towards subject specific aortic wall shear...
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Linkoping Studies in Science and TechnologyDissertation No. 1360
Towards Subject Specific AorticWall Shear Stress
- a combined CFD and MRI approach
Johan Renner
Division of Applied Thermodynamics and Fluid MechanicsDepartment of Management and Engineering
Linkoping University, SE-581 83, Linkoping, Sweden
Linkoping January 2011
Cover:Cartoon of blood flow influence on the arterial wall in form of Wall Shear Stress.The Greek letterτ (“tau”) with index wall is the common nomenclature for WallShear Stress.
Towards Subject Specific Aortic Wall Shear Stress- a combined CFD and MRI approach
Linkoping Studies in Science and TechnologyDissertation No. 1360
Printed by:LiuTryck, Linkoping, SwedenISBN 978-91-7393-244-8ISSN 0345-7524
Distributed by:Linkoping UniversityDepartment of Management and EngineeringSE-581 83, Sweden
c© 2011 Johan Renner
No part of this publication may be reproduced, stored in a retrieval system, or betransmitted, in any form or by any means, electronic, mechanic, photocopying,recording, or otherwise, without prior permission of the author.
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Qui audet adipiscitur
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To Linda!
For your patience!
Abstract
The cardiovascular system is an important part of the human body since it trans-ports both energy and oxygen to all cells throughout the body. Diseases in thissystem are often dangerous and cardiovascular diseases is the number one killerin the western world. Common cardiovascular diseases are heart attack and stroke,which origins from obstructed blood flow. It is generally important to understandthe causes for these cardiovascular diseases. The main causes for these diseasesis atherosclerosis development in the arteries (hardening and abnormal growth).This transform of the arterial wall is believed to be influenced by the mechanicalload from the flowing blood on the artery and especially the tangential force thewall shear stress. To retrieve wall shear stress information in arteries in-vivo ishighly interesting due to the coupling to atherosclerosis and indeed a challenge.
The goal of this thesis is to develop, describe and evaluate an in-vivo methodfor subject specific wall shear stress estimations in the human aorta, the largestartery in the human body. The method uses an image based computational fluiddynamics approach in order to estimate the wall shear stress.
To retrieve in-vivo geometrical descriptions of the aorta magnetic resonance imag-ing capabilities is used which creates image material describing the subject spe-cific geometry of the aorta. Magnetic resonance imaging is also used to retrievesubject specific blood velocity information in the aorta. Both aortic geometry andvelocity is gained at the same time. Thereafter the image material is interpretedusing level-set segmentation in order to get a three-dimensional description of theaorta. Computational fluid dynamics simulations is applied on the subject specificaorta in order to calculate time resolved wall shear stress distribution at the entireaortic wall included in the actual model.
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This work shows that it is possible to estimate subject specific wall shear stressin the human aorta. The results from a group of healthy volunteers revealed thatthe arterial geometry is very subject specific and the differnt wall shear stress dis-tributions have general similarities but the level and local distribution are clearlydifferent. Sensitivity (on wall shear stress) to image modality, the different seg-mentation methods and different inlet velocity profiles have been tested, whichresulted in these general conclusions:
• The aortic diameter from magnetic resonance imaging became similar tothe reference diameter measurement method.
• The fast semi-automatic level-set segmentation method gave similar geome-try and wall shear stress results when compared to a reference segmentationmethod.
• Wall shear stress distribution became different when comparing a simplifieduniform velocity profile inlet boundary condition with a measured velocityprofile.
The method proposed in this thesis have the possibility to produce subject spe-cific wall shear stress distribution in the human aorta. The method can be usedfor further medical research regarding atherosclerosis development and has thepossibility for usage in clinical work.
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Popularvetenskaplig beskrivning
Har foljer en popularvetenskaplig sammanfattning som beskriver avhandlingensinnehall och vilken betydelse resultaten kan ha for framtiden.
Detta arbete beskriver hur man med hjalp av avancerad flodesberakningsteknikoch bildinformation av en specifik manniskas stora kroppspulsader (aortan), kanuppskatta friktionskrafterna som det flodande blodet utovar pa blodaderns karlvagg.Kunskap och tillforlitliga data om denna friktionskraft kan underlatta forstaelsenkring hur aderforkalkning utvecklas och uppstar i karlvaggen. Att forsta up-pkomsten och utvecklingen av aderforkalkning ar viktigt, eftersom hjart- ochkarlsjukdomar (exempelvis hjartinfarkt och stroke) i mycket hog grad beror avaderforkalkning. Hur aderforkalkningsprocessen gar till kan inte annu fullstandigtbeskrivas och forklaras. Sedan lange vet man dock att faktorer sasom bl.a. rokn-ing, hogt blodtryck och blodfetter kan paverka denna process, men om dennatyp av globala faktorer vore de enda, sa skulle fordelningen av aderforkalkadeomraden vara jamnt utspridda i manniskans karl. Sa ar inte fallet utan omradenmed aderforkalkning ar mycket specifikt lokaliserade, till exempel i narheten avforgreningar i blodkarlssystemet. Detta har gjort att man tror och man har avenindikationer pa att friktionsbelastningen kan spela en viktig roll vid utvecklingoch lokalisering av aderforkalkning.
Teorin om sambanden mellan friktionskraft och aderforkalkning grundades franborjan pa flodesexperiment genomforda pa modeller av blodkarl och utifran ex-perimenten drog man slutsatser om kopplingar mellan olika faktorer. Dessa forsokutfordes i borjan av 1970-talet och sedan dess har man forsokt fordjupa kun-skapen. Idag ar den allmanna uppfattningen att lokala omraden med laga ochvarierande (i bade riktning och tid) friktionskrafter pa karlvaggen ar ett riskomradefor utveckling av aderforkalkning. Undersokningar av friktionskraftens paverkanhar ibland gjorts med relativt grova metoder. Till exempel har friktionskraftenfor ett tvarsnitt av blodkarlet uppskattats med hjalp av flodesteorier som blandannat baseras pa ett flode som inte varierar med tiden samt att geometrin (roret)uppstroms matningen ska vara rakt och 50-100 diametrar langt. Dessa forenklin-
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gar stammer mycket daligt med flodessituationen i ett blodkarl i manniskokrop-pen. Eftersom utveckling av aderforkalkning visar sig vara mycket lokal sa ar be-hovet av detaljerat uppskattade friktionskrafter mycket stort. Utvecklingen inomde omraden som ingar i detta arbete (exempelvis flodesberakningar och medicinskbildinsamling) har utvecklats kraftigt under de senaste aren. Genom att kombineradessa ar det nu mojligt med den foreslagna metoden att uppskatta friktionskraftenmed mycket god detaljgrad och specifikt for en enskild individ. Avhandlingenbeskriver och utvarderar kansligheten for den utvecklade metoden, som pa in-dividspecifik niva uppskattar friktionskraften pa karlvaggen. Metoden innefattar:bildinsamling med magnetresonanskamera, bildbehandling (tolkning av bildmate-rialet), dar en tredimensionell geometri av aktuellt karl skapas, och flodesberakn-ingsteknik anvands utgaende fran denna geometri for att ta fram friktionskraften.
Den direkta framtida anvandningen av denna metod ar att stodja den medicin-ska forskningen kring aderforkalkning. Framtida klinisk anvandning av metodentill nytta for enskilda individer kan vara att gora uppskattningar av friktionsfordel-ningen fore och efter kirurgiska ingrepp i karlsystemet. Vidare in i framtiden kantankbara anvandningsomraden vara att i forvag planera och optimera ingrepp samtatt bedoma eller diagnostisera en specifik individs riskomraden for karlsjukdomar.
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Acknowledgements
I deeply appreciate the valuable, numerous and refreshing ideas as well as side-tracks, which have served as motivation and fuel for me throughout this work,from my supervisor ProfessorMatts Karlsson. To my medical co-supervisor Pro-fessorToste Lanne, thank you for your valuable medical insights, knowledge andimportant comments of my work. To my colleague M.Sc.Roland Gardhagen- thank you for all fruitful discussions and teamwork as well as your stamina insolving technical issues to get forward in the research.
Another for me important person is PhDJonas Stalhand - thank you for all inspir-ing discussions regarding: research, academic issues as well as teaching and ped-agogy strategies. I also want to acknowledge Professor EmeritusDan Loyd forhis stamina in reviewing and discussing this text, and to make the engineering ap-proach to fluid mechanics and thermodynamics pragmatically clear to me. I wantto give special thanks to Associate ProfessorTino Ebbers for providing both dataand knowledge regarding magnetic resonance imaging and to PhDEinar Heiberg(Lund) for his valuable expertise in image segmentation and providing softwarefor such tasks. I also want to thank M.D.Daniel Modin for impressive staminain performing manual segmentation and M.Sc.Hossein Nadali Najafabadiforthe efforts in meshing and computational fluid dynamics setup in paper IV. I sendmy gratitude’s to all coworkers at the division of Applied Thermodynamics andFluid Mechanics for the valuable discussions as well as creating an excellent andinspiring working environment.
To be able to perform my research the hardware infrastructure needed was themagnetic resonance imaging capabilities at the Center for Medical Image Visual-ization and simulation (CMIV) and computational power at the National Super-computer Center (NSC), Linkoping.
Finally I must say that I would never have completed this work without all thevaluable moments in life which I have shared with my wonderful wifeLinda andthe furry companionLudde.
Linkoping, January 2011Johan Renner
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This work was supported by grants from:
Swedish Research Council (VR):
• VR-NT: Non-invasive estimation of Wall Shear Stress in the aorta and thelarger vessels. Dnr 2004-3803
• VR-M: Mechanical Properties of the Vessel Wall - Importance for Cardiacfunction. Dnr 2005-12661
• VR-SNIC: A framework for non-invasive estimation of wall shear stress inthe aorta and the larger vessels. SNIC 005/06-72, SNIC 025-08-7
Linkoping University (LiU):
• LiU: Strategic Research in Medical Image-Science & Visualization, SMIV,Project VAMOS-Vascular Modeling and Simulation
• LiU: Efficient methods for parameter estimation from velocity data CENIIT 99.11
Others:
• Swedish - Heart-Lung Foundation
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List of Papers
This thesis is based on the following six papers, which will be referred to by theirRoman numerals:
I. Gardhagen R, Renner J, Lanne T, Karlsson M,Subject Specific Wall ShearStress in the Human Thoracic Aorta, WSEAS Transaction on Biology andBiomedicine, ISSN 1109-9518, Issue 10, Volume 3, October 2006.
II. Svenssonτ J, Gardhagen R, Heiberg E, Ebbers T, Loyd D, Lanne T,Karlsson M,Feasibility of Patient Specific Aortic Blood Flow CFD Sim-ulation, Proceedings ofMICCAI (Medical Image Computing and Computer-Assisted Intervention), Copenhagen Denmark, ISBN 3-540-44707-5, 2-4October, 2006.
III. Modin D, Renner J, Gardhagen R, Ebbers T, Karlsson M, Lanne T,Evaluation of Aortic Geometries created by MRI data in Man,Submitted.
IV. Renner J, Nadali Najafabadi H, Modin D., Lanne T, Karlsson M,Wall Shear Stress Estimations using Semi-Automatic Segmentation,Submitted.
V. Renner J, Gardhagen R, Heiberg E, Ebbers T, Loyd D, Lanne T, Karlsson M,A Method for Subject Specific Estimation of Aortic Wall Shear Stress,WSEAS Transaction on Biology and Biomedicine, ISSN 1109-9518,Issue 3, Volume 6, July 2009.
VI. Renner J, Loyd D, Lanne T., Karlsson M,Is a Flat Inlet Profile Sufficientfor WSS Estimation in the Aortic Arch? , WSEAS Transactions on FluidMechanicsISSN: 1790-5087, Issue 4, Volume 4, October 2009.
Articles are reprinted with permission, and have been reformatted to fit the layoutof the thesis.
τ My last name was changed to Renner in August 2006.
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Contents
Abstract v
Popularvetenskaplig beskrivning ix
Acknowledgements xi
List of Papers xv
Contents xvii
Nomenclature xix
1 Introduction 1
2 Aim 7
3 Overview 93.1 Modeling cardiovascular blood flow . . . . . . . . . . . . . . . . 93.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Method 15
5 Influencing Factors 195.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Flow and Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Results and Discussion 23
7 Outlook 31
8 Review of Papers 33
xvii
CONTENTS
Bibliography 35
Paper I 43
Paper II 59
Paper III 71
Paper IV 89
Paper V 111
Paper VI 129
List of Figures 143
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Nomenclature
CFD Computational Fluid Dynamics Techniques for solving the flow equations numericallywith computer.
MRI Magnetic Resonance Imaging Imaging technique that uses magnetic fields to createimages of the inside of the human body.
MIP Maximum Intensity Projection Image view technique that combines all image slicesin one to display an overall view.
WSS Wall Shear Stress The frictional force that a flowing fluid influencesa surface with.
FSI Fluid Structure Interaction By coupling fluid and solid solvers the fluid and structuredynamic interaction behavior can be captured.
P Posterior Location in the human body which describesthat the location is to the back.
L Left Location in the human body which describesthat the location is to the left.
A Anterior Location in the human body which describesthat the location is to the front.
R Right Location in the human body which describesthat the location is to the right.
CS Cross Section3D Three-Dimensional
BMI Body Mass Index kg/m2
u Velocity component in x-direction m/su Velocity vector m/sp Pressure N/m2
m Mass flow rate kg/sµ Dynamic Viscosity kg/msρ Density kg/m3
τwall Wall Shear Stress N/m2
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Chapter 1
Introduction
The blood is the main transportation system in the human body which gives thepossibility to transport oxygen and energy to all the cells in the human body. Ar-teries and veins build a transportation network where arteries are the blood vesselsheading out of the heart and veins are the vessels heading back to the heart. Thethree main parts of this transportation system are the vessels (network), the blood(transportation fluid) and the heart (pump/motor). This transportation system isvery crucial and obstructions of the blood flow may be lethal. Occlusions in thecoronary arteries (supplying blood to the heart) can cause a myocardial infarction(heart attack). Another dangerous scenario is when carotid arteries (supplyingblood to the brain) are occluded which causes cerebral infarction (stroke). In2007, the total mortality in Sweden was 91 820 of whom 37 960 were due tocirculatory diseases, mainly myocardial infarctions and cerebral infarctions [47].In general are approximately 50% of the deaths in the Western world today areoriginating from atherosclerosis, which makes the disease to be the number onekiller in this part of the world [43]. It can clearly be stated that the function of thecardiovascular system is crucial and interferences in blood flow function are notdesirable.
Time
Figure 1: Atherosclerosis development in artery. The lumen (inner) diameter remainsunchanged in early phases and in later phases are the atherosclerotic plaques reducing thearterial lumen diameter.
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CHAPTER 1. INTRODUCTION
The most common underlying factor to circulatory diseases is atherosclerotic dis-ease and the development of atherosclerotic plaques. In early phases there will bean outward remodelling of the vessel while with time the lumen (inner) diameteris reduced, see figure 1. The atherosclerotic process is prolonged in time and earlystages are already found in children (fatty streaks) but the clinical implications willmainly be detected in middle age to elderly. Briefly described is the atheroscle-rosis development starting withfatty streakswhich can develop into next stagewhich is fibrofatty plaque. So far is the influence created by the atherosclero-sis mainly clinically unnoticed, further development including e.g. inflammation,plaque growth and calcification will create avulnerable plaque. This stage canfurther lead to three different clinical phases:I. rupture of the artery,II. occlu-sion by thrombus andIII. critical stenosis which prevents blood flow, graphicallydescribed in figure 2.
Normal Artery
Fatty Streak
Fibrofatty Plaque Vulnerable Plaque
I. Rupture II. Occlusion byThrombus
III. Critical Stenosis
Figure 2: Phases in atherosclerosis development in arteries, I-III describes clinical phases.
General risk factors are well known today e.g. smoking, high cholesterol lev-els in the blood, hypertension (high blood pressure) and lack of physical activity.All these factors are seen from an overview and/or biochemical perspective. Ifthese factors would be the only factors influencing the atherosclerosis genesis the
2
development of atherosclerotic plaques would become rather uniform in the ar-terial system. However, this is not the case. It has been shown [21, 27] that theatherosclerosis development clearly is non-uniformly distributed in the arteries.Typical atherosclerotic sites are at bifurcations in the arterial tree as well as at theinside of an arch. This fact clearly gives questions, if the flow situation locally inthese sites favours atherosclerotic development? Such sites have indeed a com-plex flow situation and especially since the heart creates a pulsating flow enteringthe arterial system. This means that the hemodynamic load on the arterial wall istime dependent (both in magnitude and direction) and indeed very complex. Inorder to define such risk sites it is interesting to look at hemodynamic factors thatinfluence the arterial wall. The wall shear stress, WSS, is such a factor, which canbe described as the friction that the flowing blood induces on the arterial wall.
Since the beginning of the seventies the WSS role in atherosclerotic developmenthas been investigated. At first the connection between WSS and atherosclerosiswere contradictory postulated as high respectively low WSS [19, 10]. Today it isrecognized that low and oscillating WSS must be taken into consideration whenlooking for atherosclerotic risk sites [7, 53, 23, 12]. The part of the arterial wallthat is influenced the of WSS is principally the innermost part of the wall, whichconsists of a singel cell layer thick, the endothelium. This is built of endothe-lial cells which are 1-2µm thick and have a size about 10-20µm. These cellsregulate the transport of substances and substances to and from the rest of thearterial wall and they are very sensitive to frictional surface loading (WSS) [37].Due to the mechanical load, the endothelium cells can remodel into different ge-
Arterial Wall
Endothelial Cells
y
x
u(y)τwall = µ
[
∂u∂y
]
wall
Figure 3: Schematic image of WSS loading (τwall) on endothelial cells at the inside ofthe arterial wall.
ometries [27]. The geometrical form of endothelial cells is normally elliptical.However, both elliptical and circular endothelial cells have been found in differ-ent kinds of frictional load situations [37]. A more circular shape, seems to bean indicator of some dysfunction as discovered by Ross and Glomset [45, 44].Further, the thin layer of cells plays an important role in atherosclerotic develop-ment. Earlier, the endothelium was generally believed to be a non-active barrier
3
CHAPTER 1. INTRODUCTION
between the blood and the rest of the arterial wall. The breakthrough by Ross andGlomset showed that this layer is very active indeed and it has a regulatory func-tion for the arterial wall, which also links atherosclerosis directly to the functionalresponse of the endothelium. This is constantly regulated due to local WSS load-ing (figure 3) [37]. The coupling between atherosclerosis and WSS was furtherinvestigated by [15, 26, 30, 56, 51, 31].The research disciplines concerning in-vivo WSS estimation continuously pro-duces new and improved methods (both experimental and computational) andtools to retrieve the details of WSS in the human arteries. This is indeed useful be-cause the research field of estimating WSS in the cardiovascular system in humanseasily becomes very complex. To be able to make estimations of the WSS insidethe human body a range of disciplines must be involved e.g. medicine, physics,fluid mechanics, non-invasive data acquisition and image processing. This chainof actions needed for the WSS estimation method will be further addressed asthe workflow. The workflow includes Magnetic Resonance Imaging (MRI) mea-surement of the artery in focus, image segmentation to produce a 3D model of theartery, Computational Fluid Dynamics (CFD) modeling, CFD simulation and postprocessing the CFD results, see figure 4.
Subject MRI Measurement CFD SimulationSegmentation WSS
Figure 4: Schematic image of the subject specific WSS estimation method
The focus of this thesis will mainly be the fluid mechanics part of the workflowwhich includes the CFD actions. The human aorta has been used in the studies.The aorta is the largest artery in the human body and it is directly attached to theheart. From the heart the aorta distributes the flowing blood through the branchesin the aortic arch that supply the head and arms with blood. Distally to the aorticarch the aorta supply the rest of the body with blood and the main branching di-vide the blood to the kidneys and the two legs. The aortic section used for WSSestimations in this work is the upper part of the human aorta starting imediatelydistal (downstream) to the coronary arteries including the aortic arch as well asthe branches to the head and arm, see figure 5.
4
Figure 5: The aorta the largest artery in the human body, dashed lines mark boundaries ofthe part in focus.
Generally speaking the arterial system is from my perspective as a fluid mechanicsengineer nothing else then a piping system with a pump (heart) which has formerlybeen analyzed in 1D. e.g. [9, 54, 55, 5, 50, 28]. From a fluid mechanics point ofview any flow even if it is around a car or in a fuel system or the cardiovascularsystem in the human body, the basic equations that describe the flow behaviorare the same. However, the cardiovascular system in detail is a very difficultand complex area of flow simulations which makes the undertaken task clearlychallenging. A special challenge is looking for detailed information such as WSSon a subject specific level.
5
Chapter 2
Aim
The overall aim of this work is to develop a method/workflow in order to estimatethe hemodynamic force wall shear stress (WSS) in the human aorta in-vivo on asubject specific level. More specified the aims are:
• Show feasibility with a combined magnetic resonance imaging (MRI) andcomputational fluid dynamics (CFD) approach in order to estimate WSSfrom blood flow simulations in the human aorta.
• Performing validation of the velocity based CFD results produced by theproposed method with MRI based in-vivo measurements of velocity on agroup of subjects.
• Determine the sensitivity of the method with regard to the geometrical de-scription due to choice of segmentation method.
• Determine the influence of simplified velocity inlet boundary conditionscompared to a subject specific boundary condition.
7
Chapter 3
Overview
This chapter will describe considerations which must be addressed when perform-ing fluid mechanics simulations in the cardiovascular system. The description willinclude both the flow situation inside the human body and how to choose and de-sign an appropriate fluid mechanics model.
3.1 Modeling cardiovascular blood flow
To be able to retrieve detailed information about hemodynamic forces in the car-diovascular system the blood flow in the artery can be mathematically modeledand calculated. This is essentially a matter of simplifying the real situation intosomething that on one hand is a physiologically relevant model and on the otherhand a simple enough simulation model due to limitations in e.g. mathematicaland computational tools, measurement methods, computer power and knowledge.Except for the computational costs, a more complex simulation model can some-times introduce larger uncertainties in e.g. parameters, which gives simulation re-sults with increased uncertainties compared to a simpler simulation model. Thus,designing a simulation model is clearly a work of carefully making appropriatesimplifications.
The flow in the cardiovascular system is pulsatile, and each cardiac cycle is unique,both between different subjects as well as during time within a subject. The geom-etry of the aorta and its branches also differs between each subject. Furthermorethe composition of blood differs by a number of individually specific parameterse.g. hematocrite (amount of red blood cells) and plasma protein levels. In order tobe fully correct in analyzing the blood flow in large arteries this complex realityhas to be carefully considered. The situation is even more complex as all thesefactors change in the long term during aging e.g. stiffening of the arterial wall ma-
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CHAPTER 3. OVERVIEW
terial [32]. In short term very much can happen in the hemodynamic world; e.g.smoking, exercise and stress which in a matter of seconds can give an influenceof the arterial wall response.
The setup to model and simulate the blood flow in the human body includes han-dling the previously discussed complexity. It is crucial to include the parameterswhich are most important (e.g. aortic geometry), and excluded those that can beneglected under certain circumstances. The studies in this thesis aim at the in-vivoestimation of the WSS and have been limited to the blood flow situation in aorta ata ”snapshot” of the examined individual. This ”snapshot” is defined as the actualaortic geometry at a certain moment with a specific heart rate and flow velocitydistribution (in this work measured at the ascending aorta). This means that anylong term influences as well as short time influences are excluded in the workpresented. The main intention is to be as subject specific as possible in order toshow the possibilities to perform individually correct analyzes of the blood flow.Another limitation is that the method should be so fast that it could be clinicallyapplicable in the future.
3.2 Fluid Mechanics
This section describes the general parts of making fluid mechanics simulationsin the cardiovascular system. The more specific approach for modeling the aorticblood flow will shortly be described in Chapter 4, Method. To model any situationincluding flowing fluids you must use some kind of flow equation that origins fromthe full Navier-Stokes equation, see equation 1 and 2 (see e.g. [1]), where mass-forces are neglected.
∂ρ
∂t+ ∇ · (ρu) = 0 (1)
∂(ρu)
∂t+ ∇ · ρuu = −∇p + ∇τ (2)
Whereτ denotes the viscous stress tensor. Depending on the flow type and sit-uation the equations are altered to fit the actual problem. Basically some simpli-fications/assumptions are made e.g. the fluid is considered incompressible andNewtonian i.e. the density and viscosity can be assumed as constants, see equa-tion 3 and 4.
∇ · u = 0 (3)
ρ(∂u
∂t+ (u · ∇)u
)
= −∇p + µ∇2u (4)
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3.2. FLUID MECHANICS
The equations will describe the behavior of a specific flowing fluid. However, thisis not enough to get results for a particular flow case. An individually definedgeometrical domain is necessary, and this domain must be limited in size (due tolimitations in the computational resources). The flow model also requires someconnection to the surroundings outside the model. An interaction between rigidwall and flowing fluid can e.g. be described as a no slip boundary condition. Thisis mathematically done by boundary conditions which describe the interactionbetween the domain and its surroundings without resolving it to the same extentas in the studied geometrical domain. The governing equations are solved insidethe geometrically defined domain which takes into account the internal fluid flowinteraction. In order to get the full flow situation the boundary conditions aresimultaneously applied on the domain boundaries, see figure 6. The flow field isthen gained in the whole geometrical domain. Properties of the fluid in focus mustalso be properly defined in order to ensure the right physics of the flow results. Themain fluid properties are density and viscosity. The system of equations, boundaryconditions and fluid properties is solved by using numerical methods included ina computational fluid dynamics (CFD) software.
SURROUNDINGS
Geometrical Domain
Flow ModelFluid Description
Bo
un
dar
yC
on
diti
on
Boundary Condition
Boundary Condition
Bo
un
dar
yC
on
diti
on
Figure 6: Schematic description of fluid mechanics model setup and interaction with thesurroundings. The analysis is performed in the geometrical domain (white box) where theflow model and fluid description influences the flow field. Flow model is also influencedby the surroundings (grey ellipse), which is reduced to the applied boundary conditions(light gray boxes) which shapes the flow field in focus.
A CFD analysis requires that the geometrically defined domain is split into a num-ber of small volumes and in each of these volumes the flow equations (e.g. equa-tios 3 and 4) are solved. This geometry splitting procedure is called meshing and
11
CHAPTER 3. OVERVIEW
is crucial to be well designed in order to ensure reliable results. Depending on thefocus of the CFD simulations the mesh needs to be appropriate created in order tofit both the flow situation and results in focus. It is important to address the overallsize of the mesh (the number of volumes) in order to ensure a good description ofthe geometry as well as ensure that the result is mesh independent. Other issuesis the quality of the mesh that needs to be sufficienly resolved in particulary inter-esting regions e.g. introducing thin prism cells near a wall to estimate WSS withgood accuracy.
The wall shear stress (WSS) is defined as the shear rate (velocity derivative) nor-mal to a wall times the viscosity of the flowing fluid, see equation 5. The shear
rate[
∂u∂y
]
describes how the flow velocity changes when moving spatially in the
flow. For WSS the shear rate at the wall[
∂u∂y
]
wallis of interest and then the spatial
direction normal to the wall. The (dynamic) viscosity (µ) is a fluid property thatdescribes the viscosity of the fluid, e.g. water has higher viscosity than air andblood has higher viscosity then water. The concepts or meaning of WSS can bedescribed as the frictional force that a flowing fluid tangentially influences a wallwith both the magnitude and direction.
τwall = µ
[
∂u
∂y
]
wall
(5)
The load in the normal direction of the wall is the pressure. A relatively easy andvery rough way to estimate WSS in an artery is to assume that the velocity profilein an artery has the form of a fully developed Poiseuille profile. The assumptionmeans that the WSS basically is just a scaling of the flow rate, cross sectionalarea and viscosity in the artery. This method gives only one value for each crosssection. Due to the high rate of simplification (described by e.g. [10, 29]) ofthis approach it is not applicable for the task of estimating WSS in arteries. De-tails of this approach and its large limitations are found in Paper I. Another wayof estimation WSS is directly from MRI spatially distributed velocity measure-ments [41, 42, 13], which could give some more information about the spatial dis-tribution around a cross-section. However, this method based on measurementslacks in the assumptions needed for the flow near the wall in order to retrieveWSS [8].
The proposed method, subject specific estimated WSS, is based on the use of CFDsimulations combined with geometry and flow information from MRI as bound-ary condition in the CFD model. This approach is generally named image basedCFD and relevant presentation of this approach is found in [48, 49, 2, 35, 40, 4].Compared to the other approaches using velocity profile/flow measurements (de-
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3.2. FLUID MECHANICS
scribed above) this approach has an other type of simplifications and is far moreadvanced e.g. the velocity profile is not assumed and the whole flow is fully re-solved with high resolution by solving the governing flow equations. WSS resultscan in principal be retrieved with a nearly unlimited resolution considering theCFD modelling part, it is basically only the computational power available thatsets the limitations.
13
Chapter 4
Method
In order to perform subject specific estimations of WSS there is a chain of actions(workflow) that is necessary. The workflow includes: subjects to analyze, mea-surement of arterial geometry and blood flow with MRI, measurement of bloodviscosity, segmentation of the MRI data into 3D geometries and boundary con-ditions, meshing of the 3D geometries, CFD model setup, CFD calculations, andpost processing of the result, see figure 7. All included parts in the workflow arein detail described in Paper V.
MRI acquisition
Geometry
MRI acquisition
Velocity
CFD Model
Set-UpMeshSegmentation
CFD
Simulation
Inflow boundary condition
Ascending aorta velocity
WSS
Subject Viscosity Measurement
Figure 7: Workflow of the in-vivo subject specific WSS estimation method. The twin lineboxes indicates where measurements is performed.
The subjects used in all studies included in this thesis is a homogeneous groupof male volunteers, age 21-26 years. The MRI measurements were performedwith the aim of retrieving measurements on the entire aortic geometry. MRI mea-surements were also performed for time resolved velocity acquisition of the bloodflow at two cross-sections at the ascending and descending aorta, see figure 8.Measurements of the abdominal aorta diameter using ultrasound and measure-ment of the viscosity of the blood were also performed for each volunteer. De-tailed data, e.g. age and BMI, about the volunteers is found in Paper V. This group
15
CHAPTER 4. METHOD
GeometryGeometry VelocityVelocity
Figure 8:Left: The MRI acquisition (volume) of the human aortic geometry and the MRIflow acquisition (plane) at ascending and descending aorta.Top Right: MRI images forboth geometry and velocity.Down Right: Trimmed and smoothed aortic model, withapplied measured velocity information at the inlet (black arrow).
of volunteers is very homogeneous when looking at the overall data e.g. age andeven BMI, which could suggest that the final findings (WSS) would be tight andhomogeneous as well.
The transformation from MRI image material to 3D geometry of the human aortais performed using a general 3D level-set segmentation algorithm implementedin the freely available software Segment [24]. The segmented geometries weresmoothed using a Gaussian smoothing filter and then were the inlet and outletsmanually trimmed to get distinctly defined surfaces to apply boundary conditionson, see the right part of figure 8. An unstructured mesh was assigned to the aorticgeometry with prism layers near the wall to resolve the velocity near the aorticwall in order to ensure accurate WSS estimations. The experience of creatingthese meshes, resulted in mesh size of2−3 ·106 cells with 6 prism-layers near theaortic wall. Meshing actions were conducted by using the commercially software
16
ANSYS ICEM (ANSYS, Inc., Pittsburgh, PA, United States). The CFD boundarycondition setup included: velocity field at the inlet based on the subject specificvelocity (MRI measurement) at the ascending aorta, mass flow fractions at the out-lets, and rigid aortic walls with no slip. The CFD equation was the Navier-Stokes´equations with the flow assumptions: time dependent, laminar, incompressiblefluid (ρ = 1060kg/m3) and Newtonian fluid (µ = 0.0044 ± 0.0005 const. sub-ject specific, see table 1 in Paper V). The CFD actions were conducted using thecommercially available software ANSYS Fluent (ANSYS, Inc., Pittsburgh, PA).Simulations were conducted on Linux cluster capabilities at the National SuperComputer center (NSC), (http://www.nsc.liu.se). A typical simulation time fortwo cardiac cycles was about 3600 cpuh (computer hours), which will be in thereach for clinical applications where the time window would be over night simu-lations. Finally the post-processing gave the WSS estimations on the aortic wall(subject 5); exemplified with a contour plot in figure 9.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-100
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Figure 9: WSS contours on a human aorta (subject 5), at different time positions markedin the cardiac cycle at the bottom.
17
Chapter 5
Influencing Factors
All WSS estimation methods include a number of simplifications and assump-tions, which can influence the results gained. The largest issues of our method areto be addressed and briefly discussed in this chapter.
5.1 Geometry
Description of the arterial geometry used in the CFD simulations is very impor-tant [6, 11, 36, 52, 40] and essential for the result. This is the main reason for ourapproach of performing simulations on subject specific aortas. The differences be-tween individuals can be very significant, see e.g. Papers III and IV. Local arterialgeometry components as curvature and smoothness will influence the flow resultsby i.a. the applied boundary conditions on the arterial wall. The flow in the humanaorta will also be influenced by the general geometry topology e.g. the numberand location of branches. The way the MRI images are interpreted and processedinto the 3D geometries is not trivial and there will always be uncertainties in thegeometry description. This can be seen in figure 10 where an aortic cross-sectionof the aortic lumen is segmented. It is a challenge to put the arterial wall locationin its best fit position, which surely can influence the gained WSS results.
5.2 Flow and Fluid Model
The important fluid parameters for the actual CFD simulations are the density andthe viscosity. The blood is assumed to be incompressible and the density constant.A typical density for blood is 1050-1060kg/m3, and in this workρ = 1060kg/m3
is used. The blood is not a homogeneous fluid and it can roughly be described asa suspension of blood plasma (fluid) and blood cells (red and white). Dependingon the fact that the blood is a suspension the viscosity becomes dependent on the
19
CHAPTER 5. INFLUENCING FACTORS
Figure 10: MRI image of an aortic cross-section with the lumen segmentation marked onthe right image.
shear rate i.e. the fluid is non-Newtonian in its general behavior. Especially insmall arteries this will be prominent due to the larger influence of the particleswhen the diameter of the artery is small. The smallest arteries have similar size asthe red blood cells and in these small arteries the blood cells must be deformed toget through. In specific ranges of (high and low) shear rate (velocity derivative)the viscosity can be considered constant. In large arteries as in the aorta the bloodcan be assumed to be Newtonian and have constant viscosity because the shearrate seldom reaches values, where the non-Newtonian effects are prominent. Thisis especially relevant when comparing non-Newtonian effects with the influenceof arterial geometry [34]. The flow situation in the healthy subjects is assumed tobe laminar because the Reynolds number rarely exceeds (in these healthy subjects)the critical value for stationary flow (2300-4000). In addition the critical Reynoldsnumber for time dependent flow is considered to be even higher [20]. However,care should be taken when considering constricted arteries where turbulent effectsmay play an important role. Details about turbulent WSS can be found in [22] anda new perspective to turbulence in blood is discussed in [3].
5.3 Boundary Conditions
To ensure relevant results from a CFD model the boundary conditions are crucialbecause they shape the flow into the aimed flow situation for the analyzed region.In the actual cardiovascular blood flow simulations the three different boundaryconditions are; inlet, outlets and the wall. This gives that the in-flowing and out-flowing blood need to be carefully taken into account as well as the interactionbetween the vessel wall and the flowing blood.
The artery wall in this work is modeled as a rigid wall with a no-slip boundarycondition (i.e. fluid velocity at the wall is zero). This assumption can influence
20
5.3. BOUNDARY CONDITIONS
the result because a real artery is distensible. The aorta as the aortic diameter canunder certain circumstances change in the range of 10%, during a cardiac cycle.One reason of the rigid wall boundary condition assumption is that the compu-tational time including wall movement is today totally unacceptable in a clinicalsetting, which is one goal with the proposed method. The coupled problem of thearterial wall response to the flowing blood (called fluid structure interaction, FSI)is out of the scope for this thesis and will be discussed in the Outlook, Chapter 7.
The measured subject specific velocity is used as inlet boundary condition. Thevelocity is both spatially and temporally resolved, which indeed will be subjectspecific and important, see Paper VI. The outflow boundary conditions are de-fined as outflow fractions of the inflowing blood, see figure 11. A rigid modelcan not store any mass so the inflow must at all times be equal to the total sum ofoutflows.
Geometrical DomainFlow Model
∇ · u = 0
ρ(
∂u
∂t+ (u · ∇)u
)
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Fluid Descriptionµ =const.
ρ =const.
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moutlet = minlet
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Figure 11: Schematic description of a CFD model of the human aorta and the bound-ary conditions used for interaction with the surroundings. Flow analysis is performed inthe geometrical domain where the flow model and fluid description influences the flowfield. The flow results is also influenced by the surroundings due to the different appliedboundary conditions which also shapes the flow field in the geometrical domain.
21
Chapter 6
Results and Discussion
This section is a short summary of the results from the different papers includedin the thesis, and also some additional results from my on-going research that en-lightens the aims and the future for the WSS estimation method.
The first question to address in the method evaluation is to see if the method pro-duces relevant WSS values when comparing the result with a reference method.Unfortunately there is lack of such methods. There are some WSS estimationapproaches that uses measured blood flow as a base directly. One such methoduses a simplified assumption, the Hagen-Poiseuille assumption, of the flow pro-file, see discussion in Paper I. This method estimates one WSS magnitude value
A4
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Figure 12:Left: Crossections A1-A3 marked at a human aortic model.Right: WSSCFD
(cross-sectional averaged from CFD simulation) and WSSHP (using Hagen-Poiseuille as-sumption) in A1, A2 and A3 during a cardiac cycle.
23
CHAPTER 6. RESULTS AND DISCUSSION
in a whole cross-section where the velocity flow profile assumes to be station-ary and fully developed (which i.a. requires approximately 50-100 diameters ofstraight pipe). This method has been intensively used to retrieve WSS data forarteries [14, 18, 38, 39, 29, 16, 17]. A comparison with results from this approachapplied on CFD generated aortic flow velocities showed a large underestimationin WSS as well as the obvious lack of details in the WSS distribution spatially. InPaper I we conclude that this simplified method is too simplified and it has not thepossibility to estimate relevant WSS values, which is exemplified in figure 12.
In order to see if our method estimates relevant WSS values, we apply it onon the whole volunteer group in order to estimate subject-specific WSS in-vivo.Validation of WSS is indeed tricky due to the lack of a gold standard estimationmethod [29]. The chosen way of estimating the validity of the gained WSS datawas done by comparing calculated (CFD) velocity profiles with MRI measuredvelocities in the descending aorta. This choice was made because the WSS ismainly retrieved from the gained velocity distribution near the vessel walls, seeequation 5. The method feasibility is presented in Paper II and the full descrip-tion of the method and velocity comparison on the whole group of volunteers is
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Figure 13: Left: Positions in the aorta of anterior (A), posterior (P), left (L) and right(R) points.Right: Velocity profiles in a thoracic ascending aorta, MRI measured velocityprofiles (circles) and CFD simulated profiles (solid line) at both anterior-posterior (A-P)and left-right (L-R) directions. The figures show four different times in the cardiac cycledefined as maximum acceleration (t1), maximum velocity (t2), maximum retardation (t3)and minimum velocity (t4).
24
shown in Paper V. An example of flow profiles in the two main directions anterior-posterior (A-P) and left-right (L-R) is compared; in figure 13. The result indicatesclear similarities in the velocity profiles. The main limitations with the velocitymeasurements is that the distribution near the wall is very hard to capture andunfortuatley this area is the most important rgion for determination of WSS. Tomeasure this region with a good resolution is very difficult due to the fact thatMRI voxels (measurement volumes) near the arterial wall will include both thewall material and the flowing blood. This is one of the main reasons to use theapproach with CFD simulations in order to retrieve WSS in detail.
The distribution of the WSS is highly dependent on the geometry and therefore thesegmentation process is very important. The process to retrieve reliable geometrydescriptions for the CFD simulations is thus very crucial. The image collectionscheme was tested to see if the geometries from two different methods give thesame geometrical results (lumen diameter). This was done by comparing our MRIapproach (manual segmentation) with ultrasound measurement of the abdominalaortic lumen diameter measurement on the volunteer group. The full work ispresented in Paper III and states that the MRI approach (manual segmentation)produces comparable abdominal aortic lumen diameters on the geometries thatare relevant for CFD simulation.
The average lumen diameters for gained were for manual segmentation of MRIimages 13.6 mm and for the ultrasound proceedure 13.8 mm so the difference
Close to Coil
Large Distance to Coil
Figure 14: MRI image slice of a human aorta with region of good (close to coil) and poor(large distance to coil) image quality marked due to distance to MRI camera coil.
25
CHAPTER 6. RESULTS AND DISCUSSION
was 0.2 mm which is considered as very low. This is even more remarkable dueto the fact that the whole aorta were examined by MRI with the coil focused onthe thoracic part of the aorta which resulted in poorer image quality in abdominalaorta due to the larger distance to the coil, see figure 14. So the MRI methodologycombined with the manual segmentation method gives relevant geometries to usein the CFD simulations, even in a region with poor image quality.
The manual segmentation approach is unfortunately very time consuming, ap-proximately 16 h for each individual. A need of faster methods is necessary. Asemi-automatic level-set segmentation method was developed and implemented
Systolic Acceleration Peak Systole
Systolic Deceleration Early Diastole
Manual
Manual Manual
Manual Semi-automatic
Semi-automatic Semi-automatic
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Figure 15: WSS magnitude contours on the aortic wall of subject 5 based on differentsegmentation methods (manual and semi-automatic). Displayed at four different cardiaccycles locations: systolic acceleration, peak systole, systolic deceleration and early dias-tole.
26
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Figure 16: WSS magnitude distribution in a thoracic cross-section for subject 5 withthe two different segmentation methods (manual and semi-automatic). Describing theinfluence of the segmentation method at four different cardiac cycle positions.
in the freely available software Segment (http://segment.heiberg.se/) [25]. Thismethod is much faster; it takes approximately 20 min to segment an aorta fromMRI images. The short segmentation time makes it possible to perform largerimage based CFD studies. In order to validate this faster segmentation method,the WSS values from this method were compared with WSS estimations from themanually created aortic geometries.
The WSS contours for the two segmentation methods are shown in figure 15for one representable subject (subject 5). The results show good similarities inthe overall WSS distribution. More detailed information about the circumferen-tial WSS distribution in the thoracic part of the aorta is shown in figure 16 for thesame subject as in figure 15. It can clearly be seen that there is similarities alsoat this detailed view of the WSS. The differences that occur indicates the sensi-tivity to the geometry description. The different behavior at different times in thecardiac cycle shows that the dependence of the geometry is greater in the decel-eration and diastolic phases of the cardiac cycle. However, generally the resultsshow good agreement and the use of semi-automatic segmented models can beused and trusted. The full set of data and the procedure are found in Paper IV.
In order to determine the influence of the inlet velocity profile the WSS distribu-
27
CHAPTER 6. RESULTS AND DISCUSSION
CS3
Measured Velocity Profile
Flat Velocity Profile
t1
t1
t2
t2
t3
t3
t4
t4
Figure 17:Top Left: Subject specific aortic geometry with the three cross-sections CS1-CS3 marked.Bottom Left:Mean inlet velocity in one subject, with time positionst1 − t4marked. The positions correspond to maximum acceleration, maximum velocity, maxi-mum retardation, and minimum velocity.Right: WSS values around CS3 for a subject(descending aorta) where P, L, A and R denotes posterior, left, anterior, and right. Solidline is WSS from the measured inlet profile and the dashed line comes from the uniforminlet velocity profile (note the different y-axis scales in the 4 diagrams).
tion was gained using two different inlet profiles and the results were compared.The hypothesis was that the aortic geometry shapes the flow to such extent that it ispossible to simplify the inlet velocity profile. The inlet velocity profiles used, werethe measured spatially distributed velocity profile and one profile with a unifiedvelocity with the same mass flow rate as the measured inlet profile. The evaluationwas made at three different cross-sections and at four different times in the cardiaccycle and the results is shown to the left in figure 17. The distribution of WSS in across-section in the thoracic descending aorta, described as cross-section 3 (CS3),is exemplified to the right in figure 17 for a representative subject. It is interestingto note the very small differences at peak systole (t2), where the WSS almost isthe same. This indicates that the importance of the spatially relevant inlet velocityprofile is more prominent at non systolic parts of the cardiac cycle where the flowvelocity is lower. The full set of results and details about this is found in PaperVI.
28
In conclusion all results and findings briefly presented in this chapter evidence thatthe presented method for WSS estimations works very well. MRI image materialcan be used with confidence as well as semi-automatic segmentation. For theCFD setup the inlet velocity profile will influence the results and thus a subjectspecific profile is strongly recommended. This is specially necessary when nonsystolic WSS is of interest, because at these times in the cardiac cycle, the WSS issensitive to both inlet boundary condition as well as the geometrical description.
29
Chapter 7
Outlook
One of the simplifications made with the proposed WSS estimation method isthat the aortic wall. This was mainly treated as a rigid wall, due to limitations incomputational speed to match clinical settings which have been the focus of thiswork. A future step can be to include the elasticity of the aortic wall in a simula-tion model with the interaction between CFD and a structure mechanics softwareand this interaction is called Fluid Structure Interaction (FSI). The basic conceptwith FSI is to determine the forces (pressure and WSS) with CFD and use that asboundary conditions on an arterial wall structural mechanics model. This strategyboth open a range of new possibilities to determine the real situation as well asgives challenges e.g. flow boundary conditions which handles pressure waves andarterial wall material descriptions and thickness. The development of both soft-
Figure 18: Left: Mesh of aortic wall (blue) and blood (red) for the FSI model.Right:WSS distribution in aortic arch for the FSI model, contours describe the magnitude andthe vectors the direction [33].
ware’s and computational power will open this possibility and pilot tests with FSIresults show that computational time is about a hundred times the computationaltime used with rigid wall approximation. Crucial to the FSI research is the need
31
CHAPTER 7. OUTLOOK
to improve boundary conditions, arterial material description, stability and com-putational efficiency. One of the largest challenges in the future is probably thestability due to the large deformations that can occur in the aorta. The main reasonof using rigid walls today is that a reasonable computational time for a clinical useof the WSS estimations is to target. This can be obtained by the use of rigid wallsand the FSI approch is not even near a clinical use today. A glance at the futureof preliminary WSS estimations using FSI is shown in figure 18, and describedin [33].
Future use of the described WSS estimation method is to apply it on both largergroups of subjects and other types of groups in order to conduct medical researchinvolving WSS. Another possibility is to use the method on animal models. Themethod has recently been used to retrieve WSS results for rats, see figure 19. Theuse of animal models enables to investigate different vessel wall compositions andinfluences on the arterial wall due to food as well as surgical procedures.
Figure 19:Left: MIP (maximum intensity projection) representation of MRI images of arat aorta.Right: WSS distribution (contours) and direction (vectors) in rat aortic arch.
To get the described method into the clinical setting the method needs to be morestreamlined especially in the segmentation and mesh generation. Newer and fasterapproaches using prototype based segmentation [46] make the segmentation fasterand easier to perform. Automatically and robust mesh generation strategies whichwill mesh an artery with appropriate mesh quality especially for the required fineresolved region near the wall which is crucial for the WSS estimate quality.
The future surely looks exciting, promising and challenging!
32
Chapter 8
Review of Papers
Paper I
In this paper a well known WSS estimation method that uses the Hagen-Poiseuilleassumptions is compared with WSS extracted from CFD simulations. The twoWSS estimating methods were applied on a relevant flow field in the human aortawhich was determined using the combination of MRI images and CFD. The re-sults clearly shows that the Hagen-Poiseuille assumption method underestimatesWSS and it is not capable to retrieve any detailed information about the WSSdistribution.
Paper II
This paper shows the feasibility of the proposed WSS estimation method on twosubjects with stationary CFD simulations. The in-vivo WSS estimation methodwas performed on the subject specific aortic geometries created from MRI images.The results indicate that the method is feasible and that the simulated velocity fieldis similar to measured velocity field in the descending aorta.
Paper III
To ensure that a correct geometry is used in the WSS estimation method the humanabdominal aortic diameter was analyzed by two image modalities. The aortic di-ameters from manually segmented geometries created from MRI were comparedwith a commonly used Ultrasound strategy to measure the abdominal aortic diam-eter. The results from both strategies show good agreement. This indicates thatmanually segmented geometries from MRI are suitable to use in WSS estimations.
33
CHAPTER 8. REVIEW OF PAPERS
Paper IV
Due to the time consuming process of manually segmented MRI images the man-ual strategy (validated in paper III) was compared to a fast semi-automatic level-set segmentation method. The aortic geometries and the WSS distribution wascompared. Results show good agreement regarding geometrical description aswell as WSS distribution for the two segmentation strategies. The result givesevidence of the validity of the semi-automatic segmentation method.
Paper V
Paper V throughly describes the WSS estimation method using the combinationof MRI images, semi-automatic segmentation and CFD simulations on the humanaorta. Time resolved for the whole cardiac cycle WSS estimations were performedon a group of nine healthy volunteers. The gained flow fields were compared withMRI measured flow fields at the descending aorta and this comparison served asvalidation of the CFD simulation flow results.
Paper VI
The influence of the inflow boundary condition flow profile on the WSS resultswas investigated. WSS results from the MRI measured inlet velocity profile (spa-tially distributed) were compared with a mass flow correct (MRI measurements)uniform velocity profile. These two sets of velocity profiles were used as inletboundary conditions in the ascending aorta. This work shows the aortic WSS dis-tribution sensitivity to the inlet boundary conditions. The main finding is that theaortic WSS distribution is sensitive to the velocity profile used as inlet boundarycondition in the ascending human aorta. The recomendation is that a measuredvelocity distribution should be used as inlet boundary condition.
34
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