Computational hemodynamics of the cerebral circulation: multiscale modeling from the circle of Willis to cerebral aneurysms

Transcription

1 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

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3 Contents Abstract 1 1 Introduction Anatomy and physiology of the cerebral circulation The circle of Willis Morphology and fluid dynamics of cerebral aneurysms The role of hemodynamics Modeling the cerebral circulation The circle of Willis Cerebral aneurysms One-dimensional models for blood flow problems Wave propagation phenomena in the cardiovascular system Modeling the vascular wall Formulation of the model A viscoelastic structural model for the vessel wall The linearized model Networks of 1D models Numerical discretization Numerical solution of the viscoelastic wall model Results and discussion Validation of the numerical model versus an analytical solution Wave propagation in a single 1-D vessel: a Gaussian pulse wave Wave propagation in a single 1-D vessel: a sinusoidal wave A 1D model network: the circle of Willis Three-dimensional models for blood flow problems Blood flow features in arteries Geometry and Flow Reynolds number Dean number Womersley number and Reduced Velocity The Navier-Stokes equations Formulation Numerical discretization Wall shear stress in the Navier-Stokes problem i

4 Contents Approximation for the velocity gradient Oscillatory Shear Index Working on regions of interest Decomposition of bifurcation branches Relating surface points to centerlines An application of three-dimensional modeling Cerebral hemodynamics The Aneurisk project Hemodynamic features of the Internal Carotid Artery Discussion Wall shear stress as a classification parameter A geometrical multiscale model of the cerebral circulation The compliant vessel problem Matching conditions in 3D rigid/1d multiscale models Numerical algorithm Matching conditions including compliance Parameters estimation Results A 1D-3D-1D coupling Results The 3D carotid model and the multiscale coupling Remarks and perspectives Computational tools An introductory note on C LifeV: a C++ finite element library Code features Implementation of networks of 1D models Building the graph Interface conditions A simple example Conclusions 109 Acknowledgements 111 ii

5 Abstract In this work we address the mathematical and numerical modeling of cerebral circulation. In particular, one-dimensional (1D) models are exploited for the representation of the complex system of cerebral arteries, featuring a peculiar structure called circle of Willis. These models, based on the Euler equations, are unable to capture the local details of the blood flow but are suitable for the description of the pressure wave propagation in large vascular networks. This phenomenon is driven by the mechanical interaction of the blood and the vessel wall, and is therefore affected by the mechanical features of the wall. Chap. 2 deals with 1D models taking into account the wall viscoelasticity. In particular, the derivation of the nonlinear model is presented in Sec. 2.2, while a linearized set of equations is presented in Sec An analytical solution is found for the latter formulation and is used to validate the adopted numerical scheme (Sec. 2.4 and Sec. 2.5). Finally, the effect of wall viscoelasticity on the wave propagation phenomena is studied in some numerical experiments representative of realistic conditions in the cardiovascular and cerebral arterial systems. The details of the blood flow can be studied by means of three-dimensional (3D) models, based on the Navier-Stokes equations for incompressible Newtonian fluids introduced in Sec These models can correctly describe blood flow patterns in medium and large arteries, and in particular allow the evaluation of the stress field in the fluid. Thus, it is possible to estimate the traction exerted by the blood flow on the vessel wall (wall shear stress, defined in Sec. 3.4). Moreover, by exploiting the representation of the vascular tree in terms of centerlines, it is possible to easily identify regions of interest in the computational domain, in which to restrict the fluid dynamics analysis: this approach is presented in Sec 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 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

6 Contents hemodynamic parameter in the classification technique improves its efficacy (Sec. 4.3). The interaction between local and global phenomena is a typical feature of the circulatory system. It is believed to be crucial in the context of cerebral circulation, since defects or diseases at the level of the circle of Willis can induce local flow conditions associated to the initiation of an aneurysm. Geometrical multiscale models are a promising tool for the modeling of this interaction. They are based on the coupling of reduced models taking into account the dynamics of the vascular network and detailed models describing the local blood features. In Sec. 5.4 a geometrical multiscale model of the cerebral circulation is presented, based on the coupling of a 1D representation of the circle of Willis and the 3D representation of a carotid artery. A novel method to describe the interface between the two models is discussed in Sec The number of potential applications of reduced models, due to their proven effectiveness in the study of vascular networks, calls for the design of efficient and robust software tools. In Chap. 6 we address this issue, by presenting some excerpts of the software specifically written in the context of this work for the simulation of the circulatory system (Sec. 6.3). 2

7 1 Introduction In this Chapter we discuss the motivation of this work, assessing the problems of interest. A description of the cerebral circulatory system and a review on the state of the art knowledge on cerebral aneurysms are presented in Sec. 1.1 and Sec. 1.2, respectively. Most of the material here presented is taken from the work by Khurana & Spetzler [65]. More details and additional references to the medical literature for these topics can be found therein. The modeling of cerebral circulation, with specific attention to the blood flow problems related to the development of vascular diseases, can enhance the comprehension of the pathology mechanisms and therefore help in devising treatment procedures. On the other hand, the complexity of the physical systems at hand calls for the definition of effective modeling strategies, balancing the need for a detailed description of the physical phenomena and the computational cost. These issues, together with a description of the original contribution of this work in the presented framework, are discussed in Sec 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 The intima of brain arteries (the innermost part of the wall) is composed of a single layer of endothelial 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 innervation 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

8 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 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 subarachnoid space. From these vessels depart smaller arterial branches such as the perforating 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 perforating 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

9 1 Introduction Middle cerebral artery Anterior communicating artery Anterior cerebral artery Ophthalmic artery Posterior communicating artery Internal carotid artery Anterior choroidal artery Posterior cerebral artery Pontine arteries Basilar artery Superior cerebellar artery Anterior inferior cerebellar artery Vertebral artery Anterior spinal artery Posterior inferior 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 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 essential 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 compression carotid test (see e.g. [60]). 5

10 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 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 mm Large mm Near-giant > 25 mm Giant 6

11 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 compressing or irritating the surrounding brain structures. However, a threshold value for the diameter has not been precisely defined, and this explains the uncertain classification 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 morphological and fluid dynamics features of pathologic vessels, with the aim of defining a classification of vascular geometries based on the probability of developing an aneurysm in specific locations [119]. Symptoms Most aneurysms are silent, and are discovered at the time of rupture. The typical symptom 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

12 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 aneurysms, 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 pulsation 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 important 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 communication, 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. Moreover, large areas of relative stagnation can cause the concentration of the dye in these regions to be low leading ultimately to undersegmentation. 8

13 1 Introduction Magnetic resonance techniques (MRI, MRA) are less invasive than cerebral angiography, 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 combination of computed tomography (CT) scanning and angiography. More precisely, an intravenous dye is injected into the patient during CT scanning. The resulting technique 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 treatment or observation is controversial. The latter implies that the patients need to undergo repeated scans to determine if the aneurysm is enlarging, therefore facing the risk of excessive postponement of the treatment and, depending on the imaging technique, 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 evaluated 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 composite 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 aneurysm, 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 comparable 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

14 1 Introduction 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 proposed 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 spatial 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 between cerebral aneurysm initiation and the effects of high-flow hemodynamic forces on the arterial wall. Studies pointed out the association of cerebral aneurysms with arterial anatomic variations and pathological conditions such as hypoplasia or occlusion of a segment of the circle of Willis [64, 81, 117]. High-flow arteriovenous malformations 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, experimental 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 degeneration 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

15 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 medial 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 usually 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 presence of high shear stresses on the wall may represent other risk factors for aneurysm rupture [22]. Moreover, narrow necks in large aneurysms geometrically induce concentrated 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 hydrocephalus 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 aneurysm 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

16 1 Introduction 1.3 Modeling the cerebral circulation The complexity of the vascular system demands for the set up of convenient mathematical 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 Navier-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 physiopathological impact of hemodynamics. Unfortunately, the high computational costs restrict their use to contiguous vascular districts only on a space scale of few centimeters 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 dimensional 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 dynamics 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 simplification 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 system, 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 simple way and with low computational costs (see e. g. [89, 97]). All these models have peculiar mathematical features. They are able to capture different 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 mechanisms 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 different 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

17 1 Introduction difficulties have been extensively discussed recently in [112] 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 different 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 comparing the results obtained with a viscoelastic model for the wall to those obtained by using a linear elastic model (see Sec ). The computational study here presented is carried out with a software tool specifically written and based on the C++ finite element 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 aneurysms, 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 computational 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, coupling 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 mechanical 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). 13

18 1 Introduction 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 capabilities 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 patientspecific 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 experimental studies based on realistic anatomical models reconstructed from images using 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 correlation 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 geometrical 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 understanding of the pathology development. 14

19 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 ). 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 discretization used to solve the equations are presented in Sec and Sec 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 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

20 2 One-dimensional models for blood flow problems Figure 2.1: Schematic representation of the human cardiovascular system. In each cardiac cycle, blood flows from the heart towards the peripheral circulation (arterioles, capillaries) and is collected back to the heart from the veins. from 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 propagating 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 reflections, 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 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

21 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: vessel 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 interaction) [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 propagation, both in physiological and pathological situations [3]. Still, an oversimplified linear mechanical model for the vessel wall structure is not able to reproduce its viscoelastic behaviour, which is observed in vivo. Several different approaches have been proposed to address the modeling of viscoelastic features of vessel vascular wall [144]. Armentano 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 standard 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. Moreover 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

22 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 t + Q x = 0 Q t + α ( ) Q 2 + A P x A ρ x + K Q R A = 0 (2.1a) (2.1b) In (2.1), α is the so-called momentum-flux correction (or Coriolis) coefficient, ρ is the fluid mass density and K R is a strictly positive quantity which represents the viscous resistance 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 P ext 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) P ext on the vessel kinematics (in turn driven by the blood flow): P (x, t) P ext = 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 A 0 and on parameters β = (β 0, β 1,..., β p ) describing the mechanical properties of the wall, a possible choice for ψ is: [ ( ) ] A β1 (x) ψ(a(x, t); A 0 (x), β 0 (x), β 1 (x)) = β 0 (x) 1. (2.3) A 0 (x) For the ease of the notation, we will hereby refer to A 0 and β, noting that in general they are to be considered as functions of the axial coordinate x. 18

23 2 One-dimensional models for blood flow problems In the previous, β 0 is an elastic coefficient, while β 1 > 0 is normally obtained by fitting the stress-strain response curves obtained by experiments. Whenever β 1 = 1 2 and β 0 = 1 A0 β = 1 πh0 E A0, (2.3) is equivalent to the following relation: 1 ξ 2 A A0 ψ(a(x, t); A 0, β) = β, (2.4) A 0 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, h 0 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 = [A Q] T. We derive a conservative form of system (2.1) [45]: where F(U) = Q α Q2 A + C 1 U t + F(U) + B(U) = 0, (2.5) x, B(U) = We denote by C 1 the following quantity C 1 = A A 0 c 2 1dτ, c 1 = K R Q A + A ρ A ρ 0 ψ A, ψ x C 1 x where c 1 is referred to as the celerity of the propagation of waves along the tube and A 0 indicates a reference value for A, here taken equal to the cross-sectional area in an unloaded configuration 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: P P ext = ãη + b 2 η t 2 + c 4 η x 4 d 2 η x 2 + g ( ) η t. (2.6) 19

24 2 One-dimensional models for blood flow problems where η = R R 0 = A A0 π ) is a generic ( is the wall radial displacement, ã, b, c, d η are positive coefficients and g 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 ãη, while the inertial effects are described by the term involving the second order time derivative of the wall displacement, where b = ρ w h 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 relation. Equation (2.6) accounts for viscoelastic effects, by describing them with term ). Following formal mathematical arguments, Quarteroni et al [113] proposed ( η g t the following formulation g ( ) η = ẽ 3 η t t 2 x, involving a third order mixed derivative of η, which shows good agreement with experimental 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: P P ext = ãη + γ η t, (2.7) where γ is the so-called viscoelastic modulus. Recalling that η = A π A0, and noting that typically in hemodynamic problems the range of variation of cross-sectional area A is small, we approximate η t = 1 2 A πa t 1 2 A πa 0 t. Moreover, the elastic response term can be recast in form (2.4), by setting ã = β A 0 π. The wall mechanics model may now be rewritten in terms of A including viscoelasticity as follows: P P ext = ψ(a(x, t); A 0, ã) + γ A (2.8) t 20

25 with γ = 2 One-dimensional models for blood flow problems γ 2 πa 0. Therefore, assuming P ext independent of x, P x = ψ A A x + ψ da 0 A 0 dx + ψ dã ã dx + γ 2 A 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: Q t + α x ( ) Q 2 + A ψ A A ρ A x γ 2 Q x 2 + K Q R A + A ρ With respect to the conservative form (2.5), we set [ ] F1 F =, F 1 = Q, F 2 = α Q2 A + C 1 F 2 ψ da 0 A 0 dx + A ψ dã ρ ã dx = 0. and, analogously, [ ] B1 B = B 2 Q, B 1 = 0, B 2 = K R A + A ψ da 0 ρ A 0 dx + A ψ dã ρ ã dx C 1 x so that system (2.1) can be rewritten as follows: A t + Q x = 0 (2.9a) Q t + F 2 x A ρ γ 2 Q x 2 + B 2 = 0 (2.9b) 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 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 ( ) Q 2 = 0; moreover, we linearize the coefficients with respect to A, setting A(x, t) A 0. The resulting linear system of first order x A partial differential equations reads: A t + Q x = 0 Q t + A 0 P ρ x + K R Q = 0. A 0 (2.10a) (2.10b) 21

26 2 One-dimensional models for blood flow problems We will consider relation (2.8) linking the pressure to the cross-sectional area, and assume for the sake of simplicity γ = 0, therefore Now (2.10b) becomes P x = ψ A A x + ψ da 0 A 0 dx + ψ dã ã dx. Q t + A 0 ψ A ρ A x + K R Q + A 0 ψ da 0 A 0 ρ A 0 dx + A 0 ψ dã ρ ã dx = 0, and system (2.10) may be written in non conservative form as follows: U t + H U L x + S L = 0, (2.11) where H L = 0 1 A 0 ψ ρ A 0, S L = K R Q + A 0 A 0 ρ 0 ψ da 0 A 0 dx + A 0 ψ ρ ã dã dx. A conservative form reads: U t + F L x + B L = 0, (2.12) where [ ] Q F L =, B L = S L C L [ 0 ] C L x, and C L = A A 0 c 2 L dτ. 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 H L are λ 1,2 = ±c L, c L = A0 ρ ψ A being 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 = Linearization of the previous relation yields ψ A ã 2 πa. ã 2 πa 0 = a. 22

27 2 One-dimensional models for blood flow problems A0 In the following we set c L = ρ a. We now denote by L, R the matrices whose rows (columns) are the left (right) eigenvectors of H L, respectively: [ ] l T L = 1, R = [ ] r 1 r 2, with the additional (non restrictive) hypothesis LR = I. Then l T 2 LH L R = Λ = diag(λ 1, λ 2 ). and the following equivalent form for system (2.11) is obtained: If there exist two quantities W 1, W 2 such that L U t + ΛL U x + LS L = 0. (2.13) W U = L, W = [W 1, W 2 ] T, then we can rewrite system (2.13) in diagonal form: where W t + Λ W x + G L = 0, (2.14) G L = LS L W da 0 A 0 dx W dã ã dx. The values W 1, W 2 are the so-called Riemann invariants for the hyperbolic system at hand. Left eigenvectors l 1,2 read [ ] ±cl l 1,2 = ζ 1 where ζ = ζ(a, Q) is an arbitrary, positive smooth function of its arguments. Therefore W 1 A = ζc L, W 1 Q = ζ, W 2 A = ζc L, W 2 Q = ζ. We now impose the integrability of the two differential forms W 1 and W 2 by choosing ζ such that 2 W i A Q = 2 W i Q A, i = 1, 2, which yields ±c L ζ Q = ζ A 23

28 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: W 1,2 = ±c L A + Q. We choose (A 0, 0) as the zero state in the (A, Q) plane, in which the characteristic variables are zero, and find after integration: W 1,2 = Q ± c L (A A 0 ). Adding viscoelasticity Let s now consider the viscoelastic term γ > 0 in (2.8): this yields P x = a A x + γ 2 A x t + ψ da 0 A 0 dx + ψ dã ã dx, and with arguments similar to those leading to system (2.9) we obtain A t + Q x = 0 (2.15a) with and B L = [ BL1 B L2 ] Q t + F L2 x A 0 ρ γ 2 Q x 2 + B L2 = 0, F L = [ FL1 F L2 ], F L1 = Q, F L2 = C L, B L1 = 0, B L2 = K R Q + A 0 ψ da 0 A 0 ρ A 0 dx + A 0 ψ dã ρ ã dx C L x. (2.15b) 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: A(x, t) = [i Â(k) exp ( ω(k)t kx )] (2.16a) Q(x, t) = ˆQ(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 Â(k), ˆQ(k) represent the wave amplitudes in x = 0, t = 0. In general, ω, k, Â(k), ˆQ(k) C, however it is understood that we will be interested in the real part of the solution. 24

29 2 One-dimensional models for blood flow problems Substituting (2.16) in the linearized Euler equations yields: ( ) [ iωâ ik ˆQ exp i ( ω(k)t kx )] = 0 ( ( ic1 k )  + ( iω + k 2 ) [ C 2 + C 3 ˆQ) exp i ( ω(k)t kx )] = 0. (2.17a) (2.17b) For the sake of simplicity, in the previous we assume that all the parameters are constant with respect to x and set C 1 = A 0a ρ, C 2 = A 0γ ρ, C 3 = K R A 0. (2.18) Moreover, we omit to indicate explicitly the dependency of  and ˆQ 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: iωâ ik ˆQ = 0 ( ic 1 k)â + (iω + k2 C 2 + C 3 ) ˆQ = 0, (2.19a) (2.19b) which yields the following condition: ω(ic 3 ω) + k 2 (C 1 + ic 2 ω) = 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 ω 1,2 (k) = i ( C 2 k 2 ) + C 3 ± (C 2 k 2 + C 3 ) 2 + 4C 1 k 2. 2 For the problem at hand, the phase velocity c p = 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 c p (k) = i (C 2k + C 3 /k) ± (C 2 k + C 3 /k) 2 + 4C 1. 2 We can conversely express the wave number k as a function of the frequency ω: k(ω) = ± ω(ω ic 3 ) C 1 + ic 2 ω, (2.21) and we note that k is in general a complex number even when ω is real. It can be therefore written as k = R(k) + ii(k). (2.22) We remark that, due to the first equation of system (2.17), the solution is such that iω(k)â = ik ˆQ, 25

30 2 One-dimensional models for blood flow problems which implies that, if ω R, then  or ˆQ (or both) are complex numbers. We may choose  R, therefore (recalling (2.22)): A(x, t) = Â(k) exp[ I(k)x ] exp [i ( ω(k)t R(k)x )] ( Q(x, t) = R ( ) ( ) ˆQ(k) ) + ii ˆQ(k) exp [ I(k)x ] [ exp i ( ω(k)t R(k)x )]. In particular, we are interested in the real part of the solution, which reads R ( A(x, t) ) = Â(k) exp[ I(k)x ] cos ( ω(k)t R(k)x ) R ( Q(x, t) ) = exp [ I(k)x ]( R( ˆQ) cos ( ωt R(k)x ) I( ˆQ) sin ( ωt R(k)x )). (2.24a) (2.24b) It follows from the previous that the imaginary part of the wave number is associated to an exponential factor modulating the wave amplitudes. When I(k) < 0, this is a damping factor corresponding to an exponential decay with variable x. We now consider the contribution of the three different terms C 1, C 2 and C 3 to the definition of k. When C 1 0 and C 2 = C 3 = 0, ω 2 k(ω) = ±, C 1 where we choose the positive root. To force C 3 = 0 means that we are considering an inviscid fluid, for which the friction term K R vanishes. On the other hand, C 2 is equal to 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 of the form (2.16). If C 1, C 3 0, the definition of k reads k(ω) = ± ω(ω ic 3 ) C 1, and k has an imaginary part. This case corresponds to the problem of a viscous fluid flowing inside a linearly elastic vessel. We can reformulate the previous definition as follows: k 2 (ω) = ω2 i C 3ω, C 1 C 1 which in polar notation reads k 2 (ω) = r exp ( i(θ + 2nπ) ), n N, ( ) ω 2 2 ( ) C3 ω 2 r = +, C 1 C 1 ( θ = arctan C ) 3. ω 26

31 2 One-dimensional models for blood flow problems Therefore k(ω) = ( ( )) θ r exp i 2 + nπ, n N, and we choose one of the two roots, such that R(k) > 0. We remark that, since R(k 2 ) > 0 and I(k 2 ) < 0, π/2 < θ < 0: this implies θ + nπ, n N 2 or [ π/4, 0] θ + nπ, n N 2 [3/4 π, π], the latter being excluded by the request R(k) > 0. This ensures that I(k) < 0 and therefore the term C 3 0 is responsible for an exponential damping of the signal amplitude. In the general case C 1 0, C 2 0, C 3 0, recalling (2.21), k 2 (ω) = ω(ω ic 3)(C 1 ic 2 ω) C1 2 + = ω2 (C 1 C 2 C 3 ) C2 2 ω2 C1 2 + C i ω(c 1C 3 + C 2 ω 2 ) 2ω C1 2 + C. 2ω With a similar procedure to that applied to the previous case it can be seen that, under the hypothesis C 1 > C 2 C 3, corresponding to assume A0 β 2 > γk R, i. e. a Young modulus large enough, both the fluid viscosity (term C 3 ) and the viscoelasticity of the wall (term C 2 ) cause an exponential decay of the travelling waves amplitudes. 2.3 Networks of 1D models Once having set up the model for a single tube, we can move towards the study of networks: from a mathematical point of view, this means to find suitable interface conditions for connected tubes. Following [49], we adopt a domain decomposition approach, and request that the solutions in interfacing domains are such that the conservation of certain physical quantities is ensured at the interface. Let s consider as an example the two tubes depicted in Fig We take as a reference system the simmetry axis x, which is the same for both vessels, and impose that the mass flow and the total pressure are conserved across the interface: { Q1 = Q 2 t > 0, at x = Γ (2.25) P t,1 = P t,2 where P t = P ϱ( Q A )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

32 2 One-dimensional models for blood flow problems Ω 1 Ω 2 x x = 0 x = Γ x = L Figure 2.3: Two connected tubes Ω 1 and Ω 2 : the interface is located at coordinate x = Γ. Formaggia et al. [45] proved that this set of interface conditions guarantees an energy inequality for the coupled problem. On the other hand, they noted that typically in blood flow problems the value of the pressure P is much greater than the kinetic energy 1 2 ϱ( Q A )2, therefore in practice the continuity of pressure can be prescribed at the interface without encountering stability problems. Other possible interface conditions can be designed to take explicitly into account the fact that the total pressure decreases as a function of the flow rate, along the flow direction, in correspondence with the interface Γ. The second of (2.25) may then then be written as follows [45]: P t,2 = P t,1 sign(q)f(q) t > 0, at x = Γ f being a positive, monotone function satisfying f(0) = 0 and referred to as dissipation function. However, appropriate formulations for f are usually not available, therefore a typical choice is f 0 corresponding to the continuity of the total pressure. 2.4 Numerical discretization Let us refer for the sake of simplicity to system 2.5. Following [45], we adopt a discretization based on a second order Taylor-Galerkin scheme which can be seen as a generalization of the classical Lax-Wendroff scheme for systems of conservation laws [71]. Given U n the approximation of the solution U(t n ) at time t n, U n+1 is obtained by solving the following system: U n+1 = U n t [ F n t ] x 2 Hn S n + t2 2 t [ S n U F n x + x ( S n + t 2 Sn US n )] (H n Fn x ), n = 0, 1,..., (2.26) 28

33 2 One-dimensional models for blood flow problems where S n U = S U (Un ) and H n, S n and F n are defined in a similar way. The Galerkin finite element method is applied to (2.26), yielding ( ( ) U n+1 h, ϕ h = (U n h, ϕ h ) + t F LW (U n h ), ϕ h x t2 2 ( H(U n h ) F x (Un h ), ϕ h x ) + t2 2 ( S U (U n h ) F(Un h ) ), ϕ h x ) t (S LW (U n h ), ϕ h), ϕ h V 0 h. (2.27) In the previous equation we used the notation F LW = F(U h ) + t 2 F U(U h )S(U h ) and S LW = S(U h ) + t 2 S U(U h )S(U h ). 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 approximation of the solution, based on P1 finite elements 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 the following equivalent form of system (2.9) [45]: Q e t A t + Q x = 0 + F 2(A, Q) x Q v t Aγ ϱ where Q is decomposed into two contributions (2.28a) = B 2 (A, Q) (2.28b) 2 Q x 2 = 0, Q = Q e + Q v, (2.28c) due to the elastic and viscoelastic behaviour of the wall mechanics, respectively. Equations (2.28a), (2.28b) compose a hyperbolic system involving the time derivative of Q e, while (2.28c) is a parabolic equation of variable Q v. On each time interval [t n, t n+1 ], n 0, the first two equations in (2.28) are solved by the Taylor-Galerkin scheme previously presented. The explicit time advancing scheme gives A n+1 and Q n+1 e as functions of A n and Q n. The third equation is used to correct the flow rate, and is solved by adopting an implicit Euler time advancing scheme and the following finite element formulation [45]: given A n+1 h and (Q e ) n+1 h, find (Q v ) h Vh 0 such that ( 1 A n+1 h (Q v ) n+1 h, ψ h ) + t γ ϱ ( ) Q n+1 h x, ψ h = 0, ψ h Vh 0 x, where we exploit the simplifying assumption that homogeneous Dirichlet boundary conditions are imposed to the correction term Q v. This corresponds to correcting the 29

34 2 One-dimensional models for blood flow problems 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 correction term vanishing on the interface between different models, there is no need of decoupling the term in the different segments. Now, knowing that Q n+1 = Q n+1 e + Q n+1 v, we can write ( 1 A n+1 h (Q v ) n+1 h, ψ h ) ( ) + t γ (Qv ) n+1 h, ψ h = ϱ x x ( ) t γ (Qe ) n+1 h, ψ h, ψ h Vh 0 ϱ x x. (2.29) 2.5 Results and discussion This section presents a set of numerical experiments designed to test the reliability of the model. Particular attention is devoted to the effects of the wall viscoelasticity on the propagation of waves in blood vessels. We start by using analytical solutions (2.24) as a benchmark case, for validating the code. Then we discuss the effect of dissipative terms associated with the fluid viscosity and the wall viscoelasticity. More precisely, following the arguments exploited for the linearized 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 can reproduce the hysteresis in the P (A) relation, which is a typical viscoelastic feature of blood vessels in vivo. Finally, a model for the circle of Willis is presented, based on the published work by Alastruey et al. [3] and modified by including a description of the viscoelastic effects in the vessel wall mechanics. A comparison of the results obtained by the two models is drawn at a qualitative level 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, Â R. We will assume ω = 2π s 1, and obtain the corresponding solution exploiting the dispersion relation (2.21). The geometrical and physical features of the simulated vessel are summarized in Tab The tube at hand is very long, in order to clearly show the damping effect of the wall viscoelasticity and blood viscosity on the wave amplitude, discussed in Sec The wall mechanical parameters are in the range of physiological values for large arteries (see e. g. [52]). In particular, the value for the viscoelastic modulus γ = dyn s cm 3 is taken from [19] and corresponds to the estimated viscous modulus of a human femoral artery. 30

35 2 One-dimensional models for blood flow problems Name Symbol Value Measurement unit length L 1000 cm radius R 0 1 cm thickness h 0.15 cm mass density ρ 1.05 g cm 3 Poisson modulus ξ Young s modulus E dyn cm 2 viscoelastic modulus [28] γ dyn s cm 3 friction parameter K R P Table 2.1: Geometrical and mechanical parameters for the simulated vessel. Based on these values, we find C 1 = dyn cm g 1, C 2 = dyn cm s g 1, C 3 = g cm 2 s 1 and k(2π) = R(k) + ii(k) = i The initial conditions for the problem are A(x, 0) = A 0 = π Q(x, 0) = exp [ I(k)x ]( R( ˆQ) cos ( R(k)x ) I( ˆQ) sin ( R(k)x )). Recalling the first equation in system 2.19 and knowing that  = A(0, 0) = π, we find that ˆQ = ω k  = R( ˆQ) + ii( ˆQ) = π + i π. The boundary conditions on the left and right boundaries prescribe two periodic flow rates: Q(x b, t) = exp [ I(k)x b ] ( R( ˆQ) cos ( ωt R(k)x b ) I( ˆQ) sin ( ωt R(k)xb ) ), x b = 0, 10 m. 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. Since we are considering a viscoelastic model for the arterial wall and blood is considered as a Newtonian fluid, the wave amplitude is damped by an exponential factor (see Sec ). This effect is clearly visible in Fig. 2.4, where red lines represent the damped amplitude of the wave: Q damp (x, t) = ±R ( ˆQ(k) ) exp [ I(k)x ]. (2.30) We remark that the operator splitting approach presented in Sec is able to recover a good approximation of the analytical solution (see Fig. 2.5). We estimated the approximation error as follows Q dx Q exact L 2 (0,T ;L 2 (Ω)) Q exact L 2 (0,T ;L 2 (Ω)) = , where we indicate by Q exact the analytical solution and by Q dx the numerical solution. 31

36 2 One-dimensional models for blood flow problems t = 0 t = 0.4 t = Q x Figure 2.4: Solution of the linearized model with the presented numerical setup. Y-axis: flow rate (in cm 3 /s); X-axis: tube axial coordinate (in m). The plot shows snapshots of the travelling wave at different time. The superimposed red lines represent the damping term (2.30) associated to blood viscosity and wall viscoelasticity analytical solution computed solution 1000 Q x Figure 2.5: Solution of the linearized model with the presented numerical setup. Y-axis: flow rate (in cm 3 /s); X-axis: tube axial coordinate (in m). The plot shows the superposition of the analytical solution (blue) and the numerical solution (red) on the whole domain, at t = 0.1s Wave propagation in a single 1-D vessel: a Gaussian pulse wave This numerical experiment describes the propagation along a vessel of a narrow, Gaussian shaped wave, a continuous approximation to a unit pulse δ(t) located at t = t 0 (i. e. δ(t 0 ) = 1 and δ(t) = 0 for t 0). The unit pulse waveform was used in [145] to track the multiple transmissions and reflections in the arterial system, while its Gaus- 32

37 2 One-dimensional models for blood flow problems sian approximation was used in [4] for the study of the effects of outflow boundary conditions in 1D blood flow simulations. Here we aim to evaluate the dissipative effects of wall viscoelasticity and blood viscosity on the travelling wave amplitude. The simulated vessel has the same characteristics as the vessel described in the previous section (see Tab. 2.1), but is described this time by the non linear model (2.1). The boundary condition prescribed on the left boundary is a flow rate of the form: [ ( )] t t0 Q(x l, t) = exp τ, with τ = 0.01 and t 0 = 0.05 s. Absorbing boundary conditions are prescribed on the right 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 When considering an inviscid flow inside an elastic shell, we can see that the shape of the wave travelling along the vessel is not altered (red line). The addition of fluid viscosity to the model attenuates the wave amplitude, in a similar way with respect to what can be seen in the linearized model (green line). If we model blood as a viscous fluid and the wall as a series of viscoelastic rings (see eq. (2.7)), we observe that the wave amplitude is extremely reduced (blue line). Again, this is qualitatively in accordance with what has already been said about the linear problem Q x Figure 2.6: Propagation of a gaussian flow rate wave in a 10m long vessel (here only the tract x [0, 1]m is represented). Viscoelasticity of the wall and blood viscosity attenuate the amplitude of the travelling wave. Red: elastic wall, inviscid fluid; Green: elastic wall, viscous fluid; Blue: viscoelastic wall, viscous fluid. X-axis: position along the tube (m); Y-axis: flow rate (cm 3 /s) 33

38 2 One-dimensional models for blood flow problems Wave propagation in a single 1-D vessel: a sinusoidal wave This numerical experiment describes the propagation of a half-sinusoidal input wave along a vessel. The inflow condition mimics a realistic cardiac output, while a lumped parameter 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 considered 1D domain are simulated by a three-element windkessel model, as proposed by Alastruey et al. [4]. They showed that by coupling this model to a 1D representation of the aorta, some features of in vivo aortic measurements can be reproduced, such as the pressure dicrotic notch and the exponential diastolic decay of the pressure. We reproduce here the same experiment, by considering a 40 cm long vessel, with the same other geometric and mechanical features as the vessel considered in previous sections (see Tab. 2.1), and described again by the non linear model (2.1). Moreover, R = dyn s cm 5, C = cm 5 GPa 1. The initial conditions for the vessel are A = A 0 = π and Q = 0. The flow rate boundary condition prescribed on the left boundary is a periodic function of time, with period T = 1s: Q(t) = 310 sin( 2π T t) 0 t < τ s 0 τ s t < T with τ s = 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 the inflow waveform. (b) Pressure waveform Figure 2.7: Flow rate and pressure waveforms, once a quasi-steady state is reached, in the middle of a 1D vessel coupled with a 0D outflow model. The wall viscoelasticity affects the wave propagation. The present model is able to capture a slightly increased wave speed associated to viscoelastic phenomena (Fig. 2.7). Moreover, Fig. 2.8 (left) shows that the contribution of the viscoelastic term to the overall pressure (see (2.8)) is not negligible. 34

39 2 One-dimensional models for blood flow problems (a) Comparison between elastic and viscoelastic contribution to the overall pressure wave. (b) A-P curve for the sin wave propagation numerical 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 viscoelastic vascular wall. In particular, it can be seen at a qualitative level that the diastolic phase, in which P is proportional to A, is clearly distinct from the systolic phase in which pressure and area waves show a significant phase shift. A more quantitative analysis of these results, based possibly on the comparison with clinically measured data or with other available models in the literature, is required to assess the accuracy of the model in the description of the physical phenomena at hand A 1D model network: the circle of Willis The proposed simulation is based on the set-up presented by Alastruey et al. [3]. The circle of Willis is immersed in a larger network of 1D models describing the main arteries bringing blood to the brain (see Fig. 2.9), and the inflow boundary condition for the whole network is provided by the heart. Peripheral circulation is accounted for by a 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). The edges of the graph correspond to the vessels, while the nodes are the junctions. Each edge 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 total pressure P t = P + 1/2ϱ( Q A )2 (see [45, 82] and Sec. 2.3). The flow rate boundary condition prescribed on the left boundary of the vessel representing the aortic arch is a periodic function of time, with period T = 1s [3]: 485 sin( 2π T t) 0 t < τ s Q(t) = 0 τ s t < T with τ s = 0.3 s. The simulation was run with dx = 0.5 cm and dt = 10 4 s. 35

40 2 One-dimensional models for blood flow problems 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 the numbered vessels in figure are reported in Fig In Fig we illustrate a snapshot of the solution in the brachial artery, comparing the pressure waveforms obtained by the elastic and viscoelastic models for the vessel wall. The viscoelastic modulus γ = 10 4 dyn s cm 3 is taken equal in all the vessels (it is in the range of values proposed in [19] for the femoral arteries, and is assumed here as a reference value for medium-size vessels). It can be seen that the wave propagation speed is increased in the viscoelastic wall, as we already noticed in previous experiments. Moreover, the dicrotic notch is more evident: indeed, faster dynamics of the wall, as found in systole and in particular at the very end of the systole, are associated to 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 P A at the systolic peak t A and pressure P B at t B, as shown in Fig These values, in the two 36

41 2 One-dimensional models for blood flow problems 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 wall t B t A 0.06 s s P A P B 7052 dyn cm dyn cm 2 We remark that the qualitative pressure waveform is not significantly affected by the wall viscoelasticity, and is comparable to the results presented in [3]. Deeper and more extensive studies are needed to quantify the role of wall viscoelasticity in specific vascular districts or under particular conditions, in which the elastic approximation may not be adequate for the correct description of the wave propagation phenomena. 37

42 2 One-dimensional models for blood flow problems P_A elastic wall viscoelastic wall P_B P t_a t_b t Figure 2.11: Pressure wave (in dyn/cm 2 ) in the middle of the brachial artery, after 10 cardiac cycles. In blue, the solution obtained with an elastic model for the wall mechanics; in red, the solution obtained with a viscoelastic model. X-axis: time (in s) 38

43 3 Three-dimensional models for blood flow problems In this Chapter we present an overview of the methods and the models that we used for the study of blood flow in three-dimensional vascular geometries. In particular, the dynamics of the velocity and pressure fields for an incompressible Newtonian fluid is described by the Navier-Stokes equations, presented in Sec When the velocity and pressure fields of a fluid are known, derived quantities such as the stress field can be obtained. In hemodynamic problems, the wall shear stress has particular importance: the force per unit area exerted by the fluid tangentially to the wall is relevant in relation to some vascular diseases, due to the reaction it induces on endothelial cells. This specific topic is discussed more carefully in Sec In general, we are interested in the analysis of flow features in specific locations of the physical system at hand: in Sec. 3.5 we present a method to select regions of interest in 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 the pumping action of the heart [92]. In particular, most locations in the arterial tree experience unsteady, pulsatile flow, which can induce flow reversals and recirculation near to the wall, with potentially pathogenic effect (such as, for instance, atherogenesis). The time pattern of pulse waves in the circulatory system, however, is not perfectly periodic, because it is adjusted in order to fit the body blood demand. The approximation to a periodic phenomenon therefore, usually accepted in hemodynamic computations, may hold only for short periods of time, in which the overall physical conditions are essentially not changing. The driving mechanism for the pulse wave propagation in the circulatory system is the mechanical interaction of the flowing blood and the vascular structure: in this respect, the compliance of the vessel wall plays a crucial role. However, an accurate computation of the fluid structure interaction problem in hemodynamics is costly, due to the strong mechanical coupling of the two systems (fluid and structure). A review of the 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 movement of the vascular wall is relatively small compared to the vessel diameter. The radius changes may be at most of the order of 15% in larger arteries, and are smaller in the peripheral arteries. Depending on which district we are interested in, the assumption 39

44 3 Three-dimensional models for blood flow problems of vascular fixed geometry may be reasonable, and allow to capture the main characteristics of the flow in a by far smaller amount of computational time, with respect to moving geometry models. On the other hand, compliance of the vascular wall is often neglected also due to the lack of measured data for specific subjects and the unavailability of a reliable mechanical characterization of the vessel structure. It has to be noted that rigid wall models may be less precise in specific applications with respect to compliant models [48, 100]. Recent works [35, 94] show that, in the case of the cerebral vasculature, hemodynamic computations based on rigid models reproduce the same flow patterns as compliant models, the latter giving slightly different estimates of the flow quantities. Therefore, the introduction of the vascular compliance seems not to be required in order to understand the main features of cerebral blood flow. This evidence, and the lower computational cost, motivated us to adopt rigid geometry models. Moreover, even in rigid wall simulations part of the compliance is accounted for by the shape of the pulsatile waveform, since it is the compliance of the system which leads to different pulsatile waveforms. The blood flow regime is usually laminar in most part of the cardiac cycle. In systole, however, the flow may become unstable in specific vascular districts, due to many different reasons: in the aortic arch, for instance, the systolic peak velocity can be very high even in physiological conditions. The presence of vessel stenosis or an increased blood demand from the organs due to physical exercise may promote in certain locations the transition of the flow regime towards instability. Some of the parameters used for the characterization of the flow regime and their physical significance are discussed in the following section. 3.2 Geometry and Flow Generally speaking, the purpose of a model is to reproduce the features of a physical system. More precisely, the model should be similar to the system from a geometrical and dynamical point of view (see [37]). Geometric similarity is provided by an accurate morphologic description of the system. In the case of vascular geometries, this depends on the quality of medical images at hand and on the effectiveness of the segmentation process and geometry reconstruction tools. Dynamic similarity is guaranteed by the identification and the control of the so-called similarity parameters, i. e. dimensionless parameters whose values in the model should match those in the real case. In the case of a fluid flow in curved pipes, important parameters are the Reynolds and Dean numbers, together with the Womersley number and/or Reduced Velocity in the case of unsteady flows. We recall hereafter their definition (see e. g. [37]). 40

45 3 Three-dimensional models for blood flow problems Reynolds number The Reynolds number Re a in an internal flow of mean sectional velocity W within a pipe or vessel of characteristic radius a is given by Re a = aw ν (3.1) where ν is the kinematic viscosity of the Newtonian fluid, while suffix a here indicates the reference length used in the definition of the Reynolds number [114]. A different possible choice is to use the pipe diameter d = 2a, thus obtaining a Reynolds number Re d = 2Re a. This parameter can be physically thought of as the ratio of inertial forces to viscous forces, as is made more evident by rearranging terms in equation (3.1): Re a = ϱw 2 µw /a, where ϱw 2 = ( ϱw ) 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 viscosity. When we have a large Reynolds number, inertial forces are dominant over viscous forces and vice versa. This makes Reynolds number the key parameter which identifies the transition of the flow to turbulence and therefore helps in determining flow stability. Moreover, flows with higher Re a are characterised by a greater persistence of geometric influences 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 coronary arteries and medium-size veins) and even less than 1 (arterioles, capillaries, venules) [48] Dean number The original form of the Dean number was defined by Dean [34]: ( a ) ( ) aw 2 K = 2 (3.2) R ν where R is the radius of curvature (see Fig. 3.1), a is the pipe radius and W is defined as a constant having the dimensions of a velocity. According to Berger et al. [15], who proposed an extensive review of the literature about flow in curved pipes, the preferable definition for Dean number is however the following: ( a ) 1/2 2aW κ = 2δ 1/2 Re a = R ν (3.3) 41

46 3 Three-dimensional models for blood flow problems (a) (b) Figure 3.1: (a) A torus (from (b) Toroidal coordinate system. The pipe is a cylinder with circular cross section of radius a. Its axis is a circle 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 first two 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 recast in terms of the angle θ defined in the plane containing the torus axis. The fluid velocity in the pipe has three components: u and v are the radial and circumferential velocity (respectively) in the cross section, while w is the axial velocity. This set up was exploited by Berger et al. in their analysis of flow in curved pipes. [15] where δ = a R. 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 take W = W in (3.2), then κ = (2K) 1/2. Other definitions for Dean number may be based on pressure gradient [15]. More precisely, in the case of fully developed flow, the following has been widely used in literature: ( ) 2a 3 1/2 ( ) G a 2 2a 1/2 D = ν 2 R µ = 4 G a 3 (3.4) R 4µν which is related to (3.2) by D = 4K 1/2. Here G represents the (constant) axial pressure gradient. This approach however has two main disadvantages: on the one hand, it is more difficult 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 (it varies with axial location and position in the cross section): this is true even in pipes of given 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

47 3 Three-dimensional models for blood flow problems ance of the forces due to inertia and centripetal acceleration versus the viscous forces: κ = 2δ 1/2 Re a = 2 ( ϱr aw a R µw a ) 2 ϱw 2 centripetal forces inertial forces viscous forces ( ) 2 In the above, we note that aw R is a measure of the angular velocity, thus ϱr aw a R is 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: G n = (D/2)τRe d where τ is a measure of the torsion; or combinations of G n and κ [37] Womersley number and Reduced Velocity/Strouhal number Figure 3.2: Representation of the laminar boundary layer (in blue) in a viscous fluid motion. The boundary layer thickness δ grows over time due to the action of viscosity. (after [37]) The (Sexl-)Womersley number [123, 147] can be physically interpreted as the ratio of the pipe diameter to the laminar boundary layer growth over the pulse period T : 2π Wo = a νt d νt where d = 2a is the pipe diameter and we exploit a dimensional argument commonly used in laminar boundary growth over flat plates, namely the fact that the boundary layer 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 motion of a Newtonian fluid in a straight circular pipe subject to a periodic pressure difference [147]. In fully developed flow, the solution is periodic with only the velocity axial component u z different from zero. If Wo is small (1 or less), the frequency of pulsations 43

48 3 Three-dimensional models for blood flow problems is sufficiently low so that a parabolic velocity profile has time to develop during each cycle. 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 the viscous 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 rather than parabolic [48]. A potential limitation of Wo is that it is related to physical scales within a given cross section of the pipe: in other words, the role of the longitudinal geometry is not represented, and this may reduce the physical significance of the parameter when the flow has significant changes in the streamwise direction. Other non-dimensional similarity parameters may be introduced in order to take into account different length scales as well, which are relevant in specific flow problems (for instance the flow in multiple bends or through stenoses). The Reduced Velocity can be useful to describe different flow regimes, and can be seen as a non-dimensionalised pulsatile period: U red = W T D Distance travelled by mean flows Diameter In particular, U red is a more appropriate parameter than Wo for unsteady flows where there are significant streamwise flow variations, since it explicitly involves an axial length scale. Note also that U red and Wo are related through Re d : U red = π The Navier-Stokes equations Re d Wo 2. 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 Formulation Let I R be a time interval and let B E be the spatial domain representing a blood vessel (with E a three-dimensional euclidean space) (see Fig. 3.3). A regular motion of points belonging to B is a function x : B I E such that x C 2 (I): in particular we may define the material velocity of points of B as the function ẋ : B I R 3 ẋ(p, t) = x(p, t) t Denoting B t := x(b, t), we may now introduce the trajectory T := {(q, t) : t I, q B t }. 44

49 3 Three-dimensional models for blood flow problems Γ out,3 Γ out,4 Γ out,2 Γ out,1 Γ w Γ in Figure 3.3: An example of spatial domain representing a basilar artery with a berry cerebral aneurysm (geometry from the Aneurisk project). The letters indicate the inflow, outflow and wall boundaries. and the spatial velocity u : T R 3 such that u(q, t) := ẋ(x 1 (q, t), t). We analogously define the fluid mass density ϱ(q, t) as a function of the spatial position and the time. By exploiting the mass conservation principle we obtain the continuity equation: ϱ + ϱ(div(u)) = ϱ,t + div(ϱu) = 0, (3.5) where ϱ is the material derivative of ϱ with respect to t, defined as follows: ϱ = d dt ϱ(q(p, t), t) = ϱ,t + ϱ 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 equation reads ϱ u = b + div(t), where b : T R 3 is a field of volume forces and T is the so-called Cauchy stress tensor: (q, t) T, T(q, t) L(R 3 ), L(R 3 ) being the space of linear tensors over R 3. 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

50 3 Three-dimensional models for blood flow problems 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 depending 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), ϱ(x(p, t), t) = ϱ 0 (p, t), p B, t I where ϱ 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-known incompressible Navier-Stokes equation: ϱ 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 incompressibility constraint. By introducing ν = µ ϱ 0 (kinematic viscosity), P 0 = P ϱ 0, b 0 = b ϱ 0, (3.8) may be rewritten in the following form: u,t + ( u)u = b 0 P 0 + 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 characteristic value for the fluid velocity modulus u. The choice of these values is arbitrary and in general related to the flow features. In blood flow problems l is generally taken equal to the blood vessel diameter, while u is the average fluid speed. The Navier-Stokes equations are modified by introducing these non-dimensionalized quantities: u = u u, t = tu l, x = x l, the pressure being non-dimensionalised as follows: P 0 = P 0 u 2. 46

51 3 Three-dimensional models for blood flow problems Moreover, the time and space derivatives are referred to the non-dimensional variables t and x : x P 0 = u2 l x P 0, x u = u l 2 x u, u,t = u2 l u,t, ( xu)u = u2 l ( x u )u. By substituting the previous definitions in (3.9a), where we set b 0 = 0 for the sake of simplicity, and after some algebraic calculations, we obtain the following non-dimensional formulation: 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 Re x u, (3.10) where we identify the Reynolds number Re. Let now l 1 and l 2 be the characteristic spatial scale of two flows, such that l 1 = λl 2, λ R. Moreover, let Re 1 and Re 2 be the Reynolds numbers for the two flows, such that Re 1 = l 1u 1 ν 1 = l 2u 2 ν 2 = Re 2 that is λ u 1 = u 2. Then (u 1 ν 1 ν, P 0,1 ) and (u 2, P 0,2 ) satisfy the same set of differential 2 equations in non-dimensional form (3.10). Two such flows are termed geometrically and dynamically similar. Boundary conditions and initial condition Considering the vascular geometry depicted in Fig. 3.3, we can identify different boundary regions, corresponding to the fixed vascular walls Γ w and to the artificial inlet and outlet 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 remaining part of the circulatory system. More precisely, we distinguish a proximal section Γ in (that is closer to the heart with respect to the mean blood flow direction) and four distal sections Γ out,i, i = 1,..., 4. They are also referred to as one inflow and four outflow sections, even if during the cardiac cycle the flow can be exiting the inflow section and entering the outflow sections, due to flow reversal which is likely to happen especially in major vessels. Typically we prescribe a velocity profile at Γ in, able to reproduce measured data (when available). We prescribe zero velocity on Γ w, corresponding to a no-slip condition for the fluid in contact with the wall. Finally, on Γ out,i the normal stress T n is prescribed, n being a vector field defined on Γ out,i and aligned to the direction normal to the section. 47

52 3 Three-dimensional models for blood flow problems 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 reduced models taking into account the remainder of the circulatory system, so that the boundary conditions on the artificial sections come from interface conditions between the different models. An example of this approach is presented in Chap. 5, where we discuss specific issues arising in the coupling of 3D models based on the incompressible Navier-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, t 0 ) = u 0 (q), q Ω, with the additional incompressibility constraint div(u 0 ) = 0. The choice of u 0 is arbitrary (usually u 0 = 0), since in general a physically relevant initial condition is not known 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 recovered only 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 the solution is dominated by the boundary conditions Numerical discretization In order to compute a numerical solution (υ, ψ) of system (3.9), we carry out a discretization 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 finite difference approximation, that means to split the time interval of interest (0, T ] into subintervals with time step t, such that t k = k t (k N) and approximate the time derivatives with suitable incremental ratios evaluating the unknowns at the instants t k. In the sequel, we will assume to discretize the equations through a backward Euler time discretization. For what concerns the space discretization, we refer to the finite element method. The discretization of the problem according to the finite element method is based on the Galerkin approximation of (3.9), that we are going to introduce. In the sequel, we denote by L 2 (Ω) the space of square integrable scalar functions in Ω, L 2 (Ω) the analogous functional space for m dimensional vector functions, H m (Ω) the functions belonging to L 2 (Ω) together with their first m spatial derivatives and H m (Ω) the functions belonging to L 2 (Ω) together with their first m spatial derivatives. In particular, H 1 0 (Ω) denotes the functions belonging to L 2 (Ω) together with their first spatial derivative and whose trace vanishes on Ω. For the sake of simplicity, we will assume Ω smooth enough 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

53 3 Three-dimensional models for blood flow problems each n 0, find υ n+1 h V h and ψ n+1 h Q h such that: m ( υ n+1 h, v h ) + a ( υ n+1 h, v h ) + g ( υ n+1 h, v h ) + +b ( v h, ψ n+1 ) h = f n+1 v h dω + m (υ n h, v h) Ω v h V h b ( υ n+1 h, q h ) = 0 qh Q h, (3.11) where {V h, h > 0} and {Q h, h > 0} are families of finite-dimensional subspaces of H 1 0 (Ω) for the velocity and of L 2 \ R for the pressure, respectively, υ k h, ψk h denote the discrete velocity and pressure computed at t k and: m (w, v) 1 t b (w, q) = w vdω, Ω a (w, v) = ν wqdω, g (w, v) = Ω w : vdω, Ω ( w) w vdω. Ω (3.12) In particular, in the finite element method, we introduce a triangulation T h of the domain Ω, i. e. a finite decomposition of Ω into tetrahedrons. Here h denotes the maximum of the diameters of the triangles of T h. Then, assume that V h is the space of piecewise polynomial functions on every element of T h, continuous in Ω and vanishing on the boundary Ω. Similarly, Q h is the space of piecewise polynomial functions on every 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 full implicit time-stepping procedures. In this work, we follow a semi-implicit strategy, based on the approximation: g ( w n+1 ) h, v h ( wh n ) wn+1 h vdω Ω However, different strategies can be considered as well (see e.g. [108]). We remark that the well posedness of problem (3.11) is ensured when the Ladyzhenskaja-Babuska-Brezzi (LBB) condition, which requires a compatibility between the choice of the polynomial degrees for the velocity and the one for the pressure, is satisfied [111]. On the basis of the presented formulation, different solution techniques can be devised to manage the resulting system of algebraic equations. We mention in particular block factorization methods, based on the splitting of the problem, for instance generating separate subproblems for the velocity and the pressure [ ]. A slightly different finite element method for the discretization of the Navier-Stokes problem has been introduced by Burman et al. [18], consisting in a stabilized Galerkin formulation using equal order interpolation for pressure and velocity and therefore reducing the dimension of the resulting algebraic problem. 49

54 3 Three-dimensional models for blood flow problems (a) Domain Ω represents an aneurysmatic Internal Carotid Artery. (b) Zoom on a part of the domain boundary Ω: the aneurysmal bleb. Figure 3.4: Visualization of an example of computational domain (a). Particular (b) of the domain boundary, with a graphical representation of the normal vectors to 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 reads σ = T n, T being the Cauchy stress tensor (see (3.6)). The tangential component of σ is referred to as the wall shear stress, and may be derived as follows: WSS = σ t = σ (σ n) n. Note that σ t by definition takes only into account the viscous component of the stress, the pressure being responsible only for a normal stress 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 necessary to retrieve a suitable approximation of the velocity gradient. This issue has been extensively treated in the literature: we mention in particular several works by Zienkiewicz & Zhu [ ] proposing a cost effective procedure for the recovery of the gradient of finite element solutions at the nodes. In the following we focus on the L 2 projection method [13, 153], which is proven to provide a superconvergent approximation of the gradient on linear elements. 50

55 3 Three-dimensional models for blood flow problems Let P be the Navier-Stokes problem defined over Ω. Suppose that we have an approximation u h of the velocity field solution for P, obtained with a Galerkin finite element method: u h [Xh r]d, u h (Ω) R d, where Xh r is the space of r-th degree piecewise polynomial functions on a tessellation T h of Ω. We compute an approximation G(u h ) of u h, such that (L 2 projection of the gradient over [Xh r]d d ): G(u h ) : vdω = u h : vdω, v [Xh r ]d d. Ω Ω The componentwise formulation reads: (u h ) k G kl (u h )vdω = vdω, v Xh r Ω Ω x. l In standard Galerkin finite element theory Xh r = span(ϕ i, i = 0,..., N), N being the number of degrees of freedom of the problem and ϕ i the scalar finite element nodal function associated to the i-th degree of freedom. Each component G kl may then be expressed as a linear combination of the finite element basis functions: G kl (u h )(x) = N j G kl (u h )(N j )ϕ j (x) where N j is the j-th degree of freedom, x Ω. It can be seen that the values G kl (u h )(N j ) are therefore solutions of the following linear system: ( ) (u h ) k ϕ i ϕ j dω G kl (u h )(N j ) = ϕ i dω, i = 0,..., N Ω Ω x l N j On the other hand, knowing that (u h ) k (x) = N j (U j ) k ϕ j (x), we can write Ω (u h ) k ϕ i dω = ϕ j (x) (U j ) k x l x l N j The overall problem reduces therefore to the solution of d d linear systems of the form Mg = f where M(i, j) = ϕ i ϕ j dω is a mass matrix while Ω g = G kl (u h )(N j ), f = D l (U) k, ϕ j (x) with D l (i, j) = ϕ i dω. Having at hand the approximate solution u h, which is Ω x l a piecewise polynomial function over the tessellation T h of the computational domain, the right hand side of these systems can be computed exactly. 51

56 3 Three-dimensional models for blood flow problems Oscillatory Shear Index Several studies focused on the effect of wall shear stress on the remodeling mechanism in blood vessels. Not only the mean WSS is associated to anatomic changes of vascular wall: the rate of change of wall shear is believed to play a role in the development of pathologies such as atherosclerosis [25]. In particular, it has been found that laminar shear stress is atheroprotective for endothelial cells, whereas nonlaminar, disturbed, or oscillatory shear stress correlates with development of atherosclerosis and neointimal hyperplasia [31]. In specific locations of the circulatory system, the blood flow may be reversed during part of the cardiac cycle: this causes the wall shear stress to significantly vary its direction. Ku et al. [68] proposed an oscillatory shear index (OSI) in order to quantify this effect, and looked for a correlation between vascular wall locations featuring high OSI and the local initiation of atheroma in the human left carotid artery. The original formulation is the following: OSI = T 0 T 0 WSS dt, (3.13) WSS dt where T is the duration of the cardiac cycle, WSS is the wall shear stress vector and WSS is defined as the stress component acting in the opposite direction with respect to the direction of the mean shear stress (in time). Taylor et al. [138] proposed a similar formulation, which encompasses a strategy to estimate the deviation of WSS from its temporal mean direction: OSI = 1 2 ( 1 WSS ) mean WSS mag, (3.14) where WSS mean is the mean shear stress, defined as the magnitude of the time-averaged stress vector σ t (see (3.6)), while WSS mag is the time-averaged magnitude of the stress vector: WSS mean = 1 T σ t dt T, WSS mag = 1 T σ t dt T 0 According to the latter formulation, OSI is minimum (and equal to 0) if WSS is constantly directed along its average direction. OSI is maximum (and equal to 0.5) if the mean WSS over the heart beat is = 0. A different approach has been followed by Steinman et al. [85], who weighted the positive values of the scalar product of WSS and the mean shear direction n: n = T 0 WSS WSS dt. 0 52

57 3 Three-dimensional models for blood flow problems The OSI is then defined as follows: OSI = T 0 WSS n H(WSS n)dt, (3.15) T 0 WSS n dt H(x) being the Heaviside unit function. In this formulation, high values for OSI indicate that WSS direction is not opposite to the mean shear direction for the most part of the cardiac cycle. 3.5 Working on regions of interest It is usually relevant to quantify the mechanical stress exerted by the blood flow on a vessel wall in locations where it may be associated to the development of vascular pathologies. This is the case of cerebral aneurysms, in which typically we want to quantify 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). vessel centerline is also represented. The On the other hand, the unknowns of a fluid dynamics problem are approximated on the whole computational domain Ω. If we are interested in evaluating some quantities on specific (sub-)regions of Ω (see Fig. 3.5), we need to define those regions and restrict there the fluid dynamics analysis. One possible approach is based on the extraction of the vessel centerline Decomposition of bifurcation branches Blood vessels can be effectively represented by centerlines, synthetic descriptors of the geometry and the topology of vascular networks. For the purposes of this work, a centerline is defined as the weighted shortest path traced between the inlet and the 53

58 3 Three-dimensional models for blood flow problems outlet of a vessel, the weight being the distance from the surface [6, 7, 103]. More precisely, a centerline is traced on the medial axis of the vascular geometry, that is the locus of 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 information about its radius. Moreover, a natural parametrization for the centerline is given by the associated curvilinear abscissa, which ranges over the line and relates each point to its Euclidean 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 the maximal inscribed spheres along the centerline defines a scalar function called tube function T, which can be expressed as follows: { } T (x) = min x c(s) 2 r 2 (s), s [0,L] where x R 3 is a point in the Euclidean space and r is the radius of the maximal inscribed sphere whose center is the centerline point c(s). The zero isosurface of T is referred to as tube or canal surface, and by construction is strictly contained inside the vascular lumen. The function T is negative inside the tube. The construction of centerlines and tubes is particularly useful in the study of bifurcating vessels, since it allows the identification and characterization of the bifurcation points [7]. For the sake of clarity, we refer to the case depicted in Fig First of all, the centerlines for the surface at hand are computed, yielding a set of curves running from the bifurcation inlet to each outlet. Then, two reference points are defined for each centerline: the first is the point in which one centerline crosses the tube surface generated by the other (and is labelled as point 1 in black ink, in Fig. 3.6); the second is the center of the nearest upstream sphere touching point 1 (and is labelled as point 2 in black ink in the figure). These four points and the associated spheres describe the position of the bifurcation and the size and shape of the vessel at the bifurcation point. Moreover, they split the centerlines into tracts corresponding to single bifurcation branches and to the bifurcation center, labelled respectively as tract 1, 3, 4 and 2 in red ink in Fig Each point of the space can then be associated to the nearest centerline tract, thus inducing a partition in regions of influence of the different tracts. These regions of the space cross the vessel surface in the bifurcation region, decomposing it in branches. We remark that the same approach can be applied to the study of aneurysms: the aneurysmal bleb may be regarded as a branch with respect to its parent vessel, and can be therefore decomposed from the vessel surface. An example of this procedure applied to the geometrical model of a pathological Internal Carotid Artery is depicted in Fig Relating surface points to centerlines By splitting each bifurcation in its branches, a simplification of the topology of the vascular network is achieved. Each branch is now topologically equivalent to a cylinder, 54

59 3 Three-dimensional models for blood flow problems Figure 3.6: Representation of a bifurcating vessel (from Left: centerlines. Middle: tubes constructed as envelopes of the maximal inscribed spheres along each centerline. Right: identification of two reference points for each centerline, for the description of the position, size and shape of the bifurcation (in black); decomposition of the centerlines in distinct tracts (in red). thus its geometrical characterization is easier. For instance, it is possible to robustly generate cross-sections of the vessel along the centerline [69]. More than that, each branch can be mapped into a rectangular parametric space: this allows the comparison among geometrical models in the same parametric space, and has been recently investigated by Antiga et al. [7]. A curvilinear reference system can be defined on each centerline branch, centered in the considered bifurcation. Each surface point y can be associated to its nearest centerline point c y, by minimizing the tube function T (y). An example of this procedure is depicted in Fig Following the same idea, regions of interest can be identified on a computational mesh, automatically selecting the nodes in which to evaluate fluid dynamic variables or derived parameters. In the context of Aneurisk project this approach was exploited for the evaluation of the spatial average wall shear stress in a specific location of the Internal Carotid Artery. In that particular case, the last tract of the ICA surface was considered, as a preferential site for aneurysm development. Having associated the centerline abscissa 55

60 3 Three-dimensional models for blood flow problems (a) Surface representing an internal carotid artery with a berry aneurysm. (b) Each point of a centerline is the center of the maximal inscribed sphere in the surface and is associated to its radius. (c) Decomposition of the surface in bifurcation branches. (d) Selection of a region of interest on one branch. Figure 3.7: From the geometrical model of a blood vessel to the selection of a region of interest. The definition of the vessel centerline allows the decomposition of the surface into branches. Undesired branches are excluded from the region of interest. to the surface points, the problem was recast into the selection of an abscissa interval on the centerlines. To this aim, two geometric landmarks were considered. The first is the center of the main bifurcation of the ICA, the second is the point delimiting the last ICA bend prior to the bifurcation. Following [119] we define a vascular bend or siphon as a segment included between two points of approximately zero curvature of the centerline. Therefore the selected region Γ comprises the last ICA siphon and the last few centimeters of the vessel, prior to the bifurcation, and the average value of WSS 56

61 3 Three-dimensional models for blood flow problems (a) Curvilinear reference system in a vessel bifurcation: the origin is placed in the center of the main bifurcation (where the aneurysm is located). (b) Each point of the surface is associated to a curvilinear abscissa. (c) A region of interest is selected (in red), by considering an interval of curvilinear abscissa and neglecting undesired branches (such as the aneurysmal bleb). (d) Wall shear stress values in the region of interest. Figure 3.8: Evaluation of wall shear stress on a region of interest selected on the vessel surface and corresponding to an interval of curvilinear abscissas on the centerline. Undesired branches are excluded from the analysis. over Γ can be defined as follows: WSS = WSSdγ Γ dγ 57 Γ. (3.16)

62 3 Three-dimensional models for blood flow problems Figure 3.8 shows the application of this technique to the study of fluid dynamics inside a realistic ICA geometry. More details on the results of this analysis in the context of Aneurisk project will be presented in Chap

63 4 An application of three-dimensional modeling to the study of cerebral aneurysms In this Chapter we focus on the study of cerebral vasculature: an introductory overview on its hemodynamic features is provided in Sec In particular, in the context of the Aneurisk project (presented in Sec. 4.2) we proposed WSS as a hemodynamic parameter for 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, so that its dynamics can be described by the three-dimensional unsteady incompressible Navier-Stokes equations (see (3.8)). It has been shown [70, 98, 101] that this approach is reasonable when looking at large arteries. The same assumption may however not hold inside the aneurysmal sac, due to the presence of slow-flow regions [14]: several studies have been presented dealing with the non-newtonian characteristics of blood flow in such conditions [22, 83]. In the following, we will not deal specifically with the flow features inside the aneurysm, hence we will refer only to Newtonian models for the blood. The wall motion is usually neglected in blood flow modeling [99, 132, 137]. For the case of intra-aneurysmal flow, the effect of moving boundaries has not yet been assessed, 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 the mechanical characterization of the wall, together with the lack of intra-arterial pressure measurements [24]: this limitation can be circumvented by using measured wall movements as boundary conditions in CFD models. Instead of solving a structural dynamics problem, driven by the stress exerted by the fluid on the structure, the position in time of the interface between the blood and the wall can be recovered for instance from dynamic angiography images through nonrigid registration algorithms. The first results obtained with this technique suggest that the main characteristics of the flow (location and size of the inflow jet, complexity and stability of the patterns) are not altered by the movement of the domain, while the computed WSS and velocity magnitude can be affected [35, 93]. Intra-aneurysmal flow patterns develop in a wide variety and complexity: vortical structures may form inside the bleb and they can be stable or move during the cardiac cycle becoming unstable. Not only the size and shape of the aneurysm determine the 59

64 4 An application of three-dimensional modeling flow structures, but the geometry of the parent artery also influences the way the blood enters 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 the inflow jet and the location of the impingement region strongly depend on the patientspecific vascular geometry [124]. 4.2 The Aneurisk project The Aneurisk research project ( ) was developed by a joint venture of different subjects: academic and non academic research centers (MOX - Department of Mathematics, Politecnico di Milano; LaBS - Department of Structural Engineering, Politecnico di Milano; M. Negri Institute for Farmacological Research, Bergamo), medical centers (Dipartimento di Neurochirurgia, Università degli Studi di Milano; Ospedale Niguarda Ca Granda di Milano; Ospedale Maggiore Policlinico di Milano), industrial partners (Siemens Medical Solutions Italy; Fondazione Politecnico di Milano). The main goal of Aneurisk project was to develop a framework for the analysis of cerebral vascular geometries. The project was based on the idea of a stream of information starting from the medical image and passing through a series of steps, each one adding a layer of knowledge to the overall process. The first step is image segmentation, together with geometry reconstruction and morphology characterization. It is followed by a modeling step for the simulation of blood flow in realistic geometries and the characterization of the wall mechanics. Statistical analyses represent the interpretive step, for the organization and the extraction of information from the complete data set. 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 (and possibly 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 project consisted in 134 three-dimensional computed rotational angiography (3D CRA) scans, obtained during clinical routine at the Neuroradiology Division of the Ospedale Niguarda Ca Granda in Milan, for assessment of cerebral aneurysms. Patient-specific models of cerebral arteries have been reconstructed from these images, after a segmentation process. Geometry characterization was performed on the models employing a set of tools part of VMTK ( an open source software project for vascular modeling [5], mainly developed at Mario Negri Institute in Bergamo. This analysis was carried out by Piccinelli et al. and showed preferential locations of the pathology in the distal upper tract of the vessel, on the outer wall of ICA bends in correspondence of local curvature maxima [102,103]. Moreover, ruptured aneurysms were found typically in more distal positions along the vessel centerline (see Fig. 4.1). A conjecture formulated by neuroradiologists at Ospedale Niguarda Ca Granda was tested, namely that some geometrical features of ICA are different according to the presence and the location of an aneurysm (see Fig. 4.2). The idea was confirmed by a classification of Aneurisk data set, proposed by Sangalli et al. [120]. They considered 60

65 4 An application of three-dimensional modeling Figure 4.1: Position of the aneurysm along the centerline of pathological ICAs: in the whole population, in patients with ruptured aneurysms and in patients with unruptured aneurysms. The point of zero abscissa is set at each ICA bifurcation, so that centerline abscissas represent distance to the bifurcation. Black dots represent mean value; crosses represent standard deviation [103]. two groups of patients. The first (blue dots in Fig. 4.3) is composed of patients with an 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 terminal bifurcation or healthy. Radius and curvature profiles were studied in the last tract of ICA prior to the bifurcation [118, 119] (see Fig. 4.2), and patients in the blue group were found to have significantly wider, more tapered and less curved ICA s. Moreover within this group there is a lower variability of radius and curvature of the ICA. On the basis of this result, a similarity index was defined to measure how the geometrical features of each vessel compare to those of the representatives of the morphological classes (see Fig. 4.3). It is well known that blood flow features strongly depend on the vascular morphology: therefore we believe that the differences in the geometry of ICA of patients belonging to the described groups induce different hemodynamic features and that these may trigger the pathologic response of the arterial wall. We then propose a CFD analysis over the Aneurisk dataset, in order to study the blood flow features in the last tract of ICA. Moreover, we look for parameters able to synthetically describe the effects of blood flow on the vessel wall, such as the spatial average of wall shear stress. This information could be used to have a better understanding of the mechanisms of aneurysm development in the vascular district at hand; on the other hand, it could be used to enhance the classification proposed in [120] by combining the mechanical and the morphological characterization of the vessels. 61

66 4 An application of three-dimensional modeling Figure 4.2: Curvature profiles along the centerline in the internal carotid artery [119]. Negative values for the curvilinear abscissa identify proximal vessel locations, with respect to the origin of the reference system which is placed at the ICA bifurcation. The top of the figure shows the estimate of the probability density function of the location of aneurysms along the ICA. Upper group membership probability Lower group membership probability Figure 4.3: Classification of Aneurisk data set based on morphological features of the Internal Carotid Artery. Red dots represent ICAs with an aneurysm before the terminal bifurcation or healthy. Blue dots represent ICAs with an aneurysm 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 classification [120] (see Fig. 4.3). Seven geometries were elicited from the group of vessels with high similarity to the representative of the red group (see Fig. 4.4), other seven from the group of vessels similar to the representative of blue group (see Fig. 4.6), the remaining were chosen among the cases whose features do not fit in either class (see Fig. 4.5). We 62

67 4 An application of three-dimensional modeling will refer to the latter as to the green group. (a) (b) (c) (d) (e) I (f) (g) Figure 4.4: Data set for the numerical simulations: ICA geometries classified as belonging to the red group. All the reconstructed geometries in Aneurisk data set were available as three-dimensional surface models represented in StL format. These models represent in general a large part of the cerebral vasculature, featuring the presence of one or more aneurysms. For this particular study, we were interested specifically in the Internal Carotid Artery, so that all the geometries had to be restricted to this region by proper trimming of the surface model. Moreover, following what has been done in [120], we focused our attention on the last tract of the Internal Carotid Artery, prior to its terminal bifurcation. In facts, this location is particularly interesting, since it is a preferential site for aneurysm development [103]. On the other hand, models of ICA reconstructed from Aneurisk 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 surface geometry. See 63

68 4 An application of three-dimensional modeling (a) (b) I (c) (d) (e) (f) (g) Figure 4.5: Data set for the numerical simulations: ICA geometries which do not belong to either group in the morphological classification. portion of the vessel, which was present in all the images. Indeed, the dimensions of the reconstructed 3D model depend on the size and the spatial position of the volume scanned during the clinical procedure, and this volume is not the same for all the considered 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 computed the integral average of WSS on a specific region of the vascular wall, corresponding to an interval of curvilinear abscissas on the vessel centerline spanning the last ICA bend 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 system (located at the bifurcation) and the point of zero curvature delimiting the nearest siphon to the bifurcation [119]. 64

69 4 An application of three-dimensional modeling (a) (b) (c) (d) (e) (f) (g) Figure 4.6: Data set for the numerical simulations: ICA geometries classified as belonging to the blue group. The choice of including the bend in the region of interest has two main reasons. On the one hand, it is an easily recognizable landmark and is present in all the geometries, 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). The latter 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 determine the hemodynamics features in all the considered region. It has also to be noted that flow pulsatility, together with the complexity of the overall vascular geometry (a sequence of sharp, non planar bends) induces strong secondary motions resulting in vortical or helical flow patterns. Evidences of this phenomenon have been recently discussed in [48, 104], and on this basis we chose to reconstruct the entire sequence of ICA siphons (to the most possible extent given the images), even if interested only in 65

70 4 An application of three-dimensional modeling the distal portion. An excessive trimming of the upstream vascular geometry could in fact lead to a non correct evaluation of the patient-specific interplay between vessel morphology and hemodynamics. (a) The original StL model. (b) The trimmed model (now mostly representing the ICA). (c) Flow extensions added to the ICA model. Figure 4.7: Definition of the computational domain: the model reconstructed from medical images (a) is trimmed (b) and flow extensions are added on the inlet and outlet sections (c). An example of a trimmed geometrical model is depicted in Fig. 4.7 (b). It represents the distal bend of an Internal Carotid Artery and its main bifurcation: the trimming procedure 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 way that 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 of these cylindrical extensions is adaptively selected as 10 times the clipped section radius. This technique allows to analytically formulate the boundary conditions, the boundary sections being circular. At the same time, the flow profile is allowed to develop in the cylindrical extensions, prior to entering the vessel domain. For each one of these geometrical models we obtained a tetrahedral grid with an average mesh size of 0.06 cm. The meshing procedure was performed by means of the software Netgen, which offers an implementation of the advancing front method together with several mesh optimization algorithms (both metric optimization and topo- 66

71 4 An application of three-dimensional modeling logical [33]). The meshes were refined on the basis of the surface curvature: an example is depicted in Fig Figure 4.8: Mesh refinement driven by the surface curvature. The computational domain was assumed to be fixed, corresponding to the hypothesis of rigid vascular walls. As pointed out above, we further assumed that blood can be modeled as a continuous incompressible Newtonian fluid, so that the blood flow problem can be described by the incompressible Navier-Stokes equations (3.8). For each vascular geometry, three cardiac cyles were simulated, in order to reduce the effects of the initial conditions and obtain the periodic solution in the last simulated heartbeat. The spatial discretization was based on the Galerkin finite element method (see Sec ), and was carried out with a P1 approximation for both the pressure and the velocity. The numerical scheme adopted is based on an edge stabilization technique [18]. The adopted time advancing scheme is a BDF of order 1, with a time step of dt = 10 3 s. The spatial integral average of WSS was computed on the arterial wall, after the removal of possible branching vessels and aneurysms, as discussed in Sec An example of the considered portion of the ICA surface is shown in Fig All the fluid dynamics simulations and the WSS computations were carried out with a 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 of partial differential equations. The treatment of vascular geometries (addition of flow extensions, splitting in branches, identification of regions of interest) has been performed by using the software VMTK. 67

72 4 An application of three-dimensional modeling Q t Figure 4.9: Flow rate (in ml/s) in an Internal Carotid Artery. The wave form of this signal reproduces measured in vivo data, the amplitude is scaled to give a time 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 is computed. Boundary conditions In Aneurisk dataset, patient-specific measurements of blood flow rates, velocity or pressures were not available. On the other hand, physiological values for ICA flow rate Q ICA can be found in the literature, and we had at hand the wave form of the blood flow rate measured in vivo in a single patient s ICA. Since our interest was to compare different vascular geometries, and to understand the effect of different morphological features on hemodynamics, starting from the available data we looked for a suitable set of boundary conditions for our geometrical models. We resorted to a non-dimensional 68

73 4 An application of three-dimensional modeling argument and imposed boundary conditions giving the same flow regime in all the computational domains. Figure 4.11: Velocity profile (in cm/s) imposed on the inflow section: only the axial component u n is non zero. This profile corresponds to (4.1) with Q in = 240 ml / min and R in = 0.2 cm. More precisely, we run a first set of simulations imposing as inflow boundary condition the wave form depicted in Fig. 4.9, scaled to give a time averaged flow rate Q ICA = 240 ml/min (a reference value for the time averaged ICA flow rate in a cardiac cycle, as found in the literature [79]). The chosen flow rate was obtained by prescribing a velocity profile on the inlet section, more precisely an axial velocity of the following form: 0 r > R in, 12 Q in 2(R in r) 1 u n = 7 R in 2 R in r R in, 12 0 r < R in, πrin 2 Q in πrin 2 where r is the radial coordinate on the inlet section while the values of R in and Q in are specified in Tab This choice corresponds to the assumption of fully developed axial flow, which can be approximated to a flat profile [48, 92] (see Fig. 4.11), and is legitimated by the use of a cylindrical boundary extension on the inlet section. It is indeed proven that the geometrical features of the vessel have a stronger influence on the solution than the presence of secondary velocities in the inlet profile. Moreover, the effects of these inlet secondary flows, even in case they are present, break down within a few diameters of the inlet [87]. We then computed the Reynolds number on the inflow boundary section, in order to classify the flow regime for each geometry. The time average over a cardiac cycle of these values is represented in Fig We found out that, for most of the considered geometries, a time average Reynolds number of Re = 350 describes with a good approximation the flow regime associated to an ICA flow rate in the range of physiological values. Therefore, we chose to scale the amplitude of the inflow datum to obtain that value in each geometry (see Tab. 4.1). In two cases (patient ID and , see Fig. 4.5 and Fig. 4.6) this flow regime was associated to blood flow rates significantly (4.1) 69

74 4 An application of three-dimensional modeling Reynolds number Figure 4.12: Boxplot of the time averaged Reynolds numbers, evaluated on the inlet section of each vascular geometry, when the imposed flow rate (averaged over a cardiac cycle) is 240 ml/min higher than in the others, due to larger ICA radius, found in both cases. Patient ID R in (cm) Q in (ml / min) Wo in U red I I Table 4.1: Inflow boundary conditions: radius R in, imposed flow rate Q in (time average over a cardiac cycle), Womersley number Wo in and Reduced Velocity U red evaluated on the inflow section. 70

75 4 An application of three-dimensional modeling A zero-stress condition was prescribed on the outlet sections through homogeneous Neumann boundary conditions. This implies that the same mechanical load was imposed on all the outflows, which is clearly not the case in vivo. However, this is a widely accepted assumption, when measures of flow rates or pressures are not available in correspondence of the boundaries of the computational domain. A possible way to avoid this strong assumption is to resort to the so-called geometrical multiscale approach, based on the coupling of detailed models (like the 3D Navier-Stokes equations (3.8)) describing the local fluid dynamics, together with reduced models (such as the 1D Euler equations (2.5)) reproducing the remainder of the circulatory system. This approach will be discussed further in Chap Discussion Figure 4.13: Integral average of the wall shear stress (in dyn / cm 2 ) over the last portion of ICA, prior to the bifurcation. The values for two different groups are shown (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 compute the spatial average WSS in the region of interest as a function of time. This will give seven time patterns, which can be averaged to find a representative time pattern for the considered quantity. If the same procedure is applied to geometries belonging to the blue group, the resulting representative time patterns can be compared as shown in Fig Moreover, the time average of these curves can be computed, yielding: Time averaged mechanical load (dyn / cm 2 ) Red group Blue group where for the sake of brevity we defined mechanical load the spatial average WSS 71

76 4 An application of three-dimensional modeling 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 mechanical load, 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 based on hemodynamics features. More precisely, we identify a threshold value, defined as the mean of the average mechanical load in the two groups: WSS = dyn / cm 2. (4.2) We then classify a vascular geometry as member of the red group if the computed mechanical load is higher than the threshold value. Otherwise, the vascular geometry is classified as member of the blue group. For the sake of clarity, we refer to the results presented in Tab. 4.2: the time average over the cardiac cycle of the mechanical load as previously defined is shown for each simulated case. It is clear from these data that in most cases the classification based on hemodynamics features agrees with the results of the morphological analysis. Indeed, as expected, vessels elicited from the red group and from the blue group are correctly classified by both strategies except for two cases which will be discussed in detail later on. On the other hand, the geometries belonging to the green group (i. e. with geometrical features not clearly associated to the red nor to the blue group, see Fig. 4.5) can now be associated to a typical red-group or blue-group hemodynamics behaviour. More precisely, 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. This means that, from the fluid dynamics view point, these cases are similar to the typical representative of the blue group, while the correct classification was not achieved on the basis of their morphological features. The hemodynamics analysis is able to correctly classify also a case of green geometry featuring an aneurysm along the ICA. The case identified as number (see Fig. 4.5) presents a small aneurysm in the very last part of the ICA, prior to the bifurcation: 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 ) is still misclassified on the basis of fluid dynamics arguments. It is indeed similar to a blue case, according to the estimated time averaged mechanical load, but it is not a typical blue case in that it has a quite big aneurysm along the Internal Carotid Artery (see Fig. 4.5). The very presence of such a big sac, however, is probably the reason why the proposed fluid dynamical analysis is not effective in this case: giant aneurysms with a large neck strongly deform locally the hosting artery, therefore calling for a more specific and detailed study than the evaluation of a spatial average value of WSS. Two cases of wrong hemodynamics classification are the red case 93817, featuring a low mechanical load, and the blue case , featuring high mechanical load. In the former case, however, the anomalous hemodynamic behaviour is due to a pathological condition of the entire vessel, namely a displasia causing the weakening of the 72

77 4 An application of three-dimensional modeling Classification Morphology Hemodynamics Patient ID Average WSS (dyn / cm 2 ) R B R R R R I R R B B B B B B R B B I B B B R B B Table 4.2: Classification of the considered dataset of ICA. The hemodynamics provides additional information with respect to the morphological analysis, and is able to properly classify uncertain cases. wall and a non physiological increase of the diameter (see Fig. 4.4). Conversely, the latter case features a particularly marked tapering in the very last tract of the ICA and is unusually tight in the bifurcation zone: this enhances the shear stress exerted by the blood on the wall in that specific region, and makes the spatial average WSS an unsuitable characterizing parameter. Finally, case deserves a comment, even if it is well classified both from the morphological and fluid dynamical point of view. It can be seen in Tab. 4.2 and in Fig that it features an extremely high mechanical load, and again the inspection of the 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 all the considered area. 73

78 4 An application of three-dimensional modeling Figure 4.14: Green group: integral average of the wall shear stress (in dyn / cm 2 ) over the last section of ICA, prior to the bifurcation. Figure 4.15: Red group: integral average of the wall shear (in dyn / cm 2 ) over the last section of ICA, prior to the bifurcation Wall shear stress as a classification parameter The results of this work suggest that high WSS, beyond the other morphological features discussed in [120], is associated to the presence of aneurysms in the Internal Carotid Artery. Starting from this observation, based on the presented data set, it may be conjectured that a high-wss environment induced by geometrical features predisposes to the development of the pathology. This conjecture is consistent with the idea of a correlation between geometrical and hemodynamical features of arteries, as assessed 74

79 4 An application of three-dimensional modeling Figure 4.16: Blue group: integral average of the wall shear stress (in dyn / cm 2 ) over the last section of ICA, prior to the bifurcation. in the literature. More precisely, elevated WSS has been associated to fragmentation of the internal elastic lamina of blood vessels [130], endothelial damage [129] and ultimately to aneurysm initiation (see also Chap. 1). 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 WSS typically features a complex spatial pattern, corresponding to a non homogeneous mechanical load distribution at the microscopic level. In most cases here considered, however, its spatial average can be a useful and synthetic indicator of the stress exerted by the blood on the arterial wall, whenever the region of interest does not feature 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 index able to discriminate the pattern of the mechanical load along the vessel curvilinear abscissa. To this aim, the approach presented in Sec for the mapping of centerline abscissas on the surface can be exploited. Each artery would then be described by a set of geometric and hemodynamics parameters regarded as functions of the curvilinear abscissa of the centerline, and a new classification could be designed following the approach presented by Sangalli et al. [120]. On the other hand, other hemodynamics parameters could be included in the analysis: in particular, vorticity is expected to be a significant flow feature in the complex ICA geometry, and could help in defining a more robust classification strategy. The in-silico setup here presented, and its further improvements, stand as a candidate tool to give a synthetic description of the mechanical solicitation exerted by blood flow 75

80 4 An application of three-dimensional modeling on the vascular wall. More than that, our results show that it can be used to characterize cerebral vascular geometries which are associated to the presence of an aneurysm. In this respect, it is worth noting that the additional information provided by CFD could have 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 prone to the development of an aneurysm in ICA. Finally, we want to remark that a strong interplay between vascular geometry and blood flow features has been clearly shown from our results. We proposed a way to couple and complement the information coming from two different analysis approaches. Further application and improvement of this twofold approach is likely to give a greater insight and comprehension of cerebral aneurysms. 76

81 5 A geometrical multiscale model of the cerebral circulation Geometrical multiscale modeling is a strategy advocated in computational hemodynamics for representing dynamics ranging over different space scales in a single numerical model. It allows to couple the description of large vascular districts given by reduced models, with the detailed analysis of blood flow in specific regions of interest. This approach is particularly interesting in the cerebral circulation, whose arteries on the one hand form a complex anastomotic vascular structure (the circle of Willis) with peculiar hemodynamics features (see Chap. 1, Chap. 2) and on the other hand are prone to 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 by means of 3D models (due to the lack of data and high computational costs), while 1D models 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 We present a multiscale model of the cerebral circulation where a one-dimensional description of the circle of Willis, relying on the Euler equations, is coupled to a fully three-dimensional model of a carotid artery, based on the solution of the incompressible Navier-Stokes equations. A similar multiscale model has been investigated in [86], where the 3D model includes compliance for avoiding spurious reflections induced by a rigid treatment of the 3D geometry in the multiscale model. Even if vascular compliance is often not relevant to the meaningfulness of 3D results (e.g. in large arteries), it is crucial in the multiscale model, since it is the driving mechanism of pressure wave propagation (Sec. 5.1). Unfortunately, 3D simulations in compliant domains still demand computational costs significantly higher than the rigid case. Appropriate matching conditions between the two models have been devised to concentrate the effects of the compliance at the interfaces and to obtain reliable results still 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 overall computational cost. This is the case of the compliance of vessels. In 3D Navier-Stokes stand-alone models compliance is quite often not relevant for bioengineering purposes. However, it is a driving mechanism of pressure wave propagation along the vascular tree. Therefore, when considering 3D/1D geometrical multiscale models in principle compliance should not be neglected in either models. 77

82 5 A geometrical multiscale model of the cerebral circulation 1D Model Γ 3D Model Figure 5.1: A simple multiscale (3D/1D) model of a cylindrical pipe The coupling between 1D and 3D compliant models has been investigated recently in [86]. The computational cost of a compliant 3D simulation is however by far higher than the rigid case. On the other hand, a naive coupling of 1D (which are intrinsically compliant) and 3D rigid problems, forcing for instance the continuity of pressure and flow rate is problematic, since the different wall modeling in the two subdomains makes the coupled problem ill conditioned and affects the numerical results. We overcome these difficulties by resorting to appropriate matching conditions that mimic the presence of the compliance by concentrating it at the interface of the two models. 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 We assume that a 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 hand side is represented by the incompressible Navier-Stokes equations (3.9). Coupling the two models requires appropriate matching conditions. In the case of a rigid 3D model, it is reasonable to prescribe the continuity of pressure and flow rate P 1D = 1 Γ Γ p 3D dγ, Q 1D = ϱ Γ u 3D ndγ, (5.1) where n is the outward normal unit vector on Γ, we added the indexes 1D and 3D for the sake of clarity and denote by Γ the area of the interface Γ. The negative sign in the second of (5.1) stems from the fact that both Q 1D and u 3D n are directed outward the 1D and 3D domains respectively. In the sequel, for easiness of notation, we set Q 3D = ϱ Γ u 3D ndγ, P 3D = 1 Γ Γ p 3D dγ. Other conditions can be considered as well, prescribing the continuity of the total pressure, of the normal stresses or of the characteristic variables (see [112]). 78

83 5 A geometrical multiscale model of the cerebral circulation Numerical algorithm When solving multiscale problems numerically it is natural to split the scheme into the iterative sequence of dimensionally homogeneous problems, which we indicate as 1D and 3D, for instance by means of the following algorithm. We assume that standard (Dirichlet or Neumann) conditions are prescribed at the boundaries of the overall 1D/3D model. Moreover, we carry out an appropriate space and time discretization of the problems. In particular, apexes n and n + 1 refer to the approximation of the solution at time steps t n and t n+1, respectively. Index k will refer to the inner iterations performed at a fixed time step, for the fulfillment of the matching conditions. For n = 0, 1,... we perform the following steps. 1) Inizialization. Set k = 0, u n+1 3D,0 = un 3D, p n+1 3D,0 = pn 3D, and P n+1 1D,0 = P 1D n, A n+1 1D,0 = ψ 1 (P n+1 1D,0 ), Qn+1 1D,0 = Qn 1D. 2) Loop on k. 2.1) Solve the 1D model with the boundary condition on Γ given by P n+1 n+1 n+1 1D,k+1 = χp3d,k + (1 χ)p1d,k, (5.2) where χ is a relaxation parameter to be set for improving the convergence rate. Solving the 1D model, pressure conditions are recasted in terms of area, thanks to the wall law A n+1 1D,k+1 = ψ 1 (P n+1 1D,k+1 ) (see Sec. 2.2). 2.2) Solve the 3D problem with the boundary conditions on Γ Q n+1 3D,k+1 = Qn+1 1D,k+1 (5.3) Set k = k ) Test. Different convergence tests can be pursued. A possibility is to check the continuity at the interface, namely terminate the iterations when P n+1 1D,k+1 P n+1 3D,k+1 ε ε being a user-defined tolerance. Swapping the role of the matching conditions in the set up of the boundary conditions for the iterative scheme, (5.2), (5.3) can be replaced by Q n+1 1D,k+1 = χqn+1 3D,k + (1 χ)qn+1 1D,k, P n+1 3D,k+1 = P n+1 1D,k+1. (5.4) The different space dependence of 1D and 3D models leads to unmatched or defective 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. The latter 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

84 5 A geometrical multiscale model of the cerebral circulation 1D Model Q 1D R 1 R 2 L Q 3D = R Γ u 3D n P 1D P C 3D Model P 3D = R Γ p 3D Figure 5.2: Representation of a multiscale 3D/1D model with a 0D element representing the compliance of the 3D model at the interface. the solution. This topic has been discussed in [112], Chap. 11, where different mathematically sound techniques for the solution of defective problems are presented. The specific method for solving the 3D problem affects the accuracy of the Navier-Stokes solution and is not relevant for the purpose of the present work, so we do not dwell upon it. 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 problems induced by the different description of the wall mechanics in the two halves of the pipe, which produces some spurious reflections at the interface and possible numerical instabilities. One could avoid this kind of problems by resorting to a compliant 3D model. As we have pointed out, this increases the computational costs strongly. More precisely, implicit coupled fluid-structure iterative schemes at each time step require to solve the Navier-Stokes and the structural problems several times. In explicit coupled fluid-structure iterative schemes, stability concerns typically require to take small time steps. In the next subsection, we present a different strategy based on the set up of an appropriate set of interface conditions Matching conditions including compliance Suppose that we give a simplified representation of the compliance of the 3D vessel in the multiscale model, by gathering its effect at the interface using a special lumped parameter model. Referring for instance to Fig. 5.2, we introduce a RCL network at the interface with the role of representing the effects of the compliance of the artery in the 3D model. In this way, we still use a 3D rigid model, which however behaves like a compliant one with respect to the system dynamics. By denoting with P the pressure associated with the capacitance C, the governing equations read P 1D R 1 Q 1D = P = P 3D L dq 3D R 2 Q 3D dt C dp dt = Q 1D + Q 3D. Taking the derivative of the first equation and using the third, we can eliminate P and finally obtain the new conditions in the iterative scheme, by replacing (5.2), (5.3) in the (5.5) 80

85 5 A geometrical multiscale model of the cerebral circulation Z 1D = P 1D R 1 Q 1D 1D Model Q 1D R 1 P 1D L P C Z 3D = P 3D R 2 Q 3D Q 3D R 2 P 3D 3D Model Figure 5.3: Alternative representation of the coupled problem: now the unknowns are z 1D, z 3D, P 1D, P 3D. algorithm with ( dq 3D,k dp 3D,k P 1D,k+1 = χ P 3D,k L 1 R 2 Q 3D,k + R 1 C 1 dt dt R 1 CL d2 Q 3D,k dt 2 R 1 CR 2 dq 3D,k dt R 1 Q 3D,k ) + (1 χ)p 1D,k (5.6) Q 3D,k+1 = Q 1D,k+1 + C dp 1D,k+1 dt R 1 C dq 1D,k+1 dt These conditions (hereafter denoted by LP (Lumped Parameter) conditions) involve time derivatives of the matching quantities to be discretized with an appropriate finite difference scheme with the same accuracy of the time advancing methods used for the time discretization of the Navier-Stokes and Euler equations. Remark As was to be expected, for C = 0, R 1 = R 2 = 0 and L = 0 we recover the coupling given by conditions (5.2), (5.3). This corresponds physically to the case of a rigid portion of artery in a network of compliant vessels, as it is the case of a stented or prosthetic 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 the derivation of an alternative formulation involving only first order time derivatives. Let s define z 1D (P 1D, Q 1D ) = P 1D R 1 Q 1D, z 3D (P 3D, Q 3D ) = P 3D R 2 Q 3D, then system (5.5) becomes L dq 3D dt C dz 1D dt = z 3D z 1D = Q 1D + Q 3D. (5.7) 81

86 5 A geometrical multiscale model of the cerebral circulation We remark that the flow rate Q 1D at interface Γ is linked to the pressure P 1D (consequently to z 1D ) through the Euler equations (2.1). We can express this relation in the following form: Q 1D = M 1D (z 1D ), (5.8) where M 1D stands for the set of equations of the 1D model. Similarly, the following relation holds between Q 3D and z 3D on Γ: z 3D = M 3D (Q 3D ), (5.9) where we indicate the 3D Navier-Stokes problem by M 3D. Equations (5.7), (5.8) and (5.9) form a non linear system in the four unknowns z 1D, z 3D, Q 1D, Q 3D, that can be rewritten in vectorial form as follows: with LP(X) = 0, C dz 1D (Q z 1D + Q 3D ) dt 1D X = z 3D Q 1D, LP(X) = z 3D M 3D (Q 3D ). Q 1D M 1D (z 1D ) Q 3D L dq 3D (z 3D z 1D ) dt Its time discretization can be retrieved for instance by means of a first order implicit Euler method: LP t (X n+1 ) = 0, (5.10) with C ( z n+1 1D LP t (X n+1 ) = L ( Q n+1 3D ) ( zn 1D t Q n+1 1D z n+1 3D M 3D(Q n+1 3D ) Q n+1 1D M 1D(z n+1 1D ) ) ( t z n+1 3D Qn 3D + Qn+1 3D zn+1 1D and with obvious meaning of the symbol X n+1. Now the iterative algorithm for the solution of the coupled problem can be formulated as a Newton scheme applied to system (5.10). Given the approximation X n+1 k of the solution at time t n+1, computed at iteration k, we can recover X n+1 k+1 by solving the following linear system in the increment δx n+1 k = X n+1 k+1 Xn+1 k : ) ) X J LP t δx n+1 k = LP(X n+1 k ), (5.11) n+1 k 82

87 5 A geometrical multiscale model of the cerebral circulation J LP t being the Jacobian matrix of function LP t : C 0 t t X M 3D J LP t = n+1 k M 1D z n+1 1D,k t t 0 L Q n+1 3D,k. (5.12) It is quite easy to explicitly compute the inverse of matrix (5.12), which reads: X J 1 LP t = n+1 k det ( 1 J LP t X n+1 k L + M 3D,k t t 2 t(l + M 3D,k t) t M 3D,k t t 2 + L(C M 1D,k t) M 3D,k t 2 M 3D,k ` tm 1D,k C, M 1D,k(L + M 3D,k t) M 1D,k t 2 C(L + M 3D,k t) + t 2 M 1D,k t t C t M 1D,k t 2 t 2 C M 1D,k t ( ) det J LP t X n+1 k where it is understood that M 1D,k = M 1D Newton iteration (5.11) finally reads: = CL + ( CM 3D,k LM 1D,k) t + (1 M 1D,k M 3D,k ) t2, X n+1 k+1 = Xn+1 k J 1 z n+1 1D,k LP t X n+1 k ) and M 3D,k = M 3D Q n+1 3D,k LP(X n+1 k ), (5.13) so that each interface variable at iteration k + 1 is expressed as linear combination of the variables at iteration k. A simplified expression for M 1D and M 3D can be obtained by considering a 0D representation of the 3D and 1D models, based again on RCL networks. We will describe the 1D model by means of an electric L-network (shown in Fig. 5.4(a)), whose dynamics is represented by the state variables Q 1D and P 1D,up, while Q 1D,up and z 1D are prescribed as boundary conditions (we recall that P 1D = Z 1D + R 1 Q 1D ):. C 1D dp 1D,up dt + Q 1D = Q 1D,up (5.14a) dq 1D L 1D + (R 1D + R 1 ) Q 1D = P 1D,up z 1D, (5.14b) dt where subscript up indicates that the subscripted quantity is evaluated in correspondence of the upstream section of the 1D model. 83

88 5 A geometrical multiscale model of the cerebral circulation Q 1D,up L 1D R 1D Q 1D Q 3D L 3D R 3D Q 3D P 1D,up C 1D P 1D P 3D P 3D,down (a) An electric L-network representing the 1D model. The model dynamics is represented by the state variables P 1D,up and Q 1D. (b) An electric RL network representing the 3D model. The model dynamics is represented by the state variable Q 3D. Figure 5.4: The electric analogy can be exploited to represent the 1D and 3D models with their 0D counterpart. If we consider the second equation in system (5.14) and approximate the time derivatives with an implicit Euler scheme, we can express Q n+1 1D as a function of zn+1 1D (given P n+1 1D,up ): Q n+1 1D ( ) 1 [ ] = L1D t + R L1D 1D + R 1 t Qn 1D + P n+1 1D,up zn+1 1D (5.15) M(z n+1 1D ), and finally we can retrieve an approximation for M 1D (z 1D): M 1D(z 1D ) z n+1 1D = dm 1D (z n+1 1D dz ) 1D 1. L 1D t + R 1D + R 1 (5.16) Since we are considering rigid wall 3D models, a simpler 0D network can be used to represent the 3D model (see Fig. 5.4(b)): the electric analogy does not feature a compliance element. Thus, we can resort to an RL network, describing the dynamics of Q 3D given z 3D and the downstream pressure P 3D,down (we recall that P 3D = Z 3D + R 2 Q 3D ): L 3D dq 3D dt + (R 2 + R 3D )Q 3D = P 3D,down z 3D. We discretize in time the previous by means of an implicit Euler scheme, obtaining: ( ) ( ) z n+1 3D = P n+1 3D,down L3D t + R 2 + R 3D Q n+1 3D + L3D t Qn 3D M 3D (Q n+1 3D ), and finally we find M 3D(Q 3D ) Q n+1 3D = dm ( ) 3D (Q n+1 3D dq ) L3D 3D t + R 2 + R 3D. 84

89 5 A geometrical multiscale model of the cerebral circulation name electric analogy expression resistance R 8πµl A 2 0 inertia L (inductance) ϱl compliance C (capacitance) A 0 3πR 3 0 l 2Eh 0 Table 5.1: Parameter estimation for the electric analog model of a cylindrical vessel (after [112]) Parameters estimation The interface lumped parameter model of Fig. 5.2 provides a physical representation to our matching conditions. A major issue in this approach is the tuning of the paramaters 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 parameter models (see e.g. [112] Chap. 10, based on a proper average of the Navier-Stokes equations) for a cylindrical vessel with length l, area A 0, with a linear elastic wall with thickness h 0 and Young modulus E, the compliance may be estimated to be C A3/2 0 l Eh 0. The resistance induced to the flow by the blood viscosity µ can be expressed as R µl, while the inertial term in the momentum equation gives rise in the 0D model to an inductance L ϱl A 0, ϱ 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 half measuring l = 5 cm, and we set A 0 = cm 2. The elastic modulus of the arterial wall is taken E = 10 6 dyn/cm 2, and the wall thickness is h 0 = 0.05 mm. Blood is assumed to be a Newtonian fluid of viscosity µ = poise. The electric analogy can be exploited to define a 0D model of the coupled problem, as depicted in Tab In particular, the lumped parameter representation of 1D and 3D models is useful for the set up of the solution strategy (5.13). The missing capacity element in the 3D model electric analogy is assigned to the interface RCL network. Physiological values of this parameter are of the order of 10 5 cm 5 /dyn. In our computation we set C = cm 5 /dyn. The other interface parameters have been properly adjusted in order to reduce spurious effects at the 1D/3D interfaces. More precisely, resistance R 1 has been introduced in [3] and, following the proposal of that paper, it is dynamically selected so that an incoming wave from the 1D model is propagated without any reflection. For R 2 and L, in this paper we have adopted an empirical trial and error approach, so that after some numerical experiments we put R 2 = 1 dyn s cm 5 and L = 0.01 g cm 4. For more complex models, A

90 5 A geometrical multiscale model of the cerebral circulation Parameters in the coupled model L 1D R 1D R 1 L R 2 L 3D R 3D C 1D C 1D model LP interface 3D model R 1D = dyn s/cm 5 R 1 = (*) R 3D = dyn s/cm 5 R 2 = 1 dyn s/cm 5 L 1D = g cm 4 L = 10 2 g cm 4 L 3D = g cm 4 C 1D = cm 5 /dyn C 1D = cm 5 /dyn - Table 5.2: A 0D model of the coupled problem depicted in Fig The numerical value for each parameter is reported. (*) R 1D is dynamically tuned to a value ensuring the wave propagation from 1D model without spurious reflections [3]. these parameters should be adapted accordingly Results The impact of the LP conditions is illustrated in Fig. 5.5 and 5.6. More precisely, in Fig. 5.5 we illustrate results obtained for the model of Fig. 5.2 when a sinusoidal waveform for the flow rate is prescribed at the inlet. We compare the values in time of the flow rate and the area (as function of the pressure) at the interface, denoted by Q 1D and A(P 1D ) in Fig. 5.2, obtained with a standard multiscale 1D/3D model, using the proposed approach and finally those obtained with a complete 1D model. In Fig. 5.6 we present similar comparisons for the case when a step waveform is prescribed at the inlet 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 description of the wall mechanics in the 1D and 3D model. These reflections change completely the profile of the solution. Observe that the complete reflection of the flow rate of Fig. 5.6 can 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 the contrary, matching conditions based on the RCL model are able to obtain a behavior similar to that of the complete 1D model, even if we are using a rigid 3D model. The same conclusions hold for the area: the RCL-based conditions allow us to find a solution significantly close to that of the complete 1D model. As pointed out, a proper tuning of the parameters is crucial to find the best RCL model. 86

91 5 A geometrical multiscale model of the cerebral circulation D 1D/3D C=0 1D/3D C D 1D/3D C=0 1D/3D C Q 1D 0 A(P 1D ) time time Figure 5.5: Comparison of dynamics of flow rate (left) and area (right) at x = 5cm of a compliant pipe simulated with a fully 1D model (thick dashed line), a multiscale 1D/3D model with direct coupling (C = 0, solid line) and with the matching conditions obtained by the lumped parameter models (C 0, thin dashed line). The input waveform of the flow rate at the tube inlet is a sine with amplitude 0.1. (time in [s], volumetric flow rate in [cm 3 /s], area in [cm 2 ]) fully 1D 1D/3D C=0 1D/3D C 0 Q 1D 0 A(P 1D ) D 1D/3D C=0 1D/3D C time time Figure 5.6: Comparison of dynamics of flow rate (left) and area (right) at x = 5cm of a compliant pipe simulated with a fully 1D model (thick dashed line), a multiscale 1D/3D model with direct coupling (C = 0, solid line) and with the matching conditions obtained by the lumped parameter models (C 0, thin dashed line). The input waveform of the flow rate at the tube inlet is a step function with amplitude 0.1 (time in [s], volumetric flow rate in [cm 3 /s], area in [cm 2 ].) 87

92 5 A geometrical multiscale model of the cerebral circulation Q 1D,l Q 3D,l Q 3D,r Q 1D,r 1D R 1 R 2 L 1 R 3 R 4 L 2 C 1 3D C 2 1D Q 1D,end P 1D,l P 3D,l P 3D,r P 1D,r Figure 5.7: Representation of a multiscale model with two 0D buffer elements at the interface. 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 the left interface we still resort to the iterative scheme with conditions (5.6). On the right interface, we adopt a similar iterative strategy where we prescribe a pressure condition to the 3D problem and flow rate conditions to the 1D Euler system downstream. More precisely, equations corresponding to the downstream interface read P 3D R 3 Q 3D = P 1D L 2 dq 1D dt C 2 dp 3D dt R 3 C 2 dq 3D dt = Q 1D + Q 3D R 4 Q 1D Consequently, the coupling conditions used in the iterative scheme are ( ) dp 3D,k dq 3D,k Q 1D,k+1 = χ Q 3D,k + C 2 R 3 C 2 + (1 χ)q 1D,k dt dt dq 1D,k+1 P 3D,k+1 = P 1D,k+1 L 2 R 4 Q 1D,k+1 + dt (5.17) R 3 C 2 dp 1D,k+1 dt R 3 C 2 L 2 d 2 Q 1D,k+1 dt 2 R 3 CR 4 dq 1D,k+1 dt R 3 Q 1D,k+1 Alternative formulation We can reformulate the interface problem (5.17) in form (5.13), by setting C ( z n+1 3D ) ( zn 3D t Q n+1 1D + ) Qn+1 3D LP t (X n+1 L ( Q n+1 1D ) = ) ( Qn 1D t z n+1 1D ) zn+1 3D z n+1 1D M 1 1D (Qn+1 1D ) Q n+1 3D M 1 3D (zn+1 3D ) 88

93 5 A geometrical multiscale model of the cerebral circulation where again X n+1 is defined as and M 1 3D,k X n+1 = X J 1 LP t = n+1 k det z n+1 1D z n+1 3D Q n+1 1D Q n+1 3D (, 1 J LP t X n+1 k t M 1 M 1 C t M 1 CL + t t L M 1 M 1 1D,k 1D,k 3D,k 3D,k 1D,k t 2 L t M 1 t t 2 t L t M 1 1D,k 1D,k t C t M 1 t C M 1 t t 2 3D,k 3D,k L t M 1 M 1 1D,k 3D,k t M 1 t 2 CL + t t M 1 3D,k 1D,k C t det ( J LP t X n+1 k t ) ) ( ( ) ( ) ) = t M 1 1D,k M 1 3D,k + ( ( ) ( ) ) C M 1 1D,k + L M 1 3D,k + CL., Results Numerical results are reported in Fig Again, we illustrate the comparison of the solutions 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 activated. The inlet waveform is sinusoidal. In the first picture we present the flow rate at the first interface (denominated Q 1D,l in Fig. 5.7), in the second picture the flow rate Q 1D,r at the second interface and finally the flow rate Q 1D,end at the outlet of the right pipe. Again, when classical matching conditions are used (corresponding to null values of the parameters) the superposition of the components induced by reflections associated to the different wall models are evident. This changes the shape of the propagating wave and affects both the amplitude and the phase at the inlet and at the outlet of the 3D model. Amplitude dissipation in the forward component of the wave is partially compensated by the superposition of the spurious reflections. In the case of LP conditions, the shape of the wave is only partially affected. Dispersion errors are remarkably small, whilst dissipation effects are present. More precisely, the dispersion error, evaluated as the difference in the occurrence of the peaks in the 1D and the multiscale LP models, are 20%, 6% and 5% in the three pictures of Fig. 5.8 respectively, while dissipation, evaluated as the difference of the peaks, are 25%, 32% and 32% respectively. The impact of finite difference schemes in the numerical implementation of matching conditions is probably the main responsible of these effects. A more accurate analysis of this aspect is one of the possible future 89

94 5 A geometrical multiscale model of the cerebral circulation Q 1D,l 0.05 Q 1D,r D 1D/3D C=0 1D/3D C D 1D/3D C=0 1D/3D C time (a) time (b) Q 1D,end D 1D/3D C=0 1D/3D C time (c) Figure 5.8: Comparison of flow rates computed by a 1D model (dash-dot line), a multiscale 1D/3D/1D model with classical matching conditions (dashed line) and with lumped parameter matching conditions (solid line) in correspondence of the first interface (a), the second (b) and the outlet of the domain (c). (time in [s], volumetric flow rate in [cm 3 /s]) 90

95 5 A geometrical multiscale model of the cerebral circulation Middle Cerebral Artery Basilar Artery Vertebral Artery External Carotid Artery Internal Carotid Artery Aorta Figure 5.9: Left: Anatomical representation of the cerebral vasculature, including the circle of Willis (after [12]). Right: Multiscale representation of the cerebral vasculature: a 3D representation of one of the carotid arteries is embedded in a 1D network of Euler problems developments of this work. Matching conditions guarantee in any case a significant reduction 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 the description of the cerebral circulation (see Sec ). The multiscale representation is depicted in Fig. 5.9, on the right. Left Internal Carotid geometry adopted is based on the realistic glass model obtained by Liepsch, see [106]. The Navier-Stokes equations in the 3D model have been solved with the code LifeV - see - based on a P 1P 1 finite element solver stabilized by means of an interior penalty approach. At the interfaces between the 3D and 1D models we prescribe conditions (5.6) at the upstream interface and conditions (5.17) at the downstream interfaces. In Fig and Fig we present the results, in comparison with the ones of a fully 1D model. Results underline that the RCL based conditions can actually obtain good solutions, in particular at the inlet. At the outlets of the carotid arteries the solution is strongly dissipated in the flow rate, while the fully 1D and the LP models are in good agreement for what concerns the phase of both flow rate and area, and the amplitude of the latter. Impact of the time discretisation of the matching conditions and the selection of the parameters on the dissipation error is under investigation. Fig illustrates velocity and pressure fields in the 3D rigid model. On the left we present a detail of the 3D velocity field, on the right a multiscale perspective coupling local and global dynamics. 91

96 5 A geometrical multiscale model of the cerebral circulation velocity flux (cm 3 /s) Flow rate at last node of CCA fully 1D lumped parameter interface time (s) area (cm 2 ) Area at last node of CCA 0.38 fully 1D lumped parameter interface time (s) Figure 5.10: Comparison of the results obtained with a fully 1D model (solid line) and the 3D rigid with RCL conditions at the inlet of Common Carotid Artery (dashed line). Left: flow rate, Right: area. (time in [s], volumetric flow rate in [cm 3 /s],area in [cm 2 ]). 7 6 Flow rate at first node of ICA fully 1D lumped parameter interface Area at first node of ICA velocity flux (cm 3 /s) area (cm 2 ) time (s) fully 1D lumped parameter interface time (s) 7 6 Flow rate at first node of ECA fully 1D lumped parameter interface 0.17 Area at first node of ECA velocity flux (cm 3 /s) area (cm 2 ) time (s) fully 1D lumped parameter interface time (s) 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 in the ECA. (time in [s], volumetric flow rate in [cm 3 /s], area in [cm 2 ]). 92

97 5 A geometrical multiscale model of the cerebral circulation Figure 5.12: Left: Representation of the 3D solution (velocity and pressure). Right: Coupling of 3D and 1D computations Remarks and perspectives For simple multiscale models, where the parameter quantification for the matching conditions is straightforwardly suggested by the mathematical derivation of the model, numerical results are really promising, showing that the multiscale 3D/0D/1D model can 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 in the cerebral vasculature, when a direct physiological quantification of the parameters is missing, results are only partially good. More precisely, at the inlet of the 3D model results still compare correctly with a fully 1D model, while downstream with respect to the 3D model dissipation effects in the flow rate are dominant. A mathematically sound fine tuning of the parameters is required. This goal can be pursued by a systematic sensitivity analysis or by extensive comparisons with standalone fully 3D models (see e.g. [4, 80]). This subject will be investigated in future works together with a validation of this approach in more complex networks. We finally point out that this approach can be extended to hydraulic networks featuring compliant pipes, beyond the specific medical applications considered here. 93

98 6 Computational tools The need for effective tools for the numerical solution of differential problems motivates 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 many years 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 the representation of the cardiovascular system as a network of vessels represented by 1D models (Sec. 6.3). 6.1 An introductory note on C++ C++ is a statically-typed general-purpose language relying on classes and virtual functions to support object-oriented programming, templates to support generic programming, and providing low-level facilities to support detailed systems programming. Bjarne Stroustrup [135] Although C++ supports different programming paradigms, as pointed out by the words 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 modeling of physical systems, as we will briefly discuss in this introduction. On the other hand, its proximity with C language has not only an historical reason, but also a philosophical motivation. It enhances the language with features allowing the programmer to 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, known in C++ as classes. The problem to be solved is modeled by a set of objects, storing privately all the data and providing the algorithms to operate on the data. Objects are considered as independent but can interact by exchanging messages. Moreover, hierarchies of objects can be defined through the concept of inheritance: generic classes can provide the shared behaviour to different specialized classes, which add custom data and algorithms for the fulfillment of specific tasks. A typical feature of C++ language allows functions and classes to operate with different 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 C++ is the fourth more popular programming language, as of January 2009, based on several indicators including an estimate of the number of job offers requiring C++ knowledge. 94

99 6 Computational tools C++ Standard Template Library (STL), enhancing the language with a set of powerful tools, in particular a framework for the definition of containers (such as vector, list, and map) and algorithms using containers. The generic-programming paradigm is indeed well illustrated by STL containers, which represent the abstract concept of a collection of items, with a set of rules to operate on them. The same general definition and the same policies (such as how to access to the data, how to add or delete elements) can apply for instance to vectors of numerical values, or to vectors of classes (even vectors of vectors). Object-oriented and generic-programming paradigms are particularly attractive for the modeling of complex physical systems: the data structures can be designed to mimic the behaviour of different interacting (physical or logical) components and their relationship can be described in a general and flexible way. The same code pattern can be 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 governing their interaction. Several software projects, indeed, propose the implementation of numerical methods for the solution of mathematical problems: an increasing number of them exploit an object-oriented programming paradigm, and they are often written in 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-exhaustive list comprehends the solution of linear systems and preconditioning problems, with support to parallel computing) and the OpenFOAM toolbox for the solution of partial differential equations. 6.2 LifeV: a C++ finite element library LifeV is a software project born from the joint collaboration of three institutions: École Polytechnique Fédérale de Lausanne (CMCS) in Switzerland, Politecnico di Milano (MOX) in Italy and INRIA (REO) in France. The Department of Mathematics and Computer Science of Emory University in Georgia (USA) started a collaboration since LifeV consists of the implementation in C++ language of algorithms and data structures for the numerical solution of partial differential equations. More precisely, LifeV provides an abstract framework for the implementation of Galerkin finite element methods, exploiting the object-oriented paradigm supported by the programming language. The development and maintainance of the core of the library are motivated by the research interests of the developers, mostly active in the numerical analysis and computer science fields. Some applications of the library include the the design and testing of efficient numerical techniques for fluid-structure interaction problems [42], algorithms for the solution of the Navier-Stokes equations [18, 54], numerical techniques for the coupling of different models in a multiscale perspective [86, 95], preconditioning strate

100 6 Computational tools gies for the Bidomain problem (arising in the modeling of the electrical activity of the heart) [55]. Moreover, software based on LifeV has been extensively used in research projects focused on the modeling of blood flow problems, such as the drug release from implantable stents [142], the design of medical procedures in cardiology [27], and the study of cerebral hemodynamics (the Aneurisk project, which motivated this work - see Chap. 4) Code features The code is hosted on a CVS server, which collects and organizes the contributions of the developers from the four different institutions. Stable releases of the code are published on the web portal which also contains the code documentation, and a gallery of applications to different modeling problems. The portability of the library is enhanced by the GNU build system, allowing LifeV 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). Moreover, LifeV is based on some of the extended C++ functionalities implemented in the Boost libraries, such as smart pointers (helping in effective memory management). Figure 6.1: LifeV code structure from A graphical representation of the code organization is presented in Fig The different parts have a hierarchical relationship based on their degree of specialization. More in detail, lifecore contains general components not directly related to scientific computing, such as the definition of the numerical types adopted in the code and a set of 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 and all the changes in a set of files, managing the concurrent contributions of the developers. also known as Autotools 96

101 6 Computational tools of array structures used in the code (vector and matrices), while a generic set of algorithms, both inherited from linear solver packages and specifically written for LifeV applications, is contained in lifealg. Mesh handling tools (for 1D, 2D or 3D meshes) are implemented in lifemesh. Most part of the data structures designed for the construction of finite element solvers is implemented on the basis of these library components. Classes representing generic finite elements, the definition of the geometrical mapping and the quadrature rules for their characterization are contained in lifefem, together with methods for matrix assembly 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 the matrices describing the linear system of equations of the discretized model, and invoke the linear solver for their solution. Methods for the data import and export from and to files are collected in lifefilters, in particular enhancing the library with the capabilities of managing different file formats. The testsuite shows examples of software built over the library, and spans the main applications from simple tasks such as mesh import from file and matrix assembly, to more complex problems such as the Navier-Stokes problem on simple geometries and with a small number of unknowns. Finally, we mention that recently the library has been extended with the support for parallel computing, mantaining the same general structure here presented. Preliminary tests show good scalability performances. 6.3 Implementation of networks of 1D models As discussed in Chap. 2, one-dimensional models offer an effective representation of the wave propagation phenomena in large parts of the circulatory system. This motivates to the design of a robust and flexible software tool, able to manage networks of 1D models of general topology in a simple and effective way. A network of one-dimensional models is here regarded as a graph, in which edges stand for the models themselves while vertices represent the interfaces between the models. 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 outflow sections, describing a reference blood flow path: this induces an orientation (and therefore a reference system) in the graph, and is useful in order to study some aspects of the fluid dynamics in the network (in particular energy balance at the interfaces, see Sec ). In this implementation of 1D model network we resorted to the data structures of boost::graph library (BGL), providing a generic interface for traversing graphs. Specific functionalities are added to the graph by associating LifeV 1D model objects with its edges. Moreover, to keep the network independent on the specific implementation of the 1D model, OneDNet has been built as a template class (with the 1D model class as template parameter). 97

102 6 Computational tools inflow section 1D model interface 1D model 1D model outflow section outflow section 1D model inflow section 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 s o l v e r class typedef typename boost : : shared_ptr < SolverType > OneDSolverPtr ; We expect to work with sparse graphs, in which the number of edges is of the same order of magnitude as the number of vertices: for this case BGL implements an adjacency-list representation. We will adopt this as the underlying data structure for OneDNet, 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 e Network _M_network ; Network is defined as the template class boost:: adjacency_list which stores a list of vertices and a list of out-edges for each vertex, that are edges oriented outwards with respect to a vertex. The first template parameter (boost:: lists ) determines what kind 98

103 6 Computational tools of container is used to store the out-edges list: this affects time complexity of adding/removing edge operations. Admitted choices include some of the container objects, as defined in the Standard Template Library (STL): in particular, an STL list turns out to be a better choice compared to an STL vector, since the latter occasionally needs reallocation in adding elements operations [128]. The second template parameter (boost::vecs) concerns vertices list: in this case vector seems to be the good choice since we do not expect the need to add or remove vertices after the initialization of the graph; moreover STL vector has a low per vertex space overhead compared to STL list, which needs to store three extra pointers per vertex [128]. The third template parameter boost:: bidirectionals selects a directed graph, which provides functions to recover in-edges and out-edges associated to each vertex. The fourth and fifth template parameters (OneDVesselsInterface and OneDVessel) are userdefined 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, and allow an easy and flexible definition of the attributes of the graph, as we will see later on. template < class SOLVER1D > class OneDNet { } ; public : typedef boost : : a d j a c e n c y _ 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 associated to an object representing a 1D model as implemented in LifeV. Each vertex is instead associated to two numbers (a numerical index and a type) and to a boolean value (internal) whose precise meaning will be explained in Sec template < class SOLVER1D > class OneDNet { public : struct OneDVesselsInterface { i n t index ; / *! < numerical 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 ; / *! < numerical l a b e l * / OneDSolverPtr onedsolver ; / *! < p o i n t e r to 1D s o l v e r class * / } ; 99

104 6 Computational tools } ; The access to edges and vertices of an adjacency_list is performed simply by subscripting the graph with the proper descriptor. Therefore the bundled properties of edges and vertices are easily set and retrieved, as is shown in the example here below: / /! D e s c r i p t o r type f o r the l i s t of v e r t i c e s typedef typename boost : : g r a p h _ t r a 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 ) ; / / set index i f o r ve rtex vd _M_network [ vd ]. index = i ; / / p r i n t out v e r t e x vd s index std : : cout << _M_network [ vd ]. index << std : : endl ; / /! D e s c r i p t o r type f o r the l i s t of edges typedef typename boost : : g r a p h _ t r a i t s < Network > : : edge_descriptor Edge_Descr ; Edge_Descr ed ; UInt t ( 0 ) ; / / set type t f o r edge ed _M_network [ ed ]. type = t ; / / p r i n t out edge ed type std : : cout << _M_network [ ed ]. type << std : : endl ; Building the graph I VII t 1 III t 6 t 2 II t 5 t 3 VI t 4 IV V Figure 6.3: A test case: 6 connected tubes. Label ti indicates tube i, while Roman numerals are associated to interfaces 100

105 6 Computational tools Consider the case depicted in Fig. 6.3: it s a simple network composed by 6 edges and 7 vertices. Vertex II is internal to the network, meaning that it is connected to more than one edge. Conversely, all the others are terminal vertices. Edges 1, 3, 5 are in-edges with respect to vertex II; edges 2, 4, 5 are out-edges. 1D model left interface right interface Figure 6.4: Left and right interfaces for 1D model. Each oriented edge has intrinsically a local reference system: in this reference system it is possible to unequivocally identify a left and a right interface, as shown in Fig Therefore, the entire Network structure can be built from a list of vertices and a connectivity 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 parameter associated to the vertices, on the basis of the network topology. This information is used to manage the prescription of boundary conditions to the 1D solvers associated to the edges: edges connected to terminal vertices need standard boundary conditions, which are managed by the attached solvers. Edges connected to an internal vertex need instead a set of interface conditions, which are managed by the OneDNet class, as we will discuss later on. The type of an internal vertex is set by the user, and determines how the 1D models interact 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 case of blood flow problems, in which networks such as the considered one would represent branching vessels or circulatory anastomosis, reasonable interface conditions would prescribe the conservation of the flowing mass and of its kinetic energy. Other possible choices include interface conditions taking into account the energy losses associated to branchings (for example depending on the flow velocity at the interface or on the branching angles of the vessels, see [49]) or the presence of a stenosis or of a valve at the interface between two vessels or two different tracts of the same vessel (which is the case of the venous system, for instance). Visiting the graph In many cases, applying an operation to the network means to recursively apply the same operation to each 1D model in the network. As an example we recall here OneDNet::timeAdvance method: template < class SOLVER1D > void OneDNet<SOLVER1D> : : timeadvance ( const Real& time_ val ) { / / impose i n t e r f a c e c o n d i t i o n s at i n t e r n a l v e r t i c e s computeinterfacetubesvalues ( ) ; 101

106 6 Computational tools } / / i t e r a t o r s to v i s i t edge l i s t Edge_ Iter ei, ei_end ; for ( t i e ( ei, ei_end ) = edges ( _M_network ) ; e i!= ei_end ; ++ e i ) { / / t e l l me what I am doing Debug ( 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 ( time_val ) ; } The first step in this method is the call to computeinterfacetubesvalues function, in order to evaluate the interface conditions in correspondence to internal vertices of the network (see Sec ). Then the edges of the graph are visited through iterators and the timeadvance method from the attached 1D models is invoked, for the prescription of the boundary conditions at each time step. We remark here that we chose to provide class OneDNet with member functions having the same name and the same parameter list as the corresponding methods in the 1D model class as implemented in LifeV. In this respect, class OneDNet can be regarded as a wrapper of LifeV :: OneDModelSolver class. On the other hand, this can be seen as a precondition on the template parameter SOLVER1D, which needs to share the same public interface as OneDModelSolver in order to work in the network class. One way to achieve this behaviour is by means of inheritance: specialized classes can be defined, providing alternative implementations of 1D models (for instance, exploiting different numerical discretization strategies) and still retaining the same shared public interface Interface conditions As previously mentioned, OneDNet class features the computeinterfacetubesvalues method, visiting the graph vertices and setting up the interface problem, when needed according 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 total pressure across the interface. In Listing 6.1, this problem corresponds to vertex type 1. The same approach applies however to different interface problems, for instance involving concentrated energy losses as discussed in Sec Listing 6.1: OneDNet::computeInterfaceTubesValues template < class SOLVER1D > void OneDNet<SOLVER1D> : : computeinterfacetubesvalues ( ) { / / v e r t e x i t e r a t o r s ( f o r v i s i t i n g the graph ) std : : pair < V e r t e x _ I t e r, V e r t e x _ I t e r > v i p ; / / v i s i t v e r t e x l i s t for ( v i p = v e r t i c e s ( _M_network ) ; v i p. f i r s t!= v i p. second ; ++ v i p. f i r s t ) { Debug ( 6330 ) << "0 Computing I n t e r f a c e " 102

107 6 Computational tools << _M_network [ * v i p. f i r s t ]. index << " \ n " ; / * In p r i n c i p l e, i t i s p o s s i b l e to implement d i f f e r e n t behaviour f o r i 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, vasculare valves etc ). ( not done yet ) * / i f ( _M_network [ * v i p. 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 i p. 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 " c o n d i t i o n s ( a c t u a l boundary c o n d i t i o n s instead ) * / 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 i p. f i r s t ) ; break ; case 99: / * o u t f l o w tube : no " i n t e r f a c e " c o n d i t i o n s ( a c t u a l boundary c o n d i t i o n s instead ) * / break ; / / other cases can be added here! } default : std : : cout << " \ n [ OneDNet : : computeinterfacevalues ] Unknown type " << _M_network [ * v i p. f i r s t ]. type << " f o r v e r t e x " << _M_network [ * v i p. f i r s t ]. index << std : : endl ; break ; } / / switch } / / i f } / / f o r Let s consider, for each 1D model in Fig. 6.3, the physical quantities total pressure P t and mass flow Q: now problem (2.25) reads having set f = (f(1),..., f(6)) and f(1) = i=1,3,5 Q i ( j=2,4,6 Q j) f = 0, (6.1) f(k) = P t,k P t,1, for k = 2,..., 6. The subscripts in the previous equations indicate that the considered quantity has to be referred to the edge marked by the corresponding numerical label. We remark that f(1) expresses the sum of the mass flows computed by 1D model solvers. In-edges give a positive contribution to the balance equation, while out-edges give a negative contribution. The continuity of mass flow follows from the balance of positive and negative flows. The continuity of total pressure is imposed by taking one edge as a reference and imposing that each one of the others features the same total pressure at the interface 103

108 6 Computational tools node. In this case, for in-edges the interface is located at the right boundary node, while for 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 order to do that, two helping structures are built in each vertex. First of all, a map associates a numerical label to 1D solvers associated to both in-edges and out-edges. This label does not necessarily correspond to the edge index, but is rather an ordinal number associated to the edges connected to the considered vertex. A second map is devised to identify the orientation of the edge with respect to the interface: the edge label is associated to a bool flag (true for positive, false for negative orientation). A graphical representation of this approach is presented in Fig template < class SOLVER1D > void OneDNet<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& v e r t e x ) { / / map the edges to t h e i r numerical l a b e l MapSolver interfacetubes ; / / map i d e n t i f y i n g in edges (+, t r u e ) and out edges(, f a l s e ) std : : 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 of in edge i t e r a t o r s std : : pair <In_Edge_Iter, In_Edge_Iter > i n _ e d g e _ i t e r _ p a i r ; / / v i s i t the l i s t of in edges for ( i n _ e d g e _ i t e r _ p a i r = in_edges ( * vertex, _M_network ) ; i n _ e d g e _ i t e r _ p a i r. f i r s t!= i n _ e d g e _ i t e r _ p a i r. second ; ++ i n _ e d g e _ i t e r _ p a i r. f i r s t ) { / / i n s e r t edges i n t o the map interfacetubes. i n s e r t ( MapSolverValueType ( i, _M_network [ * ( i n _ e d g e _ i t e r _ p a i r ). f i r s t ]. onedsolver ) ) ; 104

109 6 Computational tools } / / in edges have " + " signum ( boolean value t r u e ) signum. i n s e r t ( std : : map< int, bool > : : value_type ( i, true ) ) ; i ++; } / / p a i r of out edge i t e r a t o r s std : : pair <Out_Edge_Iter, Out_Edge_Iter > o u t _ e d g e _ i t e r _ p a i r ; / / v i s i t the l i s t of out edges for ( o u t _ e d g e _ i t e r _ p a i r = out_edges ( * vertex, _M_network ) ; o u t _ e d g e _ i t e r _ p a i r. f i r s t!= o u t _ e d g e _ i t e r _ p a i r. second ; ++ o u t _ e d g e _ i t e r _ p a i r. f i r s t ) { / / i n s e r t edges i n t o the map interfacetubes. i n s e r t ( MapSolverValueType ( i, _M_network [ * ( o u t _ e d g e _ i t e r _ p a 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 ( std : : map< int, bool > : : value_type ( i, false ) ) ; i ++; } The non linear problem (6.1) is solved by applying a Newton iterative scheme. The solution at each time step is then obtained as: x k+1 = x k J 1 f x k f(x k ), J 1 f x k being the jacobian matrix of f while x contains the interface unknowns A i, Q i, i = 1,..., 6. The code invokes the LAPACK routine dgesv, to compute J 1 f x k f through LU decomposition of the jacobian matrix, and then updates the solution at current iteration. Convergence is achieved when mass conservation is ensured by the fulfillment of the following request [45]: tol being a user-defined tolerance. f(1) < tol / / unknown of non l i n e a r equation f ( x ) = 0 Vector x ; / / non l i n e a r f u n c t i o n f Vector f ; / / jacobian of the non l i n e a r f u n c t i o n M a t r i x j a c ; / / transpose of the jacobian of the non l i n e a r f u n c t i o n M a t r i x j a c _ t r a n s ; / / tmp m a t r i x f o r lapack l u i n v e r s i o n boost : : numeric : : ublas : : vector < I n t > i p i v ; / / lapack v a r i a b l e i n t INFO [ 1 ] ; i n t NBRHS[ 1 ] ; / / nb columns of the rhs := 1. i n t NBU[ 1 ] ; / * x contains the ( unknown ) boundary values f o r edges connected to the considered i n t e r f a c e. * / x. r e s i z e ( f _ s i z e ) ; x. c l e a r ( ) ; / * prepare the data s t r u c t u r e s needed f o r s o l v i n g non l i n e a r equations 105

110 6 Computational tools ( 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 e s i z e ( f _ s i z e ) ; f. c l e a r ( ) ; j a c. r e s i z e ( f_size, f _ s i z e ) ; j a c. c l e a r ( ) ; j a c _ t r a n s. r e s i z e ( f_size, f _ s i z e ) ; j a c _ t r a n s. c l e a r ( ) ; i p i v. r e s i z e ( f _ s i z e ) ; i p i v. c l e a 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 n do { / / f i l l f and i t s jacobian m a t r i x f _ j a c ( x, f, jac, interfacetubes, signum ) ; / / transpose to pass to f o r t r a n storage ( lapack! ) j a c _ t r a n s = t r a n s ( j a c ) ; / / Compute f < ( df ( x )^{ 1} f ( x ) ) ( l u dcmp) dgesv_ (NBU, NBRHS, &j a c _ t r a n s ( 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 of y = df ( x )^{ 1} f ( x ) i s not achieved. " ) ; / / x = x df ( x )^{ 1} f ( x ) x += f ; } / / convergence check while ( ( std : : fabs ( f ( 0 ) ) > 1e 12) && (++ n i t e r < 100) ) ; / * f ( 0 ) contains the mass flow balance : by minimizing i t s value we want to ensure mass conservation. Moreover, we expect a low number of i t e r a t i o n s, since the i n i t i a l guess i s given from the boundary c o n d i t i o n s at previous time step, which are l i k e l y to be a good e s t i m a t i o n of the c u r r e n t s o l u t i o n ( time step i s small and we expect s o l u t i o n s to be continuous i n time ). * / The f_jac method fills Vector f with the expressions of interface conditions and Matrix jac with the jacobian matrix J f. For the sake of brevity we omit here the corresponding code, noting that all the computations are straightforward, given the presented data structure A simple example Running a simulation for a network of 1D models is formally the same as running a simulation for a single 1D model, since an object of class OneDNet behaves exactly like an object of class OneDModelSolver, thanks to the wrapping mechanism previously discussed. Consider again the case depicted in Fig. 6.3: the network at hand has only one internal node where we want to impose continuity interface conditions (2.25). Each 1D model is described by the Euler equations (2.1) with the assumption that the vessel wall behaves as a linear elastic solid (see (2.4)). The numerical discretization is based on the Taylor-Galerkin scheme presented in Chap. 2 (2.27). 106

111 6 Computational tools All the physical and numerical discretization parameters are the same for all the six tubes, and are summarized in Tab Moreover tubes 1, 3, 5 feature the same boundary conditions at left boundary, while tubes 2, 4, 6 feature the same boundary conditions at right boundary. More precisely, left boundary conditions for in-edges are managed by each 1D model LifeV class, and prescribe the entering characteristic variable W 1 (see [49]). Since W 1 (t) = W 1 (A(t), Q(t)), it is possible to specify W 1 (t) by selecting a function 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 outedges: namely, a value for the exiting characteristic variable W 2 (t) = W 2 (A(t), Q(t)) is imposed, 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 inedges and left boundary conditions for out-edges, as previously illustrated. Length 10 cm Radius 0.5 cm Wall thickness 0.05 cm Wall Young modulus 10 4 dyn / cm 2 Wall Poisson ratio 0.5 Blood mass density 1 g / cm 3 Blood viscosity poise Mesh 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 initial conditions A(t = 0) = A 0, Q(t = 0) = 0 in all the tubes. We report snapshots of the solution in Fig The images show that the sinusoidal signals (in terms of flow rate) enter tubes 1, 3, 5, correctly propagate in the network and exit through tubes 2, 4,

112 6 Computational tools (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