CHARACTERIZATION OF

COMMON CAROTID ARTERY

GEOMETRY AND ITS IMPACT ON

VELOCITY PROFILE SHAPE

By

Amir Manbachi

A thesis submitted in conformity with the requirements for the Degree of Master of Applied Science

The Institute of Biomaterials and Biomedical Engineering Cardiovascular Sciences Collaborative Program

University of Toronto

© Copyright by Amir Manbachi (2010)

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“Characterization of Common Carotid Artery Geometry and its impact on velocity profile shape”

MASc, 2010

Amir Manbachi Institute for Biomaterials and Biomedical Engineering

University of Toronto

Clinical and engineering studies of carotid artery disease typically assume that the

common carotid artery (CCA), proximal to the bifurcation, is relatively straight enough to

assume fully-developed flow. However, a recent study from our group (Ford et al)

showed the surprising presence, in vivo, of strongly skewed velocity profiles in mildly

curved CCAs. In this thesis we aim to understand how CCA geometry affects velocity

profile skewing.

The left and right normal CCAs of 32 participants (62±13 yrs), randomly chosen

from NIH’s VALIDATE study (N~450) were digitally segmented from aortic root to

bifurcation. It was shown that each segmented CCA could be divided into nominal

cervical and thoracic region and that each region could be approximated by planar

circular arches. Subsequent CFD simulations of CCA parametric models suggested

strong velocity profile skewing both at the inlet and outlet of cervical segment and the

effect of various geometric parameters were investigated.

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Acknowledgments

Over the course of my masters thesis, in addition to developing interest for

cardiovascular engineering, as well as learning some skills on geometric characterization

of arteries and CFD studies of blood flow, I learned something more important, which is

essentially summarized in the following quote from Albert Einstein: “Strive not to be a

success, but rather to be of value.”

This was what I mainly took from my MASc training for my scientific life, and I

owe that to my supervisor, Professor David Steinman. I thank him for giving me the

opportunity, patience and guidance throughout the duration of this project, and for

teaching me individual research. Also would like to thank Dr. Yiemeng Hoi for insightful

ideas.

I am grateful to other members in the lab group for their friendship and my family

for their unconditional love and support.

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Table of Contents

Abstract ii Acknowledgements iii

Table of Contents iv Abbreviations vi

List of Tables vii List of Figures viii

CHAPTER 1 - MOTIVATION AND BACKGROUND

1.1 Cardiovascular Disease Statistics 1

1.2 Atherosclerosis: Definition and Physiology 2

1.3 Shear Stress and Plaques in Disturbed Flow Regions 3

1.4 Essential Fluid Dynamics 5

1.4.1 Reynolds Number 7

1.4.2 Poiseuille Flow 7

1.4.3 Dean’s Flow – Dean’s Number 8

1.4.4 Pulsatile Flow - Womersley Number 11

1.4.5 Example: Pulsatile Dean Flow 12

1.5 Doppler Ultrasound and Vascular Geometry 13

1.6 Thesis Scope and Objectives 17

CHAPTER 2 – METHODOLOGY: GEOMETRY CHARACTERIZATION

2.1. Data General Info 19

2.1.1 Patient Info 19 2.1.2 Image Slice Info 20

2.2 Lumen Segmentation 21 2.2.1 Segmentation Protocol 21

2.2.2 Centerline Extraction 24

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2.3 Geometry Characterization 26 2.3.1 Few Mathematical Definitions 26

2.3.1.1 Curvature 26 2.3.1.2 Curvature in three-dimensional curves 27

2.3.1.3 Tortuosity 28 2.3.1.4 Curvature and Tortuosity Plots 29

2.3.1.5 Pivot Point 30 2.4 CCA Geometry Breakdown 31

2.5 Repeatability of geometric characterization 39

2.5 Geometry Reconstruction 44

2.4 Parametric model of a typical CCA geometry 51

CHAPTER 3 – RESULTS: CFD STUDIES ON CCA 3.1 Computational Fluid Dynamics: An Introduction 52

3.2 Fluid Dynamics of Archetypal Geometries … 56 3.3 Running ‘Newtetr’ CFD Solver 61

3.4 CFD Results of CCA Parametric Models 62 3.4.1 Influence of Thoracic Curvature on Velocity Profiles 63

3.4.2 Influence of Cervical Curvature on Velocity Profiles 66 3.4.3 Influence of Roll Angle on Velocity Profiles 67

3.4.4 Influence of Tilt Angle on Velocity Profiles 69

CHAPTER 4 – CONCLUSION, LIMITATIONS AND RECOMENDATIONS

4.1. Summary and implications of main findings 75

4.4.1 Summary 75 4.4.1 Implications 77

4.2. Assumptions and other limitations 80 4.3. Recommendations for Future Directions 83

CHAPTER 5 – REFERENCES 86

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Abbreviations

Symbol

Explanation

CVD Cardiovascular Diseases CCA Common Carotid Artery SMC Smooth Muscle Cell MRA Magnetic Resonance Angiograms CEMRA Contrast Enhanced MRA CFD Computational Fluid Dynamics VMTK Vascular Modeling Toolkit DUS Doppler Ultrasound Vmax Maximum blood velocity Vmean Mean blood velocity U Mean Sectional Velocity D Internal diameter of vessel R Inetrnal radius of vessel RC Radius of Curvature, RC, cervical Radius of Curvature, for Cervical sector RC, thoracic Radius of Curvature, for Thoracic sector ReD Reynolds Number µ Dynamic viscosity ν kinematic viscosity ρ Density Q Flow rate De Dean Number r radial distance of any given point from centre of the tube θ angular distribution of axial velocity ω angular frequency (2π x heart beats / second) α Womersely Number WSS Wall Shear Stress k Local Curvature (=1/RC) T Tortuosity L Curve length SLD Straight Line Distance

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List of Tables Table 2-1 Descriptive Statistical for CCA geometric parameters 34

Table 2-2 Reproducibility Tabulation 41

Table 3-1 CFD models investigating the impact of thoracic curvature 63

Table 3-2 Parametric model designed to investigate if nearly straight cervical implies Poiseuille flow at the outlet 66

Table 3-3 Parametric models designed to investigate impact of roll angle 67

Table 3-4 Parametric models with median curvatures designed to investigate the impact of tilt angle 69

Table 3-5 Least-curved parametric models designed to investigate the impact of tilt angle 71

Table 3-6 Tilt angle showed greater influence than curvature values 71

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List of Figures

Figure 1-1 Shear stress in vasculature 3

Figure 1-2 Vascular wall thickening at the inner wall of the bends 4

Figure 1-3 Parabolic flow and physiological ranges of WSS 8

Figure 1-4 Illustration of Rc (radius of curvature) and R (lumen radius) 9

Figure 1-5 Comparison between Poisuille flow and Dean flow 10

Figure 1-6 Pulsatile Dean velocity profiles of an oscillatory flow 12

Figure 1-7 Velocity profile skewing of a mild curvature 14

Figure 1-8 Classification of Velocity profile types 15

Figure 1-9 DUS assumes Poiseuille flow, indicating long, straight CCAs 16

Figure 2-1 Sample MRA image of CCA 20

Figure 2-2 Lumen Segmentation Protocol 23

Figure 2-3 CCA Shape Extraction 25

Figure 2-4-1 Curvature 26

Figure 2-4-2 Tortuosity 28

Figure 2-5 Cuvature / Tortuosity plots of helical tubes and planar sinusoids 29

Figure 2-6 Pivot point and peaks in the Curvature Plots 30

Figure 2-7 Best-fit-planes for Cervical and Thoracic sectors of a sample CCA 32

Figure 2-8 2D plot of the geometry breakdown 33

Figure 2-9 Box plots of Rc:R in left versus right CCAs 36

Figure 2-10 Cosine of tilt and roll angle in left versus right CCAs 37

Figure 2-11 Straight-Line Distance: left versus right CCAs 38

Figure 2-12 An anatomical view of the left and right CCAs 38

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Figure 2-13 The initial and the re-segmented lumen surfaces 40

Figure 2-14 Statistical analysis for the reproducibility 42-3

Figure 2-15 Relative configuration of Cervical and Thoracic planes 46

Figure 2-16 spline-fits based on different number of feature points 48

Figure 2-17 case-specific geometric reconstruction for a representative case 50

Figure 2-18 Median parametric Model 51

Figure 3-1 Mesh refinement analysis in CFD studies 54

Figure 3-2 ‘Time resolution’ analysis in CFD studies 55

Figure 3-3 Dean velocity profile skewing and its effect on WSS distribution 57

Figure 3-4 CFD Results showing WSS distribution of ideal helical tube 58

Figure 3-5 CFD Results for an ideal planar wiggly tube 59

Figure 3-6 Summarizing archetypal geometries and their flow profile shape 60

Figure 3-7 Parametric CFD model 62

Figure 3-8 Cycle-averaged velocity profiles of CCA parametric models with

Median Curved Cervical and various thoracic curvatures 64

Figure 3-9 Greater degree of skewing in cervical-only segment, relative to

median CCA parametric model 65

Figure 3-10 Effect of most curved thoracic on the least-curved cervical 66

Figure 3-11 Effect of the roll angle on cycle-averaged velocity profiles 68

Figure 3-12 Effect of the tilt angle on cycle-averaged velocity profiles, for

median curvatures of CCA parametric model 70

Figure 3-13 Effect of the tilt angle on cycle-averaged velocity profiles, for

the least-curved parametric model 72

Figure 4-1 examples of studies on ‘the effect of curvatures on DUS’ 78

x

“The cowards never started;

The weak died along the way;

Only the strong survived;

These were the pioneers.”

Quote by anonymous immigrant

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CHAPTER 1.

MOTIVATION AND BACKGROUND

1.1. CARDIOVASCULAR DISEASE STATISTICS

Cardiovascular diseases, CVD, are known to be the leading cause of death

worldwide. In 2009, an estimated 80 million American adults had one or more types of

cardiovascular disease. This means every one in three individuals. [1]

Among various types of CVD, stroke is the third leading cause of death and the

leading cause of serious, long-term disability in the United States. On average, every 40

seconds someone in the United States has a stroke and the estimated direct and indirect cost

of stroke for 2009 is $68.9 billion. [2]

Of all strokes, 87 percent are ischemic, meaning a decrease in blood supply to the

brain, usually related to blockage of the nearby blood vessels. Aside from cardiac

dysfunction, ischemic stroke typically occurs due to formation and rupture of plaques,

which in turn induces blood clotting, and blocks small vessels downstream. So it is of

significance in clinical studies and trials to investigate various parameters involved in such

arterial diseases. [2]

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1.2. ATHEROSCLEROSIS: DEFINITION AND PHYSIOLOGY

As introduced earlier, ischemic stroke occurs due to rupture of plaques. The deposition

of the fatty material and plaque inside the artery wall is so called, atherosclerosis.

In general, atherosclerosis occurs after the age of thirty and the incidence increases with

age. Apart from age, other risk factors such as: smoking, obesity, diabetes milieus, physical

inactivity, high blood pressure and high cholesterol, as well as genetic history play a role in

the formation and progression of the disease. [3]

Although the exact cause of atherosclerosis is unclear, many scientists think it is

initially caused due to repeated subtle injuries to the artery’s wall, i.e. endothelium. This is

done through various stress mechanisms: (i) physical stresses induced from disturbed blood

flow; (ii) inflammatory stress from the immune system; and (iii) chemical abnormalities in

the blood streams. [4]

Over time, substances traveling in the blood stream, such as cholesterol, fats and

cellular waste products accumulate inside the damaged area of the arterial wall. Oxidization

of these materials initiates an inflammatory response. The injured endothelial cells send

signals indicative of the damage and subsequently, the body responds by activating and

sending monocytes from bloodstream to the injured arterial wall. These monocytes are in

turn transformed into macrophages and foam cells, which will collect and digest fatty

material, such as cholesterol, and finally accumulate to form plaques. As the plaque

increases in size, the arterial wall thickens and hardens. In time, smooth muscle cells

(SMC) within the arterial wall begin to multiply and will move to the surface of the

atherosclerotic plaque to form a firm, fibrous cap on top of the plaque. If this cap breaks

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open over time, it can release plaque into the bloodstream, which travels downstream to

small vessels in brain and blocks the bloodstream there. This results in limited blood supply

(and hence limited oxygen and nutrition) for the neighboring brain tissue, leading to

degradation and potentially necrosis (death) of the corresponding cells. (i.e. stroke) [5]

1.3. SHEAR STRESS AND PLAQUES IN DISTURBED FLOW REGIONS

Shear stress is the tangential force per unit area. In vasculature, wall shear stress

(WSS) arises when adjacent layers of blood are moving at different speeds relative to one

another, while still tangential to the vessel wall. See Figure 1-1 for a descriptive definition

of shear stress, particularly within the cardiovascular context.

Figure 1-1 Shear stress in vasculature: (A) Shear stress is the tangential force per unit area; (B) It occurs due to adjacent layers of blood flowing tangent to one another. Image source: Slager et al. [6]

4

Blood-flow induced shear stress is known to be a key mechanical stimulator for

endothelial cells lining the wall, and so is also important in normal vascular responses such

as vasodilation. Shear stresses are also widely believed to play a central role in the genesis

and progression (natural history) of vascular diseases, notably atherosclerosis. [6,7,8]

Figure 1-2 Plaque build up or wall thickening in general, happens at the inner wall of the bends corresponding to the low shear stress regions. Blue lines show low shear regions. Panels A, B, and C panels are the chronological order of this process. Image source: Slager et al. [6].

In the presence of risk factors for atherosclerosis, low shear stress contributes to

plaque buildup. [6, 9] The lumen surface, consisting of endothelial cells, responds to a low

shear environment in a process called compensatory remodeling of the arterial wall, which

results in lumen narrowing. In Slager et al [6], it is also shown how compensatory

remodeling and endothelial dysfunction set the stage for the generation of rupture-prone,

vulnerable plaques. See Figure 1-2 for illustration of the process.

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Davies’ review article [8] takes the above argument one step further by identifying

and classifying the sites mostly prone to wall thickening. The article describes such sites as

regions of disturbed flow, and flow separation or reversal, with low/oscillatory shear

stress. It then presents a discussion classifying such regions into (i) anatomical, (ii) lesion-

related, and (iii) post-intervention sites. Examples of anatomical include curves, arches,

branches and bifurcations in arteries. Regions downstream of atherosclerotic lesions fit into

the second class. And finally examples of post-intervention sites, include sites of stent

deployment, and artery-vein attachment sites of bypass grafts.

The carotid artery bifurcation in the neck is a one of the anatomical sites mentioned

above (type I, from the locations of complex blood flow) and therefore is a region prone to

the development of atherosclerotic plaques. Considering that rupture of a plaque within the

carotid artery has the potential to cause stroke, therefore it is of clinical significance to

investigate some of the relevant blood flow quantities as related to the role of shear forces

in the development of atherosclerosis. But prior to that, it is desirable to introduce some of

the flow quantities and fluid dynamics terms typically encountered in such cardiovascular

mathematics.

1.4. ESSENTIAL FLUID DYNAMICS

In order to describe complex blood flow patterns in the regions prone to arterial disease,

use of engineering modeling tools appears easier to implement, more cost-efficient and less

time-consuming, relative to experimental approaches. To study the highly unsteady, three-

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dimensional flow that occurs in large arteries, state-of-the-art computational fluid dynamic

(CFD) techniques are typically used. Details of CFD techniques will be discussed in

chapter 3; the goal of this section is to give the reader a general perspective into basic fluid

mechanics needed for this study, as well as introducing steady flow versus pulsatile flow

and the simple, symmetric, parabolic flow in straight tubes (Poiseuille flow) versus flow

seen in planar curved tubes (Dean flow). Such fundamental insight into analytical fluid

mechanics is necessary especially when conducting CFD studies, in order to make sure the

CFD results agree with theory.

The flow regimes indicated are appropriate only for the larger arteries (diameters > 1

mm), and it is assumed that the vessel walls can be taken as rigid and the viscosity as

constant (i.e. Newtonian flow). For gaining insight into assumptions that are considered

reasonable in the context of most large artery flows, a good reference is the image-based

CFD review article. [10]

OVERVIEW

Navier-Stoke Equations, which are the system of equations describing fluid flow,

can be solved for particular values of viscosity µ, density ρ, and inflow / outflow boundary

conditions, and of the parameters specifying the geometry. However there exist certain

dimensionless parameter combinations, termed similarity parameters that will be

introduced below along with their physical significance. When modeling any fluid

mechanical environment, it is not necessary to set all the values in the model equal to the

real case, rather as long the similarity parameters (such as Reynolds number and Dean

number, described below) are set equal, the two flows are considered fluid mechanically

7

identical. Below, aside from introducing Reynolds number, Dean number will be

introduced for curved pipes and also Womersley number for unsteady (pulsatile) flows.

1.4.1. REYNOLDS NUMBER

Reynolds number can be defined as the ratio of inertial forces to viscous forces in a flow.

When simplified mathematically, it can be formulated as the following:

(Equation 1-1)

where D is the diameter of the pipe or vessel, U is the mean sectional velocity, µ is the

dynamic viscosity and ρ is the density. When prescribing values of Reynolds number, one

should note whether it is based on diameter or radius, as there exists various definitions of

Reynolds number in literature. In general, a ReD less than 2300 indicates a laminar flow

regime and values greater than 2300 for ReD suggest turbulent behavior of the fluid flow.

1.4.2. POISEUILLE FLOW

Jean Louis Marie Poiseuille (1797-1869) was a French physician / physiologist

interested in human blood flow. He was the first to scientifically describe the pressure drop

in tubes with small diameters. Later on Eduard Hagenbach (1833-1910) proved that based

on Poiseuille experiments, “parabolic” velocity profiles are seen in tubes of circular cross-

section, suggesting that the liquid in the center is moving fastest while the liquid touching

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the walls of the tube is stationary (due to friction). This results in a simple and symmetric,

flow profile, as illustrated in figure 1-3, named after Poiesuille. [11]

Figure 1-3 (A) Parabolic flow in a vessel and Poiseuille’s law for WSS, i.e. wall shear stress; (B) Physiological ranges of vascular WSS. Image source: Malek et al. [12].

1.4.3. DEAN’S FLOW – DEAN’S NUMBER

In 1920, the British Scientist W.R. Dean analytically solved the Navier–Stokes equations

for the steady, axially uniform flow of a Newtonian fluid in a toroidal pipe, and introduced

the Dean Equations by assuring R/Rc << 1 (in small ratios of tube radius over centerline’s

radius of curvature, see Figure 1-4 for illustration of these terms) and retaining just the

leading order curvature effects as an approximation. These equations were essentially

approximations to the momentum and mass balance equation for this geometry. The Dean

number is a dimensionless group of parameters appearing in this context and includes a

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combination of Reynolds number and square root of curvature [12]:

De = ρUR/µ * (R/Rc) ½ = 0.5 ReD (R/Rc) ½ (Equation 1-2)

Figure 1-4 Rc (radius of curvature) and R (lumen radius) in mildly curved conduit

Similar to Reynolds number and as noted by [12], the definition of the Dean number is not

always consistent (for example, Sherwin and Doorly [13] define De = 4(D/Rc)½ReD) and so

one should always consider the associated definitions when comparing values of the Dean

number.

For a good introduction to the fluid mechanics of flow in a toroidal bend, reference

should be made to [13]. In there, Pedley presents a truncated series solution for the axial

velocity profile u(r, θ), obtained by Dean. The series expansion is as follows, but is only

valid for small Dean numbers (De < 96):

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(Equation 1-3)

where r is the radial distance of any given point from centre of the tube and θ is the angular

distribution of axial velocity for a given radial distance. Please see Figure 1-5 below for a

sample of the Dean flow velocity profiles. Aside from axial velocity magnitude, illustrated

by color, the in-plane secondary flow velocity components are shown in the form of

arrows. These in-plane velocities are due to the centrifugal forces that appear in the flow

because of the curvature in the geometry.

REF VECTOR:

Poiseuille flow Dean flow

Figure 1-5 Axial cross-sectional velocity distribution of Poiseuille flow (left) and Dean flow (right). In Dean flow, high velocity region (red) is pushed towards one side and arrows illustrate the in-plane, secondary vortices associated with the centrifugal forces, raised due to curvature. Color Map: The bar at the top shows magnitude of the velocities relative to the mean velocity. Also shown is the reference in-plane vector for normalized magnitude of 0.1

0 1 2VEL

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1.4.4. PULSATILE FLOW - WOMERSLEY NUMBER

The Womersley number arises in the solution of the Navier Stokes equations

for sinusoidally and oscillatory flow (of laminar and incompressible nature) in a

long, straight tube and describes the level of dependence of the flow characteristics

to time. This dimensionless similarity parameter is indicative of the ratio of pulsatile

flow frequency over flow viscosity:

(Equation 1-4)

Here, R is vessel’s radius (assuming circular cross-section), ω is angular frequency

(i.e. 2π x heart beats / second) and ν is kinematic viscosity, which is the ratio of µ/ρ

(dynamic viscosity / density).

Small α (i.e. 1 or less) indicates that the frequency of pulsations is

sufficiently low such that a parabolic velocity profile has enough time to develop

during each cycle. Large values of α (i.e. 10 or more) suggest that the frequency of

pulsations is sufficiently large that the velocity profile is relatively flat or plug-like.

This is because inertia of velocity profile is such that it does not react to viscous

forces except in a thin layer near the wall, and so the profile turns out to be flat.

Next, an example is introduced that shows an application of some of the

fluid mechanics introduced here.

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1.4.5. EXAMPLE: PULSATILE DEAN FLOW

A pulsatile Dean flow is a combinatory example of the theories introduced above:

one that takes into account the curvatures of vascular geometry and the oscillatory character

of heartbeat. Below, Dean velocity profiles are shown at various time points within the

cardiac cycle, (namely the peak systole, mid-diastole and late diastole) for an oscillatory

flow of α∼3 and Re=500, representative of CCA environment (further details in chapter 3)

Figure 1-6 Time-varying axial velocity distributions of a pulsatile Dean flow shown at 3 different time points of cardiac cycle: (A) Peak Systole, (B) Mid-diastole, (C) Late Diastole. Bottom right panel illustrates the cardiac cycle and these time points.

t/T

Plotted: Pressure (normalized relative to cardiac mean pressure) vs. time (normalized to cardiac cycle period)

P/Pm

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On the figure 1-6, please note the in-plane velocity vectors representative of the

centrifugal forces. It is interesting that during the diastole (late stages of the cardiac cycle)

such effects of the centrifugal forces become stronger and greater and the velocity profiles

are skewed towards one side, whereas during the systole (early stages of the cardiac cycle)

velocity profiles tend to look rather similar to Poiseuille flow. The transition between these

two stages reveals velocity profiles of crescent shape.

1.5. DOPPLER ULTRASOUND AND VASCULAR GEOMETRY

Doppler ultrasound (DUS) is used in clinic in order to measure the pulsatile blood

velocity in the cardiovascular system. It has the ability to detect potential disturbed patterns

of flow, which is helpful in the diagnosis of atherosclerosis, as an indicator of stroke risk.

For example, spectral Doppler velocity measurements of blood flow within a vascular

narrowing have been used to quantify stenosis (i.e. ratios of velocity measurements, within

and distal to the stenosis, are an indicator of the degree of narrowing.) Furthermore, color

Doppler is typically used as a visual guide for locating the region of maximum stenosis.

[14] Wall shear stress (WSS) and time-varying blood flow rates are two of the

measurements often inferred from DUS examinations; however, clinical indices more

typically depend on ratios involving systolic peak velocity, end-diastolic peak velocity

(defined earlier on Figure 1-6) and the mean velocity throughout one complete cardiac

cycle (for example: resistive index, and pulsatility index). [15]

Typically, DUS measures the flow velocity at the centerline of the blood vessel,

which is taken as the peak velocity based on the assumption of parabolic flow. This

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assumption further suggests a peak to mean ratio of 2, i.e.

(Equation 1-5)

Subsequently, WSS and flow rates are calculated from the following equations:

Flow rate / Flux: Q = Area . Vmean (Equation 1-6)

Wall shear stress: WSS = 4µQ / πR3 (Equation 1-7)

However, this assumption of Poiseuille flow occurs only in long and straight vessels.

Poiseuille flow exhibits a symmetric velocity profile with the peak velocity at the center of

the vessel. Hence in DUS measurements, it is assumed

that the CCA, proximal to the bifurcation, is relatively

straight, or at least straight enough to consider a

symmetric distribution of blood velocities across the

lumen.

Despite the assumption mentioned above,

according to some studies such as Tortoli [16] and

Hoskins [17], CCAs do possess some curvature. The

assumption of Poiseuille flow may not be accurate as a

result, which may in turn affect the accuracy of DUS

measurements, leading to inaccurate analysis and

diagnosis of cardiovascular related diseases.

In order to investigate the effect of mild

curvatures superimposed on a straight tube, Lee SW et al

Figure 1-7 Velocity profile skewing of mild curvature.

Image source: Lee et al. [19]

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[18] used a CFD approach. It was demonstrated that relative to the simple, symmetric

velocity profiles expected from a completely straight tube, only local mild curvatures of

0.05 (R:Rc = lumen radius:centerline radius of curvature = 1:20) induces large velocity

profile skewing throughout the length of the model (See Figure 1-7). Note the “local”

curvatures superimposed on a straight tube, i.e. curvatures that change over a few diameters

or less, as oppose to a “global” curvature, which is more or less subtended over many

diameters and appears similar to a portion of a full circle. (e.g. Dean flow)

In parallel to the modeling study mentioned above, Ford et al [19] conducted an in-

vivo classification of velocity profiles observed proximal to the CCA bifurcation. The

authors used available MRI images from 45 older volunteers participating in a study of

early atherosclerosis. The study demonstrates that the widespread assumption suggesting

symmetric nature for the velocity profiles in the CCA is “the exception rather than the

rule”. [21] In figure 1-8, note that only 40% of the velocity profile shapes (which are

averaged over cardiac cycle) are simple and symmetric, along with 30% skewed velocity

profiles and 30% crescent shape velocity profiles.

Figure 1-8 In vivo Classification of Velocity profile types. Shown for each velocity type are the color-coded axial velocity distribution and the corresponding mid-horizontal and mid-vertical velocity profiles. Image source: Ford et al. [21].

Axis-symmetric skewed crescent shape 40% 30% 30%

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These studies demonstrate that while the assumption of simple, symmetric velocity

profile, proximal to the CCA bifurcation is performed routinely; it is actually non-realistic,

especially when taking into account the complex nature of the vessel geometry. Therefore

in order to fully investigate the nature of such velocity profiles, a rigorous study should be

done that takes into account the complex geometry of CCA, proximal to its bifurcation.

Figure 1-9 illustrates the use of DUS based on Poiseuille flow assumption,

indicating the unrealistic expectation of long, straight CCA geometries and demonstrating

the need for characterization of CCA shapes as a need for obtaining DUS guidelines for

anticipating and possibly correcting for skewed velocity profiles.

Figure 1-9 DUS (top right image: probe, bottom right: sample image) is front-line technology for studies of flow measurement, particularly helpful in diagnosis of atherosclerosis, as an indicator of stroke risk. DUS assumes Poiseuille flow condition (left image), due to nominally straight appearance of CCAs, a plausible assumption.

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1.6. THESIS SCOPE AND OBJECTIVES

Although the dynamics of flow in tubes having simple planar curvatures is well

known (i.e., Dean flow), less is known about how and why more complex, subtle

curvatures, such as those observed on CCA, may produce velocity profile skewing in some

cases but not others. To date, most studies have focused on the carotid bifurcation, and

none to date have considered the complexity of the CCA geometry down to the aortic root.

The objective of this thesis is to better understand how CCA geometry affects

velocity profile skewing, upstream the bifurcation. Attempts are made to determine if a

qualitative relationship exists between the nature of velocity profile skewing and the

geometric features characterizing the CCA geometry, from aortic root to bifurcation. It is

hoped that this work will ultimately lead to guidelines for anticipating, and possibly

compensating for, skewed velocity profiles in clinical and engineering studies of blood

flow in mildly curved arteries.

A unique aspect of this work is that it will rely on a small subset of in vivo magnetic

resonance images (MRI) of CCA shapes previously collected from large numbers of

subjects participating in the NIH’s ARIC (N~2000) and VALIDATE (N~450) studies of

risk factors in vascular aging. Due to the desire for large scale automated analysis in long

term, and possible clinical utility, the long-term goal of this study will be to develop

techniques for characterizing CCA geometry and velocity profile shape in as robust,

objective and automated a manner as possible.

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Specifically, my thesis has two short-term objectives:

1. Identify simple but concrete geometric parameters that can be used to quantify

CCA shape, from aortic root to bifurcation, based on available 3D MRI

angiographic images.

2. Preliminary exploration of the effect of various geometric parameters, derived

from objective 1, on the qualitative shape of the velocity profile

In other words, the primary objective of this thesis was:

‘Preliminary explorations of the factors involved in anticipating the fluid mechanics

(velocity profile skewing) of CCA from its parametric geometry.’

Chapters 2 and 3 of the thesis detail the methods and results arising from the

objective 1 and 2 studies, respectively. As will be discussed in chapter 4, such insight into

the relationship between vascular geometry and bloodstream flow could have applications

beyond correcting guidelines for Doppler ultrasound. Such knowledge could ultimately be

useful in the design of engineered vascular grafts and could help understanding the

eccentric nature of vascular wall thickening.

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CHAPTER 2

METHODOLOGY

In this chapter the intention is to describe how the

geometry characterization was done in this thesis. The

detailed steps will be introduced in the context of Patient info, Segmentation of Magnetic

Resonance Angiograms, Geometry Breakdown and Geometry Reconstruction. This chapter

will set the ground for further discussing the implications of geometry characteristics on the

shapes of CCA velocity profiles in the following chapter.

2.1 DATA GENERAL INFO

2.1.1 SUBJECT / PATIENT INFO

The left and right normal common carotid arteries from 32 human subjects (19

females, 12 males), were picked from three-dimensional contrast-enhanced MR angiograms

(CEMRA) of NIH’s VALIDATE study of factors in vascular aging, in a manner to span an

age range of 37-85 (mean ± std = 62 ± 13 yrs). The CCA lumen surfaces were digitally

segmented from contrast enhanced magnetic resonance angiograms acquired on a 3 Tesla

scanner. These subjects were chosen from a sample size resembling those who would

routinely be encountered in clinics for Doppler ultrasound neck exams, particularly, in

terms of the variety in age range, hypertension and other circumstances. Please note that

none of these subjects had developed significant atherosclerotic plaques: Stenosis < 20%

20

2.1.2 IMAGE SLICE INFORMATION:

NIH’s VALIDATE study of factors in vascular aging provided us with more than

450 three-dimensional CEMRA images, which facilitated such a variety among the 32

subjects chosen for the purpose of this thesis. The 3D CEMRA images were acquired with

a resolution of 0.8 x 0.8 x 1.0 mm, interpolated to 0.6 x 0.6 x 0.5 mm and over-contiguous

slices were used to halve the acquisition time. Figure 2-1 provides a sample of the MRA

images provided for a patient, including both the patient’s left CCA (right side of the

image) and the patient’s right CCA (to the reader’s left)

Figure 2-1 - Sample three-dimensional contrast enhanced MRA images of CCA

= 2cm

21

2.2 LUMEN SEGMENTATION

2.2.1 SEGMENTATION PROTOCOL

(FROM AORTIC ROOT TO BIFURCATION)

To reconstruct the 3D surface models (lumen surface) from the CEMRA data, the

level set method implemented within the open-source Vascular Modeling ToolKit (VMTK)

was used. [20, 21]

In VMTK, describing the implicit deformable model surface is based on a partial

differential equation (PDE). In order to define the initial conditions for this PDE, the

initialization of the deformable surface is the first step toward the surface extraction. The

operator selects the segments of interest interactively and uses deformable models to

identify the sub-voxel position of the lumen boundary.

The method used for the initialization as the first step in our segmentation protocol,

is colliding front [21], which is based on the propagation of two independent wave fronts

from two seeds, (as illustrated in figure 2-2, A and B), interactively placed at the two ends

of a vascular branches with the selection of a minimum and maximum threshold to

constrain the propagation of the wave fronts within a prescribed intensity range where the

signal-to-background ratio is high enough (more than a desirable threshold) .

22

After identification of the two appropriate seeds and initialization of the vascular

section, the fast marching method is used as the customarily numerical approach for

surface refinement purposes (see figure 2-2, C). More detailed description of the protocol is

available online at vmtk.org and also Bijari et al. [22]

To provide a good initial guess for the evolving surface, the CCA lumen (from

bifurcation to its origin) was initialized in three to four steps:

1. The first step is performed by interactively picking two points: one at the

common carotid artery and the other one at its downstream internal carotid

artery. The proper threshold would result in an initialization that satisfies

both these conditions:

a. Stays within the artery (i.e. the minimum threshold should be more

than the minimum intensity within the artery, preventing the

initialization to get outside); and

b. Is connected to both seeds. (i.e. the threshold is more than the

minimum intensity between two seeds as well.)

Based on these two criteria we initialize the first shot of the artery: from

downstream ICA to upstream CCA. (Figure 2-2, A-C)

23

2. In the second shot the same procedure is followed to segment the ECA. Two

initial surfaces are then merged as it is shown in the following figure.

(Figure 2-2, A-D)

3. Aside from the bifurcation section, typically two more pairs of seeds might

be needed to segment the upstream CCA all the way to its origin.

(A) (B) (C) (D) (E) Figure 2-2 Lumen segmentation protocol – illustrated above are the steps one needs to take in order to segment a section of CCA bifurcation. The red dots indicate the seeds chosen to initialize the segment. In this thesis, these steps have been repeated a number of times in order to segment the vessel all the way to its origin, as shown in (E). the red dots in (E) help identify three different sectors of the vascular segment, each sector has been segmented separetely and then merged with others. Scale bars show the diameter of CCA to be 5 mm: (A)-(D) share the same scale bar.

5 mm

5 mm

24

After each initialization step described above, the corresponding vascular lumen

segment shall be refined using the fast marching cube methods provided in VMTK.

Towards achieving this purpose, each segment’s initial surface is allowed to evolve for

10,000 iterations, a value chosen to ensure that the results were insensitive to the initial

threshold chosen as reported in Bijari et al. [22] This segmentation step, namely the level

set evolution, took about 1 minute per each sector. Finally, all these initialized and refined

segments shall be merged before surface refinement.

It is notable that the protocol described above is quite reliable and requires

minimum user interaction, a combination setting the stage for automation in image

segmentation. Figure 2-3 illustrates the segmented lumen surface for one of the many

CCAs chosen from the NIH’s Validate studies.

For consistency purposes, each segmented case was cut from one lumen cross-

sectional radius below the bifurcation, hereinafter refer to as CCA1, downward all the way

to the aortic arch or brachiocephalic trunk, for the left and right CCAs respectively. In 3

cases either the left or right CCA could not be reliably segmented down to its origin, and so

these cases were excluded from further geometric analysis, leaving 29 cases.

2.2.2 CENTERLINE EXTRACTION

Once the three-dimensional lumen surface models were reconstructed from the

CEMRA data, the model centerlines were generated using the VMTK centerline extraction

25

built-in feature. The resultant centerlines were stored in ASCII text format files containing

the coordinates of a number of discrete points sampling the centerline.

Figure 2-3 illustrates a sample lumen segmentation (in the right) from two 3D views

of the patient’s right CCA shown in the MRA image to its left. The segmentation also

includes the extracted centerline from CCA1, i.e. one cross-sectional lumen radius, or one

sphere radius below the bifurcation, as defined by Antiga & Steinman 2004 [21] and

commonly used in VMTK performances. This cut from CCA1 was chosen in order to avoid

the flurry region above that and also for consistency purposes.

Figure 2-3 CCA geometry and Centerline Extraction using VMTK toolkit – The left image shows the three-dimensional MRA images of a patient from the VALIDATE study and the right image shows two different 3D views of the lumen segmentation from the patient’s right CCA. As well, the extracted centerline is shown from CCA1 all the way to the CCA origin.

26

2.3 GEOMETRY CHARACTERIZATION

2.3.1 DEFINITION OF A FEW MATHEMATICAL TERMS

Before going on further, the need for aligning our definition of a few terminologies

seems inevitable; namely curvature, tortuosity, straight-line distance and pivot point (in the

context of this thesis). A brief introduction follows below.

2.3.1.1 CURVATURE

Curvature is defined as the amount, by which a geometric curve deviates from the

state of being flat, or straight. [23] There are two types of definitions for curvature:

[24]

(a) Extrinsic curvature, defined in Euclidean space based on the radius of

curvature of circles that barely touch the object;

Figure 2-4-1 Extrinsic Curvature

27

(b) Intrinsic curvature, defined at each point in a differential approach, relative

to its adjacent points, as a measure of its deviation from the straight line

connecting the neighbor points. (For more details and equations, refer to

the next section: 2.3.1.2, curvature in three-dimensional curves)

Although these definitions are two different representation of the concept, and are related to

one another: one is the inverse of the other. For instance, ideally, a ratio of a:R = 1:20

curvature defined in an extrinsic manner, corresponds to an intrinsic curvature of 0.05. For

convenience and consistency purposes, we have calculated the curvatures using the

intrinsic method, but have presented them in the extrinsic manner.

2.3.1.2 CURVATURE IN THREE-DIMENSIONAL CURVES

Given a function r (t) ∈ R3, the curvature at a given point t, can be written such as

below in the vector notation:

where and correspond to the 1st and 2nd derivatives of r (t), respectively.

Applying this definition to a parametrically defined space curve,

, the curvature can then be written as:

28

Frenet and Serret whose work is famous under their own name in vector Calculus,

have shown more rigid derivation of this formula. [25]

2.3.1.3 TORTUOSITY

Tortuosity is a property of curve being tortuous (twisted; having many turns).

There have been several attempts to quantify this property.

In mathematics, subjective estimation is often used to measure tortuosity; which

simply uses the ratio of the cumulative curve length (L) over the distance between

the curve’s initial and end point (Straight Line Distance, hereinafter referred to as:

SLD).

∀ t , T(t) = L (t) / SLD(t) - 1

t refers to spatial points as incrementing along the curve length

Figure 2-4-2 Tortuosity, Straight-line Distance (SLD) and L (Curve length)

29

2.3.1.4 CURVATURE AND TORTUOSITY PLOTS

For each segmented CCA, curvature / tortuosity plots of the extracted centerlines

were plotted versus the straight-line distance. These case-specific plots were then compared

to some of the idealized geometries such as planar sinusoids and helical tubes. Figure 2-5

illustrates the curvature and tortuosity plots versus length for three periodical cycles.

Figure 2-5 Cuvature and tortuosity plots for (A) helical curve and (B) planar sinusoid, sketched over 3 periodical cycle lengths. Green plots: curvature, k(t). Blue plots: tortuosity, T(t). Dotted plots in (A) show the helix projections on 2 planes normal to the circular cross-section; in (B) is the sinusoidal curve. (Amplitude= 1 unit)

T(t)

[non

-dim

ensi

onal

] K

(t) [1

/au]

T(

t) [n

on-d

imen

sion

al]

K(t)

[1/a

u]

t[au]

t[au]

z(t) = sin(t) x(t) = cos(t)

1st periodic cycle 2nd periodic cycle 3rd periodic cycle

1st periodic cycle

2nd periodic cycle

3rd periodic cycle

1st periodic cycle 2nd periodic cycle3rd periodic cycle

(A) Geometry:

Helix, 3 periodical cycle long

(B) Geometry:

Planar Sinusoid, 3 periodical cycle long

30

An interesting observation that led us to the rest of the proposed methodology, was

that any major geometrical bend in the geometry can be located using those major peaks on

the curvature plot versus its curve length. See Figure 2-6 below.

Figure 2-6 CCA geometry pivot point and peaks in the Curvature Plots for a representative case, R810 (meaning VALIDATE Case ID: val-1_810_00, Right CCA)

2.3.1.5 PIVOT POINT

Considering the anatomical fact that typically, and by inspection roughly the top

2/3rd of CCA is in the neck (hereinafter known as cervical), and the remaining portion in

the chest (hereinafter referred to as thoracic), observations based on 29 pairs of common

carotid arteries, suggested that each segmented CCA could be divided into nominal cervical

and thoracic segments at a point of maximum curvature found between 50-70% of the

Cur

vatu

re [1

/mm

]

CCA curve Straight Line Distance from top to bottom [mm]

31

straight-line distance from the carotid bifurcation origin to the CCA origin. (Please refer to

figure 2-6 for illustration of this pivot point and the corresponding peak in the curvature

plot, for a representative case)

Further inspection (qualitative observations on the digital segmentations) revealed

that each segment was approximately planar in nature and so the pivot point, was the

location in which the model’s centerline was transitioning from one single plane to another,

not parallel to itself.

2.4 CCA GEOMETRY BREAKDOWN

So far, it was described that each segmented CCA could be divided into nominal cervical

and thoracic segments at a point of maximum curvature (50-70% of SLD) and that each one

of these anatomical segments were found to be planar. For each anatomical segment,

namely the cervical and thoracic, a plane was fitted to that portion of the centerline using

the least-squares approach.

For each case, in order to investigate the extent of the non-planarity of the total shape of the

CCA, values of the following two angles were determined:

1. The angle between the normal vectors of each plane, aka roll; and

2. The angle between the straight lines connecting the pivot point to both

segment end points, here after named inclination angle. (See Figure 2-7)

32

Figure 2-7 Best-fit-planes for cervical and thoracic sectors of a representative CCA case, R810

For all cases, the centerline in cervical and thoracic sectors was projected to its best-

fit plane in order to establish a completely 2D curve. The distance from each point along

the curve length was compared relative to its projection, in order to obtain a measure of the

error involved in our assumption of planarity of these sectors. Root-mean-square deviations

from the best-fit planes were calculated, in order to determine whether it was reasonable to

approximate the CCA by concatenating two planar curved tubes.

The final step in order to characterize the shape of these 2D curves in each sector

was to investigate their deviation from a straight line, which connects the pivot point to the

corresponding end point in each sector. Such investigation was chosen due to its potential

in facilitating a comparison study between the shape of these 2D curves relative to the

archetypal shapes, believed in bioengineering literature to be representative of arterial

shapes (i.e. cylindrical shape, constantly curved tube, planar wiggle, and helix)

33

The proposed methodology can successfully transform the CCA’s three-

dimensional centerline coordinate system, into 2D plots illustrating the “deviation from

each sector’s straight line” versus the “curve length” for both planar sectors; and so made it

easy to analyze the geometry of the 29 pairs of CCAs listed earlier during the report. An

example of such plots is shown in the figure 2-8, below.

Figure 2-8 (A) Geometry breakdown for a representative case (L794, i.e. VALIDATE Case ID: val-1_794_00, Left CCA). Green plot shows the error associated with projecting the centerline into each segment’s best-fit plane. Blue plot represents the deviations of projected centerline from the straight- line connecting each segment’s end-point to the pivot point. For such deviations, red plot illustrates each sector’s best-fit circular arc. The circular arches appear parabolic since the axes are not to scale. (B) Similar plot, with axes to same scale. Here circular arches appear more reasonably circular.

34

Repeating the methodology proposed above, for all 29 pairs of cases, revealed that

most of these deviation plots resemble a portion of its best-fit circular arc. Thus, after

projecting each segment onto their respective planes, a best-fit circular arc was used to

define the segment’s mean radius of curvature, Rc. Please note that in panel (A) of the

above figure, the vertical and horizontal axes are not in the same scale and hence both the

deviations and their best-fit circular arcs resemble the shape of parabola in the plot. This is

remedied in panel (B) showing the axes to the same physical scale.

Also noted was the straight-line distance (SLD) connecting each sector’s end point

to the pivot point, as well as the mean lumen segmentation radius, R, and the relative

orientation of the best-fit cervical and thoracic planes, namely the roll and tilt angles. Table

2-1 below, summarizes the values of such geometric parameters.

Cervical Thoracic

Percentile R (mm) RC/R SLD (mm) R (mm) RC/R SLD (mm)

10th 1.95 26.1 42.4 1.81 7.8 29.0

Median 2.46 57.8 56.6 2.25 15.2 36.5

90th 3.05 122.3 66.7 2.64 43.8 49.0

Table 2-1 Descriptive statistics for CCA lumen radius (R, in mm), radius of curvature ratio (Rc/R) and straight-line distance (SLD, in mm) derived from the left and right CCA geometries segmented from MRI of N=29 individuals.

Based on the median values reported in Table 2-1, a parametric model of a typical CCA

includes two planar curved tubes representing the cervical and thoracic segments.

Curvature ratios (Rc/R) of the corresponding segments are approximately 60:1 and 15:1

35

respectively. The median curvature found for the cervical CCA was consistent with the

60:1 curvature ratio inferred by Tortoli et al. [17] from Doppler ultrasound-measured

velocity profile skewing, whereas Caro et al’s [30] ratio of 20:1, based on MR angiography

of a single case, was seen to be an outlier. Root-mean-square deviations from the best-fit

planes were found to be 0.3 mm on average over all cases (90th percentile range: 0.07 mm

to 0.83 mm), suggesting it was reasonable to approximate the CCA by concatenating two

planar curved tubes.

In order to reconstruct geometric parameters, approximate values were sufficient to

give us a sense of the quantities. Straight-line distance (SLD) of the median cervical and

thoracic segments were rounded to 60 mm and 40 mm, respectively. Per Table 2-1, a

typical CCA was also found to taper by about 10% along its length, which justified the

assumption of a constant diameter of 5 mm for both segments. Relative to the cervical

plane, the tilt and roll of the thoracic plane are about 25° and 90°, respectively.

Figures 2-9 to 2-11 illustrate the statistical distribution of the above-mentioned

geometric parameters, using box and whisker plots. In these figures the values of these

parameters are being compared in the left and right CCAs. Furthermore, in order to

understand whether any of such values are statistically significantly different than the

others, the paired student’s t-test was implemented and the associated probability ‘p’ values

are shown in the corresponding figures. A quick glance at these values reveals that the right

CCA is significantly shorter than the left CCA, as expected anatomically, due to the

presence of brachiocephalic artery. (See Figure 2-11)

36

Figure 2-9 Box plots of Rc:R ratio in patients’ left versus right (A) cervicals; (B) thoracics; and (C) at the pivot points. ‘p’ refers to the values of probability in paired student’s t-test. Note the different values on the y-axis in each plot.

(B)

(C)

P=0.64

P=0.099

P=0.25

Right Pivot Left Pivot

Right Thoracic Left Thoracic

Right Cervical Left Cervical

Rc

: R

[dim

ensi

onle

ss]

Rc

: R

[dim

ensi

onle

ss]

Rc

: R

[dim

ensi

onle

ss]

(A)

37

Figure 2-10 Cosine of (A) tilt and (B) roll angle in right versus left CCAs. ‘p’ refers to the values of probability in paired student’s t-test (statistical difference)

P=0.57

P=0.97

Cos

(tilt

ang

le)

(A)

Cos

(rol

l ang

le)

(B)

38

Figure 2-11 Straight-Line Distance (SLD) in the patients right versus left CCAs; ‘p’ refers to probability values in paired student’s t-test (statistical significant difference) This figure confirms that the right CCA is shorter than the left CCA.

In the context of figure above, figure below has been brought here showing an anatomical

view of the neck, containing the left and right CCA, as well as the brachiocephalic trunk.

Figure 2-12 An anatomical view of the left and right CCAs. As per figure 2-11, the geometric characterization of this thesis confirms shorter length of the right CCAs relative to the left CCAs. Image source: texasheartinstitute.org [26].

Tota

l SLD

of C

CA

s [m

m] P=0.0001

**

39

2.5 REPEATABILITYOFGEOMETRICCHARACTERIZATION Whilemost of the geometric characterization analysis is completely automated

and hence perfectly reproducible, it does depend on the segmented lumen surface

which is extractedby an operator, andhence a potential source of uncertainty. To

demonstratetheminoreffectofthis,were‐segmented,after12months,7pairs(i.e.,

14cases)selectedbyarandomnumbergenerator:VALIDATEcaseIDs773,781,787,

792, 800, 802, and 811. These were put through the same automated geometric

analysis and the results are shown in table2‐2aswell as figures2‐13and2‐14, as

scatter plots of key geometric factors (Cervical and Thoracic Rc/R, Cervical and

ThoracicSLD(Straight‐LineDistance),aswellasthetiltandrollangles.

Figure 2‐13 shows for a representative participant (both left and right CCA

arteries), the three‐dimensional visualization of the repeated versus the initial

segmented surfaces, for qualitative comparison purposes. In order to facilitate

quantitativecomparisons,table2‐2wasgeneratedwithinwhichthenumericalvalues

ofthekeygeometricparametershavebeentabulated.

Figure 2‐14 illustrates the scatter plots of key geometric factors, which

interestinglysuggestthereproducibilityofallmentionedfactors,excepttherollangle.

Consideringthat fornominallystraightgeometries,possessingsmallcurvatures,roll

angledoesn’tseemtohavearelevantdefinition,(sincefortheextremecaseofstraight

line, therearepotentially infinitenumberofplanespassing through thesecurves in

40

theory), this exception for the roll angle is not that non‐realistic. Sincemost of the

scatterinthereproducibilityplotoftherollangleoccursespeciallywithintheregion

possessingthecurvaturecriteriamentionedabove,thelackofreproducibilityforthe

roll angle seems to bemore or less justifiable. Needless tomention that perhaps a

morerobustmeasureofrollneedstobeconsidered.

Inadditiontothis,notethataspreviouslydiscussed,thevalueoftherollangleis

extremelysensitive to thechoiceofpivot.Finally,asshown in thenextchapter, roll

angleappears tohave the leasteffecton thevelocityprofile skewingandhence the

reproducibilityoftherollangleisnotasimportantasotherfactors,suchastiltangle.

R781,twoorthogonalviews L781,twoorthogonalviews

Fig2­13 Theinitialandthere­segmentedlumensurfaces,after12months,shown for a representative patient (R781 and L781). Original segmentationappears in grey and includes the bifurcations region for anatomicalclarification.TherepeatedsegmentationsareshowninredandstartfromCCA1,onesphereradiusbelowthebifurcationandcontinueall thewayto theCCA’saorticroot,exactlytheregionfocusedwithinthecontextofthisthesis.

41

lumen Cervical Pivot Thoracic y(P)­y(1) R(mm) Rc:R SLDc Rc:R Roll° Tilt° Rc:R SLDt y(N)­y(1)

L773repeat 2.38 32.51 52.89 9.09 9.46 40.42 58.45 63.39 0.65L773 2.27 51.27 54.75 9.68 11.58 44.30 49.55 59.58 0.63

L781repeat 1.95 128.87 59.21 34.65 49.14 59.93 102.54 12.88 0.50L781 1.92 107.94 59.43 46.59 195.74 46.62 156.83 11.15 0.57

L787repeat 2.42 52.09 72.57 12.81 10.38 50.22 145.01 47.52 0.66L787 2.43 28.02 62.20 8.78 13.07 58.26 29.56 49.15 0.59

L792repeat 1.99 85.62 49.44 29.40 23.91 48.42 69.53 19.02 0.51L792 1.85 111.85 48.79 32.50 23.54 50.71 69.49 24.35 0.51

L800repeat 2.18 163.55 73.14 80.46 36.87 46.96 144.38 15.72 0.62L800 2.21 156.74 70.89 80.46 43.79 43.96 140.63 11.96 0.62

L802repeat 1.87 88.48 58.56 46.04 72.29 28.22 73.28 10.50 0.67L802 1.82 99.25 59.72 52.10 41.84 50.21 43.20 16.98 0.54

L811repeat 2.40 41.89 54.14 24.81 13.42 43.10 13.86 38.07 0.61L811 2.41 40.39 53.23 21.25 13.42 45.56 21.45 39.99 0.60

R773repeat 2.52 28.85 43.61 20.89 9.68 36.92 143.22 40.22 0.60R773 2.54 28.73 44.05 20.65 9.43 36.67 144.95 41.66 0.61

R781repeat 1.90 92.90 54.08 50.38 19.48 33.17 18.04 19.85 0.63R781 1.91 84.30 54.55 53.71 19.04 30.18 25.08 20.49 0.66

R787repeat 2.91 28.98 64.76 15.26 19.84 29.75 93.81 32.38 0.70R787 2.91 31.16 66.35 12.96 25.11 29.62 14.96 27.94 0.70

R792repeat 2.19 100.47 45.66 19.24 20.63 40.79 65.52 30.86 0.55R792 2.20 97.71 49.01 17.02 18.12 34.78 137.54 32.26 0.61

R800repeat 2.22 94.54 48.66 126.97 44.24 29.04 93.96 9.95 0.63R800 2.21 101.77 49.86 101.64 42.46 26.76 118.83 10.61 0.66

R802repeat 1.85 93.13 40.12 37.84 70.18 29.46 5.78 12.97 0.58R802 1.86 100.52 42.58 81.02 87.35 30.21 12.26 10.90 0.59

R811repeat 2.43 80.24 46.48 12.97 10.54 41.58 30.67 43.62 0.59R811 2.45 197.35 42.08 12.65 12.57 45.28 48.56 43.45 0.54

Table2­2 Reproducibility Tabulation: case­specific geometriccharacterizationoftherepeatedandinitialsurfacesegmentations.Valuesofthecorresponding geometric parameters, tabulated for quantitative comparisons.RadiusandSLD(Straight­LineDistance)valuesarein‘mm’andRc:Rvaluesaredimensionless.P, thepivot index, isameasureof the positionof thepivotonthecenterlinecurvediscretelysampledtoanumberofpoints.

42

(A)

(C)

(B)

43

Figure2­14 Statistical analysis for the reproducibility of the geometriccharacterization. (A­D) show plots of Pivot indices, Cervical and Thoraciccurvaturesandtiltanglesfortherepeatedversustheinitialsegmentations,astabulatedintheprevioustable.(E)rollangleoftherepetitionsversustheinitialsegmentations,theplotsuggestsnon­reproducibilityofrollangle;(F)showsthescatter indifferencesof rollhappeningmostly fornominally straightarteries,R:Rc>0.015

roll’ vs roll (Degrees)

(D)

(E)

(F)

(Degrees)

44

2.6 GEOMETRY RECONSTRUCTION

Based on the geometry breakdown method discussed above, we propose the

following protocol in order to reconstruct an idealized geometry approximating the shape

of the actual patient-specific CCA for each subject.

For each shape, we need the following geometric parameters given:

a. Lumen radius (R)

b. Cervical segment’s radius of curvature (Rc,cervical), more specifically its ratio

over the lumen radius (i.e. Rc,cervical /R)

c. Thoracic segment’s radius of curvature (Rc,thoracic), and so Rc,thoracic/R

d. Cervical’s straight line distance, connecting pivot to CCA1 (LC)

e. Thoracic’s straight line Distance, connecting CCA origin to pivot (LT)

f. Roll angle, i.e. the angle between cervical and thoracic planes

g. Tilt angle, i.e. The angle between the straight lines connecting the pivot point to

both segment end points

These values are easily obtained when implementing the geometry breakdown

algorithm proposed in the previous sub-section. All one needs to do prior to the

reconstruction, particularly for CFD purposes, is to normalize the parameters ‘b-e’ relative

to parameter ‘a’, Lumen radius. This will result in obtaining a dimensionless parametric

model for the subject of interest.

45

We propose the following approach in order to construct a parametric model,

approximating the CCA geometry for each case:

First of all, decision making on the relative configurations of the cervical and

thoracic planes: One easy was to do this, is to fix the cervical plane and configure thoracic

plane based on parameter ‘f’, namely the roll angle. To do so, we could fix the cervical

plane arbitrarily on the XY-plane (i.e. z=0) and use roll angle to find the thoracic plane.

Since there are many potential planes in the 3D space possessing such roll angle; one might

need to use tilt angle, in order to find thoracic plane. Tilt angle could help determine LT

and consequently position of CCA origin, relative to pivot and bifurcation.

Assume LC, up-direction vector of cervical curve, connecting the pivot to

bifurcation is perfectly aligned in the positive direction of y-axis, while the cervical plane

lies on the XY-plane. Positioning pivot on the coordinate origin (i.e. the point: x=0,

y=0,z=0), one could easily obtain the coordinates of the bifurcation. Figure 2-17 depicts

the cervical plane in green and also the position of bifurcation and pivot on this plane.

Using the difference of vectors between the cervical and thoracic planes (either the

normal vectors or the up-direction vectors), available through the geometric

characterization described above, we could derive the LT (up-direction vector of thoracic’s

straight-line distance, connecting CCA origin to pivot) and NT (the normal vector of

thoracic’s plane). This will lead into the calculation of the CCA origin’s coordinates.

Figure 2-15 shows the relative configuration of the derived thoracic Plane.

46

Figure 2-15 Relative configuration of cervical and thoracic planes, as well as up-direction vectors connecting the pivot point to each sector’s end points, LC and LT. the dotted curve is a symbolic representation of the CCA centerline.

Once the plane configurations have been established, it is time for acquiring the

planar curves within each sector (centerlines of the cervical and thoracic anatomical

sectors). These curves are those discovered to be a portion of a larger planar circle, hence

fluid mechanically resembling Dean Flow within each sector. For reconstruction purposes,

spline-fit interpolation was implemented. Spline-fit is a numerical method of constructing

new smooth curves from connecting a limited number of known data points. Some of the

obvious known points would be the CCA end points as well as the pivot. Based on these

47

points and knowing the plane normal vectors and radii of curvature, other points along the

length of the CCA curve could be mathematically obtained: points such as the middle point

in each arc (corresponding to the peak in the deviation plots, as illustrated in figure 2-16,

A), the quarter points (every 1/4th of each arc’s length, figure 2-16, C), etc.

The first spline-fit implemented was based on the 5 constraining points across, one

every half way through each sector’s curve, and unfortunately appeared to be very loosely

constraining the curves (figure 2-16, A) Next, spline-fit was implemented based on the 9

constraining points across, one every quarter way through each sector’s curve (shown in

figure 2-16, C) which seemed very constraining, especially at the pivot for the extremely

curved cases, and hence not even close to physiological shapes. Therefore we had to

investigate other alternatives.

In the reconstruction of the geometry, it is important to obtain these planar curves in

a manner such that the transition from one sector to another is as smooth as possible in 3D

space (as close as possible to the in-vivo features), while preserving their shape in their

own segment. In order to achieve the criteria described above, we proposed making use of

the three dimensional cubic ‘spline-fitting method’, such that the points away from the

pivot are constrained more rigidly and the points close to pivot are relatively more relaxed,

contributing to obtain a more relaxed and smoother transition from one plane to another.

(Figure 2-16, B)

48

View 1 Spline, view 1

View 2 Spline, view 2

(A) 5 pts

(B) 7 pts

(C) 9 pts

Figure 2-16 spline-fits based on different number of feature points. Rc,cervical:R = 30:1 (10th percentile) and Rc,thoracic:R = 10:1 (25th percentile), instead of 8:1 (10th percentile) due to technical limiations in ANSYS. The associated SLD and Rc:R values are also pointed out in the left and right side of the image respectively.

SLD=27.62mm

14.52mm

24.81mm

16.04mm

24.26mm

16.94mm

Rc:R=

111.79

10.08

10.68

34.23

30.44

11.82

49

One way to fulfill such requirement is to implement the spline-fit based on 7 key

points across the centerline curve: each sector ends (1&2), the pivot (3), and the quarter

points along each sector which are closer to CCA end points (Please refer to figure 2-16,

B). This will result in fulfilling the constraint requirement above, while using a minimal

number of constraint points.

ANSYS© ICEMCFD software package was used to implement the cubic spline

fitting and finally extruding a circular surface of unit radius over the obtained centerline

curve, which resulted in the desirable parametric models. This method appeared to be

successful in capturing the in-vivo geometric features for the sample size of this study’s

focus, i.e. 29 pairs of CCAs, from VALIDATE study.

See figure 2-17 for few examples of comparison between the reconstructed

geometries for a few representative cases and their associated in-vivo lumen surfaces. In

addition to qualitative comparisons facilitated through three-dimensional visualizations of

the surface models, quantitative comparisons of these two category of geometries were

facilitated through plotting the 2D plots of ‘Deviations from straight line, vs. straight line

distance, for the reconstructed surfaces and comparing the values of the relevant parameters

relative to the surfaces obtained for the in-vivo cases. Both comparisons suggested the

promising potential of this reconstruction method, except for the events where the cervical

regions appear sinusoidal (as oppose to circular). L793 is an example in the following

figure (Fig 2-17) demonstrating such mismatch in the cervical region.

50

R795: L795:

R793: L793: lumen Cervical Pivot Thoracic y(P)-y(1)

Radius Rc:R SLDc Rc:R Roll° Tilt° Rc:R SLDt y(N)-y(1) R795ideal 2.2 127.6 51.5 46.8 128.8 21.9 17.8 26.0 0.68

R795 2.2 129.4 49.1 69.1 131.9 19.8 19.3 32.5 0.62 L795ideal 2.5 64.4 61.2 35.5 160.9 25.9 11.7 31.4 0.68

L795 2.5 65.6 65.4 30.7 162.7 28.0 9.9 31.5 0.70 R793ideal 3.2 31.4 51.2 30.0 146.4 27.7 8.7 29.1 0.66

R793 3.2 33.3 56.5 31.1 153.4 31.4 6.7 29.3 0.69 L793ideal 3.0 132.4 67.5 22.7 100.2 30.5 14.5 33.4 0.68

L793 3.0 78.3 66.6 18.8 84.5 33.0 12.4 39.9 0.65 Figure 2-17 case-specific geometric reconstructions for four representative cases; (A) three-dimensional visualizations shown at two perpendicular views, for qualitative comparisons. Red models represent the idealized models and the grey surfaces indicate the actual segmented artery. (B) Values of the corresponding geometric parameters, tabulated for quantitative comparisons. Radius and SLD (Straight-Line Distance) values are in ‘mm’ and Rc:R values are dimensionless. L793 is the only case among these cases, for which the reconstruction doesn’t match the actual surface, due to sinusoidal nature of cervical region.

(A)

(B)

51

2.7 PARAMETRIC MODEL OF A TYPICAL CCA GEOMETRY

Implementation of the previously described protocol for geometric characterization

on the summarized numerical values of the geometric parameters, shown in table 2-1,

resulted in a few parametric models of CCA geometry, obtained to approximate the variety

of geometries seen in CCAs.

In order to include a variety of curvature ranges within our study, these

parametric models were chosen in such a manner to include the median and

extreme cases (10th and 90th percentile). Figure 2-18 shows the median

parametric model, as an approximation for a typical CCA.

Root-mean-square deviations from the best-fit planes were found to be

small (details given in pg. 35), indicating it was reasonable to approximate the

CCA by concatenating two planar curved tubes.

In the next chapter, we will look into the effect of geometry on velocity profile

skewing in CCA and so aside from computational fluid dynamics models, there will be a

brief discussion on which models we thought could best help us in addressing some of fluid

mechanical questions that were central to the formation of this thesis.

Figure 2-18 Parametric model of a

typical CCA

52

CHAPTER 3:

CFD STUDIES OF CCA PARAMETRIC MODELS

In the previous chapter, we characterized the

geometry of CCAs in a sample size of 29 pairs and

proposed a method to construct a parametric model representing the shape of typical CCAs.

Based on this platform, it is now time to look into how the identified geometry features

could affect the blood flow in that region. Of particular interest is the level of asymmetry in

the velocity profiles along the CCA, especially proximal to the bifurcation.

3.1. COMPUTATIONAL FLUID DYNAMICS (CFD): AN INTRODUCTION

In order to model the blood flow inside the proposed geometry or other similar

geometries representing more curved vessels, CFD modeling was used. CFD is an

abbreviation for Computational Fluid Dynamics, a computer-based mathematical modeling

tool that incorporates numerical methods in order to solve the fundamental equations of

fluid flow, the Navier-Stokes equations. For these geometric models of the CCA, CFD

simulations of blood flow were carried out using a well-validated in-house software,

newtetr CFD solver. [27]

In running CFD models, aside from the actual numerical method executed by

computer to solve the equations, the user follows a three step protocol: (1) defining

53

geometry and inlet/outlet boundary conditions as well as fluid properties; (2) meshing the

volume; (3) visualization and post-processing. After defining the geometry, i.e. physical

boundary of the vessel, one of the most fundamental considerations in CFD is how one

treats a continuous fluid in a discretized fashion on a computer. [13, 28] One method is to

discretize the spatial domain into small cells to form a volume mesh or grid, in order to

then apply a numerical algorithm to solve the Navier-Stokes equations over each mesh

element for the flow. For doing so, various methods of meshing are available, one of those

being tetrahedral meshing.

Tetrahedral elements are 3-D elements that have the shape of a tetrahedron, defined

by four vertex nodes. A quadratic tetrahedral element has six extra nodes placed along the

six edges of a tetrahedron. Ideally, these midside nodes should be placed at the center point

between the two vertices of the edge. These elements can be quadratic in shape with curved

edges. This makes such elements appropriate for meshing geometries possessing curvature,

which is quite common in vasculature. [29]

In the course of this thesis, quadratic tetrahedral-element meshes were generated by

ICEM-CFD (ANSYS Inc Canonsburg, PA) with node spacing of 0.3 mm in a model of 5

mm diameter. This indicates a spatial resolution of 8 elements across the lumen diameter,

which was suggested to be reasonable resolution (for such geometry) by Moyle et al. [28]

Furthermore, CFD results of this geometry meshed with 6 and 12 elements across the

lumen diameter were compared. Mesh dependence was assessed based on the qualitative

appearance of the velocity profile not changing between 8 and 12 elements, and since the

54

main interest in this thesis is in the effect of geometric factors on qualitative velocity profile

shape, this was deemed sufficient. Please see Figure 3-1 for a comparison of these mesh

densities and the associated effect on CFD results.

(A) (B) (C) 6 elements / diameter 8 elements / diameter 12 elements / diameter

Figure 3-1 Mesh refinement analysis. Left column shows a coarse mesh (6 elements across lumen diameter) used for CFD studies on the CCA parametric model presented in the previous chapter, whereas the middle column shows a finer mesh (8 elements across the diameter) and the right column shows the finest mesh (12 elements across lumen diameter). Top row shows the cross-sectional mesh density. Bold lines define the quadratic element boundaries; lighter lines indicate the effective mesh density owing to the mid-side nodes. Bottom row shows qualitative appearance of the axial velocity profile, normalized to the cross-sectional mean velocity and color-coded as in Figure 1-5, not changing between 8 and 12 elements. Unless otherwise indicated, these normalizations and color scales are used in all subsequent figures.

0 1 2VEL

55

In the same manner as the Navier-Stokes equations must be solved for each

discretized spatial element, the applied numerical algorithms solve the Navier-Stoke

equations at certain discrete time fractions of a cardiac cycle and therefore one has the

option to look into the flow field at any desired fraction of cardiac cycle, only if the

‘suitable’ time resolution has been given to the solver as input. In other words, the user has

the option to specify the number of time steps used to solve the equations.

For the CFD models of interest in thesis, various time resolutions were tested,

namely 1200, 2400 and 4800 time steps per cycle, whose resultant velocity profiles are

shown in Figure 3-2 for a Dean flow of similar environment. It was found that all models

result in velocity profiles qualitatively similar to one another, but since the model with

1200 time steps, produced a lot of sub-steps in the computation (with the newteter solver),

the model with 2400 time steps per cardiac cycles was chosen for the rest of the thesis.

Midway through cardiac cycle, in CFD models with the following time-resolutions:

(A) 1200 time-steps; (B) 2400 time-steps; (B) 4800 time-steps

Figure 3-2 Time resolution analysis in CFD studies. Shown above are the flow fields and in-plane velocities corresponding to half the cardiac cycle (t = T /2) in a Dean-flow simulation. (A) in a simulation with 2400 time-steps; (B in another simulation with 4800 time-steps.

56

Before jumping into what was revealed from the mathematical modeling of blood

flow within these geometries, it is desirable to gain insight into the archetypal geometries,

typically seen in bioengineering literature, whose geometries are claimed to be

representative of capturing the geometric features of the shape of long and straight or

mildly curved vessels. Furthermore, it is worthwhile to understand the fluid dynamics of

such archetypal geometries.

3.2. FLUID DYNAMICS OF ARCHETYPAL GEOMETRIES TYPICALLY REPRESENTING VESSEL SHAPES

In bioengineering literature, four different ideal approximations to the geometry of

mildly curved vessels are seen: (1) straight cylindrical tube; (2) planar curved tube, known

as Dean Flow in fluid mechanics; (3) planar wiggly tube, such as a portion of a sinusoid

[e.g. 19]; and (4) non-planar helical tube [e.g.30, 31]. We made an effort into creating these

geometries and investigating how these "archetypal" geometric shapes impact flow. This

was essential before we could start studying the effect of pulsatile flows in more complex

shapes as in the CCA. Steady (time-invariant) models were chosen for simplicity.

The degree of curvature in all these geometries was made equal to the in vivo

degree of curvature in the CCA as estimated by Caro et al [30] to be ‘Rc:R = 20’. This

curvature was shown to be an outlier for the present study and so was picked as an extreme

value based on our analysis. The velocity profile skewness was calculated based on

constant blood viscosity of 0.035 cm2/s, mean flow rate of 381 ml/min and time period of

0.957 sec (heart rate of ~ 63 bpm), which would approximately suggest Reynolds (Re) and

Womersley numbers of 500 and 3, respectively.

57

Vascular wall shear stress (WSS) is believed to play role in the generation and

progression of intimal-media wall thickening and also development of atherosclerotic

plaques at the relevant regions. More specifically, low shear regions are thought to

correspond to locations of eccentric wall thickening. Therefore in order to gain qualitative

insight, distribution of WSS was initially investigated for these geometries.

Steady CFD modeling results, revealed the WSS distribution and velocity profile

skewness along the length, for the various archetypal geometries. Figure 3-3 illustrates the

distribution of wall shear stress on Dean-type geometries and the associated skewed

velocity profiles. As expected from theory, high shear regions (appeared in red), located

here on the outer wall of the bend, correspond to high velocity regions in the flow.

Figure 3-3 Dean flow. Left bottom: cross-sectional axial velocity profile, as well as secondary in-plane vortices. Top right: the geometry and the associated WSS distribution: the inner bend corresponds to low shear regions. Consistent with other figures, velocities and WSS color maps are shown non-dimensionalized relative to their respective inlet mean values.

Figures 3-4 and 3-5 look into planar wiggles and helices, respectively along with the

associated resultant velocity profiles and/or wall shear stress distributions. Figure 3-6

summarizes all the relevant fluid mechanical results for these steady flow models.

VEL 2 1 0

WSS 2 1 0

58

Figure 3-4 CFD Results showing normalized wall shear stress distribution on the artery walls of an ideal helical tube, rotated at four different perspective views. Notice the helical twist in the high shear regions (red areas) along the length of this helical artery. WSS magnitudes are normalized relative to cross-sectional mean shear.

On figure 3-4, please note the helical ‘ribbon-twist’ pattern in the distribution of the

high shear region along the helix. This feature is unique to helical shapes. Interestingly, as

will be discussed further in chapter 4, there is evidence in bioengineering literature that

among various shapes of engineered vascular grafts, the helices are less prone to failure

(atherosclerosis development). Therefore insight into the shape of velocity profiles and

WSS 2 1 0

59

hence WSS distribution in helical geometries could have significant potential in

engineering and clinical studies.

Figure 3-5 CFD Results for an ideal planar wiggly tube: (A) velocity magnitude profiles at various slices along the length. VEL color map: velocity magnitudes normalized relative to cross-sectional mean velocity (B-C) WSS distribution, from two views. WSS color map: WSS magnitudes normalized relative to cross-sectional mean shear. Notice the correspondence between high shear regions (red areas) and the high velocity regions shown on the slices, in (A). More than one periodic cycle of geometry is used in order to make sure the flow is fully developed.

A B C

VEL 2 1 0

WSS 2 1 0

WSS 2 1 0

60

Please note that on a wiggly geometry, depending on the shape of velocity profiles,

the high shear region could appear on one side or another.

Next, this section’s results have been summarized. In figure 3-6, various types of

velocity profile shapes have been shown for these archetypal geometries: from Poiseuille

flow, to Dean-type skewed profile to crescent shape velocity field is shown below. The

most important use of this sub-section is to illustrate how various geometric features could

lead to different velocity profile shapes at the outlet. For example whereas velocity profiles

of a straight tube are simple and axis-symmetric (with parabolic distribution), Dean

velocity profiles and those of planar wiggly tubes or helical tubes, are skewed to one side,

and appeared in crescent shape.

Straight Curved Wiggly Helical

Figure 3-6 various geometries seen in literature as representatives of vascular geometries, and their associated velocity profile shapes. Shown above, are the cycle-averaged axial flow fields as well as in-plane velocities. Color map: velocity magnitudes normalized relative to cross-sectional mean velocity

It is hoped that the velocity profiles presented here would give the reader some

insight into fluid mechanics of mildly curved conduits, prior to studying the geometric

features of typical CCAs in vivo and their potential impacts on velocity profiles.

0 1 2VEL

61

3.3. RUNNING ‘NEWTETR’ CFD SOLVER

As mentioned earlier, all the CFD models of this thesis were carried out using a

well-validated in-house software, called “newtetr” solver. Now that some of the details of

CFD studies, such as mesh refinement and time resolution issues have been introduced, it is

a good idea to briefly point out the protocol one follows for this solver. In order to run the

newtetr solver, one has to set up the input file at the beginning. Fortunately the solver

allows the user to specify various parameters. Here some of the parameters specified in my

thesis will be mentioned.

Fully developed inlet velocity boundary conditions were prescribed based on a

representative older adult flow rate waveform [32]. Rigid walls and constant blood

viscosity of 0.035 cm2/s were assumed. Based on the typical flow and heart rates at the

CCA [25], Reynolds (Re) and Womersley numbers were rounded to values of 500 and 3,

respectively. Lumen radius was set to unit, as all the models were non-dimensionalized

prior to the CFD mathematical modeling. Quadratic-tetrahedral elements was used to mesh

the geometry with a spatial resolution of 8 elements across lumen and 2400 time-steps were

used within each cardiac cycle.

62

3.4. CFD RESULTS OF CCA PARAMETRIC MODELS

Let’s review the primary objective of this thesis once again: preliminary explorations

of the factors involved in anticipating the fluid mechanics of CCA from its parametric

geometry. Based on previous chapter, some of the parameters involved in CCA geometry

were: lumen diameter, the straight line distance (SLD) as well as the radii of curvature for

each sector of CCA (i.e. cervical and thoracic), and finally the configuration of anatomical

sector’s planes relative to one another (i.e. the roll and tilt angles). Since the above-

mentioned objective is potentially so broad, it is vital to further clarify and classify the

questions we are trying to address prior to discussing the CFD results for CCA parametric

models. Consequently, after showing the CFD results of a median parametric model, the

rest of the section is broken down into few sub-sections each with a question in the title and

the relevant investigations following.

Figure 3-7 Parametric CFD model of a typical CCA geometry (left) shown with cycle-averaged, peak systolic and late diastolic axial velocity profiles near the start and end of the cervical segment. Note different contour levels for each pair of profiles. Color map: velocity magnitudes normalized relative to cross-sectional mean velocity.

VEL

63

Figure 3-7 summarizes the complex pulsatile flow patterns present in our model of a typical

CCA. The strongly curved thoracic segment introduces strong velocity profile skewing into

the cervical segment, even after averaging over the cardiac cycle. Although damped by the

more shallow curvature of the cervical segment, pronounced skewing is still evident near

the cervical CCA outlet, a location ~ 2 cm upstream of the carotid bifurcation, matching

where velocity profiles were measured by Ford et al. [21]

Now that the resultant velocity profiles of the CCA median case has been briefly

introduced, the following sections will be broken down as mentioned above, in a manner

such as to inspect the effect of each geometric parameter.

3.4.1. INFLUENCE OF THORACIC CURVATURE ON VELOCITY PROFILES

In this section, we will look into the effect of thoracic curvature on the flow. Namely,

fixing the cervical curvature on the median value, we will try various curvatures of thoracic

sector: the most curved, the median curvature and the least curved thoracic. The resultant

velocity profiles (upstream the bifurcation) will then be used to get a sense of the impact of

thoracic curvature. Table 3-1, summarizes the models used to address this question and

Figure 3-7 illustrates the CFD results, comparing the complex cycle-averaged flow patterns

in these three parametric models of CCA.

Model ID Fixed Cervical Curvature

Varying Thoracic Curvature

Tilt Angle

Roll Angle

A 60:1 10:1 25˚ 90˚ B 60:1 15:1 25˚ 90˚ C 60:1 45:1 25˚ 90˚

Table 3-1 CFD models investigating the impact of thoracic

curvature on velocity profile skewing upstream the bifurcation

64

Most Curved Thoracic Median Curved Thoracic Least Curved Thoracic

Figure 3-8 CFD results in parametric models of CCA, with median curved cervical (Rc,cervical:R = 60 mm) and various thoracic curvatures. Top row illustrates cycle-averaged velocity fields with in-plane velocities; three sphere radius upstream bifurcation (CCA3) and bottom row shows velocity magnitudes at selected slices. Color map: velocity magnitudes normalized relative to cross-sectional mean velocity.

Model A Model B Model C

Rc,

thor

acic =

10

mm

Rc,

thor

acic =

15

mm

Rc,

thor

acic =

45

mm

Rc,

cerv

ical

= 6

0 m

m

VEL 2 1 0

65

Figure 3-8 demonstrates the presence of skewed velocity profiles in model CCAs

representing typical geometries that would be encountered clinically. This main finding is

in agreement with Ford et al’s [21] observation of CCA velocity profile skewing in a

majority of cases, and so raises questions about the growing use of Doppler ultrasound for

relating WSS to intimal thickening at the CCA or other nominally straight vessels.

To see if this velocity profile skewing was a function of the shallow cervical

curvature, the more pronounced thoracic curvature, or a combination of the two, we

simulated flow in the cervical-only segment under the same pulsatile conditions, which

corresponds to an average Dean number of 64 based on its definition as 0.5*ReD (R/RC) 1/2

[14], the results of which are shown in figure 3-9. Interestingly, a greater degree of velocity

profile skewing was observed near the outlet of this cervical-only geometry.

Cervical-only segment median CCA model

Figure 3-9 The cervical-only segment (left image) produced greater degree of cycle-averaged axial velocity field skewing, relative to that of the median CCA parametric model (right image), under the same pulsatile conditions (i.e. De=64) Color map: axial velocity fields normalized relative to cross-sectional mean velocity.

Qualitatively and based on figure 3-8, varying thoracic curvature doesn’t seem to

introduce significantly different velocity fields. Next we will present results of other

simulations incorporating various permutations of the median / outlier cervical and thoracic

curvatures listed in Table 2-1 to understand better the interaction between these segments.

VEL 2 1 0

66

3.4.2. INFLUENCE OF CERVICAL CURVATURE

Stemming from our observation in the last section on the

damping effect of the more shallow curvature of the cervical

segment, we would like to check in this section whether nearly

straight cervical implies Poiseuille flow at the outlet. In order

to do so, we chose the least curved cervical curvature over this

study’s sample size (90th percentile, Rc:R = 120:1) and

attached that to the most curved thoracic (25th percentile, Rc:R

= 10:1). In this way we could check whether the most curved

thoracic could skew the velocity profile upstream the

bifurcation or whether the least curved cervical would damp

the thoracic skewing effect. Table 3-2, tabulates the parametric

details of the described geometric model and figure 3-10 shows

the velocity fields of such a geometric model. Interestingly,

even this nearly straight cervical segment does not produce the

expected Poiseuille flow at the outlet, although it should be

noted that we have not yet considered the effect of other

geometric factors: for example, in the present case the tilt angle

is 25 degrees, resulting in a relatively strong curvature as flow

enters the shallow-curved cervical segment.

Table 3-2 Parametric model designed to investigate if nearly straight cervical implies Poiseuille flow at the outlet

Model ID

Least curved Cervical

Most curved Thoracic

Tilt

Roll

D 120:1 10:1 25˚ 90˚

Figure 3-10 Cycle-averaged axial velocity profile is skewed near the outlet in a model with most curved thoracic and minimally curved cervical.

ModelD

Rc,

thor

acic =

10

mm

Rc,

cerv

ical

=12

0 m

m

VEL 2 1 0

67

3.4.3. INFLUENCE OF ROLL ANGLE ON VELOCITY PROFILES

So far, simulations incorporating various permutations of the median and outlier

cervical and thoracic curvatures (as listed in Table 2-1) suggested similar velocity fields.

Therefore the need to investigate other influential geometric parameters remains inevitable.

Among other geometric parameters, the angles associated with the relative three-

dimensional configuration of cervical and thoracic planes seem vital to consider, namely

the roll and tilt angle, as described in the previous chapter.

This section is devoted to check whether and how changes in the roll angle could

result in different types of velocity profile skewing upstream the bifurcation. In order to do

so, the median case was chosen (Rc,cervical = 60 mm, Rc,thoracic = 15 mm) and its roll angle

was changed from 90˚ to 135˚ and 180˚, while the CFD results were compared relative to

the actual median case. Due to symmetry of curved thoracic and cervicals, other angle

choices would result in redundant configurations and so the choice of these three angles

was deemed sufficient for gaining a general insight. Table 3-3, shows details of the

proposed geometric models and figure 3-11 (next page) compares the velocity fields.

Model ID Cervical Curvature Thoracic Curvature Tilt Angle Roll Angle E 60:1 15:1 25˚ 135˚ F 60:1 15:1 25˚ 180˚

Table 3-3 Parametric models designed to investigate the impact of roll angle As per figure 3-11, by changing the roll-angle from 90˚ to 135˚, the velocity profile

maintains its asymmetric shape, whereas in 180˚ the impact gets more obvious. It should be

noted that such a configuration (180˚ roll), implies a planar curve, which is likely not

physiologically relevant. Nevertheless, it is reasonable to conclude that for large variations

about the median roll of 90 degrees, changes in the roll angle will not markedly affect the

character of the velocity profile (i.e., from skewed to axis-symmetric).

68

Median case: roll of 90° roll of 135° roll of 180°

Figure 3-11 Effect of roll angle on velocity profiles in parametric models of CCA possessing median curvature in cervical and thoracic segments. Top row illustrates cycle-averaged axial velocity fields with in-plane velocities; three sphere radius upstream bifurcation (CCA3), bottom row shows velocity magnitudes at selected slices. Color map: velocity magnitudes normalized to cross-sectional mean values.

ModelB ModelE ModelF

Rc,

cerv

ical

= 6

0 m

m

Rc,

thor

acic =

15

mm

90˚ 135˚ 180˚VEL

2 1 0

69

3.4.4. INFLUENCE OF TILT ANGLE ON VELOCITY PROFILES

Further to our investigations on identifying the influential geometric parameters on

CCA’s blood flow, this section is devoted to check whether changes in the tilt angle (as

defined in chapter 2) could result in different types / levels of velocity profile skewing

upstream the bifurcation.

As an attempt to vary the tilt angle, one should note that the median parametric model

(claimed in the last chapter to represent the typical CCA encountered in clinic) possesses a

tilt angle of 25°. Therefore new parametric models possessing tilt angles with values of 0°

and 10° were designed and implemented, as shown in table 3-4, in the hope that such

models could qualitatively illustrate potential impact of tilt angle on flow fields, if any.

Model ID Cervical

Curvature Thoracic Curvature

Tilt Angle

Roll Angle

G 60:1 15:1 10˚ 90˚ H 60:1 15:1 0˚ 90˚

Table 3-4 Parametric models with median curvatures designed to investigate the impact of tilt angle on velocity profile skewing

Figure 3-12, summarizes the complex cycle-averaged blood flow patterns observed in

these models. Looking through the data, one can make the following qualitative

observations, especially in the presence of nominally straight cervical or thoracic regions:

1. Tilt has a great effect given shallow thoracic or cervical regions.

2. No tilt gives velocity profiles close to Poiseuille flow, as expected.

70

Median case (tilt of 25°) Median case, 10° tilt Median case, 0° tilt

Figure 3-12 Effect of tilt angle variation on velocity profiles in parametric models of CCA possessing median curvature in cervical and thoracic segments. Top row illustrates cycle-averaged axial velocity fields with in-plane velocities; three sphere radius upstream bifurcation (CCA3) and bottom row shows velocity magnitudes at selected slices. Dotted purple lines indicate the straight line distance (SLD) vectors used to define the tilt angle.

Model G Model B Model H

Rc,

cerv

ical

= 6

0 m

m

Rc,

thor

acic =

15

mm

25˚ 10˚ 0˚

VEL 2 1 0

71

As an attempt to compare which of the geometric parameters possesses greater

influence on velocity profile skewing, we checked the effect of tilt variation on the least

curved model, in addition to the median curved case, presented before. Table 3-5 describes

the designed models and resultant velocity fields are illustrated in fig 3-13 (next page).

Model ID Cervical Curvature Thoracic Curvature Tilt Angle Roll Angle G 120:1 45:1 10˚ 90˚ H 120:1 45:1 0˚ 90˚

Table 3-5 Tilt variation on parametric models possessing least-curved regions Table 3-6 shows the resultant velocity profile due to tilt variations, for minimally

curved segments and median curvatures. Apparently, the changes in velocity profile shape

from left to right on the top row are less pronounced than those in the bottom row, hence

one may conclude that for the least curved cases, the tilt has a stronger influence vs. the

median case, where the cervical curvature “takes over”. In other words, in the bottom row

case, where the tilt angle is the dominant curvature, it has the dominant effect, whereas in

the top row cervical curvature is more dominant.

25° tilt 10° tilt 0° tilt

Median curvatures

Rc,cervical:R = 60

Rc,thoracic:R = 15

Least-curved model

Rc,cervical:R = 120

Rc,thoracic:R = 45

Table 3-6 For the least curved cases, tilt has a stronger influence vs. the median case

72

Minimally-curved: 25° tilt Minimally-curved, 10° tilt Minimally -curved, 0° tilt

Figure 3-13 Effect of tilt angle variation on velocity profiles in parametric models of CCA possessing minimally curved cervical and thoracic segments. Top row illustrates cycle-averaged axial velocity fields with in-plane velocities; three sphere radius upstream bifurcation (CCA3) and bottom row shows axial velocity slices. Color map: velocity magnitudes are all normalized relative to cross-sectional mean velocity.

Model I Model J Model K

Rc,

cerv

ical

= 1

20 m

m

Rc,

thor

acic =

45

mm

25˚ 10˚ 0˚

VEL 2 1 0

73

Finally before finishing this section, let’s review what was achieved during the

course of this chapter:

Those geometric parameters identified in the last chapter to play a role in the

characterization of the CCA shape were investigated separately by designing various

parametric models of CCA, relevant to those parameters (models A-K). The designed

models were subsequently used to conduct CFD studies simulating the fluid mechanical

environment of CCA. Here is a list of these geometric parameters:

1. Radii of curvatures of cervical or thoracic planar sectors;

2. Angles defining the relative three-dimensional configuration of these planes:

2.1. Roll angle: angle between the planes;

2.2. Tilt angle: angle between the cervical and thoracic straight lines

connecting the both ends of each sector.

The resultant velocity profiles were cycle-averaged over the cardiac cycle and were

shown for visualization of the complex blood flow in CCA. Among these parameters, the

degree of thoracic curvature was shown to have a relatively modest influence,

whereas thegreaterofcervicalor tilt‐inducedcurvaturehas thepredominanteffect

onvelocityprofileshape.

Lastly, in order to make sure our extracted slices just upstream the bifurcation that

were used for this qualitative study of velocity profile skewing, are not affected due to

outlet effect (proximity to the model’s outlet), one of the models (the median case) was

74

extended with the same geometric features several more sphere radii and the velocity fields

were compared at similar locations. The results, reveals that there is not significant

qualitative difference in the velocity fields and hence our models are not affected by this

potential pitfall.

In the next chapter, a summary of the findings of this thesis, including the geometric

characterization of the CCA and the influence of these geometric parameters on the

velocity profile skewing will be presented. In addition to that, some of the suggested future

directions of this study will be briefly introduced.

75

CHAPTER 4 CONCLUSION, LIMITATIONS

AND RECOMMENDATIONS

In this chapter, a summary of the results and their implications will be presented, as

well as assumptions made, the limitations of the study, and recommendations for future

directions.

4.1. SUMMARY AND IMPLICATIONS OF MAIN FINDINGS

4.1.1. SUMMARY

This work relied on in vivo magnetic resonance images (MRI) of CCA shapes collected

from large numbers of subjects participating in the NIH’s VALIDATE (N~450) studies of

risk factors in vascular aging. The left and right normal common carotid arteries, from

aortic root to bifurcation, from 32 subjects (62±13 yrs; range 37-85) were digitally

segmented from contrast-enhanced magnetic resonance angiograms acquired on a 3T

scanner. In 3 cases either the left or right CCA could not be reliably segmented down its

origin at the aortic arch or brachiocephalic trunk, and so these cases were excluded from

further geometric analysis, leaving 29 cases.

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The geometric characterization studies of the vascular lumen segmentations suggested

that each segmented CCA could be divided into nominal cervical and thoracic sectors at a

point of maximum curvature found between 50-70% of the straight-line distance from the

carotid bifurcation origin to the CCA origin. For each segment, a plane was fitted to the

centerline using a least-squares approach. After projecting each sector onto their respective

planes, a best-fit circular arc was used to define the segment’s mean radius of curvature,

Rc. Also noted was the straight-line distance (SLD) from the segment end points to the

pivot point connecting them, the mean segment radius, R, and the relative orientation of the

best-fit cervical and thoracic planes.

Based on the median values reported in Table 2-1, a parametric model of a typical

CCA was constructed from two planar curved tubes representing the cervical and thoracic

segments. Curvature ratios (Rc/R) were rounded to 60:1 and 15:1 respectively. SLD of the

cervical and thoracic segments were rounded to 60 mm and 40 mm, respectively, and the

diameter was assumed to be 5 mm for both segments. Relative to the cervical plane, the

inclination and roll of the thoracic plane were set to 25° and 90°, respectively.

Using median and 90% confidence values of the above-mentioned geometric

parameters, various parametric models of typical CCA geometries were constructed from

two planar curved tubes representing the cervical and thoracic sectors. Finally CFD

simulations of pulsatile flow were carried out for these parametric models of the CCA and

the velocity profiles were extracted in each case.

77

In the CFD studies of this thesis, it was shown that for a typical carotid artery, there

is pronounced asymmetry in the velocity field as it enters the bifurcation region, further

emphasizing that in arteries appearing nominally long and straight, it doesn’t take much

curvature to cause pronounced velocity profile skewing. Rather, in order to gain insight on

the fluid dynamics of CCA, prior to bifurcation, one needs to take into account not only the

radius of curvature from the cervical region, but also the radius of curvature from thoracic

region as well as the tilt angle at the transition from one region to another. Among these

parameters, the results suggest that the dependence on thoracic curvature may be limited

while tilt has the greatest effect.

4.1.2. IMPLICATIONS

IMPLICATIONS ON DOPPLER ULTRASOUND

This thesis demonstrated the presence of skewed velocity profiles in a model CCA

representing a typical geometry that would be encountered clinically. This main finding is

in agreement with Ford et al’s [21] observation of CCA velocity profile skewing in a

majority of cases, and so raises questions about the growing use of Doppler ultrasound for

relating wall shear stresses to intimal thickening at the CCA or other nominally straight

vessels. Some studies in literature have tried to estimate the error involved in flow

measurements when assuming Poiseuille flow and have suggested an error in the order of

25-35%. [33] An ideal solution to compensate for such an error while performing the

clinical DUS exams would be to look at how far the velocity profile is skewed off the

78

centerline and then correct using an easily understandable table/formula. For example, [34]

suggests measuring the velocity profile at two locations and then uses a fit based on a

crescent-shaped profile. While this is a great step towards achieving such long-term goal,

there still remains need for more accurate estimations of the described error.

Aside from the above, another implication of this study was that contrary to

previous studies on ‘the effect of curvatures on DUS measurements’ that focus on the effect

of “global” strongly curved, Dean-type configurations, (such as those shown in Figure 4-1)

we showed that skewing can even occur on the vessels that appear relatively straight with

only mild curvature. This finding leads to potentially more realistic models to

demonstrate/correct skewing.

Figure 4-1 Shown above are few examples of many studies on ‘the effect of curvatures on DUS measurements’ that focus on the effect of strongly curved, Dean-type configurations; whereas my thesis revealed more realistic geometries and velocity profile skewings. Left to right panels: adapted from [35], [33] and [36], respectively. DESIGNING BYPASS GRAFTS

There is evidence in bioengineering literature, suggesting that relative to straight

bypass grafts, small-amplitude helical grafts are less prone to failure (Huijbregts et al [37],

79

Caro et al [30] ). Hence further studies on the geometry-induced flows of grafts possessing

mild curvatures could use some of the geometric characterization techniques and fluid

mechanical findings of this thesis to better optimize the curvatures and/or shape choice.

GLOBAL VS. LOCAL CURVATURES

It was shown during this study that the strong local curvatures of the vascular

geometry (such as those shown in Figure 1-7 [19]) produce velocity profiles different than

those with global "Dean-type" curved tubes, as commonly exhibited (e.g. Fig 1-4).

IMPACT ON CFD

Moyle [29] suggests that "memory" of the inlet velocity profile is mostly lost when

the flow enters the bifurcation (because the bifurcation geometry change is so strong it

"takes over"). Therefore it might not be necessary to include the entire length of CCA for

CFD simulations of the bifurcation. However, current work in our group is investigating the

length of CCA required to ensure this "memory loss". [38]

WSS OF CCA: Eccentric wall-thickening

As previously described, vascular wall remodeling is thought to be mediated by

WSS, and so if WSS is not circumferentially uniform, then remodeling will not be either

[41]. Our simulations suggest that cycle-averaged WSS patterns are not uniform in the

CCA, and can vary by more than +/- 50% circumferentially in the cervical CCA. This may

help explain the common finding of eccentric nature of wall thickening in CCA. [39]

80

4.2. ASSUMPTIONS AND OTHER LIMITATIONS 4.2.1. RIGID WALLS

On the CFD component of this study, the assumption of rigid walls was

implemented, since distension is expected to be on the order of about +/-5% over cardiac

cycle. From Poiseuille’s law, we could then expect the velocity and WSS magnitudes to

change by +/-10% and +/-15% or so, respectively, since area (and hence flow) and WSS

scale with R2 and R3, respectively. But since the expansion is relatively uniform over the

length of the vessel, overall qualitative patterns will remain. [40]

Yet even if the distension is relatively small, the dynamics of flow rate entering the

CCA may be somewhat different than those exiting the CCA, i.e. over the ~10 cm length

there may be changes in the shape of the flow pulse due to attenuation/dispersion effects.

Then again, we used a flow pulse based on the downstream location, which is where we

are most interested. Therefore the assumption of rigid walls is not likely to impact the

conclusions of this study on the qualitative velocity profile shapes.

4.2.2. NEWTONIAN FLOW

Shear rates in large arteries like the CCA are typically on the order of 100 s-1, where

the viscosity of blood is known to be relatively constant. Furthermore flow patterns in the

more complicated bifurcation region are shown to be largely unaffected by non-Newtonian

81

effects. [41] Consequently it seemed reasonable to assume a constant blood viscosity of

0.035 cm2/s in this study.

4.2.3. FULLY DEVELOPED INLET

It is very likely that the flow entering CCA is actually strongly skewed, as it comes

off the arch or the brachiocephalic branch. This said, given the demonstrated minor

influence of thoracic sector on the cervical flow, most probably, such skewness would not

have a strong effect. This point should be tested by attaching a CCA to a model of the

aortic arch for the median case.

4.2.4. PLANAR CURVES

During the course of this thesis, it was shown that the out-of-plane curvature

component is small relative to the planar curve. However, it was demonstrated that the

effect of minor in-plane curvatures, could be significant. Similarly, the effect of even minor

secondary curvatures could be so. From the techniques developed and data obtained in this

thesis, we have the statistical information for the nature and degree of these out-of-plane

wiggles, and that they can (and should) be incorporated into the models to determine their

relative importance. In this context, it would also be interesting to understand, from a

fundamental fluid mechanical point of view, how the effect of such minor curvatures

propagate into the bulk flow to affect, or not affect, the gross shape of the velocity profile

(e.g., development of Görtler vortices [42], [43].)

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4.2.5. CCA DIAMETER

On the CCA geometry characterization component of this thesis, tapering was

observed along the length of our lumen segments. A typical CCA was also found to taper

by about 10% along its length, which led us into the assumption of a constant diameter.

Based on the demonstrated minor effect of thoracic curvature on the cervical velocity

profiles, this assumption deemed reasonable.

4.2.5. MRI RESOLUTION

The resolution of the MRA angiograms was a potential limitation of this study.

Hoogeveen et al. [44] state that only 3 pixels are sufficient to resolve a vessel diameter to

within 10% accuracy, and the vessels in this study were about 4-6 mm diameter (90%

confidence intervals), with a resolution of 0.8 x 0.8 x 1.0 mm, we could be confident that

the accuracy of our lumen geometries, or at least the gross geometric parameters derived

from them, is minimally affected by the spatial resolution of the acquisitions.

Since the aim of thesis was to demonstrate the effect of certain geometric factors on

gross velocity profile shapes, likely none of these limitations would impact the conclusions

of the study. However, for more quantitative analyses, such as correlating velocity profile

skewing to curvatures or classifying the skewness (using effective Dean number based on a

fit to a Dean-type profile), the above assumptions may, and almost certainly the impact of

secondary curvatures will, need to be tested.

83

4.3. RECOMMENDATIONS FOR FUTURE DIRECTIONS

This study could be continued in the following several manners:

1. This study was an attempt to identify the important geometric parameters of the

CCA shape and qualitatively investigate their impact on velocity profile skewing.

Therefore, additional studies could aim to examine potential quantitative

correlations between the shape of the velocity profiles and three-dimensional

curvature features. For example, velocity profile shape could be quantified using

Ford et al's simple categorization [21], or alternatively, using effective Dean

number based on a fit to a Dean-type profile. Since geometric factors are likely not

independent, probably multiple regression analysis could be useful for such

correlation study.

2. The semi-automated techniques developed here for characterizing CCA geometry

was implemented only for a subset of VALIDATE cases. Due to the desire for

large-scale automated analysis in long term, and possible clinical utility, it should

be possible to readily apply these methods to the whole VALIDATE dataset

(N=450) to explore more comprehensively the relationship between CCA geometry,

velocity profile skewing, focal wall thickening, and their interactions with systemic

risk variables (e.g., is velocity profile skewing only relevant for hypertensive

patients and other at-risk individuals), or does it happen in all of us?

84

3. Ultimately, such knowledge may help explain the common finding of eccentric

wall thickening at the CCA [45], and also overcome errors in Doppler ultrasound

estimations of flow and wall shear rates, which typically rely on the assumption of

fully-developed flow [46]. Therefore, another possible long-term study based on

this thesis, would be to develop guidelines for anticipating, and possibly

compensating for, skewed velocity profiles in Doppler ultrasound measurements of

blood flow in clinical settings. Since it might not be feasible to measure curvatures

all the way down to the pivot in each DUS study, suggesting numbers on the

possible errors could be easier and more practical from a clinical point of view.

4. Considering that anatomically realistic local curvatures seem to capture the fluid

mechanics of vascular geometries in a much more reasonable manner, long-term

attempts should be made to introduce to some of the clinical and engineering studies

(e.g. DUS and design of bypass grafts, respectively) the difference in fluid

mechanics of a conduit with idealized “global” constant curvature (e.g. Dean’s

flow) as opposed to the fluid mechanics of conduits possessing local curvatures

(such as those shown in figures 3-4, 3-5). This is worthwhile even though

introducing this concept to individuals having little or no background in fluid

dynamics, such as clinicians, could be extremely challenging. At the very least, it is

worth reinforcing to clinicians the idea that Poiseuille flow may be the exception

rather than the rule in large arteries, and that minor curvatures are capable of

producing surprising asymmetries in the velocity profiles. For example images

taken from the biomedical research literature, shown in Figure 4-1, could leave the

85

mistaken impression that velocity profile skewing will occur only when there are

unphysiologically-shaped, long curved segments.

5. Finally, Another potential investigation based on this thesis, could be a study on the

effect of the “secondary” (out-of-plane) curvatures that were neglected in this study.

Since the associated statistical information for the nature and degree of these out-of-

plane wiggles have been quantified here, they could (and should) be incorporated

into the parametric models of the CCA, in order to determine their relative

importance. Based on the observations of this thesis, we hypothesize that these

minor out of plane wiggles will not have a dramatic effect on profile shape for

median or high curvature cervicals, but may be important for the straight or near-

straight cases similar to the demonstrated effect of tilt for the minimally curved

cases. Either way, this direction could be considered an easy and important follow-

up study.

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CHAPTER 5 –

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