MultiscaleBiomechanicalModelingof ArterialNetworks By HamidrezaGharahi ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof MechanicalEngineering-DoctorofPhilosophy 2019 ABSTRACT MULTISCALEBIOMECHANICALMODELINGOFARTERIAL NETWORKS By HamidrezaGharahi Cardiovasculardiseasesaretheleadingcauseofdeathallaroundtheworld.Withthe expansionofourunderstandinginbiomedicalsciences,avarietyoffactorsassociatedwiththe onsetandprogressionofsuchdiseaseshavebeenidenInparticular,mechanicalstresses suchaswallshearstressandcircumferentialstresshavebeenproventobeprimaryfactors forthemechanobilogy,andtheirhomeostaticconditionsareregardedasabridgebetween biomechanicsandcardiovascularbiology.Thestudyofvasculargrowthandremodeling (G&R)isathatexploitscomputationalmodelingtostudythechangesinmechanical structureandfunctionofbloodvesselsinresponsetoalteredstimuli.Duringthepast decade,vascularG&Rmodelinghasmadetcontributionstotheofbiomedical engineeringthroughallareasofcardiovascularresearch.However,thepreviousmodeling hasmostlybeendevotedtoarteries,andfewstudiesdevelopedvascularG&Rmodelsof themicrovasculature.Additionally,otherremainingtasksforthemodelinginclude:1) consolidationoftphysicalmodelsandtakingintoaccounttheirinteractions(e.g., teractions,wth)andmultiscalelevelsinspaceandtimeand2) realizationofthemodelingfortheclinicalpractice. Tothisend,wedevelopedanovelcomputationalframeworkthatincorporatesand biosolidmechanicsofarterialnetworksinphysiologicalconditionsandexpandedittomodel tvascularadaptationprocesses.Thisframeworkintegratedessentialfeaturesfrom aconstrainedmixturemodelofG&RandbloodcirculationwithanextensionofMurray's lawtoconstructaspatiallymultiscalevasculartree.Weformulatedtheframeworkasacost optimizationproblemwherethedesignofthevasculaturewasgovernedbyminimizationof themetabolicdissipationundermechanicalequilibriumasaconstraint.Subsequently,we presentedtwoimplementationsofthemodeltostudytwomultiscaleproblems:pulmonary arterialhypertension(PAH)andcoronarywregulation. InthecaseofPAH,weusedtheframeworktoestimatethehomeostaticcharacteristicsof thearterialtreeaswellastheirhemodynamics.Theresultsshowedgoodagreementwith theavailableexperimentaldatainthepulmonaryarterialvasculature.Furthermore,weused Womersley'sanalyticalsolutioncombinedwiththetheoryofsmall-on-largeinelasticity tosimulatethepulsatilehemodynamicsinthepulmonaryarterialtree.Thisstudylaysthe groundworkforfurthertemporallymultiscalestudiesofPAHwherelong-termG&Rinthe vasculature(daystoweeks)arecoupledwithshort-termhemodynamics(cardiaccycle)ina wthmodeling(FSG)framework. Inthecaseofcoronarynetwork,thebaselinepropertiesoftwomyocardialarterialtreesdistal toleftanteriordescendingcoronaryarterywereestablishedusingthepresentedmethod. Consequently,threetcoronarywregulationmechanismsw-induced,myogenic, andmetabolic)wereimplementedusingtheconstrainedmixturemodelsofsmallarteriesand arterioles.Themodelwasthencalibratedagainsttheexperimentalautoregulatorypressure- wrelations.Moreover,thepredictioncapabilityofthemodelwasevaluatedbysimulations ofexogenousadenosineinfusionandinhibitionofnitricoxidesynthesis. Inclosing,thedevelopedframeworkexhibitedgreatpromiseforapplicationsinthestudy ofvascularadaptationsinphysiologicalandpathophysiologicalconditions.Particularly, afterthehomeostaticbaselineofanarterialtreeisestablished,thekineticsofproduction andremovalofconstituentsfromstress-mediatedG&Rmodelscanbeusedtosimulate theshort-andlong-termevolutionofvasculartissuesindiseaseconditions.Furthermore, thisresearchwillsetthecornerstoneformuchneeded in-silico experimentsonpalliativeor curativemanagementsofvasculardiseases. ToMaadar&Baba, Fortheirunconditionalloveandnever-ending iv ACKNOWLEDGMENTS Firstandforemost,IwouldliketoexpressmysinceregratitudetoProfessorSeungikBaek, myPh.D.adviserandmentor,forhiscontinuoussupportovertheyears.Ihavelearned somuchfromProfessorBaekbecauseofhispatienceindiscussions,hispassionforsolving problems,andhisintegrityinconductingresearch. Iamalsogratefultomycommitteemembers:ProfessorWilliamJackson,ProfessorLik- ChuanLee,andProfessorSaraRoccabiancaforservingonmycommitteeandfortheir insightfulrecommendations. Furthermore,IwouldliketoacknowledgeourcollaboratorsfromUniversityofMichigan, ProfessorC.AlbertoFigueroaandDr.VasilinaFilonova,fortheirhelpduringthelast twoyearsofmyPh.Dstudies.IamparticularlythankfultoDr.Filonovaforherhelpin conceptualizinganddevelopingthecomputationalmodel. Iwouldalsoliketoacknowledgemycolleagues,formerandcurrentstudentsatCardiovascular andTissueMechanicsLaboratory:Dr.ByronZambrano,Dr.HailuGetachew,Zhenxiang Jiang,JosueNataren,andothers.Particularly,IamgratefultoDr.ByronZambranoand Zhenxiangfortheirhelpduringtheyearsofmygraduatestudies. Finally,Iowethegreatestdebtofgratitudetomyfamily-myextendedfamily,myin-laws, myparents,mybrother,Alireza,andmywife,Atefeh-fortheirloveandsupportthroughout theyears.Myparentsaremyteacherswhotaughtmelessonsinlifeandtheir hasbeenalwaysinspirationalforme.AlirezaismyveryscienmentorandIkeep learningfromhimeveryday.And,Atefehismyrockandherpresencegotmetothe line. v TABLEOFCONTENTS LISTOFTABLES..................................viii LISTOFFIGURES.................................ix Chapter1 Introduction......................................1 1.1Motivation.....................................1 1.2Background....................................4 1.2.1Pulmonaryarterialhypertension.....................4 1.2.2Coronarywregulation.........................13 1.3Spaimsandorganizationofthedissertation...............22 Chapter2 ATheoreticalFrameworkforEstimatingHomeostasisinVascularTrees..24 2.1Introduction....................................24 2.2Methods......................................27 2.2.1Generalwwandinitialization...................27 2.2.2Arterialmechanicsofasinglesegment.................30 2.2.3Metaboliccostofasinglesegment....................34 2.2.4Costoptimizationofasinglesegmentintermsofradius.......35 2.2.5Iterativeprocessofoptimization.....................36 2.3Summaryandconclusion.............................37 Chapter3 HomeostaticBaselineandHemodynamicsinthePulmonaryArterialTree.38 3.1Introduction....................................38 3.2Method......................................40 3.2.1Fluid-solid-growth............................40 3.2.2Pulsatilehemodynamics.........................41 3.2.3Parameterselection............................43 3.3Resultsanddiscussion..............................46 3.3.1Symmetrictree..............................46 3.3.2Asymmetrictree.............................54 3.4Summaryandconclusion.............................57 Chapter4 BaselineCharacteristicsandAdaptationsinCoronaryFlowRegulation..60 4.1Introduction....................................60 4.2Methods......................................62 4.2.1Baselineconstructionofcoronaryarterialtree.............62 4.2.2Coronarywregulation.........................63 4.2.3Modelparameters.............................66 4.3Resultsanddiscussion..............................71 vi 4.3.1Baselineoptimization...........................71 4.3.2Autoregulation..............................74 4.3.3ofadenosineandNOinhibitors.................79 4.4Summaryandconclusion.............................80 Chapter5 ConclusionandFutureWork............................83 5.1Conclusion.....................................83 5.2Futurework....................................85 APPENDICES....................................88 APPENDIXA:Formulationofsmallonlargetheory................89 APPENDIXB:OptimizationandSteadystatehemodynamicsofthearterialtree106 APPENDIXC:Axisymmetricsolutiontopulsatilehemodynamicsinonevessel..109 APPENDIXD:Recursivealgorithms:fast-timehemodynamics...........112 BIBLIOGRAPHY..................................114 vii LISTOFTABLES Table3.1Modelparametersforhomeostaticoptimizationandpulsatilehemody- namics......................................45 Table4.1Modelparametersforthehomeostaticoptimization..........68 Table4.2Estimatedconstitutiveparametersfortheconstrainedmixturemodel ofthearteries...................................71 Table4.3Estimatedcontrolparametersfortheautoregulatoryresponse......75 viii LISTOFFIGURES Figure1.1Schematicdiagramofthemodelingframework.Thestructureofthe arterialtreeisdbyabifurcationrule.Everyindividualvesselisendowed withaconstrainedmixturemodelwhichiscapableofextensiontoincludethe underlyingbiochemechanicalprocessesofvascularadaptation.........3 Figure1.2Themedialthickness(above)andtheratioofpulmonaryandaortic medialthickness(bottom)withage........................8 Figure1.3a)Male,7yearsold.b)Female,47yearsold.c)Male,78yearsold. Photographsat 192Arrowheadsshowthelimitsofmedia andblackareaareelastin.............................9 Figure1.4Intimal,medial,andadventitialthickness(percentofexternaldiame- ter)ofpulmonaryarteriesofvesselsofvarioussizes.SolidbarsrepresentPAH patientsandopenbarsrepresentcontrolsubjects................10 Figure1.5Opticalmicroscopyimagesofpulmonarytrunk(a)controland(b) hypertensivecalves.Collagenandelastinappearredandblack.Righttoleft directionisfromintimatoadventitia.Qualitatively,collagendepositioncan bedelineatedintheadventitia..........................11 Figure1.6Normalrelationshipbetweencoronarybloodwandcoronaryvenous PO 2 ,whichisanindexofmyocardialtissueoxygenation............16 Figure1.7Schematicdiagramoftheheterogeneousregulatoryresponseofvessels inresponsetopressure,shear-dependent,metabolicmechanisms........17 Figure1.8Wholeorganpressurewrelationships.................19 Figure2.1Informationexchangeforthehomeostaticoptimizationappliedforthe treeinthenestedmanner:fromthetreetobifurcationandthentoindividual vessels........................................28 Figure3.1Schematiccomponentsofthepulmonaryarterialtreeandtheircorre- spondingdataavailability:largevessels(3Dpatients-spanatomy,w andpressurewaveformsfromclinicalimagingandcatheterization)andsmall vessels(limitedmorphometricalandbiomechanicaldata)............40 Figure3.2Symmetrictree{homeostaticoptimizationresultsplottedversusgen- erationnumber:Left:diameterdistributioncomparedtoreporteddataof largervessels;Right:radiusexponentindaughter-to-parentradiirelation;.47 Figure3.3Symmetrictree{homeostaticoptimizationresultsplottedversusdiam- eters:Top-left:wallthickness-to-diameterratio(h:wallthickness);Top-right: mid-arterypressure;Bottom-left:homeostaticvalueofwallshearstress( ˝ ); Bottom-right:homeostaticvalueofcircumferentialstress ˙ h ..........48 ix Figure3.4Prescribedvariablemassfractionsofthewallconstituents:elastin, smoothmusclecellsandcollagen.Thearrowsonthetopshowthetrendin arterialcomposition................................49 Figure3.5Symmetrictree{Left:radiusexponentindaughter-to-parentradii relation;Right:homeostaticvalueofwallshearstress..............49 Figure3.6Symmetrictree{homeostaticoptimizationresultsforthewall versusgenerationnumber:Left:Young'smodulesincircumferentialandlon- gitudinaldirections;Right:Structuralnormalizedbytheunstressed radiusandcomparedtodistensibilityrelationsfromdata...........50 Figure3.7Symmetrictree{daughter-to-parentarearatiowithintherangeof open-endtypeof..........................52 Figure3.8Symmetrictree{pulsatilehemodynamicsresultsversusgeneration number:(a)pulsewavevelocityfortwoharmonics,comparedtodataand MKvelocity;(b)Womersley'snumberfortwoharmonics............52 Figure3.9Symmetrictree{Womersley'ssolutionvaliditycheckasfunctionof generationnumberfortwotharmonicmodes..............53 Figure3.10Symmetrictree:Totalinputw(top)andterminalpressure(bottom) overthegenerations................................53 Figure3.11Symmetrictree{Top:totalwandpressureattherootvesselofthe distaltree;Bottom-left:rootvesselinputimpedanceintimedomaincompared toterminalandcharacteristicimpedance;Bottom-right:inputimpedance modulusandphaseanglesinthefrequencydomain...............55 Figure3.12Asymmetrictree{homeostaticoptimizationresultsversusdistance (fromtheroot,alongthebranchpathway),dotsrepresentsoptimizationre- sultsforeachvessel,redandbluelinesindicatetheshortandlongpath, respectively:(a)logofsteady-statewrate;(b)terminalsteadypressure; (c)ratioofstructuraltounstressedradius;(d)homeostaticvaluewall shearstress.....................................56 Figure3.13Asymmetrictree{deltaparameterandpulsewavevelocityforlongest andshortestpaths.................................57 Figure3.14Symmetrictree{Top:totalwandpressureattherootvesselofthe distaltree;Bottom-left:rootvesselinputimpedanceintimedomaincompared toterminalandcharacteristicimpedance;Bottom-right:inputimpedance modulusandphaseanglesinthefrequencydomain...............58 Figure4.1Theenessofeachregulationmechanisms.............67 x Figure4.2Schematicdiagramillustratingthewwofthesimulations,with apossibleextensiontoaclosed-loopmodel.Theindependentinputsarethe coronarypressure( p in )andMVO 2 .Theconvergencetheequilibrium stateofthewregulation.Thefeedbackmechanism(notincludedinthe currentstudy)relatesthelocalwregulationincoronariestothecardiac function;heartrate,cardiacoutput,etc.....................68 Figure4.3Solidcirclesshowthepassiveresponseofthevessels.Opencircles showthefullyactivatedresponseofthearteries.Reddotsarethecomputed responsefromparameters.( R 0 :theradiusatzeropressure)......70 Figure4.4Estimatedmassfractionsforthearteriesandarteriolesoft layers.(A:Smallarteries,LA:Large,IA:Intermediate,SA:Smallarterioles.)71 Figure4.5Thebaselineoptimizationresultsplottedalongwiththegeneration number:Left:diameterdistribution;Right:radiusexponentindaughter-to- parentradiirelation( R ˘ p = R ˘ d 1 + R ˘ d 2 )......................72 Figure4.6Homeostaticoptimizationresultsplottedagainstdiameters:Top-left: wallthickness-to-diameterratioinunloadedTop-right:mid- arterypressure;Bottom-left:homeostaticvalueofwallshearstress( ˝ );Bottom- right:homeostaticvalueofcircumferentialstress ˙ h ...............74 Figure4.7wautoregulationinthecombinedtreescomparedtothe experimentaldata.................................75 Figure4.8Thepredicteddependenceofwallshearstressandactivationonthe meaninletpressure.Wallshearstressineachvesselisnormalized( ˝ )against itsvalueat p in =100mmHg...........................76 Figure4.9Theofpressurereductiononcoronaryarterialmicrovesselscom- paredtotheexperimentalobservationsonepicardialmicrovasculatureindogs.77 Figure4.10Diametersasfunctionsof p in fortvesselsofsubendocardial andsubepicardiallayers..............................78 Figure4.11Transmuraldistributionofthewduringwautoregula- tion.Theinsetshowstheobservationsindogs..................79 Figure4.12TheofadenosineinfusionandNOinhibitiononcoronaryar- terialmicrovesselscomparedtotheexperimentalobservationsonepicardial microvasculatureindogs..............................80 FigureA.1Pressureversusradiusduringatalength.Thelinearized elasticparameterswerecalculatedat90mmHg.................103 FigureA.2Linearizedparametersfortdegreesofsmoothmuscletone...104 FigureB.3Schematicofthebifurcationpoint[ k;s ]inthearterialtree.......108 xi Chapter1 Introduction 1.1 Motivation Cardiovasculardiseases(CVDs)aretheleadingcauseofdeathgloballywithanannualdeath rateofaround17.9million[1]andtheycostofalmost$1trillioninhealthcareexpenses [2].Broadly,manyCVDsareoftencharacterizedbychronicstructuralalterations,such asnarrowing,thickening,andofthearteries.Consequently,theimportanceof vascularbiomechanicsintheprogressionofCVDshasbeenextensivelystudiedoverthelast 50years.Motivatedbythebiomechanicalmarkers,vascularcomputationalmodelshave beenemployedascriticaltoolsinenhancingourknowledgeofthefundamentalprocesses involvedintheonsetandprogressionofthedisease. Animportantadvancementinsuchcomputationalmodelswastheintroductionofacon- strainedmixturemodelofarterialgrowthandremodeling(G&R)[3].Themodelingframe- workofG&R,anditslaterextensionstoincludebiochemomechanical[4]and- growth(FSG)models[5],provideapowerfultooltotestvarioushypothesesinthecardio- vasculardiseaseresearch.ThemainfocusofthepreviousapplicationsofG&Rhasbeen 1 tostudythevascularadaptationsinlargearteries.However,littleattentionhasbeenpaid tothemodelingofG&Rinarterialnetworksinthepresenceofhemodynamics.Sincethe formationandprogressionofmanyCVDsaretracedtotheinitialmalfunctionsinthedistal vasculature,thereisacrucialneedtoexpandtheG&Rframeworktocapturethemultiscale arterialnetworks. Motivatedbythisneed,theultimategoalofthecurrentstudyistoprovideamodeling frameworkthatembedstheessentialfeaturesofG&Rcoupledwiththehemodynamicswithin anarterialnetworkcharacterizedbybifurcationrules(Fig1.1).Anaccuratemultiscale modelingofG&Rrequiresthereconstructionofthearterialnetworksaswellasmechanical considerationsofindividualvessels. Thecardiovascularsystemmaintainsastateofmechanicalhomeostasisinthephysiological conditions.Theconceptofthemechanicalhomeostasis,viaquantitiessuchasstressesand/or strains,isessentialinunderstandingthebiomechanicsoflong-termresponses(i.e.,G&R). Alternatively,inthestudiesontheoptimaldesignofvascularsystems,initiatedbyMurray in1926[7],the in-vivo structureofanarterialtreehasbeenattributedtoaminimum energydissipationhypothesis(i.e.,Murray'slaw).Eachofthesetheoreticalideas,G&R andMurray'slaw,mayencapsulateinconsistentorevencontradictorynswhen comparedtoexperimentalobservations.Therefore,thereisaneedtorigorouslyevaluate theseconceptswithrespecttotheavailableexperimentaldataintheliterature.Wewill reviewthepreviousworkonG&RandgeneralizationsofMurray'slawandhighlight theshortcomingsoftheexistingframeworksinSection2.1. Inthisdissertation,forthetime,aframeworkispresentedthatintegratesfeatures fromG&RandMurray'slaw,andprovidesatheoreticalmodelwhichcanbeusedformul- tiscalemodelingofarterialnetworksbasedonavailableexperimentaldata.Thepresented frameworkcanbefurtherutilizedforanalysisofdistalarterialstructurewherenon-invasive assessmentofthearteriesisnotreadilyavailable[8].Moreover,aconstrainedmixturemodel 2 Figure1.1:Schematicdiagramofthemodelingframework.Thestructureofthearterial treeisbyabifurcationrule.Everyindividualvesselisendowedwithaconstrained mixturemodelwhichiscapableofextensiontoincludetheunderlyingbiochemechanical processesofvascularadaptation.Theinsetsareadaptedfrom[5]and[6]. ofthearterialsegmentsfacilitatesthe in-silico studyofintroducingtherapeuticmediators intothevascularnetwork.Therefore,weemployourmodeltostudytwoprominentlymul- tiscaleproblemsincardiovascularresearch:pulmonaryarterialhypertensionandcoronary wregulation. Inthenextsectionsoftheintroduction,wereviewcurrentliteratureonpulmonary arteriesinnormalandhypertensivestates(Section1.2.1).Next,areviewoftheexisting publicationsoncoronaryarteriesandtheirfunctioninsupplyingbloodtothemyocardium ispresented(Section1.2.2).Thestructureofthedissertationandspaimsofthisstudy areoutlinedattheofthischapter(Section1.3). 3 1.2 Background 1.2.1 Pulmonaryarterialhypertension Pulmonaryhypertension(PH)isacomplexdisorderassociatedwithanelevatedpulmonary arterialpressure(PAP).TheprevalenceofPH,whichstrikeswomentwiceasfrequentlyas men[9],isfairlyrarewith15incidentsoutofamillionpeople[10].DeathrateofPH, however,havebeensteadyand/orincreasingbetween1999to2008witharound5.6and5.7 deathsper100,000populationformenandwomenrespectively[11].Moreover,unlikethe systemichypertension,PHprognosisremainspoorwith > 10%mortalitywithin1yearfor forhighriskpatients[12].Ingeneral,oncePHpatientsarediagnosed,theprogressionof PHisnotreversible.PrimaryPHpatientsaretreatedpharmacologicallywithvasodilators andinotropes[13].Thepatientpopulationiscloselymonitoredusingcardiacmagneticreso- nanceimaging(CMR)andcardiaccatheterization.EarlydetectionofPHishowever becausethestandarddiagnosisisbasedonusinginvasivecatheterization,whichmightbe alreadytoolateforclinicalintervention. Inahealthyindividual,thebloodpressureinthepulmonaryarteriesisabout25/10mmHg whichisaroundsixtimeslessthanthatofthesystemicarterialpressure.Basedonclinical PHispresentwhenthearterialpressureexceedsabout40/20mmHgorthe averagepressurerisesabove25mmHg[14].Pulmonaryarterialhypertension(PAH),previ- ouslyknownasprimaryPH,istheofthevePHgroupsaccordingtotheNICEclinical [15].PAHincludesidiopathic,heritable,drug-inducedPH,orhypertensiondue tohypoxiainhighaltitude,inadditiontootherpathologicalconditionssuchascongenital heartdisease.Moreover,PAHcanbethecausetogravecomplicationssuchasright-sided heartenlargementandheartfailure(corpulmonale),bloodclots,arrhythmia,andbleeding. TheusualcauseofdeathinPAHisrightventricle(RV)failure[16]whichisrelatedtoRV 4 hypertrophyduetochronicpressureoverloadwhereeventsduringisovolumeicrelaxation aredisorganized[17].Finally,rightheartcatheterizationwiththemeasurementsofPAP, cardiacoutput(CO),andpulmonaryarterialwedgepressureremainthegoldstandardfor thediagnosisofPAH[14].Besidestheextremeinvasivenessofthisprocedure,rightheart catheterizationyieldpossiblerisksandcomplicationssuchasexcessivebleedingduetovein punctureduringcatheterinsertion,andinfection. 1.2.1.1 Morphometryofpulmonaryarteries Pulmonaryarteriesenterthelungthroughthecenterofthelobefromthehilusand,ac- companiedbytheirpairedairways(especiallyinsmallerarteries[18]),navigatetheirwayto thepleuralsurface[19].Themorphometryofthehumanpulmonaryvasculaturehasbeen studiedtoprovidemoreanatomicalinsightintothehemodynamics,geometryandbranch- ingofthepulmonaryarterialtree.Singhaletal.[20]presentedoneoftheearlierstudies onthemorphometricanalysisofthehumanpulmonaryvasculature.Theyusedresincasts andinjectionsofhumanvasculartreestomeasurethediameter,length,andorderofall branchesofbloodvesselsintherangeof13 to3 cm forarterialvessels.Theyreported anarterialtreecomprisedof17branchorderswithanestimatedtotalnumberof3 10 8 vesselsoforderoneinStrahler'sorderingsystem[21]withdiameterof13 .Moreover, theyshowedarounda60-foldincreaseinthetotalcrosssectionalareafromthepulmonary trunktothearteriolarlevel.YenandSobin[22]studiedtheelasticityandbranchingorder ofnon-capillarypulmonarybloodvesselsofthesize1785 tolessthan100 diameter. Theyobservedaslightdecreaseinthecompliance,expressedbypercentageofchangeindi- ameter,ofthearteriolesasbloodvesselsbecomesmaller.UsingStrahler'sorderingmethod, theyobservedthatarteriesofthesameorder(spbetween4-8)havethesamesize (87 to1785 ),regardlessoftheirlocationinthelung(upper,middle,orlowerlobe). 5 Huangetal.[23]publishedanextensivestudyonthemorphometryofthepulmonaryarteries. Theythepulmonaryarterialnetworkinto15ordersemploying Strahlerorderingsystem[24,25].Usingsiliconeelastomercastingtechnique,theymeasured thediameterandlengthofthepulmonaryarteriesoftwopostmortemhumansubjects.They presentedtheconnectivitymatrixwhichexpresseshowbloodvesselsofoneorderarecon- nectedtovesselsofanotherorder.Inthisstudy,thediametersoforder1haveanaverage diameterof20 increasingtoanaverageof1.4 cm (order15).Severalhemodynamic studieshaveusedtheworkbyHuangandcolleaguesasabaselineforgenerationofthe downstreamarterialtree[26,27]. 1.2.1.2 Structureofpulmonaryarteries Thepulmonaryarterialwallusuallyconsistsofthreedistinctlayersoftunicaintima,tunica media,andtunicaadventitia.Theintimaistheinnermostlayerofthearteryandconsists ofalayerofendothelialcells,asubendotheliallayerofconnectivetissue,andtheinternal elasticlamina.Themedia,whichisthemiddlelayer,consistsofconcentricallyarranged smoothmusclecells,collagenersandelastin.Theadventitia,outermostlayer,ofpul- monaryarteriesisfairlydisorganized[28]andiscomprisedofextra-cellularmatrix(ECM), orotherinterstitialcells,nerves,elastinandcollagenerbundles. Thecompositionofthevesselwall,however,tlyvariesfromtheproximalpulmonary trunktothesmallestarterioles[28].HislopandReid's[29]histologicalstudyonratsreported thatthemainpulmonaryarteryhadamuscularmediabetweeninternalandexternalelastic laminaewithfourconcentricelasticlaminainbetween.However,whilethemedialstructure oftheleftandrightpulmonaryarteriesremainedthesame,fewerlayersofelasticlaminae werepresentrelativetopulmonarytrunk.Thesmallerthevesselinthepulmonaryvascula- ture,themoremediabecameclearlydistinguishablebyasingleinternalandexternalelastic 6 lamina.Moreover,insmalldistalpulmonaryarteries,theinternalelasticlaminastarted tofragmentandeventuallydisappearwhichusuallyfacilitatesmyo-endothelialcommunica- tions[30].Asrat'sarterialtreeapproachedthealveolarlevel,thebloodvesselsbecameless muscularuntilthemuscularcoatwasnolongerseenin80%ofarterieslargerthancapil- laries[29,31].Regardless,althoughmusculararteriesinhumanadultsextendasfarasthe alveolarlevel,below1mmdiameter,arteriesinratandhumanaresimilarinstructure,with themediahavingonlyinternalandexternalelasticlamina[29]. PopulationbasedstudieshaveshowntherelationshipbetweenageandanelevatedPAP[32]. Thestudyofthepulmonaryarterialwallwithagecanshedlightonthestructuralbasisof thearterialwallinPAH.Heathetal.[33]investigatedthemediallayerofthepulmonary trunkandaortawithageinhumansubjects.Figure1.2showstheevolutionofthepul- monarytrunk'smediallayeragainstagefor71humansubjectswithoutanycardiovascular disease. Heathetal.[33]observedthatthemedialthicknessandstructuralthepul- monarytrunkandtheaortaarefairlysimilarforafetalsubjectoranewborn.However,as humansage,thepulmonarytrunktransitionstoamoredistinctivestructurewhereelastic ismoreopenandloose.Furthermore,themedialthicknessreducesbetween 40%and70%thatofaorta'smedialthicknessin6to24months.Inadults'pulmonarytrunk, theelasticlaminaeremainedmoreirregularandsparserthantheaorta. MacKayetal.[34]studiedthestructuralpropertiesofthepulmonaryvesselswithage. Theirresultsdemonstratedthattherewasanalmoststeadydecreaseinthecomplianceof pulmonaryarterywithincreasingage.Thechangesintheofthepulmonaryarteries canbetrackedqualitativelyonhistologicalimages.Histologyofa7yearoldmale(Fig.1.3- (a))showsthatthearterialelastinhadnearlystraightlamina.However,correspondinglyto thehistologyimagesin[33],thestructuralistfromthatofaortawith moredispersedelasticbers.Nevertheless,theelasticlaminabecomeswavierandfurther apart(Fig.1.3-(b)).Intheolderage,theelasticlaminacontinuestospreadandbescant, 7 Figure1.2:Themedialthickness(above)andtheratioofpulmonaryandaorticmedial thickness(bottom)withage,reproducedfromdataavailablein[33].ThelettersA,T,anPon thethicknessplotrepresenttheaortic,transitional,andpulmonarystructural ofthetrunk,respectively. whereastheintimathickensandconsistsofasmallamountofcollagen Whiletheelasticlaminaservesthefunctionofelasticrecoilofthearterialwall,theadventi- tialnetworkofcollagenerspreventsover-extensionoftheartery.Tothispoint,MacKay etal.[34]studied42arterialspecimensfromhumansaged7to87years,andshoweda1% perdecadereductionintheamountofcollagenashumangetolder. 1.2.1.3 HistopathologyofPAH PAHisassociatedwithtstructuralchangesinthemicrostructureofthearteries. Astudyon58PAHpatients[35]demonstratedsomeofthearterialhistologicalabnormal- itiesinPAH.WhilethromboticlesionsoftenapresentpatterninPAH,complexplexiform 8 Figure1.3:a)Male,7yearsold.b)Female,47yearsold.c)Male,78yearsold.Photographs at 192Arrowheadsshowthelimitsofmediaandblackareaareelastin. Picturesadaptedfrom[34]. lesions 1 andlaminarintimalwerecollectivelypresentinalmosthalfofthecases. Alternatively,laminarintimalwhilelesscommoninPAH,leadsunstabledeposition anddegradationofcollagenandelastin[40]. Undoubtedly,severaldistincthistopathologicpatternsofPAHexist.However,layer-wise remodelingseemstopredominantlyoccurindiseasedarteries.Chazovaetal.[41]observed 1 PlexiformlesionsareoneformofdilatationlesionwhichgrowinthelatestagesofPAH[36].Theyare usuallyassociatedwithfocalhypertrophyandproliferationofvascularsmoothmuscle,endothelialhyperpla- sia,andover-expressionofmatrixmetalloproteinases(MMP)whichleadtoenzymaticdegradationofECM proteins[37{39]. 9 Figure1.4:Intimal,medial,andadventitialthickness(percentofexternaldiameter)of pulmonaryarteriesofvesselsofvarioussizes.SolidbarsrepresentPAHpatientsandopen barsrepresentcontrolsubjects.Picturesadaptedfrom[41]. that,inadditiontointimalandmedialthickeningandremodeling,adventitialthickeningand remodelingareconsistentfeaturesofPAHvasculature(Fig.1.4).Particularly,theincrease intheadventitialthicknessisattributedtotheincreasedcollagendeposition.Figure1.5 reinforcesthisobservationinhypoxia-inducedPAHcalves[42].Ontheotherhand,medial thicknessisbysmoothmusclecellhypertrophythatinvolvesbothmusculararteries (70-500 diameter)andsmallerarterioles(smallerthan70 )[43]. ThepulmonaryvascularremodelingcharacteristicsinPAHcanbesummarizedasmedial hypertrophy,intimalproliferation,arteriolarmuscularization,andadventitialthickening[43, 44].Nevertheless,thepathophysiologyofPAHremainswidelyunclear[45,46].Although endothelialinjuryandexcessivevasoconstrictionwhichleadtoadecreaseintheinternal diameter(i.e.,increaseinvascularresistance),seemtobeimportantelementsinpathogenesis ofPAH[41,43,47,48],multiplestudieshypothesizedthatincreasedelastolyticactivity[37,49, 50]andcollagendeposition[42,51]contributetothepathogenesisofPAHthroughremodeling 10 Figure1.5:Opticalmicroscopyimagesofpulmonarytrunk(a)controland(b)hypertensive calves.Collagenandelastinappearredandblack.Righttoleftdirectionisfromintimato adventitia.Qualitatively,collagendepositioncanbedelineatedintheadventitia.Addition- ally,fragmentationanddispersionofelastinisclearinthepictureastheblackareasof(a) aremuchmorecondensedandstraightrelativeto(b).Picturesadaptedfrom[42]. oftheECM(i.e.,increasedss). 1.2.1.4 ofpulsatilehemodynamics AlthoughanelevatedmeanPAPisessentialindiagnosisofPAH,thecapabilityofmean PAPinprognosisofPAHhasbeenrelativelyunderperforming[52].AnimalmodelsofPAH showthatthedistalvascularremodelinginPAH,resultsin\earlyreturn"ofpres- surewavesduringRVcontractionandconsequentlytheRVsystolicfunction[53]. Motivatedbysuchanalysis,severalstudies[54{56]haveinvestigatedthewavein PAvasculature.Particularly,Hollanderandcolleagues[55]foundanegativewave incaninePAwhichfacilitatestheRVejectioninphysiologicalcondition.Quailandcowork- ers[56]observedcongruentresultsinhealthyhumansubjects.Conversely,theyfoundthat theincreasedreducedvesselarea,andpersistentvasoconstictioncreated siteswithpositivewaveonswhichresultedincompressivections. Furthermore,pulsepressurehasbeenobservedtoactinconcertwithmeanpressureandshear stressinmodulatingsmoothmusclecellsfunction[57].Particularly,experimentalstudiesof 11 cyclicstretch( ˘ 10Hz)appliedtosmoothmusclecellshasshownincreasedcollagensynthesis anddecreasedNOsynthaseexpressionwhichisavasodilationmediator[58,59].Overall,a pathologicalpulsepressuremayleadtoincreasedcollagendeposition,smoothmusclecells hypertrophy,endothelialdysfunction,etc.whichresultinand thickeningofthebloodvessels[60{66]. 1.2.1.5 Modeling Theofpulsatilehemodynamicsinthepulmonaryarterialnetworkscompels multiscalecomputationalmodelingtoincludetheinteractions(FSI)anal- ysis.Atportionofsuchisfocusedonimage-based3Dmodelscoupled withWindkesselelements(0D)attheoutlets[67].Althoughsuchmodelsarenecessaryfor patient-spstudies,theycannotcapturethesalientfeaturesoftheprogressionofPAH, forinstance,proximaltodistalwavepropagation.Alternatively,fullpulmonaryarterialtree reconstructionscanbeusedtomodelthepulmonarycirculation[27,68].Themainlimita- tionofsuchsimulations,however,isthecomputationalcostwhichrendersafullydetailed analysisimpossible. Tocircumventlimitationsofthe3Dcomputationalmodeling,one-dimensional(1D)mod- elsofthecirculationhavebeendevelopedandextensivelyusedtostudypressureandw rateintapplicationssuchasthewholesystemiccirculation[69],coronaryarter- ies[70,71],cerebralarteries[72],andpulmonaryarteries[73{75].Anappropriateutilization of1Dhemodynamicstheoriesinvascularnetworks,however,requiresaphysiologicallyreal- isticandcompletetreemorphometryofdistalvasculaturewhichischallengingduetodata acquisitionandmodelingcomplexities.Therefore,studiesbyOlufsenandcolleagues[73,74] onpulmonaryarterialnetworkshavefocusedonthefractalstructureofthearterialtree.The choiceofafractalstructure,inspiredbytheairwaystructure[76],isparticularlyattractive sinceitprovidesacomputationallyinexpensivemodelofmultiscalevascularnetworks. 12 Any1DFSImodelingt,however,hastobeendowedwithrealisticestimationofsize- andfunction-dependentmechanicalpropertiesofdownstreamvessels.Duetoin measuringsuchpropertiesofthevasculatureinsmallbloodvessels,thereisacrucialneed toutilizeaframeworkthatcanfacilitatetheparameterestimationbyexploitingall theavailabledata. 1.2.2 Coronarywregulation Coronaryarterydisease(CAD)isthemostcommontypeofCVDswithcausingmore than610,000deathsannuallyintheUnitedStates[77].CADisprimarilycharacterized byatherosclerosisintheepicardialcoronaryarterieswhichcanleadtocomplicationssuchas myocardialinfarction.However,theprevalenceofnon-obstructiveCADhasbeenreportedto beupto30%[78],wheredespitetheabsenceofstenosis,patientsexperienceadversecardio- vascularevents[79].Regardless,coronarymicrovascularabnormalitiesleadingtoischemia andanginahavebeenobservedinbothobstructiveandnon-obstructiveCADpatients.Cur- rentclinicalmanagementsofCADfocusonthetreatmentoftheatheroscleroticplaquein themajorepicardialvessels[80],whereasmyocardialbloodiscontingentonarangeoffac- torsinmicrovasculaturebesidestheseverityofthestenosisinmajorarteries.Therefore, oneofthecurrentchallengesintreatmentofCADisunderstandingthefunctional canceofmicrovascularnetworksinmyocardialperfusion.Acrucialaspectoftheresearch inCADisabetterunderstandingofthestructure-functionrelationsinthewholecoronary circulation[81]. 1.2.2.1 Anatomy,morphometry,andhemodynamicsofcoronaryarteries Thecoronaryarteriesareuniqueinthattheyareresponsiblefordeliveringoxygenandsub- strate,neededforoxidativemyocytemetabolism,tothehearttosupplythepowerrequired 13 forpumpingthebloodtotheentirecirculationsystem.Twomaincoronaryarteriessplit fromtheaortainthesinusesofValsalva.Theleftmaincoronaryarteryisashortvesselthat bifurcatesintoleftanteriordescending(LAD)andleftcir(LCx)arteries.TheLAD ismainlyresponsibleforsupplyingtheanteriorsideofleftventricle(LV)andleftatrium (LA)and2/3oftheseptum[82].Alternatively,theLCxcourseshorizontallyaroundthe heartandsuppliesthelateralsideoftheLVandLA.Rightcoronaryartery(RCA)delivers bloodthetheRVandrightatrium(RA)andtheseptumtoalesserextent[83].These mainvesselsdividetoconduitepicardialvesselsoverthesurfaceoftheheartinto17regions afterreachingasizeof1-3mm[84].Duringthecourseofthesevessels,transmuralbranches atalmostrightanglespenetratethemyocardiumwheretheygiverisetoanetworkofves- sels(i.e.,myocardialvessels)whichspreadthroughthemyocardiumfromsubepicardialto subendocardiallayers. Morphometricanalysisofswinecoronaryarterieshasbeenextensivelystudiedintheworksof Kassabandcolleagues[24,25,85{87].Theyanalyzedthemaincoronarybranchesseparately, andshowed11,11,and10ordersofvesselsStrahler)forLAD,RCA,and LCx,respectively[24].Sp,thevesselsinLADnetworkspanfrom ˘ 3000 to9 .Thisdatahasbeenusedextensivelyforhemodynamicsanalysisinthelasttwodecades. Wewillpresentasummaryofthesecontributionslaterinthischapter(Section1.2.2.4). Thetotalwincoronaryarteriesisdirectlyproportionaltocoronaryperfusionpressureand inverselyproportionaltothevascularresistance,whichresidesinbloodvesselsofsmaller than170 [86].However,adetaileddescriptionofthepulsatilenatureofthecoronary arterialhemodynamics(atrest)requiresconsiderationsoftheotherdeterminantssuchas vascular-myocardialinteractions(extravascularcompressiveforces)anddurationofdiastolic time[88].Inthisdissertation,welimitouranalysistothesteadystatehemodynamicsin thecoronaryarteries.Instead,ourstudyaimsmodelingthevascularadaptationsinthe coronarycirculationwhichoccurinamuchlargertimescalethanthatofacardiaccycle(15 14 seconds-2minutes)[89].Henceforward,thecoronaryhemodynamicsanalysisinthisstudy usesonlytotheaveragesteady-statepropertiesofbloodw. 1.2.2.2 Structurecoronaryartery Similartootherarteries,coronaryarteriesarecomposedofthreedistinctlayersincluding adventitia,media,andintimawhichconsistofcollageners,elastin,andsmoothmuscle cells[90].Chenandcoworkers[91]showedthattheratioofcollagentoelastinintheadven- titiallayerofLADandRCAis1.5and1.1,respectively,whichislikelyduetothesizeofthe vesselsandtheirtmechanicalenvironment.However,mediallayersofthecoronary arteriesaremechanicallythemostimportantlayeratnormalconditions.Mostofthesmooth musclecellsareconcentricallyarrangedinthemediallayerofthearterialwallwithinthick continuoussheetsofelastinandcollageners[92].ItisworthytomentionthatChenet al.[93]observedthatthealignmentofsmoothmusclecellsareslightlydeviatedfromthe circumferentialdirection(18.7 10.9 ).AccordingtoZoumietal.[92],theintimallayerof thecoronaryarteriesinpigsisverythinanddoesnotcontributetothemechanicalproperties ofthearterialwall.Inagedhumancoronaryarteries,Holzapfelandcoworkers[94]reported therelativethicknessofadventitia,media,andintmiaas0.4,0.36,and0.27,respectively. Regardless,Mostofthepreviousmicrostructuralstudiesoncoronaryarterieswerelimited tothelargeepicardialvessels,whilethemechanicalstructureofthecoronaryarteriesand arteriolesaremajordeterminantsoftheseverityofCADandsusceptibilityofthepatientto ischemia[95]. 1.2.2.3 Physiologyofcoronarywregulations Anormalfunctioninghearthasahighmetabolicdemandandisrichinmitochondriawhich generateupto90%oftheenergyneededforpumpingbloodviaoxidativephosphorylation 15 Figure1.6:NormalrelationshipbetweencoronarybloodwandcoronaryvenousPO 2 , whichisanindexofmyocardialtissueoxygenation.Pictureadaptedfrom[82]. [96].Becauseofthislimitedanaerobiccapacity,anyincreaseincardiacactivity(oxygen demand)isimmediatelymetwithanincreaseofavailableoxygenalmostcompletelyviaan increaseinbloodw[97]. Ananalysisoftheoxygendemandinexerciseshowsthatanaugmentedoxygenconsumption rate(MVO 2 )isinpartthedirectresultoftheheartbeatingfaster.Inaddition,arisein MVO 2 canbeattributedtotheelevationofcontractilityandventricularworkduetoactiva- tionofthesympatheticsignals[82,98].Although,aprecisedelineationofeachcontributionis therelativeroleoftheheartratehasbeensuggestedtobebetween40-60%[82,97]. Ontheoxygensupplyside,theleftventricularoxygenextractionfromarterialbloodremains almostconstantwith70-80%ofthewofoxygenabsorbedduringrestandexercise[99]. Thishighlevelofutilizationofoxygenextractioncapacitybythemyocardiumnecessitates theoxygentobesuppliedbyincreasedperfusionproportionallytoMVO 2 .Consistentwith 16 Figure1.7:Schematicdiagramoftheheterogeneousregulatoryresponseofvesselsinresponse topressure,shear-dependent,andmetabolicmechanisms.Pictureadaptedfrom[104]. theseobservations,experimentsonthecoronaryarterialresponsestothechangesincardiac activityhaveshown70%and170%increasesinthebloodwrateduringpacingandexer- cise,respectively[82].Ontheotherhand,therightventricularoxygenextractionis ˘ 45%, whichmeansamoderateriseinMVO 2 ismetwithlittleincreaseinRCAwandmore contributionfromoxygenextraction[100].Figure1.6showsthecoronarybloodwchange inLVandRV.Henceforth,ouranalysisisfocusedonthecoronarywregulationintheLV. Tofacilitateregulationofoxygensupplytomyocytes,vasodilation(orvasoconstriction) occursinthesmallarteryandarteriolarlevelviaextrinsicandintrinsicpathways[101]. Althoughamyriadofcontrolmechanismsthecoronarybloodperfusion[81,97],the primarilytialmechanismsarethemetabolic,myogenic,andshear-dependentcontrols [102,103].Figure1.7showstheheterogeneousregulationofcoronaryvasculartone. AsasympatheticresponsetoanincreaseinMVO 2 , 1 -and 2 -adrenergicreceptorsarestim- ulatedelicitingavasodilationinthearteries.Concurrently,attheonsetofexercise(orstress) 17 -adrenergicreceptorsonvascularsmoothmusclesarestimulatedwhichresultinaseemingly paradoxicalvasoconstriction[105].Itisworthytomentionthatinpatientswithcoronary arterydisease,blockadeof 1 -and 2 -adrenoceptorshasbeenprovenbforcoronary bloodwandmyocardialfunction[106].Regardless,thenetsympatheticstimulationin microvasculatureisoverriddenbylarge -adrenergicfeedforwardvasodilation.Furthermore, thefeedforwardmetabolicregulationoccursimultaneouslywithafeedbackmechanismin whichadenosinetriphosphate(ATPs)anditsmetabolites 2 releasedfromRBCs,actasva- sodilators[107].Sp,asaresponsetodesaturationoftheoxyhemoglobin,ATP, ADP,andAMPactivatepurinergicreceptorsoncapillaryendothelialcellswhichinitiatea conductedvasodilatorysignaltothearteriolesandsmallarteries( < 150 )[81]. Whilethemetabolicstimulus(i.e.,increaseinMVO 2 )isessentialinalterationofcoronary perfusion,thewrateisrelativelyedbychangesofarterialpressureintherange of50-160mmHg,demonstratingatautoregulatoryresponseinmyocardium[103]. Interestingly,thismechanismhasbeenobservedinothersystemiccirculationsexceptfor pulmonarycirculationwherethepressureandwraterelationfollowsamonotonically increasingtrend(Fig.1.8). Althoughthemetaboliccontroliscruciallyeinautoregulation,thismechanismis facilitatedthroughanintrinsicmyogenictoneinthevascularsmoothmusclecells[109]. ,studieshaveshownthatincreasedtransmuralpressureleadstodepolarizationof smoothmusclecellsandtheopeningofcalciumchannelsresponsibleforincreasedintracel- lularcalciumleadingtothecontractionofactin-myosin[110,111].Thedevelopmentand maintenanceofabasalvasculartoneviamyogenicresponseisphysiologicallyessentialfor vasodilatorsforregulationofbloodwandpressure[112].Similartothemetaboliccontrol, however,themyogenictonevariesinvesselswithtdiameter.Theconduitcoronary arteriesdonotexhibitatmyogenicactivity[81].Unsurprisingly,theautoregula- 2 adenosinediphosphate(ADP)andadenosinemonophosphate(AMP). 18 Figure1.8:Wholeorganpressurewrelationships.Pictureadaptedfrom[108]. toryresponseoccursprimarilyintheresistancevessels( < 100-150 )[113,114].Inaddi- tion,themyogenicactivityofcoronaryarteriesvariesacrossthemyocardium[103,115,116]. Kuoandcolleagues[113]showedthatathighpressure(70-100mmHg)bothsubepicar- dialandsubendocardialarteriolesdemonstratedmyogenicactivity,butsubepicardialvessels exhibitedgreaterconstrictionthansubendocardialarterioles.Inaddition,Dunckerandcol- leagues[115]determinedthatthelowerautoregulatorypressurelimitforsubendocardial vessels(40mmHg)ishigherthanthatofsubepicardialvessels(25mmHg)indicatingthat vesselsaremaximallydilatedforpressurebelow40mmHgandtheyaremoreproneto developmyocardialischemia. Thethirdmechanismthecoronarybloodwisthewallshearstressdependent(i.e., shear-orw-dependent)control.Shear-dependentdilationparticularlyhasbeenobserved inlargecoronaryarteriesfromavarietyofspecies[108].Thisvasodilatorymechanismisthe resultofshear-inducedproductionofthevasodilatorNObyendothelialcells.Theshear- dependentdilatorymechanismsin-vivohavebeenshowntobebluntedincoronarymicro- 19 circulationincomparisontoconduitarteries[117,118].However,physicaltraininghasbeen showntoaugmenttheshear-dependentvasodilationthroughoutthecoronarycirculation.In fact,studiesonexercise-trainedratsbyLaughlinandcolleagues[119,120]haveshowedthat thisimprovedenessisprimarilyaresultofincreasedendothelialexpressionofnitric oxidesynthase(NOS). 1.2.2.4 Modeling Arterialnetworkstudies- Sincethedirectmeasurementsofthehemodynamicsinthe wholecoronarynetworkisdatafrommorphometricstudieshasbeenusedtocon- structthecoronaryarterialnetworkforhemodynamicsanalysis.BeardandBassingth- waighte[121]werethesttousean\avoidance"algorithmtopresentathree-dimensional modelreconstructionoftheLADarterialnetworkinanidealizedmodeloftheheartusing statisticalmorphometricdata.Theyusedthereconstructedtreetoanalyzethefractalna- tureofspatialmyocardialbloodwdispersionandtemporalparticlewash-outandwere abletoreproducetrendssimilartoexperiments.Almostconcurrently,Smithetal.[122] providedasimilaralgorithmthatproducedthenetworkwiththeoptimalbifurcatingan- gles(basedon[123]).However,duetocomputationallimitationstheywereonlyableto generatearelativelysmallerreconstructednetwork(largestsixgenerations).Alternatively, Karchandcoworkers[124]usedaconstrainedconstructiveoptimizationmethodtogenerate thecoronaryarterialtreeinathree-dimensionalslabviasuccessivelyaddingnewsegments underasetofphysiologicalconstraints.Recently,Jaquetandcolleagues[125]expandedthis optimizationtogenerateapatientspcoronaryarterialnetworkbasedonhumancar- diacCTimagedata.Itisworthytomentionthattwo-dimensionalnetworkreconstruction attemptshavebeenmadeusingdirectuseofthestatisticaldata[86]. Kaimovitzetal.[126,127]providedastochasticframeworkformodelingtheentirecoro- naryvasculatureusingaoptimizationprocesssubjecttomorphologicalfeatures 20 from[24].Thesestudieswerelaterusedasabasisforpulsatilehemodynamicsanalysisin theworksofAlgranatiandcolleagues[88]onthestudyofpulsatilehemodynamicsofcoro- naryvasculaturecoupledwithvascular-myocardialinteractions.Finally,Namanietal.[128] recentlydevelopedanalgorithmtominimizethewdisparityinKaimovitz'smodelof thecoronaryarterialnetwork.Theyshowedthataconcurrentminimizationofwdis- persionanddiameterre-assignmentingenerationofcoronarymicrovasculartreesresulted incongruousresultswiththemeasuredwcharacteristicswhilefollowingthemeasured morphometricdata. Flowregulationmodeling- Flowandpressureregulationinthewholesystemiccircula- tionhasbeenthesubjectofseveralstudiesoverthelast50-60years.Theseminalworksof Guytonandcolleagues[129{131]usedcomplexmathematicalmodels,includingseveralhun- dredequations,aimingtounderstandtheregulationofbloodpressureandcardiacoutputin closed-loopsystem.Alternatively,themathematicalmodelsoflocalinteractionbetweenreg- ulationmechanismsinmicrovasculaturehavebeenstudiedinrentvasculature[109,132]. Coronarywregulationhasbeenunderconstantfocusoverthelast50years[97].How- ever,importantaspectsofthecoronarywregulationandtheirdiagnosticandprognostic capabilitiesremaingreatlyunknown[95].Manysuchquestionscannotbeexperimentally answeredduetofactorsincludingfailuretoreachfulldilationofvessels[133],in quantifying in-vivo hemodynamics[134],etc. Previousstudieshaveattemptedtotheoreticallyinvestigatetheinteractionandbalancebe- tweentheregulatorymechanismsincoronarycirculation[102,135].Whilethesestudiesman- agetoincorporatetheheterogeneityoftheresponseintotheirmodel,themainlimitations oftheirstudiesarecrudeofthearteriesinto4classesandignoringtheinterac- tionsofthevesselswiththesurroundingtissue.Alternatively,Pradhanetal.[99]presented adata-drivenclosedloopmodelofthecoronarywregulationwhichreliedheavilyonex- perimentaldata.Theirmodel,however,onlycapturesthetotalwandpressureregulation 21 withoutanyspatialinformation.Morerecently,Namaniandcolleagues[103,116]revisited thecomputationalmodelingofcoronarywregulationandincorporatedtwonetworksof vesselsinsubendocardialandsubepicardiallayersofthecardiacwall.Theyreconstructed thearterialtreebasedonthemorphometricdata[24,126]andincludedthemyocardialinter- actionsintheirmodel[88].Forthehemodynamicanalysis,theyuseda3-elementWindkessel modelwhichdoesnotincludethewavepropagationandinthearteries.More- over,theyusedanempiricalpressure-diameterrelationtomodelthepassiveandactivewall behavior. 1.3 Spaimsandorganizationofthedissertation Thespeaimsofthisdissertationaresummarizedasfollowing: Aim1. Todevelopacomputationalframeworkofone-dimensional(1D)arterialnet- workthatcapturesessentialfeaturesofvascularG&Randbloodcirculationusingan extendedMurray'slawtoestablishthehomeostaticbaseline. Aim2. Toconstructamodelofthepulmonarycirculation. 1. Integratetheexperimentaldataintothemodelingframeworktoestablishthe homeostaticbaselineofpulmonaryarteries. 2. Applythe1-Dmodelforsimulatingthepulsatilehemodynamicsandstudythe wavepropagationphenomenainpulmonarycirculation. Aim3. Todevelopamodelofthecoronarywregulation. 1. Applythemodelingframeworktoconstructthecoronaryarterialnetworkwith respecttotheexperimentaldataavailableintheliterature. 22 2. Formulateandmodelcoronarywregulationmechanismstounderstandtheir heterogeneousinteractionsandeness. Theremainderofthisdissertationisorganizedinthefollowingmanner:Chapter2presents anewcomputationalframeworkofarterialnetworkthatgenerateandoptimize1Darterial geometryusinganextendedMurray'slawandaconstrainedmixturemodelofarteries. Chapter3presentstheapplicationoftheoptimizationonapulmonaryarterialtreeto thehomeostaticcharacteristicsofindividualsegments.Then,theresultofapplicationofa 1Dtheorytosimulatethepulsatilehemodynamicsinanarterialtreewithphysiologically realisticpropertiesispresented.Chapter4presentstheapplicationoftheoptimizationto thephysiologyofcoronaryarteries,spwithregardstocoronarywregulation. Theestimatedbasalarterialtreeisendowedinwregulationmechanismstostudytheof vascularadaptationinthecoronaryarteries.Finally,Chapter5summarizesthesimulation resultsanddiscussesabouttheirlimitations,andfutureimprovements. 23 Chapter2 ATheoreticalFrameworkfor EstimatingHomeostasisinVascular Trees 2.1 Introduction TheoreticalstudiesonvascularG&Rdatebacktotheearly1980s[136,137].Theoretical andcomputationalmodelingofvascularG&Rviaconstrainedmixturemodels,however,only recentlyhavebeengainingattention.OriginallyproposedbyseminalworkofHumphreyand Rajagopal[3],severalstudieshaveadoptedanddevelopedG&Rcomputationalmodelsto testmultiplehypothesesbasedonexperimentalandclinicalstudiesinapplicationssuchas aneurysms,cerebralvasospasm,andhypertension[138{141]. Inessence,theG&Rmodelincludesconstitutiverelationsforthemechanicalresponseof theloadbearingconstituentsandtheirtime-varyingmasses.Thelatteraccountsforthe productionandremovalofcellsandextracellularmatrixduringadaptation.Themediation 24 processoftheadaptationisbasedontheexistenceofamechanicalhomeostaticbaseline,the deviationfromwhichcausesachangeintheturnoverrateorstructureofeachindividual constituent. InmanyapplicationsofG&Rmodeling,theconstituentsofthearterialwallaresp tohaveindividuallyprescribedmechanicalbehaviorandhomeostaticvalues.VascularG&R models,therefore,havebeendevelopedtoallowmoreinsightintothecomponent-wisemech- anismsofG&R[4,6,142].Thesemodelsfacilitatetheoreticalandnumericalsimulationsof progressivecollagenousstelastindamage/degradation,andw-orpressure-induced vasoactivetone. Previousextensiveresearchofvascularadaptationshasproposedmechanicalhomeostasis atmultiplelevels,tissue,cells,andsubcellularstructuresthataremediatedprimarilyby endothelialcells,vascularsmoothmusclecells,and[142].Therefore,processesof vascularadaptationhaveseveralintrinsiclengthandtimescales.Aniterativecouplingof G&Randinteraction(FSI),namelygrowth(FSG)modeling,has beenproposedtodealwitherenttime-scalesinvascularadaptationusingalinearization ofthenonlinearbehaviorofthearterialwalloverthecardiaccycletimescale[5,143]. Ontheotherhand,littleattentionhasbeenpaidtothespatiallymultiscaleG&Rapplications inarterialadaptation.Particularly,whiletheprincipalconceptofthestress-mediatedG&R, asproposedbyHumphreyandRajagopal[3],ispivotalforpredictingvascularadaptation andtestinghomeostatichypothesesofindividualconstituentsforthearteries,previousap- plicationswerelimitedtolargearteriesandtheconcepthasnotbeenextendedtothewhole arterialnetwork.Furthermore,althoughnotmanyexperimentalstudieshavereportedcir- cumferentialstressandwallshearstressthroughthearterialnetwork,itisnotlikelythat thehomeostaticvaluesareconstant.Forinstance,phenomenologicalstudiesonswinecoro- naryarterialnetwork[144,145]evaluatedthehomeostaticstressandshowedthatthemean circumferentialstresstlyreducesalongthearterialtree.Similartrendshavebeen 25 shownintheworksofPriesandSecombonrat'smesentericmicrovasculature[146].There- fore,anextensionofthecomputationalG&Rframeworktoanarterialnetworkrequires amoregeneralhypothesisforthewholearterialnetworkthateventuallytakesaccountof meticulousconsiderationsofthephysiologicaldatasuchaspressure,w,thicknessand radii. Asamatteroffact,themostsuccessfultheorytopredictarterialnetworkstructurewas basedonmetabolicenergyconsumption[7,87,147].Basedontheprincipleofminimum work,Murray[7]suggestedthedesignofthebloodvesselstobeanoptimizationproblem betweenthemetaboliccostofmaintainingthevolumeofbloodandthepowerneededto overcomeviscousforces.Theoptimizationresultsinacubicrelationshipbetweenthediam- eterofaparentanditsdaughtervessels( R 3 p = R 3 d 1 + R 3 d 2 )whichiscalledtheMurray'slaw. Alternatively,studiesbyZamir[123]andKassabandFung[85]havepursuedtheuniform shearhypothesisinthearterialnetworks.Itisworthytonotethattheuniformshearhy- pothesisisalsoanimplicationofMurray'swork[85].However,uniformshearhypothesishas hasbeenchallengedwhenitcomestomicrovasculature,similarlytothecircumferentialwall stress.WhileKamiyaetal.[148]observedanarrowvariationinwallshearstressthroughout thewholesystemiccirculationfromtheaortatotheprecapillaries(1-2Pa),otherstudies observeda3-to6-foldincreaseinwallshearstressfromarteries( > 100 )toprecapillary arterioles( < 100 )[118,149]. Murray'slawhasbeenusedextensivelyinstudiesfortopologyandreconstructionof in- vivo arterialnetworkstructures[27,124,125,150].Zhouandcoworkers[87]proposeda generalizedMurray'slawforanentirecoronaryarterialtreeanddiscoverednewscalinglaws thatrelatethediameter,length,andvolumeofthearterialtree.Thisgeneralizationwas laterexpandedtoincludethemetabolicdissipationofwallconstituents(e.g.,activesmooth muscletone)inswinecoronaryarteries[147]andothervasculature[151].Thesestudieswere abletosuccessfullyadaptthetheoreticalformulations,includingtheextensionofMurray's 26 law,withthemeasuredanatomicaldataandthescalinglaws.Thesescalinglawsare particularlyimportantforunderstandingthefractalnatureandlocalhemodynamicsofblood vesselsinthehomeostaticarterialtree.Murray'sideaanditsgeneralizations,however,were unabletoprovideinformationonthemicrostructureofthevesselwall.Inspiredbythework ofTaber[152],ometal.[153]incorporatedtheconventionalG&RintoMurray'slaw asamodelingframeworkthatstudiedvascularadaptationinasinglevesselmodeled usingaconstrainedmixturemodel.Lastly,astudybyPriesetal.[154]showedthetial roleofmetabolicresponsesindeterminingthewallshearstressandcircumferentialstress throughthecirculatorysystem. Motivatedbytheincompletenessofourcurrentunderstandingintheconceptofhomeostasis forarterialnetworks,theultimategoalofthischapteristoestablishacommonframework forestimationofthehomeostaticcharacteristicsofthearterialtrees.Inspiredbytheworks ofPriesetal.[154]andometal.[153],ourframeworkconstructsanarterialnetwork thatincludestheessentialfeaturesfromG&Rmodelswithmetabolicconsiderations(viaan extendedMurray'slawapproach).Moreover,thepresentedframeworkreliesheavilyonthe reconciliationofavailableexperimentaldatatopredictwrelationsinphysiological conditions.Consequently,afterweestablishthehomeostaticbaseline,anapplicationofthe modeltostudymultiscaleadaptationofarterialnetworkswillbeunchallenging. 2.2 Methods 2.2.1 Generalwwandinitialization Byion,mechanicalhomeostasisisthevascularadaptationofthearterialwallinthe longterm[155].Inotherwords,whileacardiaccycletypicallyisoftheorderofasecond, thevesselwalladaptationtakesplaceinamuchslowertimefromafewminutes(inw 27 Figure2.1:Informationexchangeforthehomeostaticoptimizationappliedforthetreein thenestedmanner:fromthetreetobifurcationandthentoindividualvessels. regulation)toweeks(ingrowthandremodeling).Therefore,theoptimizationprocessinthis chaptermustbeconsideredinviewofaslow-timescale.Inaslow-timescale,thehomeostatic conditionofthearteryisembeddedintothestrainenergyfunctiontodescribethemechanical constitutiveresponseandthenincorporatedintoslow-timestressequilibrium,expressedvia Laplacelaw(seeSection3.1formoredetailsontimescaleconsiderations). Thegeneralwwofthepresentedmodelanditsimplementationsbeginsfromaconstruc- tionofaninitialarterialtreefrombifurcationrules.Subsequently,anoptimizationprocedure isappliedtothetreewhichmayalterinitiallydiametersandbifurcationparameters. Finally,theoptimizationresultsareusedinsimulationofpulsatilehemodynamics(Chapter 3)andwregulation(Chapter4). Arterialnetworksshowcharacteristicsofafractaltreestructuretosomeextent[156].In- spiredbythis,weconstructaninitialself-similararterialtreeusingbifurcationrulesfollow- 28 ing[69].First,weassumeapowerlawdeterminestheradiuschangeacrossabifurcation R ˘ p = R ˘ d 1 + R ˘ d 2 ; (2.1) wheresubscript p referstotheparentvessel,andsubscripts d 1 and d 2 refertothedaughter vessels.Tocomputetheradiusofeachvessel,weintroducetwobifurcationparameters,area ratio( )andasymmetryratio( ),as = R 2 d 1 + R 2 d 2 R 2 p ; and = R 2 d 1 R 2 d 2 ; (2.2) where =1asymmetrictree.Theparameters ˘ , ,and arerelatedby = 1+ (1+ ˘= 2 ) 2 =˘ : (2.3) Usingtheserelations,wecanscalingparameters and as = 1 (1+ ˘= 2 ) 1 =˘ ; and = p : (2.4) Finally,theradiusofdaughtervesselscanbecomputedusingtheparentvesselusingthe scalingparametersas R d 1 = R p ; and R d 2 = R p : (2.5) Equation2.5illustratethattheradiiofvesselscanbecomputedfromtheirrespectiveparent vessel.Therefore,ifwespecifyarootradius R 0 andtwobifurcationparameters(e.g., ˘ and ),wecanconstructthewholearterialtree.Moreover,thelengthofeachsegmentin thearterialtreemustbespIneachimplementationofthemodelalength-diameter relationobtainedfrommorphometricstudieswillbeused(formoredetailsseeChapters3 and4). 29 Wepresentthehomeostasisofanarterialtreeasanoptimizationproblematindividual vesselsegmentsandthenextendittoanentirearterialtree,whichwecall\homeostatic optimization"hereafter.Thesteadystatehemodynamicsandstressequilibriumarethe constraintsappliedtotheoptimizationproblem.Thisframeworkwilldeterminetheoptimal designofthearterialtree(i.e.,changesthebifurcationrulesabove),hemodynamics(i.e., wrelations),andstructureofarterialwall.Figure2.1showsschematicdiagram ofthehomeostaticoptimizationfromatreeoutlooktoindividualvessels. 2.2.2 Arterialmechanicsofasinglesegment Asinglesegmentofthearterialtreeisconsideredasathin-walledcylindricaltubecomposed ofthreemainload-bearingconstituents:elastin,collagen,andsmoothmusclecells.Each constituentisassumedtoseparatelycontributetothestrainenergydensity: w = w e + w m + w c (2.6) wheresubscripts e , m ,and c representelastin,smoothmuscle,andcollagenrespectively. Inaddition,followingBaeketal.[157],weassumedistinctpre-stretchforeachconstituent mappingthemfromnaturaltotheoverallloadedInparticular, thepre-stretchmappingforelastincanbeexpressedas G e = 2 6 6 6 6 4 G e r 00 0 G e 0 00 G e z 3 7 7 7 7 5 (2.7) where G e and G e z arepre-stretchesassociatedwithcircumferentialandaxialdirections,and G e r = 1 G e G e z . Similarly,forcollagenersandsmoothmuscle, M i , i 2f k;m g isastheunitvector 30 inthedirectionofthecollagener( k )orsmoothmusclecell.Thepre-stretchmappings forcollagenandsmoothmusclecellsaregivenas G k = G c h M k M k ; G m = G m h M m M m : (2.8) wherethepre-stretches G c h and G m h areidenashomeostaticstretches,whicharethe stretchesoftheconstituentswhentheyareproduced[157].Weshouldnotethatinthe previousapplicationsofG&R,thepre-stretcheswereassumedtobeconstant.However,in ourgeneralizationoftheframeworktoanarterialtree,weassumethatthepre-stretches mayvaryacrossthearterialtree(formoredetailsseeSection3.2.3).Nevertheless,the pre-stretchimpliesthatthehomeostaticstateinanindividualvesselisassociatedwitha constanthomeostaticstressformaterialswithaturnoverinthevessel. Theorientationofcollagenersandsmoothmusclesintheirreferenced byangle k ,canbewrittenas M k =cos k e Z +sin k e ; M m = e : (2.9) Formodelingandextensionandofathinwallmodel,weconsideradeformation gradient F = diag [ r ; ; z ].Theincompressibilityofthewallmaterialisimposedby assuminganisochoricmotion(i.e.,det F =1),therefore, r = 1 z .Usingmembrane theory[158],themembraneCauchystress(forceperdeformedlength)canbewrittenas T = 1 J 2 D F @w @ F T ; ) T = 1 z @w @ ; and T zz = 1 @w @ z ; (2.10) 31 where J 2 D = z .Thefollowingstrainenergydensityfunctionsforthethreematerialsare usedasconstitutiverelations[159] e = c 1 2 ( e ) 2 +( e z ) 2 + 1 ( e ) 2 ( e z ) 2 3 ; (2.11) k = c 2 4 c 3 exp c 3 ( k ) 2 1 2 1 ; (2.12) m = c 4 4 c 5 exp c 5 ( m ) 2 1 2 1 : (2.13) Thetotalstrainenergyperunitareacanbewrittenas w = M e R e + X k M k R k + M m R m ; (2.14) where M i R , i 2f e;k;m g isthemassofeachconstituentperunitreferencearea.Moreover, i , i 2f e;k;m g isthestretchineachconstituent,andcanbeexpressedintermsofthe pre-stretchesusing F i = FG i e = G e ; e z = G e z z k = G c h q 2 sin 2 k + 2 z cos 2 k ; m = G m h (2.15) Therefore,thecircumferentialmembraneCauchystresscanbewrittenas T = 1 z M e R @ e @ + X k M k R @ k @ + M m R @ m @ : (2.16) Substituting2.15into2.16,andusingchainrule,wecanwrite T = 1 z M e R @ e @ e e + X k M k R d k d ( k ) 2 d ( k ) 2 2 2 + M m R d m d ( m ) 2 d ( m ) 2 2 2 (2.17) 32 Equationaboveshowsthepassivetensionresponseoftheartery.Toincludetheactivetone ofvascularsmoothmuscle,weuseapotentialfunctionasgivenby[160] m act = S ˆ + 1 3 ( M ~ ) 3 ( M 0 ) 2 ; (2.18) where M and 0 arestretchesatwhichtheactiveforcegenerationismaximumandzero, respectively,and S isthestressatthemaximumcontraction.Notethat S isgenerallya functionofwallshearstress,internalpressure,andmetabolitesbutinashorttimescaleof acardiaccyclethevalueof S isconsideredasconstant(e.g.,Chapter3).Inaddition, ~ isanactivestretchoftheSMCsinthecircumferentialdirection.Althoughthevasoactive responsecanchangeviaremodelingoftheSMCs,thecurrentformulationisappliedforthe homeostaticstatesofthearteries.Intheapplicationofthemodelinnextchapters,we consideronlythehomeostaticbaseline,incontrasttothoseofthelong-termadaptations (daystoweeks)ofthearteries.Therefore,weassume ~ = .Finally,thetotaltensionin thearterycanbewrittenas T = 1 z M e R d e e e | {z } T e + X k 1 z M k R d k d ( k ) 2 d ( k ) 2 2 2 | {z } T k + 1 z M m R d m d ( m ) 2 d ( m ) 2 2 2 + M m R d m act | {z } T m (2.19) where T i , i 2f e;c;m g isthetensionineachconstituent.Finally,foranthinwall cylinderwithpressure p s ,theforceequilibriuminthecircumferentialdirectiongives p s R = T ; (2.20) where R isthedeformedradiusofthecylinder. 33 2.2.3 Metaboliccostofasinglesegment AsproposedbyMurray[7]andlaterextendedby[152]and[153],thebloodvesselwall compositionandgeometrystrivetooptimizetheenergycostthatincludesthemetabolic costofbloodsupply,powerneededtoovercomeresistance,andthecostofmaintainingthe vesselwallmaterials.Weassumethathomeostaticstateisgovernedbysuchanoptimization rulethatcanbeforeachindividualvesselseparately. WethreecontributiontermsfortheextendedMurray'slaw.First,themetaboliccost ofarterialwallconstituentsperunitlengthisassumedtobe C wall =(2 ˇR / ˆ solid ) X i # i M i R ; (2.21) where M i R ismassperunitreferenceareaofeachconstituent(asintheprevioussection); # i isthemetaboliccostofconstituent i perunitvolume;and R isthevesselradiusin homeostaticcondition.Notethatthemetaboliccostofsmoothmusclecells(SMC; i = m ), # m ,includesboththemetaboliccostofmaintenance # m maint andactivetension # m act .Second, themetabolicpowerneededforbloodsupplyisproportionalto # b andthebloodvolume thatneedstobesustained;hence,thismetabolicpowerperunitlengthis C blood = # b ˇR 2 : (2.22) Third,thepowerperunitlengthneededtoovercometheresistanceofPoiseuillew(viscous dragforces)is C drag = 8 s 2 ˇˆ fluid R 4 ; (2.23) withaveragevolumetricwrate q s .Thus,thetotalenergycostperunitlengthatindividual 34 bloodvesselissummarizedas C ( M i R ;R )= C wall + C blood + C drag = 2 ˇ ˆ X i # i M i R R + # b ˇR 2 + 8 s 2 ˇˆ fluid R 4 : (2.24) Usingtheminimummetabolicenergyprinciple,Eq.(2.24)isminimizedsubjecttothe constraintofthemechanicalequilibrium(Eq.2.20). 2.2.4 Costoptimizationofasinglesegmentintermsofradius Nextweincorporatemasscontenttothecostfunctiontoexpressminimizationinterms ofradius.Tothisaim,wetotalmassas M t R andmassfractionsofelastin,smooth muscle,andcollagenas e = M e R M t R ; k = M k R M t R ; and m = M m R M t R : (2.25) IntroducingEq.2.25into2.19gives T = M t R (1 ˚ f ) z X i i ˙ i ) ; (2.26) where ˙ i isactingpartofthetensor ˙ ˙ ˙ i incircumferentialdirection,as ˙ ˙ ˙ i = 1 h i T i ;h i = M i R (1 ˚ f ) ˆ solid z ;i 2f e;k;m g : (2.27) Theparameter ˚ f isthevolumefractionoftheinterstitialThentheminimization problem(Eq.2.24)isconstrainedbystressequilibriumrelation(Eq.2.20)withmembrane 35 stresscomponentexpressedintermsof ˙ p s R = M t R (1 ˚ ) ˆ solid X i i ˙ i : (2.28) RearrangingEq.2.28for M t R gives M t R = (1 ˚ f ) ˆ solid p s R P i i ˙ i : (2.29) Themetaboliccostforelastincanbeneglectedforadultsubjects, # e =0,sinceitismostly producedinearlyagesandowingtoalonghalf-life,remainsrelativelyconstantovertime. Finally,forasinglevesselthemetaboliccostfunction(Eq.2.24),embeddingstressequilib- riumconstraint2.28,canbewrittenintermsofradiusforagivensteady-statew q s and pressureatthemiddleofarterialsegment p s C ( R ; p s ;q s )=2 ˇ (1 ˚ ) p s R 2 ( # c c + # m m ) P i i ˙ i + # b ˇR 2 + 8 s 2 ˆ fluid ˇR 4 : (2.30) Equation(2.30)expressestheoptimizationproblemintermsofradiusgiven p s and q s .The generalizationofthisoptimizationisdiscussedinthefollowingsection. 2.2.5 Iterativeprocessofoptimization Abifurcatingarterialtreeisinitializedbasedonthefractalruleforvesselradii(Section2.2.1). Toupdatethevesselradiioverentiretreethecostfunctionminimizationhastobeimple- mentedtogetherwithglobalhemodynamics.Algorithmically,thisimplementationcanbe donesequentiallyforeachbifurcationorgloballyforallvesselsatonce.Withoutlossofgen- eralityweproceedwiththesequentialminimization,whichisiterativelysolvingtheglobal hemodynamicsrelationbetweenpressureandw,andlocalminimizationthecostfunc- tion.TheiterativeprocessisillustratedinFig.2.1.Thepseudo-codefortheoptimization 36 procedureispresentedinAppendix5.Theinformationtransferinthisoptimizationcanbe conceivedasacouplingofaG&Rmodelwithhemodynamics,similartowhatwaspresented in[5]. 2.3 Summaryandconclusion Inthischapter,wedevelopedanumericalframeworktoestimatethehomeostaticconditions inarterialnetworks.Thisframeworkreliesonminimizingthemetaboliccosttomaintain thebloodvolumeandconstituentsofthewall,withthelocalmechanicsandglobalhemody- namicsasconstraints.Theproposedframeworkcanbeappliedtovariousvascularnetworks giventhatwehaveaspdatasetorrulesthatwouldservetoconstructthestructureof thetree(i.e.,vesselconnectivity,length,etc.).InChapters3and4wetestourframework onmodelsofpulmonaryandcoronaryarterialtrees. 37 Chapter3 HomeostaticBaselineand HemodynamicsinthePulmonary ArterialTree 3.1 Introduction Pulmonaryarterialhypertension(PAH)isacomplexdisorderassociatedwithanelevated pulmonaryarterialpressure.Progressiveandnarrowingofdistalpulmonaryvessels resultinanincreaseinpulmonaryarterialpressureinPAH,whichcanleadtofatalright heartfailure.ThepathologyofPAHisconnectedtothelong-termvascularremodelingwith prominentfeaturessuchas:smoothmusclehypertrophy,endothelialdysfunction,deposition ofcollagenandelastin,andincreasedelastolyticactivities[45].Alternatively,theshort- termhemodynamicsuchasthemagnitudeofpulsepressureandwavepropagation phenomenainthepulmonaryarterialtreehavebeenobservedtobetialindevelopment ofPAH(formoredetailsseeSection1.2.1.4). 38 ThecurrentunderstandingofG&RprocessinPAHisgreatlylimitedduetopaucityof detailedanalysesonthemannerinwhichthepulsatilenatureofhemodynamics(inthe short-termscale)inducespathologicalchangesinthevasculature(inthelong-termscales) [161].Thislimitationnecessitatesanapplicationoftemporallymultiscalecomputational modelsofthepulmonaryarteriesthatincludebothtimescalesinvascularadaptations.As explainedinChapter2,theG&Rmodelcorrespondstolong-termchangesinthestructure ofthepulmonaryarterieswherethearterialwallismodeledasaconstrainedmixtureofthe wallconstituentsendowedwiththekineticsoftheirproductionandremoval.Amongthe maindrivingfactorsinG&Rmodelsarethehemodynamicsloads(wallshearstress,and circumferentialstress)onthevascularwall.Ontheotherhand,teraction(FSI) modelsofthepulmonaryarteriesfacilitatethesimulationofthepulsatilehemodynamics inshort-timescale[74].Realisticconsiderationofdeformablewallpropertiesareintegral partofthebiomedicalFSIproblems.Therefore,exchanginginformationbetweentheG&R andFSImodelsispivotalinthestudyoftheprogressionofPAH.Thiscouplinghasbeen previouslyproposedind-growth(FSG)modelingframeworkbyFigueroaetal.[5]. Inthisstudywepresentasnapshotofsuchcouplingwherethepulmonaryarteriesarein healthycondition. Themaingoalsofthischapteristoestablishthehomeostaticbaselineandsimulatethe pulsatilehemodynamicsinapulmonaryarterialnetwork.Tothisend,thecurrentchapteris organizedasfollows:First,weexplainthewwforassimilationofthehomeostatic optimizationwitha1DFSImodel.Second,weprovidetheformulationofFSIinapulmonary arterialtree.Subsequently,theparametersusedfortheimplementationofthehomeostatic optimization(Chapter2)andFSIinthedistalpulmonaryarterialtreeareexamined.Finally, wepresentanddiscusstheresultsoftheoptimizationinsymmetricandasymmetrictrees. 39 Figure3.1:Schematiccomponentsofthepulmonaryarterialtreeandtheircorresponding dataavailability:largevessels(3Dpatients-spanatomy,wandpressurewaveforms fromclinicalimagingandcatheterization)andsmallvessels(limitedmorphometricaland biomechanicaldata). 3.2 Method 3.2.1 Fluid-solid-growth Figueroaetal.[5]illustratedhowtheFSIanalysisduringonecardiaccyclecanbecoupled withanarterialG&RmodelthroughaFSGmodelingframework.Inourstudy,weextend theFSGframeworktothedistalpulmonaryarterialtree(shownasredbinarytreesinFig. 3.1).First,weestablishedthattheresultsofhomeostaticoptimizationessentiallyconstruct aninstantofG&Rmodelingwherearteriesareinhealthycondition(Chapter2).Second, weusesmall-on-largetheorytocomputethelinearizedmechanicalresponse(i.e.,Young's modulus)fortheimplementationoftheconstrainedmixturemodelintotheFSImodel.We presentageneralformulationofthesmall-on-largeinAppendix5.Finally,weusea1DFSI model(i.e.,Womersley'stheory)forthepulsatilehemodynamicsofthepulmonaryarterial treewherethevesselwallsareconsideredlinearlyelastic(Section3.2.2). 40 3.2.2 Pulsatilehemodynamics Giventhegeometryofthetree,vesselwallandsteady-statepressureandw fromthehomeostaticoptimization,weproceedtoobtainthepulsatilesolutionsandpulse wavevelocityinatreeusingWomersley'ssolutions(seeAppendix5).Notethatwithinthe homeostaticoptimizationformulationthefast-timehemodynamics(denotedbytheirFourier domainrepresentatives P and Q inthissection)inatreeisdecoupledfromtheslow-time problem,andthusitrepresentsapost-processingoperationafterhomeostaticoptimization. Ingeneral,thisprocedurecanbefullycoupledwithslow-timeproblem. Womersley'ssolutiontopulstatilewinadeformabletubecanbefoundinAppendix5.In thissection,weexplainthegeneralizationtoanarterialnetwork.Eachbifurcationis asiteforthetravelingwave.Womersley'stheorycanbeadaptedtoincorporate suchoftravelingwaves.Duetothesystemlinearitythepressureandwcanbe decomposedtotheforwardandbackwardwaves P = P forw + P back ;P forw = H forw e i!z=c ;P back = H back e i!z=c Q = Q forw + Q back ;Q forw = H forw Z c e i!z=c ;Q back = H back Z c e i!z=c (3.1) withconstantcots H forw and H back forindividualharmonics.Thecharacteristic impedance Z c andvelocityofthewavepropagation c arethesameforbothforwardand backwardwaves[162]: Z c = P forw =Q forw = P back =Q back .Wecantheco cientas[163,164] = Z T Z c Z T + Z c = P back P forw z = L : (3.2) Thencotvariesbetween-1and1.If=0,thennowaveispresent, indicatingthatimpedancesmatch Z T = Z c .Theso-calledopen-endtypeoftionis 41 associatedwith= 1,where Z T =0or P j z = L =0.Inthiscase,thewaveis negative[165],similartoawavetravelinginastringofend.Theso-calledclosed-end typeofisrelatedto=1for Z T = 1 or P j z = L =0,wherethewaveis positive(asillustratedinanalogywithwavetravelinginastringoffreeend). Usingand Z c ,totalimpedanceatfrequency ! canbeobtainedasafunctionoflocation alongthevesseloflength L . Z ( z;! )= P Q ( z;! )= Z c 1+ e i! 2( z L ) =c 1 e i! 2( z L ) =c : (3.3) Combiningequations3.1and3.2,thepressureandwalongthevesselcanbewrittenas P ( z;! )= H forw e i!z=c (1+ e i! 2( z L ) =c ) ; Q ( z;! )= H forw Z c e i!z=c (1 e i! 2( z L ) =c ) : (3.4) Moreover,theinputimpedancecanbewrittenas Z inp = Z c 1+ e i! 2 L=c 1 e i! 2 L=c : (3.5) Consideringconservationofthetotalpressureandwatbifurcationandtheinputimpedance ateachdaughtervessel Z inp d 1 and Z inp d 2 ,weobtaintheterminalimpedanceintheparentvessels Z T p = 1 Z inp d 1 + 1 Z inp d 2 ! 1 : (3.6) Giventherootvesselw q s p , Q p ,terminalpressure p sT ,andcot T atthe terminalvessels,weusebifurcationrelationsEq.3.6tocomputetheinputimpedanceeq. 3.5,recursivelyfromthebottom-to-thetopandthenreconstructthepressureandwat eachvesselofthefractaltreefromthetop-to-the-bottom,usingEq.3.4. Usingwave-intensityanalysis,HollanderandColleagues[55]demonstratedthatthenormal 42 pulmonaryarterialcirculation(indogs)ischaracterizedbynegativewave(an open-endTheprimaryfactorforcreatingtheopen-endtypeofin pulmonaryarterialsystemislikelythelargeincreaseincross-sectionalareaoverashort distance.Asdiscussedin[55],themagnitudeofnegative,open-endtypeincreases asdaughter-to-parentratiois a d =a p > 1 : 2. 3.2.3 Parameterselection TheparametersofthemodelarelistedinTable3.1.Thefocusofthisstudyisonthe intermediate-to-smallregionofthepulmonarytree,fromtheendoftherightinterlobar arterytothearterioles.Forhemodynamicsboundaryconditionattheinlet,weprescribe inputwwaveformtakenfromhumandataatmainpulmonaryartery[67]andscaledto fourthgenerationdownstream.Thenineharmonicsareusedtorepresentthewaveform infrequencydomain.Thelength-to-radiusratioistakenfromOlufsenetal.[73]which approximateshumandatafromasinglepulmonaryarterialtreecastreportedinHuang etal.[23].However,thelengthwasscaledbyafactorofhalftobetterapproximatethe characteristicsofthetreeinthebeginningofmodelingtree.Terminalradius,similarto[73], thenumberofgenerationsinasymmetrictree(thisresultsin19generations).For theasymmetriccase,afullarterialtreeisconstructed,thentheinitialtreeispruned witharadiusthresholdof0.18 cm whichleadstontnumberofgenerationsint paths.Althoughthetreeterminatesatarteriolarlevel,weconsiderameanterminalpressure closetothecapillarypressureasthepressuredropissmallbetweenpulmonaryarterioles andcapillaries.Thebasalandactiveenergyconsumptionratesforvascularsmoothmuscle cellsaremeasuredusingthetheofadenosinetriphosphate(ATPs)intheextracellular space[166,167].Althoughthebasalmetaboliccostofvesselwallconstituentsmayvary throughoutthearterialtree,wechosethevalue1500 W=m 3 fortheentirepulmonarytree [147].ThemetaboliccostofactivetensioninSMC,asmeasuredin[166],isconsideredtobe 43 0.008721/sandproportionaltotheactivetension.Sincetheactivetensionisproportional tothemassofSMCsinthevesselwall,themetaboliccostcanbewrittenasproportionalto SMCcontent(Eq.2.30).Furthermore,theenergyrequirementforsustainingthebloodin thearteriesisadoptedfrom[168]basedonnumberofredbloodcells,whitebloodcells,and plateletsinaunitvolumeofbloodandtheiroxygenconsumptionrateforanormaladult humansubject. Themechanicalproperties(Table3.1)ofthewallfortheconstrainedmixturemodelare calibratedagainstexperimentaldataobtainedinetestsofporcineright/left pulmonaryarteries(conductedatMichiganStateUniversity[169]).Theintrinsicmaterial parametersareassumedtobeconstantforallthevesselsinthearterialnetwork. However,thepre-stretchofcollagenandelastinisadjustedineachvesseldownstreaminorder tomatchthethickness-to-diameterratioreportedin[170,171].Theunderlyingassumption hereisthatinanarterialtree,thepre-stretchofthecollagenandelastinaredependent onthehomeostaticarterialpressure.Thisisinspiredbytheexperimentalobservations showingwavierorevencompressedcollagenersandelastinsheetsinthesmallarteriesand arterioleswithactivetension[112].Themassfractionsofallconstituentsareestimatedfor theentirearterialtree:constantinCase1andvariableinCase2(Table3.1).Todetermine thecompositionofthearterialwall,theadventitiallayerofarterialwallwasassumedto have95%ofcollagenand5%ofelastininCase1.Themassfractionswereestimatedusing therelativelayerthicknessreportedin[41]andmediallayermassfractionsreportedin[34]. However,theadultpulmonaryarterialwallcompositionvariesthroughoutthearterialtree. Thevariablecompositionofthewallisinthemodel(Case2)bychangingthe contentofelastinandSMCsinthemediallayerusingdatafrom[41].Particularly,most ofthearterieslargerthan0.32 cm indiameterareelasticarteriesendowedwithmultiple layersofelasticlamina.Thearterialstructuretransitionsfromelastictomusculartypeover arangeof0.32-0.2 cm wheretheelasticlayersfragmentandarereplacedbySMC[172]. Thearteriessmallerthan0.2 cm haveamuscularmediawithtwodistinctiveinternaland 44 Table3.1:Modelparametersforhomeostaticoptimizationandpulsatilehemodynamics ParameterdescriptionRef. Initializationofthetreegeometry Rootvesselradius0.55 cm (symm);[73] 0.36 cm (asymm) Rootvesselwallthickness0.42 cm [170] Terminalradius0.005 cm (symm);[73] 0.018 cm (asymm) Radialexponent2.72 Daughter-to-parentarearatio1.2 ExtendedMurray'slaw MetaboliccostofcollagenandSMC1500 W=m 3 [147] Metaboliccostofbloodsupply51.7 W=m 3 [168] Metaboliccostofactivetension 0.008721 =s [166] Vesselwall Walldensity1060 kg=m 3 Case1:Constantmassfractions(c/m/e)77%/12%/11%[34] Case2:Variablemassfractions77-59%collagen,[29] 12-39%SMCs, 11-2%elastin Passiveandactivewallelasticityparameters c 1 =28 : 83 Pa=kg;G e = G e z =1 : 16 1 : 27 c 2 =178 : 60 Pa=kg;c 3=1 : 05 ;G k h =1 : 08 1 : 15 G k h =1 : 15 1 : 08 ; k =0 ; 45 ; 90 c 4 =24 : 51 Pa=kg;c 5 =0 : 75 ;G m h =1 : 21 S =20 kPa; M =1 : 2 ; 0 =0 : 7 Hemodynamics Inputwwaveform,scaledby1/8meanw11.65 ml=s [67] Length-to-radiusrelations,scaledby1/2 L =6 : 2 R 1 : 1 mm [73] Symmetrictree:meanterminalpressure10mmHg[74] Terminalcot-1[55] Blooddensity1060 kg=m 3 Dynamicviscosity0.0035Pa.s externalelasticlaminas[29].Whilethearteriessmallerthan0.01 cm arenotincludedinthis study,wenotethatthenumberofmusculararteriestlydropswhenthearterialsize approaches0.01 cm andbelowsothatthevesselsbecomepartiallyorfullynon-muscular. 45 3.3 Resultsanddiscussion 3.3.1 Symmetrictree Anadvantageofsymmetrictreeisthatanyvesselwithinagenerationisrepresentativeofthe wholegeneration,andthustheresultscanbeevaluatedwithrespecttogenerationsrather thanindividualvessels. 3.3.1.1 Case1:constantmassfractions Inthissectionweshowtheresultsofthehomeostaticoptimizationrelatedtotheconstant massfractionsTable3.1.ThestudybyHuangandcolleagues[23]onthebranchingpat- ternandvasculargeometryofthehumanpulmonaryarterialtreesfound15ordersinthe pulmonaryarteriesusingtheStrahlerorderingsystem.Intheirstudy,1stand15thorders correspondtotheprecapillaryvesselsandright/leftpulmonaryarteries,respectively.A comparisonofresultsobtainedinthecurrentstudytotheresultsfromHuangetal.[23] indicatesanagreementbetweenthediameterofour1stgenerationandtheir14thorder pulmonaryvesselcorrespondingtotheStrahlerorderingtechnique.Figure3.2comparesthe reportedhumandataforrightinterlobarartery[73]andlargervessels[23].Theexponent ˘ indaughter-to-parentradiusrelationalongthetree(Fig.3.2,rightpanel))remainscloseto thecubicrelation(Murray'slaw). Thewallthicknessisobtainedfromthemassoftheconstituentsresultingfromtheoptimiza- tion.Theresultingthickness-to-diameterratios,bytuningthepre-streches,areshowninFig. 3.3,top-leftpanel.Pressureinthemiddleofthearterialsegmentsalongthetreeisshownin (Fig.3.3,top-right).Pressuregradientbecomessteepertowardsthearteriolarleveltomeet theterminalpressure.Finally,homeostaticshearandcircumferentialstressesareobtained 46 Figure3.2:Symmetrictree{homeostaticoptimizationresultsplottedversusgeneration number:Left:diameterdistributioncomparedtoreporteddataoflargervessels;Right: radiusexponentindaughter-to-parentradiirelation; (Fig.3.3bottomrow).AlthoughtheoriginalMurray'slawimpliesthatthewallshearstress isconstantthroughoutthearterialtree[173],wallshearstressvalueincreasesinourmodel, whichisconsistentwiththeexperimentalstudiesforthesystemiccirculation[148,174].The decreasingtrendofcircumferentialstressvalue(Fig.3.2,bottom-right)isconsistentwith observationsinthecoronaryarterialnetwork[144]. 3.3.1.2 Case2:variablemassfractions ExperimentalstudiesbyHislopandReid[29]showedthatthecompositionofthepulmonary arteriesvariesacrossthearterialtree.Thisvariabilitywastakenintoaccountbymodifying massfractionsbasedonthecompositioninTable3.1 TwodistinctoptimizationsresultsforCase2arecomparedtoCase1inFig.3.5.During thetransitionregion(Fig.3.4)elastinisreplacedbytwomaterialsthatrequiremetabolic energyformaintenance(i.e.,metabolicallyexpensivematerials).Theoptimization,however, prefersasmallervesselwithmorepercentageofcollagenandSMCstoalargevesselwithless 47 Figure3.3:Symmetrictree{homeostaticoptimizationresultsplottedversusdiameters: Top-left:wallthickness-to-diameterratio(h:wallthickness);Top-right:mid-arterypressure; Bottom-left:homeostaticvalueofwallshearstress( ˝ );Bottom-right:homeostaticvalueof circumferentialstress ˙ h . percentageofcollagenandSMCs.Thisisinadropintheradiusexponent(Fig. 3.5,left).Thedropinresultsinastep-increaseofthewallshearstress(Fig.3.5,right). Theresultsfromthehomeostaticoptimizationareobtainedfromalinearization oforthotropicelasticmembranepropertiesinthe p mid ofeachartery(Appendix5).The axialYoung'smodulusismarkedlylowerthanthoseinsystemicarteries[160,175];andboth decreaseacrossthegenerations(forbothcases,Fig.3.6,left).Anincreaseofcompliance distallyisconsistentwiththesiteoftheopen-endtypewhichisimposedinthe 48 Figure3.4:Prescribedvariablemassfractionsofthewallconstituents:elastin,smooth musclecellsandcollagen.Thearrowsonthetopshowthetrendinarterialcomposition. Figure3.5:Symmetrictree{Left:radiusexponentindaughter-to-parentradiirelation; Right:homeostaticvalueofwallshearstress. model[55].Comparisonbetweenthetwocasesofmodelingshowsthattheaxial dropstlyinthemusculararteries.Thisresultisexpectedsincemostthesmooth musclecellsareorientedcircumferentiallyinthevesselwall. Itisexperimentallyachallengetoestimatethe in-vivo arterialofdistalpulmonary tree.However,experimentalstudieshaveemployedadistensibilityparameter whichbe 49 Figure3.6:Symmetrictree{homeostaticoptimizationresultsforthewallversus generationnumber:Left:Young'smodulesincircumferentialandlongitudinaldirections; Right:Structuralnormalizedbytheunstressedradiusandcomparedtodistensibility relationsfromdata. usedtoconstructtherelation R=R 0 =1+ withtransmuralpressure p andradiusatzero pressure R 0 .Forinstance,Yenandcolleagues[176]analyzedthemechanicsofthearteries oftsizesdissectedfromhumanlungandconcludedthatthedistensibilitydoesnot varysigtlywithsize(with =0 : 012/mmHg).Alternatively,KrenzandDawson[177] conductedameta-analysisontpulmonaryarteriesofvariousanimalswitht sizesandwhiletheythatthedistensibilityisalmostconstant,theyreporteda largervalueof =0 : 02/mmHg.Tobeabletocompareourresultstothesestudies,weuse anormalizedratioofstructuraltotheunstressedradiuswhichcanbeexpressedvia distensibilityparameter Eh=R 0 =3 = (4 )[74].RightpanelofFig.3.6showsanearconstant valueforthenormalizedforthevariablemassfractionsimulation(Case2)which isconsistentwithexperimentalstudies.Moreover,thecomputedfromthemodel isconsistentwiththatofourexperimentsonpig'spulmonaryartery(indicatedforthe generationinFig.3.6,rightpanel).However,theoptimizationispredictinglarger (smallerdistensibility)comparedtootherexperimentalstudies. 50 3.3.1.3 Pulsatilehemodynamics Beforediscussingtheresultsofthefasttimehemodynamics,itisimportanttonotethat weusedtheoptimizationresultsofCase2forthefollowinganalysissinceitcapturesthe physiologyofthedownstreamarteriesmoreaccurately.Furthermore,tovalidatethechoice of= 1fortheopen-endwaveFig.3.7showstheduaghter-to-parentarearatio comparedtoboundsproposedbyHollanderandcolleaguesin2001[55]. Oneofthemostvaluableoutcomesfromthemodelingofpulsatilewinthedistalvascu- latureisestimationofthepulsewavevelocity( c )acrossthetree(seeAppendix5).The pulsewavevelocity(shownfortwoharmonicsinFig.3.8,left)dependsontheof thewallaswellasWomersley'snumber = R p !ˆ fluid (shownfortwoharmonicsin Fig.3.8,right).Particularly,thedecreasein indicatesthedominanceofviscousforces insmallarteries(consistentwithhigherresistance)relativetotransientinertialforces.The pulsewavevelocityinthegenerationshowsexcellentconsistencywiththeexperimental measurementsof c inmainpulmonaryarteries[178,179].Moreover,heterogeneousness ofthearterialwalland resultinasteepdecreasein c afterthefewgenerations.The idealizedMoens-Korteweg(MK)( c MK = p Eh= 2 Rˆ fluid )pulsewavevelocityisalsoplotted forthecomparison.Clearly,theassumptionsofMKequation,suchasnon-viscousare voidedinsmallervessels. Tounderstandlimitationsofthemodel,wecheckthevalidityofthedeformablewallWom- ersley'stheorybycomputingtheratioofmaximumlumenoscillatoryvelocitytothepulse wavespeed, max =max t v f z =c ,whichmustbesigtlysmallerthan1forthelongwave approximationtobevalid.Figure3.9showsthatatseveralgenerationsofthetree,there mightbetnonlinearcontribution(nonlinearinertiatermforw)whichis neglectedintheWomersleytheory. TotalhemodynamicssolutionalongthevascularbedisdemonstratedinFig.3.10.The 51 Figure3.7:Symmetrictree{daughter-to-parentarearatiowithintherangeofopen-end typeofns. Figure3.8:Symmetrictree{pulsatilehemodynamicsresultsversusgenerationnumber:(a) pulsewavevelocityfortwoharmonics,comparedtodataandMKvelocity;(b)Womersley's numberfortwoharmonics. totalinputwsplitsevenlyateachgeneration.Thetotalterminalpressureiswithinthe physiologicalrange8-25mmHg.Theminimumextremeislocatedbelowthemeanpressure thatisconsistentwithnegative Thepatient-spmodelsreconstructedfrommedicalimagesoflargepulmonaryarteries canbeconnectedtothedistalvasculaturemodelviaanimpedanceboundaryconditionat 52 Figure3.9:Symmetrictree{Womersley'ssolutionvaliditycheckasfunctionofgeneration numberfortwotharmonicmodes. Figure3.10:Symmetrictree:Totalinputw(top)andterminalpressure(bottom)over thegenerations theinterface.Thewimpedanceatthelargevesselisequivalenttoinputimpedance atfollowingvasculartree.Finally,usingthepulsatilehemodynamicsattherootvesselwe computetheinputimpedanceforthearterialtree(Fig.3.11).Forcomparisonweplot 53 characteristicandterminalimpedanceattherootvessels. 3.3.2 Asymmetrictree Thissectionillustratesaasymmetrictreeexampleforthehomeostaticoptimizationand hemodynamics.Initially,anasymmetrictreeisgeneratedbythearearatio 0 =1 : 2and radialexponent ˘ 0 =2 : 72following[73].Homeostaticoptimizationonsuchatreeleads toasymmetrictreeasaresult,sincethesymmetricstructureismoreenergyt.To introduceasymmetryintheoptimization,theinitialarterialtreeisprunedwithaterminal radiusof R min =0 : 018 cm .Forthesteadystatehemodynamics,theboundaryconditions areswitchedtopressureattheinletandwattheoutlets.Homogeneousdistribution oftheperfusionwoveralltheoutletshasbeenusedinotherattemptsforanatomic reconstructionofarterialnetworks[124,180].Althoughthehomeostaticoptimizationis capableofsimulatingahighlyasymmetrictree,onlyaslightlyasymmetrictreeisconsidered inthisstudytoensurethattheanalyticalsolutionisvalidinthefast-timehemodynamic analysis.Therefore,aninitialtreewith15generationisconstructed(32,767vessels)and thenprunedwithrespecttotheterminalradius,whichresultedin4,233individualsegments in14generations.Withoutlossofgenerality,weidentifyashortandalongpath(X)along thetreeforpresentationofresults. Figure3.12showstheresultsoftheoptimization,boundedbytheshortestandlongestpaths, fortheasymmetriccasemassfractionsaredeterminedfromCase2.Attheoutletvessels thewisthesamewhileterminalpressureisnot(Fig.3.12top)asexpectedfromthe boundaryconditions.Thedistributionofresultswithintheboundscanbeexplainedby dominanceofthesymmetricsubtrees.Theratioofstructuraltounstressedradius isdistributedaround9kPaandvariesmoretowardtheoutlets(Fig.3.12bottom-right). Thewallshearstress,byvariablemassfraction(Case2)increasestowardoutlets 54 Figure3.11:Symmetrictree{Top:totalwandpressureattherootvesselofthedistal tree;Bottom-left:rootvesselinputimpedanceintimedomaincomparedtoterminaland characteristicimpedance;Bottom-right:inputimpedancemodulusandphaseanglesinthe frequencydomain. (Fig.3.12bottom-left). LeftpanelofFig.3.13showsthe parameterforthelongestandshortestpathsofthe asymmetrictree.Fornumberofgenerationssmallerthan10intheshortestpath,the parameterbecomestlylargerthan0.5whichindicatesthattheWomresleytheory isnotvalidinthosebranches(thesimulationnotshown).Furthermore,therightpanelof Fig.3.13shows c forthelongestandshortestpaths.Thespeedofthewavepropagation seemstobelargerinthelongerpathsgiventhedepthofthevesselsegmentinthearterial 55 Figure3.12:Asymmetrictree{homeostaticoptimizationresultsversusdistance(fromthe root,alongthebranchpathway),dotsrepresentsoptimizationresultsforeachvessel,red andbluelinesindicatetheshortandlongpath,respectively:(a)logofsteady-statew rate;(b)terminalsteadypressure;(c)ratioofstructuraltounstressedradius;(d) homeostaticvaluewallshearstress. tree. Finally,thepulsatilehemodynamicsattherootofthearterialtreeiscomputedandshownin Fig.3.14.Theasymmetricstructureofthearterialtreetlychangestheimpedance attherootoftheartery.Thisisparticularlycrucialsincethisimpedancecanbeusedasan outletboundaryconditionsinthepatient-spemodeling.Therefore,anaccurateanalysis ofthedistaltreeiscrucialinperformingrealistichemodynamicssimulations. 56 Figure3.13:Asymmetrictree{deltaparameterandpulsewavevelocityforlongestand shortestpaths. 3.4 Summaryandconclusion Wesuccessfullyimplementedthehomeostaticoptimizationmethodtoestimatethebaseline stateofthedistalpulmonaryarterialtree.Ourstudyincludedvasculartissuepropertiesvia aconstrainedmixturemodel(previouslyusedforG&R)aswellasthepulsatilehemody- namicsusingthemetabolicdemandconsiderations(Murray'slaw)andananalyticalblood wtheory.Particularly,thevesselsizesandmechanicalpropertieswereestimatedusing anextensionofMurray'slaw,andWomersley'stheorywasusedforsimulatingthepul- satilebloodwinanetworkofelasticvessels.Thematerialbehaviorofthevesselwall, presentedbyorthotropicmembrane,wasdescribedbynonlinearconstitutivelawatslow time-scale,andthenlinearizedatthesteadypressuretoobtainapulsatilesolutionforcar- diaccycle.Theproposedmethoddoesnotnecessitatecomputationalcostassociatedwith nonlinearproblemsolvers,nestediterativeoptimizations,andcomplextreemorphometry. Instead,ourframeworkisacomputationallyttoolthatgreatlythebiome- chanicalanalysisinvasculartrees.Indeed,itallowsfocusingprimarilyoncomplexwall tissuesprocessesandassociatedbiomechanical/biochemicalstimuli.Wehaveillustratedthe 57 Figure3.14:Symmetrictree{Top:totalwandpressureattherootvesselofthedistal tree;Bottom-left:rootvesselinputimpedanceintimedomaincomparedtoterminaland characteristicimpedance;Bottom-right:inputimpedancemodulusandphaseanglesinthe frequencydomain. frameworkfunctionalityonexamplesofsymmetricandasymmetricbinarytreesrepresenting distalintermediatepulmonaryarterialvasculature.Theresultsshowgoodagreementwith theavailableexperimentalandclinicalobservations.Thehomeostaticoptimizationprovided thewallcompositioncontent,vesselsize,andstructuralgivingvaluableestimation ofintrinsicwallpropertiesthatotherwisearenotmeasurable(attherangeof14-19genera- tionsafterthelargevessel).Inaddition,thehemodynamicssolutiongavethetotalpressure andwdistributionalongthetreerevealingtheevolutionofarterialpressurewhichisa 58 crucialmarkerforpulmonaryarterialhypertension.Whileinourexamplesweconsidered healthysubjects,theproposedframeworkcanbeappliedtopulmonaryhypertensionsub- jectsoncemoreclinicalandexperimentaldataareavailable.Anothernotmeasurabledistally butvaluablecharacteristicisthepulsewavevelocitydistributionalongthetree.Ourresults demonstratedthedecreaseofpulsewavevelocityoverthegenerationsasaconsequenceof wallandWomersleynumberchanges.Furthermore,thelowcomputationalcostof ourmodelrendersitimmenselyusefulforparametricstudies,asillustratedbyconsidering casesofconstantandvariablemassfractionsofwallconstituents.Theproposedframework isalsousefulforobtainingtheimpedancewboundaryconditionsessentialforcoupling distalvasculaturewithlargevessel3Dsimulationsinthepatient-spmodels.Were- portedsuchinputimpedanceobtainedforthesymmetricalandasymmetrictrees.Inthis study,weusedconstantbloodviscositywhichisfortheintermediaterangeofvessel size.However,constantviscosityisnotarequirementoftheformulationthoughitgreatly dynamics.Formicrovasculatureclosertocapillaries,theapparentviscosity canbeconsideredasafunctionofvesseldiameterandhematocritlevel[181].Inthiscase, thetotalhemodynamicshastobegeneralizedtoaccommodateviscosityupdates.Addi- tionaldataandworkareneededtoimprovethepulmonaryarterialtreemodel.Inpresented examples,thetreeisdescribedbyafractalstructurebasedonabifurcatingradialrule.In future,thetreestructurehastobereconstructedfromacomprehensivemorphometryof thepulmonaryarterialtree.Ideally,suchmorphometryhastobestatisticallyrepresentative forhumansubjects,spforagegroup,healthyorbypulmonaryhypertension{ thesearedataandtasksthatarecurrentlyunavailable,limited,and/orchallenging.Sim- ilarly,inthecurrentexamples,thewallparametersandconstituentcontentwasestimated fromlimitedopen-literatureresourcesandinternalexperiments.Thus,toimproveandval- idatethearterialwallmodelingthereisaneedofmoreexperimentaldataonpulmonary arterialwalltissuemechanicsforhumans. 59 Chapter4 BaselineCharacteristicsand AdaptationsinCoronaryFlow Regulation 4.1 Introduction Coronaryarteriesareresponsibleforsupplyingbloodtothemyocardium.Thecoronary arterialnetworkisinherentlytfromothercirculatorysystemsinthebodyintwo majoraspects.First,sincethehearthaslimitedanaerobiccapacity,energyproductionin myocytesishighlydependentonoxidativephosphorylation[82].Therefore,acontinuous supplyofoxygentothecardiacmyocytesisnecessaryfornormalfunctionandwithout toxygendelivery,theircontractilefunctiondeclineswithinsecondsoftheischemic insult.Thecontinuoussupplementofoxygentomyocytesisaburdenonthecoronary vasculature,i.e.,atightregulationofcoronaryvascularfunctionsisnecessaryformaintaining thebloodcirculationintheheartandeventuallythewholecardiovascularsystem(formore 60 detailsseeSection1.2.2). Second,myocardialarteriesareundertcompressiveforcesfromthemyocardium duringthesystolicphase.Theinteractionofcompressiveforcesandvascularwall(namely myocardial-vascularinteractions)arehypothesizedtobeprimarilydependentonthecavity- inducedextracellular(interstitial)pressure( p CEP ),muscleshortening-inducedpressure( p SIP ), andchangesofmyocardial(i.e.,varyingelastance)[88].Dependenceon p CEP im- pliesthatthecompressiveforcesarelargerintheinnerlayersoftheleftventricle(LV)than thoseintheouter(epicardial)layers[182].Ithasbeenlongestablishedthatthisvariation inpressurethewdistributionandstructureofthearteriesintlayers[183]. Determinationofthespatiallytialbaselinecharacteristicsofthecoronaryarteries iscrucialfortheanalysisofthecoronarywregulation,sincetheactiveresponseofthe vesselstochangesinbiomechanicaland/orbiochemicalstimulivarieswiththeirsizeand locationinthecardiacwall.Mostofthewregulationoccursinthenetworksoftree-like branchingarteriesintherentlayersofthemyocardium(subepicardial,mid-wall,and subendocardial)[81,82].Similarly,theresistanceartertiesinthecoronaryarterialnetwork arelessthan100 insizeandresideinthemyocardiallayersofthecardiacwall[184]. Thesevesselsaremainregulatorsofthewcontrolmechanismbyintrinsicand/orextrinsic moindiameterviavasoreactivityofsmoothmusclecells(SMCs). Inthischapter,weaimtoapplythehomeostaticoptimizationframeworktoestablish thebaselinecharacteristicsoftwocoronaryarterialtreesinsubendocardialandsubepicardial layers.Next,weusetheedbaselinetostudycoronarywregulation.Particularly, tialanalysisofthewautoregulationwillbeperformed.Finally,wetest thecapabilityofthemodelincapturingtheofdrugadministrations,suchasadenosine infusionandinhibitionofNOsynthesis. 61 4.2 Methods 4.2.1 Baselineconstructionofcoronaryarterialtree ThegeneralwwofarterialtreeconstructionisdescribedinChapter2.Inthischapter, theoptimizationwwisimplementedtoconstructacoronarymicrovasculartreemodel embeddedintlayersofthemyocardiumwithspscalefactors.First,twosym- metrictrees(subendocardialandsubepicardial)with12generationsofvesselsaregenerated withtheinitialradialexponent ˘ =2 : 55basedonArtsandReneman[182]andKarchet al.[124].Thischoiceof ˘ ismotivatedtheexperimentallyobservedfractalnatureofthe vasculature[121].Thelength-to-diameterratioisprescribedusingthemorphometricswine datafromKassabetal.[24].Thesubendocardialandsubepicardialtreesareassumedtobe locatedin5/6and1/6ofthemyocardium,respectively. Formodeling,weconsidertheslow-timeconditions(averageoverminutestohours),anda meanintramyocardialpressureisimposedontheindividualvesseldependingonitslocation. Sp,themechanicalequilibriumiswrittenas p tm r = T ; (4.1) where p tm isthetransmuralpressureofthebloodvesselslocatedwithinthemyocardium, T isgivenbyequation(2.19),and r istheinnerradiusofthesegment.Thetransmural pressurecanbewrittenas p tm = p p im ; (4.2) where p istheluminalpressureaspresentedbypreviouschaptersand p im istheintramy- ocardialpressure.FollowingtheanalysisbyAlgranatietal.[88]onmyocardium-coronary 62 vesselinteraction, p im isassumedas p im = p CEP + p SIP : (4.3) ForthefreewalloftheLV, p CEP varieslinearlyfromendocardium(LVpressure, p LV )to pericardium,wherethepericardialpressureisassumedtobenegligible.Therefore,forthe subendocardialandsubepicardialtreesinthischapter,theaverage p CEP valuesare5/6and 1/6of p LV ,whichisconsistentwiththeirrespectiverelativemyocardialdepth.Thepressure p SIP waschosensothatthe p im inthesubendocardiumis20%largerthanthe p LV [71].The boundaryconditionsofthebaselineoptimizationinbothsubepicardialandsubendocardial treesaretheinletandoutletpressures.Inaddition,thelayer-wisetotalwinthearterial treeisconsideredasanextraconstraint. 4.2.2 Coronarywregulation Coronarybloodwistightlyregulatedasaresponsetochangesinperfusionpressureand/or imbalancebetweenmyocardialoxygendemandandsupply.Thereactivityofthevesselwall tochangesinstimuliisdominatedbytheSMCs.Fromtheconstrainedmixturemodelin Chapter2,theactivestress T act inthesmoothmuscleis T act = S ˆ 1 M ~ M 0 2 ; (4.4) where M and 0 arestretchesatwhichtheactiveforcegenerationismaximumandzero, respectively,and S isthestressatthebasalvasoactivetone.Moreover, ~ isanactivestretch whichcanevolvebySMCremodeling.Sincewemodeltheshorttimescaleadaptations(min- utestohours),SMCremodelingisnotconsideredwhichresultsin ~ = : .Theparameter S ,thebasalactivetone,isafunctionoftransmuralpressure,andshearstressfromtheir 63 homeostaticvalues,andthecardiacactivity(MVO 2 ),whichgives S = A ( p tm p h ;˝ ˝ h ; MVO 2 ) S max ( p tm ) : (4.5) Theactivation, A ,determinestheactivationlevelfromfullydilatedtofullyconstricted.The maximumactivestress, S max ,isafunctionoftransmuralpressureaswasobservedin[102] andshowninFig.4.3.FollowingtheworkofCornelissenandcolleagues[135],weassume thatthemaximumactivetonehasasigmoidalshape,describedwithahillcurve S max ( p tm )= S 0 p tm p 0 + p tm ; (4.6) where istheslopeofthecurve, S 0 isthemaximumtone,and p 0 istheaparameterthat thecenterofthecurve.Threeprimarymechanismsthatpredominantlyregulatethe vascularreactivityaremyogenic(pressure),shearstress,andmetaboliccontrols.Thew regulationisessentiallyadynamicprocessinvolvingactivationofeachofthesemechanisms withtheirrespectivetimeresponse.Inthisstudy,however,wefocusonthesteadystateofthe vasculature,whichisreachedwithin2minutesaftertheperturbationformthehomeostatic value[89,116,132].Thistime-scaleislargeenoughforouranalysistobevalidwithrespect tothediscussioninSection3.1. MyogeniccontrolistheSMCcontractioninresponsechangesinthelocalwallstressdeter- minedbythetransmuralpressure.Thedeviationfromthebasal(homeostatic)pressure( p h ) leadstoaconstrictivestimulus.Thisstimulusineachvesselcanbewrittenas s p = a p ( p tm p h p h ) : (4.7) Contrarytomyogenictone,anincreaseinwallshearstress,inducesrelaxationoftheSMCs facilitatedbyanincreasetheNOproductionoftheendothelialcells.Thisvasodilation 64 stimuluscanbewrittenas s ˝ = a ˝ ( ˝ ˝ h ˝ h ) : (4.8) Finally,experimentalstudieshaveshownthatduringanincreasedMVO 2 (e.g.,exercise), whiletheoxygenextractioncapacityofthecardiomyocytesdoesnotchange[99].Therefore, anincreaseinMVO 2 mustbemetwithaproportionalincreaseinwrate.Althoughthe measurementofMVO 2 isnotdirectlyused,thewcanbeconceivedastherepresentative ofmetabolicdemand[116].Therefore,themetabolicmechanismofwregulationcanbe writtenas s m = a m ( ^ q (MVO 2 ) q term q term ) ; (4.9) where^ q (MVO 2 )isthetargetwasafunctionofthemyocardialoxygenconsumption,and q term isthewrateattheterminalarterioles. Towritetheintegratedstimulifromthemechanisms,twofollowingfactorsareconsidered. First,theenessofthemechanismsistforarteriesattsizes.Inpartic- ular,themyogenicreactivityisthehighestinbloodvesselswith100 diameter[102,135] whiletheshear-dependentvasodilationbecomesmostlybluntedvesselssmallerthan100 [118].Meanwhile,thesignalformetabolicresponsefromtsignalingpathways (oxygenimbalanceand/oradrenergic)isoriginatedincapillariesandisconductedupstream toprecapillaryarterioles[132].Theconductedresponse,however,decaysexponentiallyso thatitmostlythearterioles.Second,theabove-mentionedphenomenologicalequa- tionsonlydescribethestimuliwhenadeviationfromhomeostasisisoccurred(i.e.,stresscon- dition).However,smoothmusclecellsmaintainabasaltoneunderrestingconditions[185]. Thissmoothmuscletoneisexpressedasabasalstimulimediatedbythecontrolmechanisms. 65 Consideringthesefactors,wecanexpressthetotalstimulias s total = ˚ p ( r )( s p + s 0 p )+ ˚ ˝ ( r )( s ˝ s 0 ˝ )+ ˚ m ( r )( s m s 0 m ) ; (4.10) where s 0 p , s 0 ˝ and s 0 m setthebasaltoneinSMCsatrest( s 0 = ˚ p ( r ) s 0 p + ˚ ˝ ( r ) s 0 ˝ + ˚ m ( r ) s 0 m ).Furthermore, ˚ p , ˚ ˝ ,and ˚ m aretheweightsrepresentingthetivenessof eachmechanisminthetotalstimulation(Fig.4.1).Thesestimulidictatetheactivation intheSMCtonewherefullactivation( A =1)representsmaximalconstrictionandzero activation( A =0)representthefulldilation.Following[132],asigmoidalfunctionisused toconvertthestimulitotheactivationlevel A = 1 1+exp( s total ) : (4.11) Themodelingframeworkforthewregulationstartswithintroducingastimulusforau- toregulation(changeinpressure)orexercise(changeinMVO 2 ),andthesimulationsare conductedintwonestedloops.Theinnerloopdeterminesthediameters(i.e.,resistances) andtheouterloopscomputesthehemodynamicsusingPoiseuillewandupdatestheac- tivationlevels.Theprocedureiscontinueduntilconvergenceinwandpressure(Fig. 4.2). 4.2.3 Modelparameters Hemodynamics- Inthisstudy,weperformthehomeostaticoptimizationontwoarterial treeslocateddownstreamoftheLAD,andinsidethefreewallofLV.Assumingthatthe pressuredropinepicardialarteriesissmall,theinletpressureisconsideredtobethesame astheaorticpressure100mmHg.Moreover,theoutletpressureattheterminalarterioles isconsidered55mmHg[186].Thetotalratioofsubendocardialtosubepicardialwrates 66 Figure4.1:Theenessofeachregulationmechanisms,basedontheanalysesof[113, 114,117,118,135]. (ENDO/EPI)isassumedtobe ˘ 1.25[97].Theprescribedintramyocardialpressures( p im ) are47and13mmHgimposedonsubendocardialandsubepicardialvessels,respectively. Viscosity- Priesandcolleagues[149]observedthattheviscosityinthesystemicvasculature isdependentonthesizeofthevesselandthehematocritlevel( H D ).Particularly,the variationofviscosityismorepronouncedasthearteriesandarteriolesbecomesmaller.In ourstudy,weprescribetheviscosityusingthefollowing in-vivo viscositylawgivenin[149] C = 0 : 8+exp( 0 : 07 D ) 1+ 1 1+10 11 D 12 + 1 1+10 11 D 12 ; (4.12) h f = (1 H D ) C 1 (1 0 : 45) C 1 ; (4.13) 0 : 45 =6exp( 0 : 085 D )+3 : 2 2 : 44exp( 0 : 06 D 0 : 645 ) ; (4.14) vivo = 0 1+ h f ( 0 : 45 1)( D ( D 1 : 1) ) 2 ( D ( D 1 : 1) ) 2 ; (4.15) 67 Figure4.2:Schematicdiagramillustratingthewwofthesimulations,withpossible extensiontoaclosed-loopmodel.Theinputsarethecoronarypressure( p in )andMVO 2 .The convergencetheequilibriumstateofthewregulation.Thefeedbackmechanism (notincludedinthecurrentstudy)relatesthelocalwregulationincoronariestothe cardiacfunction;heartrate,cardiacoutput,etc. Table4.1:Modelparametersforthehomeostaticoptimization ParameterdescriptionReference ExtendedMurray'slaw MetaboliccostofcollagenandSMC1070 W=m 3 [167] Metaboliccostofbloodsupply51.7 W=m 3 [168] Metaboliccostofactivetension 0.008721 =s [166] Walldensity1060 kg=m 3 Hemodynamicsandgeometry Flowattheoutletarterioles(subendocardium)0.002 mm 3 =s [116,187] Inletpressure100mmHg[86] Pressureattheoutletarterioles55mmHg[186] Hematocritlevel( H D )0.45[135] Blooddensity1060 kg=m 3 Plasmadynamicviscosity 0 0.001Pa.s[149] Length-to-radiusrelation(LAD) L =0 : 145 D +20 [24,86] where D istheanatomicaldiameterinmicron,and 0 istheviscosityofthebloodplasma. 68 Vascularwallproperties- SimilartoChapter3,thebasalandtensiondependentmetabolic costofSMCsareadaptedfromenergeticconsiderationsofthevascularsmoothmusclein swinecoronaryarteries[167].Toestimatetheconstitutiveparametersforeachconstituent inthearterialwall,thevesselsegmentsinthemyocardialarterialtreesaresubdividedinto fourclasses:smallarteries( D 190 ),largearterioles(100 D< 190 ),inter- mediatearterioles(50 D< 100 ),andsmallarterioles( D 50 ),basedonthe in[102,135].Inthebiomechanicalstudyofthemicrovasculature,twofactors mustbeconsidered;First,underphysiologicalcondition,theextracellularmatrixcompo- nentsinthearteriolarwallareinastateofcompressionwhereasSMCsarecontractedto exerttone[112].Moreover,amicroscopicanalysisofthemechanicalstructureofrabbit's arterioleshasshownthattheamountofSMCsgraduallydecreasefromarteriolesof100 to30 [188].Therefore,theintrinsicpassiveandactivemechanicalpropertiesofthecon- stituents(i.e.,collageners,elastin,andSMCs)areassumedtobeconstantacrossall fourclassesinthecurrentstudy.Consequently,weestimatethemechanicalproperties,pre- strechesandmassfractionsusingthepressure-diameterrelationshipsin[102,135].Figure 4.3showsthediameter-pressurerelationshipforpassiveandfully-constrictedrepresentative vesselsinthemyocardiallayers.Furthermore,themassfractionsareestimated,asshown inFig.4.4.Themechanicalparametersarekeptconstantinthehomeostaticoptimization. However,thebasalactivationlevelofthemuscletone(viaparameter s 0 )isconsideredasa variablewhichisusedtomatchthevesselthickness-to-diameterratiofromtheoptimization resultswiththosereportedintheliterature[144]. 69 Figure4.3:Solidcirclesshowthepassivebehaviorofthevessels,digitizedfromdatain[135]. Opencirclesshowthefullyactivatedresponseofthearteries.Reddotsaretheestimated responsefromparameters.( R 0 :theradiusatzeropressure) 70 Table4.2:Estimatedconstitutiveparametersfortheconstrainedmixturemodelofthe arteries Parameterdescription Passivematerialparameters c 1 =113 : 60Pa/kg ; c 2 =333 : 70Pa/kg, c 3 =4 : 64 ; k =0 ; 45 ; 90 c 4 =78 : 02Pa/kg, c 5 =0 : 47 Activematerialparameters S 0 =2 : 45Mpa, p 0 =70mmHg, =2 : 1 0 =0 : 5, M =1 : 7 Subendocardiumpre-stretches(A,LA,IA,SA)* G e = G e z =(1 : 15 ; 1 : 12 ; 1 : 00 ; 0 : 96) G k h =(1 : 02 ; 0 : 99 ; 0 : 94 ; 0 : 9) G m h =(1 : 05 ; 1 : 05 ; 1 : 05 ; 1 : 02) Subepicardiumpre-stretches(A,LA,IA,SA)* G e = G e z =(1 : 22 ; 1 : 12 ; 1 : 06 ; 1 : 06) G k h =(1 : 05 ; 1 : 04 ; 0 : 97 ; 0 : 92) G m h =(1 : 05 ; 1 : 05 ; 1 : 05 ; 1 : 05) *A:Arteries,LA:Large,IA:Intermediate,SA:Smallarterioles. Figure4.4:Estimatedmassfractionsforthearteriesandarteriolesoftlayers.(A: Smallarteries,LA:Large,IA:Intermediate,SA:Smallarterioles.) 4.3 Resultsanddiscussion 4.3.1 Baselineoptimization Figure4.5showsthediameterandradiusexponentasresultsoftheoptimization.Suben- docardialbloodvesselsappeartobelargerthantheirsubepicardialcounterparts.Thisdif- 71 Figure4.5:Thebaselineoptimizationresultsplottedalongwiththegenerationnumber: Left:diameterdistribution;Right:radiusexponentindaughter-to-parentradiirelation ( R ˘ p = R ˘ d 1 + R ˘ d 2 ). ference,consistentwithexperimentalandsimulationresultsfrom[88,133],isduetohigher bloodperfusioninthesubendocardiallayer.Furthermore,theoptimizationresultsinin- creasingradiusexponent ˘ (2.5-2.7)withthegenerationnumber,forthebloodvesselsfrom 600 to25 .TheoriginalMurray'slawpredictedacubicradiusexponent.However, severalexperimentalstudiesonthemorphometryofvasculartreeshaveshownvarying ˘ between2inlargervesselstonear3inprecapillarylevel[85,86,189,189,190].Artsand Reneman[182]studiedthethescalinglawsindog'scoronaryvasculatureandshowedan exponentof2.55forthevesseldistalsegmentswith400 diameter.Suwaandcowork- ers[190]showedanexponentof2.7byanalyzingtvasculatureinseveralorgansin humanbody.Nevertheless,theradialexponentfromouroptimizationresultisconsistent withthoseexperimentalstudies. Figure4.6showsthestructuralandhemodynamicsresultsoftheoptimization.Theopti- mizationframeworkisabletocapturetheexperimentalthickness-to-diameterratio([144]) byslightmoofthebasalactivetone(within10%oftheestimatedactivation).Top leftpanelofFig.4.6showstheperformanceoftheoptimizationinthisregard.Mostofthe 72 coronaryvascularresistanceislocatedinarteriolesofsmallerthan100 .Thisrangeseems tobetheconstantacrosstorgansandtspecies(Fig.4.6).Similarly,Van- BavelandSpaan[191]showedthatthepressureinthecoronarymicrovasculaturedropsfrom 90to30mmHginthesmallvesselsof10- order.Ontheotherhand,thewallshearstress increasesalmost4-foldinthearterioles,consistentwithexperimentalstudies.Inparticular, Steppandcolleagues[118]analyzed ˝ incaninecoronarymicrovasculatureandobservedthat arteriolesof < 160 haveanelevatedlevelofshearstress.Insummary,thehemodynamics results,wallshearstressandpressure,theavailabledataonthemicrocirculatorysystem inaquantitativeandqualitativemanner. Furthermore,thebottom-rightpanelofFig.4.6showshomeostaticcircumferentialstress, whichiscomputedfromLaplace'slaw( ˙ h = p tm R H )where H isthe in-vivo thickness.There- fore,sincetheintramyocardialpressureislowertowardstheepicardium(highertransmural pressure),thesubepicardialhomeostaticstressislarger.GuoandKassab[194]analyzedthe circumferentialstressintheswinecoronaryarterialtreeandreportedthecircumferential stressintherangeof9.4-159kPaforarteriesof9.8-3097 .Althoughwecapturedthe sametrend,thevaluesfromoursimulationdonotcompletelymatchwiththoseoftheir study.Thisdiscrepancymaybeattributedtothefactthattheirhemodynamicsanalysis wasconductedinanarrestedheartwherethevasculartransmuralisequaltotheluminal pressure(i.e.,nointramyocardialpressure).Inamorerecentstudy,ChoyandKassab[145] evaluatedthemedialandintimalthicknessincoronaryarterialtreesandobservedthatthe thicknessinsubendocardiallayerislowerthanthethicknessinsubepicardiallayer,which leadstoahigherdiastoliccircumferentialstress( p im =0).Itisworthnotingthatthe aforementionedstudieswereconductedonvesselswithoutaantSMCtone. 73 Figure4.6:Homeostaticoptimizationresultsplottedagainstdiameters:Top-left:wall thickness-to-diameterratioinunloadedtion[144];Top-right:mid-arterypres- sure[86,186,192,193];Bottom-left:homeostaticvalueofwallshearstress( ˝ )[118,135,186]; Bottom-right:homeostaticvalueofcircumferentialstress ˙ h . 4.3.2 Autoregulation Beforepresentingtheresultsforthecoronarywregulation,itshouldbenotedthatthe outletpressureprescribedinthehomeostaticoptimizationisattheprecapillaryarterioles. Toenhancethegfortherangeofcoronarywinthearterialtree,weaddlumped parameterswithconstantresistanceattheendofeacharterialtreesegmentandprescribethe venouspressure20mmHgastheoutletpressureboundaryconditionofthetree.Tocalibrate 74 Table4.3:Estimatedcontrolparametersfortheautoregulatoryresponse. Parameter SubendocardialSubepicardial a p 0.320.23 a ˝ 1.301.10 a m 0.501.5 Figure4.7:wautoregulationinthecombinedtreescomparedtothecollected experimentaldatain[103].Thewisnormalized( q )againstthewat p in =100mmHg. thecontrollingparameters, a p , a ˝ ,and a m ,weusedthewautoregulationdata collectedin[103].TheestimatedcontrollingparametersarepresentedinTable4.3. Figure4.7showstheestimatedautoregulatoryresponseofthecoronaryvasculatureasthe averageoftheresponseinsubendocardialandsubepicardiallayers.Ourmodelthe experimentalobservationscapturingtheessentialfeaturesoftheautoregulationinrangeof inletcoronarypressures70-140mmHg.However,inthelowerinletpressures(20-50mmHg) thewratepredictionsareonaverage20%lowerthantheexperimentalmeasurements. Inthisrange,themodelpredictionsofwcurveisboundedbythefullydilated (passive)responseofthearteries.Thus,thescarcityintherangeandnumberofavailable 75 Figure4.8:Thepredicteddependenceofwallshearstressandactivationonthemeaninlet pressure.Wallshearstressineachvesselisnormalized( ˝ )againstitsvalueat p in =100 mmHg. dataonpressure-diameterrelationshipscouldbeacontributingfactorinthisdiscrepancy. Adetailedanalysisofthehemodynamicsofthearterialtreesisnecessaryinunderstanding theheterogeneousinteractionbetweentmechanismsinthecoronarywautoreg- ulation.Figure4.8showsthenormalizedwallshearstress( ˝ )andtheactivationlevelfor fourvesselsinthelayersofthemyocardium.Clearly,sincethemechanismcontrollingthe shearisalmostfullyeinthesmallestvessels,wallshearstressdoesnotshowany tregulationinthesmallandintermediatearterioles[118].Thearteries,however, 76 Figure4.9:Theofpressurereductiononcoronaryarterialmicrovesselscomparedto theexperimentalobservationsonepicardialmicrovasculatureindogs[114]. exhibitnotableofregulationofwallshearstressaroundthebaselinepressure. Activationlevels(amarkerofvasoconstriction)inthearteriolesmonotonicallyincreasefrom lowtohighinletpressures.Thisisexpected,sincethemyogenicandmetabolicmechanisms simultaneouslyplaydominantrolesinautoregulation[195,196].Furthermore,thearterioles ofthesubepicardiallayerseemtobemoresensitivetochangesinpressureandshowagreater variabilityintheactivationlevel.Alternatively,theactivationlevelinlargevesselsremains almostconstant,exhibitingnotresponsetochangesinthecoronarypressure. Toidentifythelocationoftheautoregulationinswinecoronaryarteries,twocasesofpressure reductionareconsidered.Whenthecoronarypressurewasmildlyreduced,thevesselssmaller than150 weredilatedwhereasthelargevesselsalmostremainedintheirhomeostatic value.Moreover,themagnitudeofdilationwasinwithseverelyreducingtheinlet pressure, p in =38mmHg,whilethelargervesselsshowedsomedegreeofconstriction.Similar resultswasobservedinastudybyKanatsukaetal.[114]ontheepicardialcoronariesofdogs. However,adirectquantitativecomparisonmightnotbeaccurateduetotheinterspecies inthecoronaryphysiology. 77 Figure4.10:Diametersasfunctionsof p in fortvesselsofsubendocardialandsubepi- cardiallayers. Itisworthytonotethattheextravascularpressuresactingonvesselsembeddedinthelayers ofmyocardiumaretlyent(13-47mmHg).Particularly,thesubendocardial p tm isnegativeinlargepartsofthearterialtreewhen p in islow.Consequently,thesubendocardial activationoftheSMCsisdelayedcomparedtothesubepicardiallayer.Thisdiscrepancy betweenlocationofdilateddiametersleadstoanincreaseofENDO/EPIin p in =75mmHg. Adilationofthearteriolesfollowedbyaconstrictionhasbeenobservedinautoregulationof othercirculatorysystems[109]. Alternatively,tialautoregulatoryresponseofthearteriestoachangeinpressureare analyzedintermsofENDO/EPI(Fig.4.11).NormalENDO/EPIbloodwratioshave beenreportedbetween1.09to1.49acrossditspecies[97].Themodel,however,shows thatwithaseverereductionofpressure( p in < 60mmHg),theENDO/EPIratioreducesto belowone.TheinsetinFig.4.11showsasimilartrendobservedinthetransmuralanalysis ofautoregulationincaninecoronarycirculation[197].Similarly,theexperimentsbyBalland Bache[198]showeda50%-60%decreaseintheENDO/EPIratioincanineLVasaresultof mild-severeobstructionsinlargecoronaryarteries. 78 Figure4.11:Thepredictedtransmuraldistributionofthewduringwautoreg- ulation.Theinsetshowstheobservationsindogs,from[197]. 4.3.3 ofadenosineandNOinhibitors Toevaluatethepredictioncapabilitiesofthemodel,administrationofadenosineandinhi- bitionofNOproductionareparametericallymodeledinthissection(Fig.4.12).Adenosine isametabolicvasodilatorwhichmostlythedownstreamcoronarymicrovessels[102]. Theadenosineadministrationismodeledbysettingthearteriolaractivationtozero( A =0). Alternatively,theinhibitionofNOismodeledvia1)almostfullconstrictionofsmallarteries, duetoinhibitionofshear-dependentvasodilation,and2)utilizationofthefullvasodilatory capacityinthearterioles.Theformerisbyconsideringthefactthatinthesmall arteries, ˝ istheonlymediatorofvasodilation.Theabsenceofthisdependency(viainhibi- tionofNO)leadstofullSMCtoneinsmallarteries.Thelatter,however,isbythe experimentalobservationinepicardialcoronariesofdogs,intheworkofJonesetal.[199].In fact,theyobservedthataninhibitionofNOsynthesisobstructsfurtherdilationasaresultof 79 Figure4.12:TheofadenosineinfusionandNOinhibitiononcoronaryarterialmi- crovesselscomparedtotheexperimentalobservationsonepicardialmicrovasculaturein dogs[199]. adenosineadministration.Itisworthytomentionthatinfusionofsuchexogenousagentsdid nottlychangethecoronarypressure.Figure4.12showsafairagreementbetween oursimulationresultsandtheexperimentalobservations.However,itshouldbenotedthat ourmodelreliesonthedatafromswinemyocardialvesselswhiletheobservationsarefrom canineepicardium. 4.4 Summaryandconclusion Inthischapter,thehomeostaticoptimizationframeworkofChapter2wasemployedto estimatethebaselinecharacteristicsofcoronaryarterialtree.Diverseexperimentaldataof swinecoronarieswereintegratedtoconstructtwoarterialtreeslocatedinsubendocardial andsubepicardiallayersofthemyocardium.Consequently,theestimatedhomeostaticstate ofthearterialtreeswasusedtoconstructaconstrainedmixturemodelforthecoronaryw regulation. Theimplementedmechanicalmodelwascalibratedagainstpressure-diameteravailabledata 80 intheliterature.Ourparameterestimationindicatedthatuponthe in-vivo levelofSMC tone,thecollagenersandelastinarevirtuallyinacompressedstateinthecoronary arterioles.Moreover,theestimatedmassfractionsoftheSMCsjustifytheheterogeneity oftheactiveresponseacrossthearterialtree.Inaddition,therelativemyocardialdepth ofbloodvesselsseemstobeacontributingfactorinestablishingtheirbaselinemechanical properties. Ourcomputationalpredictionsshowedexcellentcongruitycomparedtothedatafromob- servationsinthemicrocirculatorynetworks,bothquantitativelyandqualitatively.Previous modelingincoronarywregulationwerebasedonthe ex-vivo observationsofthe activeresponsesofthearteries.Thephysiologicalwregulationinthecoronaryvascula- ture,however,greatlyreliesonthedeviationsfrombasalquantitiessuchaspressureand wallshearstress.Therefore,oneofthecontributionsofourmodelisestimatingthebasal conditionsforthewcontrolmechanisms. Theconstructedarterialtreeswereusedtostudythewregulation,sppressure- wautoregulationindistalcoronaryarterialnetwork.Totheauthor'sknowledge,thisis theimplementationofaconstrainedmixturemodeltocoronaryarterialtrees.This implementationenhancesthepredictioncapabilityforcoronarywregulationaswellas arterialfunctionandvascularadaptation.Thisstudyillustratedthatthemodelwascapable ofcapturingtheessentialw-pressurerelationshipintheautoregulationofcoronaryarteries. Furthermore,wewereabletoperformantialanalysisofthesize-dependentand transmuralheterogeneityoftheautoregulatoryresponse.Themodelhighlightedthatthe autoregulationoccursmostlyinthelevelofcoronaryarterioleswhereasshearregulationis dominantonlyatthelevelofarteries.Inaddition,weobservedthattherangeofdilation andconstrictionofvesselsisafunctionoftheirrelativedepthwithinthemyocardium. Furthermore,weusedcomputationalsimulationsofadenosineinfusionandinhibitionofNO productionandcomparedtheresultswithavailableexperimentaldata. 81 Thepresentstudyhasseverallimitations.First,thecomputationalmodelisnotyetfullyval- idatedbywell-controlledanimalexperiments.Particularly,theinformationonthepressure- diameterrelationshipofvesselsinsidethemyocardiumisstillscarce.Inthisstudy,weused therelationshipsfromepicardialandsubepicardialvesselsforparameterestimationsofthe mechanicalmodel.Moreover,theavailablepressure-diameterdataextendtothevesselsof around50 ,whileagreatportionofthecoronaryvascularresistanceresidesinsmaller vessels(terminalarterioles).Moredataonarteriolessmallerthan50 willenhancethe predictioncapabilityofthemodel.Inaddition,wedidnotincludethetetheringofthearteri- olestothemyocardiuminourmodel.AlthoughasproposedbyYoungetal.[200],thesmall arteriolescanfreelyconstrict,thedilationofvesselsmightbelimitedbytheirtetheringto myocardium.Regardless,thetwo-waytetheringofthemyocardiumandcoronaryarteriesis notfullyunderstood,yet.Furthermore,ouranalysisislimitedtotheaveragedsteadystate ofthevasculature,whilethehemodynamicsofthecoronaryarteriesishighlydependent onthesystolicanddiastolicphases.Asamatteroffact,mostofthesubendocardialw hasbeenshowntooccurduringthediastolicphasewherethemyocardialpressurevanishes. Whileouranalysisshowedthatanapplicationofananalytical1Dtheory(seeChapter3)to coronaryhemodynamicsisnotphysicallyrealistic,thecurrentmodelcanbeendowedwith anon-linearelementmodelofthearterialtreetoovercomethislimitation.Lastly,an openloopanalysisofthemetabolicregulation,aswasdoneinthischapter,mayobstruct clearinterpretationoftheinteractionofthemechanisms.Therefore,extendingtheregu- lationmodeltoincludetheclosed-loopaspectsofmetabolicvasodilationwillimprovethe capabilitiesofthemodelandfacilitatethephysiologicalinterpretations. 82 Chapter5 ConclusionandFutureWork 5.1 Conclusion Inthisdissertation,wedescribedanovelcomputationalframeworkformultiscalebiome- chanicalmodelingofvascularadaptationsinarterialnetworks.Theprincipleofminimum metabolicenergyconsumptionwithlocalimportantmechanicalhomeostaticvalueswassuc- cessfullyimplementedinmodelingthearchitectureofvascularnetworks.Suchamodelpro- videsthegeneralwwofG&Rinanarterialnetworkandestablishesabaselinewhich issubstantialintheG&Rstudiesoftheonsetandprogressionofmanyvasculardiseases. ThemodeldevelopedinthisdissertationconstructsarterialtreesviaanextensionofMurray's lawwhichprovidesatightcouplingofthehemodynamicsandthearterialwallmechanics. Furthermore,eachindividualvesselinthenetworkisendowedwithaconstrainedmixture modelwhichpavesthewayformultiscalemodelingofstress-mediatedmassproduction andremovalduringgrowthandremodelingofarterialtrees.Ourmodelreliesheavilyon availabledatafromtheliteratureandprovidesauniformframeworkinwhichdatafrom tstudies(i.e.,hemodynamics,morphometric,structural,etc.)areintegratedina 83 biomechanicallyconsistentmanner.Moreover,sincemosttheavailabledataarerecorded inexternalenvironment,anothernoveltyofourmodelisinpredictingthe in-vivo geometry andpropertiesofthevasculaturewhichisifnotimpossible,tomeasure. Wehaveshownimplementationsoftheframeworktostudymutliscalevascularphenomena. Theapplicationwastoestimatethehomeostaticbaselinecharacteristicsofpulmonary arterialtrees.Theframeworkproducedanestimationofimportantpropertiesofthedistal vasculaturesuchashomeostaticmassfractions,constituent-wisestresses,andsteadystate hemodynamics.Suchpropertiesareessentialinthestudyofstress-mediatedgrowthand remodelingofthedistalvasculature,whichisaprominentfeatureoftheearlystagesofPAH. WealsoillustratedthecapabilityofthemodelinFSGsimulationviaaone-waycouplingof theresultsofhomeostaticoptimizationwithpulsatilehemodynamics.Theequivalentinput impedancewasalsocomputedforthedistalvesselswhichcanbeusedforpatient-sp modelingsofpulmonaryarterialhemodynamics. Second,wehighlightedtheversatilityofthemodelbyanimplementationofthemodelto studytheofcoronarywregulation.Weusedtwomyocardialarterialsub-trees,distally locatedtotheLADandinsidetheleftventricle.Themyocardial-vascularinteractionswere includedinourmodelviaconsiderationofaverageinterstitialpressureandmyocyte shortening.Furthermore,weutilizedthepassiveandactivepressure-diameterrelationsfrom theliteraturetoestimatethemechanicalproperties,massfractionsofconstituents,thebasal activationlevelofsmoothmusclecells,andbaselinehemodynamicsincoronarymicrovascu- lature.Theresultsofthebaselineoptimizationdemonstratedexcellentconsistencywiththe datafromtheexperimentalstudies.Furthermore,weusedtheestablishedbasalpropertiesto studythewautoregulationofcoronaryarteries.Ourmodel,consistentwithex- perimentaldataondogs,showedthattheautoregulationmostlyoccursinsmallerarterioles. Moreover,subendocardialvesselsseemtoreachfulldilationinalargerinletpressurewhen comparedtothesubepicardialvessels.Theseanalyses,coupledwithexperimentaldata,are 84 crucialinidennofthemyocardialischemicsusceptibility.Wealsoillustratedtheca- pabilityofthemodelinsimulatingthemicrovascularresponsetotheintroductionofdilatory orconstrictiveagents. Althoughsplimitationsofthedevelopedframeworkinpulmonaryandcoronaryimple- mentationshasbeenalreadydocumentedinSection3.4and4.4,itisworthytohighlighta fewremarksongenerallimitationsinthecomputationalmethodthatcanbeconsideredfor furtherinvestigation.First,thecurrentstudyofoptimizationwasoperatedusingprevious referencesofsmoothmusclecellmetaboliccosts,whichdidnottakeintoaccountoftheother constituents.Therefore,asystematicparametersensitivitystudyneedstobeconductedto distinguishthemainsourcesofuncertaintiesinthemodelingframework.Suchanalysiswill becrucialincalibrationandvalidationofthemodelagainstexperimentalobservations.Sec- ond,thestructureofthetreeinthisstudyisbasedongenerationsandbifurcations(inspired byfractaltreeideain[69])whereasexperimentalstudiesdescribethemorphometryofarterial networksbasedonorderingmethods(e.g.,Strahlerorderingmethod).Thisdisparitymay hinderaglobalcomparisonofthemorphometryofthegeneratedtreewiththeexperimental studies.Ourmodelingframework,however,canbeappliedtothepre-constructedarterial networksfordiameterassignmentandhemodynamicsasindicatedinthenextsection. 5.2 Futurework Additionalexperimentaldataonthestructureoftheproximalanddistalvessels(i.e.,thick- nessandmassfractionsofconstituents),morphometryofthearterialtrees,energeticcon- siderationsofthevascularwall,etc.arerequiredforvalidationand/orcalibrationofthe model.Withmoreavailabilityinhumandata,theframeworkcanbeimplementedonarterial networkswhicharerepresentativesforhumansubjects,healthyorbydisease. 85 Morphometricstudiesonthestructureofarterialnetworkswereconductedondissectedvas- culaturewherethepropertiesofthearterialtissuemighthavealtered(e.g.,novasoactive tone).Therefore,themainlimitationofsuchstudieswastheirinabilitytopredictthe in-vivo diametersofvesselsegmentswhereasthepresentedframeworkiscapableofmodel-baseddi- ameterre-calibrationinsuchanalyses.Thiswillgreatlyimprovethecurrentknowledgeof themicrovascularnetworksandenhancetheaccuracyofmodelingstudies.Alternatively, endowingourmodelwith\vlling"or\avoidance"algorithmsareotherpossibleex- tensionofthemodelthatmaybeexplored. Womersley'ssolutioniscapableofsimulatingthepulsatilehemodynamicsinaphysiologically realisticmanner.Themainadvantageofimplementationofthistheoryisthecomputational andalgorithmicsimplicity.Nonlinearelementmodels,however,canenhancetheme- chanicalanalysisbyincludingbiomechanicalcomplexitiessuchasnonlinearityofthevascular walland/orextravascularforces(e.g.,intramyocardialpressure).Theresultsoftheproposed frameworkcanbedirectlyusedinanon-linear1Delementmodelfordetailedanalysis ofpulsatilehemodynamicsandextensionstoFSG. WeestablishedthehomeostaticbaselineforthepulmonaryarterialnetworkinChapter 3.Consequently,ourmodelenablesustostudylong-termpathologicalconditionsinPAH suchasmmationandproliferationofSMCs,remodelingofextracellularmatrixetc. Furthermore,themodelfacilitates in-silico experimentsonintroductionexternalstimuli suchasvasodilation(viaNOinhalation[201]),elastaseinhibition(via[202]).Similarly, thedistalpathologicalconditionsofthemicrovasculatureinobstructiveandnon-obstructive CADcanbestudied.Alternatively,themodelcanbeemployedtostudycardiacallograft vasculopathy(CAV)wherethedistalremodelingispronouncedbyendothelialdysfunction, andintimalthickening. Inclosing,themaingoalofthisdissertation,todevelopaframeworktoembedtheG&Rin multiscalearterialnetworks,hasbeenachieved.Weanticipatethatourframeworkpavesthe 86 wayforfuturestudiesofvascularG&Rincharacteristicallymultiscaleproblems.Utilization ofsuchcomputationalmodelsissubstantialinunderstandingtheonsetandprogressionof diseasesandadvancingnewtherapiesfortheirmanagement. 87 APPENDICES 88 APPENDIXA:Formulationofsmallonlargetheory Thetheoryofsmalldeformationssuperimposedonlarge(namely`theoryofsmallonlarge' (SoL))hasbeenwellformulatedin1950-60s[203{205].Onlyrecently,however,theSoLwas reformulatedtolinkmodelthepulsatiledeformations invivo [5,160].Thistheoreticaltool henceservesasausefultooltoobtainalinearizedresponseofwallduringthecardiaccycle withoutcompromisingimportantmechanicalcharacteristicssuchasanisotropyandsmooth muscletone.However,thepriorformulationofSoLwasstillnotfullyunderstoodinorder toprovideasolidtheoreticalfoundationinvascularmechanics.Particularly,thepriorSoL formulationadoptedaLagrangemultiplierapproachasmanystudieshaveregularlydone forbiologicalcontinuum-mechanicsproblems.However,BaekandSrinivasa[206]illustrated thatconstraintsprescribedbytheLagrangemultipliercouldobscurethephysicalinterpre- tationofmathematicaloperations,forinstance,whentakingthesecondderivativesofthe energyfunctionsforderivingbulkmodulusandspecheat.Likewise,thisLagrangemul- tiplierapproachmayimpedeaclearphysicalinterpretationofSoLapplicationonvascular mechanics[207]. Preliminaries Letthemotionofasolid-likebody B berepresentedbymappings ˜ ofaparticlefroma reference R ( B )attimet, x = ˜ ( X ;t ) ; (A.1) where X and x arepositionvectorswithrespecttothereferenceandcurrent Furthermore,weconsiderthebodyoccupiesa 0 ( B )withthepositionvector x 0 = ˜ ( X ;t 0 )whichischaracterizedbyalargedeformationmeasuredfromthereference 89 Thus,thedeformationsofthebodyconsistoftwoconsecutiveparts:asmall displacement, u = u ( x 0 ;t ),superimposeduponthelargedeformation.Therefore,thecurrent positionofaparticlecanbewrittenas x = x 0 + u ( x 0 ;t ) : (A.2) Therefore,deformationgradientsassociatedwithmappingsfromthereference totheintermediateandcurrentare F o = @˜ ( X ;t 0 ) @ X ; F = @˜ ( X ;t ) @ X : (A.3) Similarly,forthesmalldeformationswecan F = @ x @ x 0 = I + H ; H = @ u @ x 0 ; (A.4) where F = F F o : (A.5) Inlinearelasticity,thesymmetricandskew-symmetricpartsof H canbeexpressedas = 1 2 H + H T ; (A.6) = 1 2 H H T ; (A.7) where and arethestrainandrotation,respectively.The CauchystresscanbewrittenintermsofthedeformationgradientandthesecondPiola- 90 Kircstress S withrespecttothereferenceion T = J 1 FSF T ; S =2 @W @ C ; (A.8) where J =det( F ),and W isthestoredstrainenergyfunctionasafunctionof C = F T F . ThesecondPiola-Kirchstress S withrespecttotheintermediate 0 ( B ) canbeexpressedas S = J o 1 F o SF o T ; (A.9) where J o =det( F o ).Usingapushforwardoperation,theCauchystresscanbewrittenas T = J 1 F S F T ; (A.10) where J =det( F ). LetCauchystressinanyconvenientintermediatetionberepresentedby T o whereas thatinanycurrentbedenotedas T .Wecanwritetheincrementalstressas T = T T o .Therefore,weareseekinganelasticitytensorconnecting T andthe . Toderiveanexpressionfortheincrementalstressresponseinthesmalldeformation,weuse equationsA.4andA.10togettherelation T = 1 J o f det( I + H ) g 1 ( I + H ) F o S o + @ S @ C C o C F o T ( I + H ) T ; (A.11) wheretherightCauchy-Greentensor C isas C =2 F o T F o (A.12) 91 Usingtheapproximation f det( I + H ) g 1 ˇ 1 tr( H )forsmall H andneglectingthehigher ordertermswillgive T = tr( H ) T o + HT o + T o H T + F o @ S @ C C o C F o T (A.13) = tr( H ) T o + HT o + T o H T +2 F o @ S @ C C o F o T F o F o T (A.14) Writingthelastterminindexnotationgives T ij = H kk T o ij + H ik T o kj + T o ik H jk + 2 J o F o F o j F o lp F o mq @S @C pq C = C o lm : (A.15) InsertingthesecondPiola-Kircfromequation(A.8)gives T ij = H kk T o ij + H ik T o kj + T o ik H jk + 4 J o F o F o j F o lp F o mq @ 2 W @C @C pq C = C o lm : (A.16) UsingthedecompositioninequationsA.6andA.7,thelinearizedstressresponsecanbe writtenas T ij = C ijkl kl + D ijkl kl ; (A.17) where C ijkl = kl T o ij + ik ml T o mj + jk ml T o im + 4 J o F o F o j F o lp F o mq @ 2 W @C @C pq C = C o ; (A.18) D ijkl = ik ml T o mj + jk ml T o im ; (A.19) Finitedeformationmechanicsofanartery Weassumethearterialwalltobecompressibleatandthenincompressiblityisconsid- eredasalimit.Tothisaim,acompressibilityterm k ( J 1)isaddedtothestrainenergy 92 ofthearterygivenin(2.2.2).However,beforere-introducingtheconstitutiverelations, letusexpandthemotionofthearteryfromareferencetoanintermediate as F o = 2 6 6 6 6 4 r 00 0 0 00 z 3 7 7 7 7 5 ) C o = 2 6 6 6 6 4 2 r 00 0 2 0 00 2 z 3 7 7 7 7 5 : (A.20) Wecanre-writethestrainenergyperunitareaas w = M e R e + X k M k R k + M m R m + m act )+ k ( J 1) 2 ; (A.21) where J =det( F ).Followingtheformulationin2.2.2,thedeformationgradientmapping eachconstituentformitsnaturaltiontotheintermediatecanbe writtenas F e o = F o G e ; F k o = F o G k ; F m o = F o G m ; (A.22) Withregardstotherelations @J @ F = J 1 F T ; (A.23) @J @ C = J 2 C 1 ; (A.24) @C 1 ij @C pq = 1 2 C 1 ip C 1 jq + C 1 iq C 1 jp ; (A.25) 93 itcanbeshownthat @w @C ij = M e R @ e @C ij + X k M k R @ k @C ij + M m R ( @ m @C ij + @ m act @C ij )+ kJ ( J 1) C 1 ij ; (A.26) @ 2 w @C ij @C pq = M e R @ 2 e @C ij @C pq + X k M k R @ 2 k @C ij @C pq + M m R ( @ 2 m @C ij @C pq + @ 2 m act @C ij @C pq ) 1 2 kJ ( J 1) C 1 ip C 1 jq + C 1 iq C 1 jp + 1 2 kJ (2 J 1) C 1 pq C 1 ij (A.27) TheCauchystressintheintermediatecanbewrittenas T o = 2 J o F o @w @ C C = C o F o T ; (A.28) and,thus,fromequation(A.26),wecanwrite T o ij = 2 J o M e R F o F o j @ e @C C = C o + 2 J o X k M k R F o F o j @ k @C C = C o + 2 J o M m R F o F o j ( @ m @C C = C o + @ m act @C C = C o )+2 k ( J o 1) ij = ^ T o ij +2 k ( J o 1) ij : (A.29) Forthesakeofsimplicity,wemergethethreetermsofEq.A.29into ^ T o ij ,whichdenotes theinternalstressinthearteryasaresultofdeformation.Similarly,Eq.(A.27)canbe writtenas @ 2 w @C ij @C pq C = C o = ^ C ijpq 1 2 kJ ( J 1) C 1 ip C 1 jq + C 1 iq C 1 jp + 1 2 kJ (2 J 1) C 1 pq C 1 ij ; (A.30) ^ C ijpq = M e R @ 2 e @C ij @C pq C = C o + X k M k R @ 2 k @C ij @C pq C = C o + M m R ( @ 2 m @C ij @C pq C = C o + @ 2 m act @C ij @C pq C = C o ) (A.31) 94 Constitutiverelations ThestrainenergydensityfunctionfortheconstituentsofthearterialwallaregiveninEqs. 2.11-2.13.Usingtheserelations,equations(A.20),(A.22),and(A.29)andthechainrule,we canwritethestresscomponentsattheintermediateas T o rr = c 1 J o M e R ( G e r ) 2 2 r +2 k ( J o 1) ; (A.32) T o = c 1 J o M e R ( G e ) 2 2 + c 2 J o X k M k R ( G k h ) 2 (( k ) 2 1)exp( c 3 ( k 1) 2 )sin 2 ( k ) 2 + c 4 J o ( G m h ) 2 (( m ) 2 1)exp( c 5 ( k 1) 2 ) 2 + 1 J o S ˆ M m R 1+ ( M ) 2 ( M 0 ) 2 +2 k ( J o 1) ; (A.33) T o zz = c 1 J o M e R ( G e z ) 2 2 z + c 2 J o X k M k R ( G k h ) 2 (( k ) 2 1)exp( c 3 ( k 1) 2 )cos 2 ( k ) 2 z +2 k ( J o 1) : (A.34) Fortheandextensionofanartery,weassumethattherotationalpart kl =0. Moreover,weassumethearterialwalltobeanorthotropicmaterial.Thus,theelasticity tensorinEq.A.18canbewritteninVoightnotationandthereferencesystemofprincipal direction C = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 C rrrr C rr C rrzz 000 C rr C C zz 000 C zzrr C zz C zzzz 000 000 C zz 00 0000 C rzrz 0 00000 C rr 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 4 C 1 0 0C 2 3 7 5 ; (A.35) 95 wherethecomponentsof C 1 canbeexpressedas C rrrr = T o rr +2 k; C rr = T o rr +2 k (2 J o 1) ; C rrzz = T o rr +2 k (2 J o 1) ; C rr = T o +2 k (2 J o 1) ; C = T o +2 k + 2 J o A 4 C zz = T o +2 k (2 J o 1)+ 2 J o A z 2 2 z ; C zzrr = T o zz +2 k (2 J o 1) ; C zz = T o zz +2 k (2 J o 1)++2 k (2 J o 1)+ 2 J o A z 2 2 z C zzzz = T o zz +2 k + 2 J o A zz 4 z (A.36) A = c 2 X k M k R ( G k h ) 4 (2 c 3 (( k ) 2 1) 2 +1)exp( c 3 (( k ) 2 1) 2 )sin 4 ( k ) + c 4 M m R ( G m h ) 4 (2 c 5 (( m ) 2 1) 2 +1)exp( c 5 (( m ) 2 1) 2 )+ S ˆ M m R ( M ) ( M 0 ) 2 A z = c 2 X k M k R ( G k h ) 4 (2 c 3 (( k ) 2 1) 2 +1)exp( c 3 (( k ) 2 1) 2 )sin 2 ( k )cos 2 ( k ) A zz = c 2 X k M k R ( G k h ) 4 (2 c 3 (( k ) 2 1) 2 +1)exp( c 3 (( k ) 2 1) 2 )cos 4 ( k ) (A.37) Forsimplicity,thetermsrelatedtocollagenersandpassivesmoothmuscleresponsein equationA.36aredenotedas A , A z ,and A zz ,andthetermcorrespondingtheactivetone as A act . Theelasticmoduli E i ,Poissonratios ij ,andshearmoduli ij canbeobtainedby 96 thefollowinginversematrix S = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 E r r E zr E z 000 r E r 1 E z E z 000 rz E r z E 1 E z 000 000 1 2 z 00 0000 1 2 rz 0 00000 1 2 r 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 4 C 1 1 0 0C 1 2 3 7 5 ; (A.38) Sp,wecanderivethefollowingformulasforYoung'smoduli E r = det( C 1 ) C C zzzz C zz C zz ; (A.39) E = det( C 1 ) C rrrr C zzzz C rrzz C zzrr ; (A.40) E z = det( C 1 ) C C rrrr C rr C rr : (A.41) Calculationofelasticmoduluswiththeincompressibilityconstraint asalimit Now,letusagainconsidertheandextensionofastraightsegmentofartery.We assumethatthearterycanbemodeledasathin-walledcylinder,therefore,weapproximate T o rr ˇ P o = 2where P o istheinsidepressureintheintermediateMoreover, substituting J o =det( F o )intoequation(A.32)gives T o rr = c 1 r z M e R ( G e r ) 2 2 r 2 k ( r z 1)= P o 2 : (A.42) 97 Solvingthisequationfor r gives r = z (4 k P o ) 2( C 1 +2 k 2 2 z ) (A.43) where C 1 = c 1 M e R ( G e r ) 2 forthesakeofbrevity.Equation(A.43),thus,calculatesthe J o J o = 2 2 z (4 k P o ) 2( C 1 +2 k 2 2 z ) : (A.44) Componentsoftheelasticitytensor C showthattheexpressionsforelasticmoduliinclude volumechangeratioaswellasthepre-stressesandpre-stretches. Thearterialwallisconsideredanincompressiblematerial.Thestrainenergyfunction giveninEq.A.21,however,hasacompressibilityterm k ( J 1) 2 whichvanisheswhen anincompressiblematerialistakenasalimit.Inotherwords,theterm k ( J 1)canbe treatedasapenaltytermforincompressiblematerial.ClearlyfromEq.(A.44),ifwewrite lim k !1 J o =1,wewillachievetheincompressibility.Nevertheless,undersuchconditions, theterm k ( J 1)willremainanundeterminedterm.Specifyingtheboundaryconditions willfacilitatethisparameter.Spinthecurrentapplication,thepressure P o , andpre-stretches and z canbeconsideredasboundaryconditions.Thus,wecanwrite lim k !1 k ( J o 1)=lim k !1 k ( 2 2 z (4 k P o ) 2( C 1 +2 k 2 2 z ) 1) ; (A.45) whichbringsanindeterminateformof 1 0.UsingL'Hospital'srulegives lim k !1 k ( J o 1)= 1 4 ( 2 C 1 2 2 z + P o ) ; (A.46) whichwillbringtheanundeterminedterm.Incorporatingrelation(A.46)into(A.39-A.41), willgivetheeelasticmoduliandPoisson'sratiosinthethreeprincipaldirections. 98 E r = 2( A 2 z 4 4 z +(3(2 C 1 + 2 2 z P o ) 2 ) = (4 4 4 z ) 2 A z 2 2 z T o rr + T o rr T o + A zz 4 z ( T o rr + T o )+( T o rr + T o ) T o zz + A 4 ( A zz 4 z + T o rr + T o zz ) +((2 C 1 + 2 2 z P o )( A 4 A z 2 2 z + A zz 4 z + T o rr + T o + T o zz )) = ( 2 2 z )) A 4 +(2 C 1 ) = ( 2 2 z ) 2 A z 2 2 z + A zz 4 z + P o + T o + T o zz (A.47) E = 2( A 2 z 4 4 z +(3(2 C 1 + 2 2 z P o ) 2 ) = (4 4 4 z ) 2 A z 2 2 z T o rr + T o rr T o + A zz 4 z ( T o rr + T o )+( T o rr + T o ) T o zz + A 4 ( A zz 4 z + T o rr + T o zz ) +((2 C 1 + 2 2 z P o )( A 4 A z 2 2 z + A zz 4 z + T o rr + T o + T o zz )) = ( 2 2 z )) (2 C 1 ) = ( 2 2 z )+ A zz 4 z + P o + T o rr + T o zz (A.48) E z = 2( A 2 z 4 4 z +(3(2 C 1 + 2 2 z P o ) 2 ) = (4 4 4 z ) 2 A z 2 2 z T o rr + T o rr T o + A zz 4 z ( T o rr + T o )+( T o rr + T o ) T o zz + A 4 ( A zz 4 z + T o rr + T o zz ) +((2 C 1 + 2 2 z P o )( A 4 A z 2 2 z + A zz 4 z + T o rr + T o + T o zz )) = ( 2 2 z )) (2 C 1 ) = ( 2 2 z )+ A 4 + P o + T o rr + T o (A.49) 12 = 2 C 1 + 2 2 z ( 2 A z 2 2 z +2 A zz 4 z + P o +2 T o zz ) (4 C 1 +2 2 l 3 2 ( A 4 2 A z 2 2 z + A zz 4 z + P o + T o + T o zz ) (A.50) 21 = 2 C 1 + 2 2 z ( 2 A z 2 2 z +2 A zz 4 z + P o +2 T o zz ) 4 C 1 +2 2 2 z ( A zz 4 z + P o + T o rr + T o zz ) (A.51) 13 = 2 C 1 + 2 2 z (2 A 4 2 A z 2 2 z + P o +2 T o ) 4 C 1 +2 2 2 z ( A 4 2 A z 2 2 z + A zz 4 z + P o + T o + T o zz ) (A.52) 31 = 2 C 1 + 2 2 z (2 A 4 2 A z 2 2 z + P o +2 T o ) 4 C 1 +2 2 2 z ( A 4 + P o + T o rr + T o ) (A.53) 23 = 2 C 1 + 2 2 z (2 A z 2 2 z + P o +3 T o rr T o ) 4 C 1 +2 2 2 z ( A zz 4 z + P o + T o rr + T o zz ) (A.54) 32 = 2 C 1 + 2 2 z (2 A z 2 2 z + P o +3 T o rr T o ) 4 C 1 +2 2 2 z ( A 4 + P o + T o rr + T o ) (A.55) 99 Calculationofelasticmoduluswiththeincompressibilityconstraint usingaLagrangemultiplier Themethoddescribedabove,expressestheproblemusinganincompressiblematerialusing the k ( J 1)termintheconstitutiverelation.However,thisconditioncouldberelaxedby allowingthematerialtobecompressiblebutprescribinganisochoricmotion.Intherealmof continuummechanics,thismotionischaracterizedbydet( F )=1.However,thiskinematical constraintisexpressedbyaddingtheterm p I tothestresstensor,wherethehydrostatic pressure p isaLagrangemultiplier.Therefore,theCauchystressinEq.A.8canbewritten as T = p I + FSF T ; S =2 @w @ C ; (A.56) Itisconvenienttouse ^ T = FSF T representingthestressasaresultofdeformationinthe material.FollowingEqs.(A.9toA.14),thestresscanbewrittenas T = p I +( I + H ) F o S o + @ S @ C C o C F o T ( I + H ) T ; (A.57) where p = p o + p aretheLagrangemultipliersforthelarge( p o )andsmall( p )deformations, respectively.Finally,theincrementalstresscanbewrittenas T ij = p ij + H ik ^ T o kj + ^ T o ik H jk +4 F o F o j F o lp F o mq @ 2 w @C @C pq C = C o lm ; (A.58) = p ij + C ijkl kl + D ijkl kl (A.59) 100 where C ijkl = ik ^ T o lj + jk ^ T o il + 4 J o F o F o j F o lp F o mq @ 2 w @C @C pq C = C o ; (A.60) D ijkl = ik ^ T o lj + jk ^ T o il : (A.61) Now,weconsidertheexampleofandextensionoftheartery.Followingtheprevious section,weconsidertherotationtobenegligible =0.Inaddition,letusagainsassume thatthestrainenergyfunctionisgivenbyEq.(2.14).Bytheassumptionoforthotropic material,thestresscomponentsandthetensorisgivenas ^ T o rr = c 1 M e R ( G e r ) 2 2 r ; (A.62) ^ T o = c 1 M e R ( G e ) 2 2 + c 2 X k M k R ( G k h ) 2 (( k ) 2 1)exp( c 3 ( k 1) 2 )sin 2 ( k ) 2 + c 4 ( G m h ) 2 (( m ) 2 1)exp( c 5 ( k 1) 2 ) 2 + S ˆ M m R 1+ ( M ) 2 ( M 0 ) 2 ; (A.63) ^ T o zz = c 1 M e R ( G e z ) 2 2 z + c 2 J o X k M k R ( G k h ) 2 (( k ) 2 1)exp( c 3 ( k 1) 2 )cos 2 ( k ) 2 z : (A.64) C rrrr =2 ^ T o rr ; C rr =0 ; C rrzz =0 ; C rr =0 ; C =2 ^ T o +2 A 4 C zz =2 A z 2 2 z ; C zzrr =0 ; C zz =2 A z 2 2 z ; C zzzz =2 ^ T o zz +2 A zz 4 z : (A.65) 101 DirectionalYoung'smoduluscanbecomputedbyconsideringasimpleuniaxialstretchinthe desireddirection.Forinstance,weconsiderthepre-stressedarteryincursacircumferential incrementalstress.Therefore,thelinearizedresponsecanbewritteninthefollowingreduced form 2 6 6 6 6 4 0 T 0 3 7 7 7 7 5 = 2 6 6 6 6 4 p p p 3 7 7 7 7 5 + 2 6 6 6 6 4 C rrrr 00 0 C C zz 0 C zz C zzzz 3 7 7 7 7 5 2 6 6 6 6 4 rr zz 3 7 7 7 7 5 : (A.66) Reorderingtheequationsforthestrains,wecanwrite rr = p C rrrr ; zz = p C zz C zzzz ; = C C 2 zz C zzzz 1 T + p C zz C zzzz p (A.67) Becauseofincompressibilityofthematerial,wecanwrite rr + + zz =0.Thus,using thisconditionandthesymmetries,theLagrangemultipliercanbecomputedas p = C rrrr ( C zz C zzzz ) C 2 zz + C C zzzz + C rrrr ( C 2 C zz + C zzzz ) T: (A.68) Substituting p intoequationA.67,givesthefollowinglinearrelationshipbetweenstressand strain = ( C rrrr + C zzzz ) C 2 zz + C C zzzz + C rrrr ( C 2 C zz + C zzzz ) T: (A.69) Therefore,theeeYoung'smoduluscanbewrittenas E = C 2 zz + C C zzzz + C rrrr ( C 2 C zz + C zzzz ) C rrrr + C zzzz (A.70) 102 FigureA.1:Pressureversusradiusduringatalength.Thelinearizedelastic parameterswerecalculatedat90mmHg. Finally,insertingthecorrespondingtermsfromEq.A.65 E = 2( A 2 z 4 4 z 2 A z 2 2 z ^ T o rr + A zz 4 z ^ T o rr + A zz 4 z ^ T o + ^ T o rr ^ T o (A.71) + ^ T o rr ^ T o zz + ^ T o ^ T o zz + A 4 ( A zz 4 z + ^ T o rr + ^ T o zz )) A zz 4 z + ^ T o rr + ^ T o zz (A.72) Similarly,otherdirectionalYoung'smoduliandPoisson'sratioscanbecomputed. Computationofarterialmodulusinvivo Forthepurposeofillustrationandvalidationofthepresentedlinearization,theationat alengthofanarterialsegmentfromrabbit'sbasilararteryispresented.Theparam- etersaredirectlyobtainedfrom[160].First,theelasticitysolutiontothe 103 FigureA.2:Linearizedparametersfortdegreesofsmoothmuscletone. problemisfoundusingtheequilibriumforlargedeformations.Then,usingthelineariza- tion,pressure-radiusrelationshipoverapproximatelyonecardiaccycleiscomputedbased onanintermediatestate(at p o =90mmHg).FigureA.1showsthediscrepancybetweenthe linearizationandthehyperelasticresponseoftheartery. Despiteearlyadvancesinsolvingcoupledproblems[208],largescalecomputa- tionalproblemsremainchallengingbecauseofnonlinearmaterialproperties,thecomplex wallgeometry,andthepulsatilityofthebloodwinlargearteries.Theproposedformu- lationforthetheoryofsmallonlargeshowsthatbychoosinganappropriateintermediate thelinearizationgivestheerrorinradiuswithin 1 : 32%.Thestiofthe wallhowevercanchangeduetoalterationsinpressure,smoothmuscletone,ormicrostruc- ture(fromgrowthandremodeling)[160].Toillustratethispoint,thevaluebasaltone( S ) inEq.2.18ischangedfrom0to80kPa.ThecircumferentialandaxialYoung'smoduliare plottedagainstnormalizedvalueofthebasaltoneinFig.A.2. 104 Thecalculationsshowthatasthevasoconstrictionincreases,Young'smodulusdecreases.A similarpredictionwasmadein[209]werethisincreasewasattributedtotransferoftheload tomorerigidelements(likecollagen)inthearterialwall.Nevertheless,asmentionedthe estimationof in-vivo ofthearteryhighlydependsonthecontentsoftheelasticwall (massfractions)aswellastheirdepositionstretchesandtheartery'shomeostaticcondition. 105 APPENDIXB:OptimizationandSteadystatehemo- dynamicsofthearterialtree Inthepresentmodel,thewholearterialtreeisassumedtohaveabinarytreestructure. Eachvesselinthetreeismodeledasastraightsegmentwithsteadylaminarbloodw. Thus,usingPoiseuillewineachsegment,thepressuredropalongthebloodvesselcanbe calculatedas p s = 8 ˇR 4 q s = Z 0 ( R ) q s ; (B.73) where istheviscosityofblood, l isthelengthoftheartery,and Z 0 ( R )isthehydraulic resistance.NotethattheassumptionofPoiseuillewwasusedonceinformulating C drag . Thebifurcationisassumedtooccuratonepoint.Theconservationofmassatthebifurcation requires q s p = q s d 1 + q s d 2 ; (B.74) andassumingthatpressureiscontinuousoverthebifurcation p s p = p s d 1 = p s d 2 ; (B.75) wheresubscripts p , d 1 ,and d 2 showthequantityataparentanditsdaughtervessels, respectively. Beforeconstructingthesystemofequationstosolvethehemodynamicsoftheproblem,we needtoindexeverybifurcationpointinthearterialtree. FigureB.3showsabifurcationpointinthearterialtreestructure.Index k the 106 generationnumberoftheparentvesselrangingfrom1toprevioustolastgenerationnumber N 1.Alternatively,valueof s showstheindexofthebifurcationpointacrossonegeneration, therefore,rangingbetween1tothenumberofbifurcationsinthe k thgeneration.The arrowsonshowthedirectionofthenumbering.Inaddition,thesegmentsarecounted successively,inaccordwiththefashionusedfor k and s .Therefore,theblockofmatrix relatedtothebifurcationpoint[ k;s ]canbeconstructedas 2 6 6 6 6 4 2 k +2 s 32 k +2 s 22 k +1 +4 s 52 k +1 +4 s 42 k +1 +4 s 32 k +1 +4 s 2 3(2 k 1 + s 2)+1 10 10 10 3(2 k 1 + s 2)+2 01 Z 0 ( R j +1 ) 100 3(2 k 1 + s 2)+3 0100 Z 0 ( R j ) 1 3 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 q s ( k;s ) p s ( k;s ) q s ( k +1 ; 2 s 1) p s ( k +1 ; 2 s 1) q s ( k +1 ; 2 s ) p s ( k +1 ; 2 s ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 4 0 0 0 3 7 7 7 7 5 (B.76) where q s ( k;s ) isessentiallythewrateintheparentvesselatthebifurcation.Thesystem ofequations(B.76)calculateswrateandpressureateverylocationinthearterialtree withprescriptionofappropriateboundaryconditions(pressureandwrateattheinlet andoutlets). Thecostfunctionminimization,min C ( R ; p s ;q s )inEq.2.30,ateachindividualvessel[ k;s ] issolvedtogetherwiththesteadystatehemodynamicsactingontheentiretree(B.76). Suchminimizationisimplementedinthenestedloop.Attheexternalloop,foracurrent treegeometry(i.e.,given R and L )thetreehemodynamics p s and q s isresolved.Thenfor currenthemodynamicsstate(i.e.,given p s and q s )intheinternalloop,Newton-Raphson 107 FigureB.3:Schematicofthebifurcationpoint[ k;s ]inthearterialtree. methodsolvedtheoptimizationproblemtoupdate R ateachvesselsequentially. 108 APPENDIXC:Axisymmetricsolutiontopulsatilehemo- dynamicsinonevessel Herewedemonstratethesolutiontothefasttime(cardiaccycle)hemodynamics(velocity v andpressure p )foraxisymmetricincylindricalcoordinates( r;z; ).Consideracylindrical, straight,longvesselwithawalldescribedaselasticmembranewithnotorsionandrotation: h<>R and c>> v f z >>v f r ,where c isthepulsewavepropagationspeedand ! isangularfrequency.Second, weconsiderthearterytobelongitudinallyconstrainedwiththepressureandwallshearstress actingasfast-timebodyforces,asproposedby[210].Finally,weuseanorthotropicrelation forthemembraneTherefore,usingtheanalyticalsolutionpresentedin[208,210], wecandrivethehemodynamicssolutionofperturbationateachgivenfrequency ! p f = Pe i! ( t z=c ) ;q f = Qe i! ( t z=c ) Q = ˇR 2 P cˆ fluid (1 g ) ; v f r = iP!R 2 c 2 ˆ fluid r R J 1 r=R ) J 0 e i! ( t z=c ) ; v f z = P cˆ fluid 1 J 0 r=R ) J 0 e i! ( t z=c ) ; u f r = PR 2 c 2 ˆ fluid (1 g ) e i! ( t z=c ) ;u f z =0 ; (C.80) withtheBesselfunctionsofthekind J 0 , J 1 , g =2 J 1 = J 0 = i 3 = 2 ,andthe Womersleynumber = R q !ˆ fluid (C.81) Thewavepropagationspeed c (i.e.,pulsewavevelocity)isaclinicallyimportantcardiovas- cularmetric,usedtoinferthevascularwall[165].Thepulsewavevelocityforthe longitudinallytetheredvesselwallmotioncanbeexpressedas c = s (1 g ) h A j p s 2 Rˆ fluid : (C.82) Sincethevesselwallislongitudinallyconstrained,themaincontributiontothewave propagationequationscomesfromthecircumferentialcomponentofthematrix, A whichislinearizedatcurrentsteadypressure p s (seeChapter2formoredetail).Thetotal pressureandwrateattimedomainareobtainedbyapplyingFourierseriesforsolution 110 (C.80)with P n and Q n atmultiplefrequencies ! n =2 ˇn=T ,andusingslow-timesolution (C.79)atzero-frequency p = p s +Re 1 X n =1 P n e i! n ( t z=c n ) ! ;q = q s +Re 1 X n =1 Q n e i! n ( t z=c n ) ! ; (C.83) whererealvaluesofoscillatorypressureandwratearetaken.Furthermore,wecompute thecharacteristicimpedance Z c ( ! )[165]infrequencydomain,usingWomersley'ssolution (C.80),andhydraulicresistanceusingPoiseuillew(C.79) Z c ( ! )= P Q = cˆ fluid ˇR 2 (1 g ) = 1 ˇR 2 s hˆ fluid A j p s 2 R (1 g ) : (C.84) Similarlywealsocomputetheinputimpedance Z inp ( ! )= P=Q j z =0 andterminalimpedance Z T ( ! )= P=Q j z = L .Thewrelationintimedomain(hereatvesselend)canbe expressedviaconvolutionintegralofterminalimpedanceandwas p ( L;t )= 1 T Z t t T q ( L;t 1 ) z T ( L;t t 1 ) dt 1 ; (C.85) whichcanbeusedaswimpedanceboundaryconditionforpatient-spcgeometries [211]. 111 APPENDIXD:Recursivealgorithms:fast-timehemo- dynamics Herewepresenttherecursivealgorithmthatforagiventreegeometryanddiscretefrequency computestheimpedanceateachbifurcationfromthebottomtothetopforusingequations (3.3-3.6). Algorithm1 Fast-time:StepI(backward): Given: ˆ fluid ;h;T; R [ k;s ] ;L [ k;s ] ; A [ k;s ]= A j p s [ k;s ] ;s =1 ;:: 2 k 1 ; Z inp n [ N;s ] ;s =1 ; 2 N 1 Find Z inp n [ k;s ]for k 1 Q n inp [ k;s ]= P n inp [ k;s ] Z inp n [ k;s ] H n forw = P n inp [ k;s ] 1+ n [ k;s ] e i 2 ! n L [ k;s ] =c n [ k;s ] P n T [ k;s ]= H n forw e i! n L=c n [ k;s ] (1+ n [ k;s ]) end end 113 BIBLIOGRAPHY 114 BIBLIOGRAPHY [1] GBDMortalityandCausesofDeathCollaborators,\Global,regional,andnational lifeexpectancy,all-causemortality,andcause-spcmortalityfor249causesofdeath, 1980-2015:asystematicanalysisfortheGlobalBurdenofDiseaseStudy2015.," Lancet (London,England) ,vol.388,pp.1459{1544,oct2016. [2] WorldHeartFederation-ChampionAdvocatesProgramme,\Thecostof CVD,"2019.[Online].Available: http://www.championadvocates.org/en/ champion-advocates-programme/the-costs-of-cvd . 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