The effect of flow-diverter stent porosity on the intrasaccular blood flow according to a 2D cerebral aneurysm model

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Bachelor Informatica

The effect of flow-diverter stent

porosity on the intrasaccular

blood flow according to a 2D

cerebral aneurysm model

Nicole Sang-Ajang

July 2019

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Abstract

Cerebral aneurysms are pathological malformations on brain arteries that have a signi-ficant morbidity and mortality [34] [28]. A relatively new effective treatment is flow-diverter stent (FDS) treatment. The effectiveness of this treatment is associated with local hemo-dynamics, i.e. the dynamics of blood flow inside the aneurysmal sac. Hence, this thesis aims to study the effect of varying FDS porosity on the intrasaccular blood flow in terms of streamline structure and velocity. To be able to carry out this parametric study a valid and reliable 2D in silico system is modeled in Palabos based on the Lattice Boltzman method. The model and simulation are configured to properly represent cerebral aneurysms with a vessel diameter of 2.5-4 mm. Simulation experiments are conducted for two different geo-metries, i.e. a presumed posterior cerebral artery (PCA) and a presumed middle cerebral artery (MCA). The results show a geometry dependent relationship between FDS poros-ity and intrasaccular streamline structure, and a geometry dependent relationship between FDS porosity and intrasaccular velocity. In the PCA a denser FDS leads to more intrasac-cular streamline structure reduction and distortion and more average intrasacintrasac-cular velocity reduction, whereas, in the MCA the opposite effect is observed. In addition, the results suggest an interrelationship between FDS porosity and intrasaccular streamline structure and intrasaccular velocity. Further research could study these relationships on large scale.

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3.2.1 Reynolds number . . . 24 3.2.2 Mach number . . . 24 4 Experiments 25 4.1 Porosity experiment . . . 25 4.1.1 Geometry 1 . . . 25 4.1.2 Geometry 2 . . . 26 5 Results 27 5.1 Geometry 1 . . . 27 5.2 Geometry 2 . . . 30 6 Conclusion 33 7 Discussion 35 Appendices 41 A 41 A.1 General . . . 41 A.2 Method . . . 41

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CHAPTER 1

Introduction

Blood rheology is considered to be a very active area of research. Whereas collecting data in vivo would be ideal, limitations have driven science to find alternatives such as in vitro experimental systems resulting in hemorheology1 _{as it is known today [18]. However, the possibilities and}

research development in this area has not stagnated. Besides in vivo and in vitro experiments a third powerful tool can be found in the area of Computational Science, also known as Computa-tional Fluid Dynamics (CFD). Modeling blood flow systems offers great benefits besides taking away the problem of limited resources and research subjects. The power of computation can overcome obstacles on ethical grounds, complexity of systems and otherwise dependent variables such as physical time. The development of modeling blood on cellular level can therefore be considered very valuable. This case study focuses on an already existing context where further development and research is required.

A cerebral aneurysm2 _{is a common abnormal bulge emerging in an artery wall during the}

course of life which can lead to death or disability if ruptured [19][32]. Fortunately, rupture rate is low. Nevertheless, rupture has a significant morbidity and mortality and it is preferred to prevent rupture [28]. On the other hand, the risk of complications due to conventional surgical treatment can be significant, and furthermore the effectiveness leads to difficult medical-decision making [33][34]. However, the rise of a relatively new technology regarding endovascular treatment of aneurysms gives a promising prospect in this matter. It was not until recently that flow-diverters are found to be applicable as an innovative treatment [35], which focuses on treatment of the artery harboring the aneurysm instead of the aneurysm itself [10].

These flow-diverting stents (FDS) are a new generation of stents whose type can vary in properties such as material, thickness, weave pattern, porosity and pore density [26]. They are ideally designed to fit the demands. Its design choices are heavily subjected to the hemodynamics inside the aneurysm geometry, as characteristic intra-aneurysmal hemodynamics are associated with cerebral aneurysm pathophysiology according to previous studies [33]. Hence, underlying aneurysmal flow physics play a significant role in cerebral aneurysm remodelling, dilation and rupture; moreover, disruption of flow near the aneurysm neck allows for normalization of the aneurysmal artery wall [10][26] which is partially the concept of FDS treatment. Analyzing intra-aneurysmal hemodynamics can therefore contribute to determine the extent by which the FDS demands are met. This thesis focuses on analyzing intra-aneurysmal hemodynamics following placement of a type PED3 _{FDS when varying its porosity.}

As mentioned previously it is preferable for various reasons that hemorheology utilizes models. Models exist inside other contexts covering other cases, using traditional CFD methods or using specific software such as the commercial FEA program Abaqus/Explicit 6.9 [24]. This thesis implements the model using the widely used scientific open source software Palabos, which is based on the relatively new Lattice Boltzmann method (LBM). LBM is reported to be a valid CFD method to simulate transient fluid flow in a 3D cerebral aneurysm geometry [34].

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Implementing this model will not only enable this study, but might facilitate further research and development.

Hence, it is aimed to use LBM to implement a model based on a 2D geometry to be able to study the effects of FDS placement on the aneurysmal flow. Subsequently, this model will be used to conduct simulation experiments to be able to answer the main question: ‘How is blood flow inside the aneurysmal sac in terms of flow velocity and streamline structure affected by varying porosity of the FDS?’. To be able to answer this question the following sub-questions will be answered:

1. What is blood, what are its properties and how does blood flow originate? 2. How can the geometry and the FDS be modeled using LBM and Palabos? 3. How can a proper model and simulation be configured?

4. How does varying the porosity of the FDS effects the intrasaccular flow according to the simulation experiments?

To evaluate the effects a parametric study is performed by modelling, model validation and verification, simulation experiments and analysis. First of all relevant theory will be elucidated referring to literature. Secondly, required in vivo and in vitro data is gathered using literature. Subsequently, the former and the latter will be combined to construct and implement a reliable and valid model using the Palabos framework. Afterwards, the model will be validated and verified. Second-last, simulation experiments will be ran with the model to provide data for analysis. At last, the data will be analyzed in ParaView to answer the research question.

Introductory this thesis will discuss the theoretical background providing insights on the con-text to be able to substantiate the model’s implementation. To proceed, the method discusses the approach to implementing the model and configuring the simulation. Thereafter, the exper-iments document the conducted simulation experexper-iments with the subsequent chapter presenting their results. In the last chapter the results are analyzed and a conclusion is drawn. Finally, suggestions for further research will be proposed.

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CHAPTER 2

Theoretical background

In order to implement the model it is necessary to study the circulatory system and the mechanics behind blood flow. This requires getting into physiology, hemorheology and hemodynamics. This chapter discusses the basics of the circulatory system, the mechanics involved in blood flow and both in the context of FDS treatment. Lastly, this chapter briefly explains the Lattice Boltzman method used to model and simulate complex fluid systems.

2.1 The circulatory system

The very fact that humans stay alive results from complex (control) systems inside the human body [17]. One of those systems is the circulatory system. The circulatory system consists of the heart, a network of vessels (i.e. vascular system) and blood, and is responsible for the transport of substances to and from cells inside the human body. Cells form the basic living unit of the body, moreover, larger components e.g. human tissue consists of it. Cells are dependent on the transport of substances as this provides them with required substances such as oxygen, nutrients and hormones, and disposes cells of (waste) products such as CO2. Without this mechanism cells will dysfunction. Many systems subsist due to the circulatory system and blood is considered a fundamental component due to it being the medium responsible for transport. Although subsystems of the circulatory system can be classified from a physiological point of view, as the emphasis is on rheology, the circulatory system is divided into the heart, the arteries, the microcirculation and the veins.

Throughout the vascular system different types of vessels occur which’ anatomical character-istics depend on their functionality. For instance, the heart is the engine pumping blood through the circulatory system causing a relatively high pressure also known as aortic pulse pressure. This requires the aorta (artery of the heart) to withstand the expansion accompanied with the aortic pressure. Hence, these vessels have a relatively thick vessel wall and are rather compliant. Contrary, the outermost vessels, i.e. the microcirculation, are relatively small and have walls delicate enough for substances to pass from blood to surrounding tissues and vice versa. They carry a distinct name that is capillaries. Capillaries are usually between 4 − 8 · 10−4 cm in diameter. Whereas, the typical artery diameter is 0.2 − 2.4 cm1 in diameter. Aside from the differences in certain characteristics a shared characteristic can be observed in the distensible nature of all artery walls inside the vascular system. This distensible nature enables them to average out the pressure pulsations of the heart, supplying a smooth steady-state flow of blood in the microcirculation [17]2_{. Details of circularly functioning are rather complex.}

Notwith-standing, characterizing flow properties can be observed as will be discussed in the subsequent section.

1_{September 17, 2018, Lecture 5: Blood flow in the human body, Jrgen Arendt Jensen, Department of Electrical}

Engineering (DTU Elektro), Biomedical Engineering Group, Technical University of Denmark.

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2.2 Hemodynamics and hemorheology

Fluid flow is generally described as the flow velocity u at any spatial point x at any time t written as the expression:

u = u(x, t)

Fluid flow thus provides information about all fluid elements their state [3] at any time and consequently their behaviour e.g. their motion [1]. Blood is rather than a pure fluid a colloid3 _{consisting of a fluid base containing several different particles (cells, cell fragments and}

macromolecules). The fluid base is called plasma and consists of 90% water, 1% electrolytes and various molecules making up the remainder composed of macromolecules. Plasma is perceived as the carrier fluid carrying particles such as red cells, white cells, platelets and macromolecules. Due to the fact that beside the fluid base these particles should be considered while studying blood flow, hemodynamics can be said to be rather complex. Even though all particles are subjected to several forces red blood cells are of special interest in hemodynamics due to hematocrit4

typically ranging between 40-50%, causing red blood cells to have a significant effect on local hemodynamics [29]. To put it briefly, hemodynamics studies the physical principles of blood flow with special interest in the interrelationships among pressure, flow and resistance [7]. The sections below briefly discuss these physical principles while integrating them into the context of hemodynamics and hemorheology.

2.2.1 Gravity

The long-range gravity forces on planet earth are structured in a gravitational field affecting all particles. Particles in a fluid are affected by two forces arising in the gravity field (see 2.1).

1. Gravitational force: the particle’s weight causes it to fall in a gravitational field. Thus weight is perceived as the measurement of the pull of gravity. Weight is determined by mass which on its turn involves density and volume.

2. Buoyant force: the hydrostatic pressure (difference in pressure in the fluid due to height difference) causes the particle to rise in a gravitational field.

Figure 2.1: Different forces on a static particle in a stationary fluid [18].

Assuming a stationary fluid and a static particle the counter effects of the two forces yield a net force. Using the fact that force is considered a vector quantity which has both a magnitude and a direction, the magnitude and direction of both vectors determine the height position of the particle. This height can be perceived as a particle sinking, rising or to be ’neutrally buoyant’ inside a fluid. Since density of an average red cell is slightly higher than that of plasma5red cells have a minor tendency to sink [18].

3_{‘A mixture in which one substance of microscopically dispersed insoluble particles is suspended throughout}

another substance.’.

4_{Volume percentage (vol%) of red blood cells in blood.}

5_{Note that fluid and plasma are used interchangeably, since the aim is to relate fluid dynamics to the context}

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2.2.2 Pressure

Naturally speaking fluid in living organisms is not stationary and the particles are not static. The differences in pressure inside the vessel ∆P will cause a pressure gradient force ∇P to press blood from a point with higher pressure to a point with lower pressure. Contrary, vascular resistance, i.e. friction between the flowing blood and the intra-vascular endothelium6_{, opposes}

to this motion [17]. Overall, a dominating pressure gradient force will result in blood flow (see 2.2). If no additional force balances the pressure gradient force the fluid and the particles will accelerate according to Newton’s second law of motion. Hence, ∇P can be perceived as a driving force.

Figure 2.2: Pressure, resistance, and blood flow in context of vessels [17].

2.2.3 Reynolds number

Besides the previously mentioned forces allowing for blood flow to arise, depending on the flow conditions inside the vessel other forces will arise [18]. Assuming pressure forces drive flow another force might attempt to maintain this fluid motion, i.e. inertial, whereas oppositely yet another one resists the fluid motion, i.e. viscous [9]. The importance of these two type of forces, i.e. inertial and viscous, can be quantified as a single measure called the Reynolds number (Re). Plainly speaking Re is the ratio of inertial to viscous forces. Hence, a relatively high Re indicates a predominance of inertial forces. This aspect makes Re meaningful regarding the behaviour of fluid flow.

The Reynolds number is a dimensionless number in fluid mechanics that can serve several purposes such as predicting the fluid flow [27] as will be elucidated hereafter. Re is context (e.g. varying (surface) geometries and flow types) dependant. Therefore, the context determines Re’s definition. Re can be defined for several different circumstances involving a fluid relative in motion to a surface. The geometry presumed in the subsequent sections is that of a circular duct corresponding with the common vessel geometry.

Inertial force

An inertial force results from forces acting on a particle and inertia7 of the particle. Assuming the pressure gradient corresponds with the fluid flow as fluid flow is driven by it, inertia can be exemplified as fluid layers acting upon a particle (see 2.3) causing the particle to accelerate when the pressure forces exceed the resistive forces. This acceleration generates inertia, i.e. the particle and the fluid want to maintain their momentum, hence, blood is pushed forward. This forward motion will continue as long as overcoming the viscous forces [9].

6_{The cell layer of the inner vessel wall.}

7_{Resistance to change in velocity. Since velocity is a vector, change can imply both a change direction and}

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Figure 2.3: Different types of inertial forces arising from the motion of fluid layers with respect to the particle [18].

Viscous force

Fluids are substances that continually deform due to shear forces [9]. Shear deformation, i.e. deformation of a fluid without changing volume, is opposed by viscous forces. However, these resistive viscous forces only become apparent when fluid is in motion. As blood flow occurs viscous forces arise between fluid layers and adjacent surfaces. This interaction can involve a fluid layer, a particle or a wall of the vessel sliding against each other. The ability to resist the growth of shear deformation is perceived as a physical property of a fluid and is termed viscosity. Hence, viscosity quantifies the viscous forces [18].

As viscous forces act between adjacent components, i.e. their common interface, elements in motion will decelerate. To put it simply, inertial force attempts to push fluid and particles forward whereas viscous force opposes the forward motion. These opposing effects influence fluid and particles’ motion and therefore blood flow. The state of blood in terms of velocity at a given time t can be spatially visualized as so called streamlines (see 2.4 B, C) consisting of the derivatives of u(x, t) for each spatial point x. One can imagine that a large ratio between inertial forces and viscous forces will cause unstable velocity gradients of fluid and particles at spatial points, resulting in chaotic vortical motion8. Consequently, a chaotic pattern emerges in the streamlines perceived as turbulent flow (see 2.4 C). Conversely, a smooth pattern of streamlines is considered laminar flow.

Figure 2.4: [17]

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Re is thus useful when predicting whether flow will behave in a turbulent or a laminar manner. Since Re comprises both static (e.g. duct geometry) and dynamic properties (e.g. characteristics of the fluid, velocity and viscosity) [27]. It can formally be defined as:

Re = umax· d

ν (2.1)

Figure 2.5: Reynolds number. umax= maximum velocity in parabolic Poiseuille profile9, d =

vessel diameter and ν = kinematic blood viscosity.

A larger velocity causes greater inertia resulting in a higher Re mostly prevalent in larger vessels. Whereas in microcirculation viscous forces dominate causing a lower Re [18]. The char-acterizing common Reynolds number in vessels therefore ranges between values corresponding with the vessel’s physiology. Although there is no strict threshold between laminar and turbulent flow as in between there exists a grey area partially due to Re’s dependence on the context, the previously mentioned distensible nature of artery walls generally results in a non-turbulent flow at which an Re below 2000 is commonly observed10 _{[18]. Overall, blood flow can form very}

complicated flow patterns.

2.2.4 Vortex

A special flow pattern can be observed as fluid flow revolving around an axis line. This typical flow region in a fluid is known as a vortex (see 2.6). Vortical motion can revolve around a straight or curved axis. A vortex can be stable, however, once emerged a vortex can move, remodel (e.g. expand/shrink), deform (e.g. twist) and interact in complex ways. Complex vortical flow with multiple vortices inside the aneurysmal sac is associated with aneurysm rupture [33]. Depending on the context, e.g. turbulent flow, vortices can be a major flow component.

Figure 2.6: A stable vortex occuring in the aneurysmal sac [18].

2.2.5 Poiseuille flow

While there occurs blood flow all fluid and particles are in motion but velocity varies among fluid layers. This is due to the viscosity of blood which is predominantly explained by red blood cells exerting frictional drag, i.e. a viscous force, against adjacent cells and against the intra-vascular endothelium. Hematocrit causes the effect of other particles such as proteins to be insignificant in determining blood viscosity [17]. This leads to red blood cells vastly determining blood viscosity and consequently the velocity profile. As a result, blood has close to zero velocity near the walls, whereas, at the center between the opposite walls velocity will be at its largest. The parabolic profile (see 2.4 B) that results from this is formally defined by Poiseuille’s law. Note however, that this equation applies to straight ducts.

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2.2.6 Flow rate

The circulatory system is a closed system, i.e. the flow has nowhere else to go. The rate of blood flow, i.e. the volume of blood passing through a given cross sectional area per unit time denoted as Q, is closely controlled in relation to the tissue need [17]. Tissues require a relatively constant flow. Hence, the flow rate has to more or less remain consistent. To maintain this consistency pressure and resistance are altered to satisfy the hemodynamic relationship called Ohm’s law [17] (see eq. 2.2).

F = ∆P

R (2.2)

Figure 2.7: Ohm’s law. F = blood flow, ∆P = pressure drop and R = resistance. According to Poiseuille’s law the flow rate for a given ∆P is proportional to the 4th power of the vessel radius [17]. This implies that a very slight change in vessel diameter can change its ability to conduct blood considerably. Simultaneously, the circulatory systems still obeyes the continuity equation (i.e. Ohm’s law).

2.2.7 Incompressible Newtonian fluid

There are two more classifications of importance, that is: Newtonian vs non-Newtonian fluid and compressible vs incompressible fluid. In the previous section a quantification for the importance of intertial and viscous forces was introduced. In addition, a quantification for the amount of force per unit area is termed stress [9]. Stress emerges due to the forces in blood flow and is calculated by dividing the force by the area upon which the force is applied [9]. Shear stress is important to determine whether a fluid can be quantified as Newtonion or non-Newtonion. Shear stress incorporates shear forces which, recalling from section Viscous force previously, are described by ’viscosity effects’. The relationship between these components is as follows:

τ = µ · γ (2.3)

Figure 2.8: τ = shear stress, µ = dynamic viscosity and γ = shear rate.

Shear rate is the rate at which shear deformation occurs. If the relationship between shear stress and shear rate is linear and independent of the shear rate, the fluid is considered Newto-nian [9]. In general it is reasonable to treat blood as a NewtoNewto-nian fluid with a viscosity equal to that from the high-shear region [18]. Nevertheless, when the shear rate of blood is small the non-Newtonian aspects become apparent [9]. Though, this only applies to regions in the circulatory system where small shear rate is the case (e.g. capillaries). Secondly, blood is con-sidered incompressible since compression and expansion do not significantly affect blood density ρ. Hence, density is treated as a constant. One measure of the degree of compressibility is the Mach number which is a dimensionless quantity. A low Mach number thus corresponds with the presumed incompressibility of blood. To conclude, blood is generally presumed to be an incompressible Newtonion fluid.

2.3 FDS treatment

Cerebral aneurysms are lesions occurring in the wall of an artery in the brain [4]. Whereas a normal artery wall consists of three layers (see 2.9) the aneurysmal wall is thin and weak due to abnormal loss or absence of the muscular middle layer. Subsequently, only two layers will remain.

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Figure 2.9

Having become too weak to oppose hemodynamic pressure the artery wall distends resulting in an abnormal geometry, i.e. aneurysmal sac emerging in the artery wall. Saccular aneurysms are the most common geometry emerging [13] and moreover the type of cerebral aneurysm studied in this thesis. It is characterized by the aneurysmal sac being exclusively connected with the artery through a neck (see 2.10). Despite arising more frequently in arterial branching at the base of the brain [32] aneurysms can develop throughout any artery in the brain.

Figure 2.10: Saccular cerebral aneurysm

The majority of aneurysms is relatively small and 50 to 80% does not rupture during the course of life [4]. Nevertheless, a significant risk of death is associated with cerebral aneurysm rupture as rupture leads to subarachnoid haemorrhage, i.e. bleeding into the space surrounding the brain [32]. Prior conventional treatment has been adhering to the paradigm that treatment ought to revolve around treating the aneurysm itself, e.g. by clipping the neck of the aneurysm sac. However, the introduction of FDS caused a paradigm shift towards treatment of the segment bearing the aneurysm instead [10], i.e. the FDS is placed inside the artery itself (see 2.11).

Given the significant treatment effect of FDS [26], FDS treatment has become a well-accepted treatment option for particularly small aneurysms [14], e.g. anterior cerebral artery aneurysms [6]. The treatment provides cerebral aneurysm occlusion via flow diversion. The FDS mechanic-ally redirects blood flow which allows for intra-aneurysmal stagnation of blood, intra-aneurysmal thrombosis, remodeling and resorption of the aneurysmal sac, and ultimately endothelial11

growth, while preserving flow into parent vessel and adjacent branches (see 2.11). This process has been reported to lead to long-term normalization of the artery wall [30].

11_{The cells that line the interior surface of blood vessels and lymphatic vessels, involved in vessel formation}

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Figure 2.11: Schematic simplification of the therapeutic mechanism following FDS treatment [8].

2.4 Lattice Boltzmann method & Palabos

Section Hemodynamics and hemorheology elucidated that blood in this case can be defined as a viscous incompressible Newtonian fluid. For this reason, solving the Navier-Stokes equations for an incompressible viscous fluid is considered to be a reliable model for blood flow [5]. Relat-ively new is the Lattice Boltzmann method (LBM) which instead of numerically integrating the Stokes equations solves the discrete Boltzmann equation. Whereas solving the Navier-Stokes equations can at most numerically describe the behavior of a fluid (i.e. on the macroscopic scale12_{) (see 2.12), LBM enables to study fluid flow also taking the microscopic scale, i.e. the}

molecular level, into account. This chapter explained that the circulatory system and blood being a colloid, results in a complex flow system where the molecular level plays a significant role. In this apparent particle chaos the behaviour of blood seems rather unpredictable (and unexpected), however, it incorporates macroscopic information. In essence LBM links the micro-scopic and macromicro-scopic scale to allow for (visual) simulation of fluids where the molecular level is of importance [25]. Hence, LBM can be used to properly model blood flow.

Figure 2.12: Left: the fluid at macroscopic scale. Right: the fluid at microscopic scale. Lattice Boltzmann method is a quasi-incompressible method which represents fluid in a spatial

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grid geometry, where each spatial point is a node that contains a cluster of particles, and which number of particles depends on the local density. The geometric space, however, is not modeled in solely the Cartesian space (i.e. (x, y, z) coordinates) but also includes velocity and time spaces. All are imbedded in the so-called phase space which defines the distribution of particles by a function f (see eq. 2.4) [25].

f (x, e, t) (2.4)

Figure 2.13: The probability for a particle at a position [x] to possess a velocity [e] at a time [t]. The discretized spatial grid constructed is not oriented completely free due to the presence of velocities. Since this thesis implements a 2D model, a grid of type D2Q9 is used which implies a 2D spatial grid with each node having 9 discrete velocities (see 2.14). I.e. all particles have 9 possible directions to move towards, as in one time step a particle can only move to its nearest neighbors sites or remain its position. Each possible velocity direction is assigned a probability by function f (see eq. 2.4) that determines whether a particle remains or moves during a time step (advection). From this the number of particles moving in a certain direction with a certain speed is obtained. Once a cluster of particles has moved to one of the neighbouring sites they will collide with another cluster of particles being present due to having remained there, or having moved there at the same time. This process of advection and collision outlines the fundamental steps of the Lattice Boltzmann equation (LBE) in LBM without going into the mathematical details. After every time step (advection and collision) LBM computes the new state of the grid. So, these steps are realized as the collide and stream processes inside the Palabos framework.

Figure 2.14: Top: probability distribution of possible discrete velocities. Bottom: spatial grid and speed of possible discrete velocities.

This grid structure allows for LBM to be parallelized due to the locality of dynamics, e.g. sub-domains whose computation is independent of the adjacent sub-domains. Moreover, Palabos is provided with the (underlying) feature for sub-domains to communicate.

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CHAPTER 3

Method

To model the cerebral aneurysm geometry a 2D model is implemented using the Palabos frame-work. Palabos is an open-source CFD solver based on the lattice Boltzmann method. This chapter begins with exemplifying the implementation of the model while referring to the theor-etical background. Lastly, the model and simulation fitting is discussed.

3.1 Implementation

The previously discussed LBM grid is termed a lattice in Palabos. For Palabos to construct the model the implementation must properly process the following transitions:

The physical system (P ) ↔ the model (LB1_{) ↔ the simulation}

The program therefore begins with collecting data about the physical system by reading the geometry from a .ppm input image, and P information from a .xml input configuration file2. P as a LB model, is represented by creating a lattice of equal dimension as the image and consists of a node at each spatial point. Hence, each pixel corresponds with a node3 in the lattice, and each node type can be detected according to the mapping of pixel color to node type: black → wall, white → fluid, red → inlet, green → outlet and blue → porous (i.e. FDS)4_{. Due to}

this mapping the dynamics and initial value of each node can be set. However, proper setup requires information about P . E.g. measuring the artery diameter in P yields 4 mm, whereas, an identical measurement in LB will yield a value in lb units as the computer can only ’measure’ the lattice which is not necessarily the same value. Scaling values from one system to another can be accomplished through the conversion factor C.

valueP = valueLB· Cvalue (3.1)

Ensuring all conversion factors are correct and consequently the same measurement yields values implying an identical matter in both systems is exemplified in the subsequent subsection.

3.1.1 Unit systems

In order to construct a valid model it is important the simulation’s discrete system properly represents the physical system. This requires valid conversion from units in the physical system (P ) to units in the simulation system (LB). To transition from one to another the Law of Similarity is applied using the Re due to it being an independent dimensionless number, thus,

1_{Lattice Boltzman}

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being valid in both systems. From this the theorem is derived that: ‘Two flows obeying the Navier-Stokes equations are equivalent if they are embedded in the same geometry (except for a scaling factor) and have the same Reynolds number.’ [22]. Aiming for equivalence in both systems to ensure validity, invariance of the Reynoldsnumber in P and LB is used to scale values from P to LB.5 Re = Re∗ (3.2) Re = umax· w ν , Re ∗₌u∗max· w∗ ν∗ (3.3) umax· w ν = u∗max· w∗ ν∗ (3.4)

Figure 3.1: See appendix A.1 table A.1 for the symbols legend.

Using the fact that for each value there exists a conversion factor C (eq. 3.1) and inequality of the systems (eq. 3.2), leads to the equation to be satisfied as follows:

ν ν∗ = umax u∗ max w w∗ =⇒ Cν = Cumax· Cw (3.5)

Of the required parameters the following are explicitly defined through the configuration file: ν, umax, u∗max and ∆x (i.e. Cw). The missing parameter ν∗ is calculated satisfying:

ν∗= _u ν

max

u∗

max· ∆x

(3.6)

Regarding LB parameters, the model only allows for ∆x and u∗

max to be chosen freely, as

other values depend on their value when obeying equations. As a double verification whether LB represents P , the program checks Re and Re∗ on equality.

3.1.2 Model implementation

After the program has calculated the required parameters the program proceeds with initializing the whole lattice with default values. Since a complex system is modelled, certain attributes such as speed of particles are not exactly known (yet) at each spatial point. Hence, the model allows for a self-regulating LB system to evolve into a representation of the P system desired. The lattice is therefore initialized setting all cells to equilibrium state (i.e. there are no net forces, and no acceleration) and zero velocity. Consequently, every particle initially has equal probability of its velocity to be in a certain direction. The geometrical space is therefore isotropic. In addition the density is set to 1 , given that the density in an incompressible fluid takes a constant value resulting in a homogeneous distribution invariant in time and space [22]. By doing so a static fluid at rest is set up.

As the relevant blood fluid is solely present inside the geometry, the program continues with distinguishing fluid from non-fluid nodes. Using the previously mentioned one-to-one mapping of pixel to node type, non-fluid nodes are deemed vessel walls and subsequently set to having bounce-back dynamics, i.e. they will act like walls.

The program then proceeds to allow for the fluid initially at rest to undergo motion, i.e. to enable blood flow to emerge. This requires proper conditions for in- and outflow. In chapter 2 it was stated that flow is relatively constant, implying that the equation of continuity must be satisfied (see eq. 3.7).

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Q = A¯v (3.7)

Figure 3.2: Equation of continuity. Q = volume flow rate, A = cross sectional area and ¯v = average velocity.

The cross sectional area of the inlet remains constant. In addition, ¯v is enforced to remain constant by setting the dynamics at the inlet to be of type Dirichlet boundary condition. This means that all velocity components, i.e. fluid layers, have an imposed value. In doing so con-tinuity of flow is ensured.

Secondly, the velocity values are set. Chapter 2 explained that blood flow has a parabolic Poiseuille profile6. The inflow is therefore set to be a parabolic Poiseuille profile as at least the inlet is perceived as a straight duct, regardless of the further geometry of the vessel (see eq. 3.8). v = − 1 4µ ∆P ∆x(R 2_{− r}2₎ _(3.8)

Figure 3.3: Hagen-Poiseuille equation. v = velocity, µ = dynamic viscosity, ∆P = pressure drop, ∆x = vessel length, R = internal vessel radius and r = fluid layer radius.

The Poiseuille equation defines the velocity v of liquid moving through the inlet as a function of the distance r from the fluid layer, to the center of the inlet’s cross sectional area A. The inlet is found through detecting the inlet in the grid and saving its domain. From this the program acquires the radius R of A, and for each fluid layer in the domain the distance r to the center of A (i.e. the relative spatial position). The remaining unknown variables are found using u∗max.

Because u∗max is defined, the equation for u∗max7 can be solved such that a constant C equal

to the unknown variables is found:

u∗max= − 1 4µ ∆P ∆x(R 2 − r2) (3.9) u∗max= C · (R2− r2) (3.10) C = u ∗ max (R2_{− r}2₎ (3.11)

The program sets the velocity at the nodes in the inlet domain by v = C · (R2− r2_{). Since}

the program enforces inlets to be located on the edges of the grid, the inlets can be positioned at the bottom, top, left or right edge. Consequently, the velocity v is adapted if necessary (−1 · v) such that inflow will always be in the correct direction, i.e. flow from inlet to outlet.

The outlets are set to be constant pressure outlets, i.e. the dynamics are set to be of type density boundary condition and the density value is set to 1. This is done since the exact details of the flow distribution are unknown.

Lastly, the FDS is set. The FDS modeled is a type PED FDS, which is a self-expanding cylindrical device composed of 48 braided strands [26]. Due to this design it acts like a porous medium. Flow in porous media can be considered on three scales: the pore scale, the represent-ative elementary volume (REV) scale and the domain scale [15]. The model ignores the detailed structure of the FDS. Hence, the program implements the porous medium on REV scale, i.e. the fluid and medium properties are averaged out over a representative elementary volume [31]. The representative elementary volume is denoted by the FDS layer. The presence of this porous medium can be modeled by adding an external resistive force to LBM. This so-called Darcy force

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(see eq. 3.12) is based on Darcy’s law (see eq. 3.13) which describes the flow of a fluid through a porous medium and holds for flows obeying the Navier-Stokes equations [31].

f = −C · v (3.12)

Figure 3.4: Darcy force. f = force density, C = constant related to geometric features of the porous material (e.g. porosity) and v = local flow velocity [15].

q = φ · v (3.13)

Figure 3.5: Darcy’s law. The relationship between: q = Darcy flux/velocity, φ = porosity and v = fluid velocity.

The Darcy force is applied after each collide and stream using the method proposed by Guo and co-workers [16]. Even though Guo’s method is very accurate it is also more computationally intensive. Computation is therefore done in parallel.

3.1.3 Simulation implementation

After the program has constructed the desired model it will run the simulation. The simulation configuration is specified by the configuration file (see appendix A.2 listing A.1). The program simulates the specified physical duration by iterating a number of time steps until the corresponding physical duration is met. During each iteration the collide and stream steps compute the next state of the lattice and Darcy force is applied. At the specified frequency a gif image of the lattice is saved in the specified output directory. Idem dito for saving lattice data in a VTK file representing the lattice. This VTK file contains for each spatial point the velocity norm and velocity vector in physical units. The data can be used for post processing and analysis in ParaView.

3.2 Model and simulation fitting

The previous section mentioned that the model and simulation can be configured. The subsequent section discusses the configuration process applied to acquire a proper model and simulation while compromising between accuracy, stability and efficiency.

The first geometry modeled represents a cerebral aneurysmal artery 3-4mm in diameter, i.e. an average cross sectional area of 9.6 mm2. PED FDS thickness varies between the range of 50-100 µm8. The configuration parameters and variables were set to correspond as follows:

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Table 3.1: Parameter configuration legend

# Parameter Value Description

1 ν 3.5·10−6 m2/s Experimental value. Used in equation to calculate LB parameters.

2 umax 0.28 m/s Experimental value. Used in equation to calculate LB parameters.

3 u∗max 0.01 lu/lt

Chosen value. Used in equation to calculate LB parameters. Important for the Mach number.

4 ∆x 5 · 10−5 m

Chosen value. Discrete space ∆x is the physical width represented by the smallest unit in the simula-tion (i.e. a lattice node). Hence, it defines the resolusimula-tion.

5 h∗ 2 lu Chosen value. Height of the FDS layer in the provided geometry in

pixels. Serves to calculate and output the P value of the FDS height.

6 t 1 s Chosen value. Physical time simulated.

1. Physical value obtained in vitro.

2. Physical value obtained in vivo from cerebral arteries with approximately an equal cross sectional area within the mean ± SE. I.e. posterior cerebral arteries (PCA) [11]. The value obtained is the average of the peak velocity in the left and right cerebral hemisphere. 3. LB value small enough for stability9_{and < 0.03-0.1 for accuracy [21].}

4. Conversion factor Cw determined by being accurate enough to properly represent the

geo-metry. I.e. the cross sectional area and the FDS thickness. The discrete space step is therefore chosen focusing on accuracy rather than computational efficiency.

5. Pixel thickness, i.e. LB value, of the FDS layer placed in in the geometry. The previous section Lattice Boltzmann method & Palabos exemplified the traversal path of particles in the lattice. Hence, the layer is 2 pixels thick for every adjacent fluid node to prevent a leaky FDS.

6. Physical value determined by being sufficient for the simulation to have reached steady-state flow, i.e. stationary state, but not unnecessarily large taking efficiency, i.e. computation time, into consideration.

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Table 3.2: Variable configuration

# Variable Value Description

1 φ 0.1, 0.5 or 0.9

Chosen value. The void fraction is the measure of the void spaces in the FDS layer with respect to the total layer volume. May range between 0 and 1.

1. Independent variable varied to observe the effect on the dependent/outcome variable i.e. the intrasaccular blood flow.

3.2.1 Reynolds number

Chapter two exemplified how Re can be used to check for equality of flow systems. Besides ensuring the equality of Re it is also manually verified to be within a range corresponding to the human arterial system in vivo. I.e Re < 500 which corresponds to being lower than the peak Re in the smallest arteries [18] and lower than the transition Re to turbulence under steady inlet conditions [2] [23].

Listing 3.1: Output Re check 1 P h y s i c a l Re : 0 . 2 8 ∗ 0 . 0 0 4 9 / 3 . 5 e −06 = 392

2 Lbm Re : 0 . 0 1 ∗ 9 8 / 0 . 0 0 2 5 = 392 3

4 Lbm s y s t e m matches p h y s i c a l s y s t e m !

3.2.2 Mach number

Chapter two stated that a low Mach corresponds with this thesis’ definition of blood. I.e. an incompressible Newtonion fluid. LB is mostly used for the simulation of incompressible fluids where the Mach number is small. If the Mach number is too large fluid waves will emerge that carry a numerical error. Hence, the simulation will be numerically unstable. The method will become stable if this error gradually vanishes. Nevertheless, it can be stated that LB simulations are usually run in the limit of small Mach numbers [21]. The Mach number is calculated as follows [21]: M a∗= u ∗ max c∗ s (3.14)

Figure 3.6: Mach number. M a∗ = the Mach number, u∗max = the local flow velocity with

respect to the boundaries and c∗s = the speed of sound in the medium.

The speed of sound for D2Q9 grids is fixed and has the value √1

3. Thus, the Mach number

can only be adjusted by modifying u∗max. Even though it is recommended that u∗max< 0.4 for

stability [21], it has been found that with this model configuration the threshold of numerical instability lies at 0.017. The value of u∗max is set within a safe range to satisfy a low mach

number, i.e. Mach number << 1 (see listing 3.2).

Listing 3.2: Output Mach number check 1 Lbm Ma : 0 . 0 1 7 3 2 0 5

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CHAPTER 4

Experiments

To be able to answer the question: ‘How are the hemodynamics inside the aneurysmal sac in terms of flow velocity and stream line structure effected by varying porosity of the FDS?’, a parametric study was conducted to research the effects of varying the FDS porosity on the velocity and the streamline structure inside the aneurysmal sac. This chapter reports the conducted simulation experiments and the post-processing using ParaView afterwards.

4.1 Porosity experiment

For two different saccular cerebral aneurysm geometries four simulations were conducted with a fitting model and simulation configuration obtained as described in chapter 3.

4.1.1 Geometry 1

Table 4.1: Parameter configuration geometry 1

# Parameter Value 1 ν 3.5·10−6 m2_/s 2 umax 0.28 m/s 3 u∗max 0.01 lu/lt 4 ∆x 5 · 10−5 m 5 h∗ 2 lu 6 t 1 s

Geometry presumed to be a posterior cerebral artery (PCA) [11].

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4.1.2 Geometry 2

Table 4.2: Parameter configuration geometry 2

# Parameter Value 1 ν 3.5·10−6 m2/s 2 umax 0.305 m/s 3 u∗max 0.01 lu/lt 4 ∆x 5 · 10−5 _m 5 h∗ 2 lu 6 t 1 s

Geometry presumed to be a middle cerebral artery (MCA) [11].

See appendix A.3.2 for the (calculated) simulation parameters output.

The simulation experiments were conducted with porosity as independent variable and in-trasaccular blood flow as dependent variable. The studied FDS porosity values are: 0.1, 0.5 and 0.9 plus a control simulation i.e. without FDS. Each simulation experiment simulated a physical time of 1.0 s such that the simulation has reached a stationary state1. For every simulation the VTK file of the last frame was taken to provide consistent data for analysis. The VTK file was post-processed in ParaView as follows:

1. In order to be able to visualize the 2D data it was converted to 3D by setting the z-coordinate of each spatial point to zero. Secondly, the range of the velocity color legend is set low enough for the distinguishable intrasaccular velocity magnitudes to become appar-ent.

2. Streamlines. For each simulation the streamlines are visualized by placing a constant amount of seeds (i.e. N = 50) along a line with the same start and end point from which the streamlines with a constant maximum length will sprout.

3. Velocity. The intrasaccular velocity magnitudes are visualized using the previously men-tioned color mapping range. Since only the effect, i.e. the difference, of varying porosity is of interest it is accepted that velocities of the control experiment might exceed this range. 4. Lastly, the geometry is clipped such that only the intrasaccular geometry of interest

re-mains.

Per geometry for each experiment the streamline structure and velocity magnitudes are eval-uated to find the relationship between porosity and streamline structure, and the relationship between porosity and velocity magnitudes.

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CHAPTER 5

Results

In this chapter the results from the simulation experiments are presented and evaluated.

5.1 Geometry 1

Con trol φ = 0.1 φ = 0.5 φ = 0.9

Figure 5.1: Geometry 1. Intrasaccular streamline structure of the 2D saccular cerebral an-eurysm model with 3 different FDS porosity values at t = 1.0 s. Stent thickness = 100 µm. a control has no FDS; b φ = 0.1; c φ = 0.5; d φ = 0.9;

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The result of the control simulation a shows one smooth vortex emerging. From b it is noticed that the FDS layer results in a noticeable distortion of the vortical motion. Besides, the amount of streamlines seems to be slightly less as more void space is observed. c does not show visibly more void space compared to control simulation a. In addition, it shows very little distortion visible near the aneurysmal intra-vascular wall. The streamline structure of d resembles the one of c as there seems to be no major evident difference. For this geometry a lower FDS porosity leads to fewer1_{intrasaccular streamlines and more intrasaccular streamline distortion.}

Con trol φ = 0.1 φ = 0.5 φ = 0.9

Figure 5.2: Geometry 1. Intrasaccular velocity of the 2D saccular cerebral aneurysm model with 3 different FDS porosity values at t = 1.0 s. FDS layer thickness = 100 µm. a control has no FDS; b φ = 0.1; c φ = 0.5; d φ = 0.9;

The control simulation a shows considerably larger velocity magnitudes. b indicates a tre-mendous decline in velocity magnitudes due to FDS placement. Again between c and d no evident difference is observed. Moreover, at both larger velocity magnitudes than at b are observed. For this geometry a lower FDS porosity leads to smaller intrasaccular velocity magnitudes. The relationship between porosity and average intrasaccular velocity magnitude is shown below:

1_{Note that with amount and fewer/more not the concrete number of streamlines but the area occupied by the}

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Figure 5.3: Geometry 1. Relationship between FDS layer porosity [1] and average instrasaccular velocity [m/s] at t = 1.0 s. FDS layer thickness = 100 µm.

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5.2 Geometry 2

Con trol φ = 0.1 φ = 0.5 φ = 0.9

Figure 5.4: Geometry 2. Intrasaccular streamline structure of the 2D saccular cerebral an-eurysm model with 3 different FDS porosity values at t = 1.0 s. FDS layer thickness = 100 µm. a control has no FDS; b φ = 0.1; c φ = 0.5; d φ = 0.9.

The result of the control simulation a shows one smooth vortex emerging. b maintains an overall smooth streamline structure pattern with very little distortion visible at the top of the aneurysmal sac. The area of streamlines has decreased as more void space is observed near the aneurysmal intra-vascular wall. Similarly, at c a further decrease of the area of streamlines is observed. In addition, more distortion starts to emerge. d results in the most noticeable distortion of the vortical motion. Besides, the streamline structure seems to have stretched sideways compared to b and c. At last, the amount of streamlines appears to be lesser than the control simulation. For this geometry a higher FDS porosity leads to fewer intrasaccular streamlines and more intrasaccular distortion.

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Con trol φ = 0.1 φ = 0.5 φ = 0.9

Figure 5.5: Geometry 2. Intrasaccular velocity of the 2D saccular cerebral aneurysm model with 3 different FDS porosity values at t = 1.0 s. FDS layer thickness = 100 µm. a control has no FDS; b φ = 0.1; c φ = 0.5; d φ = 0.9.

The control simulation a shows considerably larger velocity magnitudes. b indicates a tre-mendous decline in velocity magnitudes due to FDS placement. However, the velocity magnitudes are still larger than observed in c and d. Again between c and d no evident difference is observed. For this geometry a higher FDS porosity leads to smaller intrasaccular velocity magnitudes. The relationship between porosity and average intrasaccular velocity magnitude is shown below:

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Figure 5.6: Geometry 2. Relationship between FDS layer porosity [1] and average instrasaccular velocity [m/s] at t = 1.0 s. FDS layer thickness = 100 µm.

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CHAPTER 6

Conclusion

To be able to answer the question: ‘How is blood flow inside the aneurysmal sac in terms of flow velocity and streamline structure affected by varying porosity of the FDS?’, a parametric study is carried out using the implemented 2D cerebral aneurysm model.

The results indicate that FDS placement has a diminishing effect on the intrasaccular amount of streamlines and velocity magnitudes. Secondly, a relationship is observed between FDS poros-ity and intrasaccular streamline structure and a relationship is observed between FDS porosporos-ity and intrasaccular velocity magnitudes. These relationships do not hold among different geomet-ries. In geometry 1 a higher FDS porosity leads to lesser streamline structure distortion and lesser velocity magnitude reduction, whereas, in geometry 2 a higher FDS porosity leads to more streamline structure distortion and more velocity magnitude reduction. Thus, both relationships are geometry dependent.

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CHAPTER 7

Discussion

This thesis studied the effect of varying FDS porosity on the instrasaccular blood flow in terms of velocity and streamline structure. To study this effect a parametric study was conducted using a reliable and valid model implemented in Palabos. This model presumed that blood is an incompressible Newtonion fluid, which is perceived as a generally accepted blood flow model under most circumstances [20]. Four simulation experiments were carried out with varying porosity and a control experiment. The data was post-processed and evaluated using Paraview afterwards.

From the results it appears there is a relationship between FDS porosity and intrasaccular streamline structure and a relationship between FDS porosity and intrasaccular velocity. This relationship seems to be geometry dependent. In geometry 1 increasing the FDS porosity leads to lesser streamline distortion and lesser velocity reduction. Contrary, in geometry 2 the exact opposite effect is observed. The results indicate a interrelationship between FDS porosity and intrasaccular streamline structure and velocity, as in both the consequential variation correspond. Still this needs to be validated by experimenting with more geometries.

Even though the considered validity of the research design it can be mentioned that this study has its limitations. First of all, the reader should bear in mind that the geometries used are primarily theoretical. More realistic will be to use a geometries obtained in vivo. Morever, not the complete cylindrical FDS device was modeled. Albeit it might seem that the absent parts of the FDS only affect blood flow in the adjacent branches, blood flow remains a complex system and the effect on the intrasaccular blood flow can only be verified by an experiment. So, solely modeling the FDS segment occluding the aneurysm neck may have had a significant effect on the intrasaccular blood flow. Lastly, the model is 2D whereas the physical system in vivo is 3D. Dimension reduction might have caused loss of information as particles lack the possibility of in vivo discrete velocites, e.g. along the z-axis. Hence, the results are known to be limited by this 2D model.

It is beyond the scope of this study to examine the effect on large scale. Therefore, further research could examine larger data sets to determine till which extent these findings are gener-alizable to e.g. different type of geometries, whether the (inter)relationship holds and whether a possible correlation is significant. Secondly, further research might take time into account as this thesis solely studied the intrasaccular blood flow as soon as stationary state was reached. E.g. one might be more interested in the effect on intrasaccular blood flow over a given period of time, as FDS treatment has been reported to lead to intra-aneurysmal thrombosis [10]. Besides, the effect of other independent variables such as FDS thickness can be studied.

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APPENDIX A

A.1 General

Table A.1: Symbols legend

# Symbol Name Meaning thesis’ context System Unit

ν Kinematic viscosity Kinematic blood viscosity P m2/s

µ Dynamic viscosity Dynamic blood viscosity P m2_/s

umax Maximum velocity Peak velocity in Poiseuille profile at inlet P m/s

u∗max Maximum velocity Peak velocity in Poiseuille profile at inlet LB lu/lt

w Width Characterizing reference length in P (e.g. width of the inlet) P m

∆xb _{Delta x} _{Resolution (i.e. w/w}∗₎ _P _m

φ Porosity Porosity of FDS layer - [1]a

h∗ Height Height FDS layer (i.e. thickness) LB lub

a_{Dimensionless.}

b_{Lattice Boltzmann units.}

A.2 Method

Listing A.1: .xml configuration file 1 <?xml v e r s i o n=” 1 . 0 ” ?> 2 3 <s i m u l a t i o n> < !−− G l o b a l s i m u l a t i o n p a r a m e t e r s −−> 4 <o u t p u t D i r>tmp</ o u t p u t D i r> 5 <simLength>1</ simLength> < !−− P h y s i c a l d u r a t i o n [ s ] t o be s i m u l a t e d 6 ( r e m a r k : d o e s n o t mean run t i m e ) −−> 7 <s a v e G i f F r e q u e n c y>0 . 0 5</ s a v e G i f F r e q u e n c y>

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11 <g e o m e t r y> < !−− Geometry i n f o r m a t i o n −−>

12 < f i l e >o r i g i n a l g e o m e t r y P . ppm</ f i l e > < !−− I n p u t image f i l e n a m e −−> 13

14 <phys> < !−− P h y s i c a l system p a r a m e t e r s −−> 15 < v i s c o s i t y>3 . 5 e−6</ v i s c o s i t y>

16 17 <uMax>0 . 1</uMax> 18 19 </ phys> 20 21 <sim> < !−− LB system p a r a m e t e r s −−> 22 <dx>5 e−5</ dx> 23 24 <uMax>0 . 0 1</uMax> 25 26 <s t e n t> 27 0 . 0 28 <t h i c k n e s s>2</ t h i c k n e s s> 29 </ s t e n t> 30 </ sim> 31 </ g e o m e t r y>

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A.3 Experiments

A.3.1 Geometry 1 simulation experiment

1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ S i m u l a t i o n p a r a m e t e r s ∗ 3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 4 f i l e : o r i g i n a l g e o m e t r y . ppm 5 s i z e : 454 x632 6 t [ s ] : 1 7 dx [m ] : 5 e −05 8 d t [ s ] : 1 . 7 8 5 7 1 e −06 9 t a u : 0 . 5 0 7 5 10 omega : 1 . 9 7 0 4 4 11 v i s c o s i t y [mˆ2/ s ] 3 . 5 e −06 12 U max , [ m/ s ] 0 . 2 8 13 U max , lbm : 0 . 0 1 14 15 E s t i m a t e d a v e r a g e v e s s e l d i a m e t e r [m ] : 0 . 0 0 3 3 1 6 6 7 16 − 17 − 18 − 19 20 P h y s i c a l Re : 0 . 2 8 ∗ 0 . 0 0 4 9 / 3 . 5 e −06 = 392 21 Lbm Re : 0 . 0 1 ∗ 9 8 / 0 . 0 0 2 5 = 392 22 23 Lbm s y s t e m matches p h y s i c a l s y s t e m ! 24 25 P h y s i c a l Ma : 0 . 0 0 0 1 7 8 3 4 4 26 Lbm Ma : 0 . 0 1 7 3 2 0 5

Listing A.2: a = control

27 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 28 ∗ S i m u l a t i o n p a r a m e t e r s ∗ 29 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 30 f i l e : o r i g i n a l g e o m e t r y P . ppm 31 s i z e : 454 x632 32 t [ s ] : 1 33 dx [m ] : 5 e −05 34 d t [ s ] : 1 . 7 8 5 7 1 e −06 35 t a u : 0 . 5 0 7 5 36 omega : 1 . 9 7 0 4 4 37 v i s c o s i t y [mˆ2/ s ] 3 . 5 e −06 38 U max , [ m/ s ] 0 . 2 8 39 U max , lbm : 0 . 0 1 40 41 E s t i m a t e d a v e r a g e v e s s e l d i a m e t e r [m ] : 0 . 0 0 3 3 1 6 6 7 42 S t e n t p o r o s i t y , lbm : 0 . 1 43 S t e n t t h i c k n e s s , lbm : 2 44 S t e n t t h i c k n e s s [m ] : 0 . 0 0 0 1 45 46 P h y s i c a l Re : 0 . 2 8 ∗ 0 . 0 0 4 9 / 3 . 5 e −06 = 392 47 Lbm Re : 0 . 0 1 ∗ 9 8 / 0 . 0 0 2 5 = 392 48 49 Lbm s y s t e m matches p h y s i c a l s y s t e m ! 50 51 P h y s i c a l Ma : 0 . 0 0 0 1 7 8 3 4 4 52 Lbm Ma : 0 . 0 1 7 3 2 0 5 Listing A.3: b φ = 0.1 53 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 54 ∗ S i m u l a t i o n p a r a m e t e r s ∗ 55 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 56 f i l e : o r i g i n a l g e o m e t r y P . ppm 57 s i z e : 454 x632 58 t [ s ] : 1 59 dx [m ] : 5 e −05 60 d t [ s ] : 1 . 7 8 5 7 1 e −06 61 t a u : 0 . 5 0 7 5 62 omega : 1 . 9 7 0 4 4 63 v i s c o s i t y [mˆ2/ s ] 3 . 5 e −06 64 U max , [ m/ s ] 0 . 2 8 65 U max , lbm : 0 . 0 1 66 67 E s t i m a t e d a v e r a g e v e s s e l d i a m e t e r [m ] : 0 . 0 0 3 3 1 6 6 7 68 S t e n t p o r o s i t y , lbm : 0 . 5 69 S t e n t t h i c k n e s s , lbm : 2 70 S t e n t t h i c k n e s s [m ] : 0 . 0 0 0 1 71 72 P h y s i c a l Re : 0 . 2 8 ∗ 0 . 0 0 4 9 / 3 . 5 e −06 = 392 73 Lbm Re : 0 . 0 1 ∗ 9 8 / 0 . 0 0 2 5 = 392 74 75 Lbm s y s t e m matches p h y s i c a l s y s t e m ! 76 77 P h y s i c a l Ma : 0 . 0 0 0 1 7 8 3 4 4 78 Lbm Ma : 0 . 0 1 7 3 2 0 5 Listing A.4: c φ = 0.5 79 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 80 ∗ S i m u l a t i o n p a r a m e t e r s ∗ 81 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 82 f i l e : o r i g i n a l g e o m e t r y P . ppm 83 s i z e : 454 x632 84 t [ s ] : 1 85 dx [m ] : 5 e −05 86 d t [ s ] : 1 . 7 8 5 7 1 e −06 87 t a u : 0 . 5 0 7 5 88 omega : 1 . 9 7 0 4 4 89 v i s c o s i t y [mˆ2/ s ] 3 . 5 e −06 90 U max , [ m/ s ] 0 . 2 8 91 U max , lbm : 0 . 0 1 92 93 E s t i m a t e d a v e r a g e v e s s e l d i a m e t e r [m ] : 0 . 0 0 3 3 1 6 6 7 94 S t e n t p o r o s i t y , lbm : 0 . 9 95 S t e n t t h i c k n e s s , lbm : 2 96 S t e n t t h i c k n e s s [m ] : 0 . 0 0 0 1 97 98 P h y s i c a l Re : 0 . 2 8 ∗ 0 . 0 0 4 9 / 3 . 5 e −06 = 392 99 Lbm Re : 0 . 0 1 ∗ 9 8 / 0 . 0 0 2 5 = 392 100 101 Lbm s y s t e m matches p h y s i c a l s y s t e m ! 102 103 P h y s i c a l Ma : 0 . 0 0 0 1 7 8 3 4 4 104 Lbm Ma : 0 . 0 1 7 3 2 0 5 Listing A.5: d φ = 0.9

The effect of flow-diverter stent porosity on the intrasaccular blood flow according to a 2D cerebral aneurysm model

Bachelor Informatica