Particle Tracking in the Circle of Willis

Transcription

1 Particle Tracking in the Circle of Willis Pim van Ooij A report of work carried out at the Centre of Bioengineering University of Canterbury TU Eindhoven BMTE05.40 June

2 Summary An ischemic stroke is most commonly caused by a blood clot that blocks an artery in the brain, resulting in oxygen deficit in brain tissue and accompanying brain damage. To examine the route a blood clot travels in the Circle of Willis, the ring-like structure of blood vessels that distribute the blood flow to the cerebral mass, the influence of outlet diameter, bifurcation angle and mass flux of five different geometries on particle trajectory is studied, with the intention to predict particle trajectories in the Circle of Willis. The finite volume package Fluent is used, which supports two particle models: the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM). The difference between the models is the degree of interaction between the fluid and the particles. The results show a larger effect of outlet diameter and bifurcation angle on the particle trajectory for MPM than for DPM. For DPM, there is a linear relation between mass flux and density of particles. The higher the mass flux, the higher the density of particles in the flow. Since momentum transfer between fluid and particles for MPM is larger, the relation between mass flux and amount of particles is not as obvious. A simulation using DPM of particle tracks in the Circle of Willis produces predicted results. Unfortunately it was not possible to verify the prediction of particle tracks in the Circle of Willis for MPM.

3 Contents 1 Introduction 3 2 Theory and Methodology Emboli Circle of Willis Geometry model Geometries Circle of Willis Fluids model Geometries Circle of Willis Particle model Discrete Phase Model Macroscopic Particle Model Results Geometries General Outlet diameter Bifurcation angle Mass flux Circle of Willis General Outlet diameter Bifurcation angle Mass flux Discussion Geometries Outlet diameter Bifurcation angle Mass flux Differences between DPM and MPM Circle of Willis Conclusion 43 6 Recommendations 44 1

4 CONTENTS CONTENTS A Discrete Phase Model Manual 46 A.1 Introduction A.2 Limitations A.3 Overview of Discrete Phase Modeling Procedures A.4 Equations of Motion of Particles A.5 Coupling Between the Discrete and Continuous Phases B Macroscopic Particle Model Manual 54 2

5 Chapter 1 Introduction The cardiovascular disease called stroke is the third largest cause of death, after heart diseases and all forms of cancer. There are two types of strokes: the ischemic stroke and the hemorrhagic stroke. Although the latter type of stroke has a much higher fatality rate than the former, this project will focus on the former type of stroke which causes percent of all strokes. An ischemic stroke occurs when a blood vessel to the brain is blocked by an abnormal blood clot, called embolus. Due to the following deprivation of oxygen, brain tissue dies which causes irreparable brain damage or death. The embolus most often blocks an artery in the Circle of Willis, a ring-like structure of blood vessels found beneath the hypothalamus at the base of the brain, which main function is to distribute oxygen-rich blood to the cerebral mass. To know more about the route a blood clot travels in the brain, simulations of particle trajectories in the Circle of Willis are needed, which is the main goal of this project. Five different geometries are used to study the influence of artery diameter, bifurcation angles and mass flux on the particle trajectories. Mass flux is varied by prescribing different boundary conditions on the outlets of the geometries. With this information the particle tracks in the Circle of Willis are predicted. This prediction is then verified by a simulation of particle tracks on the Circle of Willis. The null hypothesis of this project is that there is no relationship between the geometry of the Circle of Willis and the distribution of the particles. However, it is expected that more mass flux through an artery causes more particle flow through this artery. Also, for equal pressure gradients, a large diameter of an artery is supposed to cause more mass flux through this particular artery, bringing more particles with it, then through an artery with a smaller diameter. Furthermore, it is assumed that a smaller bifurcation angle causes more mass flux through the adjacent artery, causing more particle flow. By comparing the resulting particle distribution on the outlets with the particle distribution on the inlet of the geometries, it is possible to predict particle tracks in the Circle of Willis, so that more knowledge is obtained about the trajectory of blood clots in the brain. 3

6 Chapter 2 Theory and Methodology 2.1 Emboli Bloods clots are created in so-called thromboembolic conditions, which are usually caused twofold [1]. First, a roughened surface, caused by arteriosclerosis, infection or trauma, is likely to initiate the clotting process. Secondly, clotting occurs in blood that flows very slowly through blood vessels because small quantities of thrombin and other procoagulants are always being formed. These are generally removed from the blood by the macrophage system, mainly the Kupffer cells of the liver. The concentration of procoagulants rises often high enough to initiate clotting, but when blood flows rapidly, they are mixed with large quantities of blood and are removed during passage through the liver. When an abnormal blood clot develops in a blood vessel, it is called a thrombus. Continued flow of blood past the clot is likely to break it away from its attachment resulting in a clot that flows along with the blood. These free flowing clots are known as emboli, which do not stop flowing until they come to a narrow point in the circulatory system. Thus, emboli originated in the venous system and in the right side of the heart flow into the vessels of the lung and cause pulmonary arterial embolism. Emboli originated in large arteries or in the left side of the heart will plug smaller arteries of arterioles in the brain, kidneys or elsewhere. 2.2 Circle of Willis The Circle of Willis is a ring-like structure of blood vessels in the brain, displayed in figure 2.1 and distributes oxygenated blood throughout the cerebral mass. It is estimated that among the general population, 50% have a complete Circle of Willis [2]. Variations include underdeveloped of even absent blood vessels. This can present a health risk, mostly for ischaemic stroke, while an individual with one of these variations may suffer no ill effects. Examples of these variations are the fetal P1, where the P1 section of the Posterior Cerebral Artery (PCA) is underdeveloped, and the missing A1, where the A1 section of the Anterior Cerebral Artery is missing, as displayed in figure 2.2. In these figures are displayed the abbreviations of the names of the arteries in the Circle of Willis. These will be briefly discussed. In figure 2.2, R stands for right and L for left. The Internal Carotid Arteries 4

7 Theory and Methodology 2.2 Circle of Willis Figure 2.1: The Circle of Willis, frontal (a) Fetal P1 (b) Missing A1 Figure 2.2: Variations in the Circle of Willis (ICA), the Vertebral Arteries (VA) and the Basilar Artery (BA) are the vessels that transport blood into the Circle of Willis and are called afferent arteries. The Middle Cerebral Artery (MCA), Posterior Cerebral Artery (PCA) and the Anterior Cerebral Artery (ACA) are the vessels that transport blood from the Circle of Willis and are called efferent arteries. The ICA is connected with the PCA via the Posterior Communicating Artery (PCoA) and the Left ACA is connected with the Right ACA via the Anterior Communicating Artery (ACoA). 5

8 Theory and Methodology 2.3 Geometry model 2.3 Geometry model Geometries To study the influence of outlet diameter, bifurcation angle and mass flux on particle trajectories, five different geometries are used. The geometry meshes are created by the meshing software package GAMBIT and comprise of approximately 300,000 tetrahedral volumes. The different geometries all consist of one inlet and two outlets. The diameter of the inlet is 5 millimeters. From inlet to outlet the geometries measure approximately 45 millimeters and from outlet to outlet the geometries measure approximately 30 millimeters, see figure Figure 2.3: Measures of geometry 1 The diameters of the outlets vary, as well as the bifurcation angles of the outlets. The geometries are displayed in figure 2.4. The different features of the geometries are summarized in table 2.1. For bifurcations of vessels in the human body, a physiological relation exist between outlet diameter and bifurcation angle and is given by Zamir [3]: with cos θ 1 = (1 + α3 ) 4/3 + 1 α 4 2(1 + α 3 ) 2/3 (2.1) cos θ 2 = (1 + α3 ) 4/3 + α 4 1 2α 2 (1 + α 3 ) 2/3 (2.2) α = a 2 a 1 (2.3) where θ 1 and θ 2 are the bifurcation angles, a 1 is the diameter of outlet 1 and a 2 the diameter of outlet 2. The diameters of the geometries are randomly chosen. These equations are used to calculate the bifurcation angles of geometry 5. Then geometry 4 is taken as the opposite of geometry 5. The properties of geometry 5 are calculated by equations 2.1 and 2.2 and can therefore actually be found in the human body, while geometries 1, 2, 3 and 4 are hypothetical geometries. In figure 2.4 the right outlets of the geometries are outlets 1, the left ones outlets 2. 6

9 Theory and Methodology 2.3 Geometry model (a) Geometry 1 (b) Geometry 2 (c) Geometry 3 (d) Geometry 4 (e) Geometry 5 Figure 2.4: Geometries used for particle tracking 7

10 Theory and Methodology 2.3 Geometry model Table 2.1: Properties of the five different geometries Geometry D outlet 1 D outlet 2 Angle outlet 1 Angle outlet 2 mm mm o o Circle of Willis The 3D mesh of the Circle of Willis comprises of approximately 1 million tetrahedral volumes. The mesh is displayed in figure 2.5. The diameters of the various arterial segments of the Circle of Willis were obtained from a population study of retrospective MRA scans [2]. The measurements and standard deviations are displayed in table 2.2. Figure 2.5: Mesh of the Circle of Willis 8

11 Theory and Methodology 2.4 Fluids model Table 2.2: Circle of Willis Measurements Artery Diameter Std Dev mm mm ACA - A1 Anterior Cerebral Artery - A ACA - A2 Anterior Cerebral Artery - A MCA Middle Cerebral Artery PCA - P1 Posterior Cerebral Artery - P PCA - P2 Posterior Cerebral Artery - P ACoA Anterior Communicating Artery PCoA Posterior Communicating Artery BA - B1 Basilar Artery - B BA - B2 Basilar Artery - B ICA Internal Carotid Artery Since the bifurcation angles in the Circle of Willis are not measured, a rough estimation of 90 o is made for the bifurcation between the ICA and the PCoA, and between the ICA and the ACA. There is no angle between ICA and MCA. This estimation is done by rotating the Circle of Willis in Fluent and study its features by eye. 2.4 Fluids model Geometries The blood flow through the geometries is modeled as unsteady, incompressible and viscous. This means that the governing transport equations to be solved by Fluent are the continuity equation and the momentum equation: ρu da = 0 (2.4) V A δu δt + ρuu da = pida + η( ɛ) ɛ da + FdV (2.5) A A A V where u is the velocity vector, ρ is the blood density, with a value of 1410 kg/m 3, p is the pressure, η is the fluid viscosity and F is the body forces vector on the fluid. V is a closed volume and A is the edge of a closed volume. ɛ is the strain rate tensor and is represented as: ɛ = 1 2 ( u + ut ) (2.6) In this project blood is simulated as a non-newtonian fluid and therefore the Carreau-Yasuda model for the viscosity is implemented: η η η 0 η = (1 + (λ γ) a ) n 1 a (2.7) where η is the infinite shear viscosity, set at P a s and η 0 is the zero shear viscosity, taken as P a s. λ is 0.11 s, a is taken as and n 9

12 Theory and Methodology 2.5 Particle model as [2]. γ is the strain rate magnitude, derived from the second invariant of the strain rate tensor, which for an incompressible fluid becomes: γ = 2 ɛ ij ɛ ij (2.8) These equation properties are taken from Moore [2]. To study the influence of different mass fluxes on the particle trajectories, for each geometry, three different pressure profiles are prescribed on the outlets. In the first situation, the pressure on outlet 1 is the same as the pressure on outlet 2: 98 mm Hg. In the second situation, a pressure of 95 mm Hg is prescribed on outlet 1, while the pressure on outlet 2 remains 98 mm Hg. Finally, the pressure on outlet 1 returns to 98 mm Hg, while the pressure on outlet 2 is set to 95 mm Hg. The pressure on the inlet remains constant at 100 mm Hg at all times. These values are an estimation of pressure differences in arteries of equal size in the human body. The boundary conditions on the outlets of the geometries are summarized in tabel 2.3. In this report, they will be referred to as boundary condition 1, boundary condition 2 and boundary condition 3. Table 2.3: Boundary conditions on the geometries Boundary outlet Pressure condition mm Hg A time step of 0.01 seconds is used in all the simulations. The Reynolds number of the flows are defined on the inlets and are calculated by: Re = 4ṁ πdη (2.9) with ṁ the mass flux through the inlet and d the diameter of the inlet Circle of Willis All the equations mentioned in subsection apply to the Circle of Willis as well. The boundary conditions on the inlet and outlets of the geometries are chosen as described in the previous subsection, thus they differ from the boundary conditions on the inlets and outlets of the Circle of Willis, since these are chosen as the mean values of the pressures in systole and diastole of arteries (afferent arteries) and veins (efferent arteries). These are summarized in table 2.4 and are equal for left and right arteries. 2.5 Particle model Two models to simulate particle tracks are implemented in Fluent: the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM). In this section 10

13 Theory and Methodology 2.5 Particle model Table 2.4: Boundary conditions on the Circle of Willis Artery Pressure mm Hg VA 100 ICA 100 ACA 4 MCA 4 PCA 4 both models will be described. Unfortunately, it is not yet possible to apply MPM to the Circle of Willis, since the MPM algorithms are not fully developed yet. To study particle tracks in geometries, first of all, particles need to be injected in a certain prescribed distribution. The distribution of the particles in the injection is the same for DPM and MPM at the inlet of the geometries as for DPM in the Circle of Willis. In the latter, the particles are injected in the RICA, see figure 2.2. The injection is displayed in figure 2.6. The distribution of the injection is chosen after the fact that emboli develop attached to vessels and thus are transported at the sides of a flow. No initial velocity is chosen for the particles. The particles used both in DPM and MPM, and in the geometries and the Circle of Willis, are of the material anthracite, with a density of 1550 kg/m 3 and a diameter of 0.5 mm. The value of the density is chosen after the assumption that a particle has a higher density than a liquid of the same material. The value of the diameter chosen so that multiple particles in an injection can be simulated. Figure 2.6: Injected particle distribution in all geometries Discrete Phase Model In addition to solving the transport equations for the continuous phase, Fluent is able to simulate a discrete second phase that consists of spherical particles dispersed in the continuous phase. To calculate the discrete phase trajectories, Fluent uses a Lagrangian formulation that includes the discrete phase inertia and hydrodynamic drag [4]. This formulation is only suited for a continuous phase flow with a well-defined entrance and exit. It contains the assumption that the second phase is sufficiently dilute, so that particle-particle interactions are negligible. This implies that the volume fraction of the discrete phase must be sufficiently low, usually less than 10-12%. At a total particle volume of approximately 5 mm 3 and a fluid volume around 500 mm 3, this is the case for the geometries. Since the total volume of the Circle of Willis is larger than the 11

14 Theory and Methodology 2.5 Particle model volume of a geometry, the volume fraction of the discrete phase of the Circle of Willis is lower than the volume fraction of a geometry. The trajectory of a discrete phase particle is calculated by use of integration of the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written as: du p dt = F D (u u p ) + g x(ρ p ρ) ρ p + F x (2.10) Since gravity is omitted, and the additional forces, F x only play a role when ρ > ρ p [4], equation A.1 reduces to: du p = F D (u u p ) (2.11) dt with F D (u u p ) the drag force per unit particle mass and F D defined as F D = 18µ C D Re ρ p d 2 p 24 (2.12) with Reynolds number Re defined as Re = ρd p u p u µ (2.13) and drag coefficient C D as C D = a 1 + a 2 Re + a 3 Re 2 (2.14) where u is the continuous phase (fluid) velocity, u p the particle velocity, µ the kinematic viscosity of the fluid, ρ the density of the fluid, ρ p the density of the particle, d p the particle diameter and a 1, a 2 and a 3 constants that apply for smooth spherical particles over several ranges of Re given by Morsi and Alexander [5]. The trajectory equations are solved by stepwise integration over discrete time steps. Integration in time of equation 2.11 yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by: dx dt = u p (2.15) With y and z in place of x for every coordinate direction. Equation 2.11 can be rewritten in simplified form as: du p dt = 1 τ p (u u p ) (2.16) where τ p is the particle relaxation time. Then a trapezoidal scheme is used for integrating 2.16: u n+1 p u n p t where n represents the iteration number and = 1 τ p (u u n+1 p ) +... (2.17) 12

15 Theory and Methodology 2.5 Particle model u = 1 2 (un + u n+1 ) (2.18) u n+1 p = u n p + tu n p u n p (2.19) These final two equations are solved simultaneously to determine the velocity and position of the particle at any given time. In these simulations the coupled approach is used, which means that the continuous flow pattern is impacted by the discrete phase and vice versa. The calculations of the continuous and discrete phase are alternated until a converged solution is achieved. The momentum transfer from the continuous phase to the discrete phase is computed in Fluent by examining the change in momentum of a particle as it passes through each control volume in the model, as illustrated in figure 2.7. Figure 2.7: Momentum transfer between the discrete and the continuous phases The momentum transfer from the continuous phase to the discrete phase is computed by examining the change in momentum of a particle as it passes through each control volume in the Fluent model: F = ( 18µC DRe ρ p d 2 p24 (u p u) + F other ) m p t (2.20) where F other are other interaction forces, in this case 0, m p is the mass flow rate of the particles and t is the time step. When a particle strikes a wall, it slides along the wall depending on the particle properties and the impact angle [4] Macroscopic Particle Model The Macroscopic Particle Model takes into account: The blockage and the momentum transfer of the fluid by particles. 13

16 Theory and Methodology 2.5 Particle model Evaluation of the drag force and the torque experienced by the particles. Particle-particle as well as particle-wall collision. In MPM, the particles are treated in a Lagrangian frame of reference as well. Rigid body velocity of a particle is imposed on the fluid cells that are touched by the particle, as displayed in figure 2.8 [6]. Figure 2.8: Particle velocity of a particle imposed on touched cells Firstly, this means that the particles add momentum to the fluid. The momentum is integrated and the particle drag and particle torque vectors are calculated for each particle. From these vectors the new position, velocity and angular velocity of the particles are calculated. Secondly, this means that momentum can also diffuse from touched cells to the particles. This momentum represents the hydrodynamic forces on the particles. Particle-wall collision works in the following manner. First, the boundary faces that are intersected by the particle are identified. Secondly, the particle velocity onto the normal and tangential vector of the wall is projected. Finally, the restitution coefficient is applied to calculate the outgoing velocity of the particle. Particle-particle collision is determined in a similar manner. At first, the particles which are going to collide are detected. The line-of-action of the collision is found next, which identifies the normal direction. Then, incoming particle velocities are projected onto the line-of-action to get the normal and tangential components. Finally, the coefficient of restitution and conservation of momentum to the normal components of the incoming velocities are applied to obtain the final velocities of the particles. The coefficients of restitution in the simulations are set to 0.8. For MPM, it is necessary to define the mass and the moment of inertia of the particles. The mass of the particles is set to 1e-5 kg, the moment of inertia to 1e-3 kg/m 2, sufficiently small to neglect. Furthermore, particle-particle attraction force and particle-wall attraction force are set to 1. For simulations of flow and pressure on the Circle of Willis, Fluent needs a library containing user-defined functions that describes the behavior of these properties. However, the MPM code for Fluent is also written in a library of user-defined functions and unfortunately, for this version of Fluent, it was not 14

17 Theory and Methodology 2.5 Particle model possible to use multiple libraries for the same simulation. Nor was it possible to combine both the libraries, since the user-defined functions of MPM are not known, as it is a part of the Fluent program. In future versions of Fluent and MPM, it will be possible to use multiple libraries and the simulation using MPM on the Circle of Willis can be done. 15

18 Chapter 3 Results 3.1 Geometries General On each geometry, three different boundary condition situations are set. Since 5 geometries are used, 15 simulations are done for DPM and 15 simulations for MPM. A total of 30 simulations. The DPM results are obtained using Fluent. It is possible in DPM to trap the particles on the outlets, so that the positions of the particles on the outlets are easy to determine. For MPM, trapping particles on outlets is not possible. The positions of the particles in MPM are saved in files, the final positions are isolated, and displayed using Matlab. Therefore, the results of DPM and MPM look slightly different. Unfortunately, it was not possible in Matlab to display the particles at real size. An example of a result is given in figure

19 Results 3.1 Geometries (a) DPM, Geometry 1, equal boundary conditions (b) MPM, Geometry 1, equal boundary conditions Figure 3.1: Example of DPM results and MPM results 17

20 Results 3.1 Geometries From these results, it is easy to determine the amount of particles on outlet 1 and how many on outlet 2. This data in combination with the geometry and boundary conditions is used to evaluate particle trajectories in various circumstances. In tables 3.1 and 3.2 duration of the simulations is summarized. Table 3.1: Time for all particles to be trapped on the outlets for DPM in seconds Geometry boundary conditions 98 outlet 1 98 outlet outlet 1 98 outlet outlet 1 95 outlet Table 3.2: Time for all particles to exit the outlets for MPM in seconds Geometry boundary conditions 98 outlet 1 98 outlet outlet 1 98 outlet outlet 1 95 outlet First, the influence of diameter of the outlets of the geometries and bifurcation angle on the particle trajectories is evaluated. Then, the influence of mass flux on the particle trajectories is evaluated Outlet diameter To evaluate the influence of outlet diameter on the distribution of the particles on the outlets, first, a comparison is made between geometry 1 and geometry 2, since the diameter of outlet 1 of geometry 2 is half the size of outlet 1 of geometry 1, while outlets 2 of both geometries remain the same size. Secondly, a comparison is made between geometries 4 and 5, where geometry 5 is the inverse is of geometry 4, see figure 2.4 and table 2.1. The results for DPM are displayed in figure 3.3, the results for MPM are displayed in figure

21 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 1 where the diameters of outlets 1 and 2 both are 3 mm. (b) Percentage of particles against different mass fluxes on geometry 2 where the diameter of outlet 1 is 1.5 mm and the diameter of outlet 2 is 3 mm. Figure 3.2: Percentage of particles against the diameters of geometries 1 and 2 for DPM 19

22 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 4 where the diameter of outlet 1 is 2 mm and the diameter of outlet 2 is 3 mm. (b) Percentage of particles against different mass fluxes on geometry 5 where the diameter of outlet 1 is 3 mm and the diameter of outlet 2 is 2 mm. Figure 3.3: Percentage of particles against the diameters of geometries 4 and 5 for DPM. 20

23 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 1 where the diameters of outlets 1 and 2 both are 3 mm. (b) Percentage of particles against different mass fluxes on geometry 2 where the diameter of outlet 1 is 1.5 mm and the diameter of outlet 2 is 3 mm. Figure 3.4: Percentage of particles against the diameters of geometries 1 and 2 for MPM. 21

24 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 4 where the diameter of outlet 1 is 2 mm and the diameter of outlet 2 is 3 mm. (b) Percentage of particles against different mass fluxes on geometry 5 where the diameter of outlet 1 is 3 mm and the diameter of outlet 2 is 2 mm. Figure 3.5: Percentage of particles against the diameters of geometries 4 and 5 for MPM. 22

25 Results 3.1 Geometries Bifurcation angle To evaluate the influence of the bifurcation angles of the geometries, the results of geometry 1 are compared with the results of geometry 3. This is because both geometries have equal diameters on the outlets, while the bifurcation angles differ. There is, however, a difference between the diameters of the outlets of geometry 1 and geometry 3, but this is not considered as of influence on the particle distribution, since equal diameter is thought to cause equal distribution of particles. Furthermore, a comparison is made between the outlets with the same diameter of geometry 4 and 5. This means that outlet 1 of geometry 4 is compared with outlet 2 of geometry 5 and the opposite. See figure 2.4 and table 2.1 for details. 23

26 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcation angles of outlets 1 and 2 both are 40 o. (b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcation angle of outlet 1 is 30 o and where the bifurcation angle of outlet 2 is 50 o. Figure 3.6: Percentage of particles against the bifurcation angles of geometries 1 and 3 for DPM. 24

27 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is o and where the bifurcation angle of outlet 2 is o. (b) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is o and where the bifurcation angle of outlet 2 is o. Figure 3.7: Percentage of particles against the bifurcation angles of geometries 4 and 5 for DPM. 25

28 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcation angles of outlets 1 and 2 both are 40 o. (b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcation angles of outlet 1 is 30 o and where the bifurcation angle of outlet 2 is 50 o. Figure 3.8: Percentage of particles against the bifurcation angles of geometries 1 and 3 for MPM. 26

29 Results 3.1 Geometries (a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is o and where the bifurcation angle of outlet 2 is o. (b) Percentage of particles against different mass fluxes on geometry 5 where the bifurcation angles of outlet 1 is o and where the bifurcation angle of outlet 2 is o. Figure 3.9: Percentage of particles against the bifurcation angles of geometries 4 and 5 for MPM. 27

30 Results 3.1 Geometries Mass flux Different boundary conditions on the outlets will cause different mass fluxes through the artery. For every geometry, three different pressure situations is applied, see table 2.3. To examine the influence of mass flux on particle trajectory, the percentage of particles on the outlets is plotted against the mass flux for every geometry. This means that each plot contains six points: the percentage of particles on outlet 1 for three different mass fluxes, and the percentage of particles on outlet 2 for three different mass fluxes. First, the results for DPM are displayed. The results of DPM can be found in figure 3.10, the results of MPM in figure In figure 3.12(a) and figure 3.12(b) all percentages of the particles on the outlets of all simulations are plotted against every mass flux on the outlets. The mass fluxes on the inlets are used to calculate the Reynolds numbers of the different flows. These are displayed in table 3.3 for DPM and in table 3.4 for MPM. (a) Geometry 1 (b) Geometry 2 (c) Geometry 3 (d) Geometry 4 28

31 Results 3.1 Geometries (e) Geometry 5 Figure 3.10: Percentage of particles on the outlets as a function of mass flux in DPM 29

32 Results 3.1 Geometries (a) Geometry 1 (b) Geometry 2 (c) Geometry 3 (d) Geometry 4 (e) Geometry 5 Figure 3.11: Percentage of particles on the outlets as a function of mass flux in MPM 30

33 Results 3.1 Geometries (a) Percentage of particles as a function of mass flux of every simulation in DPM (b) Percentage of particles as a function of mass flux of every simulation in MPM Figure 3.12: The percentage of particles as a function of mass flux for every simulation in DPM and MPM 31

34 Results 3.1 Geometries Table 3.3: Reynolds numbers for different boundary conditions of the geometries based on the inlet for DPM Geometry boundary conditions 98 outlet 1 98 outlet outlet 1 98 outlet outlet 1 95 outlet Table 3.4: Reynolds numbers for different boundary conditions of the geometries based on the inlet for MPM Geometry boundary conditions 98 outlet 1 98 outlet outlet 1 98 outlet outlet 1 95 outlet

35 Results 3.2 Circle of Willis 3.2 Circle of Willis General In figure 3.13 the end state of the the results of DPM on the Circle of Willis is displayed. The whole simulation can be found in.mpeg file called sequence-1 or ParticlesCoW. The development of each particle from injection to end state can be studied in this movie. From the figure only the final distribution on the outlets of the Circle of Willis can be derived. Since some particles are trapped in the same positions on the MCA-outlet, it s not clear to see that there are 6 particles trapped on the the MCA-outlet. Two particles are trapped on the ACA, and one on the PCA. Figure 3.13: Result of particle trajectory on the Circle of Willis using DPM Outlet diameter In figure 3.14 the percentage of the particles on the outlets is plotted. The blood flow through the PCA is an addition of the blood flow through the BA and the 33

36 Results 3.2 Circle of Willis blood flow through the PCoA. Since the particle trajectory of the particle found on the PCA contains the PCoA and not the BA, the contribution of blood flow from the BA is not considered. Therefore, instead of displaying the percentage of particles on the PCA, the percentage of particles through the PCoA is displayed. The diameters of the displayed outlets can be found in table 2.2 in subsection Figure 3.14: Percentage of particles as a function of diameter outlet of the Circle of Willis using DPM Bifurcation angle For the results of bifurcation angle, figure 3.14 can be used. The angles between the arteries are mentioned in subsection

37 Results 3.2 Circle of Willis Mass flux Figure 3.15: Percentage of particles as a function of mass flux through the outlets of the Circle of Willis using DPM The Reynolds number defined on the inlet of the right ICA is 769. simulation time is 1.09 seconds. The 35

38 Chapter 4 Discussion 4.1 Geometries Outlet diameter DPM The diameter of outlet 1 of geometry 2 is two times as small as the diameter of outlet 1 of geometry 1. In comparison of figure 3.2(a) and 3.2(b) it shows that, for all three boundary conditions, the percentage of particles on outlet 1 of geometry 2 is significantly less than the percentage of particles on outlet 1 of geometry 1. Since there is no difference in bifurcation angle, this indicates that a smaller diameter of an outlet is of significant influence on the particle distribution on the outlets for DPM. This is supported by comparing figure 3.3(a) with figure 3.3(b) where outlet 1 of geometry 4 traps a significant smaller percentage of particles than outlet 1 of geometry 5 for all pressures submitted on the outlets. It is obvious that if a smaller diameter of an outlet causes a smaller percentage of particles on this outlet, a larger diameter of an outlet causes a larger percentage of particles on this outlet. This can also be found by observing figure 3.3. MPM When comparing figure 3.4(a) with 3.4(b), it is seen that for geometry 2, the diameter outlet of outlet 1 has a large effect on the particle distribution, since the percentages of particles are significantly lower than the percentages on outlet 1 of geometry 1. Furthermore, when figure 3.5(a) is compared with 3.5(b), it can be seen that for the three boundary conditions, the percentage of particles on outlet 1 of geometry 5 is significantly higher than the percentage of particles on outlet 1 of geometry 4. The hypothesis that a smaller diameter of an outlet causes a smaller percentage of particles on the outlet is thus supported by the results of MPM. 36

39 Discussion 4.1 Geometries Bifurcation angle DPM Note that figure 3.6(a) is exactly the same as figure 3.6(b). The same amount of particles is found on the outlets of geometry 3 as on the outlets of geometry 1, while the bifurcation angles of the geometries differ. This result does not support the hypothesis that a larger bifurcation angle causes a smaller percentage of particles through the artery. The comparison between figure 3.7(a) and figure 3.7(b) is as follows. Outlet 1 of geometry 4 needs to be compared with outlet 2 of geometry 5, since these outlets both have a diameter of 2 mm, while the bifurcation angles differ. Outlet 2 of geometry 4 needs to be compared with outlet 1 of geometry 5, since these outlets have a diameter of 3 mm. This means that boundary condition 2 of geometry 4 needs to be compared with the boundary condition 3 of geometry 5 and the opposite. Now, it can be seen that the results for boundary condition 1 for both geometries and boundary condition 3 for geometry 4 and boundary condition 2 for geometry 5 are equal. The only indication that a different bifurcation angle might be of influence on the distribution of particles on the outlets can be found in the comparison between boundary condition 2 of geometry 4 and boundary condition 3 of geometry 5. Here it can be seen that more particles are found on outlet 1 of geometry 5, where the bifurcation angle is o, than on outlet 2 of geometry 4, where the bifurcation angle is o, and less particles on outlet 2 of geometry 5, where the bifurcation angle is o, than on outlet 1 of geometry 4, where the bifurcation angle is o. It is from these results, however, not clear that, for DPM, a smaller bifurcation angle causes a larger percentage of particles on the outlet. MPM From figure 3.8(a) and figure 3.8(b) it is observed that a higher percentage of particles, for all three boundary conditions, is found on outlet 1 of geometry 3, where the bifurcation angle is 30 o, than on outlet 1 of geometry 1, where the bifurcation angle is 40 o. Consequently, a lower percentage of particles is found on outlet 2 of geometry 3, where the bifurcation angle is 50 o, than on outlet 2 of geometry 1, where the bifurcation angle is 40 o. Furthermore, figures 3.9(a) and 3.9(b) show that a difference in bifurcation angle, for MPM, does cause a difference in particle distribution. There are significantly more particles found on outlet 1 of geometry 5 than on outlet 2 of geometry 4, when a pressure of 98 mm Hg is applied on both outlets. This is also the case for the other two boundary condition situations, when outlet 1 of geometry 4 is compared with outlet 2 of geometry 5 and vice-versa. It can be seen that when boundary condition 2 is applied on geometry 4, a larger amount of particles is found than on outlet 2 of geometry 5, when boundary condition 3 is applied. This is also the case for boundary condition 2 on geometry 5 compared with boundary condition 3 on geometry 4. Thus, the smaller the bifurcation angle, the larger the amount of particles. This result indicates that bifurcation angle is of significant influence on particle trajectories in MPM. 37

40 Discussion 4.1 Geometries Mass flux DPM From figure 3.10(a), it can be derived that when boundary condition 1 on geometry 1 is applied, this results in a nearly equal mass flux through both outlets. There is a slightly larger mass flux through outlet 2, which causes one particle more on this outlet than on outlet 1. The small difference in mass flux is probably due to the mesh, which is unstructured and thus not totally symmetrical. From this figure, it can also be derived that a pressure of 95 mm Hg on an outlet causes a larger mass flux through this outlet and consequently a smaller mass flux through the other outlet where a pressure of 98 mm Hg is applied. As a result, a larger amount of particles is found on the outlet where the mass flux is higher. This indicates that there exists a linear relation between mass flux and particle percentage. Since the Reynolds number is a measure for the mass flux through the inlet of the flow, in table 3.3, for geometry 1, it can be seen that the different boundary conditions have an obvious effect on the total mass flux of the flow. When a pressure of 95 mm Hg is applied on one on the outlets, and 98 mm Hg on the other, this causes a higher total mass flux of the flow than when a pressure of 98 mm Hg on both outlets is applied. When these Reynolds numbers are compared with the time of the simulations found in table 3.1, it can be seen that a higher mass flux through the flow causes a higher particle velocity for geometry 1, since the travel time of the particles is shorter than the travel time of the particles when a lower Reynolds number is found. In figure 3.10(b) it can be seen that the mass flux through outlet 1 of geometry 2, where the diameter is 1.5 mm, is for every boundary condition less than the mass flux through outlet 1 of geometry 1, where the diameter is 3 mm. This results in a small amount of percentages of particles on outlet 1, even 0 when a pressure of 95 mm Hg is applied on outlet 2. The mass flux through outlet 2 of geometry 2 does not differ much from the mass flux through outlet 2 of geometry 1. This means that the total mass flux of the flow through geometry 2 is lower than the total mass flux of the flow through geometry 1. This indicates that there exists a linear relation between mass flux and outlet diameter. The above can be verified by observing the Reynolds numbers in table 3.3. As expected, since total mass flux is lower, the Reynolds number of the flow through geometry 2 is lower than the Reynolds number of the flow through geometry 1. Also, when a pressure of 95 mm Hg is applied to outlet 1 of geometry 2, it has a smaller effect on the mass flux through the flow than when a pressure of 95 mm Hg is applied to outlet 2. These results support the hypothesis that a smaller diameter of an outlet causes less mass flux, and consequently less particles, to flow through this outlet. It can be seen in table 3.1 that the simulation times between the first two boundary conditions situations are nearly equal. This is as expected, since the corresponding Reynolds numbers do not differ much. When the Reynolds number increases in boundary condition 3, the time for the particles to be trapped on the outlet decreases. There is, however, a large difference between simulation times of geometry 1 and geometry 2. The simulation time of equal pressures of geometry 2 is lower 38

41 Discussion 4.1 Geometries than the simulation time of equal pressures of geometry 1, while the Reynolds number of the flow through geometry 2 is smaller than the Reynolds number of the flow through geometry 1. The opposite is expected. It is not known why this is the case. Apparently, for different geometries, different relations between Reynolds number and duration of simulation apply. A remarkable result is found in figure 3.10(c). For boundary condition 1, the mass flux through outlet 1 is slightly higher than the mass flux through outlet 2, as expected by bifurcation angle difference. On outlet 1, however, a lower amount of particles is found than on outlet 2. This indicates that the interaction between continuous phase and discrete phase can cause unexpected results in DPM. This result can be explained by a momentum transfer of fluid to the particles, which causes a decrease in mass flux of the flow, while the velocity of the particles increases. Since the difference in mass flux between the outlets is extremely small, 3.2e-5 kg/s, which is 0.12 % of the mass flux through the inlet, it is assumed that in DPM, this unexpected mass flux difference does not occur often. The results of the other boundary conditions is as expected. When figure 3.10(c) is compared with figure 3.10(a), it is seen that the mass fluxes are smaller for geometry 3 than for geometry 1, in all three boundary conditions. This is not due to the difference in bifurcation angle, as discussed in the previous section, but to the diameter of 2.5 mm of the outlets of geometry 3, instead of 3 mm for the outlets of geometry 1. Although the Reynolds numbers are smaller than the Reynolds numbers of geometry 1, they show a similar pattern. It is possible that the smaller Reynolds number of geometry 3, in boundary condition 3, in comparison with the Reynolds number of boundary condition 2, is caused by the larger bifurcation angle of outlet 2. This can, however, not be known for certain, since the Reynolds number for this simulation in geometry 1 is also smaller. In table 3.1 a similar result for geometry 3 as for geometry 1 can be found. The simulation time when equal pressures on both outlets are applied, however, is extremely long. This is due to one particle that interacted with the wall of the geometry. It probably lost some of its velocity during this interaction and therefore it took longer to travel to the outlet. After the interaction the velocity of the particle increases, which means that momentum of the fluid is transferred onto the particle, causing less mass flux on the outlet. In figure 3.10(d) it can be seen that the mass flux through outlet 1, where the diameter is 2 mm and the bifurcation angle o, for the three circumstances (see table 2.3), is smaller than the mass flux through outlet 2, where the diameter is 3 mm and the bifurcation angle is o. This indicates that the diameter of the outlet is of larger influence on the mass flux than the bifurcation angle, which means, as seen before, that the outlet with the larger diameter traps more particles than the outlet with the smaller diameter, although the bifurcation angle of the latter is smaller than the former. As a consequence the Reynolds number, in boundary condition 2, is larger than the Reynolds number when both pressures are equal, but is smaller than the Reynolds number of boundary condition 3, as can be seen in table 3.3. The simulation times for geometry 4 in table 3.1 are not in correspondence with the Reynolds numbers of geometry 4. If it is assumed that the times of boundary condition 1 and boundary condition 3 are correct, than a simulation time of approximately 0.45 seconds is expected for the simulation with boundary condition 2. There is no suitable explanation for this. 39

42 Discussion 4.1 Geometries As expected, the mass flux through outlet 1 of geometry 5, where the diameter is 3 mm is, for all three boundary conditions, higher than the mass flux through outlet 2, where the diameter is 2 mm, as can be seen in figure 3.10(e). Since the mass fluxes through geometry 5 are of approximately the same size as the mass fluxes through geometry 4, Reynolds numbers are of same size as well, as can be seen from 3.3. For the simulation time, the same problem arises as for geometry 4. The simulation time of geometry 5 with boundary condition 3, is expected to be lower than the time of boundary condition 1, and higher than the time of boundary condition 3. Here, however, the long simulation time can be explained by a particle losing velocity while interacting with the wall. MPM Note in figure 3.11(a) that for boundary condition 1, on outlet 1 a higher percentage of particles is found, while the mass flux is lower than on outlet 2, where the mass flux is higher, as is seen before in the previous section. This can only be explained by assuming that momentum of the fluid is imposed on the particles that travel towards an outlet, resulting in a lower mass flux through this outlet. The mass flux difference of the outlets is e-5 kg/s, which is 1.8 % of the mass flux through the inlet. This particular result, obviously, causes a decrease in linearity in the graph. The results of boundary conditions 2 and 3 are as expected: a higher mass flux causes a higher percentage of particles, a lower mass flux causes a lower percentage. In table 3.4 a similar pattern for the Reynolds numbers in MPM is found as in DPM. The Reynolds numbers for boundary conditions 2 and 3 are higher than the Reynolds number for boundary condition 1. Furthermore, the former two Reynolds numbers do not differ significantly, as expected. In table 3.2 it can be seen that the pattern for simulation time in MPM is not as obvious as it is in DPM. This is due to the mutual momentum interaction between the particles and the flow. Since, for a particular flow, the exact momentum transfers are not known, it is not possible to verify this effect on simulation time. The results presented in figure 3.11(b) show that mass fluxes through outlets with small diameters are small for the three boundary conditions, and subsequently large through the other outlet, as expected and discussed in the previous section. The Reynolds numbers show the same pattern as for DPM, where boundary condition 3 causes a larger mass flux through the geometry than boundary condition 2. The time of the simulation does not correspond with the Reynolds numbers. Here, mutual momentum interaction is assumed to be the cause as well. In figure 3.11(c) it is shown that for boundary condition 1, a lower mass flux through outlet 1 causes a higher particle percentage, where the opposite is expected. The momentum of the fluid has a large effect on the particles, as described earlier. The difference between mass fluxes is 9.5e-5 kg/s, which is 3.95 % of the mass flux through the inlet. The other boundary conditions produce the expected results. There is a slightly higher Reynolds number of geometry 4 when boundary condition 3 is applied, than when boundary condition 2 is applied. The opposite is expected. It can be explained by the fluid transferring more momentum to the particles in outlet 2 for boundary condition 3, than fluid transferring momentum 40