand, the rupture itself may cut off the supply of oxygen and othernutrients to the cells in the wall, thus further weakening it and predisposing it to asubsequent rupture.

Complications

Rehemorrage is one of most frequent and severe complications of cerebral aneurysms.Multiple SAHs may occur from the same aneurysm, especially in patients sufferingfrom hypertension, since the walls are weakened after the initial rupture. Moreover,the risk of rebleeding increases with time, therefore an early treatment is mostly impor-tant for the patient outcome. Another feared complication is vasospasm, a temporaryovercontraction of cerebral arteries which can result in a stroke. It may be triggered bya SAH, due to the presence of blood in the subarachnoid space, and can last few days tothree weeks. Less frequent or severe complications include hydrocefalus, seizures, cardiacstunning and sodium and fluid imbalance [65].

Detection

Cerebral angiography is frequently used to detect brain aneurysms. One of its maindisadvantages is invasivity, since it requires the femoral artery to be punctured and acatheter to be inserted and navigated through the arterial tree to inject an opaque dyenear to the observed region. Radiographs are taken while the dye is advected by theblood flow. This technique can show the course of arteries, their pattern of communi-cation, their length and diameter and the presence of abnormalities such as aneurysms.However, in presence of a clot it may not show the real extent of the aneurysm. More-over, large areas of relative stagnation can cause the concentration of the dye in theseregions to be low leading ultimately to undersegmentation.

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1 Introduction

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 toperioperatory risks associated to the chosen procedures.

The general criterium associating a risk of rupture to aneurysms based on their size isnot practically accepted, since it is believed that each brain aneurysm should be evalu-ated on an individual basis, with consideration of patient’s 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 recommendedas 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 thebrain 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 Introduction

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 of structural 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 alignedin 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 promotethe 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 ruptureof 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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1 Introduction

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 elastinand 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 of low 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 of high 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 Introduction

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 threeclasses 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 equationsand 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 evidentin 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

12

1 Introduction

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, andis 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 ring per 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 ofthe 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).

∗http://www.lifev.org

13

http://www.lifev.org

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.

†http://www2.mate.polimi.it:9080/aneurisk

14

http://www2.mate.polimi.it:9080/aneurisk

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 thecirculatory 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 abilityof 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 describedby 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 cardiovascularsystem

The circulatory system is responsible for the distribution of blood flow through thehuman body. Blood is pumped by the heart into the network of arteries, reaches thecapillaries where most of the biochemical phenomena associated to the tissue nutritiontake place, and is finally collected by the network of veins 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 heart’s ventricles, and a late phase (diastole), in which blood motion

15

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 ofthe 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 abetter insight on the behaviour of the wall under the effect of stresses exerted by the

16


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 constitutivelaw 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 canbe 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 simplerKelvin-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 bloodflow 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 Ω ⊂ 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 areaof S, P (x, t) is the mean pressure on S and Q(x, t) is the fluid velocity flux through S.For all x ∈ Ω 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

17


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

equations [40]:∂A

∂t+∂Q

∂x= 0 (2.1a)

∂Q

∂t+ α

∂

∂x

(Q2

A

)+A

ρ

∂P

∂x+KR

Q

A= 0 (2.1b)

In (2.1), α is the so-called momentum-flux correction (or Coriolis) coefficient, ρ 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 ψ 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 = ψ(A(x, t);x, t). (2.2)

We may define ψ in a simple yet rather general way, as a function of A (together withits derivatives) and of a set of parameters which may depend on x, t or A. Under thehypothesis that the pressure depends onA, on the reference cross-sectional areaA0 andon parameters β = (β0, β1, . . . , βp) describing the mechanical properties of the wall, apossible choice for ψ is:

ψ(A(x, t);A0(x), β0(x), β1(x)) = β0(x)

[(A

A0(x)

)β1(x)

− 1

]. (2.3)

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

18


In the previous, β0 is an elastic coefficient, while β1 > 0 is normally obtained byfitting the stress-strain response curves obtained by experiments. Whenever β1 = 1

2

and β0 = 1√A0β = 1√

A0

√πh0E1−ξ2 , (2.3) is equivalent to the following relation:

ψ(A(x, t);A0, β) = β

√A−

√A0

A0, (2.4)

which is derived from the linear elastic law for the wall mechanics of a cylindricalvessel and where E(x) is the Young’s modulus, h0 the wall thickness and ξ 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

αQ2

A+ C1

, B(U) =

0

KRQ

A+A

ρ

∂ψ

∂x− ∂C1

∂x

.

We denote by C1 the following quantity

C1 =∫ A

A0

c21dτ , c1 =

√A

ρ

∂ψ

∂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 = aη + b∂2η

∂t2+ c

∂4η

∂x4− d

∂2η

∂x2+ g

(∂η

∂t

)(2.6)

19


where

η = R−R0 =√A−

√A0√

π

is the wall radial displacement, a, b, c, d are positive coefficients and g(∂η

∂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 aη, while theinertial effects are described by the term involving the second order time derivative ofthe wall displacement, where b = ρwh 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 viscoelasticnature. 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

(∂η

∂t

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

the following formulation

g

(∂η

∂t

)= −e ∂3η

∂t∂2x,

involving a third order mixed derivative of η, 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 η:

P − Pext = aη + γ∂η

∂t, (2.7)

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

√A−

√A0√

π, and noting that typically in hemodynamic problems the

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

∂η

∂t=

12√πA

∂A

∂t' 1

2√πA0

∂A

∂t.

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

A0

√π.

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

P − Pext = ψ(A(x, t);A0, a) + γ∂A

∂t(2.8)

20


with γ = γ2√

πA0. Therefore, assuming Pext independent of x,

∂P

∂x=∂ψ

∂A

∂A

∂x+

∂ψ

∂A0

dA0

dx+∂ψ

∂a

da

dx+ γ

∂2A

∂x∂t,

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

Substitution of the previous in (2.1b) gives:

∂Q

∂t+ α

∂

∂x

(Q2

A

)+A

ρ

∂ψ

∂A

∂A

∂x− γ

∂2Q

∂x2+KR

Q

A+A

ρ

∂ψ

∂A0

dA0

dx+A

ρ

∂ψ

∂a

da

dx= 0 .

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

F =[F1

F2

], F1 = Q , F2 = α

Q2

A+ C1

and, analogously,

B =[B1

B2

], B1 = 0 , B2 = KR

Q

A+A

ρ

∂ψ

∂A0

dA0

dx+A

ρ

∂ψ

∂a

da

dx− ∂C1

∂x

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

∂A

∂t+∂Q

∂x= 0 (2.9a)

∂Q

∂t+∂F2

∂x− A

ργ∂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 ψ 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 α∂

∂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

ρ

∂P

∂x+KR

A0Q = 0 . (2.10b)

21


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

∂P

∂x=∂ψ

∂A

∂A

∂x+

∂ψ

∂A0

dA0

dx+∂ψ

∂a

da

dx.

Now (2.10b) becomes

∂Q

∂t+A0

ρ

∂ψ

∂A

∂A

∂x+KR

A0Q+

A0

ρ

∂ψ

∂A0

dA0

dx+A0

ρ

∂ψ

∂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

ρ

∂ψ

∂A0

, SL =

0

KR

A0Q+

A0

ρ

∂ψ

∂A0

dA0

dx+A0

ρ

∂ψ

∂a

da

dx

.

A conservative form reads:

∂U∂t

+∂FL

∂x+ BL = 0 , (2.12)

where

FL =[QCL

], BL = SL −

[0

∂CL

∂x

],

and CL =∫ AA0c2Ldτ .

System (2.11) is said to be strictly hyperbolic if H is similar to a diagonal matrixand its eigenvalues are real and distinct. In particular, the eigenvalues of matrix HL

are λ1,2 = ±cL, cL =√A0

ρ

∂ψ

∂Abeing the wave celerity in the linearized problem. A

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

∂A> 0,

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

∂ψ

∂A=

a

2√πA

.

Linearization of the previous relation yields

∂ψ

∂A' a

2√πA0

= a .

22


In the following we set cL =√A0

ρa.

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

L =[lT1lT2

], R =

[r1 r2

],

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

LHLR = Λ = diag(λ1, λ2) .

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

L∂U∂t

+ ΛL∂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

+ Λ∂W∂x

+ GL = 0 , (2.14)

whereGL = 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 = ζ

[±cL1

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

∂W1

∂A= ζcL ,

∂W1

∂Q= ζ ,

∂W2

∂A= −ζcL ,

∂W2

∂Q= ζ .

We now impose the integrability of the two differential formsW1 andW2 by choosingζ such that

∂2Wi

∂A∂Q=

∂2Wi

∂Q∂A, i = 1, 2 ,

which yields

±cL∂ζ

∂Q=∂ζ

∂A

23


thus we may simply choose ζ = 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 = ±cL∂A+ ∂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

Let’s now consider the viscoelastic term γ > 0 in (2.8): this yields

∂P

∂x= a

∂A

∂x+ γ

∂2A

∂x∂t+

∂ψ

∂A0

dA0

dx+∂ψ

∂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+∂FL2

∂x− A0

ργ∂2Q

∂x2+BL2 = 0 , (2.15b)

with

FL =[FL1

FL2

], FL1 = Q , FL2 = CL

and

BL =[BL1

BL2

], BL1 = 0 , BL2 =

KR

A0Q+

A0

ρ

∂ψ

∂A0

dA0

dx+A0

ρ

∂ψ

∂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 of harmonic waves:

A(x, t) = A(k) exp[i(ω(k)t− kx

)](2.16a)

Q(x, t) = Q(k) exp[i(ω(k)t− kx

)]. (2.16b)

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

24


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

)exp[i(ω(k)t− kx

)]= 0 (2.17a)((

−iC1k)A+

(iω + k2C2 + C3

)Q)

exp[i(ω(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

ρ, C2 =

A0γ

ρ, C3 =

KR

A0. (2.18)

Moreover, we omit to indicate explicitly the dependency of A 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:

iωA− ikQ = 0 (2.19a)

(−iC1k)A+ (iω + k2C2 + C3)Q = 0 , (2.19b)

which yields the following condition:

ω(iC3 − ω) + k2(C1 + iC2ω) = 0 . (2.20)

We can now study the dispersion relation ω(k), linking the angular frequency to thewave number: solutions to system (2.17) are travelling waves with angular frequency

ω1,2(k) =i(C2k

2 + C3

)±√− (C2k2 + C3)

2 + 4C1k2

2.

For the problem at hand, the phase velocity cp = ω(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, givenby

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 ω:

k(ω) = ±

√ω(ω − iC3)C1 + iC2ω

, (2.21)

and we note that k is in general a complex number even when ω 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

iω(k)A = ikQ ,

25


which implies that, if ω ∈ 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

]exp[i(ω(k)t−<(k)x

)]Q(x, t) =

(<(Q(k)

)+ i=

(Q(k)

))exp[=(k)x

]exp[i(ω(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(ω(k)t−<(k)x

)(2.24a)

<(Q(x, t)

)= exp

[=(k)x

](<(Q) cos

(ωt−<(k)x

)−=(Q) sin

(ωt−<(k)x

)). (2.24b)

It follows from the previous that the imaginary part of the wave number is associatedto 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 of k. When C1 6= 0 and C2 = C3 = 0,

k(ω) = ±

√ω2

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).

If C1, C3 6= 0, the definition of k reads

k(ω) = ±

√ω(ω − 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(ω) =ω2

C1− i

C3ω

C1,

which in polar notation reads

k2(ω) = r exp(i(θ + 2nπ)

), n ∈ N ,

r =

√(ω2

C1

)2

+(C3ω

C1

)2

,

θ = arctan(−C3

ω

).

26


Therefore

k(ω) =√r exp

(i

(θ

2+ nπ

)), n ∈ N ,

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

θ

2+ nπ , n ∈ N ∈ [−π/4, 0]

orθ

2+ nπ , n ∈ N ∈ [3/4 π, π] ,

the latter being excluded by the request <(k) > 0. This ensures that =(k) < 0 and there-fore the term C3 6= 0 is responsible for an exponential damping of the signal amplitude.

In the general case C1 6= 0, C2 6= 0, C3 6= 0, recalling (2.21),

k2(ω) =ω(ω − iC3)(C1 − iC2ω)

C21 + C2

2ω2

=ω2(C1 − C2C3)C2

1 + C2ω− i

ω(C1C3 + C2ω2)

C21 + C2ω

.

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

A0β

2> γKR ,

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 wavesamplitudes.

2.3 Networks of 1D models

Once having set up the model for a single tube, we can move towards the study of net-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.

Let’s 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 = ΓPt,1 = Pt,2

(2.25)

where Pt = P + 12%(

QA )2 stands for the total pressure and the subscripts 1, 2 indicate that

the subscripted quantity is relative to tube 1 or 2, respectively.

27


x = Γ

x

x = L

Ω1 Ω2

x = 0

Figure 2.3: Two connected tubes Ω1 and Ω2: the interface is located at coordinate x = Γ.

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 inblood 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 accountthe fact that the total pressure decreases as a function of the flow rate, along the flowdirection, in correspondence with the interface Γ. The second of (2.25) may then thenbe written as follows [45]:

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

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 atypical 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

[Sn

U

∂Fn

∂x+

∂

∂x

(Hn∂F

n

∂x

)]−∆t

(Sn +

∆t2

SnUSn

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

28


where SnU =

∂S∂U

(Un) and Hn, Sn and Fn are defined in a similar way. The Galerkinfinite element method is applied to (2.26), yielding

(Un+1

h , ϕh

)= (Un

h, ϕh) + ∆t(FLW (Un

h),∂ϕh

∂x

)+

∆t2

2

(SU(Un

h)∂F(Un

h)∂x

, ϕh

)− ∆t2

2

(H(Un

h)∂F∂x

(Unh),

∂ϕh

∂x

)−∆t (SLW (Un

h), ϕh) , ∀ϕh ∈ V0h. (2.27)

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

SLW = 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)

∂Qe

∂t+∂F2(A,Q)

∂x= B2(A,Q) (2.28b)

∂Qv

∂t− Aγ

%

∂2Q

∂x2= 0 , (2.28c)

where Q is decomposed into two contributions

Q = 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 of Qe,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+1

e as functions of An 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+1

h and (Qe)n+1h , find (Qv)h ∈ V 0

h

such that (1

An+1h

(Qv)n+1h , ψh

)+ ∆t

γ

%

(∂Qn+1

h

∂x,∂ψh

∂x

)= 0, ∀ψh ∈ V 0

h ,

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

29


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 +Qn+1

v , we can write

(1

An+1h

(Qv)n+1h , ψh

)+ ∆t

γ

%

(∂(Qv)n+1

h

∂x,∂ψh

∂x

)=

−∆tγ

%

(∂(Qe)n+1

h

∂x,∂ψh

∂x

), ∀ψh ∈ V 0

h . (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 ofblood vessels in vivo.

Finally, a model for the circle of Willis is presented, based on the published work byAlastruey 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 ω, A ∈ R. We will assume ω = 2π s−1, 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 γ = 6· 104 dyn s cm−3 is taken from [19] and corresponds to the estimated viscous modulusof a human femoral artery.

30


Name Symbol Value Measurement unitlength L 1000 cmradius R0 1 cmthickness h 0.15 cmmass density ρ 1.05 g cm−3

Poisson modulus ξ 0.5 -Young’s modulus E 4 · 106 dyn cm−2

viscoelastic modulus [28] γ 6 · 104 dyn s cm−3

friction 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 g−1 , C2 = 2.8571 104 dyn cm s g−1 , C3 = 0.83811 g cm−2 s−1

andk(2π) = <(k) + i=(k) = 0.0093286− i0.0027481 .

The initial conditions for the problem are

A(x, 0) = A0 = π

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) = π, we findthat

Q =ω

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

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

Q(xb, t) = exp[=(k)xb

](<(Q) cos

(ωt−<(k)xb

)−=(Q) sin

(ωt−<(k)xb

)), xb = 0, 10 m .

The wave propagation is simulated on the time interval t ∈ [0, 1] seconds, with atime step of dt = 10−4 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 −Qexact‖L2(0,T ;L2(Ω))

‖Qexact‖L2(0,T ;L2(Ω))

= 1.2122 · 10−4 ,

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

31


0 2 4 6 8 10−2000

−1500

−1000

−500

0

500

1000

1500

2000

x

Q

t = 0t = 0.4t = 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 showssnapshots 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 10−1000

−500

0

500

1000

1500

2000

x

Q

analytical solutioncomputed 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 δ(t) located at t = t0(i. e. δ(t0) = 1 and δ(t) = 0 for t 6= 0). The unit pulse waveform was used in [145] totrack the multiple transmissions and reflections in the arterial system, while its Gaus-

32


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− t0τ

)],

with τ = 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 of dt = 10−4 s. The mesh size is dx = 0.5 cm.

The results of the numerical experiment of propagation are shown in Fig. 2.6. Whenconsidering 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 canbe seen in the linearized model (green line).

If we model blood as a viscous fluid and the wall as a series of viscoelastic rings (seeeq. (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 1−0.2

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)

33


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 proposedby 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 cm−5 , C = 6.3 · 10−4 cm5 GPa−1. The initial conditions for thevessel are A = A0 = π 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(2π

T t) 0 ≤ t < τs

0 τs ≤ t < T

with τs = 0.3 s. The simulation was run with dx = 0.5 cm and dt = 10−4 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.

34


(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%(Q

A )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(2π

T t) 0 ≤ t < τs

0 τs ≤ t < T

with τs = 0.3 s. The simulation was run with dx = 0.5 cm and dt = 10−4 s.

35


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 γ = 104 dyn s cm−3 is taken equal in all the vessels (itis in the range of values proposed in [19] for the femoral arteries, and is assumed hereas a reference value for medium-size vessels). It can be seen that the wave propagationspeed is increased in the viscoelastic wall, as we already noticed in previous experi-ments. Moreover, the dicrotic notch is more evident: indeed, faster dynamics of thewall, as found in systole and in particular at the very end of the systole, are associatedto a more significant contribution of the viscoelastic term to the overall pressure (see(2.8)). The amplitude of the notch can be quantified by considering pressure PA at thesystolic peak tA and pressure PB at tB , as shown in Fig. 2.11. These values, in the two

36


Figure 2.10: Physiological data used in the model shown in Fig. 2.9 (from [3]).

different models, are shown in the following table:

Elastic wall Viscoelastic walltB − tA 0.06 s 0.078 sPA − PB 7052 dyn cm−2 8826 dyn cm−2

We remark that the qualitative pressure waveform is not significantly affected bythe wall viscoelasticity, and is comparable to the results presented in [3]. Deeper andmore extensive studies are needed to quantify the role of wall viscoelasticity in specificvascular districts or under particular conditions, in which the elastic approximationmay not be adequate for the correct description of the wave propagation phenomena.

37


0 0.5 0.75 1t_A t_B

120.000

140.000

160.000

P_A

P_B

t

P

elastic wallviscoelastic wall

Figure 2.11: Pressure wave (in dyn/cm2) in the middle of the brachial artery, after 10cardiac cycles. In blue, the solution obtained with an elastic model for thewall mechanics; in red, the solution obtained with a viscoelastic model.X-axis: time (in s)

38

3 Three-dimensional models for bloodflow problems

In this Chapter we present an overview of the methods and the models that we usedfor the study of blood flow in three-dimensional vascular geometries. In particular, thedynamics of the velocity and pressure fields for an incompressible Newtonian fluid isdescribed by the Navier-Stokes equations, presented in Sec. 3.3. When the velocity andpressure fields of a fluid are known, derived quantities such as the stress field can beobtained. In hemodynamic problems, the wall shear stress has particular importance: theforce per unit area exerted by the fluid tangentially to the wall is relevant in relation tosome vascular diseases, due to the reaction it induces on endothelial cells. This specifictopic is discussed more carefully in Sec. 3.4.

In general, we are interested in the analysis of flow features in specific locations ofthe physical system at hand: in Sec. 3.5 we present a method to select regions of interestin the computational domain.

3.1 Blood flow features in arteries

One of the most evident features of blood flow in the arteries is pulsatility due to thepumping action of the heart [92]. In particular, most locations in the arterial tree experi-ence unsteady, pulsatile flow, which can induce flow reversals and recirculation near tothe wall, with potentially pathogenic effect (such as, for instance, atherogenesis). Thetime pattern of pulse waves in the circulatory system, however, is not perfectly peri-odic, because it is adjusted in order to fit the body blood demand. The approximationto a periodic phenomenon therefore, usually accepted in hemodynamic computations,may hold only for short periods of time, in which the overall physical conditions areessentially not changing.

The driving mechanism for the pulse wave propagation in the circulatory systemis the mechanical interaction of the flowing blood and the vascular structure: in thisrespect, the compliance of the vessel wall plays a crucial role. However, an accuratecomputation of the fluid structure interaction problem in hemodynamics is costly, dueto the strong mechanical coupling of the two systems (fluid and structure). A review ofthe available techniques for the study of this problem is presented in [48].

It is worth observing that, in most part of the cardiovascular system, the movementof the vascular wall is relatively small compared to the vessel diameter. The radiuschanges may be at most of the order of 15% in larger arteries, and are smaller in theperipheral arteries. Depending on which district we are interested in, the assumption

39

3 Three-dimensional models for blood flow problems

of vascular fixed geometry may be reasonable, and allow to capture the main charac-teristics of the flow in a by far smaller amount of computational time, with respect tomoving geometry models. On the other hand, compliance of the vascular wall is oftenneglected also due to the lack of measured data for specific subjects and the unavail-ability of a reliable mechanical characterization of the vessel structure.

It has to be noted that rigid wall models may be less precise in specific applicationswith respect to compliant models [48, 100]. Recent works [35, 94] show that, in the caseof the cerebral vasculature, hemodynamic computations based on rigid models repro-duce the same flow patterns as compliant models, the latter giving slightly differentestimates of the flow quantities. Therefore, the introduction of the vascular complianceseems not to be required in order to understand the main features of cerebral bloodflow. This evidence, and the lower computational cost, motivated us to adopt rigidgeometry models. Moreover, even in rigid wall simulations part of the compliance isaccounted for by the shape of the pulsatile waveform, since it is the compliance of thesystem which leads to different pulsatile waveforms.

The blood flow regime is usually laminar in most part of the cardiac cycle. In sys-tole, however, the flow may become unstable in specific vascular districts, due to manydifferent reasons: in the aortic arch, for instance, the systolic peak velocity can be veryhigh even in physiological conditions. The presence of vessel stenosis or an increasedblood demand from the organs due to physical exercise may promote in certain loca-tions the transition of the flow regime towards instability. Some of the parameters usedfor the characterization of the flow regime and their physical significance are discussedin the following section.

3.2 Geometry and Flow

Generally speaking, the purpose of a model is to reproduce the features of a physicalsystem. More precisely, the model should be similar to the system from a geometricaland dynamical point of view (see [37]).

Geometric similarity is provided by an accurate morphologic description of the sys-tem. In the case of vascular geometries, this depends on the quality of medical imagesat hand and on the effectiveness of the segmentation process and geometry reconstruc-tion tools.

Dynamic similarity is guaranteed by the identification and the control of the so-calledsimilarity parameters, i. e. dimensionless parameters whose values in the model shouldmatch those in the real case. In the case of a fluid flow in curved pipes, importantparameters are the Reynolds and Dean numbers, together with the Womersley num-ber and/or Reduced Velocity in the case of unsteady flows. We recall hereafter theirdefinition (see e. g. [37]).

40


3.2.1 Reynolds number

The Reynolds number Rea in an internal flow of mean sectional velocity W within apipe or vessel of characteristic radius a is given by

Rea =aW

ν(3.1)

where ν is the kinematic viscosity of the Newtonian fluid, while suffix a here indicatesthe reference length used in the definition of the Reynolds number [114]. A differentpossible choice is to use the pipe diameter d = 2a, thus obtaining a Reynolds numberRed = 2Rea.

This parameter can be physically thought of as the ratio of inertial forces to viscousforces, as is made more evident by rearranging terms in equation (3.1):

Rea =%W

2

µW/a,

where %W 2 =(%W

)W can be interpreted as the flux of the momentum over the pipe,

while µW/a is an estimate of the wall shear stress, where µ is the dynamic viscos-ity. When we have a large Reynolds number, inertial forces are dominant over viscousforces and vice versa. This makes Reynolds number the key parameter which identifiesthe transition of the flow to turbulence and therefore helps in determining flow stability.Moreover, flows with higher Rea are characterised by a greater persistence of geometricinfluences downstream of a bend or other disturbances [37].

Typical values for Re in the human cardiovascular system range from few thousands(in bigger arteries such as aorta, iliac arteries, brachial arteries and in bigger veins)to less than 1000 in medium-size vessels (such as carotid arteries, the main coronaryarteries and medium-size veins) and even less than 1 (arterioles, capillaries, venules)[48].

3.2.2 Dean number

The original form of the Dean number was defined by Dean [34]:

K = 2( aR

)(aWν

)2

(3.2)

whereR is the radius of curvature (see Fig. 3.1), a is the pipe radius andW is defined asa constant having the dimensions of a velocity. According to Berger et al. [15], who pro-posed an extensive review of the literature about flow in curved pipes, the preferabledefinition for Dean number is however the following:

κ = 2δ1/2Rea =( aR

)1/2 2aWν

(3.3)

41


(a) (b)

Figure 3.1: (a) A torus (from www.wikipedia.org). (b) Toroidal coordinate system.The pipe is a cylinder with circular cross section of radius a. Its axis is acircle of radius R, centered in O and belonging to a plane normal to z axis.A point P inside the pipe is identified by the coordinates r′, α, s′: the firsttwo are polar coordinates defined on the cross section which P belongs to;s′ is the position of that cross section along the pipe axis, which can be recastin terms of the angle θ defined in the plane containing the torus axis. Thefluid velocity in the pipe has three components: u′ and v′ are the radial andcircumferential velocity (respectively) in the cross section, while w′ is theaxial velocity. This set up was exploited by Berger et al. in their analysis offlow in curved pipes. [15]

where δ = aR . This definition is in fact based on W , the mean axial velocity, which has

the advantage of being readily measured in most cases. If we takeW = W in (3.2), thenκ = (2K)1/2.

Other definitions for Dean number may be based on pressure gradient [15]. Moreprecisely, in the case of fully developed flow, the following has been widely used inliterature:

D =(

2a3

ν2R

)1/2G a2

µ= 4

(2aR

)1/2 G a3

4µν(3.4)

which is related to (3.2) by D = 4K1/2. Here G represents the (constant) axial pressuregradient.

This approach however has two main disadvantages: on the one hand, it is more dif-ficult to measure the pressure gradient than the mean axial velocity; on the other hand,when the flow is not fully developed, the pressure gradient is generally not constant (itvaries with axial location and position in the cross section): this is true even in pipes ofgiven cross section and fixed flow conditions, where instead W is constant.

A physical interpretation for the Dean number can be provided in terms of the bal-

42

www.wikipedia.org


ance of the forces due to inertia and centripetal acceleration versus the viscous forces:

κ = 2δ1/2Rea = 2

√%Ra

(aWR

)2× %W

2

µWa

≈√

centripetal forces× inertial forcesviscous forces

In the above, we note that aWR is a measure of the angular velocity, thus %R

a

(aWR

)2is an

approximation of the force producing the centripetal acceleration.If we consider helical pipes, having a torsion, additional similarity parameters may

be introduced such as the Germano number:

Gn = (D/2)τRed

where τ is a measure of the torsion; or combinations of Gn and κ [37].

3.2.3 Womersley number and Reduced Velocity/Strouhal number

Figure 3.2: Representation of the laminar boundary layer (in blue) in a viscous fluidmotion. The boundary layer thickness δ grows over time due to the actionof viscosity. (after [37])

The (Sexl-)Womersley number [123, 147] can be physically interpreted as the ratio ofthe pipe diameter to the laminar boundary layer growth over the pulse period T :

Wo = a

√2πνT

∝ d√νT

where d = 2a is the pipe diameter and we exploit a dimensional argument commonlyused in laminar boundary growth over flat plates, namely the fact that the boundarylayer growth (due to viscous forces) is proportional to

√νT [37] (see Fig. 3.2).

The Womersley number is used in the definition of an exact solution for the motionof a Newtonian fluid in a straight circular pipe subject to a periodic pressure differ-ence [147]. In fully developed flow, the solution is periodic with only the velocity axialcomponent uz different from zero. If Wo is small (1 or less), the frequency of pulsations

43


is sufficiently low so that a parabolic velocity profile has time to develop during eachcycle. Conversely, if Wo > 10 the velocity profile is relatively flat.

Larger arteries and, less markedly, larger veins feature high Womersley numbers(Wo = 5 ∼ 10), whereas in smaller vessels Wo ≈ O(1). Indeed, in larger arteries theviscous layer has not enough time to grow to such an extent to dominate the solution(as is conversely seen in Poiseuille flow), and the velocity profile tends to be flat ratherthan parabolic [48].

A potential limitation of Wo is that it is related to physical scales within a given crosssection of the pipe: in other words, the role of the longitudinal geometry is not repre-sented, and this may reduce the physical significance of the parameter when the flowhas significant changes in the streamwise direction. Other non-dimensional similarityparameters may be introduced in order to take into account different length scales aswell, which are relevant in specific flow problems (for instance the flow in multiplebends or through stenoses).

The Reduced Velocity can be useful to describe different flow regimes, and can beseen as a non-dimensionalised pulsatile period:

Ured =W T

D≈ Distance travelled by mean flows

Diameter

In particular, Ured is a more appropriate parameter than Wo for unsteady flows wherethere are significant streamwise flow variations, since it explicitly involves an axiallength scale.

Note also that Ured and Wo are related through Red:

Ured =π

2Red

Wo2 .

3.3 The Navier-Stokes equations

In general, mainly due to the presence of red cells, blood exhibits a complex rheology.However in larger vessels it can be approximated to a homogeneous Newtonian fluid[48]. In this case, the flow is governed by the classical Navier-Stokes equations.

3.3.1 Formulation

Let I ⊂ R be a time interval and let B ⊂ E be the spatial domain representing a bloodvessel (with E a three-dimensional euclidean space) (see Fig. 3.3). A regular motion ofpoints belonging to B is a function x : B × I → E such that x ∈ C2(I): in particular wemay define the material velocity of points of B as the function x : B × I → R3

x(p, t) =∂x(p, t)∂t

.

Denoting Bt := x(B, t), we may now introduce the trajectory

T := (q, t) : t ∈ I, q ∈ Bt

44


Γout,2

Γin

Γout,1

Γw

Γout,4

Γout,3

Figure 3.3: An example of spatial domain representing a basilar artery with a berry cere-bral aneurysm (geometry from the Aneurisk project). The letters indicate theinflow, outflow and wall boundaries.

and the spatial velocity u : T → R3 such that u(q, t) := x(x−1(q, t), t).We analogously define the fluid mass density %(q, t) as a function of the spatial po-

sition and the time. By exploiting the mass conservation principle we obtain the conti-nuity equation:

%+ %(div(u)) = %,t + div(%u) = 0 , (3.5)

where % is the material derivative of % with respect to t, defined as follows:

% =d

dt%(q(p, t), t) = %,t +∇% · u .

We indicate by •,t the partial derivative with respect to time of the considered variable,the spatial position being fixed.

We recall now that, thanks to the Cauchy theorem, the momentum balance equationreads

%u = b + div(T) ,

where b : T → R3 is a field of volume forces and T is the so-called Cauchy stress tensor:∀ (q, t) ∈ T, ∃ T(q, t) ∈ L(R3), L(R3) being the space of linear tensors over R3.

For Newtonian fluids, T is thought to depend linearly on the velocity gradient L =∇u (and more precisely, according to the objectivity principle, on the symmetric part

45


D = sym(L) and its first invariant trD = div(u)):

T = −P I + λ(div(u))I + 2µD , (3.6)

where P is the fluid pressure, while λ and µ are constant viscosity coefficients depend-ing on the fluid characteristics.

When considering incompressible fluids, for which div(u) = 0, (3.6) becomes

T = −P I + 2µD . (3.7)

We remark also that, under the same assumptions and recalling equation (3.5),

%(x(p, t), t) = %0(p, t), ∀p ∈ B, t ∈ I

where %0(p, t) is the fluid density in the reference configuration.Let’s now focus our attention on a fixed spatial domain Ω which for all the time of

interest is inside the portion of space filled by an incompressible Newtonian fluid, i. e.Ω ∈ T. By applying the momentum conservation principle we recover the well-knownincompressible Navier-Stokes equation:

%0(u,t + (∇u)u) = b−∇P + µ∆u , in Ω , (3.8)

where we exploited the fact that 2div(D) = ∆u +∇(div(u)) and the incompressibilityconstraint.

By introducing ν = µ%0

(kinematic viscosity), P0 = P%0

, b0 = b%0

, (3.8) may be rewrittenin the following form:

u,t + (∇u)u = b0 −∇P0 + ν∆u , (3.9a)

together with the incompressibility constraint

div(u) = 0 . (3.9b)

Non-dimensional form

A fluid flow problem can be characterized by a typical spatial scale l and a characteristicvalue for the fluid velocity modulus u. The choice of these values is arbitrary and ingeneral related to the flow features. In blood flow problems l is generally taken equalto the blood vessel diameter, while u is the average fluid speed.

The Navier-Stokes equations are modified by introducing these non-dimensionalizedquantities:

u∗ =uu, t∗ =

tu

l, x∗ =

xl,

the pressure being non-dimensionalised as follows:

P ∗0 =

P0

u2.

46


Moreover, the time and space derivatives are referred to the non-dimensional variablest∗ and x∗:

∇xP0 =u2

l∇x∗P

∗0 , ∆xu =

u

l2∆x∗u∗ ,

u,t =u2

lu∗,t∗ , (∇xu)u =

u2

l(∇x∗u∗)u∗ .

By substituting the previous definitions in (3.9a), where we set b0 = 0 for the sake ofsimplicity, and after some algebraic calculations, we obtain the following non-dimensionalformulation:

u∗,t∗ + (∇x∗u∗)u∗ = −∇x∗P∗0 +

ν

ul∆x∗u∗ .

Remembering (3.1) we can write

u∗,t∗ + (∇x∗u∗)u∗ = −∇x∗P∗0 +

1Re

∆x∗u∗ , (3.10)

where we identify the Reynolds number Re. Let now l1 and l2 be the characteristicspatial scale of two flows, such that l1 = λl2, λ ∈ R. Moreover, let Re1 and Re2 be theReynolds numbers for the two flows, such that

Re1 =l1u1

ν1=l2u2

ν2= Re2

that is λu1

ν1=

u2

ν2. Then (u∗1, P ∗

0,1) and (u∗2, P ∗0,2) satisfy the same set of differential

equations in non-dimensional form (3.10). Two such flows are termed geometrically anddynamically similar.

Boundary conditions and initial condition

Considering the vascular geometry depicted in Fig. 3.3, we can identify different bound-ary regions, corresponding to the fixed vascular walls Γw and to the artificial inlet andoutlet sections for the fluid.

The latter do not correspond to a physical interface between the fluid and the exterior,but they are introduced in order to separate the region of interest from the remainingpart of the circulatory system. More precisely, we distinguish a proximal section Γin

(that is closer to the heart with respect to the mean blood flow direction) and four distalsections Γout,i, i = 1, . . . , 4. They are also referred to as one inflow and four outflowsections, even if during the cardiac cycle the flow can be exiting the inflow section andentering the outflow sections, due to flow reversal which is likely to happen especiallyin major vessels.

Typically we prescribe a velocity profile at Γin, able to reproduce measured data(when available). We prescribe zero velocity on Γw, corresponding to a no-slip con-dition for the fluid in contact with the wall. Finally, on Γout,i the normal stress T · n isprescribed, n being a vector field defined on Γout,i and aligned to the direction normalto the section.

47


Other possible choices for the prescription of boundary conditions can be exploited,in particular in the context of multiscale modeling [48, 141] (see Chap. 5). In that case,the Navier-Stokes equations describing specific vascular districts are coupled with re-duced models taking into account the remainder of the circulatory system, so that theboundary conditions on the artificial sections come from interface conditions between thedifferent models. An example of this approach is presented in Chap. 5, where we dis-cuss specific issues arising in the coupling of 3D models based on the incompressibleNavier-Stokes equations and 1D models based on the Euler equations.

The initial status of the fluid velocity is prescribed through suitable initial conditions,typically in the form

u(q, t0) = u0(q) , q ∈ Ω ,

with the additional incompressibility constraint div(u0) = 0. The choice of u0 is ar-bitrary (usually u0 = 0), since in general a physically relevant initial condition is notknown in hemodynamic computations. This influences the computed solution u(q, t):provided that the boundary conditions are correct, the dynamics of u(q, t) is recoveredonly after a transient in which the effect of the initial condition fades away. In practice,this transient is commonly assumed to last for two or three heart beats, after which thesolution is dominated by the boundary conditions.

3.3.2 Numerical discretization

In order to compute a numerical solution (υ, ψ) of system (3.9), we carry out a dis-cretization of such equations with respect both to the space and to the time variables.For what concerns the time discretization, a typical approach is based on the finitedifference approximation, that means to split the time interval of interest (0, T ] intosubintervals with time step ∆t, such that tk = k∆t (k ∈ N) and approximate the timederivatives with suitable incremental ratios evaluating the unknowns at the instants tk.In the sequel, we will assume to discretize the equations through a backward Euler timediscretization. For what concerns the space discretization, we refer to the finite elementmethod.

The discretization of the problem according to the finite element method is based onthe Galerkin approximation of (3.9), that we are going to introduce. In the sequel, wedenote by L2(Ω) the space of square integrable scalar functions in Ω, L2 (Ω) the anal-ogous functional space for m−dimensional vector functions, Hm (Ω) the functions be-longing toL2(Ω) together with their firstm spatial derivatives and Hm (Ω) the functionsbelonging to L2 (Ω) together with their first m spatial derivatives. In particular, H1

0 (Ω)denotes the functions belonging to L2 (Ω) together with their first spatial derivativeand whose trace vanishes on ∂Ω. For the sake of simplicity, we will assume Ω smoothenough and homogeneous Dirichlet conditions υ = 0 on ∂Ω for equations (3.9).

Then, the backward Euler-Galerkin approximation of the problem (3.9) reads: for

48


each n ≥ 0, find υn+1h ∈ Vh and ψn+1

h ∈ Qh such that:

m(υn+1

h ,vh

)+ a

(υn+1

h ,vh

)+ g

(υn+1

h ,vh

)+

+b(vh, ψ

n+1h

)=∫

Ωfn+1 · vhdω +m (υn

h,vh) ∀vh ∈ Vh

b(υn+1

h , qh)

= 0 ∀qh ∈ Qh,

(3.11)

where Vh, h > 0 and Qh, h > 0 are families of finite-dimensional subspaces of H10 (Ω)

for the velocity and of L2 \ R for the pressure, respectively, υkh, ψ

kh denote the discrete

velocity and pressure computed at tk and:

m (w,v) ≡ 1∆t

∫Ωw · vdω, a (w,v) = ν

∫Ω∇w : ∇vdω,

b (w, q) = −∫

Ω∇ ·wqdω, g (w,v) =

∫Ω

(∇w)w · vdω.(3.12)

In particular, in the finite element method, we introduce a triangulation Th of the do-main Ω, i. e. a finite decomposition of Ω into tetrahedrons. Here h denotes the max-imum of the diameters of the triangles of Th. Then, assume that Vh is the space ofpiecewise polynomial functions on every element of Th, continuous in Ω and vanishingon the boundary ∂Ω. Similarly, Qh is the space of piecewise polynomial functions onevery element of the decomposition, not necessarily continuous.

Due to the presence of the nonlinear convective term in the momentum equation(3.9a), (3.11) yields the solution of a system of non linear equations when using fullimplicit time-stepping procedures. In this work, we follow a semi-implicit strategy, basedon the approximation:

g(wn+1

h ,vh

)≈∫

Ω(∇wn

h)wn+1h · vdω

However, different strategies can be considered as well (see e.g. [108]). We remark thatthe well posedness of problem (3.11) is ensured when the Ladyzhenskaja-Babuska-Brezzi(LBB) condition, which requires a compatibility between the choice of the polynomialdegrees for the velocity and the one for the pressure, is satisfied [111].

On the basis of the presented formulation, different solution techniques can be de-vised to manage the resulting system of algebraic equations. We mention in particularblock factorization methods, based on the splitting of the problem, for instance generat-ing separate subproblems for the velocity and the pressure [109–111].

A slightly different finite element method for the discretization of the Navier-Stokesproblem has been introduced by Burman et al. [18], consisting in a stabilized Galerkinformulation using equal order interpolation for pressure and velocity and thereforereducing the dimension of the resulting algebraic problem.

49


(a) Domain Ω represents an aneurysmatic Internal CarotidArtery.

(b) Zoom on a part of the domainboundary ∂Ω: the aneurysmal bleb.

Figure 3.4: Visualization of an example of computational domain (a). Particular (b) ofthe domain boundary, with a graphical representation of the normal vectorsto the surface.

3.4 Wall shear stress in the Navier-Stokes problem

Let u represent the fluid velocity in a domain Ω ⊂ E and let n be the normal on ∂Ω(see Fig. 3.4): the stress exerted by the fluid over the boundary of the domain readsσ = −T · n, T being the Cauchy stress tensor (see (3.6)). The tangential component of σis referred to as the wall shear stress, and may be derived as follows:

WSS = σt = σ − (σ · n) n.

Note that σt by definition takes only into account the viscous component of the stress,the pressure being responsible only for a normal stress.

3.4.1 Approximation for the velocity gradient

It follows from the definition of T (see (3.7)) that in order to estimate the stress field(and in particular the wall shear stress) over the computational domain it is necessaryto retrieve a suitable approximation of the velocity gradient. This issue has been exten-sively treated in the literature: we mention in particular several works by Zienkiewicz& Zhu [150–152] proposing a cost effective procedure for the recovery of the gradientof finite element solutions at the nodes. In the following we focus on the L2 projectionmethod [13, 153], which is proven to provide a superconvergent approximation of thegradient on linear elements.

50


Let P be the Navier-Stokes problem defined over Ω. Suppose that we have an ap-proximation uh of the velocity field solution for P , obtained with a Galerkin finite ele-ment method: uh ∈ [Xr

h]d , uh(Ω) ∈ Rd, where Xrh is the space of r-th degree piecewise

polynomial functions on a tessellation Th of Ω.We compute an approximation G(uh) of ∇uh, such that (L2 projection of the gradient

over [Xrh]d×d): ∫

ΩG(uh) : vdω =

∫Ω∇uh : vdω, ∀v ∈ [Xr

h]d×d .

The componentwise formulation reads:∫ΩGkl(uh)vdω =

∫Ω

∂(uh)k

∂xlvdω, ∀v ∈ Xr

h .

In standard Galerkin finite element theory Xrh = span(ϕi, i = 0, . . . , N), N being

the number of degrees of freedom of the problem and ϕi the scalar finite element nodalfunction associated to the i-th degree of freedom. Each component Gkl may then beexpressed as a linear combination of the finite element basis functions:

Gkl(uh)(x) =∑Nj

Gkl(uh)(Nj)ϕj(x)

where Nj is the j-th degree of freedom, x ∈ Ω. It can be seen that the values Gkl(uh)(Nj)are therefore solutions of the following linear system:∑

Nj

(∫Ωϕiϕjdω

)Gkl(uh)(Nj) =

∫Ω

∂(uh)k

∂xlϕidω, i = 0, . . . , N

On the other hand, knowing that (uh)k(x) =∑

Nj(Uj)kϕj(x), we can write∫

Ω

∂(uh)k

∂xlϕidω =

∑Nj

(Uj)k∂ϕj(x)∂xl

The overall problem reduces therefore to the solution of d × d linear systems of theform

Mg = f

where M(i, j) =∫

Ωϕiϕjdω is a mass matrix while

g = Gkl(uh)(Nj) , f = Dl(U)k ,

with Dl(i, j) =∫

Ωϕi∂ϕj(x)∂xl

dω. Having at hand the approximate solution uh, which is

a piecewise polynomial function over the tessellation Th of the computational domain,the right hand side of these systems can be computed exactly.

51


3.4.2 Oscillatory Shear Index

Several studies focused on the effect of wall shear stress on the remodeling mechanismin blood vessels. Not only the mean WSS is associated to anatomic changes of vascularwall: the rate of change of wall shear is believed to play a role in the development ofpathologies such as atherosclerosis [25]. In particular, it has been found that laminarshear stress is atheroprotective for endothelial cells, whereas nonlaminar, disturbed, oroscillatory shear stress correlates with development of atherosclerosis and neointimalhyperplasia [31].

In specific locations of the circulatory system, the blood flow may be reversed dur-ing part of the cardiac cycle: this causes the wall shear stress to significantly vary itsdirection. Ku et al. [68] proposed an oscillatory shear index (OSI) in order to quantifythis effect, and looked for a correlation between vascular wall locations featuring highOSI and the local initiation of atheroma in the human left carotid artery. The originalformulation is the following:

OSI =

∫ T

0|WSS∗|dt∫ T

0|WSS|dt

, (3.13)

where T is the duration of the cardiac cycle, WSS is the wall shear stress vector andWSS∗ is defined as the stress component acting in the opposite direction with respectto the direction of the mean shear stress (in time).

Taylor et al. [138] proposed a similar formulation, which encompasses a strategy toestimate the deviation of WSS from its temporal mean direction:

OSI =12

(1− WSSmean

WSSmag

), (3.14)

where WSSmean is the mean shear stress, defined as the magnitude of the time-averagedstress vector σt (see (3.6)), while WSSmag is the time-averaged magnitude of the stressvector:

WSSmean =∣∣∣∣ 1T∫ T

0σtdt

∣∣∣∣, WSSmag =1T

∫ T

0|σt|dt

According to the latter formulation, OSI is minimum (and equal to 0) if WSS is con-stantly directed along its average direction. OSI is maximum (and equal to 0.5) if themean WSS over the heart beat is = 0.

A different approach has been followed by Steinman et al. [85], who weighted thepositive values of the scalar product of WSS and the mean shear direction n:

n =∫ T

0

WSS|WSS|

dt .

52


The OSI is then defined as follows:

OSI =

∫ T

0|WSS · n|H(WSS · n)dt∫ T

0|WSS · n|dt

, (3.15)

H(x) being the Heaviside unit function. In this formulation, high values for OSI indi-cate that WSS direction is not opposite to the mean shear direction for the most part ofthe cardiac cycle.

3.5 Working on regions of interest

It is usually relevant to quantify the mechanical stress exerted by the blood flow ona vessel wall in locations where it may be associated to the development of vascularpathologies. This is the case of cerebral aneurysms, in which typically we want toquantify the WSS only in the neighbourhood of the aneurysmal sac.

Figure 3.5: Schematic representation of a region of interest in a vessel (in red). Thevessel centerline is also represented.

On the other hand, the unknowns of a fluid dynamics problem are approximated onthe whole computational domain Ω. If we are interested in evaluating some quantitieson specific (sub-)regions of Ω (see Fig. 3.5), we need to define those regions and restrictthere the fluid dynamics analysis. One possible approach is based on the extraction ofthe vessel centerline.

3.5.1 Decomposition of bifurcation branches

Blood vessels can be effectively represented by centerlines, synthetic descriptors of thegeometry and the topology of vascular networks. For the purposes of this work, acenterline is defined as the weighted shortest path traced between the inlet and the

53


outlet of a vessel, the weight being the distance from the surface [6, 7, 103]. More pre-cisely, a centerline is traced on the medial axis of the vascular geometry, that is the locusof the centers of the maximal inscribed spheres in the shape of the vessel. Therefore,each point of the centerline is the center of such a sphere and carries the informationabout its radius. Moreover, a natural parametrization for the centerline is given by theassociated curvilinear abscissa, which ranges over the line and relates each point to itsEuclidean distance from a point chosen as the origin (see Fig. 3.8(a)).

In formal terms, a centerline can then be seen as a parametric curve c(s), s ∈ [0, L]being the curvilinear abscissa and L the centerline arc length. The envelope of themaximal inscribed spheres along the centerline defines a scalar function called tubefunction T , which can be expressed as follows:

T (x) = mins∈[0,L]

|x− c(s)|2 − r2(s)

,

where x ∈ R3 is a point in the Euclidean space and r is the radius of the maximalinscribed sphere whose center is the centerline point c(s). The zero isosurface of T isreferred to as tube or canal surface, and by construction is strictly contained inside thevascular lumen. The function T is negative inside the tube.

The construction of centerlines and tubes is particularly useful in the study of bifur-cating vessels, since it allows the identification and characterization of the bifurcationpoints [7]. For the sake of clarity, we refer to the case depicted in Fig. 3.6. First of all, thecenterlines for the surface at hand are computed, yielding a set of curves running fromthe bifurcation inlet to each outlet. Then, two reference points are defined for each cen-terline: the first is the point in which one centerline crosses the tube surface generatedby the other (and is labelled as point 1 in black ink, in Fig. 3.6); the second is the centerof the nearest upstream sphere touching point 1 (and is labelled as point 2 in black inkin the figure).

These four points and the associated spheres describe the position of the bifurcationand the size and shape of the vessel at the bifurcation point. Moreover, they split thecenterlines into tracts corresponding to single bifurcation branches and to the bifurca-tion center, labelled respectively as tract 1, 3, 4 and 2 in red ink in Fig. 3.6. Each point ofthe space can then be associated to the nearest centerline tract, thus inducing a partitionin regions of influence of the different tracts. These regions of the space cross the vesselsurface in the bifurcation region, decomposing it in branches.

We remark that the same approach can be applied to the study of aneurysms: theaneurysmal bleb may be regarded as a branch with respect to its parent vessel, andcan be therefore decomposed from the vessel surface. An example of this procedureapplied to the geometrical model of a pathological Internal Carotid Artery is depictedin Fig. 3.7.

3.5.2 Relating surface points to centerlines

By splitting each bifurcation in its branches, a simplification of the topology of the vas-cular network is achieved. Each branch is now topologically equivalent to a cylinder,

54


Figure 3.6: Representation of a bifurcating vessel (from http://vmtk.org). Left: cen-terlines. Middle: tubes constructed as envelopes of the maximal inscribedspheres along each centerline. Right: identification of two reference pointsfor each centerline, for the description of the position, size and shape of thebifurcation (in black); decomposition of the centerlines in distinct tracts (inred).

thus its geometrical characterization is easier. For instance, it is possible to robustly gen-erate cross-sections of the vessel along the centerline [69]. More than that, each branchcan be mapped into a rectangular parametric space: this allows the comparison amonggeometrical models in the same parametric space, and has been recently investigatedby Antiga et al. [7].

A curvilinear reference system can be defined on each centerline branch, centered inthe considered bifurcation. Each surface point y can be associated to its nearest cen-terline point cy, by minimizing the tube function T (y). An example of this procedureis depicted in Fig. 3.8. Following the same idea, regions of interest can be identifiedon a computational mesh, automatically selecting the nodes in which to evaluate fluiddynamic variables or derived parameters.

In the context of Aneurisk project this approach was exploited for the evaluationof the spatial average wall shear stress in a specific location of the Internal CarotidArtery. In that particular case, the last tract of the ICA surface was considered, as apreferential site for aneurysm development. Having associated the centerline abscissa

55

http://vmtk.org


(a) Surface representing an internal carotidartery with a berry aneurysm.

(b) Each point of a centerline is the center of themaximal inscribed sphere in the surface and isassociated to its radius.

(c) Decomposition of the surface in bifurcationbranches.

(d) Selection of a region of interest on onebranch.

Figure 3.7: From the geometrical model of a blood vessel to the selection of a region ofinterest. The definition of the vessel centerline allows the decomposition ofthe surface into branches. Undesired branches are excluded from the regionof interest.

to the surface points, the problem was recast into the selection of an abscissa intervalon the centerlines. To this aim, two geometric landmarks were considered. The firstis the center of the main bifurcation of the ICA, the second is the point delimiting thelast ICA bend prior to the bifurcation. Following [119] we define a vascular bend orsiphon as a segment included between two points of approximately zero curvature ofthe centerline. Therefore the selected region Γ comprises the last ICA siphon and thelast few centimeters of the vessel, prior to the bifurcation, and the average value of WSS

56


(a) Curvilinear reference system in a vessel bi-furcation: the origin is placed in the center ofthe main bifurcation (where the aneurysm is lo-cated).

(b) Each point of the surface is associated to acurvilinear abscissa.

(c) A region of interest is selected (in red), byconsidering an interval of curvilinear abscissaand neglecting undesired branches (such as theaneurysmal bleb).

(d) Wall shear stress values in the region of inter-est.

Figure 3.8: Evaluation of wall shear stress on a region of interest selected on the ves-sel surface and corresponding to an interval of curvilinear abscissas on thecenterline. Undesired branches are excluded from the analysis.

over Γ can be defined as follows:

WSS =

∫Γ

WSSdγ∫Γdγ

. (3.16)

57


Figure 3.8 shows the application of this technique to the study of fluid dynamicsinside a realistic ICA geometry. More details on the results of this analysis in the contextof Aneurisk project will be presented in Chap. 4.

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4 An application of three-dimensionalmodeling to the study of cerebralaneurysms

In this Chapter we focus on the study of cerebral vasculature: an introductory overviewon its hemodynamic features is provided in Sec. 4.1. In particular, in the context of theAneurisk project (presented in Sec. 4.2) we proposed WSS as a hemodynamic parameterfor the classification of internal carotid artery geometries (Sec. 4.3).

4.1 Cerebral hemodynamics

A typical assumption is that blood is a continuous incompressible Newtonian fluid, sothat its dynamics can be described by the three-dimensional unsteady incompressibleNavier-Stokes equations (see (3.8)). It has been shown [70, 98, 101] that this approachis reasonable when looking at large arteries. The same assumption may however nothold inside the aneurysmal sac, due to the presence of slow-flow regions [14]: severalstudies have been presented dealing with the non-Newtonian characteristics of bloodflow in such conditions [22, 83]. In the following, we will not deal specifically with theflow features inside the aneurysm, hence we will refer only to Newtonian models forthe blood.

The wall motion is usually neglected in blood flow modeling [99, 132, 137]. For thecase of intra-aneurysmal flow, the effect of moving boundaries has not yet been as-sessed, though recently some works moved towards this direction [24,127,137,148,149].One of the major issues when considering fluid-structure interaction problems is themechanical characterization of the wall, together with the lack of intra-arterial pressuremeasurements [24]: this limitation can be circumvented by using measured wall move-ments as boundary conditions in CFD models. Instead of solving a structural dynamicsproblem, driven by the stress exerted by the fluid on the structure, the position in timeof the interface between the blood and the wall can be recovered for instance from dy-namic angiography images through nonrigid registration algorithms. The first resultsobtained with this technique suggest that the main characteristics of the flow (locationand size of the inflow jet, complexity and stability of the patterns) are not altered bythe movement of the domain, while the computed WSS and velocity magnitude can beaffected [35, 93].

Intra-aneurysmal flow patterns develop in a wide variety and complexity: vorticalstructures may form inside the bleb and they can be stable or move during the cardiaccycle becoming unstable. Not only the size and shape of the aneurysm determine the

59

4 An application of three-dimensional modeling

flow structures, but the geometry of the parent artery also influences the way the bloodenters and flows into the aneurysm. In particular, an inflow jet is usually generated,which impacts the wall producing a region of locally elevated WSS. The size of theinflow jet and the location of the impingement region strongly depend on the patient-specific vascular geometry [124].

4.2 The Aneurisk project

The Aneurisk research project (2005-2008) was developed by a joint venture of differentsubjects: academic and non academic research centers (MOX - Department of Math-ematics, Politecnico di Milano; LaBS - Department of Structural Engineering, Politec-nico di Milano; “M. Negri” Institute for Farmacological Research, Bergamo), medicalcenters (Dipartimento di Neurochirurgia, Università degli Studi di Milano; OspedaleNiguarda Ca’ Granda di Milano; Ospedale Maggiore Policlinico di Milano), industrialpartners (Siemens Medical Solutions Italy; Fondazione Politecnico di Milano).

The main goal of Aneurisk project was to develop a framework for the analysis ofcerebral vascular geometries. The project was based on the idea of a stream of infor-mation starting from the medical image and passing through a series of steps, each oneadding a layer of knowledge to the overall process. The first step is image segmen-tation, together with geometry reconstruction and morphology characterization. It isfollowed by a modeling step for the simulation of blood flow in realistic geometries andthe characterization of the wall mechanics. Statistical analyses represent the interpre-tive step, for the organization and the extraction of information from the complete dataset. The final product of this process is intended to be an “enhanced” medical image,analysed in its more significant features which are then synthetized in a diagnostic (andpossibly prognostic) perspective.

A particularly interesting case study for Aneurisk was the Internal Carotid Artery(ICA), a renowned site for cerebral aneurysm development. The data set of the projectconsisted in 134 three-dimensional computed rotational angiography (3D CRA) scans,obtained during clinical routine at the Neuroradiology Division of the Ospedale Ni-guarda Ca’ Granda in Milan, for assessment of cerebral aneurysms. Patient-specificmodels of cerebral arteries have been reconstructed from these images, after a segmen-tation process. Geometry characterization was performed on the models employing aset of tools part of VMTK (www.vmtk.org), an open source software project for vas-cular modeling [5], mainly developed at “Mario Negri” Institute in Bergamo. Thisanalysis was carried out by Piccinelli et al. and showed preferential locations of thepathology in the distal upper tract of the vessel, on the outer wall of ICA bends in cor-respondence of local curvature maxima [102,103]. Moreover, ruptured aneurysms werefound typically in more distal positions along the vessel centerline (see Fig. 4.1).

A conjecture formulated by neuroradiologists at Ospedale Niguarda Ca’ Granda wastested, namely that some geometrical features of ICA are different according to thepresence and the location of an aneurysm (see Fig. 4.2). The idea was confirmed by aclassification of Aneurisk data set, proposed by Sangalli et al. [120]. They considered

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Figure 4.1: Position of the aneurysm along the centerline of pathological ICAs: in thewhole population, in patients with ruptured aneurysms and in patients withunruptured aneurysms. The point of zero abscissa is set at each ICA bifurca-tion, so that centerline abscissas represent distance to the bifurcation. Blackdots represent mean value; crosses represent standard deviation [103].

two groups of patients. The first (blue dots in Fig. 4.3) is composed of patients withan aneurysm located at or after the terminal bifurcation of the ICA; the second group(red dots in Fig. 4.3) is composed by patients having an aneurysm before the terminalbifurcation or healthy. Radius and curvature profiles were studied in the last tract ofICA prior to the bifurcation [118, 119] (see Fig. 4.2), and patients in the blue groupwere found to have significantly wider, more tapered and less curved ICA’s. Moreoverwithin this group there is a lower variability of radius and curvature of the ICA. Onthe basis of this result, a similarity index was defined to measure how the geometricalfeatures of each vessel compare to those of the representatives of the morphologicalclasses (see Fig. 4.3).

It is well known that blood flow features strongly depend on the vascular morphol-ogy: therefore we believe that the differences in the geometry of ICA of patients be-longing to the described groups induce different hemodynamic features and that thesemay trigger the pathologic response of the arterial wall. We then propose a CFD analy-sis over the Aneurisk dataset, in order to study the blood flow features in the last tractof ICA. Moreover, we look for parameters able to synthetically describe the effects ofblood flow on the vessel wall, such as the spatial average of wall shear stress. Thisinformation could be used to have a better understanding of the mechanisms of aneu-rysm development in the vascular district at hand; on the other hand, it could be usedto enhance the classification proposed in [120] by combining the mechanical and themorphological characterization of the vessels.

61


Figure 4.2: Curvature profiles along the centerline in the internal carotid artery [119].Negative values for the curvilinear abscissa identify proximal vessel loca-tions, with respect to the origin of the reference system which is placed atthe ICA bifurcation. The top of the figure shows the estimate of the proba-bility density function of the location of aneurysms along the ICA.

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

10.

90.

80.

70.

60.

50.

40.

30.

20.

10

Upp

er g

roup

mem

bers

hip

prob

abili

ty

Low

er g

roup

mem

bers

hip

prob

abili

ty

Figure 4.3: Classification of Aneurisk data set based on morphological features of theInternal Carotid Artery. Red dots represent ICAs with an aneurysm beforethe terminal bifurcation or healthy. Blue dots represent ICAs with an aneu-rysm at or after the terminal bifurcation.

4.3 Hemodynamic features of the Internal Carotid Artery

We chose 21 ICA geometry models, based on their score in the morphological classifi-cation [120] (see Fig. 4.3). Seven geometries were elicited from the group of vessels withhigh similarity to the representative of the red group (see Fig. 4.4), other seven from thegroup of vessels similar to the representative of blue group (see Fig. 4.6), the remainingwere chosen among the cases whose features do not fit in either class (see Fig. 4.5). We

62


will refer to the latter as to the green group.

(a) 81256 (b) 93817 (c) 100170 (d) 146842

(e) 149198I (f) 187618 (g) 218122

Figure 4.4: Data set for the numerical simulations: ICA geometries classified as belong-ing to the red group.

All the reconstructed geometries in Aneurisk data set were available as three-dimen-sional surface models represented in StL format ∗. These models represent in generala large part of the cerebral vasculature, featuring the presence of one or more aneu-rysms. For this particular study, we were interested specifically in the Internal CarotidArtery, so that all the geometries had to be restricted to this region by proper trimmingof the surface model. Moreover, following what has been done in [120], we focusedour attention on the last tract of the Internal Carotid Artery, prior to its terminal bifur-cation. In facts, this location is particularly interesting, since it is a preferential site foraneurysm development [103]. On the other hand, models of ICA reconstructed fromAneurisk dataset of medical images could only be compared by looking at the distal∗StL stands for Stereo Lithography, and indicates a triangular representation of a three-dimensional sur-

face geometry. See http://www.ennex.com/~fabbers/StL.asp

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(a) 97930 (b) 148385I (c) 179174 (d) 183983

(e) 184480 (f) 184773 (g) 198273

Figure 4.5: Data set for the numerical simulations: ICA geometries which do not belongto either group in the morphological classification.

portion of the vessel, which was present in all the images. Indeed, the dimensions ofthe reconstructed 3D model depend on the size and the spatial position of the volumescanned during the clinical procedure, and this volume is not the same for all the con-sidered patients. According to the surgeon’s choice, based on the aneurysm location,different specific districts of the cerebral vasculature were captured in medical images.Therefore, it was not possible to reconstruct the ICA to the same extent in all the cases.

The hemodynamic quantity of main interest was wall shear stress (WSS): we com-puted the integral average of WSS on a specific region of the vascular wall, correspond-ing to an interval of curvilinear abscissas on the vessel centerline spanning the last ICAbend and the last few centimeters prior to the bifurcation (see Sec. 3.5). More precisely,we considered the tract of centerline comprised between the origin of the reference sys-tem (located at the bifurcation) and the point of zero curvature delimiting the nearestsiphon to the bifurcation [119].

64


(a) 12438 (b) 145573 (c) 146495 (d) 188801

(e) 205752 (f) 209834 (g) 215056

Figure 4.6: Data set for the numerical simulations: ICA geometries classified as belong-ing to the blue group.

The choice of including the bend in the region of interest has two main reasons. Onthe one hand, it is an easily recognizable landmark and is present in all the geome-tries, therefore the selected regions are the same in all the data set. On the other side,the flow features are strongly affected by the presence of bends in the vessel geometry(see [37]), especially in flows with high values of the reduced velocity (see Tab. 4.1). Thelatter is associated to the persistence of flow structures along the streamwise direction,so that mixing effects and vortical patterns induced by the bend are expected to deter-mine the hemodynamics features in all the considered region. It has also to be notedthat flow pulsatility, together with the complexity of the overall vascular geometry (asequence of sharp, non planar bends) induces strong secondary motions resulting invortical or helical flow patterns. Evidences of this phenomenon have been recentlydiscussed in [48, 104], and on this basis we chose to reconstruct the entire sequence ofICA siphons (to the most possible extent given the images), even if interested only in

65


the distal portion. An excessive trimming of the upstream vascular geometry couldin fact lead to a non correct evaluation of the patient-specific interplay between vesselmorphology and hemodynamics.

(a) The original StL model. (b) The trimmed model(now mostly representingthe ICA).

(c) Flow extensions addedto the ICA model.

Figure 4.7: Definition of the computational domain: the model reconstructed from med-ical images (a) is trimmed (b) and flow extensions are added on the inlet andoutlet sections (c).

An example of a trimmed geometrical model is depicted in Fig. 4.7 (b). It representsthe distal bend of an Internal Carotid Artery and its main bifurcation: the trimmingprocedure excluded the downstream circulation, but not the upstream part of the artery.Cylindrical prolongations are added to each extremity of the surface, in such a waythat the geometrical model features circular inlet and outlet sections (see Fig. 4.7 (c)),corresponding to the proximal and the distal boundaries respectively. The length ofthese cylindrical extensions is adaptively selected as 10 times the clipped section radius.This technique allows to analytically formulate the boundary conditions, the boundarysections being circular. At the same time, the flow profile is allowed to develop in thecylindrical extensions, prior to entering the vessel domain.

For each one of these geometrical models we obtained a tetrahedral grid with an av-erage mesh size of 0.06 cm. The meshing procedure was performed by means of thesoftware Netgen †, which offers an implementation of the advancing front method to-gether with several mesh optimization algorithms (both metric optimization and topo-

†http://www.hpfem.jku.at/netgen/

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http://www.hpfem.jku.at/netgen/


logical [33]). The meshes were refined on the basis of the surface curvature: an exampleis depicted in Fig. 4.8.

Figure 4.8: Mesh refinement driven by the surface curvature.

The computational domain was assumed to be fixed, corresponding to the hypothe-sis of rigid vascular walls. As pointed out above, we further assumed that blood can bemodeled as a continuous incompressible Newtonian fluid, so that the blood flow prob-lem can be described by the incompressible Navier-Stokes equations (3.8). For eachvascular geometry, three cardiac cyles were simulated, in order to reduce the effects ofthe initial conditions and obtain the periodic solution in the last simulated heartbeat.

The spatial discretization was based on the Galerkin finite element method (seeSec. 3.3.2), and was carried out with a P1 approximation for both the pressure andthe velocity. The numerical scheme adopted is based on an edge stabilization tech-nique [18]. The adopted time advancing scheme is a BDF of order 1, with a time step ofdt = 10−3 s.

The spatial integral average of WSS was computed on the arterial wall, after theremoval of possible branching vessels and aneurysms, as discussed in Sec. 3.5. Anexample of the considered portion of the ICA surface is shown in Fig. 4.10.

All the fluid dynamics simulations and the WSS computations were carried out witha software specifically realized for Aneurisk project and based on LifeV (see Sec. 6.2),a C++ implementation of algorithms and data structures for the numerical solution ofpartial differential equations. The treatment of vascular geometries (addition of flowextensions, splitting in branches, identification of regions of interest) has been per-formed by using the software VMTK ‡.

‡www.vmtk.org

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0 0.2 0.4 0.6 0.8 12

3

4

5

6

7

8

9

t

Q

Figure 4.9: Flow rate (in ml/s) in an Internal Carotid Artery. The wave form of thissignal reproduces measured in vivo data, the amplitude is scaled to give atime average value of 240 ml/min, in the range of physiological values [79].

Figure 4.10: The portion of the vessel wall over which the integral average of WSS iscomputed.

Boundary conditions

In Aneurisk dataset, patient-specific measurements of blood flow rates, velocity or pres-sures were not available. On the other hand, physiological values for ICA flow rateQICA can be found in the literature, and we had at hand the wave form of the bloodflow rate measured in vivo in a single patient’s ICA. Since our interest was to comparedifferent vascular geometries, and to understand the effect of different morphologicalfeatures on hemodynamics, starting from the available data we looked for a suitable setof boundary conditions for our geometrical models. We resorted to a non-dimensional

68


argument and imposed boundary conditions giving the same flow regime in all thecomputational domains.

Figure 4.11: Velocity profile (in cm/s) imposed on the inflow section: only the axialcomponent un is non zero. This profile corresponds to (4.1) with Qin = 240ml / min and Rin = 0.2 cm.

More precisely, we run a first set of simulations imposing as inflow boundary con-dition the wave form depicted in Fig. 4.9, scaled to give a time averaged flow rateQICA = 240 ml/min (a reference value for the time averaged ICA flow rate in a cardiaccycle, as found in the literature [79]). The chosen flow rate was obtained by prescribinga velocity profile on the inlet section, more precisely an axial velocity of the followingform:

un =

0 r > Rin ,127Qin

πR2in

2(Rin − r)Rin

12Rin ≤ r ≤ Rin ,

127Qin

πR2in

0 ≤ r <12Rin ,

(4.1)

where r is the radial coordinate on the inlet section while the values of Rin and Qin

are specified in Tab. 4.1. This choice corresponds to the assumption of fully developedaxial flow, which can be approximated to a flat profile [48, 92] (see Fig. 4.11), and islegitimated by the use of a cylindrical boundary extension on the inlet section. It isindeed proven that the geometrical features of the vessel have a stronger influence onthe solution than the presence of secondary velocities in the inlet profile. Moreover, theeffects of these inlet secondary flows, even in case they are present, break down withina few diameters of the inlet [87].

We then computed the Reynolds number on the inflow boundary section, in orderto classify the flow regime for each geometry. The time average over a cardiac cycleof these values is represented in Fig. 4.12. We found out that, for most of the consid-ered geometries, a time average Reynolds number of Re = 350 describes with a goodapproximation the flow regime associated to an ICA flow rate in the range of physio-logical values. Therefore, we chose to scale the amplitude of the inflow datum to obtainthat value in each geometry (see Tab. 4.1). In two cases (patient ID 183983 and 215056,see Fig. 4.5 and Fig. 4.6) this flow regime was associated to blood flow rates significantly

69


250

300

350

400

Reyn

olds

num

ber

Figure 4.12: Boxplot of the time averaged Reynolds numbers, evaluated on the inletsection of each vascular geometry, when the imposed flow rate (averagedover a cardiac cycle) is 240 ml/min

higher than in the others, due to larger ICA radius, found in both cases.

Patient ID Rin (cm) Qin (ml / min) Woin Ured12438 0.229 263.62 3.072 5881256 0.178 206.88 2.38 97.89793817 0.215 249.14 2.879 66.57997930 0.224 258.11 3.002 60.853

100170 0.189 217.51 2.534 85.336145573 0.21 240.04 2.81 68.989146495 0.22 253.21 2.95 62.9146842 0.201 235.37 2.699 76.403148385I 0.181 208.20 2.422 93.48149198I 0.196 224.47 2.628 78.911179174 0.196 221.93 2.621 78.608183983 0.284 326.57 3.802 37.916184480 0.241 273.53 3.23 51.765184773 0.185 213.76 2.473 90.16187618 0.203 234.13 2.718 74.433188801 0.229 264.95 3.075 58.161198273 0.215 245.02 2.887 64.964205752 0.223 259.31 2.995 61.583209834 0.213 245.65 2.848 67.838215056 0.283 326.36 3.788 38.323218122 0.17 198.23 2.276 107.29

Table 4.1: Inflow boundary conditions: radius Rin, imposed flow rate Qin (time averageover a cardiac cycle), Womersley number Woin and Reduced Velocity Uredevaluated on the inflow section.

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A zero-stress condition was prescribed on the outlet sections through homogeneousNeumann boundary conditions. This implies that the same mechanical load was im-posed on all the outflows, which is clearly not the case in vivo. However, this is a widelyaccepted assumption, when measures of flow rates or pressures are not available in cor-respondence of the boundaries of the computational domain. A possible way to avoidthis strong assumption is to resort to the so-called geometrical multiscale approach, basedon the coupling of detailed models (like the 3D Navier-Stokes equations (3.8)) describ-ing the local fluid dynamics, together with reduced models (such as the 1D Euler equa-tions (2.5)) reproducing the remainder of the circulatory system. This approach will bediscussed further in Chap. 5.

4.3.1 Discussion

Figure 4.13: Integral average of the wall shear stress (in dyn / cm2) over the last portionof ICA, prior to the bifurcation. The values for two different groups areshown (red group with red line, blue group with blue line)

Let us consider all the geometries elicited from the red group (see Fig. 4.4), and com-pute the spatial average WSS in the region of interest as a function of time. This willgive seven time patterns, which can be averaged to find a “representative” time patternfor the considered quantity. If the same procedure is applied to geometries belongingto the blue group, the resulting representative time patterns can be compared as shownin Fig. 4.13. Moreover, the time average of these curves can be computed, yielding:

Time averaged mechanical load (dyn / cm2)Red group Blue group

24.097 17.124

where for the sake of brevity we defined “mechanical load” the spatial average WSS

71


in the considered ICA region.This analysis shows that arteries of patients belonging to the red group (that is, with

narrower, less tapered and more curved vessels) typically undergo a higher mechanicalload, with respect to geometries belonging to the blue group, in the same flow regime.We use this result to define a simple classification strategy of vascular geometries basedon hemodynamics features. More precisely, we identify a threshold value, defined asthe mean of the average mechanical load in the two groups:

WSS = 20.611 dyn / cm2 . (4.2)

We then classify a vascular geometry as member of the red group if the computedmechanical load is higher than the threshold value. Otherwise, the vascular geometryis classified as member of the blue group.

For the sake of clarity, we refer to the results presented in Tab. 4.2: the time averageover the cardiac cycle of the mechanical load as previously defined is shown for eachsimulated case. It is clear from these data that in most cases the classification based onhemodynamics features agrees with the results of the morphological analysis. Indeed,as expected, vessels elicited from the red group and from the blue group are correctlyclassified by both strategies except for two cases which will be discussed in detail lateron.

On the other hand, the geometries belonging to the green group (i. e. with geometricalfeatures not clearly associated to the red nor to the blue group, see Fig. 4.5) can now beassociated to a typical “red-group” or “blue-group” hemodynamics behaviour. Moreprecisely, low values of the spatial average WSS (with respect to the threshold value(4.2)) are found in carotid arteries which do not feature the presence of aneurysms. Thismeans that, from the fluid dynamics view point, these cases are similar to the typicalrepresentative of the blue group, while the correct classification was not achieved onthe basis of their morphological features.

The hemodynamics analysis is able to correctly classify also a case of “green” geom-etry featuring an aneurysm along the ICA. The case identified as number 184773 (seeFig. 4.5) presents a small aneurysm in the very last part of the ICA, prior to the bifurca-tion: correspondingly, its time averaged mechanical load is similar to the typical “red”case value (i. e. higher than the threshold value).

One “green” case (identified as number 184480) is still misclassified on the basis offluid dynamics arguments. It is indeed similar to a “blue” case, according to the esti-mated time averaged mechanical load, but it is not a typical “blue” case in that it has aquite big aneurysm along the Internal Carotid Artery (see Fig. 4.5). The very presenceof such a big sac, however, is probably the reason why the proposed fluid dynamicalanalysis is not effective in this case: giant aneurysms with a large neck strongly deformlocally the hosting artery, therefore calling for a more specific and detailed study thanthe evaluation of a spatial average value of WSS.

Two cases of wrong hemodynamics classification are the “red” case 93817, featuringa low mechanical load, and the “blue” case 205752, featuring high mechanical load. Inthe former case, however, the anomalous hemodynamic behaviour is due to a patho-logical condition of the entire vessel, namely a displasia causing the weakening of the

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ClassificationPatient ID Average WSS (dyn / cm2)

Morphology Hemodynamics

R

R 81256 20.246B 93817 11.003R 100170 23.870R 146842 24.504R 149198I 20.096R 187618 20.523R 218122 48.434

B

B 12438 13.260B 145573 16.615B 146495 16.609B 188801 18.997B 204552 19.366R 205752 25.372B 215056 11.556

–

B 148385I 15.591B 179174 16.979B 183983 14.791B 184480 14.757R 184773 20.949B 198273 12.956B 97930 16.595

Table 4.2: Classification of the considered dataset of ICA. The hemodynamics providesadditional information with respect to the morphological analysis, and is ableto properly classify uncertain cases.

wall and a non physiological increase of the diameter (see Fig. 4.4). Conversely, thelatter case features a particularly marked tapering in the very last tract of the ICA andis unusually tight in the bifurcation zone: this enhances the shear stress exerted by theblood on the wall in that specific region, and makes the spatial average WSS an unsuit-able characterizing parameter.

Finally, case 218122 deserves a comment, even if it is well classified both from themorphological and fluid dynamical point of view. It can be seen in Tab. 4.2 and inFig. 4.15 that it features an extremely high mechanical load, and again the inspection ofthe anatomy of the artery provides a reasonable explanation: as can be seen in Fig. 4.4,this vessel is very narrow in the distal part, so that high WSS has to be expected in allthe considered area.

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Figure 4.14: Green group: integral average of the wall shear stress (in dyn / cm2) overthe last section of ICA, prior to the bifurcation.

Figure 4.15: Red group: integral average of the wall shear (in dyn / cm2) over the lastsection of ICA, prior to the bifurcation.

4.3.2 Wall shear stress as a classification parameter

The results of this work suggest that high WSS, beyond the other morphological fea-tures discussed in [120], is associated to the presence of aneurysms in the InternalCarotid Artery. Starting from this observation, based on the presented data set, it maybe conjectured that a high-WSS environment induced by geometrical features predis-poses to the development of the pathology. This conjecture is consistent with the idea ofa correlation between geometrical and hemodynamical features of arteries, as assessed

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Figure 4.16: Blue group: integral average of the wall shear stress (in dyn / cm2) overthe last section of ICA, prior to the bifurcation.

in the literature. More precisely, elevated WSS has been associated to fragmentationof the internal elastic lamina of blood vessels [130], endothelial damage [129] and ulti-mately to aneurysm initiation (see also Chap. 1).

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

WSS typically features a complex spatial pattern, corresponding to a non homoge-neous mechanical load distribution at the microscopic level. In most cases here consid-ered, however, its spatial average can be a useful and synthetic indicator of the stressexerted by the blood on the arterial wall, whenever the region of interest does not fea-ture abrupt changes in the geometric features, such as localized stenosis or narrowing,or the presence of big aneurysms with large necks.

One interesting extension of this analysis technique could be the definition of an in-dex able to discriminate the pattern of the mechanical load along the vessel curvilinearabscissa. To this aim, the approach presented in Sec. 3.5.2 for the mapping of centerlineabscissas on the surface can be exploited. Each artery would then be described by aset of geometric and hemodynamics parameters regarded as functions of the curvilin-ear abscissa of the centerline, and a new classification could be designed following theapproach presented by Sangalli et al. [120]. On the other hand, other hemodynamicsparameters could be included in the analysis: in particular, vorticity is expected to bea significant flow feature in the complex ICA geometry, and could help in defining amore robust classification strategy.

The in-silico setup here presented, and its further improvements, stand as a candidatetool to give a synthetic description of the mechanical solicitation exerted by blood flow

75


on the vascular wall. More than that, our results show that it can be used to characterizecerebral vascular geometries which are associated to the presence of an aneurysm. Inthis respect, it is worth noting that the additional information provided by CFD couldhave a prognostic value, helping to assess the evolution trend of the studied vessels:geometries featuring high WSS in the region near to the bifurcation could be more proneto the development of an aneurysm in ICA.

Finally, we want to remark that a strong interplay between vascular geometry andblood flow features has been clearly shown from our results. We proposed a way to cou-ple and complement the information coming from two different analysis approaches.Further application and improvement of this twofold approach is likely to give a greaterinsight and comprehension of cerebral aneurysms.

76

5 A geometrical multiscale model of thecerebral circulation

Geometrical multiscale modeling is a strategy advocated in computational hemody-namics for representing dynamics ranging over different space scales in a single nu-merical model. It allows to couple the description of large vascular districts given byreduced models, with the detailed analysis of blood flow in specific regions of interest.This approach is particularly interesting in the cerebral circulation, whose arteries onthe one hand form a complex anastomotic vascular structure (the circle of Willis) withpeculiar hemodynamics features (see Chap. 1, Chap. 2) and on the other hand are proneto develop localized diseases such as cerebral aneurysms (see Chap. 1, Chap. 4).

The modeling of the entire circulatory system of the brain would be unfeasible bymeans of 3D models (due to the lack of data and high computational costs), while 1Dmodels are not suitable for the modeling of the microscopic features of the blood flow.A coupled approach can be effectively used in this context, as we discuss in Sec. 5.4.We present a multiscale model of the cerebral circulation where a one-dimensional de-scription of the circle of Willis, relying on the Euler equations, is coupled to a fullythree-dimensional model of a carotid artery, based on the solution of the incompress-ible Navier-Stokes equations. A similar multiscale model has been investigated in [86],where the 3D model includes compliance for avoiding spurious reflections induced bya rigid treatment of the 3D geometry in the multiscale model.

Even if vascular compliance is often not relevant to the meaningfulness of 3D re-sults (e.g. in large arteries), it is crucial in the multiscale model, since it is the drivingmechanism of pressure wave propagation (Sec. 5.1). Unfortunately, 3D simulations incompliant domains still demand computational costs significantly higher than the rigidcase. Appropriate matching conditions between the two models have been devised toconcentrate the effects of the compliance at the interfaces and to obtain reliable resultsstill solving a 3D rigid problem (Sec. 5.2).

5.1 The compliant vessel problem

A practical difficulty arises when some features, that at a certain scale can be neglected,become relevant in the coupled model, inducing a significant increase of the overallcomputational cost. This is the case of the compliance of vessels. In 3D Navier-Stokesstand-alone models compliance is quite often not relevant for bioengineering purposes.However, it is a driving mechanism of pressure wave propagation along the vasculartree. Therefore, when considering 3D/1D geometrical multiscale models in principlecompliance should not be neglected in either models.

77

5 A geometrical multiscale model of the cerebral circulation

1D Model

3D Model

Γ

Figure 5.1: A simple multiscale (3D/1D) model of a cylindrical pipe

The coupling between 1D and 3D compliant models has been investigated recentlyin [86]. The computational cost of a compliant 3D simulation is however by far higherthan the rigid case. On the other hand, a naive coupling of 1D (which are intrinsi-cally compliant) and 3D rigid problems, forcing for instance the continuity of pressureand flow rate is problematic, since the different wall modeling in the two subdomainsmakes the coupled problem ill conditioned and affects the numerical results.

We overcome these difficulties by resorting to appropriate matching conditions thatmimic the presence of the compliance by concentrating it at the interface of the twomodels. This allows to simulate the overall dyamics still solving a 3D rigid model,getting reliable results with relatively low CPU times.

5.2 Matching conditions in 3D rigid/1D multiscale models

To fix the ideas, let us refer to the simple model represented in Fig. 5.1. We assume thata cylindrical pipe has been split at section Γ into two halves.

The left one is described in terms of the 1D Euler equations (2.1), while the right handside is represented by the incompressible Navier-Stokes equations (3.9).

Coupling the two models requires appropriate matching conditions. In the case of arigid 3D model, it is reasonable to prescribe the continuity of pressure and flow rate

P1D =1|Γ|

∫Γp3Ddγ, Q1D = −%

∫Γu3D · ndγ, (5.1)

where n is the outward normal unit vector on Γ, we added the indexes 1D and 3D forthe sake of clarity and denote by |Γ| the area of the interface Γ. The negative sign in thesecond of (5.1) stems from the fact that both Q1D and u3D · n are directed outward the1D and 3D domains respectively. In the sequel, for easiness of notation, we set

Q3D = %

∫Γu3D · ndγ , P3D =

1|Γ|

∫Γp3Ddγ .

Other conditions can be considered as well, prescribing the continuity of the totalpressure, of the normal stresses or of the characteristic variables (see [112]).

78


5.2.1 Numerical algorithm

When solving multiscale problems numerically it is natural to split the scheme into theiterative sequence of dimensionally homogeneous problems, which we indicate as 1Dand 3D, for instance by means of the following algorithm.

We assume that standard (Dirichlet or Neumann) conditions are prescribed at theboundaries of the overall 1D/3D model. Moreover, we carry out an appropriate spaceand time discretization of the problems. In particular, apexes n and n + 1 refer to theapproximation of the solution at time steps tn and tn+1 , respectively. Index k will referto the inner iterations performed at a fixed time step, for the fulfillment of the matchingconditions.

For n = 0, 1, . . . we perform the following steps.1) Inizialization. Set k = 0,

un+13D,0 = un

3D , pn+13D,0 = pn

3D , and

Pn+11D,0 = Pn

1D , An+11D,0 = ψ−1(Pn+1

1D,0) , Qn+11D,0 = Qn

1D .

2) Loop on k.2.1) Solve the 1D model with the boundary condition on Γ given by

Pn+11D,k+1 = χPn+1

3D,k + (1− χ)Pn+11D,k, (5.2)

where χ is a relaxation parameter to be set for improving the convergence rate. Solvingthe 1D model, pressure conditions are recasted in terms of area, thanks to the wall lawAn+1

1D,k+1 = ψ−1(Pn+11D,k+1) (see Sec. 2.2).

2.2) Solve the 3D problem with the boundary conditions on Γ

Qn+13D,k+1 = −Qn+1

1D,k+1 (5.3)

Set k = k + 1.3) Test. Different convergence tests can be pursued. A possibility is to check thecontinuity at the interface, namely terminate the iterations when∣∣∣Pn+1

1D,k+1 − Pn+13D,k+1

∣∣∣ ≤ ε

ε being a user-defined tolerance.Swapping the role of the matching conditions in the set up of the boundary condi-

tions for the iterative scheme, (5.2), (5.3) can be replaced by

Qn+11D,k+1 = −χQn+1

3D,k + (1− χ)Qn+11D,k, Pn+1

3D,k+1 = Pn+11D,k+1. (5.4)

The different space dependence of 1D and 3D models leads to unmatched or defec-tive conditions (step 2 of the loop) and in particular (5.3) (or the second condition (5.4))do not prescribe sufficient conditions for the closure of the Navier-Stokes problem. Thelatter needs to be solved in the framework of the so-called defective boundary problems,the data being available at the boundary not enough to guarantee the uniqueness of

79


3D Model

L

P3D =RΓ p3D

Q3D =RΓ u3D · n

P1D

P

R1 R2

C

Q1D

1D Model

Figure 5.2: Representation of a multiscale 3D/1D model with a 0D element representingthe compliance of the 3D model at the interface.

the solution. This topic has been discussed in [112], Chap. 11, where different mathe-matically sound techniques for the solution of defective problems are presented. Thespecific method for solving the 3D problem affects the accuracy of the Navier-Stokes so-lution and is not relevant for the purpose of the present work, so we do not dwell uponit. Any reasonable technique can be used in the context of our multiscale modeling.

The iterative approach given by the previous three steps suffers from numerical prob-lems induced by the different description of the wall mechanics in the two halves of thepipe, which produces some spurious reflections at the interface and possible numeri-cal instabilities. One could avoid this kind of problems by resorting to a compliant 3Dmodel. As we have pointed out, this increases the computational costs strongly. Moreprecisely, implicit coupled fluid-structure iterative schemes at each time step require tosolve the Navier-Stokes and the structural problems several times. In explicit coupledfluid-structure iterative schemes, stability concerns typically require to take small timesteps. In the next subsection, we present a different strategy based on the set up of anappropriate set of interface conditions.

5.2.2 Matching conditions including compliance

Suppose that we give a simplified representation of the compliance of the 3D vesselin the multiscale model, by gathering its effect at the interface using a special lumpedparameter model. Referring for instance to Fig. 5.2, we introduce a RCL network at theinterface with the role of representing the effects of the compliance of the artery in the3D model. In this way, we still use a 3D rigid model, which however behaves like acompliant one with respect to the system dynamics.

By denoting with P the pressure associated with the capacitance C, the governingequations read

P1D −R1Q1D = P = P3D − LdQ3D

dt−R2Q3D

CdP

dt= Q1D +Q3D.

(5.5)

Taking the derivative of the first equation and using the third, we can eliminate P andfinally obtain the new conditions in the iterative scheme, by replacing (5.2), (5.3) in the

80


1D Model

P1D

P C

Z3D = P3D −R2Q3D

Q3D

L

R1Q1D

R2P3D

3D Model

Z1D = P1D −R1Q1D

Figure 5.3: Alternative representation of the coupled problem: now the unknowns arez1D, z3D, P1D, P3D.

algorithm with

P1D,k+1 = χ

(P3D,k − L1

dQ3D,k

dt−R2Q3D,k +R1C1

dP3D,k

dt

−R1CLd2Q3D,k

dt2−R1CR2

dQ3D,k

dt−R1Q3D,k

)+ (1− χ)P1D,k

Q3D,k+1 = −Q1D,k+1 + CdP1D,k+1

dt−R1C

dQ1D,k+1

dt

(5.6)

These conditions (hereafter denoted by LP (Lumped Parameter) conditions) involve timederivatives of the matching quantities to be discretized with an appropriate finite dif-ference scheme with the same accuracy of the time advancing methods used for thetime discretization of the Navier-Stokes and Euler equations.Remark As was to be expected, for C = 0, R1 = R2 = 0 and L = 0 we recover thecoupling given by conditions (5.2), (5.3). This corresponds physically to the case of arigid portion of artery in a network of compliant vessels, as it is the case of a stented orprosthetic segment (see [45]).

Alternative formulation

Equations (5.6) involve the second order time derivatives of the interface variables,whose accurate numerical approximation is in general not trivial. This motivates thederivation of an alternative formulation involving only first order time derivatives.

Let’s define

z1D (P1D, Q1D) = P1D −R1Q1D , z3D (P3D, Q3D) = P3D −R2Q3D ,

then system (5.5) becomes

LdQ3D

dt= z3D − z1D

Cdz1D

dt= Q1D +Q3D .

(5.7)

81


We remark that the flow rate Q1D at interface Γ is linked to the pressure P1D (conse-quently to z1D) through the Euler equations (2.1). We can express this relation in thefollowing form:

Q1D = M1D(z1D) , (5.8)

where M1D stands for the set of equations of the 1D model.Similarly, the following relation holds between Q3D and z3D on Γ:

z3D = M3D(Q3D) , (5.9)

where we indicate the 3D Navier-Stokes problem by M3D.Equations (5.7), (5.8) and (5.9) form a non linear system in the four unknowns z1D,

z3D, Q1D, Q3D, that can be rewritten in vectorial form as follows:

LP(X) = 0 ,

with

X =

z1D

z3D

Q1D

Q3D

, LP(X) =

Cdz1D

dt− (Q1D +Q3D)

z3D −M3D(Q3D)

Q1D −M1D(z1D)

LdQ3D

dt− (z3D − z1D)

.

Its time discretization can be retrieved for instance by means of a first order implicitEuler method:

LP∆t(Xn+1) = 0 , (5.10)

with

LP∆t(Xn+1) =

C(zn+11D − zn

1D

)−∆t

(Qn+1

1D +Qn+13D

)zn+13D −M3D(Qn+1

3D )

Qn+11D −M1D(zn+1

1D )

L(Qn+1

3D −Qn3D

)−∆t

(zn+13D − zn+1

1D

)

and with obvious meaning of the symbol Xn+1.

Now the iterative algorithm for the solution of the coupled problem can be formu-lated as a Newton scheme applied to system (5.10). Given the approximation Xn+1

k ofthe solution at time tn+1, computed at iteration k, we can recover Xn+1

k+1 by solving thefollowing linear system in the increment δXn+1

k = Xn+1k+1 −Xn+1

k :

JLP∆t

∣∣∣∣Xn+1

k

δXn+1k = −LP(Xn+1

k ) , (5.11)

82


JLP∆tbeing the Jacobian matrix of function LP∆t:

JLP∆t

∣∣∣∣Xn+1

k

=

C 0 −∆t −∆t

0 1 0 −M′3D

∣∣∣Qn+1

3D,k

−M′1D

∣∣∣zn+11D,k

0 1 0

∆t −∆t 0 L

. (5.12)

It is quite easy to explicitly compute the inverse of matrix (5.12), which reads:

J−1LP∆t

∣∣∣∣Xn+1

k

=1

det

(JLP∆t

∣∣∣∣Xn+1

k

) ·

L +M′3D,k∆t ∆t2 ∆t(L +M′

3D,k∆t) ∆t

M′3D,k∆t ∆t2 + L(C −M′

1D,k∆t) M′3D,k∆t2 M′

3D,k

`∆tM′

1D,k − C´

M′1D,k(L +M′

3D,k∆t) M′1D,k∆t2 C(L +M′

3D,k∆t) + ∆t2 M′1D,k∆t

−∆t C∆t−M′1D,k∆t2 −∆t2 C −M′

1D,k∆t

,

det

(JLP∆t

∣∣∣∣Xn+1

k

)= CL+

(CM′

3D,k − LM′1D,k

)∆t+ (1−M′

1D,kM′3D,k)∆t

2 ,

where it is understood that M′1D,k = M′

1D

∣∣∣zn+11D,k

and M′3D,k = M′

3D

∣∣∣Qn+1

3D,k

.

Newton iteration (5.11) finally reads:

Xn+1k+1 = Xn+1

k − J−1LP∆t

∣∣∣∣Xn+1

k

LP(Xn+1k ) , (5.13)

so that each interface variable at iteration k + 1 is expressed as linear combination ofthe variables at iteration k.

A simplified expression for M′1D and M′

3D can be obtained by considering a 0Drepresentation of the 3D and 1D models, based again on RCL networks. We will describethe 1D model by means of an electric L-network (shown in Fig. 5.4(a)), whose dynamicsis represented by the state variablesQ1D and P1D,up, whileQ1D,up and z1D are prescribedas boundary conditions (we recall that P1D = Z1D +R1Q1D):

C1DdP1D,up

dt+Q1D = Q1D,up (5.14a)

L1DdQ1D

dt+ (R1D +R1)Q1D = P1D,up − z1D , (5.14b)

where subscript “up” indicates that the subscripted quantity is evaluated in correspon-dence of the “upstream” section of the 1D model.

83


P1D

L1D R1D

C1D

Q1D,up Q1D

P1D,up

(a) An electric L-network repre-senting the 1D model. The modeldynamics is represented by thestate variables P1D,up and Q1D .

P3D

Q3D Q3D

P3D,down

L3D R3D

(b) An electric RL network repre-senting the 3D model. The modeldynamics is represented by thestate variable Q3D .

Figure 5.4: The electric analogy can be exploited to represent the 1D and 3D modelswith their 0D counterpart.

If we consider the second equation in system (5.14) and approximate the time deriva-tives with an implicit Euler scheme, we can express Qn+1

1D as a function of zn+11D (given

Pn+11D,up):

Qn+11D =

(L1D

∆t+R1D +R1

)−1 [L1D

∆tQn

1D + Pn+11D,up − zn+1

1D

]≈M(zn+1

1D ) ,(5.15)

and finally we can retrieve an approximation for M′1D(z1D):

M′1D(z1D)

∣∣∣zn+11D

=dM1D

dz1D(zn+1

1D )

≈ − 1L1D

∆t+R1D +R1

.(5.16)

Since we are considering rigid wall 3D models, a simpler 0D network can be usedto represent the 3D model (see Fig. 5.4(b)): the electric analogy does not feature acompliance element. Thus, we can resort to an RL network, describing the dynam-ics of Q3D given z3D and the “downstream” pressure P3D,down (we recall that P3D =Z3D +R2Q3D):

L3DdQ3D

dt+ (R2 +R3D)Q3D = P3D,down − z3D .

We discretize in time the previous by means of an implicit Euler scheme, obtaining:

zn+13D = Pn+1

3D,down −(L3D

∆t+R2 +R3D

)Qn+1

3D +(L3D

∆tQn

3D

)≈M3D(Qn+1

3D ) ,

and finally we find

M′3D(Q3D)

∣∣∣Qn+1

3D

=dM3D

dQ3D(Qn+1

3D ) ≈ −(L3D

∆t+R2 +R3D

).

84


name electric analogy expression

resistance R8πµlA2

0

inertia L (inductance)%l

A0

compliance C (capacitance)3πR3

0l

2Eh0

Table 5.1: Parameter estimation for the electric analog model of a cylindrical vessel (af-ter [112]).

5.2.3 Parameters estimation

The interface lumped parameter model of Fig. 5.2 provides a physical representationto our matching conditions. A major issue in this approach is the tuning of the para-maters featuring the LP model. In particular, we started from a classical RCL network,advocated for representing the capillary circulation (see [2, 134]).

We remind that, following classical arguments for the derivation of lumped param-eter models (see e.g. [112] Chap. 10, based on a proper average of the Navier-Stokesequations) for a cylindrical vessel with length l, area A0, with a linear elastic wall with

thickness h0 and Young modulus E, the compliance may be estimated to be C ∝ A3/20 l

Eh0.

The resistance induced to the flow by the blood viscosity µ can be expressed asR ∝ µl

A20

,

while the inertial term in the momentum equation gives rise in the 0D model to an in-

ductance L ∝ %l

A0, % being the blood density (see Tab. 5.1).

In the case at hand, depicted in Fig. 5.2, we consider a 10 cm long tube, each halfmeasuring l = 5 cm, and we set A0 = 0.785 cm2. The elastic modulus of the arterialwall is taken E = 106 dyn/cm2, and the wall thickness is h0 = 0.05 mm. Blood isassumed to be a Newtonian fluid of viscosity µ = 0.035 poise. The electric analogy canbe exploited to define a 0D model of the coupled problem, as depicted in Tab. 5.2. Inparticular, the lumped parameter representation of 1D and 3D models is useful for theset up of the solution strategy (5.13).

The missing capacity element in the 3D model electric analogy is assigned to the in-terface RCL network. Physiological values of this parameter are of the order of 10−5

cm5/dyn. In our computation we set C = 5.8910−5 cm5/dyn. The other interface pa-rameters have been properly adjusted in order to reduce spurious effects at the 1D/3Dinterfaces. More precisely, resistance R1 has been introduced in [3] and, following theproposal of that paper, it is dynamically selected so that an incoming wave from the1D model is propagated without any reflection. For R2 and L, in this paper we haveadopted an empirical trial and error approach, so that after some numerical experi-ments we put R2 = 1 dyn s cm−5 and L = 0.01 g cm−4. For more complex models,

85


Parameters in the coupled model

R1DL1D R3DL3D

C1D

R1

C

R2L

1D model LP interface 3D modelR1D = 7.1301 dyn s/cm5 R1 = (*) R3D = 7.1301 dyn s/cm5

R2 = 1 dyn s/cm5

L1D = 6.3662 g cm−4 L = 10−2 g cm−4 L3D = 6.3662 g cm−4

C1D = 5.8910−5 cm5/dyn C1D = 5.8910−5 cm5/dyn -

Table 5.2: A 0D model of the coupled problem depicted in Fig. 5.2. The numerical valuefor each parameter is reported. (*) R1D is dynamically tuned to a value en-suring the wave propagation from 1D model without spurious reflections [3].

these parameters should be adapted accordingly.

5.2.4 Results

The impact of the LP conditions is illustrated in Fig. 5.5 and 5.6. More precisely, inFig. 5.5 we illustrate results obtained for the model of Fig. 5.2 when a sinusoidal wave-form for the flow rate is prescribed at the inlet. We compare the values in time of theflow rate and the area (as function of the pressure) at the interface, denoted by Q1D

and A(P1D) in Fig. 5.2, obtained with a standard multiscale 1D/3D model, using theproposed approach and finally those obtained with a complete 1D model. In Fig. 5.6we present similar comparisons for the case when a step waveform is prescribed at theinlet of the domain.

The impact of interface conditions is evident. In the case based on classical matching,the solution is dramatically affected by reflections induced by the different descriptionof the wall mechanics in the 1D and 3D model. These reflections change completely theprofile of the solution. Observe that the complete reflection of the flow rate of Fig. 5.6can be justified by a linear analysis of the reflection coefficient considered e.g. in [89].For a rigid downstream pipe, this coefficient corresponds to total reflection. On thecontrary, matching conditions based on the RCL model are able to obtain a behaviorsimilar to that of the complete 1D model, even if we are using a rigid 3D model. Thesame conclusions hold for the area: the RCL-based conditions allow us to find a solutionsignificantly close to that of the complete 1D model. As pointed out, a proper tuning ofthe parameters is crucial to find the best RCL model.

86


0 0.05 0.1 0.15 0.2 0.25

−0.1

−0.05

0

0.05

0.1

0.15

time

Q1

D

1D1D/3D C=0

1D/3D C≠ 0

0 0.05 0.1 0.15 0.2 0.250.785

0.7851

0.7852

0.7853

0.7854

0.7855

0.7856

0.7857

0.7858

time

A(P

1D

)

1D1D/3D C=01D/3D C ≠ 0

Figure 5.5: Comparison of dynamics of flow rate (left) and area (right) at x = 5cm ofa compliant pipe simulated with a fully 1D model (thick dashed line), amultiscale 1D/3D model with direct coupling (C = 0, solid line) and withthe matching conditions obtained by the lumped parameter models (C 6= 0,thin dashed line). The input waveform of the flow rate at the tube inlet is asine with amplitude 0.1. (time in [s], volumetric flow rate in [cm3/s], area in[cm2])

0 0.5 0.1 0.15 0.2 0.25 0.3−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

time

Q1

D

1D1D/3D C=0

1D/3D C ≠ 0

0 0.05 0.1 0.15 0.2 0.25 0.30.7853

0.7854

0.7854

0.7855

0.7855

0.7856

0.7856

0.7857

0.7857

0.7858

0.7858

time

A(P

1D

)

fully 1D1D/3D C=0

1D/3D C ≠ 0

Figure 5.6: Comparison of dynamics of flow rate (left) and area (right) at x = 5cm ofa compliant pipe simulated with a fully 1D model (thick dashed line), amultiscale 1D/3D model with direct coupling (C = 0, solid line) and withthe matching conditions obtained by the lumped parameter models (C 6= 0,thin dashed line). The input waveform of the flow rate at the tube inlet is astep function with amplitude 0.1 (time in [s], volumetric flow rate in [cm3/s],area in [cm2].)

87


Q1D,endR1 R2 L1 R3 R4 L2

C2C1

Q1D,l Q3D,l

P1D,lP3D,l P3D,r

P1D,r

Q3D,r Q1D,r

3D 1D1D

Figure 5.7: Representation of a multiscale model with two 0D buffer elements at theinterface.

5.3 A 1D-3D-1D coupling

Let us consider now the case represented in Fig. 5.7, where we show a sequence of 1D-3D-1D model with appropriate LP conditions. From the numerical point of view on theleft interface we still resort to the iterative scheme with conditions (5.6). On the rightinterface, we adopt a similar iterative strategy where we prescribe a pressure conditionto the 3D problem and flow rate conditions to the 1D Euler system downstream. Moreprecisely, equations corresponding to the downstream interface read

P3D −R3Q3D = P1D − L2dQ1D

dt−R4Q1D

C2dP3D

dt−R3C2

dQ3D

dt= Q1D +Q3D

Consequently, the coupling conditions used in the iterative scheme are

Q1D,k+1 = χ

(−Q3D,k + C2

dP3D,k

dt−R3C2

dQ3D,k

dt

)+ (1− χ)Q1D,k

P3D,k+1 = P1D,k+1 − L2dQ1D,k+1

dt−R4Q1D,k+1+

R3C2dP1D,k+1

dt−R3C2L2

d2Q1D,k+1

dt2−R3CR4

dQ1D,k+1

dt−R3Q1D,k+1

(5.17)

Alternative formulation

We can reformulate the interface problem (5.17) in form (5.13), by setting

LP∆t(Xn+1) =

C(zn+13D − zn

3D

)−∆t

(Qn+1

1D +Qn+13D

)L(Qn+1

1D −Qn1D

)−∆t

(zn+11D − zn+1

3D

)zn+11D −M−1

1D(Qn+11D )

Qn+13D −M−1

3D(zn+13D )

88


where again Xn+1 is defined as

Xn+1 =

zn+11D

zn+13D

Qn+11D

Qn+13D

,

and

J−1LP∆t

∣∣∣∣Xn+1

k

=1

det

(JLP∆t

∣∣∣∣Xn+1

k

) ·

−∆t“M−1

1D,k

”′ “M−1

1D,k

”′ “C −∆t

“M−1

3D,k

”′”CL + ∆t

“∆t − L

“M−1

3D,k

”′”−

“M−1

1D,k

”′∆t2

L −∆t“M−1

1D,k

”′∆t ∆t2 ∆t

“L −∆t

“M−1

1D,k

”′”−∆t C −∆t

“M−1

3D,k

”′∆t

“C −

“M−1

3D,k

”′∆t

”−∆t2“

M−13D,k

”′ “L −∆t

“M−1

1D,k

”′” “M−1

3D,k

”′∆t

“M−1

3D,k

”′∆t2 CL + ∆t

“∆t −

“M−1

1D,k

”′C∆t

”

,

det

(JLP∆t

∣∣∣∣Xn+1

k

)= ∆t2

(1 +

(M−1

1D,k

)′ (M−1

3D,k

)′)+

−∆t(C(M−1

1D,k

)′+ L

(M−1

3D,k

)′)+ CL .

5.3.1 Results

Numerical results are reported in Fig. 5.8. Again, we illustrate the comparison of the so-lutions obtained with a 1D model, and the multiscale models corresponding to Fig. 5.7,where all the lumped parameters are null (classical conditions) and when they are acti-vated. The inlet waveform is sinusoidal.

In the first picture we present the flow rate at the first interface (denominatedQ1D,l inFig. 5.7), in the second picture the flow rate −Q1D,r at the second interface and finallythe flow rate Q1D,end at the outlet of the right pipe. Again, when classical matchingconditions are used (corresponding to null values of the parameters) the superpositionof the components induced by reflections associated to the different wall models areevident. This changes the shape of the propagating wave and affects both the ampli-tude and the phase at the inlet and at the outlet of the 3D model. Amplitude dissipationin the forward component of the wave is partially compensated by the superpositionof the spurious reflections. In the case of LP conditions, the shape of the wave is onlypartially affected. Dispersion errors are remarkably small, whilst dissipation effects arepresent. More precisely, the dispersion error, evaluated as the difference in the occur-rence of the peaks in the 1D and the multiscale LP models, are 20%, 6% and 5% in thethree pictures of Fig. 5.8 respectively, while dissipation, evaluated as the difference ofthe peaks, are 25%, 32% and 32% respectively. The impact of finite difference schemesin the numerical implementation of matching conditions is probably the main responsi-ble of these effects. A more accurate analysis of this aspect is one of the possible future

89


0 0.05 0.1 0.15 0.2 0.25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

time

Q1

D,l

1D1D/3D C=0

1D/3D C ≠ 0

(a)

0 0.05 0.1 0.15 0.2 0.25−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

timeQ

1D

,r

1D1D/3D C=0

1D/3D C ≠ 0

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.1

−0.05

0

0.05

0.1

time

Q1

D,e

nd

1D1D/3D C=01D/3D C ≠ 0

(c)

Figure 5.8: Comparison of flow rates computed by a 1D model (dash-dot line), a mul-tiscale 1D/3D/1D model with classical matching conditions (dashed line)and with lumped parameter matching conditions (solid line) in correspon-dence of the first interface (a), the second (b) and the outlet of the domain(c). (time in [s], volumetric flow rate in [cm3/s])

90


Cerebral Artery

Artery

Vertebral Artery

Middle

External Carotid

Basilar Artery

Internal Carotid Artery

Aorta

Figure 5.9: Left: Anatomical representation of the cerebral vasculature, including thecircle of Willis (after [12]). Right: Multiscale representation of the cerebralvasculature: a 3D representation of one of the carotid arteries is embeddedin a 1D network of Euler problems

developments of this work. Matching conditions guarantee in any case a significantreduction of spurious reflections.

5.4 The 3D carotid model and the multiscale coupling

The proposed model is based on the set-up presented by Alastruey et al. [3] for thedescription of the cerebral circulation (see Sec. 2.5.4). The multiscale representation isdepicted in Fig. 5.9, on the right. Left Internal Carotid geometry adopted is based onthe realistic glass model obtained by Liepsch, see [106]. The Navier-Stokes equations inthe 3D model have been solved with the code LifeV - see www.lifev.org - based on aP1P1 finite element solver stabilized by means of an interior penalty approach. At theinterfaces between the 3D and 1D models we prescribe conditions (5.6) at the upstreaminterface and conditions (5.17) at the downstream interfaces.

In Fig. 5.10 and Fig. 5.11 we present the results, in comparison with the ones of a fully1D model. Results underline that the RCL based conditions can actually obtain goodsolutions, in particular at the inlet. At the outlets of the carotid arteries the solution isstrongly dissipated in the flow rate, while the fully 1D and the LP models are in goodagreement for what concerns the phase of both flow rate and area, and the amplitude ofthe latter. Impact of the time discretisation of the matching conditions and the selectionof the parameters on the dissipation error is under investigation.

Fig. 5.12 illustrates velocity and pressure fields in the 3D rigid model. On the left wepresent a detail of the 3D velocity field, on the right a multiscale perspective couplinglocal and global dynamics.

91


0 0.25 0.5 0.75 1−2

0

2

4

6

8

10

12

14Flow rate at last node of CCA

time (s)

veloc

ity flu

x (cm

3 /s)

fully 1Dlumped parameter interface

0 0.25 0.5 0.75 10.36

0.38

0.4

0.42

0.44

0.46Area at last node of CCA

time (s)

area (

cm2 )


Figure 5.10: Comparison of the results obtained with a fully 1D model (solid line) andthe 3D rigid with RCL conditions at the inlet of Common Carotid Artery(dashed line). Left: flow rate, Right: area. (time in [s], volumetric flow ratein [cm3/s],area in [cm2]).

0 0.25 0.5 0.75 1−1

0

1

2

3

4

5

6

7Flow rate at first node of ICA

time (s)

veloc

ity flu

x (cm

3 /s)


0 0.25 0.5 0.75 10.195

0.2

0.205

0.21

0.215Area at first node of ICA

time (s)

area (

cm2 )


0 0.25 0.5 0.75 1−1

0

1

2

3

4

5

6

7Flow rate at first node of ECA

time (s)

veloc

ity flu

x (cm

3 /s)


0 0.25 0.5 0.75 10.15

0.155

0.16

0.165

0.17Area at first node of ECA

time (s)

area (

cm2 )


Figure 5.11: Comparison of the results obtained with the fully 1D model (solid line)and the 3D rigid with RCL conditions in the branches (dashed line). Top:Left: flow rate in the Internal Carotid Artery (ICA), Right: area in the ICA.Bottom: Left: flow rate in the External Carotid Artery (ECA), Right: area inthe ECA. (time in [s], volumetric flow rate in [cm3/s], area in [cm2]).

92


Figure 5.12: Left: Representation of the 3D solution (velocity and pressure). Right: Cou-pling of 3D and 1D computations.

5.4.1 Remarks and perspectives

For simple multiscale models, where the parameter quantification for the matchingconditions is straightforwardly suggested by the mathematical derivation of the model,numerical results are really promising, showing that the multiscale 3D/0D/1D modelcan both capture the correct wave propagation (in comparison with a fully 1D model)and compute the local 3D flow. In more complex situations, like the circle of Willis inthe cerebral vasculature, when a direct physiological quantification of the parametersis missing, results are only partially good. More precisely, at the inlet of the 3D modelresults still compare correctly with a fully 1D model, while downstream with respect tothe 3D model dissipation effects in the flow rate are dominant.

A mathematically sound fine tuning of the parameters is required. This goal can bepursued by a systematic sensitivity analysis or by extensive comparisons with stand-alone fully 3D models (see e.g. [4,80]). This subject will be investigated in future workstogether with a validation of this approach in more complex networks.

We finally point out that this approach can be extended to hydraulic networks fea-turing compliant pipes, beyond the specific medical applications considered here.

93

6 Computational tools

The need for effective tools for the numerical solution of differential problems moti-vates the development of efficient and application-specific algorithms and techniques.In this Chapter we present the LifeV software project (Sec. 6.2) and we mention its manyyears’ application to the study of blood flow problems. As an example of its features,we discuss here in particular the implementation of the data structures which allow therepresentation of the cardiovascular system as a network of vessels represented by 1Dmodels (Sec. 6.3).

6.1 An introductory note on C++

C++ is a statically-typed general-purpose language relying on classes and virtualfunctions to support object-oriented programming, templates to support genericprogramming, and providing low-level facilities to support detailed systems pro-gramming. Bjarne Stroustrup [135]

Although C++ supports different programming paradigms, as pointed out by thewords of its very creator, it is usually thought as an object-oriented language. Indeed,this aspect is particularly interesting for software applications dealing with the mod-eling of physical systems, as we will briefly discuss in this introduction. On the otherhand, its proximity with C language has not only an historical reason, but also a philo-sophical motivation. It enhances the language with features allowing the programmerto control the software behaviour at a very low level (near to the machine language).Versatility is perhaps the key to C++ success in the software industry. ∗

The object-orented programming paradigm is based on abstract data types, knownin C++ as classes. The problem to be solved is modeled by a set of objects, storingprivately all the data and providing the algorithms to operate on the data. Objectsare considered as independent but can interact by exchanging messages. Moreover,hierarchies of objects can be defined through the concept of inheritance: generic classescan provide the shared behaviour to different specialized classes, which add customdata and algorithms for the fulfillment of specific tasks.

A typical feature of C++ language allows functions and classes to operate with dif-ferent data types through a unique interface: they are defined as (function and class)templates. Important examples of the use of templates are the classes provided by the

∗According to www.langpop.comC++ is the fourth more popular programming language, as of January2009, based on several indicators including an estimate of the number of job offers requiring C++knowledge.

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C++ Standard Template Library (STL), enhancing the language with a set of powerfultools, in particular a framework for the definition of containers (such as vector, list, andmap) and algorithms using containers. The generic-programming paradigm is indeedwell illustrated by STL containers, which represent the abstract concept of a collectionof items, with a set of rules to operate on them. The same general definition and thesame policies (such as how to access to the data, how to add or delete elements) canapply for instance to vectors of numerical values, or to vectors of classes (even vectorsof vectors).

Object-oriented and generic-programming paradigms are particularly attractive forthe modeling of complex physical systems: the data structures can be designed tomimic the behaviour of different interacting (physical or logical) components and theirrelationship can be described in a general and flexible way. The same code pattern canbe effectively adapted to describe different problems with a similar general structure.In particular, this ideas can be applied to the representation of mathematical objects,such as functions or functional spaces, and to the definition of general sets of rules gov-erning their interaction. Several software projects, indeed, propose the implementationof numerical methods for the solution of mathematical problems: an increasing numberof them exploit an object-oriented programming paradigm, and they are often writtenin C++. We want to mention here some important examples, such as the Trilinos project† including a huge set of packages for several different applications (a non-exhaustivelist comprehends the solution of linear systems and preconditioning problems, withsupport to parallel computing) and the OpenFOAM toolbox ‡ for the solution of partialdifferential equations.

6.2 LifeV: a C++ finite element library

LifeV § is a software project born from the joint collaboration of three institutions: ÉcolePolytechnique Fédérale de Lausanne (CMCS) in Switzerland, Politecnico di Milano(MOX) in Italy and INRIA (REO) in France. The Department of Mathematics and Com-puter Science of Emory University in Georgia (USA) started a collaboration since 2008.LifeV consists of the implementation in C++ language of algorithms and data structuresfor the numerical solution of partial differential equations. More precisely, LifeV pro-vides an abstract framework for the implementation of Galerkin finite element meth-ods, exploiting the object-oriented paradigm supported by the programming language.

The development and maintainance of the core of the library are motivated by the re-search interests of the developers, mostly active in the numerical analysis and computerscience fields. Some applications of the library include the the design and testing of ef-ficient numerical techniques for fluid-structure interaction problems [42], algorithmsfor the solution of the Navier-Stokes equations [18, 54], numerical techniques for thecoupling of different models in a multiscale perspective [86,95], preconditioning strate-

†http://trilinos.sandia.gov/‡http://www.opencfd.co.uk/openfoam/§www.lifev.org

95

http://trilinos.sandia.gov/

http://www.opencfd.co.uk/openfoam/

www.lifev.org


gies for the Bidomain problem (arising in the modeling of the electrical activity of theheart) [55]. Moreover, software based on LifeV has been extensively used in researchprojects focused on the modeling of blood flow problems, such as the drug release fromimplantable stents [142], the design of medical procedures in cardiology [27], and thestudy of cerebral hemodynamics (the Aneurisk project, which motivated this work -see Chap. 4).

6.2.1 Code features

The code is hosted on a CVS ¶ server, which collects and organizes the contributionsof the developers from the four different institutions. Stable releases of the code arepublished on the web portal www.lifev.org, which also contains the code documen-tation, and a gallery of applications to different modeling problems.

The portability of the library is enhanced by the GNU build system ‖, allowingLifeV to compile on all Unix-like systems (also on Windows systems running Cygwin).Third party required libraries are the standard linear algebra packages BLAS and LA-PACK and linear solver packages (such as Aztec, Trilinos, PetSC, UMFPACK). More-over, LifeV is based on some of the extended C++ functionalities implemented in theBoost ∗∗ libraries, such as smart pointers (helping in effective memory management).

Figure 6.1: LifeV code structure from www.lifev.org

A graphical representation of the code organization is presented in Fig. 6.1. The dif-ferent parts have a hierarchical relationship based on their degree of specialization.More in detail, lifecore contains general components not directly related to scientificcomputing, such as the definition of the numerical types adopted in the code and a setof assertion macros to help code correctness. lifearray mainly deals with the definition

¶The Concurrent Versions System (CVS), is a revision control system keeping track of all the work andall the changes in a set of files, managing the concurrent contributions of the developers.

‖also known as Autotools∗∗www.boost.org

96

www.lifev.org

www.lifev.org

www.boost.org


of array structures used in the code (vector and matrices), while a generic set of algo-rithms, both inherited from linear solver packages and specifically written for LifeVapplications, is contained in lifealg. Mesh handling tools (for 1D, 2D or 3D meshes) areimplemented in lifemesh.

Most part of the data structures designed for the construction of finite element solversis implemented on the basis of these library components. Classes representing genericfinite elements, the definition of the geometrical mapping and the quadrature rules fortheir characterization are contained in lifefem, together with methods for matrix assem-bly and boundary conditions management. lifesolver is a collection of several classes,each dealing with the set up and the solution of a specific problem (e. g. the Navier-Stokes problem or the Darcy problem): these objects take care of the construction of thematrices describing the linear system of equations of the discretized model, and invokethe linear solver for their solution. Methods for the data import and export from and tofiles are collected in lifefilters, in particular enhancing the library with the capabilities ofmanaging different file formats.

The testsuite shows examples of software built over the library, and spans the mainapplications from simple tasks such as mesh import from file and matrix assembly, tomore complex problems such as the Navier-Stokes problem on simple geometries andwith a small number of unknowns.

Finally, we mention that recently the library has been extended with the support forparallel computing, mantaining the same general structure here presented. Preliminarytests show good scalability performances.

6.3 Implementation of networks of 1D models

As discussed in Chap. 2, one-dimensional models offer an effective representation ofthe wave propagation phenomena in large parts of the circulatory system. This moti-vates to the design of a robust and flexible software tool, able to manage networks of1D models of general topology in a simple and effective way.

A network of one-dimensional models is here regarded as a graph, in which edgesstand for the models themselves while vertices represent the interfaces between themodels. As shown in Fig. 6.2, an interface can correspond to the center of a branching.Note that when studying vascular networks, it is natural to identify inflow and outflowsections, describing a reference blood flow path: this induces an orientation (and there-fore a reference system) in the graph, and is useful in order to study some aspects ofthe fluid dynamics in the network (in particular energy balance at the interfaces, seeSec. 6.3.2).

In this implementation of 1D model network we resorted to the data structures ofboost::graph library (BGL), providing a generic interface for traversing graphs. Specificfunctionalities are added to the graph by associating LifeV 1D model objects with itsedges. Moreover, to keep the network independent on the specific implementation ofthe 1D model, OneDNet has been built as a template class (with the 1D model class astemplate parameter).

97


interface

outflow section

inflow section

inflow section

1D model

1D model

outflow section

1D model1D model

Figure 6.2: 1D model networks can be seen as oriented graphs

template < class SOLVER1D >class OneDNet

public :

typedef SOLVER1D SolverType ;/ / ! Boost shared p o i n t e r to 1D so lve r c lasstypedef typename boost : : shared_ptr < SolverType > OneDSolverPtr ;

;

We expect to work with sparse graphs, in which the number of edges is of thesame order of magnitude as the number of vertices: for this case BGL implements anadjacency-list representation ††. We will adopt this as the underlying data structure forOneDNet, providing it with a private member _M_Network of class Network.

template < class SOLVER1D >class OneDNet

private :

/ / ! boost : : a d j a c e n c y _ l i s t v a r i a b l eNetwork _M_network ;

;

Network is defined as the template class boost::adjacency_list which stores a list ofvertices and a list of out-edges for each vertex, that are edges oriented outwards withrespect to a vertex. The first template parameter (boost:: listS ) determines what kind

††http://www.boost.org/libs/graph/doc/index.html

98

http://www.boost.org/libs/graph/doc/index.html


of container is used to store the out-edges list: this affects time complexity of adding/re-moving edge operations. Admitted choices include some of the container objects, asdefined in the Standard Template Library (STL): in particular, an STL list turns out tobe a better choice compared to an STL vector, since the latter occasionally needs reallo-cation in adding elements operations [128].

The second template parameter (boost::vecS) concerns vertices list: in this casevector seems to be the good choice since we do not expect the need to add or removevertices after the initialization of the graph; moreover STL vector has a low per vertexspace overhead compared to STL list , which needs to store three extra pointers pervertex [128].

The third template parameter boost:: bidirectionalS selects a directed graph, whichprovides functions to recover in-edges and out-edges associated to each vertex. Thefourth and fifth template parameters (OneDVesselsInterface and OneDVessel) are user-defined classes containing properties to be attached respectively to vertices and edges.This features of the boost::adjacency_list class are referred to as bundled properties, andallow an easy and flexible definition of the attributes of the graph, as we will see lateron.template < class SOLVER1D >class OneDNet

public :

typedef boost : : ad jacency_ l i s t <boost : : l i s t S ,boost : : vecS ,boost : : b i d i r e c t i o n a l S ,OneDVesselsInterface ,OneDVessel>

Network ;

;

In particular, we want each edge to be identified by a numerical index and associatedto an object representing a 1D model as implemented in LifeV. Each vertex is insteadassociated to two numbers (a numerical index and a type) and to a boolean value (inter-nal) whose precise meaning will be explained in Sec. 6.3.1.template < class SOLVER1D >class OneDNet

public :

struct OneDVesselsInterface i n t index ; / * ! < numer ica l l a b e l * /bool i n t e r n a l ; / * ! < i s i t an i n t e r n a l i n t e r f a c e ? * /i n t type ; / * ! < the type of the i n t e r f a c e * /

;

struct OneDVessel i n t index ; / * ! < numer ica l l a b e l * /OneDSolverPtr onedsolver ; / * ! < p o i n t e r to 1D so lve r c lass * /

;

99


;

The access to edges and vertices of an adjacency_list is performed simply by sub-scripting the graph with the proper descriptor. Therefore the bundled properties of edgesand vertices are easily set and retrieved, as is shown in the example here below:

/ / ! Desc r i p to r type f o r the l i s t o f v e r t i c e stypedef typename boost : : g raph_ t ra i t s < Network > : : v e r t e x _ d e s c r i p t o r Vertex_Descr ;Vertex_Descr vd ;UInt i ( 0 ) ;/ / se t index i f o r ve r tex vd_M_network [ vd ] . index = i ;/ / p r i n t out ve r tex vd ’ s indexs td : : cout << _M_network [ vd ] . index << std : : endl ;

/ / ! Desc r i p to r type f o r the l i s t o f edgestypedef typename boost : : g raph_ t ra i t s < Network > : : edge_descr ip tor Edge_Descr ;Edge_Descr ed ;UInt t ( 0 ) ;/ / se t type t f o r edge ed_M_network [ ed ] . type = t ;/ / p r i n t out edge ed ’ types td : : cout << _M_network [ ed ] . type << std : : endl ;

6.3.1 Building the graph

I

VII

VI

II

t 1

t 4

t 3

t 2

t 5

t 6

III

IV

V

Figure 6.3: A test case: 6 connected tubes. Label ti indicates tube i, while Roman nu-merals are associated to interfaces

100


Consider the case depicted in Fig. 6.3: it’s a simple network composed by 6 edgesand 7 vertices. Vertex II is internal to the network, meaning that it is connected to morethan one edge. Conversely, all the others are terminal vertices. Edges 1, 3, 5 are in-edgeswith respect to vertex II; edges 2, 4, 5 are out-edges.

left interface right interface

1D model

Figure 6.4: Left and right interfaces for 1D model.

Each oriented edge has intrinsically a local reference system: in this reference systemit is possible to unequivocally identify a left and a right interface, as shown in Fig. 6.4.Therefore, the entire Network structure can be built from a list of vertices and a connec-tivity list whose elements are left-right interface pairs, each one describing an edge.

The constructor of the OneDNet class visits the graph and sets the internal bool pa-rameter associated to the vertices, on the basis of the network topology. This infor-mation is used to manage the prescription of boundary conditions to the 1D solversassociated to the edges: edges connected to terminal vertices need standard boundaryconditions, which are managed by the attached solvers. Edges connected to an internalvertex need instead a set of interface conditions, which are managed by the OneDNetclass, as we will discuss later on.

The type of an internal vertex is set by the user, and determines how the 1D modelsinteract in the corresponding interface. In other words, according to the interface type,different sets of interface conditions are imposed to the connected models. For the caseof blood flow problems, in which networks such as the considered one would representbranching vessels or circulatory anastomosis, reasonable interface conditions wouldprescribe the conservation of the flowing mass and of its kinetic energy. Other possiblechoices include interface conditions taking into account the energy losses associatedto branchings (for example depending on the flow velocity at the interface or on thebranching angles of the vessels, see [49]) or the presence of a stenosis or of a valve atthe interface between two vessels or two different tracts of the same vessel (which isthe case of the venous system, for instance).

Visiting the graph

In many cases, applying an operation to the network means to recursively apply thesame operation to each 1D model in the network. As an example we recall hereOneDNet::timeAdvance method:

template < class SOLVER1D >void OneDNet<SOLVER1D> : : timeAdvance ( const Real& t ime_va l )

/ / impose i n t e r f a c e cond i t i ons a t i n t e r n a l v e r t i c e scomputeInterfaceTubesValues ( ) ;

101


/ / i t e r a t o r s to v i s i t edge l i s tEdge_I ter e i , ei_end ;

for ( t i e ( e i , ei_end ) = edges ( _M_network ) ; e i != ei_end ; ++ e i ) / / t e l l me what I am doingDebug ( 6330 ) << " [ OneDNet : : timeAdvance ] 0− Time advancing tube "

<< _M_network [ * e i ] . index << " \ n " ;/ / c a l l OneDModelSolver method_M_network [ * e i ] . onedsolver−>timeAdvance ( t ime_va l ) ;

The first step in this method is the call to computeInterfaceTubesValues function, inorder to evaluate the interface conditions in correspondence to internal vertices of thenetwork (see Sec. 6.3.2). Then the edges of the graph are visited through iterators andthe timeAdvance method from the attached 1D models is invoked, for the prescriptionof the boundary conditions at each time step.

We remark here that we chose to provide class OneDNet with member functionshaving the same name and the same parameter list as the corresponding methods in the1D model class as implemented in LifeV. In this respect, class OneDNet can be regardedas a wrapper of LifeV :: OneDModelSolver class. On the other hand, this can be seen asa precondition on the template parameter SOLVER1D, which needs to share the samepublic interface as OneDModelSolver in order to work in the network class. One wayto achieve this behaviour is by means of inheritance: specialized classes can be defined,providing alternative implementations of 1D models (for instance, exploiting differentnumerical discretization strategies) and still retaining the same shared public interface.

6.3.2 Interface conditions

As previously mentioned, OneDNet class features the computeInterfaceTubesValuesmethod, visiting the graph vertices and setting up the interface problem, when neededaccording to vertex type. For the sake of simplicity, we will refer here only to problem(2.25), namely the prescription of the mass conservation and the continuity of the totalpressure across the interface. In Listing 6.1, this problem corresponds to vertex type1. The same approach applies however to different interface problems, for instanceinvolving concentrated energy losses as discussed in Sec. 2.3.

Listing 6.1: OneDNet::computeInterfaceTubesValuestemplate < class SOLVER1D >voidOneDNet<SOLVER1D> : : computeInterfaceTubesValues ( )

/ / ve r tex i t e r a t o r s ( f o r v i s i t i n g the graph )s td : : pa i r < Ver tex_ I te r , V e r t e x _ I t e r > v ip ;/ / v i s i t ve r tex l i s tfor ( v ip = v e r t i c e s ( _M_network ) ;

v ip . f i r s t != v ip . second ;++ v ip . f i r s t )

Debug ( 6330 ) << "0− Computing I n t e r f a c e "

102


<< _M_network [ * v ip . f i r s t ] . index << " \ n " ;/ *

In p r i n c i p l e , i t i s poss ib le to implement d i f f e r e n t behaviour f o ri n t e r f a c e s ( e . g . energy d i s s i p a t i o n due to branching angles ,vascu lare valves etc ) . ( not done yet )

* /i f ( _M_network [ * v ip . f i r s t ] . i n t e r n a l ) / / i n t e r n a l i n t e r f a c e

switch ( _M_network [ * v ip . f i r s t ] . type ) case 0:

/ *i n f l o w tube : no " i n t e r f a c e " cond i t i ons( ac tua l boundary cond i t i ons ins tead )

* /break ;

case 1:i n t e r f a c e _ c o n t i n u i t y _ c o n d i t i o n s ( v ip . f i r s t ) ;break ;

case 99:/ *

ou t f l ow tube : no " i n t e r f a c e " cond i t i ons( ac tua l boundary cond i t i ons ins tead )

* /break ;

/ / o ther cases can be added here !

defaul t :s td : : cout << " \ n [ OneDNet : : computeInter faceValues ] Unknown type "

<< _M_network [ * v ip . f i r s t ] . type<< " f o r ve r tex " << _M_network [ * v ip . f i r s t ] . index<< std : : endl ;

break ; / / sw i tch

/ / i f / / f o r

Let’s consider, for each 1D model in Fig. 6.3, the physical quantities total pressure Pt

and mass flow Q: now problem (2.25) reads

f = 0 , (6.1)

having set f = (f(1), . . . , f(6)) and

f(1) =∑

i=1,3,5Qi − (∑

j=2,4,6Qj)

f(k) = Pt,k − Pt,1, for k = 2, . . . , 6 .

The subscripts in the previous equations indicate that the considered quantity has to bereferred to the edge marked by the corresponding numerical label.

We remark that f(1) expresses the sum of the mass flows computed by 1D modelsolvers. In-edges give a positive contribution to the balance equation, while out-edgesgive a negative contribution. The continuity of mass flow follows from the balance ofpositive and negative flows.

The continuity of total pressure is imposed by taking one edge as a reference andimposing that each one of the others features the same total pressure at the interface

103


node. In this case, for in-edges the interface is located at the right boundary node, whilefor out-edges it’s on the left boundary.

−−

+

+

in-edge

out-edge

in-edge

out-edge

Figure 6.5: Edges connected to a vertex have a label and a signum.

Method interface_continuity_conditions takes care of setting up system (6.1). In orderto do that, two helping structures are built in each vertex. First of all, a map associatesa numerical label to 1D solvers associated to both in-edges and out-edges. This label doesnot necessarily correspond to the edge index, but is rather an ordinal number associatedto the edges connected to the considered vertex. A second map is devised to identifythe orientation of the edge with respect to the interface: the edge label is associated to abool flag (true for positive, false for negative orientation). A graphical representationof this approach is presented in Fig. 6.5.

template < class SOLVER1D >voidOneDNet<SOLVER1D> : : i n t e r f a c e _ c o n t i n u i t y _ c o n d i t i o n s ( V e r t e x _ I t e r const& ver tex )

/ / map the edges to t h e i r numer ica l l a b e lMapSolver in te r faceTubes ;/ / map i d e n t i f y i n g in−edges (+ , t r ue ) and out−edges(− , f a l s e )s td : : map< int , bool > signum ;

/ / take i n t o account i n t e r f a c e type/ / you expect t h i s i n t e r f a c e to have both in−edges and out−edges/ / p a i r o f in−edge i t e r a t o r ss td : : pa i r < In_Edge_I ter , In_Edge_I ter > i n_edge_ i t e r_pa i r ;/ / v i s i t the l i s t o f in−edgesfor ( i n_edge_ i t e r_pa i r = in_edges ( * ver tex , _M_network ) ;

i n_edge_ i t e r_pa i r . f i r s t != i n_edge_ i t e r_pa i r . second ;++ in_edge_ i t e r_pa i r . f i r s t )

/ / i n s e r t edges i n t o the mapi n te r faceTubes . i n s e r t

( MapSolverValueType ( i ,_M_network [ * ( i n_edge_ i t e r_pa i r ) . f i r s t ] . onedsolver ) ) ;

104


/ / in−edges have "+" signum ( boolean value t rue )signum . i n s e r t ( s td : : map< int , bool > : : value_type ( i , true ) ) ;i ++;

/ / p a i r o f out−edge i t e r a t o r ss td : : pa i r <Out_Edge_Iter , Out_Edge_Iter > ou t_edge_ i te r_pa i r ;/ / v i s i t the l i s t o f out−edgesfor ( ou t_edge_ i te r_pa i r = out_edges ( * ver tex , _M_network ) ;

ou t_edge_ i te r_pa i r . f i r s t != ou t_edge_ i te r_pa i r . second ;++ ou t_edge_ i te r_pa i r . f i r s t )

/ / i n s e r t edges i n t o the mapi n te r faceTubes . i n s e r t

( MapSolverValueType ( i ,_M_network [ * ( ou t_edge_ i te r_pa i r ) . f i r s t ] . onedsolver ) ) ;

/ / in−edges have "−" signum ( boolean value f a l s e )signum . i n s e r t ( s td : : map< int , bool > : : value_type ( i , fa lse ) ) ;i ++;

The non linear problem (6.1) is solved by applying a Newton iterative scheme. Thesolution at each time step is then obtained as:

xk+1 = xk − J−1f

∣∣xk f(xk) ,

J−1f

∣∣xk being the jacobian matrix of f while x contains the interface unknowns Ai, Qi,

i = 1, . . . , 6. The code invokes the LAPACK routine dgesv, to compute J−1f

∣∣xk f

through LU decomposition of the jacobian matrix, and then updates the solution atcurrent iteration. Convergence is achieved when mass conservation is ensured by thefulfillment of the following request [45]:

f(1) < tol

tol being a user-defined tolerance./ / unknown of non l i n e a r equat ion f ( x ) = 0Vector x ;/ / non l i n e a r f u n c t i o n fVector f ;/ / jacob ian o f the non l i n e a r f u n c t i o nMat r i x j ac ;/ / t ranspose of the jacob ian o f the non l i n e a r f u n c t i o nMat r i x j ac_ t rans ;/ / tmp mat r i x f o r lapack l u i n v e r s i o nboost : : numeric : : ublas : : vector < In t > i p i v ;/ / lapack v a r i a b l ei n t INFO [ 1 ] ;i n t NBRHS[ 1 ] ; / / nb columns of the rhs := 1 .i n t NBU[ 1 ] ;/ *

x conta ins the ( unknown ) boundary values f o r edges connectedto the considered i n t e r f a c e .

* /x . res i ze ( f _ s i z e ) ;x . c l ea r ( ) ;/ *

prepare the data s t r u c t u r e s needed f o r so l v i ng non l i n e a r equat ions

105


( c o n t i n u i t y + c o m p a t i b i l i t y )* /f . r es i ze ( f _ s i z e ) ;f . c l ea r ( ) ;j ac . res i ze ( f_s ize , f _ s i z e ) ;j ac . c l ea r ( ) ;j ac_ t rans . res i ze ( f_s ize , f _ s i z e ) ;j ac_ t rans . c l ea r ( ) ;i p i v . res i ze ( f _ s i z e ) ;i p i v . c l ea r ( ) ;INFO [ 0 ] = 0 ;NBRHS[ 0 ] = 1 ; / / nb columns of the rhs := 1 .NBU[ 0 ] = f _ s i z e ;i =0;/ / newton raphson i t e r a t i o ndo

/ / f i l l f and i t s jacob ian mat r i xf _ j a c ( x , f , jac , in ter faceTubes , signum ) ;/ / t ranspose to pass to f o r t r a n storage ( lapack ! )j a c_ t rans = t rans ( jac ) ;/ / Compute f <− ( d f ( x )^−1 f ( x ) ) ( l u dcmp)dgesv_ (NBU, NBRHS, &jac_ t rans ( 0 , 0 ) , NBU , & i p i v ( 0 ) , & f ( 0 ) , NBU, INFO ) ;ASSERT_PRE ( ! INFO [ 0 ] ,

" Lapack LU r e s o l u t i o n o f y = df ( x )^−1 f ( x ) i s not achieved . " ) ;/ / x = x − df ( x )^−1 f ( x )x += − f ;

/ / convergence checkwhile ( ( s td : : fabs ( f ( 0 ) ) > 1e−12) && (++ n i t e r < 100) ) ;/ *

f ( 0 ) conta ins the mass f low balance : by min imiz ing i t s value we wantto ensure mass conservat ion .Moreover , we expect a low number o f i t e r a t i o n s , s ince the i n i t i a lguess i s given from the boundary cond i t i ons a t prev ious t ime step ,which are l i k e l y to be a good es t ima t ion o f the cu r ren t s o l u t i o n( t ime step i s smal l and we expect s o l u t i o n s to be cont inuousi n t ime ) .

* /

The f_jac method fills Vector f with the expressions of interface conditions and Matrixjac with the jacobian matrix Jf . For the sake of brevity we omit here the correspondingcode, noting that all the computations are straightforward, given the presented datastructure.

6.3.3 A simple example

Running a simulation for a network of 1D models is formally the same as running asimulation for a single 1D model, since an object of class OneDNet behaves exactlylike an object of class OneDModelSolver, thanks to the wrapping mechanism previouslydiscussed.

Consider again the case depicted in Fig. 6.3: the network at hand has only one in-ternal node where we want to impose continuity interface conditions (2.25). Each 1Dmodel is described by the Euler equations (2.1) with the assumption that the vessel wallbehaves as a linear elastic solid (see (2.4)). The numerical discretization is based on theTaylor-Galerkin scheme presented in Chap. 2 (2.27).

106


All the physical and numerical discretization parameters are the same for all the sixtubes, and are summarized in Tab. 6.1. Moreover tubes 1, 3, 5 feature the same bound-ary conditions at left boundary, while tubes 2, 4, 6 feature the same boundary conditionsat right boundary. More precisely, left boundary conditions for in-edges are managedby each 1D model LifeV class, and prescribe the entering characteristic variable W1

(see [49]). Since W1(t) = W1(A(t), Q(t)), it is possible to specify W1(t) by selecting afunction A(t) and a function Q(t). In this simulation we set:

A(t) = A0 (at left boundary for tubes 1, 3, 5)Q(t) = sin(4πt)

Absorbing boundary conditions (see [49]) are prescribed at right boundary for out-edges: namely, a value for the exiting characteristic variable W2(t) = W2(A(t), Q(t)) isimposed, setting:

A(t) = A0 (at right boundary for tube 2, 4, 6)Q(t) = 0

The network class manages the prescription of right boundary conditions for in-edges and left boundary conditions for out-edges, as previously illustrated.

Length 10 cmRadius 0.5 cm

Wall thickness 0.05 cmWall Young modulus 104 dyn / cm2

Wall Poisson ratio 0.5Blood mass density 1 g / cm3

Blood viscosity 0.035 poiseMesh spacing 0.1 cm

Time step 10−5 s

Table 6.1: Physical and discretization parameters for the considered numerical set up.

We simulate the dynamics of the network in a 2 s time interval, starting from initialconditions

A(t = 0) = A0 , Q(t = 0) = 0

in all the tubes. We report snapshots of the solution in Fig. 6.6. The images show thatthe sinusoidal signals (in terms of flow rate) enter tubes 1, 3, 5, correctly propagate inthe network and exit through tubes 2, 4, 6.

107


(a) t = 1.0 s (b) t = 1.1 s

(c) t = 1.2 s (d) t = 1.3 s

(e) t = 1.4 s (f) t = 1.5 s

Figure 6.6: Solutions of 6 tubes test case.

108

7 Conclusions

The study of biological systems offers many different subjects, attaining to differentresearch areas. On the one hand this motivates researchers to devise refined tools for theaccurate modeling of the phenomena of interest. On the other hand, it may promote co-operation and stimulate the creation of inter-disciplinary teams and research projects.Of course, these two aspects are bound together, since many different individual skillsmake a research team stronger and more prompt to fulfill its objectives.

The experience of Aneurisk project is representative in this respect, since it promoteda balance between the development of the technical knowledge on the side of the per-sonal research interests of each involved researcher, but at the same time forced theexchange of information, a continuous update of the group activities, and a commonvision.

The present study on the mathematical and numerical modeling of cerebral circu-lation is mostly beholden to the interactive work environment realized by Aneurisk.Several suggestions from different fields were merged to define novel methods andresearch approaches. Indeed, the study of cerebral circulation (as of many other bi-ological systems) is thwarted by the lack of data for the full characterization of theavailable models or for the validation of new ones. On the other hand, medical knowl-edge, based on long time experience on the evolution of the pathologies, can suggestthe “right questions” and stimulate the researchers to try to answer them. We expe-rienced this phenomenon through the collaboration with neurosurgeons at OspedaleNiguarda Ca’Granda in Milan, often discovering that the most interesting issues for amedical doctor were not the most hard to solve, from a technical point of view: theywere instead the more intensive to face, requiring all the members of the project toexpand their specific professional skills to find a way to co-operate. In this context,many results here presented do not stand as goals, but rather as milestones, tracking atrail towards a deeper understanding of the physiology and the pathology of the braincirculation.

We discussed in Chap. 2 about the flexibility of one-dimensional models, suitable forthe description of large and complex vascular networks, in different physiological anpathological conditions. In particular, we studied the effects of the mechanical featuresof the vascular wall on the wave propagation phenomena typical of the circulatorysystem. The number of potential applications of reduced models, due to their proveneffectivity in the study of vascular networks, calls for the design of efficient and robustsoftware tools. In Chap. 6 we addressed this issue, by presenting some excerpts ofthe software specifically written in the context of this work for the simulation of thecirculatory system.

In Chap. 3 we presented 3D models for blood flow, being aware that in order to

109

7 Conclusions

understand the deepest mechanisms of vascular pathology initiation and progression,the detailed description of the mechanical action of blood flow on the vessel wall isrequired. For the application of these models to the study of cerebral aneurysms, dis-cussed in Chap. 4, we resorted to an integrated approach able to exploit at most theexpertise of the different members of project Aneurisk and the available data. Start-ing from medical images, the accurate geometry reconstruction of a large number ofcerebral vessels was performed, and some patterns in the location of aneurysms in thedata set were observed [103]. Advanced statistical techniques allowed the definition ofa classification of the vascular geometries, correlating morphological features to aneu-rysm position [120]. In the present work, new classification criteria were suggested bythe definition of a hemodynamic parameter for the estimation of the mechanical load onthe peri-aneurysmal region of the arterial wall. This work could lead to the formulationof a novel risk index for the cerebral aneurysm in the internal carotid artery.

Accurate modeling of the blood flow features in specific districts of the vascular sys-tem and the understanding of the mechanisms of the wave propagation in the entirenetwork form a prelude to an integrated research on the correlation between the me-chanic features of the vascular tree as a whole and the localization of the wall diseases.Chap. 5 describes the two different perspectives embodied in 1D and 3D models andtheir use in a single, coupled model. This approach has been applied to the descriptionof the blood flow in a complex vascular network, including a 1D model of the circle ofWillis and a 3D model of a carotid bifurcation.

In the vision of Aneurisk project, and in the intentions of the author, the modelingtechniques developed and presented in this thesis are to be regarded as components ofa more general stream of information, which connects the medical image to the clinicalpractice, through added layers of knowledge from different interacting sources.

110

Acknowledgements

First of all, I would like to acknowledge the Aneurisk project, for funding my researchactivity during the period May 2005 - April 2008 with a grant provided by FondazionePolitecnico di Milano and Siemens Medical Solutions Italy.

My gratitude goes in particular to prof. Alessandro Veneziani, head of the Aneuriskproject and tireless supervisor and guide of my research activity.

To all of the other researchers involved in the project, for their friendship, besidestheir professional skills, which made Aneurisk an exciting and gratifying environmentfor a young researcher like me.

To all the friends and colleagues at MOX, the laboratory of Modeling and ScientificComputing of the Department of Mathematics, Politecnico di Milano, for being the bestgroup in Italy where to enjoy mathematics and numerical modeling.

To all the special people I met at the Department of Mathematics and Computer Sci-ence of Emory University, Atlanta, for having welcome me last year in an exciting en-vironment and having offered me a gratifying experience abroad.

To Conferenza dei Rettori delle Università Italiane (CRUI) and British Council, forpartially supporting my research activity with the grant awarded to the project “Nu-merical Modelling of Cerebral Blood Flow and Auto-regulation”, in collaboration withDr. Jordi Alastruey, Dr. Carlo D’angelo, prof. Joaquim Peiró and prof. Luca Formaggia.

Special thanks go to Dr. Jordi Alastruey for his professional support and friendlymind in the collaboration on the study of 1D models for the cerebral circulation.

To prof. Luca Formaggia for having always kept his door open to visits and ques-tions.

To Prof. Spencer Sherwin and Dr. Rod Hose for the detailed review of my thesis andthe many stimulating comments on my work.

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