NUMERICAL ANALYSIS OF BLOOD FLOW IN
AN AORTOCORONARY BYPASS MODEL
A Dissertation by
Foo Kok
Master of Science, Wichita State University, 2010
Bachelor of Engineering, Wichita State University, 2008
Submitted to the Department of Aerospace Engineering
and the faculty of the Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
May 2015
© Copyright 2015 by Foo Kok
All Rights Reserved
NUMERICAL ANALYSIS OF BLOOD FLOW IN
AN AORTOCORONARY BYPASS MODEL
The following faculty members have examined the final copy of this dissertation for form and
content, and recommend that it be accepted in partial fulfillment of the requirement for the
degree of Doctor of Philosophy, with a major in Aerospace Engineering.
___________________________________
Myose Roy, Committee Chair
___________________________________
Klaus A. Hoffmann, Committee Member
___________________________________
Walter Horn, Committee Member
___________________________________
Linda Klimant, Committee Member
___________________________________
Hamid Lankarani, Committee Member
Accepted for the College of Engineering
_________________________________________
Royce Bowden, Dean
Accepted for the Graduate School
_________________________________________
Abu S. M. Masud, Interim Dean
iii
ACKNOWLEDGEMENTS
I would like to thank my advisor, Roy Myose, for his relentless help and guidance
throughout my time in Wichita State University as a graduate student. Thanks to all my
committee members who have given me valuable feedback to accomplish this work. I would also
like to acknowledge Kristy Bixby who has spent many days helping me more clearly
communicate my ideas throughout this dissertation.
Thanks to my family for their unconditional support, love, and prayers. They are part of
me. Thanks to Christ, my God for His forgiveness, acceptance, and love to allow me to pursue
my interest. As I learn more about the mechanisms involved in the human vascular system, I am
aware of my limitations and feel the boundaries of human knowledge. I truly have enjoyed this
journey.
iv
ABSTRACT
Aortocoronary bypass (ACB) surgery is a treatment to bypass a blocked artery using a
graft. Approximately 500,000 bypass surgeries are performed in the U.S. each year. Of these,
about 15% to 20% have an early-phase failure, typically in the proximal region. A natural
question which arises is whether fluid dynamic geometry plays a role in the patency. Since only
a few studies have considered proximal region geometry, this motivates the present study.
Blood flow in a coronary artery bypass often involves very complex fluid dynamic
behavior. Numerical simulation is employed to investigate the flow field environment of the
proximal region of a bypass. A series of different branching models with laminar inflow are
considered to systematically investigate the geometrical effect under different flow conditions.
Model types (with branch diameter to host diameter ratio D2/D1 < 1, with or without blend at the
junction) include T-junction and host to branch junction with a radius of curvature with and
without helical pitch. Some flow conditions included non-Newtonian blood and non-steady flow.
Non-blended T-junction is subjected to flow separation at the inner wall of the branch
unless the flow rate ratio (𝑚̇2 /𝑚̇1 ) is high and D2 is small. By employing a blend radius at the Tjunction, there is a critical flow rate ratio where separation near the inner wall of the junction can
be diminished under steady state flow condition. This parameter is known as (𝑚̇2 /𝑚̇1 )crit. Given
a blended T-junction, there is a common branch radius rB which corresponds to (𝑚̇2 /𝑚̇1 )crit,min
that gives the minimum separation scale under steady state flow condition or delays the onset of
separation under non-steady flow conditions. This parameter was found to be independent of Re.
A geometric correlation that corresponds to non-separated flow in blended T-junction was
developed and can be expressed as rB = 0.146 D21.74. This correlation is independent to flow rate
ratio.
v
TABLE OF CONTENTS
Chapter
1.
INTRODUCTION
1.1.
1.2.
2.
Page
1
Aortocoronary Bypass Surgeries .............................................................................1
1.1.1 SVG and Types of Anastomosis ..................................................................3
1.1.2 Locations of SVG Complication ..................................................................5
1.1.3 Motivation and Limitation ...........................................................................7
1.1.4 Area of Study .............................................................................................10
Dissertation Outline ...............................................................................................11
BACKGROUND AND SIGNIFICANCE .........................................................................12
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
Vascular Wall Structures and Diseases..................................................................12
2.1.1 Thrombosis ................................................................................................13
2.1.2 Neo-intimal Hyperplasia ............................................................................14
2.1.3 Atherosclerosis ...........................................................................................14
Role of Hemodynamics on Vascular Biology/Disease ..........................................15
2.2.1 Effects of Hemodynamics Forces on Endothelial Dysfunction .................16
2.2.2 Relationship between Endothelial Cells and Nitric Oxide ........................16
2.2.3 Relationship between Nitric Oxide and Hemodynamic Forces .................17
2.2.4 Shear Stress on Endothelial Cells ..............................................................17
Types of Hemodynamics Shear Stress in Literature ..............................................18
2.3.1 Steady Laminar and Turbulent Shear Stress ..............................................19
2.3.2 Unsteady Laminar Shear Stress (Spatial and Temporal) ...........................20
2.3.3 Unsteady/Pulsatile Turbulent Shear Stress ................................................21
Indicators of Shear Stress for Biological Response of Vascular Wall ...................22
2.4.1 Steady Laminar Wall Shear Stress .............................................................23
2.4.2 Time-Averaged Wall Shear Stress .............................................................26
2.4.3 Wall Shear Stress Gradient ........................................................................28
2.4.4 Oscillatory Stress Index .............................................................................29
2.4.5 Relative Residence Time ...........................................................................32
Typical Flow Patterns in Vascular System ............................................................34
2.5.1 Separation Flow and Recirculation Zone ...................................................34
2.5.2 Dynamic View of Unsteady Secondary Flow Behavior ............................36
2.5.3 Dynamic View of Unsteady Straight-Tube Flow Behavior .......................36
2.5.4 Dynamic View of Unsteady Curved-Tube Axial Flow Behavior ..............37
2.5.5 Dynamic View of T-Junction/Branching Channel Flow Behavior............39
2.5.6 Dynamic View of Human Aortic Flow Behavior ......................................40
Disturbed Flow on Vascular Disease .....................................................................42
2.6.1 Flow Separation on Vascular Disease........................................................42
2.6.2 Secondary Flow on Vascular Disease ........................................................42
vi
TABLE OF CONTENTS (continued)
Chapter
2.7.
3.
Review of Related Work ........................................................................................45
2.7.1 Existing Numerical Work on ACB Model.................................................45
APPROACH AND THEORIES ........................................................................................48
3.1.
3.2.
3.3.
3.4.
4.
Page
Numerical Approach ..............................................................................................48
Governing Equations for Fluid Motion..................................................................50
3.2.1 Newtonian vs. Non-Newtonian Fluids .......................................................52
3.2.2 Viscous Models for Newtonian and Non-Newtonian Fluids .....................55
Solver Background and Theories ...........................................................................56
3.3.1 Solver Background ....................................................................................56
3.3.2 Solver Theories .........................................................................................58
Hemodynamics Indicators .....................................................................................60
3.4.1 TAWSS, OSI, and RRT .............................................................................60
VALIDATION STUDIES .................................................................................................62
4.1.
Validation Approach ..............................................................................................62
4.1.1 Newtonian vs. Non-Newtonian (Steady Laminar Flow) ...........................63
4.1.2 T-Junction Flow (Steady Laminar Flow) ...................................................64
4.1.3 Straight Tube (Pulsatile Laminar Flow) ....................................................67
4.1.4 Curved Tube (Pulsatile Laminar Flow) .....................................................70
4.1.5 Turbulent Flow in Curved Tube ................................................................72
5.
NUMERICAL MODELS ..................................................................................................76
6.
RESULTS ..........................................................................................................................84
6.1.
6.2.
6.3.
6.4.
6.5.
Test Model Series 1: Non-Blended T-Junction (Steady Laminar Flow) ...............85
6.1.1 Grid Independence Study...........................................................................86
6.1.2 Branch Diameter and Flow Rate Ratio on Flow Separation Length .........89
6.1.3 Effects of Branch Diameter on Hemodynamic Indicators .........................94
Test Model Series 2: Blended T-Junction (Steady Laminar Flow) .......................96
6.2.1 Flow Rate Ratio and Blend Radius on Onset of Flow Separation .............99
6.2.2 Effect of Reynolds Number on Critical Minimum Mass Flow
Rate Ratio.................................................................................................102
6.2.3 Secondary Flow Characteristics ...............................................................105
Newtonian vs. Non-Newtonian Models on T-Junction Flow (Steady-State Flow) ....107
Test Model Series 3: Helical Curved Tube (Steady-State Flow) .........................113
Non-Blended vs. Blended T-Junctions (Unsteady-State Flow) ...........................116
6.5.1 Flow Separation Characteristics ..............................................................118
vii
TABLE OF CONTENTS (continued)
Chapter
Page
6.5.2
6.5.3
6.6.
7.
Secondary Flow Characteristics ...............................................................123
Hemodynamic Parameters .......................................................................126
Non-Blended ACB vs. Blended ACB models (Unsteady-State Flow) ................132
6.6.1 Flow Separation Characteristics in ACB models.....................................134
6.6.2 Secondary Flow Characteristics ...............................................................141
6.6.3 Hemodynamic Parameters .......................................................................144
CONCLUSIONS..............................................................................................................150
BIBLIOGRAPHY ........................................................................................................................157
APPENDICES .............................................................................................................................179
A. Velocity, Pressure, Streamline, and Secondary Flow Plots for Non-blended
T-junction with Different Flow Rate Ratios ..............................................................180
B. Velocity, Pressure, Streamline, and Secondary Flow Plots for Blended
T-junction with Different Blend Radiuses .................................................................184
C. Inlet and Outlet Pressure Conditions for Non-blended T-Junction with
Different Flow Rate Ratios ........................................................................................188
viii
LIST OF TABLES
Table
Page
4.1
Validation Models Adopted in Present Study ..................................................................63
4.2
Numerical Model Geometry and Flow Conditions for Validation Case 2 ......................65
4.3
Numerical Model Geometry and Flow Conditions for Validation Case 3 ......................68
4.4
Numerical Model Geometry and Flow Conditions for Validation Case 4 ......................71
4.5
Numerical Model Geometry and Flow Conditions for Validation Case 5 ......................73
5.1.
Geometry and Parameters of Non-Blended T-Junction ...................................................79
5.2.
Geometry and Parameters of Blended T-Junction ...........................................................80
5.3.
Geometry and Parameters of Curved-Tube Model ..........................................................82
5.4.
Geometry and Parameters of ACB Models .....................................................................83
ix
LIST OF FIGURES
Figure
Page
1.1
Reconstructed aortic section with two bypass grafts .........................................................3
1.2
External saphenous vein support (eSVS) ..........................................................................9
1.3
Focus of area of study......................................................................................................10
2.1
Normal vascular wall structure. .......................................................................................12
2.2
Effect of hemodynamic shear stress on vascular graft ....................................................18
3.1
Approach of numerical study ..........................................................................................49
3.2
Relationship between shear rate and viscosity of normal human blood .........................53
4.1
Validation of different viscosity models .........................................................................64
4.2
Numerical grid for case study 2.......................................................................................65
4.3
Comparison of velocity profiles for different flow conditions ........................................66
4.4
Numerical grid for case study 3.......................................................................................69
4.5
Comparison of oscillatory axial velocity distributions at different frequencies..............70
4.6
Schematic of flow domain of case study 4 ......................................................................72
4.7
Comparison of instantaneous axial velocity profiles at  = 90°. .....................................72
4.8
Numerical grid for case study 5.......................................................................................74
4.9
Comparison of longitudinal velocity profiles (u-component) with experimental
results at different axial locations at symmetry plane .....................................................74
4.10
Comparison of circumferential velocity profiles (v-component) with experimental
results ...............................................................................................................................75
5.1
Schematic diagram and geometrical parameters of numerical model .............................77
x
LIST OF FIGURES (continued)
Figure
Page
6.1
Baseline models of study—non-blended T-junction .......................................................85
6.2
Comparison of velocity profiles with different grid sizes (
6.3
Sample grid structures applied on non-blended T-junction ............................................88
= 1e-5, D2 = 3 mm) ......87
6.4 (a) Distribution of y-component wall shear stress at inner wall of branch of nonblended T-junction model (Re = 1817, D2 = 5mm) .........................................................90
= 0.00075,
6.4 (b) Length of separation LSP based on negative WSSy (Re = 1817,
D2 = 5 mm) .....................................................................................................................90
6.5
Effects of flow rate ratio on the length of flow separation for different branch
diameters at Re = 1817 ....................................................................................................91
6.6
Distribution of: (a) pressure coefficient and (b) friction coefficient at inner wall of
non-blended T-junction branch (Re =1817 and D2 = 5mm) ............................................93
6.7
Correlation between minimum length of separation and branch diameter of nonblended T-junction...........................................................................................................94
6.8
Distribution of WSS at (a) inner wall and (b) outer wall of the non-blended
T-junction with different branch diameters (Re = 1817, 𝑚̇2 /𝑚̇1 = 0.004) .....................95
6.9
Comparison of velocity profiles of two different blended models measured at
5 mm from entrance of branch ........................................................................................97
6.10
Sample grid structures applied on blended T-junction ....................................................98
6.11
Determination of critical mass flow rate ratio for non-separation of blended
T-junction (Re = 1817, D2 = 5 mm, rB = 4 mm).............................................................99
6.12 (a) Determination of minimum critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit min of blended
T-junction (Re = 1817, D2 = 5 mm, rB = 5 mm)...........................................................100
6.12 (b) WSSy distribution for determination of critical mass flow rate ratio of blended
T-junction (Re = 1817, D2 = 5 mm, rB = 1 to 8 mm) ....................................................100
6.13
Effect of blend radius rB on distribution of critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit
at different branch diameters for Re = 1817 ..................................................................101
xi
LIST OF FIGURES (continued)
Figure
Page
6.14
Correlation between rB and D2 for (𝑚̇2 /𝑚̇1 )crit,min of blended T-junction at
Re = 1817 .......................................................................................................................102
6.15
Distribution of (𝑚̇2 /𝑚̇1 )crit for blended T-junction at Re = 227, 454, 909, 1817…. .....104
6.16
Comparison of pressure coefficient distribution for non-blended T-junction
(solid line) and blended T-junction (dashed line). .........................................................104
6.17
Comparison of secondary flow for non-blended and blended T-junctions…. ..............106
6.18 (a) Comparison of velocity distribution for Newtonian and nonNewtonian fluids at symmetric plane of model .............................................................108
6.18 (b) Distribution of velocity-u at: (a) Re = 454, (b) Re = 909, and (c) Re = 1817 for
Newtonian (solid line) and non-Newtonian (dashed line) fluids…...............................109
6.19
Distribution of velocity-v at: (a) Re = 454, (b) Re = 909, and (c) Re = 1817 for
Newtonian (solid line) and non-Newtonian (dashed line) fluids...................................110
6.20
Distribution of velocity-v for D2/D1: (a) 0.100, (b) 0.133, (c) 0.167, and (d) 0.200…. 112
6.21
Distribution of vy,max predicted by Newtonian and non-Newtonian models for
non-blended T-junction with D2 = 0.100, 0.133, 0.167, and 0.200 at Re = 1817..........113
6.22
Comparison of averaged steady WSS along four different line segments of tube
with different parameters: (a) D2 and (b) rC ..................................................................115
6.23
Physiological waveforms of aorta and graft ..................................................................116
6.24
Waveforms replicated with Fourier series for inlet and outlet boundary conditions ....117
6.25
Time-step independence test based on velocity of monitor point in junction of
non-blended T-junction (Repeak = 1819,  = 23.13, D1 = 30 mm, D2 = 3 mm) ............118
6.26
Comparison of temporal streamline at symmetry plane: (a) non-blended
T-junction and (b) blended T-junction during acceleration phase ................................119
6.27
Comparison of temporal streamline at symmetry plane: (a) non-blended
T-junction and (b) blended T-junction during deceleration phase ................................120
6.28
Comparison of separation of length for non-blended and blended T-junctions ............121
xii
LIST OF FIGURES (continued)
Figure
Page
6.29
Corresponding mass flow rate ratio for the onset of flow separation: (a) nonblended T-junction and (b) blended T-junction .............................................................122
6.30
Comparison of temporal secondary flow at y = 1.0 mm: (a) non-blended and (b)
blended T-junctions .......................................................................................................124
6.31
Comparison of temporal secondary flow at y = 2.5 mm: (a) non-blended and (b)
blended T-junctions .......................................................................................................125
6.30
Comparison of TAWSS distribution: (a) side view plots (b) axial line segments
for non-blended model (dashed line) and blended model (solid line) ...........................128
6.31
Comparison of OSI distribution: (a) side view plots and (b) axial line segments
for non-blended model (dashed line) and blended model (solid line) ...........................129
6.32
Comparison of: (a) TAWSS and (b) OSI for non-blended and blended
T-junctions .....................................................................................................................131
6.33
Schematic diagram and numerical grid structure of: (a) non-blended ACB model
and (b) blended helical ACB model ..............................................................................133
6.36
Comparison of temporal streamline at the symmetry plane during acceleration phase
with HP = 3 cm and rB = 1 mm ......................................................................................135
6.35
Comparison of temporal streamline at the symmetry plane during deceleration phase
with HP = 3 cm and rB = 1 mm ......................................................................................136
6.36
Comparison of length of separation for non-blended and blended ACB model ...........138
6.37
Corresponding mass flow rate ratio at onset of flow separation with rB = 1 mm and HP
= 3.0 cm for ACB models..............................................................................................138
6.38
Three-dimensional temporal streamline distribution plots with rB = 1 mm and HP = 3.0
cm for ACB models : (a) non-blended and (b) blended ...............................................140
6.39
Comparison of temporal secondary flow at y = 1.0 mm for ACB models:
(a)
non-blended and (b) blended .........................................................................................142
6.40
Comparison of temporal secondary flow at y = 2.5 mm for ACB models:
(a)
non-blended and (b) blended .........................................................................................143
xiii
LIST OF FIGURES (continued)
Figure
Page
6.43 (a) Comparison of TAWSS distribution at junction of ACB models: (a) non-blended
and (b) blended ..............................................................................................................145
6.43 (b) Comparison of TAWSS distribution at branch of non-blended (solid line) and
blended (dashed line) ACB models at different line segments .....................................146
6.44 (b) Comparison of OSI distribution at the junction of ACB models: (a) non-blended and (b)
blended ..........................................................................................................................147
6.44 (b) Comparison of OSI distribution at branch of non-blended (solid line) and
blended
(dashed line) ACB models at different line segments ...................................................147
6.45
Comparison of TAWSS and OSI at junction of host channel for ACB models:
(a) non-blended and (b) blended....................................................................................148
xiv
NOMENCLATURE
Cf
Friction coefficient
CP
Pressure coefficient
D
Diameter
De
Dean number, De = (uD/)(D/rC)0.5
Dg
Diameter of graft
DP
Diameter of aortic punch hole
f
Body force per unit mass
Hp
Helical pitch of branch
J0, J1, J2, J3
Bessel functions of the first kind
L
Length
L1
Length of host channel (Chapter 6)
L2
Length of branch channel (Chapter 6)
Le
Length of entrance
LSP
Length of flow separation
𝑚̇
Mass flow rate
p
Pressure
r
Radial location in the pipe
rB
Blend radius of branch
rC
Radius of curvature of branch
R
Pipe radius
Re
Reynolds number, Re = ux /
Reosc
Oscillatory Reynolds number
xv
NOMENCLATURE (continued)
Repeak
Peak Reynolds number
Rest
Steady Reynolds number
Sij
Strain rate
u
Velocity (x-component)
U1
Host inlet velocity (i.e., general inlet velocity)
U2
Branch outlet velocity (note: "U" is used, although it is in the y direction)
U3
Host outlet velocity
Ue
Steady inlet velocity
Uosc
Oscillatory inlet velocity
v
Velocity (y-component)
w
Velocity (w-component)
x
Host channel downstream direction (Chapter 6)
XB
Length of branch channel (Chapter 6)
XH
Length of host channel (Chapter 6)
y
Branch channel downstream direction (Chapter 6)
YB
Length of branch channel (Chapter 6)
YH
Length of host channel (Chapter 6)
z
Normal axis to the x-y plane (Chapter 6)

Womersley number,  = r(/)0.5

Velocity ratio (Uosc/Ue)

Shear strain rate

Kronecker delta
xvi
NOMENCLATURE (continued)
ij
Shear rate tensor

Bulk viscosity

Successive positive roots of J2() = 0

Dynamic viscosity

Density

Stress tensor

Shear stress

Bend angle

Frequency
xvii
ABBREVIATIONS
ACB
Aortocoronary Bypass
BAEC
Bovine Aortic Endothelial Cell
BPM
Beats per Minute
CABG
Coronary Artery Bypass Graft
CAD
Coronary Artery Disease
CPT®
Current Procedural Terminology
EC
Endothelial Cell
eNOS
Endothelial Nitric Oxide Synthase
eSVS
External Saphenous Vein Support
FDM
Finite Difference Method
FEM
Finite Element Method
FVM
Finite Volume Method
HUVEC
Human Umbilical Vein Endothelial Cell
LAD
Left Anterior Descending
LDL
Low Density Lipoprotein
LDV
Laser Doppler Velocimetry
LITA
Left Internal Thoracic Artery
LVAD
Left Ventricular Assist Device
MEMS
Microelectromechanical System
MRI
Magnetic Resonance Image
m-SVG
Multiple Distal Anastomoses of SVG
NO
Nitric Oxide
OSI
Oscillatory Shear Index
xviii
ABBREVIATIONS (continued)
PGI2
Prostacyclin
PIV
Particle Image Velocimetry
PTFE
Polytetrafluoroethylene
RCA
Right Coronary Artery
RBC
Red Blood Cell
RRT
Relative Residence Time
SIMPLE
Semi-Implicit Method For Pressure-Linked Equations
SMC
Smooth Muscle Cell
s-SVG
Single Distal Anastomosis of SVG
SV
Saphenous Vein
SVG
Saphenous Vein Graft
SWSSG
Spatial Wall Shear Stress Gradient
TAWSS
Time-Averaged Wall Shear Stress
TEE
Transesophageal Echocardiography
TWSSG
Temporal Wall Shear Stress Gradient
WSS
Wall Shear Stress
WSSx
x-component Wall Shear Stress
WSSy
y-component Wall Shear Stress
WSSz
z-component Wall Shear Stress
WSSG
Wall Shear Stress Gradient
xix
CHAPTER 1
INTRODUCTION
1.1
Aortocoronary Bypass Surgeries
Coronary artery bypass graft (CABG) surgery is a revascularization treatment performed
to relieve coronary artery disease (CAD), the most common form of heart disease [Bhatt et al.
(2013)]. Aortocoronary bypass (ACB) surgery is the revascularization treatment for coronary
artery disease in which the occluded artery is bypassed using a graft that connects the aorta and
the coronary arteries to create an access channel for blood flow to the myocardium. Early ACB
surgery using a saphenous vein graft (SVG) to bypass the occluded coronary artery of a human
was recorded in the work of Sabiston (1963) and Garrett et al. (1973). In 1962, Sabiston (1963)
was the first surgeon to perform ACB surgery with an SVG. However, the patient died three days
after surgery due to the formation of thrombus at the proximal anastomotic region of the graft.
The first successful ACB surgery was documented several years later in the work of Garrett et al.
(1973). The SVG that was sutured between the ascending aorta and the blocked artery of a male
patient was reported to be able to remain functioning during seven years of follow-up procedure.
Today, various types of grafts are available for CABG surgery. Several autologous grafts
have been selected as the standard choice of conduits for coronary revascularization surgery.
This includes the saphenous vein (SV), mammary artery (internal thoracic artery), and radial
artery. Among these analogous grafts, the left internal thoracic artery (LITA) and the SV have
been reported as the most common grafts used in current CABG surgery, their selection
depending on the situation. According to Sabik and Blackstone (2008), two critical questions are
considered by surgeons when choosing between arteries or saphenous veins as bypass conduits
in CABG surgery: (a) Is there a degree of coronary artery stenosis (abnormal narrowing of a
1
blood vessel) below which a saphenous vein graft will more likely remain patent (open and
unobstructed) and therefore be more effective than an arterial graft? and (b) How does this
comparison change over time as vein grafts fail from development of vein graft arteriosclerosis?
Although a number of studies have indicated that the LITA graft is more effective and reliable
than the SV graft in terms of patency rate [Kitamura et al. (1996); Arima et al. (2005); Gardner
(2007)], the latter is still the typical choice of conduit used in current CABG surgery [Adlam et
al. (2011)], especially in elderly patients or in emergency situations because these veins are
easier to access and harvest, and they are more readily available than arterial grafts [Tappainer
2007]. Some recent studies have indicated that, except for the revascularization of the left
anterior descending (LAD) coronary artery, the SVG is the most commonly used graft for CABG
surgery [Souza et al. (2009); Nikolaos et al. (2012)]. In traditional coronary bypass grafting
surgical procedures, the LITA is usually used to bypass the LAD coronary artery, with the graft
connected between the aortic branch and the coronary artery, while the SVG is generally used for
aortocoronary bypass surgery [Bojar (2011)].
Due to the fact that the operative risk of mortality is relatively high in coronary
revascularization surgery, this surgery is always conducted using an inpatient procedure
according to standard coding guidelines for CABG procedures, i.e., Current Procedural
Terminology (CPT®) codes [Rhonda et al. (2004)]; Smith (2011)]. Although CPT® codes have
been the standard guidelines for CABG surgery, the guidance of such coding is in fact limited to
a case-by-case basis, but not patient-by-patient basis, due to the different levels of stenosis and
different locations of each patient’s occlusion. Moreover, the pathophysiological effects and
relation between grafts and arteries are still not fully understood today. Therefore, the question of
how best to construct bypass grafts to other coronary arteries remains open [Gardner (2007)].
2
The location of anastomosis and geometry of the bypass graft used in CABG surgery is still
largely based on the experience of the surgeon and conventional clinical procedures.
1.1.1 SVG and Types of Anastomosis
Saphenous veins are those blood vessels located in the legs and are perhaps the most
widely used conduit in CABG surgery to bridge blood from the ascending aorta to the coronary
arteries distal to the occlusion, excluding the LAD coronary artery. The CPT® codes (33510–
33516) are the standard reference guidelines used for venous grafting [Smith (2011)]. The
specific code is selected by the number of grafts applied in the CABG surgery [Bowie and
Schaffer (2012)]. Figure 1.1 illustrates the locations of the human aorta, and the typical setting of
proximal and distal anastomosis of venous grafts in CABG surgery.
Ascending Aorta
Proximal Anastomosis
Bypass Grafts
Blockage
Coronary Artery
Distal Anastomosis
Figure 1.1. Reconstructed aortic section with two bypass grafts.
Since the 1960s, when the first bypass graft was applied in CABG surgery, a massive
effort has been made to understand their physiological functions. With all the results and years of
experience and observation, it is now well recognized that the success of a CABG surgery is
3
directly related to the graft’s patency rate (i.e., the state of it being wide open) [Fitzgibbon et al.
(1996)]. A considerable number of results based on clinical, numerical, and experimental studies
have already been collected to evaluate the efficiency of different types of bypass conduit
applications in terms of alternative material, anastomosis setting, and geometry. The advanced
medical imaging techniques developed in recent years have tremendously improved the
assessment of the status of different grafts more specifically and completely in post-surgery.
Nevertheless, the controversy over the patency rate of saphenous vein grafts on single distal
anastomoses (s-SVGs) vs. multiple distal anastomoses (m-SVGs) has remained for the past two
decades.
Although the short- and long-term patency rates of SVGs used in CABG surgery have
been well documented in the past few decades, no agreement has been reached on the patency
difference observed between s-SVG and m-SVG settings. Results from angiographic and
pathological studies have been rather inconsistent and inconclusive. Opinions on the qualitative
correlation of the graft settings are still divided. For example, a number of clinical studies have
shown that m-SVGs have a better patency rate than s-SVGs [Eschenbruch et al. (1981); Meurala
et al. (1982); Christenson (1998); Vural et al. (2001); Oz et al. (2006)], while other clinical
studies have suggested no difference [Meeter et al. (1991); Cho (2006)] or worse [Kieser et al.
(1986); Goldman (1997); Mehta et al. (2011)].
Discrepancies between these clinical studies have been discussed in detail in the work of
Mehta et al. (2011) and Sabik (2011). For example, to compensate for the limitations of these
findings, Mehta et al. (2011) conducted a rather complete and systematic study to reevaluate the
patency difference based on the two different types of graft settings. This study was conducted
on 3,014 patients undergoing primary CABG surgery with at least two planned SVGs. Results
4
from the study have shown that the failure rate of the m-SVG was higher than that of the s-SVG
in a short-term (one-year) follow-up evaluation, and was significantly higher in a long-term
(five-year) follow-up evaluation. Therefore, the authors suggested that s-SVG be the conduit of
choice in multiple bypass surgery when situations allow it.
One of the noteworthy aspects from this assessment of SVGs is that the geometrical
effects, such as the location of anastomosis, graft diameter, angle of anastomosis, helical pitch,
and radius of curvature, were not taken into account while the assessments were performed. This
raises a question of whether geometry of the graft in ACB surgery plays a role in the patency
rates of the graft and, hence, results of the assessments. This question has driven the present
study’s area of focus to be on the geometrical effects of the bypass graft in the ACB model.
1.1.2 Locations of SVG Complication
Successful CABG surgery has been proven to extend the function of a cardiovascular
system with coronary artery disease by more than 10 to 15 years. However, statistics have shown
that only 50% of SVGs remain patent after ten years. Also, 15% to 20% of venous grafts are
reported to fail within the first year, which is known as early-phase graft failure [Mehta et al.
(1997); Canos et al. (2004)].
Although prevailing studies have been conducted to understand the pathological process
of the bypass graft applied in CABG surgery, the majority of these have largely concentrated on
the geometric effects of distal anastomoses with single or multiple grafts sutured end-to-side to
the coronary arteries [Frauenfelder et al. (2007)]. One of the common reasons is due to the wellrecognized subsequent late-phase graft failure caused by intimal hyperplasia, which is typically
found at the distal anastomotic site of the graft [Weyman et al. (1975); Echave et al. (1979);
LoGerfo et al. (1983)]. Studies that focus on the proximal region of a bypass graft (i.e., the
5
intersection region of the graft that is attached to the aorta) are very limited. It is necessary to
clarify that early-phase and intermediate-phase complications, such as graft thrombosis, graft
aneurysm, and graft spasm, are commonly found at other sites of the graft (i.e., proximal, distal,
and main body) [Leaper and Whitaker (2010)]. A clinical study by Canos et al. (2004) on earlyphase SVG failure (within a year) of 100 patients with SVGs applied in CABG surgery has
shown that the locations of early SVG lesions that lead to graft failure are mostly found at the
proximal or ostial region of the graft [Canos et al. (2004)]. A large amount of coronary
angiographic and radiographic assessment data that show the complications of the SVG at the
proximal region can be found in many recent clinical reports.
1.1.3 Motivation and Limitation
Nearly a half million CABG surgeries are performed in the U.S. each year. A successful
graft may last from about 10 to 15 years, or even longer. However, clinical studies have revealed
that about 15% to 20% of grafts fail within the first year after surgery due to vascular disease
caused by surgical technical errors or hemodynamics in general. The so-called early-phase graft
failure (less than one year) generally occurs at the proximal region of the graft. The present study
is motivated by two major needs: (a) a clinical need for understanding the effects of the proximal
anastomotic geometry of a bypass graft on hemodynamics in ACB surgery (i.e., ACB model
setting), and (b) a need for systematic analysis of bifurcated curved-tube flow analogous to the
ACB model, due to the lack of information on flow behaviors correlated with major geometric
parameters of settings such as size, angle, location of anastomosis, and radius of curvature of the
graft and aorta.
Flow behaviors associated with vascular disease have been widely investigated by many
researchers. The stagnation point, recirculation zones, reversed flow, and wall shear stress have
6
been implicated as the major hemodynamic factors contributing to the pathogenesis of
thrombosis and atherosclerosis. As discussed in the preceding section, the majority of effort has
centered on the distal portion of the graft, particularly on the end-to-side anastomosis [Steinman
et al. (1996); Bertolotti and Deplano (2000); Haruguchi and Teraoka (2003); Pousset et al.
(2006)]. An understanding of the relationship between blood flow and the proximal anastomotic
setting of the graft (i.e., region between graft and aorta) is rather limited; however, it is believed
that this is significantly important in reducing the early-phase failure rate of aortocoronary
bypass surgery. Current studies related to the ACB model are normally based on either a straight
host artery with aortic punch hole diameter equal or close to the branch diameter
[Sankaranarayanan et al. (2005) and (2006); Morbiducci et al. (2007); Zhang et al. (2008b)], or
subject-specific models in which the geometry of the models are highly complex in dimension
and geometry, which vary from patient to patient [for example, Dur et al. (2011)]. One of the
common findings from these studies is that the flow at the proximal region of the bypass graft
has shown to be highly dependent on the geometry at the junction. However, local flow
dynamics at the proximal location originating from the mismatch between the aortic punch hole
diameter and the proximal bypass graft diameter have not received much attention. From the
research finding in this dissertation, it is speculated that the flow behavior associated with the
size mismatch between graft and target host channel in ACB surgery may predispose the grafts
to early graft failure.
In addition to size mismatch in ACB surgery, the relationship between geometry of the
graft body (in terms of radius of curvature, helical pitch, and diameter) and flow dynamics in
ACB surgery has not yet been well recognized. Through recently invented novel surgical
devices, it is believed that recognizing the relation between such geometry and flow, and
7
identifying the critical parameters may help to improve blood flow in ACB surgery. For
example, the recently invented external mesh support saphenous graft, such as the external
saphenous vein support (eSVS) by Kibs Bay Medical Inc. (Minneapolis, MN, USA) for CABG
surgery has shown the need for more detail understanding of the relation between the
geometrical effects and blood flow behavior in ACB surgery.
Several recent experimental studies have indicated the advantages of using this device
because it has been shown that such support is able to provide a preferable hemodynamics
environment to the SVGs and hence the patency rate [Zilla et al. (2008)]. Genoni et al. (2013)
evaluated the patency rate of eSVS in 20 patients using the computed tomography angiography
technique and concluded that such support does not compromise the early graft patency rate.
However, the complete effect of such a device on the outcome of CABG surgery has yet to be
answered [Genoni et al. (2013)]. Some studies have indicated that the merits of such a device
could not be reproduced in clinical studies involving human CABG surgery. Trying to verify the
experimental findings on eSVS, Schoettler et al. (2011) performed the first in-man clinical study
using the angiographic technique to assess the use of eSVS in CABG surgery. Ninety-five
bypass grafts implanted in 20 patients were studied, and the patency rates of mesh-supported
grafts, conventional venous grafts, and arterial grafts were compared over nine months. Results
show that the conventional venous graft has a better patency rate than the mesh-supported graft,
with a patency rate of 85.7% compared to 27.8%. The authors of this study raised the issue of
diameter selection, anastomotic method, and fixation of the mesh tube to the venous graft as
being the critical determinants of the outcome of grafts [Schoettler et al. (2011)]. Figure 1.2
illustrates the external saphenous vein support mesh made by Kibs Bay Medical Inc.
8
Figure 1.2. External saphenous vein support (eSVS)
(Kibs Bay Medical Inc., Minneapolis, MN, USA, http://www.kipsbaymedical.com/esvs_mesh).
1.1.4 Area of Study
A substantial body of evidence has shown the complexity of fluid dynamics associated
with a curved channel under steady entry flow. The flow field of curved tubes with unsteady
entry flow is even more complex. The fluid dynamics of the ACB model, which is composed of
bifurcated tubes with a high curvature flow path and a significant difference in the crosssectional area, will be more complicated. The configuration of the graft on the ascending aorta
(size, angle of anastomosis, location of anastomosis, and radius of curvature) is believed to
strongly influence flow behavior at the proximal region. Random locations of proximal
anastomosis of the graft may impose different impacts on its functioning and lead to a different
progression rate of vein graft diseases. If this is the case, then clinical assessment based on such
settings may not fully reflect the actual condition of the case. The overall goal in this dissertation
is to better understand the effect of the ACB model geometry on local flow characteristics by
using a noninvasive approach. In the present work, the area of study is on the flow analysis of the
9
ACB model, specifically aimed at the proximal territory of the bypass graft. This direction of
focus is presented in Figure 1.3.
Figure 1.3. Focus of area of study.
1.2
Dissertation Outline
This dissertation is organized as follows: This first chapter has introduced the general
problems of vascular disease related to aortocoronary bypass surgery and the motivation behind
the work in this study. Chapter 2 presents background information and the significance of blood
flow behaviors and patterns, followed by the introduction of the relationship between vascular
disease and local hemodynamics of the vascular system, and the corresponding shear stress
indicators on the biological and pathological responses of the vascular system. The sections that
follow this cover a review of the typical flow environment in the human vascular system and
existing work related to the study of ACB flow, upon which the approach of numerical analysis
in this work is based. Chapter 3 continues with numerical methods applied in this study for the
hemodynamic analysis of the ACB model. The first section explains the governing equations for
fluid motion and assumptions applied in the present analysis. The second section describes the
10
different kinds of shear stress indicators that have been commonly used to quantify and correlate
with the initiation and development of vascular disease. Chapter 4 follows with a series of
validations aimed at different purposes of the numerical methods applied in this dissertation. The
capability and limitation of the present solver in this study are discussed. All results are
evaluated and compared against experimental results and analytical solutions by different
authors. Chapter 5 presents this study’s proposed numerical models for the analysis of blood
flow in ACB surgery. Chapter 6 discusses results of the study, and Chapter 7 presents
conclusions.
11
CHAPTER 2
BACKGROUND AND SIGNIFICANCE
2.1
Vascular Wall Structures and Diseases
The integrity of the vascular wall is basically defined by the particles carried by the blood
and local hemodynamic environment. The major factor that initiates vascular disease and leads to
graft failure is a dysfunction of the vascular wall. Understanding the biological response of the
vascular wall relative to hemodynamics still remains a challenging area of study. The
relationship between hemodynamics and vascular wall function is generally understood through
structures of the normal vascular wall, which are composed of three tissue layers—intima,
media, and adventitia—as shown in Figure 2.1.
Endothelial Cells
Intima
Smooth Muscle Cells
Media
Adventitia
Figure 2.1. Normal vascular wall structure [Lilly (2010)].
These tissue layers are formed with unique structures that have different physiological
properties. The intima, the innermost layer of the arterial wall, consists of a monolayer of
endothelial cells (ECs) known as the vascular endothelium. Damage to endothelial cells is found
to be responsible for early development of vascular disease. The media is the thickest of the three
layers and is formed as the main structural component in the middle of the wall. This layer
consists of multiple layers of smooth muscle cells embedded in an extracellular matrix. The
12
adventitia is the outermost layer surrounding the media, which contains mast cells, nerve
endings, and micro-vessels [Libby et al. (2011)].
Because of the different properties possessed by these layers, each responds differently to
their inner-outer environmental conditions, such as physical injuries, inflammatory stimuli, and
hemodynamic forces [Michel (2007); Verrier and Morgan (1998)]. The intima and media are
generally more vulnerable than the adventitia [Lilly (2010)]. It has been well recognized today
that thrombosis, neo-intimal hyperplasia, and atherosclerosis are the three major stages of vein
graft complications [Motwani and Topol (1998)]. Each stage is pathophysiologically interrelated.
The following sections provide an overview of information relative to the three major pathologic
stages that are responsible for vein graft disease.
2.1.1
Thrombosis
Thrombosis has been found to be the most common vein graft complication within the
first postoperative month of surgery [Bourassa (1991); Fitzgibbon et al. (1996)]. The location of
this disease is typically found in regions with focal endothelial damage [Fitzgibbon et al. (1996)],
which normally occur during surgery with the anastomosis procedure [Sabik and Blackstone
(2008)]. The factors purported to cause vascular thrombosis include the alteration of normal
blood flow, endothelial injury due to shear stress or hypertension, and alteration of blood
coagulation [Armstrong and Golan (2008)]. A number of studies have also identified the relevant
molecules that correspond to the formation of thrombosis. For example, thrombomodulin, tissue
and urokinase-type plasminogen activators, heparan sulfate proteoglycans, prostacyclin (PGI2),
and nitric oxide (NO) have been indicated as important antithrombotic molecules that resist the
formation of thrombosis [Libby (2009)]. Low production of NO and PGI2 have been indicated as
pro-thrombotic molecules that promote the formation of thrombosis [(Motwani and Topol
13
(1998)]. In the role of hemodynamics, shear stress on thrombosis has also been reported by a
number of investigators. Among these studies, low shear stress has been widely indicated to
reduce the shear independent release of NO and PGI2 [Davies (1995); Allaire et al. (1997);
Hanada et al. (2000)].
2.1.2
Neo-Intimal Hyperplasia
Within the first year after bypass surgery, intimal hyperplasia is the major disease process
responsible for venous graft failure or occlusion [Domanski et al. (2000)]. The process of vein
graft intimal hyperplasia is characterized by the abnormal migration and proliferation of smooth
muscle cells associated with the deposition of the extracellular matrix in the intimal layer of the
graft [Clowes (1993); Bhardwaj et al. (2005)]. Almost all venous grafts experience intimal
thickening between the first and second postoperative months. Although this process rarely
induces significant occlusive stenosis [Motwani and Topol (1998)], it has laid down the
fundamental morphologic feature for later vascular disease, which is the major factor of late vein
graft failure, and at this stage, the disease is known as an atherosclerotic lesion. Extensive
experimental work has been carried out to understand the underlying causes of intimal
hyperplasia. Injury of the vascular wall and hemodynamic stimuli have been indicated to be
important factors behind the development of intimal hyperplasia.
2.1.3 Atherosclerosis
The formation of atherosclerosis in vein grafts typically occurs beyond a year after
bypass surgery. Follow-up studies have shown that major stenosis with significant increase of
atheroma occurs normally between five and six years after surgery [Neitzel et al. (1986); Kalan
and Roberts (1990)]. Atherosclerosis is an inflammatory disease characterized by vessel wall
thickening and hardening due to the accumulation of plaque made up of substances like calcium,
14
fibrin, cholesterol, fatty materials, and cellular waste products in the inner lining of an artery
[Ross (1999)]. This primary disease affects the intimal layer. The favored locations for
atherosclerosis lesions are usually found at the origins of tributaries, branches, bifurcations, and
curvatures [Thubrikar (2007)]. Blood flow in these regions tends to form recirculation flow with
separation, reversed flow, stagnation, and turbulence. Some findings have indicated that the
development of atherosclerosis in aortocoronary bypass grafts are correlated with risk factors
associated with atherosclerosis in coronary arteries. Bypass grafts associated with atherosclerosis
are also susceptible to the formation of aneurysms [Neitzel et al. (1986)].
2.2
Role of Hemodynamics on Vascular Biology/Disease
The role of blood dynamics (hemodynamics) in vascular disease has been well
recognized and identified as playing a large part in determining the fate of a bypass graft.
However, due to the presence of pulsatile, non-Newtonian, and geometrical effects in arterial
circulation, the dynamics of blood flow in human vessels is complex, highly disturbed, and
varied with time. The analysis of blood flow becomes more complicated because the arterial and
blood properties, such as wall properties, blood pressure, density, and flow frequency, have
different magnitudes and vary spatially, temporarily, by age of the patient, and from individual to
individual. Although modern techniques and treatments for coronary artery bypass grafting
surgery have been well established, the outcome of ACB surgery still largely depends on how
successful the graft adapts to arterial circulation, which has a very different hemodynamic
environment than the original graft [Casey et al. (2001)].
Considerable effort has been made in the past five decades to better understand the
relationship between blood flow dynamics and the biological response of the human vascular
system. The role that hemodynamics plays in vascular biology has become clearer and consistent
15
in the literature today. However, vast uncertainties about its biological mechanisms remain to be
understood. The following sections provide some important background information for
understanding hemodynamic forces on the pathogenesis of vascular disease.
2.2.1 Effects of Hemodynamics Forces on Endothelial Dysfunction
The impairment of endothelial cells in the vein graft has been found to be the initial and
primary response of the vascular wall to the mechanisms and development of vascular diseases
[Ross (1993); Wu et al. (1996)]. Across the literature pertaining to vascular hemodynamics,
shear stress exerted on the vascular wall from blood flow is perhaps by far the most supported
determinant of vascular remodeling and pathology [Jadlowiec and Dardik (2013)]. The processes
of vascular wall remodeling and endothelial dysfunction due to hemodynamic forces have also
been very well documented in the literature. Therefore, detailed analysis of the distribution of
hemodynamic forces in the vascular system has become one of the most important and helpful
approaches in improving the outcome of bypass graft surgery. The influence of hemodynamics
on endothelial cells is discussed in the following section.
2.2.2 Relationship between Endothelial Cells and Nitric Oxide
As shown previously in Figure 2.1, the vascular endothelium is a thin layer that lies
between the flowing blood and the inner layers of the vascular wall. This is the outermost layer
that comes in direct contact with the dynamics of blood. The general function of normal
endothelium includes producing or releasing antithrombotic molecules to prevent clotting,
secreting substances to inhibit proliferation of smooth muscle cells into the intima, and
modulating the immune response to resist local inflammation. The general processes involved in
endothelial dysfunction include the up-regulation of adhesion molecules, increased chemokine
secretion and leukocyte adhesion, increased cell permeability, enhanced low-density lipoprotein
16
oxidation, platelet activation, cytokine elaboration, and smooth muscle cell (SMC) proliferation
and migration [Lilly (2010); Davignon and Ganz (2004)].
Nitric oxide (NO) formed in endothelial cells is recognized as the major substance
responsible for normal endothelial function [Garg and Hassid (1989); Kubes et al. (1991)].
Several studies have shown that the reduction of NO is a key factor leading to endothelial
dysfunction [Guzman et al. (1997); Spiecker et al. (1998); Davignon and Ganz (2004)].
2.2.3 Relationship between Nitric Oxide and Hemodynamic Forces
Several major physiological stimuli have been identified with NO production in
endothelial cells. These stimuli can generally be categorized as humoral factors and mechanical
forces. Humoral factors include the platelet-derived growth factors, angiotensin II, acetylcholine,
and cytokines, whereas mechanical forces include hydrostatic pressure, circumferential tension,
and wall shear stress induced by blood flow. Growing evidence shows that shear stress exerted at
the luminal surface of blood vessels appears to be most relevant activator of NO, thereby making
it the most important determinant in vascular integrity and hemostasis [Davies (1995); Corson et
al. (1996); Li (2005); Deanfield et al. (2007); Loot et al. (2008); Djordjević et al. (2011)].
2.2.4 Shear Stress on Endothelial Cells
Findings from in vitro studies have proven that the monolayer endothelium is highly
sensitive to hemodynamic forces from blood flow [Rubanyi et al. (1986); Davies et al. (2009)].
Theoretically, these forces are comprised of two components: normal force and shear force
(along the blood flow direction). The shear stress (shear force per unit area) generated on
vascular walls from blood flow has been found to be the most influential determinant of
endothelial function, vascular remodeling, and vascular pathology [van Thienen et al. (2006);
Loufrani and Henrion (2008); Jadlowiec and Dardik (2013)]. The direct response of endothelial
17
cells to different forms of hemodynamic forces has been very well tested. For example, animal
experimental studies [Mattsson et al. (1997)] with polytetrafluoroethylene (PTFE) grafts inserted
into the aortoiliac of baboons have shown that higher shear stress may induce the regression of
intimal hyperplasia. Grafts exposed to blood flow with two months of high shear stress after two
months of normal flow exposure (total of four months flow exposure) were shown to suppress
not only the growth of intimal hyperplasia but also induce intimal regression significantly,
whereas grafts with four months of normal shear stress exposure were shown to have higher
intimal thickening than that of high shear stress exposure. Figure 2.2 illustrates the influence of
hemodynamic shear stress on a vascular graft, as demonstrated by Mattsson et al. (1997).
Figure 2.2. Effect of hemodynamic shear stress on vascular graft: (A) two months of normal shear
stress, (B) two months of normal shear stress followed by two months of normal shear stress, and
(C) two months of normal shear stress followed by two months of high shear stress
[Mattsson et al. (1997)].
2.3
Types of Hemodynamics Shear Stress in Literature
Different approaches have been employed to quantify the shear stress exerted on an
arterial wall in order to understand the biological responses of the vascular wall and the
correlation between hemodynamic shear stress and the initiation and development of vascular
disease. Shear stress induced by blood flow on the vascular wall is typically calculated from
velocity profiles extracted from the vascular system. These profiles are generally obtained from
either experimental measurement or computational simulation. In the past few decades, blood
18
flow in the carotid bifurcation, coronary arteries, aorta, and bypass grafts have received the most
attention for investigation. In these studies, the shear stress applied can be generally categorized
as follows: (a) steady vs. unsteady (pulsatile), and (b) laminar vs. turbulent. Section 2.3.1
describes relationships between different types of shear stress and the biological response of the
vascular wall.
2.3.1 Steady Laminar and Turbulent Shear Stress
The physical deformation of endothelial cells due to hemodynamic shear stress is
microscopically small. However, the influence of shear stress on the biological response of the
vascular wall is significant. The proliferation and apoptosis (process of cell death) of endothelial
cells is greatly dependent on the local hemodynamic shear stress of the vessel. The major
pathway of shear force affecting the vascular pathobiology is through the shear stress sensing
elements in the endothelial cells. A number of studies on cultured cells have demonstrated that
shear stress from a steady laminar flow condition is the critical and preferable hemodynamic
state in order to maintain vascular tone by regulating and stimulating the atheroprotective genes
of the vascular system to inhibit the development of vascular disease [Girard and Nerem (1993);
Cunningham et al. (2005)]. Under a steady shear stress condition, the shape of the endothelial
cells exposed to high laminar shear stress are normally elongated, and the network structure of Factin fibers (major cell property in maintaining endothelial barrier function [Ehringer et al.
(1999)] may undergo reorientation and realignment with the flow direction [Nerem et al. (1981);
Levesque et al. (1986); Kim et al. (1989)], which are found to be an important atheroprotective
state for the vascular system [Vartanian et al. (2008)]. In contrast, vascular cells subjected to
non-laminar flow, especially flows associated with oscillation, low shear, and turbulence,
generally result in cell turnover and random distribution, which predisposes the cells to vascular
19
disease. An earlier in vitro study by Davies et al. (1986) with turbulent and laminar flows applied
on endothelial cells has shown that EC turnover is highly sensitive to low shear turbulent flow
compared to high shear laminar flow. Cells under high laminar shear flow were shown to align
with the flow direction, while cells under low shear turbulent flow experienced cell loss and
misalignment. A study by Dardik et al. (2005) with different flow conditions applied on bovine
aortic endothelial cells (BAECs) has also shown that steady laminar shear stress increases the
cell survival rate and decreases the proliferation rate of ECs.
2.3.2 Unsteady Laminar Shear Stress (Spatial and Temporal)
Blood flow in human arteries is not steady but rather pulsatile in nature. Although steady
laminar flow does not represent the real environment of blood flow in the human vascular
system, the study of steady flow effects on endothelial cells has established essential
fundamental information about the biological response of the vascular wall to hemodynamic
shear stress. The study of unsteady laminar shear stress can generally be categorized under flow
with a reversing condition, or non-reversing condition. Work done by Helmlinger et al. (1991)
involved the application of different pulsating flow conditions to investigate the influence of
pulsating shear stress on cellular morphology. Results from this experiment demonstrate the
different responses of cultured endothelial cells under steady and unsteady flow conditions in
terms of their orientation and shape. ECs experienced cell detachment as they were exposed to an
unsteady reversing flow with a relatively high shear stress condition. Using cultured cells of
human endothelium, Chappell et al. (1998) studied the relationship between oscillatory shear
stress and the biological response of endothelial cells. Their results show that low shear
oscillatory flow up-regulated the adhesion molecules of the ECs and down-regulated the
production of endothelial nitric oxide. A similar study by Dardik et al. (2005) using orbital shear
20
stress on cultured endothelial cells also shows that low orbital shear stress increased the
proliferation and the apoptosis rates of the cells.
In vivo work by Kohler et al. (1991) with different flow rates applied on a series of
endothelialized vascular grafts has shown that the increase of blood flow reduced the formation
of the neo-intimal area of grafts implanted in baboons. A similar observation was reported in the
work of Mattsson et al. (1997), which showed that the increased blood flow induced the
regression of intimal hyperplasia of PTFE grafts implanted in baboons. An animal experimental
study by Langille et al. (1989) also shows agreement with the effect of shear stress on neo-intima
in the carotid artery of rabbit. In this study, a reduction in the flow rate by 70% to 80% was
shown to cause the internal diameter of the arterial wall to grow about 21%. This inverse effect
of shear stress on intimal hyperplasia is also supported in the clinical follow-up study by Wentzel
et al. (2001). Here, results based on 14 patients with stented coronary arteries over a period of six
months showed that the regions with low shear stress had a higher average intimal thickness than
those regions with high shear stress.
2.3.3 Unsteady/Pulsatile Turbulent Shear Stress
The existence of turbulent flow in the human vascular system has been well recognized.
Although such disturbed flow conditions in the human circulation system have been confirmed,
the information is scanty, especially on the responses of endothelial cells to turbulent shear stress
compared to laminar shear stress. Unlike laminar flow, the structure of turbulent flow is, in
general, highly irregular, with random motion relative to space and time. Blood flow in the
cardiovascular system associated with turbulent flow has been thought to be an abnormal
condition by most cardiologists [Bluestein and Einav (1994)]. Either turbulent or disturbed flow
has been indicated as an important risk factor correlated with vascular disease. Regions
21
associated with turbulent flow are susceptible to atherosclerosis [Constantinides (1989); Davies
(2009)].
An experiment performed by Davies et al. (1986) on BAECs shows that cells exposed to
turbulent flow with high shear stress may exhibit cell retraction and cell loss conditions. Jared et
al. (2004) studied the relation of adhesive molecule of ECs with turbulent flow and found that Eselectin {one of the major inflammatory mediators that increases the risk of vascular disease
[Poredos (2011)]} was down-regulated in the presence of laminar flow but not in the presence of
turbulent flow. Endothelial cells exposed to turbulent flow also fail to up-regulate the level of
NO [Noris et al. (1995)]. Brooks et al. (2002) employed both laminar and disturbed pulsating
flow conditions on cultured human aortic endothelial cells and identified more than 100 genes
stimulated in the cells during the experiment. The majority of genes induced by laminar flow
were found to be atheroprotective. Under disturbed flow condition, results show that the level of
pro-atherosclerotic genes increases with molecules that promote inflammation, apoptosis, and
coagulation.
In summary, it is generally accepted now that disturbed flow and low shear stress
particularly associated with a high-oscillation condition are correlated with the occurrence of
vascular lesions. These findings are also consistent with results shown in the work of Caro et al.
(1971), Glagov et al. (1988), Zarins et al. (1987), Guzman et al. (1997), and Traub and Berk.
(1998).
2.4
Indicators of Shear Stress for Biological Response of Vascular Wall
Endothelium has been proven to be a shear-sensitive layer. It responds to hemodynamic
shear stress differently by activating different biological processes to achieve the equilibrium
state for which it was designed. Several measurement methods of shear stress have been
22
proposed in the past to elucidate the effects of pulsatile shear stress on the biological and
morphological responses of the vascular system. Different indicators of shear stress have been
used by different researchers in an attempt to describe the relationship between shear stress
(steady or unsteady) and vascular wall function. For example, Zarins et al. (1987) applied a
magnitude of instantaneous wall shear stress instead of stress directions (stress vector) in his
studies to correlate the intimal thickening of the aorta of a monkey. Kohler et al. (1991)
suggested that the intimal thickening is more sensitive to the absolute magnitude of shear stress
rather than the amount of shear stress, which varies over time. Work done by Nagel et al. (1999)
found that the spatial shear stress gradient may be an important variable to representing the
expression of local modulators of endothelium. Other studies have indicated that the temporal
gradient wall shear stress induces endothelial cell proliferation, while the effect of the spatial
shear stress gradient on endothelial cell proliferation is found to be similar to the effect of
laminar shear stress [White et al. (2001)]. The different types of indicators and the threshold of
shear stress from the literature will be discussed in the next section.
No single theory is sufficient to elucidate the biological response of the vascular wall to
hemodynamic shear stress. Theories that widely abound in the present literature include steady
wall shear stress (WSS), time-averaged WSS (TAWSS), WSS gradient (WSSG), oscillatory
shear index (OSI), and relative residence time (RRT). These different types of indicators are
discussed in the following sections along with their thresholds on the risk of vascular disease.
2.4.1
Steady Laminar Wall Shear Stress
The study of steady flow in the human vascular system is the essential step to
understanding the direct response of vascular wall function to hemodynamic shear stress. Several
decades ago the effect of hemodynamic shear stress was suspected as a major factor in
23
manipulating the function of the human vascular system. Early in vivo experimental work by Fry
(1976) provided the first insight into endothelium damage due to steady shear stress induced by a
high rate of blood flow. In this study, the endothelial histology of the aorta was accessed by
exposing a dog’s aorta to a wide range of shear stress levels. Results showed that the critical
steady shear stress of the endothelial cells was found to be around 30–40 Pa. Significant cell
damage was observed as the cells were exposed to this level of stress for one hour. Cells below
this level of shear stress were described as being in a normal condition. Afterwards, more efforts
were made to better understand the response of endothelial cells to different levels of shear stress
based on different forms of blood flow, including shear stress induced by pulsating flow,
oscillating flow, and turbulent flow. It is now well recognized that shear stress is a major
stimulus that controls the expression of endothelial genes, the condition of vascular wall
morphology, and the quiescence of endothelial cells [Pohl et al. (1986); Lansman (1988); Dekker
et al. (2006)]. Research efforts today have more accurately pinpointed the range of critical shear
stress associated with vascular lesions.
As discussed in the previous section, wall shear stress from steady laminar flow is the
preferable dynamic condition that is found to be important in protecting and maintaining the
integrity of vascular tone. A number of studies have shown that under steady laminar flow
condition, the proliferation of the endothelial cells was completely inhibited, compared to the
unsteady flow condition [Levesque et al. (1990); White et al. (2001)]. Depending on the
magnitude and form of shear stress, researchers have found that the shear sensitive layer of the
endothelium responds differently to shear stress by expressing different genes and molecules that
have different roles in the development and prevention of vascular diseases. For example, a study
by Topper et al. (1996) on human umbilical vein endothelial cells (HUVECs) with laminar WSS
24
of 1 Pa has shown that the atheroprotective gene of nitric oxide was up-regulated approximately
threefold compared to the static (non-shear) condition. This result is in agreement with a
previous study by Nishida et al. (1992) on cultured bovine aortic endothelial cells based on a
steady shear stress of 1.5 Pa. Other similar studies have also demonstrated that the level of the
atheroprotective gene of NO was increased several fold on cultured endothelial cultured cells
subjected to unidirectional laminar WSS ranges between 1.5 Pa and 2.5 Pa [Uematsu et al.
(1995); Carson et al. (1996); Davis et al. (2001)]. A recent study by Wang et al. (2013) involving
different flow directions on cultured endothelial cells (unidirectional flow and perpendicular
flow with respect to endothelial cell orientation) also indicated that NO synthase was activated
the most under laminar flow and was undetectable under flow perpendicular to endothelial cells.
In terms of cell proliferation, Akimoto et al. (2000) employed a cultured cell method
using bovine aortic and human umbilical vein endothelial cells and observed that the
proliferation of these cells was shown to be able to be inhibited under steady laminar shear stress
of 0.5 Pa and 3 Pa. In similar work by Conway et al. (2010) on human umbilical endothelial
cells, cells exposed to a steady shear stress of 0.1 Pa were compared with those exposed to a
steady shear stress of 1.5 Pa. Results showed that the cell proliferation rate was higher on those
subjected to the lower shear stress condition than the higher shear stress condition. Cells exposed
to 0.1 Pa were also shown to have a higher up-regulation of gene expression than those exposed
to higher shear stress. Using the orbital shaker and parallel-plate flow-generation techniques,
Dardik et al. (2005) conducted an in vitro experiment with BAECs to study their response under
laminar and non-laminar WSS effects. In this study, ECs subjected to lower shear stress were
found to be highly vulnerable compared to cells with a higher WSS condition. The proliferation
rate of regions with shear stress of 0.5 Pa was approximately 30% higher than that of the static
25
condition. However, regions with WSS of 1.1 Pa only showed about 16% of the proliferation rate
compared to the static condition.
An in vitro experiment by Gonzales and Wick (1996) with human umbilical vein
endothelial cells has also demonstrated that the endothelial cell subjected to low shear stress
(0.2–1 Pa) exhibited approximately 350% of monocyte adherence to the endothelium compared
to cells subjected to the static condition, while the adhesion of monocyte molecules on cells
subjected to 3 Pa of shear stress were shown to be similar to that of the static condition.
2.4.2 Time-Averaged Wall Shear Stress
Assessment of the biological response of endothelial cells subjected to shear stress
induced by unsteady flows (pulsating or oscillating) is in fact a very challenging task in either in
vivo or in vitro experiments. Different approaches have been proposed by different researchers to
correlate the biological response and process of endothelium to unsteady WSS in vivo or in vitro.
Early in vivo experimental work on baboons based on PTFE prosthetic grafts by Geary et al.
(1994), Kraiss et al. (1996), and Mattsson et al. (1997) has provided valuable insight into the
mechanisms of graft disease associated with shear stress. By exposing prosthetic grafts with high
WSS and low WSS with mean shear stress ranges of 5–7.5 Pa and 0.08–2.2 Pa, Geary et al.
(1994) observed that both the intimal area and the number of smooth muscle cell subjected to
low WSS after 28 days was approximately fivefold larger than that involving high WSS. A
similar study by Kraiss et al. (1996) with mean shear stress ranges of 0.5–1.4 Pa and 2.3–3.6 Pa
for low and high mean shear stress, respectively, has shown that grafts subjected to low shear
stress showed an approximate twofold proliferation than that of the high shear stress after four
days of observation. Mattsson et al. (1997) further carried out a study to investigate if high flowinduced shear stress is capable of inhibiting the establishment of neointima. Their results reveal
26
that the elevation of blood flow rate to inhibit the growth of the developing neointima thickness
through the loss of smooth muscle cells and matrix is possible. In this experiment, a marked
decrease of the cross-sectional area of the grafts with established neointima was observed after a
four-month exposure to a high flow rate.
Recent in vivo assessments by Samady et al. (2011) on the human coronary artery
focusing on TAWSS effects have indicated that low TAWSS (< 0.1 Pa) enhances the progression
rate of coronary plaque in patients, while high TAWSS (> 2.5 Pa) is suspected to be associated
with plaque destabilization. The work of Chatzizisis et al. (2011), which focuses on swine
coronary arteries, also agrees with the finding that regions of low shear stress correlate to the
promotion of plaque formation due to the deformation of endothelial cells and the accumulation
of inflammation cells. In this study, results show that the majority of plaque (72%) was formed at
regions under low TAWSS (0.64–0.70 Pa), compared to regions with high TAWSS (1.52–1.66
Pa). The hypothesis of low TAWSS associated with vascular disease is also supported by
previous in vivo findings based on animal studies involving the iliac artery [Zarins et al. (1987)]
and the carotid artery [Mondy et al. (1997); Cheng et al. (2006)], in which it was suggested that
regions exposed to low TAWSS correlated with the formation of atherosclerotic plaques. These
observations and results relative to the influence of TAWSS are consistent with earlier
experimental studies by Ku et al. (1985) on a subject-specific model of human carotid arteries,
which showed that atherosclerotic plaque in the human vascular system correlates with regions
of low wall shear stress, thus suggesting that marked oscillatory shear stress may enhance the
development of atherosclerotic lesions.
27
2.4.3
Wall Shear Stress Gradient
A number of studies have shown that the rate of change of shear stress, or wall shear
stress gradient, may yield better information on certain atherogenetic genes in endothelial cells
subjected to pulsatile flow. The two most commonly used indicators based on the WSSG are the
temporal WSSG (TWSSG) and the spatial WSSG (SWSSG), in which the WSSG is calculated
with respect to the rate of change of the time and the location (distance), respectively. Biological
and morphological responses of endothelial cells induced by both indicators have been reported
by a number of investigators. However, the correlation of these two indicators on atherogenetic
risk is still controversial. Several in vitro studies have employed a steady step flow technique to
create both a temporal and a spatial effect on cells through the recirculation zone that is
generated between the separation and reattachment points.
For instance, Tardy et al. (1997) investigated the effect of the SWSSG on human
umbilical vein endothelial cell monolayers by comparing cells subjected to SWSSG ranges
between –0.3 and +0.5 Pa per mm (which corresponds to local WSS ranges between –1 Pa and
+10 Pa) against those with a steady laminar shear stress of 1.4 Pa. Results show that the
endothelial layer exposed to a high SWSSG (particularly at the reattachment zone where the
mean WSS is very low and close to zero) exhibited a higher proliferation, migration, and
detachment rate than that of laminar flow regions. This observation is also supported in prior
work by DePaola et al. (1992) who applied a similar method on bovine aortic endothelial cells
with an SWSSG that varied between 0 and +0.6 Pa per mm (which corresponds to a local WSS
ranging between –1.5 and +2.5 Pa). The endothelial cells under a high SWSSG were shown to
have more morphological changes and a higher division rate than those cells under a laminar
shear stress condition.
28
In terms of endothelial cell response to the TWSSG, several studies have indicated that a
large TWSSG may cause endothelial injury or damage [Ojha (1994)] and stimulate certain proatherogenetic genes and adhesion molecules in the endothelial monolayer [Bao et al. (1999);
Hsiai et al. (2003)].
Using the step flow technique, White et al. (2001) further extended the method to
investigate the difference between the effects of the TWSSG and SWSSG on endothelium cells.
In this study, the effect of the TWSSG was achieved by applying a ramping method to induce
temporal variation of shear stress on the specimens. Results show that significant cell
proliferation was observed at the reattachment regions subjected to a TWSSG of approximately
300 Pa/s, but such behavior did not occur because the effect of the TWSSG was absent and the
cells were only subjected to the SWSSG (0–40 Pa/mm). Cells in those regions were described as
behaving similarly to those subjected to steady laminar flow. The authors suggest that the spatial
WSSG does not play a role in the proliferation of endothelium.
Work done by Dusserre et al. (2004) using a syringe pump technique shows that, under
steady flow alone, the endothelium absent the TWSSG does not induce the dissociation of the
PECAM-1 molecule (an important mediator of atherosclerosis, which is most abundantly
expressed in endothelial cells), while the presence of the TWSSG was found to be accompanied
by the dissociation of the molecule, a common behavior observed in atheroprone regions. Further
work done by Otte et al. (2009) also reported the same observation in their study on HUVECs.
2.4.4 Oscillatory Shear Index
The region subjected to low WSS with profound oscillating flow is perhaps the mostrecognized condition for atherosclerotic lesions. An early experimental study by Ku et al. (1985)
on a subject-specific model of human carotid bifurcation has shown that regions with low and
29
prominent oscillatory shear stress were strongly correlated to the development of intimal plaque.
This observation is in agreement with the prior experimental work of Bharadnaj et al. (1982) and
Zarins et al. (1983) who employed visualization methods to investigate vascular lesions on
anatomical models of human carotid bifurcation. In order to quantify the cyclical variation of
WSS vectors, Ku et al. (1985) introduced the oscillatory shear index to account for the change in
WSS vectors that are exerted on the vascular wall in the unsteady pulsating condition. Such
indicators can be used to identify flow reversal that exists in a vascular system.
The hypothesis made by Ku et al. (1985) on low and oscillatory shear stress was later
supported by many researchers in either animal or human studies. For example, Bassiouny et al.
(1992) accessed the intimal hyperplasia of venous and prosthetic grafts implanted in animals to
investigate the impact caused by the mechanical and hemodynamic factors. Results show that
intimal hyperplasia, which formed on the host arterial wall, is correlated to low shear oscillatory
flow. The concept of low and oscillatory shear stress is also supported in the work by Moore et
al. (1994) who studied blood flow in the human abdominal aorta using a magnetic resonance
imaging (MRI) velocimetry technique. By comparing the measurement results with the
probability of occurrence maps of atherosclerosis in 1,378 patients and other clinical data
reported by others, it was shown that the regions associated with low mean WSS and oscillatory
flow were highly correlated to locations with atherosclerotic lesions. It was also pointed out from
the same study that shear stress stimulated under an exercise condition was able to provide a
favorable hemodynamic environment that may increase the low mean WSS magnitude and
decrease the oscillatory and flow reversal effects.
Similar to TAWSS, TWSSG, and SWSSG, oscillatory flow has been identified to be able
to regulate different genes and molecules that have different impacts on vascular tone. Work
30
done by Helmlinger et al. (1995) has demonstrated that calcium on cultured BAECs responded
differently because the cells were subjected to different types of shear stresses induced by steady,
pulsatile, and pure oscillatory flows. The authors showed that oscillatory flow with shear stress
ranging between –2 and +2 Pa totally failed to induce any (Ca2+)i (a major element in the
activation of NO synthase), compared to steady WSS ranges between 0.02 and 7 Pa and pulsatile
WSS ranges between 2 and 6 Pa and between –2 and 6 Pa. A similar study by Ziegler et al.
(1998) showed that regions with low oscillatory shear stress ranging between –0.3 and +0.3 Pa
down-regulated expression of the endothelial nitric oxide synthase (eNOS) level, whereas
regions with steady WSS as low as 0.03 Pa induced as much as sixfold eNOS, compared to the
static condition. Unidirectional pulsatile WSS ranges with mean shear stress of 0.6 Pa were also
shown to have induced the expression of eNOS approximately thirteenfold, compared to cells
under the static WSS condition.
A different approach proposed by Hsiai et al. (2003) using a real-time shear stress sensor
coupled with cell velocity tracking by microelectromechanical system (MEMS) sensors has also
demonstrated that under low OSI with shear stresses in the range of ±2.6 Pa, the monocyte
adhesion of endothelium cells was up-regulated approximately sevenfold under the static
endothelium, whereas cells under unidirectional pulsatile WSS with a TAWSS of 5 Pa was
shown to be able to weaken the adhering effect of monocytes on the endothelium cells
significantly to a level close to the static condition. This observation is also supported by the
experimental study of Hwang et al. (2003) on endothelial aortic cells of mice.
Other recent findings have demonstrated that regions exposed to low and oscillatory
shear stress may induce proatherogenetic molecules that promote the initiation of atherosclerosis
[Himburg et al. (2004); Takabe et al. (2011)] and the growth of aneurysms [Kojima et al.
31
(2012)]. A recent clinical study using a 4D-MRI technique by Markl et al. (2013) quantified the
WSS and OSI among 70 patients and 12 healthy volunteers, and reported that most of the regions
associated with complex plaque are co-located with high OSI. These regions have been shown to
have higher permeability of endothelium to macromolecules and disruption of the alignment of
endothelial cells [Himburg et al. (2004)].
2.4.5
Relative Residence Time
The use of relative residence time in correlating vascular lesions began in 2004. This
indicator was introduced by Himburg et al. (2004) who investigated the hemodynamic WSS
effects on the permeability of endothelial cells to macromolecules based on experimental and
computational approaches. Due to the fact that the OSI indicator is insensitive to WSS, it is
difficult to correlate it to the permeability variation of endothelial cells. Regions with low
TAWSS may not necessarily have a high OSI, and vice versa. To reconcile the gap between the
indicators of OSI and TAWSS, the authors introduced the concept of relative residence time
based on the non-adhesive particle flow concept to emphasize the low WSS and high OSI
condition of an area. Based on the low WSS and high OSI hypothesis, the larger the metric, the
higher risk of atherosclerotic lesions.
A number of studies have indicated that regions with prolonged RRT are vulnerable and
susceptible to vascular diseases. Work done by Hoi et al. (2011) compared the distribution of
aortic plaques of mice with three different indicators—OSI, RRT, and TAWSS individually. The
authors showed that plaque forming in the vessels is correlated to the OSI and RRT but not
TAWSS. Some studies also suggest that TAWSS and OSI can be replaced by the single metric
RRT alone [Lee et al. (2009)]. The work of Soulis et al. (2011) shows that the aortic arch
exposed to high RRT is correlated to low shear high oscillatory flow, suggesting that high RRT
32
can be an appropriate indicator to correlate the concentration of atheroma. A more recent study
by Sugiyama et al. (2013) has shown that the atherosclerotic lesions formed in the intracranial
aneurysmal walls of 30 patients correlated with maximum RRTs, which were computed using
computational methods.
It should be noted that, at the present time, none of the theories fully explain or correlate
the initiation and development of arterial disease due to hemodynamic shear stress. Each of these
indicators has been shown to have limitations. For example, the cell culture study with human
endothelial cells by White et al. (2001) has shown that the temporal wall shear stress gradient is
the major determinant in cell proliferation, whereas the proliferation induced by the spatial wall
shear stress gradient is found to be similar to that of steady uniform shear stress. Rikhtegar et al.
(2012) extracted 30 subject-specific geometries of the left coronary artery from 30 patients to
investigate the relationship between different indicators (TAWSS, TAWSSG, OSI, RRT) and
plaque locations. Using the reconstructed models with plaque removed from the models,
numerical results show that TAWSS has significant higher sensitivity than the other indicators.
More recent review work by Peiffer et al. (2013), which focused on the impact of low and
oscillatory WSS, has also pointed out the limitations of these indicators. A complete theory to
elucidate the mechanism and the pathology of vascular disease due to blood flow is still under
investigation. Even so, the role of wall shear stress on vascular biological response, especially on
endothelial cell function, has been very well established, accepted, and supported in a large body
of literature. In the present study, the WSS, TAWSS, and OSI are used to investigate the relation
between hemodynamics and different study case models and flow conditions.
33
2.5
Typical Flow Patterns in Vascular System
Although hemodynamic shear stress is recognized as the most influential determinant of
endothelial function and phenotype in terms of hemodynamic factors, the mechanisms that
govern the distribution of shear stress lie in the local flow field of the vascular system, which is
generally characterized by the geometry and fluid dynamics of the physical system’s flow field.
The flow field in the vascular system experiences a wide range of alterations in terms of
magnitude and direction due to the presence of pulsation, fluid viscosity, wall friction, and
geometrical effects. Several regional disturbed flow characteristics in the vascular system have
been identified and recognized as triggering factors in the abnormal biological function of blood
vessels. Major flow characteristics include the recirculation zone, stagnation and reattachment
zones, separation flow, secondary flow, and turbulent flow. Since the distribution of shear stress
is based on local flow patterns in the system, it is this understanding of geometrical effects on
local flow field of the vascular system that becomes the major approach in improving the
hemodynamic environment in the vascular system, especially relative to bypass surgery.
The following section focuses on the definition of different types of disturbed flows that
have been found to correlate with vascular lesions. The significance of each of these flow
behaviors on vascular lesions along with findings from the literature are discussed.
2.5.1
Separation Flow and Recirculation Zone
Flow separation in blood vessels can be referred to as a process when fluid elements
adjacent to the artery detach from their wall and no longer move with the main streamline flow
but rather move into the interior of the boundary layer with an unsteady motion. Under a steady
laminar condition, flow separation occurs when the momentum of the boundary layer close to the
surface is insufficient to overcome the local adverse (positive) pressure gradient experienced by
34
the fluid. This results in a decrease of kinetic energy in the boundary layer, which causes the
flow to decelerate, cease, and eventually reverse and detach from the surface.
In two-dimensional flow, the separation point is defined as the region where the shear
stress induced by the blood flow on the vascular wall is zero [Granger (1995)]. As laminar flow
separation occurs, the downstream flow may reattach to the surface again as a turbulent
boundary layer or it may remain separated (bursting) [Jones et al. (2008)]. The region between
these two stagnant points is known as the laminar separation bubble (or recirculation zone)
where the fluid dynamics in this region is predominated by the inflectional nature of the flow that
consists of a group of large- and small-scale vortices [Alam and Sandham (2000); Robinet
(2013)]. The fluid properties in the separation zone are normally classified into two major areas
of study: short bubble and long bubble [Tani (1964)]. The physical factors that determine the
structure, dynamics, and dimensions of these bubbles remain open for discussion today. The
presence of a pulsatile nature of blood flow increases the complexity level of this process.
Recirculation flow in a vascular system normally is observed in the aorta arch, aortic Tjunction, and bifurcations zones during the systolic phase (period when heart is contracting and
pressure is maximum) of the cardiac cycle [Karino (1986); Schwenke and Carew (1989); Zand et
al. (1999)]. Regions associated with recirculation flow are normally referred to as dead zones in
which the kinetic energy of particles is low and the particle residence time is high. Due to the
formation of adverse pressure gradient, the local velocity gradient in the separation region is
reduced, and the wall shear stress in this region generally experiences relatively low shear stress
effect [Bernard et al. (1998)]. Vascular regions exposed to a disturbed flow condition may
encounter on and off flow separation due to the pulsatile effect. Therefore, the magnitude and
amplitude of shear stress at these regions are varied throughout the cardiac cycle. Thus,
35
numerical simulation based on physiological waveform is important to reveal and capture such
effects in the physical domain.
2.5.2
Dynamic View of Unsteady Secondary Flow Behavior
Takayoshi and Katsumi (1980) studied the velocity distribution of a circular tube through
theoretical and experimental methods. Applying the Navier-Stokes equation in a cylindrical
form, an analytical equation was derived and applied to compare with experimental results
obtained from steady, oscillating (zero mean), and pulsating (non-zero mean) flow conditions
with Reynolds numbers (Re) in the range of 520, < 561, and < 312, respectively. The Womersley
number () (a dimensionless parameter commonly used to relate the viscous effect to the
pulsating flow frequency) for oscillatory and pulsating conditions ranged between 1.3–23 and 2–
12, respectively. Results show that under the oscillating flow condition, the maximum velocity
profiles were observed to be flattened out when the Womersley number increased from 3.9 to
12.7. This observation is in agreement with McDonald (1974) who explained that the unsteady
flow field is dominated by an inertia force when the Womersley number of the flow is high,
whereas the flow field is dominated by viscous flow when the Womersley number is low.
2.5.3
Dynamic View of Unsteady Straight-Tube Flow Behavior
Attention on unsteady flow behavior of a curved tube generally began in the early 1970s
as Lyne (1971) applied theoretical and experimental methods to investigate the unique structure
of secondary motion induced by pulsatile flow. Using a sinusoidal oscillatory flow condition, the
maximum axial velocity in the curved section of the tube was found to be skewed toward the
inner wall of the curve, which is opposite that of the steady-state condition. On the secondary
flow structure, the circulation direction in the core region of the curved section was also found to
be opposite to that predicted for the steady-flow condition, but at the near-wall region, the
36
secondary flow direction was found to be similar to that predicted for the steady-state condition.
Such a phenomenon for the unsteady secondary flow was then distinguished by the author using
two different flow regions: (a) core region in which the flow was described as being dominated
by the inertial effect where the viscous effect is negligible, and (b) boundary region in which the
flow is dominated by the viscous effect.
Using laser Doppler velocimetry (LDV), Timité et al. (2010) investigated the
development of the axial velocity profiles in a 90-degree curved pipe in order to understand the
influence of pulsating flow on the structure of the secondary flow. Flow conditions were
characterized by steady-flow Reynolds number and flow-rate ratio, or Dean number (De),
ranging between 300 and 1200, and between 0.5–1.85, respectively, with Womersley numbers
varying from 1 to 20. This study showed that the intensity of the secondary flow was inversely
proportional to the frequency of the flow in the accelerating phase of the cycle, whereas the
Lyne-type secondary flow structure was observed and intensified during the decelerating phase.
Similar studies with different ranges of Re, De, and  based on experimental, numerical,
and analytical methods have also confirmed observations in other work [Lyne (1971); Zalosh
and Nelson (1973); Munson (1975); Chandran et al. (1979); Gammack and Hydon (2001)].
2.5.4 Dynamic View of Unsteady Curved-Tube Axial Flow Behavior
The distribution of axial velocity of unsteady curved-tube flow was later studied by many
investigators. In order to understand the time-dependent flow of a curved tube analogous to the
human vascular system, Chandran et al. (1979) employed a single-component hot-film probe to
investigate the development of axial velocity profiles in a 180-degree solid curved pipe
(Plexiglas) with a uniform entry flow condition analogous to the physiological condition in the
human aorta (time-averaged Re = 1019, De = 332, and  = 21.89). Results indicate that the
37
maximum axial velocity during the systolic phase was observed at the apex of the inner wall and
continued with reversed flow during the diastolic phase (period when the heart is relaxing and
pressure is minimum), whereas flows near the outer wall continued their motion downstream.
Further study by the same authors a few years later using a three-dimensional hot film probe
quantitatively verified these observations [Chandran and Yearwood (1981)]. The adverse
pressure gradient during the diastolic phase was found to cause the same reversed flow behavior
along the inner wall as reported previously. The wall shear stress exerted on the inner and outer
walls of the tube was also calculated using the first-order finite-difference approximation
method. It was shown that the major difference in the shear-stress distribution between the walls
was due to flow reversal along the inner wall during the diastolic period. Experimental results
published by Talbot and Gong (1983) based on a laser-velocimetry technique on a curved tube
with a sinusoidal waveform inlet flow have also shown the same observation of reversed flow
during the diastolic phase along the inner wall of the curved section.
More recently, Huang et al. (2011) employed a particle image velocimetry (PIV)
technique on a 180-degree curved tube to investigate the pulsatile flow effect on curved tubes
constructed with and without stenosis blocks at the inner arch region. From this study, the
recirculation and reverse flow behavior during the diastolic phase, as shown in the prior study,
were also observed in the model that had no stenosis.
It is worth noting that for the curved tube without stenosis, reversed flow was present
along the inner wall of the inner curve during the entire diastolic phase, and a recirculation zone
was observed during the late diastolic phase at the outer wall of the ascending straight tube.
Across the literature involving a curved tube with unsteady entrance flow, one common
observation is the occurrence of reverse flow along the inner wall of the curve during the
38
diastolic phase [Lyne (1971); Zalosh and Nelson (1973); Chandran et al. (1979); Talbot and
Gong (1983); Naruse and Tanishita (1996)].
2.5.5
Dynamic View of T-Junction/Branching Channel Flow Behavior
The present study of the ACB model focuses on the recommended angle (i.e., 90 degrees)
of anastomosis as suggested by ACB surgeons [Jatene et al. (2003); Frazier et al. (2005)], which
is also the required angle for most proximal graft connector devices [Ramchandani and Bedeir
(2011)] and also for the prevention of graft kinking [Izutani et al. (2005)]. Even though the angle
of proximal anastomosis varies from patient to patient, a 90-degree angle is one of the common
settings applied in ACB surgery, second only to the 45-degree angle. Today, the question of
whether malpositioning of a graft may predispose the area to postoperative lesion and graft
failure is still under investigation [Frazier et al. (2005)]. Results of clinical studies have shown
that many proximal anastomoses of bypass grafts in ACB surgery were not positioned at the
recommended angle. In some cases, the take-off angle could be as low as ≤ 45 degrees [Campeau
et al. (1975); Poston et al. (2004)]. In a rather large angiographic study on postoperative
saphenous vein graft status of ACB surgery, Campeau et al. (1975) reported that at least half of
the early-stage graft failure (i.e., < one year) was found to be related to surgical techniques such
as kinking and angulation problems, which normally lead to vascular lesions occurring at the
anastomotic site of the graft.
Several complex flow features around the 90-degree junction have already been
recognized under a steady fully developed flow condition. Typical flow patterns associated with
branching flow include separation, recirculation, flow reversal, secondary flow, stagnation zone,
and reattachment zone. The flow field at this junction is even more complex under the unsteady
39
flow condition, because the flow can never be fully developed, and the inertia force varies
periodically.
The human vascular system consists of many junction networks that deliver blood to
different parts of the body. Extensive evidence from previous bifurcated vascular flow studies
have shown that atherosclerosis is often found at the bifurcation regions that experience low
mean shear stress and prevailing oscillatory shear stress [Caro et al. (1971); Bharadvaj et al
(1982); Zarins et al. (1983); Ku et al. (1985)]. The biological and pathological responses of
endothelium to these hemodynamic environments have also been well documented by many
researchers, as discussed in the previous section.
2.5.6 Dynamic View of Human Aortic Flow Behavior
Several experimental studies based on human aortic geometry have been proposed in
order to investigate the physiological pulsatile flow conditions in the human aorta. Yearwood
and Chandran (1982) employed hot film anemometry to investigate the pulsatile flow effects in a
subject-specific human aortic model without including the major branches of the aorta. Results
show that significant reversed flow was generated in the inner wall of the curve during the
diastolic phase, and the helical effect that formed during the systolic phase was dissipated in the
diastolic phase. Profound secondary flow was also observed at the inner wall of the arch region
Using the laser Doppler anemometer technique, Chandran et al. (1985a, 1985b)
conducted a similar study on three major branches of the aorta included in their subject-specific
model. From their results, velocity profiles were shown to be highly dependent on the type and
orientation of the aortic valve. For example, in their first experiment with caged ball valves
mounted on the aortic root, no reversed flow was observed in the midsection of the arch;
40
however, a fully reversed flow was observed during the diastolic phase when the entry flow was
subjected to a set of tiling disc valves.
The effect of wall elasticity on the pulsating flow field in a human aortic model was
considered by Rieu et al. (1985) who employed Doppler ultrasonic velocimetry to study the flow
field in an elastic aortic model that included the iliac bifurcation portion. In contrast to the results
obtained by Chandran (1985a and b) on the caged ball valve-mounted model, reversed flow was
observed in the inner wall of the arch region in the experimental results by Rieu et al. (1985).
The velocity profiles during the systolic phase were found to be qualitatively similar to the solidbased models, as later described in the work by Chandran (2001).
The advent of transesophageal echocardiography (TEE) and magnetic resonance velocity
mapping has further improved the understanding of flow in the human aorta. The in vivo and in
vitro studies performed by Frazin et al. (1990) using the echocardiography technique coupled
with color Doppler ultrasound on a healthy human aorta have revealed the rotational flow
component that forms in the arch and descending portions during systolic and diastolic phases of
the cardiac cycle. It was shown that during the systolic phase, a clockwise rotational flow was
formed in the arch region of the aorta, and the flow continued to the descending part of the
artery, whereas a counterclockwise rotational flow was observed in the majority of subjects
during the diastolic phase. These observations are consistent with results obtained by Yearwood
and Chandran (1982) on their subject-specific model of the human aorta.
Using magnetic resonance velocity mapping technique, Kilner et al. (1993) were able to
show the helical and retrograde streams that are present in the aortic arch section of healthy
patients. During the late systolic phase, the aortic arch was predominated by a clockwise helical
41
flow, while a retrograde flow was found to be established in the inner wall of the arch section
during the end of the systolic phase.
2.6
Disturbed Flow on Vascular Disease
2.6.1
Flow Separation on Vascular Disease
The disruption of endothelial cells due to external factors may permit the influx of other
pro-inflammatory molecules into the vascular wall [Hunt et al. (2002)]. The exposure of
endothelial cells to separated flow has been observed to disturb the cell alignment [Björkerud
and Bondjers (1973); Fry (1976)] and increase the rate of cell motility, migration, detachment,
and proliferation [Tardy et al. (1997)]. The separated flow region associated with recirculation
flow has also been shown to elevate the level of low density lipoprotein (LDL) cholesterol,
which has been identified as one of the major risk factors of cardiovascular disease
[Schoenhagen (2006)]. In vivo experiment studies by Berceli et al. (1990) have investigated the
metabolism of LDL on the vascular wall by exposing the aortoiliac bifurcation of a rabbit to
three different flow conditions: separated, transitional, and laminar. Results show that the
elevation of LDL incorporation and catabolism rate were only observed in regions that were
exposed to separated flow but not in the transitional or laminar zones.
2.6.2 Secondary Flow on Vascular Disease
The formation of secondary flow in vascular disease has been the subject of many studies
due to its advantage of being a controllable mixing process, which is of significant importance in
many industrial applications. However in the human vascular system, the presence of secondary
flow can be either beneficial or harmful. The velocity component of a secondary flow is
perpendicular to the axial flow direction. A secondary motion is formed when there is a change
in the flow direction. Therefore, in a unidirectional laminar flow, such as a straight-tube flow
42
with a uniform inlet flow condition, the secondary flow motion or structure is generally small or
negligible. The profile shapes of the mainstream flow are characterized by the strength and
pattern of the secondary flow. Under the steady-state condition, Dean (1927) observed that
secondary flow in a U-bend curved tube is composed of two counter-rotating vortices caused by
the centrifugal force effect. In subsequent literature, these vortical structures are referred to as
Dean cells. Due to these secondary vortical flow structures, the distribution of shear stress is
varied along the wall. The structure of secondary motion in steady-state flow can generally be
expressed by the Dean number and Reynolds number. However, under an unsteady flow
condition, such as a pulsating flow, this structure becomes extremely complicated. The flow
structure can be formed with multiple different cell structures with highly irregular forms at the
same time, and this can vary throughout the pulse cycle [Timité et al. (2010); Jarrahi et al.
(2010)]. In order to quantify the structures of these secondary flows, the Womersley number is
typically included in the characterization. Due to the complexity of the secondary flow structure,
it is important to know that the effects of secondary flow on the biological and pathological
responses of endothelial cells are not solely determined by the location of occurrence, but also by
its structure and strength at the specific location throughout the cardiac cycle. A survey of
literature suggests that although the presence of secondary motion may affect the axial flow
condition in the human vascular system, the flow structure can be beneficial or harmful to the
human vascular system. In the human vascular system, secondary flow is a common
phenomenon that is normally found in curved and bifurcated vessels such as the aortic arch
section and the carotid artery, which normally develops during the systolic phase of the cardiac
cycle [Kilner et al. (1993)].
43
The distribution of wall shear stress is varied along the curvature of the tube based on the
local velocity distribution of flow field which can be characterized by the De, Re, and 
numbers. Studies have shown that an increase in pipe curvature may increase the strength of the
secondary flow in both steady-state and unsteady-state conditions [Gammack and Hydon
(2001)], while using the torsional effect of a tube with a helical geometry is shown to be able to
reduce the secondary flow and the area of minimal wall shear stress and stagnation zone
[Huijbregts et al. (2006)].
From a biological perspective of endothelium on secondary flow, depending on the local
flow condition, the presence of a secondary movement exhibits different impacts to the vascular
system. Caro (1966) indicated that secondary flow may promote the growth of an arterial wall
with an expanded cross-sectional area where separated flow is expected. However, secondary
flow alone has been shown to inhibit wall growth. Several studies have also indicated that the
formation of secondary flow in curved tubes under steady laminar flow condition was shown to
be able to stabilize the transition process of a flow to turbulence and hence delay the onset of a
turbulent flow [White (1929); Taylor (1929)].
Several studies have also reported that a secondary flow may lead to the dysfunction of
the vascular wall. For example, in an animal study, Ohyama et al. (1988) reported that the region
associated with secondary flow in a rabbit aorta is susceptible to atherosclerosis due to the
alteration of endothelial cells found at those regions.
In addition to the distribution of WSS, the presence of a secondary flow may also alter
the particle residence time of blood particles. Evidence has shown that the region exposed to a
long particle residence time poses higher potential risk for plaque formation through the
44
deposition of atherosclerotic particles and the expression of adhesive molecules on the contact
surface [Malinauskas et al. (1998)].
Using a subject-specific model of rabbit aorta, Malinauskas et al. (1998) investigated the
distribution of subendothelial macrophages induced by steady and pulsatile flow effects. This
study suggested that secondary flow may promote the accumulation of macrophages and the
expression of leukocyte adhesion molecules on the endothelium. These regions were described
as those associated with the initiation of vascular lesion. Work done by Pfeiffer et al. (1998) has
indicated that secondary flow has shortened the residence time of such soluble molecules, which
decreases the formation of thrombosis and platelet adhesion.
2.7
Review of Related Work
2.7.1
Existing Numerical Work on ACB Model
The present study focuses on the effects of geometry, fluid properties, and flow condition
on the hemodynamics of the aortocoronary bypass model. In the past few decades, an extensive
research effort has concentrated on the distal region of grafts. Although the status of quality
grafts has been extracted from clinical assessments, understanding the effect of geometry on the
hemodynamic environment at the proximal and middle region of the bypass graft (i.e., region
where grafts are anastomosed to the aorta) are still very limited. Studies on multiple-vein
grafting on the aorta are also relatively few. Several investigators have considered this territory
in their study models, but systematic analysis of the effects of graft geometry interaction with the
host artery, or aorta, is not available. Attention given to the local hemodynamics of bypass grafts
in the ACB model is limited.
Study models that relate to ACB hemodynamics can generally be separated into two
categories: models including single proximal anastomosis, and models including multiple
45
proximal anastomoses. Models used for these studies are generally either simplified, subjectspecific models, which are reconstructed based on medical image data, or circular tubes with
geometries analogous to normal aortas and grafts.
The first model was studied by a number of researchers using experimental and
numerical methods. Redaelli et al. (2004) applied a circular cross-sectional tube as the host artery
for the ACB model to study the flow field at the proximal region of a single graft connected
using a hand-sewn technique and aortic connector. Results show that the aortic connector with a
90-degree take-off angle provides a better hemodynamic environment than the hand-sewn
technique with a lower OSI and higher WSS.
Morbiducci et al. (2007) studied hemodynamic variables at proximal anastomosis based
on a numerical method for a single graft sutured on the aorta with different anastomotic angles.
The computational domain consisted of an idealized vascular model including an aorta, and a
single graft attached at the anterior wall of the ascending aorta was used to assess the local
hemodynamics at the proximal anastomosis territory. A similar numerical model including distal
anastomosis was performed by Sankaranarayanan et al. (2005 and 2006) in which the flow
dynamics at specific instants of time in the cardiac cycle was assessed. Numerical results were
then validated by Zhang et al. (2008a) based on an in vitro experiment utilizing the PIV
technique. A numerical study by Quaini et al. (2011) for a left ventricular assist device (LVAD)
also provided preliminary insight into the effects of anastomosis on the hemodynamics of a
proximal territory. An idealized two-dimensional model consisting of a single cannula tube and
an ascending aortic section was used to investigate the hemodynamic conditions near the aortic
valve. However, due to the different inlet and outlet boundary conditions applied on the LVAD
46
domain, limited information of the proximal flow behavior of venous bypass grafts can be
extracted from the study.
For the second type of model, a recent study by Dur et al. (2011) evaluated the
hemodynamics in multi-scale cardiovascular geometries based on a rather complete subjectspecific aortocoronary model, which included the aorta, major coronary arteries, and multiple
bypass grafts, based on typical settings for the saphenous vein graft and the left internal thoracic
artery. In the results, the WSSG was shown to be reduced significantly by optimizing the shape
of the graft through a computer-aided design method. Even though multiple vein grafts were
included in the model, the primary focus of the study was on LITA and sequential grafts settings.
No direct comparison was made to understand the effect of graft geometry on local
hemodynamics at the proximal anastomosis of the grafts.
47
CHAPTER 3
APPROACH AND THEORIES
3.1
Numerical Approach
Despite the fact that qualitative data extracted from MRI, Doppler, or ultrasound sound
techniques have provided a rather in-depth insight to the hemodynamics of the human vascular
system, obtaining detailed quantitative information of the human vascular system flow field is
still a challenging task, especially with the current technology. Present-day imaging devices and
measurement technologies have their limitations when it comes to high-resolution results of
time-dependent flow features such as the recirculation zone, stagnation zone, and near-wall
reverse flow on the micro-level scale of the vascular system. The measurement of the flow field
around the vicinity of sharp corners, such as 90-degree junctions or bifurcations, is difficult to
perform experimentally. It is costly to obtain or reproduce a collection of experimental data for
models with a variety of complex geometry and flow conditions. Therefore, it would be of great
advantage if numerical methods could be employed to reproduce the flow field environment in
the vascular system. In this regard, the focus of the present work will be on a non-invasive
method of numerical analysis of the human vascular system.
The general approach in this study is to apply numerical methods to simulate the flow
environment in a series of models analogous to aortocoronary bypass surgery, which consists of
a set of bifurcated curved tubes. A summary of the methodology of a numerical approach for the
present study is shown in Figure 3.1. Numerical methods are first validated individually with a
series of existing experimental and analytical results that are close to the conditions of the
present model of study in terms of geometry and flow condition. These validations include the
blood properties (viscosity models), pulsating effect (boundary conditions), specific flow
48
features in branching and curved tube flows (separation, recirculation, secondary, etc.), and
numerical schemes of the solver (laminar and turbulent models). These methods are then
integrated and applied to the case models proposed in this study.
•
Blood Properties
•
Pulsating Effect
Viscosity Models
̇2
Case Study
̇
•
Branching Flow
̇1
•
Curved-Tube Flow
Figure 4.2. Numerical grid for validation case 2.
Figure 3.1. Approach of numerical study.
49
The governing equations for fluid motion will be discussed in the following section. The
fluid models and numerical schemes used to solve for the governing equations from the solver
will be discussed after that. The last section of this chapter outlines the formulae of the different
hemodynamic indicators selected for analysis in this study.
3.2
Governing Equations for Fluid Motion
The target flow domain in the present study is in the human vascular system, with a focus
on the aorta and bypass grafts. The flow field to be investigated is unsteady and threedimensional, and assumed to be isothermal, homogeneous, and incompressible. The fluid is
assumed to be made up of continuous rather than discrete particles. The flow domain is solved
by using a numerical approach based on the classical governing equations derived from the
principles of conservation of mass, momentum, and energy equations. Since the flow field in this
study is assumed to be isothermal, the conservation of energy is not required to be solved in this
study. Therefore, the flow field in this study will be obtained based on sets of equations derived
from the principles of mass conservation and momentum conservation only. With this
assumption, the equation of the conversation of mass and momentum can be expressed as
(3.1)
(3.2)
for a three-dimensional flow field i, j = 1, 2, 3, where u is velocity,  is stress tensor, f is body
force per unit mass, and  and  are the fluid properties of density and dynamic viscosity,
respectively.
For incompressible and isotropic Newtonian fluids, the shear stress in the fluid is
assumed to be a linear function of the velocity gradient (shear strain rate ). With this
50
assumption, the stress tensor in equation (3.2) for Newtonian fluid can be written in terms of the
normal stress  and shear stress  as
(3.3)
(3.4)
where p is the thermodynamic pressure,  is the bulk viscosity,  is the dynamic viscosity,  is
the Kronecker delta, and ·
is the volumetric dilation rate.
Since for incompressible flow the density is constant ( = constant, 0), equation (3.1)
can be simplified as
(3.5)
(3.6)
Substituting equations (3.4) and (3.6) into (3.2) yields the following general form of the
equations of motion for incompressible Newtonian fluid:
(3.7)
Equation (3.7) is also referred as the Navier-Stokes equation for incompressible Newtonian
viscous fluid. Due to the convective term in equation (3.7), for most of the fluid problems, the
equation will be, in general, a set of nonlinear equations in which the exact solution for the
equations does not exist. To solve for the variables, the equations can be discretized into a set of
algebraic equations using a numerical scheme and solved numerically to obtain approximate
solutions.
51
Practically, for the convenience of differentiating the difference between Newtonian fluid
and non-Newtonian fluid, the shear rate tensor in equation (3.7) can be expressed in terms of
fluid strain rate Sij as
(3.8)
And the shear stress tensor term in equation (3.7) can then be rewritten as
(3.9)
The fluids that exhibit a linear relationship between shear stress and shear rate are generally
classified as Newtonian fluids, whereas fluids with shear stress that is not linearly proportional to
shear rate are classified as non-Newtonian fluids. The proportionality for these two quantities
(i.e.,  is defined as the coefficient of the fluid viscosity. For Newtonian fluids, is a constant,
whereas for non-Newtonian fluids,is a non-constant.
3.2.1
Newtonian vs. Non-Newtonian Fluids
The viscosity of blood can be determined through the viscosity of plasma, the amount of
red blood cells (RBCs) (i.e., hematocrit) in the blood and their mechanical properties [Pasquini et
al. (1983); Baskurt and Meiselman (2003)]. The major determinant of blood viscosity is the
aggregation of red blood cells, which is inversely proportional to the velocity of blood flow.
Because blood flow is relatively slow, aggregation is high so part of the shear stress is applied to
break up the red blood cell aggregation. Depending on the amount of aggregation, shear stress
applied on disaggregation is varied with the aggregation rate. Under this condition, blood
viscosity will respond nonlinearly to the ratio of shear stress and shear rate. A blood rheology
study by Merrill (1969) has demonstrated that normal human blood only exhibits non-Newtonian
behavior when the shear rate of the blood is approximately below 100 sec-1. The viscosity of
52
blood between 0 and 100 sec-1 is shown to decrease with an increase in shear rate. This behavior
is now known as the shear-thinning effect of pseudoplastic fluid. As the shear rate exceeds 100
sec-1, the viscosity of the blood is shown to be independent of shear rate (Figure 3.2). Blood flow
under this condition can therefore be assumed as Newtonian fluid.
Figure 3.2. Relationship between shear rate and viscosity of normal human blood
(from Merrill 1969).
In large human arteries (aorta, carotid artery), the shear rate is normally higher (rate >100
sec-1) compared to small arteries (microcirculation). Therefore, in most of the literature, blood
flow in large arteries is commonly assumed and modeled based on a Newtonian fluid flow
model. For blood flow in small arteries in which the shear rate is low (<100 sec-1), the
assumption of Newtonian fluid will not be able to reproduce the flow field accurately.
Even with the above argument that the shear rate in large arteries is normally higher, the
assumption of Newtonian behavior may not be reasonable all of the time. It is important to note
that the shear rate in a vascular system is a time-dependent quantity. The blood in arteries may
53
undergo a wide range of shear rate during a cardiac cycle. The use of a Newtonian model may
exclude the shear-thinning properties of blood in a low shear rate regime. The assumption of
Newtonian behavior of blood in large arteries may be appropriate to describe the viscosity of the
blood in a steady flow condition, but for an unsteady flow condition, the shear-thinning effect of
the blood must be taken into account. The experimental work of Mann and Tarbell (1990)
compared the shear rate in a rigid tube based on a Newtonian analogous fluid and three blood
analogous fluids under a steady flow condition. Results show that under a steady flow condition,
the influence of fluid properties on shear rate was insignificant. However, the difference in shear
rate was significant when the flow was subjected to an oscillating condition.
Several experimental studies with focus on the difference between Newtonian and nonNewtonian fluid have indicated that the difference in the distribution of velocity profiles in large
arteries can be significant between the two models. Work done by Moravec and Liepsch (1983)
and Liepsch and Moravec (1984) on Newtonian and non-Newtonian fluids has shown that the
difference in velocity profiles for branching models is significant under steady and pulsating
flow conditions. Marked differences of reverse flow and a separation zone at the regions distal to
the branch were observed. Ku and Liepsch (1986) performed a similar study on a 90-degree
bifurcated elastic tube and reported a similar observation at the regions distal to the branch.
Experimental and numerical studies by Gijsen (1998) have also demonstrated that the
shear-thinning effect is very sensitive with a small variation of shear rate. Using a 90-degree
curved-tube model, the authors investigated flow features of the tube under an unsteady flow
condition with both Newtonian and non-Newtonian fluids. Results show that the axial velocity
profiles predicted using the Newtonian model failed to model the non-Newtonian flow when
high shear viscosity was used in the model, but satisfactory results were obtained when the
54
average shear rate of the non-Newtonian fluid viscosity was used as the viscosity of the
Newtonian model.
3.2.2
Viscous Models for Newtonian and Non-Newtonian Fluids
To differentiate the difference between Newtonian and non-Newtonian fluids, it is
customary to further simplify equation (3.7) using the shear rate tensor ij as
(3.10)
Therefore, the shear stress tensor of the fluid can be expressed as
(3.11)
For Newtonian fluid, the dynamic viscosity  is a constant with a given temperature and
concentration. The dynamic viscosity in the present study for Newtonian blood flow is adopted
from the typical viscosity applied to large vessels, such as the aorta, as
(3.12)
For non-Newtonian fluid, the dynamic viscosity shown in equation (3.11) is referred to as
the apparent viscosity—a nonlinear variable that depends on the shear rate of the flow. Several
models have been proposed by different authors to model non-Newtonian blood flow behavior.
Models commonly applied in the literature are as follows: Casson, power-law, Bingham plastic,
and Carreau-Yasuda. The Carreau-Yasuda model has been found to be more accurate in
predicting the shear-thinning effect of blood and fits the complex rheology of blood for a wider
range of shear rates by several researchers [Gijsen et al. (1999); Coelho and Pinho (2004); Nouar
et al. (2007)]. The viscosity for blood flow in the present study will be based on the CarreauYasuda model in order to model the non-Newtonian shear-thinning effect of the blood. The
viscosity of Carreau-Yasuda for blood is given as
(3.13)
55
where
is the viscosity at an infinite shear rate,
relaxation time,
is the zero shear rate viscosity,  is the
is the shear rate, and a and n are parameters obtained from the experiment to
fit the shear stress-rate curve of the blood. The parameters for human blood based on the
experimental data that cover shear rate ranges between 1 and 10,000 s–1 are as follows [Gijsen
(1998)]:
= 0.022 Pa·s,
= 0.0022 Pa·s, a = 0.644, n = 0.392, and  = 0.110 s. It is
important to note that the models discussed above do not include the viscoelasticity (deformation
behavior of RBCs) and thixotropy (change of viscosity with time of shearing) properties of
blood. Thus, only the shear-thinning effect was included in the comparison of Newtonian and
non-Newtonian blood flow.
With regard to the significant effects of non-Newtonian fluid on the flow field outlined
above, a non-Newtonian fluid model is used for all simulations in this study. However,
simulations based on Newtonian fluid are also carried out to compare the difference between
Newtonian and non-Newtonian fluids in the study models.
3.3
Solver Background and Theories
3.3.1
Solver Background
As discussed in the previous section, equation (3.7) will be, in general, a set of nonlinear
equations whose exact solutions do not exist. To solve for variables in the equations, the
equations can be discretized and solved by using a numerical method to obtain the approximate
solutions. Three of the commonly used numerical schemes for the discretization will be
discussed below.
In general, the physical domain of the solution can be divided into a finite number of
small control volumes with structured or unstructured cell shapes. The variables in each of the
volume cells can be evaluated by solving a system of algebraic equations that are discretized
56
from the governing equations for fluid motion using different finite approximation numerical
schemes. The most common numerical schemes to solve the Navier-Stokes equations are the
finite element method (FEM), finite difference method (FDM), and finite volume method
(FVM). Among these methods, FVM is perhaps the most popular method adopted in current
commercial fluid mechanics solvers.
One of the reasons for the FVM’s popularity is its flexibility on the discrete volume of
the physical domain. Unlike the FDM method, in which a structured grid domain is required to
solve for the governing equations, the FVM can be implemented on an unstructured grid domain,
which is of great advantage and importance when it comes to the physical domains with complex
geometries or irregular shapes. Studies have shown that implementing the FDM for a domain
with a highly irregular shape would oftentimes lead to serious problems affecting the accuracy
and convergence of the solutions due to a numerical discontinuity issue caused by the
smoothness of the grid system. This issue can generally be overcome by using the block grid
system; however, the applicability of this method is very limited. In addition, since the variables
are integrated directly on each of the control volumes in the FVM, the conservation of mass,
momentum, and energy is always guaranteed, regardless of the cell shape. Therefore, an
unstructured grid can be implemented for the physical domain of a solution.
Using the iteration method over the discretized governing equations, the solution of the
variables in the discretized equations (velocity, pressure, and temperature) can be obtained based
on finite approximation convergence in the iterative solution. The solutions are justified to be
converged when the residuals in the conservation equations meet the specific convergence
criterion. The residuals for convergence in most practices are typically set at 1e-5 or lower,
depending on the problem.
57
3.3.2
Solver Theories
The commercial software ANSYS Fluent v.12.1.4 (ANSYS Inc.) was employed to
perform the iterative solution for the blood flow simulations in the present study. The fluid
condition of the blood is viscous, and the density, temperature, and genetic structure of the blood
are assumed to be constant throughout the flow field. Therefore, the results in the present study
are based on the assumptions of incompressible, isothermal, and homogeneous blood. The
variables solved were the velocity and pressure fields of the blood. The procedure for solving the
variables is discussed below.
The commercial software ANSYS Fluent from ANSYS, Inc. employs the FVM (i.e.,
discretizing the governing equations based on integral method) to convert the governing
equations of fluid flow equations (3.1) and (3.2) to a set of algebraic equations. Using the
divergence theorem, the integral form of equations (3.1) and (3.2) can be written in terms of
volume and surface integrals, respectively, as follows:
(3.14)
(3.15)
For incompressible flow, the Newtonian fluid in equation (3.5) and equations (3.11) and
(3.12) can be written, respectively, as follows:
(3.16)
(3.17)
58
Since equation (3.17) is, in general, non-linear, in order to solve for the variables of a cell p in a
computational domain, equations (3.16) and (3.17) are discretized and expressed in a linearized
form as
(3.18)
where ap and anb are the liberalized coefficients for the scalar quantities in the governing
equations, subscript nb denotes the neighbor cells of p, and b is the scalar quantity of the
variables in the governing equations. By applying this equation over each element of the
subdivided physical domain, the governing equations will result in a system of algebraic
equations that can be arranged in a matrix form and solved iteratively.
One of the major issues in solving the equations for incompressible flow is the lack of an
explicit equation for pressure due to the constant density applied in the conservation of mass
equation. A number of coupling algorithms are available to resolve this issue by deriving a
discrete pressure correction equation from the continuity equation and momentum equations to
solve the pressure field. In this study, the pressure-velocity coupling is handled with the default
algorithm SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) by Patankar and
Spalding (1972).
Since the transient (unsteady) solution of blood flow is considered in the present study,
the governing equation (3.17) will be discretized in terms of space (spatial discretization) and
time (temporal discretization). For the spatial discretization of the momentum equation (3.17), a
second-order upwind scheme is applied. For the temporal discretization of equation (3.17), the
second-order implicit backward difference scheme is applied. For the SIMPLE equation, the
equation is discretized by using a second-order difference scheme. To solve for the scalars at the
cell faces and the secondary diffusion terms and velocity derivative, the gradients of the scalars
59
need to be computed. A Green-Gauss node-based model is applied in this study for computing
the scalar. This system of equations is solved by the segregated solution procedure in which the
conservation equations of each variable are solved separately in a sequential manner.
For the solutions, the convergence criterion for the residuals of the continuity and
momentum equations in the present study is set at 1e-5. In order to reduce the transient effect
originating from the initiation condition, all simulation results in this study are based on the
fourth period of the cardiac cycle. Results from the sensitivity study of grid and time-step
independence in the present study is discussed in the following chapter.
3.4
Hemodynamics Indicators
The aim of this study was to apply the numerical method to investigate the velocity
distribution flow patterns of blood flow in models analogous to the ACB model. The major
purpose here was to investigate how the geometry, fluid properties, and flow condition will
affect the distribution of the local flow field, which governs the hemodynamic environment of
the blood vessel. Three out of the five indicators discussed in Chapter 2 were selected to
illustrate the influence of flow field to the hemodynamic environment of the study models. These
indicators are the WSS, TAWSS, and OSI. Their equations are summarized below.
3.4.1
TAWSS, OSI, and RRT
The time-averaged wall shear stress indicates the distribution of the averaged shear stress
exerted on the vascular wall over a cardiac cycle. This is calculated by integrating the absolute
shear stress w over a cardiac cycle and dividing it by the pulse period T. TAWSS is defined as
(3.19)
Since this indicator is based on the absolute magnitude of shear stress, the indicator does
not specify any information about the direction of the shear stress. As discussed in Chapter 2,
60
reversed flow is commonly found in the vascular system, especially during the diastolic phase of
the cardiac cycle. In order to include the directional effect of shear stress in the vascular system,
the non-dimensional parameter, oscillatory shear index, proposed by Ku et al. (1985), is selected
in this study to indicate the oscillatory scale of the shear stress in the study models. OSI is
defined as
(3.20)
The scale of OSI ranges from 0 to 0.5. The maximum OSI (i.e., 0.5) indicates the condition of
pure oscillatory flow without zero net forward flow. For unidirectional WSS, the OSI equals 0.
To incorporate the effects of both OSI and TAWSS induced by pulsatile or unsteady
flow, the relative residence time proposed by Himburg et al. (2004) is selected to reconcile the
gap between the indicators of OSI and TAWSS due to the insensitivity of OSI to TAWSS. RRT
is defined as
(3.21)
61
CHAPTER 4
VALIDATION STUDIES
4.1
Validation Approach
The flow field in a curved tube with pulsatile entry flow is highly complex, with multiple
forms of potential flow structures appearing at the same time and varying periodically. Extensive
numerical work has been done to elucidate the hemodynamic environment in the human vascular
system. Research has shown that, due to complexity of the vascular system’s flow field, it may
be easier to underestimate the errors that result from numerical methods such as numerical
schemes, grid and time-step resolutions, and boundary conditions.
To avoid potential errors from simulations, numerical schemes of the solver and models
with physical and flow conditions related to the present study were first validated with
experimental and analytical results before they were integrated and applied to the models in this
study. Validation work was also performed to ensure that the potential flow features involved in
the study models could be resolved and captured appropriately using the same numerical
methods in this section. The experimental and analytical results from the literature also serve to
provide a preliminary understanding of the flow structures that may be involved later on in this
study.
The validation study was separated into five different areas of focus: (a) blood properties,
(b) branching flow, (c) unsteady/pulsating flow, (d) curved-tube flow, and (e) turbulent
branching flow. Numerical results are compared with experimental and analytical results from
previous investigations, based on the geometries and flow conditions that are found to be closely
related to the present study. Table 4.1 summarizes the models, flow conditions, and methods
applied in the work of different authors, followed by the purposes of validation in this study.
62
TABLE 4.1
VALIDATION MODELS ADOPTED IN PRESENT STUDY
Validation
Model
Flow
Type
Straight Tube
(Case 1)
Steady:
Laminar
Flow
Regime
Method
Purpose
Blood Properties
Re = 270
(Newtonian vs. nonNewtonian fluid)
Re = 469,
Flow Division
T-Junction
Steady:
Experimental
1062
(angle suggested by ACB
Laminar
(Liepsch et al. 1982)
(Case 2)
surgery cardiologist)
Re = 280,
Analytical
Pulsating Flow
Straight Tube
Pulsatile:
520
(Takayoshi and
(physiological blood
Laminar
(Case 3)
Katsumi 1980)
flow conditions)
 = 5–28
o
Re = 600
90 Curved Tube Pulsatile:
Experimental
Pulsating Flow
Laminar
(Timité et al. 2010)
(curved tube)
(Case 4)
 = 17.17
180o Curved Tube Steady:
Experimental
Turbulent Flow
Re = 57,400
Turbulent
(Costa et al. 2006)
(curved tube)
(Case 5)
4.1.1
Experimental
(Gijsen et al. 1999)
Straight-Tube Model Validation Case 1—Newtonian vs. Non-Newtonian Fluid
(Steady Laminar Flow)
The first model to be validated was for Newtonian and non-Newtonian flow based on a
constant dynamic viscosity and the Carreau-Yasuda model discussed in the previous section.
Numerical results of case 1 were compared against the experimental results of Gijsen et al.
(1999) who investigated the shear-thinning effect of blood in a carotid bifurcation model based
on a blood analogy fluid (KSCN-X solution). Viscosity was set at  = 0.0029 Pa·s for the
Newtonian fluid, and density was set at = 1,410 kg/m3 in both cases. The numerical model was
a circular tube with a diameter of 0.4 cm. The corresponding Reynolds number was based on
constant viscosity at Re = 270. For the flow to be fully developed, it was applied at the inlet, with
the entrance length Le estimated based on the correlation equation of Granger (1995) for a
circular pipe.
A constant velocity of 0.139 m/s and zero outlet pressure were applied at the inlet and
outlet of the model, respectively. Detailed information about the experiment can be found in
63
Gijsen et al. (1999). Figures 4.1(a) and (b) show velocity profiles measured at 0.45 m from the
inlet of both models and compared with experimental results. The numerical result was
normalized with the local maximum velocity. A red line is used to represent the numerical results
for all cases in the present study, which indicate that the Carreau-Yasuda model satisfactorily
describes the shear-thinning effect of blood in this model. The velocity profile for the non-
u/umax
u/umax
Newtonian model appears to be more viscous than that for the Newtonian model.
r/R
r/R
(a)
(b)
Figure 4.1. Validation of different viscosity models: (a) Newtonian model and
(b) non-Newtonian model (numerical data = red line; experimental data = symbols).
4.1.2
T-Junction Flow Model Validation Case 2 (Steady Laminar Flow)
Flow features in the branching channel are always complex - involving separation,
recirculation, secondary motion, and stagnation and reattachment zones. Since the anastomosis
angle of 90 degrees is by far one of the most common angles seen in ACB surgery, flow with a
90-degree branching duct model was selected to validate this study. For simplicity, only twodimensional flow is considered here.
Two different examples with steady Reynolds numbers of 496 and 1062 were selected for
the T-junction flow model from the experimental studies of Liepsch et al. (1982). Table 4.2
summarizes the fluid properties and flow conditions for validation case 2. A fully developed
flow was applied for the inlet, and both outlets were prescribed with a mass flow rate condition.
64
Figure 4.2 shows the numerical model constructed for both simulations. A symmetrical plane
flow domain was constructed with a structured grid composed of 452,485 nodes and 435,840
elements. The resolution of the grid was tested to reach grid independence.
TABLE 4.2
NUMERICAL MODEL GEOMETRY AND FLOW CONDITIONS FOR VALIDATION CASE 2
Cross Section
Height, H
Length, L
Width, W
Branch Angle
Flow Condition
Geometry
Rectangular
10 mm
10 mm
80 mm
90 deg
Flow Conditions
Fluid
Water
Density, p
997 kg/m3
Dynamic Viscosity, µ
8.899e-4 Pa·s
Reinlet—Model 1
496
Reinlet—Model 2
1062
0.44
Mass Flow Rate Ratio (𝑚̇2 /𝑚̇1 ) —Model 1
0.58
Mass Flow Rate Ratio (𝑚̇2 /𝑚̇1 ) —Model 2
𝑚̇2
𝑚̇
𝑚̇1
Figure 4.2. Numerical grid for validation case 2.
65
Figures 4.3(a) and (b) show a comparison of the axial velocity profiles for validation case
2 with Reynolds numbers of 496 and 1062, respectively. Two recirculation zones can be clearly
seen from the streamline plot on the upper right of Figure 4.3(a). Good agreement can be seen on
the velocity profiles in the host pipe where the stagnation and recirculation zones form at the
bottom of the host pipe wall. The velocity profile in the bifurcation also shows satisfactory
agreement. Even though the velocity profiles were slightly skewed towards the right end wall of
the junction, the profile shapes are, in general, identical to the experimental results. The
recirculation and stagnation regions are well captured by the numerical method, with minor
differences at the beginning of the dividing zone of the branch.
v [m/s]
u [m/s]
(a)
Figure 4.3. Comparison of velocity profiles for different flow conditions:
(a) Re = 496, U2/U1 = 0.44; (b) Re = 1062, U2/U1 = 0.58 (continued on next page).
66
v [m/s]
Experimental Data
Not Available
u [m/s]
(b)
Figure 4.3. Comparison of velocity profiles for different flow conditions:
(a) Re = 496, U2/U1 = 0.44; (b) Re = 1062, U2/U1 = 0.58.
4.1.3 Straight-Tube Model Validation Case 3 (Pulsatile Laminar Flow)
In case 3, the theoretical equation proposed by Takayoshi and Katsumi (1980) and
discussed previously in Chapter 2 for laminar pulsating flow, was selected to validate the
straight-tube pulsating flow by using the numerical method. Transient velocity is given as
∞
𝑈=
𝑈(𝑟,𝑡)
𝑈𝑒
′2
= [2(1 − 𝑅 ) + 2 ∑
𝐽0 (𝑛 𝑅 ′ )−𝐽0 (𝑛 )
{𝑛
𝐽0 (𝑛 )
𝑛=1
3
+𝛽 [Im {
∞
−4𝛽 ∑
𝑛=1
{𝑛
exp (
𝑛 2 𝜐
𝑟0 2
𝑡) }]
3
𝐽0 (𝑖 2 𝑅 ′ )−𝐽0 (𝑖 2 )
3
𝐽2 (𝑖 2 )
𝑒 𝑖𝜔𝑡 }]
𝑛 2 𝐽0 (𝑛 𝑅 ′ )−𝐽0 (𝑛 )
𝑛 4 +4 𝐽1 (𝑛 )−𝐽3 (𝑛 )
67
𝑛 2 𝜔
exp (
2
𝑡) }
(4.1)
where r is the pipe radius, R’ = r/R, U1 is the steady flow velocity of the inlet,  is the
dimensionless parameter (i.e., Womersley parameter  = r(/)0.5), J0, J1, J2, J3, are Bessel
functions of the first kind, n is the successive positive roots of J2() = 0, and  is the kinematic
viscosity. Readers are referred to the work of Takayoshi and Katsumi (1980) for more detailed
information about this equation.
The same equation was used for Newtonian blood with constant density and viscosity to
compare with results obtained from the simulations. The kinematic viscosity  was 3.3019e-6
m2/s for Newtonian blood, with density  and dynamic viscosity  of 1,060 kg/m3 and 0.0035
Pa·s, respectively. Table 4.3 summarizes the geometry and flow conditions in validation case 3.
TABLE 4.3
NUMERICAL MODEL GEOMETRY AND FLOW CONDITIONS FOR VALIDATION CASE 3
Cross Section
Pipe Radius, r
Length, L
Geometry
Flow Conditions
Circular
2 cm
250 cm
Fluid
Newtonian Blood
1,060 kg/m3
Density, 
0.0035 Pa·s
Dynamic Viscosity, 
Inlet Velocity (Steady), U1
0.4292 m/s
Inlet Velocity (Oscillatory), Uosc
Ue(1 + 𝛽sin(.t))
1
Velocity Ratio, = Uosc,max/U1
Reynolds Number (Steady), Rest
520
Reynolds Number (Oscillatory), Reosc
0–1040
–1
0.2231 rads , 1.8573 rads–1, 6.2832 rads–1
Frequency, 
5.20, 15.00, 27.59
Womersley Parameter, 
Three different cases based on different flow frequency ( ) ranges from 0.2231 to 6.2832 rad/s
with identical amplitudes were considered. The highest frequency was selected according to the
typical resting human heart rate (60–72 beats per minute). This simulation was carried out using
68
a circular cross-section pipe with a diameter of 4 cm and length of 2.5 m. Figure 4.4 shows the
grid applied for the physical domain.
Figure 4.4. Numerical grid for validation case 3.
Figure 4.5 shows a comparison of the development of velocity profiles from three
different oscillatory inflow conditions. Results indicate that the numerical results agree very well
with the theoretical results, with maximum errors less than 0.6%. A flattened effect of the
velocity profile on the low frequency flow during the peak oscillation shown in the work of
Takayoshi and Katsumi (1980) can be observed from the three profiles presented here as the
Womersley parameter increases (from equation [4.1]), from 3.9 to 12.7. The three profiles based
on same amplitude also clearly demonstrate the increase of reversed flow condition due to the
frequency increase.
69
u [m/s]
Uosc,max
u [m/s]
Uosc [m/s]
Uosc [m/s]
Uosc [m/s]
-Uosc,max
u [m/s]
r [m]
(a)
u [m/s]
r [m]
r [m]
(b)
(c)
Figure 4.5. Comparison of oscillatory axial velocity distributions at different frequencies:
(a)  = 0.2231 rad/s, (b)  = 1.8573 rad/s, (c)  = 6.2832 rad/s (numerical results = red lines;
analytical results = symbols).
4.1.4
90-Degree Curved-Tube Model Validation Case 4 (Pulsatile Laminar Flow)
The experimental results of Timité et al. (2010), discussed previously in Chapter 2, were
selected to verify the curved-tube pulsating flow behavior verified in case 4 by the numerical
method used in this study. Table 4.4 summarizes the geometry and flow conditions applied in the
simulation. More detailed information about the experimental model and flow conditions can be
found in the work of Timité et al. (2010).
70
TABLE 4.4
NUMERICAL MODEL GEOMETRY AND FLOW CONDITIONS FOR VALIDATION CASE 4
Geometry
Cross Section
Circular
2 cm
Pipe Radius, r
Length, L
250 cm
0.22 cm
Radius of Curvature, rC
90 deg
Bend Angle, 
Flow Conditions
Fluid
Water
997 kg/m3
Density, 
Dynamic Viscosity, µ
8.899e-4 Pa·s
Inlet Velocity (Steady), U1
0.01339 m/s
Inlet Velocity (Oscillatory), Uosc
Ue(1 + sin(.t))
1
Velocity Ratio,
= Uosc,max/U1
Reynolds Number (Steady), Rest
600
Reynolds Number (Oscillatory), Reosc
0–1200
0.6579 rads–1
Frequency, 
17.17
Womersley Parameter, 
Figure 4.6 illustrates the physical domain of the numerical model and the boundary
conditions applied on both the inlet and outlet. Comparisons of the development of axial velocity
profiles from different cross-sectional planes at = 90 degrees of the curved tube are shown in
Figure 4.7 for the flow condition with Reynolds number, velocity ratio, and Womersley number
of Re = 600,  = 1, and  = 17.17, respectively. Qualitatively, the numerical results, in general,
are in good agreement with the experimental results, except for the profile in the y-z plane at .t
= 180°, where the velocity was over-predicted with an average error of 6.5%. Results also show
that the reversed flow during the late decelerating phase on the y-z and x-z planes are well
captured by the present numerical results.
71
u/umax
u/umax
Figure 4.6. Schematic diagram of physical flow domain of validation case 4 model.
r [m]
r [m]
Figure 4.7. Comparison of instantaneous axial velocity profiles at  = 90o: axial velocity profiles
on y-z plane (left) and axial velocity comparison on x-z plane (right).
4.1.5 180-Degree Curved-Tube Model Validation Case 5 (Steady Turbulent Flow)
The simulation of the curved-tube flow in a turbulent regime was also studied to
understand the capability of the present methods for solving the curved channel associated with a
turbulent flow. The validation result for comparison was adopted from the work of Azzola et al.
(1986) who measured the streamwise and circumferential velocity profiles in a 180-degree Ubend tube using the laser-Doppler technique for Reynolds numbers of 57,400 and 110,000.
Readers are referred to the work of Takayoshi and Katsumi (1980) for detailed information about
the experiment.
72
The validation study in this section is based on Re = 57,400. Flow conditions are
summarized in Table 4.5, and the numerical grid is shown in Figure 4.8. The fully developed
flow is applied at the entrance to the pipe. The distance of the first layer’s cell from the wall is
calculated according to the y+ value obtained from the k- turbulent model with a standard wall
function, which was found to be 1.178.
TABLE 4.5
NUMERICAL MODEL GEOMETRY AND FLOW CONDITIONS FOR VALIDATION CASE 5
Geometry
Cross Section
Pipe Radius, r
Radius of Curvature/Normalized Diameter, rC/D
Flow Conditions
Fluid
Density, 
Dynamic Viscosity, 
Inlet Velocity (Steady), U1
Reynolds Number, Re
Dean Number, De
Circular
2.225 cm
3.375 cm
Water
997 kg/m3
8.899e-4 Pa·s
1.29 cm
57,400
31,300
Unlike results obtained for the laminar cases, the accuracy of the numerical models used
in modeling the turbulent flow in this case is relatively weak. As shown in Figure 4.9, the axial
velocity profiles from the entrance of the bend to the central bend at 135° are predicted well by
the k- model; however, the velocity profiles further downstream of the bend do not agree well
with the experimental results. Similar to the axial velocity profiles, the circumferential velocity
profiles, shown in Figure 4.10, were shown to be rather weak in capturing the secondary motions
at the cross sections from 45° to 180° and 1D (i.e., one diameter beyond the 180° location). Such
discrepancies in the results were also reported in the work of Moulinec et al. (2005) who
employed a large eddy simulation method to compare with the same experimental results by
Azzola et al. (1986).
73
rC
u/U1
Figure 4.8. Numerical grid for validation case 5.
center
r/R
r/R
outer wall center
outer wall
Figure 4.9. Comparison of longitudinal velocity profiles (u-component) with experimental results
at different axial locations on symmetry plane.
74
v /U1
center
r/R
r/R
r/R
outer wall center
outer wall
Figure 4.10. Comparison of circumferential velocity profiles (v-component) with experimental
results at different axial locations on symmetry plane.
r/R
75
CHAPTER 5
NUMERICAL MODELS
5.1
Models of Study
The primary area of focus in this dissertation is on the flow field at the proximal vicinity
of an ACB-like model. Five different models were generated for investigating the geometrical
effect, flow condition, and fluid properties on the flow field in ACB model. Relative to
geometrical effects on the ACB model, the area of focus includes the branch diameter D2, blend
radius rB, radius of curvature rC, and helical pitch Hp. The effect of shear-thinning properties of
blood in the numerical analysis of the ACB-like model was also studied by comparing the flow
field computed based on the two commonly used viscous models, namely the Newtonian model
and the non-Newtonian model, which accounts for the shear-thinning properties of blood.
The geometry of these models was basically generated according to the typical
anastomotic setting of the bypass graft observed from angiography data. The numerical models
in the present study consisted of three major flow domains: host channel, junction, and branch.
The host channel represents the ascending aorta, the junction represents the proximal
anastomosis of the bypass graft, and the branch represents the graft. To reduce the complexity of
the analysis, the ascending aorta was simplified to a smooth straight tube. Figure 5.1 illustrates
the overall geometry and parameters of the numerical model considered in the present study. The
baseline model used to study the ACB model was a simple T-junction tube, which was found to
be very similar to the domain of interest in this study. Since the flow field in the ACB model is
extremely complicated, understanding the flow patterns at the proximal vicinity of ACB model
via a simple T-junction could be a better way of providing an understanding of the root causes of
the dynamics behaviors associated with the geometry, flow conditions or fluid properties.
76
rC
Parameter
Description
D1
Diameter of Host Channel
D2
Diameter of Branch
DP
Diameter of Aortic Punch Hole
Radius of Curvature of Branch
rC
Hp
Helical Pitch of Branch
L1
Axial Length of Host Channel
Radius of Blend of Branch
rB
Branch Angle

Figure 5.1. Schematic diagrams (top and middle) and geometrical parameters (bottom)
of numerical model.
Dimensions of the numerical model are relative to that of a normal human aorta, as
documented in the work of DeRubertis et al. (2011). The dimensions are based on 350 patients
ranging in age from 20 to 80 years old. In this work, the diameter of the host channel D1 (i.e., the
77
ascending aorta) is based on the average diameter for that of patients who are 75 years old (D1 =
3.0 cm). The dimension of the graft (i.e., branch diameter D2) is based on the typical crosssectional diameter of the adult saphenous vein used in bypass surgery (3 mm < D2 < 6 mm). The
relatively broader range of the aortic punch hole (0.5 cm < DP < 2.8 cm) was selected for the
investigation in this study.
A number of clinical investigations on the study of graft-host hemodynamics have shown
that a larger graft diameter [Qiao and Liu (2006)] and high graft-to-host artery ratio [Sui et al.
(2009)] are the preferable geometries for CABG surgery [Ghista and Kabinejadian (2013)]. At
present, understanding the definition of graft size is still rather obscure in terms of
hemodynamics. Also, from several clinical studies, it is common practice for surgeons to apply
an aortic punch device with the same punching diameter to perform proximal anastomosis for
bypass grafts with different proximal diameters. Whether such mismatch of proximal
anastomosis has any impact on the proximal flow field as well as the early phase failure of grafts
at the proximal region in ACB surgery has yet to be answered.
Two series of simple T-junctions were first modeled for investigating the geometrical
effects on the flow field at the proximal vicinity of the branch (i.e., junction). Table 5.1
summarizes the geometry and flow conditions of the first series models in which the diameter of
the branch is set the same as the diameter of the aortic punch hole (DP = D1) (the hole on the host
artery for anastomosis). The geometry and flow conditions of the second series models are
summarized in Table 5.2 in which the diameter of the branch is less than or equal to that of the
aortic punch hole (DP ≤ D1). Models from the two different series are referred to as non-blended
junctions and blended T-junctions, respectively. Knowing that the mass flow rate ratio between
the ascending aorta and the coronary artery where the graft is bypassed varies from patient to
78
patient, the critical relation between geometry and flow field of these two models is evaluated on
the basis of the relationship between the flow rate ratio, branching geometry, and local flow
patterns.
TABLE 5.1
GEOMETRY AND PARAMETERS OF NON-BLENDED T-JUNCTIONS
Geometry
Cross Section
Circular
Cross Section
Circular
Diameter (Host), D1
30 mm
Diameter (Branch), D2
3, 4, 5, 6 mm
Diameter (Aortic Punch Hole), DP
D2
Length (Host), L1
6.4 cm
Length (Branch), L2
4.0 cm
90 deg
Branch Angle, 
Flow Conditions (Steady)
Fluid
Blood (Newtonian)
1,060 kg/m3
Density, 
0.0035 Pa·s
Dynamic Viscosity, 
Inlet (Fully Developed Flow)
0.2 m/s
Reynolds Number, Re
1817
Outlet (Host)—Static Pressure, p
0 Pa
kg/s
Outlet
0.0003–0.0030 kg/s
Outlet(Branch)—Mass
(Branch)—MassFlow
Flow Rate,
Rate, 𝑚
𝑚̇ ̇ 22 0.0003–0.0030
0.002–0.020
Mass
0.002–0.020
MassFlow
FlowRate
RateRatio,
Ratio,𝑚
𝑚̇ ̇2 /𝑚
/𝑚̇ ̇1
2
1
Outlet
Outlet
(Branch)
(Branch)
Inlet
Inlet
(Host)
(Host)
Outlet
Outlet
(Host)
(Host)
79
TABLE 5.2
GEOMETRY AND PARAMETERS OF BLENDED T-JUNCTIONS
Geometry
Cross Section
Cross Section
Diameter (Host), D1
Diameter (Branch), D2
Length (Host), L1
Length (Branch), L2
Branch Angle, 
Branch Radius rB
Diameter (Aortic Punch Hole), DP
Circular
Circular
30 mm
3, 4, 5, 6 mm
6.4 cm
4.0 cm
90 deg
0.5, 2, 3, 4, 6, 8 mm
D2 + 2rB
Flow Conditions (Steady)
Fluid
Blood (Newtonian)
1,060 kg/m3
Density, 
0.0035 Pa·s
Dynamic Viscosity, 
Inlet (Fully Developed Flow)
0.025, 0.05, 0.10, 0.20 m/s
Reynolds Number, Re
227, 454, 909, 1817
Outlet (Host)—Static Pressure, p
0 Pa
TBD
Outlet (Branch)—Mass Flow Rate,
TBD
Mass Flow Rate Ratio, 𝑚̇2 /𝑚̇1
Blended model
Outlet (Branch)—Mass Flow Rate, 𝑚̇2
Mass Flow Rate Ratio, 𝑚̇2 /𝑚̇1
Outlet
(Branch)
Outlet
(Branch)
L2
0.0003–0.0030 kg/s
0.002–0.020
D2

Inlet
(Host)
D1
Inlet
(Host)
L1
Outlet
(Host)
Outlet
(Host)
Outlet (Branch)
80
Outlet (Host)
Depending on the location of anastomoses on the proximal and distal regions of the graft,
the bypass graft in the ACB model may be in a helical condition where the graft is turned in a
helical way. In vitro and in vivo studies have shown that helical geometry improves the
hemodynamic conditions on the arterial wall of a graft by enhancing the wall shear stress level
through the swirling flow effect induced by the helical geometry of the graft [Caro et al. (2005);
Huijbregts et al. (2006)]. The merits of helical geometry on vascular hemodynamics have also
been reported in the experimental and numerical work of several investigators [Sherwin et al.
(2000); Papaharilaou et al. (2002)]. To investigate the effects of branch diameter D2, helical pitch
Hp, and radius of curvature rC in ACB models as well as the relationship among these
parameters, a series of curved-tube models based on the possible geometrical range of the graft
in the ACB model were generated. The geometry range considered in the present work is as
follows: 3 mm < D2 < 5 mm, 0 cm < Hp < 8 cm, and 25 mm < rC < 35 mm. Table 5.3
summarizes the overall geometry range and flow condition of the two curved-tube models.
Two additional models are considered in this work to investigate the overall geometrical
effects on the flow field and hemodynamic parameters of ACB-like models based on the major
parameters investigated in the previous section (i.e., D2, rB, rC, and Hp). The first model is
composed of a non-blended bifurcated curved branch with symmetrical planar flow domain on
the host and branch channels (i.e., no helical pitch on the branch). The second model is
composed with a blended helical branch that is attached on the host channel. The diameter of the
host channel and branch of both models are set with D1 = 3 cm and D2 = 3 mm. The
determination of these parameters is discussed in Chapter 6. Table 5.4 summarizes the geometry
and flow conditions of the two different models. The first model is called the non-blended ACB,
and the second model is called the blended helical ACB.
81
TABLE 5.3
GEOMETRY AND PARAMETERS OF CURVED-TUBE MODELS
rC
Case 1
Case 2
(Varies D2 and Hp) (Varies rC and Hp
Geometry
Cross Section
Circular
Circular
30 mm
25, 30, 35 mm
Radius of Curvature, rC
Diameter, D2
3, 4, 5, mm
3 mm
Helical Pitch, Hp
0, 2, 4, 6, 8 cm
0, 2, 4, 6, 8 cm
Flow Conditions (Steady)
Fluid
Blood (Newtonian) Blood (Newtonian)
1,060 kg/m3
1,060 kg/m3
Density, 
0.0035 Pa·s
0.0035 Pa·s
Dynamic Viscosity, 
Inlet Velocity, U1
0.08 m/s
0.08 m/s
Reynolds Number, Re
72.69, 96.91, 121.14
72.69
Dean Number, De
16.25, 25.02, 34.97
8.90, 8.13, 7.52
Outlet (Host)—Static Pressure, p
0 Pa
0 Pa
82
TABLE 5.4
GEOMETRY AND PARAMETERS OF ACB MODELS
Blended
ACB Model
Non-Blended
ACB Model
Blended
Non-Blended
ACB Model
ACB Model
Geometry
Cross Section (Host)
Circular
Circular
Cross Section (Branch)
Circular
Circular
Diameter (Host), D1
30 mm
30 mm
Diameter (Branch, D2
3 mm
TBD
Length (Host), L1
6.4 cm
6.4 cm
2.5 cm
TBD
Radius of Curvature (Branch), rC
0 mm
TBD
Blend Radius, rB
Diameter (Aortic Punch Hole), DP
D2 + 2rB
D2 + 2rB
Helical Pitch, Hp
0 cm
TBD
Flow Conditions (Unsteady)
Fluid
Blood (Newtonian) Blood (Newtonian)
1,060 kg/m3
1,060 kg/m3
Density, 
0.0035 Pa·s
0.0035 Pa·s
Dynamic Viscosity, 
7.854 rad/s
7.854 rad/s
Frequency, 
Inlet (Host)—Velocity, U1
Aorta Waveform
Aorta Waveform
Outlet (Host)—Static Pressure, p
0 Pa
0 Pa
Graft Waveform
Outlet (Branch)—Mass Flow Rate, 𝑚̇2 Graft Waveform
Physiological waveforms of aorta and graft.
(data from Transonic System, Inc.)
83
CHAPTER 6
RESULTS
In this chapter, the flow fields of models discussed in previous chapters are investigated
based on the simulation results. The results can be separated into two categories: (i) steady-state
flow condition (Sections 6.1–6.4) and (ii) transient flow condition (Sections 6.5–6.6). In the first
part of the results, the non-blended and blended T-junction model series are first studied in
sections 6.1 and 6.2 on the basis of Newtonian blood. Through these sections, the critical
parameters and the relations between flow rate ratio and branching flow patterns under steadystate flow condition are able to be identified. Different correlations for flow separation in nonblended and blended T-junction models are developed. In section 6.3, results obtained based on
Newtonian and non-Newtonian models are compared for a non-blended T-junction under the
laminar steady-state flow condition in order to investigate the effect of shear-thinning properties
of blood in the models of the present work. The relationship between the geometrical effects and
flow field at the branch section of the T-junction is discussed in section 6.4. Geometries studied
include helical curve, radius of curvature, and diameter of the branch. The distribution of
hemodynamic indicators is used as the comparison parameter to study these effects. For all cases
under the steady-state flow condition, the flow condition is based on the typical averaged flow
rate in the bypass graft of the ACB model. For the second part of the results (sections 6.5–6.6),
the correlation for flow separation developed in the first part of the results are further
investigated under the transient flow condition through relevant physiological flow to understand
its applicability under the transient flow condition. Finally, in the last section, the critical
parameters gathered from sections 6.1–6.5 are integrated and applied on two ACB-like models to
obtain an overall understanding on the effects of these parameters in the ACB model. These
84
parameters include the branch diameter D2, blend radius rB, radius of curvature RC, and helical
pitch HP.
6.1
Test Model Series 1: Non-Blended T-Junction (Steady-State Flow)
The focus in this section is on the flow field at the junction of the branch of a simple T-
junction model series. The flow field at the junction is studied based on the effect of flow rate
and diameter on the flow separation near the junction. The first model series is the non-blended
T-junction where the diameter of the branch matches the diameter of the aortic punch hole of the
host channel, as shown in Figure 6.1. The separated flow at the junction is quantified based on
the flow separation length LSP near the upstream junction of the branch. The hemodynamic shear
stress induced by different mass flow rate ratios and diameters are used to investigate the
geometrical effects from the biological aspect. As shown previously in Table 5.1, four similar
models with a varying branch diameter of 3 mm ≤ D2 ≤ 6 mm were evaluated with flow rate ratio
ranges between 0.020 ≤ 𝑚̇2 /𝑚̇1 ≤ 0.002. To avoid the differences caused by the underdeveloped
flow at the junction of the branch, a fully developed flow profile with a constant velocity of u =
0.2 m/s (Re = 1817) was applied at the inlet of all models.
Non-blended model
Figure 6.1. Baseline model of study – non-blended T-junction.
85
6.1.1
Grid Independence Study
A grid independence test of the first series models was first carried out to ensure that the
grid size was sufficient to reach solution independence and capture details of the separation zone
at the junction. Figure 6.2 shows the velocity comparison at the symmetry plane of the junction
based on different grid sizes ranging from 0.2 million to 4.9 million. From these results, the grid
size of the model is shown to achieve an independent state as the cell number reaches
approximately 1.5 million. The difference between the 4.0 million and 1.5 million models is only
0.42% and 2.0%, respectively, for the local maximum and minimum of the curve with an
absolute maximum difference of velocity of v < 5.1%. Since the difference between 1.5 million
and 4.0 million is insignificant, the grid size that corresponded to the model with a cell number
of 1.5 million was therefore selected as the baseline setting (in terms of boundary layer inflation
rate, face size, and volume size) for the other models in this section—non-blended T-junctions.
Element sizes that corresponded to the 1.5 million cells model for the host, junction, and branch
sections were found at 5.0e-4 m, 8.0e-5 m, and 2.5e-4 m, respectively. Figure 6.3 shows a
sample grid structure of the different parts of the models with different grid sizes based on the
element sizes described above. The same grid size was applied to other models discussed in this
section.
86
1mm
Velocity v (m/s)
x/XB
Velocity v (m/s)
Velocity v (m/s)
x/XB
x/XB
Figure 6.2. Comparison of velocity-v profiles with different grid sizes (
87
= 1e-5, D2 = 3 mm).
Non-blended model
D2
L2

D1
L1
Outlet (Branch)
Outlet (Host)
Wall (Solid)
Inlet
Figure 6.3. Sample grid structure applied for non-blended T-junction (D2 = 3 mm).
88
6.1.2
Branch Diameter and Flow Rate Ratio on Flow Separation Length
With the element size discussed above, the numerical models of the four non-blended T-
junctions were generated for models with diameter ranges between 3 and 6 mm. Simulations
were conducted based on the mass flow rate ratio in the range of 0.02 ≤ 𝑚̇2 /𝑚̇1 ≤ 0.002 and inlet
Reynolds number of Re = 1817. Figures 6.4(a) and (b) illustrate the method used to quantify the
flow separation at the junction of the branch. First, the simulations were conducted with different
mass flow rates ṁ2 for each model. The y-component wall shear stress vector WSS𝑦 at the inner
wall of the branch was then measured and plotted against the y-axis, as shown in Figure 6.4(a).
Since WSS𝑦 is a function of the gradient of the y-component velocity near the boundary of the
wall WSS𝑦 = 𝜇 (𝜕v⁄𝜕𝑧), negative WSS𝑦 hence implies that there is a reversed or negative
velocity gradient which can be used as an indication of reversed flow in the separation zone.
Therefore, the length of the negative WSS𝑦 is used as a reference for the comparison of flow
separation at the junction of all models in this dissertation. The length of flow separation LSP is
denoted as the parameter for the total length of negative WSS𝑦 , as shown in Figure 6.4(b).
Figure 6.5 shows a comparison of the length of separation measured from the four nonblended T-junctions with branch diameters ranging from 3 to 6 mm with a mass flow rate ratio in
the range of 0.02 ≤ 𝑚̇2 /𝑚̇1 ≤ 0.002 and inlet Reynolds number of Re = 1817. The local
minimum of each curve is connected through a parabolic curve, as shown in red color.
89
-1
0.012
𝑚̇2 (kgs )
0.00015
Series2
0.00030
Series1
0.00045
Series3
0.00075
Series5
0.00105
Series7
0.00120
Series6
Series4
0.00150
y- axis [m]
0.010
0.008
0.006
0.004
0.002
-1
0.000
-0.5
0
0.5
WSSy [Pa]
Figure 6.4 (a). Distribution of y-component wall shear stress at inner wall of branch
for non-blended T-junction model (Re = 1817, D2 = 5mm).
0.012
y- axis [m]
0.010
= 5 mm
̇ 2 = 0.00075
0.008
0.006
0.004
0.002
-1
-0.5
0.000
0
0.5
WSSy [Pa]
Figure 6.4(b) Length of separation LSP based on negative WSS𝑦
(Re = 1817, D2 = 5 mm, 𝑚̇2 = 0.00075).
90
5.00
D2 (mm)
3
Series6
4
Series1
5
Series2
6
Series3
Series4
Local
LSP [mm]
4.00
3.00
Minimum
2.00
1.00
0.00
0
0.005
0.01
0.015
0.02
Flow Rate Ratio (𝑚̇ 2 /𝑚̇ 1 )
Figure 6.5. Effects of flow rate ratio on length of flow separation for different branch diameters
at Re = 1817 (red line indicates local minimum of each curve).
Based on the four LSP curves shown in Figure 6.5, the distribution of the length of
separation under the steady-state flow condition for the non-blended T-junction is, in general, not
identical for different diameter. However, each of the curves generally exhibits a parabolic-like
form, with its sensitivity of flow rate becoming smaller and smaller as the diameter of the branch
increases. This implies that the sensitivity of LSP or the rate of LSP decreases as the diameter of
the branch increases.
In addition, the parabolic shape curves also indicate that for each of the models under a
given flow condition, there is a minimum point of LSP,min. For all LSP curves, it is clear that below
a certain flow rate ratio (i.e., local minimum of each curve), the separation at the junction
decreases with the increase of flow rate ratio and begins to increase as the flow rate ratio passes
the local minimum. One possible explanation could be that below the local minimum flow rate,
the increase of the adverse pressure gradient at the junction of the branch due to increase of the
flow rate ratio is relatively slow compared to the adverse pressure gradient above the trigger flow
91
rate ratio (local minimum). Under this condition, the increase of momentum of the fluid near the
boundary due to the increase of the flow rate at the branch outlet is faster than the increase of the
adverse pressure gradient; thus, a portion of the adverse pressure gradient that is able to be
overcome by the fluid momentum in the boundary layer increases as the outflow rate increases,
hence reducing the total length of separation. As the flow rate passes the local minimum, the rate
of the adverse pressure near the boundary increases faster than that below the local minimum,
and the momentum gained by the fluid due to the increase of the outflow rate is no longer
sufficient to catch up with the increase in pressure. Thus, the separation zone increases with the
increase of flow rate ratio.
Figures 6.6(a) and (b) show a sample of the pressure and friction distribution near the
inner wall of the branch for the non-blended T-junction (D2 = 5 mm) with different flow rate
ratios. The pressure and friction are expressed in a non-dimensional form by pressure coefficient
, where P is the static pressure,
and friction coefficient
reference pressure in which the outlet pressure is used (0 Pa),
is the
is the inlet velocity at the inlet
(0.2 m/s), and WSSy is the y-component wall shear stress vector. The static pressure is measured
from the near-wall boundary layer (1e-5m normal to the wall). From Figure 6.6(a), it can be seen
that the increased rate of the adverse pressure gradient at the inner wall of the branch can be
observed as the flow rate increases. The effect of the flow rate ratio on the separation length is
even more obvious from the friction distribution plot. Taking a closer look at the intersection
locations of each curve from which the reattachment zone of the recirculation bubble is
determined, the length of separation is shown to decrease as the flow rate ṁ2 increases from
0.00030 kg/s to 0.00075 kg/s and start to increase as the flow rate continues to increase from
0.00075 kg/s to 0.00150 kg/s. The details of the velocity, pressure, streamline, and secondary
92
flow conditions for models are summarized in appendix A and B. Appendix C shows the
averaged static pressure at the inlet and outlets of a sample non-blended T-junction model. The
loss of pressure in the host and branch sections was within the reasonable range.
0.00
𝑚̇2 (𝑘𝑔/𝑠)
-0.50
0.00030
0.00045
0.00075
0.00105
0.00120
0.00150
CP
-1.00
-1.50
y
-2.00
x
-2.50
-3.00
0.0
0.5
y/YB
(a)
1.0
0.0
0.5
y/YB
(b)
1.0
0.14
0.12
0.10
0.08
Cf
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
Figure 6.6. Distribution of (a) pressure coefficient and (b) friction coefficient at the inner wall of
the branch of non-blended T-junction for Re =1817 and D2 = 5mm.
93
As shown in Figure 6.5, the flow rate ratio that corresponds to the local minimum
separation length LSP,min exhibits a parabolic trend with the increase in branch diameter. The
length of separation for each case shown in Figure 6.5 is plotted against the corresponding
branch diameter in Figure 6.7. The result is curve fitted with a parabolic curve. The increase of
/
D2 from 3 mm to 4 mm increases the LSP,min from 0.98% to 6.41%
[mm]
Figure 6.7. Correlation between minimum lengths of separation and branch diameters
of non-blended T-junction.
6.1.3
Effects of Branch Diameter on Hemodynamic Indicators
To understand the influence of flow separation on the distribution of wall shear stress in
the branch, the magnitude of WSS on both sides of the wall of the branch is calculated based on
WSS𝑥 , ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
WSS𝑦 , and ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
WSS𝑧 . Figure 6.8 illustrates the
the shear stress vector components: ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
distribution patterns of WSS at both sides of the wall (i.e., inner wall and outer wall) for
different diameters under a constant mass flow rate ratio of 𝑚̇2 /𝑚̇1 = 0.004. As expected, the
recirculation flow at the inner wall of the branch has led to a local minimum WSS at this
location, as shown in Figure 6.8(a), followed by a recovery of WSS after the reattachment zone.
The recovery area of the branch is shown to decrease with the increase in diameter. And the
magnitude of WSS at the full-recovery section is shown to decrease exponentially with the
94
increase in diameter. In contrast to the inner wall, the outer wall experiences a sudden rise of
WSS at the entrance of the junction, due to the sharp corner, which forces a drastic turn on the
incoming flow and leads to a steep velocity gradient
near the entrance of the outer wall.
The high WSS is followed by a gradual drop until it reaches a constant state where the flow
reaches its fully developed state, as shown in Figure 6.8(b). The distribution of the velocity
profiles of the branch is presented in Figures 6.9 and 6.10.
Inner wall
x
(a)
(b)
2.5
2.5
Re = 1817
𝑚̇ 2 /𝑚̇ 1 = 0.004
𝑫𝟐 (mm)
Series1
3
Series2
4
Series6
5
Series3
6
1.5
1.0
Re = 1817
𝑚̇ 2 /𝑚̇ 1 = 0.004
2.0
WSS [Pa]
WSS [Pa]
2.0
0.5
0.0
Outer wall
x
𝑫𝟐 (mm)
3 = 3 mm
D2
4 = 4 mm
D2
5 = 5 mm
D2
D2
6 = 6 mm
1.5
1.0
0.5
0
0.2
0.4
0.6
0.8
0.0
1
y/YB
0
0.2
0.4
0.6
0.8
1
y/YB
Figure 6.8: Distribution of WSS with different branch diameters of non-blended T-junction:
(a) inner wall and (b) outer wall (Re = 1817, 𝑚̇2 /𝑚̇1 = 0.004).
In general, the overall WSS on both sides of the wall decreases with the increase of D2.
At the entrance of the junction, the inner wall of the branch experiences a drastic drop of WSS,
whereas the outer wall experiences a drastic raise of WSS. As the flow travels far from the
junction, the WSS on both sides of the wall decreases gradually and settles down to a constant
value. Given a constant mass flow rate ratio, the results above suggest that, for a non-blended
circular T-junction, the length of separation LSP and the minimum length of separation LSP,min are
95
inversely proportional to the branch diameter, whereas the distribution of WSS on both sides of
the wall is directly proportional to the diameter of the branch.
6.2
Test Model Series 2: Blended T-Junction (Steady-State Flow)
The previous section has shown that separation flow in an non-blended T-junction is
associated with the flow rate ratio and the geometry of the junction. From a hemodynamics point
of view, it is of importance for a graft to be exposed to more laminar flow than a separated or
disturbed flow environment. In order to investigate the effect of the junction geometry and the
size mismatch of proximal anastomosis on the local flow behavior in the ACB model, this
section investigates a blended T-junction model with size analogous to the ACB model. This
model is then used to compare with the non-blended models discussed above. The focus of this
section is to investigate the relationship between the branch diameter D2, blend radius rB , mass
flow rate ratio (𝑚̇2 /𝑚̇1 ), and Reynolds number Re on a blended T-junction and their influence
on the flow patterns and hemodynamic parameters under a steady flow condition. There are total
of 24 models with branch diameters (D2) of 3–6 mm and blend radiuses (rB) of 0.5–8.0 mm
simulated. A fully developed flow profile with a constant velocity of u = 0.2 m/s (Re = 1817)
was applied at the inlet of all models.
Using a similar approach as in the previous section, the minimum element sizes that were
found to achieve solution independence for the host, branch, and junction were selected and
applied to the rest of the models. Figure 6.9 shows the mesh independence test results for two
different blended models. The independence of mesh size was justified by comparing the results
obtained from the models with the largest and smallest D2 and rB under a constant mass flow rate
ratio of 𝑚̇2 /𝑚̇1 = 0.002. The element sizes that are found to be sufficient for the simulation of
the non-blended models are 5.5e-4 m, 1.7e-4 m, and 2.2e-4 m for the host, junction, and the
96
branch s ections, respectively. Figure 6.10 shows one of the numerical models of the blended Tjunction used for the simulation.
= 3 mm
rB = 8.0 mm
v [m/s]
v [m/s]
= 3 mm
rB = 0.5 mm
No. of grid
281,034
1,188,317
2,261,143
No. of grid
373,224
1,705,183
2,473,763
x/XB
x/XB
(a)
No. of grid
468,936
1,407,372
2,521,504
= 6 mm
rB = 8.0 mm
v [m/s]
v [m/s]
= 6 mm
rB = 0.5 mm
x/XB
No. of grid
613,702
2,282,683
3,610,524
x/XB
(b)
Figure 6.9. Comparison of velocity profiles of two different blended models measured
at 5 mm from entrance of branch.
97
Blended model
D2
L2

D1
L1
Outlet (Branch)
Outlet (Host)
Wall (Solid)
Inlet
Figure 6.10. Sample meshed model in for blended T-junction model (D2 = 5mm, rB = 3mm).
98
6.2.1
Flow Rate Ratio and Blend Radius at Onset of Separation Flow
In this section, the relationship between the flow rate ratio (𝑚̇2 /𝑚̇1 ) and the blend radius
rB of the blended T-junction is determined by imposing different flow rate ratios at the branch
outlet. Unlike the non-blended T-junction, when a blended T-junction is exposed to a certain
mass flow rate ratio, the length of separation of the blended T-junction is shown to decrease
nonlinearly with respect to the flow rate ratio. As the flow rate continues to increase, separation
at the junction disappears and flow near the junction eventually reaches a fully laminar state. The
flow rate ratio that corresponds to the non-separation condition is denoted as the critical flow rate
ratio (𝑚̇2 /𝑚̇1 )crit. The flow rate is adjusted manually until the fully laminar state flow is achieved
at the junction, as shown in Figure 6.11.
3.5
Re = 1817
= 5 mm
rB = 4 mm
3.0
LSP [mm]
2.5
2.0
1.5
1.0
𝑚̇2 /𝑚̇1
0.5
0.0
0
0.001
0.002
0.003
Flow Rate Ratio (𝑚̇ 2 /𝑚̇ 1 )
0.004
Figure 6.11. Determination of critical mass flow rate ratio for non-separation
of blended T-junction (Re = 1817, D2 = 5 mm, and rB = 4 mm).
To investigate the relationship between (𝑚̇2 /𝑚̇1 )crit and rB, the (𝑚̇2 /𝑚̇1 )crit that
corresponds to different rB’s is obtained through a series of simulations by adjusting the outflow
rate of the branch for models with different rB’s. Figure 6.12(a) illustrates the transition of
separation flow at the junction to non-separated flow of a sample blended T-junction model with
D2 = 5 mm. The corresponding WSSy of the inner wall of the branch is shown in Figure 6.12(b).
99
𝑚̇ 2 /𝑚̇ 1 = 0.0060
(𝑚̇2 /𝑚̇1 )crit
Non-separated
𝑚̇ 2 /𝑚̇ 1 = 0.0035
Re = 1817
D2 = 5mm
Separated
𝑚̇ 2 /𝑚̇ 1 = 0.0020
rB [mm]
Figure 6.12(a). Determination of the minimum critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit, min of
blended T-junction for Re = 1817, D2 = 5mm, and rB = 5 mm.
0.035
rSeries1
B = 1mm
rSeries2
B = 2mm
rSeries3
B = 4mm
rSeries4
B = 6mm
rSeries5
B = 8mm
y- axis [m]
0.030
0.025
0.020
0.015
rB (mm) (
0.010
0.005
-0.2
rB
0.000
0
0.2
0.4
WSSy [Pa]
1
2
4
6
8
)crit, min
5.4053
3.2432
3.5968
4.8714
6.4062
Figure 6.12(b). WSSy distribution for determination of critical mass flow rate ratio
of blended T-junction (Re = 1817, D2 = 5 mm, and rB = 1 to 8 mm).
100
The same method is repeated for models with different branch diameters D2 ranging from
3 to 6 mm. Figure 6.13 shows the (𝑚̇2 /𝑚̇1 )crit corresponding to the blended T-junction with rB
ranges between 0.5 and 8 mm and diameter D2 ranges between 3 and 6 mm at Re = 1817. The
local minimum of each curve is connected with a red curve and these points are denoted as the
critical minimum flow rate ratios (𝑚̇2 /𝑚̇1 )crit,min, which reveals the important relationship
between 𝑚̇2 /𝑚̇1 , rB, and D2 for the pulsatile flow condition. From Figure 6.13, it can be seen
that for every (𝑚̇2 /𝑚̇1 )crit curve, there exists an rB that corresponds to the minimum of the curve,
which is denoted as the (𝑚̇2 /𝑚̇1 )crit,min, since under the pulsatile flow condition, the flow rate
ratio is subject to variation, with a wide range of magnitude. The (𝑚̇2 /𝑚̇1 )crit,min shown in Figure
6.13 indicates that for a given rB, the recirculation zone can be reduced to the lowest level at a
certain mass flow rate ratio, or vice versa.
0.0100
D2 = 6 mm
Re = 1817
0.0080
(𝑚̇ 2 /𝑚̇ 1 )crit
D2 = 5 mm
0.0060
D2 = 4 mm
0.0040
D2 = 3 mm
0.0020
0.0000
0
2
4
6
8
10
rB [mm]
Figure 6.13. Effect of blend radius rB on distribution of critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit
at different branch diameters for Re = 1817 (red line shows blend radius rB that corresponds to
(𝑚̇2 /𝑚̇1 )crit,min).
.
101
The rB corresponding to the (𝑚̇2 /𝑚̇1 )crit,min of blended T-junctions with different branch
diameters shown in Figure 6.13 are plotted against D2 in Figure 6.14. The correlation between rB
and D2 in this figure can be approximated with the following power series:
rB = 0.146D21.74
(6.1)
3.5
3.0
rB [mm]
2.5
rB = 0.146D21.74
2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
D2 [mm]
Figure 6.14. Correlation between rB and D2 for (𝑚̇2 /𝑚̇1 )crit,min of blended T-junction
at Re = 1817.
6.2.2
Effect of Reynolds Number on Critical Minimum Mass Flow Rate Ratio
Equation (6.1) was further studied to investigate its relationship with four different
Reynolds numbers: 227, 454, 909, and 1817 (U1: 0.025 m/s, 0.05 m/s, 0.1 m/s, and 0.2 m/s). The
blended model with branch diameter D2 = 5 mm was chosen as the baseline model for
simulation. Figure 6.15 shows the distribution of critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit for the
blended model of D2 = 4 mm at different Reynolds numbers. In general, the increase in Reynolds
numbers increases the overall distribution of the critical mass flow rate ratio. However the rB that
corresponds to the minimum (𝑚̇2 /𝑚̇1 )crit,min was shown to be independent of Re. This indicates
that, given a known branch diameter, equation (6.1) can be used to predict the optimal blend
102
radius rB for minimum recirculation of a blended T-junction. It must be noted that, equation (6.1)
was approximated based on D2 ranges between 3 and 6 mm, which is the typical range of a
bypass graft applied in CABG surgery. Further study needs to be performed for a larger range of
D2. In this dissertation, only the relevant range of D2 for the ACB model was considered.
From the surgical viewpoint, given that the proximal diameter of a graft is known (i.e.,
D2) and the blood flow rate ratio is constant, the results above suggest that different aortic punch
hole-to-branch diameters, i.e., (2rB + D2)/D2 ratios, induces different scale-of-separation zones
at the proximal region of the graft. The separation zone is shown to be proportional to the blend
radius in a parabolic form. The results shown in this section suggest that the recirculation zone at
the proximal region may be reduced when the aortic punch hole diameter (2rB) coincides with
the minimal critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit,min of the (𝑚̇2 /𝑚̇1 )crit curve, as shown in
Figure 6.13. This may be explained by the fact that the transition of the pressure gradient
introduced by the curvature at the blended junction that flows in the boundary layer is able to be
overcome when (𝑚̇2 /𝑚̇1 ) is greater than the (𝑚̇2 /𝑚̇1 )crit,min. To understand this, Figure 6.16
shows an example of the comparison of the pressure coefficient (CP) distribution between a nonblended T-junction and a blended T-junction (rB = 3 mm, D2 = 6 mm, and Re = 1817) at (𝑚̇2 /𝑚̇1
= 0.008. One can see that, although an adverse pressure gradient is present at the intersection of
the blended T-junction, the transition of the pressure gradient at the junction is relatively smooth
compared to the drastic rise of CP near the sharp corner of the inner wall of the non-blended
model. The strength of the adverse pressure gradient of the blended T-junction is also small, with
an exposure length (Ladv_p) approximately 63% smaller than that of the non-blended T-junction.
As discussed in section 6.2.1, since the (𝑚̇2 /𝑚̇1 ) of the blended model in this case is greater than
the (𝑚̇2 /𝑚̇1 )crit,min, no separation is initiated at the inner wall of the branch, and the flow in the
103
boundary layer is shown to have sufficient momentum to overcome the adverse pressure
gradient.
0.0050
Re = 1817
D2 = 4mm
(𝑚̇ 2 /𝑚̇ 1 )crit
0.0040
0.0030
0.0020
Re = 909
0.0010
Re = 454
0.0000
Re = 227
0
2
4
rB [mm]
6
8
Figure 6.15. Distribution of (𝑚̇2 /𝑚̇1 )crit for blended T-junction at Re = 227, 454, 909, 1817.
CP
Blended T-junction
Lower adverse
pressure gradient
Higher adverse
pressure gradient
-1
-0.5
x/(XH/2)
0
0.5
y/YB
l/
1
Non-blended Tjunction
Figure 6.16. Comparison of pressure coefficient distribution for non-blended
T-junction (solid line) and blended T-junction (dashed line)
(Re = 1817, 𝑚̇2 /𝑚̇1 = 0.008, D2 = 6 mm, rB = 3 mm).
104
6.2.3
Secondary Flow Characteristics
In this section, the secondary flow of both the non-blended and blended T-junctions is
studied. Figures 6.17(a) and (b) show the secondary flow structure at different cross-sectional
planes measured at y = 1.0 mm and y = 2.5 mm from the entrance of the branch of both models.
At y = 1.0 mm, both models exhibit a pair of symmetrical vortices, as shown in Figure 6.17(a).
However, near the circumferential region of the inner wall of the branch, the non-blended model
experiences a marked weak secondary motion compared to the blended model. Details of the
flow structure can be seen from the streamline and vector plots displayed. Unlike the blended
model, vortical cells near the inner-wall region of the non-blended model are separated as they
get closer to the inner wall (shown by red arrows). One possible explanation could be that the
smooth curved junction of the branch introduces an outward centrifugal force on the incoming
fluid elements. To maintain the equilibrium of the fluid’s momentum, the radial pressure at the
junction is adjusted in such a way that the maximum pressure is switched to the outer wall and
the minimum pressure is switched to the inner wall. The combination effect of centrifugal force
and the pressure difference across the cross-sectional area results in a shift of the planar velocity
at the center towards the outer wall of the cross section. To balance the pressure difference
across the plane, the slower velocity fluid in the boundary layer has to move from the outer wall
towards the inner wall. The net motion effect at the cross-section eventually results in two
counter rotation cells and skews the maximum component of the axial velocity profile to the
outer wall. In this case, due to the sharp turning angle of the non-blended model, a large low
pressure area develops near the inner-wall region. Fluid in this area is predominated by the
pressure force instead of the centrifugal force. Hence, as the fluid gets closer to the top of the
inner wall, the strong pressure force causes the fluid to separate from turning back to the center
of the cross section, as shown in the blended model.
105
y = 1.0 mm
Outer wall
Velocity
Streamline
Secondary Flow Velocity
Inner wall
Vector
(a)
y = 2.5 mm
Inner Wall
Outer Wall
Velocity
Streamline
Vector
(b)
Figure 6.17. Comparison of secondary flow for non-blended and blended T-junctions:
(a) y = 1.0 mm and (b) y = 2.5 mm (Re = 1817, D1 = 30 mm, D2 = 3 mm, rB = 1 mm).
106
As the flow moves farther away from the junction, the secondary motion of both models
becomes weaker in magnitude due to the diminishing centrifugal force. This results in a weaker
drag from the centrifugal force on the flow near the top of the inner wall and leads to a relatively
weak secondary effect at this region of both models, as shown in Figure 6.17(b). However, the
secondary zone of the non-blended model is shown to be larger compared to that of the blended
model. The separation of the vortical cells is still present near the top of the model but not on the
unblended model.
From a surgical viewpoint, the region with low secondary flow as shown in the nonblended model may be exposed to a higher risk of vascular lesion compared to the blended
model, because the shear stress exerted on the local endothelium may not be sufficient to trigger
the vascular cells to modulate an adequate expression of molecules that maintain the integrity of
the vascular wall.
6.3
Newtonian vs. Non-Newtonian Models and T-Junction Flow (Steady-State Flow)
For simplicity, the numerical analysis in this dissertation was conducted based on the
simulation results obtained from the Newtonian blood model. With this model, the dynamic
viscosity-dependent parameters, such as Re, , and D, can therefore be calculated based on the
commonly used constant dynamic viscosity for blood ( = 0.0035 Pa·s). As discussed in section
3.2.1, the shear rate in small arteries generally exhibits non-Newtonian behavior; therefore, the
shear-thinning effect must be taken into account in order to accurately reproduce the flow field in
the artery. However, the primary focus in this dissertation is on the comparison of geometrical
effects on the branching flow field in T-junction models and ACB models instead of reproducing
the actual flow field in the vascular system. Therefore, the assumption of Newtonian blood will
be reasonable to reflect the geometrical effects on the flow features in these models. To obtain
107
insight on the difference between Newtonian ( = 0.0035 Pa·s) and non-Newtonian ( =
Carreau-Yasuda) models, this section shows results of the effects of Reynolds number and
branch diameter based on flow condition, with Reynolds numbers in the range of 454 < Re <
1817 and T-junction models with diameters in the range of 0.1 < D2/D1 < 0.2.
Figure 6.18 shows the x-component velocity distribution in the symmetry plane of the
host channel for Re = 454, 909, and 1817 at a constant flow rate ratio (𝑚̇2 /𝑚̇1 = 0.004). The
velocity profiles from the Newtonian and non-Newtonian models are plotted against each other
for comparison purposes. In general, the velocity profiles of the host channel obtained from the
non-Newtonian model are almost identical with that of the Newtonian model. The maximum
difference between these three different plots is less than 1%, which is in line with experimental
findings that show the shear-thinning effect is negligible in large arteries.
y
Outer wall
Inner wall
y/YB = 0.80
y/YB = 0.40
y/YB = 0.26
y/YB = 0.10
x/XH = -0.625
y/YB = 0.04
x
Figure 6.18(a). Comparison of velocity distribution for Newtonian and non-Newtonian fluids
on symmetric plane of model.
108
u/u,max
u/u,max
u/u,max
y/YH
y/YH
y/YH
(a)
(b)
(c)
Figure 6.18(b). Distribution of velocity-u for Newtonian (solid line) and non-Newtonian
(dashed line) fluids: (a) Re = 454, (b) Re = 909, and (c) Re = 1817.
The distribution of the y-component velocity at the symmetry plane of the branch section
for Re = 454, 909, and 1817 is shown in Figure 6.19. Here, the difference between the
Newtonian and non-Newtonian velocity profiles is evident. The non-Newtonian profile shapes,
in general, are flatter compared to the Newtonian profiles. At y/YB ≈ 0.9, the fully developed
condition of the flow can be observed for all velocity profiles. At the inner wall, the effect of
shear thinning is shown to increase with the increase of Re. The average difference of the local
maximum of v at y/YB = 0.9 (i.e., fully developed flow profile) is approximately 9.5%. At the
outer wall, the velocity gradient for all cases is almost identical. The local maximum v predicted
by the Newtonian model tends to be higher than that of the non-Newtonian model. The velocity
gradient at the inner wall of the branch is also shown to be smaller before the stage of fully
developed flow.
109
v/U1
v/U1
v/U1
y/YB = 0.80
y/YB = 0.40
y/YB = 0.26
y/YB = 0.10
y/YB = 0.04
x/XB
x/XB
x/XB
(a)
(b)
(c)
Figure 6.19. Distribution of velocity-v for Newtonian (solid line) and non-Newtonian
(dashed line) fluids: (a) Re = 454, (b) Re = 909, and (c) Re = 1817.
110
Figure 6.20 shows the distribution of the y-component velocity at the symmetry plane of
the branch with D2 = 3, 4, 5, and 6 mm at Re = 1817 and (𝑚̇2 /𝑚̇1 = 0.004). Again, all the
velocity profiles are shown to achieve fully developed at y/YB ≈ 0.9. The non-Newtonian profiles
are generally flatter than the Newtonian profiles. The average difference of the local maximum vy
for the four cases at the fully developed region is approximately 9.25%. For the profiles under
the fully developed condition, the Newtonian model, in general, over predicts the velocity at the
outer wall and under predicts flow at the inner wall of the branch. The velocity profile at the
inner wall of the non-Newtonian model is less steep compared to the Newtonian model. The flow
reversal zones at the inner wall near the junction predicted by the non-Newtonian model for D2 =
5 mm and 6 mm are also shown to be smaller than that of the Newtonian model. For a small
branch diameter, i.e., D2 = 3 mm (D2/D1 = 0.1), the Newtonian velocity profiles are shown to be
almost identical to that of the non-Newtonian model, with the maximum difference of v
approximately 9.5% higher than that of the non-Newtonian model. The difference between the
velocity gradient predicted by both models at the inner wall generally increases with the increase
of the diameter.
To obtain a quantitative scale on the effect of D2 on the flow of the T-junction branch
under constant Re and 𝑚̇2 /𝑚̇1 , the local maximum v of each profile is plotted as a function of the
axial flow distance from the entrance of the branch (Figure 6.21). Interestingly, the difference of
the local maximum of v (i.e., vmax) predicted by the two different models is shown to increase
with the D2 under a constant flow rate ratio. The highest difference of the local maximum of the
profile is located at around 30% from the entrance of the branch (y/YB ≈ 0.28).
111
v/U1
v/U1
v/U1
v/U1
y/YB = 0.80
y/YB = 0.40
y/YB = 0.26
y/YB = 0.10
y/YB = 0.04
x/XB
x/XB
x/XB
(a)
(b)
(c)
x/XB
(d)
Figure 6.20. Distribution of velocity-v for Newtonian (solid line) and non-Newtonian (dashed
line) fluids for D2/D1: (a) 0.100, (b) 0.133, (c) 0.167, and (d) 0.200
(Re = 1817, 𝑚̇2 /𝑚̇1 = 0.004).
112
vmax [%]
40
35
D2 /D1
0.200
30
0.167
25
0.133
0.100
20
15
10
5
0
-5
0
0.2
0.4
0.6
0.8
1
y/YB
Figure 6.21. Distribution of vmax predicted by Newtonian and non-Newtonian models
for non-blended T-junction with D2/D1= 0.100, 0.133, 0.167, and 0.200 at Re = 1817.
The direct relation between D2 and vmax and the difference between velocity gradient at
the inner wall predicted by both of the viscous models may be attributed to the flattened velocity
profile caused by the shear-thinning properties of non-Newtonian fluid and the direct relation
between the separation length of a T-junction to the branch diameter as discussed in the previous
section. Further examination is required for the full explanation of this relation.
6.4
Test Model Series 3: Helical Curved-Tube (Steady-State Flow)
This section investigates the effects of different geometrical parameters on helical
curved-tube flow. Parameters include the helical pitch Hp, diameter D2, and radius of curvature
rC. The effects of these parameters are evaluated based on the distribution of wall shear stress.
The results are separated into two different model series: the first series evaluates the effects of
D2, and the second series evaluates the effects of rC. Each of the model series is tested with
helical pitch Hp in the range of 0 – 8 cm with steady laminar flow simulation. A total of 30
simulations were performed. A constant velocity that corresponds to the peak mass flow of the
113
graft waveform shown in Chapter 5 was imposed at the inlet of each model. The information and
the geometry as well as the corresponding De and the Re are summarized in Table 5.3.
Figures 6.22(a) and (b) show the distribution of WSS plotted as a function of Hp for
models with different D2 and rC, respectively. The WSS is calculated based on the average WSS
measured at different axial line segments of the graft as shown in the top of the figure with four
different colors. Figure 6.22(a) shows the relationship between Hp and D2 for the tube with
constant radius of curvature (rC = 3.0 cm). The results show that the WSS on the outer and inner
walls is inversely correlated to HP, while directly correlated to the left and right sides of the wall.
This relation may be attributed to the swirling effect caused by the helical geometry which
partially increases the velocity gradient in the boundary layer of the wall. Although the WSS on
the outer and inner walls reduce due to the increase of Hp, it is a relatively small decrease
(maximum decrease of 3% from Hp = 0 to 8 cm) in comparison to the marked significant
increase of WSS on the left and right side of walls (~23% increase from Hp = 0 to 8 cm).
Figure 6.22(b) shows the relationship between Hp and rC for tubes with a constant
diameter (D2 = 3 mm). Just like the trend for WSS shown in Figure 6.22 (a), the WSS on the
outer and inner walls is inversely proportional to the Hp and directly proportional to the left-side
and right-side walls. The reduction of WSS on the upper and lower walls due to the increase of
Hp is also relatively small compared to that on the left- and right-side walls. Due to the complex
relation between the geometry and flow structure of the helical tube and the objective of this
dissertation, the relationship between rC, Hp, D2, and Re was not examined further. A more
thorough study on the parameters needs to be conducted in order to obtain a theoretical
explanation of the WSS distribution of the helical tube.
114
Upper Wall
Right-Side Wall
Left-Side Wall
Lower Wall
rC
Vin
D2
Hp
rC = 3 cm
WSS (Pa)
D2 = 3, 4, 5 mm
HP (cm)
HP (cm)
HP (cm)
a
D2 = 3 mm
b
D2 = 4 mm
c
D2 = 5 mm
(a)
HP (cm)
D2 = 3 mm
rC = 2.5, 3.0, 3.5 cm
arC
= 3.5 cm
WSS (Pa)
brC = 3.0 cm
crC = 2.5 cm
HP (cm)
HP (cm)
HP (cm)
HP (cm)
Figure 6.22. Comparison of averaged steady WSS along four different line segments of tube with
different parameters: (a) D2 and (b) rC.
115
6.5
Non-Blended vs. Blended T-Junctions (Unsteady-State Flow)
Sections 6.1 and 6.2 showed the relationship between the separated flow and the
geometrical effects of a T-junction branch under a steady-state flow condition. Results indicate
that the blended junction, or the ratio between the aortic punch hole and the branch diameter,
plays a significant role on the separated flow region at the inner wall of the junction. To link the
geometrical effect of the T-junction on the separated flow and hemodynamic environment of the
ACB model, in this section, physiological boundary conditions are imposed at the inlet and outlet
of the branch, based on a patient’s specific waveforms measured from the ascending aorta and
bypass graft (Figure 6.23). For analysis purposes, these waveforms were simplified into a
periodic Fourier series function with a frequency 1.25 Hz ( =7.854 rad/s) based on the normal
resting heartbeat rate of an adult human (i.e., ~ 1.00–1.67 Hz) with a corresponding Womersley
number of  = 23.13 and a peak Reynolds number of Repeak = 1819. Figure 6.24 shows the
replicated waveforms shown in the profiles in Figure 6.23.
Figure 6.23. Physiological waveforms of aorta and graft (from Transonic System, Inc.)
116
Host Inlet
𝑚̇1 (kg/s)
Branch Outlet
𝑚̇2 (kg/s)
Aorta
Branch Outlet
Host Outlet
Host Inlet
Graft
Solid Wall
𝑡 (s)
Figure 6.24. Waveforms replicated with Fourier series for inlet and outlet boundary conditions.
Two models generated in the previous section with
= 3 mm were selected for the
investigation of pulsatile flow in T-junction models. The first model was the non-blended model
tested in section 6.1, and the second model was the blended T-junction from section 6.2 with rB =
1 mm. The rB applied on the second model is based on the (𝑚̇2 /𝑚̇1 )crit,min determined in section
6.2.1 (Figure 6.13), which corresponds to the minimum flow separation. Although the
corresponding rB from the results for (𝑚̇2 /𝑚̇1 )crit,min is approximately 1.35 for
2
= 3 mm, the
difference of the (𝑚̇2 /𝑚̇1 )crit is indeed insignificant. Also, for analysis and simplicity purposes,
the blended model where rB = 1 mm is chosen as the model for comparison in this section.
Figure 6.25 shows the independence test results for the time-step size of the first model
based on the instantaneous velocity at a monitor point located near the inner wall of the junction
where reverse flow or flow separation is expected. Since the maximum absolute difference of
velocity v between time-steps of 0.01s and 0.001s is 3.49%, t = 0.01s is therefore considered to
be sufficient for the time-step independent solution in the simulations here.
117
Monitor Point
0.05
Velocity
v [m/s]
v [m/s]
0.04
0.03
t
0.02
0.100 s
0.050 s
0.010 s
0.001 s
0.01
0
-0.01
-0.02
0
0.2
0.4
t [s]
t [s]
0.6
0.8
Figure 6.25. Time-step independence test based on velocity of monitor point in junction
of non-blended T-junction (Repeak = 1819,  = 23.13, D1 = 30 mm, D2 = 3 mm).
6.5.1
Flow Separation Characteristics
Figures 6.26 and 6.27 show the temporal streamline plots of the non-blended and blended
T-junction models during the systolic and diastolic phases of the cardiac pulse, respectively.
Three major observations can be seen in the flow characteristics of these models. First, a
stagnation zone is formed near the outer wall of the junction in both models right after the
acceleration phase and continues to remain until the late deceleration phase of the aortic pulse.
Second, a marked flow separation is initiated in the non-blended T-junction during the early
acceleration phase. This separation zone starts to diminish during the mid-deceleration phase of
the aortic pulse, whereas the flow separation zone in the blended model is insignificant compared
to the non-blended one throughout the cardiac cycle. Third, the end of flow separation is
followed by a stagnation zone near the inner wall of the junction and continues to fluctuate in a
small amplitude until the peak of the RCA pulse.
118
Time (s)
0.00
0.04
0.08
0.12
0.16
0.20
0.24
(a)
(b)
Figure 6.26. Comparison of temporal streamline at symmetry planes during acceleration
(systolic) phase: (a) non-blended T-junction and (b) blended T-junction (Repeak = 1819,
 = 23.13, D1 = 30 mm, D2 = 3 mm, rB = 1 mm).
119
Time (s)
0.28
0.32
0.36
0.40
0.44
0.48
0.52
(a)
(b)
Figure 6.27. Comparison of temporal streamline at symmetry planes during deceleration
(diastolic) phase: (a) non-blended T-junction and (b) blended T-junction (Repeak = 1819,
 = 23.13, D1 = 30 mm, D2 = 3 mm, rB = 1 mm).
120
Since the previous sections have shown that flow separation in the T-junction is
correlated to the mass flow rate ratio of the host inlet and the branch outlet, for quantitative
analysis, the length of separation LSP of both models is plotted against the mass flow rate
ratio 𝑚̇2 /𝑚̇1 . Figure 6.28 shows the influence of 𝑚̇2 /𝑚̇1 on the LSP of both models over the
complete cycle of the pulse. For both models, it can be seen that the flow separation is initiated
during loss of the mass flow rate ratio. A closer look at Figure 6.28 is presented in Figure 6.29 to
show the corresponding mass flow rate ratio where the flow separation is initiated near the
junction. A significant difference can be observed in the separation length between the nonblended and blended models. For the blended model, the separation flow occurs at a transient
flow rate ratio (𝑚̇2 /𝑚̇1 )crit,trn approximately 80% lower than that of the non-blended model. The
local maximum of the LSP curve LSP,max for the blended model is approximately 2.3 times smaller
than that of the non-blended model. In addition, the cyclic exposure time of the separated flow at
the junction is shown to be about 2.4 times smaller than that of the non-blended model.
3.5
3.5
0.12
0.12
LSP,max
3.0
3.0
Non-blended Model
0.04
0.04
Flow Rate Ratio
1.5
1.5
/
[mm]
Blended Model
2.0
2.0
0.00
0.00
LSP,max
1.0
1.0
-0.04
-0.04
0.5
0.5
0.0
0.0
0.08
0.08
𝒎̇𝟐 /𝒎̇𝟏
Lsp [mm]
2.5
2.5
0
0.0
0.2
0.2
0.4
0.4
tt [s]
[s]
0.6
0.6
0.8
0.8
-0.08
-0.08
Figure 6.28. Comparison of length of separation for non-blended and blended T-junctions
(Repeak = 1819, = 23.13, D1 = 30 mm, D2 = 3 mm, rB = 1 mm).
121
LSP [mm]
t [s]
(𝑚̇2 /𝑚̇1)crit,trn = 0.0055
𝑚̇2 /𝑚̇1
𝑚̇2 /𝑚̇1
LSP [mm]
(𝑚̇2 /𝑚̇1)crit,trn = 0.0272
t [s]
Figure 6.29. Corresponding mass flow rate ratio for onset of flow separation: (a) non-blended
T-junction and (b) blended T-junction (Repeak = 1819, , = 23.13, D1 = 30 mm, D2 = 3 mm,
rB = 1 mm).
In general, results obtained from the pulsatile flow condition in this section are still in
line with results of the steady-state flow condition in the previous section. Using the rB
correlation developed in section 6.2 for minimum flow separation, simulation results show that
the recirculation zone caused by separation at the junction of the blended model is reduced about
80% compared to the non-blended model under the same cardiac pulse condition. The temporal
separation formed in the blended model is shown to occur at a mass flow rate higher than that
predicted in section 6.2 for the steady-state flow condition. This phenomenon could possibly be
attributed to the transient dynamic effect of the flow field. Due to the complexity of the fluid
structure involved in the transient condition, details of the difference between the (𝑚̇2 /𝑚̇1 )crit,min
predicted in steady-state and transient state conditions are not included in the present work.
From a surgical viewpoint, the blended T-junction may have several important
implications on flow at the proximal vicinity of a bypass graft under a pulsatile flow condition.
122
Based on the findings here, it is strongly believed that different ratios of the diameter of the
aortic punch hole and bypass graft may impose different scale flow separations at the proximal
region in terms of length, strength, onset condition, and exposure time of the separation flow.
However, such a condition may be reduced with the blended connecting approach with rB that
corresponds to the minimal critical mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit,min applied at the proximal
junction.
6.5.2
Secondary Flow Characteristics
The secondary velocity of the non-blended and blended T-junction models measured at
the cross-sectional plane located at y = 1.0 mm and y = 2.5 mm from the junction of the branch is
shown in Figures 6.30 and 6.31, respectively. Due to the symmetrical nature of the physical flow
domain, the secondary flow is generally composed of a pair of symmetrical cells throughout the
cycle for the non-blended and blended models, at y = 1.0 mm and 2.5 mm. At y =1.0 mm, a
relatively large area of low magnitude velocity is seen to be concentrated at the circumferential
region of the inner and outer walls of the non-blended T-junction throughout the cycle. This
characteristic of secondary flow is similar to that explained in section 6.2.3 for the non-blended
T-junction under steady flow condition, where the secondary flow is predominated by a strong
adjustment of radial pressure due to the junction’s sharp corner. Separation at the outer wall
caused by the pressure force effect and the formation of the symmetrical cells has significantly
reduced the velocity at the outer and inner walls. In general, the flow structure of the blended
model is shown to be similar to that of the non-blended model. However, the overall kinetic
energy of the flow structure is shown to be high throughout the cycle. Instead of having two lowspeed flow regions as can be seen in the non-blended model, there is only one area with lowspeed secondary flow, which is concentrated in a small corner of the inner wall of the cross-
123
y = 1.0 mm
Velocity
Time [s]
(a)
(b)
Inner Wall
Outer Wall
0.00
0.11
0.23
0.34
0.46
0.57
0.69
Figure 6.30. Comparison of temporal secondary flow at y = 1.0 mm:
(a) non-blended and (b) blended T-junctions.
124
y = 2.5 mm
Velocity
Time [s]
(a)
(b)
Inner Wall
Outer Wall
0.00
0.11
0.23
0.34
0.46
0.57
0.69
Figure 6.31. Comparison of temporal secondary flow at y = 2.5 mm:
(a) non-blended and (b) blended T-junctions.
125
section of the blended model, as shown in Figure 6.30(b). Further downstream of the junction (y
= 2.5 mm), the secondary effect is weakened with the kinetic energy of the secondary flow
(Figure 6.31(a) and (b)). As expected, the flow energy further downstream of the branch is
getting dominated by the mainstream flow of the branch. The flow structures of both models
appear to be very similar.
6.5.3 Hemodynamic Parameters
In order to understand the transient effect from the biological aspect on the non-blended
and blended models, the time-average wall shear stress TAWSS and the OSI are calculated for
both of the models presented above. Figures 6.32(a) and (b) show the distribution of the TAWSS
of the non-blended and blended T-junctions for the D1 = 30 mm, D2 = 3 mm at  = 23.13, Repeak
= 1819. In order to provide an overall visualization on the distribution of the indicators near the
junction area of the branch, color plots are applied in Figures 6.32(a) and 6.33(a). As shown in
Figure 6.32 (a), the overall TAWSS at the junction is shown to have a significantly large area
covered by high TAWSS near the entrance of the junction (i.e., proximal site of the branch). This
result is expected, as shown in the previous section under steady laminar flow, where the wall
area exposed to flow separation zone experiences a great reduction of WSS. Under transient flow
conditions, the streamline plots shown in Figure 6.26 also demonstrate that the separation zone
of the non-blended T-junction is reduced in the blended T-junction when the rB correlation
developed in section 6.2.1 is applied. The boundary layer near the entrance of the junction under
this condition has less separation than that of the non-blended model.
Figure 6.32(b) shows the comparison of the distribution of TAWSS at different line
segments along the y-axis of both models. From the three line segments, it can be clearly seen
that about 10% of the branch of the blended T-junction has significant high TAWSS compared
126
with the non-blended model. The local maximum TAWSS at the proximal vicinity of the three
line segments (i.e., Line 1, Line 2, and Line 3) are found to be approximately 2.1, 2.0, and 1.6
times higher than that of the non-blended model.
Figure 6.33(a) and (b) show the distribution of OSI of the branch of both models. Marked
OSI is shown can be seen at the inner wall of the non-blended model with approximately 12% of
the total length of the branch exposed to high OSI of 0.2. The higher OSI of the non-blended
model is due to the significant separation zone as shown in Figure 6.26. Figure 6.33(b) shows
that significantly high OSI is present at Line 3 (i.e., outer wall) of the blended model with OSI ≈
0.5. This is due to the formation of a small oscillating stagnation zone that is initiated during the
early systolic phase of the cardiac pulse, as shown in the streamline plots in Figures 6.26 and
6.27. It should be noted that in order to take into account the OSI on the blended section of the
branch, the length of the blended model used to measure the OSI here includes part of the host
channel area. Wall shear stress in this region is close to a pure oscillation condition. Long
exposure of the oscillating stagnation zone causes the net shear stress ≈ 0 at the outer wall, hence
the high OSI. Although the Line 3 segment of the blended model shows significantly high OSI,
the area of exposure is relatively small, ~2% of the total length of the branch, compared to the
non-blended model in which about 12% of length at Line 1 is exposed to an OSI of 0.2.
127
TAWSS [Pa]
2.0
Non-Blended
T-Junction
1.5
1.0
0.5
Blended
T-Junction
0.0
y/YB
Line 2
y/YB
TAWSS [Pa]
Line 1
TAWSS [Pa]
TAWSS [Pa]
(a)
Line 3
y/YB
(b)
Figure 6.32. Comparison of TAWSS distribution: (a) side view plots and (b) axial line segments
for non-blended model (dashed line) and blended model (solid line) (Repeak = 1819, = 23.13,
D1 = 30 mm, D2 = 3 mm).
128
OSI
0.5
Non-Blended
T-Junction
0.4
0.3
0.1
Blended
T-Junction
0.0
(a)
y/YB
Line 3
OSI
Line 2
OSI
OSI
Line 1
y/YB
y/YB
(b)
Figure 6.33. Comparison of OSI distribution: (a) side view plots and (b) axial line segments for
non-blended model (dashed line) and blended model (solid line) (Repeak = 1819, = 23.13, D1 =
30 mm, D2 = 3 mm)..
129
Figure 6.34 shows the distribution of TAWSS and OSI from the top view for two
different models. A close-up look at the junciton of the branch (right figures) shows that the
blended model is surrounded by a higher TAWSS compared to the non-blended model. A similar
observation (of low OSI) is also shown on the OSI distribution plots, Figure 6.34 (b). The area
where OSI ≈ 0 of the blended model is found to be slightly larger than that of the non-blended
model, especially at the outer wall of the branch (right side of the circle).
It is perhap worthwhile to point out here that the simulations above are based on the
actual physiological cardiac pulse with geometry close to the physical geometry of ACB model.
From the aspect of biological response of vascular system to WSS, the findings on the marked
high TAWSS and low OSI at the junction of the branch of the blended model suggests that,
under identical transient flow conditions, blended anastomosis at the proximal site of the bypass
graft based on the rB correlation may help to reduce the risk of vascular disease better than the
non-blended one, provided that the ratio between the aortic punch hole and graft diameter (2rB
/D2+ 1) corresponds to the (𝑚̇2 /𝑚̇1 )crit,min.
130
TAWSS [Pa]
(a)
OSI
(b)
Figure 6.34. Comparison of non-blended and blended T-junction with rB = 1 mm: (a) TAWSS
and (b) OSI.
131
6.6
Non-Blended ACB vs. Blended Helical ACB Models (Unsteady-State Flow)
To better understand the correlation developed in section 6.2 on the ACB model, the non-
blended and blended T-junction models presented in the previous section were further
investigated by imposing a radius of curvature on both non-blended and blended (ACB) models
plus helical pitch geometry on the blended version. The effect of helical pitch tube with different
diameters and radiuses of curvature under the steady flow condition were covered previously in
section 6.4. In this section, two models with selected helical pitch and radius of curvature are
considered. The first model, called the non-blended ACB model, is constructed with a straight
tube as the host channel and a semi-circle planar curved tube with rC = 2.5 cm. The second
model, the blended ACB model, is composed of a blended helical graft with rB = 1 mm, HP = 3
cm, and rC = 2.5 cm. Again, the blend radius of the second model is determined based on the rB
correlation for the critical minimum mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit,min criterion developed in
section 6.2.1 for minimum flow separation under the steady-state flow condition. The geometry
and flow conditions for these two models were summarized previously in Table 5.4. The inlet
and outlet boundary conditions of both models are based on the same transient waveforms
applied in the previous section. The corresponding Womersley number and Reynolds number
are = 23.13, Repeak = 1819, respectively.
Figures 6.35(a) and (b) show the schematic views and grid structure of the two numerical
models. Since the only difference between the two models is the curved geometry of the branch,
the mesh size that was applied in the previous section for the non-blended T-junction is applied
on the mesh construction of both models. The mesh sizes are 5.0e-4 m, 8.0e-5 m, and 2.5e-4 m
for the host, junction, and branch sections, respectively. The total number of elements for the
non-blended ACB model is approximately 1.6 million, and for the blended ACB model is 1.4
132
million. Simulation results are based on the same time-step size as applied in the previous
section, i.e., t = 0.010 s. All results are taken from the third cycle of the pulse, which has been
shown in this study to be sufficient to avoid the transient effect from the initial condition.
Non-Blended ACB
(a)
Blended Helical ACB
(b)
Figure 6.35. Schematic diagrams (left) and numerical grid structures (right) of:
(a) non-blended and (b) blended helical ACB models.
133
6.6.1
Flow Separation Characteristics in ACB Models
Figures 6.36 and 6.37 show the temporal streamline distribution of the two ACB models
1 and 2 under the acceleration and deceleration phases, respectively, of the cardiac pulse. As
expected, the flow characteristics of the non-blended and the blended ACB models are
qualitatively similar to that of the T-junction models discussed in section 6.5. The stagnation
zone that appears near the outer wall of the T-junction models during the acceleration phase can
still be seen here in both ACB models. Stagnation zones that formed near the inner wall of the
branch entrance of the non-blended and blended T-junctions in the previous section are shown at
the inner wall of both models during the deceleration phase. In terms of the streamline flow
pattern on the symmetry plane, the streamline structures of both models are rather similar to that
of the T-junctions.
134
Time (s)
0.00
0.04
0.08
0.12
0.16
0.20
0.24
(a)
(b)
Figure 6.36. Comparison of temporal streamline at the symmetry plane during acceleration phase
with HP = 3 cm and rB = 1 mm: (a) non-blended and (b) blended helical ACB models
(Repeak = 1819, = 23.13, D1 = 30 mm, D2 = 3 mm, rC = 2.5 cm).
135
Time (s)
0.28
0.32
0.36
0.40
0.44
0.48
0.52
(a)
(b)
Figure 6.37. Comparison of temporal streamline at the symmetry plane during deceleration phase
with HP = 3 cm and rB = 1 mm: (a) non-blended and (b) blended helical ACB models
(Repeak = 1819, = 23.13, D1 = 30 mm, D2 = 3 mm, rC = 2.5 cm).
136
Figure 6.38 shows the effect of the mass flow rate ratio 𝑚̇2 /𝑚̇1 on the length of
separation LSP of both ACB models. For the non-blended ACB model (model 1), the effect of
𝑚̇2 /𝑚̇1 on the LSP curve is very similar in terms of the magnitude of LSP, onset of flow
separation, and exposure time to that of the non-blended T-junction (discussed earlier and shown
in Figure 6.28). The difference in the local maximum of the LSP curve LSP,max between the nonblended ACB model (Figure 6.38) and the non-blended T-junction (Figure 6.28) is
approximately 7%.
For model 2 (blended helical ACB model), a marked difference is observed on the
separation characteristic at the junction of the branch. Considerable flow improvement is shown
on the separation length LSP compared to that of the non-blended ACB model as well as the
previous blended T-junction. The most obvious difference would have been the local maximum
of the LSP curve where the LSP,max of the blended helical ACB is about 8.5 times smaller than that
of the non-blended ACB model compared to that of the T-junction models where the difference
of the LSP,max between the non-blended and blended T-junctions is about 2.3 times. In addition,
improvement is also seen on the critical (crit) transient (trn) mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit,trn of
the blended ACB model. The corresponding (𝑚̇2 /𝑚̇1 )crit,trn of both models is 0.0272 for the nonblended ACB model and 0.0047 for the blended model, which means that the (𝑚̇2 /𝑚̇1)crit,trn of
the blended helical ACB model is about 83% smaller than that of the non-blended ACB model.
The (𝑚̇2 /𝑚̇1)crit,trn of the blended ACB model is about 15% smaller than the blended T-junction.
A closer look at the onset of flow separation and exposure time of separation for both models, as
can be seen in Figures 6.39(a) and (b), shows that the onset of separation of the blended ACB
model is delayed, with a time approximately four times greater than that of the non-blended
137
model. The exposure time of separation of the blended ACB model is also found to be
approximately four times smaller than the non-blended ACB model.
3.5
0.12
3.0
0.08
,max
2.0
0.04
1.5
0.00
/
Lsp [mm]
[mm]
2.5
1.0
-0.04
0.5
0.0
,max
0
0.2
0.4
tt [s]
[s]
0.6
0.8
-0.08
Figure 6.38. Comparison of length of separation for non-blended and blended ACB models
(Repeak = 1819,  = 23.13, D1 = 30 mm, D2 = 3 mm, rB = 1 mm).
LSP [mm]
LSP [mm]
(𝑚̇2 /𝑚̇1 )crit,trn = 0.0272
(𝑚̇2 /𝑚̇1 )crit,trn = 0.0047
t [s]
t [s]
(a)
(b)
Figure 6.39. Corresponding mass flow rate ratio at onset of flow separation with rB = 1 mm and
HP = 3.0 cm for ACB models: (a) non-blended and (b) blended (Repeak = 1819, = 23.13,
D1 = 30 mm, D2 = 3 mm, rC = 2.5 cm).
138
The pronounced difference in the separation characteristic shown by the blended ACB
model can be attributed to the additional swirling effect induced by the helical effect of the
branch geometry. As shown in sections 6.2 and 6.3, the blended T-junction with rB
corresponding to the (𝑚̇2 /𝑚̇1 )crit,min significantly reduces both the strength and exposure length
of the adverse pressure gradient in the boundary layer at the inner wall of the junction, and thus
reduces the potential energy of the fluid and the separation zone. This separation zone is further
weakened by the circumferential velocity component imparted from the swirling flow induced by
the helical flow past the branch. This effect is illustrated with color velocity plots in Figure 6.40
which shows the temporal three-dimensional streamline of the non-blended and blended ACB
model. At t = 0.20 s which corresponds to the peak of the systolic phase, the swirling effect
formed at the junction of the blended branch can be clearly seen. Figure 6.40 also clearly shows
an indication of flow reversal near the junction of the non-blended model.
The effect of the curved branch of the non-blended ACB model on the flow separation
zone of < 7% and the critical transient mass flow rate ratio (𝑚̇2 /𝑚̇1 )crit,trn of < 2% is found to be
small compared to that of the non-blended T-junction model. This implies that the blended
helical branch applied on the ACB model plays a crucial role in reducing the LSP and (𝑚̇2 /
𝑚̇1 )crit,trn.
139
Time (s)
0.00
0.20
0.40
0.60
Velocity
Figure 6.40. Three-dimensional temporal streamline distribution plots with rB = 1 mm
and HP = 3.0 cm for ACB models: (a) non-blended and (b) blended (Repeak = 1819,
 = 23.13, D1 = 30 mm, D2 = 3 mm, rC = 2.5 cm).
140
6.6.2
Secondary Flow Characteristics
The cross-sectional planar flow near the junction of both models under a transient effect
is evaluated in this section. The secondary velocity of the non-blended and blended ACB models
measured at the cross-sectional plane located at y = 1.0 mm and y = 2.5 mm from the junction of
the branch is shown in Figures 6.41 and 6.42, respectively. Throughout the cycle for the nonblended model, at y = 1.0 mm, the secondary flow is generally composed of a pair of
symmetrical cells, and a relatively large area of low magnitude velocity is concentrated at the
circumferential region of the inner and outer walls of the cross section. This characteristic of
secondary flow is similar to that explained in section 6.2.3 for the non-blended T-junction where
the secondary flow is predominated by a strong adjustment of radial pressure due to the
junction’s sharp corner. Separation at the outer wall caused by the pressure force effect and the
formation of the symmetrical cells has significantly reduced the velocity at the outer and inner
walls. The flow structure of the blended model is shown to be highly complex, and the overall
kinetic energy of the flow structure is shown to be high throughout the cycle. Instead of having
two low-speed flow regions as can be seen in the non-blended model, there is only one area with
low-speed secondary flow, which is concentrated in a small corner of the inner wall of the cross
section, as shown in Figure 6.41(b). Although the structure of the flow is complex, the small
region of low-speed velocity can be understood through the centrifugal force caused by the
helical and curve effect of the branch geometry. The major cause that leads to the low-speed
zone at the inner wall can be attributed to the centrifugal force due to the radius of the branch
curvature. However, due to the helical pitch effect, an additional centrifugal force is added into
the flow path of the branch, which switches the low-speed zone of the flow from the center of the
inner wall.
141
y = 1.0 mm
(a)
Velocity
(b)
Time [s]
0.00
Inner Wall
Outer Wall
0.11
0.23
0.34
0.46
0.57
0.69
Figure 6.41. Comparison of temporal secondary flow at y = 1.0 mm for ACB models:
(a) non-blended and (b) blended helical.
142
y = 2.5 mm
(a)
Velocity
(b)
Time [s]
0.00
Inner Wall
Outer Wall
0.11
0.23
0.34
0.46
0.57
0.69
Figure 6.42. Comparison of temporal secondary flow at y = 2.5 mm for ACB models:
(a) non-blended and (b) blended helical.
143
Further downstream of the branch at y = 2.5 mm, both models exhibit a minor change in
the overall distribution of the flow structure. For the non-blended model, the symmetrical
vertical cells still remain throughout the cycle. However, the strength of the low-speed zone that
surrounds the circumferential area is reduced. The flow structure of the blended model is quite
similar compared to the flow measured at y = 1.0 mm with a significant elevation of the bulk
velocity. However, the low-speed zone at the inner wall of the branch at y = 1.0 mm is shown to
have disappeared during the deceleration phase, and the area is occupied by a flow with
significant high velocity. The magnitude of the secondary velocity in the blended model is
approximately double the non-blended model.
From a biological aspect, one of the features of secondary flow in this section worth
noting is that the low-speed zone near the inner and outer walls of the non-blended model is
reduced considerably in the blended model, based on the rB correlation defined in section 6.2.1.
As mentioned in the work of Caro 1966, the region with secondary flow alone without the
presence of flow separation may inhibit the growth of the vascular wall. Since the previous
section has shown that the separation flow is relatively small in the blended ACB model, it is
believed that the profound increase of secondary flow at the junction of the blended ACB model
may benefit the vascular wall function by imparting additional circumferential shear stress and
inhibiting the growth of the arterial wall caused by low shear flow.
6.3.2
Hemodynamics Parameters
Using the same comparison method as in the previous section, the TAWSS and OSI of
the ACB models under the cardiac transient flow condition are evaluated in this section. Figures
6.43(a) and (b) show the distribution of TAWSS on the branch of both models. The color plots
are applied in Figures 6.43(a) and 6.44(a) to provide an overall visualization of the distribution
144
of TAWSS and OSI near the junction area of the branch. As can be seen in Figure 6.43(a), the
most obvious difference of both models is in the significant high TAWSS that is distributed
around side wall 1 of the blended branch. The difference of the TAWSS distribution around the
whole branch can be better understood from the TAWSS distribution plots measured from
different axial line segments of the branch, as shown in Figure 6.43(b). Results show that the
overall distribution of TAWSS of the blended junction is generally higher than that of the nonblended model, particularly at the region close to the entrance of the branch. The local maximum
of the TAWSS curves of the blended model at line segments 1, 2, 3, and 4 is approximately 14%,
16%, 8%, and 16% higher than that of the non-blended model.
TAWSS
[Pa]
(Pa)
Side Wall 1
(a)
(b)
Side Wall 2
Side Wall 1
Side Wall 2
Figure 6.43(a). Comparison of TAWSS distribution at junction of ACB models:
(a) non-blended and (b) blended.
145
y /YB
y /YB
Line 3
y /YB
TAWSS [Pa]
Line 2
TAWSS [Pa]
TAWSS [Pa]
TAWSS [Pa]
Line 1
Line 4
y /YB
Figure 6.43(b). Comparison of TAWSS distribution at branch of non-blended (solid line) and
blended (dashed line) ACB models at different line segments.
The overall elevation of TAWSS in the blended model can be explained by the swirling
effect of the axial velocity discussed in section 6.6.1, which significantly reduces the temporal
recirculation zone formed at the junction during the cycle. The marked higher TAWSS at side
wall 1 of the blended model can be attributed to the helical pitch effect, which introduces a
centrifugal force that skews the direction of the axial flow and changes the structure of the
secondary flow, as discussed in section 6.6.2.
Figures 6.44(a) and (b) show the distribution of OSI at the branch of the ACB models. In
Figure 6.44(a), the most obvious difference is seen at the inner wall (Line 3 of the branch) of the
non-blended branch where a relatively high OSI is observed. Figure 6.44(b) shows that
approximately 12% of the Line 3 branch section is covered with an OSI of 1.4. A relatively large
OSI is also observed at location Line 2 of the blended model with a coverage range of
approximately 1% of the upstream branch section.
146
OSI
(a)
Side Wall 1
(b)
Side Wall 2
Side Wall 1
Side Wall 2
Figure 6.44(a). Comparison of OSI distribution at junction of ACB models:
(a) non-blended and (b) blended.
Line 2
y /YB
Line 3
y /YB
Line 4
OSI
OSI
OSI
OSI
Line 1
y /YB
y /YB
Figure 6.44 (b). Comparison of OSI distribution at branch of non-blended
(solid line) and blended (dashed line) ACB models at different line segments.
Figure 6.45 shows the distribution of the TAWSS and OSI taken from the upper view of
the two models. Quite similar to the T-junction models, a slightly larger area of high TAWSS
147
and low OSI can be observed near the inner wall and the outer wall of the blended model’s
junction. A close-up view of the parameter distribution is shown on the right-hand side of the
figure.
TAWSS [Pa]
OSI
(a)
(b)
Figure 6.45. Comparison of TAWSS and OSI at junction of host channel for ACB models:
(a) non-blended and (b) blended.
148
The findings from this section have provided some implications for flow field control in
ACB surgery. A comparison of the hemodynamic parameters for the non-blended and blended
ACB models indicates that a better hemodynamic environment at the proximal site of the graft
may be obtained by controlling the blended junction geometry and helical pitch effect, as shown
in this section. Improvement of the flow field through the rB correlation defined in this study is
shown to delay the onset of flow separation, and reduces the separation length and exposure
time. The helical graft with a mild radius of curvature rC (short graft) may further reduce the
separation zone at the proximal region of the graft and increase the overall distribution of
TAWSS of the graft body.
149
CHAPTER 7
CONCLUSIONS
In this dissertation, the flow field of Newtonian blood in a branching flow with one inlet
and two outlets are investigated numerically. To explore the effect of branch geometry on flow
separation length, two major series of T-junction models—non-blended and blended—as well as
curved-tube series under steady-state flow conditions are investigated. The effect of blend radius,
radius of curvature, and helical pitch of the branch are then evaluated separately under transientstate flow conditions.
Aortocoronary bypass (ACB) surgery is a treatment performed to relieve coronary artery
disease in which the occluded (i.e., blocked) artery is bypassed using a graft that connects the
aorta and the coronary arteries to create a bypassed access channel for blood flow.
Approximately 500,000 bypass surgeries are performed every year in the United States. Clinical
and experimental data have highlighted an unsolved latent complication issue during the early
phase of ACB post-surgery which oftentimes leads to a graft occlusion or failure within a year.
About 15% to 20% of these surgeries have been reported to suffer from early-phase failure. The
location of failure is oftentimes found at the proximal region of the graft. Results obtained from
in vivo (i.e., internal animal experiments) and in vitro (external) studies suggest that vascular
wall function is greatly dependent on the hemodynamics of the vascular system, which in turn
depends on fluid dynamics and the geometry of the physical domain. This then raises the
question of whether geometry also plays a role in patency for the proximal region. Since only a
few studies have considered early-phase failure in the proximal region, this motivates the present
study.
150
Blood flow in a coronary artery bypass often involves very complex dynamic behavior.
Current imaging devices and measurement technologies have their limitations when it comes to
high-resolution results of time-dependent flow features. Thus, it is difficult to obtain high-fidelity
experimental results. Numerical simulation is employed in the present study to investigate the
flow field environment focused on the proximal site of the bypass graft on aortocoronary bypass
(ACB) model. A series of different branching models with laminar inflow are considered to
systematically investigate the geometrical effect under different flow conditions. Some of the
geometries considered can be categorized as follows: non-blended and blended T-junction,
helical pitch tube, ACB model (host to branch junction with a radius of curvature) with and
without helical pitch. The goal was to determine the critical criterion that governs the flow field
in ACB model, particularly at the proximal site of the graft so that early phase failure in ACB
surgery may be reduced.
The first model series (i.e. non-blended T-junction) were initially tested to investigate the
relation between flow rate ratio (𝑚̇2 /𝑚̇1 ) and flow separation length LSP at the inner wall of Tjunction model with different branch diameter D2 (D2/D1 < 1) under steady state flow condition.
The critical flow rate ratio (𝑚̇2 /𝑚̇1 )crit which minimized the flow separation length as a function
of branch diameter was first determined for the non-blended T-junction. Then, another
simulation series was performed for the second model series (i.e., blended T-junction) with
different blend radiuses rB and D2 to investigate the relationship between rB, D2, and (𝑚̇2 /𝑚̇1 ).
Different inflow regimes with Re = 227, 454, 909, and 1817 were then employed to investigate
the Re effect on the blended T-junction. For comparison purposes, all calculations in this work
were based on the Newtonian blood with viscosity of 0.0035 Pa·s. However, in order to obtain a
general sense of the difference between Newtonian and non-Newtonian viscous models on the
151
prediction of branching flow behavior, a non-Newtonian model (Carreau Yasuda model) was
applied on one of the non-blended T-junction models and the results were compared against the
results from the Newtonian model. The third model series covered the curved tube effects due to
helical pitch Hp and radius of curvature rC of the branch tube. The effect was evaluated based on
the distribution of WSS induced on the branch. The critical geometric relation and parameters
(i.e. rB – D2 relation, Hp, and rC) obtained from the results were integrated for the evaluation of
the branching flow field in ACB model under non-steady (pulsatile) state flow condition and the
selected hemodynamic parameters (TAWSS and OSI). To better differentiate the effects of these
parameter, the test started from the non-blended and blended T-junction models. Finally, the
study was extended to the fourth and fifth models (i.e. non-blended ACB and blended ACB
models) to investigate the effects of rB – D2 relation, helical pitch HP, and radius of curvature rC
geometries on the non-steady flow field of ACB surgery.
The results presented in this dissertation are based on the simulations, with numerical
methods validated based on experimental and analytical results adopted from the literature. In
terms of general flow behavior, separated flow involving a recirculation zone forms in the inner
wall of the branch. In the first part of the study involving non-blended T-junction models,
(𝑚̇2 /𝑚̇1 ) was found to be the critical determinant for the LSP. For constant inflow condition, the
LSP curve as a function of (𝑚̇2 /𝑚̇1 ) is parabolic-shaped with the minimum value (LSP,min)
requiring increased (𝑚̇2 /𝑚̇1 ) for increasing values of D2. On the blended T-junction, the LSP was
found to decrease with an increase in 𝑚̇2 /𝑚̇1 . This length eventually diminished (to zero) as a
certain mass flow rate ratio was reached, referred to as the critical mass flow rate – (𝑚̇2 /𝑚̇1 )crit
in this dissertation. Given the test range of rB covered in this study, the (𝑚̇2 /𝑚̇1 )crit is shown to
be nonlinearly proportional to rB in a parabolic-like form. With this relation, a local minimum of
152
the (𝑚̇2 /𝑚̇1 )crit – rB curve can be determined, which is known as the critical minimum mass flow
rate – (𝑚̇2 /𝑚̇1 )crit,min of the blended T-junction. This is a very important criterion for blended Tjunction as it determines the scale of separation in the branch of T-junction under unsteady or
pulsatile flow conditions. Through the parabolic curves obtained from the (𝑚̇2 /𝑚̇1 )crit and rB
relation at different D2, it is evident that the blended branch is subject to flow separation at
varying conditions depending upon the values for D2 and rB. Through tests on different Re, the
(𝑚̇2 /𝑚̇1 )crit,min was found vary with Re. However, the parameter rB that corresponds to (𝑚̇2 /
𝑚̇1 )crit,min was found to be independent of Re. Therefore, given Re and D2/D1 as constants, the
results affirm that there must be a local minimum corresponds to the (𝑚̇2 /𝑚̇1 )crit versus rB curve
in which the rB that corresponds to the (𝑚̇2 /𝑚̇1 )crit,min is independent to Re. For given rB and D2,
a general relationship for (𝑚̇2 /𝑚̇1 )crit,min which results in non-separation in blended T-junction
was then developed. This relationship can be expressed as rB = 0.146D21.74, which is independent
to flow rate ratio. This technique was then applied to non-steady flow condition in order to
determine the (𝑚̇2 /𝑚̇1 )crit,min, which minimized or eliminated LSP in non-blended and blended Tjunctions. Results show that flow separation in the blended model occurs at a (𝑚̇2 /𝑚̇1 ) about
80% lower than that of the non-blended one. Both the LSP and the cyclical exposure time of
separation zone in the blended model was shown to be about two times smaller than that of the
non-blended model under the same flow condition. The structure of the secondary flow for both
models was complex; however a pair of vortical cells was observed during the cycle in both
models. Under steady flow condition, the vortical cells near the inner-wall region of the nonblended model were separated as they moved closer to the inner wall whereas this behavior was
absent in the blended model for mass flow rate ratios above (𝑚̇2 /𝑚̇1 )crit,min. The strength of the
secondary motion was shown to be directly proportional to the cardiac pulse. Due to the sharp
153
corner of the non-blended T-junction, the secondary flow was predominated by a strong
adjustment of radial pressure compared to that of the blended T-junction. Coupled with the effect
from the vortical cells, the strength of the secondary motion near the inner wall and the outer
wall of the non-blended T-junction was generally smaller than the blended T-junction. The
insignificant separation nature of the blended T-junction caused the model to be able to gain
higher TAWSS and smaller OSI near the inner wall of the branch.
Results showed that the Newtonian model in general over predicts the velocity at the
outer wall and under predicts the flow at the inner wall of the branch. The flow reversal zone at
the inner wall near the junction predicted by the non-Newtonian model was also shown to be
smaller than that of the Newtonian model. Since the primary focus of this work is not on the
biological aspects or replication of actual blood flow in vascular system, but on the geometric
and flow conditions in ACB model, Newtonian blood was considered the most appropriate
viscous model to investigate the root causes of the flow characteristics in this study. From the
analysis of Hp, and rC, the computational results obtained from the helical tube models under
steady state flow condition show that an increase in D2 was shown to have significant impact on
the distribution of WSS compared to that of rC.
The results from the non-blended ACB models show that the effect of rC does not pose a
significant influence on the flow field compared to the non-blended T-junction model. The
blended T-junction with rB corresponding to the (𝑚̇2 /𝑚̇1 )crit,min significantly reduced both the
strength and exposure length of the adverse pressure gradient in the boundary layer at the inner
wall of the junction, and thus weakened the separation zone. This separation zone is further
weakened by the circumferential velocity component imparted from the swirling flow induced by
the helical flow past the branch. The blended helical pitched ACB model had an onset of flow
154
separation delayed about 4 times more than the non-blended ACB model. The exposure time of
flow separation for the blended ACB was only about 1/4th of the non-blended model and the
maximum LSP for the blended ACB was 1/8th the length of the non-blended ACB model.
Given a T-junction model composed of one inflow and two outflows where D2/D1 < 1, it
may be concluded that:
i.
Non-blended T-junction is subjected to flow separation at the inner wall of the
branch unless (𝑚̇2 /𝑚̇1 ) is high and D2 is small.
ii.
By employing a blend radius at the T-junction, there is a critical flow rate ratio
where separation near the inner wall of the junction can be diminished under
steady state flow condition. This parameter is known as (𝑚̇2 /𝑚̇1 )crit.
iii.
Given a blended T-junction, there is a common rB which corresponds to (𝑚̇2 /
𝑚̇1 )crit,min that gives the minimum separation scale under steady state flow
condition or delays the onset of separation under non-steady flow conditions. This
parameter is independent to Re.
Results from this study suggest that proximal size match (i.e., proximal geometry
mismatch between graft and aortic punch-hole diameters) and branch geometry (i.e., radius of
curvature and helical pitch) play an important role in the vascular flow environment. If the
geometry of the bypass graft was able to be arranged in such a way that the conditions fall below
the critical criteria demonstrated in this study, flow separation will be reduced along with an
increase in TAWSS and a reduction in OSI. This will result in a much better quiescent
hemodynamic state in the vascular wall of ACB graft in maintaining vascular tone. Although
applying the correlation developed in this work on proximal anastomosis of a graft requires
highly precise tools and extensive skill of the surgeon, this procedure will be more feasible when
155
an external mesh support is used. Therefore, the design of an external mesh support and the
geometry of a non-autologous graft is believed to be an area worthwhile to be considered in
future work.
The correlation developed in this work is considered to be an empirical solution based on
numerical solutions for a specific range of flow conditions and geometry. Future work could be
conducted to extend the range of study covered in this work through experimental data for the
exploration of theoretical and analytical solutions.
156
BIBLIOGRAPHY
157
BIBLIOGRAPHY
Adlam, D., Antoniades, C., Lee, R., DiesVol.ch, J., Shirodaria, C., Taggart, D., Leeson, P., and
Channon, K. M., “OCT Characteristics of Saphenous Vein Graft Atherosclerosis,” Journal of the
American College of Cardiology: Cardiovascular Imaging, July 2011, Vol. 4, No. 7, pp. 807–
809.
Akimoto, S., Mitsumata, M., Sasaguri, T., and Yoshida, Y., “Laminar Shear Stress Inhibits
Vascular Endothelial Cell Proliferation by Inducing Cyclin-Dependent Kinase Inhibitor
p21Sdi1/Cip1/Waf1,” Circulation Research, Feb. 2000, Vol. 86, No. 2, pp. 185–190.
Alam, M., and Sandham, N., “Direct Numerical Simulation of Short Laminar Separation Bubbles
With Turbulent Reattachment,” Journal of Fluid Mechanics, 2000, Vol. 410, pp. 1–28.
Allaire, E., and Clowes, A. W., “Endothelial Cell Injury in Cardiovascular Surgery: The Intimal
Hyperplastic Response,” The Annals of Thoracic Surgery, 1997, Vol. 63, pp. 582–591.
American Heart Association, Dallas, Texas, http://www.americanheart.org/, 2001.
Arima, M., Kanoh, T., Suzuki, T., Kuremoto, K., Tanimoto, K., Oigawa, T., and Matsuda, S.,
“Serial Angiographic Follow-Up beyond 10 Years after Coronary Artery Bypass Grafting,”
Circulation Journal, 2005, Vol. 69, No. 8, pp. 896–902.
Armstrong, A. W., and Golan, D. E., “Pharmacology of Hemostasis and Thrombosis,” in
Principles of Pharmacology. The Pathophysiologic Basis of Drug Therapy, ed. D. E. Golan, A.
H. Tashjian, Jr., E. J. Armstrong, and A. W. Armstrong, 2nd ed., Wolters Kluwer/Lippincott
Williams and Wilkins, Baltimore, MD, 2008, p. 396.
Azzola, J., Humphrey, J. A. C., Iacovides, H., and Launder, B. E., “Developing Turbulent Flow
in A U-Bend of Circular Cross-Section: Measurement and Computation,” Transactions of the
ASME, June 1986, Vol. 108, pp. 214–221.
Bao, X., Lu, C., and Frangos, J. A., “Temporal Gradient in Shear but Not Steady Shear Stress
Induces Pdgf-A and Mcp-1 Expression in Endothelial Cells: Role of NO, NF Kappa B, and egr1,” Arteriosclerosis, Thrombosis, and Vascular Biology, 1999, Vol. 19, pp. 996–1003.
Baskurt, O. K., and Meiselman, H. J., “Blood Rheology and Hemodynamics,” Seminars in
Thrombosis and Hemostasis, Oct. 2003, Vol. 29, No. 5, pp. 435–450.
Bassiouny, H. S., White, S., Glagov, S., Choi, E., Giddens, D. P., and Zarins, C. K.,
“Anastomotic Intimal Hyperplasia: Mechanical Injury or Flow Induced,” Journal of Vascular
Surgery, April 1992, Vol. 15, No. 4, pp. 708–16.
Bastarrika, G., and Schoepf, U. J., “Evolving CT Applications in Ischemic Heart Disease,”
Seminars in Thoracic and Cardiovascular Surgery, 2008, Vol. 20, No. 4, pp. 380–392.
158
Berceli, S. A., Warty, V. S., Sheppeck, R. A., Mandarino, W. A., Tanksale, S. K., and Borovetz,
H. S., “Hemodynamics and Low Density Lipoprotein Metabolism. Rates of Low Density
Lipoprotein Incorporation and Degradation along Medial and Lateral Walls of the Rabbit AortoIliac Bifurcation,” Arteriosclerosis. Sept.-Oct. 1990, Vol. 10, No. 5, pp. 686–694.
Bernard, P. S., Crouch, J. D., Choudhari, M., Bogard, D. G., and Thole, K. A., “Transition and
Turbulence,” Chapter 13 in The Handbook of Fluid Dynamics, ed. R.W. Johnson, CRC Press,
Boca Raton, FL, 1998, pp. 13–60.
Bertolotti, C., and Deplano, V., “Three-dimensional Numerical Simulations of Flow through a
Stenosed Coronary Bypass,” Journal of Biomechanics, 2000, Vol. 33, No. 8, pp. 1011–1022.
Bharadvaj, B. K., Mabon, R. F., and Giddens, D. P., “Steady Flow in a Model of the Human
Carotid Bifurcation. Part I—Flow Visualization,” Journal of Biomechanics, 1982, Vol. 15, No.
5, pp. 349–362.
Bhardwaj, S., Roy, H., Heikura, T., and Ylä-Herttuala, S., “VEGF-A, VEGF-D and VEGFD(DeltaNDeltaC) Induced Intimal Hyperplasia in Carotid Arteries,” European Journal of
Clinical Investigation, 2005, Vol. 35, pp. 669–676.
Bhatt, D. L., Underwood, A., Leighton, S., and Linkinhoker M., “Harvard Medical School
Diagnosis: Heart Failure (Harvard Medical School Special Health Reports),” Harvard Health
Publications, Sept. 2013.
Björkerud, S., and Bondjers, G., “Endothelial Integrity and Viability in the Aorta of the Normal
Rabbit and Rat as Evaluated with Dye Exclusion Tests and Interference Contrast Microscopy,”
Atherosclerosis, May-June 1972, Vol. 15, No. 3, pp. 285–300.
Bluestein, D., and Einav, S., “Transition to Turbulence in Pulsatile Flow through Heart Valves –
A Modified Stability Approach,” Journal of Biomechanics, 1994, Vol. 116, No 4, pp. 447–487.
Bojar R. M., Manual of Perioperative Care in Adult Cardiac Surgery, Wiley-Blackwell, 2011.
Bourassa, M. G., “Fate of Venous Grafts: The Past, the Present and the Future,” Journal of the
American College of Cardiology, 1991, Vol. 17, No. 5, pp. 1081–1083.
Bowie, M. J., and Schaffer, R. M., Understanding Procedural Coding: A Worktext, 3rd ed.,
Delmar Cengage Learning, Clifton Park, NY, 2012, p. 208.
Brooks, A. R, Lilkes, P. I., and Rubanyi, G. M., “ Gene Expression Profiling of Human Aortic
Endothelial Cells Exposed to Disturbed Flow and Steady Laminar Flow,” Physiological
Genomics, 2002, Vol. 9, No. 1, pp. 27–41.
Budoff, M. J., Achenbach, S., and Duerinckx, A., “Clinical Utility of Computed Tomography
and Magnetic Resonance Techniques for Noninvasive Coronary Angiography,” Journal of the
American College of Cardiology, 2003, Vol. 42, pp. 1867–1878.
159
Campeau, L., Crochet, D., Lespérance, J., Bourassa, M. G, and Grondin, C. M. “Postoperative
Changes in Aortocoronary Saphenous Vein Grafts Revisited: Angiographic Studies at Two
Weeks and at One Year in Two Series of Consecutive Patients,” Circulation, Sept. 1975, Vol.
52, No. 3, pp. 369-377.
Canos, D. A., Mintz, G. S., Rock, B. Y., Escolar, E., Gevorkian, N., Pichard, A. D., Satler, L. F.,
Kent, K. M., Neil, J., and Weissman, N. J., “Early Saphenous Vein Graft Failure: A Predictor of
Poor Outcomes after Percutaneous Coronary Intervention,” Journal of The American College of
Cardiology, Jan. 2004, Vol. 43, No. 5s1, pp. A76–A76.
Caro, C. G., “The Dispersion of Indicator Flowing through Simplified Models of the Circulation
and Its Relevance to Velocity Profile in Blood Vessels,” Journal of Physiology, Aug. 1966, Vol.
185, No. 3, pp. 501–519.
Caro, C. G., Fitz-Gerald, J. M., and Schroter, R. C., “Atheroma and Arterial Wall Shear.
Observation, Correlation and Proposal of a Shear Dependent Mass Transfer Mechanism for
Atherogenesis,” Proceedings of the Royal Society B: Biological Sciences, Feb. 1971, Vol. 177,
No. 1046, pp. 109–159.
Caro, C., Cheshire, N., and Watkins, N., “Preliminary Comparative Study of Small Amplitude
Helical and Conventional ePTEE Arteriovenous Shunts in Pigs,” 2005, Journal of The Royal
Society Interface, Vol. 2, No. 3, pp. 261–266.
Casey, P. J, Dattilo, J. B, Dai, G., Albert, J. A., Tsukurov, O. I., Orkin, R. W., Gertler, J. P., and
Abbott, W. M., “The Effect of Combined Arterial Hemodynamics on Saphenous Venous
Endothelial Nitric Oxide Production,” Journal of Vascular Surgery, June 2001, Vol. 33, No. 6,
pp. 1199–1205.
Chandran, K. B., “Flow Dynamics in the Human Aorta: Techniques and Applications,” in
Cardiovascular Techniques: Biomechanical Systems: Techniques and Applications, Vol. II, ed.
Cornelius Leondes, CRC Press LLC, London, 2001, pp. 194–215.
Chandran, K. B., Khalighi, B., and Chen, C. J., “Experimental Study of Physiological Pulsatile
Flow Past Valve Prostheses in a Model of Human Aorta—I. Caged Ball Valves,” Journal of
Biomechanics, 1985a, Vol. 18, No. 10, pp. 763–772.
Chandran, K. B., Khalighi, B., and Chen, C. J., “Experimental Study of Physiological Pulsatile
Flow Past Valve Prostheses in a Model of Human Aorta—II. Tilting Disc Valves and the Effect
of Orientation,” Journal of Biomechanics, 1985b, Vol. 18, No. 10, pp. 773–780.
Chandran, K. B., and Yearwood, T. L, “Experimental Study of Physiological Pulsatile Flow in a
Curved Tube,” Journal of Fluid Mechanics, 1981, Vol. 111, pp. 59–85
Chandran, K. B., Yearwood, T. L, and Wieting, D. W., “An Experimental Study of Pulsatile
Flow in a Curved Tube,” Journal of Biomechanics, 1979, Vol. 12, pp. 793–805.
160
Chappell, D. C., Varner, S. E., Nerem, R. M., Medford, R. M., and Alexander, R. W.,
“Oscillatory Shear Stress Stimulates Adhesion Molecule Expression in Cultured Human
Endothelium,” Circulation Research, 1998, Vol. 82, pp. 532–539.
Chatzizisis, Y. S., Baker, A. B., Sukhova, G. K., Koskinas, K. C., Papafaklis, M. I., Beigel, R.,
Jonas, M., Coskun, A. U., Stone, B. V., Maynard, C., Shi, G. P., Libby, P., Feldman, C. L.,
Edelman, E. R., and Stone, P. H., “Augmented Expression and Activity of Extracellular MatrixDegrading Enzymes in Regions of Low Endothelial Shear Stress Colocalize with Coronary
Atheromata with Thin Fibrous Caps in Pigs,” Circulation, Feb. 2011, Vol. 123, No. 6, pp. 621–
630.
Cheng, C., Tempel, D., van Haperen, R., van der Baan, A., Grosveld, F., Daemen, M. J., Krams,
R., and de Crom, R., “Atherosclerotic Lesion Size and Vulnerability are Determined by Patterns
of Fluid Shear Stress,” Circulation, June 2006, Vol. 113, No. 23, pp. 2744–2753.
Cho, K. R., Kim, J. S., Choi, J. S., and Kim, K. B., “Serial Angiographic Follow-Up of Grafts
One Year and Five Years after Coronary Artery Bypass Surgery,” European Journal CardioThoracic Surgery, 2006, Vol. 29, pp. 511–516.
Christenson, J. T., Simonet, F., and Schmuziger, M., “Sequential Vein Bypass Grafting: Tactics
and Long-Term Results,” Cardiovascular Surgery, 1998, Vol. 6, pp. 389–397.
Clowes, A., “Intimal Hyperplasia and Graft Failure,” Cardiovascular Pathology, 1993, Vol. 2,
No. 3, pp. 179–186.
Coelho, P. M., and Pinho, F. T., “Vortex Shedding in Cylinder Flow of Shear-Thinning Fluids.
III—Pressure Measurements,” Journal of Non-Newtonian Fluid Mechanics, 2004, Vol. 121, pp.
55–68.
Conway, D. E., Williams, M. R., Eskin, S. G., and McIntire, L. V., “Endothelial Cell Responses
to Atheroprone Flow are Driven by Two Separate Flow Components: Low Time-Average Shear
Stress and Fluid Flow Reversal,” American Journal of Physiology—Heart and Circulatory
Physiology, Feb. 2010, Vol. 298, No. 2, pp. H367–H374.
Constantinides, P., “The Role of Arterial Wall Injury in Atherogenesis and Arterial
Thrombogenesis,” Zentralbl Allg Pathol, 1989, Vol. 135, No. 6, pp/ 517–530.
Corson, M. A., James, N. L., Latta, S. E., Nerem, R. M., Berk, B. C., and Harrison, D. G.,
“Phosphorylation of Endothelial Nitric Oxide Synthase in Response to Fluid Shear Stress,”
Circulation Research, 1996, Vol. 79, pp. 984–991.
Cunningham, K. S., and Gotlieb, A. I., “The Role of Shear Stress in the Pathogenesis of
Atherosclerosis,” Lab Investigation, Jan. 2005, Vol. 85, No. 1, pp. 9–23.
161
Dardik, A., Chen, L., Frattini, J., Asada, H., Aziz F., Kudo, F. A., and Sumpio, B. E.,
“Differential Effects of Orbital and Laminar Shear Stress on Endothelial Cells,” Journal of
Vascular Surgery, May 2005, Vol. 41, No. 5, pp. 869–880.
Davies, P. F., “Flow-Mediated Endothelial Mechanotransduction,” Physiological Reviews, July
1995, Vol. 75, No. 3, pp. 519–560.
Davies, P. F., “Hemodynamic Shear Stress and the Endothelium in Cardiovascular
Pathophysiology,” Nature Clinical Practice Cardiovascular Medicine, Jan. 2009, Vol. 6, No. 1,
pp. 16–26.
Davies, P. F., Remuzzi, A., Gordon, E. J., Dewey, C. F., Jr., and Gimbrone, M. A., Jr.,
“Turbulent Fluid Shear Stress Induces Vascular Endothelial Cell Turnover In Vitro,”
Proceedings of the National Academy of Sciences USA, April 1986, Vol. 83, No. 7, pp. 2114–
2117.
Davignon, J., and Ganz, P., “Atherosclerosis: Evolving Vascular Biology and Clinical
Implications,” Circulation, 2004, Vol. 109, pp. III-27–III-32.
Davis, M. E, Cai, H., Drummond, G. R., and Harrison, D. G., “Shear Stress Regulates
Endothelial Nitric Oxide Synthase Expression through C-Src by Divergent Signaling Pathways,”
Circulation Research, Nov. 2001, Vol. 89, No. 11, pp. 1073–1080.
Dean, W. R., “Note on the Motion of Fluid in a Curved Pipe,” Philosophical Magazine and
Journal of Science, 1927, Vol. 4, No. 20, pp. 208–223.
Deanfield, J. E., Halcox J. P., and Rabelink T. J., “Endothelial Function and Dysfunction:
Testing and Clinical Relevance,” Circulation, March 2007, Vol. 115, No. 10, pp. 1285–1295.
Dekker, R. J., Boon, R. A, Rondaji, M. J., et al., “KLF2 Provokes a Gene Expression Pattern that
Establishes Functional Quiescent Differentiation of the Endothelium,” Blood, 2006, Vol. 107,
No. 11, pp. 4354–4363.
DePaola, N., Gimbrone, M. A., Jr., Davies, P. F., and Dewey, C. F., Jr., “Vascular Endothelium
Responds to Fluid Shear Stress Gradients,” Arteriosclerosis and Thrombosis, Nov. 1992, Vol.
12, No. 11, pp. 1254–1257.
DeRubertis, B. G., Alktaifi, A., Finn, J. P., Farley, S., Saleh, R. S., Moore, W. S., and Lawrence,
P. F., “RR7. Anatomic Assessment of the Aortic Arch: Relationship between Arch Anatomy and
Age,” Journal of Vascular Surgery, June 2011, Vol. 53, No. 6, Supplement, p. 95S.
Djordjević, V. B., Ćosić. V., Zvezdanović-Čelebić. L., Djordjević. V. V., and Vlahović. P.,
“Endothelial Cells and Vasculitis: Endothelial Cells and Vasculitis,” in Advances in the Etiology,
Pathogenesis and Pathology of Vasculitis, ed. Luis M. Amezcua-Guerra (Hrsg.), InTech 2011.
162
Domanski, M. J., Borkowf, C. B., Campeau, L., et al., “Prognostic Factors for Atherosclerosis
Progression in Saphenous Vein Grafts: The Postcoronary Artery Bypass Graft (Post-Cabg) Trial.
Post-CABG Trial Investigators,” Journal of the American College of Cardiology, 2000, Vol. 36,
pp. 1877–1883.
Dur, O., Coskun, S. T., Coskun, K. O., Frakes, D., Kara, L. B., and Pekkan, K., “ComputerAided Patient-Specific Coronary Artery Graft Design Improvements Using CFD Coupled Shape
Optimizer,” Cardiovascular Engineering and Technology, March 2011, Vol. 2, No. 1, pp. 35–47.
Dusserre, N., L’Heureux, N., Bell, K. S., Stevens, H. Y., Yeh, J., Otte, L. A., Loufrani, L., and
Frangos, J. A., “PECAM-1 Interacts with Nitric Oxide Synthase in Human Endothelial Cells:
Implication for Flow-Induced Nitric Oxide Synthase Activation,” Arteriosclerosis, Thrombosis,
and Vascular Biology, Oct. 2004, Vol. 24, No. 10, pp. 1796–1802.
Echave, V., Koornick, A. R., Haimov, M., and Jacobson, J. H., “Intimal Hyperplasia as a
Complication of the Use of the Polytetrafluoroethylene Graft for Femoral-Popliteal Bypass,”
Surgery, Dec. 1979, Vol. 86, No. 6, pp. 791–798.
Ehringer, W., Yamany, S., Steier, K., Farag, A., Roisen, F., Dozier, A., and Miller, F.,
“Quantitative Image Analysis of F-Actin in Endothelial Cells,” Microcirculation, 1999, Vol. 6,
pp. 291–303.
Eschenbruch, E. M., Pabst, F., Tollenaere, P., Roskamm, H., and Schuziger, M., “The
Significance of Coronary Topography for Operative Technique and Tactics in Multiple
Myocardial Revascularization with Jump-Grafts,” Journal of Thoracic and Cardiovascular
Surgery, 1981, Vol. 29, pp. 206–211.
Feuchtner, G. M., Schachner, T., Bonatti, J., Friedrich, G. J., Soegner, P., Klauser, A., and zur
Nedden, D., “Diagnostic Performance of 64-Slice Computed Tomography in Evaluation of
Coronary Artery Bypass Grafts,” American Journal of Roentgenology, 2007, Vol. 189, pp. 574–
580.
Fitzgibbon, G. M., Kafka, H. P., Leach, A. J., Keon, W. J., Hooper, G. D., and Burton, J. R.,
“Coronary Bypass Graft Fate and Patient Outcome: Angiographic Follow-Up of 5,065 Grafts
Related to Survival and Reoperation in 1,388 Patients During 25 Years,” Journal of the
American College of Cardiology, 1996, Vol. 28, pp. 616–626.
Frattini, J. C., Sessa, W., Sumpio, B., and Dardik, A., “Effects of Turbulent Flow on Cell
Adhesive Molecule Expression in Endothelial Cells,” Journal of the American College of
Surgeons, 2004, Vol. 199, No. 3, pp. 112.
Frauenfelder. T., Boutsianis, E., Husmann, L., Leschka, S., et al., “Flow and Wall Shear Stress in
End-to-Side Anastomosis of Venous Coronary Artery Bypass Grafts,” BioMedical Engineering
Online, Sept 2007, Vol. 6: 35.
163
Frazier, A. A., Qureshi, F., Read, K. M., Gilkeson, R. C., Poston, R. S., and White, C. S.,
“Coronary Artery Bypass Grafts: Assessment with Multidetector Ct in the Early and Late
Postoperative Settings,” Radiographics, July-Aug. 2005, Vol. 25, No. 4, pp. 881–896.
Frazin, L. J., Lanza, G., Vonesh, M., Khasho, F., Spitzzeri, C., McGee, S., Mehlman, D.,
Chandran, K.B., Talano, J., and McPherson, D., “Functional Chiral Asymmetry in Descending
Thoracic Aorta,” Circulation, Dec. 1990, Vol. 82, No. 6, pp. 1985–1994.
Fry, D. L., Hemodynamic Forces in Atherogenesis in Cerebrovascular Diseases, ed. P. Steinberg
Raven Press, New York, 1976, 77–95.
Gammack, D., and Hydon, P. E., “Flow in Pipes with Non-Uniform Curvature and Torsion,”
Journal of Fluid Mechanics, 2001, Vol. 433, pp. 357–382.
Gardner T. J., “Searching for the Second-Best Coronary Artery Bypass Graft: Is It the Radial
Artery?” Circulation, 2007, Vol. 115, pp. 678–680.
Garg, U. C., and Hassid, A., “Nitric Oxide-Generating Vasodilators and 8-Bromo-Cyclic
Guanosine Monophosphate Inhibit Mitogenesis and Proliferation of Cultured Rat Vascular
Smooth Muscle Cells,” Journal of Clinical Investigation, May 1989, Vol. 83, No. 5, pp. 1774–
1777.
Garrett H. E., Dennis E. W., and DeBakey M. E., “Aortocoronary Bypass with Saphenous Vein
Graft. Seven-Year Follow-Up,” Journal of the American Medical Association, 1973, Vol. 223,
pp. 792–794.
Geary, R. L., Kohler, T. R., Vergel, S., Kirkman, T. R., and Clowes, A. W., “Time Course of
Flow-Induced Smooth Muscle Cell Proliferation and Intimal Thickening in Endothelialized
Baboon Vascular Grafts,” Circulation Research, Jan. 1994, Vol. 74, No. 1, pp. 14–23.
Genoni, M., Odavic, D., Loblein, H., and Dzemali, O., “Use Of the eSVS Mesh: External Vein
Support Does Not Negatively Impact Early Graft Patency,” Innovations (Phila), May-June 2013,
Vol. 8, No. 3, pp. 211–214.
Ghersin, E., Keidar, Z., Rispler, S., Litmanovich, D., Bar-Shalom, R., Roguin, A., Soil, A., and
Israel, O., Engel A., “Images in Cardiovascular Medicine. Hybrid Cardiac Single Photon
Emission Computed Tomography/Computed Tomography Imaging with Myocardial Perfusion
Single Photon Emission Computed Tomography and Multidetector Computed Tomography
Coronary Angiography for the Assessment of Unstable Angina Pectoris after Coronary Artery
Bypass Grafting,” Circulation, Aug. 2006, Vol. 114, No. 6, pp. e237–e239
Ghista, D. N., and Kabinejadian, F., “Coronary Artery Bypass Grafting Hemodynamics and
Anastomosis Design: A Biomedical Engineering Review,” Dec. 2013, BioMedical Engineering
Online, Vol. 12, No. 129.
164
Gijsen, F. J., “Modeling of Wall Shear Stress in Large Arteries,” Ph.D. thesis, Eindhoven
University of Technology, 1998.
Gijsen, F. J., Allanic, E., van de Vosse, F. N., and Janssen, J. D., “The Influence of the NonNewtonian Properties of Blood on the Flow in Large Arteries: Unsteady Flow in a 90 Degrees
Curved Tube,” Journal of Biomechanics, July 1999, Vol. 32, No. 7, pp. 705–713.
Girard, P. R. and Nerem, R. M., “Endothelial Cell Signaling and Cytoskeletal Changes in
Response to Shear Stress,” Frontiers of Medical and Biological Engineering, 1993, Vol. 5, No.
1, pp. 31–36.
Glagov, S., Zarins, C., Giddens, D. P., and Ku, D. N., “Hemodynamics and Atherosclerosis.
Insights and Perspectives Gained from Studies of Human Arteries,” Archives of Pathology &
Laboratory Medicine, Oct 1988, Vol. 112, No. 10, pp. 1018–1031.
Goldman, S., Zadina, K., Krasnicka, B., Moritz, T., Sethi, G., Copeland, J., Ovitt, T., and
Henderson, W., “Predictors of Graft Patency 3 Years after Coronary Artery Bypass Graft
Surgery,” Journal of the American College of Cardiology, 1997, Vol. 29, pp. 1563–1568.
Gonzales, R. S., and Wick, T. M., “Hemodynamic Modulation of Monocytic Cell Adherence to
Vascular Endothelium,” Annals of Biomedical Engineering, May-June 1996, Vol. 24, No. 3, pp.
382–393.
Granger R. A., Fluid Mechanics, Dover, New York, 1995, pp. 484, 649, 726.
Gurevitch, J., Gaspar, T., Orlov, B., Amar, R., Dvir, D., Peled, N., and Aravot, D. J.,
“Noninvasive Evaluation of Arterial Grafts with Newly Released Multidetector Computed
Tomography,” The Annals of Thoracic Surgery, 2003, Vol. 76, No. 5, pp. 1523–1527
Guzman, R. J., Abe, K, and Zarins, C. K., “Flow-Induced Arterial Enlargement is Inhibited by
Suppression of Nitric Oxide Synthase Activity in Vivo,” Surgery, Aug. 1997, Vol. 122, No. 2,
pp. 273–279
Hanada, T., Hashimoto, M., Nosaka, S., Sasaki, T., Nakayama, K., Masumura, S., Yamauchi,
M., and Tamura, K., “Shear Stress Enhances Prostacyclin Release from Endocardial Endothelial
Cells,” Life Sciences, 2000, Vol. 66, pp. 215–220.
Haruguchi, H., and Teraoka, S., “Intimal Hyperplasia and Hemodynamic Factors in Arterial
Bypass and Arteriovenous Grafts: A Review,” Journal of Artificial Organs, 2003, Vol. 6, pp.
227–235
Helmlinger, G., Berk, B. C., and Nerem, R. M., “Calcium Responses of Endothelial Cell
Monolayers Subjected to Pulsatile and Steady Laminar Flow Differ,” American Journal of
Physiology, Aug. 1995, Vol. 269, No. 2, Part 1, pp. C367–C375.
165
Helmlinger, G., Geiger, R. V., Schreck, S., and Nerem, R. M., “Effects of Pulsatile Flow on
Cultured Vascular Endothelial Cell Morphology,” Journal of Biomechanical Engineering, 1991,
Vol. 113, No. 2, pp. 123–134.
Himburg, H. A., Grzybowski, D. M., Hazel, A. L., LaMack, J. A., Li, X. M., and Friedman, M.
H., “Spatial Comparison between Wall Shear Stress Measures and Porcine Arterial Endothelial
Permeability,” American Journal of Physiology—Heart and Circulatory Physiology, May 2004,
Vol. 286, No. 5, pp. H1916–H1922.
Hoi, Y., Zhou, Y. Q., Zhang, X., Henkelman, R. M., and Steinman, D. A., “Correlation between
Local Hemodynamics and Lesion Distribution in a Novel Aortic Regurgitation Murine Model of
Atherosclerosis,” Annals of Biomedical Engineering, May 2011, Vol. 39, No. 5, pp. 1414–1422.
Hsiai, T. K., Cho S. K., Wong, P. K., Ing M., Salazar, A., Sevanian, A., Navab, M., Demer L. L.,
and Ho, C. M., “Monocyte Recruitment to Endothelial Cells in Response to Oscillatory Shear
Stress,” Federation of American Societies for Experimental Biology, Sep 2003, Vol. 17, No. 12,
pp. 1648–1657.
Huang, R. F., Ho, C. Y., and Chen, J. K., “Pulsatile Flow Patterns and Wall Shear Stresses in
Arch of a Turn-Around Tube with/without Stenosis,” Journal of Mechanics, March 2011, Vol.
27, No. 1, pp. 79–94.
Huijbregts, H. J., Blankestijn, P. J., Caro, C. G., Cheshire, N. J., Hoedt, M. T., Tutein
Nolthenius, R. P., and Moll, F. L., “A Helical PTFE Arteriovenous Access Graft to Swirl Flow
Across the Distal Anastomosis: Results of a Preliminary Clinical Study,” European Journal of
Vascular and Endovascular Surgery, April 2007, Vol. 33, No. 4, pp. 472–475.
Hunt, B. J., Poston, L., Schachter, M., and Halliday A., eds., An Introduction to Vascular
Biology: From Basic Science to Clinical Practice, Cambridge University Press, New York,
2002, p. 55.
Hwang, J., Saha, A., Boo, Y. C., Sorescu, G. P., McNally, J. S., Holland, S. M., Dikalov, S.,
Giddens, D. P., Griendling, K. K., Harrison, D. G., and Jo, H., “Oscillatory Shear Stress
Stimulates Endothelial Production of O2—from p47phox-Dependent NAD(P)H Oxidases,
Leading to Monocyte Adhesion,” The Journal of Biological Chemistry, 2003, Vol. 278, pp.
47291–47298.
Izutani, H., Yoshitatsu, M., Kawamoto, J., and Katayama, K., “A Case of Ostial Stenosis with
the Pas-Port Proximal Anastomosis System in Off-Pump Coronary Artery Bypass Grafting,”
Interactive Cardiovascular and Thoracic Surgery, Aug. 2005, Vol. 4, No. 4, pp. 341–343.
Jabara, R., Chronos, N., Klein, L., Eisenberg, S., Allen, R., Bradford, S., and Frohwein, S.,
“Comparison of Multidetector 64-Slice Computed Tomographic Angiography to Coronary
Angiography to Assess the Patency of Coronary Artery Bypass Grafts,” American Journal of
Cardiology, 2007, Vol. 99, No. 11, pp. 1529–1534.
166
Jadlowiec, C., and Dardik, A., “Shear Stress and Endothelial Cell Retention in Critical Lower
Limb Ischemia,” in Inflammatory Response in Cardiovascular Surgery, ed. E. A. Gabriel and S.
A. Gabriel, 2013, Springer-Verlag, London, 2013, pp 107–116
Jarrahi, M., Castelain, C., and Peerhossaini, H., “Secondary Flow Patterns and Mixing in
Laminar Pulsating Flow through a Curved Pipe,” Experiments in Fluids, 2010, Vol. 50, pp.
1539–1558
Jatene, F. J., Dallan, L. A., Hueb, A. C., Lisboa, L. A., Monteiro, R., Takeda, F. R., Marin, J. F.,
and Oliveira, S. A., “Initial Results on the Use of Mechanical Devices for Proximal Saphenous
Vein Graft Anastomoses: A Clinical and Angiographic Evaluation,” Revista Portuguesa de
Cirurgia Cardio-Torácica e Vascular, Oct.-Dec. 2003, Vol. 10, No. 4, pp. 171–176.
Johnson, R. W., The Handbook of Fluid Dynamics, CRC Press, Boca Raton, FL, 1998, pp. 18–
23.
Jones, L. E., Sandberg, R. D., and Sandham, N. D., “Direct Numerical Simulations of Forced and
Unforced Separation Bubbles on an Airfoil at Incidence,” Journal of Fluid Mechanics, 2008,
Vol. 602, pp.175–207.
Kalan, J. M. and Roberts, W.C., “Morphologic Findings in Saphenous Veins Used as Coronary
Arterial Bypass Conduits for Longer than 1 Year: Necropsy Analysis of 53 Patients, 123
Saphenous Veins, and 1865 Five-Millimeter Segments of Veins,” American Heart Journal,
1990, Vol. 119, pp. 1164–1184.
Karino, T., “Microscopic Structure of Disturbed Flows in the Arterial and Venous Systems, and
Its Implication in the Localization of Vascular Diseases,” International Angiology, 1986, Vol. 5,
No. 4, pp. 297–313.
Kieser, T. M., FitzGibbon, G. M., and Keon, W. J., “Sequential Coronary Bypass Grafts: LongTerm Follow-Up,” Journal of Thoracic and Cardiovascular Surgery, 1986, Vol. 91, pp. 767–
772.
Kilner, P. J., Yang, G. Z., Mohiaddin, R. H., Firmin, D. N., and Longmore, D. B., “Helical and
Retrograde Secondary Flow Patterns in the Aortic Arch Studied by Three-Directional Magnetic
Resonance Velocity Mapping,” Circulation, Nov. 1993, Vol. 88, No. 5, Part 1, pp. 2235–2247.
Kim, D. W., Gotlieb, A. I., and Langille, B.L., “In Vivo Modulation of Endothelial F-Actin
Microfilaments by Experimental Alterations in Shear Stress,” Arteriosclerosis, 1989, Vol. 9, p.
439.
Kips Bay Medical, Inc., “The Future of Cardiac Surgery—eSVS Mesh,” URL:
http://www.kipsbaymedical.com/the_future_of_cardiac_surgery (cited Jan. 1, 2013).
Kitamura, S., Kawachi, K., Kawata, T., Kobayashi, S., Mizuguchi, K., Kameda, Y., Nishioka,
H., Hamada, Y., and Yoshida, Y., “Ten-Year Survival and Cardiac Event-Free Rates in Japanese
167
Patients with the Left Anterior Descending Artery Revascularized with Internal Thoracic Artery
or Saphenous Vein Graft: A Comparative Study,” Nippon Geka Gakkai Zasshi, 1996, Vol. 97,
No. 3, pp. 202–209
Kohler, T. R., Kirkman, T. R., Kraiss, L. W., Zierler, B. K., and Clowes, A. W., “Increased
Blood Flow Inhibits Neointimal Hyperplasia in Endothelialized Vascular Grafts,” Circulation
Research, 1991, Vol. 69, pp. 1557–1565.
Kojima, M., Irie, K., Ikeda, S., Fukuda, T., Arai, F., Hirose, Y., and Negoro, M., “The
Hemodynamic Study for Growth Factor Evaluation of Rupture Cerebral Aneurysm Followed Up
for Five Years,” Journal of Biomedical Science and Engineering, 2012, Vol. 5, No. 12, pp. 884–
891
Kraiss, L. W., Geary, R. L., Mattsson, E. J., Vergel, S., Au, Y. P., and Clowes, A. W., “Acute
Reductions in Blood Flow and Shear Stress Induce Platelet-Derived Growth Factor-A
Expression in Baboon Prosthetic Grafts,” Circulation Research, July 1996, Vol. 79, No. 1, pp.
45–53.
Ku, D. N., Giddens, D. P., Zarins, C. K., and Glagov, S., “Pulsatile Flow and Atherosclerosis in
the Human Carotid Bifurcation. Positive Correlation between Plaque Location and Low
Oscillating Shear Stress,” Arteriosclerosis, May-June 1985, Vol. 5, No. 3, pp. 293–302.
Ku, D. N., and Liepsch, D., “The Effects of Non-Newtonian Viscoelasticity and Wall Elasticity
on Flow at a 90 Degrees Bifurcation,” Biorheology, 1986, Vol. 23, No. 4, pp. 359–370.
Kubes, P., Suzuki, M., and Granger, D. N., “Nitric Oxide. An Endogenous Modulator of
Leukocyte Adhesion.” Proceedings of the National Academy of Sciences, 1991, Vol. 88, pp.
4651–4655.
Langille, B. L., Bendeck, M. P., and Keeley, F. W., “Adaptations of Carotid Arteries of Young
and Mature Rabbits to Reduced Carotid Blood Flow,” American Journal of Physiology, 1989,
Vol. 256, pp. H931–H939.
Lansman, J. B., “Endothelial Mechanosensors: Going With the Flow,” Nature, 1988, Vol. 331,
No. 6156, pp. 481–482.
Laynez-Carnicero, A., Estornell-Erill, J., Trigo-Bautista, A., et al., “Non-Invasive Assessment of
Coronary Artery Bypass Grafts and Native Coronary Arteries Using 64-Slice Computed
Tomography: Comparison with Invasive Coronary Angiography,” Revista Española de
Cardiología, 2010, Vol. 63, No. 2, pp. 161–169.
Leaper, D., and Whitaker, I., Editors. Post-Operative Complications, 2nd edition, Oxford
Specialist Handbooks, Oxford University Press, 2010.
168
Lee, S. W., Antiga, L., and Steinman, D. A., “Correlations among Indicators of Disturbed Flow
at the Normal Carotid Bifurcation,” Journal of Biomechanics Eng. 2009 June, Vol. 131, No. 6, p.
061013.
Levesque, M. J., Liepsch, D., Moranec, S., and Nerem, R. M., “Correlation of Endothelial Cell
Shape and Wall Shear Stress in a Stenosed Dog Aorta,” Arteriosclerosis, 1986, Vol. 6, pp. 220–
229.
Levesque, M. J., Nerem, R. M., and Sprague, E. A., “Vascular Endothelial Cell Proliferation in
Culture and the Influence of Flow,” Biomaterials, Nov. 1990, Vol. 11, No. 9, pp. 702–707.
Li, S., Huang, N. F., and Hsu, S., “Mechanotransduction in Endothelial Cell Migration,” Journal
of Cellular Biochemistry, Dec. 2005, Vol. 96, No. 6, pp. 1110–1126.
Libby P., “Lipoproteins: Mechanisms for Atherogenesis and Progression of Atherothrombotic
Disease,” in A Companion to Braunwald's Heart Disease, ed. C. Ballantyne, Saunders,
Philadelphia, PA, 2009, pp. 130–143.
Libby, P., Ridker, P. M., and Hansson, G. K., “Progress and Challenges in Translating the
Biology of Atherosclerosis,” Nature, 2011, Vol. 473, No. 7347, pp. 317–325.
Liepsch, D., and Moravec, S., “Pulsatile Flow of Non-Newtonian Fluid in Distensible Models of
Human Arteries,” Biorheology, 1984, Vol. 21, No. 4, pp. 571–586.
Liepsch, D., Moravec, S., Rastogi, A. K., and Vlachos, N. S., “Measurement and Calculations of
Laminar Flow in a Ninety Degree Bifurcation,” Journal of Biomechanics, 1982, Vol. 15, No. 7,
pp. 473–485.
Lilly, L. S., Pathophysiology of Heart Disease: A Collaborative Project of Medical Students and
Faculty, 5th ed. Lippincott Williams and Wilkins, 2011, p. 114.
LoGerfo, F. W., Quist, W. C., Nowak, M. D., Crawshaw, H. M., and Haudenschild, C. C.,
‘‘Downstream Anastomotic Hyperplasia,’’ Annals of Surgery, 1983, Vol. 197, pp. 479–483.
Loot, A. E., Popp, R., Fisslthaler, B., Vriens, J., Nilius, B., and Fleming, I., “Role of Cytochrome
P450-Dependent Transient Receptor Potential V4 Activation in Flow-Induced Vasodilatation,”
Cardiovascular Research, Dec. 2008, Vol. 80, No. 3, pp. 445–452.
Loufrani, L., and Henrion D., “Role of the Cytoskeleton in Flow (Shear Stress)-Induced Dilation
and Remodeling in Resistance Arteries,” Medical and Biological Engineering and Computing,
May 2008, Vol. 46, No. 5, pp. 451–460.
Lyne, W. H., “Unsteady Viscous Flow over a Wavy Wall,” Journal of Fluid Mechanics, 1971,
Vol. 50, pp. 33–48.
169
Malinauskas, R. A., Sarraf, P., Barber, K. M., and Truskey G. A., “Association between
Secondary Flow in Models of the Aorto-Celiac Junction and Subendothelial Macrophages in the
Normal Rabbit,” Atherosclerosis, Sept. 1998, Vol. 140, No. 1, pp. 121–34.
Mann, D.E., and Tarbell, J. M., “Flow of Non-Newtonian Blood Analog Fluids in Rigid Curved
and Straight Artery Models,” Biorheology, 1990, Vol. 27, No. 5, pp. 711-33.
Markl, M., Brendecke, S. M., Simon, J., Barker, A. J., Weiller, C., and Harloff, A., “Coregistration of the Distribution of Wall Shear Stress and 140 Complex Plaques of the Aorta,”
Magnetic Resonance Imaging, Sept. 2013, Vol. 31, No. 7, pp. 1156-62.
Mattsson, E. J., Kohler, T. R., Vergel, S. M., and Clowes, A. W., “Increased Blood Flow Induces
Regression of Intimal Hyperplasia,” Arteriosclerosis, Thrombosis, and Vascular Biology, 1997,
Vol. 17, pp. 2245–2249.
McDonald, D. A., “Blood Flow in Arteries,” 2nd edition. Edward Arnold, London, 1974.
Meeter, K., Veldkamp, R., Tijssen, J. G., van Herwerden, L. L., and Bos, E., “Clinical Outcomes
of Single Versus Sequential Grafts in Coronary Bypass Operations at Ten Years’ Follow-Up,”
Journal of Thoracic and Cardiovascular Surgery, 1991, Vol. 101, pp. 1076–1081.
Mehta, R. H., Ferguson, T. B., Lopes, R. D., Hafley, G. E., Mack, M. J., Kouchoukos, N. T.,
Gibson, C. M., Harrington, R. A., Califf, R. M., Peterson, E. D., and Alexander, J. H.,
“Saphenous Vein Grafts with Multiple Versus Single Distal Targets in Patients Undergoing
Coronary Artery Bypass Surgery: One-Year Graft Failure and Five-Year Outcomes from the
Project of Ex-Vivo Vein Graft Engineering via Transfection (Prevent) IV Trial,” Circulation,
2011, Vol. 124, pp. 280–288.
Merrill, E. W., “Rheology of Blood”, Physiology Reviews, Vol. 40, No. 4, 1969, pp. 863–867.
Meurala, H., Valle, M., Hekali, P., Somer, K, Frick, M. H., and Harjola, P. T., “Patency of
Sequential Versus Single Vein Grafts in Coronary Bypass Surgery,” Journal of Thoracic and
Cardiovascular Surgery, 1982, Vol. 30, pp. 147–151.
Meyer, T. S., Martinoff, S., Hadamitzky, M., Will, A., Kastrati, A., Schömig, A., and Hausleiter,
J., “Improved Noninvasive Assessment of Coronary Artery Bypass Grafts with 64-Slice
Computed Tomographic Angiography in an Unselected Patient Population,” Journal of the
American College of Cardiology, 2007, Vol. 49, No. 9, pp. 946–950.
Michel, J. B., Thaunat, O., Houard, X., Meilhac, O., Caligiuri, G., and Nicoletti, A.,
“Topological Determinants and Consequences of Adventitial Responses to Arterial Wall Injury,”
Arteriosclerosis, Thrombosis, and Vascular Biology, 2007, Vol. 27, pp. 1259–1268
Mondy, J. S., Lindner, V., Miyashiro, J. K., Berk, B. C., Dean, R. H., and Geary, R. L., “PlateletDerived Growth Factor Ligand and Receptor Expression in Response to Altered Blood Flow In
Vivo,” Circulation Research, Sept. 1997, Vol. 81, No. 3, pp. 320–327.
170
Moore, J. E., Xu, C., Glagov, S., Zarins, C. K., and Ku, D. N., “Fluid Wall Shear Stress
Measurements in a Model of the Human Abdominal Aorta: Oscillatory Behavior and
Relationship to Atherosclerosis,” Atherosclerosis, 1994, Vol. 110, pp. 225–240.
Moravec, S., and Liepsch, D., “Flow Investigations in a Model of a Three-Dimensional Human
Artery with Newtonian and Non-Newtonian Fluids. Part I,” Biorheology, 1983, Vol. 20, No. 6,
pp. 745–759.
Morbiducci, U., Ponzini, R., Grigioni, M., and Redaelli, A., "Helical Flow as Fluid Dynamic
Signature for Atherogenesis Risk in Aortocoronary Bypass. A Numeric Study,” Journal of
Biomechanics, 2007, Vol. 40, pp. 519–534.
Motwani, J. G., and Topol, E. J., “Aortocoronary Saphenous Vein Graft Disease: Pathogenesis,
Predisposition, and Prevention,” Circulation. 1998, Vol. 97, No. 9, pp. 916–931.
Moulinec, C., Benhamadouche, S., Laurence, D., and Peric, M., “LES in a U-Bend Pipe Meshed
by Polyhedral Cells,” Engineering Turbulence Modelling and Experiments, 2005, Vol. 6, pp.
237–246.
Munson, B. R., “Experimental Results for Oscillating Flow in a Curved Pipe,” Physics of Fluids,
1975, Vol. 18, pp. 1607–1609.
Nagel, T., Resnick, N., Dewey, C. F., Jr., and Gimbrone, M. A., Jr., “ Vascular Endothelial Cells
Responds to Spatial Gradient in Fluid Shear Stress by Enhanced Activation of Transcription
Factors, Arterioscler Thromb Vasc Biol, 1999, Vol. 19, No. 8, pp. 1825–1834.
Naruse, T., and Tanishita, K., “Large Curvature Effect on Pulsatile Entrance Flow in a Curved
Tube: Model Experiment Simulating Blood Flow in an Aortic Arch,” Journal of Biomechanics
Engineering, May 1996, Vol. 118, No. 2, pp. 180–186.
Naveed, T., Ayub, M., Nazeer, M., Mallick, N. H., Mohydin, B. S., and Ali, Z., “Role of 64Slice Multi Detector Computed Tomography for Non-Invasive Visualization of Coronary Artery
Bypass Grafts for Follow Up in Post CABG Patients,” Journal of Ayub Medical College
Abbottabad, Apr-Jun. 2010, Vol. 22, No. 2, pp. 13–19
Neitzel, G. F., Barboriak, J. J., Pintar, K., and Qureshi, I., “Atherosclerosis in Aortocoronary
Bypass Grafts: Morphologic Study and Risk Factor Analysis 6 to 12 Years after Surgery,”
Arteriosclerosis, 1986, Vol. 6, pp. 594–600.
Nerem, R. M., Levesque, M. J., and Cornhill, J. F., “Vascular Endothelial Morphology as an
Indicator of the Pattern of Blood Flow,” Journal of Biomechanical Engineering, 1981, Vol. 103,
pp. 172–176.
Nikolaos, K. and Georgios, S., “Post-Coronary Artery Bypass Grafting—Degenerated
Saphenous Vein Graft Intervention, or Native Vessel Coronary Chronic Total Occlusion
Recanalisation?” Interventional Cardiology, 2012, Vol. 7, No. 1, pp. 66–70.
171
Nishida, K., Harrison, D. G., Navas, J. P., Fisher, A. A., Dockery, S. P., Uematsu, M., Nerem, R.
M., Alexander, R. W., and Murphy, T. J., “Molecular Cloning and Characterization of the
Constitutive Bovine Aortic Endothelial Cell Nitric Oxide Synthase,” Journal of Clinical
Investigation, Nov. 1992, Vol. 90, No. 5, pp. 2092–2096.
Noris, M., Morigi, M., Donadelli, R., Aiello, S., et al., “Nitric Oxide Synthesis by Cultured
Endothelial Cells is Modulated by Flow Conditions,” Circulation Research, 1995, Vol. 76, No.
4, pp. 536–543.
Nouar, C., Bottaro, A., and Brancher, J. P., “Delaying Transition to Turbulence in Channel Flow:
Revisiting the Stability of Shear-Thinning fluids,” Journal of Fluid Mechanics, 2007, Vol. 592,
pp. 177–194.
Ohyama, N., Nishiyama, A., and Okamura, T., “Secondary Flow and Atherogenesis,” in Role of
Blood Flow in Atherogenesis, ed. Y.Yoshida, T. Yamaguchi, G. C. Caro, S. Glagov, and R. M.
Nerem, 1988, pp. 55-59
Ojha, M., “Wall Shear Stress Temporal Gradient and Anastomotic Intimal Hyperplasia,”
Circulation Research, June 1994, Vol. 74, No. 6, pp. 1227–1231.
Otte, L. A., Bell, K. S., Loufrani, L., Yeh, J. C., Melchior, B., Dao, D. N., Stevens, H. Y., White,
C. R., and Frangos, J. A., “Rapid Changes in Shear Stress Induce Dissociation of a G Alpha
(q/11)- Platelet Endothelial Cell Adhesion Molecule-1 Complex,” Journal of Physiology, 2009,
Vol. 587 Part 10, pp. 2365–2373
Oz, B. S., Iyem, H., Akay, H. T., Bolcal, C., Yokusoglu, M., Kuralay, E., Demirkilic, U., and
Tatar, H., “Mid-Term Angiographic Comparison of Sequential and Individual Anastomosis
Techniques for Diagonal Artery,” Journal of Cardiac Surgery, 2006, Vol. 21, pp. 471–474.
Papaharilaou, Y., Doorly, D. J., and Sherwin, S. J., “The Influence of Out-of-Plane Geometry on
Pulsatile Flow Within a Distal End-to-Side Anastomosis,” Journal of Biomechanics, 2002, Vol.
35, No. 9, pp. 1225–1239.
Pasquini, G., Albanese, B., Manescalchi, P. G., and Morini, R., “Relation of Blood Viscosity,
Plasma Viscosity and Hematocrit,” La Ricerca in clinica e in laboratorio, 1983, Vol. 3 Suppl. 3,
pp. 3273–3231.
Patankar, S. V., and Spalding, D. B., (1972), “A Calculation Procedure for Heat, Mass and
Momentum Transfer in Three-Dimensional Parabolic Flows,” International Journal of Heat and
Mass Transfer, Oct. 1972, Vol. 15, No. 10, pp. 1787–1806.
Poredos, P., “Markers of Preclinical Atherosclerosis and Their Clinical Relevance,” The Open
Atherosclerosis & Thrombosis Journal, 2011, Vol. 4, pp. 1–10.
172
Peiffer, V., Sherwin, S. J., and Weinberg, P. D., “Does Low and Oscillatory Wall Shear Stress
Correlate Spatially with Early Atherosclerosis? A Systematic Review,” Cardiovascular
Research, July 2013, Vol. 99, No. 2, pp. 242–250.
Pfeiffer, N., Mandrusov, E., Vroman, L., and Leonard, E. F., “Effects of Secondary Flow Caused
by a Curved Channel on Plasma Protein Adsorption to Artificial Surfaces,” Biotechnology
Progress, 1998, Vol. 14, No. 2, pp. 338–342.
Pohl, U., Holtz, J., Busse, R., and Bassenge, E., “Crucial Role of Endothelial in the Vasodilator
Response to Increased Flow In Vivo,” Hypertension, 1986, Vol. 8, No. 1, pp. 37-44.
Poston, R., White, C., Read, K., Gu, J., Lee, A., Avari, T., and Griffith, B., “Virchow Triad, but
Not Use of an Aortic Connector Device, Predicts Early Graft Failure after Off-Pump Coronary
Bypass,” Heart Surgery Forum, 2004, Vol. 7, No. 5, pp. E428–E433.
Pousset, Y., Lermusiaux, P., Berton, G., Le Gouez, J. M., and Leroy, R., “Numerical Model
Study of Flow Dynamics through an End-to-Side Anastomosis: Choice of Anastomosis Angle
and Prosthesis Diameter,” Annals of Vascular Surgery, 2006, Vol. 20, No. 6, pp. 773–779.
Pulling, T. M., and Uyesugi, W. Y., “Aneurysm of an Autologous Aorta to Right Coronary
Artery Reverse Saphenous Vein Graft Presenting as a Mediastinal Mass: A Case Report, Cases
Journal, Nov. 2008, Vol. 1, No. 1, p. 340
Qiao, A., and Liu, Y., “Influence of Graft-Host Diameter Ratio on the Hemodynamics of
CABG,” Biomed Mater Eng, 2006, Vol. 16, No. 3, pp. 189–201.
Quaini, A., Canić, S., and Paniagua, D., “Numerical Characterization of Hemodynamics
Conditions Near Aortic Valve after Implantation of Left Ventricular Assist Device,”
Mathematical Biosciences and Engineering, July 2011, Vol. 8, No. 3, pp. 785–806.
Ramchandani, M., and Bedeir, K., “Vascular Connector Devices Increase the Availability of
Minimally Invasive Cardiac Surgery to Ischemic Heart Patients,” HSR Proceedings in Intensive
Care and Cardiovascular Anesthesia, 2011, Vol. 3, No. 4, pp. 239–243.
Redaelli, A., Maisano, F., Ligorio, G., Cattaneo, E., Montevecchi, F. M., and Alfieri, O., “Flow
Dynamics of the St. Jude Medical Symmetry Aortic Connector Vein Graft Anastomosis Do Not
Contribute to the Risk of Acute Thrombosis,” Journal of Thoracic and Cardiovascular Surgery,
July 2004, Vol. 128, No. 1, pp. 117–123.
Replogle, R. L., Meiselman, H. J., and Merrill, E. W., “Clinical Implications of Blood Rheology
Studies,” Circulation, July 1967, Vol. 36, No. 1, pp. 148–160.
Rhonda R. B., Robert L. M., Thelma M. G., Richard F. A., and Barbara A. S., ICD-10-PCS
Reference Manual (ICD-10-PCS), Centers for Medicare and Medicaid Services, 2004.
173
Rieu, R., Friggi, A., and Pelissier, R., “Velocity Distribution along an Elastic Model of Human
Arterial Tree,” Journal of Biomechanics, 1985, Vol. 18, No. 9, pp. 703–715.
Rikhtegar, F., Knight, J.A., Olgac, U., Saur, S. C., Poulikakos, D., Marshall, W., Jr., Cattin, P.
C., Alkadhi, H., and Kurtcuoglu, V., “Choosing the Optimal Wall Shear Parameter for the
Prediction of Plaque Location—A Patient-Specific Computational Study in Human Left
Coronary Arteries,” Atherosclerosis, April 2012, Vol. 221, No. 2, pp. 432–437.
Robinet, J. C., “Instabilities in Laminar Separation Bubbles,” Journal of Fluid Mechanics, Oct.
2013, Vol. 732, pp. 1–4.
Ross, R., “The Pathogenesis of Atherosclerosis: A Perspective for the 1990s,” Nature, 1993, Vol.
362, pp. 801–809.
Ross, R., “Atherosclerosis—An Inflammatory Disease,” New England Journal of Medicine,
1999, Vol. 340, pp. 115–126.
Rubanyi, G. M., Romero, J. C., and Vanhoutte, P. M., “Flow-Induced Release of EndotheliumDerived Relaxing Factor,” American Journal of Physiology, 1986, Vol. 250, pp. H1145–H1149.
Sabik, J. F., “Understanding Saphenous Vein Graft Patency,” Circulation, 2011, Vol. 124, No. 3,
pp. 273–275.
Sabik J. F., and Blackstone E.H., “Coronary Artery Bypass Graft Patency and Competitive
Flow,” Journal of the American College of Cardiology, 2008, Vol. 51, pp. 126–128
Sabiston, D. C., Jr., Ross, R. S., Criley, J. M., Gaertner, R. A., Neill, C. A., and Taussig H. B.,
“Surgical Management of Congenital Lesions of the Coronary Circulation,” Annals of Surgery,
1963, Vol. 157, pp. 908–924.
Salm, L. P., Bax J. J., Jukema, J. W., Schuijf, J. D., Vliegen, H. W., Lamb, H. J., Wall, E. E., and
De Roos, A. “Comprehensive Assessment of Patients after Coronary Artery Bypass Grafting by
16-Detector-Row Computed Tomography,” American Heart Journal, 2005, Vol. 150, p. 775.
Samady, H., Eshtehardi, P., McDaniel, M. C., Suo, J., Dhawan, S. S., Maynard, C., Timmins, L.
H., Quyyumi, A.A., and Giddens, D.P., “Coronary Artery Wall Shear Stress is Associated with
Progression and Transformation of Atherosclerotic Plaque and arterial Remodeling in Patients
with Coronary Artery Disease,” Circulation, Aug. 2011, Vol. 124, No. 7, pp. 779–788.
Sankaranarayanan, M., Chua, L. P., Ghista, D. N., and Tan, Y. S., “Computational Model of
Blood Flow in the Aortocoronary Bypass Graft,” BioMedical Engineering OnLine, Mar. 2005,
Vol. 4, pp. 14
Sankaranarayanan, M., Ghista, D. N., Poh, C. L., Seng, T. Y., and Kassab, G. S., “Analysis of
Blood Flow in an Out-of-Plane CABG Model,” American Journal of Physiology—Heart and
Circulatory Physiology, July 2006, Vol. 291, No. 1, pp. H283–H295.
174
Schoenhagen, M., ed., “Current Developments in Atherosclerosis Research,” Nova Science Inc.,
Hauppauge, NY, 2006, p. 43
Schoettler, J., Jussli-Melchers, J., Grothusen, C., Stracke, L., Schoeneich, F., Stohn, S.,
Hoffmann, G., and Cremer, J., “Highly Flexible Nitinol Mesh to Encase Aortocoronary
Saphenous Vein Grafts: First Clinical Experiences and Angiographic Results Nine Months
Postoperatively,” Interactive Cardiovascular and Thoracic Surgery, 2011, Vol. 13, pp. 396–400.
Schwenke, D. C. and Carew, T. E., “Initiation of Atherosclerotic Lesions in Cholesterol-Fed
Rabbits. II. Selective Retention of LDL vs. Selective Increases in LDL Permeability in
Susceptible Sites of Arteries,” Arteriosclerosis, 1989, Vol. 9, No. 6, pp. 908–918.
Sherwin, S. J., Shah, O., Doorly, D. J., Peiro, J., Papaharilaou, Y., et al., “The Influence of Outof-Plane Geometry on the Flow within a Distal End-to-Side Anastomosis,” Journal of
Biomechanical Engineering, 2000, Vol. 122, No. 1, pp. 86–95.
Smith, G. I., Coding Surgical Procedures: Beyond the Basics, Delmar Cengage Learning,
Clifton Park, NY, 2011.
Soulis, J. V., Lampri, O. P., Fytanidis, D. K., and Giannoglou, G. D., “Relative Residence Time
and Oscillatory Shear Index of Non-Newtonian Flow Models in Aorta,” Proceedings of IEEE
10th International Workshop on Biomedical Engineering, Kos Island, Greece, October 5–7,
2011, pp. 1–4.
Souza D. S. R., Arbeus, M., Pinheiro, B. B., and Filbey, D., “The No-Touch Technique of
Harvesting the Saphenous Vein for Coronary Artery Bypass Grafting Surgery,” Multimedia
Manual of Cardiothoracic Surgery, July 2009.
Spiecker, M., Darius, H., Kaboth, K., Hübner, F., and Liao, J. K., “Differential Regulation of
Endothelial Cell Adhesion Molecule Expression by Nitric Oxide Donors and Antioxidants,”
Journal of Leukocyte Biology, Jun. 1998, Vol. 63, No. 6, pp. 732-739.
Steinman, D. A., Frayne, R., Zhang, X. D., Rutt, B. K., and Ethier, C. R., “MR Measurement and
Numerical Simulation of Steady Flow in an End-to-Side Anastomosis Model,” Journal of
Biomechanics, April 1996, Vol. 29, No. 4, pp. 537–542.
Sugiyama, S., Niizuma, K., Nakayama, T., Shimizu, H., Endo, H., Inoue, T., Fujimura, M., Ohta,
M., Takahashi, A., and Tominaga, T., “Relative Residence Time Prolongation in Intracranial
Aneurysms: A Possible Association with Atherosclerosis,” Neurosurgery, Nov. 2013, Vol. 73,
No. 5, pp. 767–776.
Sui, Y., Ma, B., Chu, B., and Qiao, A., “Optimization of Anastomotic Configuration in CABG
Surgery,” Communications in Numerical Methods in Engineering, 2009, Vol. 25, No. 11, pp.
1097–1106.
175
Summit Medical Group, “Coronary Artery Bypass Graft Surgery (CABG),” URL:
http://www.summitmedicalgroup.com/library/adult_care/ac-cabg_tx/ (cited Jan. 1, 2014).
Sundaram, B., Patel, S., Agarwal, P. and Kazerooni, E. A., “Anatomy and Terminology for the
Interpretation and Reporting of Cardiac MDCT: Part 2, CT Angiography, Cardiac Function
Assessment, and Noncoronary and Extracardiac Findings,” American Journal of Roentgenology,
2009, Vol. 192, No. 3, pp. 584–598.
Takabe, W., Jen, N., Ai, L., Hamilton, R., Wang, S., Holmes, K., Dharbandi, F., Khalsa, B.,
Bressler, S., Barr, M. L., Li, R., and Hsiai, T. K., “Oscillatory Shear Stress Induces
Mitochondrial Superoxide Production: Implication of NADPH Oxidase and C-Jun NH2Terminal Kinase Signaling,” Antioxidants and Redox Signaling, Sept. 2011, Vol. 15, No. 5, pp.
1379–1388.
Takayoshi, M., and Katsumi, N., “Unsteady Flow in Circular Tube (Velocity Distribution of
Pulsating Flow),” Bulletin of Japan Society of Mechanical Engineers, 1980, Vol. 23, No. 186,
pp. pp. 1990–1996.
Talbot, L., and Gong, K. O., “Pulsatile Entrance Flow in a Curved Pipe,” Journal of Fluid
Mechanics, 1983, Vol. 127, pp. 1–25.
Tani, I., “Low Speed Flows Involving Bubble Separations,” Progress in Aerospace Sciences,
1964, Vol. 5, pp. 70–103.
Tappainer, E., “New Device for Saphenous Vein-to-Aorta Proximal Anastomosis without SideClamping,” Journal of Cardiothoracic Surgery, April 2007, p. 31.
Tardy, Y., Resnick, N., Nagel, T., Gimbrone, M. A., Jr., and Dewey, C. F., Jr., “Shear Stress
Gradients Remodel Endothelial Monolayers In Vitro via a Cell Proliferation-Migration-Loss
Cycle,” Arteriosclerosis, Thrombosis, and Vascular Biology, Nov. 1997, Vol. 17, No. 11, pp.
3102–3106.
Tariq, R., Kureshi, S. B., Siddiqui, U.T., and Ahmed, R., “Congenital Anomalies of Coronary
Arteries: Diagnosis with 64 Slice Multidetector CT,” European Journal of Radiology, 2012, Vol.
81, pp. 1790–1797.
Taylor, G. I., “The Criterion for Turbulence in Curved Pipes,” Proceedings of the Royal Society,
1929, Vol. 124, pp. 243–249.
Thubrikar, M. J., Vascular Mechanics and Pathology, Springer, New York, 2007, pp. 20–21.
Timité, B., Castelain, C., and Peerhossaini, H., “Pulsatile Viscous Flow in a Curved Pipe: Effects
of Pulsation on the Development of Secondary Flow,” International Journal of Heat and Fluid
Flow, Oct. 2010, Vol. 31, No. 5, pp. 879–896.
176
Topper, J. N., Cai, J., Falb, D., and Gimbrone, M. A., Jr., “Identification of Vascular Endothelial
Genes Differentially Responsive to Fluid Mechanical Stimuli: Cyclooxygenase-2, Manganese
Superoxide Dismutase, and Endothelial Cell Nitric Oxide Synthase are Selectively Up-Regulated
by Steady Laminar Shear Stress,” Proceedings of the National Academy of Sciences, Sept. 1996,
Vol. 93, No. 19, pp. 10417–10422.
Traub, O., and Berk, D. C., “Laminar Shear Stress: Mechanisms by Which Endothelial Cells
Transduce and Atheroprotective Force, Arteriocscler Thromb Vasc Biol 1998, Vol. 18, No. 5,
pp. 677-685.
Uematsu, M., Ohara, Y., Navas, J. P., Nishida, K., Murphy, T. J., Alexander, R. W., Nerem, R.
M., and Harrison, D. G., “Regulation of Endothelial Cell Nitric Oxide Synthase mRNA
Expression by Shear Stress,” American Journal of Physiology, Dec. 1995, Vol. 269, No. 6, Part
1, pp. C1371–C1378.
van Thienen, J. V., Fledderus, J. O., Dekker, R. J., Rohlena, J., van Ijzendoorn, G. A., Kootstra,
N. A., Pannekoek, H., and Horrevoets, A. J., “Shear Stress Sustains Atheroprotective Endothelial
KLF2 Expression More Potently than Statins through mRNA Stabilization,” Cardiovascular
Research, Nov. 2006, Vol. 72, No. 2, pp. 231–240.
Vartanian, K. B., Kirkpatrick, S. J., Hanson, S. R., and Hinds, M. T., “Endothelial Cell
Cytoskeletal Alignment Independent of Fluid Shear Stress on Micropatterned Surfaces,”
Biochemical and Biophysical Research Communications, July 2008, Vol. 371, No. 4, pp. 787–
792.
Verrier, E. D., and Morgan, E. N., “Endothelial Response to Cardiopulmonary Bypass Surgery,”
Annals of Thoracic Surgery, 1998, Vol. 66, pp. S17–SS19.
Vural, K. M., Sener, E., and Tasdemir, O., “Long-Term Patency of Sequential and Individual
Saphenous Vein Coronary Bypass Grafts,” European Journal Cardio-Thoracic Surgery, 2001,
Vol. 19, pp. 140–144.
Wang, H., Wang, A.X., Aylor, K., and Barrett, E. J., “Nitric Oxide Directly Promotes Vascular
Endothelial Insulin Transport,” Diabetes, Dec. 2013, Vol. 62, No. 12, pp. 4030–4042.
Wentzel, J. J., Krams, R., Schuurbiers, J. C., Oomen, J. A., Kloet, J., van der Giessen, W. J.,
Serruys, P. W., and Slager C. J., “Relationship Between Neointimal Thickness and Shear Stress
after Wallstent Implantation in Human Coronary Arteries,” Circulation, 2001, Vol. 103, pp.
1740–1745.
Weyman, A. K., Plume, S. K., and DeWeese, J. A., “Bovine Heterografts and Autogenous Veins
as Canine Arterial Bypass Grafts,” Archives of Surgery, June 1975, Vol. 110, No. 6, pp. 746–
750.
White, C. M., “Streamline Flow through Curved Pipes,” Proceedings of the Royal Society, 1929,
Vol. 123, pp. 645–663.
177
White, C. R., Haidekker, M., Bao, X., and Frangos, J. A., “Temporal Gradients in Shear, but Not
Spatial Gradients, Stimulate Endothelial Cell Proliferation,” Circulation, May 2001, Vol. 103,
No. 20, pp. 2508–2513.
Wu, Y. P., van Breugel, H. H., Lankhof, H., Wise, R. J., Handin, R. I., de Groot, P. G., and
Sixma, J. J., “Platelet Adhesion to Multimeric and Dimeric Von Willebrand Factor and to
Collagen Type III Preincubated with Von Willebrand Factor,” Arteriosclerosis, Thrombosis, and
Vascular Biology, 1996, Vol. 16, pp. 611–620.
Yearwood, T. L., and Chandran, K. B., “Physiological Pulsatile Flow Experiments in a Model of
the Human Aortic Arch,” Journal of Biomechanics, 1982, Vol. 15, No. 9, pp. 683–704.
Zalosh, R. G., and Nelson, W. G., “Pulsating Flow in a Curved Tube,” Journal of Fluid
Mechanics, 1973, Vol. 59, pp. 693–705.
Zand, T, Hoffman, A. H, Savilonis, B. J., Underwood, J. M., Nunnari, J. J., Majno, G., and Joris,
I., “Lipid Deposition in Rat Aortas with Intraluminal Hemispherical Plug Stenosis: A
Morphological and Biophysical Study,” American Journal of Pathology, 1999, Vol. 155, pp. 85–
92.
Zarins, C. K., Giddens, D. P., Bharadvaj, B. K., Sottiurai, V. S., Mabon, R. F., and Glagov, S.,
“Carotid Bifurcation Atherosclerosis. Quantitative Correlation of Plaque Localization with Flow
Velocity Profiles and Wall Shear Stress,” Circulation Research, Oct. 1983, Vol. 53, No. 4, pp.
502–514.
Zarins, C. K., Zatina, M. A., Giddens, D. P., Ku, D. N., and Glagov, S., “Shear Stress Regulation
of Artery Lumen Diameter in Experimental Atherogenesis,” Journal of Vascular Surgery, March
1987, Vol. 5, No. 3, pp. 413–420.
Zhang, J. M., Chua, L. P., Ghista, D. N., Zhou, T., and Tan, Y. S., “Validation of Numerical
Simulation with PIV Measurements for Two Anastomosis Models,” Medical Engineering and
Physics, 2008a, Vol. 30, No. 2, pp. 226–247.
Zhang, J. M.,, Chua, L. P., Ghista, D. N., Yu, S. C., and Tan, Y. S., “Numerical Investigation and
Identification of Susceptible Sites of Atherosclerotic Lesion Formation in a Complete Coronary
Artery Bypass Model,” Medical and Biological Engineering and Computing, Jul. 2008b, Vol.
46, No. 7, pp. 689–699.
Ziegler, T., Silacci, P., Harrison, V. J., and Hayoz, D., “Nitric Oxide Synthase Expression in
Endothelial Cells Exposed to Mechanical Forces,” Hypertension, Aug. 1998, Vol. 32, No. 2, pp.
351–355.
Zilla, P., Human, P., Wolf, M., Lichtenberg, W., Rafiee, N., Bezuidenhout, D., Samodien, N.,
Schmidt, C., and Franz, T., “Constrictive External Nitinol Meshes Inhibit Vein Graft Intimal
Hyperplasia in Nonhuman Primates,” Journal of Thoracic and Cardiovascular Surgery, 2008,
Vol. 136, pp. 717–725.
178
APPENDICES
179
APPENDIX A
VELOCITY, PRESSURE, STREAMLINE, AND SECONDARY FLOW
FOR NON-BLENDED T-JUNCTION WITH DIFFERENT FLOW RATE RATIOS
D2 = 3 mm
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3; (b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2; (d) 𝑚̇2 /𝑚̇1 = 1.602e-2;
(e) 𝑚̇2 /𝑚̇1 = 2.002e-2
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
180
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX A (continued)
D2 = 4 mm
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3; (b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2; (d) 𝑚̇2 /𝑚̇1 = 1.602e-2;
(e) 𝑚̇2 /𝑚̇1 = 2.002e-2
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
181
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX A (continued)
D2 = 5 mm
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3; (b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2; (d) 𝑚̇2 /𝑚̇1 = 01.602e-2;
(e) 𝑚̇2 /𝑚̇1 =2.002e-2
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
182
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX A (continued)
D2 = 6 mm
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3; (b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2; (d) 𝑚̇2 /𝑚̇1 = 01.602e-2;
(e) 𝑚̇2 /𝑚̇1 = 2.002e-2
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
183
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX B
VELOCITY, PRESSURE, STREAMLINE, AND SECONDARY FLOW
FOR BLENDED T-JUNCTION WITH DIFFERENT BLEND RADIUSES
D2 = 3 mm
(a) rB = 1 mm, (b) rB = 2 mm, (c) rB = 4 mm, (d) rB = 6 mm, (e) rB = 8 mm
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
184
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX B (continued)
D2 = 4 mm
(a) rB = 1 mm, (b) rB = 2 mm, (c) rB = 4 mm, (d) rB = 6 mm, (e) rB = 8 mm
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
185
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX B (continued)
D2 = 5 mm
(a) rB = 1 mm, (b) rB = 2 mm, (c) rB = 4 mm, (d) rB rB = 6 mm, (e) rB = 8 mm
Velocity
(Symmetry Plane)
Velocity [m/s]
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure [Pa]
186
Secondary Flow
(y = 8 mm)
Secondary Flow [m/s]
APPENDIX B (continued)
D2 = 6 mm
(a) rB = 1 mm, (b) rB = 2 mm, (c) rB = 4 mm, (d) rB = 6 mm, (e) rB = 8 mm
Velocity
(Symmetry Plane)
Velocity
Static Pressure
(Symmetry Plane)
Streamline
(Symmetry Plane)
Static Pressure
187
Secondary Flow
(y = 8 mm)
Secondary Flow
APPENDIX C
INLET AND OUTLET PRESSURE CONDITIONS FOR
NON-BLENDED T-JUNCTION WITH DIFFERENT FLOW RATE RATIOS
The pressure loss in the host and branch channels for this study was within the reasonable
range of pressure loss for a vascular system. This can be justified through the corresponding
WSS (WSS = P*D/4L) exerted on the host and branch channels of the T-junction models in
this study. The average WSS of the test models induced by different flow rate ratio in this study
ranges between 0.01 – 2.5 Pa while typical mean WSS in straight arteries range between 0.5 –
2.0 Pa). The table below shows the pressure conditions and the average WSS of a sample nonblended T-junction model.
D2 = 5 mm
D2 = 5 mm
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3; (b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2;
(a) 𝑚̇2 /𝑚̇1 = 6.010e-3;
(b) 𝑚̇2 /𝑚̇1 = 1.001e-2; (c) 𝑚̇2 /𝑚̇1 = 1.401e-2; (d) 𝑚̇2 /𝑚̇1 = 1.602e-2;
(d) 𝑚̇2 /𝑚̇1 = 1.602e-2;
(e) 𝑚̇2 /𝑚̇1 = 2.002e-2
(e) 𝑚̇2 /𝑚̇1 = 2.002e-2
Host inlet Host outlet Branch outlet Host channel Branch channel
Pin[Pa]
Pout[Pa]
Pout[Pa]
WSS [Pa]
WSS [Pa]
(a)
1.12
10.44
0
0.13
0.27
0.84
20.88
0
0.10
0.49
0.58
33.72
0
0.07
0.73
0.46
40.95
0
0.05
0.87
0.22
Static Pressure [Pa]
(Symmetry Plane)
56.91
0
0.03
1.15
(b)
(c)
(d)
(e)
Static Pressure [Pa]
(Symmetry Plane)
188