Biomechanics and FE Modelling of Aneurysm: Review and Advances in Computational Models

24 Pages·2012·2.13 MB·English

CChhaapptteerr 10
Biomechanics and FE Modelling of Aneurysm:
Review and Advances in Computational Models
SimonaCeliandSergioBerti
Additionalinformationisavailableattheendofthechapter
http://dx.doi.org/10.5772/46030
1.Introduction
Abdominal aortic aneurysm (AAA) disease is the 18th most common cause of death in all
individualsandthe15thmostcommoninindividualsagedover65[48]. Clinicaltreatment
forthisdisordercanconsistofopensurgicalrepairor,morerecently,ofminimallyinvasive
endovascular repair procedures [71]. However, both clinical treatments present significant
risks and, consequently, require specific patient selection. Given the risks of current repair
techniques, during the course of an aneurysm it is important to determine when the risk
ofrupturejustifiestheriskofrepair. Inthisscenario, howtodeterminetheruptureriskof
an aneurysm is still an open question. Currently, the trend in determining the severity of
an AAA is to use the maximum diameter criterion. Unfortunately, this criterion is only a
generalruleandnotareliableindicatorsincesmallaneurysmscanalsorupture,asreported
in autopsy studies, while many aneurysms can become very large without rupturing [16].
The maximum diameter criterion, in fact, is based on the law of Laplace that establish a
linearrelationshipbetweendiameterandwallstress. However,thelawofLaplaceissimply
based on cylindrical geometries, where only one radius of curvature is involved, whereas
aneurysmsarecomplexstructures,andthereforethelawfailstopredictrealisticwallstresses.
Fromabiomechanicalpointofview, ruptureeventsoccurwhenactingwallstressesexceed
thetensilestrengthofthedegeneratedaorticabdominal(AA)wall. Biomechanicsrelatesthe
functionofaphysiologicalsystemtoitsstructureanditsobjectiveistodeducethefunctionof
asystemfromitsgeometry,materialpropertiesandboundaryconditionsbasedonthebalance
lawsofmechanics(e.g.conservationofmass,momentumandenergy).Consequently,froma
more general and extensive perspective, the stress state in a body is determined by several
factors such as geometry, material properties, load and boundary conditions. In order to
understandthecapabilitytoestimatethepotentialrupturerisk,itisfundamentaltocapture
themechanicalresponseoftheaortictissueanditschangesduringaneurysmalformation.In
fact,while,todate,theprecisepathogenesisofAAAispoorlyunderstood,itiswellknown
thatthischangesignificantlyimpactonthestructureoftheaorticwallandonitsmechanical
behavior.
©2012CeliandBerti,licenseeInTech.Thisisanopenaccesschapterdistributedunderthetermsofthe
CreativeCommonsAttributionLicense(http://creativecommons.org/licenses/by/3.0),whichpermits
unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
4 2Aneurysm Will-be-set-by-IN-TECH
Thischapterwillreviewthestateofliteratureonthemechanicalpropertiesandmodellingof
AAAtissueandwillpresentadvancedcomputationalmodels.Thefirstpart(Sec.2)includes
adescriptionofthemechanicaltestcurrentlyused(2.1),theaorticmechanicalproperties(2.2)
andareviewoftheliteratureonmaterialconstitutiveequations(2.3)andgeometricalmodels
(2.4). Tostressoutthemorphologicalcomplexityoftheaorticsegment,inSec3theregional
variationsofmaterialpropertiesandwallthicknessreportedinliteratureformexperimental
investigationsarereported. Thesecondpart(Sec. 4)describesouroriginalcontributionwith
adescriptionofourFiniteElement(FE)modelsandourprobabilisticapproachimplemented
intoFEsimulationstoperformsensitivityanalysis(Sec.4.1).Themainresultsarereportedin
Sec.4.2anddiscussedindetailinSec.5.
2.Review
Inordertounderstandthebiomechanicalissuesintheetiologyandtreatmentofabdominal
aortic aneurysms, it is important to understand the structures of the aortic wall and how
theyaffectthemechanicalresponse. Biologicaltissuesaresubjecttothesamebalancelaws
ofconservationofmass,momentumandenergyoftheclassicalengineeringmaterial. What
distinguishbiologicaltissuesfrommaterialsofthefieldofclassicalengineeringmechanicsis
theiruniquestructure. Softbiologicaltissues,infact,haveaverycomplexstructurethatcan
beregardedaseitheractiveandpassive. Theactivecomponentsarisefromtheactivationof
thesmoothmusclecellswhilethepassiveresponseisgovernedprimarilybytheelastinand
collagenfibres[15].Thedistributionandthearrangementofthecollagenfibres,inparticular,
have a significant influence on the mechanical properties due to they attribute anisotropic
properties[49]tothetissue. Differentstudieshaveshownthatthisstructuralarrangementis
verycomplexandvariesaccordingtotheaorticsegment(thoracicorabdominal)[20].Aswell
asbeinganisotropic,thematerialresponseofsoftbiologicaltissueisalsohighlynon-linear.
2.1.Experimentaltest
To determine mechanical properties of AAA, studies have used both in-vivo “tests” and
ex-vivo/in-vitro testing. As reported by Raghavan and da Silva [53], both of them offer
advantages and disadvantages. In particular, in the first case the main difficult is to
accuratelydeterminethetrueforceandthedisplacementdistributionascertainingstress-free
configurationofthebiologicalentity.Ontheotherside,isolatingsamplesmayintroduceasyet
unknownchangestotheirbehavioraffectingtheresultsofsuchtests.Invivomeasurementare
oftenperformedbyusingimagingmodality.Byusingultrasoundphase-lockedecho-tracking,
Lanne et al. [43] reported that the pressure-strain elastic modulus (Ep), Eq. 1, was higher
on average and more widely dispersed in aneurysmal abdominal aorta compared to the
non-aneurysmalaortagroup. TheEpmoduluswascalculatedbasedonthediameter(Ds,Dd)
andpressure(Ps,Pd)atthesystolicanddiastolicvaluesasfollow:
−
Ep =Dp Ps−Pd (1)
Ds Dd
Using similar consideration, MacSweeney et al. [44] founded that Ep was higher in
aneurysmalabdominalaortacomparedtocontrols.
Morerecently,van’tVeeretal.[75]estimatedthecomplianceanddistensibilityoftheAAAby
meansofsimultaneousinstantaneouspressureandvolumemeasurementsobtainedwiththe
BiomechanicsandFEModBeliliongmofeAcnehuarynsmic:sR eavinewda nFdEA MdvaoncdeseilnliCnogmp outfa tiAonnaleMuodreylssm: Review and Advances in Computational Models3 5
magneticresonanceimaging(MRI).ByusingtimeresolvedECG-gatedCTimagingdatafrom
67patients,Gantenetal.[27]foundthatthecomplianceofAAAdidnotdifferbetweensmall
andlargelesions. In2011Molacekandco-authors[47]didnotfindanycorrelationbetween
aneurysmdiameteranddistensibilityofAAAwallandofnormalaorta. Howeveritisworth
tonoticethatallthesestudiesdonotprovideintrinsicmechanicalpropertiesofthetissuebut
moregeneralextrinsicAAAbehavior. Asfarastheex-vivotesting,thereareseveraltypesof
mechanicalteststhatcanbecarriedoutonmaterialstoobtaininformationontheirmechanical
behavior.Suchtestsincludesimpletensiontest,biaxialtension,conductedonthinsamplesof
material,andextension/inflationtestofthin-walledtubes.
Uniaxialtest. Uniaxialextensiontestingisthesimplestandmostcommonofex-vivotesting
methods. Here, a rectangular planar sample is subjected to extension along its length at a
constantdisplacement(orload)ratewhiletheforce(orthedisplacement)isrecordedduring
extension. Under the assumption of incompressibility (zero changes of volume during the
tensiletest,Eq.2)therecordedforce-extensiondataareconvertedtostress/strain:
=
A L AL (2)
0 0
where A and L aretheinitialcrosssectionalareaandtheinitiallengthwhile Aand Lare
0 0
thevaluesinthecurrentconfiguration.InterestedreadercanrefertoDiPuccioetal.[19]fora
recentreviewontheincompressibilityassumptiononsoftbiologicaltissue.
Biaxialtest. Duetothepresenceofthecollagenfibers,theuniaxialtestingisnotsufficientfor
highlightingtheaortatissueandthestressdistributiondoesnotfullyconformtophysiological
conditions. Therefore, biaxial tension tests should be performed. During biaxial test, an
initial square thin sheet of material is stress normally to both edges. Even if, theoretically,
thebiaxialtestarenotsufficienttofullycharacterizedanisotropicmaterials,[40,50]theyare
able to capture additional information regarding the mechanical behavior of the specimens
withrespecttouniaxialone. Bycontrast, biaxialtestsprovidesacompletecharacterization
ofthematerialpropertiesforisotropicmaterial. Tosomeextentsoftbiologicaltissuescanbe
consideredasisotropicwithincertainlimitation,however,intheirgeneralformulation,they
respond anisotropically under loads. Figure 1 depicts as example the mechanical test and
responseofasofttissueunderuniaxial(a)andbiaxial(b)test.
As we can observe, a distinctive mechanical characteristic of soft tissue in tension tests is
itsinitialflatresponseandrelativelylargeextensionsfollowedbyanincreasedstiffeningat
higherextension.Asitiswellknown,thisbehavioristheresultofcollagenfibresrecruitment
as proposed by Roach and Burton [61]. The non-linear stress strain curve arises from the
phenomenon of the fibres recruitment. As the material is stretched, the fibres gradually
becomeuncrimpedandbecomemorealignedwiththedirectionofappliedload.
Theresultsofuniaxialandbiaxialtestsareusedtocharacterizethemechanicalbehaviorof
softtissueunderinvestigation. Duetothelargedeformationthatcharacterizesthistypeof
tissue, from a mathematical point of view, a Strain Energy Function (SEF) denoted byW is
introduced.TheCauchystresstensor(σ)iscalculatedas:
σ= J−1F∂W (3)
∂F
6 4Aneurysm Will-be-set-by-IN-TECH
Figure1.Schematicofuniaxial(a)andbiaxial(b)testandcurves.Thetvalueisarappresentative
tensionvalueandeatypicalextensiondimension.
whereFisthedeformationgradienttensor,definedasF = ∂x/∂X,i.e. thederivativeofthe
currentposition x asregardstotheinitialposition X duringadeformationprocessand J is
thedeterminantofF. Undertheassumptionofincompressibility(J=1), theSEFissplitina
volumetric(W )andisochoric(W )component.Incaseofuniaxialtestwehave:
vol isoch
∂W
σ =λ isoch (4)
11 11 ∂λ
11
whereλ isthestretchinthe1-1direction(seeFig. 1(a)). Forbiaxialtestbothcomponents
11
canbecalculated:
∂W
σθθ =λθθ ∂λisoch (5)
θθ
∂W
σzz =λzz ∂λisoch (6)
zz
Equations 5-6 represent the stress components used in the follow sections. With respect to
Fig. 1direction1-1and2-2arenowdefinedasthecircumferentialσθθ andtheaxialσzz ones,
respectively.
2.2.Mechanicalpropertiesofhealthyandpathologicalaortictissue
AAA development is multifactorial phenomenon. A mechanism postulated for AAA
formationfocusesoninflammatoryprocesseswheremacrophagesrecruitmentleadstoMMP
production and elastase release. The biomechanical change associated with enzymatic
degradation of structural proteins suggests that AAA expansion is primarily related to
elastolysis[21]: adecreasingquadraticrelationshipwasfoundbetweenelastinconcentration
and diameter for normal aortas and for pathological increasing diameter [65]. Despite
universal recognition of the importance of wall mechanics in the natural history of AAAs
[2,38,81],therearefewdetailedstudiesofthemechanicalproperties.
Early studies focused on simple uniaxial tests. He and Roach [32] obtained rectangular
specimen strips during surgical resection of eight AAA patients and subjected them to
uniaxial extension tests up to a pre-defined maximum load rather than until failure. They
showedthatthestress-strainbehaviorofAAAtissuewasnon-linear. Later, intworeports,
BiomechanicsandFEModBeliliongmofeAcnehuarynsmic:sR eavinewda nFdEA MdvaoncdeseilnliCnogmp outfa tiAonnaleMuodreylssm: Review and Advances in Computational Models5 7
Vorp et al. [82] and Raghavan et al. [57] reported on uniaxial extension testing of strips
harvestedfromtheanteriormidsectionof69AAA.Thespecimenswereextendeduntilfailure.
Inmostcases, therectangularspecimens’lengthwasintheaxialorientation, butinasmall
population,theywereorientedcircumferentially. Resultshavefoundthataneurysmaltissue
issubstantiallyweakerandstifferthannormalaorta[18,57,70].
Todate,themostcompletedataonboththebiaxialmechanicalbehaviorofaortaandAAAs
comesfromVandeGeestetal. [76,77]. ThesourceofthesespecimensbecomesfromAAA
ventraltissueavailableduringtheopensurgicalrepairofunrupturedlesions. Theyreported
biaxialmechanicaldataforAAA(26samples)andnormalhumanAAasafunctionofage:
less than 30, between 30 and 60 and over 60 years of age. In particular Vande Geest and
co-workersconfirmedthattheaortictissuebecomeslesscompliantwithageandthatAAA
tissueissignificantlystifferthannormalabdominalaortictissue,Figure2.
Figure2.Stress-stretchplotcomparingtheequibiaxialresponseforAAAandHAAforfourpatient
groups,forcircumferentialdirection(a)andaxialdirection(b).Modifiedfrom[22].
The specimen was subjected to force-controlled testing with varying prescribed forces
betweenthetwoorthogonaldirections.ACCDcamerawasusedtotrackthedisplacementof
markersforminga5x5mmsquareplacedonthespecimen. Itisworthtostressoutthatthe
useofopticalextensometer(markerstrackingwithCCDcamera)isfundamentaltomeasure
the deformation during test avoiding the potential tissue slippage from the clamps. Figure
3 depicts a representative biaxial stress-stretch data for healthy (a-b) and pathological (c-d)
samplesconsideringthreedifferenttensionratios(Tθ :Tz)equalto1:1,0.75:1and1:0.75.
2.3.Materialmodels
Equationsthatcharacterizeamaterialanditsresponsetoappliedloadsarecalledconstitutive
relationssincetheydescribethegrossbehaviorresultingfromtheinternalconstitutionofa
material.Constitutivemodellingofvasculartissueisanactivefieldofresearchandnumerous
descriptionshavebeenreported.Constitutivemodelsforbiologicaltissuescanbeestablished
followingaso-calledphenomenologicalorstructuralapproach. Thefirsttypeofformulation
[14, 26, 36, 73] does not take into account any histological constituents and attempt to
describe the global mechanical behavior of the tissue without referring to its underlying
microstructure. Thephenomenologicalapproachiscommonlyusedbuthasledtoanumber
of difficulties in describing the mechanical behavior of tissues. Among phenomenological
8 6Aneurysm Will-be-set-by-IN-TECH
Figure3.Experimentalbiaxialdataforbothhealthy(a-b)andpathological(c-d)sampleswithdifferent
tensionratio.Opendiamonds,1:1;opensquares,0.75:1andopencircles,1:0.75.Modifiedfrom[9].
SEFs,VandeGeestandco-authors[77]foundthataconstitutivefunctionalformusedearlier
byChoiandVito[12],Equation7,wouldbestsuittheirexperimentaldata:
(cid:2) (cid:3)
Wisoch =b0 e12b1Eθ2θ +e12b2Ez2z+e12b3EθθEzz−3 (7)
whereb0,b1,b2andb3arethematerialparametersandEθθandEzzarethecomponentsofthe
Green-straintensor(Eq.8)definedasfollows:
(cid:2) (cid:3)
1 1
E= (C−I)= FFT−I (8)
2 2
whereIistheidentitymatrixandCistherightCauchy-Greenstraintensor.
Alternatively,structuralconstitutivedescriptions[5,28,34,35]overcomethislimitationand
integrate histological and mechanical information of the arterial wall. In particular, the
contributions of constitutive cells, fibers and networks of elements are added together to
depictthewholetissuebehavior. Thestructural-basedapproachhasbecomecommonwith
theadventofmicrostructuralimagingmethods[64,80]. Infact,softbiologicaltissueshavea
verycomplexmicrostructure,consistingofmanydifferentcomponentsandincludingelastin
fibres,collagenfibres,smoothmusclecellsandextracellularmatrix.
ThesameexperimentaldataobtainedbyvandeGeestetal. [77]werethenfittedbyusingan
invariantbasedconstitutiveequationwithtwofibrefamilies(2FF)byBascianoetal. [5],Eq.
BiomechanicsandFEModBeliliongmofeAcnehuarynsmic:sR eavinewda nFdEA MdvaoncdeseilnliCnogmp outfa tiAonnaleMuodreylssm: Review and Advances in Computational Models7 9
9,andRodriguezetal.[62,63],Eq.10.
(cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5)
W =α I −3 2+β I −1 6+γ I −1 6 (9)
isoch 1 4 6
(cid:6) (cid:2) (cid:3) (cid:7)
W =C (cid:4)I −3(cid:5)+∑4 ki1 eki2 (1−ρ)(I1−3)2+ρ(I4−I0)2 −1 (10)
isoch 1 1 i=32ki2
whereI isthefirstinvariantoftheisochoricportionoftherightCauchy-Greenstretchtensor
1
(Eq.11)andI andI aremixedinvariantsoftheisochoricportionoftherightCauchy-Green
4 6
deformationtensor(Eq.12-13),introducedfromembeddedfibers[6,33,69].
=
I trC (11)
1
= ·
I a Ca (12)
4 0 0
= ·
I b Cb (13)
6 0 0
where a , b are the direction of the fibers as reported in Figure 4(a-b). In Eq. 9, α is the
0 0
coefficientsfortheisotropicpartwhile βandγfortheanisotropiccomponent. Inthesame
manner,inEq. 10,C isastress-likematerialparameterforthepurelyelastincontribute,and
1
ki arematerialparameterscorrespondingtothefibers(k3 = k4 andk3 = k4). Theparameter
j 1 1 2 2
ρ ∈ [0;1] is a (dimensionless) measure of anisotropy, I > 1 is dimensionless parameters
0
regardedastheinitialcrimpingofthefibers(Fig.4(b)).
Figure4.Collagenfiberorientation(a0,b0)inasquarespecimenoftissueforthe2FFmodel(a),the2FF
modelwithdispersion(b)andforthe4FFmodel(c).Notethecrimpandfibresdispersionincase(b)and
thetwoadditionalfibrefamily(c0,d0)in(c).
Aconstitutiverelationbasedonfourfibresfamily(4FF)(Fig. 4(c))wasproposedbyBaeket
al.[4],includingtwoadditionallyfibresfamily(inlongitudinalandcircumferentialdirection,
[86]),Equation14: (cid:8) (cid:9)
(cid:2) (cid:3)
W = c (cid:4)I −3(cid:5)+∑4 c1i ec2i I4i−1 2−1 (14)
isoch 2 1 i=14c2i
wherec,ci andci arematerialparametersforthisspecificSEF.Ferruzzietal. [22]assumed
1 2
thatdiagonalfamiliesofcollagenwereregardedasmechanicallyequivalent,hencec3 = c4,
1 1
c3 = c4. Byfittingthebiaxialdata, themodelparameterassociatedwiththeisotropicterm
2 2
decreasedwithincreasingageforAAspecimensanddecreasedmarkedlyforAAAspecimens
[22,31].Thesefindingareingoodagreementwithhistopathologicalresultsofreducedelastin
inageing[30,51]andAAAs,e.g.[32,60].
−
Forallmodels,thediagonalfibresareaccountedforbya = b ;inthe4FFmodel,axial(c )
0 0 0
◦ ◦
andcircumferential(d )fibresarefixedat90 and0 ,respectively.
0
10 8Aneurysm Will-be-set-by-IN-TECH
Among the variety of constitutive equations reported in literature, the most significant
differenceinstructuralformulationwereincludedbyHolzapfelgroup[62]andbyBaekand
co-authors[4].Foramoredetailedanalysisontheeffectofthisassumptionandacomparison
betweenthetwoconstitutivemodel, interestedreadercanreferto[17]. Itisworthtostress
outthat, asobservedbyZeinali-Davaraniandco-authors[87], inparameterestimation, the
largernumberofparametersforamodelprovidesmoreflexibilityandgenerallygivesbetter
fitting,i.e.,decreasestheresidualerror. Acommonassumptioninallpreviousmodelsisto
assumethesamefibersdistributionandmechanicalresponsethroughoutthethickness.More
recentlySchriefletal. [66]hasobservedthatinthecaseoftheintimalayer,duetothehigher
fibers dispersion, the number of fiber families varying from two to four. However, not all
intimas investigated had more than two fiber families while two prominent fibers families
werealwaysvisible. Thenumberoffiberfamiliesequaltotwowaspreviouslyreportedby
Haskettetal.[31]byanalyzing207aorticsamples.
Finally, it is worth to notice that it is fundamental to define a constitutive model and its
material constants over some specific range, from experiments that replicate conditions
(physiologicalorpathological), [17], inordertoprovidemoreaccurateresponse. Infactall
constitutiveformulationarebasedonspecificassumptionsandhypotheses.Thecomplexities
of the artery wall poses several new conceptual and methodological challenges in the
cardiovascular biomechanics. There exist several recent frameworks, in fact, to develop
theoriesofarterialgrowthandremodeling(G&R)ofsofttissues. Interestedreadercanrefer
to a more complete and detailed review by Humphrey and Rajagopal [38, 39] and in [41].
However,inthisstudy,werestrictourattentiontostructuralbasedformulationstoemphasize
theirparticulareffects.
2.4.Geometricalmodel
ByusingFiniteElementanalyses,Fillingeretal. [23]showedthatpeakwallstressisamore
reliableparameterthanmaximumtransversediameterinpredictingpotentialruptureevent.
These findings appear to be supported by the results obtained by Venkatasubramaniam et
al. [79],whoindicatedthatthelocationofthemaximumwallstresscorrelateswellwiththe
siteofruptureand,additionally,bytheobservationthatAAAformationisaccompaniedby
an increase in wall stress [55, 83], and a decrease in wall strength [84]. Simulation on 3D
patient-specificmodelsareaimedtoanalyzethedistributionofthewallstresstoestimatethe
ruptureriskduringtheevolutionofthepathology[23],theeffectofthethrombus[29,85]or
calcification[42,45,68]onthepeakstress.Integrationofgeometrydatawithsolidmodelling
is used for estimation of vessel wall distension, strain and stress patterns. Studies, to date,
havetypicallyused3Dgeometriesusuallyobtainedfromcomputertomography(CT)[52]or
MRI[7]scansorhaveusedsimplifiedmorphologies[17,62]. Figure5reportsasexamplethe
phases from a CT reconstruction. However, both approaches present some limitations. In
particular, itisworthpointingoutthat3Dsimulationsarenotfullypatient-specificmodels
butonlybasedon3Dpatient-specificgeometrieswhilethematerialpropertiesareassumed
as mean population values due to the difficulty of assessing in-vivo material properties.
Consequently,todate,nofullypatient-specificmodelhasbeenperformed. Additionally,due
tothecomplexityofthestructureandthehighcomputationalcostrequiredbypatient-specific
models, sensitivity analyses have not been performed on 3D real geometries, and only
univariateinvestigationshavebeenperformedonidealizedshapes,toestimatetheinfluence
ofasingleparameteronthewholestressmap[63].
BiomechanicsandFEModBeliliongmofeAcnehuarynsmic:sR eavinewda nFdEA MdvaoncdeseilnliCnogmp outfa tiAonnaleMuodreylssm: Review and Advances in Computational Models9 11
Figure5.ExampleofAAA(a),segmentationofaCTcrosssection(b)and3DreconstructionofaAAA
(c),from[10].
3.Regionalvariationsinwallthicknessandmaterialproperties
As reported in previous section, starting from the observation that most AAAs are
characterizedbyacomplexnotaxisymmetricgeometriesagrowingamountofliteraturehas
beenpublishedontheinfluenceofthegeometricalfeatures.Howeveronelimitationinallthe
studiespublishedsofarisaconstantwallthicknessandhomogeneousmaterialassumedin
theFEmodels.
Wallthickness. Whilethesegmentationofthearteriallumenisawellestablishedtechnique
andhasbeenperformedwithdifferentmodalitiesinlivingsubjects,thesegmentationofthe
wallanditsconnectivecomponentsisnotafeasibleprocessduetothelowcontrastbetween
thewallandthesurroundingtissues. Theconventionalimagingtechniques, infact, donot
provide sufficient spatial resolution to assess the wall thickness measurement and variant
in-vivo.
DuringtheAAAformationthearterywallissubjectedtotheremodellingprocess[77]and,
as a consequence, the ratio between AA and AAA wall thickness changes. Di Martino
et al. [18] noted a significant difference in wall thickness between ruptured and elective
± ±
AAAs(3.6 0.3mmvs2.5 0.1mm,respectively). Bycomparingthewallthicknessbetween
healthyandpathologicalsamples,VandeGeestetal. [77]reportedthatthemeanmeasured
± ±
thickness values were 1.49 0.11 and 1.32 0.08 mm for the AA and AAA specimens,
respectively. In all these studies, samples were measured only in the anterior area and
consequently no information regarding regional variation between ventral and dorsal was
reported. Thubrikar et al. [70] obtained five whole unruptured AAA specimens during
surgical resection. Raghavan et al. [54] performed similar measures on three unruptured
and one ruptured AAA, harvested as a whole during necropsy. More recently Celi et al.
[10] performed measurements on 12 harvested unrupture ascending segments. In Table 1,
themainresultsoftheseexperimentalmeasuresarereportedforbothanteriorandposterior
±
region(mean sd).
Itisworthtonoticethatthethicknessdistributionseemstobeoppositeofthatinthenormal
abdominalaortawherethewallisthickerthantheposteriorwallin64%ofcases[74].
From the computational point of view, in literature only few authors have investigated the
effectonwallthicknessreduction. Scottietal. [67]usedanonuniformwallthicknessinan
12 1A0neurysm Will-be-set-by-IN-TECH
N.ofsamples District Thk (mm) Thk (mm) Ref
anterior posterior
± ±
5 AAA 2.09 0.51 2.73 0.46 [70]
± ±
4 AAA 2.25 0.37 2.34 0.48 [54]
± ±
12 aTAA 1.63 0.48 2.18 0.35 [10]
Table1.Wallthicknessmeasurements,reportedinliterature,categorizedbycircumferentiallocationas
anteriorandposterior
idealizedisotropicmodeltoperformedFSIsimulations. Theirresultsshowthatthemodels
with a non uniform wall thickness have a maximum wall stress nearly four times that of
auniformone. Startingfromexperimentalmeasurementon12humanharvestedascending
aorticsamples,Celietal.[10]developedstructural3DmodelsofascendingAAAbyincluding
wallthicknessregionalvariationbetweendorsalandventralareas.
Material properties. As far as the material properties, to date, different behavior has funded
between healthy (HAA) and pathological samples. However, due to the lack of sufficient
biaxialdata,afullcharacterizationinregionalvariationsarenotprovided(incircumferential
directioninparticular),andmechanicaltestshavebeenperformedmainlyintheventralarea
where the bulge was formed. As well as the material properties change during the AAA
progression,alsothewallstrengthvaluechanges.Thisaspectplaysafundamentalroleinthe
rupture phenomenon. In fact, the concept is that AAA rupture follows the basic principles
ofmaterialfailure;i.e.,ananeurysmruptureswhenthemuralstressesordeformationmeets
anappropriatefailurecriterion. Inthefiledoftheclassicalmechanics,thisconceptisdefined
bymeansofthepotentialrupturerisk(RPI)parameterandquantifyastheratiooflocalwall
stresstolocalwallstrength:
localstress
=
RPI (15)
localstrength
Inthesamemannerthesafetyfactor(SF)canbeusedastheinverseoftheRPI.
Thubrikaretal. [70]performeduniaxialtensiletestsinbothlongitudinalandcircumferential
direction, on samples from five aneurysms. To study the regional variation they obtained
samplesfromanterior,lateral(withoutdistinctionbetweenleftandrightside)andposterior
regions. Inthisstudy,however,authorsdidnotperformtestsuntilfailureandtheyrecorded
theyieldstresstodefinetheinitialpointofapermanentdamage. Thubrikaretal. observed
that in both directions, the yield stress was greater in the lateral region with respect to
the anterior and the posterior region, Figure 6(a). Experimental values regarding ultimate
stresswerereportedby Raghavanetal. [54], Figure 6(b). Theycutmultiple longitudinally
oriented rectangular specimen strips at various locations from three unruptured AAA and
onerupturedAAAforatotalof48strips. Samplesweretesteduniaxiallyuntilfailure. They
observedthatthefailuretension(ultimate)ofspecimenstripsvariedregionallyfrom55kPa
(neartherupturesite)to423kPaattheundilatedneck. Howevertheyreportthattherewas
noperceptiblepatterninfailurepropertiesalongthecircumference.
Using multiple linear regression, Vande Geest et al. [78] proposed a mathematical model
toestimatethewallstrengthbyincludingseveralmixedparameterssuchasthegender,the
presenceoftheintraluminalthrombus(ILT)andthefamilyhistory.Thefinalstatisticalmodel
forlocalCauchywallstrength(Eq.16,dimensioninkPa)wasgivenby:
(cid:2) (cid:3)
σu =719−379 ILT12 −0.81 −156(DNORM−2.46)−213HIST+193SEX (16)