Blood Flow in Curved Blood Vessels on a Cellular Level

(1)

Bachelor Informatica

Blood Flow in Curved Blood

Vessels on a Cellular Level

Vuong Ngo

June 17, 2019

Supervisor(s): dr. G´abor Z´avodszky

Inf

orma

tica

—

Universiteit

v

an

Ams

terd

am

(2)

(3)

Abstract

A necessary component for sustaining the human body is the circulatory system. The circulatory system is an organ system that permits the circulation of blood and transports substances within the body. Blood is a suspension of a number of different particles such as cells, cell fragments and macromolecules. The amount of each particle affects the behaviour and flow properties of the blood suspension. Blood rheology is the area of research that studies the flow properties of blood. With the addition the modern disciple computational fluid dynamics, a plethora of cases in blood rheology can be researched and evaluated, especially with the rising performance of computers. One such case is the blood flow in the blood vessels of a capillary. Most researches consider a straight vessel section with Poiseuille flow. This has been a standard in blood rheology research in blood vessels. However, most blood vessels in a capillary are curved instead of straight. Therefore, it is more interesting to research the blood flow in a curved vessel and how exactly the curvature of the vessel effects certain aspects of the blood. This thesis researches a blood flow in a curved vessel section of the capillary on a cellular level, as the capillary vessels are among the smallest vessels in the body. In this research, the focus is on the blood viscosity of the blood flow. The blood viscosity in a curved vessel is measured and calculated through simulations and compared to the blood viscosity of a blood flow in a straight vessel calculated with the Poiseuille equation.

(4)

(5)

Introduction

1.1 Context

A necessary system for sustaining the human body is the circulatory system. The essential com-ponents of the circulatory system are the heart, blood and blood vessels. This system permits blood flow through out the body. Blood flow is peculiar as its properties are dependent on a plethora of factors. This means that blood flow and its properties are highly subject to modeling, evaluation and research. Blood rheology for that matter is the study of blood flow that has been developed for decades and continues to be a highly active field of research [6].

Fluid dynamics, like blood rheology, is an active field of research. It focuses on the effects of forces on fluid motion. It typically is quite mathematically complex and as such has many problems partly or fully unsolved. These problems are best addressed by numerical methods, commonly utilizing computers. Fortunately, a modern disciple devoted to this approach of using computers is computational fluid dynamics. Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that utilizes numerical analysis and data structures to analyze and solve math-ematical problems involving fluid motion. It is an important addition to blood rheology as it allows researchers to observe and evaluate the blood flow via computational performance and simulations to achieve a better understanding of its properties and behaviour. With the increas-ing performance of computers, simulations that requires substantial computation, such as blood flow with particles through a blood vessel, can be simulated more viably.

Many of the peculiar properties of blood come from its cellular nature [12]. It is therefore important to model the blood flow on the cellular level in order to improve the accuracy. Hemo-Cell is a computation framework that has been developed for modeling the flow of blood on a cellular level [12]. It is written in C++. Conveniently, HemoCell models particles that are found in blood such as red blood cells (RBC’s) and platelets. By utilisation of the Hemocell framework, a blood vessel can be simulated on a cellular level and it will thus form the basis of this thesis on which the blood flow simulations are implemented.

A particularly important property of blood, is its viscosity. Blood viscosity is the measure-ment of the thickness of the blood. It affects the magnitudes of a lot of other phenomena in the body, such as the degree to which the heart must work and the quantity of oxygen transferred to tissues and organs. It is a direct measurement of the ability to flow of blood [7].

This thesis will research the viscosity of blood in a blood vessel. However, in this thesis a curved vessel section of a capillary is considered. Capillaries are the smallest vessels in the circu-latory system that convey blood between the arterioles and venules. Microvessels from sections in a capillary are suitable for evaluation since their size allow better managability and oversight in simulations or experiments. The motivation of considering a curved vessel section instead of a straight vessel section is that blood viscosity is almost exclusively researched in straight vessel

(8)

sections. For example, a prior research that considers a straight vessel section with Poiseuille flow is [9]. Since then, it has been the standard of blood rheology in vessels. Thus, it is interesting to simulate and evaluate the blood flow in a curved vessel section and see how its blood viscosity compares to the blood viscosity in straight vessel sections.

Thus, in this thesis blood flow in a curved vessel section is modeled and simulated on a cellular level. The blood viscosity is calculated and evaluated. The research question of the thesis is: How does the curved vessel section affect the blood viscosity in comparison to a straight vessel section.

The structure of this thesis is as follows. In chapter 2 relevant theoretical background information for this thesis, such as the constituents of blood, fluid dynamics and the Hemocell framework, is addressed and elaborated on. In chapter 3 the implementation of the simulation is described and visualized. Chapter 4 is where the results of the experiment are examined and visualized. At last, the choices made for this thesis is elaborated on and future work suggestions are discussed, as well as certain limitations of this thesis.

(9)

CHAPTER 2

Theoretical background

Relevant background information for this thesis will be addressed in this chapter. First a basic understanding of the constituents of blood and blood viscosity will be provided. Also, aspects of fluid dynamics are elaborated on. Furthermore, the HemoCell framework will be briefly explained.

2.1 Blood

Blood is the fluid which flows in the circulatory system. However, it is not exactly a pure fluid but rather a suspension of a number of different particles, such as cells, in a fluid base [6]. Since it is a rather complex suspension of particles instead of a pure fluid, it has fluid behaviour and properties that are dependent on those particles. Some relevant components of blood and blood viscosity are addressed in the following sections.

2.1.1 Blood Plasma

Blood plasma is the fluid base that makes up the liquid component of blood in which the cells and other constituents of blood are suspended. It makes up about 55% of the body’s total blood volume. It is a straw coloured fluid that is composed of 90% water , 1% electrolytes and various molecules making up the remainder [6].

(10)

2.1.2 Red Blood Cells

The principle particle in blood is the red blood cell (RBC) or erythrocyte. The volume percentage of RBC’s in blood is called the hematocrit. For example, if a blood volume of 100 mm3 _is

considered, a hematocrit of 30% means that there is a RBC volume of 30 mm3_{. The hematocrit}

has a typical range of 41-52 % in men and 36-48 % in women [6]. The RBC is unique as it has no nucleus and has a biconcave shape, unlike other cells. Human RBC’s have a diameter of 7.5 µm and a thickness of 2 µm as shown in figure 2.1 [6].

Figure 2.1: The shape and dimensions of a RBC. from; [6]

The rheological properties of RBC’s are among the key factors for the flow resistance of blood in microvessels such as capillaries. Because of their shape and large surface area, RBC’s are easily deformed by external forces. Some diseases ,such as diabetus mellitus and sickle cell anemia, alter the mechanical properties of a RBC [8]. This results in a reduction of deformability of a RBC, which leads to an increased flow resistance of blood [3].

2.1.3 Blood Viscosity

Blood viscosity is the thickness of blood, as mentioned before. This viscosity has consequences on numerous mechanics in the body, so it is important to see what factors determines the viscosity of blood. There are five primary of factors that determine blood viscosity [7].

Hematocrit

The hematocrit is an evident determinant of blood viscosity. The higher the hematocrit, the higher the viscosity of the blood. Hematocrit accounts for approximately 50% of the difference between normal blood viscosity and high blood viscosity [7].

Eryhtrocyte Deformability

Erythrocyte deformability refers to the ability of RBC’s to change shape. RBC’s elongate at high velocity and they can bend and fold in order to pass through small vessels such as capillaries. More flexible RBC’s result in less viscous blood, while a reduction in deformability of RBC’s results in more viscous blood [7].

Plasma Viscosity

Plasma Viscosity refers to the thickness of the fluid component of blood. It is highly affected by hydration and plasma proteins [7].

Erythrocyte Aggregation

Erythrocyte aggregation refers to the tendency of RBC’s to be attracted to each other and stick together. Both plasma proteins and erythrocyte deformability play a role in erythrocyte aggregation [7].

(11)

Temperature

Blood flows more easily at higher temperatures, as with most fluids. It is approximated that a 1°C increase in temperature results in a 2 % decrease in blood viscosity [10].

2.2 Fluid Dynamics

Fluid dynamics is the branch of fluid mechanics that studies the effect of forces on fluid motion. In this section relevant aspects of fluid dynamics are considered.

2.2.1 Viscosity

The viscosity of a fluid (µ) is a measure of its resistance to deformation under a force. This corresponds to the informal concept of ”thickness” for liquids. In a simple example, water has a low viscosity, while honey and tar have a high viscosity. Thus, viscosity as a concept is described as quantifying the frictional force that arises between adjacent layers of fluid that are in relative motion. The SI unit for kinematic viscosity is the (m2/s).

2.2.2 Poisseuile Flow

In fluid dynamics, the Poiseuille equation is a physical law that provides the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a cylindrical pipe of constant cross section. Essentially, it gives the difference in pressure between the two ends of a pipe. It has application for numerous cases, such as liquid flow through a drinking straw or blood flow in a blood vessel. The assumptions of the equation are that the fluid is incompressible and Newtonian, the laminar flow is through a cylindrical pipe that is substantially longer than its diameter and there is no fluid acceleration in the pipe. The original form of this equation is

∆P =8µLQ

πR4 , (2.1)

where ∆P is the pressure drop (N

m3), µ is the viscosity of the fluid (

m2

s ), L is the length of

the pipe (m), Q is the volumetric flow rate (m_s3) and R is the radius of the pipe (m).

In this thesis, the pressure drop of the simulation is compared to the calculated pressure drop of Poisseuile flow. For this, the original equation 2.1 needs to be rewritten. The average velocity of the fluid needs to be calculated in order to calculate the pressure drop. This is done in a few steps. First, the equation of the Poiseuille flow in 2.1 is rewritten to

v = − 1 4µ ∗

∆P L ∗ (R

2_{− r}2_), _(2.2)

where v is the fluid velocity in the direction of the pipe axis and r is the distance of the point to the center of the pipe. v is dependent on what r is picked, for example in the center of the pipe r = 0 or at the wall r = R.

Then, to calculate the maximum value of the fluid velocity vmax, the following equation is

used, which is derived from equation 2.2:

vmax= − 1 4µ ∗ ∆P L ∗ (R 2_). _(2.3)

After that, the mean of fluid velocity ¯v is then calculated with

¯

v = vmax

2 . (2.4)

Finally, the pressure drop between the two ends of the pipe is calculated with

∆P =8µL ∗ ¯v

(12)

2.2.3 Wall Shear Stress

In a fluid flow in a pipe, fluid at one point does not move with the same speed as every other point in the pipe. Fluid flows slowest near the wall and fastest in the center. This is due to the friction of the fluid with the pipe wall. This friction is referred to as the wall shear stress [11] and is a shear force. The SI unit for shear stress is Pa or _m∗skg2. The equation for the wall shear stress is given by

τw=

R∆P

2L . (2.6)

This equation, with help of equation 2.5, can be rewritten to

τw=

4µ¯v

R . (2.7)

2.2.4 Body Force

A body force (Fbody) is a force that is exerted throughout a volume of a body, contrary to surface

forces such as shear forces, which are only exerted at the surface of a body. An example of a body force is gravity. The SI unit of body force is _mN3. The body force can be calculated by the dividing the total force (Ftot) by the volume of the body. The total force in this research can be

calculated with

Ftot= 2 ∗ R ∗ π ∗ L ∗ τw. (2.8)

The volume of a cylinder is used as the volume in the body force equation. The body force equation is then defined as

Fbody = Ftot volume = 2 ∗ R ∗ π ∗ L R2_{∗ π ∗ L} ∗ τw= 2 ∗ τw R , (2.9)

which can be rewritten to following formula using equation 2.7:

Fbody=

8 ∗ µ ∗ ¯v

R2 . (2.10)

2.2.5 Reynolds Number

The Reynolds number (Re) is a valuable dimensionless quantity in fluid mechanics that is used to help estimate the flow patterns in different fluid flow settings. It is the ratio of intertial to viscious forces. Inertial force is simply the force due to the momentum of the fluid. The higher the density and velocity is of a fluid , the higher the momentum or inertia. A force that inhibits this inertial force is shear stress, which is a viscous force. Thus, the Reynolds number can be defined as

Re = uL

ν , (2.11)

where u is the velocity of the fluid relative to the object (m/s), L is the diameter of the pipe (m) and ν is the kinematic viscosity of the fluid (m2/s). Kinematic viscosity is the ratio between the viscosity of fluid µ and the density of the fluid p.

The Reynolds number has a wide range of applications, from the air passage over an aircraft wing to fluid flow in a pipe. Therefore, it is adequate for application in the blood flow in a vessel, since that is comparable to a fluid flow in a pipe.

Effects due to inertia of the fluid are most common in larger vessels. These effects are rele-vant at high Reynolds numbers. Effects relerele-vant at low Reynolds numbers, where viscous forces are the dominant forces, are more common in microcirculations [6]. In this thesis curved sections in a capillary are considered, as such, low Reynolds numbers are to be used, since capillaries are microcirculations.

(13)

2.2.6 Lattice Boltzmann Method

Conventionally, the Navier-Stokes equations were used to model the behaviour of fluids. The Navier-Stokes equations describe the motion of viscous fluids at macroscopic level. However, the lattice Boltzmann method has shown to be a promising method of simulating fluids in the last decades.

The lattice Boltzmann method (LBM) is a parallel and efficient algorithm for simulating fluid flows [2]. The LBM approach is advantageous for parallel computations due to its local dynam-ics and relative ease of application on systems with complex boundaries [13]. The fundamental concept of the LBM is to create simplified models that incorporate the essential physics of micro-scopic or mesocopic processes so that the macromicro-scopic properties correspond with their respective macroscopic equations. The premise for utilizing these simple kinetic-type methods for macro-scopic fluid flows is that the macromacro-scopic dynamics of a fluid are the result of the collective behaviour of the microscopic particles in the system and that the macroscopic dynamics are insensitive to underlying details of microscopic physics [2].

Essentially, the lattice Boltzmann method is derived from the kinetic theory of gases of Lud-wig Boltzmann, where fundamentally the idea is that fluids can be imagened as consisting of numerous small particles moving with random motions. The exchange of momentum and energy is accomplished through particle streaming and billiard-like particle collision [1]. This is modeled by the Boltzmann transport equation, which is defined as

Ω =∂f

∂t + ~u ∗ ∇f, (2.12)

where f (~x, t) is the particle distribution function, ~u is the particle velocity and Ω is the collision operator [1]. The lattice Boltzmann method further simplifies this idea by reducing the number of particles and confining them to the nodes of a lattice. In a two dimensional LBM model, a particle, that is inhibited in a node of the lattice, is limited to stream in 9 different directions or nodes, including the node it is currently at. These 9 velocities are referred to as microscopic velocities, denoted by ~ei, where i = 0..8 [1]. If a particle is at node (x, y) then the

microscopic velocities ~ei can be defined by

~ ei =                                    (x, y), if i = 0 (x + 1, y), if i = 1 (x, y + 1), if i = 2 (x − 1, y), if i = 3 (x, y − 1), if i = 4 (x + 1, y + 1), if i = 5 (x − 1, y + 1), if i = 6 (x − 1, y − 1), if i = 7 (x + 1, y − 1), if i = 8 (2.13)

This model is known as the D2Q9 model, as it is two dimensional and involves 9 velocity vectors. A typical D2Q9 model is illustrated in figure 2.2.

(14)

~

e

0

~

e

7

~

e

4

~

e

8

~

e

1

~

e

5

~

e

2

~

e

6

~

e

3

Figure 2.2: Schematic overview of the D2Q9 model. As illustrated, a particle can move in 9 directions, including the node it is currently resting at.

For each particle on the lattice, a discrete probability distribution function f (~x, ~ei, t), i = 0..8

is associated to it, which describes the probability of a particle streaming in a certain direction. Afterwards, the macroscopic density and velocity are calculated. The essential steps in the lattice Boltzmann method are the streaming and collision processes. The streaming and collision models are computed separately and are carefully attended when dealing with boundary nodes [1], which in case of a blood vessel represents the vessel wall. By repeating the streaming and collision processes, a motion of fluid can be modeled.

(15)

2.3 HemoCell

In this section relevant information about HemoCell is addressed.

2.3.1 Blood Component Models

The constituents of blood, such as blood plasma and particles, are modeled in HemoCell. In this section the modeling of blood plasma, RBC’s and blood vessels are briefly presented.

Plasma

In HemoCell blood plasma is modeled as an incompressible Newtonian fluid using the lattice Boltzmann method implemented in the Palabos library, which has proven to be competent at creating accurate flow results in vascular settings [12].

Red Blood Cells

For RBC’s, HemoCell has two different material models. One is the course-grained spectrin-link model of [5]. This model however has several shortcomings, so HemoCell now has its own material model of a RBC [12]. The surfaces of RBC’s are described as boundary layers immersed into the blood plasma. The discretization of these layers is achieved by using Nv vertices connected by

Ne edges and holding Ntsurface triangles, as visualised in figure 2.3 [12].

Figure 2.3: Visualisation of the membrane meshes of a RBC in HemoCell. from; [12]

The model used in HemoCell allows the RBC’s to deform when forces are exerted on them and the model is able to handle strong deformations [12].

Blood Vessels

A blood vessel section, which is the simulation domain, can be specified in multiple ways in HemoCell. For simple cases, such as a blood flow in one straight direction, one could simply create a cube with one periodic direction. However, a STL file can be loaded for more complex cases where more complex geometry is simulated, such as a curved blood vessel. A STL file describes only the surface geometry of a three dimensional object without texture, color or other properties. An example of a pipe STL file is shown in figure 2.4.

(16)

Figure 2.4: Visualization of a pipe STL file.

In HemoCell the terms inlet and outlet are used. The inlet is simply the opening in which the fluid flows in the simulation domain. The outlet is the opening where the fluid flows out of the simulation domain. This is illustrated in figure 2.5.

Figure 2.5: Visualization of a pipe with its inlet and outlet.

2.3.2 Simulation Process in HemoCell

When simulating a case in HemoCell, such as a fluid flow in a pipe, a series of actions need to be taken to achieve the simulation. The simulation program is a file, which is written in C. In this file, the necessary header files, which are provided by the Hemocell framework, are included. The main function is defined, as it is the main program. The simulation process in the main program can be structured into sections.

Creating a Configuration File

The configuration file is a XML file to be passed as an argument to the main program when running the simulation. The main program will check for this configuration file, otherwise the simulation will abort. Important variables that are used in the simulation, such as Reynolds number and fluid density, are saved in this file.

Simulation Domain & Fluid Setup

The simulation domain, where the blood flow simulation takes place, needs to be implemented. The geometry of this domain can be specified via a STL file. The location of this STL file is specified in the configuration file. The STL file is voxelized in order to get a voxelized domain.

(17)

Figure 2.6: Visualization of voxelized geometry. In this case, a pipe is voxelized.

As shown in figure 2.6, the geometry is voxelized into atomic blocks. Afterwards, the fluid is set-up via the Palabos library that implements the lattice Boltzmann method to model the fluid. A fluid lattice will be initialized in HemoCell. The lattice units have a length which is specified as dx in the configuration file and is equal to 5 ∗ 10−7 m or 0.5 µm. The warm-up parameter is defined in the configuration file as the amount of LBM iterations to prepare the fluid field. The rest of the parameters for the lattice Boltzmann method simulation are read from the configuration file. Then the boundaries in the Palabos fluid field are set-up and the periodicity of the fluid flow is defined. Furthermore, the inlet and outlet of the simulation domain are specified.

Adding Red Blood Cells

The RBC’s are generated in the simulation, with their locations and rotations in the simulation specified in a file. This file can be generated by a script in HemoCell. This script is run with arguments such as domain size and hematocrit. The domain size is the size of the domain where the cells are generated and specified as 3 values that represent the x, y and z axis. The result of running the script is a POS file. The contents of an example POS file is shown in 2.7.

2

1 2 . 2 9 5 7 . 0 1 5 9 2 1 9 . 6 3 3 6 −72.4841 −9.34123 3 6 . 2 3 8 0 . 8 0 3 7 7 2 8 . 7 3 0 1 7 3 1 . 9 2 5 2 −28.0439 5 . 6 4 1 0 2 −25.1471

Figure 2.7: A POS file that specifies 2 cells with their respective location and rotation values on each line.

The first line in the POS file represent the amount of cells to be generated. Each line after the first line represent a cell. The first 3 values of these lines represent the coordinates of cell and the remaining 3 lines represent the rotation of the cell.

When RBC’s are to be generated in a simulation domain, the simulation will read the data of the RBC’s from the POS file and generate them accordingly in the specified simulation do-main. If the coordinates of the POS file aren’t within the domain size of the simulation domain then the RBC’s aren’t generated. Thus, it is essential to specify the correct domain size values when passing them as arguments to the script.

(18)

Running The Simulation

The simulation is run for an amount of time steps. The amount of time steps is specified as tmax in the configuration file. The time step is specified as dt in the configuration file and is equal to 1 ∗ 10−7 s or 0.1 µs. In each time step, the statistics of the fluid, such as velocity and force are calculated and displayed. The location and rotation of each cell is calculated and the statisics of the cells, such as the number of cells, are also displayed. After a specified interval, named tmeas in the configuration file, the data of cells of the current time step are written into a CSV file. The CSV file contains data such as the coordinates of the cell, the cell ID and the velocity of the cell. A blood flow simulation in a straight vessel is shown in figure 2.8.

(19)

CHAPTER 3

Implementation

In order to properly simulate blood flow in a curved vessel, considerations are made on the implementation. This chapter discusses the implementation of the blood flow simulation in a curved vessel and clarifies some choices made for implementation.

3.1 Simulation Domain

The first considered component of the implementation is the simulation domain. Since a curved vessel section of a capillary is simulated, the simulation domain has to have a geometry that matches a curved vessel. The curved vessel STL file that specifies this geometry is based on the straight tube STL file in figure 2.4, but it has been extended with a circular tube. The curved vessel STL file is shown in figure 3.1.

Figure 3.1: Visualization of the curved vessel STL file.

This STL file will be read by the simulation and voxelized. The diameter of the voxelized vessel contains 40 voxels and since a voxel has a length of 0.5 µm, the diameter is equal to 20 µm. The boundary walls, inlet and outlet are then defined. However, there is a problem with this simulation domain if it were to be the sole simulated environment, because of the simulation process of HemoCell. The RBC’s are generated in the domain according to the location and rotation values in the POS file as previously shown in figure 2.7. This means that if the simulation were to only simulate the simulation domain, the red blood cells would be generated in the simulation domain. The purpose of the simulation is to simulate a blood flow in the curved vessel, so the RBC’s need to flow into the simulation domain, rather than be generated into it without velocity. In order to let blood flow into the simulation domain, a preinlet is implemented.

(20)

3.2 Preinlet

The preinlet is the domain where blood flows prior to the inlet of the simulation domain, hence the name preinlet. The purpose of the preinlet is to be the initial environment, where RBC’s can be generated and to supply the simulation domain with constant fluid flow and RBC’s. The preinlet achieves a steady flow of the fluid and RBC’s by being periodic. What flows out of the preinlet, flows back in the other opening, which means that there is a continuous fluid flow in the preinlet. When RBC’s reach the outlet of the preinlet, they are deleted after being copied over to the inlet of the simulation domain and the inlet of the preinlet. This results in the preinlet constantly supplying itself and the simulation domain with blood. The blood flow simulation with the preinlet is schematically illustrated in figure 3.2.

Preinlet Simulation

domain

Blood flow

Outletsim Inletsim Outletpre Inletpre

Figure 3.2: Schematic overview of the blood flow simulation.

It should be noted that RBC’s that reach the outlet of the simulation domain are also deleted. Since there is a steady blood flow flowing into the simulation domain, the simulation setup, extended with the preinlet, is complete and adequate for simulation and experimentation. The simulation setup is shown in figure 3.3.

Figure 3.3: Visualization of the complete simulation setup for blood flow in curved vessels. The green and red areas represent the inlets and outlet respectively. Blood flows from the preinlet to the simulation domain. (a) Preinlet. (b) Simulation domain.

(21)

3.3 Blood

Since the simulation is set-up as shown in figure 3.3, the blood in the preinlet will flow in the negative x direction. The Reynolds number, defined in the configuration file, will be used to predict the velocity of the fluid, in this case blood, for the simulation. The body force of the fluid is calculated in this process. In earlier iterations of the simulation, the default Reynolds number was 0.5, but it had proven to be too high, since the RBC’s were getting flattened by the velocity of the blood flow. This phenomenon of highly deformed and flattened RBC’s is shown in figure 3.4 .

Figure 3.4: Flattened RBC’s caused by high velocity of the blood that is determined by an inadequately high Reynolds number.

The Reynolds number has been adjusted to 0.1, since the blood vessel sections in a capillary are smaller than other blood vessels and thus, the Reynolds number needed to be decreased. The RBC’s in the blood are generated by a HemoCell script and the output POS file is read in by the simulation. A hematocrit of 40% will be used in simulations that include RBC’s as it is within the typical range of hematocrit in humans.

3.4 Simulation Parameters

For this research, two different simulations will be conducted. Both simulations simulate a blood flow in a curved vessel section and both will be using the same parameters . The difference between the two simulations is that the first simulation will be run with RBC’s, while the second simulation will be run without RBC’s. These simulations are conducted in order to observe how RBC’s affect the viscosity of the blood in the curved vessel section. Relevant parameters of the simulations are shown in the in table 3.1.

Parameter Value Description

tmax 500,000 Total amount of iterations for the simulation to run. tmeas 500 Iteration interval after which data is written.

rhoP 1025 Density of the fluid, in this case blood plasma (kg/m3_).

nuP 1 ∗ 10−6 Viscosity of the fluid, in this case blood plasma (m2/s). dx 5 ∗ 10−7 Length of 1 lattice unit (m).

dt 1 ∗ 10−7 Duration of 1 iteration time step (s). Re 0.1 Reynolds number.

(22)

3.5 Simulation With RBC’s

The first simulation will be simulating a blood flow in a curved vessel section of a capillary. In this simulation, the aforementioned hematocrit of 40% will be used to generate the amount of RBC’s. The simulation will run for 500, 000 iterations or time steps, which is equal to 50 ms. The simulation is run in parallel on multiple cores to speed up the process. The resulting simulation is shown in 3.5.

(a) Time step 0 (0 ms). (b) Time step 87,500 (8.75 ms).

(c) Time step 250,000 (25 ms). (d) Time step 500,000 (50 ms).

Figure 3.5: The curved vessel blood flow simulation with RBC’s at different time steps.

The RBC’s are first generated in the preinlet as can be observed in figure 3.3. As the simulation continues, the RBC’s experience the force of the fluid flow exerted on them and stream in the direction of the fluid flow. As the RBC’s flow into the simulation domain, they are copied over to the inlet of the preinlet to achieve a steady flow of blood. When a RBC reaches the outlet of the simulation domain, they are deleted permanently, as stated in previous sections.

(23)

3.6 Simulation Without RBC’s

The second simulation simulates a blood flow in a curved vessel section of a capillary without the RBC’s. The hematocrit of this simulation is consequently 0%. Since there are no RBC’s, the simulation will look exactly the same in every time step, as shown in figure 3.6.

(24)

(25)

CHAPTER 4

Results

With the execution of the two simulations, the resulting data can be evaluated in order to achieve the conclusion for the thesis.

4.1 Blood Velocity Evaluation

The average velocity ¯v of the blood in the preinlet in the simulation with RBC’s can be measured in the simulation. It can be compared to a theoretical velocity, denoted ¯vt. In theory, ¯v = ¯vtif

there are no RBC’s, since there are no RBC’s to affect the blood flow. Consequently, if there are RBC’s, then ¯v < ¯vt. The theoretical velocity of the blood will be calculated based on the

Reynolds number. The Reynolds number equation 2.11 is recalled and filled in with parameters from the simulation. The resulting equation is

Re = v¯t∗ L νp

, (4.1)

where L is the diameter of the pipe and νp the viscosity of the blood plasma. To calculate

the theoretical velocity ¯vt, the equation is rewritten to

¯ vt=

Re ∗ ν

L . (4.2)

The Reynolds number and blood plasma viscosity are taken from the configuration file that is used in the simulations, as displayed in table 3.1. The Reynolds number is equal to 0.1 and the blood plasma viscosity is equal to 1 ∗ 10−6m2/s. The diameter of the blood vessel is the amount of voxels in the diameter multiplied by the voxel size dx. There are 40 voxels in the diameter and dx equals 0.5 µm, therefore, the diameter equals 20 µm or 2 ∗ 10−5 m. The theoretical velocity of the blood is then calculated using equation 4.2:

¯ vt=

0.1 ∗ (1 ∗ 10−6)

2 ∗ 10−5 = 5 ∗ 10

−3_m/s. _(4.3)

The average velocity ¯v of the simulation is plotted over the first 2 milliseconds of the simu-lation, since the velocity stabilizes during that duration. It is also compared to the theoretical velocity plot. This is shown in figure 4.1.

(26)

Figure 4.1: ¯v and ¯vtof both simulations compared.

As can be seen in the plot, the average velocity rises substantially before it reachs a stationary state. This beginning is a startup phenomenon and is attributed to the fact that the simulation takes some time before it reaches a steady flow. The maximum average velocity ¯v is measured at 2.6 ∗ 10−3m/s. As can be seen, 2.6 ∗ 10−3< 5 ∗ 10−3, therefore the theory of ¯v < ¯vtis satisfied.

This result can be explained by the fact that the RBC’s increase the the viscosity of the blood, which limits the blood flow at low shear rates. This is due to the aggregation of RBC’s that is prevalent at low shear rates, such as in capillary vessels, while at high shear rates the RBC’s deform to optimally adapt to the blood flow conditions [4].

4.2 Pressure Drop Evaluation

The pressure drop of the simulation with RBC’s, denoted ∆Prbc, is the difference in pressure

between the inlet and outlet of the curved vessel domain. It is compared to the pressure drop of the simulation without RBC’s, denoted ∆Pnonrbc, which is calculated using equation 2.5, which

leads to ∆Pnonrbc= 8µL ∗ ¯v R2 = 8 ∗ (1 ∗ 10−6) ∗ (6 ∗ 10−5) ∗ 5 ∗ 10−3 (1 ∗ 10−5₎2 = 0.0024 = 2.4 ∗ 10 −3_N/m3_{, (4.4)}

where µ = 1 ∗ 10−6m2/s, since the simulation has no RBC’s and therefore the blood viscosity is the same as blood plasma, L = 6 ∗ 10−5m, because that is approximately the curved vessel domain length, ¯v = ¯vt= 5∗10−3m/s, because the simulation has no RBC’s, so the blood velocity

is the same as ¯vtand R = 1∗10−5m, since the radius of the vessel is half the diameter of the vessel.

∆Prbc is measured over the duration of the simulation with RBC’s. The duration of the whole

simulation, which has 500,000 iterations, is equal to 50 milliseconds. An interval of 25,0000 iterations between data points will be used, so there is a 2.5 millisecond interval between data points. To measure ∆Prbc, the density gradient between the inlet and outlet of the curved vessel

is taken, because in the lattice Boltzmann method, pressure is proportional to density, so their gradients should be approximately equal. The plot of ∆Prbc over time is shown in figure 4.2.

(27)

Figure 4.2: Pressure drop ∆P over time in simulation with RBC’s.

The maximum pressure drop of the simulation is measured at 2.87 ∗ 10−3N/m3. The pressure drop of ∆Prbc is greater than of ∆Pnonrbc. This is can be explained by the presence of RBC’s

in the blood. The frictional forces, that are caused by the resistance to flow, is exerted on the blood as it flows through the vessel. Since the hematocrit is higher in the simulation with RBC’s, the viscosity increases as well. The resistance to flow is therefore greater in the simulation with RBC’s than in the simulation with no RBC’s. This means that the pressure at the inlet of the simulation with RBC’s is much higher than the pressure at the outlet compared to the simulation with no RBC’s. Thus, ∆Prbc > ∆Pnonrbc.

4.3 Blood Viscosity in Curved Vessel

From the previous results, the viscosity of the blood in the curve vessel, denoted νc can be

approximated by taking the ratio of νc and νp and equate it to the ratio of the pressure drop

of the simulation with RBC’s ∆Prbc and the pressure drop of the simulation without RBC’s

∆Pnonrbc. This is shown in equation 4.5.

νc νp = ∆Prbc ∆Pnonrbc , (4.5) which is rewritten to νc= ∆Prbc∗ νp ∆Pnonrbc . (4.6)

For ∆Prbc the measured result of of the simulation of the previous section is used, which is

2.87 ∗ 10−3_N/m3_{. For ∆P}

nonrbcthe value 2.4 ∗ 10−3N/m3is used, which is calculated in equation

4.4. Lastly, for νp the viscosity of the blood in the preinlet can be used, since that is basically a

straight vessel. Therefore, νpis equal to 1 ∗ 10−6m2/s. From there, the viscosity of the blood in

the curved vessel νc is calculated:

νc=

(2.87 ∗ 10−3) ∗ (1 ∗ 10−6)

(2.4 ∗ 10−3₎ = 1.12 ∗ 10

−6_m2_/s. _(4.7)

The viscosity in the curved vessel νc = 1.12 ∗ 10−6m2/s is higher than in the viscosity in the

(28)

(29)

CHAPTER 5

Conclusion

In this thesis, the process of simulating a blood vessel in HemoCell has been explored. Each step in the implementation has shown the flexibility of HemoCell in terms of configuration. The subject of this research is cellular level blood flow in a curved blood vessel and the research ques-tion is: How does the curved vessel secques-tion affect the blood viscosity in comparison to a straight vessel section. Two different curved blood flow simulations have been implemented and executed and the resulting data of those simulations have been used to answer the research question.

To answer the research question, the viscosity of the blood in the curved vessel is compared to the theoretical viscosity of blood in a straight vessel. The theoretical blood viscosity in a straight vessel is represented by the blood viscosity in the preinlet, that has been calculated with the Poiseuille equation.

In comparison to the straight vessel blood viscosity, it is found that the curved vessel blood viscosity is higher than that of the straight vessel. This is found by rewriting the Poiseuille equa-tion and substituting the variables with values extracted from the simulaequa-tions. It has also been found that simulations with the presence of RBC’s resulted in a higher pressure drop than in simulations without RBC’s. In addition, it has been found that the average velocity of the blood in the simulation with RBC’s was much lower than of the blood in the simulation without RBC’s.

Thus, it can be concluded that the effect of a curved blood vessel section on the blood vis-cosity is that the visvis-cosity is increased in comparison to a straight vessel section.

(30)

(31)

CHAPTER 6

Discussion

6.1 Accuracy of Blood Viscosity Approximation

In previous sections, the viscosity of the blood in the curved vessel is approximated. However, it should be noted that the Poiseuille equation is used in this approximation. The Poiseuille equation is used to calculate the pressure drop in straight vessels or tubes. Since there are no equations to our knowledge that consider curved vessels to calculate the pressure drop, the pressure drop is approximated instead with the Poiseuille equation.

6.2 Limited Particle Diversity in Simulation

For the simulation, RBC’s are generated into the preinlet before flowing into the curved vessel domain. The hematocrit of the blood is an important factor for the behaviour of the blood flow. Blood contains a lot more particles than RBC’s however. Particles such as proteins, platelets and white blood cells also affect the flow properties of blood. In the beginning, some simulations included platelets as shown in figure 6.1.

Figure 6.1: A blood flow simulation with the platelets colored yellow.

The choice to limit the particles to RBC’s was made due to the increased need of computing in the processing of the simulation, which already took a substantial amount of time. This in conjunction with the limited amount of time provided for the thesis were the causes to this choice.

(32)

6.3 Future Work

6.3.1 Extension of Blood Particles

As discussed in a previous section, the limited use of only RBC’s do affect the results of the simulation and calculations. For future work, it would be ideal to extend the amount of different particles to enhance the accuracy of this thesis.

6.3.2 Different hematocrits

In this research, a hematocrit of 40% is used for the simulation with RBC’s. The hematocrit of the blood in humans is not static however. It usually ranges from 41-52 % in men and 36-48 % in women [6]. It is therefore interesting to simulate the blood flow at different hematocrits and see how the viscosity scales with the hematocrit in curved vessels.

6.3.3 Evaluation of Other Properties of Blood

Since most of our capillary veins aren’t straight but curved, other properties of blood are likely not the same as in straight vessels. This thesis is focused on the viscosity of blood in curved vessels, however, properties as density are not homogenous in a curved vessel. For example, the simulation allow for observation of the density values as shown in figure 6.2.

Figure 6.2: Visualization of the density values of the fluid in the curved vessel simulation with RBC’s.

Therefore, it would be interesting to take the density into consideration of research.

6.3.4 Geometry of Real Capillary Vessels

In this thesis, a relatively simple model of a curved vessel is used for the simulation. It would be interesting to use the geometry of a real capillary vessel as simulation domain and evaluate sections of that geometry. Multiple sections, denoted secti for i = 0, .., N where N is the total

amount of sections, can be evaluated. The average section case, denoted sec could then be¯ calculated by taking the average of all the other sections. This would result in a more stable well-rounded research.

(33)

6.3.5 Utilization of a More Tortuous Vessel

As previously mentioned, a simple model of a curve vessel is used for the simulation. This vessel has only one curve. In a tortuous vessel that contains more curves, the blood viscosity might be even higher. It is therefore interesting to see how the implementation of such a vessel would affect the blood flow.

6.3.6 Heart Pulsatility

The simulations simulate a steady blood flow. However, the heart continuously pumps blood through the blood vessels of the body. The blood therefore is not a steady flow but more like a pulsating flow. It would be interesting to add some level of pulsalility to the simulations as that might also influence the results.

(34)

(35)

CHAPTER 7

(36)

(37)

Bibliography

[1] Yuanxun Bill Bao and Justin Meskas. “Lattice Boltzmann method for fluid simulations”. In: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York (2011).

[2] Shiyi Chen and Gary D Doolen. “Lattice Boltzmann method for fluid flows”. In: Annual review of fluid mechanics 30.1 (1998), pp. 329–364.

[3] Shu Chien. “Red cell deformability and its relevance to blood flow”. In: Annual review of physiology 49.1 (1987), pp. 177–192.

[4] M Ercan and C Koksal. “The relationship between shear rate and vessel diameter”. In: Anesthesia & Analgesia 96.1 (2003), p. 307.

[5] Dmitry A Fedosov, Bruce Caswell, and George Em Karniadakis. “Systematic coarse-graining of spectrin-level red blood cell models”. In: Computer Methods in Applied Mechanics and Engineering 199.29-32 (2010), pp. 1937–1948.

[6] Peter R Hoskins, Patricia V Lawford, and Barry J Doyle. Cardiovascular biomechanics. Springer, 2017.

[7] Pushpa Larsen. Blood Viscosity. https://ndnr.com/cardiopulmonary-medicine/blood-viscosity/. Accessed: 2010-09-30.

[8] Hiroshi Noguchi and Gerhard Gompper. “Shape transitions of fluid vesicles and red blood cells in capillary flows”. In: Proceedings of the National Academy of Sciences 102.40 (2005), pp. 14159–14164.

[9] Axel R Pries, D Neuhaus, and P Gaehtgens. “Blood viscosity in tube flow: dependence on diameter and hematocrit”. In: American Journal of Physiology-Heart and Circulatory Physiology 263.6 (1992), H1770–H1778.

[10] Peter W Rand et al. “Viscosity of normal human blood under normothermic and hypother-mic conditions”. In: Journal of Applied Physiology 19.1 (1964), pp. 117–122.

[11] Wall Shear Stress. https://www.corrosionpedia.com/definition/1157/wall-shear-stress.

[12] G´abor Z´avodszky et al. “Cellular level in-silico modeling of blood rheology with an im-proved material model for red blood cells”. In: Frontiers in physiology 8 (2017), p. 563. [13] Junfeng Zhang, Paul C Johnson, and Aleksander S Popel. “Red blood cell aggregation

and dissociation in shear flows simulated by lattice Boltzmann method”. In: Journal of biomechanics 41.1 (2008), pp. 47–55.

Blood Flow in Curved Blood Vessels on a Cellular Level

Bachelor Informatica