Nationai Library Bibliothèque nationale ($1 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services senrices bibliographiques 395 Wellington Street 395, rue Wellington OîiawaON K1AON4 Ottawa ON KIA ON4 Canada Canada
The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distriiute or seU reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique.
The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Numerical Modelling of the Carotid Artery Bifurcation with a Mild Stenosis
by
Ronald Gin
Graduate Program in the Department of Mechanical and Materials Engineering
Submitted in Partial fiil filment of the requirements for the degree of Master of Engineering Science
Faculty of Graduate Studies University of Western Ontario March 24,2000
9 Copyright by Ronald Gin 2000 Atherosclerosis is a disease characterized by the hardening and thickening of the anerial walls due to the formation of plaque. As this disease progresses, the formation of plaque reduces the arterial passage area creating uncharacteristic blood flow patterns. As a rcsult. this restricrion. if severe enough, cm cause individuals to suffer cardiac arrest or strokr. The plaque deposition is focal and usually occurs in sharp curvatures and bifurcations in the cardiovascular system such as the carotid artenes. The main factor of the disease's drvelopment is related to the hemodynamics of the blood flow field.
Numencal simulations are provided in mildly constricted vessels under physiological conditions of the cardiac wave. Simulations are carried out using a modified commercial Cornputational Fluid Dynamics soRware package to solve the goveming equations. Modifications to the source code are made to introduce a novel technique of imposing two pressure specified boundary conditions to solve pulsatile simulations. The final simulations gcncrated for the mildly constncted vessels are discussed, providing conclusions to the kritures created from the tlow field. The accuracy of the simulations perforrned in this study is discussed. The approach of limiting the errors associated with numencal niodelling of pulsatile simulations is presented.
.A technique of simulating the two outlet boundary conditions under pulsatile conditions \vas show to be a good addition in providing accurate and efficient simulations by rcducing the ovrrall computational resources and computation time. To provide results for pulsatilc simulations in the mildly constricted vessels substantial small element sizes are required which are less than 0.016 times the Common Carotid Artery diameter. To obtain fully developed pulsatile solutions a minimum of two cardiac cycles must be carried out with no less than 100 uneven time steps for each cardiac cycle. The simulations carried out in the mildly constricted artenes present complex flow patterns that shows relevance to the drvelopment of the atherosclerotic disease. The discussion involves cornparison between three rnodels of the carotid artery conceming the velocity and vorticity field as well as the recirculation zones and wall shear stresses. DEDICATION For Mom, Dad and the big sisten.
I thank Dr. Straatman for providing his geat knowledge and his excellent guidance to my studies throughout my Masten degree. t also thank Dr. Steinman for his contribution and suggestions made through several discussions conceming this work. 1 would like to thank Jaques Milner for his help and assistance in several pre- and post-processing tasks. I would like to thank Bob Saunders for his technical support in relation to the computer resources. I would also hke to thank Dr. Brain Rutt and Dr. Alastair Martin for their assistance in acquiring the XlRi velocity data.
1 would like to thank my fellow colleagues and kiends for their help in my studies. A special thnnks goes to Shelley Sunderland for her continued support and friendship. .. CERTIFICATE OF E-Y,4iiIlNATION ...... ii
.. . .AEISTFUCT ...... iii
... LIST OF TAE3LES ...... viii
.*. NOMENCLATC'RE. ABBREVIATIONS AND DEFNTIONS ...... xi11
CHAPTER 1 : INTRODUCTION ..... + ...... 1 1.1 Scope and Objective ...... 4 1.2 OutIine ...... 5
CHAPTER 2: LITERATURE REWEW ...... 6 2.1 Introduction ...... +...... 6 2.1 Physiological Geometries of the Carotid Artery ...... 6 2.2.1 The Bharadvaj's Model: The Average Normal Model ...... 8 2.2.2 Geometric Variations: Smith's Models and Other Modeis ...... 9 2 -3 Hemodynarnic Theones of the Atherosclerotic Disease ...... 14 2.4 Review of Relevant Experiments and Numerical Simulations ...... 15 2.4.1 Experiments and Flow Visualization ...... 15 2-42 Steady, 2-dimensional Analysis in Bifùrcating Georneiries ...... 17 2.43 Strady. 3-dimensional Analysis in Bifurcating Geometries ...... 18 7.1.4 2-D and 3-D Pulsatile Flow in Bifurcating Geometries...... 19 2.4.5 Flows in Constricted Artenes ...... 22 2 .5 Surnrnary of Review ...... 24
CHAPTER 3 : COMP UTATIONAL MODELLING OF THE CAROTID .4 RTERY B IFC'RC..\TION ...... 26 3.1 htroduc tion ...... 26 3.2 The Geometry of the Carotid ktery ...... 26 3.3 Mathematical FormuIation ...... 27 3 4 The Computational Mode1 ...... 30 3.5 [rnplementation of Boundary Conditions ...... 33 3-51 Steady Flow Simulations ...... 33 3.5. The Cardiac Waveform and Pulsatile Flow Simulations ...... 35 3 -6 CFX-Tfc Code Modifications ...... 39 3.6.1 Inlet Velocity Conditions ...... 39 3.6.2 TimeStepIncrementsforPulsatileFlow...... 40 3.6.3 Implementation of the Two-Pressure Dirichlet Condition ...... 41 3.7 Surnmary ...... 49
CHAPTER 1: CONVERGENCE OF RESULTS ...... 50 4.1 Introduction...... 50 4.2 Spatial Convergence ...... 50 4.3 Temporai Convergence...... 57 4.3.1 Required Nurnber of Time Steps ...... 58 1.3.2 Required Number of Cycles ...... 60 1.4 Residual Convergence ...... 62 1.5 Dual Pressure Boundary Condition Convergence ...... 62 4.6 Grid Independence for 30% Models ...... 64 4.7 Conclusion ...... 67 CHAPTER 5: RESULTS hW DISCUSSION ...... 69 5.1 introduction ...... 69 5 2 General Flow Characteristics ...... 70 5.3 Systolic Portion OFthe Cardiac Wave: Acceleration Portion ...... 73 5.4 Systolic Portion of the Cardiac Wave: Deceleration Portion ...... 76 5 .5 Diastolic Portion of the Cardiac Wave: Small Secondary Pulse ... 78 5.6 Diastolic Portion of the Cardiac Wave: Steady Portion ...... 79 5.7 Conclusion ...... 80
CHAPTER 6: CONCLUSIONS .&ND RECOWlMATIONS ...... 106 6.1 Introduction ...... 106 6.2 Conclusions ...... 106 6.3 Contributions ...... 108 6.4 Future Work ...... 1 O9
APPEND [CS ...... 118 Appendix -4 1: Portions of the modified source cod From CFX-Tfc ...... 118 Appendix .42 . Flow chart for Two-pressure boundary condition ...... 150 Table 4.1 Mesh information for normal carotid artery ...... 53
Table 4.2 Velocity and wall shear stress convergence erron for the normal mode1 at Re=.?50 ...... 54
Table 4.3 Velocity and wall shear stress convergence erron for the nonnal mode1 at Re=I 100 ...... 57
Table 4.4 Erron hmincreasing the number of time steps ...... 59
Table 3.5 Element information for the 30% eccentric meshes ...... 65
Table 4.6 Gnd Convergence for the 30% eccentric mesh at Re=/ 100 ...... 67 Figure 1.1 The carotid artery bihircation (Callo, 1996)...... ,7
Figure 2.1 Bhiüadvaj's model of the nomal carotid artery bifiucation
( B haradvaj et al., 1982)...... 9
Figure 2.1 The NASCET stenosis index versus an area reduction ...... 1 1
Figure 2.3 Ll'irefnme drawings of the normal and stenosed carotid biiùrcation generated by Smith et al. (a) Modified Bharadvaj normal geometry; concentric geomenies: (b) 30?/0, (c) JO%, (ci) 60%. (e) 70%. and (f) 80% stenosis. and eccentric yeometries: (g) 30%. (h) JO!%, (i) 60%. (j)70%, and (k) 80% stenosis. (Smith et al., 1996) ...... 12
Figure 2.4 Standard template used by Smith et ai. (1996) to compare angiogaphic data. (Smith et al. 1996) ...... 13
Figure 2.5 Positions in the carotid artery susceptible to the disease. (Lei ri al., 1995)...... 2 1
Figure 3.1 Smith models. (a) Normal model, (b) 30% concentric model, md (c) 30% eccentric model...... 27
Figure 3.2 2-D fuiite element-based finite volume mesh ...... 30
Figure 3.3 Boundary conditions for steady simulations ...... 34
Figure 3.4 The cardiac wave of a normal human subject ...... 36 Figure 3.5 2-D hlly developed profiles for the cardiac wave...... 37
Figure 3.6 Boundary conditions for pulsatile simulations ...... 38
Figure 3.7 Time step marching with (a) 100 even time steps and (b) 100 uniform cord length time steps...... 42
Figure 3.5 Pressure field development for steady simulations through the normal carotid artery for the ECA face. (a) Pressure vs. Flow Division. and (b) Pressure vs. Time Step Number ...... 46
Figure 3.9 Pressure field development for pulsatile simulations through the normal, 30% concentric, and 30?6 eccentric model of the carotid artery for the (a) ECA. and (b) ICA...... 18
Figure 4.1 The mode1 of normal carotid artery bifurcation...... 5 1
Figure 4.7 (a) Velocity profiles for the normal model at Re-30 (b) Wall shear stress contours for the normal modei Re=2j0 ...... 54
Figure 4.3 (a) Cornparison for the normal mode between MRI data and the computational data 1 with the 800K grid at Re=2j0. (b) Numerical results at Re4100 for the normal rnodel...... 56
Figure 4.4 Profiles for error of incrernenting the number of time steps...... 59
Figure 4.5 Error be~eenfirst and second pulse ...... 61
Figure 4.6 Long and short rnodels of the normal carotid artery...... 63
Figure 4.7 The 30% eccentnc geometry of the carotid artery bifurcation...... 65 Figure 4.8 Velocity and wall shear stress for the 30% eccentric mode1 at Re = 1100...... 66
Figure 5.t The cardiac pulse in the normal carotid artery. (Holdsworth er
(11.. 1999) ...... 70
Figure 5.2 Converged Pressure differences for (a) the ECA and (b) ICA ...... 72
Figure 5.3 Velocity profiles dunng the acceleration phase at (a) T=6/60. (b) T=S/60. (c)T=9/60, and (d) T= IV60 ...... 53
Figure 5.4 Reverse flow dunng the acceleration phase at (a) T=8/60and (b)T=11/60...... 85
Figure 5.5 (a) 2-D view of pathlines during the acceleration phase at T=9/60. and (b) Isomebïc view of padilines dunng the ricceleration phase at T=9/60...... 86
Figure 5.6 Velocity magnitude plot during the acceleration phase at (a) T=6/60, (b) T=9/60and T=l1/60...... 88
Figure 5.7 Wall shear stress during acceleration at (a) T=8/60, (b) T=9/60 and (c) T= 11/60...... 90
Figure 5.8 Velocity profiles during the deceleration phase at (a) T= 14/60 and (b) T=18/60...... 92
Figure 5.9 Reverse flow during the deceleration phase at (a) T=14/60 and (bj T=Z8/60 ...... 93 Figure 5.10 Vorticity dunng the deceleration phase at (a) T=10/60.(b) T=14/6O and jc) T=-O/6O ...... ,.. 94
Figure 5.1 1 (a) Isometric view of pathlines during the deceleraiion phase at T=i3/60.(b)2-D view of pathiines dunng the deceleration phase at t=ij/6. and (c) 7-D view of pathlines during the deceleration phase at r=70/60 ...... 96
Figure 5.12 Vorticiry field during small secondary pulse for (a) the normal geometry. (b) the 3096 concentric geometry and (c) the ilO?O cccentric geometry...... 99
Figure 5-13 Wall shear stress during small secondary pulse ...... 10 1
Figure 5.14 Rccirculation during small secondary pulse at T = 27/60 ...... 10 1
Figure 5.15 Vorticity 1 eld for the (a) 30% eccentnc model and (b) 30% concentric modei during acceleration of the small secondary pulse ......
Figure 5.16 Vorticity field t'or the 30% eccentric mode1 during the end flow rates of the small secondary pulse ...... 103
Figure 5.17 Vrlocity magnitude field during the end of the cardiac wave for the 30% eccentric model ...... 104
Figure 5.1 8: Vorticity field dunng the end of the cardiac wave for the 30% eccentric modei ...... 104
Figure 5.19 Reverse flow at the end of the cardiac wave ...... ,.,.. 105 -4s surface area CCA Common Carotid Anery CFD Computational Fluid Dynamics 4 diameter of ICA at maximum severity D diameter
DiC.4 ICA diarneter EC .A Extemal Carotid Artery ICA Intemal Carotid Artex-y
JI. Bessel function of the v order 0 tlow rate LDV Laser Doppler Velocimetry
/Pl mass of finite volume rrr mass flow rate .MM .Magnetic Resonance Imaging n number of patients i, unit vector normal in the j direction
NAC Numerical Advection Correction XASCET Nonh American Symptomatic Carotid Endwterectomy TraiI P pressure R radius Re Reynolds Number Riil S Root-Mean-Square S stenosis severity SQ source term for scalar # SUDS Skewed Upstream differencing Scheme f time T time (non-dimensionalized)
-XI distance in the i direction
xiii Ut velocity in the i direction Ü , L',, average velocity hatv time averaged velocity UDS Cpwind-Di fferencing Scheme
volume of control volume weighting factor ratio of diameters (Le. d/D) Womersely number angle between centerline of CCA and ICA angle behveen centerline of CCA and ECA di fisivity fluid density scalar quantity 4~ scalar qiiantity at integration point 4 scalar quantity at new time 4 scalar quantity at previous time 4) scalar quantity at p node (center of volume) P dynarnic viscosity L' L' kinernatic viscosity
xiv 1.0 INTRODUCTION
Stroke is the 3"l leading cause of death in Nonh America, the leading cause being hean attacks (Gorelick, 1995). The risk of death from saoke is the most important çonsrqurnce but in the event that an individual survives a stroke, there are other consequcncrs. One issue is the high economic cost associated with the initial care and trsatmcnt of patients at high risk of stroke occumng. In addition to this cost is the cost of the continucd care and treatment required for patients after the occurrence of the fint stroke. The draths caused by stroke are associated with one of the most frequently existing cardiov;iscular diseases. known as atherosclerosis. Atheroscierosis is a disease that affects practicall y every individual. however. the degree to which an individual is affecred depends on factors such as the individual's health and circulatory system. For stroke. this disease rakcs place in the carotid artenes and for heart attacks. mainly in the coronary arteries.
The mere existence of atherosclerosis does not alone put an individual at high risk. nor dors this individual necessarily suffer from any of the consequences of the disease. [t is artually the influence that the disease has on the cardiovascular system that is the driving force that an cause death or senous disability in terms of vital motor functions. .-\theroscltxosis is characterized by a build-up of "gruniel" (mainly cornposed of Fatty deposits) that penetrates the artenal walls and collects undemeath the endothehum layer (White, 1989). -An atherosclerotic plaque is fomed. which consists of fatty material and blood platelets. As a result of this plaque. the normally distensible anenal walls tend to thicken and stiffen in the vicinity of the disease. The disease affects the cardiovascular system by causing a reduction in the lumen and transformations to the geometric shape of the arteries. The constriction in the vesse1 can reduce or, in an extreme case. prevent oxygenated blood from passing to vital organs. However, more comrnonly, thrombus occurs whereby small emboli detach from the anerial walls and travel downstream clogging smaller vessels and arterioles, again leading to starvation of essential organs of the nutrients required to Function. Figure 1.1 : The carotid artery bifurcation (Callo. 1996).
Rrsearch on atherosclerosis has been carried out since the early 1960's but became a topic of intense snidy in the eariy to mid 1970's. The past research has established essential information about the disease. The disease is known to affect specific locations in the cardiovascular tree. localizing mainly in areas of cumed and bifurcating vessels (Caro et ai.. 1969). .As mentioned earlier, one of the vital locations where this disease consistently drvelops is in the carotid anery bifurcation. The carotid anenes are located on both sides of the neck and they are the primary arteries that transport oxygenated blood to the brain and facial tissues (Figure 1.1). The carotid arteries consist of the Common Carotid Artery (CC.-\). which originate fiom the aona (for the left carotid anery) and from the innominate artery (for the right carotid anery). As the CCA travels up the neck, the artery branches into two smaller vessels. the Internai Carotid Artery (ICA) and the Extemal Carotid Anery (EC;\). The ICA. the artery of greater importance, provides oxygenated blood to the brain. while the ECA provides oxygenated blood to the facial tissues. It is observed in many asymptomatic patients that as the two branches bifurcate fiom the CCA, the ICA develops a buib-like structure. known as the sinus. with the artery lumen widening as the blood enten into the ICA and decreasing shortly downstrearn. It is the sinus where the disease consistently devrlops.
The exact mechanism for the progression of atherosclerosis is unclear and. therefore. difticult to examine. Srveral hypothesises have been developed to explain the disease usine biological theories (White. 1989). The majority of the theories deal with the concept of the influences caused by the injury of the artery's endothelium layer, resulting in the exposure of rhr sub-cndothelium (Ross. 1986). Many theories on the development of the disease discuss and rrlatc the revealing of the sub-endothelium to the biood stream. such as the adhesion of the platelets from the blood stream and the proliferation of the smooth muscle cells positionsd under thri endothelium layer (Mustard er al., 1983. Gono er al.. 1977).
In addition to research on the disease itself, accurate techniques to access the nsk of complications From the disease need to be developed. Currently. the risk of stroke is assesscd mainly on the ICA stenosis severity. which is not an accunte indication. i.r. individuais with high stenosis grades (severe constrictions) may never suffer from a stroke and individuals with low stenosis grades (mild constrictions) may be at high risk. Some physicians use S-rays to observe the amount of constriction present in the artery and then recommend surgery for patients having a stenosed artery beyond a critical level. Other physicians rnay listen to the sound of flow disturbances in the artery (bruit) and determine the patient's risk by the presence of any abnormal sound. However. the diagnoses for risk are nor based solrly on these techniques, they also include biological symptoms of the patient. Hypertension is the most important factor with increasing blood pressure resulting in rlrvated risk of stroke (Gorelick, 1995. Nerem, 1984). Nevertheless, the intluence of a stenosed anery presents the introduction of several hypotheses for the pathology of the disease in relation to the hernodynamics. Numerous theories are summarized by Glagov et ai. ( 1989) and Gessner (1973) and discuss existence of the cornplex fiow feanires in the normal carotid artery and their possible relation to the development of the disease. Improvemenr in the evaluation of the disease will enabie early detection and accurate assessment of the disease and thus, result in improved treatment techniques. Many researchen believe that the influence of the fluid dynamics within the cardiovascular system, the hemodynamics, plays an important role in the localization and progression of this cardiovascular disease (Gessner, 1973). This was mainly hypothesised for the reason that the disease is focused in areas of curved or bifurcating vessels, as previously mentioned. For the case of the carotid artery bifurcation. the expansion and then narrowinp of the vrççel in the vicinity of the sinus contributes to the complexity of the flow field. In addition to geometry effects. the flow is pulsatile. which tunher adds to the complexity of the flow. The fluid shear stresses applied to the vesse1 walls may be a factor in the progression of the disease. Thus. the predicti~nof the stresses in both normal and dissasrd vrssels may provide much-needed insight for assessing the nsk.
The present work focuses on the numerical prediction of pulsatile tlow in the carotid bifurcation. The main motivation is to explore fluid dynarnical effects that may provide insight into the assessrnent of nsk of cardiovascular disease. This study explores several issues related to the numencal prediction of a time-periodic flow.
1.1 Scope and Objective
The current study deals with the numerical simulation of the fluid tlow through the carotid anrries undrr the development of the disease. The main geometric configurations will invoive the 30?6 stenosed artenes with eccentnc and concentric positioning of the plaque. The objective of the present research is to.
( i) introduce a novel technique to impose boundary conditions in numerical simulations of transient flows with bifurcating geometries, such as the carotid artery. The results will show that with the use of the proposed boundary condition implementation. the technique will be beneficial to numerical simulations.
( ii) present and discuss the approach to provide numerically accurate results. The procedure will include the discussion of the spatial and temporal convergence for penodic simulations. (iii) perfom a detailed study and analysis of pulsatile blood flow through mildly constricted geometries created from pliysiological data of symptornatic patients.
1.2 Outline
The chaptsrs in this thesis will consist of the followin~discussions. Chapter 2 will discuss [lis relevant material that has been carried out by earlier researchers and summarize tlieir coiitributions and theories to the atherosclerotic disease. Chapter 3 will discuss the mathsrnaticri1 techniques and tools used to solve the flow through the complex geometry. Chaptttr 3 will also discuss boundary condition irnplementation as weil as the techniques iiscd io impose of the novel boundary condition mentioned above. Chapter 4 will discuss and describe the convergence criterion required to pnerate accurare solutions. Simulations will be prowird under both steady and tnnsient conditions showing the difficulties of generating convcrpcd solutions. .A detailed description and cornpanson of pulsatile results generated is prrscntcd in Chapter 5 for two rnodels of the constricred carotid artery. Finally, Chapter 6 will enclose a summary of the conclusions and recommendations for future work. 2.0 LITERATURE REVIEW
2.1 Introduction
This section will discuss and review the relevant materia1 that has been carried out in the past. First, the discussion tvill involve the geometry or shape of the carotid artery and the importance of selecting an appropriate geometry. ARer an accurate representation of the -ceomet- has been discussed. the discussion will continue by summarizing the results that have bren prrformed using both experimental and numerical techniques. In the expenrnental section. borh Ni vivo and in vivo results measured by a variety of techniques will be discusscd. The numerical section will discuss the results from .?-dimensional and 3- dimensional numerical simulations with both steady and pulsatile conditions.
2.2 Phvsiological Geometries of the Carotid Arterv
The shape of thc carotid mery is very unique and difficult to accurately charactenze. due to the physiological variability encountered among individuals and the difficulties in extract in- the geometry frorn patients. The first studies of bi furcating or branching vessels bey with studies on "T" and "Y' joined piping networks in straight pipes. The main applications For these studies were for analyzing the flow field in industry processes. The analysij of these industrial processes has provided some information on the flow field that esists in bifurcating vessels. however, the geometry of the carotid artery is much different from thosr rncountered in simple "T" and "Y" joints, thus, producing very different and complrx flows. Due to the geometric differences and the associated differences in the flow field. an accurate and appropriate representation of the carotid Seometry must be established to simulate the carotid anery bifurcation under realistic conditions.
Studies cm be perforrned on geometries distinct to each individual (Liepsch et ai.. 1998. Salzar et al.. 1995, Motomiya ei al.. 1984) or be performed on averaged geometric models (Bharadvaj er al., 1982, Forster et al., 1985) that would characterize the carotid artery bifurcation of a typical individual. There are benefits and drawbacks of both procedures. The models of distinct individuals can take on distinct geometric characteristic features of that individual and can generate finer detail in the results. A simulation could produce results that acnially occur in living specimens and could generate al1 the flow details that would anse in that specitic geometric shape. Data from \ive patients can be extracted and used to make direct compansons andior validate either experimental or numerical simulations. The simulatrd results could be usrd to compare with other individual's distinct carotid anenes and possi bly establish a correlation to the development and progression of the disease. Howvcr. this may be difficult to accomplish since every individual has a distinct geometry and it is practically impossible to simulate every geometry that would bc encountered. Also. the results grnerated from such simulations would be specific to that individual and no other. The detail of the flow field would be highly dependent on the geometric shape of the carotid mep. Thus. it would be very difficult to extract distinct features in a tlow field for cornparison against other geometnc models unless extracted under a qualitative approach.
Average geometric models can be created, which would allow one model to classify a croup of individuals based on a specific cntenon. b For example. a category of healthy individuals with no occlusion of the artery could be used to create a model that is representative of the normal geometric mode1 of a carotid artery. Averaged models can also br created for geometnes categorized by a specific geometric chancteristic or any other characteristic that could be relevant to the disease. In doing so, an appropriate characteristic rnust be seiected. as well as a technique for categorizing the specific characteristic. The results iiom such an analysis would produce a model that would represent a group of individuals. however, the finer details in the desired results would not be visible due to the averaging of the models. In the average model approach, the neglected finer details would be considered as non-relevant noise in the flow field and thus, valid to filter out. With this average model method, a specific geometic characteristic can be analyzed under a controlled environmeni. The results can be used to associate or disregard the characteristic in question. in relation to the development and progression of the occlusive disease. Currently, a majority of the research has been perforrned on average models using both experimental and numencal techniques. The remainder of this section is used to introduce two popular eeornetric models. The first model will discuss the normal configuration created by Cr Bharadvaj et al. 1982 that corresponds to patients of asymptomatic carotid arteries. The eeornrtry of interest to the current study will involve the geometries created by Smith et ri!. C (1996). Smith's geomemes allow for observations to be made in carotid arteries that incorporate mild and severe stenosed geometnes depicting diseased arteries.
1.2.1 The Bharadvai's Model: The Average Normal Model
Many different averaged models of the carotid artery bifurcation have been utilized to study the tlow field within the lumen of the carotid artery (Forster ri al., 1985. Reneman et ul.. 1984). however. the most popular and well studied rnodel is the geometry described by
Bharadvaj gr al. (1983). Bharadvaj's rnodel represents an average mode1 of a healthy. asymptomatic individual with no occlusion of the ICA bulb. The mode1 was derived by using pairs of two-dimensional angiograms depicting the lateral and anterioposterior projections of the same carotid artery. A total of 124 angiograms were used with 57 of the angiograrns takrn from twenty-two adult patients between the age of 34 and 77 yean. The rcmaining angiograrns were exnacted from 50 children under the age of 18 years. A substantial number of angiograms were taken from children for the reason that atherosclerosis begins early in life; thus, taking angiograms from children would reduce the amount of growh present in the carotid artery (McGill et al.. 1968).
The final geometry was created by extracting measurements of the lumen geometry from 15 different locations deemed to be crucial dimensions in describing the carotid artery. The angiopms were measured using the CCA diameter as the unit length scale, since there was no common dimension availabie, thus, producing dimensionless rneasurement quantities. Bharadvaj3 model of the carotid artery bifurcation can be seen in Figure 2.1. Upon analyzing the angioprns and extracting the required measurements, Bharadvaj made some assumptions to the presented angiograms. Fint, the CCA was assumed to be parallel to the angiograms. thus. paralle1 to the major structures of the neck. Second, the artery bad circular cross sections throughout the whole dornain of the geometry. Third, the CCA and the two ECA and ICA branches were al1 aligned on one cornrnon plane. And lastly, al1 portions of the same angiogram were displayed at the one magnification.
Figure 2.1: Bharadvaj's model of the normal carotid artery bifurcation (Bharadvaj et al.. 1982).
The model created by Bharadvaj has been used in many experirnental and numerical investigations into the development of the atherosclerosis, however. ihere are other ceometries that may be considered to influence disease. One major development in the b models of the carotid anery are the geometries created by Smith er al. (1996). where the models are based on symptomatic patients, which will be discussed in the subsequent section.
7.2.2 Geometric Variations: Smith's Models and Other Models
Since Bharadvaj's model, there has been an interest in looking at variations in the geometry of the normal carotid artery. This interest has been provoked by the highly unusual flow field encountered as the shape of the geometry changes. This led to the creation of other new average geometries, or other models denved from Bharadvaj's model, that incorponte a specific geometry variation from the nom. ïhe normal geometry of the carotid anery is beneficial in the investigation of the initial development of disease but, other
c.reometnc configurations such as occluded geometries are valuable in giving insight to not only the development, but also the progression of disease. Perktold et al. ( 199 t b) looked at geometries with small variations in the shape of the sinus but mainly focused on simulations involving different bifurcation angles between the ICA and the ECA. Wells et al. (L996) made slight modifications to Bharadvaj's model to examine 3 parch reconstructed endarterectomy and a gradually tapered model (no carotid bulb). However. one of the most significant geometric variations in the carotid artery and arteries where the disease consistently develops, such as the coronary artery, is the reduction in lumen diameter. In the carotid artery. the reduction in the vesse1 diameter occurs in the sinus of the ICA and is observed as the disease evolves. This reduction in lumen diameter is mrasured by the ilorth American Symptomatic Carotid Endanerectomy Trail (NASCET) stènosis index. which is detined as.
~vhcreS is the percentage of stenosis, d, is the diameter measured in the ICA at maximum stsnosis and DTCAis the ICA diameter measured weil downstream from the bulb. Some researchers have characterized the grade of stenosis by an area reduction instead of a diametrr reduction as done so by NASCET stenosis index. This stenosis index is directly related to the reduction in the cross sectional area by the gaph shown in Figure 2.2. Thus, a 30% diameter reduction would be equivalent to a 51% area reduction. In the present study, al1 stenosis gades will be referred to by the NASCET stenosis index.
There has been research performed in stenosed geometries, however, a majority of the studies wcre perfomed on straight tubes, which do not incorporate the complex nature of the bifurcating vessels observed in the carotid artenes (Deshpande et al., 1976, O'Brien el al.. 1985). Due to the complex nature of the geometry and the dificulties involved in describing the diseased vessels, there have not been many models that accurately describe the development of stenosis. One of the initial studies perfomed on stenosed geometries of the carotid artery was by Fei et al. (1988)' however, the work did not provide stenosed eeometrirs that are representative of physiological shapes that would be encountered. A CI discussion involving stenosed geometries will be provided in a later section of this cbapter.
I
80 +--
------
I -- - - 1 I 1 O 20 40 60 80 100 I
NASCET Stenosis Index (O/p) Il
Figure 2.2: The NASCET stenosis index venus an area reduction.
Smith er al. ( 1996) created a family of models that physiologically correspond to the shape of the carotid artery at different stages of the atherosclerotic narrowing, staning from the normal carotid artery, and evolving to a 90% stenosis (see Figure 2.3). Smith created the modrls by using 67 biplane angiograms showing the anteropostenor and lateral views from symptomatic patients. The angiograms were categorized by using the NASCET stenosis index and divided into 3 distinct classifications ranging from mild (S < 30, n = 12), moderate (S -= 70. n = 'a), and severe (S < 99, n = 22).
Al1 the angioprns were digitized and manually traced out on a high-resolution cornputer monitor. The data generated from the angiograms were used to create several unes to define the centerlines and radius of the outer geornetry. The curves were then manipulated onto a standard template in such a way as to maintain the shape of the geometry (Figure 2.4). The standard template, which Smith used as the disease free model, was generated from the averaged geometry created by Bharadvaj et al. (1982), presented in Figure 2.3: Wireframe drawings of the normal and stenosed carotid bifurcation çenerated by Smith et al. (a) Modified Bharadvaj normal geometry; çoncentnc pometries: (b) 30%. (c) jU%, (d) 60%, (e) 70%. and If) 80% stenosis, and eccentnc geometries: (g) 30%, (h) 30%. (i) 60%. 0') 70%. and (k) 80% stenosis. (Smith et al.. 1996)
Figure 2.1. The main difference between the two normal geometries is that the standard ternplate incorporates a more physiologically realistic description of the downstream angle benveen ICA and the EC-A branches. The ICA and the ECA branches were bent inwards to be paralle1 with the common carotid artery (CCA). This implies that afier the CCA divides, the two branches will proceed venically at a physiologically realistic inclination, instead of as continuously diverging branches, as modelled by Bharadvaj.
The stenosed models were avenged to create several models with varying sizes of lumen reduction. The models were created by using the radius data from the diseased models R interna1 , int
external ' Re xt
Figure 2.4: Standard template used by Smith et al. ( 1996) to compare angioçpphic data. (Smith et al. 1996) and the centerlinr data frorn the normal geometry. Another important factor to consider in cenerating discased geometries is the position of the stenosis since the plaque can build up C rvrni y around the inner surface of the artery walls or on a specific side of the artery. Smith's mode1 incorporates the plaque position by creating concentric and eccentric models. The concentric stenosis represents the growth of the disease such that the artenal lumen is positioncd in the middie of the artery. The eccentric stenosis describes the case where the disease is tocused to one side of the anery. most commonly on the outer wall of the sinus. opposite to the apex.
This family of models, created by Smith et al. (1996). is one of the first models to incorporate rnild and severe stenosed geometries that are based on patient data. The assumptions made whilr creating these models are sirnilar to the ones imposed by Bharadvaj. The geometry of the ECA and ICA branches is assumed to be coplanar with the CCA and all aneries having circular cross-sections. These models are not intended to exactly mode1 the eeometrirs of syrnptomatic patients but to mode1 a geometry that provides a physiological C description of the nanowing e ffect encountered as the disease develops. Many researchers have concluded that the vesse1 geometry is a main factor in the fiow field (Young, 1979). Thus. with the use of these models vital information can be established concerning the development and progression of atherosclerosis, in relation to stenosed arteries. By prrsenting observations in stenosed geomerries, interpretation of the reçults may dari@ factors that influence the progression of the disease and provide detailed descriptions of the tlow distinct tlow patterns. The use of the models created by Smith et al. (1996) will allow thrse observations to be considered and will benefit investigations on atherosclerosis.
2.3 Hemodvnamic Theories of the Atherosclerotic Disease
The main goal of research perfomed in this area is to detemine factors that influence the dcvrlopment and progression of atherosclerosis. Research from Fry ( 1968) concluded that atherosclerosis originated from areas of high shear stresses on the vesse1 walls, thus. on the inner wall of the ICA in the vicinity of the apex. Fry believed that due to the high shear stress. the rndothelial lining of the artery wall would be damaged and cause interactions to occur between the rxposed mery and the blood flow. However, according to the work perfomed by Caro er al. (1969) it was the exact opposite of Fry's theory that the atherosclrrotic lesions anse in areas of low shear stress because of the shear-dependent mass transfer mechanism of the disease. Thus, implying that the disease develops on the outer wal1 of the ICA in the region of the sinus. The theory of Caro er al.. has since been confirmeci by rnany other researchers (Ku et ai., 1985, Zarins et al.. 1983, LoGerfo et al..
198 1 ) and by observations in autopsy studies of cadavers diagnosed with the disease (Solberg et d..197 1).
However. other researchers were not satisfied with just the low shear phenornenon. Some researchers (Lei et al., 1995, Giddens et al., 1987) believed that in addition to the low shear stress. it is an oscillating shear stress that causes increased and rapid endotheliai injury. This hypothesis suggests that back and forth shear motion produces a higher imtation of the rndothelium layer allowing for rapid degeneration of the artery wall, thus, enhancing the penetration of abnomal materials and perhaps the development of atherosclerosis.
Other researchen (Ma et al., 1997. Glagov et a!., 1988) believed that it is the prolonged residence time that certain amounts of fats and lipids are allowed re-circulate in one specific area. With the prolonged residence time, the fats and lipids are given an extended oppomnity to penetrate the artery wall. The recirculation zone also decreases the ability of fresh nutients to enter the recirculated stream for the exchange of vital nutrients between the artery walls. Friedman (1986, 1989) believed that aged vessels are more susceptible to the disease because of the elevated shear rates encountered. Friedman ( 1989) hypothrsised that intima! thickening occurs because of an inappropriate correction process that the artery cames out to maintain a favourable shear rate on the artery walls.
These theories have provided good dialog on the disease but it could actually be any combination of these theories that could result in such a disease or perhaps none of these. XI1 rhe above hypotheses discuss different features for the mechanism of the disease. however. rhcy al1 belirve that the wall shear stress, and thus the flow field. are important considentions in the development and progression of disease. Thus, an in-depth analysis of the hernodynamics in areas of bifùrcating vessels is of great importance.
2.4 Review of Relevant Ex~erirnentsand Numerical Simulations
2.4.1 Experirnents and Flow Visualization
The main rxperimental studies of fluid dynamics in the carotid artery began with sxperiments in flow visualization. Bharadvaj et al. (1982) provided steady simulations at various Re and flow divisions between the ECA and [CA with the main emphasis on Re=400. The qualitative results were generated by using the method of hydrogen bubbles and dye injection. These techniques allow bubbles of hydrogen gas or dye to be released in the flow stream. To observe the 80w patterns, the hydrogen bubbles are visually recorded to describe the main flow features. The results Frorn this study showed the existence of cornplex fiow patterns with the development of secondary flows. Helical flow patterns where observed in the sinus with separation occurring at the Front of the bulb and recirculation zones within the sinus. The complex secondary flows indicate the iimited benefit of observations in two-dimensional simulations for both nurnencal and experimental flows. These secondary flows are created by centrifuga1 forces as a result flow through the cumed geometry. The results also indicate that areas vulnerable to the development of the diseasc occur in regions of low, oscillating flows and not in areas of high shear stress.
LoGerfo et al. (1981) also provided visualizations of steady simulations. but in a modified Bharadvaj model. This modified model was shaped to provide sirnilar angles of the ECA and ICA branches of the angiogram for a particular symptomatic patient. This çonstructrd geometry was intended to represent the disease-free artery of that particular patient. thus. enabling cornpansons to be made between the localization of the disease and the flou. patterns that exist before the development of the atheroscierotic plaque. Dye injection was used to provide visualization of the flow field. The steady simulations were done at a Re between 200 and 1200 at various flow divisions between the ECA and ICA. The results genentrd from the steady simulations showed that the maximum size of the separation region in the sinus occiirred at a flow division of M:7O (ECAKA) and the region of separarion corresponded to the position of the development of the disease.
Ku er ai. ( 1953) presented flow visualizations for pulsatile conditions. The imposrd wavcfom was characterized by using a half sine wave for the systolic portion of the pulse and a constant tlow for the diastole portion. This wave doesn't accurately describe a typical waveform that would occur under physiological conditions but for the purpose of visualization. provided useful information on pulsatile fiow patterns in the carotid artery. This investigation employed Bharavdvaj's model with the hydrogen bubble technique for tlow visualization. The presented results showed marked differences in the tlow parterns from steady simulations. The separation zone varied in size and position with reverse flow varyin3 in magnitude and direction. The results indicated a prolonged residence of hydrogen bubbles in the bulb opposite of the apex that remained for several pulse cycles. Palmen et al. (1994) also showed experimental results but in a mildly constncted model that used an unrealistic stenosed geometry. The expenment was conducted under pulsatile conditions but with a simplified cardiac wave. With the use of the hydrogen bubble technique, the cornpanson between the normal and mildly constncted artery showed greater influence on the tlow regime in the shear layer with small differences in the main Stream. Experirnents based solely on visualization techniques are not able to provide specific information on the development of the disease. However, these visualizations aid in describine the complex features of the flow and may provide limited insight to the developrnent of the disease. To provide vital information conceming the disease other cxperimental and numencai techniques must be used. For a better analysis of the flow brtween normal and constricted geometries quantitative results must be incorporated to make direct comparisons. With the addition of quantitative measurements particular propenies of the tlow tidd cm be used to establish the differences behveen the normat and the diseased 1ir-terit.s.
1A.7 S teadv. ?-dimensionai Analvsis in Bifurcatine Geometries
Expcrimental and numerical studies of the carotid artery have been camed out that involve using a 3-dimensional assurnption. Femandez et ai. (1976) provided steady rxptirimentiil simulation in a straight sectioned. bnnching channel at a low Re number (Re=IJ6)and showed ihat even at low Re nurnbers a separation region exists along the outer wall opposite of the apex. Wille (1980) provided a numerical simulation. also using straight bitùrcating channels but presented conflicting results showing separation occumng at the innrr wall of smali angled bifurcations at vanous Re numben. This can be explained by the
.. .. ecometry Wille used. Wille incorporated a larger cwed apex at the bifurcation. which caused the separation region to occur at the inner wall instead of at the wall adjacent to the apex. These two simulations show the importance of including the generai features of the carotid geometry, such as the bulb positioned in the ICA and the shape of the apex. Plane bifurcating channels do not incorporate the complex geometry of the carotid artery. .Alrhough these results do provide insight to the flow pattems, results from accurate representation of the carotid geometry would be beneficial.
Rindt et al. ( 1987) provided a .-dimensional, numerical and experimental analysis in the carotid geometry created by Bharadvaj et al. (1982a). The results presented demonstrate sipificant differences with simple pipe geometries. Differences were observed in the overall flow pattems. Ieading to variations in the distribution of the wall shear stress. Similar steady, ?dimensional experimental results for the normal carotid artery were found by Nazemi et al.
( 1990). Even though the results were distinct in some aspects, al1 the discussed 2- dimensional simulations showed similarities in the existence of separation and reattacbment points with the addition of recirculation zones. However, these simulations were carried out undrr the assumption of ?-dimensional flow, which is not realistic. From the expenmental visualkation techniques (discussed above), it was found that the fiow afier the bifurcation consists of highly secondary effects. In 2-dimensional simulations, these secondary flows are unattainable and present an uncenainty in data obtained from 2-dimensional simulations.
7.4.3 Steadv. 3-dimensional Analvsis in Bifircatine. Geometies
Based on the previous 7-dimensional analysis, the nent logical step was to camy out 3-dimensional numencal and experimental simulations. allowing the results to predict comples secondary flows. Rindt er al. (1989) provided numerical. 3-dimensional. steady simulations that confirmed the capabilities of numerical simulations to predict these secondary flows and made direct cornpanson with experirnental LDV data. The simulations provided results with various combinations of flow division ratios. bnnching angles, and a pair of relatively low Re nurnbers. The results concluded these affects to be small in the rçgion of reverse flow. However. simulations were not provided at highrr Re numbers that would correspond to flow at the average or peak systole. Lee et al. ( 1996) also provided low Re number numerical rimulations but included intimal thickening under elevated leveis of shear stress. thus allowing the shape of the carotid bulb to adapt to maintain an appropnate level of shear stress (Friedman et al., 1986). The results showed that the artery tends to thicken to fil1 the recirculation region, and thus, minimize the amount of reverse flow. However. the results were highly dependent on the approximation used to mode1 the aneries ability to thicken (Lee et al. 1992). Zarins et al. (1983) provided LDV measurements under steady conditions in the mode1 created by Bharadvaj. The measurements supported the obsemations and arguments proposed earlier by Bhmdvaj et al. (1982), conceming the helical flow patterns and complex 80w structures. In addition the LDV measurements provided quantitative results that support intimal thickening occurring on the outer walls opposite of the apex where low shear stress with oscillating flow and flow separation exist, as done so by Lee et al. ( 1996). Pingping et al. ( 1997) showed steady numerical simulations of the prolonged residence time of fluid particles in the buib and suggested a possible correlation of plaque accumulation.
These steady numerical simulations provided little indication of the spatial convergence or resolution of the results. They were dont: essentially to make qualitative obsrnxion in çomparison to experirnental results. This concem rnay not be of great importance in low Rr number simulations under steady flow. but may be important in pulsaile simulations. Furthemore. to make crucial conclusions in relation of the disease. an addcd complexity to the problem must be incorporated and that is the puisatile nature of the t1 O LL..
2.4.4 2-D and 3-D Pulsatile Flow in Bifurcating Geometries
The above steady simulations are not actual flows that occur under physioiogical conditions. To provide results comparable to the flows observed under physiological conditions. pulsatile flow must be incorporated. Nazemi et al. ( 1990) numerically showed that steady tlow simulations are not useful for describing the flow behaviour and the wall shear stresses that result from transient simulations; rven simulations that are perfomed at vanous Re numben that would chancterize the cardiac wave. In pulsatile flow, the recirculation zones increase in size during the deceleration phase and can create multiple recirculations zones. Rindt et al. (1957) confinned these results, showing considerable differenses in the size of the recirculation zone between numerical steady and pulsatile simulations. The 2-dimensional results showed the maximum diameter reduction created by the recirculation zone was about 25% for steady simulations cornpared with a 50% diameter reduction during flow decelention of pulsatile flows. Liepsch et al. (1984) also provided supponing evidence in pulsatile LDV measurements but in different bifurcating vessels. Apain. the results ernphasize the importance of the flow division ratio, which vaiy the size and position of the recirculation zone, as mentioned by LoGerfo et al. (1981), and the eeometry of the branching vesseis. The results showed increase in the size of the Ci an recirculation region in pulsatile results in cornparison to steady simulations. Frmandez et al. ( 1976) reported growth and disappearance of the separation zone in bifurcating tubes at low Re numben with the use of a penodic sinusoidal forcing function.
Ehrlich et al. ( 1977) atso used a sinusoidal curve to extract results and found that the significmces of particle stasis (or residence time) rnay not be an important factor in athttrogenrsis. However, in applying pulsatile flow. Lee et al. (1992a) confirmed the importance of the description of the forcing wave to be relatively imponant. Imposing a siniisoidal wave superimposed on a mean velocity does not provide an adequare curve to cxtract conclusions concerning physiological flow patterns because of the great differences presentrd in the results. There are many differences encountered in the physiologie wavefom arnong individuals but certain fundamental features of the curve must be present. The cxdiac wave must consist of the a high accelerating and deceleratinj fiow occumng ovrr a shon tirne frame and a secondary pulse with the remainder of the wave having reasonabiy constant flow.
From the analyses perfonned by Perktold et al. ( 1986) and later by other researchers (Rindt. 1996. Perktold et al., 199 lc, Van de Vosse et al.. 1990). the pulsatile flow provides complra velocity patterns throughout the carotid wave. With the sharp acceleration and dscelrration of the flow field. it creates temporal changes in the position and length of the recirculation zone. And thus. temporal variations are presented in the magnitude and direction of the shear stress that is induced on the walls of the carotid sinus. Nazemi et al.
( 1990) indicated that the upper region of the bulb, the region aligned with the i~erwall of the ICA. and regions in the lower end of the bifurcation may be susceptible to the disease because of low and high shear stress oscillation (Figure 2.5). Lei et al. (1995) confirmed the possibility of plaque formation in these locations by ernploying a new dimensionless parameter il. which "correlates the physics of abnonnal hemodynamics with the biology of abnormal events" (see Lei et al., 1995 for the exact description and the equations). Ku et ai.
( 1985) provided LDV measurements in the Bhavadvaj mode1 and used a physiological pulse for the inlet and outlet conditions. Also. carotid arteries obtained from human cadavers were used to directly measure intima thickness. Ku et al. (1985) provided conclusive cornparisons that correlated areas of high intima1 thickening with regions of high, inverse time averaged Figure 1.5: Positions in the carotid artery susceptible to the disease. (Lei er al.. 1995). shear stress. high regions of the inverse muimurn shçar stress and positions where oscillating shear stress is abundant. Later, several researchers (Friedman et al. 1989, 1986. Ku er al.. 198 7, Rindt er al.. 1988) provided supponing evidence showing that regions of low and oscillating shear stress are associated with the developrnent of atherosclerotic plaque as opposed to areas of unidirectional high shear stress.
Currently, the discussion has mainly involved the analysis of normal carotid artenes, however. there are other aspects that cm be analyzed. Perktold et al. (1990b) discussed 3- dimensional geometric variations of the normal carotid artery. taking into consideration the stiape of the bifurcation and the branch angles of the ECA and ICA. The solution incorporated the flow features of the cardiac wave and the results conciuded that an elevated width of the carotid bulb is a favourable factor in atherogenesis. Thus, adding to the many hypotheses that regions of reversed and stagnant flow with recirculation zones are influential to the developrnent of the disease. In relation to the study concerning the different bifùrcating angles, the results indicated an influence on the position of the recirculation zone, which conflicts with steady, 3-dimensional results provided by Rindt et al. (1989). These results showed the recirculation region, that exists in the carotid sinus, is positioned further downstream in large angled bifurcations and lower in decreased bifurcation angles. Rindt et al. ( 1996) adds that a decrease in the sinus angle (the angle between the extended CCA and KA). creates a smaller region of reversed flow. Ku et al. (1985) also discussed the dependence on the branch angle between the ECA and ICA to the size of the recirculation zone. as discussed by others above. With increased branch angle. an enlargement of the recirculation zone occurs creating an increased region, which rnay be more susceptible to the growth of disease.
2.4.5 Flows in Constricted Arteries
Substantial evidence has been found to correlate particular tlows patterns with the drvrlopment of the disease in the normal carotid artery. however additional information can be acquired hmstudying other parameters that may influence disease. As mentioned in other sections. diseased vesseis with the stenotic geornetric configuration would be one of the most significant and appropriate parameters to provide future conclusions. The stenosis would be the next feature of interest because of the high frequency of occurrence in relation to the existence of the disease. Also, diseased artenes may provide information pertinent to not only the development of atherosclerotic lesions but in addition. to the mechanisms for the crowth and continued evolution of the disease. CI
Solzbach er al. (1987) studied low Re number experiments under steady conditions, but in straight circular tubes. Constrictions were incorporated by introducing sudden obstructions instead of a smooth transition, wtiich occurs in the ICA sinus. The results indicated elevated flow instabilities with increased cross sectional area reduction but decreased tlow instabilities with increases in the length of the stenosis. Liepsch et al. ( 1992) performed a similar experimental study but with a broader stenosis range and an elevated range of Re numben resembling a physiological wave. In addition to showing increased tlow disturbances with increased Re numbers, the results illustrated an increased size of the recirculation zone. Ahmed et al. (1983a & 1983b) provided steady LDV measurements in a smooth constriction and showed the intensity of the flow disturbances were relatively low with a 50% area reduction. Young et al. (1973) expenrnentally observed influences based on steady flow through non-symmetric and symrnetric stenosed tubes. It was observed in non- symmetric stenoses that higher maximum watl shear stress Ievels were created and non- symme tric stenoses are more susceptible to instabilities. Rosenfeld ( 19%) and Rosenfeld et al. ( 1995) provided an in-depth snidy and review of numerical pulsatile flow in constricted nibes with discussions of vortex generation and stenosis severity. Additional experiments (Van de Vosse et al. (1990). Ahrned et al., 1983a&b, Khalifa et al., 198 1. Azuma er al., 19%) and numencal computations (Hung et al., 1997 & 1997, Wong et al.. 1991, O'Brien et al.. 1985. Siouffi et al.. 1984. Deshpande et al., 1976) have been performed showing sirnilar conclusions.
The simulations discussed above are camied out in straight pipe tlow and have limited insight to the development of the disease because of the complex configuration of the carotid -crometry. Currently. very few have provided experimental and numerical simulations in relevant geometries under physiological conditions. Fei et al. (1988) provided steady experirnental results on a stenosed geornetry of the carotid anery but provided no evidence of the sonstnction in relation to physiological geometries. The results, obtained from Doppler pulse ultrasound. showed that stable flow occurred below a 2096 diameter reduction with oscillating jet flow at 40% stenoses. The laminar-turbulent transition was found to occur with a 60O0 diameter reduction and hlly turbulent flow at an 80% constriction. As rnentioned radier Palmen et al. (1994) provided tlow visualization in a stenosed carotid model. The results were observed in a 75% area reduction geometry showing the existence of vortex structures at the base of the systolic portion of the pulse. Clark el al. (1983) canied out nurnerical simulations in a bihrcating geometry but positioned the stenosis in front of an apex in what would represent the CCA. Steinman et al. (1999) provide experimental and numerical simulations on stenosed geornetries showing the differences in the flow patterns and wall shear stress between eccennic and concentrïc geometries. Currently, accurate experimental and nurnerical studies are required for cardiac flow of physiological geometries of mild to severe stenosis. 2.5 Summary of Review
The information in this chapter has provided sufficient evidence that the hernodynamics in the carotid anery plays a vital role in the development of atherosclerosis. Little reseürch has been perfomed on stenosed modeis of the carotid artery. and because of the importance of hemodynamics in relation to the development of the disease. numerical simulations would be beneficial. A detailed description of the flow field in mildly stenosed geomrtrirs rnay provide new flow features to develop in relation the atherosclerotic disease or enhance the mechanisms of previous theories. The geometnc shape of the carotid anery under the development of the disease is an important consideration and in this study the rnildly constricted models will be taken from the study carried out by Smith er al. ( 19%). The rnudels created by Smith were generated based on data of symptomatic patients and provide a wide range of stenosis severity. which would correspond to evolution of the disease with continued development. This family of models will allow for direct cornparisons to be made conceming continued growth of the disease. The results from such study will aid in enhancins previously discussed hypothesis conceming the atherosclerosis or create new compelline relations between particuiar flow features and the growth of the disease.
The information given in this chapter provided detailed descriptions of relevant numerical and experimental studies conceming the carotid artery and atherosclerosis. The numerical simulations have provided information on several factors on the flow patterns. However. for numerical studies to be a continued benefit to studies in this area other features or improvemenrs must be snidied. With increased cornplexity of both geomemc models and numerical algorithms, the Iimiting factor is currently the computational resources and computational tirne for these simulations. Thus, numerical techniques must be improved upon to provide accurate results with limited computational time and resources. Because of the continued advancements in the complexity of models created for the carotid artery these new numerical techniques will be beneficial to those studying the disease as well as the CFD community. Thus, the current study will provide one possible technique to reduce the required computational resources. Existing experimental results will aid in the validation of the results created from these simulations and the technique employed to limit computational dependence.
In providing numencal simulations, attention will be given to the accuracy and conversence of the numerical results. Rindt et al. (1996) provided pulsatile results and discussed the ability of providing estimates for the accuracy of the numerical results by providing simulations at finer grid densities. This task was not camed out at that time brcause of the limited cornputer resources. Many oiher researchers have also ignored this fundamental principle, thus. suggesting that a majority of the results may be under-resolved and have high levels of numerical error. The current snidy will provide numerical simulations with accurate estimates of the spatial and temporal convergence. A detailed description of the results will be provided. as well as the procedure to obtain such results. With this addition to the numerical simulations. results cm be provided that are independent of significant numencal error. 3.0 COMPUTATIONAL MODELLING OF THE CAROTID ARTERY BIFURATION
3.1 Introduction
To prrfonn a numencal simulation. the governing equations for isothermal fluid tlow mut be formulated and solved based on the conservation of mass and the conservation of mornentum. To allow for the numencal simulation of the fluid flow in the ciirotid anery bifurcation, sevenl factors and assurnptions must be considered and justiticd. In this chapter. the numencal techniques will be discussed showing the discrrrizcttion and implernentation of schemes used to solve the flow field. Funhermore. an in-dspth discussion of boundary condition implementation using both steady and transient conditions is given. The periodic waveform that will be used to implement the physiological inlet boundary condition will also be discussed. as well as the innovative technique for implementing the two tnnsient outlet boundary conditions.
3.2 The Geometrv of the Carotid Arterv
In this investigation. the geornetries to be studied are the same models created by Smith et ai. (1996) (see Figure 3.1). Simulations were camed out in the normal geometry (Figure 3.14, however. these results were mainly used for validation of the numerical method in both puisatile and steady simulations. The main ernphasis of the curent snidy is to sirnulate the geornetries of symptomatic patients with rnildly constricted vessels. Simulations will genente outcomes for the 30% stenosed geometries with both concentric and eccenaic build up of plaque (Figure 3.lb and 3.1~). Currently, only the 30% stenosed models will be examined because, based on steady experimental data of constricted models (Fei et al., 1988, Solzbach et al., 1987, Giddens ei al., 1976), no turbulence structures are expected to be present Beyond a 30% stenosis, it is not yet known whether the flow remains Iaminar. Figure 3.1: Smith rnodels. (a) Normal model. (b) 30% concentric model, and (c) 30% eccentric model.
3.3 Mathematical Formulation
The numerical simulations perforrned in this study will involve a laminar tlow regime under pulsatile conditions. Thus, the solution is obtained by solving the conservation of mass and momentum equations. which are presented in strongly conservative form as:
where U,is the velocity in the x, directions and P is the pressure. The fluid properties p
and p are the density and dynamic viscosity, respectively. In solving these equations, several assumptions must be made concerning the flow of blood in artenes under the development of atherosclerotic plaque. The first and most argued assurnption by researchers is the consideration of the fluid medium, blood, to be eithrr a Yewtonian or non-Newtonian fluid. There has been research performed that models blood as a non-Newtonian fluid (Perktold et al., 199 1a & 199 1b, Ku et al., 1986. Liepsch er ai.. 1984), however, a rnajority of the studies have assumed a Newtonian fluid. This indecision is prompted because the properties of blood differ depending on the scale at which it is obsenied. In microcirculation, flow through anerioles and capillaries. blood is considered to be a NO-phase fluid. with formed elements consisting of platelets. and red and white blood cells suspended in plasma. Plasma is a Newonian tluid. but with the existence of formed elements, blood has non-Newtonian fluid properties in the mircocirculation. The non-Newtonian properties of blood are termed 'pseudoplastic ' or shrar thinning. meaning that with elevated levels of çhear. blood is allowed to flow easirr and while esposed to low values of shear, the texhire of blood thickens. In maçrocirculation. tlow in larse arteries, blood can be assumed to behave as a Newtonian tluid (Young.- 1979). Based on the fact that the size of the carotid artenes (approximately 8 wnin diametrr) are considered to be large in cornparison to anerioles and capillaries (Iess than 390 ,unin diameter), the flow of blood in the carotid arteries can be considered as 3, 'c:wtonian tluid. Researchers have shown that the effect of the non-Newtonian behaviour of blood is negligible in the carotid meries (BaiIyk, 1994 Perktold et al., 199 1b & 199 1c. LoGerfo et al.. 198 1). Thus, the assumption of biood as having Newtonian properties will be used throughout this study. With this assumption, the viscosity of blood is constant and can be used to sirnpliQ the goveming equations by moving viscosity. p, outside of the differential (Equation 3.2). Furthemore, the density, p cm be withdrawn frorn the differential because blood is considered to be an incompressible fluid. ïhe working equations are then: Some additional assumptions that must be considered do not really relate to the goveming equations. but instead relate to the modelling of blood flow through diseased aneries. First, an assumption must be made conceming the elastic nature of the artery walls. Healthy artery walls behave as though having an elastic property by dilating and çonstricting depending on factors in the circulatory system, such as the periodic flow Md and the îlow requirements of the downstream organ. However, with the existence and growth of the disease. the elastic property of the artery diminishes and the artery hardens in the vicinity of the disease. There have been both numerical (Perktold et al.. i99Z. Perktold et al.. 1994) and experimental (Anayiotos et al.. 1994, Ku et al., 1985) studies that mode1 compliant artery walls. however. these studies investigated healthy anrrics. while the current study deals with artenes under the intluence of diseasc. Xlso. the arteries ofrldrrly patients. who suffer more from atherosclerosis. are more inclined to becorne hardened (Reneman et al.. 1985). Employing rigid vessels. in the extreme case. is rxpçcted to overestimate the wall shear stress (Liepsch et al., 1984) and thus, provide a sakty factor from conclusions derived Etom values of wall shear stress. Even though the discase dors not affect the full geomerry of the artery. it would be more accurate to çonsidcr rigid vessels since the domain of interest, the ICA sinus, contains the existence of the disease. It would be dificult to develop a numerical mode1 having vessels that are panially clastic and partially rigid.
..\ second assumption must be made conceming the flow division that occurs betwecn the ECA and the ICA. In simulating pulsatiie flows in cornplex bifurcating vessels. it is difficult to detemine the flow ratio between the ECA and ICA required to duplicate the flow in a typical patient. In actual fact, the flow of blood through the branches changes throughout the whole cardiac pulse and is very dificult to quantify. Many studies have imposed constant flow divisions because of the difficulty involved in generating experimental resuits that could be used for cornparison. Measurements have found that the flow ratio between the ECA and ICA was 45.35 at peak systole (Ku et al., 1987). During diastole the ratio was reduced substantially to 10:90. Since there is more activity occumng during the systolic portion of the wave, the current study will maintain
3 constant mas flow division at peak systole. Expenments carried out by Steinman et al. ( 1998) used a flow division ratio of 44.-56. This division is extremely close to the values found by Ku et al. ( 1987) and due to the direct availability of the expenmental results from Steinman et of. (1998). the current study used a constant flow division ratio of 44?6 through the EC.4.
3.4 The Computational Mode1
To solve the consemation equations, a commercial Cornputational Fluid Dynamics (CFD) package. CFX-Tfc (Advanced Scientific Computing Ltd.). was used. Intemal parameters rvere adjusted such that the discretized equations were representative of n transisnt. 3-dimensional. incompressible and Newtonian nuid flow (Equation 3.3 and 3.In CFX-Tfc the partial differential equations are converted to algebraic equations by discretization in space and time. To discretize the equations, CFX-Tfc utilizes an slement-based. finite volume technique that operates with hybrid. unstmctured grids. An element-based. finite volume technique ernploys a finite element mesh. but solves the goveming equations by supenmposing a volume mesh as show in Figure 3.2.
Finitc rvdumc
Figure 3.2: 2-D finite element-based finite volume mesh. The conservation equations are integrated over each volume and time. For a general scalar. #. with ü transport equation of the form,
the equation for integration is.
Csing a tiilly implicit scheme (first order in time) the inteption of each tenn is camed out ris follo~vs.
I .) For the transient term.
2.1 For the convection tenn,
otav =C(m#Y+% ", S
3.) For the diffusion tenn, 4.) For the source term.
Thus. transport terms are convened from volume to surface integrals by employing the divergence theorem and the transient and source terms are lefi as volume integrals. Hence. combining al1 the terms. the final discretized equation is as follows,
To evaluate the surface integrals (surnrnation over each surface), approximations are made for each desired variable at integration points (ip),which are positioned on each sub-volume face (see Figure 3.2). There are several techniques that can be used to detemine the quantities at these integration points (&). CFX-Tfc approximates the advective transport throqh the integration points by ernploying a second-order advection scheme based on a Skewed Upstrearn Differencing Scheme (SUDS), with the addition of Numencal Advection Correction (NAC). SUDS is similar to the technique of Upwind- Differencing Scheme (UDS), except that instead of approximating hP from an upstream 2nd point. SUDS obtains an approximation for hP by integrating upstream along a C streamline. This scheme provides a better approximation of the advecting quantities at the surfaces. Incorporating the use of NAC provides an additional level of accuracy in the approximation of the bp values. NAC helps by including the discretization of the advection and source term to provide a scheme that is second-ordet accurate, where UDS is first-order accurate and SUDS alone is only slightly better. With the addition of the convection and source terms, SLDS is not only based on the upstream position but is also intluenced by the downstream position. This weighting of the upstream position is executed by NAC.
Whilr advection is not the only process that must be rnodelled, it is the most important. It is well-known that a minimum second-order advection scheme is required to rninimize the effects of false diffiision. False diffusion occurs when the direction of flow through an element face is oblique to the element gnd lines. This error rnainly anses whrn a low order technique is used to approximate the values of hP for the sub- volume faces. For instance. UûS assumes the upstream nodal point dong the grid line tor 4,. thus. treating the flow behveen control volumes to be locally 2-dimensional and first-ordrr accurate. As a result of this, ( would be nurnerically smeared throughout the tlow field due to what is termed false diffusion. The error occurs throughout the fluid domain. however. the amount of error is dependent on the severity of the angle present. .A substantial amount of error can be reduced by using small elements and, wherever possible. amnging the rlement gnd Iines in the direction of the flow field. However, since we do not always know what the flow field looks like a priori, a much better approach is to use a second-order method to approximate the values of hPthe sub-volume faces. The method currently employed, SUDS with NAC, is a proficient technique to minimize the generation of false difhsion.
3.5 Implementation of Boundarv Conditions
3 3.1 Steady Flow Simulations It is relatively straightforward to perform steady flow simulations using the CFX- Tic software, however, in its present form a specified profile cannot be appticd on boundary faces; only uniform or plug-like profiles can be implemented. To remedy this. one could impose a plug-like flow at the inlet of a long straight tube and ailow the flow to develop weli before the flow reachcs the carotid artery test section. Irnplcmrnting such an enirance section would significantly increase the number of elernents rcquircd and would result in an increased solving the. Thus, an irnproved technique has been employed.
Parabolic Profile (spifi,
Pressure Specified Specified
Figure 3.3: Boundary conditions for steady simulations.
To resolve this inconvenience, the source code to CFX-Tfc was modified to aliow velocity profis to be prescribed on the boundary faces. To do so, nodal locations were found and used to calculate the appropriate velocity to be specifed. A more elaborate explanation about these modifications can be found in Section 3.6, CFX-Tfc Code Modifications. For the outlet boundary conditions, pressure and mass flow rate can be specificd (see Figure 3.3). -4 Dirichlet pressure condition is imposed by setting the average pressure at the specified boundary face equal to zero, which then also acts as a refrrence pressure level for the rest of the domain. The velocity components are set to have zero velocity gradients across the face. The mass fiow rate boundary condition is considered to be a Neuman boundary condition and is imposed by setting the sum of the mass tlows through a set of boundary faces to a specified value. This boundary condition t.stablishsd the flow division between the ECA and the ICA.
To impose wall conditions on the boundaries of the geometry. zero velocities
wre set 31 sach wall nodal point. which would correspond to the no-slip. zero penetration boundary conditions. Since. the geometry is constructed from circuiar cross sections with the center of rach circle positioned on one plane. a symmetry plane was used to allow for the rnodelling of only half the full volume. The symmetric boundary condition was imposed by setting al1 axial velocity gradients in the cross plane direction cquai to zero. thus. preventing flow from crossing the plane.
3 5.3 The Cardiac Waveform and Pulsatile Flow Simulations
Blood tlow through the cardiovascular system is transient with highly varying îlow rates throughout the cardiac wave. The cardiac cycle is periodic and consists of two phases. the npid acceleration and deceleration portion or systolic phase and the relaxation portion or diastolic phase. where the tlow rate is reasonably steady. To characterize the wave, two parameten are required. The two parameters that are commonly used in combination are the Reynolds Number (Re) and Womenely Number
( 7).The Re is defined as the ratio of inertial forces to viscous forces. and xcorresponds to the ratio of unsteady forces to viscous forces. The two panmeten can be calculated by using fundamental fluid variables as show below. Figure 3.4: The cardiac wave of a normal human subject.
Measurements of the tlow field waveform are difficutt to obtain and diffrcult to charactrrizr dur to the highly variable flow fields between different individuals and the complrx procedures of cxtracting the tlow field. Many researchers have chancterised the physiological waveform. with each researcher showing differences between their own results and others. These differences are expected due to the highly flexible flow fields sncountered from individual to individual. However. these researches have concluded that the actual waveform is not of major concem, as long as the key features are present. That is. the high acceieration and deceleration occumng in a short time over the systolic portion. and then relaxation in diastole. Another important feature is the srnail secondary pulse. which is believed to be caused by the aona, acting as a reservoir to decrease the initial high-pressure gradients. The cardiac waveform that was used in the current study is taken from the results extracted by Holdsworth et al. (1999) (see Fibpre 3.4). Holdswonh el al. ( 1999) used a total of 3560 cardiac cycles From 17 normal volunteen benveen the age of 14 to 34, from both the left and the right CCA. The rneasurements of the waveform were taken using a non-intrusive technique (pulsed-Doppler ultrasound). This technique allows the fiow field to not be disturbed when extracting the required measurements. The cardiac wave has a Re that varies from approximately 19 to a masimum of approxirnately 1100. with an average of 775. The Womersely number that was used throughout this study was taken from the experiments performed by Steinrnan et al. ( 1998) and has the value of 5.6.
To impose a pulsatile waveform at the inlet of the CCA. the fully developed, time prriodic profiles must be estabiished from the cardiac waveform (Figure 3.4). The analptical squation for hlly developed. time periodic flow in a circular pipe was derived by McDonald ( 1955), where the equation is presented in section 3.6 (Equation 3.9). The calculaird profiles cmbe seen in Figure 3.5 at several time steps throughout the cardiac cycir. in imposing these protiles, the inlet face has to be supplied with an ample number of nodrs to describe the highly complex shape of each profile created. From Figure 3.5. the protilcs show regions of reverse flow at sevenl stages over the cardiac cycle. thus. a direct inlet condition can not be imposed. In CFX-Tfc an inlet condition only allows for tlow to enter into the computational domain. The condition irnposed can still be applied
Figure 3.5: 2-D hlly developed profiles for the cardiac wave. as a Dirichlet condition and can be implemented the sarne way as the steady Flow simu1ations. However, to allow the fluid to enter and leave the domain at the same boundary lace of the CCA, the imposed condition is classified as an opzning. In imposing an open boundary condition, fluid is aUowed to enter and leave at the same face.
Using the same analogy as used on the hlet face, the outlet faces, which also exhibit reverse flow, can be set to opening boundary conditions. Since there are two outlet faces, the ECA and the ICA, a dirichlet pressure condition may br set an one facc, the ICA, and a velocity profile on the other. the ECA. However, in using this sri of boundary conditions, a substantiaily long ou!let section must be employcd so that a Fully dcveloped velocity protile can be re-established and specified on the ECA face. This extended outlet channel wili aîiow the flow to redevelop into the profiles presented above (Figure 3.5) but at a smalier magnitude, which is related to the flow split beiween the ECA and ICA and the area of the outlet face. Many researchers use this technique or
Fully Developed Pulsatile Profie Specified -z
Rigid
Constant Pressure Adjusting S pecified
------Figure 3.6: Boundary conditions for pulsatiie simulations. sirnilar techniques resulting in speciQing a Dirichlet velocity boundary condition at one of the outlet faces (Perktold et at., 199 lc). Use of this technique would again result in a substantial increase in elements and an increase in computational tirne. nus, a new technique was derived and used that would allow for a decrease in the total nurnber of rlrrnrnts and a decreased simulation time without sacrificing the accuracy of the flow Md. This innovative boundary condition technique was to impose dirichlet pressure conditions on both outlet faces with zero velocity gradients. One face. the ICA face. would constantly be set to zero, as done so in the previous technique and in the steady simulations. The other Face would set pressure to fluctuate rvith the tlow field while dlowing the appropriate desired flow division between the ECA and ICA. In this manncr. the velocity profiles can be developed naturally at both ourlets and full development of the velocity field is not necessary. The exact numerical algorithm will be discussed in a later section. as well as the results showing confirmation of the technique. This set of boundary conditions not only simplifies the present cornputations. but may also br used in studies invoiving turbulent smictures in the flow dornain. Since the profiles in turbulent tlows cannot be directly calculated for the outiet faces of this ceomrtry. the two-pressure dirichlet boundary condition is ideal. C
3.6 CFX-Tfc Code Modifications
To perform the required pulsatile simulations several modifications were made to the source code of CFX-Tfc. An in-depth understanding of the code was required, which was a difficult task because of the complicated structure of the code. A general layout of the main branches of the code cm be seen in Appendix Al with a brief description of rach routine, as well as the new routines added. In this section, the discussion will begin with the irnplementation of the inlet velocity profiles and then move on to imposing the tuo-pressure dirichlet conditions.
3.6.1 Inlet Veloci~Conditions To imposing the required boundary conditions for the inlet face, several new routines were incorporated in the existing CFX-Tfc software. The new routines recognizrd the inlet Face by the name specified for the boundary face. Actually, in al1 the new routines. the boundary faces are distinguished by the name of each boundary face. This dlou~ssimulations to run using the original code. To speciQ a profile on the boundary hce. information concerning the nodal positions are found for the CCA face. From this information the center of the circular face is extracted and used to calculate the local radius of rach point. (y..r/R), on the boundary. The radial coordinates allow a specifisd velocity cm be imposed based on an equation for the desired profile. The squation used for steady and pulsatile flows are given in Equations 3.5 and 3.9. respectively. These routines for imposing the inlet boundary conditions were used for both steiidy and pulsatile simulations. but with the pulsatile code having slight modifications to allow the profile to change at each new time step.
3.6.1 Time Steo Increments for Pulsatile Flow
In sirnulaiing steady runs, the solution is obtained by mnning a pseudo-tnnsient simulation. This is done by marching in time, solving one intemal loop of mass and momentum (tenned coefficient loops) for each tirne loop until a fully developed steady- state solutions is reached. However, for pulsatile simulations the solutions must solve several coefficient loops to obtain converged solutions at each time step. In doing so, if constant tirne increments where used, a substantial number of time steps would .be required to obtain detailed solutions of the systole portion of the pulse. Also, a sipificant number of time steps would be wasted on the diastole portion of the pulse and thus. increasing simulation time (see Figure 3.7a & 3.7b). To remedy this problem a routine was added to march in tirne using uneven time sreps; time steps that are based on havins constant cord lengths between each time step taken. This allowed for a decrease in number of time steps required while maintaining an ample number of time steps in both the systole and diastole portions of the pulse.
3.6.3 Implementation of the Two-Pressure Dirichlet Condition
To cany out the pulsatile simulations, dirichlet pressure conditions were imposed on hotli outlet faces. This allowed for a decrease in the lengths of the outlet sections bçcausc no fixrd. hlly developed profile was required to be imposed for each time step. The conditions were imposed by having one face. the ICA. set to impose a constant average pressure and the other face. the ECA, to have a tluctuating average pressure that changes for rach tirne step. By imposing a constant pressure value at the ICA face for rach timr step. this condition acts as a reference pressure for the fluid domain and causes the pressure field throughout the carotid domain to be re-established at each new time incremrnt. The changing pressure specified on the ECA outlet face is rnanipulated until a specificid crireria for the flow division is met. Thus, the pressure field would be altered until the desired flow ratio reached to within 1% accuracy while allowing the solution to converge. It is difiicult to accuntely pinpoint the correct pressrire value to specify on the EC.4 hce because the pressure camot be directly calculated or approximated. An exact pressure value for the ECA face cannot be calculated and directly imposed based on the information available: inlet velocity profile, flow split ratio and geometric configuration. This is because of the difficulties associated with the modelling of the compiex geometry and the constant fiow split ratio. Also. for pulsatile simulations. the pressure field is not in constant phase with the pulse wave and the pressures Vary in magnitude and direction throughout the pulse. The pressure field is highly dependent on the pattern of cardiac wave and the associated time step size for each increment in time. For instance, in the systolic portion of the cardiac wave, the high acceleration and deceleration that occurs in a short time frame will produce drastic changes in the pressure field that is required to maintain the desired flow split. ------. ------1 ù 1 O 2 3 3 O 1 G 5 9 6 9 7 0 8 O 9 Timc (s)
- -- Figure 3.7~Time step marching with 100 even time steps.
Figure ?.7b: Time step marching with 100 uniform cord length time steps. Consider the primitive technique shown below.
whrrr PtEji. and PorD are the new and old pressure vrilues. respectively. and R,,u and RA.,,,.,, lire the current and desired flow percentage through the ECA. This technique uses the pressure tield and flow split ratio of the previous time step solution to directiy calculate the corrected pressure value for the ECA face that is required to maintain the dcsired tlow split ratio. This technique causes the solution to quickly diverge for both srrady and pulsatile simulations. This is because of the accuracy of the initial gurss for the solution of the current timr step. For the first step. the initial guess is the initial coriditions for both steady and pulsatile simulations. This divergence at the initial time step occurs as a result of the initial guess being largely incorrect. If a very good estimate were provided for the next time step then the method may converge. In essence. at anytime level where changes are substantial. the primitive technique given in Equation 3. IO causes difficulty. Thus. in our pulse cycle. the technique would be appropriate for the later pan of the pulse. but during the initial accelention~deceIerationphase, the primitive technique would surely $il.
To provide a robust technique, the method of the dual pressure conditions must carefully take inio consideration the validity of the previous solution. This will allow for extraction of a new pressure value that will steer the flow regime in the correct direction aliowing the flow to converge and produce the desired flow split ratio. The pressure field must approach the correct solution by increasing or decreasing the pressure value at the ECA face to maintain the desired flow division. To overcme the problerns of the previous technique (Equation 3.10), a new relation was developed that will provide a good first approximation for the pressure increment. This relation was developed to C substantially under-predict the first new pressure value so that the flow is allowed to develop to the new solution. Recall that large pressure increments drastically change the tlow field. thus. resulting in divergence of the solution. Since the solver is based on an itcratiw technique it is besr to lag the pressure so that small changes occur in the tlow field whilc still approaching a converged solution. The equation developed to descnbe the movement of the specified pressure value was based on a power-law relation as,
where .\/,.,., is the rnass tlow error described as,
and E is an rxponent that controls the direction of the pressure increment by changing froni ci ther positive or neyative quantities basrd on the pressure field. E also controls the size of the increment by increasing or decreasing in magnitude depending on the required incremrnt nreded to sustain the tlow split ratio. Currently E is based on the desired flow split ratio and changing ratio ai each loop of thc conservation equations. The hnction for E map br basrd on othrr parnrneters. however. it was found that the function was c;ucct.sstùl with.
This equation creates an exponent that increases wher. the flow ratio is far From that desired. and is small as the solution approaches the correct flow division. This power-law relation could possibly be replaced by other more advanced functions that ma. provide better equations to approximate the solution. However, the advanced functions may only provide solutions to a specific geometry and the current method cm be made even more flexible with the addition of relaxation parameters. Relaxation parameters are incorporated because using Equations 3.11-3.12 alone provides a slow conversence to the solution. With the aid of relaxation the solution cm be accelerated. Relaxation parameten are, in this case, used to multiply the approximated pressure P.v~w calculated from Equation 3.10. to increase or decrease the pressure value (see Equation
3.13 ). The relaxation parameter, fi incorporates an additional consideration to accelerate the convergence. In this study. the relaxation parameter was divided into three catcgories. high. medium and low. A panicular category is employed when a specified range of the difference in current tlow division and the desired flow division is met. Aiso. the rate of change of the update flow divisions from the previous iterations are considcred. For example. when the tlow division difference is large (lO?G from desired), a large multiplier (0.3or 1.5) will be used to increase or decrease the pressure to maintain the dcsircd tlow division. Or. for esample. when pressure values are moving in the correct direction but the tlow division is moving away from or slowly to that desired, then a highrr relaxation parameter is used. Several other factors could be incorporated into the relaxation parameters. however, to maintain a simple approach. no other additions where made. To provide a better understanding of this two-pressure boundary condition technique. a tlow chan is provided in Appendix B.
In using the present dual pressure conditions for steady simulations it was found that high relaxation parameters were allowed. while for pulsatile simulations relaxation parameters were used but to a lesser extent. This was expected because for steady simulations. the final solution evolves to become fully developed. while in pulsatile simulations. the solution must be arrive at for srnall increments in time, thus requiring finer tuning of the pressure conditions. An in-depth description of the relaxation parameters used can be seen in Appendix Al, where the subroutine is presented with detailed comrnents. For a typical steady simulation of flow through the carotid artery, the evolution of the pressure field can be seen in Figure 3.8. Figure 3.8a shows the path the pressure solution traveis in relation to the flow division and is observed to follow a l I j Flow Division (96) (3)
1 ;P l l
< b
*
Time Step Number
Figure 3.8: Pressure field development for steady simulations through the normal carotid mery for the ECA face. (a) Pressure vs. Flow Division, and (t)Pressure vs. Time Step Number. circulating spiral shape. Figure 3.8b displays the pressure as the solution moves from each time step. The pressure values evolve for approximately ?O iterations until the final pressure value where the solution continues to develop the flow field. For pulsatile simulations. the final pressure adjustments are show in Figure 3.9, with the cardiac wave superimposed for both the ECA and the ICA face. The pulsatile simulations create pressure values that seem to flow smoothly with each step in tirne. For Figure 3.9b. the stenosed geometries required a higher pressure dunng acceleration to maintain the fiow division ratio than that found for the normal carotid artery.
This scherne works with the cunent set of simulations, however. the scherne doss have limitations. For instance. when the pressure approaches a value of zero. indicating that the pressure value requires a sign reversal. the current scheme requires intemention. This task is currently cûmed out by a sepante portion of the code: when pressure reaches a very srnaIl value in proximity to zero. the sign is autornatically tlipped. Whrn the sign on the pressure is flipped, the convergence toward the correct pressure is slow brcause of the smail error multiplier and the relaxation parameten. Also. if small pressure values are required at the ECA boundary face that are within the limits set for sign tlipping, small oscillations of the pressure value will occur. Thus, the region bounding zero that is set as the critenon for sign reversa1 must be small unless it is known in ndvance that small pressure values are not required. To overcome this problem. the value specified on the ICA face may be set to a high positive or negative value, thus. preventing the pressure from requiring a sign change. This again requires pnor knowledge of the pressure field and perhaps the adjustment of the relaxation pararneters to adapt large pressure values. As for selecting the appropriate values for the relaxation parameters. this is another in-depth process that requires tuning. Currently, the equations and relaxation pararneters descnbed in Appendix Al work well with steady pulsatile simulations through the normal and 30% stenosed aneries. The values chosen are optimized to provide quick and reliable convergence for simulations of the current It is currently not known the abilities of the function and Cgeometries used in this study. pararneters for other geometries or further reductions of the stenosis grade. 3 O 1 ,zZ 53 04 0.5 06 07 08 09 t (s) (a) fime
Figure 3.9: Pressure field development for pulsatile simulations through the normal, 30% concentric, and 30% eccenrric mode1 of the carotid for the (a) ECA, and (b) [CA. 3.7 Summarv
Numerical flow simulations are camed out in the geometries created by Smith et dl. ( 1996) for the normal and both the 30% concentrically and 30% eccentrically stenosed -oeometries. Thesc reometnc configurations were simulated by numerically solving the eoveming equations for laminar flow under pulsatile conditions. The assumption of a C Newronian fluid and rigid vessels were used throughout this study. Thcse conservation squations were solved using a commercial CFD package and boundary conditions were imposed by modifications made to the source code. A direct velocity profile was imposed on the CCA at each time step corresponding to the cardiac wave presented by
Holdswonh et ai. ( 1999). The ICA face was set to have a constant pressure value for al1 times of the cardiac wave. The ECA boundary condition was managed by imposing a new two-pressure technique. such that the pressure value adjusted to maintain a specified tlow division betwren the two aneries while still allowing the tlow to converge. This technique will later be show to be an advantage in decreasing the required cornputational resources and computational time. 4.0 Convergence of Results
4.1 Introduction
For computational simulations the presence of numerical error may exist that cause the final solution to deviate creating impractical solutions. Thus, to decrease or limit inaccuracies. an error analysis is canied out which is based on the numencal convergence of consecutive simulations. This chapter will deal with the errors encountered from computational studies and present the method to enhance the precision of the final numerical solution. In obtaining accurate solutions fiom numerical simulations, the results must be converged to a reasonabie degree of accuracy. The simulations are usually solved to the best possible solution. but are often limited by the amount of computational time and the computational resources available. AI1 simulations are carried out on a Bin Mcroqvstem. Lrhr 3 where for steady simulations, the approximate computational time is 30 minutes for rvery 100,000 elements and for pulsatile simulations, 10 minutes for every 100,000 elements pcr time step and per cycle. In computational simulations there are three general critena required to gensnte accurate solutions.
(i) the spatial resolution of the mesh, (ii) the temporal resolution and, (iii) the convergence of the conservation of mass and momenturn equatioos.
To obtain accuratz results for pulsatile flows al1 three of these convergence criterion must be satisfied. In this chapter, these cntena wiil be discussed. Results will also be presented to verify the converged solutions and the numerical technique.
4.2 Spatial Convergence
The results of a numerical simulation are strongly dependent upon the resolution and quality of the mesh used to define the fluid domain. Simulations based solely on the convergence of the conservation principles are not sufficient to justiS the accuracy of the results. Sirnularions perfonned on course meshes cm generate false results because the elernents are too large. For large elements, where high fluid variable gradients occur, the nodal positions are too far apart to accurately describe the real gradients in the fiow. However, the solver will still be able to conserve mass and momentum over the volume. Tlius, inaccuracies occur, which degrade the overall solution. To overcome this dependence on the mesh size, a grid-independence study must be executed to achieve spatial convergence.
-- - pp - - .- Figure 4.1: The mode1 of normal carotid artery bifurcation.
The geometry that was selected for the initial grid-independence study was the normal carotid artery, reproduced in Figure 4.1. The normal geometric mode1 was chosen for two reasons. Fust, on the basis of experimental evidence (Ku et al., 1983), the normal carotid flow is not turbulent. Second, MRI data exists for this case so that the grid converged results of velocity cm be validated. To ensure that the resulting grid is satisfactory for both steady and pulsatile simulations, the grid-independence is based on steady flow calculations at both average and peak Reynolds numbers observed in a characteristic, pulsatile blood flow (see Figure 3 -4). On the bais of experimental measurements (Holdsworth et al., 1999), the expected average and peak Reynolds number for pulsatile flow in the normal carotid artery is approxirnately Re=275 and 1100, respectively. Because of physiological variability associated with calculating the average Reynolds number, and based on the availability of experimental data at Re=ZSO, a Reynolds number of 250 was used for the average in this case. The resolution of the mesh created for the normal carotid geometry may not nrcessanly be applicable for mildly stenosed cases. With the incorporation of a stenosed geometry. higher gradients are produced, which may require an increased grid density. The convergence of the stenosed models will be discussed and handled in a later section of this r hapter.
In simulatine the steady computations, a parabolic profile was imposed on the inlet of the CC.\ and an out-tlowing mass specified boundary condition was applied on the ECA. The mass tlow split between the ECA and ICA was set to have 44?6 of the inlet mass tlow Iraving the ECA. The ICA boundary condition was implemented u~inga pressure specified optrning which would act as a pressure reference within the carotid anery domain. The Re number \vas set based on the inlet radius of 1 cm. To implement Re=ljO. the kinematic viscosiry was çhosrn to be 8.0*10' cm2!s. therefore producing an average inlet velocity of 1 cm S. For simulations at Re=! 100. the same average inlet vrlocity was employed with the appropriate kinrmatic viscosity.
The initial grid was designed to contain approximately 50.000 tetrahrdral elcments and subsrquent gids were obtained by sequrntially doubling the number of elements until the drsired accuracy was ottained. In al1 the models. elements were generated based on a specified element size: the height of a symmetric or perfect tetrahednl. Due to the difficulty in generating smooth transitions from coarse to fine elements. owing to the octree-based rnesh generator used (ICEM CFD Engineering), al1 elements were generated based on the same rlement size. This increased the grid density in areas where fewer elements would have been sufficient. such as areas upsueam in the CCA and perhaps, downsaeam in the two branches. However. such simulations will aid in determining the required elernent size or the number of elements that would accurately mode1 the area in the vicinity of the bifurcation. The accuracy was assessed by companng predicted results for the wall shear stresses at al1 points on the surface of the carotid artexy. The desired accuracy is obtained when the stresses from two subsequent grids (one twice as dense as the other) are predicted to within the prescribed accuracy. The wall shear stress was chosen over the velocity field as the gid independence panmeter because it is the wall shear stress that is the hemodynamic parameter of greatest interest. Furthemore, recent work by Ethier et al. (1998), showed that wall shear stress independence is a more difficult convergence critena to achieve, owing to the lower order of interpolation for the velocity gradients.
Simulations were computed for Reynolds nurnbers of Re=XO (average) and 2100 (peak) with varying mesh densities, as described in Table 4.1. For simplicity, this study will reîer to rach mesh density by the approximate nurnber of elements, however, in actuality, the total number of nodes is more important because this is where each of the conservation equations are solvsd and thus. providing a better cornparison behveen other studies. Also, because of the many differences that can be found for the overall size of the geometry, the non-dimensionalized element size is reponed. This number is non-dimensionalized with the CCA diameter and would enable direct cornparison between the mesh densities for different geomrtries and results generated by others.
Table 4.1: Mesh information for normal carotid artery.
- -- Numberof Numberof Name Element Sizem 1 1 elements 1 Noder /
The densities range f?om approximately jOK to SOOK fint order tetrahedral elements. Cornparisons were made between the converged results of each set of subsequent grids (one twice as dense as the other), however, particular attention was directed at the high density meshes (IOOK and up). Velocity and wall shear stress grid independence were compared by computed errors. Also, compatisons at Re=2SO are made with expenmental MRI data.
For the Re=7SO, the results of the velocity profiles and the wall shear stress contours cmbe seen in Figure 4.2. The velocity profiles, Figure 4.2%are plotted in several positions along the symmetry plane and display the profiles for the grid densities containing 200K, 400K and 800K elements. The Root-Meamsquare (RUS)errors between each subsequent grid density, for both velocity and wall shear stress, are presented in Table 4.2. The errors G
- SOOK
Figure 4.2: (a) Velocity profiles for the normal model at Re=?50 (b) Wall shear stress contours for the normal mode1 Rs=250.
Table 4.2: Velocity and wall shear stress convergence mors for the normal model at Re=250.
50K-100K 100K-200K 200K-4OOK JOOK-800K Velocity 9.2 1% 7.36% 5.23% 3.87% WSS 14.88% 10.83% 8.36% 6.99% aere computed by comparing the data at each nodal point of the lower resolution mesh within the 3-dimensional geometry. The data from higher resoiution mesh were interpolated on to the iower density mesh by using an inverse-distance rnethod presented in the Tecplot post processing software (Amtec Engineering, Inc.). In comparing the velocity fields of the -aid densities 200K. 400K and 800K elements (see Figure J.Za). qualitatively the curves coincide with minute deviations. Quantitatively, the errors that resulted from increasing the density from 200K to 400K and from 400K to BOOK were found to be approximately 5.23% and 3.57%, respectively. However, recognizing that 5.23% is an adequate level of accuracy considering normal physiological variations, the 400K mesh was considered grid independent based on velocity. In considenng wall shear stress grid independence for the velocity converged grid additional inaccuracies produce an estimated error of approximately 8.36°0. Cpon incrementing the grid density to SOOK, the wall shear stress error reduces to approximately 6.99?4 compared to the 3.87% detemined in the velocity independence test. The contour plot of wall shear stress at the SOOK level. Figure 4.7b. indicates good agreement. especially in close proximi ty to the bifurcation. Thus. from the results presented, -crid independence using wall shear stress should be considered when reporting quantitative results for wall shear stress. Caution should be advised when reportin5 velocity grid independence results for these values.
The cxperimental velocity data for the normal carotid artery were acquired using a whole-body clinical rnagnetic resonance imaging (MM) scanner (Signa 1.5T: GE Medical Systems. Milwaukee WI) fitted with a custorn high-resolution gradient coi1 set. An agarose gel tlow-through model of the normal carotid bifurcation (Figure 4.1) was perfused with a C 60.4 (by volume) water:glycerol blood rnimic having a viscosity of 3.6 cstokes. A centrifugai pump provided a steady flow of 6.0 ml/s, producing an inlet Reynolds number of approximately 250. A 30 phase contrast pulse sequence was used with the following imaçing parameters: ?j degree flip angle; 35 ms repetition time: 7.2 cm field-of-view; 40 cnvs encoding velocity; 10 mm thick slab parallel to model symmetry plane; 32 slab phase encodes: 36.~756acquisition matrix; 4 signal averages; 76 minute total scan time. The 44.36 flow division benveen the ECA and ICA was imposed as reproduced in the computational simulations.
A plot showing the axial velocity profiles for MRI data cm be seen in Figure 4.3a with the computational grid independent solution superimposed. The plot shows good agreement but with a srnaIl deviation in the sinus area. Within the sinus, the one velocity profile of the MM data, indicates a rapid increase in the magnitude of reverse flow proximal to the outer wall opposite to the apex. This could possibly be caused by an unsteadiness of Figure 4.3: (a) Cornparison for the normal mode behveen MFü data and the computational data 1 with the 800K grid at Re=XO. (b) Numerical results at Re=1100 for the normal mode[. the recirculation zone when obtaining the expenmental data. The recirculation zone could shift back and forth in the axial direction about its center.
The numerical simulation with a Reynolds number of 1100 generated somewhat different velocity profiles due to the complexity of the flow field (see Figure 4.3b j. In the bulb, a considerable increase in the amount of reverse flow that occun was observed along the sinus wall. thus' increasing the wall shear stress. The recirculation zone expanded in size, gowing outwards in the direction toward the apex and causing a highly skewed profile along the inner wall of the apex. These predictions were similar to the laser-doppler memorneter (LDA) velocity measurements generated by Bharadvaj et al. (198?b), who ais0 reported a growth in the circulation region with increments in Reynolds number. Due to the increase in the Reynolds number, an expected increase in the grid convergence error was observed within the CCA (see Table 4.3). However, qualitatively rxamining the velocity profiles at several locations along the symmetry plane the curves coincide with only small deviations. The wall shear stress error will be employed to distinguish convergence for the Re=I IO0 simulations since it was shown from the grid conver_eence at Re=.?jU. to be a more severe convergence critenon. The wall shear stress rrrors computed in proximity to the bifurcation are less than 3%,where the error based on velocity is 16.25%. The solution at Re=I IO0 is not fùlly wall shear stress grid independent. but for the purpose of puisatile simulations, the Reynolds number of 1IO0 only exists for a vrry small fraction of the waveform seen in Figure 3.4. This suggests that a grid indrprndencr study based on Re=ljO or perhaps between 2-70 and 1100 may provide an appropriate grid density. Due to the limitation of the computational resources available. simularions for an additionai doubling of the 800K mesh was not currently permiaed.
Tabie 4.3: Velocity and wall shear stress convergence errors for the normal mode1 at Re=1100.
4.3 Temporal Convergence
To senerate converged results from pulsatile or transient simulations, additional factors must be takrn into consideration from steady, non-transient simulations. For steady simulations. the converged results are not dependent of time. and thus, only spatial resolution and residual convergence are required to achieve accurate steady results. However, in tirne dependent solutions temporal convergence must be achieved as well as the above two criterion. With the incorporation of time, two distinct additional factors must be detennined; the number of time steps to be used in each pulse and the number of pulse cycles required to reach the time periodic solution. For both studies, the normal carotid artery was chosen with the use of the grid independent mesh density from the previous section. Using this mesh dçnsity would rnsure that both temporal convergences would occur without the dependence on the quanrity and resolution of mesh. The inlet face was specified with fully developed, pulsatilr velocity profiles from the cardiac wave show in Figure 3.4. At the outlet faces the dual pressure condition explainrd in Chapter 3 was imposed. The initial guess used for the al1 the pulsatilr simulations was based on a steady simulation of the first time step in the cardix wave.
4.3.1 Required Number of Time Steps
It rvas nccessary to determine the number of time steps in each pulse so that the cardiac wavs could br accurately described throughout the simulation. As mentioned in the prrvious chaptcr. uneven increments of the timr steps were taken such that an even cord leneth alone the pulse cuve was achieved. With this uneven movement in time the required number of time steps was significantly reduced. However. an adequate number of tirne steps must still br made so that the jump to the next time does not produce a drastic chanse in the flou field producing unstable convergence and possible divergence of the solution. The solution from the previous time step must be a good approximation of the new time step solution. thus, enabling a rapid convergence. The temporal rrror is analysed by simulating an initial number of tirne steps for one pulse cycle and subsequently doubling the number of time steps until a reasonable degree of agreement is achieved between each pulse.
The pulsatile simulations were performed in the normal carotid artery initially with 50 uneven time steps over the cardiac wave. Additional simulations were performed at 100 and 390 time steps. The results were compared from each set of simulations by calculating the error from each subsequent run (50 to 100 and 100 to 700). Since the mesh configurations were identical in each set of simulations, the errors were calculated by comparing the velocity field at al1 the nodes and time steps of the simulation with the lower number of time steps. The erras were evaiuated by calculating the maximum RMS error at any time throughout the pulse and time-averaged RMS error over the entire pulse cycle. The - errors where non-dimensionalized by the tirne-averaged inlet velocity ( UT,. = 27.5 cm l s ). The RiMS error from 50 to 100 time steps is on average 2.8976 with the maximum error , rime sttp = 0 b1 1 O
1 7 Sold - 200 rime srcps - 100 rime sreps "/ '.J Dashed -LOO tirne srcps d1 Dashd - 100 time swpi 1 Dotted - 050 timc sicps i Duttd - O50 timr iieps I
Figure 4.4: Profiles for error of incrementing the number of time sreps.
Table 4.4: Erron from increasing the number of time steps. Time-averaged Time Steps Maximum Time of RMS error RMS error maximum error 50 to 100 2.89% 8.57% 0.6211 .O sec 100 to 200 2.38% 7.1 9% 0.6011 .O sec
occurring at T=0.62/1.0 with a value of 8.57%. Increasing the number of time steps to 200 did not show much improvement. quantitatively or qualitatively, in cornparison to the 100 rime step run (see Table 4.4 and Figure 4.4). The average RMS error decreased by an amount of 0.5276 and the maximum error by an additional 1.35%. Thus, it did not seem justified to simulate 200 time steps for the minimal increase in accuracy especially since the 100 time step run required nearly double the CPU time. Thus. 100 uneven time steps were used in al1 pulsatile simulations. 100 time steps were chosen because the generated
- Figure 4.5: Error between first and second pulse. the computed errors plotted in Figure 4.5, the error decreases with each increment in time and tiom pulse to puise. except in the regions of sharp accrleration and deceleration. where the mors fluctuate because of the complexity of the flow field. The error is also erratic near the end of the wave because the time step size is larger in this area due to the flat diastole portion of the wave. Thus, with larger time steps, the error increases. The error that occurs ai the end of the pulse is substantially smaller than in any other portion of the wave and there is no relevance associated with the fluctuating errors. From the first-order trend line the rrror is observed to decrease over time, with the maximum RMS error of 1.84% occumng at the first tirne step and the time-averaged RMS error over the entire pulse of 0.63%. The solution generated from the first pulse was invalid because the errors associated with the initial guess for the first time step are still present. The good initial guess used in the first time step of the tirst pulse may give good reason for the small errors associated from simulating the first pulse to the second pulse. Due to the small error associated with the increment to the second pulse, no Merpulses were simulated and the results from the second pulse were considered to the hlly developed pulsatile solutions in ail simulations. Perktold et al. ( 199 1c) reported the requirement of two-and-a-half pulse cycles, however, the results were provided on a lower resolution mesh, which would influence the accuracy of the solution and the number of pulses required.
4.4 Residual Convergence
CFX-Tfc is based on a finite volume scheme and because of this method's discretization technique. the convergence of the mass and momennim equations is based on the residual of each conservation balance. In the finite volume technique each discrete equation corresponds to a conservation balance and is considrred for each volume element. Since. mass and momentum are conservation principles. the summation of each volume sirment's conserved quantities must ciearly sum to zero. In numencal simulations. the summation of the conserved quantities is denoted as the residual and describes the quantity lcft over or the accuracy of the results. For numerical simulations the residual should approach zero.
The quantitirs of mass and momentum are solved and conserved for each volume slrment to a drgree of accuracy. To ensure convergence of the results, a set of criteria must be established based on the average residual of al1 elements in the domain or the maximum residual in any elernent. For al1 simulations performed in this study the cniena for the mass and momentum residual was set to lirnit the maximum volume element residual to LUE-4. Wi th this degree of accuracy high-quality results are produced.
4.5 Dual Pressure Boundary Condition Convergence
To ensure the dual outlet boundary conditions are valid in the carotid anery simulations. a test case was studied. To verifjt the duai pressure technique the test involved limiting the length of the outlet channels and thus, the number of rlements required to simulate the _eeometry. This test will ensure the profiles on the outlet faces of the cut mode1 wi!l match profiles in the original, longer model. In shonening the lengths of both the outlet channels. different pressure values are acquired fiom the added boundary condition routines to maintain the set flow division between the ECA and the ICA, To conf~rmthat both the results and the procedure for imposing the pressure conditions are accurate, cornparisons are made for the fluid dornain between both simulations.
For this test, simulations were performed in the normal carotid artery under pulsatile conditions. The original geometry of the normal carotid artery had lengths for the CCA of 15 uînits (1 unit = 1 CCA radii) measured from the inlet face to the apex of the bihircation. The ECA and ICA had lengths of 25 units, which were measured fiom the apex of the bifurcation to the outlet faces. For the reduced model, the inlet channel was condensed by 7 units to decrease the total number of elements (see Figure 4.6). Since the imposed inlet profiles are calculated fkorn fully developed boundary conditions no differences are to be encountered in the CCA (Rindt et al., 1987, Bharadvaj et al., 1982). The imposed inlet boundary conditions were still far enough upstream such that no influence was observed in the flow field. The outlet charnels were decreased by 13 units to a Iength 12 inlet radii. The original and shortened geomeûies were simulated with the same element size to lhit the dependence on the mesh density. The generated mesh produced a reduction in the mesh size fiom 802,000 elements for the original geometry to approximately 435,000 elements for the shortened configuration.
Figure 4.6: Long and short models of the normal carotid artery. The pulsatile simulations for both geornetnc configurations were simulated and compared. To make cornparisons of the flow field, qualitative observations for each velocity component and the wall shear stress magnitude were made. Upon observation only small deviations were observed and it is though that these deviations are a result of repositioning of the elements and nodes. The difference may also be associated with the srnail difference in the convergence of the flow division ratio. As mentioned earlier the flow division ratio is set to limit the maximum flow split error to 1% of the flow split through the ECA at al1 times throuehout the pulse. The solutions were compared by plotting velocity profiles at several locations dong the symmetry plane and for several time steps in the pulse. There were vinually no observable differences indicating that the pressure condition can be safely applird well upstream of the original boundary.
Further reduction of the ECA and [CA branches was not simulated because the results would not provide rnough ponion of the downstream channels for visualization. Frorn the current set of simulations. the dual pressure boundary was shown to be beneficial for simulatine the carotid anery. The greatest benefit of the shonened geometry is the reduction in the simulation time required: the simulation time was reduced by approximately half of that for the original configuration.
4.6 Grid Independence for 30% Models
It is suspected that the 300, constncted geometries require a separate grid independence study from that performed on the normal carotid artery because of the presence of the stenosis. With the constncted geometries an additional concem is brought on by the creation of the complex flow patterns and especially, the higher velocity gradients. Based on the previous tests. it was shown that the dual boundary conditions provide accurate results as well as a reduction in the overall geometry size and the total number of required elements. This allows for an increase in the mesh density that was not permitted in the longer models. a constraint based on computer resources. To ensure that results are grid independent for the 30?6 stenosed geometries, a grid independence study was performed. The grid convergence for 30% stenosed models was performed only on the 30% eccenmc mode1 (reproduced in Figure 4.7: The 30% eccenûic geometry of the carotid artery bifurcation.
Figure 4.7) due to the increased complexity of the eccentric geometry in cornparison to the concentric model. Since the eccenbic rnodel would htroduce greater velocity gradients due to the shifting of the stenoses, it is understood that the element size acquired for the 30% eccentric mesh would also be valid for the 30% concentic rnesh.
This gnd independencc snidy was perfonned with approximately SOOK, 750K and 1000K element meshes. The relative element sizes for the 3 rneshes used can be found in Table 4.5, were the vaIues are non-dimensionalized with the inlet face diameter. Different rlement sizes frorn those used in the normal model grid convergence snidy were used to mavimize the cornputer tesources available to the simulations. The simulations used in the
c.end independence study for the 30% stenosed models only occurred for the maximum Reynolds nurnber found in the in the cardiac wave (Re4100). Also, fkom the previous grid- independence study perfonned on the normal model it was found that at the average Reynolds number (Re=ZSO) the results were grîd converged on coarser grids than those rrquired for the Re=1100 simulations. Thus, it would be valid to base the present grid independence snidy oniy on the maximum Reynolds Nurnber.
Table 4.5: Elernent information for the 30% eccentric meshes. umber of Nurnber of Element Name nodes elements size/D Figure 4.8: Velocity and wall shear stress for the 30% eccentric mode1 at Re = 1100.
For the 30% eccentnc geometry a plot is provided of the velocity profiles along the synmetry plane as well as for the wall shear stresses (see Figure 4.8). In this plot only the - - ZJOKand lOOOK clement meshes are shown. From the velocity profiles, it is evident that the cuwcs coincide with good agreement. Quantitative errors for each incremented set of mesh densities can be seen in Table 4.6 for both velocity and wall shear stress. The method used to analyze the error was identical to the approach used in the previous grid independence study for the normal carotid artery. Here again, the erron are considered from the more demanding convergence critena of wall shear stress values. The erron calcuiated from the wali shear stress values are 23.476 and 26.4% for the increment of jOOK to 750K and 750K to IOOOK, respec tively. Table 4.6: Gnd Convergence for the 30% eccentnc mesh at Re=l100.
1 ' 500K-750K 750K-1000K 1 Velocity 7.76% 11.10% 1 WSS . 23.36% 1 26.37% ,
Conclusion
.A gcneral approach has been provided in this chapter for carrying out pulsatile simulations and generating Fully converged results. First a spatial grid test is used to dctcrminc lin appropriate grid density so that the generated solutions are independent of the drnsity of the mrsh. Nnt. two temporal critena must be established: the total number of time steps in each pulse and the number cycles of the pulse. An added benetit to computational simulations is to utilize uneven movements in time while creating pulsatile jirnulations of coniplex waveforms. This aids in the temporal convergence and reduces the overall simulation time required.
Other resccirchers that have provided numencal simulations have not provided a detailed description of the process to provide convereed resiilis. The present convergence methods shows that n rnajority of simulations performed in pulsatile flow in a 3-dimensional rnodrls of the carotid geornerry may be unresolved. Since, a direct comparison cannot be made with the element size used by others. only the number of elements will be discussed. Rindt et al. ( 1996) used 11.763 nodes in a mode1 of the carotid geometry. Perktold et al.
( 1990a) used a total of 27.934 nodes in a T-junction, which in this geometry, may require a large increase in the nurnber of' elements because of the abrupt change in flow direction. Perktold et al. ( 199 1a & 199 1b) used 34.198 nodes in a non-Newtonian study. Perktold et al. (1995) used 84.337 nodes in a compliant artery study. From these studies, the number of elernents used are still well below the amount that was used in the present study; 184.586 for a shonened constricted geometry. However, the presented results may still include some error. but as mentioned. the presented results are based on wall shear stress convergence. A rnajority of the othen studies are based on infenor techniques involving the velocity field or qualitative observations. Rindt et al. ( 1990) provided steady runs in a normal carotid artery at a lower Re numben and concluded that a substantial amount of computing power would be required to generate My converged results but the time required for the simulations would be a limiting factor. Thus, the erors are acceptable given physiological conditions and based on the approximations used. 5.0 RESULTS AND DISCUSSION
5.1 Introduction
Numerical simulations have been carried out for the normal, 30% concentric and 30% eccentric models from the carotid artery geometnes created by Smith et al. (1996). The main discussion will be the presentation of the mildly constricted models with cornparisons made to the nomla1 carotid mode1 and with each other. The simulations were camed out under pulsatile conditions wi th the implementation of the cardiac wave of a normal patient created by Holdsworth et 111. i 1999). To present the results, first a general description will be presented, giving a detailed explmation of the converged pressure field. Furthemore. observations and descriptions of the hemodynarnics in the carotid artery bifurcation will be presented according to positions dong the cardiac pulse. For the systolic portion of the pulse. the discussion will be divided into two sections: the acceleration and the deceleration region. The diastolic portion of the pulse will also be divided by presenting details in the small secondary puise and the steady region. Figure 5.1 shows a plot of the cardiac wave showing the divisions of each section to be discussed. Due to the large amount of data
weenerated from pulsatile simulations, plots will only be provided showing specific points throughout the cardiac cycle.
This chapter will give a detailed description of pulsatile flow of the normal and rnildly diseased carotid artrry bifurcations. Discussion of the flow field will begm with the examination of the velocity field based on extracted velocity profiles and the relative magnitudes of the velocity field. Discussions will also focus on the recirculation regions including the sepuation and reanachrnent points, and the intensity of these recirculation regions. Calculated vorticity fields will also be included in this discussion. The vorticity fields provide information ro relate increased levels of flow disturbances to the possible onset of transitional flow fiom larninar to turbulent flow regimes. The shear stress created on the walls of the carotid artery will aiso be discussed for several locations throughout die pulsatile flow with emphasis on areas of low and oscillating shear. A - Acceleration B - Deceleration C - Secondan, Pulse 1
Figure 5.1: The cardiac pulse in the normal carotid artery. (Holdsworth ei ai., 1999).
The rcsults shows in detail the differences encountered in al1 the models stuc These differences may provide vital information for diagnosing an individuals' risk of stroke and the intlurnce of the amount of atherosclerotic plaque. The differences associated with the position of the stenosis will also be discussed.
5.2 General Flow Characteristics
Plots are provided for the symmetry plane showing velocity profiles and quantities of velocity magnitude, wall shear stress, vorticity and recirculation zones for several temporal locations during the pulsatile wave. Pathlines will also be provided showing the general flow patterns throughout the complex 34mensionai domain. Al1 of these plots are provided at the end of this chapter. Throughout the pulse cycle observations of the flow field through the CCA and the ECA of the carotid domain showed Iittle difference from one mode1 to another. In the CC.\ the profiles maintained the shape of the profiles specified on the iniet boundary until the geometry began to bifurcate. As for the flow in the ECA, the fluid flows through the curvature and has it's own distinct Bow patterns but were sirnilar to the results created fiom al1 geometries. It was expected to observe this little difference in the ECA flow field From model to mode1 because the same specified flow division was to be maintained for each mode1 and also because no differences were made to the geometry for the ECA. As the flow divides from the CCX into the ICA, the varied geomeh-y of the ICA created disturbances in the fluid domain that are distinct for each geometric configuration and will be discussed in derail in the following sections of this chapter.
Calculations were made for the pressure differences required to maintain the flow divisions between the ECA and ICA. Figure 5.2 shows the converged pressure differences for the ECA (Figure 5.h) and the ICA (Figure 5.Zb). The pressure differences represent the difference be~eenthe average pressure over the ECA or ICA boundaries and the average pressure over the inlet boundary. The pressure values are non-dimensionalized with the average pressure calculated from the force on the ICA face at the average Re number. Benveen the ECA and the CCA, the pressure difference does not Vary much fiom one geometry to another. This is related to the small deviations observed in the flow field. However. when observing the flow field through the ICA, the pressure was cornparably higher requiring an elevated pressure gradient to compensate for the narrowing of the artery. With the higher pressure gradients more work must be provided by the purnp (the heart) to sustain the required flow of oxygen-rich biood to the brain. Based on the position of the stenosis, the two models present slight differences between the pressure fields. The 30% eccentnc model required slightly higher pressure gradients than the 30% concentric model during the acceleration portion, thus implying, that the position of the stenosis also effects the effort required by the heart to sustain the 80w division. Thus, with increased stenosis grades, the pressure gradient across the constriction increases, requiring higher levels of work. If this work is not available, then a reduction in the amount of circulation to the brain occurs, which may have additional effects on the patient. Even though in physiological flows the flow division does not maintain a constant ratio, there would stiII be reduced flow to the brain if elevated pressures gradients are not applied. il O 1 J 2 3 3 0.4 0.5 O 6 a.? 9 a 0 9 1 (a) nme (s)
Figure 5.2: Converged Pressure differences for (a) the ECA and (b) the ICA. The forth-coming plots created to characterize the flow field are provided with non- dimensionalized quantities. The non-dimensional velocity field was obtained by dividing the local velocity with the average velocity at the average Re number for the carotid pulse cycle. The wall shear stress is non-dimensionalized with the wall shear stress created at the average Re number for hlly developed Poiseuille flow. The voxticity field $ots were maed from the non-dimensionalized velocity field and then non-dimensionalized by the radius of the CCA inlet face. Non-dirnensionalized plots will allow direct cornparisons to other data by using an appropriate scaling factor.
The following sections of this chapter will present and discuss the flow patterns of the pulsatile tlow through the carotid artery focusing on the ICA, providing cornparisons bstween the three models simulated: the normal, 30% concentric and 30% eccentric models. The detailrd descriptions will be provided as plots for the veiocity profiles, pathlines, velocity magnitude. wall shear stress, vorticity and recirculation contours.
5.3 Svstolic Portion of the Cardiac Wave: Acceleration Portion
Plots are provided at several times throughout the acceleration portion of the cardiac wave. Velocity profiles are shown in Figure 5.3, showing the shape of the velocity profiles at several cuts along the symetry place. Figure 5.4 indicates regions of reverse Bow denoted by shaded areas in the carotid domain. Figure 5.5 shows plots of the pathlines during acceleration with the increasing levels of the velocity in the forward direction corresponding to increasing intensity of the red pathlines and vice versa for increasing intensities of the blue pathlines. Full coloured contour plots are show for the velocity magnitude and wall shear stress in Figure 5.6 and Figure 5.7, respectively.
In the acceleration portion of the pulse (Figure 5.3 and 5.7), the average Re number changes hom approximately ?O to 1100. During the initial acceleration, the normal carotid artery begins to create the greatest amount of stagnant flow in the carotid bulb (see Figure 5.3a and 5.6a). The 30% eccentric model also created regions of stagnant flow on the outer wall and the 30% concentric model created a large stagnant flow region on the berwail, but distal to the apex with a small stagnant region on the outer wall. These stagnant flow regions should not be confused with regions of reverse ffow because in the initial acceleration of the pulse, the mount of recirculation is smali.
-4s the flow further accelerates. the stagnant regions decrease in size and an increased tluid motion begins to fiIl the domain (see Figure 5.3b and 5.3~). Due to the increased motion. the recirculation regions increase in size and magnitude for the rnildly constncted models. much different from the normal model where little, if any, reverse flow is obseried (sec Figure 5-43and 5.lb). The 30?6 concentric model creates small areas of reverse flow dong the inner and outer walls with separation cccumng distal to the maximum narrowing and reattachment occuring not far downstream. The 30% eccentric model creates only a large region of reverse flow on the outer wall in the carotid sinus with separation occurring proximal to the maximum stenosis and recirculation occtming in the middle of the bulb.
With the initial widening and curvature of the normal carotid aery bulb. the flow enters the ICA causing the flow to expand and slow down with a majority of itç flow travelling in the forward direction (Figure 53). This is different fiorn the flow patterns observed in stenosed geometries. For the stenosed models, the fluid fills the domain as the geornetry nrtrrows and secondary flows are observed downstream of the constnction. These initial secondary flows are observed as helical flow patterns and recirculation zones. The helical flow patterns are more visible in the 30% concentric model than with the 30% eccentric mode! having large regions of recirculation (see Figure 5.5b).
The 3076 concentrically stenosed model creates fluid acceleration fkom the constriction with a plug-like profile. Because of the symrnetric nature of the stenosis, the reverse flow regions form along both walls of the artery. For the 30% eccentric model. the flow accelerates out of the constriction with more of a skewed profile because of the obstruction created by the narrowing artery (see Figure 5.6). As the flow accelerates through the constriction, the veiocity field is skewed with higher velocities adjacent to the imer wall of the IC14and an intense recirculation zone is created on the outer wall. At peak systolic flow, the profiles are more pronounced for the plug-like profile of the 3046 concentric model and the skewness of the 3096 eccentric model (Figure 5.3d & 5.6~).The tlow through the constricted models created high speed Bows that extended the entire lengrh of the carotid bulb section, while still preserving the secondary helical flow patterns. The normal model did begin to create a region of reverse flow dong the outer wall of the ICA. Separation occurred at the leading edge of the bifurcation on the outer wall and sxtended into the middle of the bulb. The recirculation regions for the stenosed models intensi. with increasing amounts of the reverse flow. For the 30% concentnc model, the flow created a thin, elongated recirculation region on the outer walls of the sinus with an increased recirculation zone on the imer wall Further downstream of the narrowing. These recirculation patterns are much different from the 30% eccentric model where a very large reverse flow region is created that fills the carotid bulb with a small zone located on the imer wall. Separation occurs at the leading edge of the maximum stenoses with the recirculation zone extending to the reattachment point positioned at the end of the sinus. The intensity of the reverse tlow in the 30% eccentnc model is much larger than that of the other hvo models. creating the largest values of reverse flow near the back end of the sinus (see Figure 5.6~).
The wall shear stresses maintain a positive direction duough a majonty of the acceleration phase (Figure 5.7). The shear stresses increase with increased flow rate and reach a maximum level at peak systolic flow. The normal mode1 only reaches it's maximum shear stress for a short time on a small potion of the imer wall of the ECA near the apex. The stenosed models also have this high shear stress region on the ECA wall, but also exhibit high shear stresses in the ICA. For the stenosed models the maximum shear stresses are more than double the values presented in the normal geometry. The high shear stress exists around the whole circumference of the constriction and gows in magnitude as the flow reaches peak systolic flow. in the early part of the acceleration phase, the high shear stress regions only occurred around the circumference of the constriction, but at peak flow, a region of high stress was found near the outer wall, distal to the bulb in both constricted models. The 3006 eccentric model created large regions of high shear stress on the imer wall that stretched almost two diameten fhm the apex. This large region of high shear stress was not observed in the other two models (Figure 5.7~). 5.4 Svstolic Portion of the Cardiac Wave: Deceleration Portion
For the deceleration portion of the cardiac wave, velocity profile plots are provided in Figure 5.3 and regions of reverse flow are found in the contour plots of Figure 5.9. Contour plots of the vorticity field are observed in Figure 5.10 at several locations of the decelerating portion of the wave. Figure 5.1 1 provide pathline plots of the flow field throughout the carotid artery domain.
As the flow begins to decelerate frorn peak systole, several disturbances occur in the flow field. Due to the rapid transition from acceleration to deceleration, the flow pattems are very disturbed and disorganized. The deceleration causes the existence of complex velocity profiles throughout the carotid domain (Figure 5.8a and 5.8b). The profiles for the stenosed grornetries become sharp downstrea.of the constriction with regions of reverse flow. These regions of reverse flow pwas the flow decelerates and the sharp velocity profiles become blunted to a point where a majority of the flow is in the reverse direction. The profiles produced in the normal geometry showed more of a smooth transition fiom accelerating flow to decelerating flow. The gowth of the reverse flow region for the normal mode1 develops mainly in the carotid buib on the outer wall where in the constncted models these zones build up on both inner and outer walls (Figure 5.9a and 5.9b). The velocity gradients which occur on the artery walls of the stenosed carotid arteries are much more severe than those observed for the normal geornetry. These profiles would be directly related to the levels of unfavourable wall shear stress.
Due to the complexity of the fiow field generated during deceleration, if transitional or turbulent fiows were to occur, this decelerating region is likely where it would occur. However, since turbulence modelling is not inciuded in this computational snidy, only the vorticity field will be considered. The vorticity field was used as an indication of the potential onset of transition because with generation of high levels or complex pattems of vorticity, turbulent structures usually follow. The vorticity was not of major interest up to the decelention portion because no distinct features were observed during acceleration. Ln the normal carotid artery smooth vorticity streaks were produced and nothing of interest was observed (Figure 5.10), however, on the stenosed models interesting features were found. In the concentnc model complex vorticity streaks were observed in the carotid bulb. This highly disrupted flow may have some relation to the development of atherosclerosis. The structures observed in the bulb were unusual with complex vorticity patterns and may cause increased amounts of circulating flow. In the eccentric model. during the initial decelention. distinct flow pattems were observed within the carotid bulb. The vorticity field created complex flow patterns where intense vortex structures propagated from the back sinus edge to the front of the narrowing artery. As the flow approached the end of the deceleration phase an intense region oivorticity was observed on the inner wall distal to the sinus. This concentrated vortex structure would later be released as the flow continues to decelerate during the srnail secondary pulse.
To provide visualization of the 3-dimentional domain pathline plots are provided in Figure 5. 11. The isometric view of the pathlines at t=13/60 (Figure 5.1 la) show the normal and the 30% eccentric model having regions of recirculating flow in the sinus and the 30% concentnc model having recirculation regions not only in the sinus but recirculation on the imer wall created from the helical flow pattems moving along the circumference of walls. Thrse helicai flow panerns were only observed in the stenosed geometries. At t=I5/60 (Figure 5.1 lb), strong helical flow pattems were observed in both 30% concentric and 30% eccentric models after the constriction and continued downstream beyond the carotid bulb. The normal model only provided recirculation and helical flow patterns in the carotid sinus and none further downstream. As the flow continued toward the end of the deceleration portion of the wave (Figure 5.1 lc), the recirculation zones and helical fiow patterns begin to diminish in the stenosed geometries, however, for the normal geometry, a large circulating pattern was still present in the sinus dong the outer wall. The complex helical 80w patterns continued to vanish as the flow moves through the remaining portion of the pulse. 5.5 Diastolic Portion of the Cardiac Wave: Small Secondary Pulse
The plots provided for the small secondary pulse are observed in Figure 5.12 through Figure 5-16. Figure 5.12, Figure 5.15 and Figure 5.16 are plots of the vorticity field along the symrnetry plane. A contour plot of the wall shean stress along the walls of the carotid domain is provided in Figure 5.13. In Figure 5.14, a plot showing the regions of reverse flow are provided.
Dunng the small secondary pulse, many cornplex flow pattems become apparent. One of the most inreresting features, which occun at the beginning of this small puise is the pattern presented in the vonicity field. For the 30% eccentric model, the high Ievel of vorticity generated during deceleration, created the detachment of a circulating vortex. However. sincr the flow continued to decelerate, the vortex was not able to travel far downstream. Instead the vortex structure collided with imer wall shortly downstream (Figure 5.12~). Due to this collision and it's proximity to the artery wall, high ievels of reverse shear stress werr generated at this distinct location (Figure 5.13)
For the 30% concentnc model, no such features were observed, however, complex vorticity pattems tvere present in the carotid bulb behind the constriction (Figure 5. tZb). The normal carotid artery, however, did produce a weak vortex structure during the same tirne of the occurrence in the 30% eccentric model. However, in the normal artery, the vortex structure did not occur in proxirnity to the artery wall. Instead, the vortex was released into the middle of the ICA and continued downstream with no obstructions, but disappeared shortly downstrearn, again because of the decelerating flow (Figure 5.12a).
As the flow reaches it's minimum flow rate, the arnount of reverse flow increases substantially and also reaches its maximum (Figure 5.14). As the flow, again, begins to accelerate the 80w is quite unstable and complex flow patterns occur. Here again the vonicity field is of interest. in the 30% eccentric model, the flow field caused another vortex shedding occurrence at the same location as the fint but at a much lower intensity (Figure j.15a). Because of this Iow intensity vortex, no elevatcd levels wall shear stress were observed in this location. In observing the 30% concentric model, again no vortex shedding was observed, however, the pattern of the vorticity field was quite unstable with the vorticity streams oscillating through the carotid bulb (Figure 5.15a). As for the normal carotid geometry the vorticity field was reasonably stable with no development of peculiar stmc tures.
As tlow again reaches a region of transition From accelerating to decelenting tlow. the nature of the vorticity is unstable. In the both the 30% concentnc and more so in the eccentric model. the vonicity plots show oscillation of the vorticity strearns. This leads to a suggestion that the flow may be on the verge of transition; more so for the eccentric model (see Figure 5.1 6).
5.6 Diastolic Portion of the Cardiac Wave: Steadv Portion
The steady portion of the pulse showing the velocity field is presented in Figure 5.17 and the corresponding vorticity field in Figure 5.18. A plot of the reverse 80w is also provided at the end of the pulse in Figure 5.19.
During the steady portion of the diastolic wave, the flow is reasonably stable for the normal and 30% concentric model. However, fur the 30% eccennic rnodel the flow is initially unstable from the oscillation of the path taken to follow the cardiac wave (Figure 5.16). As the flow continues with moderate variations for the inlet velocity profile, the flow beings to stabilize as with the other two geometries. By observing the vorticity field this cm be seen (Figure 5.17 & Figure 5.18).
The regions of reverse flow are maintained at approximately the same locations and size through this portion of the pulse for the stenosed geometries (Figure 5.19). The 30% concentnc sustains a region of reverse flow on both the inner and outer wdls, simi1a.r to the flow approaching peak systole. The eccentnc model also had similarities of reverse flow near peak systole creating recirculation on the outer wall in the carotid sinus. The remaining velocity profiles and wall shear stress field for this portion of the wave maintained a reasonably constant flow field and created no distinguishing features as the flow reaches the end of the pulse.
With the flow stabilizing, it caused the next cycle to repeat the presented flow features and characteristics. However, if further consnictions were imposed the flow may not be able to stabilize, thus continuing to create vortex structures. causing transitional flow or turbulent regirnes.
5.7 Conclusions
Numerical rnodelling of mildly constricted carotid arteries provide vital information for deterrnining the pathology of the atherosclerotic disease. The simulations also aid in diagnosing the patient's risk of suffenng from a stroke. It was shown that with the existence of a 30?6 constriction, elevated pressure gradients were required to maintain flow to the brain, thus. requinng the heart to work harder. As well as the occurrence of high pressure jradients. substantial increases of the intensity of the velocity field was round throughout the flow domriin.
Due to the existence of the ICA narrowing, high speed flow moved through the constriction. The velocity profiles showed very distinct shapes from geornetry to geornetry and from time step to time step. Due to the constricted geometnes, increased velocity levels were created. These increased velocities created sharp velocity gradients. Due to this high speed fiow and because of the complexity of the cardiac wave, increased regions of reverse flow were observed with elongation and widening of the recirculation zone. These recircdation zones created elevated wall shear stresses on the carotid bulb and intensified the amount of oscillating shear stress and the magnitudes of these stresses. By observing the vorticity field several distinct features were present. In the normal carotid artery one distinct vortex shedding event was observed as the flow approached the small secondary pulse of the wave and onginated Erom within the carotid bulb. Due to the decelerating tlow, the vortex did not travel fx downstrearn. The 30% concentric mode1 did not producc vonex separation but seemed to be on the verge of doing so, showing the existence of cornpiex vorticity streaks following the constnction. The 3046 eccentric model presentcd the most interesting features from al1 the geometries studied. First. one vortex shedding event was observed at approxirnately the same time as the occurrence of the one in the normal geornetry. however, the vortex was generated and released from a position shonly ciotvnstie;im of the constriction on the inner wall. The vortex in the 30% eccentric model was generatrd with very strong intensity, however, was not able to travei downstrearn. due to the interaction of the artrry wall. thus genrrating a region of high shear stress.
.A second. lower intensity vortex shedding event occurred in the deceleration portion of the small secondary pulse. This vortex occurred at the same location as the iirst. however. it broke down shortly downstream with little influence on the artery wall. Dunng the steady portion of the wave. rhr flow field seems to be unstable, with oscillations in the flow strearn, however. the flow quickly recovered and becarne sirnilar to that before the begiming of the next cycle.
The secondary flow patterns that are observed throughout the cardiac pulse are
observed 3s complex helical flow pattems. The presence of these helical fiows were observed in ail the geomeaic models studied, however, the patterns were substantially different frorn one mode1 to mother. During the acceleration portion of the pulse distinct and intense helical patterns were observed in the 30% concentric and only small arnounts were observed in the 3076 eccentnc model. As the flow continued into the deceleration portion of the pulse the intensity of these helical patterns grew and travelled downstream into the ICA. However. in the nomal model, the helical flow patterns were confined to the recirculation zone in the carotid bulb and did not rnove downstream. Al1 the presented plots also indicate the levels of maximum and minimum quantities of the featured plot at the bonom of each figure. For the velocity magnitude, the maximum velocity is more than 30% larger for the stenosed rnodels than the normal geometry. This increased velocity field may have a significant effect on the development of atherosclerosis. With the increased velocity field, elevated levels of reverse flow exist creating additional oscillating waii shear stress. The maximum increase of reverse flow occurred on the 30% eccentric yeometry with the 3074 concentric geometry following close by. in cornparison with the normal geornetry, the maximum arnount of reverse Bow increased by approxirnately 117% and 56OC with the eccentric and concentric model, respectively. With increased velocity cornes increased areas and quantities of shear stress exerted on the artery walls. The rnmimurn wall shear stress for the stenosed geometnes increased by approxirnately 13?6 from the nomal geometry. For the eccentric geometry the wall shear stress increases by more than 100?h. These increased values of shear stress may have significant influences in the development and progression of atherosclerosis as well as the dependence on the position of the stenosis. Velocity P rof ile Plots ln let Profile
Figure 5.3a: Velocity profiles during the acceleration phase at t=6/60.
Velocity Normal P rof ile Plots lnlet Profile
Figure 5.3b: Velocity profiles during the acceleration phase at t=8/60. Velocity Normal Profile Plots lniet Profile
Figure 5.3~:Velocity profiles du~gthe acceleration phase at r= 9/60.
Ve locity P rof ile Plots Iniet Profile
Figure 5.3d: Velocity profiles during the acceleration phase at t=11/60. Reverse Flow lnlet Profile
Figure 5.4a: Reverse flow during the acceleration phase at r=8/60.
Reverse 30% Conœntric Flow lnlet Profile
Figure 5.4b: Reverse flo w during the acceleraiion phase at t= 11/60. Normal 30% Concentric 30% Eccentric
Figure 5.5a: 2-D view of pathlines during the acceleration phase at t=9/60. Figure 5.5b: Isomevic view of pathhes during the acceleration phase at t=9/60. Vel. Mag. 30% Concentric P lots
Inlet Profile
vd. rnag. rrtax. = 8.274 vei. mag.rrrau. = 1 1 .O58 vd. mag. min. = 0.0 vd. mag. min. = 0.0
Figure 5.6a: Velocity magnitude plot during the acceleration phase at t=6/60.
Vel. Mag. 30% Concentric Plots lnlet Profile
vel. mag. max. = 8.274 vel. mag.max. = 11.058 vd. mag. = 10.797 vel. rnag. min. = 0.0 MI. rrag. min. = 0.0 vd. rnag. mn. = 0.0
Figure 5.6b: Velocity magnitude plot during the acceleration phase at t=9/60. Vel. Mag. Normal 30% Concentnc Plots
lnlet Profile
vel. W.m. = 8274 vel. mag.max. = 11 .O58 vel. rriag. F. = 10.797 vd. mag. min. = 0.0 vel. mg. min. = 0.0 vel. mag. min. = 0.0
Figure 5.6~:Velocity magnitude plot during the acceleration phase at t=11/60. Wall shear stress plot
lnlet Profile
shear stress rrrw. = 33.97 shear stress ma. = 75.84 shear sttsss rr\ax = T1.37 shear stress min. = -1.45 shear slm min. = -1 -61 shear stmmin. = -2.83
Figure 5.7~Wall shear stress during acceleration at t=8/60.
Wall shear Normal stress plot
Iniet Profile
shear stress m = 33.97 shear strass m~<.= 75.84 shear stress rrrax = T1.37 shear stress min. = -1 -45 sheer stress min. = -1.61 shear striass min. = -283
Figure 5.7b: Wall shear stress during acceleration at t=9/60. Wall shear 30% Concentric stress plot
Inlet Profile
shear stress max. = 33.97 shear stress mpc. = 75.84 shear stress max. = n.37 shear stress min. = -1.45 shear stress mtn. = -1.61 shear stress min. = -2.83
Figure 5.7~:Wall shear stress during acceleration at t=11/60. Velocity Normal P rofile Plots lnlet Profile
Figure 5.8a: Velocity profiles during the deceleration phase at t= lMO.
-- Velocity P rofile Plots Inlet Profile
Figure 5.8b: Velocity profiles during the deceleration phase at t=18/60. Reverse Flow
Inlet Profile
Figure 5.9a: Reverse flow during the deceleration phase at t=14/60.
Reverse Flow
Inlet Profile
Figure 5.9b: Reverse 80w during the deceleration phase at t=18/60. Vorticity Plots
Inlet Profile
voniaty rnax. = 26.60 vorticity m.= 27.68 vorticity ma. = 45.19 vortidty min. = -24.36 vorticity min. = -33.94 vortidy min. = -39.92
Figure 5.10a: Vorticity during the deceleration phase at t = 1MO.
Vorticity 30% Eccentric Plots
vorticity m. = 26.M vortiaty rrrpc = 45.1 9 vorüQtY min. = -24.36 vorticity min = -39.92
Figure 5.lob: Vorticity during the deceleration phase at t=14/60. V orticity 30% Concenûic Plots lnlet Profile
vorüaty max. = 26.60 vorücity m.= 27.68 vurticiiy ma^. = 45.19 vortidîy min. = -24.36 vorticity min. = -33.94 vorticity min. = -39.92
Figure 5.10~:Vorticity during the deceleration phase at t=20/60. Figure 5.1 1 a: isornetric view of pathlines during the deceleration phase at t= 13/60. Normal 1
Figure 5.1 Lb: 2-Dview of pathlines during the deceleration phase at t=15/60. 1 Normal p30% Concentric
Figure 5.1 Ic: 2-D view of pathlines during the deceleration phase at t=20/60. Figure 5.12a: Vorticity field during srnall secondary puise for the normal geometry.
Figure 5.12b: Vorticity field during small secondary pulse for the 30% concenttic model. Figure 5.12~:Vorticity field during srnall secondary pulse for the 30% concentric modef. Wall shear stress plot
Inlet Profile
shwr stress m. = 33.97 shear strass m.= 75.84 shear stress ma%. = 77.37 shetar stress min. = -1.45 sheer stress min. = - 1.6 1 shear ares min. = -2.83
Figure 5.13 : Wall shear stress during small secondary pulse.
Reverse Normal Flow
lnlet Profile
Figure 5.14: Recirculation during mal1 secondary pulse at t = 2 7/60. I Figure 5.15a: Vorticity field for the 30% eccentnc model during- acceleration of the small secondary pulse.
t = 30160 t = 31i60 t = 32/60 t = 33160
Figure 5.15b: Vorticity field for the 30% concentric model during acceleration of the smail secondas, pulse. Figure 5.16: Vorticity field for the 30% eccentnc mode1 during the end tlow rates of the small secondary pulse. Figure 5.17: Velocity magnitude field during the end of the cardiac wave for the 30% eccentric model.
Figure 5.18: Vorticity field during the end of the cardiac wave for the 30% eccentric model. Reverse Normal Flow
Inlet Profile
b igue 5.19: Reverse flow at the end of the cardiac wave. 6.0 CONCLUSIONS AND RECOMMENDATIONS
6.1 Introduction
The influence caused by the existence of the cardiovascuiar disease, atherosclerosis, has sevcre consrquences to an individual's health. This disease has the ability to cause serious dysfunction or death. The disease mainly occurs in curved and bihrcating vessels such as the carotid anery. The development of the disease in these areas may cause the occurrence stroke. The main cause for the development of atherosclerosis is currently unknown. however. srvenl hypotheses have been formulated to describr its existence. From such hypothescs. the main factor. which is shared among these theories. is the basis that the influence thar hemodynamics plays is an important factor in the development of the disease. The role of hemodynamics has been linked to such theones as regions of recirculation zones. oscillating shear stresses and blood residence tirne. In addition to the unknown nature of the disease. accurate techniques to access the rkk of srrious affects frorn the disease are not adequate to warrant high reliance on c linical diagnoses.
To acquire information conceming the local hemodynamics within mildly constricted vessels of the carotid artery, numerical simulation were used to solve the 3- dimensional tlow field. The assurnptions of a Newtonian flow were simulated with rigid vessrls under pulsatile conditions. The effects of using a Newtonian fluid have been discussed by other researchers, showing the validity of assuming a Newtonian property for blood. The assumption of rigid vessels is a reasonable assumption because of the hardening of the anery as the atherosclerotic plaque devrlops.
6.2 Conclusions
The research goals set out to be accomplished, which were outlined in Chapter 1 were met. The introduction of a novel technique for imposing boundary conditions was provided as well as a detailed analysis to reduce the numerical error. Information pertaining to the local hemodynamics of the flow patterns in the mildly constricted geomeûies of the carotid anery bifurcation were provided.
An effective technique of imposing the two outlet boundary conditions was introduced in Chapier 3. The technique required the use of two pressure specified outlet boundary faces: one constant pressure condition and the othrr a varying pressure condition. W i th the implementation of such boundary conditions. irnprovements in the abiiity of providing the numerical solution were achieved. The use of the two-pressure boundary condition removrd the dependence of speciQing analytically calculated profiles at one of the outlet faces. In addition. if cornplex outlet profiles were to exist in the simulations. which cannot be analytically calculated. then speciS>ing direct profiles would not be an appropriate condition and the two-pressure condition must be employed. The two-pressure condition also allows the required profiles to develop to maintain the specified flow division between the EC.4 and the ICA.
Due to the complexity of the geometric configuration of the carotid artenes. which cause complrx fiow fields to be solved, a hi& density mesh is required. With the use of the pressure conditions. the geometry of the outlet sections was shown to be accurately solved for the flow field with shortened outlet charnels instead of elongated tubes, wiiich would be required when specifying direct profiles on the face. In addition, this assisted in creating higher density meshes to be employed without the aid of wasted elements. The shortened
2ueornetry was also beneficial in decreasing the overall simulation time, which would have been required to simulate the original lengthy geometry. To ensure the new boundary condition implernentation functioned accurately, test simulations were canied out and presented in Chapter 4 showing positive correlations.
Chapter 4 also provided the method of generating numerically accurate simulations of mildly constricted vessels under pulsatile conditions. The process requires that first an appropriate grid density be acquired by caqing out consecutive simulations of various grid spacings. After an appropriate level of accuracy is achieved showing independence of the mesh density, temporal convergence was next to be obtained. In doing so, two aspects of temporal convergence were considered: the number of time steps to use in one pulse cycle and the number of pulse cycles to simulate. Obtaining the nurnber of time steps to use in one pulse ensured that enough solutions are acquired to properly simulate and consider the complex nature of the cardiac pulse. In addition to this, sirnulating unequal time steps but cven cord length increments provided the requirement of 100 time step increments. The benrfit of employing uneven time step increments enabled less time steps to be used in describing the ovenll pattern of the cardiac wave. Obtaining the nurnber of cycles of the cardiac wavr to simulate ensures the resuits are fully time-periodic. This was achirved after the second cycle was simulated.
In providing simulations of the rnildly stenosed arteries a detailed description of the tlow tield was given in Chapter 5. With the incorporation of the 30% constriction. the results were drastically different From the geometry of a normal patient. The stenosis caused high speed t'fows to occur and elevated concentrations of recirculation zones and wall shear stresses. Also. with the modelling of a concentric and an eccentric geomeûy with the same stenosis srverity, again variations were observed between the flow patterns. The results show the eccentnc mode1 introducing considerably more complex flow pattems. Distinct occurrences of vortex shedding were observed at two specitic instances in time, where in the concentric mode1 none were developed. The influences of these tlow pattems may be directly related to the theories proposed by other researchers concerning the development of the disease. The flow field information provided by this study may assist medical researchers in making new or upholding existing theones conceming the developrnent of the disease.
6.3 Contributions
1.) Presented a novel technique of imposing the two-pressure dirichlet condition for pulsatile flow through bifurcating geornetries. This boundary condition led to the reduction in the required length of the ECA and ICA branches, which reduced the necessary computational rescurces and computational time. 7.) Provided a thorough investigation in limiting the numerical error for simulations of the pulsatile flow in the carotid artery. For the constricted geometries. an element size of 0.0157j per CCA diarneter created a grid dependence error of less than 11.1% based on velocity and less than 23.4% based on wall shear stress. To obtain fully developed time periodic solutions ? or more cycles of the cardiac wave must be carried out with a minimum of 100 even cord length time increments.
3 Detailed observations were made conceming pulsatile flow in mildly constricted models of the carotid artery bifurcation with direct cornparisons made with the normal carotid artery. The position of the stenosis in the mildly constricted models present additional differences in the flow field.
6.4 Future Work
Numencal simulations provided in the carotid artery are beneficial in providing information conceming the hemodynamics. To continue the snidy. information can be provided penaining the simulating varying flow ratios between the ECAACA split throughout the cardiac wave. Because of the variability of the flow ratio through one cycle, changing from as low as 44.36 and as high as 10:90 split may cause distinct variations in the flow patterns. The ability of providing this information is simple in computational studies, however. more difficulty to achieve rrrperimentally. Also, due to the presents of the constricted artery, it would be reasonably sound to impose an additional variation in the 80w division because of the elevated flow resistance caused by the consuiction and observed in the required pressure gradients. In addition to altering the flow division ratio, the changing frequency of the flow oscillation may provide fiirther significant features in the flow field.
The ability of using other eiement types to mesh the fluid domain rnay provide better mesh resolutions. Currently, tetrahedral elements are employed and the full capabilities of the hybrid CFX-Tfc code was not utilized. With the use of a hybrid mesh a better quality rnesh can be generated to descnbe the flow patterns. Currently, not available in the ccde, but may give some benefit to these simulations is an adaptive mesh algonthrn. This would eliminate the need of canying out a gnd independence study, however, to do so appropriate algorithms mus: be used to ensure proper element subdivision or growing. Providing an adapted mesh that is valid for al1 simulation times would be highiy beneficial, however, the sost of doing so would increase computational time and perhaps computational resource to camy out such a task. Substantial time would be wasted to optimise the quality of the mesh.
The possibility of simulating further stenosis severity cannot be justified to impose a laminar flow assumption. Thus, research pertaining to the prediction of regions of transitional tlow would benrfit further numerical simulations. In doing, this aould enable the possibility of predictine the required stenosis severity to cause transitional flow. In addition. turbulence modelling cmbe employed in severely constncted arteries. Advanced Scirnti fic Computing Ltd., MATechnoiogy. CFX-Tfc.
.Uuned, S.A.. Giddens, D.P., 1983a Velocity Measurements in Steady Flow through Axisymmetric S trnoses at Moderate Reynolds Numbers. Jor~rnulof Biomechanics, Vol. 16, pp. 505-5 16.
..vin;ed. S.A.. Giddens, D.P.. 1983b, Flow Disturbance Measurements Through a Constricted Tube at Moderate Reynolds Numbers. Journal of Biomecltonics, Vol. 16, pp. 955-963.
Amtec Engineering, Inc.. Bellevue. WA, 1999. hayiotos. A.. Jones. S.. Giddens. D.. Glagov. S., Zarins, C. 1994, Shear Stress at a Cornpliant Mode1 O lthe Human Caro tid Bifurcation. Joicnzai of' Biomrchu~iicalEngineering, Vol. 1 16, pp. 9s- 106.
Azurna, T.. Fukushima, T. 1976. Flow Patterns in Stenotic Blood Vesse1 Models. Biorheoiogy, Vol. 13, pp.337-354.
Baaijens. J.P.. van Steenhoven. A.;\.. Janssen, I.D. 1993, Numerical Analysis of Steady Generalized Newtonian blood flow in a 2-0 Mode1 of the Carotid Artery Bifurcation. Biorlirologi. Vol. 30, pp. 63-74.
Bharadvaj B.K.. Mabon. R.F.. Giddens, D.P.,1982a. Steady Flow in a iModel of the Human Carotid Bifurcation. Part [--Flow Visuaiization. Jozrmal of Biomechonics, Vol. 133, pp. 349- 3 63.
Bharadvaj, B.K.. Mabon. R.F., Giddens, D.P., 1982b, Steady flow in a mode1 of the human Carotid bitiircation. Part II-Laser-doppler anemometer Measurernents. Journal of Bionzecltanics. Vol. 15. pp. 363-378.
Callow, AD..1996, Surgery of the Carotid and Vertebnl Artenes for the Prevention of Stroke. Wiilzanis CG WiiA-ins, Balt irnore.
Caro. C.G., Fitz-Gerald. J.M., Schroter, R.C., 1971, Atheroma and Artenal Wall Shear Observations, Correlation and Proposa1 of a Shear Dependent Mas Transfer Mechanism for Atherogenesis. Proc. R. Soc. London Series B. Vol. 177, pp. 109-159.
Caro, CG., Fitz-Gerald, J.M., Schroter, R.C., 1969, Arterial wall shear and distribution of early Atherorna in man. Narure (Land), Vol. 223, pp. 1 159- 1961.
Clark M.E., Robertson, J-iM., 1983, Stenosis Severity Effects for Unbalanced Simple-Pulsatile Bifurcation Flow. Journal of Bioflzrid Mechanics, Vol. 16, pp. 895-906. Deshpande, M.D., Giddens, D.P.,Mabon, R.F. 1976, Steady laminar flow through modelled vascular stenoses. Joirrnal of Biomechanics, Vol. 9, pp. 165-174.
Ehrlich, L.W., Friedman, M.H., 1977, Particle Paths and Stasis in Unsteady Flow Through a Bifurcation. Joiirnnl of Biorneclza~zics,Vol. 10, pp. 56 1-568.
Ethier, CR., Steinman. DA.. Couch G., Ohja, M., 1998, Validation of complex separation patterns in an snd-to-side anastomosis model. 1998 Advurices in Bioengineering 39:77-78.
Fei, D., Billian. C., Rittgen, S.E. Flow dynamics in a stenosed carotid bifurcation model - Part i : Basic velocity measurements. Ultrasound in Medicine and Biologv, 1988, Vol. 14, pp. 2 1-3 1.
Femandez. R.C.. De Witt. KI.. Botwin, M.R., 1976, Pulsatile Flow Through a Bifurcation with Appl kat ions to Artenal Disease. Jorirnal of Biornechanics, Vol. 9, 575-580.
Friedman. hW.. 1989. .A biologically plausible rnodel of Thickening of the Arterial Intima Cndrr Shear. .~rteriosclerosis.Vol. 9. pp. 5 1 1-522.
Friedman, M.H., Deiers, O.J.. Bargeron, C.B., Hutchins, G.M., Mark, F.F., 1986, Shear- Dependent Thickening of the Human Artenal Intima. .4tlterosclerosis, Vol. 60, pp. 16 1- 17 1.
Fry. D.L.. 1976. Haemodynamics forces in atherogenesis. Cerebrovasnrlar Disease, pp. 77-95.
Fry. D.L., 1965. Acutr vascular endothelial changes associated wiih increased blood velocity gradients. Circihrion Resemch. Vol. 22, pp. 165- 197.
Forster, F. K.. Chikos, P.M.. Frazier, J.S.. 1985, Geometric Modelling of the Carotid Bifurcation in Humans: Implications in Ultrasonic Doppler and Radiologic investigations. KU,Vol. 13, pp. 385-390.
Gessner, F.B ., 1973, Hemodynamic Theones of Atherogenesis. Circulation Research, Vol. 33, pp. 259-266.
Giddens, D.P..Ku, D.N., 1987, A Note on the Relationship Between inout Flow Waveform and Wall Shear Rate in Pulsatile, Separating Flows. Journal of Biomechanical Engineering, Vol. 109. pp. 109-1 10.
Giddens, D.P.. Wabon, R.F. Cassanova, R.A., 1976, Measurements of Disordered Flow Distal to Subtotal Vascular Stenosis in the Coronary Aortas of Dogs. Circulation Research, Vol. 39, pp. 112-1 19.
Glagov, S.G., Zarins, C., Giddens, D.P.,Ku, D.N., 1988, Hemodynamics and Atherosclerosis: Insights and persepectives gained kom studies of hurnan arteries. hch Pathol Lab Med, Vol. 112, pp. 1018-1031. Gotto, A.M., Robertson, AL., Epstein, S.E., McCollum. C.H., 1977, The Upjohn Company: KaIrnazoo, Michigan.
Gorelick P.B., 1995, Stoke Prevention. [Review]. Archives of Neurolugy, Vol. 52, pp. 347-355.
Holdsworth, D.W., Norley C.D., Frayne, R. Steinman, DA., Rua, B.K.,1999, Characterization of the Comrnon Carotid Blood-flow Waveforms in Normal Hurnan Subject. Physiologicnl .Ileasure»zenf. Vol. 20(3), pp. 2 19-240.
Hung, Tin-Kan. Tsai, T., 1997, finematic and Dynarnic Charactenstics of Pulsatile Flows in Stenotic Vessels. Jolinid of Engirreeri~zgiIfechanics, Vol. 123, pp. 247-259.
Hung. Th-Kan. Tsai, T.. 1996. Pulsatile Blood Flows in Stenotic Artery. Journal of Etzgineering Meclzanics, Vo 1. 122, pp. 590-896.
ICE hI CFD Enjineering. TETRA. Berkeley CA.
Khatifa, -4.M.. Giddens, D.P.. 1981, Characterization and Evolution of Postenotic Flow Disturbances. Jozirtlai oj'Bionreclianics, Vol. 14. pp. 279-296.
Ku. D., Giddens. D.P., 1987, Laser Doppler Anemometer Measuements of Pulsatile Flow in a Mode1 Carotid Bifurcation. Jolirnal oj'Biontecahincs, Vol. 20, pp. 407-42 1.
Ku, D., Liepsch D., 1986, The Effects of Non-Newtonian Viscoelasticity and Wall Elasticity on the Flow at a 90' Bifurcation. Biorheology, Vo1.23, pp. 359-370.
Ku, D., Giddens. D.P.. Zarins. C.K.. Glûgov, S., 1985, Pulsatile flow and atherosclerosis in the hurnan carotid bifurcation. .4rteriosclerosis, Vol. 5, pp. 193-302.
Ku, D., Giddens, D.P., 1983, Pulsatile Flows in a ;Mode1 Carotid Bifurcation. Arteriosclerosis, Vol. 3. pp. 3 1-39.
Lee, D., Chiu, 3.J.. 1996. Intima1 Thickening Under Shev in a Carotid Bifurcation - A Nurnencal Study. Jouniai of Biomechanics, Vo1.29, pp. 1- 1 1.
Lee, D., Chiu, J.J., 1 99îa. Computation of Physiologicai Bifurcation Flows using a Patched Grid. Compiiters and Fhids, Vol. 2 1, 5 19-53 5.
Lee, D., Chiu. J.J., 1992, A Numencal Simulation of Intima1 Thickening Under in Artenes. Biorheology, Vol. 29, pp. 33 7-35 1.
Lei, M.. Kleinstreuer, C., Truskey, G.A., 1995, Numencal Investigation and Prediction of Atherogenic Sites in Branching Arteries. Journal of Biomechonical Ë'ngineering, Vol. 1 17, pp. 350-357. Liepsch. D.. Pflugbeil, G., Matsuc, T., Lesniak, B., 1998, Flow Visualization and 1- and 3-D Laser-Doppler-Anemometer Measurements in modles of hurnan carotid artenes. Clinical Hemorheologi and Microcircularion, Vol. 18, pp. 1-30.
Liepsch, D., Singh, M., Lee, M. Expenmental haiysis of the Influence of Stenotic Geometry on Steady Flow. Biorheology, 1992, Vol. 29, pp. 419-43 1.
Liepsch, D., Moravec, S., 1984, Pulsatile Flow of Non-Newtonian Fluid in Distensible Models of Human Arteries. Biorheology. Vol. 7 1, pp. 57 1-586.
LoGrrfo, F.W. Nowak, M.D., Quist, W.C., Crawshaw H.M., Bharadvaj, B.K., 1981, Flow studies in a model carotid bifurcation. .4rteriosclerosis, Vol. 1, pp. 235-241.
.McDonald. DA.,1955, Method for the calculation of the velocity, rate of flow and viscous drag in aneries when the pressure gradient is known. Journal of Ph~vsioiog.Vol. 127, pp. 553-563.
MerriIl. E. W.. 1965, " Rheology of Human Blood and Some Speculations on Its Role in Vascular Homeostasis". Biomechunicul Mecltonisms in Vasatlar Homeostasis and I)itravascttlar Tlrrombosis, pp. 121-137.
Motomiya, M.. Karino, T., 1984, Flow Patterns in the Human Carotid Arte~Bihrcation. Stoke. Vol. 15, pp. 50-56.
Mustard, J.F.. Packharn, MA., Kinlough-Rathbone, R.L. 1993, Platelets and Atherosclerosis. Xtherosclerosis: Mechanisms and Approaches to Therapy: Raven Press, New York, pp. 29-43.
Nazemi. M.. Kleinstreuer, C., Archie, J.P., 1990, Pulsatile Two-dimensional Flow and Plaque Formation in a Carotid Artery Bifurcation. Journal of Biomechanics. Vol. 23, pp. 103 1- 103%
Nerem, R.M. Atherogenesis: Hemodyamics, vascular geornetry and the endothelium. Biorheology, Vol. 2 1, pp. 565-569.
O'Brien V., Ehrlich, L.W.. 1985, Simple pulsatile fiow in an artery with a constriction. Journal of Biornechanics. Vol. 18, pp. 117- 127.
Palmen, D.E.,van de Vosse, F.N., Janssen, J.D., van Dongen, M.E., 1994, halysis of the Flow in Stenosed Carotid Artery Bifûcation Models - Hydrogen Bubble Visualization. Journal of Biornecllunics, Vol. 27, pp. 58 1-590.
Perktold, K.. Rappitsch. G.. 1995, Cornputer simulation of local blood flow and vesse1 mechanics in a cornpliant carotid artery bifurcation model. Journal of Biomechanics, Vol. 28, pp. 845-856.
Perktold, K., Thumer, E., Kemer, T. Flow and Stress Characteristics in Rigid Walled and Cornpliant carotid Artery Bifurcation Models. Medical & Bzological Engineering & Computing. Vol. 32, pp. 19-26. Perktold, K.. Peter R., Resch M., Langs G., 1991a, Pulsatile non-Newtonian blood flow in three- dimensional carotid bifurcation models: a numerical sîudy of flow phenornena under diferent bifurcation angles. Jozmal of Biomedical Engineering, Vol. 13, pp. 507-5 15.
Pertold, K., Resch M., Florian H., 1991b, Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model. Jozrmal of Bioniechanical Engineering, Vol. 113, pp. 161-475.
Pertold, K.. Resch M., Peter R., 199 1c, Three-dimensional Nurnencal Analysis of Pulsatile Flow and Wall Shear Stress in the Carotid Artery Bifurcation. Joiirnal of Biomechanics, Vol. 24, pp. 409-320.
Perktold, K., Peter, R.. 1 WOa, Numerical 3D-Simulations of Pulsatile Wall Shear Stress in an Werial T-bifurcation model. Journal of Biomedical Engineering, Vol. 12. pp.2- 12.
Perktold. K.. Resch. M., 1990b. Nurnerical Flow Studies in Human Carotid Mery Bifurcation: Basic discussion on the geometric factors in atherogenesis. Journal oJBiomedical Engineering, Vol. 12, pp. 11 1-123.
Perktold, K., Hilbert, D., 1986, Numerical Simulation of Pulsatile Flow in a Carotid Bifurcation Model. Jorirnril oj'Bionzedical Engineering, Vol. 8, pp. 193-199.
Ma, P.. Li, ,Y.. Ku, D., 1997, Convective Mass Transfer at the Carotid Bihircation. Jolrrnai of Bioniecltunics. Vol. 30, pp. 565-571.
Reneman, R., Merode. T., Hick, P., Hoeks, A., 1985, Flow velocity patterns in and distensibility of the carotid artery bulb in subjects of various ages. Circziiation, Vol. 71. 500-509.
Rindt, C.C.. van Steenhoven, A.A., 1996, Unsteady Flow in a Rigid 3-D Model of the Carotid .Uery Bifurcation. Jolinial of Biomechunical Engineeritig, Vol. 1 18, pp. 90-96.
Rindt, CC. Steenvhoven, A.A., Janssen, I.D., Reneman, R.S., 1990, A Nurnencal Analysis of Steady Flow in a Three-dimensional Model of the Cmtid Artery Bifurcation. Journal of Biomechanics. Vol. 23, pp. 36 1-473.
Rindt, C.C.. Steenvhoven, A.A., 1988, An Expenmental Analysis of the Flow Field in a ïhree- Dimensional Model of the Human Carotid Artery Bifurcation. Journal of Biomechanics, Vol. 21, pp. 985-99 1.
Rindt, CC, van de Vosse, F.N., van Steenhoven, A.A., Janssen, J.D., 1987, A Nurnencal and Expenmental Analysis of the Flow Field in a Two-dimensional Model of the Human Carotid Artery Bifurcation. Journal of Biornechanics, Vol. 20,499-509.
Rosenfeld, M.' 1995, A Numencal Study of Pulsatile Flow behind a constriction. Journal of Fluid Mechanics, Vol. 30 1, pp. 203-223. Rosenfeld, M., Einav, S., 1995, The Effect of Constriction Size on the Pulsatile Flow in a Channel. Jozmal of Fitrici Mechanics, Vol. 1 17, pp. 571-576.
Ross, R., 1986. Pathologies of Atherosclerosis - an update. New England Journal of Medicine, Vol. 314, pp. 488.
Salzar. R.S., Thubrikar. M. J., Eppink, R.T., 1995, Pressure-induced Mechanical Stress in the Carotid hery Bifurcation: -4 Possible Correlation to Atherosclerosis. Joirnml of Biomechanics. Vol. 18(11), pp. 1333- 1340.
Siouffi. M., Pelissier. R., Fluahi far, D., Rieu, R., 1984, The effect of Unsteadiness on the Flow Through Stenoses and Bifurcations. Journal of Biomechanics, Vol. 17. pp. 299-3 15.
Smith. R.F.. Rurt, B.K., Fox, A.J.. Rankin, RN., Holdsworth. D.W.. 1996, Geometric characterization of stenosed hurnan carotid arteries. Acad Radiol3:898-9 1 1.
Steinman. DA.. Rutt. B.K.. 1998. On the nature and reduction of plaque-mimicking fiow artifacts in black blood MRI of the carotid bifurcation. :C.fagnelicResona~lce in Medicine. Vol. 39(4, pp. 635-641.
Solberg, LA.. Eggen. D.A..197 1, Localization and sequence of development of Aiherosclerotic lesions in the carotid and vertebral arteries. Circulation, Vol. 43. pp. 71 1-723.
Solzbach. L'., Wollschlager. H., Zeiher, A., Just, H., 1987, Effect of Stenotic Geometry on Flow Behaviour Across Stenotic Models. iblerlical and Biofogtcai Engineeri~rgand Comprrters, Vol. 255, pp. 543-550.
Taylor. M.G.. 1959, "The Influence of the Anomalous Viscosity of Blood Upon Oscillatory Flow", Phpics in Medicine and Biology, Vol. 3. pp. 273-290.
Van de Vosse. F.N.. Van Steehoven. AA, Janssen, J.D.. 1990, A Two-dimensional Numerical halysis of Unsteady Flow in the carotid Artery Bifurcation. Biorheolop, Vo1.27, pp. 163- 189.
Wells, D.R., Archie, J.P. Kieinstreuer, C., 1996, Effect of the carotid artery geometry on the magnitude and destmction of the wall shear stress gradients. Jotrmal of Vascular Surgery, Vol. 23, pp. 667-678.
White. R.A., 1989, Atherosclerosis and Arteriosclerosis: Human pathology and experimental animal methods and models. CRC Press. Inc.: Flonda.
Wille S.O., 1980, Pressure and Flow in Two-dimensional Numerical Bifurcation Models. Applied ~ClarhematicalModelling, Vol.4, pp. 35 1-356. Wong, P.K., Johnston, K.W., Either, C.R., Cobbold, R.S., 1991, Computer Simulation of Blood Flow Patterns in Artenes of' Various Geometries. Journal of Vasnilar Szrrgery, Vol. 14. pp. 658- 667.
Young, D.F. Fluid Mechanics of herial Stenoses., 1979, Jorirnal of Bioniechanical Engineering, Vol. 101, pp. 157-175.
Young, D.F.. Tsai, F.Y., 1973, Flow Charactenstics in Models of Artenal Stenosis. 1. Steady Flow. Journal of Biomeclirrnics. Vol. 6, pp. 393-410.
Zarins, C.K.. Giddens. D.P.. Bharadvaj, B.K., Sottiurai, V.S.. Mabon, R.F.. 1983, Carotid Bifurcation .4therasclerosis-Quantitative Correlation of Plaque Localization with flow velocity Profiles and Wall Shear Stress. Cirailation Research, Vol. 53, pp. 502-5 14. Appendix A 1
Portions of the modified source code fkom CFX-Tfc SUBROUTINE MAINLN - the main routine - added routine to caiculate time step increments - added output routines (OUTPUT2& OUTPUTZ) - added routine to pass variables to new routines (SEW) CALL IMTTAL - initiakation - added routine to find radius for dl UO faces (MAXRAD) CONTINüE - time step loop f CONTINUE - coefficient loop / CALL FLUIDS - assemble and solve of mass-momentum CALL FLUIDB - impose boundaxy conditions - added routine to calculate new / pressure condition for ECA (BCPRES). - added routine to impose velocity field (BCOPEN).
CALL FBCUPD - update boundary conditions for fluids
CALL SCALAR - assemble and solve dl scalar equations
CALL SBCUPD - update boundary conditions for scalars l CALL PROPER - new material properties CALL MODELC - new mode1 coefficients
\\ End of coefficient loop
CALL VARUPD - move to next time step \ End of time step loop CALL FNU - finalization
REW
Figure Al: Main branches in the CFX-Tfc source code. This routine is calied by many of the new routines to initialize some of the variables instead of passing variables through the routines.
Variables: LSS logic for steady-state or pulsatile simulations LPRINT lo@c to print pulsatile output data MAXTSL maximum number of time step loops WCFL maximum number of coefficient loops SINGLE iogic to write dl data to a single file NPWLD number of pulses desired SMbTLT NOT USED FIRSTLO NOT USED PI =3.1415926535 RH0 density PERCENT flow division through ECA Nu dynamics viscosity REY Reynolds number twskip output writing: l -allT2-skip1, etc. FREQ f?equency of puise LONE logic to calculate SS results of TS=! of pulse NPUL number of pulses (cycles of the pulse)
IM'EGER MAXTSLJ!WCFL,WRNUMTNPULD
REAL PI~MO,PERCENT,NU,REYtSMULT$~LO~Q
LOGICAL LSS,LPRINT,SINGLETLONE
C EXECUTABLE STATEMENTS
C- __1_ C**NOTES** C -change the Buid properties to match C -change timestep size C -mut set MAXTSL & MAXCFL to be same as in prm file C -check TS limitationinrnainln.F (at endofroutine) C -if running 1 puise and it is not time4.0, watch the setMg of MAXTSL (changes the C pulse time?therefore,the profile). C -if adjusting the &equenci(FREQ) change the maximum time (TiMAX) in the C mainln.F routine.
C--Lnitialize constants
PI =3.1415926535 RH0 = 1.0 FREQ =1.0
PERCENT = 0.30 PERCENT = 0.44
LSS =.mm. LSS =.FALSE.
Lob3 = .TRIE. LONE = .FAUSE.
C ->Paramete~sfor pulsatile nui IF( .NOT. LSS) THEN LPRINT = .TRIE. SINGLE = .T'RUE. m'M=1 FIRSLO = 0.0 SM[ILT = 1.0 WTSL= 400 MAXCFL = 300 NPULD = 2 REY = 275
C ->Parameters for single puise run IF(L0NE) THE% LPRINT = .FALSE. MAXTSL = 200 MAXCFL = 1 ENDIF C ->Parameters for steady mn ELSEIF(LSS) THEN MAXTSL = 400 MAXCFL = 1 SINGLE = .TRUE. SMULT = 0.0
C REY = 250.0 C NU = 0.008
C REY =400.0 C NU = 0.005
REY = 1100.0 MJ = O.OOl8l81818l818l C LOCATION: C maintn.F + initial.F i upfini.F i rnawad.F C C DESCRIPTION: C This routine is calleci by upfi.F to search for the maximum and C minumim x-coordinate on the face of velocity specified faces (works for C in-corn & out-ext). The values are storaged in XMAXO and XMINO. C The comesponding max. & min radius is calculated and storaged in RMAX() C and RMINO-
C Variables: C NNODE number of nodes in current element type C NNODED total number of nûdes C NOT node number not on inletIoutlet surface of element C FACE curent face number of element on miet/outlet surface C NF NOT USE C CHF.4CE logic to change face for ECA and ICA C XN,YN,ZN x,y,z-coordinates of current node c w,m ma.and min. x value of face C RMAX maximum radius C RMIN minuniim radius C XYZL local array of x,y,zîoordinates CRMAXRMIN calulated Max. and min. radius of node w/XMAX and ZMIN C SETNAM name of cwrent face set C LASNAM NOT USED C MSMB mberof face; 1->inam, 2-aut-ext C
C Subroutine Arguments c----- C CHFACE = O C LASTNAM = DEFAULT c rn(l&2)= 1000 c XMAX(1&2)=0
C Get nodes on face and radius of center
IF (SETNAM .EQ.'in-com') THEN WRITE(90,') SETNAM CHFACE = .T'RUE. m=1 ELSEIF(SETNAM .EQ.'out-extf)THEN WRITE(90,*) SETNAM CHFACE = .FALSE. NUMB=2 ENDIF
C Set X & Y totals to zero m.0 m4.0 m4.0
DO 1 1 1 NN=I ,NNODE IF(NN .EQ.NOT) GûTO 11 1 XN = XYZL(1,NN) YN = XYZL(Z,.NN) IF (SETNAM .EQ.'in-corn') THEN -xMrN@rmB) = 0.0 RMIN(MJMB) = 0.0 ENDIF
WRITE(90,*) 'XN = 'JN,'YN = ',YN,' ZN = ',ZN WUTE(90,*) 'RMAX(1)= ',RMAX(1),' RMIN(1) = ',RMIN(1) WRITE(90,*) 'RMAX(2)= ',RMAX(Z),' RMN(2) = ',RMIN(Z) WRITE(90,*) 111 CONTrn'UE C*************** C CALTIME CS************** C C caitime.F c------
LOCATION: mainln.F + caitime.F
Calculates equally-spaced time steps base on cord Iength. Given a mean velocity waveform in the fom of Fourier coefficients this routine discretizes the waveform into approximately 'nt' thesteps, separated by approximately the same arc lengrh.
author: David S teinman Modified by R.GIN
Variables: RO out radius of inlet face DTIME array of calculated thesteps FREQ fkequency of puise NPULD nurnber of pulses desired nf number of Fourier coefficients ccoef cosine coefficients of Fourier senes scoef sinusoidal coefficients of Fourier series nt number of time steps tstart start time tend end the
REAL RO,DTIME(I000),PI~O,PERCENT,NU,REY,SMULT,FIRSLO,FREQ7 & tend, tstart
implicit real (a-h,o-y), complex (z), integer(i-n) dimension ccoef(0: 13),scoef(O: 13) LOGICAL LONE,SINGLE,LSS,LPN C LOCAL PARAMETERS and DATA c ------
C--Coeficients for pulsatile waveform
DATA nf,ccoeefcoel3,1 ,O.179325,-0.385872,-0.4508 1 1,-0.106449, & 0.295767,O.19463 1 ,-0.0336471,-0.0444237,-0.0253872, & 0.0242532,O.O1 97074,0.00928437,O.OO 1657 14, & 0,0.498146,OS 1 1875,-0.329866,-0.263848,-0.317235, & O. 177449,0.0964081,0.0253 163,-0.0337126, & -0.0221 089,0.00370957,0.0083496,0.0120435/
CALL SETLTP(LSS,LPRINT,~sL,MAX~,SINGLE,S~LO. & PI~O,PERCENT,NU,REY,WRNUMJREQ,LONE,NPULD)
C-reaci print option, pop: C popW gives output suitable for inclusion in ftirne block C popt= 1 dumps Mievs. waveform (usefûl for plotting) C popt=2 dumps rime vs. wavefom at each test point (useful for plotting)
popt = 2
C read desired number of hesteps, start/end times
nt = MAXTSWNPULD tstart = 0.0 tend = 1.0
C Calculate womenley number OMEGA = 2*PI*FMQ ALPHA = RO*SQRT(OMEGA/NLT)
C First establish the maximum and minimum of the waveform, to dow for C proper scaling (Le so that Iq(4since /tf N = 10000 dt = l.O/N qmax = -le30 qrnin = 1 e30 do 10 i=l,N t = i*dt cd1 avewomer(nf,ccoef,scoef,t,q) if (q.gt.qmax) qmax = q if (q.lt.qmin) qmin = q IO continue qrange = abs(qmaxqmin) C qrange=l.O C Calculate the total arclength, which is then divided by the desired C number of timesteps to give the correct equd-arclength spacing. N = 100000 dt = 1.ON s = 0.0 cail aveworner(nf,ccoef,scoef,O.O,ci) do 30 i=l,N qold = q t = ifdt cal1 avewomer(nf,ccoef,scoetSq) dq = (qqold)/qrange cls = sqrt(dq*dq+dt*dt) s=s* 20 continue dsmax = dnt if (popt.gt.0) write (250,*) Total length = ',s if (popt.gt.0) wite (250,*) Max. indiv. length = ',dsmax C N controls the resolution of the calculations. Large N means that C each individual ds is closer to dsmax, at the cost of a longer C execution the. Smailer N nins fabut increases the variation C of ds. N = t 00000 dt = LO/N ds = 0.0 t = tstart told = 0.0 C loop until dFdsmax TLOOP = O 30 continue qold = q t = t+dt cd1 avewomer(nf,ccoef,scoeçSQ) C calculate the change in flowrate over dt dq = (q-qold)/qrange C calculate the arclength ds = ds+sqrt(dq*dq+dt*dt) if (popt.gt.2) wite(250,*) t,q,t-told,nt*(t-told),ds C quit if we have reached the end of the cycle if (t.gt.tend) goto 100 C dump output and reset quantities if ds==ax if (ds.ge.dsmax) then TLOOP = TLOOP + 1 if @opt.eq.O) write(ZO,g 10) t if (popt.gt.0) write(250,500) TLOOP,t,q,t-told,nt*(t-told),ds if (popt.gt.0) write(25 1,'([email protected]))') t,t-told DTlME(TLOOP)= t-told ds = 0.0 told = t endif goto 30 100 continue TLOOP = TLOOP+l mend if (popt.eq.0) then wite(250,g 10) tend elseif (popt.gt.0) then write(250,500) nOOP,ten&q,tend-tol4nt*(t-told),& wite(25 1,'(2(F 10.7))') tend,tend-told endif DIIME(TLOOP)=tend-told IF(NPULD .GT.1)THEN DO 200 I=l,NP~~-1 DO 300 J=l,nt TLOOP = TLOOPH DTIME(TLO0P) = DTIME(J) t-ttDTIME(TLOOP) WRITE(25 1,'(2(F 1 O. 7))') t,DTTME(TLOOP) 300 CONTIMJE 200 corn ENDIF CLOSE(üNIT=750) CLOSE(t,'NIT=ZS 1) end real ccoef(0: 1OOO),scoef(O: 1000),t,q integer nf * ********************************************************************** complex zi,n,~q real pi q = ccoef(0) do 10 k=l,nf a = -*pi*n*k*t zq = (ccoef(k)-zi*scoefCk))*cexp(zt) q = q+real(zq) 10 continue end C BCPRES C******S******** C LOCATION: C mainln.F + f3uids.F + 0uidb.F i bcpres.F C C DESCPRTION: C Pressure vaiues are calculated to maintain the specified flow division C Input C MKOM massfiow through iniet C MOEXT massflow through extemal outlet " MOINT massflow through interal outlet number of timestep loop number of coefficient loop Modified PSPEC calucated pressure for exterai face Local MAXTSL maximum number if hmestep Ioops .ViAXCFL maximum number of coefficient loops PERDES percentage desired (split berneen EXï:COM) PRV old pressure values PER old percentage split NPRV new pressure value PERD11 percentage ciifference between desired & curent PPDIF percentage difference between older values LSS logical for steady nias LPRINT NOT USED METHOD character seing for output -> EE = use enor method (*Es8) -> 1s = fint pressure correction -> L* = limit smaU vaiues of error -> *F = fiip hm-ve to +ve or +ve or -ve -> *S = accelerate for smali change -> *D = accelerate for wrong direction C PERPER error: limits of percentage of percentage flow split C .VIINPER minimum percentage change ailowed REAL MICOM,.MOEXT,MOINT,PSPEC,PERPER,WER, & MAXTSL~CFL*PI~OyPERDES7rUv7REY, & MGH,MED,LOW,DrvD LOGICAL LSS,LPRINT,SINGLE,LONE7FCOEFL C--Get initial parameten CALL SETUP(LSS,I,PRJNTyMAXTSL4MAXCFL,SINGLE7SMULT,FIRSLO~ & PIIRHO,PERDES,NV~Y,~W~Q,LONE~ULD) C--Shp hst coef loop and use set values fiom GUI C-lnitialize variables for RELAXATION C Default for pulsaile nuis C PERPER = 0.01 C WER=0.03 C HIGH =OS C AMED = 0.3 C LOW =0.1 C DlVD = 3.0 C De fault for steady nuis PERPER = 0.01 MDPER = 0.00 HIGH = 0.25 MED = 0.15 LOW = 0.05 DIVD = 2.0 C Initialize additional variables METHOD = 'EE' PER( 1) = MOEXT/(MOEXT+MOINT) CPER = PER(1) PRV(1) = PSPEC PERD11 = PERDES - PER(1) PERDU = PERDES - PER(2) PPDIF = PER(2) - PER(1) .rnT = 1.0 FCOEFL = .FALSE. C--No change in pressure if % difference fiom desired is within error limits IF (ABS(PERDI1) .LE.PERDES*PERPER) THEN METHOD = WC' NPRV =PRV(l) PSPEC = NPRV FCOEFL = .TRIE. GOTO 700 ELSE FCOEFL = .FALSE. ENDIF C-Compare massflow rate (for fïrst massflow correction) (2nd loop) c--- IF ( ((KNTTSL.EQ. 1 .AND. KNTCFL.EQ.2) .OR. & ((LSS .OR. LONE) .AND. KNTTSL .EQ.2)) & .AND. ABSPSPEC) .EQ.0.0 ) THEN NPRV = -5.0 ELSE hTRV = 5.0 ENDIF C-Compare massfiow rate (for after nnt rnassflow correction) c ------ ELSE C-Calculate new pressure by using mon IF(PRV(1) .LT.0.0) THEN RATIO = ABS((PERDES*(MOEXT+MODJT))/MOEXT) ELSE RATIO = ABS(MOEXT/(PERDES*(MOEXT+MOINT))) ENDIF C limit how small the ratio cmbe IF(L4TIO .GT.1 .O .AND. RATIO .LE.l+MINPER) THEN RATIO = l+.WER METHOD( I :1) = 'L' ELSEIF(RATI0 .LT.1 .O ..AND. RATIO .GE.1 -iVIINPER) THEN RATIO = 1-MINPER hETHOD(1: 1) = 2' ENDIF EXPT = (PERDES+ABS(PERDIl))/PERDES NPRV = PRV(l)*RAnO**EXPT C NPRV = PRV(l)*RATIO C-- Weighting factor for C i) mail pressure value (fiip to opposite sig) C ii) accelerate change in flow % to move in the correct direction. C iii) accelerate movement in the correct direction (current direction). C i) sign Bip -> won? work for ABS@rv(l)) < 0.125 C and for pressure close to this value C IF (ABS(PRV(l)).LT.O.15) ïHEN MULT = 4.0 METHOD(2:2) = T' C ENDIF ELSE C If percentage far away f?om desired then increase IF(ABS(PERD1I) .GT.O. 1O)THEN METHOD(2:2) = 73' IF(PER(1) .GT.PERDES)THM MULT = 1 -HIGH ELSE MULT = l+HIGH ENDlF C ii) Direction chanage for percentage greater than desired C - if percentage ciifference between old and new is srnail and the pressure ciifference between the desired is greater than ##% then increase lots else ktincrease ELSEIF(PER(1) .GT.PERDES .AND. PER(2) .GT.PERDES)THEN IF(ABS(PER(1)) .GT.ABS(PER(2)))THEN IF(ABS(PPDIF) .LE.0.001)THEN IF(ABS(PERDI1) .GT.PERDES*(PERPER+PERPER/DrvD))THEI; MULT = 1-HIGH ELSEIF(ABS(PERDI1) .GT.PERDES*PERPER+û.0005 .AND. ABS(PERDI1) .LT.PERDES*(PERPER-+PERPER/DIVD))THE'iu' MULT = 1-MED ELSE MULT = 1-LOW ENDIF METHOD(2:Z) = 'S' ELSE MULT = 2-MINPER METHOD(2:2) = Dr ENDIF C Increase mail rnovements in the comtdirection for percentage C greater than desired IF(ABS(PERDI1) .GT.PERDES*(PERPER+PERPER/DIVD))THEN MULT = 1-MED-tO.05 & ABS(PERDI1) .LT.PERDES*(PERPER+PERPEWDIVD))THEN MULT = 1-LOW-0.05 ENDIF METHOD(2:2) = 'S' ENDE C Direction chanage for percentage less than desired C - if percentage difference between old and new is mal1 and the C pressure différence between the desired is greater than #% C then increase lots else ktincrease ELSEIF(PER(1) .LT.PERDES .AND.PER(2) .LT.PERDES)THEN IF(.4BS(PER(l)) .LT.ABS(PER(2)))THEN IF(ABS(PPDIF) .LE.0.0010)THEN ïF(ABS(PERDI1) .GT.PERDES *(PERPER+PERPER/DIVD))THm MULT = l+HIGH ELSEIF(ABS(PERDI1) .GT.PERDES*PERPER+0.0005 .AND. & ABS(PERDI1) .LT.PERDES*(PERPER+PERPER/DIVD))THEN MULT = l+MED ELSE MULT = l+LOW ENDIF -METHOD(2:2)= 'Sv ELSE klfIlLT = l+MINPER ETHOD(2:2) = D' ENDIF C lncrease small movements in the correct direction for percentage C less than desired ABS(PERDI1) .LT.PERDES *(PERPER+PERPERIDIM))THEN rnT= l+LOW+o.O5 ENDIF METHOD(2:2) = 'S' ENDIF ENDIF ENDIF C For positive pressure values (change mult. value) IF(PRV(1) .GT.0.0 .AND.MLTLT .GT.0.O)THEN 700 CONTINUE C--Write to screen WRiTE(*,*) 'TSLCFL MICOM -> MOEXT MOINT', & ' CPER CPRV NPRV CH MULT' WRITE(*, 10) KNTTSL,KN~CFL,LWCOM, & MOEXT,MOINT,PER(1)* 100,PRV(1),NPRV,METHOD,MLJLT C-Update old variables 800 CONTINUE CS************ C BCOPEN CS************ C C bc0pen.F c------ LOCATION: mW.F + flu.ids.F + fluidb.F + facf1u.F + bc0pen.F DESCRITION This routines is the main routine to calculate the velocities for each face node for both steady and pulsatile simulations. INTEGER NODTD,NNODE,NNODED,NF,WRNUM INTEGER NOT,FACE,KNTTSL,KNTCFL,MAXTSL,MAXCFL,NPbZD REAL RC,XC,YC,GNNL(NNODED),RMAX(2),RMIN(2),TTJM&ALPHA, & XYZL(3,NNODED),REYPI,NU~0TpERCENT9Y,VEL,SMULTJRSL09 FREQrnW LOGICAL PRESLV,LSS,INLET,LPm,SINGLE,LONE CHPJIACTER*40 SETNAM C EXECUTABLE STATEMENTS c -*---- CALL SETUP(LSS,LPRINT,MAXTSL,MAXCFL,SINGLE,SMULT,FIRSLO, & PlWO,PERCENT,WSY,WRNUM~Q,LONE,) C-Calculate Radius w.r.t. whole geometry CALL RADIUS(XYZL,NODTD,FACE,NNODE,GNNL.,RC,XC7YC, & NOTJWODED,NFPRESLV) C-Calucalate Radius w.r.t to face center CALL RADCEN(SETNAMJXESLV, & ~,RCBOXC,YC, & PImm C--Calculate veiocities for faces in steady and pulsatile simulations IF (LSS)THEN CALL STEmY(RC&O,UAVE,VEL,INLET,Y, & REY,W,PERCENT) ELSE IF(KNTTSL .EQ.MAXTSL/NPULD)THEN MTIM= TT ELSEIF(KNTTSL .GT.WTSL/NPULD)TEN TT=TT-MA)(;TIM ENDE CALL WOMER(?T,VEL,RC,ALPHA,PERCENT,INLET,Y, & P~jiO,MJrnOrnQ,REY) IF(KNTTSL .GT.MAXTSL'NPWLD)THEN TT=TT"Wrn ENDIF EhTDIF C--Wnte statements to BCOPEN.OUT IF(PRESLV .AND. KNTCFL .EQ.1) THEN IF(LSS) THEN WRITE(50,10)SETNAM&O,UAVE,RC,Y,VEL ELSE LOCATION main1n.F f1uids.F 0uidb.F + facflaF -3 bc0pen.F + radius.F DESCPTION This routine computes the radius of the face center of the velocity specified openhg faces. The oripin of coordinate system is with respect to the whole geometry. Routine RADCEN will calculate the center radius with respect to the face origin. .WZL - Local nodal numben; input NODTD - Totai number of nodes; input FACE -Numberofcurrent face; input NNODE - Total nurnber of nodes for local element; input GNNL - Global nodal number for local element; input RC* - Radius of face center, output XC* - x-coordinate for center of face; output YC - y-coorhate for center of face; output NOT - local node number NOT on face; output C Subroutine Arguments C Used for loop Counter C INTEGER NN,NOT,FACE Used for calcuiation face center REAL RC,GNNL(NNODED),XN,YN,ZN,XT,YT,ZT,XC,YC, & XYU(3,NNODED) LOGICAL PRESLV C Get nodes on face and radius of center C Set X & Y totals to zero (used cd. face center) m=O.O YN=O .O m4.0 DO 11 i NN=l,NNODE F(NN .EQ.NOT) GOTO 11 1 XN = XYZL(1JN) YN = XYZL(2,NN) ZN = XYZL(3,NN) C RADCEN CS************* C LOCATION: C mainh.F fluids.F * fluidb.F + facfluI i bc0pen.F + radius.F C DESCRIPTION C This is used to calculate the radius w.r.t to the center of the face so that a velocity C can be caiucalted. C C UAVE* - Average velocity; output C RO* - Outer radius; output; output C RC* - Radius of face center; output (modified) P LOGICAL PRESLV,INLET C EXECUTABLE STATEMENTS C-- _U___ C--set logic if inlet face IF (SETNAM .EQ.'in-corn') THEN INLET = .T'RUE. m=1 ELSEIF(SETNAM .EQ.'out-ext')THEN MET= .FALSE. NUMB=2 ENDIF C-Calculate RO and recdculate RC if required IF (INLET) THEN RO = RMAX(NUMB) ELSE XCEN = OS*(RMAX(NUMB)+ RO = O.~*(RMAX(N'WMB)- RMIN(NLIMB)) XC = ABS(XC) - ABS(XCEN) MOD=' ' IF (RC ET.RO) THEN RC = RO MOD ='M' EMDE C--Calculate average velocity fiom Reynolds number for specimg profile on ECA C IF(INLET)THEN C UAVE = RED*NU/(2.0*RO) C MASSIN = RHO*UAVE*PI*RO**2 C ELSE C MASSOUT = PERCENT*MASSIN C UAVE = MASSOUT/(RHO*PI*RO**2) C ENDIF C c ---- Exit. --- C RETURN END C STEADY C********S**** C LOCATION: C m&.F fluids-F 3 fluidb.F * fac8u.F i bc0pen.F + steady.F C C DESCRIPTION C This routine calculates the normal velocity for the inlet conditions. To C calulate this velocity the calculations are based on the inputted radius of C the center of the face. C Inputs: C RC :centerradiusofcu~~entface C STW : current time C TO : total time C IOP 1 : output parameter to print hard coded properties C IOP2 : output parameter to print calculated values C Outputs: C VEL : normal velocity for current radius C Local: C RO : outer radius C Y : dimensionless radius C NU : kinematic viscosity C C LOCAL VPLRBLAE3LES c ------.-- LOGICAL NET c--- C EXECUTABLE STATEMENTS C--Dimensionless parameter FWET)THEN UAVE = REY*NU/(2.0*RO) RIN=RO VEL = 2.O*UAVE*(l .O-Y**2) ELSE VEL = 2.0*UAVE*(l .O-Y**-) WRITE(*,*) UAVEJUXJL0,Y VEL = PERCENT*VEL*R[N**2/R0**2 END Cf********** C WOMER C*********** LOCATION: rnainln.F + fiuids.F + fluidb.F + facf1u.F i bc0pen.F i radius.F DESCRIPTION: ïhis routine calculates the normai veIocity for the inlet conditions. To caldate this velocity the calculations are based on the inputted radius of the center of the face. Inputs: RC : center radius of curent face RO : outer radius TT : nondimensional tirne scale (STIME/TMAX) RH0 : density hV : kinematic viscosity VN : normai velocity for curent radius ALPHA :womersley number Local: Y :dimensionlessradius C : anay of cosine coefficients for Fourier series S :arrayofsinecoefficientsforFouriersenes NC : nurnber of Fourier coefficients OMEGA : circuiar kequency FREQ : kequency in Hz C LOCAL VARBIABLES c----- REAL VNyTT&~,Y,C,S&PHA,OMEG~QN2I REAL RCJOJ2ERCENTyREY COMPLEX Z~,ZBES,ZT,ZFlO,ZUBAR LOGICAL XNLET C LOCAL PAUMETERS and DATA C--Coefficients for pulsatile waveform DATA NC,C,S/13,1,O. 179325,-0.385872,-0.4508 1 1 ,-O.106449, & 0.295767,O.19463 1 ,-0.0336471,-0.0444237,-0.0253872, & 0.0242532,0.0197074,0.00928437,0.00165714, & 0,0.398146,0.5 1 1875,-0.329866,-0.263848,-0.31723 5, & O. 177449,0.096408 1 ,0.0253163,-0.0337 126, & -0.0221 089,0.00370957,0.0083496,0.0 120435/ constants C Note: times area by rn se AREAT is hdf of the actual area C Calculate wornersley number OMEGA = 2*PI*FREQ ALPHA = RO*SQRT(OMEGW C Normalize radius C ------Begin womersley calculation C Loop over cos and sin coefficients C Function statement to calculate bessel fùnction c ------ COMPLEX mcnoN ZBES(N,Y) COMPLEX Z,URG,Y mGERI,N ZARG = -0.25*YSY z = 1.0 ZBES = 1.O DO 10 I=1,10000 Z=Z*ZARG/V(I+N) IF (CABS(Z).LE.1 .E-20)GOTO 20 ZBES=ZBES+Z 10 corn 20 ZBES = ZBES*(OS*Y)**N END Appendix A?: Flow chart for Two-pressure boundary condition LEGJ34-D inro pressure romne Pliid Old pressure value P,, P,, Yew pressure Value V blass error (pg 44) E Exponent (Eq.3.12) P Relaxation panrneter Run with ' uiitial values \ 1 cdculrite initial l vririabIrs1 Ianci E for Eq 3.13. Xsxt page I Next page 4 /' /' \ Set .Li, to ' minimum value. \ J no be tort. roiaxation. r f Set P for P,,, 0.0 + Set p to aip sien 1 L L-r'Set p for P,, > 0.0 'T' l-7Cpdate Puid= P,, out of pressure routine 1