Computational Fluid Dynamics of Steady Blood Flow Through a Fusiform Aneurysm by Bradley Campbell 2016 MPhys Physics Supervisor: Dr Shane O’Hehir Abstract The work within this project is an investigation of the current assessment of the risk of rupture of an aneurysm, which is solely based on the size of the aneurysm. Investigations will be made on whether a new approach could be used to determine the risk of rupture of an aneurysm, which in turn would lower the current fatality rate of patients with aneurysms. This project was conducted by reviewing the hemodynamics of blood in an aneurysm, specifically how the hemodynamics are affected by the size of the aneurysm. The study found that an alternative factor, other than the diameter of the aneurysm, could cause an increased risk of rupture in patients. This factor is the wall shear stress of the aneurysm and was found to be highest in ‘medium’ sized aneurysms, that under the current assessment criteria would be left untreated. These medium sized aneurysms have been found to rupture in 5% of cases of aneurysms leading to the consensus that the wall shear stress should be a factor considered when assessing the risk of rupture.
Transcript
Computational Fluid Dynamics of Steady BloodFlow Through a Fusiform Aneurysm
by Bradley Campbell
2016
MPhys Physics
Supervisor: Dr Shane O’Hehir
Abstract
The work within this project is an investigation of the current assessment of the risk of rupture of ananeurysm, which is solely based on the size of the aneurysm. Investigations will be made on whethera new approach could be used to determine the risk of rupture of an aneurysm, which in turn wouldlower the current fatality rate of patients with aneurysms. This project was conducted by reviewingthe hemodynamics of blood in an aneurysm, specifically how the hemodynamics are affected bythe size of the aneurysm. The study found that an alternative factor, other than the diameter of theaneurysm, could cause an increased risk of rupture in patients. This factor is the wall shear stressof the aneurysm and was found to be highest in ‘medium’ sized aneurysms, that under the currentassessment criteria would be left untreated. These medium sized aneurysms have been found torupture in 5% of cases of aneurysms leading to the consensus that the wall shear stress should be afactor considered when assessing the risk of rupture.
1 A detailed image of one of the main arteries of the human body showing details of each ofthe layers [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 An image showing the two shapes of aneurysms typically found [5]. . . . . . . . . . . . . 23 An image showing the location of an AAA in the arterial network [10]. . . . . . . . . . . . 24 A flow chart showing the pathogenesis of an aneurysm [5]. . . . . . . . . . . . . . . . . . 35 An image of a stent graft inserted into an AAA [15]. . . . . . . . . . . . . . . . . . . . . 46 A schematic of a 2-D pipe used to create a simple model of an aneurysm. . . . . . . . . . . 67 A schematic showing the direction of a particle in a laminar flow and the direction of a
particle in a turbulent flow [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 A simple model of a flow through a 2-D pipe to show the effect of a gradual enlargement on
the velocity [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 A simple model of a flow through a 2-D pipe to show the effect of a gradual enlargement on
the pressure [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 A schematic showing the parameters for the geometry of the models [35]. . . . . . . . . . . 1211 The geometry of model F in two different orientations. . . . . . . . . . . . . . . . . . . . 1312 The mesh of model F shown on the left before the virtual topography was applied and on the
right after the addition of virtual topography. . . . . . . . . . . . . . . . . . . . . . . . . 1413 The mesh of model F shown on the left without an inflation layer and on the right with an
inflation layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 Image showing two different orientations of the aneurysm to show where the velocity was
measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515 A graph to show the convergence test of model F in order to determine a suitable size mesh. . 1616 A graph to show the residual of the continuity equation of model F, along with x, y and z
velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917 Images of the pressure profile of models B (left), D (centre)and F (right). . . . . . . . . . . 1918 A graph to show the pressure gradient between the inlet and outlet in all models. . . . . . . 2019 An image of the pressure profile of model F highlighting the region of low and high pressure
at the proximal and distal end of the aneurysm respectively. . . . . . . . . . . . . . . . . 2120 Images of the velocity profile of models B (left), D (centre)and F (right). . . . . . . . . . . 2221 Images of velocity streamlines of models B (left), D (centre)and F (right). . . . . . . . . . . 2222 Images of velocity vectors of models B (left), D (centre)and F (right). . . . . . . . . . . . . 2323 An image of the vector plots of models B and C highlighting the formation of vortices in
model C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2324 Images of the WSS of models B (left), D (centre)and F (right). . . . . . . . . . . . . . . . 2425 A graph of the average value of WSS against the diameter of the aneurysm. . . . . . . . . . 2426 An image highlighting the region of low WSS at the distal end of model B and the region of
high WSS at the distal end of model D. . . . . . . . . . . . . . . . . . . . . . . . . . . 2527 A graph of the maximum value of WSS against the diameter of the aneurysm. . . . . . . . . 2528 A graph showing a two aneurysm system which shows a pressure increase at the distal end
of the aneurysm [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2629 A graph showing the relationship between the WSS and the diameter (cm) of an aneurysm [47]. 2830 A graph to show the convergence test of model A in order to determine a suitable size mesh. . 3331 A graph to show the convergence test of model B in order to determine a suitable size mesh. . 33
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32 A graph to show the convergence test of model C in order to determine a suitable size mesh. . 3433 A graph to show the convergence test of model D in order to determine a suitable size mesh. . 3434 A graph to show the convergence test of model E in order to determine a suitable size mesh. . 3535 A graph to show the convergence test of model G in order to determine a suitable size mesh. . 3536 A graph to show the convergence test of model H in order to determine a suitable size mesh. . 3637 Images of the pressure profile of models A-D. . . . . . . . . . . . . . . . . . . . . . . . 3638 Images of the pressure profile of models E-H. . . . . . . . . . . . . . . . . . . . . . . . 3739 Images of the velocity profile of models A-D. . . . . . . . . . . . . . . . . . . . . . . . 3740 Images of the velocity profile of models E-H. . . . . . . . . . . . . . . . . . . . . . . . 3841 Images of the velocity streamlines of models A-D. . . . . . . . . . . . . . . . . . . . . . 3842 Images of the velocity streamlines of models E-H. . . . . . . . . . . . . . . . . . . . . . 3943 Images of the velocity vectors of models A-D. . . . . . . . . . . . . . . . . . . . . . . . 3944 Images of the velocity vectors of models E-H. . . . . . . . . . . . . . . . . . . . . . . . 4045 Images of the wall shear of models A-D. . . . . . . . . . . . . . . . . . . . . . . . . . . 4046 Images of the wall shear of models E-H. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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List of Tables
1 Estimation of the risk of rupture of an AAA with respect to diameter [14]. . . . . . . . . . 42 Table to show the different models that will be used. . . . . . . . . . . . . . . . . . . . . 113 Mathematical equations that can be used to model the curvature of the aneurysm wall [36]. . 124 Table showing the effect of increasing the number of elements in the mesh on the velocity at
the point (0,-0.02,0) cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Table showing the optimum number of elements needed for each model, determined from
the results of the convergence tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Table showing the convergence of the velocity at different residual values of the continuity
Arteries and veins are responsible for carrying blood around the body. The primary role of arteriesis to carry oxygenated blood away from the heart and into the limbs and organs. The veins then takethe de-oxygenated blood back to the heart [1]. The artery of interest in this project is the aorta whichis the main artery of the human body, passing through the chest and abdomen eventually splitting atthe iliac bifurcation to take blood to either leg [2].
The main arteries in the human body are made up of three layers. These are the tunica intima, thetunica media, and the tunica adventitia [3]. The tunica intima is the inner most layer of the arterialwall that comes into direct contact with the blood flow. The tunica media is the layer sandwichedbetween the intima and the adventitia which is responsible for the elasticity of the artery. The tunicaadventitia is the outer most layer of the artery. All three of these layers consist of different types ofcells which are responsible for various functions [4]. An image of one of the main arteries of thebody, such as the aorta, is shown in figure 1.
Figure 1: A detailed image of one of the main arteries of the human body showing details of each of thelayers [5].
An aneurysm can occur in both arteries and veins however they are much more common in arteries.A first definition of an aneurysm was established in 1991 by ‘the Society of Vascular Surgery andthe International Society of Cardiovascular Surgery’, as a permanent dilation of an artery/vein,having 50% increase in diameter compared with the normal diameter of the artery/vein [6]. Arterialaneurysms are usually split up into three different forms, these are pseudo-aneurysms, saccular andfusiform aneurysms. Which can then be categorised into ‘true’ or ‘false’ aneurysms, a pseudo-aneurysmis a haematoma that forms as a result of a hole in the arterial wall and is known as a false aneurysm.The other two forms of aneurysm are then categorised based on there shape, an image of these isshown in figure 2 [7].
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A saccular aneurysm is a berry like sac extending from the arterial wall of an artery. Saccularaneurysms most commonly occur in curved arteries or at bifurcation angles, especially at the circleof willis [8]. A fusiform aneurysm is spindle like shaped and are most predominant in the arterybehind the knee (popliteal artery) or in the abdominal aorta [5].
Figure 2: An image showing the two shapes of aneurysms typically found [5].
The particular aneurysms of interest in this project are abdominal aortic aneurysms (AAAs). Theseare a type of fusiform aneurysm located in the abdominal region of the aorta, with 90% of cases ofAAAs occurring below the renal arteries (the arteries that lead to the kidneys) and above the iliacarteries of the legs [9]. Figure 3 shows the location of an AAA within the arterial network of theabdomen.
Figure 3: An image showing the location of an AAA in the arterial network [10].
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The pathogenesis of an AAA is still not fully understood, however it is thought that a combinationof factors can lead to the formation of an AAA [5]. Figure 4 is a flow chart of some of the factorsthat are thought to lead to the formation of such an aneurysm.
Figure 4: A flow chart showing the pathogenesis of an aneurysm [5].
A combination of these factors can lead to the weakening of the arterial wall. This weakening ofthe arterial wall means that the wall can no longer sustain the pressures it is put under by the bloodflow. This causes fatigue of the arterial wall and leads to the irreversible bulging of the artery and theformation of an AAA [5].
The need for treatments of AAAs is important, as AAA rupture is the 15th leading cause of deathin the United States, in most cases affecting patients over the ages of 55 [11]. There are currentlytwo approaches to take when an AAA is discovered. The first of which is watchful waiting, this issuggested if the aneurysm is small. Watchful waiting means that the aneurysm will be continuallymonitored to see if there is any or significant growth. The aneurysm is monitored by screening viaa method such as ultrasonography or computed tomography (CT) among other techniques [12]. Ifit is discovered that the aneurysm is stable and no or minimal growth is present then it will be leftuntreated.
On the other hand, if a significantly large aneurysm is found then surgery is offered to patientsthat are at significant risk of rupture. There are two types of surgery for the treatment of AAAs, theseare open surgery and endovascular aneurysm repair (EVAR). Open surgery involves cutting open theabdomen and completely removing the AAA. The removed part of the artery is then replaced by anartificial artery known as a stent graft [13].
EVAR emerged in the early 1990s as an alternative approach to open surgery [14]. EVAR is aless invasive method of surgery as it is a type of keyhole surgery where the stent graft is inserted intothe artery of the leg and manoeuvred through the artery into the abdominal area where the aneurysmis present.
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A stent graft is a tube of material usually made of Teflon that is inserted into the aneurysm to mimicthe normal abdominal aorta [15]. In EVAR the aim is to reduce the pressure on the arterial wall. Thisdecreased pressure on the wall stops the aneurysm from growing any larger and decreases the risk ofrupture. An example of a stent graft inserted into an AAA can be seen in figure 5.
Figure 5: An image of a stent graft inserted into an AAA [15].
Currently the risk of rupture of an AAA is assessed by the diameter of the aneurysm and the growthrate. Surgery is recommended for AAAs exceeding 5 cm in diameter and those with expansionrates >0.5 cm/year [16]. Many studies have being conducted to try and assess the risk of rupture,particularly the work of Brewster et al [14].Where they have made an estimate of the annual risk ofrupture based on the diameter of the AAA this can be seen in table 1.
Table 1: Estimation of the risk of rupture of an AAA with respect to diameter [14].
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However the current assessment of the risk of rupture of an aneurysm has come under some scrutinyas many AAAs with diameter smaller than 5 cm have been found to rupture and be fatal [17]. A studynamed ‘The UK Small Aneurysm Trail’ monitored 2257 patients with AAAs of diameter less than5.5 cm and found that 103 of these aneurysms ruptured (5%) and of these ruptured aneurysms 79were fatal (77%) [18]. It has been suggested that consideration of the wall shear stress (WSS) on thewall of an AAA could be an important factor in determining the risk of rupture, alongside the currentassessment criteria [19]. This is supported by the work of Fillinger et al, were they determined therelationship between WSS and aneurysm diameter [20]. Other factors have also been considered tohave an affect on the risk of rupture of an AAA [21], however only the WSS will be assessed in thisproject.
This project will include an investigation of how the hemodynamics of blood vary with increase in thesize of an AAA. This will include assessment of the pressure, velocity and WSS of an axisymmetricAAA. The aim of which is to gain an understanding about whether factoring in the WSS into the riskof rupture would be advantageous.
2 Theory
2.1 Theoretical Model of an Aneurysm
A simple theoretical model of an aneurysm can be written in order to determine the general principlesof a flow such as that of blood moving through a pipe with varied diameter, which is a generalisationof an aneurysm. This theoretical model can be made simple by assuming that blood is an ideal fluid.An ideal fluid is one which is incompressible and has no viscosity [22].
The geometry of the theoretical problem can be modelled as a 2-D pipe of inlet diameter d1= 2cm that has a gradual expansion to a larger pipe that has an outlet diameter d2= 6 cm. A schematicof this can be seen in figure 6.
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Figure 6: A schematic of a 2-D pipe used to create a simple model of an aneurysm.
To be able to determine the characteristics of the fluid at the outlet of the pipe some initial conditionsmust be set. These conditions will be the inlet parameters, assume that the velocity at the inletV1 = 8 cm/s and that the pressure at the inlet P1 = 16000 Pa. These inlet parameters were chosenas they are typical values of pressure and velocity in the aorta averaged between systole and diastole[7]. The equations which are used to determine the flow of blood through the 2-D pipe are theconservation of mass flow rate and Bernoulli’s equation. The conservation of mass flow rate statesthat the mass per unit time entering the pipe must be equal to the mass per unit time leaving the pipe.The mass flow rate can be applied if a closed system is assumed meaning that there are no othersources of fluid [22]. Bernoulli’s equation is a form of conservation of energy and is true for laminarflow only. The reason for this is because Bernoulli’s equation is only true for a streamline and inturbulent flow the particles do not follow a streamline, this is shown in figure 7. By creating the pipeas a gradual enlargement to a larger diameter rather than a sudden step to a larger diameter a laminarflow can be assumed [23].
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Figure 7: A schematic showing the direction of a particle in a laminar flow and the direction of a particle ina turbulent flow [24].
The mass flow rate equation is given by:
m = ρA1V1 = ρA2V2 (1)
Where m is the mass flow rate, ρ is the density of blood, A is cross-sectional area and V is thevelocity of the fluid. This equation can be generalised for the 2-D pipe and used to determine thevelocity of the blood at the outlet. For a 2-D pipe the equation reduces to:
d1V1 = d2V2 (2)
Where d is the diameter of the pipe. Solving equation (2) for V2 it is found that the velocity ofblood at the outlet, V2 = 0.8 cm/s, showing that blood velocity decreases as the diameter of the pipeincreases. Once the velocity was known at the outlet it can be used to determine the pressure at theoutlet. This is found by applying Bernoulli’s equation (neglecting effects due to gravity), Bernoulli’sequation is given by:
P1
ρ+V 21
2=P2
ρ+V 22
2(3)
P2 = P1 +1
2ρ(V 2
1 − V 22 ) (4)
The density of blood is given by ρ= 1050 kgm−3 [25] and from equation (4) it was found that thepressure at the outlet, P2 = 16003.03 Pa, meaning that there is a slight increase in pressure at theoutlet of the pipe compared to the inlet. This was expected as can be seen by inspection of Bernoulli’sequation pressure is inversely proportional to velocity.
The results of this have also been visualised in the work of Satish et al, were they have produceda graphic of a flow progressing through a gradual enlargement [23]. The software used withinthese simulations was a mixture of ‘Gambit’ to make the geometry and ‘Ansys fluent’ to run thecalculations. The parameters used within this model are an inlet diameter d1= 2 cm, outlet diameterd2= 4 cm and an inlet velocity of V1= 5.5 cm/s. Although these parameters are somewhat differentfrom those used in the calculations above the simulations still show the same nature of the flow as itpasses through the enlargement.
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Figure 8: A simple model of a flow through a 2-D pipe to show the effect of a gradual enlargement on thevelocity [23].
Figure 9: A simple model of a flow through a 2-D pipe to show the effect of a gradual enlargement on thepressure [23].
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In figure 8 and 9 it shows that as the flow passes through the enlargement the velocity of the flowdecreases and that the pressure increases. This result is the same as that found using the applicationof the conservation of mass flow rate and Bernoulli’s equation.
Although this simple model of a flow through a 2-D pipe displays the characteristics of a flow throughan enlargement which is what happens when blood flows through an aneurysm. It is not an accuraterepresentation of the hemodynamics of blood. The reason for this is because blood is not an idealfluid therefore the hemodynamics of blood are a lot more involved than a simple increase or decreasein pressure and velocity. To be able to determine an accurate representation of the hemodynamics ofblood computational fluid dynamics (CFD) must be used.
2.2 Computational Fluid Dynamics
Fluid dynamics is a discipline of fluid mechanics which deals with the flow of fluids. As fluiddynamics problems have evolved and become more involved, via the introduction of new parameterssuch as viscosity, a new way in which they could be solved had to be found. With the advent ofmodern computers in the early 1950s, along came modern CFD a numerical analysis method ofsolving fluid dynamics problems [26]. One of the main contributors to modern CFD was Prof. DavidCaughey who published many papers in the late 20th century and paved a way for the CFD techniquesthat are used today [27]. Most relevant for this project was his work on the Navier-stokes equationswhich can be used to model blood flow.
The software that was used to produce the CFD simulations within this project was ‘Ansys’. Ansyswas developed in 1970 and is computer aided engineering software. The software has the abilityto simulate problems ranging from electronics, structural analysis and fluid dynamics. The fluiddynamics part of the software is used in this project and is called ‘Ansys fluent’.
Ansys fluent contains within it 5 different parts. These are geometry, mesh, set-up, solution andresults. The geometry section of the software allows for computer aided design (CAD) models of theaneurysm to be made. The CAD model can be made in Ansys itself or on an external CAD softwareand imported into Ansys. The meshing part of Ansys allows for the geometry of the aneurysm to bemeshed. The need for a mesh is important as it breaks down the geometry into a finite set of elementsfrom which calculations can be made. The set-up of the simulation and the solution run under thesame piece of software within Ansys fluent. This part of the software allows for the parameters ofthe simulation to be set which includes the model used, the materials and the boundary conditions.The final part of the software is the results section. This section imports the results of the simulationsinto a post CFD analysis workbench. The workbench allows for many aspects of the simulations tobe analysed by plotting images of the solution, creating tables of data and plotting graphs.
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2.3 Governing Equations of Blood Flow
The equations governing a flow such as that of blood are the Navier-Stokes equations, which werefirst established in the 1840s based on the conservation laws [28]. The equations are a generalizationof Euler’s equation to include effects due to viscosity, which is an important factor when assessingthe hemodynamics of blood. The Navier-Stokes equations are made up of the continuity equationand the momentum conservation equation. The general form of the continuity equation is given by:
∂ρ
∂t+∇ · (ρ~V ) = 0 (5)
Where ρ is the density of the fluid and ~V is the velocity vector of the fluid [29].
The momentum conservation equation is given by:
ρD~V
Dt= p~g −∇p+ µ∇2~V (6)
Again ρ is the density of the fluid and ~V is the velocity vector. P defines the pressure of the system,
µ the dynamic viscosity [29] andD
Dtis the substantial derivative given by:
D
Dt=
∂
∂t+ ~V · ∇ (7)
Where ~V is the velocity vector again [19].
2.4 Wall Stresses
The law of Laplace states that the WSS in a cylindrical pipe is directly related to the diameter of thepipe [30], suggesting that the WSS in a pipe is directly related to its size. As previously mentioned,further investigation of the WSS will be carried out to determine the effect WSS has on the risk ofrupture of an aneurysm. If the law of Laplace can be applied to an aneurysm then to only considerthe diameter when assessing the risk of rupture would be appropriate.
WSS is defined as the tangential force that blood applies on the arterial wall as it flows [7]. Thisforce is in the direction of the blood flow and is defined by:
τw = −µ∂V∂x
(8)
Where τw is the WSS, µ is the viscosity of the fluid, V is the velocity of the fluid, x is the distancefrom the boundary (wall of the aneurysm) and ∂V
∂x is the velocity gradient of the fluid in the xdirection. [31].
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3 Effect of Size of an Aneurysm
A CFD study has been conducted to determine what effect the size of an aneurysm has on thehemodynamics of blood. The reason for carrying out this study is to verify that the current assessmentof the risk of rupture of an aneurysm being solely based on the diameter of the aneurysm is notsufficient.
This study involves the modelling of a steady, homogeneous, incompressible and Newtonian flow.In the human body blood flow is a transient flow (or unsteady flow), meaning that it varies over timeas the heart beats. However for the purpose of this project a steady flow is assumed by taking anaverage pressure and velocity of blood in the aorta between systole and diastole. The flow of bloodis also a non-Newtonian flow, meaning that the viscosity of blood changes under different values ofstress. In all of the simulations within this project the diameter of the aorta is larger than 0.5 mm,this allows for the blood viscosity to be assumed to be constant. The reason for this is that bloodviscosity is relatively constant at the high rates of shear found in the aorta. If blood viscosity can beassumed to be constant then it is possible to assume blood is a Newtonian flow [32].
The following sections go through the building of a model to simulate the effect of size of ananeurysm using Ansys. This will include the making of the geometry of an aneurysm, how anappropriately accurate mesh is produced and the set-up procedure i.e choosing the model and initialparameters. Then the results of each of the models will be compared to determine how the hemodynamicsof blood vary with the size of an aneurysm. Which will include a review of the pressure, velocityand WSS.
3.1 Geometry
The geometry of the aneurysm was made using DesignModeler, an integrated part of Ansys. CADmodels can be made in DesignModeler or uploaded from other software [33]. In this project themodels will be drawn in DesignModeler.
Eight different models of the aneurysm will be made that vary in size. The parameter which will bevaried is the maximum diameter D of the aneurysm, this will range from D= 2 to 9 cm in incrementedsteps.
Model D (cm)A 2B 3C 4D 5E 6F 7G 8H 9
Table 2: Table to show the different models that will be used.
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All models are constrained to an inlet diameter d= 2 cm, making model A a control model as it doesnot contain an aneurysm [34]. The total length of the model h= 15 cm and an aspect ratio of L/d=4 cm was used as this is typical of fusiform aneurysms [35]. Thus from these parameters the lengthof the aneurysm L, can be calculated to be L= 8 cm. A schematic of these parameters is shown infigure 10.
Figure 10: A schematic showing the parameters for the geometry of the models [35].
The wall curvature of an aneurysm can be modelled using various different equations, examples ofsome of the equations that have been previously used can be seen in table 3 [36].
Table 3: Mathematical equations that can be used to model the curvature of the aneurysm wall [36].
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However in this project the curvature of the aneurysm wall will be modelled based on an ellipse. Theequation of an ellipse is given by:
x2
b2+y2
a2= 1 (9)
The coordinates of the ellipse are x and y, b is the radius of the x-axis and a is the radius of the y-axis.The orientation of the ellipse is also defined by a and b, in this instance a is the major axis and b isthe minor axis.
The geometry of the aneurysm is created by solving equation (9) at a fixed number of points onthe y-axis. This was chosen to be in the range y= -4 to 4 cm in incremented steps. At each of thesepoints a circle of radius x was drawn, the radius being calculated via equation (9). The results of thiscan be seen in figure 11 which shows the geometry of model F.
Figure 11: The geometry of model F in two different orientations.
3.2 Mesh
Once the geometry of the aneurysm had been created for each of the models in table 2 a mesh of eachwas produced. A mesh is needed to break the geometry down into a finite number of elements forwhich calculations are made giving each element a numerical value. Therefore the finer the mesh themore accurate the results. The mesh is created by first using the virtual topology tool, this tool allowsfor different surfaces and edges to be merged. The need for merging the surfaces comes from the fact
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that the geometry was created by solving equation (9) at numerous points, each point creating a newsurface. These different surfaces can be seen by inspection of figure 11. The merging of the surfacesallows for the mesh to better approximate an aneurysm, as an aneurysm is one complete surface [37].Figure 12 shows the mesh of model F before and after virtual topology was applied. The image onthe left clearly has horizontal lines that break up the mesh, caused by the presence of the differentsurfaces. The image on the right has no horizontal lines and acts as one surface.
Figure 12: The mesh of model F shown on the left before the virtual topography was applied and on the rightafter the addition of virtual topography.
The addition of an inflation layer was also introduced into the mesh near the wall of the aneurysm. Aninflation layer is a type of boundary layer placed near the wall of the aneurysm to allow for refinementof the mesh in that region. This then allows for a denser mesh near the wall of the aneurysm andhence improves the accuracy of results in that region [7]. An inflation layer of 0.1 cm was chosen foreach model, which covers 10% of the inlet. An area of 10% of the diameter of the inlet was chosenas this is the region of highest velocity gradient from the wall [7].
Figure 13: The mesh of model F shown on the left without an inflation layer and on the right with an inflationlayer.
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The body sizing tool could then be used to either increase or decrease the number of nodes andelements in the mesh, this is done by changing the size of the elements. An optimum number ofnodes and elements can be determined by measuring the velocity of blood within the aneurysm ata given point for various densities of elements. It was chosen that the velocity of blood would bemeasured at the point (0, -0.02, 0) cm, this point was chosen as it is in a region of high velocitychange. Figure 14 shows where this point is located within the aneurysm and is represented by theyellow cross.
Figure 14: Image showing two different orientations of the aneurysm to show where the velocity wasmeasured.
A convergence test was then carried out for each of the models to find an optimum number ofelements in each. The results of the convergence test for model F can be seen tabulated in table 4.
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Nodes Number of elements (Thousands) Velocity (cm/s)7036 19.433 7.088913990 46.061 9.058224895 91.124 9.457932298 121.593 9.504638972 149.882 9.706053475 212.943 9.8678
Table 4: Table showing the effect of increasing the number of elements in the mesh on the velocity at thepoint (0,-0.02,0) cm.
The data in table 4 was plotted to determine the optimum number of elements needed in the mesh, agraph of this can be seen in figure 15.
Figure 15: A graph to show the convergence test of model F in order to determine a suitable size mesh.
From figure 15 it can be seen that after about 100k elements there is only a small variation in thevelocity. This means that a mesh of around 100k elements would be suitable to use to produce anaccurate result. Using a mesh any denser than this would increase the computational time withoutsignificant gain in accuracy of results. The convergence tests for the other models can be found in theappendix in figure 30 to 36. A table has been produced for the optimum number of elements neededfor each model, this is shown in table 5.
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Model Number of elements (Thousands)A 108,124B 102,544C 164,562D 180,572E 136,138F 167,533G 192,900H 142,282
Table 5: Table showing the optimum number of elements needed for each model, determined from the resultsof the convergence tests.
3.3 Set-up
The process of setting up the simulations can then be carried out, this firstly involves setting theinitial boundary conditions of the problem. The inlet velocity was ~V = 8 cm/s in the y-directionperpendicular to the inlet and the outlet pressure was p= 120 mm/Hg or 16000 Pa [34]. A boundarycondition was also set at the walls of the aneurysm this was the no slip condition meaning that thevelocity at the walls was set to zero. This boundary condition is needed to differentiate the fluidregime from the rigid walls of the aneurysm [9].
The material that is being used must also be set, this is done by setting the density and dynamicviscosity of the material. For blood the average density is given by ρ= 1050 kg m−3 and the dynamicviscosity is given by µ = 0.0035 Pa s [38].
The model by which the simulations will be calculated using must then be determined. The correctequations to use for the simulations are found by finding the Reynolds number of each of the models.The Reynolds number is used to determine whether a flow is laminar or turbulent [39]. The Reynoldsnumber in an aneurysm is given by:
Re =ρV D
µ(10)
This is where ρ is the density of fluid, V is the velocity of the fluid, D is the diameter of the aneurysmand µ is the dynamic viscosity of the fluid [34]. Therefore the Reynolds number of models A-H liein the range 480 ≤ Re ≤ 2160. The criterion for laminar flow is that the Re < 2100, thus all modelsother than one lie within this criterion. However the region 2000≤ Re≤ 4000 is a transitional regionto turbulent flow [40]. Knowing this, it is possible to assume that all the models can be accuratelysimulated using laminar flow, as only one model lies slightly within the transitional region.
As blood is incompressible the equation of continuity (equation (5)) can be simplified and theequation can be reduced to:
∇ · ~V = 0 (11)
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Also the effect due to gravity is not considered within the simulations so the momentum conservationequation can also be reduced, this equation reduces to:
ρD~V
Dt= −∇p+ µ∇2~V (12)
This is where ρ is the density of the fluid, ~V is the velocity vector, p is the pressure of the systemand µ is the dynamic viscosity [29]. Once the correct model has been decided upon the simulationcan be initialized, there are two options for this in Ansys either hybrid or standard initialization.Standard allows for the solution to be initialized based on a single parameter, whereas hybrid allowsfor multiple parameters to be used [41]. For the simulations within this project hybrid initializationwas used so that both the inlet velocity and outlet pressure are used to initialize the problem.
A test must then be carried out on the continuity equation to determine an optimum residual valuethat the simulation should be left to reach. This test was carried out my decreasing the value ofresidual for the continuity equation and measuring the velocity at the point (0,-0.02,0) cm, wherethis point is located in the aneurysm shown in figure 14. The results of this test can be seen tabulatedin table 6.
Residual of continuity equation Velocity (cm/s)1x10−1 9.83461x10−2 9.75631x10−3 9.76161x10−4 9.7616
Table 6: Table showing the convergence of the velocity at different residual values of the continuity equation.
By inspecting table 6 it can be seen that a residual of 1x10−3 is sufficient. Again using a higher valueof residual would simply increase the computational time without any significant gain in accuracy.The solution of the residual of the continuity equation for model F can be seen in figure 16, thisgraphic also shows that the residual of the x,y and z velocities was also monitored. However as theresidual values of these was much lower than that of the continuity equation it was chosen that theresidual of the continuity equation would be the determining factor in this test.
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Figure 16: A graph to show the residual of the continuity equation of model F, along with x, y and z velocities.
3.4 Results
Once the simulations have been run for all models the results of each can be viewed in the postCFD package. The parameters of interest when analysing the results of the models are the pressure,velocity and WSS, and how the size of the aneurysm affects these parameters. The pressure profileof models B, D and F can be seen in figure 17, the pressure profile of all other models are found infigure 37 and 38 in the appendix.
Figure 17: Images of the pressure profile of models B (left), D (centre)and F (right).
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The first thing that is noticed about the pressure within the aneurysm is that there is a small pressuregradient between the inlet and outlet in each of the models, with a large pressure at the inlet andsmaller pressure at the outlet. In each of the models the variation in pressure is much less than 0.1%of the total pressure. It was also found that the pressure gradient decreased in magnitude upto adiameter of 4 cm and then seemed to reach a plateau as the aneurysm grew larger, the results of thisare shown graphically in figure 18.
Figure 18: A graph to show the pressure gradient between the inlet and outlet in all models.
Another interesting thing to note about the pressure is that there is a region of low pressure at thepoint at which the aorta begins to become larger, the proximal end of the aneurysm and a region ofhigh pressure at the exit of the aneurysm, the distal end.
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Figure 19: An image of the pressure profile of model F highlighting the region of low and high pressure atthe proximal and distal end of the aneurysm respectively.
These regions of high and low pressure are present in all sizes of aneurysm. However at small sizesof aneurysm the region of high pressure is not focused at the distal end rather it extends half way upthe aneurysm, as shown in model B in figure 17.
By observing the velocity profile of the models it can be seen that there is a slight gradient in thevelocity with low velocity at the inlet and high velocity at the outlet. This is the inverse of thatfound in the pressure profile with high pressure at the inlet and low pressure at the outlet. This resultis expected as it was shown in the theoretical model that there is an inverse relationship betweenpressure and velocity expressed by equation (5). It was also shown that the velocity has two maincomponents, a stream of high velocity fluid going straight through the centre of the aneurysm and aslow moving fluid surrounding this. Figure 20 shows the velocity profile of models B, D and F, fromthis figure the two components of velocity can be seen. The velocity profile of the other models arein figure 39 and 40 of the appendix.
21
Figure 20: Images of the velocity profile of models B (left), D (centre)and F (right).
The slow moving fluid near the walls of the aneurysm increases in volume as the aneurysm grows andcan be shown to be a recirculation of the flow. This is represented in figure 21 which is a streamlineplot of models B, D and F. From this it can be seen that the fluid at the distal end of the aneurysmseparates from the main stream and recirculates around the edge of the aneurysm reconnecting to themain stream. The streamline plots of all models is shown in figure 41 and 42 of the appendix.
Figure 21: Images of velocity streamlines of models B (left), D (centre)and F (right).
This recirculation of the flow can be shown to contain vortices in some aneurysms. These vorticescan be seen in the streamline plots but to be able to see them in more detail vector plots of the velocitycan be made. Figure 22 shows the vector plots of models B, D and F, the vector plots of all models
22
are in the appendix in figure 43 and 44.
Figure 22: Images of velocity vectors of models B (left), D (centre)and F (right).
The vector plots were then looked at in more detail to determine the size of aneurysm which vorticesfirst appear. By closer inspection of models B and C it was found that vortices first appear in model Cwhich represents a diameter of aneurysm of 4 cm. A zoomed in image of the vector plots of modelsB and C can be seen in figure 23 which shows the development of these vortices. It has also beenfound that at sizes of aneurysm greater than 4 cm vortices appear in all cases.
Figure 23: An image of the vector plots of models B and C highlighting the formation of vortices in modelC.
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The WSS within the aneurysm was then looked at and it was found that at the inlet upto the proximalend of the aneurysm and at the distal end of the aneurysm upto the outlet there is a region of highWSS. Between these regions there is an area of low WSS across the bulge of the aneurysm. This canbe observed in figure 24 which shows models B, D and F. The WSS of the other models can be foundin the appendix in figure 45 and 46.
Figure 24: Images of the WSS of models B (left), D (centre)and F (right).
For each model of aneurysm the volume averaged WSS was found and a plot of this average WSSagainst the size of the aneurysm was made. This is shown in figure 25 which shows that as the sizeof the aneurysm increases the volume averaged WSS decreases.
Figure 25: A graph of the average value of WSS against the diameter of the aneurysm.
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Another aspect of the WSS to observe is that there is a region of high WSS at the distal end ofthe aneurysm in all models other than A and B. These are the only two models which do notcontain vortices so there could be a link between WSS and the presence of vortices, this link will beinvestigated further in the discussion section.
Figure 26: An image highlighting the region of low WSS at the distal end of model B and the region of highWSS at the distal end of model D.
The maximum value of the WSS for each model was then obtained and the results of this can beplotted, shown in figure 27. This graph shows that the maximum value of WSS occurs at a value ofaneurysm diameter of 4 cm, again this could be linked to the formation of vortices.
Figure 27: A graph of the maximum value of WSS against the diameter of the aneurysm.
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4 Discussion
In the work of both Leung et al [42] and Kelly et al [43], a small pressure gradient was found in theirstudy of aneurysms. This pressure gradient had a variation of less than 0.1% along the length of theaneurysm, which is around the same variation found within this project. However in their work therewas no comparison between the size of the aneurysm and the pressure gradient as conducted in thisproject. In figure 18 it shows that the pressure gradient decreases upto an aneurysm diameter of 4 cmand then reaches a plateau as the aneurysm grows. The only difference between the aneurysms thatwere smaller than 4 cm and those which were larger was the formation of vortices which could beattributed to this plateau.
Within this project it was also shown that at the proximal end of the aneurysm there was a region oflow pressure and at the distal end there was a region of high pressure that can be seen in figure 19.There is also evidence of a region of low pressure within the theoretical model in figure 9, at thepoint at which the pipe begins to enlarge as found in the computational model. The region of higherpressure at the distal end of the aneurysm has also been observed in the work of others such asFinol et al [44]. The results of their work have been plotted as a graph of pressure against streamwise direction. The stream wise direction represents the length of the aneurysm, with a solid blackline to represent the surface features of the aneurysm. The system used in their model is a doubleaneurysm system however for comparison with the pressure distribution in this model that differencedoes not matter. The different models plotted are represented as Reynolds number with an increase inReynolds number representing an increase in size of aneurysm. This graph can be seen in figure 28,the graph also shows a lowering of the pressure at the proximal end of both aneurysms this can beenseen at a stream wise direction of 0 and 2 cm.
Figure 28: A graph showing a two aneurysm system which shows a pressure increase at the distal end of theaneurysm [44].
It has been suggested in the work of Altuwaijri, that the increase in pressure at the distal end of theaneurysm is caused by the formation of the vortices [7]. This hypothesis is supported by figure 37
26
and 38 in the appendix which shows the pressure profiles of all models. The reason for this is thatin models A and B there are no vortices and in neither of these is there an increase in pressure atthe distal end of the aneurysm. Whereas in models D to H there are vortices present in all cases,along with a region of increased pressure at the distal end of the aneurysm, supporting the hypothesissuggested by Altuwaijri. An exception to this hypothesis is model C which is the smallest aneurysmto contain vortices. In this model there is an increase in pressure towards the distal end of theaneurysm but this increase extends half way up the aneurysm and is not primarily focused at thedistal end.
The findings of the velocity profile of the aneurysm show that there was a velocity gradient in eachof the models, with an increase in velocity at the outlet of the aneurysm compared to the inlet in allmodels. This same trend in velocity was found in the work of Altuwaijri [7], who suggested that theincrease in velocity was due to the recirculation of the flow which rejoins the main stream down thecentre of the aneurysm and causes an increase in the velocity at the outlet.
It was shown that the recirculation of the flow occurs in all models of aneurysm except for the controlmodel, therefore this suggests that recirculation occurs in all aneurysms. This is because model B isthe smallest the aneurysm can be by the definition of an aneurysm and this contains a recirculationof the flow. It was also shown that the recirculation contains vortices in aneurysms with a diameter≥ 4 cm. These vortices occur at the distal end of the aneurysm in all models as found in the workof Patel [34]. Patel also states that the intensity and the size of the vortices increased as the size ofthe aneurysm increased [34], by inspection of figure 41 to 44 in the appendix it can be seen that thevortices do increase in size as the aneurysm grows. However there is no numerical data to supportthis statement and no way to determine the intensity of the vortices.
In the results of the WSS of the aneurysm it was found that most of the region of the aneurysm wasunder a low WSS. However there was a region of high WSS at the inlet and outlet of the aneurysm,this result was also found in the work of Papaharilaou et al [19]. The region of low WSS in the bulgeof the aneurysm is attributed to the recirculation of the flow [34]. This recirculation decreases thevelocity gradient near the walls of the aneurysm and thus decreasing the WSS, equation (8) representsthis link.
Within this region of low WSS in the bulge of the aneurysm, there is an area at the distal end thatwas found to be at a higher WSS in all models that contained vortices. Which would suggest thatthis region of high WSS was linked to the formation of vortices, this theory is back up by the workof Patel [7] who found that vortices cause this increase in WSS.
The effect that the size of the aneurysm has on the volume average WSS was then found, whichwas represented in figure 25. This graph showed that as the size of the aneurysm increased thevolume averaged WSS decreased, this seems intuitive as when the aneurysm grows there becomes alarger volume of flow which is recirculating, seen in figure 39 and 40 in the appendix. This largervolume of recirculating flow decreases the velocity gradient near the wall and thus decrease the WSSvia equation (8). This contradicts the law of Laplace which many thought to be a good theoreticalbasis for using the diameter of the aneurysm to assess the risk of rupture [30]. However there are tworeasons why using the law of Laplace to predict the risk of rupture is not sufficient, the first of which
27
is that an aneurysm is far from a simple cylinder rather an aneurysms wall is complexly shaped [45].The second reason is that the law of Laplace only considers the diameter of the pipe as a cause ofWSS [46]. In an AAA the diameter is not the only determinant of WSS, one other factor previouslymentioned which contributes to the WSS is the recirculation of the flow.
The maximum value of the WSS of the aneurysm was then found and this contradicted what wasfound in the volume averaged WSS as there was no inverse relationship between maximum WSSand size of aneurysm which might have been expected. It was actually found that the maximumvalue of WSS occurred in an aneurysm of diameter 4 cm, this is represented in figure 27. This sameresult was also found in the work of Peattie et al [47] who also observed that the maximum value ofWSS occurred at a diameter of 4 cm. The results of their work can be seen in figure 29.
Figure 29: A graph showing the relationship between the WSS and the diameter (cm) of an aneurysm [47].
From the findings of the work of Peattie et al, they concluded that ‘medium’ sized aneurysmswere more prone to maximum values of WSS. This conclusion would support the statement thatconsidering the WSS as a determinant factor in the risk of rupture would be advantageous. As ifthe WSS in these medium sized aneurysms exceeds the strength of the walls then the aneurysm willrupture. From the data of ‘The UK Small Aneurysm Trail’ it was found that 5% of these mediumsized aneurysms rupture and 77% of these were fatal. The results found within this project alsosupport this hypothesis as it was found that the maximum WSS also occurred in ‘medium’ sizedaneurysms. In this project there lacks any evidence other than the WSS, that there are any otherfactors that could be contributing to the risk of rupture. So it can be said that the regions of highWSS localized at the distal end of the aneurysm are the most likely cause of the risk of rupture ofmedium sized aneurysms.
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5 Conclusion
Within this project the specific hemodynamics of blood have been found for various sizes of AAA.It has been shown in both the theoretical model and the computational model that in an AAA thereis a pressure and velocity gradient, which are inversely related. Another property of the blood flowwhich was found was that all aneurysms contain a recirculation of blood which occurs at the distalend of the aneurysm and recirculates around the edge of the aneurysm reconnecting to the mainstream down the centre of the aneurysm. This recirculation increases in volume as the aneurysmgrows and has been shown to have an inverse relationship with the volume average WSS. Thus thevolume average WSS decreases as the aneurysm grows.
The recirculation of blood has been shown to contain vortices in aneurysms that have a diameter≥ 4 cm. In the cases where vortices are present there is a localization of a high pressure and WSS atthe distal end of the aneurysm, which has been linked with the formation of such vortices. This haslead to the conclusion that vortices play an important role in the hemodynamics of AAAs and have acontribution to the maximum value of WSS found in an AAA.
There is significant evidence to support the hypothesis that WSS should be a factor considered whenassessing the risk of rupture of a patient with an AAA. The reason for this is that the maximum WSSin an AAA was not found to be in an aneurysm with the largest diameter. It was actually found in amedium sized aneurysm which by the current assessment criteria for the risk of rupture would havebeen left untreated. This could have lead to the rupture of this aneurysm and most probably the deathof the patient. For this reason alone it is concluded that the WSS should be a factor considered whenassessing the risk of rupture of an AAA.
6 Further Work
To further develop this project the next stage would be to use a model which was not axisymmetric,rather use a realistic geometry of an AAA which has been developed via an imaging technique suchas a CT scan. By implementing this it would cause many other factors to affect the hemodynamicsof blood such as the asymmetry, neck angle and the bifurcation angle of the abdominal aorta. Somework has already been conducted in this area [9], [34] and [48].
7 Acknowledgements
I would like to thank my supervisor Dr Shane O’Hehir for continual guidance throughout the projectand for useful discussion when problems were found. I would also like to thank Dr Tim Cawthornefor useful discussion about the fluid dynamics involved in the flow of blood.
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References
[1] J.W.Hurst, The heart arteries and veins, 4th edition, 1979, New York, McGraw-Hill.
[2] R.Uflacker, and C.J.Feldman, Atlas of vascular anatomy: an angiographic approach, 2007,Philadelphia: Lippincott Williams & Wilkins.
[3] Y.Fung, Biomechanics: mechanical properties of living tissues, 1993, San Diego, Springer.
[5] J.C.Lasheras, The Biomechanics of Arterial Aneurysms, Annu. Rev. Fluid Mech, 2007,39:293-319
[6] N.Filipovic, Hemodynamic flow model through an abdominal aortic aneurysm using data miningtools, 2011, BioMed, Vol 15, no 2
[7] O.Altuwaijri, Advanced Computer Modeling of Abdominal Aortic Aneurysms to Predict Riskof Rupture, Unpublished PhD thesis, The university of Hull, 2012
[8] G.N.Foutrakis, H.Yonas, and R.J.Sclabassi, Saccular Aneurysm Formation in Curved andBifurcating Arteries, 1999, Am J Neuroradiol 20:1309-1317.
[9] E.Soudah, CFD Modelling of Abdominal Aortic Aneurysm on Hemodynamic Loads Using aRealistic Geometry with CT, 2013, Computational and Mathematical Methods in Medicine.
[12] M.D.Silverstein et al. Abdominal aortic aneurysm (AAA): cost-effectiveness of screening,surveillance of intermediate-sized AAA, and management of symptomatic AAA, BUMCPROCEEDINGS 2005;18:345-367.
[13] C.M.Brady, Open Repair of Abdominal Aortic Aneurysms, 2012, Wiley-Blackwell.
[14] D.C.Brewster et al, Guidelines for the treatment of abdominal aortic aneurysms, 2003, J. Vasc.Surg, Volume 37, Number 5.
[15] J.C.Parodi, Endovascular repair of abdominal aortic aneurysms and other arterial lesions, J.VASC. SURG 1995;21:549-57.
[16] P.M.Brown, D.T.Zelt, B.Sobolev, The risk of rupture in untreated aneurysms: The impact ofsize, gender, and expansion rate, 2003, J. Vasc. Surg, 37:280-84.
[17] S.C.Nicholls et al, Rupture in small abdominal aortic aneurysms. J. Vasc. Surg, 1998;28:884-8.
[18] L.C.Brown and J.T.Powell, Risk Factors for Aneurysm Rupture in Patients Kept UnderUltrasound Surveillance, 1999, ANNALS OF SURGERY, Vol. 230, No. 3, 289-297.
[19] Y.Papaharilaou et al, A decoupled fluid structure approach for estimating wall stress inabdominal aortic aneurysms, 2007 J. Biomech, 40(2): 367-377.
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[20] M.F.Fillinger et al, Prediction of rupture risk in abdominal aortic aneurysm during observation:Wall stress versus diameter, J Vasc Surg, 2003;37:724-32.
[21] D.A.Vorp et al, Mechanical wall stress in abdominal aortic aneurysm: Influence of diameterand asymmetry, J Vasc Surg, 1998;27:632-9.
[22] G.K.Batchelor, An introduction to fluid dynamics, 2014 Cambridge university press.
[23] G.Satish et al, Comparison of flow analysis of a sudden and gradual change of pipe diameterusing fluent software, 2013, IJRET Volume: 02 Issue: 12.
[31] A.V.Salsac et al, Evolution of the wall shear stresses during the progressive enlargement ofsymmetric abdominal aortic aneurysms, J. Fluid Mech, (2006), vol. 560, pp. 19-51.
[32] C.M.Scotti, E.A.Finol. Compliant biomechanics of abdominal aortic aneurysms: Afluid–structure interaction study. Computers and Structures 85 (2007) 1097-1113.
[34] P.Vashishth, Impact of geometry on blood flow patterns in abdominal aortic aneurysms, 2011,Phd Thesis, Graduate School-New Brunswick Rutgers, The State University of New Jersey.
[35] C.J.Egelhof, R.S. Budwig, et al, Model studies of the flow in abdominal aortic aneurysmsduring resting and exercise conditions, J Biomech 32 (1999) 1319-1329.
[36] D.Elger, The influence of shape on the stresses in model abdominal aortic aneurysms, 1996, Jbiomech eng, 118: 326-332.
[37] T.J. Tautges, Automatic Detail Reduction for Mesh Generation Applications, Proceedings, 10thInternational Meshing Roundtable, SANDY2001-2976C, 2001, Sandia National Laboratories, pp.407-418.
[38] M.W.Siebert, Newtonian and Non-Newtonian Blood Flow over a Backward-Facing Step – ACase Study, 2009, Physics Department, Cleveland State University.
[39] A.R.Paterson. A first course in fluid dynamics, 1997, Cambridge university press.
[41] Ansys,Inc, Ansys 14.5 Help, 2012, Canonsburg, Pa
[42] J.H.Leung, Fluid structure interaction of patient specific abdominal aortic aneurysms: acomparison with solid stress models, 2006, Biomedical Engineering Online, 5:33.
[43] S.Kelly, M.O’Rourke, A two-system, single-analysis, fluid–structure interaction techniquefor modelling abdominal aortic aneurysms, 2009, Journal of Engineering in Medicine, 224:0954-4119.
[44] E.A.Finol, Flow-induced Wall Shear Stress in Abdominal Aortic Aneurysms: Part I – SteadyFlow Hemodynamics, Computer Methods in Biomechanics and Biomedical Engineering, 2002Vol. 5 (4), pp. 309-318
[45] M.S.Sacks et al, In vivo three-dimensional surface geometry of abdominal aortic aneurysms,Annals of Biomedical Engineering, 1999;27:469-79.
[46] J.P.Vande Geest et al, A noninvasive method for determination of patient-specific wallstrength distrubtion in abdominal aortic aneurysms, Annals of Biomedical Engineering,2006a;34:1098-1106.
[47] A.Peattie, J.Riehle, I.Bluth, Pulsatile flow in fusiform models of abdoiminal aortic aneurysms:flow fields, velocity patterns and flow-induced wall stresses. 2004, J. Biomech. Eng, 126: 438-46.
[48] M.L.Raghavan et al, Wall stress distribution on three dimensionally reconstructed models ofhuman abdominal aortic aneurysm, Journal of Vascular Surgery, 2000;31:760-9.
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8 Appendix
Figure 30: A graph to show the convergence test of model A in order to determine a suitable size mesh.
Figure 31: A graph to show the convergence test of model B in order to determine a suitable size mesh.
33
Figure 32: A graph to show the convergence test of model C in order to determine a suitable size mesh.
Figure 33: A graph to show the convergence test of model D in order to determine a suitable size mesh.
34
Figure 34: A graph to show the convergence test of model E in order to determine a suitable size mesh.
Figure 35: A graph to show the convergence test of model G in order to determine a suitable size mesh.
35
Figure 36: A graph to show the convergence test of model H in order to determine a suitable size mesh.
Figure 37: Images of the pressure profile of models A-D.
36
Figure 38: Images of the pressure profile of models E-H.
Figure 39: Images of the velocity profile of models A-D.
37
Figure 40: Images of the velocity profile of models E-H.
Figure 41: Images of the velocity streamlines of models A-D.
38
Figure 42: Images of the velocity streamlines of models E-H.
Figure 43: Images of the velocity vectors of models A-D.
39
Figure 44: Images of the velocity vectors of models E-H.
Figure 45: Images of the wall shear of models A-D.
40
Figure 46: Images of the wall shear of models E-H.
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Project Proposal
Bradley Campbell
Project supervisor: Dr Shane O’Hehir
February 24, 2016
Project Title
Computational Fluid Dynamics Model of Blood Flow Through a Fusiform Aneurysm.
Arrangements for contact with supervisor and 2nd supervisor
Regular meetings have been scheduled with my supervisor every Tuesday at 5:00pm. A progressmeeting has also been scheduled with my secondary supervisor Dr Tim Cawthorne on Wednesday16th March at 3:00pm.
Brief description of project
The aim of this project is to produce a computational fluid dynamics (CFD) model of the flow ofblood through an axisymmetric abdominal aortic aneurysm (AAA). The reason for this will be toassess the effect that different size aneurysms have on the characteristics of blood flow.
Ansys 14.5 is the software that will be used to produce the CFD model. This will allow the flowof blood to be visualised through the AAA. The relationship between the maximum diameter of theaneurysm and the blood flow can then be determined. The link between the diameter of the abdomi-nal aorta and the blood flow will also be investigated.
A theoretical model of the flow of blood through an AAA will be derived for comparison with resultsof the CFD model. This theoretical model will encompass a flow which is viscous, incompressibleand non-Newtonian.
The results of the CFD model will also be compared to those from literature. This will includecomparison with computational models, in vivo and in vitro measurements.
The different factors which cause the genesis and continual growth of an aneurysm will be reviewed.The treatments of AAAs will also be investigated whether these be the best treatment at present orthose still in development.
1
Reason for undertaking the work
The main reason for undertaking this work is to understand the pathophysiology of aneurysms. Thiswill include the study of the velocity of blood and wall shear stress through AAAs. The aim of whichis to understand how these factors affect the growth of an aneurysm. The effect that treatments ofAAAs have on the blood flow will also be investigated.
Review of literature
A definition of an aneurysm was established in 1991 by the Society of Vascular Surgery and the In-ternational Society of Cardiovascular Surgery as “a permanent localized dilation of an artery having50% increase in diameter compared with the normal diameter of the artery.” [10] One of the causesof this dilation of the artery has been shown to be due to the pressure exerted on the arterial wallby the flow of blood. Over time this pressure causes fatigue of the arterial wall which can lead to apermanent dilation of the wall. [1]
Another factor that could contribute to the further enlargement of an AAA is the recirculation ofthe flow in the form of vortices. [11] Due to this recirculation, a build up of plaque forms in morethan 90% of all cases of AAAs. This covers the wall of the aneurysms and causes the destructionof the endothelial layer by hypoxia. [2] This further weakens the arterial wall causing growth of theaneurysm.
The Navier-Stokes equation is the fundamental equation for the flow of a viscous incompressiblefluid (such as blood) this can be used to derive a theoretical model. [3] The Navier-Stokes equationin vector notation is given by:
ρD~u
Dt= ρ~F −∇p+ µ∇2~u (1)
This is where ρ is the density of fluid, ~u is the velocity vector, ~F is external force per unit mass, p ispressure and µ is viscosity.[24]
Many CFD models have been made to determine the flow of blood through an aneurysm. An exam-ple of one such model using Ansys is shown in figure 1, this shows the velocity profile through anAAA. Many other CFD models have been made to model the flow of blood through aneurysms. [5]- [11]
2
Figure 1: An image of the velocity profile of blood through an AAA showing the recirculation of blood nearthe arterial wall. [4]
Models of fluid flow through an aneurysm can also be determined experimentally using a pumpingsystem to pump a blood-like fluid through a glass chamber which is shaped like an aneurysm. Anexample of such set up is shown in figure 2. [12]
Figure 2: A schematic of the experimental set-up to mimic the flow through an aneurysm. [12]
The flow patterns through the set-up in figure 2 can then be found using particle image velocimetry.Similar experimental studies can be found in [13] - [17].
3
In vivo measurements of the flow of blood through aneurysms have been imaged using techniquessuch as CT scans and MRI scans. Also an intra-arterial dual-sensor guide-wire can be used, theresults of this can be seen in [18]. Results from other imaging techniques can be found in [19] and[20]. An example of results from the reconstruction of a CT scan can be seen in figure 3.
Figure 3: An image of the stress profile through an AAA reconstructed from CT scan results. [21]
AAA rupture is the 15th leading cause of death in the united states, affecting patients over 55 yearsof age, typically 2-4% of elderly males.[22] Current treatments for AAAs are offered based on twocriteria; it must either have a maximum diameter of 55mm or a growth rate of over 5mm in a 6 monthperiod. [23]
There are currently two techniques for the treatment of AAAs. [25] The first of which is endovascu-lar surgery. A type of keyhole surgery in which a metal graft is inserted into the artery of the leg andmoved up into the abdominal area where the aneurysm is located. The addition of this graft reducesthe stress on the arterial walls. The second technique is open surgery in which the stomach is cutopen and the aneurysm is removed and completely replaced by a metal graft.
4
Where the work will be carried out and software used
The main bulk of the work will be carried out in the library or in LE203 in Leighton building of theUniversity of Central Lancashire. The reason for this is that the software needed to perform the workis available in these locations. This software will be Ansys 14.5 which is a piece of computer-aidedengineering software which can be used to model CFD models.
Action plan
A CFD model of the flow of blood through various sizes of AAA should be finished by 6th March,along with a theoretical model of the blood flow. This will allow for nearly two weeks for them tobe analysed and compared before a progress meeting with my secondary supervisor on Wednesday16th March. After this there is some time for the models to be refined and improved before writingthe report, for which a draft will be handed in on the 20th April. Leaving two weeks to get feedbackand modify before the final deadline on the 11th April 2016.
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