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A Study of the Effect of Atherosclerosis in the Left
Coronary Artery Bifurcation Using 3D Numerical
Simulation
By Andrew Keaveney
Project Supervisor: Dr Adel Nasser
Submitted to the Faculty of Mechanical, Aerospace and Civil Engineering in April of 2015, in
Partial Fulfilment of a Masters in Aerospace Engineering

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Abstract
This report has been compiled with a view to offering members of the medical profession an insight and
reference for their understanding of the effect of atherosclerosis in blood vessels, with the hope that the
research will facilitate an improvement in the quality of their prognosis when diagnosing patients, and allow
for earlier detection of the manifestation of the disease. The primary focus of the study is to analyse the
effects of varying degrees of stenosis on flow conditions in a model of a diseased left coronary artery
bifurcation using CFD analysis, and to link these conditions back to theories related to the onset and
development of atherosclerosis. The study then proceeds to analyse the effects of using a turbulence model,
and explores the use of the endocardial viability ratio (EVR) as a new method for determining the level of
disease at which a heart attack occurs. The main findings of a thorough literature review, as well as data from
reliable and credible sources will be used to justify and validate the models used in this report so that the data
and conclusions drawn can be regarded as trustworthy. The project takes a tried and tested stance in its setup
which replicates real hemodynamic behaviour well, incorporating the Non-Newtonian Carreau-Yasuda
model, 3D geometries and transient pulsatile inlet conditions into a CFD finite volume analysis. It is hoped
that by calculating the wall shear stresses (WSS), velocity profiles, pressure distributions and mass fluxes
through the domain, the flow can be visualised so that flow phenomena such as recirculation and stagnation
can be easily identified and located.
The study has found that recirculation and altering WSS becomes increasingly prominent within the domain
as the disease progresses, showing an accelerated progression after the initial blockage. The apex of the
stenosis profile is subject to very high WSS, which indicates that plaque deposit would further develop at this
point. This high WSS increases the likelihood of the plaques rupturing and the development of thrombosis
(blood clot). The LAD branch is also subject to varying WSS, so plaque build-up would be likely to develop
here in a scattered manner, while the LCX branch remains in conditions that indicate that no stenotic build up
would occur. However, due to recirculation behind the stenosis, it is likely that plaque deposit would occur at
the entrance to the LCX at the bifurcation point. High velocities through the stenosis indicate that the artery
in this region would suffer damage to the endothelial layer in the 60% and 80% cases. The LCX branch also
suffers from a 22% reduction in mass flow rate for 60% stenosis, and reversed flow is a prominent feature in
this branch for the 80% case due to the development of a strong negative pressure region just past the
stenosis. In general, conditions in the model up to 60% stenosis would be tolerated by an actual artery, but
not past this point. Using the newly proposed EVR analysis, it was found that subendocardial ischaemia
would occur in this geometry at around 67% blockage, and in these conditions, a heart attack would be likely.

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Table of Contents
Abstract...........................................................................................................................................................2
List of Figures ................................................................................................................................................4
List of Tables..................................................................................................................................................6
Glossary of Terms..........................................................................................................................................7
Nomenclature .................................................................................................................................................9
1 – Introduction............................................................................................................................................11
1.1 –Atherosclerosis................................................................................................................................11
1.2 –Motivation for Study.......................................................................................................................14
1.2 - Project Scope and Objectives..........................................................................................................16
2 - Literature Review....................................................................................................................................17
2.1 – Biological Background...................................................................................................................17
2.1.1 - The Cardiovascular System ....................................................................................................17
2.1.2 - Arteries....................................................................................................................................18
2.1.3 – The Left Coronary Artery ......................................................................................................19
2.1.4 – Treatment of Atherosclerosis.................................................................................................20
2.2 – History of Numerical Simulation in Biological Networks ............................................................22
2.2.1 - Overview.................................................................................................................................22
2.2.2 – History of Work......................................................................................................................23
3 - Applied Theory.......................................................................................................................................28
3.1 – Navier - Stokes Equations..............................................................................................................28
3.2 - The Womersley Number.................................................................................................................29
3.3 - Reynolds Number ...........................................................................................................................29
3.4 - Wall Shear Stress (WSS) ................................................................................................................31
3.5 – Non – Newtonian Carreau - Yasuda Model...................................................................................31
4 - Methodology...........................................................................................................................................33
4.1 – Geometries......................................................................................................................................33
4.1.1 - Vessel Dimensions..................................................................................................................33
4.1.2 - Models.....................................................................................................................................34
4.1.3 - Meshing...................................................................................................................................36
4.2 - Simulation Setup.............................................................................................................................38
4.2.1 - Assumptions............................................................................................................................38
4.2.3 - Boundary Conditions ..............................................................................................................39
4.2.2 – Discretisation..........................................................................................................................40

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5 – Results and Discussions .........................................................................................................................41
5.1 – Validation.......................................................................................................................................41
5.2 - Mass Flow Rates.............................................................................................................................43
5.3 – Velocity Analysis...........................................................................................................................46
5.4 - Wall Shear Stress Analysis.............................................................................................................52
5.5 - Pressure Analysis............................................................................................................................58
5.6 – Turbulence Modelling....................................................................................................................59
5.7 - Endocardial Viability Ratio (EVR).................................................................................................61
6 – Conclusions............................................................................................................................................65
7 - Future Work............................................................................................................................................68
Project Planning and Management...............................................................................................................69
References ....................................................................................................................................................72
Appendices...................................................................................................................................................76
Appendix A – Pulsatile Flow Conditions................................................................................................76
Appendix B – Blood Properties ..............................................................................................................77
Appendix C – K-Omega Turbulence Model Constants..........................................................................77
Appendix D – Meshes.............................................................................................................................78
Appendix E – Mesh Independence Tests................................................................................................80
Appendix F – Animation Link ................................................................................................................80
List of Figures
Figure 1: Visualization of Atherosclerotic Plaque Formation (Kanyanta et al. 2014) ................................11
Figure 2: Human Case of Atherosclerosis in the Carotid Artery Bifurcation (Uthman, 2006)...................12
Figure 3: Comparison of Pressure Gradients in Diseased and Normal Conditions (Chalyan. 2008)..........13
Figure 4: Atherosclerosis Development with a Blood Clot (Buy-toniclife.com, 2015)..............................14
Figure 5: Breakdown of Cardiovascular Disease Subsections (American Heart Association. 2011).........14
Figure 6: The Cardiovascular System (Digihealer. 2014) ...........................................................................17
Figure 7: Heart Pump Cycle for Large Arteries (Quarteroni. 2006)............................................................18
Figure 8: Artery Anatomy (Quarteroni. 2006).............................................................................................18
Figure 9: 3D CT Visualisation of a Diseased Left Coronary Artery (Chaichana et al’s (2011))................20
Figure 10: The Process of Surgical Angioplasty (Chalyan. 2008) ..............................................................21
Figure 11: Variation of Viscosity with Shear Rate in the Carreau Model (Arc.vt.edu, 2015)....................32

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Figure 12: Front View of CAD for Standard Geometry..............................................................................34
Figure 13: Side On View of CAD for Standard Geometry..........................................................................34
Figure 14: Bottom View of CAD for Standard Geometry...........................................................................34
Figure 15: Top View of CAD for Standard Geometry ................................................................................34
Figure 16: 30% Stenosis Case Study............................................................................................................35
Figure 17: 60% Stenosis Case......................................................................................................................35
Figure 18: 80% Stenosis Case Study............................................................................................................36
Figure 19: Near Wall Inflation Propagation for Standard Geometry Mesh.................................................37
Figure 20: Velocity Profile of Blood Due to Wall Shear Stresses (Klabunde. 2011) .................................39
Figure 21: Theoretical Inlet Conditions (Sinnott et al. 2006)......................................................................39
Figure 22: Comparison of Velocity Fields Through a Central Plane for Validation (Case Study – Left)..41
Figure 23: Comparison of Wall Shear Stresses for Validation....................................................................42
Figure 24: Inlet Velocity Profile (Taken from an Area-Weighted Surface Monitor)..................................42
Figure 25: Transient Mass Flow Rate Through the LCX ............................................................................45
Figure 26: Transient Mass Flow Rate Through the LAD ............................................................................45
Figure 27: Percentage of Mass Flow Through the LCX with Disease Progression ....................................46
Figure 28: Boundary Layer Profiles at 1mm (Left) and 48mm (Right) into the Domain ...........................47
Figure 29: Velocity Through a Central Plane @ Peak Sytole for 1) Standard Geometry 2) 30% Stenosis
3) 60% Stenosis 4) 80% Stenosis................................................................................................................48
Figure 30: Velocity Through a Central Plane @ 0.5 Seconds for 1) Standard Geometry 2) 30% Stenosis
3) 60% Stenosis 4) 80% Stenosis.................................................................................................................49
Figure 31: Velocity Contours Through the Area of Stenosis @ Peak Systole for 1) Standard Geometry
2) 30% Stenosis 3) 60% Stenosis 4) 80% Stenosis....................................................................................50
Figure 32: Streamlines and Velocity Vectors @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis
3) 60% Stenosis 4) 80% Stenosis.................................................................................................................51
Figure 33: Wall Shear Stresses @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60%
Stenosis 4) 80% Stenosis..............................................................................................................................52
Figure 34: Wall Shear Stresses @ 0.5 Seconds for 1) Standard Geometry 2) 30% Stenosis 3) 60%
Stenosis 4) 80% Stenosis..............................................................................................................................53
Figure 35: Trendline for Wall Shear Stress Development (at Mid-Point of Stenosis Profile @ Peak
Systole) with Level of Stenosis....................................................................................................................56
Figure 36: Transient Profile of Wall Shear Stress at Mid-Point of Stenosis...............................................56

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Figure 37: 3D Plot of Wall Shear Stresses Along the Bottom Half of the Stenosis Profile for 30%
Restriction.....................................................................................................................................................57
Figure 38: Pressure Contours @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis
4) 80% Stenosis............................................................................................................................................58
Figure 39: Pressure Contours @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis
4) 80% Stenosis............................................................................................................................................60
Figure 40: Velocity Contours (Left) and WSS (Right) @ Peak Systole for 80% Stenosis with a..............60
Figure 41: Endocardial Viability Ratio with Level of Stenosis...................................................................64
Figure 42: The User Defined Function Used as Inlet to LM (Jiang, 2015).................................................76
Figure 43: C Source Script for User Defined Function at Inlet (Provided by Jiang (2015)).......................76
Figure 44: Mesh for Standard Geometry......................................................................................................78
Figure 45: Mesh for 30% Stenosis Case Study............................................................................................78
Figure 46: Mesh for 60% Stenosis Case Study............................................................................................79
Figure 47: Mesh for 80% Stenosis Case Study............................................................................................79
List of Tables
Table 1: Considerations Made in Simulation Set Up During Studies .........................................................22
Table 2: Standard Geometry Dimensions ....................................................................................................33
Table 3: Comparison of the Mass Flow Rates Through Each of the Outlets @ 0.5 Seconds .....................43
Table 4: Comparison of the Mass Flow Rates Through Each of the Outlets @ 0.11 Seconds ...................43
Table 5: Standard Geometry EVR Analysis ................................................................................................62
Table 6: Stenosis EVR Analysis ..................................................................................................................62
Table 7: 60% Stenosis EVR Analysis..........................................................................................................62
Table 8: Stenosis (Turbulence Model Results) EVR Analysis....................................................................63
Table 9: Newtonian Blood Model Parameters (Shanmugavelasyudam et al (2010))..................................77
Table 10: Non – Newtonian Blood Model Parameters (Ma and Turan. 2011)............................................77
Table 11: K-Omega Turbulence Model Constants ......................................................................................77
Table 12: Standard Geometry Mesh Attributes ...........................................................................................78
Table 13: 30% Stenosis Case Study Mesh Characteristics..........................................................................78
Table 14: 60% Stenosis Case Study Mesh Characteristics..........................................................................79
Table 15: 80% Stenosis Cases Study Mesh Characteristics ........................................................................79
Table 16: Mesh Independence Tests............................................................................................................80

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Glossary of Terms
Aneurysms – A serious physical dilation of an artery into an almost balloon like shape
Angina – Chest pains caused by Atherosclerosis
Anti-Coagulants – Drugs that prevent the clotting of blood
Anti-Platelets – Drugs that reduce the aggression of platelets and prevent thrombus formation
Bifurcation – The branching of a main body into two separate bodies
CFD - Computational Fluid Dynamics, Type of Numerical Simulation
Claudication – Pain, tiredness and discomfort in legs during physical work due to lack of oxygenated
blood supply. Can also mean impairment of the legs
Coagulates – When blood clumps together and becomes solid
DICOM – Digital Imaging and Communications in Medicine
Fibrinolysis – The process of breaking down blood clots
FEM – Finite Element Method, Type of Numerical Simulation
Hemodynamics – Hemodynamics is an important part of cardiovascular physiology, dealing with the
forces the heart has to develop to circulate blood throughout the cardiovascular system
In Vitro – Studies undergone on living organisms outside of the body
In Vivo – Studies undergone on living organisms inside the body
Ischaemia – A condition that occurs when oxygen demand outstrips supply
Lumen – Central space in blood vessels where blood flows
MRI – Magnetic Resonance Imaging
Myocardium – The muscle mass of the heart
Occlusion – Obstruction of a passage
Pathological – The study of an organism in diseased conditions

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Physiological – The study of an organism in normal conditions
QUICK Spatial Discretization - Quadratic Upwind Interpolation for Convective Kinematics
Rheology – The flow of matter, normally in liquid form
SIMPLE Pressure Velocity Coupling - Semi- Implicit Method for Pressure-Linked equations
Stenosis - The narrowing of blood vessels due to atherosclerotic build up
Thrombosis – Blood clots
Vasoconstrictors – Cause the muscle in blood vessels to contract, reducing the diameter of the lumen
Vasodilators – Cause the widening of the blood vessels

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Nomenclature
D - Diameter of Blood Vessel Et – Total Energy
𝝆 - Density T - Time
𝝎 - Angular Frequency 𝝉 – Shear Stress
𝝁 - Blood Viscosity F – Force Applied
P - Pressure A – Cross Sectional Area of Material
q – Heat Flux 𝝁 – Newtonian Dynamic (Absolute) Viscosity
𝛈- Non- Newtonian Dynamic (Absolute) Viscosity 𝝉 𝒘 – Wall Shear Stress Tensor
Re – Reynolds Number y – Distance to the Vessel Wall
Pr – Prandtl Number U – Flow Velocity Parallel to the Wall
𝛈 𝒆𝒇𝒇 – Non – Newtonian Effective Viscosity 𝝉 𝒚-Yield Stress
𝜸̇ - Shear Rate 𝛈∞- Non – Newtonian Infinite Shear Viscosity
𝛈 𝟎- Non – Newtonian Zero Shear Viscosity l – Length
u, v and w - Velocity Components in x, y and z x, y and z - Standard Cartesian Coordinates

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t - Time p - Pressure
E - Energy q – Heat Flux
Pr - Prandtl Number
𝜶 - Womersley number
D - Diameter
𝛚 – Angular Frequency
𝝂 - Kinematic Viscosity 𝜹 – Boundary Layer Thickness
𝑹 𝑫
̅̅̅̅ – Rate of Deformation Tensor λ - Time Constant
𝒏 - Power Index

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1 – Introduction
The use of computational numerical simulation in fluid dynamics contexts is wide spread in engineering
environments, and is a pivotal process in acquiring a detailed knowledge of how flow conditions alter
during transit. These techniques are employed so that the expense of experiments can be negated, or when
it is either inconvenient or impossible to collect valid results in real life terms. Using the fundamental
continuity and momentum equations, well known as the Navier-Stokes equations, it is possible to
decipher the flow characteristics of a vast range of flows using numerical simulation, stretching from the
micro analysis of flow around hairs to the macro analysis of flow around buildings.
There are endless applications where computational fluid dynamics can provide a platform for
development, and this project concerns itself with its usefulness in biological systems – namely, the
human cardiovascular system. The primary purpose of the project is to calculate blood flow in a
generalised model of the left coronary artery (LCA) bifurcation, and investigate the effect of
atherosclerosis on the flow domain. The study also analyses the use of a turbulence model on the flow
domain, and explores a new method of determining whether a heart attack will occur using the
endocardial viability ratio (EVR).
1.1 –Atherosclerosis
Atherosclerosis is the hardening of the artery walls and narrowing of the central lumen, and is a condition
that develops due to the congestion of arteries with fatty substances known as plaques or atheroma, as
well as degenerative materials such as lipids, calcium and proliferating cells. (Kanyanta et al. 2014)
Visualisations of this build up can be seen in Figures 1 and 2.
Figure 1: Visualization of Atherosclerotic Plaque Formation (Kanyanta et al. 2014)

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Atherosclerosis is a disease that can have seriously detrimental effects on a person’s health. Stenotic build
up alters the hemodynamic behaviour of blood, and can lead to difficulties regarding increased blood
pressure, thrombosis, strokes and heart attacks. The left coronary artery (LCA) supplies blood to
approximately 70% of the total myocardium (Shanmugavelasyudam et al. 2010), so severe stenosis in this
area is likely to cause a heart attack due to the lack of oxygenated blood supply. As the myocardium has
the highest oxygen consumption per tissue mass of all human organs (Ramanathan & Skinner. 2005),
problems with oxygen and glucose deficiency will occur sooner in the manifestation of atherosclerosis in
the LCA than for other arteries.
Naturally, an artery responds to plaque build-up by dilating by up to 150% of its regular diameter to
maintain the necessary mass flow of blood and regulate its wall shear stress exposure, but atherosclerosis
hardens the artery wall, making it less compliant with mass flow and shear stress needs. As the artery can
no longer dilate to maintain luminal patency, the lumen becomes occluded and the mass flow rate of
blood is indefinitely restricted. (Ai et al. 2010) Figure 3 is an example of the adverse effect of blockages,
depicting the large difference in the pressure gradients of a 65% blockage in an artery when compared
with a reference artery with no stenosis. (Chalyan. 2008)
Figure 2: Human Case of Atherosclerosis in the Carotid Artery Bifurcation (Uthman, 2006)

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In diseased conditions where the blockages exceed 70%, clinical symptoms begin to occur as a result of
restricted blood flow delivery to the essential organs. The main symptom is known as angina – a recurring
pain in the chest caused by the heart muscles overworking to maintain the required mass flow of blood.
Due to atherosclerosis, the heart cannot always supply enough blood to the essential organs, and a patient
may consequently suffer a heart attack or stroke (depending on whether the cerebral arteries to the brain
or the coronary arteries to heart are blocked). Ischemia occurs when the heart muscles require more
oxygen than is being supplied, and this is a precursor to a heart attack. Up to the point at which angina
occurs, the disease is largely symptomless, making early diagnosis of the disease difficult. Often, the
disease is allowed to develop until a resulting heart attack or stroke. Disease of the larger arteries in the
network such as the aorta also reduces blood flow to the extremities, resulting in conditions such as
erectile dysfunction and claudication, as well as muscular pains and tissue damage. (Schwartz & Kloner.
2011)
Another dangerous consequence of the disease is the possibility of developing thrombosis, whereby blood
coagulates on the top layer of the plaque formation due to the plaques becoming unstable and rupturing.
(Muschealth.com. 2015) This thrombosis furthers the narrowing of the lumen and restriction of blood
flow, as shown by Figure 4. Plaque rupture normally occurs when high stresses are concentrated in
vulnerable areas of the plaque surface, where the vulnerable plaques are normally composed of soft
extracellular lipids. This is why an understanding of stress exposure along the stenosis profile is very
important. (Falk. 1992)
Figure 3: Comparison of Pressure Gradients in Diseased and Normal Conditions (Chalyan. 2008)

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1.2 –Motivation for Study
In the developed world, atherosclerosis is the single most prominent cause of death, with one in three
accountable for by it. In the UK alone, there are an estimated 124,000 deaths annually as a result of the
disease, and for every death, there are 2 instances of non-fatal but life changing strokes or heart attacks
(NHS. 2014). In the current social climate of increasing obesity, these numbers are expected to grow, so
an extensive understanding of how atherosclerotic lesions affect the parameters of blood flow in a
multitude of conditions is essential in preventing the disease becoming wider spread (NHS. 2014).
Coronary Heart Disease (CHD) is the most prominent cardiovascular disease subsection, as shown by
Figure 5, and this occurs as a direct result of atherosclerotic lesions blocking the coronary arteries
(American Heart Association. 2011). It is for this reason that this study is looking into a coronary artery.
Figure 4: Atherosclerosis Development with a Blood Clot (Buy-toniclife.com, 2015)
Figure 5: Breakdown of Cardiovascular Disease Subsections (American Heart Association. 2011)

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It is generally well accepted in the field that recirculation zones and extreme or oscillatory flow induced
wall shear stresses within blood vessels can significantly contribute to the development of atherosclerosis
(Sousa et al. 2011) (Seo. 2013) (Kanyanta et al. 2014), so a good knowledge of how these hemodynamic
conditions alter during transit is vital in understanding how to diagnose, treat and prevent the disease.
In terms of measuring these parameters through experimental methods, the technology available at this
point in time is restrictive, with traditional diagnostic methods that focus on blood flow only capable of
detecting plaques in the latter stages of the disease. (Kanyanta et al. 2014) The in vivo measurement of
flow velocity is possible through methods such as Doppler Velocimetry (LDV), the ultrasound Doppler
method and tracking of red blood cells using video monitoring (Sang Joon Lee & Ho Jin Ha. 2012).
However, the poor spatial resolution achieved when measuring the velocity profiles using the Doppler
ultrasound method (the most common method) limits its usefulness (Hamid et al. 1988), and ultrasound
methods interfere with the flows, effecting results. (Ai et al. 2010)
These limitations, combined with the fact that wall shear stresses cannot be directly measured in vivo
(Shanmugavelasyudam et al. 2010), illustrates the importance of the accurate CFD modelling of arterial
response to physiological and pathological conditions in the understanding of the disease. The process
gives an insight to the conditions leading to the initiation and progression of the disease, thus serving as a
tool for early prediction and diagnosis. (Hollander et al. 2011) Members of the medical profession are
unlikely to have the skills required to undergo the computational analysis necessary to depict these flows,
so it is up to engineers who have used CFD and FEM techniques in other applications to branch out and
contribute to this medical research.
It has already been suggested that in the case of an aneurysm under the size of 1 cm, computational
analysis could be completed on the same day as the diagnosis of an acute haemorrhage, as this process
generally only lasted around 8 hours. This means patient specific CFD results could become available
before surgery has even begun, giving surgeons a much greater understanding of the task at hand (Bai-
Nan et al. 2011). The technology also has the capacity to offer predictions for the outcomes of surgeries,
and is a vital process in the design of devices that mimic or alter blood flow in reconstruction and
revascularization operations (Sousa et al. 2011) (Su et al. 2005). This really proves the power of the
technology and how useful it can be in medical services.

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1.2 - Project Scope and Objectives
In compiling this report, it is hoped that a further understanding of the characteristics of blood flow in
pathological conditions will be obtained, to offer members of the medical profession an insight and
reference that could facilitate an improvement in the quality of their prognosis when diagnosing patients.
Although the stance of the project is somewhat generalised, it may offer a platform on which fast and
reputable diagnostic techniques may be developed. This could allow for the prediction and earlier
detection of blockages in arteries, so that informed decisions can be made regarding the subsequent
actions of medical staff.
The characteristics of hemodynamic flow are examined through a generalized, non-patient specific model
of a left coronary artery (LCA) that is subject to varying degrees of blockage. As artery geometry varies
between humans, and a CAT scan of a real geometry has been unobtainable, it is a close approximation of
the average geometry. Pulsatile inlet conditions are applied that replicate realistic conditions at 120 bpm
(moderate workout), and the non-Newtonian Carreau-Yasuda model is used to replicate the material
properties of blood. The blood is assumed incompressible.
Important flow parameters such as pressure, wall shear stress and velocity are calculated and analysed in
four geometries with progressive blockage, in order to obtain an understanding of the downstream
characteristics of the flow in pathological conditions. These findings are then linked back to theories
related to the onset and development of atherosclerosis. Wall shear stresses are examined along the
surface of the stenosis profile to see where the plaque could be subject to rupture, and the mass flux of
blood is analysed to see how the mass flow distribution between the two bifurcation branches varies as
the disease develops. The endocardial viability ratio (EVR) has been calculated for each geometry, as a
new method in this type of study for detecting whether the myocardium would receive enough oxygen to
continue to function when subject to increasing blockage, and the degree of blockage at which a heart
attack occurs has been found. Also, the effect of using a turbulence model in the 80% stenosis case study
has been analysed and compared to the laminar case.
The domain has been calculated using the commercial CFD package Ansys-Fluent, and data is displayed
using a series of contours developed using Ansys Post Processing software. Graphs have been created by
exporting data and plotting using Mathworks Matlab. The results of a simple Newtonian study have been
validated against a case study, so that the results and conclusions drawn subsequently may be considered
trustworthy and reliable. Recommendations on how to improve the study have been given at the end of

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the report, alongside a description of the mistakes made and difficulties endured in the process.
Predictions on how the assumptions made in the simulation set up have affected results have also been
provided.
Firstly, a thorough analysis of the prior literature has been assembled that describes and analyses both the
biological background of the disease and the computational endeavours in its simulation already
undergone in the field. This has been done so that justified decisions could be made in regards to the
simulation setup and incorporated assumptions. An overview of the applied theory employed throughout
the project has then been given, in the hope that further studies can easily incorporate the methods used.
2 - Literature Review
2.1 – Biological Background
2.1.1 - The Cardiovascular System
The cardiovascular system is a complex network that supplies blood to nearly all of the body’s tissues, as
shown by Figure 6. It is made up of 3 components – the heart, the blood vessels and the blood itself. The
heart works in cycles to pump blood containing vital oxygen, nutrients, hormones and glucose to the
necessary destinations, as well as transporting cellular waste. As the heart contracts, it pushes blood out
through the arteries –a stage known as systole. The second stage is called diastole, when the heart is at
rest and drawing blood back to the heart through the venous system. Figure 7 describes the flow rate
through a large artery at time intervals along the cycle. The pulsatile nature of blood flow will be critical
when modelling the flow in this project, as this characteristic induces recirculation zones in the domain,
which is a key part of the development of atherosclerosis.
Figure 6: The Cardiovascular System (Digihealer. 2014)

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Blood is a composite material made up of many components and particles that are suspended in an
aqueous polymer solution called plasma. This property is what makes blood a non-Newtonian material
rather than Newtonian. Approximately 45% of the volume consists of formed elements (mostly red blood
cells), whilst the remaining 55% is the surrounding plasma. (Sousa et al. 2011)
2.1.2 - Arteries
As this project deals with blood flow through a coronary artery, it is important to understand the anatomy
and functionality of this blood vessel. Artery walls are very thick and muscular, so they can withstand the
pressure waves induced by the blood flow after the contraction of the heart. The muscle and elastic fibres
in the thick outer wall enable the arteries to stretch and contract, pushing blood through the vessel even
when the heart is at rest during diastole. One way valves ensure correct flow direction, so during
simulations, the blood can modelled to flow in only one direction, rather than backwards and forwards.
Figure 8: Artery Anatomy (Quarteroni. 2006)
Figure 7: Heart Pump Cycle for Large Arteries (Quarteroni. 2006)

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The distinct layers that make up artery anatomy are described in Figure 8. The endothelium is a thin layer
of cells which lines the inside of the artery, and is thought to be the key mediator of any hemodynamic
effect. (Nerem et al. 1992) The purpose of this lining is to regulate vascular tone and structure, and to
exert anticoagulant, antiplatelet and fibrinolytic properties. The endothelium layer produces
vasoconstrictors (endothelin) and vasodilators (nitric oxide), but should endothelial dysfunction or
damage arise, this can impair the balance in production and lead to further atherosclerosis or the
development of aneurysms. Experimental data shows that arteries that suffer with damage to the
endothelium layer lose their capacity dilate and constrict when compensating for the reduced blood flow
that occurs due to the development of atherosclerosis, and blood flow in inhibited because of this. (Sousa
et al. 2011) A study by Ludmer et al (1986) observed that paradoxical constriction in coronary artery
disease patients indicates that endothelial dysfunction is present in the early stages of atherosclerosis.
The three main layers of the blood vessel are as follows (Knipe & D’Souza. 2014):
- The Intima : Composed of the endothelial cells and a small amount of subendothelial
connective tissue
- The Media: This is the thick, muscular part of the anatomy, providing structural support,
vasoreactivity (i.e. Vasodilation and Vasoconstriction) and elasticity.
- The Adventitia: The connective tissue, nutrient vessels and autonomic nerves
Each layer of the artery contains different amounts of elastin, collagen, vascular smooth muscle cells and
extracellular matrix that characterise the properties and purpose of the layer. (Sousa et al. 2011)
2.1.3 – The Left Coronary Artery
As shown by Figure 9, the LCA descends from the aortic sinus before bifurcating into the left circumflex
artery (LCX) and the left anterior descending artery (LAD). This bifurcation is the key area under
investigation, as atherosclerosis and the corresponding restriction in blood delivery at this point can be
very dangerous. Due to increased blood flow density close to the bifurcation in the LAD, this is an area
prone to atherosclerotic lesion localization (Shanmugavelasyudam et al. 2010), and the arrow in Figure 9
points to a patient specific example of this build-up. Also, studies have found that atherosclerosis occurs
in areas of high curvature and branching, which are two inherent and defining features of the left coronary
artery. (Giddens et al. 1993)

20. 20 | P a g e
The left coronary arteries (and its branches) purpose is to supply oxygenated, nutritious blood to the
myocardium, as well as the heart ventricles and the left atrium. The left circumflex artery (LCX) branch
supplies blood to the muscle mass at the back of the heart, whilst the left anterior descending artery
(LAD) supplies blood likewise to the front of the heart (Heart and Vascular Institute at the George
Washington University. 2014).
The LAD is a direct continuation of the left coronary artery, and branches from this enter the septal
myocardium and supply the anterior two-thirds of the interventricular septum. The LAD is the most
commonly occluded of the coronary arteries, and as it provides the major blood supply to the
interventricular septum, blockage here can be extremely dangerous. The LCX branches off from the LCA
and feeds most of the left atrium. In 40 – 50% of hearts, the LCX also supplies blood to the SA node - the
part of the cardiac conduction system that controls heart rate. (The University of Minnesota, 2015) It is
clear to see why blood flow restriction in this artery is so life threatening.
2.1.4 – Treatment of Atherosclerosis
Treatment of atherosclerosis can be as simple as undergoing lifestyle changes such as eating a healthier
diet and exercising regularly. In more severe cases, medicines can be taken such as Angiotensin-
converting enzyme (ACE) inhibitors, which widen arteries and reduce the amount of water in blood, thus
reducing blood pressure. Alternatives are calcium channel blockers and thiazide diuretics, which also lead
Figure 9: 3D CT Visualisation of a Diseased Left Coronary Artery (Chaichana et al’s (2011))

21. 21 | P a g e
to the widening of blood vessels, as well as anti-platelets, which can be used to thin the blood so it is
easier for it to pass an area of stenosis (NHS Choices. 2014)
In more serious cases, surgery may be necessary to widen the lumen. The most common of these
surgeries is a coronary/carotid angioplasty, whereby a catheter with a small balloon on the end is fed to
the area of interest, at which point the balloon is inflated to increase the diameter of the lumen, as shown
by Figure 10. Small metal tubes called stents are then used to keep the artery open. Other methods include
coronary artery bypass grafts, where segments of blood vessel from other parts of the body are taken and
used to bypass the diseased section, and endarterectomy, whereby the inner lining of the artery is
physically removed, along with the plaque build-up (NHS Choices. 2014).
This biological overview incorporates the main and relevant aspects of the disease. Should further
research be required, some recommended reading is Franklin et al’s (1999) New England Journal of
Medicine entry named “Atherosclerosis — An Inflammatory Disease”.
Figure 10: The Process of Surgical Angioplasty (Chalyan. 2008)

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2.2 – History of Numerical Simulation in Biological Networks
2.2.1 - Overview
Extensive work has been undergone in the analysis of blood flow using numerical simulation, with
varying degrees of success. The simulation setup and assumptions used in the methodologies is pivotal in
acquiring a coherent and realistic set of data.
The factors shown in Table 1 have been considered, manipulated and employed varyingly in each of the
reports studied in order for their specific objectives to be achieved. By analysing the consequences of the
decisions made in an array of studies, and comparing the benefits and pitfalls of each, it is hoped that a
realistic analysis can be undergone with sound justification for every choice. However, the achievement
of realistic results must be balanced with the scope of the project, given the limited completion time and
skillset. This literature review will proceed to analyse the use of each of the considerations given in Table
1, so a clear knowledge of how these variables affect the study can be obtained for any future work in the
field. As well as this, a background to the biological implications of atherosclerosis will be provided.
Type of Vessel Studied, including
where in vessel (such as
bifurcation)
Methods of Compiling Geometry
(e.g. CAT Scan for Patient
Specific)
2D or 3D Model
Newtonian or Non-Newtonian Which Type of Model to Employ
(e.g. Carreau, Power Law, Casson)
Laminar or Turbulence Model
Pulsatile or Non-Pulsatile Flow
and Corresponding Velocity
Distribution
Constant or Transient
Pressure Outlets
Rigid or Elastic Walls
FEM or CFD Fluid-Structure Interaction (FSI)
Analysis or Simply Fluid Analysis
Solving Technique (e.g. finite
difference, finite volume etc.)
Table 1: Considerations Made in Simulation Set Up During Studies
MaterialBoundaryTechniqueGeometric

23. 23 | P a g e
2.2.2 – History of Work
Conflicting arguments regarding the effect of wall shear stress on the development of atherosclerosis are
apparent in Fry’s (1976) and Caro et al’s (1969, 1971, 1973) research. According to Fry, early
atherosclerotic lesions are to be expected in regions with high wall shear stresses, which were found to
induce an increasing endothelial surface permeability. However, according to the contradictory work by
Caro and co-workers, early lesions can develop in regions with low wall shear stresses, due to the shear
dependent mass transport mechanism for Atherogenesis.
It is however well understood that wall shear stresses (WSS) are extremely important in arterial flow, and
are inherently affiliated with atherosclerotic build up. Shanmugavelasyudam et al (2010) supported Caro
et al’s research, agreeing that low WSS results in the formation of atherosclerotic lesions, and research by
Bai-Nan et al (2011) has shown that increased WSS caused by increased flow velocity initiates the release
of endothelium-derived nitrous oxide. This nitrous oxide is a strong vasodilator and also weakens the
arterial wall, thus leading to aneurysms.
Experiments by Fry (1968) also indicated that the maintenance of high levels of WSS for short periods
causes irreversible damage to the endothelial surface, which enhances the permeability of the wall. Fluid
shear stresses, flow reversal and stagnation zones have also been shown to induce vascular oxidation
stress, pro-inflammatory states and arterial internal thickening (Sousa et al. 2011). Many researchers,
including Quarteroni et al (2000), have made findings that indicate that the formation of a mild stenosis
within an artery leads to the accelerated development of atherosclerosis, and this is generally well
accepted in the field.
From these studies, it is clear to see that arteries are very sensitive to extreme and changing wall shear
stresses. Experimentally, wall shear stresses are extremely hard to accurately measure. The in vivo
measurement of the WSS had been tried in a canine aorta using flush mounted hot film probes, but this
proved to be unsuccessful. There was a need for a new approach to calculating this important parameter.
A computational approach was then pursued by numerous members of the profession, with Hamid et al
(1988) simulating the blood flow through 3 different diseased artery conditions – a straight stenosis, a
straight aneurysm and a curved stenosis. The reports main conclusion is that the use of computational
numerical methods is a very promising and useful technology, although realistic results had not been
achieved at the time. The downfalls of the methodology were the Newtonian blood and non-pulsatile inlet

24. 24 | P a g e
assumptions employed in the model, and in the closing remarks, it is stated that the research must be
developed to include bifurcations and branches also.
Using in vitro experiments, Friedman et al (1993) investigated the relationship between the bifurcation
angle in the left main coronary artery and the formation of atherosclerotic lesions. The results showed a
small branching angle may be a geometric risk factor for proximal atherosclerotic disease in daughter
vessels, but a similar research by Perktold et al (1991) and Wells et al (1996) contradicted these finding
and pointed towards a large bifurcation angle amplifying the phenomenon. These findings were then
further supported by Nguyen et al (2008). However, work by Lee et al (2008) disagreed with this
research, and derived that the bifurcation angle has no effect on atherosclerotic build up. The general
opinion in the field is that large bifurcation angles lead to increased atherosclerotic development due to
the increased size of recirculation zones, showing that the in vitro study conducted by Friedman was not
sufficient to analyse this type of flow.
Morega et al (2009) used FEM analysis to compare the results of steady and pulsatile flow through a
bifurcation. The eighth period of the pulsating flow was analysed, so that a quasi-steady flow regime was
sufficiently implemented. The blood was considered Newtonian and laminar. It was found that the steady
flow case did not take in to account the considerable wall stresses inherent in pulsatile flow, so it is clear
that the incorporation of this boundary condition into the case studies is extremely important, as wall
shear stresses are a key mediator in the development of atherosclerosis.
Work undergone by Shanmugavelayudam et al (2010) compares the use of 2D and 3D models in the
numerical solution of the LCA bifurcation using turbulent models. It was found that the 2D model
predicted fairly similar hemodynamic properties under normal arterial conditions, but in diseased
conditions such as atherosclerosis, the model underestimated the shear stress distribution inside the
recirculation zone. It was deemed that a 2D model was sufficient for normal arterial conditions, but not
for diseased conditions.
Blood rheology is a very complex property, and a standard model to depict the non-Newtonian
characteristics of it has not been agreed upon in the field. Buick et al (2007) measured significant changes
in the flow behaviour when comparing the use of two non-Newtonian models with that of a Newtonian
model in their solutions, and concluded that in order to calculate the flow correctly you must employ a
Non-Newtonian model. Three popular choices of model are the Casson, Power law and Carreau-Yasuda.

25. 25 | P a g e
In the modelling of the femoral artery, Dabiri et al (2005) researched the use of the Newtonian model, the
Casson model and the power law model in numerical simulation. It was found that the power law model
did not adequately represent the behaviour of the blood, as it underestimated the shear force, whereas the
non-Newtonian Casson model represented the rheological behaviour well. Also, experiments by Shibeshi
and Collins (2006) found that the power law model does not take into account the fact that blood at rest
requires a yield stress in order to start flowing, whereas the Casson model does account for this
characteristic feature of blood flow. These finding also state that at low shear rates, the Carreau model
coincides with the Power Law model, but at larger shear rate ranges it behaves more like the Casson and
Newtonian models.
Mustapha and Amin (2008) further derived that as there was a significant similarity in the Newtonian and
Casson models, blood flow through large arteries (greater than 1mm) could be considered Newtonian,
whereas in smaller arteries, a non-Newtonian approach is necessary in order to produce realistic results.
Charm and Kurland (1965) made experimental findings that pointed towards the Casson model as the best
representative for blood flow in narrow arteries, and Merrill et al (1965) validated that for vessels
between 130 and 1000 micro metres in diameter, the Casson model satisfactorily depicts the flow
behaviour.
Chaniotis et al (2010) also studied blood flow through coronary segments and found that using a non-
Newtonian assumption had little effect on the WSS at Reynolds numbers typical for blood flow,
somewhat supporting Mustapha and Amin’s (2008) findings that support the use of the non-Newtonian
assumption only in narrow arteries. Fan et al (2009) also conducted a study into the carotid artery
bifurcation, and found that the Casson model is only necessary when depicting the properties of blood
when the shear rate is below 10 s-1
, as when the shear is larger than this, the blood exhibits mainly
Newtonian fluid properties with no difference to the flow characterisation.
Neofytou and Drikakis (2003) similarly compared the Newtonian, Power Law and Casson models, but
also looked at the non-Newtonian Quemeda model on top of this. It is reported that eddy breaking and
eddy doubling develops to a similar extent in the Quemada and Casson models, but to a lesser extent in
the Power Law model. In terms of vorticity generation, the Quemada and Casson models again agree
quite closely, while the Power Law and Newtonian models exhibit higher vorticity levels.
Another important model in blood flow simulation is the non-Newtonian Carreau-Yasuda model. Buick et
al’s (2007) study compared this model with the Casson model for simple steady flow and oscillatory flow

26. 26 | P a g e
in straight and curved pipe geometries, and found significant differences in the steady flow situation.
They derived that “in straight pipe oscillatory flows, both models exhibit differences in velocity and
shear, with the largest differences occurring at low Reynolds and Womersley numbers” when compared
to analogous Newtonian flows, with larger differences occurring in the Casson model. It is stated by
Buick et al (2007) that the differences in these models could prove important in the study of
atherosclerotic development.
Exemplary work that significantly contributed to the further development of the CFD analysis of blood
flow was undergone in separate works by Chuchard et al (2011), Xu et al (2010), Morega et al (2009) and
Kaazempur-Mofrad et al (2003). In these works, computer imaging techniques are used to develop patient
specific geometries upon which boundary conditions are implemented. This process provided enhanced
and accurate geometries that were used to greatly increase the understanding of blood flows in both
general and patient specific applications. Each project makes use of different technologies for geometric
generation; Xu et al took the approach of using CT angiography and 3D digital subtraction angiography in
DICOM format to obtain images that were subsequently processed by MIMICS software (Bai-Nan et al.
2011). Merega et al’s (2009) geometries came from DICOM image sets that comprise angio-MRI 3D
images, whilst Kaazempur-Mofrad et al’s (2003) research involved using MRI imaging and 3D
reconstruction to obtain the patient specific geometries. All of these approaches led to accurate
geometries being obtained, supplying these researchers with a much more viable platform on which to
conduct their research, when compared with other research that simply used average geometries. These
reports can be applied for clinical use.
The inclusion of a turbulence model in simulations is sometimes of interest, as in certain conditions,
blood flow can drift into the turbulent region. Klabunde (2007) states that generally blood flow is laminar,
but in areas of high flow rate (particularly in the ascending aorta), turbulent regions can arise. It is also
stated that turbulent flow can develop at bifurcations, in diseased and stenotic conditions, and across
stenotic heart valves, due to the disruption of the flow. High flow rates caused by physical exercise can
also induce the transition of the flow from laminar to turbulent. The development of turbulent flow has a
significant effect on the pressure-flow relationship in blood vessels, as turbulence increases the coronary
perfusion pressure required to achieve the necessary flow conditions. The perfusion pressure is the
pressure gradient that pushes blood out of the heart, and is an important parameter in the process of
cardiac arrest (Klabunde. 2007) (Sutton. 2014).

27. 27 | P a g e
Compressibility effects in blood have been analysed by Wang et al (2001), and the findings pointed
towards a slight degree of compressibility, with the density ranging from 1,010 to 1,060 g/l. However, all
of the literature reviewed for this project has assumed incompressibility, and it is generally well accepted
that this assumption is valid.
Fluid Structure Interaction (FSI) analysis is becoming increasingly popular in the bioengineering
community, as it simultaneously models the blood flow and wall deformations, and takes into account the
elasticity of arteries. Wood et al (2009) studied the effects of wall compliance on a patient-specific right
coronary artery, and found that there is a significant difference in the results obtained from this when
compared to rigid wall models. Huo et al. (2009) underwent a similar research and found that the time
averaged WSS predicted by the compliant FSI model was smaller than in rigid vessels. The incorporation
of FSI interaction into simulations is very important in the future development of this kind of study, and it
has been shown by many researches that it is essential in depicting the WSS distribution correctly.

28. 28 | P a g e
3 - Applied Theory
3.1 – Navier - Stokes Equations
The flow characteristics in this project are calculated using the compressible, unsteady Navier-Stokes
equations, as shown by Equations 1-6. These equations describe the relationship between the velocity,
pressure, temperature and density of a moving fluid, whilst taking into account the effects of viscosity.
They contain coupled partial differential equations that extend from the Euler equations, and include a
time dependent continuity equation for conservation of mass (taking into account compressibility effects),
three time dependent equations for conservation of momentum (for 3D analysis) and a time dependent
equation for conservation of energy. The terms on the left hand side of the momentum equations are the
convection terms, whilst the terms on the right hand side are the diffusion terms. (NASA. 2014)
Equation 1: Continuity Equation
𝝏(𝒖)
𝝏𝒙
+
𝝏(𝒗)
𝝏𝒚
+
𝝏(𝒘)
𝝏𝒛
= 𝟎
𝝏(𝝆𝒖)
𝝏𝒕
+
𝝏(𝝆𝒖 𝟐
)
𝝏𝒙
+
𝝏(𝝆𝒖𝒗)
𝝏𝒚
+
𝝏(𝝆𝒖𝒘)
𝝏𝒛
= −
𝝏𝒑
𝝏𝒙
+
𝟏
𝑹𝒆
[
𝝏𝝉 𝒙𝒙
𝝏𝒙
+
𝝏𝝉 𝒙𝒚
𝝏𝒚
+
𝝏𝝉 𝒙𝒛
𝝏𝒛
]
𝝏(𝝆𝒗)
𝝏𝒕
+
𝝏(𝝆𝒖𝒗)
𝝏𝒙
+
𝝏(𝝆𝒗 𝟐
)
𝝏𝒚
+
𝝏(𝝆𝒗𝒘)
𝝏𝒛
= −
𝝏𝒑
𝝏𝒚
+
𝟏
𝑹𝒆
[
𝝏𝝉 𝒙𝒚
𝝏𝒙
+
𝝏𝝉 𝒚𝒚
𝝏𝒚
+
𝝏𝝉 𝒚𝒛
𝝏𝒛
]
𝝏(𝝆𝒘)
𝝏𝒕
+
𝝏(𝝆𝒖𝒘)
𝝏𝒙
+
𝝏(𝝆𝒗𝒘)
𝝏𝒚
+
𝝏(𝝆𝒘 𝟐
)
𝝏𝒛
= −
𝝏𝒑
𝝏𝒛
+
𝟏
𝑹𝒆
[
𝝏𝝉 𝒙𝒛
𝝏𝒙
+
𝝏𝝉 𝒚𝒛
𝝏𝒚
+
𝝏𝝉 𝒛𝒛
𝝏𝒛
]
Equation 2: X-Momentum Equation
Equation 3: Y-Momentum Equation
Equation 4: Z-Momentum Equation

29. 29 | P a g e
Equation 6
Equation 7
𝝏(𝑬)
𝝏𝒕
+
𝝏(𝒖𝑬)
𝝏𝒙
+
𝝏(𝒗𝑬)
𝝏𝒚
+
𝝏(𝒘𝑬)
𝝏𝒛
= −
𝝏(𝒖𝒑)
𝝏𝒙
−
𝝏(𝒗𝒑)
𝝏𝒚
−
𝝏(𝒘𝒑)
𝝏𝒛
−
𝟏
𝑹𝒆×𝑷𝒓
[
𝝏𝒒 𝒙
𝝏𝒙
+
𝝏𝒒 𝒚
𝝏𝒚
+
𝝏𝒒 𝒛
𝝏𝒛
] +
𝟏
𝑹𝒆
[
𝝏(𝒖𝝉 𝒙𝒙+𝒗𝝉 𝒙𝒚+𝒘𝝉 𝒙𝒛)
𝝏𝒙
+
𝝏(𝒖𝝉 𝒙𝒚+𝒗𝝉 𝒚𝒚+𝒘𝝉 𝒚𝒛)
𝝏𝒚
+
𝝏(𝒖𝝉 𝒙𝒛+𝒗𝝉 𝒚𝒛+𝒘𝝉 𝒛𝒛
𝝏𝒛
]
These equations are extremely difficult to solve analytically without using assumptions and
simplifications that will alter the accuracy of the results. Therefore computers are used to solve them,
using techniques such as finite difference, finite volume (used by Fluent), finite element and spectral
methods. (NASA. 2014) In blood flow, the assumption that the flow is incompressible can be made to
simplify these equations
3.2 - The Womersley Number
The Womersley number is vital in depicting the pulsatile nature of blood flow. It is a dimensionless
number that is very prominent in bioengineering, and is described by Equation 6. The expression relates
the frequency of the pulses to viscous effects, and can be used when scaling experiments (due to its non-
dimensionality) and calculating the size of the boundary layer in piped flow. The Womersley number
obtained for this project in Equation 7 is a typical value for this blood vessel. The blood is assumed
Newtonian for this calculation, and the frequency is derived from the inlet conditions of 120 bpm.
𝜶 =
𝑫
𝟐
√
𝝆𝝎
𝝁
=
𝑫
𝟐
√
𝝆𝟐𝝅𝒇
𝝁
For this project:
𝜶 =
𝟗×𝟏𝟎−𝟑
𝟐
√
𝟏𝟎𝟓𝟎×𝟏𝟐.𝟓𝟕
𝟎.𝟎𝟎𝟑𝟓
= 𝟖. 𝟕𝟒
3.3 - Reynolds Number
The transition between laminar and turbulent flow arises when the Reynolds number surpasses a critical
value. The Reynolds number is a dimensionless number that is dependent on flow parameters, and relates
the velocity, dynamic viscosity and density of the flowing fluid with the diameter of the vessel it is
Equation 5: Energy Equation

30. 30 | P a g e
passing through. The Reynolds number is defined in Equation 8 as inertial force divided by the viscous
force.
𝑹𝒆 =
𝝆𝑼𝑫
𝝁
For the standard geometry used in this project, the maximum velocity is 0.62 m/s, the average diameter is
9mm and the density is 1050 kg/m3
. As viscosity will change during the transient solution due to the non-
Newtonian property of blood, the Newtonian value of 0.0035 kg/m-s has been assumed for this
calculation. The result is shown in Equation 9.
𝑹𝒆 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 =
𝟏𝟎𝟓𝟎×𝟎.𝟔𝟐×𝟗×𝟏𝟎−𝟑
𝟎.𝟎𝟎𝟑𝟓
= 𝟏𝟔𝟕𝟒
As this is in the within the limits of the laminar region, it is expected that no turbulent flow will be
encountered for the Standard Geometry.
The maximum Reynolds number for each of the geometries have been calculated and listed below in
Equations 10 - 12, taking the max velocity and diameter of the vessel at the point of stenosis to see
whether the flow enters the turbulent region in this area.
𝑹𝒆 𝟑𝟎% =
𝟏𝟎𝟓𝟎×𝟏.𝟑𝟒×𝟔.𝟑×𝟏𝟎−𝟑
𝟎.𝟎𝟎𝟑𝟓
= 𝟐𝟓𝟑𝟐
𝑹𝒆 𝟔𝟎% =
𝟏𝟎𝟓𝟎×𝟑.𝟔𝟏×𝟑.𝟔×𝟏𝟎−𝟑
𝟎.𝟎𝟎𝟑𝟓
= 𝟑𝟖𝟗𝟖
𝑹𝒆 𝟖𝟎% =
𝟏𝟎𝟓𝟎×𝟏𝟒×𝟏.𝟖×𝟏𝟎−𝟑
𝟎.𝟎𝟎𝟑𝟓
= 𝟕𝟓𝟔𝟎
For a pipe, the boundary layer thickness is given by Pradlt’s equation, as shown in Equation 13. Based on
this calculation, the degree of inflation in the mesh was decided to capture the whole of this boundary
layer development in detail.
Laminar Region Re<2000
Transitional Region 2000<Re<4000
Turbulent Region Re>4000
Equation 9
Equation 8
Equation 12
Equation 10
Equation 11
It is expected that the 30% and 60% case studies will
enter the transitional region at the point of maximum
constriction, and in the 80% case study, the flow will
become fully turbulent

31. 31 | P a g e
𝜹 = 𝟔𝟐. 𝟕 ×
𝑫 𝒂𝒗𝒆𝒓𝒂𝒈𝒆
𝑹𝒆
𝟕
𝟖
= 𝟔𝟐. 𝟕 ×
𝟔.𝟕𝟓×𝟏𝟎−𝟑
𝟏𝟑𝟓𝟎
𝟕
𝟖
= 𝟎. 𝟕𝟕 𝒎𝒎
This is the maximum size of the boundary layer, so the inflation size of around 0.8 mm used in all the
meshes will more than satisfactorily capture the boundary layers in each of the geometries.
3.4 - Wall Shear Stress (WSS)
The wall shear stress is a vital parameter in blood flow. It depicts the shear stresses involved in the layer
of fluid next to the vessel wall, and for Newtonian, incompressible fluids, is defined by Equations 14 and
15.
𝝉 𝒘 = 𝝁𝑹 𝑫
̅̅̅̅
Where 𝑹 𝑫
̅̅̅̅ is defined by 𝑹 𝑫
̅̅̅̅ =
𝝏𝒖 𝒋
𝝏𝒙 𝒊
+
𝝏𝒖 𝒊
𝝏𝒙 𝒋
For non-Newtonian fluid cases, the shear stress is proportional to the rate of deformation tensor, as shown
by Equations 15 and 16.
𝝉 𝒘 = 𝛈𝑹 𝑫
̅̅̅̅
The shear stress can also be written in terms of a non – Newtonian viscosity, as in Equation 17.
𝝉 𝒘̅̅̅̅ = 𝛈(𝑹 𝑫
̅̅̅̅) 𝑹 𝑫
̅̅̅̅
(ANSYS FLUENT 12.0 User's Guide. 2015)
Giddens et al (1993) found that arteries adapt their diameters in order to regulate and keep the wall shear
stresses they are subject to in a narrow range around 15 dynes/cm2
(1.5 Pascals). The results achieved are
analysed with reference to this value.
3.5 – Non – Newtonian Carreau - Yasuda Model
To account for the non-Newtonian characteristics of blood, the Carraeu – Yasuda model for depicting
Pseudo-Plastics has been employed, as this is a popular choice that captures the shear thinning behaviour
of human blood well. (Ma and Turan. 2011) It also depicts the viscoelasticity inherent in blood due to the
Equation 14
Equation 15
Equation 16
Equation 17
Equation 13

32. 32 | P a g e
elastic characteristics exhibited when undergoing deformation. However, the non-Newtonian behaviour of
the blood changes with temperature and this model does not account for this in its equations. (Sousa et al.
2011) For this project, this is not a problem, as temperature is assumed constant.
Viscosity in the model in limited by η0 and η∞ , as shown by Figure 11. Viscosity is dependent on
shear stresses in non-Newtonian fluids, whereas Newtonian fluids have a constant viscosity. The effective
viscosity according to the Carraeu – Yasuda model is calculated by Equation 18.
𝛈 𝒆𝒇𝒇 = 𝛈∞ + (𝛈 𝟎 − 𝛈∞)[𝟏 + (𝛌𝜸̇ ) 𝟐]
𝒏−𝟏
𝟐
At low shear rate (𝜸̇ << 𝟏), the model depicts the flow as a Newtonian fluid, but when the shear rate
increases (𝜸̇ >>
𝟏
𝝀
), Carreau fluids behave as a Power Law fluid. (ANSYS FLUENT 12.0 User's
Guide. 2015) The relaxation time constant and the power law index control the respective transitions and
slope in the Power Law region.
From the literature review, it was found that the non-Newtonian assumption is only necessary in small
vessels and in diseased conditions, so it is completely justifiable and necessary to use this assumption in
this project. The properties of the blood model have been taken from Ma and Turan’s (2011) paper, and
are as shown in Appendix B.
Figure 11: Variation of Viscosity with Shear Rate in the Carreau Model (Arc.vt.edu, 2015)
Equation 18

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4 - Methodology
The processes involved in the implementation of the biologically similar conditions in the setup of the
simulations are detailed herein.
4.1 – Geometries
4.1.1 - Vessel Dimensions
The dimensions used for the standard geometry are based on dimensions used by Shanmugavelasyudam
et al in their 2010 study entitled “Effect of Geometrical Assumptions on Numerical Modelling of
Coronary Blood Flow Under Normal and Disease Conditions”, as well as from considerations of a large
database of patient specific data compiled by Funabashi et al (2003). The geometry has been made to
incorporate characteristics that would be seen in the LCA, after the study of numerous patient specific
CAT scans showed that meandering walls and multi-direction changes are defining characteristics of the
vessel. The geometry also has a relatively long run up to the bifurcation, to allow for the flow to fully
develop before the bifurcation, and this is verified in the results section. The bifurcation angle between
the LCA and the LCX for this study is 51 degrees, and the rest of the dimensions are as shown in Table 2.
The geometries were constructed using the commercial CAD software Solidworks 2013, and imported
into Ansys workbench to undergo the meshing and simulation process. The standard geometry and the
further case studies are shown by Figures 12 to 18 in Section 4.1.2
Inlet
Diameter
(mm)
Outlet Diameter
(mm)
Length (mm)
Main Branch (LM) 9 9 ~ 90
Left Anterior Descending (LAD) 9 5.2 ~ 80
Left Circumflex (LCX) 11.25 4.5 ~ 65
Total Length ~ 170
Table 2: Standard Geometry Dimensions

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4.1.2 - Models
Figure 12: Front View of CAD for Standard Geometry
Figure 15: Top View of CAD for Standard Geometry
Figure 13: Side On View of CAD for Standard Geometry
Figure 14: Bottom View of CAD for Standard Geometry
LCX
LAD
LM

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Figure 16: 30% Stenosis Case Study
Figure 17: 60% Stenosis Case

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From initial simulations, it was found that an area of low wall shear stress formed on the left hand side of
the geometry, just under the point of bifurcation, so it was assumed that atherosclerotic build up manifests
at this point as this complies with the literature. The case study geometries (Figures 16, 17 and 18) were
based around a percentile reduction in lumen size at this point.
4.1.3 - Meshing
To correctly capture the complex and highly sensitive flows involved in arterial blood flow, much care
was taken to create geometric meshes that were both fine enough to capture small details, as well as
having good quality cells to ensure correct solutions. A limiting factor in the creation of the meshes was
the fact that the educational version of Fluent only allows 512,000 elements in simulations. Fortunately,
the mesh independence tests (Appendix E) found that all the results were independent of the mesh before
the max limit on elements was reached. The completed meshes are shown in Appendix D
Figure 18: 80% Stenosis Case Study

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Inflation was used to capture the boundary layer near to the wall, as a no slip condition has been applied
in the simulation setup. This is shown by Figure 19. A gradient of increasing cell size was created
propagating from the edge of the artery wall, in order to balance the need for a fine mesh near the wall
and computational efficiency within the simulation. The size of this boundary layer was calculated as 0.77
mm in Equation 13, so the inflation was implemented to stretch to 0.8 mm to capture this correctly.
The meshing was completed using an advanced size function that depended on the curvature of the
geometry, so that the flow could be correctly captured at the key areas in the geometries such as the
bifurcation point and the area of stenosis. This meant that refinement occurred at points at points of high
curvature, which is an important feature as atherosclerosis is prone to development in these areas due to
significant flow changes. The LCX was also refined for the case studies, as the flow in this branch was
expected to become volatile as the pathological conditions escalated. A patch independent method was
also employed in two of the cases, so that only the geometry is used to associate the boundary faces of the
mesh to the regions of interest, to ensure that gaps, overlaps and other geometric issues are negated.
Mesh independence studies are very important in ensuring accuracy in these types of simulation. Once a
preliminary mesh was created for each geometry, each of the meshes was checked for changes in the
solution with increasing amounts of elements (specifically at the points of highest interest), until the
change was considered negligible. This study was done under the same conditions as were to be used in
the final simulations. The number of elements in the meshes was changed by altering the element sizing
and normal curvature angle, and doing iterative studies to determine the best mesh in terms of quality and
accuracy. Computational time was not a limiting factor in the decision, as accuracy has been deemed
more important that the length of simulation. The final two iterations of the mesh independent study for
Figure 19: Near Wall Inflation Propagation for Standard Geometry Mesh

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each geometry are shown in Appendix E. The minimum percentage difference was decided to be a 0.25%
change, in order to ensure very accurate solutions.
The quality of the meshes was ensured by monitoring the minimum orthogonal quality and max skewness
in the network, and keeping the values within the recommended bounds. Meshes are only as good as their
worst cell, so as a general rule of thumb, the skewness was kept below 0.9, and the minimum orthogonal
quality was kept above 0.01.
4.2 - Simulation Setup
4.2.1 - Assumptions
Below is a list of the assumptions employed in simulations:
1) Incompressible Blood– Assumed despite blood being slightly compressible due to its inherent
composition of floating components
2) Non – Newtonian Blood Property Assumption Using the Carreau-Yasuda Model - Viscous
forces vary with shear rates for non-Newtonian fluids (u = f (𝛾̇)). It has been derived from the
literature that this model accurately describes blood viscosity - Section 3.5.
3) Rigid Walls – Assumed despite the elastic nature inherent in vessel walls, due to the
surrounding tissues. Assumption becomes more valid with further disease as artery walls
become hardened as atherosclerosis develops in pathological conditions.
4) No-slip condition at walls - Assume parabolic boundary layer formation, as in Figure 20.
5) Artery wall is Homogenous and Isotropic – Despite the layered structure of arteries.
6) Biomechanical and Mechanical Interactions between the Blood and the Artery Tissues are
Assumed Negligible – FSI analysis required to incorporate this feature.
7) Constant Velocity Inlet During Diastole Stage of Inlet Cycle – Despite the velocity varying in
the systolic stage of the cycle due to the muscles in the artery wall pushing the flow along.
8) Constant Pressure Outlets for the LCX and LAD – Despite the pressure varying over the
cardiac cycle.

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4.2.3 - Boundary Conditions
Velocity Inlet to LM
The inlet condition applied to the geometry is shown in Figure 21, and represents a human’s pulsatile
behaviour during moderate exercise at a heart rate of 120 bpm. While real life pulse characteristics will
alter between patients, and not be as mathematically coherent as this, it is a good description of the
general behaviour of the flow velocity through systole and diastole. Only one cardiac cycle is analysed in
this study, with a period of 0.5 seconds. The UDF function is described in Appendix A.
Pressure Outlets of LAD and LCX
Due to the varying outlet pressures during systole and diastole, an average of the two pressures was
assumed for calculations. During systole in a healthy human, the pressure is around 120 mmHg, whilst in
diastole it is around 80 mmHg. Taking the average as 100 mmHg gives the constant static gauge pressure
Figure 21: Theoretical Inlet Conditions (Sinnott et al. 2006)
Figure 20: Velocity Profile of Blood Due to Wall Shear Stresses (Klabunde. 2011)

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used for the outlets of the domain, (Sinnott et al. 2006) which converts to 13332 Pascals. This assumption
is unlike the hemodynamic characteristics present in arterial blood flow, as the pressure varies transiently
through the cycle of systole and diastole, but for ease of calculation, this assumption has been
implemented.
4.2.2 – Discretisation
Ansys-Fluent is a finite volume code in which the Navier-Stokes equations are discretised and calculated
for each node within the domain. Suitable convergence schemes and discretisation methods have been
incorporated to correctly solve the flow parameters and ensure boundedness, accuracy and convergence.
The decisions made have been listed below:
- Fractional Step Scheme for Pressure – Velocity Coupling
- Least Squares Cell-Based Gradient Discretisation
- Second Order Upwind Momentum Spatial Discretisation
- Second Order Implicit Transient Formulation for the Transient Flow
- Second Order Pressure Spatial Discretisation
A non-iterative time advancement algorithm has been used in conjunction with a Fractional Step
Pressure-Velocity coupling, as the convergence behaviour was deemed far better than that of other
coupling schemes during initial simulations. This fractional step scheme decouples the continuity and
momentum equations to make the simulation less computationally expensive. Second order spatial
discretisation is used to ensure accuracy in the results and to prevent numerical diffusion occurring within
simulations.
The transient cycle was split into very small time steps that take minimal iterations to converge, as is the
common practice in the field. A time step size of 0.0005 seconds was chosen, with 1000 time steps in the
whole simulation to capture the entire cardiac cycle time of 0.5 seconds. The convergence criterion for the
convergence of residuals was set to 0.0001 to ensure very accurate solutions.

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5 – Results and Discussions
5.1 – Validation
As no comprehensive and viable experimental data has been obtainable, validation of the coherence of
work has been achieved through comparison with Chaichana et al’s (2011) work entitled ‘Computation of
Hemodynamics in the Left Coronary Artery with Variable Angulations’. This is a good case to study as
CAT scans of patient arteries in healthy conditions have been used to obtain the wall shear stresses,
making the results a good representation of physical conditions. Whilst the geometries employed in this
project are different, the boundary conditions applied in validation are the same, and the blood was
considered Newtonian and laminar to replicate the case study. Therefore a good comparison can be made.
The defining characteristics of the contour plots are very similar to those obtained by Chaichana et al, so
it can be concluded that no abnormal results are obtained by the mesh and simulation set up used in the
extension of this work.
In Figure 22, it is observed that there is a common area of separation and recirculation at the start of the
LCX branch at the point at which the bifurcation occurs, and a region of higher velocity through the
centre of the geometry before the bifurcation point. The numerical values achieved in validation
correspond well with what was expected from this simulation, with the scales of each agreeing well.
Figure 22: Comparison of Velocity Fields Through a Central Plane for Validation (Case Study – Left)

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It can be seen that the LCX in both cases shown in Figure 23 is subject to higher wall shear stresses, and
that a region of higher wall shear stress develops just before the bifurcation point. Both the simulations
correspond well for the domain displayed in the case study, with the numerical boundaries and contour
trends matching well.
Validation of the inlet conditions has been achieved by comparing the theoretical inlet conditions (Figure
21) to the actual velocity profile measured at the inlet (Figure 24), so it is verified that the UDF
interpreted in Fluent is performing as expected. Also, Tables 3 and 4 depicts the mass flow rates passing
through the inlet and outlets at two different time steps, and verifies that continuity is being satisfied over
all of the simulations, as the total mass flux is either very small or zero in each of the cases. The fact that
the mesh independence test has proven successful also indicates that highly accurate solutions that are
independent of mesh quality have been achieved.
Figure 24: Inlet Velocity Profile (Taken from an Area-Weighted Surface Monitor)
Figure 23: Comparison of Wall Shear Stresses for Validation

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5.2 - Mass Flow Rates
In Tables 3 and 4, negative values for mass flow rate are defined as leaving the domain, whereas positive
values are mass flow into the domain. For the 80% case study, the flow is reversed in the LCX for both of
the time steps stated, with flow entering the domain through the outlet. This is due to the high pressure
gradient between the constant 13332 Pascal’s prescribed at the outlet and the formation of a negative
pressure region just past the bifurcation point, as shown by the pressure analysis in Figure 38.
Table 3: Comparison of the Mass Flow Rates Through Each of the Outlets @ 0.5 Seconds
Table 4: Comparison of the Mass Flow Rates Through Each of the Outlets @ 0.11 Seconds
Model
Net Result For
Verification of
Continuity
Inlet Mass
Flow Rate
(kg/s)
LAD Mass
Flow Rate
(kg/s)
LCX Mass
Flow Rate
(kg/s)
Percentage of
Flow Out of
LCX
Standard
Geometry
0 0.00664938 -0.00496773 -0.00168165 25.29%
30% Stenosis
Case
0 0.00665856 -0.00530151 -0.00135705 20.38%
60% Stenosis
Case
1 e-09 0.00664904 -0.00634159 -0.000307449 4.62%
80% Stenosis
Case
5 e-08 0.00661557 -0.0121462 0.00553068 Reversed Flow
Model
Net Result For
Verification of
Continuity
Inlet Mass
Flow Rate
(kg/s)
LAD Mass
Flow Rate
(kg/s)
LCX Mass
Flow Rate
(kg/s)
Percentage of
Flow Out of
LCX
Standard
Geometry
0 0.0332441 -0.0226038 -0.0106403 32.01%
30% Stenosis
Case
-3 e-08 0.0332899 -0.0244755 -0.00881443 26.48%
60% Stenosis
Case
-3 e-08 0.0332423 -0.0297928 -0.00344953 10.38%
80% Stenosis
Case
4.4 e-07 0.033075 -0.0347385 0.00166394 Reversed Flow

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The tables show the extent to which flow through the LCX outlet is reduced as atherosclerosis takes hold,
with the percentage mass flow rate through the LCX dropping by around 22% between the healthy
geometry and the 60% blockage case for both of the cycle times shown. This decline in mass flow would
result in a reduced supply of oxygen and vital nutrients to left atrium and, in 40 – 50% of hearts, the SA
node, which are both vital parts of the cardiac system. EVR analysis will be shown later to indicate
whether this lack of mass flow would result in a heart attack for each of the cases studied.
For 0.5 and 0.11 seconds into the cycle, the flow becomes reversed at 63% and 73% stenosis respectively,
as described by Figure 27. This figure illustrates the exponential drop off in percentage flow out of the
LCX as the disease progresses. Both of the trends follow the same profile, with increasingly steep
negative gradients developing with disease progression. It is interesting to note that the percentage of flow
through the LCX is increased at the higher speeds of systole when compared to diastole, by a degree of
around 5% for every level of stenosis up to 73%. This is due to the increased recirculation in the area
behind the stenosis, and the pressure gradient consequently sucking the blood in this area up and through
the LCX.
Figure 26 reiterates the negative correlation between flow through the LCX and disease progression, and
describes how reversed flow in the LCX is a major characteristic for both the 60% and 80% cases, with
the feature present for 58% and 82% of the cycle time respectively. The 30% stenosis case also has a
small degree of reversed flow in the LCX, but only about 16% of the cycle time during the transition
between systole and diastole.
The drop in LCX mass flow rate is correspondingly balanced with an increase in the LAD mass flow rate,
as shown by Figure 25. It should be noted that the peak mass flow rate corresponds with the peak velocity
at 0.11 seconds for the standard geometry, but moves increasingly later as lumen occlusion increases.
This is due to the turbulent characteristics that develop in the LAD branch, as shown by Figure 39. The
mass flow through both of the branches settles quite quickly upon the initiation of diastole for the
Standard, 30% and 60% case studies, but there is still a significant gradient in the mass flow for the 80%
case study once the full cycle has been completed.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
Time (seconds)
MassFlowRateThroughLAD(kg/s)
Standard Geometry
30% Stenosis Case Study
60% Stenosis Case Study
80% Stenosis Case Study
Figure 26: Transient Mass Flow Rate Through the LAD
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Time (seconds)
MassFlowRateThroughLCX(kg/s)
Standard Geometry
30% Stenosis Case Study
60% Stenosis Case Study
80% Stenosis Case Study
Figure 25: Transient Mass Flow Rate Through the LCX

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Figure 27: Percentage of Mass Flow Through the LCX with Disease Progression
5.3 – Velocity Analysis
It is recommended that the animations created in this project are used in conjunction with the discussion
of velocities and wall shear stresses, so a clear idea of the transient characteristics of these parameters can
be obtained. The link for the animations can be found in Appendix F.
It was important to ensure and validate that the boundary layer was fully developed within the domain
before it reached the bifurcation point in the geometry. A long run up to the bifurcation point was created
in the geometry to allow for this development. If this were not to happen, the results would not have been
an accurate representation of the actual hemodynamic behaviour inside the artery, as actual flow would
fully develop shortly after leaving the heart, and remain in this state for its entire passage through the
network. The boundary layer develops due to the no slip condition applied to the domain walls, and the
frictional effects passed through each of the layers of fluid propagating from the wall. Theoretically, the
profile should replicate that shown in Figure 20.
0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
60
70
80
90
100
Level of Stenosis (% Reduction of Lumen Size)
PercentageofFlowThroughtheLCX
0.5 Seconds
0.11 Seconds

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Figure 28 shows the profile of the boundary layer at two different stages along the geometry – just after
the inlet, and at a point 48 mm into the domain. The flat profile between the two parabolas in the
measurements near the inlet indicate that the central flow has not yet had time to fully develop. However,
at 48 mm, the boundary layer is fully developed, as no change in this profile occurs at further points in the
geometry. It is noted that that the skewness in the velocity profile in the right hand graph is due to the
curvature of the geometry.
The velocity contour plots on a central 2D plane are shown in Figures 29 and 30 for the main artery stem,
up to and partially past the bifurcation point at peak systole and 0.5 seconds respectively. For the standard
geometry at peak systole, a small recirculation zone can be identified as forming at the point at which the
LCX bifurcates. This recirculation zone gets progressively bigger as the stenosis escalates, extending up
the LAD as the flow of blood becomes increasingly tighter to the opposite wall to the stenosis in a jet like
manner. The severe reduction in mass flow rate through the LCX is emphasized in these figures due to the
large areas of blue. Figures 31 and 32 also emphasize how the separation zone grows as the disease takes
hold. Stenotic build up is likely to manifest in this area, furthering the disease. An area of separation also
develops just before the stenosis point, which means that the further build up could develop on both sides
of the stenosis on the same wall. Velocity characteristics are largely similar for systole and diastole
conditions, meaning the conditions in which stenosis development will occur remain present for the
whole cycle, but be amplified during systole. There is a sharp velocity gradient in each of the case studies
as the flow enters the stenosis area. In the 80% case at peak systole, the flow reaches a maximum of 14
m/s, which is a significant increase compared to the peak velocity of 1.248 m/s that occurs in the standard
geometry. This kind of velocity would have a severe effect on the lining of the arterial wall.
0 1 2 3 4 5 6 7 8 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Distance Across Inlet (mm)
Velocity(m/s)
0 1 2 3 4 5 6 7 8 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Distance Across Inlet (mm)
Velocity(m/s)
Figure 28: Boundary Layer Profiles at 1mm (Left) and 48mm (Right) into the Domain

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Figure 29: Velocity Through a Central Plane @ Peak Sytole for 1) Standard Geometry 2) 30% Stenosis
3) 60% Stenosis 4) 80% Stenosis
1
3 4
2

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Figure 30: Velocity Through a Central Plane @ 0.5 Seconds for 1) Standard Geometry 2) 30% Stenosis
3) 60% Stenosis 4) 80% Stenosis
1 2
3 4

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Figure 31 shows how the flow becomes increasingly concentrated in the area next to the wall on the
opposite side of the stenosis as the restriction in increased, and portrays the increasing size of the
separation region behind the stenosis. Consistent concentric fluid layers can also be seen in the velocity
profiles in the first two circles through the main stem of the standard geometry, confirming a fully
developed boundary layer.
50 Streamlines were initiated from the inlet to acquire the data shown in Figure 32. It can be seen that as
the level of stenosis increases, the flow begins to act in a turbulent manner through the LAD, with the
60% and 80% case studies showing the most erratic behaviour. The reduction in number streamlines
through the LCX also emphasises the point made about reduced mass flow through this branch. The
velocity vectors shown in the figures show that there is a lot of recirculation within this area, as well as a
strongly directional, narrow and fast jet of bulk flow that stays close to the artery wall.
Figure 31: Velocity Contours Through the Area of Stenosis @ Peak Systole for 1) Standard Geometry
2) 30% Stenosis 3) 60% Stenosis 4) 80% Stenosis
1 2
3 4

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Figure 32: Streamlines and Velocity Vectors @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis 4) 80%
Stenosis
1 2
3 4

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5.4 - Wall Shear Stress Analysis
Figure 33: Wall Shear Stresses @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis 4) 80% Stenosis
1 2
3 4

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Figure 34: Wall Shear Stresses @ 0.5 Seconds for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis 4) 80% Stenosis
4
1 2
3

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The contour plots shown in Figure 33 depict the significant changes that occur in the WSS at peak systole
as the disease develops. In the standard geometry, an area of higher WSS develops just before the
bifurcation point, making it likely that the disease would manifest in this region. Hence, the stenosis was
placed here for the case studies. As the disease develops, the apex of the stenosis profile is exposed to
increasing amounts of WSS, and in real life conditions, this would accelerate the progression of the
disease. This is derived from the findings of the literature review, which stated that large changes in WSS
causes atherosclerotic build up.
The most prominent changes in the contours occur in a scattered manner at various points past the
bifurcation, as well as on the wall opposite the stenosis, where the flow is subject to higher amounts of
WSS exposure. These changing WSS patterns mean that plaque build-up could manifest at various points
along the LAD, as well as on the side of the wall opposite the stenosis. It is noticeable that the areas in
which flow separation occurs (as described by the velocity analysis) is where the WSS is at its lowest, and
that in areas of minimal curvature, the WSS remains consistently low throughout the case studies. The
lower bound of WSS decreases with increased blockage, but at a much slower rate than the higher bound,
which increases significantly at the apex of the stenosis. During both systole and diastole for the 60% and
80% case studies, the LCX is subject to minimal exposure to changing WSS, which indicates that this
area will not be prone to atherosclerosis, unlike the LAD.
Figure 34 shows the conditions in the diastole phase at 0.5 seconds, and it is noticeable that the contour
patterns show peaks and troughs in similar places, with just the magnitudes of the stresses reducing. The
distribution of WSS is however less erratic through the LAD branch during diastole, especially when
comparing the 60% case study. Again, it can be derived from this that the WSS conditions for
atherosclerosis to manifest develop not just during the systole, but over the entire cycle, but with the
biggest developments in the disease arising during systole.
Although the exact value at which endothelial damage occurs is not known, and would vary between
patients, the WSS is very high in the final two case studies at peak systole, so it is expected that damage
would occur on the inside lining of the artery. Giddens et al (1993) made findings that state that arteries
dilate and constrict to keep the WSS at around 15 dynes/cm2
(1.5 Pascal’s) for every part of the geometry
when subject to pathological conditions. For cases one and two at peak systole (Figure 33), the values for
WSS are either around or below this benchmark for a majority of the geometry, with the exception of the
point at which build-up occurs, where it rises to around 11 and 13 Pascal’s respectively. This means the
artery would not have to transform much to maintain constant WSS exposure for these levels of

55. 55 | P a g e
atherosclerosis. However, in cases 3 and 4, the artery would have to work much harder to maintain a
constant WSS in the geometry, as the lower bound becomes very small (0.1 and 0.07 Pascal’s
respectively) for most of the geometry, meaning the artery would have to constrict in the areas of blue
shown. At the stenosis, the WSS becomes very large (185 and 2265 Pascal’s respectively), so the artery
would have to dilate to maintain optimal conditions. The importance of elastic walls in correctly defining
hemodynamic conditions becomes apparent when looking at this, as the WSS with elastic walls would be
much lower with this feature incorporated.
The trend between the amount of WSS present in the geometry and the reduction in lumen size is
described by Figure 35. It is clear to see that the relationship is linear up until around 55% restriction, at
which point the WSS increases significantly in an exponential manner. This validates the theory offered
by many researchers, including Quarteroni et al (2000), that state that there is accelerated progression of
the disease once minor occlusion occurs. This sudden increase in WSS indicates that 55% restriction in
the arteries may be the limit to normal functionality in the artery, and that past this point, the artery cannot
control the WSS it is exposed to, even by changing its luminal patency.
Figure 36 shows how the WSS changes over time at the apex of the stenosis profile for each of the
studies. A logarithmic scale has been employed on the y axis due to the large differences in magnitude
over the different case studies. The transient WSS profiles for the Standard, 30% Stenosis and 60%
Stenosis cases are largely similar. The velocity at the inlet acts in a sinusoidal manner over the systolic
period, and the WSS follows this profile, with the max velocity and max WSS correspondingly peaking at
around 0.11 seconds. The 80% case differs slightly during the systolic stage, with two peaks in WSS
occurring. This is due to the increased amount of reversed flow coming back from the LCX in the latter
stages of systole, as described by the mass flow rate analysis. The WSS magnitude levels off at a later
stage in the cycle as stenotic conditions increase. As WSS is therefore changing for a longer period of
time as the disease progresses, this predicts that the rate at which fatty plaques are deposited on the wall
would increase, agreeing well with the literature.
Once diastole has been fully implemented after 0.22 seconds, the WSS levels off at around 0.22, 0.23,
0.25 and 0.32 seconds for the Standard, 30%, 60% and 80% geometries respectively. Once it has levelled
off, the conditions remain largely constant, apart from in the standard geometry case, where the WSS
drops further at around 0.35 seconds

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0 10 20 30 40 50 60 70 80
0
500
1000
1500
2000
2500
Level of Stenosis (% Reduction of Lumen Size)
WSSatMidPointofStenosis(Maximum)(Pascals)
WSS with Laminar Assumption
WSS with Turbulence Model
Figure 35: Trendline for Wall Shear Stress Development (at Mid-Point of Stenosis Profile @ Peak Systole) with
Level of Stenosis
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
10
-1
10
0
10
1
10
2
10
3
10
4
Time (seconds)
WSSatMidPointofStenosis(Maximum)(Pascals)
Standard Geometry
30% Stenosis Case Study
60% Stenosis Case Study
80% Stenosis Case Study
Figure 36: Transient Profile of Wall Shear Stress at Mid-Point of Stenosis

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Figure 37: 3D Plot of Wall Shear Stresses Along the Bottom Half of the Stenosis Profile for 30% Restriction
The development of WSS along the bottom half of the 30% stenosis profile is shown by Figure 37, with
the WSS increasing fairly rapidly over the central part and peaking once the apex has been reached. The
WSS distributions for each part of the profile are consistently symmetrical, with flat but angled circles
formed that represent a steady gradient of increasing WSS around the wall from one side of the artery to
the other. It is noted that at the start of the stenosis profile (yellow and purple rings), the WSS on the
opposite side of the wall to the build-up is higher than that on the same side. However, further along the
profile (blue and red rings), the difference between the WSS swaps sides, with the stenosis profile
exposed to much higher stresses. The difference between the stress at the bottom of the stenosis profile
and that at the top is around 23 Pascal’s, growing from normal physiological conditions to high stress
levels. These high stresses on the stenosis profile could lead to the rupturing of the plaques and the
development of thrombosis that would further block the artery. The analysis for further disease
progression has not been shown, but the WSS trends are the same, but with increasing gradients between
circles and higher overall magnitudes.
1 2 3 4 5 6 7 8 9 10 11
x 10
-3
-5
0
5
x 10
-3
0
5
10
15
20
25
30
X Coordinate (m)
Z Coordinate (m)
WallShearStress(Pa)
At Peak Stenosis
Peak Stenosis Minus 0.001 metres
Peak Stenosis Minus 0.002 Metres
Peak Stenosis Minus 0.003 Metres
Peak Stenosis Minus 0.004 Metres
Start of Stenosis Profile

58. 58 | P a g e
5.5 - Pressure Analysis
1 2
3 4
Figure 38: Pressure Contours @ Peak Systole for 1) Standard Geometry 2) 30% Stenosis 3) 60% Stenosis 4) 80% Stenosis