Evolving mechanical properties of a model of abdominal aortic aneurysm

18 Pages·2007·0.8 MB·English

BiomechanModelMechanobiol
DOI10.1007/s10237-007-0115-9
ORIGINAL PAPER
Evolving mechanical properties of a model of abdominal
aortic aneurysm
P.N.Watton · N.A.Hill
Received:5November2007/Accepted:19November2007
©Springer-Verlag2007
Abstract Thenovelthree-dimensional(3D)mathematical ciatedwithdilationofthearterialwallandthepossibilityof
model for the development of abdominal aortic aneurysm rupture;80–90%ofrupturedaneurysmswillresultindeath
(AAA)ofWattonetal.BiomechModelMechanobiol3(2): (Wilminketal.1999).Surgerytoremovetheaneurysmisan
98–113,(2004)describeshowchangesinthemicro-structure option,butitisahigh-riskprocedurewitha5%mortalityrate
ofthearterialwallleadtothedevelopmentofAAA,during (Raghavan and Vorp 2000). Statistically, it is observed that
whichcollagenremodelstocompensateforlossofelastin.In theriskofruptureexceedstheriskoftheoperationwhenthe
thispaper,weexaminetheinfluenceofseveralofthemodel’s diameterexceeds5.5cm(PowellandBrady2004).However,
materialandremodellingparametersongrowthratesofthe thiscriterionfailstoidentifysmallaneurysmswithhighrisk
AAAandcomparewithclinicaldata.Furthermore,wecalcu- ofruptureandlargeaneurysmswithlowrisk.Thusthereisa
latethedynamicpropertiesoftheAAAatdifferentstagesin generalneedforimproveddiagnosticcriteriatoaidclinical
itsdevelopmentandexaminetheevolutionofclinicallymea- decisions.
surable mechanical properties. The model predicts that the Several models have analysed the stress distributions in
maximumdiameteroftheaneurysmincreasesexponentially AAAsutilisingidealised(Elgeretal.1996)orphysiological
andthattheratioofsystolictodiastolicdiameterdecreases geometries(RaghavanandVorp2000).Theseaimtoyieldan
from1.13to1.02astheaneurysmdevelops;thesepredictions improveddiagnostictoolforpredictingrupture.However,a
areconsistentwithphysiologicalobservationsofVardulaki detailedknowledgeoftheAAAwallthicknessandhistology
etal.BrJSurg85:1674–1680(1998)andLanneetal.EurJ of the tissue needs to be non-invasively determined to suc-
VascSurg6:178–184(1992),respectively.Weconcludethat cessfullypredictstresses(andthestrength)oftheaneurysmal
mathematicalmodelsofaneurysmgrowthhavethepotential tissueonapatientspecificbasis.Hencecurrentstate-ofthe-
tobeuseful,noninvasivediagnostictoolsandthusmeritfur- artbiomechanicalmodelsofAAAarenotreadyforclinical
therdevelopment. applicationandpatientmanagement(Vorp2007)andthusto
date,ruptureriskisstillbasedonthediameterofanAAA.
Modellinggrowthofananeurysmwillultimatelyleadto
1 Introduction a greater understanding of the pathogenesis of the disease
and may yield improved criteria for the prediction of rup-
Abdominal aortic aneurysm (AAA) is characterised by a ture. However, we emphasize that growth models of AAA
bulgeintheabdominalaorta.DevelopmentofAAAisasso- (Wattonetal.2004),andsimilarlyintracranialcerebralaneu-
rysms (Baek and Humphrey 2006; Kroon and Holzapfel
B 2007),stillrequiresignificantdevelopmenttorealiseclinical
P.N.Watton( )
application.
DepartmentofEngineeringScience,UniversityofOxford,
ParksRoad,OxfordOX13PJ,UK Inthehealthyartery,themainloadbearingconstituentsin
e-mail:[email protected] thearterialwallareelastinandcollagen.Collagenisconsid-
erably stiffer than elastin; however, at physiological strains
N.A.Hill
elastinbearsmostoftheload(Armentanoetal.1995).This
DepartmentofMathematics,UniversityofGlasgow,
UniversityGardens,GlasgowG128QW,UK is because collagen is tortuous in nature (Shadwick 1999;
123
P.N.Watton,N.A.Hill
Raghavan et al. 1999). In fact, at each point in unloaded et al. 2004; Gleason and Humphrey 2004, 2005; Baek and
tissue, there is a population of collagen fibrils which dis- Humphrey2006).
playvaryingdegreesofundulation.Itisonlyatphysiological DuringthegrowthofanAAA,itisobservedthatthereis
pressuresthatthefibrilsbegintostraightenoutandcontrib- anaccompanyinglossofelastin(HeandRoach1993).How-
utetoloadbearing(SverdlikandLanir2002;Sacks2003). ever, the degradation of elastin alone does not explain the
Theprogressiverecruitmentofapopulationoffibrilsgives large dilatations observed in aneurysms because the colla-
risetothehighlynonlinearmechanicalresponseofthecol- genisverystiffandhasastronglynonlineartensileresponse
lagen. curve. Therefore models of aneurysm growth must address
Several recent microstructural models of collagen des- boththedegradationofelastinandtheremodellingofcolla-
cribethedetailedbehaviourofspecimensofcollagenoustis- gen.
sue under repeated loading in vitro. For example, Sverdlik Theunloadedartery(nointernalpressure,noaxialforce)
and Lanir (2002) examined pre-conditioning of sheep dig- isnotstressfree.Ifaradialcutismadeintheaxialdirection,
ital tendons assuming that individual collagen fibrils have itspringsopenintoanapproximatecircularsectorwhichis
quasi-linearviscoelasticproperties,whileSacks(2003)stud- characterised by an opening angle which can be measured
iednativebovinepericardiumandtookthefibrilstobelin- experimentally(RachevandGreenwald2003).Thisgeome-
early elastic and assumed that it is the progressive recruit- tryisoftenadoptedasthestress-freeconfiguration.However,
ment of a population of fibrils that gives rise to the highly thegeometryisnotactuallyacircularsectorandmoreover
nonlinearmechanicalresponseofthecollagenoustissue.In thearterymaystillcontainresidualstresses.Consequently,
both these models, the waviness of individual fibrils in the twoalternativehypothesesexistthatenableanunstressedref-
tissuevariesaccordingtoaprobabilitydensityfunction,and erence configuration to be defined. It may be hypothesised
thetissuestress–strainrelationshipsareobtainedbyintegrat- thatthephysiologicalstateofahealthyarteryhaseither:
ingoverthepopulationoffibrils.Suchmodelsoftendefine
thestraighteningstretchratio(SSR)ofeachindividualfibril (i) constant circumferential stress in each of its layers
(e.g.SverdlikandLanir2002)andassumethattheindividual (OgdenandSchulze-Bauer2000)
fibrilsdonotbearloaduntiltheundulationdisappears.(Note (ii) uniformstraindistributionthroughthethicknessofthe
thoughthatHansenetal.(2002)reportsasmallbutdetectable wall.(TakamizawaandHayashi1987).
forcewhilstthereisstillanobservablecrimp).Thesemodels
are complicated and require a number of parameters to be Althoughalltheprevious(three)hypothesesarenotequiva-
fittedtoexperimentaldatausingstatisticalmodelsensitivity lent,theyallyieldstressandstraindistributionsthroughthe
tests.Moreoverthesemodelsdonotaddressthechangesin thickness of the wall that are more uniform at physiologi-
the tissue stress–strain relationship that arise when the tis- cal pressures than if the unloaded artery had been adopted
sueremodelsinvivo.Henceweutilizeaconstitutivemodel asthestress-freereferenceconfiguration.Forexample,Ster-
(Holzapfeletal.2000)thataccountsforthegrossmechanical giopulosetal.(2001)modeltheporcineaorticmediaastwo
responseofthecollagen. stress free circular sectors and observe that the radial dis-
Collagenisinacontinualstateofdepositionanddegrada- tributionofcircumferentialstretchisclosetouniformwith
tionwitharelativelyfastturnoverof3–90days(Humphrey a difference between stretch ratios at the inner and outer
1999).Morespecifically,Nissenetal.(1978)determinedthe surfacesofthearterialwallatphysiologicalpressurestobe
collagenhalf-lifeoftheaortaandmesentericarteriesofarat lessthan10%.StalhandandKlarbring(2005)determinedthe
to be 60–70days in normotensive animals and reduced to materialandresidualstrainparametersforahumanabdom-
17days in hypertensive conditions. The collagen fibres are inal aorta (assuming a stress-free configuration of a circu-
secreted by fibroblasts. These cells work on the collagen, larsector)usinginvivopressure–diameterdataappliedtoa
crawlingoveritandtuggingonitinordertocompactitinto Fungtypestrainenergydensityfunction.Theyalsofindthat
sheetsanddrawitoutintocables.Indoingthis,thefibroblasts the circumferential stress and the circumferential strain are
attachthecollagenfibrestotheextra-cellularmatrixinastate approximatelyuniformatphysiologicalpressures.
of strain (Alberts et al. 1994, p. 984). This ensures that, at At physiological pressures the radius of the abdominal
thehigherendofthephysiologicalrangeofpressures,colla- aorta is approximately 10mm and the thickness is 1mm.
gencontributestotheloadbearingofthearterialwall.The Neglecting thrombus formation, the ratio of the thickness
processoffibredepositionanddegradationnaturallyactsto of the wall to the diameter of the aneurysm will decrease
remodelthearterialwallinresponsetochangingphysiolog- as the aneurysm enlarges, therefore the deformation of the
ical conditions. The theoretical study by Humphrey (1999) three-dimensional(3D)arterialwallofthedevelopinganeu-
hasservedasthefoundationforanumberoftheoreticalstud- rysmiscloselyrelatedtothedeformationofitsmidplane.The
iesthataddresstheremodellingofcollageninpathological residual strain that is present in the unloaded configuration
conditions (Humphrey and Rajagopal 2002, 2003; Gleason gives rise to an approximately uniform strain field through
123
Evolvingmechanicalpropertiesofamodelofabdominalaorticaneurysm
thethicknessofthearterialwallatphysiologicalpressures. icalstudies(Lanneetal.1992).Itisseenthatrealisticrates
Ifitisassumedthatthephysiologicalmechanismbywhich of dilation can be achieved for a range of physiologically-
collagen fibres attach to the artery is independent of both consistentgeometric,materialandremodellingparameters.
thecurrentconfigurationofthearteryandtheradialposition Moreover, the evolution of the compliance, pressure-strain
inthearterialwall,thentheremodellingprocessmaynatu- elasticmodulusandstiffnessoftheAAAmodelarefoundto
rallymaintainauniformstrainfield(inthecollagenfibres) beconsistentwithpublisheddata.
throughthethicknessofthearterialwallastheAAAdevel-
ops. These considerations thus support the suitability of a
membranemodel(Wempner1973;Heil1996)tomodelthe 2 Methods
deformationoftheabdominalaortaanddevelopment ofan
aneurysmatphysiologicalpressures. 2.1 Therecruitmentstretchλ
R
Wattonetal.(2004)modeltheabdominalaortaasacylin-
dricalnonlinearlyelasticmembranesubjecttoanaxialpre- AsinWattonetal.(2004),weassumethatthegrossmechan-
stretch and a constant internal systolic pressure. The distal ical response of a population of collagen fibrils of varying
andproximalendsoftheabdominalaortaarefixedtosimulate undulation can be represented by a nonlinear function of
vasculartetheringbytherenalandiliacarteries.Formation stretch,whichisdefinedrelativetotheonsetofrecruitment
and development of AAA is assumed to be a consequence ofthefibrilstoloadbearing.Effectivelyonecanpicturean
ofthematerialconstituentsofthearteryremodelling,whilst idealised collagen fibre which depicts the mechanical res-
subject to a constant systolic pressure. The constitutive ponse of a population of collagen fibrils. We assume that
relationsforthetissueinthemodelofWattonetal.(2004) themechanicalpropertiesoftheidealisedfibreremaincon-
incorporate variables that relate to the concentration of the stant as the tissue remodels, i.e. the undulation distribution
elastin, and to the density and waviness of collagen fibres. of the population of collagen fibrils does not change. This
This enables the remodelling of the arterial microstructure approachallowsustosimulatethetissuestress–strainlawof
duringAAAdevelopmenttobeaddressed.Thedegradation apopulationoffibrilswithjustafewmaterialparameters.
ofelastinisprescribedusingatime-dependentaxisymmetric Thestretchinthecollagenisdefinedwithrespecttothe
function.Collagenfibresareassumedtobeinacontinuous configurationinwhichitisrecruitedtoloadbearing,whilst
stateofdegradationanddeposition,andtoattachtotheartery the stretch in the elastin is always defined with respect to
such that their strain at systole is a constant. As the AAA the initial (undeformed) reference configuration. Consider
develops, the deposition and degradation of collagen fibres Fig.1b,whichdepictsaparallelarrangementofelastinand
acttomaintainthestraininthefibreswithinanequilibrium
range.
Wattonetal.(2004)demonstratedthatusingasetofphys- (a)
L
iologicalmaterialparameterstomodeltheabdominalaorta
and realistic remodelling rates for its constituents, the pre-
dicteddilationsoftheaneurysmmodelwereconsistentwith
thoseobservedinvivo.Anasymmetricaneurysmwithspi-
nalcontactisalsomodelledandthestressdistributionsare
broadlyconsistentwithpreviousstudies.However,thefact
λL
that their model of AAA growth achieves realistic dimen- R
sions is a factor of its construction. Indeed, the proportion (b)
F
L
of load initially borne by the collagen, the functional form Unstrained Material Elastin
ofthespatialdegradationofelastin,theturnoverrateofthe Length L Collagen
collagenfibres,therateatwhichadditionalcollagenisdepos- L x
Onset of
ited/lostandthemechanicalnonlinearityofthecollagen,all collagen
dwiereecxtlaymininfleuethneceefthfeecgtsroowftvhaoryfitnhgetahneesueryinsmflu.eInntitahlisppaarapmer-, rSeycsrtuoitlimcent F0 L λRL x0 F0
Configuration
eters on the growth rates of the AAA model and compare
withclinicalobservations(Vardulakietal.1998;Bradyetal. (Equilibrium strain) λRL y0
F
0
2004).Wealsoapplyaphysiologicalpressurepulsetocalcu-
latethesteadytemporalevolutionofthesystolicanddiastolic Fig. 1 aCollagenfibresarerecruited,i.e.begintobearload,when
geometriesoftheAAAmodel.Parametersthatcharacterise unstrainedtissueisstretchedbyafactorλR.bIntheunstrainedstate,
thecollageniscrimpedandthelengthofelastinisL.Aconstantforce
thestiffnessintermsofthesystolicanddiastolicconfigura- isappliedtobothends.Inequilibrium,thestretchinthecollagenisλA
tionscanthenbedeterminedandcomparedwithphysiolog- andthustheG–LstraininthecollagenisEA=(λ2A−1)/2
123
P.N.Watton,N.A.Hill
collagen.Thelength L denotesthelengthoftheunstrained fibrilstretchduringthecardiaccycle(whichwilloccur
tissueand x itsextension.Inunstrainedtissue,thestrainin atsystole)isequaltothemaximumattachmentstretch.
theelastinisequalto0,andthusthestrainintheelastinis
measuredwithrespecttotheundeformedconfiguration.The Fibrils that achieve maximum attachment stretches will
stretchintheelastinis haveminimumundulationandthusarethefirsttoberecruited
toloadbearingasthetissueisstretched.Ifthereisadistribu-
λ=[(L+x)/L]. (1)
tioninattachmentstretchesthatthefibroblastscanachieve
Collagenfibresarerecruitedtoloadbearingwhentheunstrai- thiswouldnaturallyaccountforthevariationinundulation
ned tissue is stretched by a factor λ . The collagen stretch ofthefibrilsinunloadedtissue.Notealsothatthedefinition
R
is defined with respect to the configuration at which it is ofamaximumattachmentstretchisconsistentwiththedef-
recruitedtoloadbearing(seeFig.1),thus inition of recruitment stretch, which relates to those fibrils
(of minimum undulation) which are recruited first to load
λ =(λ L+y)/λ L (2)
C R R bearing.
Now y = L +x −λ L,hencefromEq.(1)and(2)asim-
R
2.3 Collagenremodellingviatherecruitmentstretch
plerelationshipbetweentheelastinstretchandthecollagen
stretchisobtained:
The remodelling of the recruitment stretch is subtle and it
λ
λ = (3) is helpful to visualise what is occurring on the scale of the
C λ
R collagenfibres.Figure2portraystheeffectsoffibredeposi-
Thesignificanceofthisrelationship,i.e.Eq.(3),isthatthe tionanddegradationforatissuethatisstretchedandheldat
recruitmentstretchλRcanremodeltomaintainthestretchin fixedlengthwhilstremodellingoccurs.Notethatthefigure
thecollagentoanequilibriumvaluewhilstthestretchinthe depictshypotheticalcollagenfibres,eachofwhichrepresents
elastin increases as the aneurysm dilates. Equivalently: the the mechanical response of a population of collagen fibrils
modeladoptsdistinctreferenceconfigurationsfortheelastin- ofvaryingundulation.Thefigureillustratesthatthemicro-
ousandcollagenousconstituents;thereferenceconfiguration structuralchangestothetissue,whichariseasaresultofthe
forthecollagenousconstituentscanadapt. physiological turnover of collagen fibres, can be captured
byremodellingtherecruitmentstretchλ .Itisimportantto
R
2.2 TheattachmentstretchλA appreciatethatthetimeforthetissuetoremodelfromstates
(3)to(4)(inFig.2)isdependentontheturnoverrateofthe
AnequilibriumstretchλA(orattachmentstretch)forthecol- collagen fibres. Equivalently, the rate at which the recruit-
lagenneedstobedefinedtoenabletheremodellingofthecol- ment stretch remodels is dependent on the turnover rate of
lagentobeaddressed,asthearteryenlarges.Theattachment thefibres.
stretch is defined to be the stretch in the collagen (defined Inadditiontotherecruitmentstretches,variablesmaybe
withrespecttotheonsetofrecruitmenttoloadbearingofa introducedwhichrelatetothedensityofthecollagenfibres.
population of collagen fibrils) at systole at t = 0. The jus- This enables the collagen remodelling that occurs during
tificationforthisdefinitionofλA isbasedonthefollowing aneurysmdevelopmenttobemodelled.
hypotheses:
2.4 Mathematicalmethod
(i) Initially the arterial wall is in a homeostatic state.
Although collagen fibrils are in a state of continual Ageometricnonlinearmembranetheory(e.g.Wempner1973;
deposition and degradation, the gross structural and Heil1996)isadoptedtomodelthedeformationofthearte-
mechanicalpropertiesofthecollagenareconstant. rialwall.Theabdominalaortaistreatedasathincylinderof
(ii) Newlydepositedcollagenfibrilsareactedonbyfibro- undeformedradius R ,length L,andthickness H.Itissub-
0
blast cells to attach them to the extra-cellular matrix ject to a physiological axial pre-stretch, λ0, and a constant
z
(ECM)inastateofstretchsothattheycontributeto systolicpressure p (= 16kPa = 120mmHg)whichcauses
0
theloadbearingatphysiologicalpressures.Weassume a circumferential stretch of λ0 he initial in vivo configura-
θ
thereisamaximumattachmentstretchthatthefibro- tion is thus a cylindrical tube of length λ0L, radius λ0R ,
z θ 0
blastscanachieve. andthicknessH/λ0λ0.Theendsoftheinitialdeformedcon-
z θ
(iii) we assume that the time taken for the fibroblasts to figurationarespatiallyfixedtosimulatevasculartetheringby
configurethecollagenfibrilsismuchlongerthanthe the renal and iliac arteries. The length of the section of the
durationofacardiaccycle.Thisimpliesthatthemaxi- abdominalaortabetweentherenalandiliacarteriesistaken
mumfibrilattachmentstretchwilloccuratthesystolic to be 16cm which is consistent with physiological values
(peak)stretchofthecardiaccycle.Thusthemaximum (Raghavanetal.2006).
123
Evolvingmechanicalpropertiesofamodelofabdominalaorticaneurysm
Fig. 2 Attachmentofcollagenfibresinalteredconfigurations.1The fibreswillproceedtorestorestraininallthefibrestoequilibriumlevels.
reference state is the undeformed configuration of the tissue. In the 4Alloftheoldfibreshavedecayedandhavebeenreplacedbynewfibres
undeformedstate,thefibreshaveacharacteristicwaviness.2Theini- ofstrainEA.Thearteryreachesanewequilibriumconfiguration.5If
tialphysiologicalstateissuchthatthestraininthecollagenfibresisthe thetissueiscontractedbacktothereferenceconfiguration,thecrimp
equilibriumvalue,i.e.atsystolicpressuretheG–Lstraininthefibres ofthecollagenwillhaveincreased.Equivalently,thefactorthetissue
isEA.Thetissueiscurrentlyinequilibrium.Althoughthefibresarein mustbestretchedforthecollagentoberecruitedhasincreased.Hence
acontinualstateofdegradationanddeposition,newfibreswillattach theeffectsofdepositionanddegradationinalteredconfigurationscan
withidenticallevelsofstraintothosethatdecayandthusnochanges becapturedbyremodellingtherecruitmentstretchλR—whichrelates
occurinthemechanicalpropertiesofthetissue.3Supposethetissue tothewavinessofcollagenintheundeformedconfiguration.Notethat
isstretchedfurtherand,forthepurposesofthisexample,heldatfixed thetimeforthetissuetoremodelfromstate(3)tostate(4)isdependent
length.Newfibresattachtothetissuesothatatthenewsystolicconfig- ontheturnoverrateofthecollagenfibres,i.e.thehalflife.Equivalently,
urationtheirG–Lstrainis EA.Theoldfibresdecay,andthedistribu- therateatwhichtherecruitmentstretchremodelsisdependentonthe
tionofcollagenfibreschanges.Thenaturallyoccurringturnoverofthe turnoverrateofthefibres
Detailsofthemathematicalformulationusedtomodelthe sity functions (SEDFs) for the media (W ) and adventitia
M
developmentofanAAAandnumericalimplementationcan (W )needtobespecified.
A
befoundinWattonetal.(2004).Herewegiveanoutlineof
themethodologytoavoidunnecessaryrepetition.Thesteady 2.5 Strain–energydensityfunctionsforheterogeneous
deformationofthearteryisgovernedbytheprincipleofvir- aneurysmaltissue
tualdisplacements,
Thefirststepinthedevelopmentofthemodelistoaccurately
δ(cid:4) −δ(cid:4) =0, (4)
strain load modelthehealthyabdominalaorta.Theconstitutivemodel
proposedbyHolzapfeletal.(2000)isutilised.Thearterial
where δ(cid:4) represents the variation of the strain energy
strain
wall is modelled as two layers. The inner layer models the
stored in the arterial wall and δ(cid:4) is the work done by
load
mechanical response of the media, with contributions from
thepressureduringavirtualdisplacementaboutanequilib-
groundsubstance,elastinandadoublehelicalpitchofcolla-
riumdisplacementfield.Thevariationoftheinternalelastic
genfibres.Theouterlayermodelsthemechanicalresponse
energyfora2Dcylindricalmembraneis
oftheadventitia,withmechanicalcontributionsfromground
2(cid:1)πR(cid:1)L substanceandadoublehelicalpitchofcollagenfibres.The
δ(H W +H W )dx1dx2, (5) mechanicalresponseofeachlayerismodelledasthesumof
M M A A
a neo-Hookean SEDF and a highly nonlinear SEDF which
0 0
representsthemechanicalresponseofthecollagen.Gundiah
where H and H denotethethicknessesofthemediaand etal.(2007)confirmthesuitabilityoftheneo-HookeanSEDF
M A
adventitia,respectively,andtheintegralisevaluatedonthe torepresentthemechanicalbehaviourofelastin.Themodel
midplaneofthemembrane:0≤ x ≤ Land0≤ x <2πR ofHolzapfeletal.(2000)isadaptedtoincorporateadegrada-
1 2 0
aretheaxialandazimuthalLagrangianmidplanecoordinates, tionfunctionfortheelastin,andmicrostructuralrecruitment
respectively.Thefunctionalformsforthestrain–energyden- anddensityvariablesforthecollagenfibres.
123
P.N.Watton,N.A.Hill
We assume that the neo-Hookean response in the media exhibitednoappreciablediseaseitwasfromanelderlyspec-
has an elastinous and a non-elastinous part, k , due to the imen(80-year-oldfemalecadaver)andthustheorientations
g
groundsubstance.Theelastinouscontributioninthemediais maybeunrepresentativevaluesforyoungerarteries.Infact,
multipliedbyanormalisedspatiallyandtemporallydepen- fibreorientationsarepossiblyalsospeciesandvesseldepen-
dent concentration of elastin function, c (x ,x ,t), where dent,e.g.:theorientationofcollagenfibresinhumanaortic
E 1 2
c (x ,x ,t =0)=1throughoutthemembrane.Weassume tissueis±8.4◦inthemediallayerand±41.9◦intheadven-
E 1 2
that the contribution from the ground substance is equal in titiallayer(Holzapfeletal.2002)whilstfibreorientationsin
themediaandadventitia.TheSEDFsarethus thecarotidarteryofarabbitare±29◦inthemediallayerand
W =(K +c (x ,x ,t)K )(E +E +E ) ±62◦intheadventitiallayer(Holzapfeletal.2000).Forour
M g E 1 2 E 11 22 33
(cid:2) model of the human abdominal aorta fibre orientations are
+ nMp(x1,x2,t) (6) chosentobeγM =30◦, γA =60◦.
εMp,p=± Thethicknessofthearterialwalloftheabdominalaortais
C
approximately2mm(MohanandMelvin1982).Following
×K {exp[A (EMp(E ,λMp))2]−1}
M x C Mp R Holzapfeletal.(2000)weassumethemediaoccupies2/3of
forx ∈[−H,−H+2HM],and thethicknessofthearterialwall.Notethatalthoughthatthis
3 2 2
ratioisbasedondataforthecarotidarteryofarabbitanditis
WA = Kg(E11(cid:2)+E22+E33) assumedrepresentativeofotherarteries,e.g.Holzapfeletal.
+ n (x ,x ,t) (2000) assumed this to be true for the human left anterior
Ap 1 2
descendingcoronaryartery.WealsofollowHolzapfeletal.
εAp,p=±
C (2000)andassumethattheneo-Hookeancontributionfrom
×K {exp[A (EAp(E ,λAp))2]−1} (7) the adventitia is an order of magnitude lower than that for
A x C Ap R
(cid:3) (cid:4) themedia,i.e.weset K = K /10;thisisbasedonphysi-
g E
for x3 ∈ H−22HA, H2 .ECJp(x1,x2,t) denotes the Green– ologicalobservationsfortheYoungsmodulusofthemedial
Lagrange(GL)strainsinthecollagenfibresandλJp(x ,x ,t) andadventitiallayersofporcinethoracicaortaintherefer-
andn (x ,x ,t),arethecollagenfibrerecruitmeRnta1ndd2en- encestate(whereelastinousconstituentsdominatemechan-
Jp 1 2 ical behaviour—see Holzapfel et al. (2000) for discussion)
sityvariables,respectively.JtakesvaluesMandA,referring
andthuswouldseemsuitablefortheabdominalaorta.
tothemedia(M)andadventitia(A).Thefibresareorientated
at angle of γ to the azimuthal axis, where p denotes the The most recent models of the abdominal aorta do not
pitch±γ . Jp explicitlymodeltheindividualcontributionsfromthemedial
J
TheGLstrainsofcollagen EC areafunctionoftheGL and adventitial layers. Stalhand et al. (2004) determined
Jp materialandresidualstrainparametersforaonelayeredcon-
strainsintheelastinresolvedinthedirectionsofthecollagen
fibres,i.e. E ,andtherecruitmentstretchesλJp(x ,x ,t), stitutivemodel(Holzapfeletal.2000)tofittheinvivopres-
Jp R 1 2 sure diameter data of Sonneson et al. (1994). Stalhand and
where
Klarbring (2005) adopted a Fung type constitutive model
EC =(E +(1−(λJp)2)/2)/(λJp)2, (8) to the same in vivo data and enforced the constraint that
Jp Jp R R
theaxialforceisinvarianttothepressure(withinthephys-
and
iological range) to assist parameter determination. Van de
E =E sin2γ +E cos2γ +2E sinγ cosγ , Geest et al. (2006) have performed biaxial testing of aneu-
Jp 11 Jp 22 Jp 12 Jp Jp
rysmal and non-aneurysmal abdominal aorta however they
(9)
didnottestthemedialandadventitiallayersindependently.
FortheaxisymmetricmodelE12 =0,thuswecandropthe Consequently, there is insufficient guidance in the litera-
subscript panddenote EJ ≡ EJ+ = EJ−,i.e.ateachpoint turetodeterminethematerialparametersforthemedialand
fibresofpositiveandnegativepitch(inboththemediaand adventitialcollagen.Inthispaper,wehaveassumedthatthe
adventitia) have identical values of GL strain. This implies ratioofthemedialandadventitialcollagenmaterialparam-
that there are only two independent recruitment variables, etersforthecarotidarteryofarabbit(Holzapfeletal.2000)
i.e.λJ ≡λJ+ =λJ−(J = M,A),andtwoindependentfibre are representative for the human abdominal aorta, i.e. we
R R R
densityvariablesn ≡n =n .Att =0,n (x ,x ,0)= specify K = K /4. We emphasize this is a purely arbi-
J J+ J− M 1 2 A M
n (x ,x ,0)=1. trary choice: mechanical testing of the individual layers of
A 1 2
Theorientationsofthecollagenfibresinthemedialand theabdominalaortaisrequiredtoguidethedeterminationof
adventitial layers need to be specified. Holzapfel (2006) suchparameters.
determined mean orientations of the medial and advential Threeindependentmaterialparameters,namely K ,K
E M
collagen fibres for an abdominal aorta to be ±37.8◦ and and A remain to be determined so that the SEDFs model
x
±58.9◦, respectively. However, whilst the tissue sample the mechanical behaviour of the human abdominal aorta.
123
Evolvingmechanicalpropertiesofamodelofabdominalaorticaneurysm
Thepressure–diametercurveofahealthyabdominalaortais cementfieldforacylindricalmembrane.Fulldetailsofthe
utilised(Lanneetal.1992)forthispurpose.Thediastolicand derivationcanbefoundinWattonetal.(2004).
systolic diameters (15 and 17.2mm) and pressures (80 and Nowforanarterysubjecttoafixedaxialpre-stretchand
120mmHg) can be observed directly. However, the experi- subjecttoradialinflation,Eq.(10)canbewrittenmoresuc-
mentswereperformedinvivoandthusthereferenceconfig- cinctlyas
uration dimensions are unavailable. Hence we take typical
physiological values of axial and circumferential stretches p(λθ)= PE:Cp(λθ)+(1− PE:C)p(λθ) (11)
fromtheliterature. = fE(KE,λθ)+ fC(KC,Ax,λθ)
Learoyd and Taylor (1966) analysed the retraction of
where PE:Cdenotestheproportionofloadbornebytheelas-
human abdominal aortas and found that older and younger tinous constituents att=0; and thus (1− PE:C) represents
vesselsretractedbyapproximately30and20%,respectively. the proportion of load borne by the collagen at t = 0. The
Consistentwiththesefindings,Holzapfeletal.(2007)found functions fE(KE,λθ) and fC(KC,Ax,λθ) represent the
the average in situ axial stretch of seven human abdominal
mechanicalcontributiontoloadbearingfromtheelastinand
aortastobe1.19±0.084.Wechooseλ0 =1.3—whichrep-
z collagen,respectively;theycanbeexplicitlyidentifiedfrom
resentsanupperboundfortheaxialpre-stretchofthehuman
thefirst-andsecond-termsontheright-handsideofEq.10.
abdominal aorta. Langewouters et al. (1984) analysed the Given a value for PE:C at systolic pressure (p = p0), this
staticelasticpropertiesof20abdominalaortasinvitro:from
enables K tobedeterminedimmediately:
E
their data the average circumferential stretch of 11 abdom-
inalaortasfrompatientsofagesrangingfrom30to59can PE:Cp0 = fE(λ0θ,KE)⇒ KE = fE−1(PE:Cp0,λ0θ). (12)
bedeterminedtobe1.29.Thisisconsistentwiththefindings
Anadditionalequationisrequiredtodeterminetheremaining
of Stalhand and Klarbring (2005) who used an axial force
materialparameters, K and A .Weconsiderthediameter
M x
constraint to identity in vivo parameters of the abdominal
atasecond,greaterpressure(p = p200 =200mmHg,λθ =
aorta—they predicted a circumferental stretch of 1.27 at a
λ200).Thisyieldstwoequations,
θ
systolicpressureof16kPa.
Assuming a circumferential stretch of 1.3 and utilising (1− PE:C)p0 = fC(KC,AX,λ0θ),
the pressure–diameter data of Lanne et al. (1992), i.e. sys- p200 = fE(λθp=200)+ fC(KC,AX,λθp=200) (13)
tolicdiameterof17.2mm,thisimpliesadiameterofapprox-
imately13.2mminthereferenceconfiguration.Weassume thus enabling the two remaining constants, K and A , to
M x
that the onset of collagen recruitment occurs between the bedetermined.
diastolic and systolic diameters (Armentano et al. 1995), A value for the parameters PE:C needs to be specified.
and arbitrarily estimate this to occur at a diameter close Armentano et al. (1995) proposed a simple 1-D model to
tothediastolicdiameter,i.e.15.2mm.Initialvaluesforthe accountfortheindividualmechanicalcontributionstoload
recruitmentvariablesarethencalculatedtobeλM(t =0)= bearingoftheelastinousandcollagenousconstituentsforthe
R
1.19, λA(t = 0) = 1.19 and the equilibrium attachment aortaofaconsciousdog.Theirmodelpredictsthatatsystolic
R
G–Lstrainis EA =(λ2A−1)/2=0.098. pressure,theelastinbears88%oftheload,i.e.PE:C ≈0.9for
TodeterminethethreeunknownmaterialparametersK , anaorta.Giventhathumanarteriesnaturallyloseelasticityas
E
K andA weanalysethegoverningforce-balanceequation theyage(Lanneetal.1992;Wuytsetal.1995),thissuggests
M x
for a cylindrical membrane of fixed axial stretch subject to that 0.9 is a suitable upper bound for PE:C. For the default
radialinflation: model of the abdominal aorta, we assume a slightly lower
value,i.e. PE:C = 0.8.However,lowervaluesof PE:C may
(cid:5) (cid:6) (cid:7) (cid:8)(cid:9)(cid:10)
bemoreappropriategiventhatAAAgenerallyoccursinolder
1 1
p= R0λz (HM+HA)Kg+HMcEKE 1−λ2zλ4θ individuals.Thereforeweconsidertheinfluenceofthispara-
+⎧⎪⎨ 1 (cid:2) 4nJHJKJAxECJ exp[Ax(ECJ)2]cos2γJ⎫⎪⎬ meFteirguorneg3roiwlluthstorafttehseaApAreAssmuroed–edlifaomre0t.e1r≤relPatEi:oCn≤shi0p.9f.or
⎪⎩R0λzJ=M,A;εC≥0 (λRJ)2 ⎪⎭ ourmodeloftheabdominalaortaandillustratestheindivi-
J
dualcontributionstoloadbearingoftheelastinandthecol-
(10)
lagen.Note,thematerialparameterssummarisedinTable1
where the first- and second-terms on the right-hand side (and correspondingly the pressure–diameter relationship
of Eq. (10) correspond to the contribution to load bearing depicted in Fig. 3) are based on PE:C(λθ = λ0θ) = 0.8.
from the elastinous and collagenous constituents, respecti- Notethatforourmodeloftheabdominalaorta,thesystolic
vely.Equation(10)isderivedfromthegoverningvarational (diastolic)axialandazimuthalCauchystressesare102.9kPa
equation[i.e.Eq.(4)],thefunctionalformofthestrainenergy (87.4kPa) and 116.5kPa (78.3kPa), respectively. The axial
densityfunctions(Eqs.6,7)andconsiderationofthedispla- forceis3.97Natdiastoleand3.34Natsystole.
123
P.N.Watton,N.A.Hill
2.6 Elastindegradation
Elastin + Collagen
Elastin
ThedilationofanAAAisaccompaniedbylossofelastin(He
200 Collagen
andRoach1993;Shimizuetal.2006).Thehalf-lifeofelas-
tinisapproximately50years,consequentlythelossofelastin
inadultsalmostcertainlyresultsfromincreasedelastolysis
)g 150 ratherthaninsufficientsynthesis(Shimizuetal.2006).How-
H
m ever,thephysiologicalprocessesthatleadtothedegradation
m
( e of elastin in AAA are not clearly understood. One sugges-
ru tion is that the degradation may be linked to the uptake of
ss 100 oxidisedcholesterolthroughthearterialwall(Davies1998).
e
rp Thissuggestsaprogressivedegradationofelastinovertime.
However, the functional form of the spatial degradation is
unknown.Weassumethatthereisapointinthecentralregion
50
of the domain where elastin degradation begins, and in the
developedaneurysmthisregionhasbeendegradedthemost.
Thisfunctionalformshouldgiverisetoafusiformdilatation
0 oftheabdominalaortacentredinthecomputationaldomain.
13 14 15 16 17 18 19
Onesuchsuitablecandidateforthedegradationfunctionis
D (mm)
p
c (x ,t)=1−(1−(C )t/T)exp[−µ(1−2x /L)2],
E 1 min 1
Fig. 3 Pressure–diameterrelationshipforthemodeloftheabdominal
(14)
aorta.Thediameteroftheabdominalaortainthereferenceconfiguration
istakentobe13.2mm.ThediastolicandsystolicdiametersareDDIAS=
15mmandDSYS=17.2mm,respectively.Recruitmentofcollagento where L denotes the axial Lagrangian length of the mem-
loadbearingisestimatedtobeginatDDIAS=15.2mm.Collagenbears brane,µ≥0controlsthewidthofthedegradation,andCmin
one-fifthoftheloadatsystole istheminimumconcentrationofelastinattimet =T.Spec-
ifyingµ = 0 yieldsauniformdegradationthroughoutthe
domain.Increasingthevalueofµ increasesthespatiallocal-
Table1 Materialconstantsandphysiologicaldatausedformodelling
isationofthedegradationaboutapointofminimumconcen-
thehumanabdominalaorta
trationinthecentreoftheaxialdomain.
Wallthickness:
The timescale of aneurysm development may vary from
Media HM 1.33mm
person toperson. Given thataverage growth rateis 0.4cm/
Adventitia HA 0.67mm
year(Humphrey2002),andthedecisionwhethertooperate
Fibreorientation on an aneurysm occurs when the diameter reaches 5.5cm,
Media γM 30◦ thenthetimescaleofdevelopmentisoftheorderof10years.
Adventitia γA 60◦ Estimates suggest that 63–92% of the elastin is lost in
Systolicpressure p0 120mmHg aneurysmaltissue(HeandRoach1993),i.e.0.08≤Cmin≤
Undeformedradius R0 6.6mm 0.37. Consequently, we choose default values for these
Axialpre-stretch λ0 1.3 parameterswithinthisrange,specificallyC =0.2, T=10.
z min
Circumferentialstretch: Thelocalisationofdegradationofelastinisvariedbycon-
Atonsetofrecruitment λrθec 15.2/13.2 sideringarangeofvaluesfortheparameterµ,namely0 ≤
AtSystole λ0θ 17.2/13.2 µ≤160,andtheresultingaxialprofilesofdevelopedaneu-
At p= p200=200mmHg λ2θ00 18.5/13.2 rysmsareanalysed.(Itisobservedthatµ=20yieldssatis-
factoryresults—seeSection3)
MaterialParameters:
Elastin KE 97.6kPa
Groundsubstance Kg=KE/10 9.76kPa 2.7 Collagenremodelling
Collagenmedia KM 3.52kPa
Collagenadventitia KA=KM/4 0.88kPa Theturnoverofcollagenissimulatedbyproposingremod-
Exponentialconstantforcollagen Ax 40 elling equations that act to maintain the G-L strain in the
collagen fibres to an equilibrium value E (equivalently
A
Thisdataisestimatedusingthepressure–radiuscurvefortheabdominal
remodellingthestretchofthecollagenfibrestoλ ).Thisis
aortain(Lanneetal.1992)andphysiologicalmeasurementsofaxial A
(LearoydandTaylor1966)andcircumferentialstretches(Langewouters achieved by remodelling the reference configuration of the
etal.1984)atsystole collagenfibres,i.e.remodellingtherecruitmentstretches.In
123
Evolvingmechanicalpropertiesofamodelofabdominalaorticaneurysm
addition,itisassumedthatchangesinthedensityofthecol- Thepressure–strainelasticmodulus, E ,isgivenby
p
lagen fibres are driven by deviations of the collagen strain
P − P
fromequilibriumvalues. E =k sys dias , (17)
p (D −D )/D
Fortheaxisymmeticmodel,twoindependentrecruitment sys dias dias
variables and two independent fibre density variables are wherek =133.3isaconversionparametersothatE ismea-
p
required. In the absence of physiological data, linear dif- suredinN/m2.P (=120mmHg)andP (=80mmHg)are
sys dias
ferential equations are proposed for the remodelling of the thesystolicanddiastolicbloodpressure.Duetothenonlin-
recruitmentanddensityvariables,i.e. earpressure–diameterrelationshipofthearterialwall,E is
p
pressuredependent.Astiffnessparameter,γ,thatappearsto
∂λJ ∂n
R =α(EC−E ), J =β(EC−E ), (15) characterisetheentiredeformationofthearterialwallwith-
∂t J A ∂t J A
outpressuredependence(Lanneetal.1992)is
where α,β > 0 and E = (λ2 −1)/2. The remodelling
A A ln(P /P )
parameterα canbedeterminednumericallysothatitcorre- γ = sys dias . (18)
spondstoaprescribedhalf-life,α (months),ofthecollagen (Dsys−Ddias)/Ddias
H
fibres, i.e. α = α(αH) (Watton 2002). Given that collagen Note,highervaluesofEpandγ implythatthearteryisstiffer
in the arterial wall typically has a half-life of 60days, we andlessdistensibleandthushasalowercompliance.
assumeadefaultvalueofα = 2.However,turnoverrates
H
mayincrease(Nissenetal.1978)ordecrease(Carmoetal. 2.9 Overviewofthemodel
2002)inpathologicalconditions.Consequentlyweexplorea
rangeofcollagenhalf-lives,viz.0.1≤α ≤6,toexplorethe
H Tosummarisethemodel:Eqs.(6)and(7)arethe(spatially
influenceofthisphysiologicalparameteronAAAgrowth. and temporally) heterogeneous SEDFs of the medial and
Thegrowthrateparameterβ relatestotherateatwhich
adventitiallayersofthearterialwall,whichaccountforinde-
additional collagen is deposited and configured within the
pendentreferenceconfigurationsfortheelastinandcollagen
arterialwall.Itisnumericallydeterminedsothatthedilation
ateachpointinthetissue.Equation(14)representsthecon-
isphysiologicallyconsistentoveratimescaletypicalofaneu-
centrationofelastinwhichdegradesinaspatiallyprescribed
rysmdevelopment:weassumethatthediameterincreasesby
manner over time. Equation (15) (left) acts to maintain the
afactorof2–3overaperiodof10years. GL strain in the collagen to E by remodelling the refer-
A
Notethereisadistinctionbetweengrowthandremodel- enceconfigurationofthecollagenfibres,i.e.itsimulatesthe
ling.Heretherecruitmentvariablescapturetheremodelling
consequence of fibre deposition and degradation in altered
ofthetissue,i.e.thegrossmicrostructuralchangesthatoccur
configurations. The second of equations (15) controls the
withinthetissueduetothenaturalphysiologicalprocessof remodellingofthecollagenfibredensity.Equation(4)gov-
fibre deposition and degradation, in altered configurations,
erns the equilibrium displacement field and is solved by a
forafixedmassofcollagenconstituents.Thecollagenden-
finiteelementmethodcodedinFORTRAN77.
sityvariablesrelatetothegrowth/atrophyofthecollagenous
constituentswithinthearterialwall.
3 Results
2.8 AnalysisofmechanicalpropertiesoftheAAAmodel
3.1 AxisymmetricAAAgrowth
Toanalysetheevolutionofclinicallymeasurablemechani-
calparameters, thediastolicdeformations arecalculated as
WeillustratethedevelopmentofanAAAthatarisesdueto
theaneurysmevolves.Ateachtimestepofthecomputational
an axisymmetric degradation of elastin. The default set of
simulation,thepressureisincrementallydecreasedfromsys-
parameters we use for the AAA growth model to achieve
tolictodiastolicvaluestocalculatethegeometryoftheAAA realisticdilationare:α = 2, β = 25, C = 0.2, µ =
H min
atallpressures.Aclinicalmeasure(Lanneetal.1992)ofthe 20, PE:C =0.8.1Figure4ashowstheconcentrationofelas-
strainintheAAAwall,ε,iscalculated,i.e. tin(asafunctionoftheLagrangiancoordinate,x(cid:8) ∈ [0,L],
1
D −D seeEq.23)andFig.4bitsevolvinggeometryevery2years.
ε = p dias (16)
D Thegridintheillustration(Fig.4b)enablesdeformationof
dias
materialpointsonthearterialwalltobevisualised:noticethe
where D is the maximum diameter at a pressure p, and
p
D the maximum diameter of the AAA at diastolic pres-
dias
sure. The maximum fractional diameter change, ε , or 1 NotethatinWattonetal.(2004)thestatedhalf-lifeusedwas6months,
max howeveranumericalmistakeinthecomputationalcodeactuallymeant
maximumarterialstrain(Lanneetal.1992)isthusε =
max thatthehalf-lifeusedwas0.6months.Thisshouldbeborneinmind
ε|Dp=Dsys,where Dsys isthemaximumdiameteratsystole. whencomparingtheirresultswiththispaper.
123
P.N.Watton,N.A.Hill
diameter of approximately 45mm at t = 10 (see Fig. 5a).
Weincludethecaseα =6inthisanalysisasthisachieves
H
arealisticdilationwithnoremodellingofthefibredensity,
and thus represents an upper bound for the collagen half-
lifepredictedbyourmathematicalmodel.Figures5bandc
illustratetherateofgrowthasafunctionofthesystolicdiam-
eter and time, respectively. It is seen that as the half-life is
decreasedtoachieverealisticdilationatt = 10,thegrowth
rateprogressivelydecreasesatearliertimesandismorerapid
atlatertimes.Interestingly,inallcasesconsideredherethe
rate of growth evolves approximately linearly with respect
to the diameter (Fig. 5c). For an AAA of diameter 45mm,
the predicted growth rates range from 4 to 26mm/year for
α =6toα =0.1,respectively.
H H
Next,considertheevolutionofthediameterastheremod-
elling parameters α and β are varied independently. First
H
α isvariedwhilstβ =0,i.e.thereisnoincreaseincollagen
H
contentwithinthewallastheAAAdevelops.Increasingthe
turnoverrateofthefibres,i.e.decreasingα ,increasesthe
H
rateofdilationandincreasesthenonlinearityofthedilation
(seeFig.6a).Figure6billustratesevolving systolicdiame-
ters for α = 2 and β = {0,5,15,25,50,100}. It can be
H
seenthatincreasingtherateatwhichthearterycandeposit
newcollagen,i.e.increasingthevalueofβ,reducestherate
ofdilation.
Toillustratethedependencyofgrowthontheelastindeg-
radation function (Eq. 23), the parameter µ is varied from
µ = 0(thespecialcaseofuniformdegradationthroughout
the domain) to µ = 320 a very localised degradation in a
centralregionofthedomain:Fig.7ashowsthedegradation
(asafunctionoftheLagrangiancoordinate)att = 10for7
cases(µ=0, 0.25, 0.5, 15, 20, 80, 320)andFig.7cthe
Fig. 4 a Prescribed concentration of elastin as a function of the correspondingaxialgeometries.Itisobserved:forµ≤1the
Lagrangiancoordinate x1 (seeEq.23)usingµ = 20, Cmin = 0.2, geometryoftheAAAisconstrainedbythefixedboundary
L = 160/λ0, T = 10, x‘ ∈ [0,L] (Note: no circumferential
variation). bzA developing ax1isymmetric aneurysm, t = 2, 4, 6, conditionsattheendofthedomain;for5 ≤ µ ≤ 80,aneu-
8, 10 years from left to right. Notice the axial dilation of the cen- rysmsofsimilardiametersdevelopandthegeometryofthe
tral region of the aneurysm and retraction of the ends. Remodelling AAAdoesnotappeartodependonthefixedradialposition
parametersareαH = 2,i.e.ahalf-lifeof2months,andβ = 25.The of the boundaries (but it still depends on the axial position
gridisforvisualisationonlyandindicatestherelativedeformationof
oftheboundaries);forµ≥160,theaneurysmsbecomepro-
pointsonthearterialwall
gressively smaller. It can be seen that 5 ≤ µ ≤ 80 yields
aneurysms of maximum diameters in the range 42–44mm
modelpredictsthatAAAgrowthisaccompaniedbyanaxial that display a characteristic bulge in the central region of
dilationofthecentralregionofthearteryandaretractionof thedomain.Forourdefaultmodel,wechooseµ = 20,this
itsdistalandproximalends. yields an appreciable bulge which is not unduly influenced
However, the fact that our model of AAA development bytheboundaryconditionsattheendsofthedomain.
achieves physiological dimensions (Fig. 4b) can partly be TheparameterPE:Cdefinestheproportionofloadinitially
attributed to its construction and selection of parameters. bornebytheelastinatsystolicpressure.Theinfluenceofthis
Indeed,thereexistsasetofpairsofremodellingparameters, parameterontheevolvingdiameteroftheAAAisshownin
(αH,β) which can be numerically determined to achieve Fig.8.Astobeexpected,increasing PE:Crenderstheartery
identicalsystolicdiametersatt =10.Assumingaphysiolog- moresensitivetothedegradation ofelastinandfasterrates
icalrangeforthecollagenhalf-life,i.e.0.1–3months,thefol- ofdilationareobserved;theconverseofthisisclearlytrue.
lowingremodellingparameterpairs,(α ,β) = (0.1,950), Lastly we consider the effect of varying the parameter
H
(0.5, 195), (2, 25), (3, 10) and (6, 0) all yield a systolic A ,whichgovernsthemechanicalnonlinearityofthecolla-
x
123