POLITECNICO DI MILANO Dipartimento di Matematica F. Brioschi Ph. D. course in Mathematical Engineering XXI cycle
Computational hemodynamics of the cerebral circulation: multiscale modeling from the circle of Willis to cerebral aneurysms
Ph. D. candidate: Tiziano PASSERINI
Milano, 2009
Supervisor: Prof. Alessandro VENEZIANI To Lucia Contents
Abstract 1
1 Introduction 3 1.1 Anatomy and physiology of the cerebral circulation ...... 3 1.1.1 The circle of Willis ...... 4 1.2 Morphology and fluid dynamics of cerebral aneurysms ...... 6 1.2.1 The role of hemodynamics ...... 10 1.3 Modeling the cerebral circulation ...... 12 1.3.1 The circle of Willis ...... 13 1.3.2 Cerebral aneurysms ...... 14
2 One-dimensional models for blood flow problems 15 2.1 Wave propagation phenomena in the cardiovascular system ...... 15 2.1.1 Modeling the vascular wall ...... 16 2.2 Formulation of the model ...... 17 2.2.1 A viscoelastic structural model for the vessel wall ...... 19 2.2.2 The linearized model ...... 21 2.3 Networks of 1D models ...... 27 2.4 Numerical discretization ...... 28 2.4.1 Numerical solution of the viscoelastic wall model ...... 29 2.5 Results and discussion ...... 30 2.5.1 Validation of the numerical model versus an analytical solution . 30 2.5.2 Wave propagation in a single 1-D vessel: a Gaussian pulse wave 32 2.5.3 Wave propagation in a single 1-D vessel: a sinusoidal wave . . . 34 2.5.4 A 1D model network: the circle of Willis ...... 35
3 Three-dimensional models for blood flow problems 39 3.1 Blood flow features in arteries ...... 39 3.2 Geometry and Flow ...... 40 3.2.1 Reynolds number ...... 41 3.2.2 Dean number ...... 41 3.2.3 Womersley number and Reduced Velocity ...... 43 3.3 The Navier-Stokes equations ...... 44 3.3.1 Formulation ...... 44 3.3.2 Numerical discretization ...... 48 3.4 Wall shear stress in the Navier-Stokes problem ...... 50
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3.4.1 Approximation for the velocity gradient ...... 50 3.4.2 Oscillatory Shear Index ...... 52 3.5 Working on regions of interest ...... 53 3.5.1 Decomposition of bifurcation branches ...... 53 3.5.2 Relating surface points to centerlines ...... 54
4 An application of three-dimensional modeling 59 4.1 Cerebral hemodynamics ...... 59 4.2 The Aneurisk project ...... 60 4.3 Hemodynamic features of the Internal Carotid Artery ...... 62 4.3.1 Discussion ...... 71 4.3.2 Wall shear stress as a classification parameter ...... 74
5 A geometrical multiscale model of the cerebral circulation 77 5.1 The compliant vessel problem ...... 77 5.2 Matching conditions in 3D rigid/1D multiscale models ...... 78 5.2.1 Numerical algorithm ...... 79 5.2.2 Matching conditions including compliance ...... 80 5.2.3 Parameters estimation ...... 85 5.2.4 Results ...... 86 5.3 A 1D-3D-1D coupling ...... 88 5.3.1 Results ...... 89 5.4 The 3D carotid model and the multiscale coupling ...... 91 5.4.1 Remarks and perspectives ...... 93
6 Computational tools 94 6.1 An introductory note on C++ ...... 94 6.2 LifeV: a C++ finite element library ...... 95 6.2.1 Code features ...... 96 6.3 Implementation of networks of 1D models ...... 97 6.3.1 Building the graph ...... 100 6.3.2 Interface conditions ...... 102 6.3.3 A simple example ...... 106
7 Conclusions 109
Acknowledgements 111
ii Abstract
In this work we address the mathematical and numerical modeling of cerebral circu- lation. In particular, one-dimensional (1D) models are exploited for the representation of the complex system of cerebral arteries, featuring a peculiar structure called circle of Willis. These models, based on the Euler equations, are unable to capture the lo- cal details of the blood flow but are suitable for the description of the pressure wave propagation in large vascular networks. This phenomenon is driven by the mechanical interaction of the blood and the vessel wall, and is therefore affected by the mechanical features of the wall. Chap. 2 deals with 1D models taking into account the wall 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 is found for the latter formulation and is used to validate the adopted numerical scheme (Sec. 2.4 and Sec. 2.5). Finally, the effect of wall viscoelasticity on the wave propaga- tion phenomena is studied in some numerical experiments representative of realistic conditions in the cardiovascular and cerebral arterial systems. The details of the blood flow can be studied by means of three-dimensional (3D) 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 medium and large arteries, and in particular allow the evaluation of the stress field in the fluid. Thus, it is possible to estimate the traction exerted by the blood flow on the vessel wall (wall shear stress, defined in Sec. 3.4). Moreover, by exploiting the representation of the vascular tree in terms of centerlines, it is possible to easily identify regions of inter- est in the computational domain, in which to restrict the fluid dynamics analysis: this approach is presented in Sec. 3.5. Cerebral aneurysms are a disease of the vascular wall causing a local dilation, which tends to grow and can rupture, leading to severe damage to the brain. The mechanisms of initiation, growth and rupture have not been completely explained yet, but the effects of blood flow on the vascular wall are generally accepted as risk factors, as discussed in Sec. 1.2. In the context of Aneurisk project, an extensive statistical investigation has been conducted on the geometrical features of the internal carotid artery, finding that certain spatial patterns of radius and curvature are associated to the presence and to the position of an aneurysm in the cerebral vasculature (Sec. 4.2). Starting from this observation, a classification strategy for vascular geometries has been devised. In the present work, blood flow has been simulated in the patient-specific vascular geometries reconstructed in the context of the Aneurisk project, and an index of the mechanical load exerted by the blood on the vascular wall near the aneurysm has been defined. Finally, it has been shown that certain values of the mechanical load are associated to the presence and the location of an aneurysm in the cerebral circulation. Adding this
1 Contents hemodynamic parameter in the classification technique improves its efficacy (Sec. 4.3). The interaction between local and global phenomena is a typical feature of the cir- culatory system. It is believed to be crucial in the context of cerebral circulation, since defects or diseases at the level of the circle of Willis can induce local flow conditions 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 reduced models 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 of the cerebral circulation is presented, based on the coupling of a 1D representation of the circle of Willis and the 3D representation of a carotid artery. A novel method to describe the interface between the two models is discussed in Sec. 5.2. The number of potential applications of reduced models, due to their proven effec- tiveness in the study of vascular networks, calls for the design of efficient and robust software tools. In Chap. 6 we address this issue, by presenting some excerpts of the software specifically written in the context of this work for the simulation of the circu- latory system (Sec. 6.3).
2 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 art knowledge on cerebral aneurysms are presented in Sec. 1.1 and Sec. 1.2, respectively. Most of the material here presented is taken from the work by Khurana & Spetzler [65]. More details and additional references to the medical literature for these topics can be found therein. The modeling of cerebral circulation, with specific attention to the blood flow prob- lems related to the development of vascular diseases, can enhance the comprehension of the pathology mechanisms and therefore help in devising treatment procedures. On the other hand, the complexity of the physical systems at hand calls for the definition of effective modeling strategies, balancing the need for a detailed description of the phys- ical phenomena and the computational cost. These issues, together with a description of the original contribution of this work in the presented framework, are discussed in Sec. 1.3.
1.1 Anatomy and physiology of the cerebral circulation
Cerebral vasculature is a complex structure, ensuring the adequate perfusion to all the brain districts [39]. Cerebral blood vessels are responsible for feeding the brain with oxygen and nutrients (brain arteries) and for the draining of metabolic waste products from the brain (brain veins). To illustrate the typical features of a cerebral artery, we refer for the sake of clarity to the schematic representation of its cross section, depicted in Fig. 1.1. The intima of brain arteries (the innermost part of the wall) is composed of a single layer of endothe- lial cells (represented as light blue cells in the figure), resting on a protein-rich layer called the basal lamina (inner part of the black circle). The outer part of the black circle represents the elastic lamina, whose main component is elastin protein, while smooth muscle cells (large red cells) form the media. Fibroblasts (thin green cells) and nerve fibers (orange fibers) are located in the adventitia (the outermost layer of the wall) and are 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 as a dark blue cell, are present only at the level of the smallest brain vessels (the brain capillaries) and provide biochemical support to the endothelial cells [65].
3 1 Introduction
Figure 1.1: Cross-section of a brain artery, showing the layers and components of the wall. The innermost part is a hollow space (the lumen) containing serum and blood cells. The cells here illustrated are not to scale for the vessels in and around 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 internal carotid arteries enter at the front, while the two vertebral arteries enter at the back. All the 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 and left posterior communicating arteries (rPCoA and lPCoA), the right and left anterior cerebral arteries (rACA and lACA) and the anterior communicating artery (ACoA). The two internal carotid arteries (rICA and lICA) feed the anterior circulation, delivering blood in the anterior part of the brain, while the two vertebral arteries (rVA and lVA) join into the basilar artery (BA), feeding the posterior circulation which delivers blood in the 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 latter course over the brain surface (cortex) and into the brain valleys (sulci), originating per- forating arterioles feeding the deeper cerebral tissue. The arterioles end in capillaries, which drain first into venules and then into larger veins. A high-volume, low-pressure venous system (the dural venous sinuses) collects blood and empties into the jugular veins 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 it can supply blood to the brain even when one or more vessels are occluded or missing.
4 1 Introduction
Anterior communicating Anterior Middle artery cerebral cerebral artery artery Ophthalmic artery
Internal carotid Anterior artery choroidal artery
Posterior communicating artery
Posterior cerebral artery
Superior Pontine cerebellar arteries artery Basilar artery
Anterior inferior cerebellar artery
Vertebral artery
Anterior Posterior spinal inferior artery cerebellar artery
Figure 1.2: Representation of the circle of Willis. Left: overview of the undersurface of 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 of the circle is absent or partially developed [74], but this finding is regarded as a normal variation of brain vessels anatomy. On the other hand, the circle protects the brain from disuniform 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 pathologies such as aneurysms develop, and is relevant in clinical practice for many intracranial or extracranial procedures like the endoarterectomy, the carotid stenting or the compres- sion carotid test (see e.g. [60]).
5 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-like structure which forms where the blood vessel wall weakens, ballooning outwards (see Fig. 1.3). The most common type of cerebral aneurysm is the saccular or berry aneurysm, similar to a sack sticking from the side of a blood vessel wall. It is usually characterised by 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 expanded in 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 an injury or break in the arterial wall. From now on, we will focus our attention on berry aneurysms, 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 Large 20 - 24 mm Near-giant > 25 mm Giant
6 1 Introduction
Small and large aneurysms behave actually in similar ways in that they tend to grow and 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 for the 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 main branches. Moreover, most tend to occur in the anterior circulation, preferentially in regions where arteries branch. Indeed brain blood vessels could be naturally weaker in such locations, which are also preferential sites for fatty plaques deposition [65]. An extensive statistical investigation of the location of cerebral aneurysms has been one of the goals of the Aneurisk research project which motivated the present work. We will discuss this point more thoroughly in Chap. 4.
Risk factors
Aneurysms may be congenital, but most of them are nowadays thought to be acquired. The main risk factors for aneurysm formation are listed in the following table: The main risk factors for aneurysm formation Hypertension Previous aneurysm Family history of brain aneurysm Connective tissue disorder Older than 40 years Female Blood vessel injury or dissection Some inherited genetic defects may predispose to the forming of aneurysms and be compounded 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 in Chap. 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 a classification 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 of cases, 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
7 1 Introduction aneurysm large size. In this case, symptoms include chronic headaches, nausea, loss of functions in the nerve bundles in the brain causing disturbs such as double vision. In other 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 brain attack, 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: this event is referred to as a warning leak. The bleed occurring after the rupture is known as subaracnoid hemorrage (SAH). In case the flow slows down in the aneurysm lumen, a thrombus may form, either before or after the rupture. Thrombosis can stop the bleeding after a rupture, but may also 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 channels of blood (recanalization), which can be associated to the growth, rupture and rerupture of the aneurysm. On the other hand, the rupture itself may cut off the supply of oxygen and other nutrients to the cells in the wall, thus further weakening it and predisposing it to a subsequent 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 suffering from 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 temporary overcontraction of cerebral arteries which can result in a stroke. It may be triggered by a SAH, due to the presence of blood in the subarachnoid space, and can last few days to three weeks. Less frequent or severe complications include hydrocefalus, seizures, cardiac stunning and sodium and fluid imbalance [65].
Detection
Cerebral angiography is frequently used to detect brain aneurysms. One of its main disadvantages is invasivity, since it requires the femoral artery to be punctured and a catheter to be inserted and navigated through the arterial tree to inject an opaque dye near to the observed region. Radiographs are taken while the dye is advected by the blood 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 these regions to be low leading ultimately to undersegmentation.
8 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 well as 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, an intravenous dye is injected into the patient during CT scanning. The resulting tech- nique is quicker, cheaper and less invasive than the traditional cerebral angiography and able to produce high-resolution, color and 3D images. Ultrasound techniques and common radiography have no role in the detection of aneurysms [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 the risk of excessive postponement of the treatment and, depending on the imaging tech- nique, the exposition to multiple invasive procedures. The former exposes patients to perioperatory risks associated to the chosen procedures. The general criterium associating a risk of rupture to aneurysms based on their size is not practically accepted, since it is believed that each brain aneurysm should be evalu- ated on an individual basis, with consideration of 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 recommended as early as possible after a rupture. Most of the different types of open surgery are based 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 blood pathway or to make it clot off and eventually shrink. Another therapeutic choice, less certain than the clipping, is the surgical reconstruction of the aneurysmal part of the wall. On the other hand, endovascular intervention requires the insertion of a catheter, typically into the femoral artery, which is navigated through the aorta and up into the brain to the region of the aneurysm. Then platinum microcoils or a “glue” or other com- posite materials can be placed in the lumen of the aneurysm in order to slow the flow of 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 its blood supply. Even combinations of the presented procedures can be performed. In all cases, 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 which would 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].
9 1 Introduction
1.2.1 The role of hemodynamics
It is accepted in the literature that hemodynamics plays a major role in the process of aneurysm formation, progression and rupture. This introduction briefly summarizes the 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 shear stress (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 vessel enlargement in order to reduce WSS to physiological values [38, 76]. However, this kind of structural remodeling can be potentially destructive, when triggered by locally increased WSS: in this situation, a damage to the arterial wall and a subsequent focal enlargement may take place [124]. On the other hand, endothelial cells can sense WSS and consequently adapt their spa- tial organization: uniform shear stress fields cause the cells to be stretched and aligned in the direction of the flow, while irregular shapes and orientation are assumed under the action of low and oscillatory wall shear stress. The latter situation promotes intimal wall thickening and potentially atherogenesis [31, 43, 50, 68], however in the particular case of cerebral aneurysms could be a protective factor against wall weakening and rupture [124]. Many clinical and experimental observations support the theory of a relation be- tween cerebral aneurysm initiation and the effects of high-flow hemodynamic forces on the arterial wall. Studies pointed out the association of cerebral aneurysms with ar- terial anatomic variations and pathological conditions such as hypoplasia or occlusion of a segment of the circle of Willis [64, 81, 117]. High-flow arteriovenous malforma- tions inducing a local increase of blood flow in the cerebral circulation [96] can promote the disease. Furthermore, aneurysms usually localize in sites of flow separation and elevated WSS such as bifurcations. These conditions were found to be associated in animal 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 systemic hypertension 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 specimens show that the aneurysmal wall is mostly composed by collagen and that it can tolerate stresses in the range of those imposed in vivo by the mean blood pressure. The rupture of an aneurysm is thought to be the result of a process of weakening of the wall, whose mechanisms have not been explained yet. In particular, it is not clear if either low or high 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 weaken under the action of abnormal shear stress fields, due to biochemical processes leading ultimately to apoptosis of the smooth muscle cells and loss of arterial tone [51]. There-
10 1 Introduction fore, the prevalence of blood pressure forces over internal wall stress forces may cause a local dilation, which then grows under the action of non physiological blood shear stresses. The wall stiffens, because of stretching of elastin and collagen fibers in the me- dial and adventitial layers. Eventually an equilibrium can be reached, in which elastin and collagen are constantly under a non physiologically large mechanical load: in this situation wall remodeling may take place.
Low blood flows in the aneurysm can cause blood stagnation in the dome, and this is believed to be the major responsible for wall damage in the low-flow theory. Stagnation promotes the aggregation of red cells, the accumulation and the adhesion of platelets and leukocytes along the intimal surface [57]. This may be a cause of inflammation, due to the infiltration of white blood cells and fibrin in the intimal layer [29]. The wall tissue then degenerates and becomes unable to support blood pressure with physiological tensile forces. In this situation the aneurysmal wall progressively thins and may finally rupture.
As previously discussed, a strong correlation between the size of aneurysms and their rate of rupture has been documented in literature. This led to the definition of a clinical measure termed aspect ratio (defined as the depth of the aneurysm divided by the neck width): it has been found that an aspect ratio bigger than 1.6 is correlated to a risk of rupture [139]. On the other hand, it is known that flow velocities in aneurysms depend 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 rupture mechanism.
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 dome of 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 aneurysm rupture [22]. Moreover, narrow necks in large aneurysms geometrically induce concen- trated inflow jets and localized impact zones: the correlation between big aspect ratio and 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 pressure exerted 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 evolution is 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 interactions with the PAE can cause non uniformly distributed or unbalanced contact, with either protective or detrimental effect on the evolution of the aneurysm [116].
11 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 three classes of models, featuring a different level of detail in the space dependence. Fully three-dimensional models (3D, see Chap. 3) are based on the incompressible Na- vier-Stokes equations possibly coupled to appropriate models that describe the blood rheology and the deformation of the vascular tissue. These models are well suited for investigating the effects of the geometry on the blood flow and the possible phys- iopathological impact of hemodynamics. Unfortunately, the high computational costs restrict their use to contiguous vascular districts only on a space scale of few centime- ters or fractions of meter at most (see e.g. [8], [56], [107]). By exploiting the cylindrical geometry of vessels, it is possible to resort to one dimen- sional models (1D), reducing the space dependence to the vessel axial coordinate only (see Chap. 2). These models are basically given by the well known Euler equations and provide an optimal tool for the analysis of wave propagation phenomena in the vascular 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 inhibits their use for the whole circulatory system. In fact, it would be unfeasible to follow the geometrical 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 and electrical circuits. The fundamental ingredient of these lumped parameter models (0D) are the Kirchhoff laws, which lead to systems of differential-algebraic equations. These models 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 presence of 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 to districts affected by a vascular occlusion thanks to the development of compensatory dynamics, strongly relies on this coupling of different space scales. Feedback mecha- nisms essential to the correct functioning of the vascular system work over the space scale of the entire network, even if they are activated by local phenomena such as an occlusion or the local demand of more oxygen by an organ. This is particularly evident in the cerebral vasculature, as mentioned earlier in this Chapter. To devise numerical models able to cope with coupled dynamics ranging on differ- ent space scales a geometrical multiscale approach has been proposed in [47]. Following this approach, the three different classes of models are mathematically coupled in a unique numerical model. Despite the intuitiveness of this approach, many difficulties arise 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 blood flow in the circle of Willis. After the first works based on hydraulic or electric analog models [11,26,41,88,115], most of the research has been based on modeling the circle of Willis 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 a feedback mechanism driving an appropriate blood supply into the circle on the basis of oxygen demand by the brain [3]. Furthermore, a complete 3D image based numerical model of the circle of Willis has been presented in [20]. This model, however, requires medical data that are currently beyond the usual availability in common practice, and is computationally intensive compared with the 1D counterpart. In the present work, the modeling of the circle of Willis is addressed from several dif- ferent viewpoints. The features of the arterial ring per se are discussed in Chap. 2, where a one-dimensional model (previously published by Alastruey et al. [2]) is studied with particular attention to the problem of correctly modeling the mechanical behaviour of the arterial wall. Its viscoelastic features affect indeed the time and space pattern of pressure 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 by using a linear elastic model (see Sec. 2.5.4). The computational study here presented is 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 network of interacting vessels, each one described by a 1D model. The design of algorithms and data 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 geometrical features of the vessels which need to be studied in detail. Reduced models (such as 1D models) are not suitable for this task: on the other hand, a full 3D modeling of a large and complex system of arteries can be unaffordable, both because of high compu- tational costs and because of the lack of medical data to completely set up the problem. In Chap. 5 we present a geometrical multiscale model for the cerebral circulation, cou- pling a detailed 3D model of a carotid bifurcation together with a reduced 1D model of the circle of Willis. The different models entail different assumptions on the mechan- ical behaviour of the vascular wall: its compliance is the driving mechanism for the propagation of pressure and flow rate waves, and is differently modeled at different geometric scales. Proper matching conditions have been devised to retrieve the correct description of the dynamics of the coupled system (see Sec. 5.2).
∗http://www.lifev.org
13 1 Introduction
1.3.2 Cerebral aneurysms In the last years, the study of the blood flow dynamics of cerebral aneurysms has been carried out with different tools. Experimental and clinical studies, focused on idealized aneurysm geometries or on surgically created aneurysms on animals, were able to show the complexity of intra-cerebral hemodynamics [121]: however, they did not explain the relation between hemodynamics and clinical events. The same limitation holds for in vitro studies, which on the other hand can give a very detailed description of the flow mechanics inside idealized geometries [73]: the main drawback for this approach is the unfeasibility 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 to recent advances in medical imaging tools, it is relatively easy to obtain accurate patient- specific geometrical models of cerebral circulation. The blood motion inside arteries and 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 mainly their validation, since the in vivo correct estimation of blood flow patterns is still an open problem within nowadays imaging technology. However, employing virtual or simulated angiography, it has been shown that CFD models are able to reproduce the flow patterns observed in vivo during angiographic examinations [23, 44]. In the context of the Aneurisk project † (see Chap. 4) a study of the internal carotid artery as a preferential site for aneurysms formation has been proposed. More precisely, starting from patient-specific geometrical modeling based on medical images [103], the parent 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 the location 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 than that, we show that by considering fluid dynamics parameters together with geometri- cal parameters for the description of the considered cerebral vessels, the classification can be enhanced. It is indeed our belief that an integrated approach, starting from the medical image and systematically collecting different sources of information for the characterization 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 2 One-dimensional models for blood flow problems
Reduced models for blood flow problems prove to be effective in capturing the main features of the wave propagation phenomena in the human cardiovascular system [19, 45, 126]. In particular, one-dimensional models based on the Euler equations offer a reliable description of the mechanics of blood-vessel interaction under the assumption of cylindrical arteries, the direction of the cylinder axis being the main direction of flow considered in the model. This approximation easily applies to large parts of the circulatory system, whenever we are not interested in the detailed description of flow features 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 problems and 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 way to 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 Euler equations can be obtained. Its derivation and the validation of the numerical discretiza- tion used to solve the equations are presented in Sec. 2.2.2 and Sec. 2.5.1 respectively. The fully non linear problem is solved in some test cases (Sec. 2.5), showing the ability of the model at hand to capture the main features of the studied problems. In the spirit of 1D representation, the circulatory system as a whole can be seen as a network of interconnected vessels. By this representation we build one-dimensional models of large regions of the circulatory system (Sec. 2.3), each vessel being described by Euler equations. The application of this approach to the study of cerebral circulation is discussed in Sec. 2.5.4.
2.1 Wave propagation phenomena in the cardiovascular system
The circulatory system is responsible for the distribution of blood flow through the human body. Blood is pumped by the heart into the network of arteries, reaches the capillaries where most of the biochemical phenomena associated to the tissue nutrition take 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 ejection of 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. from http://www.williamsclass.com/ is driven by the compliance of the vascular wall. In systole, the contraction of the heart induces a pressure wave which travels along the arterial tree causing the dilation of the vessels. In diastole, arteries deflate and push blood towards the capillaries and the venous 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 local pathologies and systemic features. An interesting case in this respect is the effect of arterial 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 hemodynamic overload [75]). This condition can determine left ventricular hypertrophy and altered coronary 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 the functionality of cardiovascular system. Indeed, the mechanical properties of the wall determine the wave propagation, and this suggests that pathologies which affect the wall may be associated to non physiological pressure waveforms. Besides giving a better insight on the behaviour of the wall under the effect of stresses exerted by the
16 2 One-dimensional models for blood flow problems blood flow, an accurate mechanical modeling of the vessels could in principle allow the detection of vascular diseases from information on the pulse propagation patterns of pressure and flow rate in the circulatory system [30]. The mechanical modeling of blood vessels requires the definition of a constitutive law describing the relationship between stress and strain fields in the vessel structure. The latter being a complex layered tissue, its mechanical characterization is still an open problem. 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 heterogenous model taking into account the micro-structure (cells, fibers and their mechanical inter- action) [144]. Hereafter we will focus our attention on homogeneous models, since in the spirit of 1D representation the local detail of the physical phenomena at hand can be foresaken. In the simplest approach, the wall can be treated as a linearly elastic membrane [36,45,125]. This leads to a reliable description of the main features of the wave propa- gation, both in physiological and pathological situations [3]. Still, an oversimplified lin- ear mechanical model for the vessel wall structure is not able to reproduce its viscoelas- tic behaviour, which is observed in vivo. Several different approaches have been pro- posed to address the modeling of viscoelastic features of vessel vascular wall [144]. Ar- mentano et al. showed that even a simple Kelvin-Voigt type model can be used to obtain a good agreement between in vivo measured data and numerical experiments [9,28]. A similar approach was followed by Canic et al. [19], who exploited a linearly viscoelastic cylindrical Koiter shell model for the arterial wall, based again on a Kelvin-Voigt type description of the structure viscoelastic features. A slightly more complex model was employed 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 et al. [36], who also noted that the strain relaxation, which is not modeled by the simpler Kelvin-Voigt model, can be relevant in the study of large arteries [140]. In the following we extend the analysis on a previously published model for blood flow in viscoelastic vessels [46], with the aim of validating the numerical scheme there proposed 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 not captured from linearly elastic structural models, as we show in some test cases. Finally we use the presented model to devise a one-dimensional description of the cerebral circulation, 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 vessel depicted in Fig. 2.2 and let I ⊂ R be a time interval. Given S(x, t) a cross section located along the vessel at axial coordinate x, considered at time t, A(x, t) is the area of 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 2 One-dimensional models for blood flow problems
Figure 2.2: A cylindrical compliant vessel. The shaded plane highlights a section S at axial coordinate x and at time t. equations [40]: ∂A ∂Q + = 0 (2.1a) ∂t ∂x ∂Q ∂ Q2 A ∂P Q + α + + K = 0 (2.1b) ∂t ∂x A ρ ∂x R A In (2.1), α is the so-called momentum-flux correction (or Coriolis) coefficient, ρ is the fluid mass 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, P and 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 the pressure external to the vessel: the wall mechanics can therefore be given in terms of a function ψ establishing the dependence of the transmural pressure P (x, t) − Pext on the vessel 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 with its derivatives) and of a set of parameters which may depend on x, t or A. Under the hypothesis that the pressure depends on A, on the reference cross-sectional area A0 and on parameters β = (β0, β1, . . . , βp) describing the mechanical properties of the wall, a possible choice for ψ is: " # A β1(x) ψ(A(x, t); A0(x), β0(x), β1(x)) = β0(x) − 1 . (2.3) A0(x)
For the ease of the notation, we will hereby refer to A0 and β, noting that in general they are to be considered as functions of the axial coordinate x.
18 2 One-dimensional models for blood flow problems
In the previous, β0 is an elastic coefficient, while β1 > 0 is normally obtained by β = 1 fitting the stress-strain√ response curves obtained by experiments. Whenever 1 2 1 1 πh0E and β0 = √ β = √ 2 , (2.3) is equivalent to the following relation: A0 A0 1−ξ √ √ A − A0 ψ(A(x, t); A0, β) = β , (2.4) A0 which is derived from the linear elastic law for the wall mechanics of a cylindrical vessel and where E(x) is the Young’s modulus, h0 the wall thickness and ξ the Poisson ratio [49]. The adoption of a linearly elastic model for the vessel wall mechanics is convenient since it simplifies the derivation of the equations, and is still able to capture the main features of the wave propagation phenomena in vascular system [16, 46, 82]. However, more accurate and complex mechanical models can be exploited, accounting for the vessel wall inelastic behaviour which is verified in vivo [144]: in the following we will discuss this aspect more thoroughly. Remark Set U = [AQ]T . We derive a conservative form of system (2.1) [45]: ∂U ∂F(U) + + B(U) = 0 , (2.5) ∂t ∂x where Q 0 F(U) = , B(U) = . Q2 Q A ∂ψ ∂C1 α + C KR + − A 1 A ρ ∂x ∂x
We denote by C1 the following quantity
Z A s 2 A ∂ψ C1 = c1dτ , c1 = , A0 ρ ∂A where c1 is referred to as the celerity of the propagation of waves along the tube and A0 indicates a reference value for A, here taken equal to the cross-sectional area in an unloaded 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 interplay of 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 into account inner longitudinal actions, in a way similar to what is done in the classical vibrating rod equation: ∂2η ∂4η ∂2η ∂η P − P =aη ˜ + ˜b +c ˜ − d˜ + g (2.6) ext ∂t2 ∂x4 ∂x2 ∂t
19 2 One-dimensional models for blood flow problems where √ √ A − A η = R − R = √ 0 0 π ∂η is the wall radial displacement, a˜, ˜b, c˜, d˜are positive coefficients and g is a generic ∂t function of the time derivative of the displacement. According to (2.6), the transmural pressure on the vascular wall is balanced by five terms, describing different mechanical features of the structure. The elastic response of the material is represented by the first term aη˜ , while the inertial effects are described by the term involving the second order time derivative of the wall displacement, where ˜b = ρwh is the product of the wall mass density and the wall thickness. Resistance to bendings is expressed in this model by the term involving a fourth order space derivative, while resistance to traction is taken into account by the term involving the second order space derivative. One of the most interesting mechanical features of the vascular wall is its viscoelastic nature. Arteries exhibit creep, stress relaxation and hysteresis in the stress-strain re- lation. Equation (2.6) accounts for viscoelastic effects, by describing them with term ∂η g . Following formal mathematical arguments, Quarteroni et al [113] proposed ∂t the following formulation ∂η ∂3η g = −e˜ , ∂t ∂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 simpler term, based on the Voigt viscoelastic model [52]. Moreover, we will neglect for the sake of simplicity the other non elastic effects, setting ˜b =c ˜ = d˜= 0. This leads to the following differential equation linking the transmural pressure to the wall radial displacement η: ∂η P − P =aη ˜ +γ ˜ , (2.7) ext ∂t where γ˜ is the so-called√ viscoelastic√ modulus. A√− A0 Recalling that η = π , and noting that typically in hemodynamic problems the range of variation of cross-sectional area A is small, we approximate ∂η 1 ∂A 1 ∂A = √ ' √ . ∂t 2 πA ∂t 2 πA0 ∂t β √ Moreover, the elastic response term can be recast in form (2.4), by setting a˜ = π. A0 The wall mechanics model may now be rewritten in terms of A including viscoelas- ticity as follows: ∂A P − P = ψ(A(x, t); A , a˜) + γ (2.8) ext 0 ∂t
20 2 One-dimensional models for blood flow problems
γ˜ with γ = √ . Therefore, assuming Pext independent of x, 2 πA0 ∂P ∂ψ ∂A ∂ψ dA ∂ψ da˜ ∂2A = + 0 + + γ , ∂x ∂A ∂x ∂A0 dx ∂a˜ dx ∂x∂t and we note that the second order mixed derivative of A can be recast into a second order derivative of Q by exploiting the mass conservation equation (2.1a). Substitution of the previous in (2.1b) gives:
2 2 ∂Q ∂ Q A ∂ψ ∂A ∂ Q Q A ∂ψ dA0 A ∂ψ da˜ + α + − γ 2 + KR + + = 0 . ∂t ∂x A ρ ∂A ∂x ∂x A ρ ∂A0 dx ρ ∂a˜ dx With respect to the conservative form (2.5), we set
2 F1 Q F = ,F1 = Q,F2 = α + C1 F2 A and, analogously, B1 Q A ∂ψ dA0 A ∂ψ da˜ ∂C1 B = ,B1 = 0 ,B2 = KR + + − B2 A ρ ∂A0 dx ρ ∂a˜ dx ∂x so that system (2.1) can be rewritten as follows: ∂A ∂Q + = 0 (2.9a) ∂t ∂x 2 ∂Q ∂F2 A ∂ Q + − γ + B = 0 (2.9b) ∂t ∂x ρ ∂x2 2 Clearly, the introduction of the viscoelastic term makes this system of equations no longer hyperbolic. However, we may assume that the elastic response function ψ plays a leading role in determining the wall mechanics. On the basis of this assumption, an operator splitting approach can be devised [45]. More details on this technique will be presented 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 ∂ Q2 neglect the nonlinear term, therefore α = 0; moreover, we linearize the coef- ∂x A ficients with respect to A, setting A(x, t) ' A0. The resulting linear system of first order partial differential equations reads: ∂A ∂Q + = 0 (2.10a) ∂t ∂x
∂Q A0 ∂P KR + + Q = 0 . (2.10b) ∂t ρ ∂x A0
21 2 One-dimensional models for blood flow problems
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 ∂ψ ∂A ∂ψ dA ∂ψ da˜ = + 0 + . ∂x ∂A ∂x ∂A0 dx ∂a˜ dx Now (2.10b) becomes ∂Q A ∂ψ ∂A K A ∂ψ dA A ∂ψ da˜ + 0 + R Q + 0 0 + 0 = 0 , ∂t ρ ∂A ∂x A0 ρ ∂A0 dx ρ ∂a˜ dx and system (2.10) may be written in non conservative form as follows:
∂U ∂U + H + S = 0 , (2.11) ∂t L ∂x L where 0 1 0 HL = A0 ∂ψ , SL = . 0 KR A0 ∂ψ dA0 A0 ∂ψ da˜ ρ ∂A Q + + A0 ρ ∂A0 dx ρ ∂a˜ dx
A conservative form reads:
∂U ∂FL + + B = 0 , (2.12) ∂t ∂x L where " # Q 0 FL = , BL = SL − ∂CL , CL ∂x
A and C = R c2 dτ. L A0 L System (2.11) is said to be strictly hyperbolic if H is similar to a diagonal matrix and its eigenvalues are real and distinct. In particular, the eigenvalues of matrix HL r A0 ∂ψ are λ = ±c , c = being the wave celerity in the linearized problem. A 1,2 L L ρ ∂A ∂ψ necessary and sufficient condition for the eigenvalues to be real and distinct is > 0, ∂A 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 . ∂A 2 πA0
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r A0 In the following we set c = a. L ρ We now denote by L, R the matrices whose rows (columns) are the left (right) eigen- vectors of HL, respectively:
T l1 L = T , R = r1 r2 , l2 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:
∂U ∂U L + ΛL + LS = 0 . (2.13) ∂t ∂x L
If there exist two quantities W1, W2 such that
∂W = L , W = [W ,W ]T , ∂U 1 2 then we can rewrite system (2.13) in diagonal form:
∂W ∂W + Λ + G = 0 , (2.14) ∂t ∂x L where ∂W dA0 ∂W da˜ GL = LSL − − . ∂A0 dx ∂a˜ dx
The values W1, W2 are the so-called Riemann invariants for the hyperbolic system at hand. Left eigenvectors l1,2 read ±c l = ζ L 1,2 1 where ζ = ζ(A, Q) is an arbitrary, positive smooth function of its arguments. Therefore
∂W ∂W ∂W ∂W 1 = ζc , 1 = ζ , 2 = −ζc , 2 = ζ . ∂A L ∂Q ∂A L ∂Q
We now impose the integrability of the two differential forms W1 and W2 by choosing ζ such that ∂2W ∂2W i = i , i = 1, 2 , ∂A∂Q ∂Q∂A which yields ∂ζ ∂ζ ±c = L ∂Q ∂A
23 2 One-dimensional models for blood flow problems 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 differential form: ∂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 ∂A ∂2A ∂ψ dA ∂ψ da˜ = a + γ + 0 + , ∂x ∂x ∂x∂t ∂A0 dx ∂a˜ dx and with arguments similar to those leading to system (2.9) we obtain
∂A ∂Q + = 0 (2.15a) ∂t ∂x
2 ∂Q ∂FL2 A0 ∂ Q + − γ + B = 0 , (2.15b) ∂t ∂x ρ ∂x2 L2 with FL1 FL = ,FL1 = Q,FL2 = CL FL2 and BL1 KR A0 ∂ψ dA0 A0 ∂ψ da˜ ∂CL BL = ,BL1 = 0 ,BL2 = Q + + − . BL2 A0 ρ ∂A0 dx ρ ∂a˜ dx ∂x
An analytical solution
The linearized equations (2.15) describe the propagation of area and flow rate waves in the space-time domain. We may look for solutions in the form of harmonic waves: h i A(x, t) = Aˆ(k) exp i ω(k)t − kx (2.16a) h i Q(x, t) = Qˆ(k) exp i ω(k)t − kx . (2.16b)
In the previous, k is the wave number, defined as the number of complete oscillations in the range x ∈ [0, 2π]; ω is the (angular) frequency and Aˆ(k), Qˆ(k) represent the wave amplitudes in x = 0, t = 0. In general, ω, k, Aˆ(k), Qˆ(k) ∈ C, however it is understood that we will be interested in the real part of the solution.
24 2 One-dimensional models for blood flow problems
Substituting (2.16) in the linearized Euler equations yields: h i iωAˆ − ikQˆ exp i ω(k)t − kx = 0 (2.17a)
2 h i −iC1k Aˆ + iω + k C2 + C3 Qˆ exp i ω(k)t − kx = 0 . (2.17b)
For the sake of simplicity, in the previous we assume that all the parameters are constant with respect to x and set
A0a A0γ KR C1 = ,C2 = ,C3 = . (2.18) ρ ρ A0
Moreover, we omit to indicate explicitly the dependency of Aˆ and Qˆ 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:
iωAˆ − ikQˆ = 0 (2.19a) 2 (−iC1k)Aˆ + (iω + k C2 + C3)Qˆ = 0 , (2.19b) which yields the following condition:
2 ω(iC3 − ω) + k (C1 + iC2ω) = 0 . (2.20)
We can now study the dispersion relation ω(k), linking the angular frequency to the wave number: solutions to system (2.17) are travelling waves with angular frequency q 2 2 2 2 i C2k + C3 ± − (C2k + C3) + 4C1k ω (k) = . 1,2 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, given by q 2 i (C2k + C3/k) ± − (C2k + C3/k) + 4C1 c (k) = . p 2 We can conversely express the wave number k as a function of the frequency ω: s ω(ω − iC3) k(ω) = ± , (2.21) C1 + iC2ω and we note that k is in general a complex number even when ω is real. It can be therefore 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)Aˆ = ikQ,ˆ
25 2 One-dimensional models for blood flow problems which implies that, if ω ∈ R, then Aˆ or Qˆ (or both) are complex numbers. We may choose Aˆ ∈ R, therefore (recalling (2.22)): h i A(x, t) = Aˆ(k) exp=(k)x exp i ω(k)t − <(k)x h i Q(x, t) = <