On Aortic Blood Flow Simulations Scale-Resolved Image

Linköping Studies in Science and Technology
Dissertation No. 1493
On Aortic Blood Flow Simulations
Scale-Resolved Image-Based CFD
Jonas Lantz
Division of Applied Thermodynamics and Fluid Mechanics
Department of Management and Engineering
Linköping University
On Aortic Blood Flow Simulations
Scale-Resolved Image-Based CFD
Linköping Studies in Science and Technology
Dissertation No. 1493
Department of Management and Engineering
Linköping University
SE-581 83, Linköping, Sweden
Printed by:
LiuTryck, Linköping, Sweden
ISBN 978-91-7519-720-3
ISSN 0345-7524
c 2013 Jonas Lantz, unless otherwise noted
Copyright No part of this publication may be reproduced, stored in a retrieval system, or be
transmitted, in any form or by any means, electronic, mechanic, photocopying,
recording, or otherwise, without prior permission of the author.
Cover: ColorFul Display of turbulent kinetic energy in an aortic coarctation
Nobody climbs mountains for scientific reasons.
Science is used to raise money for the expeditions,
but you really climb for the hell of it.
Sir Edmund Hillary (1919-2008)
iii
iv
Abstract
This thesis focuses on modeling and simulation of the blood flow in the aorta, the
largest artery in the human body. It is an accepted fact that abnormal biological
and mechanical interactions between the blood flow and the vessel wall are involved in the genesis and progression of cardiovascular diseases. The transport of
low-density lipoprotein into the wall has been linked to the initiation of atherosclerosis. The mechanical forces acting on the wall can impede the endothelial cell
layer function, which normally acts as a barrier to harmful substances. The wall
shear stress (WSS) affects endothelial cell function, and is a direct consequence
of the flow field; steady laminar flows are generally considered atheroprotective,
while the unsteady turbulent flow could contribute to atherogenesis. Quantification of regions with abnormal wall shear stress is therefore vital in order to
understand the initiation and progression of atherosclerosis.
However, flow forces such as WSS cannot today be measured with significant accuracy using present clinical measurement techniques. Instead, researches
rely on image-based computational modeling and simulation. With the aid of
advanced mathematical models it is possible to simulate the blood flow, vessel
dynamics, and even biochemical reactions, enabling information and insights that
are currently unavailable through other techniques. During the cardiac cycle, the
normally laminar aortic blood flow can become unstable and undergo transition to
turbulence, at least in pathological cases such as coarctation of the aorta where the
vessel is locally narrowed. The coarctation results in the formation of a jet with a
high velocity, which will create the transition to turbulent flow. The high velocity
will also increase the forces on the vessel wall. Turbulence is generally very difficult to model, requiring advanced mathematical models in order to resolve the
flow features. As the flow is highly dependent on geometry, patient-specific representations of the in vivo arterial walls are needed, in order to perform an accurate
and reliable simulation.
Scale-resolving flow simulations were used to compute the WSS on the aortic wall and resolve the turbulent scales in the complex flow field. In addition to
WSS, the turbulent flow before and after surgical intervention in an aortic coarctation was assessed. Numerical results were compared to state-of-the-art magnetic
v
resonance imaging measurements. The results agreed very well, suggesting that
that the measurement technique is reliable and could be used as a complement to
standard clinical procedures when evaluating the outcome of an intervention.
The work described in the thesis deals with patient-specific flows, and is, when
possible, validated with experimental measurements. The results provide new insights to turbulent aortic flows, and show that image-based computational modeling and simulation are now ready for clinical practice.
vi
Populärvetenskaplig beskrivning
Den vanligaste dödsorsaken i Sverige och övriga delar av västvärlden är hjärtoch kärlsjukdomar. Förutom riskfaktorer som t.ex. rökning, diabetes och bukfetma finns det även en koppling mellan hjärt-kärlsjukdomar och blodflöde. De
delar i kärlsystemet där blodet har en onaturlig effekt på kärlväggen sammanfaller
ofta med områden med åderförfettning, ateroskleros, dvs. inlagring av fett och
kolesterol i kärlväggen. Vad är då en onaturlig effekt?
Kraften som blodet påverkar kärlväggen med kan delas upp i två komponenter:
en som är vinkelrät mot kärlväggen och en som är riktad längs med kärlväggen.
Den första komponenten är blodtrycket och den andra är en kraft som uppkommer på grund av friktionen mellan blodet och kärlväggen. Denna friktionskraft
är avsevärt mindre än blodtrycket, men flera decenniers forskning har visat att
denna kraft trots detta är av stor betydelse för var uppkomsten av ateroskleros
sker. Då kraften är riktad längs med kärlväggen kan den dra ut eller trycka ihop
det yttersta cellagret på väggen, vilket normalt fungerar som en barriär mot olika
skadliga mekanismer. En hög konstant kraft har visat sig vara bättre än en låg
och oscillerande. Det är därför mycket intressant att undersöka hur och var i kärlsystemet dessa krafter är onaturliga och man får på sätt en bättre förståelse för
uppkomsten av ateroskleros. Blodtryck mäts vanligtvis med en manschett runt
ena överarmen, men dessvärre går det inte med dagens teknik att direkt mäta friktionskraften mellan blodet och kärlväggen.
Det är här modellering och simulering av blodflöden kommer in. Denna
avhandling beskriver hur man med hjälp av avancerade matematiska modeller
kan bestämma hur det yttersta cellagret påverkas av blodflödet. Vanligtvis är
blodflödet laminärt, dvs. välordnat och effektivt, men vid sjukdomsfall där förträngningar av kärl eller hjärtklaffar lokalt minskar tvärsnittsarean kan flödet övergå till att bli turbulent. Ett turbulent flöde karaktäriseras av oregelbundenhet och
är, sett ur ett energiperspektiv, ineffektivt. Dessutom kommer ett turbulent flöde
att resultera i komplexa friktionskrafter på väggen, med både varierande riktning
och storlek. En noggrann och korrekt kvantifiering av dessa krafter är mycket
viktiga för att förstå uppkomst och utveckling av olika hjärt- och kärlsjukdomar.
Parallellt med beräkningar krävs mätningar på patienter. En patient som unvii
dersöktes led av en förträngning på aortan som påverkade både blodflödet och
krafterna på kärlväggen. Patienten undersöktes både före och efter operation för
att kunna utvärdera ingreppet. Förträngningen hade tvingat fram en övergång
från laminärt till turbulent flöde och beräkningar visade bland annat att det turbulenta flödet minskade efter en operation där förträngningen vidgades. Resultaten
jämfördes med en ny experimentell teknik för turbulensmätningar och överensstämmelsen mellan beräkningar och mätningar var mycket god. Detta innebar att
mätmetoden är mycket lovande och att den efter ytterligare studier skulle kunna
användas i en klinisk tillämpning som komplement till traditionella undersökningsmetoder.
Arbetet som är beskrivet i avhandlingen visar potentialen av att använda modellering och simulering av biologiska flöden för att få kliniskt relevant information för diagnos, operationsplanering och/eller uppföljning av ingreppet på en patientspecifik nivå.
viii
Acknowledgements
This thesis was carried out at the Division of Applied Thermodynamics and Fluid
Mechanics, Department of Management and Engineering, Linköping University.
I would like to thank my main advisor Matts Karlsson for introducing me to
the wonderful field of image-based computational fluid dynamics, and for being
a never-ending source of new thoughts and bright ideas. Somehow we managed
to cherry-pick the best ones, and I am both proud and pleased with the outcome.
Thank you!
Writing scientific research articles is a team effort and none of the articles
in this thesis would have been published without my skilled colleagues and coauthors. A big thank you goes to Roland Gårdhagen, Fredrik Carlsson, Johan
Renner, Tino Ebbers and Jan Engvall for their hard work and valuable input. Dan
Loyd is greatly acknowledged for his thoughts and comments on the draft of this
thesis. I would also like to take the opportunity to express my gratitude to my
friends and colleagues at the Division of Applied Thermodynamics and Fluid Mechanics for the valuable discussions and good company during these five years. In
addition, I would like to acknowledge the people I got to know at IEI, IMT and
CMIV who in different ways also contributed to this work.
Besides doing fancy research and colorful pictures, I have also been teaching
in courses ranging from basic thermodynamics to aerodynamics and computational fluid mechanics. Teaching and interacting with more than 500 students
from all over the world has been both fun and enlightening, and for that I thank
you all.
Last, but not least, I would like to express my sincere gratitude to my family
and friends for always being there when I need them, and for reminding me that
there is another reality outside the university. A very special thank you goes to
my wife Karin for your love and affection - ’Without You I’m Nothing’.
Jonas Lantz
November 2012
ix
Funding
This work was supported by The Swedish Research Council (Vetenskapsrådet)
under grants:
• VR 2007-4085
• VR 2010-4282
Computational resources were provided by The Swedish National Infrastructure
for Computing (SNIC), under grant:
• SNIC022/09-11
Simulations were run on the Neolith, Kappa, and Triolith computer clusters at National Supercomputer Centre (NSC), Linköping, Sweden, and the Abisko cluster
at High Performance Computing Center North (HPC2N), Umeå, Sweden.
xi
List of Papers
This thesis is based on the following five papers, which will be referred to by their
Roman numerals:
I. Quantifying Turbulent Wall Shear Stress in a Stenosed Pipe Using Large
Eddy Simulation
Roland Gårdhagen, Jonas Lantz, Fredrik Carlsson and Matts Karlsson
Journal of Biomechanical Engineering, 2010, 132, 061002
II. Quantifying Turbulent Wall Shear Stress in a Subject Specific Human
Aorta Using Large Eddy Simulation
Jonas Lantz, Roland Gårdhagen and Matts Karlsson
Medical Engineering and Physics, 2012, 34, 1139-1148
III. Wall Shear Stress in a Subject Specific Human Aorta - Influence of
Fluid-Structure Interaction
Jonas Lantz, Johan Renner and Matts Karlsson
International Journal of Applied Mechanics, 2011, 3, 759-778
IV. Large Eddy Simulation of LDL Surface Concentration in a Subject
Specific Human Aorta
Jonas Lantz and Matts Karlsson
Journal of Biomechanics, 2012, 45, 537-542
V. Numerical and Experimental Assessment of Turbulent Kinetic Energy
in an Aortic Coarctation
Jonas Lantz, Tino Ebbers, Jan Engvall and Matts Karlsson
Submitted for publication
Articles are reprinted with permission.
xiii
Abbreviations
α
φ
CAD
CFD
CFL
CT
DNS
FSI
KE
LDL
LES
MIP
MRI
OSI
PC-MRI
Pe
PWV
RANS
Re
RMS
RSM
Sc
SGS
Ti
TAWSS
TKE or k
US
WALE
WSS
Womersley number
Generic flow variable
Computer Aided Design
Computational Fluid Dynamics
Courant-Friedrichs-Lewy condition
Computed Tomography
Direct Numerical Simulation
Fluid-Structure Interaction
Kinetic Energy
Low-Density Lipoprotein
Large Eddy Simulation
Maximum Intensity Projection
Magnetic Resonance Imaging
Oscillatory Shear Index
Phase Contrast Magnetic Resonance Imaging
Péclet Number
Pulse Wave Velocity
Reynolds Averaged Navier Stokes
Reynolds Number
Root-Mean-Square
Reynolds Stress Model
Schmidt Number
Subgrid-Scale Model
Turbulence intensity
Time-Averaged Wall Shear Stress
Turbulent Kinetic Energy
Ultrasound
Wall-Adapting Local Eddy-Viscosity Subgrid-Scale Model
Wall Shear Stress
xv
Contents
v
Abstract
Populärvetenskaplig beskrivning
vii
Acknowledgements
ix
Funding
xi
List of Papers
xiii
Abbreviations
xv
Contents
xvii
1
Introduction
1
2
Aims
3
3
Physiological Background
3.1 The Circulatory System . . . . . . . . .
3.2 Anatomy of the Aorta . . . . . . . . . .
3.3 Cardiovascular Disease and Blood Flow
3.4 Medical Imaging Modalities . . . . . .
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Modeling Cardiovascular Flows
4.1 Governing Equations . . . .
4.2 Flow Properties . . . . . . .
4.3 Flow Descriptors . . . . . .
4.4 Geometrical Representation .
4.5 Blood Properties . . . . . .
4.6 Boundary Conditions . . . .
4.7 Fluid-Structure Interaction .
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xvii
CONTENTS
4.8
4.9
Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Modeling Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . 27
5
Results
35
5.1 Quantification of Aortic Wall Shear Stress . . . . . . . . . . . . . 35
5.2 Aortic Mass Transfer and LDL . . . . . . . . . . . . . . . . . . . 42
5.3 Turbulent Flow in an Aortic Coarctation . . . . . . . . . . . . . . 45
6
Discussion
47
7
Review of Included Papers
51
xviii
Chapter 1
Introduction
The purpose of the human cardiovascular system is to transport oxygenated blood
from the lungs to the rest of the body, and in addition, transport nutrients, hormones, waste products and other important substances around the blood stream.
Diseases related to the cardiovascular system is the most common cause of death,
both in Sweden and worldwide [1]. During 2010, 41% of women and 39% of men
in Sweden had cardiovascular disease as the underlying cause of death [2].
There is a close connection between some cardiovascular diseases and blood
flow [3], and in order to understand the genesis and progression of these diseases,
accurate description and assessment of blood flow features are crucial. While noninvasive measurement techniques are getting more and more advanced, accuracy
and resolution are still a limiting factor. Flow features such as wall shear stress,
which depend on the velocity gradient at the arterial wall, cannot be measured
with a significant accuracy using present measurement techniques. Additionally,
complex biological systems and individual variability makes it difficult to use
imaging and experiences from larger groups to provide information on a single
individual patient [4].
This is where modeling and simulation of physiological flows come in. Where
today’s measurement techniques are limited in spatial and temporal resolution,
mathematical models representing physiological flow situations are, in essence,
only limited by computer power. With computational fluid dynamics (CFD) it is
possible, at least in theory, to simulate not only healthy and diseased conditions
but also what-if scenarios, for example to determine the optimal location of a
stent or to predict the outcome of a surgery, on an individual basis. As noted by
Taylor et al [5], CFD could be a powerful tool, ”[...] surpassing experimental
fluid mechanics methods to investigate mechanisms of disease, and design and
evaluation of medical devices and therapeutic interventions”. These opportunities
are still in its infancy, and necessary steps in terms of accuracy and validation are
required before it can be used on a clinical basis. Robust methods are essential,
1
CHAPTER 1. INTRODUCTION
not only for reliable results but also for convincing the (sometimes conservative)
physician about the possibilities of CFD modeling and simulation.
Modeling physiological flows is difficult. The difficulties arise from both
physics and physiology; the flow may be transitional or even turbulent in some
cases, which calls for the need of advanced turbulence models to accurately predict the flow features. At the same time, each patient is unique in terms of vascular geometry and flow, introducing the need for patient-specific geometries and
boundary conditions in the flow model. Thus, in order to use modeling tools to
simulate flows on a clinical level, measurements of each patient has to be made.
This can be done with magnetic resonance imaging (MRI) or computer tomography (CT). While both image modalities are non-invasive, CT uses ionizing radiation and is unable to quantify flow, making MRI the natural choice in cardiovascular flow research. Image-based CFD refers to the use of image material from a
measurement technique, such as MRI, in a numerical flow simulation.
Blood flow features are affected by the geometry, but not only in the vicinity
of the area of interest, but also upstream and downstream the blood vessel. Effects such as wave reflections, wall distensibility, and flow branching all affect the
flow, and may need to be considered in a flow model. Simplifications can sometimes be made; the assumptions of a rigid arterial wall and that blood behaves
like a Newtonian fluid are the two most common. However, it is important to
understand how these simplifications affect the result. Recent advances in computer hardware and numerical methods has made it possible to simulate complex
flow problems, including transitional and turbulent flows that are related with cardiovascular disease [5]. The goal of any simulation determines the amount of
simplifications that can be made, and thus, requires a thorough understanding of
both the fluid mechanics and the physiology of the cardiovascular system.
This thesis focuses on modeling and simulation of blood flow in the human
aorta. The aorta has a complex shape, characterized by curvature, bending and
tapering. With the addition of branching vessels, a highly complicated threedimensional flow is obtained, even in healthy subjects. In a diseased environment
the flow may become even more complex with the transition to turbulent flow.
Accurate modeling of these types of flows may be crucial for understanding the
progression and genesis of cardiovascular diseases, or when evaluating different
surgical options. Modeling may also be used for intervention planning and evaluation of the outcome of a surgery. Obviously, computational modeling can and will
provide a useful tool in clinical practice. The challenge is to translate the opportunities and possibilities available in computational simulations to the clinic [3].
2
Chapter 2
Aims
The goal of the research described in this thesis was to model and quantify the
blood flow in the human aorta, considering both healthy subjects and patients. It is
believed that much can be gained in a clinical environment if image-based computational modeling and simulation can be used for diagnose, intervention planning,
and/or treatment follow-up. Specifically, the following aims were addressed:
• Use advanced computational models, in particular large eddy simulation
(LES) for simulating turbulent flows and fluid-structure interaction (FSI)
for simulating wall motion, in order to quantify aortic blood flow in both
healthy subjects and patients.
• Simulate the flow-dependent transport of a passive scalar, e.g. low-density
lipoprotein (LDL) and correlate its accumulation on the arterial wall to flow
features and locations prone to develop cardiovascular disease.
• Apply the gained knowledge on a patient with an aortic coarctation before
and after surgery to evaluate the change in blood flow due to the intervention, and by doing so take computational modeling one step closer to clinical
practice.
3
Chapter 3
Physiological Background
3.1
The Circulatory System
The circulatory system in humans include three important parts: a heart, blood and
blood vessels. The heart pumps the blood through the vessels in a loop, and the
system is able to adapt to a large number of inputs as the demand on circulation
varies throughout the body, day and life. The circulatory need is e.g. different
between rest and exercise, and in different body positions.
During systole, the left ventricle in the heart contracts and ejects the blood
volume into the aorta. The blood pressure in aorta increases and the arterial wall
is distended. After the left ventricle has relaxed, the aortic valve closes and maintains the pressure in the aorta while the blood flows throughout the body. The
blood continues to flow through smaller and smaller arteries, until it reaches the
capillary bed where water, oxygen, and other nutrients and waste products are being exchanged, and is then transported back to the right side of the heart through
the venous system. The right side of the heart pumps the blood to the lungs for
oxygenation, which then enters the left side of the heart again, closing the loop [6].
3.2
Anatomy of the Aorta
The blood leaves the left ventricle of the heart during systole and is ejected through
the aortic valve into the ascending aorta. After the ascending aorta the blood
deflects into (normally) three larger branching vessels in the aortic arch which
supplies the arms and head, or makes a 180-degree turn and continues through the
descending and thoracic aorta towards the abdomen.
The parts of the aorta all have different shapes, in terms of bending, branching
and tapering, creating different flow fields. The flow behavior in the ascending
aorta is characterized by the flow through the aortic valve, and the curvature can
5
CHAPTER 3. PHYSIOLOGICAL BACKGROUND
Figure 1: Schematic figure of the largest artery in the human body, the aorta.
create a skewed velocity profile. The flow in the arch is highly three-dimensional,
with helical flow patterns developing due to the curvature, and unsteady flows can
be created as a result of the branching vessels. The flow patterns that are created
in the ascending aorta and arch are still present in the descending aorta, where
local recirculation regions may appear as a result of the curvature and bending of
the arch.
The aortic wall is elastic in its healthy state, and will deform due to the increase
or decrease in blood pressure. The wall consists of three layers: intima, media,
and adventitia. Regardless of the contents of each layer, the arterial wall is made
up out of four basic building blocks: endothelial cells, elastic fibers, collagen
fibers, and smooth muscle cells [6]. All blood vessels are lined with a single
layer of endothelial cells that are in direct contact with the blood flow. The elastic
fibers are mainly made up of elastin and are, as the name suggests, responsible for
the elastic properties of the vessel; elastin fibers are capable of stretching more
than 100% under physiological conditions. Collagen fibers on the other hand,
are only capable of stretching 3-4% and together with the elastin, determines the
compliance and distensibility of the artery. Finally, the smooth muscle cells are
muscle fibers which, when activated can contract the wall to change the vessel
diameter and, thus, change blood pressure. When they are relaxed they do not
contribute significantly to the elastic properties. Arteries are thicker than veins
due to a larger amount of smooth muscle cells in the walls.
6
3.3. CARDIOVASCULAR DISEASE AND BLOOD FLOW
3.3
Cardiovascular Disease and Blood Flow
Blood flow characteristics are involved either directly or indirectly in the initiation
and progression of some cardiovascular diseases. In particular, highly oscillating,
disturbed, or turbulent flows, which are uncommon in normal healthy persons, can
introduce adverse effects to the heart or blood vessel [7–10]
It is now common knowledge that blood flow affect the endothelial structure,
which, in turn, may initiate vascular diseases such as atherosclerosis or aneurysms.
Atherosclerosis is an ongoing inflammatory response to local endothelial dysfunction initiated by one or several factors, such as abnormal wall shear stress levels,
hypertension, oxidative stress, and elevated low-density lipoprotein levels [3, 11–
13]. Research on the importance of blood flow in the development of atherosclerosis have been performed since the late 1960’s to early 1970’s [14, 15], but a
complete understanding of the disease is still lacking. The influence of flow on
the endothelial cell layer is believed to be correlated to the development and progression of atherosclerotic disease [9, 10, 15, 16]. Formation and development
of aortic aneurysms are highly dependent on the structural integrity of the arterial
wall, making hemodynamics an important factor when characterizing the biomechanical environment [17]. Aortic dissection is another disease that is highly flow
dependent; the blood flow creates a fake lumen between the intima and media
layers in the wall, causing the formation of a stenosis or even occlusion of the
vessel. Carotid artery dissection is a common cause of stroke among young and
middle-aged persons [18].
Flow characteristics can also be used as an indicator of cardiovascular disease;
a common example is the turbulent blood flow through an aortic valve stenosis,
where the fluctuating pressure levels produce sounds (heart murmurs) that can be
heard in a stethoscope. Normally, turbulent or highly disturbed flow are considered abnormal and are often an indication of a narrowed blood vessel or a stenotic
heart valve, which by decreasing the cross-sectional area increases the flow velocity and triggers a transition to turbulence. The turbulent kinetic energy is a
measure of the amount of turbulent fluctuations, and high values indicates a very
energy ineffective flow, as energy from the mean flow is lost to feed the turbulent fluctuations, which in turn increases the heart work load to maintain the flow
rate [19]. This also applies to constrictions such as coarctations or stenoses, which
introduce additional pressure losses over the constriction. In essence, any flow that
departs from the energy efficient laminar characteristics to a disturbed turbulent
flow, will introduce a higher workload on the heart and vessels.
The force that affects the vessel wall consists of two components: the blood
pressure and wall shear stress. The blood pressure acts in the normal direction to
the wall, while the wall shear stress acts tangentially. Blood pressure is normally
on the order of 1000 times larger than the wall shear stress. However, endothelial
7
CHAPTER 3. PHYSIOLOGICAL BACKGROUND
cells are much more susceptible to the frictional shear force than the pressure,
making them very sensitive to local flow conditions. They have been shown to
align with flow direction if the shear magnitude is steady and large enough, while
they become randomly orientated and take on a cobblestone shape in low or oscillating wall shear regions [8, 20, 21].
Although there are several risk factors (including both environmental, genetic
and biological) linked to the development of atherosclerosis, the disease is often
localized to certain vascular regions, such as in the vicinity of branching or highly
curved vessels and arterial stenoses [22]. These are locations where nonuniform
blood flows are present, creating a locally very complex wall shear stress pattern. Regions experiencing low and/or oscillating shear stress has been shown to
be more prone to develop atherosclerotic lesions [8, 11, 15, 22–27], possibly due
to the fact that the permeability of the endothelial cell layer can be shear dependent [11, 28, 29]. High levels of wall shear stress has been found to be atheroprotective, but a strict definition of high and low values is difficult to define [22].
In addition to low and oscillating shear stress, elevated low-density lipoprotein (LDL) surface concentration and increased particle residence time of the flow
field could promote mass transport into the vessel wall, especially if the wall
permeability is enhanced due to abnormal wall shear stress. Increased particle
residence time occurs in regions with recirculating flow or very slowly moving
fluids [30, 31]. Increased levels of low-density lipoprotein has been shown to
promote the accumulation of cholesterol within the intima layer of large arteries [32, 33]. There is a small flux of water from the blood to the arterial wall,
driven by the arterial pressure difference, which can transport LDL to the arterial wall. The endothelium presents a barrier to LDL, creating a flow-dependent
concentration boundary layer. This concentration polarization is interesting, as regions of elevated LDL are co-located with low shear stress regions [13], suggesting a relationship between accumulation and flow dynamics. Studies in humans
and animals indicate that the flux of LDL from the plasma into the arterial wall
depends both on the concentration and the permeability at the plasma-arterial wall
interface [34].
3.4
Medical Imaging Modalities
In clinical practice there are, in general, three different techniques used for cardiovascular imaging: Computed Tomography (CT), Magnetic Resonance Imaging
(MRI), and Ultrasound (US).
CT has the advantage of providing a very high resolution of lumenal geometries, and also has the ability to detect different materials due to the fact that the
absorption of x-rays change with the material. However, CT is based on ionizing
8
3.4. MEDICAL IMAGING MODALITIES
radiation and uses contrast agents to distinguish between the lumen and surrounding tissue. Also, the technique is unable to measure flow, and is therefore not the
method of choice for studies in blood flow research [11].
Ultrasound is non-invasive, has an excellent temporal resolution and is perhaps
the most widely available clinical imaging technique. A high-frequency beam
is transmitted into the body and the resulting echoes are collected and used to
produce an image. Normally, flow velocity measurements only yield one value
per lumenal cross-section. Thus, flow wave forms are normally obtained with the
assumption of a known velocity profile (normally Hagen-Poiseuille), which may
be far from in vivo flow conditions [3, 35]. Additionally, the image quality is very
dependent on the proximity of the transducer to the vessel, making US limited
to superficial vessels. Despite these drawbacks, it is commonly used in clinical
practice as it is a relatively cheap method compared to CT and MRI, making it
useful for screening examinations.
The method of choice for studying cardiovascular flows is MRI [3, 11]. There
are no ionizing radiation and both flow and geometry can be measured. The technique is based on the detection of magnetization arising from the nucleus of hydrogen atoms in water. Radio frequencies are used to change the alignment of the
magnetization vector, and the time it takes for the magnetization vector to return
to its original position after the radio frequency signal has been shut off will indicate what kind of tissue that returns the signal to the receiver. Phase Contrast
MRI (PC-MRI) is a technique that images the flowing blood inside the vessel,
and thereby gives both velocity and geometrical information. The name comes
from the fact that the velocity signal is encoded in the phase of the complex MRI
signal. A common approach is to measure the velocities within a thin slice perpendicular to the blood vessel, and thereby acquiring a spatially resolved velocity
profile. The technique can measure all three velocity components [36], and has
also recently been shown to being able to quantify turbulent kinetic energy [37].
The image material obtained from MRI is useful when making computer models of cardiovascular flows, as it provides both the geometry and proper flow
boundary conditions. Additionally, image data for validation of simulation results
are also made available. However, compared to numerical models the resolution is
coarse, especially near the walls. Wall shear stress would be desirable to measure
with MRI, see e.g. [38], but as the near-wall velocity gradient cannot be resolved
with sufficient accuracy, estimations of MRI-based wall shear stress will be inaccurate [39, 40]. Instead, numerical models are used extensively to resolve local
wall shear stress patterns and other hemodynamic parameters.
9
Chapter 4
Modeling Cardiovascular Flows
4.1
Governing Equations
The motion of a fluid is governed by the following conservation laws [41, 42]:
• Conservation of mass
• Conservation of momentum
• Conservation of energy
The first law states that mass cannot be created nor destroyed. The second law
states that the rate of change of momentum equals the sum of the forces on a fluid
particle, and is described by Newton’s second law. The third law states that the
rate of change of energy is equal to the rate of change of heat addition and work
done on a fluid particle, and is the first law of thermodynamics. If the motion
of a fluid is only affected by phenomena on a macroscopic scale and molecular
effects can be ignored, it is regarded as a continuum. A fluid element therefore
represents an average of a large enough number of molecules in a point in space
and time. On a continuum level, the fluid element is the building block on which
the conservation of mass, momentum, and energy apply to.
Throughout the text an incompressible, isothermal fluid is assumed. This assumption is often justifiable for liquids under normal pressure levels, such as water
and blood. Conservation of mass for an incompressible fluid results in the following volume continuity equation for a fluid element:
∇·u=0
(1)
where u is the velocity vector. Equation 1 implies that an equal amount of mass
that enters a volume also must leave it. Conservation of energy states that total
amount of energy is constant over time within a system or control volume. In an
11
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
isothermal system the conservation law balances the amount of energy lost due to
work done by the system with the change in internal energy of the system.
The rate of change of momentum of a fluid element can be described as:
∂u
ρ
+ u · ∇u
(2)
∂t
where ρ is the density of the fluid and t time. The rate of change of momentum
balances the forces acting on the fluid element. The forces are usually divided into
two types: body and surface forces. Body forces could e.g. be gravity, centrifugal,
Coriolis, or electromagnetical forces, while surface forces are typically pressure
and viscous forces. Body forces are introduced through a source term S, while
surface forces are included through the stress tensor σ. For a Newtonian fluid,
where the viscosity is constant, the stress tensor becomes:
σ = −∇p + µ∇2 u
(3)
which is the sum of pressure and viscous forces. Here p is the pressure and µ the
viscosity of the fluid. The stress tensor and body forces balances the rate of the
change of momentum [42], yielding the Navier-Stokes equations:
∂u
+ u · ∇u = −∇p + µ∇2 u + S
(4)
ρ
∂t
The first term on the left hand side of Equation 4 describes the transient acceleration while the second term is the convective acceleration. The terms on the right
hand side are a pressure gradient, a viscous term, and a source term accounting for
body forces. Together with the continuity equation, Equation 1, and proper initial
and boundary conditions they form a complete description of a fluid’s velocity
and pressure fields, u(x, t) and p(x, t).
4.2
Flow Properties
Flows can be categorized as either steady or transient, and laminar or turbulent. A
steady (time-independent) type of flow can be present in predominantly smaller
arteries in the human body, far away from the pumping heart. But, even in larger
arteries a steady flow assumption can be useful, as it can provide initial insights
and information when modeling physiological flows. However, the flow in the
aorta and other larger vessels are pulsatile due to the pumping motion of the heart,
and a transient approach is therefore needed to accurately capture time-dependent
blood flow features.
12
4.2. FLOW PROPERTIES
An important parameter in fluid mechanics is the Reynolds number (Re),
which is a measure of the ratio of inertial to viscous forces, defined as:
ρU L
Re =
(5)
µ
where ρ and µ have been defined earlier and U and L are characteristic velocity
and length scales in the flow, respectively. In hemodynamic flows, U is the mean
velocity while L is the diameter of the blood vessel. The Reynolds number is
very often used in dimensional analysis when determining dynamic similarities
between two flows, but can also be used to quantify flow regimes. Empirical studies have found that steady, fully developed flow in circular pipes is laminar when
the critical Reynolds number is below a value on the order of 2300 [43]. There is
no well defined limit when the flow is fully turbulent, but it is normally assumed
to be above the critical Reynolds number. For pulsating flows the transition to
turbulence is often at higher critical Reynolds numbers [44], as the accelerating
phase in the pulse tend to keep the flow structured, while the flow breaks up in
disturbed and chaotic features during the decelerating phase.
The flow in larger arteries is often assumed to be laminar [45–48], based on
both experience from measurements and a discussion on the range of Reynolds
numbers present. Fully developed turbulence is rarely seen in healthy humans [49].
However, the flow in the aorta can be in the transitional regime between laminar
and turbulent flow, especially during the deceleration phase where flow instabilities can occur. Measurements based on both hot-wire anemometry and MRI supports the idea that healthy subjects can have transitional or slightly turbulent flows
in the aorta [50–52]. In patients with certain cardiovascular diseases, fully turbulent flows are not uncommon. Normally, an aortic stenosis or stenosed heart valve
will introduce turbulence as direct consequence of the narrowing of the crosssectional area, which, in turn makes the flow velocity increase and trigger turbulence. Heart murmurs are a consequence of turbulent flow, where the severity of
the murmurs is directly linked to the amount of narrowing of the cross-sectional
area in the heart valve.
Despite an intuitive feeling of turbulence and several decades of research on
the topic, a precise definition of turbulence is not easy to define [53, 54]. A turbulent flow contains eddies which are patterns of fluctuating velocity, vorticity and
pressure. These eddies exists over in wide range of scales in both space and time,
where the larger eddies contains the most energy, which is passed down to smaller
and smaller eddies through the cascade process. At the smallest eddies the effect
of viscosity becomes dominant and the energy finally dissipates into heat. Some
of the characteristics of turbulent flow are [53, 54]:
• High Reynolds number: transition to turbulence occur at large Reynolds
numbers, due to flow instability. The non-linear convective term in the
13
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
Navier-Stokes equations becomes dominant over the viscous term, increasing the sensitivity to instability which otherwise is damped by the viscous
term. This is evident in high Reynolds-number flows, which are in essence
inviscid.
• Randomness: a turbulent flow has a very large number of spatial degrees of
freedom and is unpredictable in detail, but statistical properties can be reproducible if the turbulence is considered ergodic, i.e. that statistical properties
can be calculated from a sufficiently large realization.
• Wide range of scales: turbulent flows contain a wide range of spatial and
temporal scales, with spatial scales superimposed in each other. Energy
is transferred from the large energy-carrying scales through the cascadeprocess to the small scales where it is dissipated into heat by viscosity. The
smallest scales are several orders of magnitude larger than the molecular
free path, making turbulence a continuum phenomenon. The dynamic behavior of the flow involves all scales.
• Dissipation: As energy is dissipated (smeared out) at the smallest scales by
viscosity, a continuous supply of energy from the larger scales is needed to
feed the turbulence. If the energy is cut off the flow will return to a laminar
state as the Reynolds number decrease.
• Diffusivity: Turbulent flow is highly diffusive, as indicated by the increase
of mixing and diffusion of momentum and heat transfer.
• Nonlinearity: small disturbances in a well-structured flow can grow fast and
result in an unstable flow.
• Small-scale vorticity: as vorticity is defined as the curl of the velocity field,
the derivatives will depend on the smallest scales of velocity, making the
spatial scale of vorticity fluctuations the smallest in the turbulent range of
scales. This scale is called the Kolmogorov scale and here the energy input
from the larger scales are in exact balance with the viscous dissipation.
Due to the random behavior of turbulence, it usually needs to be treated with
statistical tools [54]. For non-laminar but not fully turbulent flows, it can be said
to be disturbed, and a laminar flow assumption may not be suitable, as transitional
effects can play a major role in the flow behavior. In addition, cardiovascular
flows are often pulsatile and the effects from both inertial and viscous forces are
important. The flow in large vessels is highly three-dimensional and can have
strong secondary flows, especially in diseased vessels [11].
14
4.3. FLOW DESCRIPTORS
0
10
20 [mm]
Figure 2: CFD simulation of a turbulent flow field in an aortic coarctation. Flow is from
left to right and strong velocity gradients are present as a result of the turbulent fluctuations.
4.3
Flow Descriptors
Flow can be described and quantified in a large number of ways, and here a few
descriptors for turbulent flows are presented. A flow variable φ can be decomposed into a mean and a fluctuating part, as:
φ(x, t) = φ(x) + φ0 (x, t)
(6)
where the overbar represents the mean value and the prime the fluctuating part.
Further, the mean or time average of the flow variable is defined as:
1
φ(x) =
∆t
∆t
Z
φ(x, t)dt
(7)
0
where ∆t is a sufficiently long time. The (time) average of the fluctuating part, by
definition, is zero:
Z ∆t
1
0
φ (x) =
φ0 (x, t)dt ≡ 0
(8)
∆t 0
The spread of the fluctuating part φ0 (x, t) around the mean φ(x) can be described
by the variance and root-mean-square (RMS) values:
(φ0 (x, t))2
1
=
∆t
Z
∆t
(φ(x, t) − φ(x))2 dt
(9)
0
15
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
φ0RM S (x, t) =
q
s
(φ0 (x, t))2 =
1
∆t
Z
∆t
(φ(x, t) − φ(x))2 dt
(10)
0
where the decomposition in Equation 6 has been used. A turbulent velocity fluctuation in a steady flow is plotted in figure 3, where it is obvious that while the
mean value of the fluctuations is zero (by the definition, Equation 8), the RMS
values are not.
Velocity [m/s]
1
0
u’
u’± urms
−1
0
0.05
0.1
Time [s]
0.15
0.2
Figure 3: Example of a fluctuating velocity u0 in a steady flow. The mean value is zero,
while the amount of fluctuations is described by the RMS values.
In a pulsating flow, the decomposition in Equation 6 instead becomes:
φ(x, t) = hφi(x, t) + φ0 (x, t)
(11)
where hφi(x, t) represents phase-average instead of the time-average. The phase
average operator h.i is defined as:
hφi(x, t) =
N −1
1 X
φ(x, t + nT )
N n=0
(12)
where N is the number of cardiac cycles and T the (constant) period of the cardiac cycle. Thus, the phase-average is the mean value of φ over N number of cycles, at each time during the cardiac cycle. Note that the decompositions defined
here are valid for any flow variable, including wall shear stress. A purely laminar flow does not exhibit any random fluctuations and the decomposition would
therefore not render any fluctuating components. Besides variance and RMS, the
third and fourth order moments (known as skewness and kurtosis) can be obtained
by changing the exponents in Equation 9 from 2 to 3 or 4, respectively.
The RMS of a velocity fluctuation can be measured experimentally, and in
cardiovascular applications it can be performed in vivo using hot-film anemometry [50, 55] which is a very invasive process. More recently, MRI techniques have
been used to estimate the RMS values [56]. Other descriptors used to quantify
16
4.3. FLOW DESCRIPTORS
Velocity [m/s]
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
Figure 4: Example of a temporal velocity signal during a cardiac cycle in a point in a
constricted aorta. Notice how the velocity fluctuates in the deceleration phase in systole
and the early parts of diastole, while being essentially undisturbed in the other parts of the
cardiac cycle. In order to quantify the amount of disturbances in a numerical model, several cardiac cycles are needed to compute a phase-average, as described by Equations 11
and 12.
the flow are the turbulent kinetic energy k (or sometimes referred to as TKE) and
turbulence intensity Ti . The turbulent kinetic energy is defined as:
1 k ≡ ρ u02 + v 02 + w02
(13)
2
and represents the mean kinetic energy of the turbulent fluctuations in the flow. In
the turbulent jet after an aortic coarctation, the turbulent kinetic energy can locally
be on the order of 1000 Pa, while the kinetic energy is on the order of 5000 Pa, see
e.g. Paper IV. The turbulence intensity is defined as the magnitude of the velocity
fluctuations to a reference velocity:
q
2
k
3
(14)
Ti =
uref
Values on the order of 1% is considered low, while 10% is considered high. Finally, the Womersley number α is often used as a measure of the unsteadiness of
the flow, and it is defined as:
r
ω
α=r
(15)
ν
where r is the vessel radius, ω the frequency of the cardiac cycle and ν the kinematic viscosity of blood. For large vessels such as the aorta the Womersley number is in the range of 10-20 [57, 58], while it decreases significantly in the smaller
vessels. For a large Womersley number the velocity profile is blunt, as the effect
of viscosity does not propagate far from the wall. Therefore, in a highly pulsatile flow or in large vessels, the velocity profile takes on a plug shape, while for
17
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
smaller vessels where the Womersley number is small the velocity profile more
closely resembles the classical Poiseuille profile [57].
4.4
Geometrical Representation
Image segmentation can be defined as the transformation of image material into
a geometrical representation, such a CAD surface. A variety of segmentation
methods exist, which ranges in complexity from simple thresholding of image
intensity to advanced pattern-recognition methods based on images of anatomical
features.
In this thesis, the image material obtained from MRI measurements was transformed into a CAD surface using a cardiac image analysis software (Segment,
http://segment.heiberg.se) [59]. It uses a level-set algorithm, with seed points
placed at a few locations to increase segmentation speed and accuracy. Filtering
functions were introduced to get a smooth surface, but was used with care; the final geometry closely resembled the original geometry and volume was conserved.
Smaller vessels were not included, as the resolution was not sufficient to resolve
them, and it was assumed that they did not contribute to the overall flow field.
Figure 5: Left: Maximum intensity projection (MIP) of the human heart, aorta and connecting arteries. Center: Resulting CAD surface after segmentation, including the three
largest vessels leaving the aortic arch. Right: Close-up on the computational mesh in the
inlet and ascending aorta. Notice the dense mesh resolution in the near-wall region.
When the CAD surface was created, it was discretized into control volumes. The
size of each control volume (or mesh cell) determines the spatial resolution in the
computational model and is therefore arbitrary. The more volumes in an area, the
higher resolution. This comes of course with a prize, as the computational cost
18
4.4. GEOMETRICAL REPRESENTATION
increases with each volume. In general, a fine resolution is needed in areas with
large gradients, such as near walls where the velocity profile goes from zero at the
wall to the free stream velocity, or in highly disturbed flow regions. Accurate treatment of the near-wall flow is essential, as the shape of the velocity profile at the
wall directly determines the wall shear stress and species concentration boundary
layer [5]. Figure 5 shows an example of the near-wall resolution of a hexahedral
mesh on the inlet in the ascending aorta, ensuring that the velocity gradient at the
wall is resolved.
The choice of flow model also put demands on the mesh resolution; for scaleresolving models (such as LES) both the local y + value and CFL number must
be below 1, for accurate resolution of the turbulent flow features. The y + is a
dimensionless distance from the wall to the first mesh cell, and a value less than
unity ensures that the first mesh cell is inside the viscous sub-layer in the boundary layer. With a reasonable growth factor on the near-wall mesh elements, the
entire boundary layer becomes resolved. The shape of mesh elements also affect solution accuracy; tetrahedral meshes are easy to create but might suffer from
excessive numerical dissipation in shear layers or if they are highly skewed. Hexahedral mesh elements are harder to create in a complex geometry such as the
aorta, but the mesh quality is greatly improved and errors due to numerical diffusion is decreased compared to a tetrahedral volume. Also, in general, the memory
requirement and calculation time per tetrahedral mesh element is 50% more than
a hexahedral cell [60].
When fluid-structure interaction was simulated (Paper III), the mesh resolution
was coarser compared to the meshes when using a LES turbulence model. This
was because a RANS approach was employed (see Section 4.9), which would not
benefit from a finer resolution, as the turbulent effects were modeled instead of
resolved. In addition, the arterial wall was meshed with 65 000 elements, with
three elements dividing the wall thickness, putting serious demands on computer
resources. Table 1 summarizes the details of the meshes used in this thesis.
Table 1: Details about the meshes used in each of the papers in this thesis.
y+
Paper
Application
Type of mesh
# of cells
I
II
III
IV
V
LES
LES
RANS+FSI
LES
LES
Hexahedral
Hexahedral
Hexahedral
Hexahedral
Hexahedral
6 000 000
<0.2
5 000 000
0.1-1
500 000 + 65 000 <1.5
5 000 000
0.1-1
7 000 000
0.05-0.5
19
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
4.5
Blood Properties
From an engineering point of view, blood can be a very complex fluid. It consists
of platelets and red and white blood cells suspended in a plasma. There has been
significant research aimed at developing a constitutive model for all features of
blood flow, but to date a full description is still not complete. The plasma acts
as a Newtonian fluid with a constant viscosity, but due to the cellular content
the whole blood acts as a non-Newtonian fluid, at least in small vessels and low
shear rates. However, in large arteries such as the aorta, non-Newtonian effects
are small and can generally be ignored. In general, blood behaves as a homogeneous Newtonian fluid in vessels with a diameter larger than 1 mm and shear rates
over 100 s−1 [6, 58]. Some popular non-Newtonian models are the Power-law,
Casson, and Carreau-Yashuda models, which describes the relationship between
blood viscosity and shear thinning as nonlinear, see Figure 6. For shear rates above
100 s−1 , which are normally found in the aorta, the Casson and Carreau-Yashuda
models approaches the Newtonian viscosity, which is normally set to 0.0035 Pa.s.
Blood can be considered as incompressible with a density in the range of 10501060 kg/m−3 [61]. In this thesis the blood has been considered as Newtonian, as
the shear rates were high enough and the cross-section of the aorta is large enough
to prevent non-linear effects.
−1
Apparent Viscosity [Pa s]
10
Power Law
Casson
Carreau−Yasuda
Newtonian
−2
10
−3
10
−1
10
0
10
1
10
Shear rate [1/s]
2
10
3
10
Figure 6: Examples of shear rate dependency in three non-Newtonian blood viscosity
models and a Newtonian viscosity for comparison. Notice how the Casson and CarreauYashuda models approaches the Newtonian viscosity at high shear rates.
20
4.6. BOUNDARY CONDITIONS
Boundary Conditions
4.6
The flow in a CFD simulation is driven by the boundary conditions, and accurate
treatment is one of the most challenging problems when modeling cardiovascular
flows [11, 62]. The boundaries of the computational domain are usually represented by inlets, outlets, and walls.
Inlets
Inlet velocity [m/s]
Inlets are often prescribed with a time-dependent mass flow rate or velocity profile. Fully developed velocity profiles has been used extensively in the past, but
this assumes a very long straight vessel upstream the location of the inlet, making undeveloped profiles very likely to exist in vivo [5]. Instead, as MRI can
measuremen both geometry and flow, velocity profiles can be used as input for a
patient-specific model [11]. Figure 7 shows three examples of measured velocity
profiles in the ascending aorta, at maximum systolic acceleration, peak systole,
and maximum deceleration. In this measurement the spatial resolution was about
25x25 pixels per cross-section and the temporal resolution 40 frames per cardiac
cycle, or about 25 ms per frame. As the resolution in a CFD simulation usually
is much smaller than that, some sort of interpolation technique in space and time
must be employed.
1
1
1
0.5
0.5
0.5
0
0
0
Figure 7: Velocity profiles measured by MRI in the ascending aorta. From left to right:
maximum systolic acceleration, peak systole, and maximum deceleration.
It is obvious that when patient-specific flows are of interest, measured velocity
profiles should be included in addition to vessel geometry, to ensure that the correct flow field is being simulated. Normally only velocity profiles are specified,
allowing turbulent quantities or unsteadiness develop from the transient velocity
profile.
Outlets
A very common boundary condition is to use pressure on outlets. Since it is only
the gradient of the pressure that is present in the incompressible Navier-Stokes
21
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
equations, Equation 4, the absolute pressure level in the model has to be specified.
Therefore, pressure boundary conditions with the static pressure equal to zero is
often used, making the pressure inside the model relative to that on the outlet, i.e.
the pressure inside the domain will be an implicit result of the pressure on the
outlet. If there are several outlets, the pressure values will determine the amount
of flow through each outlet [11], and physiological waveforms should therefore be
specified. In an incompressible model with rigid walls the wave speed becomes
infinite, as opposed to a finite wave speed if the walls are allowed to deform due
to the flow. Then, the wave speed can be approximated with the Moens-Korteweg
equation:
s
PWV =
Eh
2rρ
(16)
where P W V is the pulse wave velocity, E is the Young’s modulus, h the wall
thickness, r the radius of the vessel and ρ the fluid density. The underlying assumptions of the equation are that the vessel wall is thin, that the fluid is incompressible and that the wall stiffness is constant [63].
Prescribed pressure boundary conditions are only suitable when the wall is
assumed to be rigid and the actual pressure level can be ignored. For FSI simulations, or when trying to predict the outcome of an intervention where the pressure
is not known a priori, other methods are needed [5]. The use of Windkessel models [64, 65] or more advanced arterial tree models [66] can be used to take into
account the effects of wave reflections and attenuation of the pressure pulse from
downstream locations. These lumped-parameter or 1-D models can account for
both the peripheral and systemic resistance together with compliance effects of
the downstream vessels [11].
A simple model is the 3-element Windkessel model, represented by an electrical analog as a resistance in series with a resistance and a capacitance in parallel. Using a mass flow pulse as input, the Windkessel model can respond with a
physiological pressure pulse. The values for the resistances and the capacitance
determines the pressure pulse wave form and magnitude, and some sort of tuning
or optimization is often needed [5, 63, 67].
Walls
The boundary conditions for the arterial walls can be assumed rigid, have a prescribed motion, or deform as a consequence of the fluid pressure. The walls can
also be modeled as either smooth or rough, to account for surface roughness.
Common for all wall boundary conditions in cardiovascular applications is that a
no-slip condition is obeyed, meaning that frictional forces will create a boundary
layer along the wall. Vessel walls are also normally assumed impermeable, at least
22
4.7. FLUID-STRUCTURE INTERACTION
Figure 8: Left: electrical analog of a 3-element Windkessel model with two resistances R1
and R2 representing peripheral and systemic resistance and a capacitance CS representing
vessel compliance. Right: Schematic figure of where the windkessel models can be used
as outlet boundary conditions.
when only the flow is of interest. One exception is when studying mass transfer
and the effect from blood flow, as in Paper IV. There, a passive (non-reacting)
scalar representing low-density lipoprotein (LDL) was transported in the blood
and allowed to exit the flow domain through the wall. In that way it was possible
to correlate flow features to LDL surface accumulation.
When performing FSI-analysis, extra treatment of the wall boundary is needed,
as it needs to be constrained in space. It is common to allow wall motion normal
to the surface while preventing axial motion at the in- and outlets [67]. The outer
side of the wall is affected by the surrounding tissue and organs [3, 68, 69], which
may been needed to taken into account in a model. As a first approximation a
linear spring support can be used, which at the same time acts as a damper, see
Paper III. The influence from larger parts of the body such as organs and the spine
should be included, but is often not. One exception is the work of Moireau et al.
who demonstrated a FSI-model which took into effect from the spine, surrounding
tissue and organs in the thoracic cavity [70].
4.7
Fluid-Structure Interaction
The coupling of fluid and solid models is referred to as fluid-structure interaction,
or FSI. Solving the FSI-equations can be done either in a monolithic or iterative
way. In a monolithic solution, both the fluid and solid equations are solved in a
single matrix, while in an iterative solution, forces and displacements are passed
between two solvers through a common interface. The coupling can be one-way
or two-way, where the former means that data is transferred from one solver to
23
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
the other, while in the latter data is transferred both ways between the solvers in a
loop.
An example of an 1-way FSI case is when the fluid forces is passed to the solid
model to calculate stresses and strains, but the deformations can be considered
small enough to not influence the fluid. Therefore, information is only passed
from the fluid to the solid case. In a two-way FSI case, the fluid forces is passed
to the solid model which, in turn, responds by passing the resulting displacement
back to the fluid model. In that way the fluid domain deforms and the new forces
are passed to the solid model in an iterative manner.
Force
Force
Fluid solver
Solid solver
Fluid solver
Solid solver
Displacement
Figure 9: Schematic figure illustrating the data transfer in one-way and two-way fluidstructure interaction.
The governing equation for the solid can be described by the equation of motion:
ρs
∂ 2 ds
= ∇ · σ s + fs
∂t2
(17)
where subscript s denote solid, and ρs is the wall density, ds represents the displacement vector, σs is the Cauchy stress tensor, and fs is an externally applied
body force vector. Boundary conditions on the FSI interface states that the velocities of the fluid and solid must be compatible and that the traction at the boundaries
are in equilibrium. These boundary conditions are formulated as:
us = uf
σ s · n̂s = σ f · n̂f
(18)
(19)
where u are velocities, σ are stress tensors and n̂ surface normals. Together with
a constitutive relationship that relates the stress to the strain, the equation system
can be solved.
Fluid-structure interaction has recently begun being used when modeling cardiovascular diseases. Some common applications are computing the flow in aortic
aneurysms [71–79], stenotic arteries [80–84] and heart valves [85, 86]. Modeling
deformation of the arterial wall requires knowledge about the wall structure. As
described in Chapter 3, the arterial wall consists of several layers, each of which
has different mechanical properties. Stresses and strains in the wall are related
through a constitutive equation, where the most simple relationship is Hooke’s
24
4.7. FLUID-STRUCTURE INTERACTION
Law which relates the stress to the strain times a material constant called the
Young’s Modulus. Researches have used both this linear relationship [83, 87–89],
and also non-linear constitutive models such as Mooney-Rivlin [77, 86, 90–92],
Ogden [81, 93], and Fung [94].
However, the non-linear constitutive models require knowledge about the residual stress that is present in real arterial walls, even in an unpressurized state. As
the models often are based on experimental testing of actual real arteries, the (often unknown) residual stresses should be included in the model to yield reliable
results. A shrink-stretch process can be employed, as in [95], where the wall is
first shrunk and then pressurized with a diastolic blood pressure until it matches
the original geometry. Besides problems with residual stresses, the wall material parameters should be considered patient-specific with different behaviors in
healthy and diseased locations, making constitutive modeling a complex task.
In large healthy vessels such as the aorta, the wall is often modeled with an linear constitutive law [96], which, despite the approximations made, might be better
than a rigid wall assumption. In this way, wall motion is included in the model
without the problems that comes with residual stresses in a non-linear model.
Therefore, as a first approximation the material mechanics of the wall was considered linear in this thesis, with a Young’s modulus on the order of 1 MPa. Paper III
investigates the effect on the flow field from the wall motion using different values
of the wall stiffness.
Another way of including wall motion in the simulation is to not model it at all,
but instead prescribe a measured displacement [46, 97]. In that way the difficulties
associated with residual stresses can be overcome and the correct wall motion is
directly incorporated into the model. Besides the fact that stresses in the wall
is not obtained with this method, there are other difficulties; the motion must be
based on measurements (typically MRI, CT or US) all of which has a lower spatial
and temporal resolution compared to the numerical model. Therefore, some sort
of interpolation in time and space has to be employed, as the location of the wall
between measurements is unknown.
Besides prescribing the wall motion, the effect of compliant vessels can be
approximated by modeling the fluid as a compressible fluid, while keeping the
walls rigid, as in [67]. They modeled the flow in a healthy aorta with FSI, rigid
walls and as a compressible fluid. The compressible fluid was tuned to get the
correct wave speed and their results showed that modeling the fluid to account
for compliance yielded very similar results to a full-scale FSI simulation. The
computational overhead was very low compared to the FSI simulation, suggesting
that this kind of modeling might be useful in clinical practice.
25
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
4.8
Mass Transfer
Besides blood flow, the transport of species can be included in a flow model.
Blood can be modeled as multi-phase fluid where the interaction between all of
the different components are accounted for. Particle flow can be either interacting
with each other or non-interacting. A straightforward model for mass transport is
to treat the species as a passive non-reacting scalar and solving a transport equation in addition to the Navier-Stokes equations:
∂C
+ ∇ · (uC) = ∇ · (D∇C)
∂t
(20)
where C is the species concentration, u is the computed fluid velocity which is
now known from the CFD simulation, and D the kinematic diffusivity. In this
manner the mass transport is assumed to not affect the flow features, but is instead
an implicit result of it - the species are just transported with the flow. Depending
on the values of u and D the transport can be either dominated by convection or
diffusion, and similar to the Reynolds number, the Péclet number quantifies the
ratio of convection to diffusion:
Pe =
LU
D
(21)
where L is a characteristic length and U the velocity magnitude. When the Péclet
number is larger than unity, convective effects dominate the mass transfer while
for smaller values, diffusion controls the mass transfer. Species concentration is
therefore a function of P e [98]. In large blood vessels the transport is mainly
dominated by convection, while in smaller arteries the diffusion can have a significant impact. The relative thickness of the hydrodynamic boundary layer to the
concentration boundary layer can be quantified with the dimensionless Schmidt
number, defined as:
ν
(22)
Sc =
D
It is a measure of the ratio of momentum (viscous) diffusivity to the mass (molecular) diffusivity and is a function of fluid properties only. For gases the Schmidt
number is on the order of unity, meaning that the hydrodynamic and concentration
layers are equally thick, while for liquids the Schmidt number is several orders of
magnitude higher, making the concentration layer very thin [99]. This puts additional constraints on the near-wall computational mesh resolution, as both boundary layers needs to be resolved. The concentration and hydrodynamic boundary
layers are illustrated in Figure 10. The Schmidt number is analogous to the Prandtl
number in convection heat transfer.
26
4.9. MODELING TURBULENT FLOW
In paper IV the transport of low-density lipoprotein (LDL) is modeled in a
human aorta and the transport from the blood and to the arterial wall was investigated. As the actual aortic wall was not included in the simulation, a boundary
condition which allows for mass transfer through the wall was set on the arterial
surface. The net transport of LDL from the blood to the wall was modeled as the
difference between the amount of LDL carried to the wall as water filtration and
the amount diffusing back to the bulk flow. Due to the filtration flow of water
into the wall and LDL rejection at the endothelium, a concentration polarization
effect appears at the luminal side of the vessel wall. This boundary condition was
modeled as:
∂C = Kw Cw
(23)
Cw V w − D
∂n w
where Cw is the concentration of LDL at the wall, Vw the water infiltration velocity, D the kinematic diffusivity, ∂C/∂n the concentration gradient normal to
the wall, and Kw a permeability coefficient for the wall. Numerical values for the
coefficients were found in literature [100–102].
δ
δc
Figure 10: Schematic figure illustrating the hydrodynamic boundary layer thickness δ and
the concentration boundary layer thickness δc of LDL particles. Relative boundary layer
thickness is not to scale.
4.9
Modeling Turbulent Flow
The Navier-Stokes equations can fully describe all the details of any flow situation, making them very powerful even though they look relatively simple. But,
as explained by Pope [43], their power is also their weakness, as all of the flow
details are described, starting from the largest energy carrying turbulent scale governed by geometry, to the smallest scale where dissipation of energy occurs. Given
the importance of turbulent flows in engineering applications, substantial research
has been put into the development of numerical methods to capture the effects of
turbulent flow features without needing to resolve all of the details. In general,
the methods can be ordered in three groups: RANS, LES, and DNS, where the
27
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
two first models employ different approaches to avoid resolving all flow features,
while in DNS all turbulent scales present in the flow are computed. Therefore, the
computational cost also increases with the amount of details; RANS models are
significantly cheaper to run compared to LES, while LES is significantly cheaper
to run compared to DNS [54].
RANS Turbulence Models
In a RANS turbulence model the effort is put into modeling the effect of the turbulence on the mean flow quantities. This can be useful if the goal of a simulation
is the mean or time-averaged values of e.g. velocity, but details about the turbulent fluctuations are not resolved. The decomposition introduced in Equation 6 is
applied to the velocity and pressure as:
ui = ui + u0i
p = p + p0
(24)
(25)
Again, overbar represents an averaged quantity, while the prime denotes a fluctuating variable. The Navier-Stokes equations are then averaged, which yields an
additional term called the Reynolds stress tensor that needs to be modeled in order
to close the equations. In tensor notation, the Reynolds-Averaged Navier-Stokes
(RANS) equations are:
∂u0i u0j
∂
1 ∂p
∂
∂ui ∂uj
∂ui
+
(ui uj ) = −
+ν
+
−
(26)
∂t
∂xj
ρ ∂xi
∂xj ∂xj
∂xi
∂xj
The right hand side can be rewritten as:
1 ∂
∂ui ∂uj
0 0
−pδij + µ
+
− ρui uj
ρ ∂xj
∂xj
∂xi
(27)
where δij is the Kronecker delta. The first term inside the square brackets represents the mean pressure stress, while the second term is the mean viscous stress
tensor. The last term is the Reynolds stress tensor, normally denoted τij . It is a
fictitious stress tensor and represents the average momentum flux due to turbulent
velocity fluctuations. In fully developed turbulence, the Reynolds stresses can be
several orders of magnitude larger than the mean viscous stress tensor [54]. As τij
is unknown, additional information needs to be introduced in order to close the
equations. In a RANS approach, this is usually done through the eddy-viscosity
hypothesis proposed by Boussinesq in 1877, which relates the Reynolds stresses
to the mean rates of deformation and a eddy-viscosity, as:
∂ui ∂uj
2
τij = kδij − νt
+
(28)
3
∂xj
∂xi
28
4.9. MODELING TURBULENT FLOW
where k = τkk /2 or the turbulent kinetic energy, and νt the eddy (or turbulent)
viscosity. Combining Equations 26 and 28 yields:
∂ui
∂
∂
∂ui ∂uj
1 ∂pt
+
(ui uj ) = −
+
(ν + νt )
+
(29)
∂t
∂xj
ρ ∂xi ∂xj
∂xj
∂xi
where pt is a modified pressure: pt = p + 23 k. Closure is obtained by relating the
eddy viscosity to the turbulent kinetic energy k and the turbulent dissipation rate
or the turbulence frequency ω, as:
k2
k
νt =
ω
νt = C µ
(30)
(31)
where Cµ is a dimensionless constant. This means that two additional transport
equations are introduced for either k and or k and ω, forming the k − [103, 104]
and k − ω [105] turbulence models. The additional transport equations contains
model constants that have been adjusted to be as general as possible. Each of the
models exists in a variety of versions, all with different strengths and weaknesses.
The basic k − model has been shown to be useful in the free-stream but does
not perform well in regions with adverse pressure gradients. Contrary, the k − ω
model performs well close to the wall but is very sensitive to the value of ω in
the free-stream [42]. Menter [106] and Menter el al. [107] proposed a hybrid
turbulence model called SST k − ω that combined the best features from both the
k − and k − ω models. In the inner parts of the boundary layer it uses the k − ω
formulation, while it blends gradually to the k − formulation in a free-stream.
A more advanced type of turbulence model that also use the RANS equations
is the Reynolds stress model (RSM), but it does not make use of the eddy-viscosity
approach to close the equations. Instead, the Reynolds stresses are computed directly using additional equations for each Reynolds stress component and with
an equation for either or ω [43, 108]. The RSM model is particularly useful
in highly swirling flows with anisotropic turbulence, which can be found in e.g.
turbomachinery and compressors. However, the additional equations makes the
RSM model more computationally expensive and harder to converge compared to
the standard two-equation RANS models, and is therefore rarely used in simulations of cardiovascular flows.
Large Eddy Simulation
Contrary to RANS, in a LES model the large scale turbulent motion of the flow
is resolved, while the flow scales smaller than a filter width, e.g. the grid size,
29
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
is handled with a subgrid-scale model (SGS). The rationale behind LES is that
the large scales transport most of the momentum, mass and energy, and are also
more problem-dependent, i.e. more sensitive to the geometry and boundary conditions compared to the smaller scales, which become more universal and isotropic
and thus easier to model [109]. Therefore most of the effects of the turbulent
fluctuations can be resolved, and, in addition, LES models are better at handling
transitional flows compared to RANS models [54]. The Navier-Stokes equations
are filtered instead of averaged, and in a finite volume approach the scales smaller
than the control volume are normally handled by the SGS model. Filtering the
Navier-Stokes equations yields:
∂
1 ∂ pe
∂
∂e
ui ∂e
uj
∂e
ui
+
(ug
+ν
+
(32)
i uj ) = −
∂t
∂xj
ρ ∂xi
∂xj ∂xj
∂xi
where the tilde represents filtering and the term ∂x∂ j (ug
i uj ) contains terms that are
unknown. The following decomposition can be made [42]:
ug
ei u
ej + τij
i uj = u
(33)
where τij denotes the subgrid-scale stress. Applying Equation 33 to Equation 32
yields
∂e
ui
∂
1 ∂ pe
∂
∂e
ui ∂e
uj
∂τij
+
(e
ui u
ej ) = −
+ν
+
−
(34)
∂t
∂xj
ρ ∂xi
∂xj ∂xj
∂xi
∂xj
Unlike the Reynolds stresses in a RANS formulation, the sub-grid stresses in a
LES formulation contains further information [42]. The stresses can be decomposed into a resolved and a unresolved part [110]:
τij = Lij + Cij + Rij
(35)
where Lij are the Leonard stresses due flow features on a resolved scale, Cij are
the cross stresses that arise from interaction between sub-grid eddies and resolved
flow, and Rij are the LES Reynolds stresses that arise from convective momentum transfer due to sub-grid eddies. However, while Rij are invariant with respect
to Galilean transformation, Lij and Cij are not [111], making the decomposition
dependent on coordinate system. Therefore, in most practical applications the
Leonard and cross-stresses are lumped together with the LES Reynolds stresses,
and τij is solved in an eddy-viscosity approach similar to the RANS formulation [42, 109]. The subgrid-stresses are modeled using the filtered strain rate
tensor and the sub-grid eddy viscosity νsgs as:
1
∂e
ui ∂e
uj
τij = τkk δij − νsgs
+
(36)
3
∂xj
∂xi
30
4.9. MODELING TURBULENT FLOW
Combining Equations 34 and 36 yields the LES equations with sub-grid viscosity:
∂e
ui
∂
1 ∂ Pe
∂
∂e
ui ∂e
uj
+
(e
ui u
ej ) = −
+
(ν + νsgs )
+
∂t
∂xj
ρ ∂xi ∂xj
∂xj
∂xi
(37)
Here the isotropic part of τkk is not modeled, but instead added to the filtered static
pressure: Pe = pe + τkk /3. The SGS model is needed to introduce an additional
physically correct dissipation rate (in forms of eddy viscosity) into the system,
as the smallest scales are not resolved. Therefore, LES does not model the actual influence of the unresolved scaled, but instead the dissipation of turbulence
into heat [54, 109]. Additionally, the SGS model must account for the interaction
between filtered and unfiltered scales, which require some sort of empirical information, as a general description of turbulence is unavailable [43]. Thus, a true
separation of scales is not possible, as there is interaction between the smallest
and largest scales. In order to solve Equation 36, models for νsgs are needed. In
this thesis two different SGS models have been used: the Dynamic SmagorinskyLilly model [112, 113] and the Wall-Adapting Local Eddy-Viscosity (WALE)
model [114]. Both correctly provides zero eddy-viscosity in laminar shear layers,
which is important as cardiovascular flows can undergo transition from a purely
laminar state to a disturbed and transitional behavior, and the model should not
affect the laminar flow.
The eddy-viscosity in the WALE model is formulated as:
νsgs = (Cwale ∆)2
(Sijd Sijd )3/2
(Seij Seij )5/2 + (Sijd Sijd )5/4
(38)
where Cwale is a model constant set to 0.5 based on results from homogeneous
isotropic turbulence, ∆ is the cube root of the computational cell volume, and Sijd
is the traceless symmetric part of the square of the velocity gradient tensor. The
e ij as:
Sijd tensor can be rewritten in terms of filtered strain-rate Seij and vorticity Ω
e ik Ω
e kj − 1 δij (Semn Semn − Ω
e mn Ω
e mn )
Sijd = Seik Sekj + Ω
3
(39)
Details about the Dynamic Smagorinsky-Lilly model are presented in Paper I.
As a comparison between the RANS and LES models, notice how similar Equations 26 and 28 looks to Equations 34 and 36. The differences are the averaging or
filtering operation and the formulation of the eddy-viscosities νt and νsgs . These
subtle differences may greatly affect the resulting flow field, as LES is more likely
to resolve flow features that are instead modeled in a RANS model. While νt accounts for the entire influence of turbulence on the mean flow, νsgs is only relevant
for the scales smaller than the filter size. The numerical value of the eddy viscosity
31
CHAPTER 4. MODELING CARDIOVASCULAR FLOWS
in a LES model is very small compared to the eddy viscosity in a RANS model,
making it confined to the smaller scales, leaving the larger scales to be resolved
by numerics [54]. The sub-grid scale stresses only accounts for a fraction of the
total stresses, making the overall flow field less sensitive to any modeling effect
in the sub-grid scale model, compared to the modeling in a RANS turbulence
model [111]. The eddy viscosity ratio, defined as the ratio of the eddy viscosity
to fluid viscosity, can interesting to compute; for values less than unity, the dissipation in the flow model is handled mainly by actual viscous dissipation, while
larger values indicates the need of additional dissipation obtained from the SGS or
RANS turbulence model. In this thesis, eddy viscosity values were on the order of
0.5-10 for LES simulations, while it was on the order of 100-1000 for the RANS
simulations in Paper III.
Given the increased resolution and smaller amount of turbulence modeling,
LES may be well suited for computing cardiovascular flows [115], whereas common two-equation RANS models may be unsuitable or even unable to capture
transitional and relaminarizing flows [3, 5]. However, modeling flows with LES
comes at a high prize; the mesh and time step requirements in a LES simulation far
exceeds the requirements needed for a RANS simulation, and, as a result the computational cost for a LES simulation can be orders of magnitude higher in terms of
memory and computational time. The highest mesh requirements lie in resolving
the boundary layers, which limits the use of LES in wall-bounded high-Reynolds
number flows [43]. LES will likely not be able to give exactly the same results
as DNS due to the sub-grid scale modeling, which affects higher-order statistics
more than lower-order. Thus, compared to DNS, LES is normally reliable for first
and second order moments and is able to reproduce basic flow structures [111], to
a significantly lower computational cost.
Direct Numerical Simulation
In a direct numerical simulation, DNS, all flow scales are computed and there is
no modeling performed at all. The Navier-Stokes equations are solved on meshes
that are fine enough to resolve all relevant length scales, and the temporal resolution is fine enough to capture all transient fluctuations. Besides spatial and temporal resolution, a DNS normally use high-order numerical schemes to minimize
numerical errors. Altogether, this comes with an enormously large computational
cost, making it unpractical for most engineering types of flows, at least with the
current supercomputer capacity. There are a few exceptions, where one example is
Varghese et al. who performed DNS simulations of both steady and pulsatile flows
in an idealized stenotic pipe [116, 117]. High-order DNS with patient-specific geometries and boundary conditions might become a reality in the future, but with
the present computer power and numerical methods, only simple idealized models
32
4.9. MODELING TURBULENT FLOW
can be solved.
To conclude, there is little hope of finding an analytical theory of turbulence
and current research is therefore aimed at developing numerical methods to calculate the relevant properties of turbulent flow [43].
33
Chapter 5
Results
5.1
Quantification of Aortic Wall Shear Stress
Papers I-III
Low and/or oscillating wall shear stress has been established as predictors of
increased risk for the development of atherosclerosis [9, 10, 30, 118, 119]. In addition, turbulent flow has been shown to affect endothelial cell function [8], making turbulent flow and abnormal WSS interesting when studying cardiovascular
diseases. In these papers, computational fluid dynamics were used to resolve the
WSS on the aortic surface, as clinical measurement techniques have been shown
to be unable to measure the near-wall velocity gradient with sufficient resolution [39, 40].
First, steady flow in an idealized circular vessel with a stenosis was investigated using LES. It is an well-known geometry (see Figure 11 and [120] for details) where both experimental measurements and DNS data are available. In an
article by Gårdhagen et al. [121], it was shown that a LES model could replicate
results from both DNS and experimental Laser Doppler Velocimetry, indicating
that LES is suitable for this kind of physiological flows.
Figure 11: The geometry of the idealized vessel used in Paper I. Flow is from left to right
and the length downstream the stenosis is 20 diameters to ensure that outlet effects does
not affect the flow.
The fluid enters the idealized vessel as a laminar well-structured flow, but in the
throat of the stenosis a transition to turbulence is triggered due to the increased
35
CHAPTER 5. RESULTS
velocity. The flow situation is similar to in vivo flows such as aortic coarctations
or stenotic heart valves, where turbulent flow often is present. To take cardiovascular flow quantification a step further, the turbulent WSS signal on the walls was
decomposed into a mean and a fluctuating part (denoted with overbar and prime,
respectively). Sampling was performed at every diameter downstream the stenosis
and the decomposition reads:
WSS = WSS + wss0
(40)
It was found that the magnitudes of the mean and the fluctuating parts could be
almost as large in the reattachment region, where most disturbed flow was found.
The fluctuations decreased in magnitude further downstream, while the mean part
approached a constant value, as turbulent effects reduced. Conversely, a RANSmodel showed no or very low WSS magnitude in the reattachment region, reflecting the RANS models inability to predict turbulent parameters, as well as the need
for a scale-resolving turbulence model in these types of flows. Also, as fluctuations are modeled and not resolved in RANS models, a decomposition into a mean
and a fluctuating part was not feasible.
The mean and fluctuating parts were further decomposed into axial and circumferential components. It is evident from the right plot in Figure 12 that there
is a recirculation region from D =1-6, where the mean axial component is negative,
then changes sign in the reattachment point and becomes directed in the main flow
direction. The fluctuating components cannot discern the flow direction, as they
fluctuate around the mean value. As expected from a well converged simulation
the mean circumferential part was zero, i.e. there was no preferred circumferential flow direction that would influence the magnitude. However, the fluctuating
circumferential component was always larger than the axial component showing
that even though, on average, the mean value is zero the fluctuating values can be
significant.
Clearly, turbulent flow creates complex WSS patterns with significant fluctuations, which can impair endothelial function. It was therefore decided to apply the
WSS decomposition to the flow field in a real human aorta. Here, the flow was pulsating and phase-averages (denoted with <.>) were computed to get statistically
convergent results. Surface renderings of the phase-averaged WSS magnitude
<WSS>, and the corresponding fluctuations WSS’ at maximum systolic deceleration are plotted in Figure 13. Large mean WSS values are obtained around the
branches and locally in the descending aorta, while fluctuating WSS are present
on the inner curvature of the descending aorta and locally in the branching vessels.
However, there are large spatial (and temporal) gradients of mean and fluctuating
WSS on the aorta, and as discussed in [122], these gradients can injure the endothelial cells by causing e.g. high cell turnover, leaky cell junctions, enhanced
permeability, or even cell-cell bond rupture.
36
5.1. QUANTIFICATION OF AORTIC WALL SHEAR STRESS
15
15
WSS ax
|WSS|
|wss|’
|WSS| RANS
WSScirc
10
wss’ax
wss’circ
10
WSS [ − ]
WSS [ − ]
5
0
5
−5
0
1
5
10
D
15
20
−10
1
5
10
D
15
20
Figure 12: Left: Normalized WSS magnitude of mean and fluctuating parts of the LES
model together with normalized WSS magnitude of the RANS model at the downstream
region of the stenosis. The reattachment point is located close to D = 5 diameters downstream the throat of the stenosis, which is where the LES model predicts the maximum
values, while the RANS model predicts the lowest value. Right: Decomposition into axial and circumferential mean and fluctuating WSS components. As expected is the mean
circumferential component zero.
In order to quantify the WSS during a complete heart beat, two locations upstream and downstream the third branching vessel on the aortic arch were selected
and the decomposition were plotted over time together with the time-averaged
WSS, see Figure 14. The results illustrate that while instantaneous mean WSS values can take on values that are significantly higher than the time-averaged WSS,
there are times during the cardiac cycle when the fluctuations also can become
elevated, especially during systole.
The oscillatory shear index (OSI) is a common measure of the cyclic departure
of the WSS vector from its predominant alignment [123]. It is an integrated quantity, and here it was used in a novel way together with the time-averaged WSS as
locations with elevated values of both variables were mapped back onto the aorta.
In general, the OSI parameter only reached the maximum values when the timeaveraged WSS was low, which is in line with other studies [48]. However, there
were a few locations that experienced both elevated TAWSS and OSI values, and
these points were mapped back onto the aorta. It was found that regions exhibiting
both high TAWSS and OSI values were located in the vicinity of the branching
vessels in the aortic arch, and on the inner curvature of the descending aorta.
These locations are common sites of development of atherosclerosis [16, 123].
Here it was assumed that TAWSS values higher than 2 Pa was considered high if
there was elevated OSI values at the same location.
37
CHAPTER 5. RESULTS
Figure 13: Surface rendering of phase-averaged WSS magnitude (upper row) and fluctuating WSS (lower row) at maximum systolic deceleration. Notice the different color scale
between magnitude and fluctuation.
38
5.1. QUANTIFICATION OF AORTIC WALL SHEAR STRESS
<WSS>
WSS’
<TAWSS>
25
20
WSS [Pa]
WSS [Pa]
20
15
15
10
10
5
5
0
0
<WSS>
WSS’
<TAWSS>
25
0
0.2
0.4
0.6
Time [s]
0.8
1
0
0.2
0.4
0.6
Time [s]
0.8
1
Figure 14: Two examples of instantaneous mean WSS (phase average, <WSS>), the
corresponding fluctuating WSS (WSS’) and the time-averaged WSS (<TAWSS>) at two
locations in a healthy human aorta. Data were sampled over several cardiac cycles to get a
statistically convergent result. Left plot is located in front of one of the branching vessels
in the aortic arch, while the right plot is located behind the same vessel.
Figure 15: Left: Time-averaged WSS plotted against oscillatory shear index. Values
inside the dotted box were mapped back onto the aorta. Right: Locations on the aorta
where elevated TAWSS and OSI values are present.
39
CHAPTER 5. RESULTS
The effect of wall compliance on the blood flow and WSS was investigated by
performing FSI simulations on a healthy aorta by changing the wall stiffness. FSI
modeling is a very computational intensive process due to the iterative matter in
which information is passed between between a fluid and a solid solver. Therefore,
turbulent effects were modeled with a RANS turbulence model. The goal was not
to resolve all the flow details, but instead try to assess whether how wall motion
and propagating pressure/mass waves affected the overall WSS distribution. Three
FSI cases were simulated with a Young’s modulus of 0.5, 0.75 and 1.0 MPa. These
cases where compared with two rigid wall models which were extracted from the
0.5 MPa model at peak systole and late diastole, to fully investigate the effect of
wall compliance, but also to assess whether the rigid wall assumption is sensitive
to the measurement technique. As the geometry reconstruction is created from an
average of several cardiac cycles during the MRI measurement, the wall motion
will be blended in the MR images. Ideally, MRI measurements of the geometry
would be performed at different times during the cardiac cycle, yielding a specific
geometry at each time point. As this was not the case, the acquired geometry will
most probably be diastolic dominant, as approximately 2/3 of the cardiac cycle is
during diastole. In the FSI models the effect from the tissue surrounding the outer
part of the aortic wall was modeled as an elastic spring.
As seen in Figure 16, time time-averaged WSS are almost identical between
the five models, indicating that wall motion and finite wave speeds does not significantly affect time-averaged WSS, which is in line with other studies, see e.g. [67].
On the other hand, instantaneous WSS values were different between the rigid
wall and FSI models, indicating that if instantaneous values are important to assess
endothelial function, FSI modeling might be needed to fully capture and resolve
near-wall details.
40
5.1. QUANTIFICATION OF AORTIC WALL SHEAR STRESS
(a)
(d)
(b)
(c)
(e)
Figure 16: Time-averaged WSS on five different models. Three FSI simulations with
Young’s modulus 0.5, 0.75 and 1.0 MPa correspond to cases a-c, while cases d and e are
two rigid wall models at sampled at peak systole and late diastole, respectively. The effect
of wall motion has little effect on the time-averaged WSS.
41
CHAPTER 5. RESULTS
5.2
Aortic Mass Transfer and LDL
(Paper IV)
The transport of low-density lipoprotein (LDL) was simulated as a passive
(non-reacting) scalar using LES. The walls were permeable, allowing LDL to be
transported from the blood and to the wall. With this boundary condition, a concentration polarization effect appeared, where regions of elevated LDL surface
concentration were found to be co-located with areas of low shear stress, suggesting a relationship between LDL accumulation and flow dynamics. The results
Figure 17: Upper row: WSS distribution on the arterial surface. Lower row: Normalized
LDL distribution on the arterial surface. From left to right: maximum acceleration, max
deceleration, beginning of diastole, and mid diastole. Regions of elevated WSS seems to
correspond to a decrease in LDL surface concentration.
indicated that LDL accumulation was inversely proportional to the WSS magnitude, so in order to futher assess the relation between WSS and LDL surface
concentration, surface data from 50 consecutive cardiac cycles were plotted in the
same figure, yielding approximately 2.5 billion data points to be post-processed,
see Figure 18. Clearly, the WSS is inversely proportional to the LDL surface
42
5.2. AORTIC MASS TRANSFER AND LDL
concentration, as low WSS values can give elevated LDL levels, and vice versa.
As the concentration levels are normalized with the inlet value, the figure shows
that the surface has a higher LDL concentration compared to the bulk blood flow.
The increase is approximately 5-10% of LDL on the surface, with some regions
experiencing almost 25% higher concentration levels.
Figure 18: Normalized LDL values plotted against Instantaneous WSS magnitude on the
entire aortic arch. In total, 50 cardiac cycles were used, yielding approximately 2.5 billion
surface data points which are all present in the figure.
In addition, near-wall flow flow affects both the velocity gradient (and thus
the WSS) and the LDL concentration, as laminar flow regimes tend to create a
stable concentration boundary layer while disturbed flow creates fluctuations in
concentration levels. This was visualized by plotting the velocity magnitude along
a line 5 mm normal to the surface into the flow domain for each time point in the
cardiac cycle. By doing so, a spatio-temporal map of the near-wall flow was
created. It was concluded that while the velocity closest to the wall was almost
stationary, flow features at fractions of a mm from the wall greatly affected the
WSS and LDL distribution, see Figure 19. Thus, the concentration boundary
layer is sensitive to both the wall shear stress and flow effects from the bulk flow.
43
CHAPTER 5. RESULTS
Figure 19: Upper figure: LDL surface concentration and wall shear stress as a function
of time, at a point on the outer side of the descending aorta. Lower figure: The near-wall
velocity along a line directed normal to the surface at the same location. X-axis legend
reads MA: systolic maximum acceleration, PS: peak systole, MD: systolic maximum deceleration, BD: beginning of diastole.
44
5.3. TURBULENT FLOW IN AN AORTIC COARCTATION
Turbulent Flow in an Aortic Coarctation
5.3
(Paper V)
The blood flow in a patient with an aortic coarctation was investigated before
and after intervention. The intervention increased the cross-sectional area of the
coarctation, resulting in a decreased pressure drop and increased flow rate. The
turbulent kinetic energy (TKE) was evaluated using both MRI measurements and
LES simulations, to complement traditional diagnostic tools when assessing the
severity of the coarctation. Large TKE values indicate that energy is drawn from
the mean flow to feed the turbulent fluctuations. In order to assess whether the
current MRI methodology is able to estimate TKE in vivo, a comparison between
MRI and CFD was performed. Integrated values in a volume distal the coarctation was considered, and a very good agreement was found; the increase and
decrease in TKE levels matched almost perfectly for both both the pre- and postintervention models, while peak values differed slightly. The peak TKE values
did not occur at peak flow rate when the Reynolds number reached its maximum
value, but instead during the systolic deceleration phase, when the flow breaks up.
A discussion whether a flow is turbulent or not based solely on Reynolds number
may therefore not be ideal - features of the flow should considered be instead.
Differences between the MRI and CFD results can be a result of measurement erPre−op
Post−op
0.4
10
0.4
CFD
MRI
Mass flow
0.25
0.1
2.5
0
0
−0.05
0.2
0.4
0.6
Time [s]
0.8
−0.2
7.5
∫ TKE [mJ]
5
Mass Flow rate [kg/s]
∫ TKE [mJ]
7.5
CFD
MRI
Mass flow
0.25
5
0.1
2.5
0
0
Mass Flow rate [kg/s]
10
−0.05
0.2
0.4
0.6
Time [s]
0.8
−0.2
Figure 20: Comparison of integrated TKE values between MRI measurements and CFD
results in the pre- and post-intervention model. The mass flow in the ascending aorta is
included as reference.
rors and noise in the MRI-signal, and uncertainties in segmentation and boundary
condition specification in the CFD model. Despite the two different methods to
obtain TKE, the close agreement between CFD and MRI results give confidence
45
CHAPTER 5. RESULTS
that it can be used in a clinical context. However, further clinical studies are
needed to quantify abnormal or dangerous TKE levels and whether integrated or
local peak values are most important. CFD-computed TKE can provide additional
information about the flow field, as the resolution is significantly higher compared
to MRI measurements.
Figure 21: Volume renderings of a CFD simulation (left) and MRI measurement (right)
of the turbulent kinetic energy in the pre-intervention case at peak flow rate.
46
Chapter 6
Discussion
The work described in this thesis focuses on disturbed, transitional and turbulent
flow in the human aorta. The goal was to take image-based CFD of cardiovascular
flows closer to clinical practice, as it is believed that it can provide additional information which is unavailable in traditional measurement techniques. Advanced
models such as large eddy simulation and fluid-structure interaction were employed, to resolve the flow and compute hemodynamic parameters that can be
useful for the understanding of cardiovascular diseases such as atherosclerosis or
aortic coarctations.
Image-based CFD consists of two parts: acquisition of flow and geometry using a clinical image modality, and the actual CFD model. The transformation from
image material to a geometrical representation is today routinely made, and flow
measurements are normally accurate enough to be used as boundary conditions
in the model. Even though fluid-structure interaction and scale-resolving flow
modeling are being used more and more in cardiovascular studies, researchers
still choose from one of the two; the computational cost for a unification of a
scale-resolving turbulence model and FSI is still extremely expensive. This is the
reason why a rigid wall assumption and/or a laminar flow or RANS-approach are
common in the biofluid research community, even if there can be severe modeling
errors and flow effects that are not resolved.
The decomposition of the wall shear stress vector into a mean and a fluctuating
component was first investigated in an idealized stenotic vessel, and then applied
to a subject-specific human aorta. Locations of turbulent WSS could be deducted,
and it was discovered that the fluctuations could be as large as the time-averaged
values during some parts of the cardiac cycle. The described methodology provides additional knowledge about the nature of the stress the endothelial cells are
subjected to, as regions of turbulent WSS correlates with locations that are prone
to atherosclerotic development. However, the relationship between fluctuating
WSS and the development of vascular disease is not fully understood, and require
47
CHAPTER 6. DISCUSSION
further clinical trials. The WSS, and especially its fluctuating component, cannot be measured with significant accuracy using present measurement techniques,
which calls for the need of image-based modeling and simulation. Note that a
scale-resolving turbulence method is needed, as it was shown in Paper I that a
simpler RANS method could yield unphysical results.
Another flow-related feature that cannot be measured with any present measurement technique is the flow-dependent transfer of LDL in the blood stream
and accumulation on the luminal surface. Assuming that LDL can be seen as
a non-reacting passive scalar in the blood, it was found that the blood flow features determine the accumulation and distribution of LDL on the arterial surface.
The near-wall flow (and as a consequence, the WSS) affects the concentration
boundary layer and, again, regions prone to atherosclerotic development seem to
co-locate with regions of high LDL accumulation.
The turbulent kinetic energy before and after intervention was investigated using both CFD simulations and MRI measurements in an aortic coarctation. The
numerical results agreed very well with experimental measurements, indicating
that the measurement technique is robust. The CFD model was able to provide
high resolution details of the turbulent flow field, and with the results it was possible to assess the impact of the intervention from a fluid dynamics point of view. It
was seen that while the pressure drop decreased as a result of the increased crosssectional area, the amount of turbulence (as quantified by TKE) did decrease, but
it did not disappear. Turbulent flow is generally undesirable in the human body,
and should therefore be avoided if possible. Thus, when evaluating the outcome
of the intervention the measured change in turbulence level can be assessed, in
addition to the pressure drop and mass flow rate that is normally considered.
Image-based CFD can provide parameters and variables that cannot be measured. Furthermore, as opposed to all measurement techniques, CFD can also
be used as a predictive tool for treatment planning and intervention optimization.
The combination of simulations and in vivo measurements provide information
that would not otherwise be possible to obtain. Ideally, modeling and simulation
could be used for diagnose, intervention planning and follow-up studies, which
would then require both clinical studies and further development of numerical
methods and models. The two latter, intervention planning and follow-up studies,
are possible today, albeit with some limitations; optimization of surgical procedures using CFD are being done, even though it is still on a small scale. Followup studies would be able to provide data such as pressure and flow distributions,
WSS maps, and TKE renderings that could be useful when assessing the outcome
of a surgery. A very important factor to consider is uncertainty quantification, i.e.
how a small uncertainty in the input to the model (e.g. noisy image data, boundary
conditions, model resolution, etc.) affect the result. This can be done by perform48
ing a large number of simulations with small differences in input to assess the
sensitivity of the output. Hence, a close collaboration in a team of physicians and
experts in fluid mechanics and image modalities will be valuable, to fully exploit
the possibilities of image-based CFD in cardiovascular flows.
The work described in this thesis takes image-based CFD from initial studies
on idealized geometries, to the assessment of the outcome of an intervention on
a patient. The methods have been shown to be robust and accurate, and with the
tight integration of physics and physiology, simulated cardiovascular flows are
now ready for the clinic.
Outlook
The use of image-based CFD in clinical environments is still in its infancy, but
current research shows its importance and the use will certainly increase in the
future. In order to gain confidence that numerical models are both reliable and
provide important information, larger clinical studies should be made, in close
cooperation with clinical staff. Relatively simple follow-up studies can already
be performed, where the flow field can be assessed in detail. Recently, intervention planning using CFD has attracted attention, see e.g. [124–127] for numerical
modeling of intervention planning of extra-cardiac Fontan surgery. The long-term
goal of being able to predict the risk of developing a disease using CFD may become possible in the future, but it will certainly require further understanding (and
mathematical modeling!) of physiological features such as growth and remodeling of arterial walls.
The models discussed in this thesis are mechanical and not biomechanical
models. To be able to predict the initiation and progression of cardiovascular diseases, closer links between chemistry, biology, physiology, and mechanics are
needed [5]. The vascular response in terms of growth and remodeling is important to fully understand pathophysiological mechanisms, calling for the need
to increase the biological relevance in image-based computational modeling and
simulation. Besides introducing more physics and physiology in the models, a
solid understanding of the numerics and requirements for performing a reliable
simulation is, and will continue to be crucial. As eloquently put by Roache [128]:
it is a matter of solving the equations right, while solving the right equations.
49
Chapter 7
Review of Included Papers
Paper I
Quantifying Turbulent Wall Shear Stress in a Stenosed Pipe Using Large
Eddy Simulation, Roland Gårdhagen, Jonas Lantz, Fredrik Carlsson, and Matts
Karlsson, Journal of Biomechanical Engineering, 2010, vol 132, 061002
This paper focuses on non-pulsating turbulent flow through an idealized stenotic artery. Both RANS and LES models were used and the WSS vector was decomposed in a novel way into a mean and a fluctuating part. It was found that the
turbulent WSS fluctuations were almost as large as the mean WSS in the immediate post-stenotic region, while the fluctuations decreased gradually further downstream. Another finding was that a simpler RANS-approach predicted completely
different results compared to the LES model, indicating that RANS modeling of
stenotic flows might not be suitable.
Paper II
Quantifying Turbulent Wall Shear Stress in a Subject Specific Human Aorta
Using Large Eddy Simulation, Jonas Lantz, Roland Gårdhagen, and Matts Karlsson, Medical Engineering & Physics, 2012, 34, 1139-1148
This paper uses the decomposition introduced in Paper I in a subject-specific
human aorta, and investigates how the mean and fluctuating WSS components
changes at various locations in the aorta. Large eddy simulation was used and
50 cardiac cycles were simulated to ensure a statistically reliable result. Both elevated mean and fluctuating WSS values were found in the vicinity of the branches
on the aortic arch, and the time-averaged WSS was found to be inversely proportional to the oscillatory shear index.
51
CHAPTER 7. REVIEW OF INCLUDED PAPERS
Paper III
Wall Shear Stress in a Subject Specific Human Aorta - Influence of FluidStructure Interaction, Jonas Lantz, Johan Renner, and Matts Karlsson, International Journal of Applied Mechanics, 2011, Vol. 3, No. 4, 759-778
In this paper the WSS distribution in a healthy human aorta was investigated
using fluid-structure interaction. Three different values of the wall stiffness were
used, to investigate the influence of the wall motion. In addition, two rigid wall
models with different sizes were also included, to be able to estimate the influence
of the rigid-wall geometries. Comparisons between simulation results and MRI
measurements of descending aortic velocity profiles indicated a very good agreement between the simulations and measurements, while both the time-averaged
WSS and the oscillatory shear index seemed unaffected by the wall motion. However, instantaneous WSS values were affected by wall motion, indicating that if
instantaneous WSS values are important, it might be necessary to include fluidstructure interaction.
Paper IV
Large Eddy Simulation of LDL Surface Concentration in a Subject Specific
Human Aorta, Jonas Lantz and Matts Karlsson, Journal of Biomechanics, 2012,
vol. 45, 537-542
The mass transport of low-density lipoprotein (LDL) from the blood to the
arterial wall was simulated and the influence of near-wall velocity patterns were
investigated. An inverse relationship between time-averaged WSS and LDL surface concentration was found, where regions with high shear had a low surface
concentration. It was also found that there were large temporal changes of LDL
during a cardiac cycle, with decreasing values during systolic acceleration and a
build-up phase during systolic deceleration and systole.
Paper V
Numerical and Experimental Assessment of Turbulent Kinetic Energy in an
Aortic Coarctation, Jonas Lantz, Tino Ebbers, Jan Engvall, and Matts Karlsson,
(submitted for publication)
The turbulent kinetic energy in an aortic coarctation before and after intervention was assessed using both MRI and CFD. It was shown that while the mass
flow rate increased (and, thus, also the Reynolds number) after intervention as
a result of decreased pressure drop, global TKE levels decreased. This suggests
52
that peak Reynolds number might not be suitable for estimation of the amount of
turbulence in cardiovascular flows, but instead a quantity such as TKE could be
considered. In addition, integrated values of TKE distal the coarctation showed
very good agreement between the experimental data and numerical results. This
means that MRI-measured TKE might be useful when determining the outcome
of an intervention, while CFD results complement the experimental data, as the
resolution is significantly higher in the numerical model.
53
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