1-dimensional blood flow modeling with varying viscoelastic wall properties along a network of

156 Pages·2015·9.66 MB·English

by  Christina Battista

ABSTRACT
BATTISTA,CHRISTINA.ParameterEstimationofViscoelasticWallModelsinaOne-Dimensional
CirculatoryNetwork.(UnderthedirectionofMetteS.OlufsenandMansoorA.Haider.)
Flow and pressure waves originate from the contraction of the heart and propagate
alongdeformablevesselswherethewavesarereflected,dampened,anddispersedwithin
smaller sub-networks of vessels. Wave propagation in the circulatory system has been
studiedfrommanydifferentangleswiththemostsuccessfulbeingfromafluiddynamics
approach.Thisthesisdevelopsandappliesaone-dimensionalnonlinearfluiddynamics
modelforpulsewavepropagationinlargesystemicovinearterieswiththegoalofenhanc-
ingtheunderstandingofcardiovasculardiseaseandpotentiallyimpactingdiagnostictech-
niquesrelatedtosystemichypertension.Hypertension,highbloodpressure,isassociated
withthestiffeningoflargeorsmallarteries,andbystiffeningthearteriesinournetwork,
weshowtheimpactthateachhasonthepressurewaveform.
TheNavier-Stokesequationsthatgovernbloodflowinthelargearteriesarehighlyde-
pendentupontheparametersspecifiedbyboththein-andoutflowboundaryconditions
aswellasthecoupledarterialwallmodel.Themostcommonoutflowboundarycondition,
the three-element Windkessel model, requires diligent estimation of parameters to pro-
ducephysiologicallyrelevantandsensibleresults.Simultaneously,biomechanicalproper-
tiesofthearterialwallchangealongtheaxialdirection,resultinginstifferarteriesandless
viscoelasticitywithprogressivelysmallervesselsorasdiseasesprogress.Withachangein
mechanicalpropertiesofthearterialwall,inparticularviscoelasticity,variousamountsof
energyarelostthroughoutthesystem.Energylossmustthenbecompensatedforinthe
downstream vasculature via means of the outflow boundary conditions. Thus the Wind-
kesselmodelreliesnotonlyonpressureandflowbutalsothewallmodelanditsparame-
ters.
While the Windkessel model has been implemented and studied for many years, the
currentapproachfordeterminingtheparametershasbeenbasedonelasticwallmodels
whereminimalamountsofenergyarelostasthepulsewavespropagatealongthenetwork.
When incorporating viscoelastic walls where larger energy losses are evident, it is a non-
trivial task to estimate the outflow boundary condition parameters. This thesis presents
asystemicapproachfordeterminingWindkesselparametersbasedonvesselradius,stiff-
ness,andviscoelasticity.
©Copyright2015byChristinaBattista
AllRightsReserved
ParameterEstimationofViscoelasticWallModelsinaOne-DimensionalCirculatory
Network
by
ChristinaBattista
AdissertationsubmittedtotheGraduateFacultyof
NorthCarolinaStateUniversity
inpartialfulfillmentofthe
requirementsfortheDegreeof
DoctorofPhilosophy
AppliedMathematics
Raleigh,NorthCarolina
2015
APPROVEDBY:
MetteS.Olufsen MansoorA.Haider
Co-chairofAdvisoryCommittee Co-chairofAdvisoryCommittee
BrookeN.Steele PierreGremaud
SharonLubkin
DEDICATION
Forallofmygrandparents....
ThankyouforbeingmybiggestfansandalwaysremindingmeIcouldaccomplishthis.
“Whatchildrenneedmostaretheessentialsthatgrandparentsprovideinabundance.They
giveunconditionallove,kindness,patience,humor,comfort,lessonsinlife...andmost
importantly,cookies.”
–RudyGiuliani
ii
BIOGRAPHY
ChristinawasborninBuffalo,NYandgrewupinasmallsuburbcalledAlden.Upongrad-
uating high school in 2007, she attended the Rochester Institute of Technology (RIT) in
Rochester,NYwhereshedecidedtomajorinAppliedMathematics.AfterattendingGeorge
Washington University’s Summer Program for Women in Mathematics at the end of her
sophomoreyear,Christinaknewshedidn’twanthereducationtoendafterherbachelor’s
degree.SheregisteredfortheBS/MSprograminAppliedandComputationalMathematics
atRITandaddedtwominors,AppliedStatisticsandCriminalJustice,andaconcentration
inPhysics.
WhileatRIT,sheparticipatedinnumerousresearchprojectswithmanyprofessorswho
helpedpavethewayforherloveofappliedmathematics(Dr.DarrenNarayan,Dr.Tamas
Wiandt,andDr.DavidS.Ross).SheworkedwithDr.DavidS.RossatRITonhermaster’s
research project which was entitled “Parathyroid hormone and cell signaling in bone re-
modeling.”ThishadbeenthefirsttimeChristinaappliedmathematicstobiology-related
topicsandtrulyembracedherinnernerd.Realizingthatshelovedthebiologyaswellasthe
mathematicsbeingappliedtoit,shedecidedtoapplyforPh.D.programswheretherewere
opportunitiestocontinueappliedmathematicsresearchwithbiologicalapplications.
InFebruaryof2011,ChristinareceivedacallwithnewsthattheDepartmentofMath-
ematics at North Carolina State University was inviting her into their graduate program.
ShemovedtoRaleighthatJuneandMansoorHaiderofferedheraresearchassistantship
forthefirstsemesterinthe2011schoolyear.MansoorHaiderandMetteOlufsenadvised
Christina through four years of research and a successful dissertation defense in August
2015.ChristinahasacceptedapositionatTheHamnerInstituteforHealthSciencesasa
postdoctoralfellowwithintheDILI-simgroup.
iii
ACKNOWLEDGEMENTS
Firstandforemost,Iwouldliketoexpressmysincereappreciationtomyadvisorycommit-
tee: Mette S. Olufsen, Mansoor A. Haider, Brooke L. Steele, Pierre Gremaud, and Sharon
Lubkin.Thankyouallforprovidingmewithsupportandinsightfulcriticismthroughout
thisresearch.Aspecialthankstomyadvisors,MetteandMansoor,fortheirpatienceand
guidancethesepastfouryears.Afiniteamountofwordscannotdojusticetothankthem
for helping me grow as a researcher and letting me have a little fun along the way. Their
confidenceinmeneverfaltered,evenwhenminedid,andtheyalwayspushedmetolook
atthebiggerpicture.Withouttheirhelpandsupport,Iwouldn’thavemadeitwhereIam
today.
AspecialthankstoallthefriendsImadeduringgraduateschool.Iwasfortunateenough
tohavebeensurroundedbysomanyfriendsfromthemomentImovedtoNorthCarolina.
TherearetoomanypeopletolistbutIamgratefulforthedistractionsprovidedbysporting
,
events,trivia,craftnights,cardgames,crosswords,andgeocaching.
I am especially thankful for my officemates over the years–Nakeya Williams, Gregory
Mader, Christian Olsen, Jacob Sturdy, and Reneé Brady–and to the rest of the Cardiovas-
cularDynamicsGroup(#wedonothing).Thankyoufornotonlybeingclassmatesandcol-
leagues,butalso friends,andfor makingmy workspacesucha funplaceto be.Youguys
maderesearchtolerableonthedayswhenIdidn’tthinkIcoulddoitandprovidedmewith
enoughdistractionsthroughouttheprocesstokeepmesane.Someofmyhappiestandmy
saddestmomentsingraduateschoolhappenedinourofficeandI’mgladtohavehadyou
theretosharethemwith.
Thankstothemostadorable(andsupportive!)four-leggedfriendsintheworld,Tippy
andZoë,andtotherecentfamilyaddition,Gus.
Lastly, and most importantly, I want to thank my family both near and far. Graduate
school is hard enough as it is and I couldn’t imagine doing it without the support of my
family. They have been there for me every step of the way, keeping me sane via phone
calls and visits, loving me unconditionally, and standing by me through all of my tough
decisions.
iv
TABLEOFCONTENTS
ListofTables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Summaryofthedissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter2 CardiovascularPhysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Thecardiovascularsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Thesystemicandpulmonarycircuits . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Thecirculationofblood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Vasculature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Walltissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Biomechanicsofwalltissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Vascularpathology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter3 ExperimentalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Experimentalsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Surgicalpreparationandacquisitionofsegments . . . . . . . . . . . . . . 18
3.1.2 Exvivoexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Datapreprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Availabledatainliterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Pressure-areadata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Pressure-flowdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter4 ModelingBloodFlowintheArteries . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 One-dimensionalfluidsmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.1 Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.2 Derivationofconservationlaws . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Arterialwallmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.2 Elasticmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.3 Viscoelasticmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.4 Quasilinearviscoelasticitytheory. . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.2 Outflowboundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 Inflowboundaryconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.4 Junctionsconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
v
Chapter5 WaveIntensityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 CalculatingWIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter6 NumericalMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1 Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Finiteelementmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Strongform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.4 WeakformandFEMdiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 Convergenceofthesolver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.5.1 Non-taperedvessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.5.2 Taperedvessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Chapter7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.1 Parameterestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.1.1 Vesselstiffnessandunstressedvesselradii . . . . . . . . . . . . . . . . . . . 75
7.2 Networkgeometryextractedfromexvivoexperimentalmeasurements . . . 77
7.3 Elasticnetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3.1 Modelparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3.2 Networksimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.4 Viscoelasticnetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.4.1 Singlevesselnetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4.2 Symmetricbranchingnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.5 WIAresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.1 Limitationsandfuturework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
AppendixA Conservationofmomentumnondimensionalization . . . . . . . . . . 127
AppendixB Wallmodelderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
AppendixC HyperbolicityofPDEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
C.1 Elasticwallmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
C.2 Kelvinlinearviscoelasticwallmodel . . . . . . . . . . . . . . . . . . . . . . . . . 136
AppendixD TheNelderMeadMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
vi
LISTOFTABLES
Table2.1 Percentcompositionofthemediaandadventitiaofthreearteriesat
invivobloodpressure.Valuesgivenaremean(cid:6)standarddeviation.
Adaptedandreproducedwithpermissionfrom[55].. . . . . . . . . . . . . 14
Table3.1 Summaryofdataavailableinliteratureforinvivoorexvivopressure
p,areaA,andflowq.Alsospecifiedarethespeciesandarteriesfrom
whichthedataismeasured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Table4.1 Generalsummaryofbloodflowmodelingin1-Dand3-Dplusthose
thathavestudiedandcomparedresultsinbothdimensions. . . . . . . 28
Table4.2 Summaryofwallmodelsstudiedbydifferentgroups. . . . . . . . . . . . . 37
Table4.3 TwolinearwallmodelswritteninQLVformulationwithcreepfunc-
tionK(t)andinverseelasticresponsefunctions(e)[p].Notethatthe
creepfunctiondistinguishesbetweenelasticandviscoelastic. . . . . . 47
Table5.1 WIAtableshowingtypeanddirectionofdominatingwaveforms.Cor-
respondingdiscretizedWIAterminologyisgiveninparenthesis. . . . . 57
Table6.1 Summaryofnumericalmethodsimplementedbyreferencestosolve
1-Dfluidmodels.MoC:methodofcharacteristics,FDM:finitediffer-
encemethod,FEM:finiteelementmethod,DG:discontinuousGalerkin. 60
Table7.1 Averagegeometricandmechanicaloptimizedparametersfortheelas-
tic and viscoelastic wall models. Results are presented as mean (cid:6)
standarddeviation.Forallsegments,n =11,h isthewallthickness,
r isthezero-strainradius,E istheYoung’smodulus,andA andb
0 1 1
aretheviscoelasticrelaxationparameters.Parametersnotedn.d.are
nondimensional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Table7.2 Vessel dimensions and flow distribution. For each vessel segment
shown in Figure 7.4, the table specifies length, proximal and distal
unstressedradius(allincm),aswellasthedistributionofflowtothe
vesselsegment[31,82,146].Thetotalcardiacoutputwassetat66.9
mL/s[40].Unstressedvesselradiiwereaveragedfromoptimalvalues
presentedinTable7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Table7.3 Modelparameters.Fluiddynamicsparameterswereusedinboththe
singlevessel(SV)andthenetworksimulations.Allvesselsarelisted
inthelargenetworkbutonlyterminalvesselsrequireoutflowbound-
ary conditions (RCR values). Parameters noted n.d. are non dimen-
sional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vii
Table7.4 ResultsforeachofthesymmetricnetworkswhereEh=r isconstant
0
throughoutthevasculature.TheincreasingR trendobservedinthe
1
singlevesseland3-vesselnetworkchangesasmoregenerationsare
added.However,thetrendsforR andR =R remainthesameforthe
2 1 t
symmetricbifurcatingnetworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Table7.5 Parameter values and units for (7.4.4) and (7.4.5). The notation n.d.
indicatesnon-dimensionalquantities. . . . . . . . . . . . . . . . . . . . . . . . 96
Table7.6 ResultsforeachofthesymmetricnetworkswhereEh=r variesbased
0
on r throughoutthevasculature.R nowincreasesforallnetworks
0 1
as viscoelastic degree (A ) is increased. However, the trends for R
1 2
andR =R remainthesameforthesymmetricbifurcatingnetworks. . 98
1 t
Table7.7 Parametervaluesandunitsfor(7.4.6)and(7.4.7)relatingR andR
1 2
toA andr forsymmetricnetworkswhereEh=r varieswithr . . . . . 99
1 0 0 0
viii