Wall stress and flow dynamics in abdominal aortic aneurysms

22 Pages·2008·2.85 MB·English

ComputerMethodsinBiomechanicsandBiomedicalEngineering
Vol.11,No.3,June2008,301–322
Wall stress and flow dynamics in abdominal aortic aneurysms: finite element analysis vs.
fluid–structure interaction
ChristineM. Scottia,Jorge Jimenezb, Satish C. Mulukc and Ender A.Finola,d*
aBiomedicalEngineeringDepartment,CarnegieMellonUniversity,Pittsburgh,USA;bDepartmentofMechanicalEngineering,
CarnegieMellonUniversity,Pittsburgh,USA;cDivisionofVascularSurgery,AlleghenyGeneralHospital,Pittsburgh,USA;dInstitute
forComplexEngineeredSystems,CarnegieMellonUniversity,Pittsburgh,USA
(Received11May2007;finalversionreceived25November2007)
Abdominalaortic aneurysm (AAA) ruptureistheclinical manifestationofan inducedforce exceedingtheresistance
providedbythestrengthofthearterialwall.Thisforceismostfrequentlyassumedtobetheproductofauniformluminal
pressureactingalongthediseasedwall.HoweverfluiddynamicsisaknowncontributortothepathogenesisofAAAs,
andthedynamicinteractionofbloodflowandthearterialwallrepresentstheinvivoenvironmentatthemacro-scale.
The primaryobjective ofthis investigation istoassess thesignificance ofassuming an arbitrary estimated peakfluid
pressureinsidetheaneurysmsacfortheevaluationofAAAwallmechanics,ascomparedwiththenon-uniformpressure
resultingfromacoupledfluid–structureinteraction(FSI)analysis.Inaddition,afiniteelementapproachisutilisedto
estimatetheeffectsofasymmetryandwallthicknessonthewallstressandfluiddynamicsoftenidealisedAAAmodels
andonenon-aneurysmalcontrol.Fivedegreesofasymmetrywithuniformandvariablewallthicknessareused.Eachwas
modelledunderastaticpressure-deformationanalysis,aswellasatransientFSI.Theresultsshowthattheinclusionof
fluid flow yields a maximum AAAwall stress up to 20% higher compared to that obtained with a static wall stress
analysiswithanassumedpeakluminalpressureof117mmHg.Thevariablewallmodelshaveamaximumwallstress
nearlyfourtimesthatofauniformwallthickness,andalsoincreasingwithasymmetryinbothinstances.Theinclusionof
anaxialstretchandexternalpressuretothecomputationaldomaindecreasesthewallstressby17%.
Keywords: abdominalaorticaneurysm;fluid–structureinteraction;asymmetry;wallthickness
2000MathematicsSubjectClassifications: 74F10;92C10
Nomenclature r Radius fromcentreline toanteriorwall of AAA
b Asymmetryparameter R Radius fromcentreline toposterior wall of AAA
C Best-fit material parameters of the data examined r Material density (f denotes fluid,s denotes solid)
i f,s
inRaghavan andVorp(2000) s Principal stresses
1,2,3
d Displacementvector s Failurestress
s F
d€ Local acceleration of the solid t Fluidstress tensor
s f
d Kronecker delta t Solidstress tensor
ij s
d Undilated diameter n Velocity vector
D Maximumdiameter uh,vh Velocity generatedfrom subspaces h
1 Strain rate w Movingreferencevelocity
ij
F Coupled system ofequations tW Strainenergydensityfunction
0
fB Body force per unit volume X Nodalsolutionvariables(Batheand Zhang 2004)
I ith variant of the left Cauchy–Green tensor VF;S Computationaldomain(Fdenotesfluid,Sdenotes
i
J Reduced invariants of the left Cauchy–Green solid)
i
tensor
k Bulk modulus
L Overallarterial length
l Wall thicknessofmodel 1. Introduction
m Molecular viscosityofthe fluid Abdominal aortic aneurysms (AAA) are local enlarge-
n^ Normalvector ments of the aorta that occur preferentially below the
p Fluid pressure renal bifurcation. They are commonly described as a
*Correspondingauthor.Email:fi[email protected]
ISSN1025-5842print/ISSN1476-8259online
q2008Taylor&Francis
DOI:10.1080/10255840701827412
http://www.informaworld.com
302 C.M. Scotti et al.
50% increase in aortic diameter, an infrarenal aortic susceptibility of the aneurysm to rupture. For example,
diameter of 3cm, or a ratio of infrarenal to suprarenal fluidflowhasbeencorrelatedwithchangesatthecellular
aortic diameters of 1.5:1 (Upchurch and Schaub 2006). level,anincreaseinwallstress,reduced nutrientsupply
The continued expansion of the aorta, coupled with a to the wall, and the development or stabilization of ILT
multitude of other factors, places the patient at an and atherosclerotic plaque (Adolph et al. 1997; Taylor
increased risk for aneurysm rupture, a deleterious event et al. 1998; Bonert et al. 2003; Thubrikar et al. 2003;
that is the 10th leading cause of death in white men Zohdi2005). Furthermore, previous studieshave shown
65–74 years of age in the US (Upchurch and Schaub thatILTcontainsanumberofinflammatoryandimmune
2006). To prevent rupture, diagnosed AAAs are response agents such as macrophages, neutrophils, and
characterised by their suitability for surgical repair cytokines(Adolphetal.1997),anditwasfoundnearor
based on maximum diameter or aneurysm growth rate. at the site of rupture in 80% of autopsies reviewed
Those that are smaller than 4cm are placed under (SimaodaSilvaetal.2000).Therefore,itisapparentthat
clinical surveillance while those greater than 5 or 6cm theflowdynamicsspecifictotheAAAgeometryshould
or growing at a rate of 1cm/year are recommended for beincludedinthecomputationalmodel.Tothisend,the
surgical or endovascular intervention (Longo and use of fluid–structure interactions (FSIs) to analyse the
Upchurch 2005). While these criteria capture a mechanical stress within AAAs may represent the most
significant portion of at-risk aneurysms, it has also realistic mechanical environment, yet it has yielded
been reported that as many as 33% of ruptured AAAs conflicting results.
have diameters smaller than 5cm, and other retro- InitialFSIworkdemonstratedonlyslightdifferences
spective studies have found limited correlations with between the CSS and FSI-predicted wall stresses (Di
AAA diameter and incidence of rupture (Darling et al. Martino et al. 2001; Finol et al. 2003a). However, the
1977; Simao da Silva et al. 2000). This has led to continual investigation of both decoupled and fully
ongoing computational efforts to examine other coupledFSIresultshaveshownanincreaseinmaximum
predictors of AAA rupture, such as wall stress, that wall stress by as much as 21% for linearly elastic wall
canthenbepotentiallyimplementedintoaclinicaltool. material properties (Scotti et al. 2005a, 2005b;
Wall stress is frequently quantified as either the Papaharilaou et al. 2007). Therefore, fluid dynamics
maximum principal stress or the Von Mises stress may contribute to an improved prediction of the AAA
resultingfromaforceexertedalongthediseasedarterial wall stress and facilitate the identification of additional
wall. From a mechanical perspective, AAA rupture is patient features which will correspond to a quantifiable
theclinicalmanifestationofaninducedforceexceeding risk of rupture.
the strength of the AAA wall. As the impinging normal FSIsareincreasinglybeingimplementedinthestudy
force exerted on the arterial wall, luminal pressure is of vascular diseases and AAAs (Di Martino et al. 2001;
presumably the dominant stress producing the dis- Figueroa et al. 2005; Finol et al. 2005; Wolters et al.
tributed wall deformation. Consequently, the majority 2005; Leung et al. 2006; Li and Kleinstreuer 2006;
of published computational studies in AAAs look to Papaharilaou et al. 2007); however the advantage of
quantify and compare the wall stress of an aneurysm coupling fluid flow with the arterial wall in comparison
with the reported failure strength of the aneurysmal with a pressure driven wall-mechanics-only approach
wall, utilizing a uniform peak systolic luminal pressure remains unclear. While the addition of blood flow is
withacomputationalsolidstress(CSS)approach.These physiologically more realistic, the benefit of accurately
analyses have identified several morphological features predicting AAA wall stress vs. the increased compu-
which affect wall stress, including tortuosity (Sacks tational time has yet to be proven useful in a clinical
et al. 1999; Li and Kleinstreuer 2006), wall thickness setting. In the present investigation we model the
(Raghavan et al. 2004; Scotti et al. 2005a; Di Martino interaction of blood flow and the AAA wall under
et al. 2006), asymmetry (Vorp et al. 1998; Scotti et al. physiologically realistic flow conditions and non-linear
2005a), presence of intraluminal thrombus (ILT) (Di hyperelastic wall material properties, comparing the
Martino and Vorp 2003; Kleinstreuer and Li 2006), resulting wall stress distributions to a non-aneurysmal
luminal pressure gradients (Scotti and Finol 2007), and control and to the AAA wall mechanics predicted by
saccular index (D /L ) (Kleinstreuer and Li quasi-static solid stress analyses. The objectives of this
AAA,max AAA
2006). However, when these factors are quantified and study are to: (i) assess the significance of assuming an
applied to patient-specific geometries with known estimated fluid pressure inside the aneurysm sac for
clinical outcomes, they fail to reliably and accurately evaluatingstaticAAAwallmechanics,and(ii)performa
predict aneurysm rupture. complete parametric evaluation of geometrical features
Intheabsenceofpatient-specificcriteriaforrupture, and computational approaches for their effect on wall
attempts have been made to identify additional features stressasacontributortotheassessmentofAAArupture
which may affect the wall stress and contribute to the potential.
Computer Methods in Biomechanics and Biomedical Engineering 303
2. Methods
2.1 AAA geometry
Ten idealised aneurysm models used in this study were
generated with the CAD software ProEngineer Wildfire
(ParametricTechnologyCorporation,Needham,MA)and
have been described previously (Scotti et al. 2005a).
Inshort,themodelsdifferindegreeofasymmetryandwall
heterogeneity and are characterised by circular cross
sectionsperpendiculartothez-axisofthegeometry,which
coincideswithitscentreline.A12cmlongdilatedsegment
of the geometry represents the aneurysmal abdominal
aorta, while proximal and distal undilated segments (of
totallengthequaltothelengthoftheAAAsac)areadded
for convenience of applying the numerical boundary
conditions as to not affect the global flow characteristics
withintheAAAsac.Inaddition,anon-aneurysmalaorta
withauniformwallthicknesswasalsoincludedinthestudy
asthecontrolgeometry,withaconstantdiameterof2cm
andalengthof24cm.Thelengthofthecontrolgeometryis
chosentomatchthelengthoftheAAAmodels,suchthat
appropriateproximalflowentryanddistaloutflowlengths
areprovided.
Each model is comprised of a fluid domain, VF,
representing the aortic lumen and a solid domain, VS,
representing the AAA wall. Figure 1(a) shows b¼0.6
with fluid domain VF entities in a uniform thickness
model. Figure 1(b) illustrates qualitatively the variable
Figure 1. CAD geometry for b¼0.6 model: (a) fluid and
thicknessdomainVSatthreedifferentwalllocationsfor
solid domains with uniform wall thickness, and (b) solid
b¼0.6. The asymmetry of the models is given by domainwithvariablewallthickness.Availableincolouronline.
b[{1.0, 0.8, 0.6, 0.4, 0.2}, defined by Equation (1) as
originally postulated in (Vorp et al. 1998). b¼1.0
corresponds to azymuthal symmetry and b¼0.2 is an thicknessoftheaneurysmasmaterialextrudingnormally
AAAforwhichonlytheanteriorwallisdilatedwhilethe fromthesurfaceenclosingthelumen.TheUWmodelhasa
posteriorwall is nearlyflat. thickness given by l¼1.5mm, while the VW model
variesbetween0.5and1.5mm(withameanof1.0mmfor
r theAAAsac),inverselyproportionaltothelocaldiameter
b¼ ð1Þ
R of the cross-section. For a ruptured AAA, wall thickness
canbeaslowas0.23mmattherupturesitewithasurface-
LetrandRbetheradiimeasuredatthemidsectionofthe
wideaverageof1.45mm(Raghavanetal.2004).
AAAsacfromthelongitudinalz-axistotheposteriorand
anterior walls, respectively, as shown in the inset of
Figure 1(a).
Theundilateddiameterofthefluiddomainisd¼2cm, 2.2 Governingequations – fluid
atthe inletandoutletsectionsand a maximumdiameter, ThegoverningequationsforbloodflowaretheNavier–
D¼3d,atthemidsectionoftheAAAsac.AcriticalAAA Stokes formulations with the assumptions of an
transverse diameter of 5–6cm is the most common incompressible, laminar flow that is homogeneous.
threshold value used clinically to recommend surgical Because the aorta has a diameter greater than 0.5mm,
repair or endovascular intervention (Galland et al. 1998; anassumptionofNewtonianflowisreasonableduetothe
Scottetal.1999).Therefore,amaximumdiameterof6cm factthatbloodviscosityisrelativelyconstantatthehigh
waschosenforthisstudysinceitisthelargesttransverse ratesofsheartypicallyfoundinthehealthyaorta(Milnor
dimensionforassessmentofrupturepotentialfoundinthe 1989;Fournier1998).ForFSI,thefluiddomaindeforms
literature. The length (L) of the model is 24cm. The inresponsetothecompliantarterialwallandtheluminal
geometryofthesoliddomainisgivenbyanAAAwallwith forces. The Arbitrary Lagrangian–Eulerian (ALE)
either a (i) uniform thickness (UW) or a (ii) variable formulation for the fluid domain used in ADINA (v8.2,
thickness (VW). Both types of wall designs model the ADINA R&D, Inc., Watertown, MA) is shown in
304 C.M. Scotti et al.
Equation (2) (Donea et al. 1982; Zhang et al. 2003; with the bulk modulus and including the volumetric
Scotti and Finol 2007). strain energydensity function.
rf››vt þrfððv2wÞ·7Þv27·tf ¼fB ð2Þ t0W ¼C1(cid:1)t0J1(cid:2)þC3(cid:1)t0J1(cid:2)2 ð7Þ
p¼2kðJ 21Þ ð8Þ
Thefluidstresstensor(t)andstrainrate(1 )areshown 3
f ij
inEquations(3)and(4),wheret isthefluiddensityand
f
vthevelocityvector.Thequantitypisthefluidpressure,
d theKroneckerdelta,andmthemolecularviscosityof
ij 2.4 Boundary conditions
the fluid. Blood is modelled to have a density of
r ¼1.05g/cm3andamolecularviscosityofm¼3.5cP The boundary of the fluid domain is divided into the
f
following regions as shown in Figure 1(a) for the
(Finoletal.2003b).Forthefluiddomainthebodyforce
term (fB) is neglected and gravitational acceleration in assignment of boundary conditions: inlet (GFinlet), outlet
(GF ), and the FSI interface (GF ). The applied
the streamwise direction is considered negligible as an outlet FSI
boundary conditions on the non-FSI regions are: (i) a
AAApatientliesonahospitalbedduringCTdiagnosis.
time dependent parabolic velocity profile on GF and
inlet
t ¼2pd þ2m1 ð3Þ (ii) a time dependent normal traction (due to luminal
f ij ij
pressure) on GF and shown in Scotti et al. (2005a).
1 outlet
1 ¼ ð7nþ7nTÞ ð4Þ The inlet velocity waveform was generated from
ij
2 ultrasound measurements conducted on an AAA patient
with an extended period and greater peak velocity than
thatusedinourpreviousstudy(Scottietal.2005a).Peak
systolic flow is achieved at 0.4s with a velocity of
2.3 Governing equations – solid
43.9cm/s, as shown in Figure 2(a), while the fluid
The governing equation for the solid domain is the
velocityis18.2cm/swhenpeaksystolicpressureoccurs
momentumconservationgiven by:
(t¼0.5s). The time-average Reynolds number is 410
7·t þfB ¼rd€ ð5Þ and the Womersley number is 13.1, both values typical
s s s s for the human abdominal aorta under resting conditions
where r is the AAAwall density, t is the solid stress (Milnor1989;NicholsandO’Rourke1990).Thepressure
s s
tensor,fB arethebodyforcesperunitvolume,andd€ is waveform is a triphasic pulse appropriate for normal
s s
the localacceleration of the solid. A typical Lagrangian hemodynamics conditions in the infrarenal segment of
formulation of the solid domain (Donea et al. 1982; the human abdominal aorta first reported by Mills et al.
Bathe et al. 1999; Zhang et al. 2003; Bathe and Zhang (1970).Ano-slipboundaryconditiononthewallsofthe
2004; Scotti et al. 2005a) was implemented by utilizing fluiddomain GF was alsoapplied for all analyses.
FSI
ADINA. The AAAwall is assumed to be a non-linear, The boundary of the solid domain is divided into
isotropic, hyperelastic material with a density inlet (GS ), outlet (GS ) and the fluid–structure
inlet outlet
r ¼1.2g/cm3. The hyperelastic formulation interface (GS ) regions, shown in Figure 1(b).
s FSI
implemented in this work represents a tissue of average The FSI interfaces GF and GS are identical, coupling
FSI FSI
characteristics for the aneurysmal abdominal aorta, i.e. the fluid and solid domains. The customary boundary
the experimental fit of the stress–stretch curve as conditionsoffixedrotationandtranslationontheinletand
reported in Raghavan and Vorp (2000). Within ADINA, outlet were imposed for this study, as in previous
this hyperelastic formulation is represented as a investigations (Finol et al. 2005). However, additional
simplified, general Mooney–Rivlin material model for setsofboundaryconditionswereimposedfortheb¼0.6
the strain energy density function (tW) shown in (UW) model in an effort to compare their effects. These
0
Equation(6)whereI representstheithvariantoftheleft boundary conditions included the addition of an external
i
Cauchy–Greentensor andC arebasedonthemeans of pressure load acting on the outer surface of the wall, to
i
the best-fit material parameters of the data examined in account for the surrounding tissue and organs that affect
Raghavanand Vorp (2000). aortic pulsatility. There is limited published data on the
exact magnitude of the abdominal pressure, however
tW ¼C (cid:1)tI 23(cid:2)þC (cid:1)tI 23(cid:2)2 ð6Þ Hinnen et al. (2005) quantify it to be about 12mmHg.
0 1 0 1 3 0 1
The inclusion of this boundary condition to the solid
For the three-dimensional solid elements used in this domain has a limited affect on computational time, with
study,Equation(6)issolvedusingthereducedinvariants preliminaryestimatesshowingasmallerthan1%increase.
(J)andvolumeratioformulationshowninEquations(7) To better simulate the tethering experienced by the
i
and (8), respectively. Incompressibility is maintained abdominalaortafromsurroundingbranchesoftheaorta,
Computer Methods in Biomechanics and Biomedical Engineering 305
Figure2. Invivoluminalpulsatile(a)velocitywaveformand(b)pressurewaveformreproducedfromMillsetal.(1970).Inletpeak
systolicflowoccursatt¼0.4sandoutletpeaksystolicpressureatt¼0.5s.
an axial stretch can also be applied as a boundary external pressure (F/EX/SP), and SP with external
condition. Previous analyses have used either a pinned pressure and axial stretch (AX/EX/SP). The remainder
(Fillingeretal.2002)orfixed(notranslationorrotation) of the AAA models and the non-aneurysmal control
boundaryconditionontheinletandoutlettoapproximate implementedonlyfixedboundaryconditions.
this tethering. Since it has been established that the Forthe CSS analyses, blood flow is disregarded and
arterialwallisintensionwithinthebody(Holzapfeletal. only the solid domain VS is considered. A spatially-
2000), one suggested alternative to the tethering of the uniform, peak systolic pressure function is typically
arterial wall is to apply an axial stretch at the inlet and appliedtotheinnerwall,aspreviousworkhassuggested
outlets (Tang et al. 2005; Stylianopoulos and Barocas thisisthedominantforceinAAAbiomechanics.Forthe
2006). Within the present study, an axial stretch of 5% current investigation, this analysis (CSS ) was
U,pk
was employed asrepresented inEquation (9). completed under a pressure of 116.8mmHg, the peak
systolicpressureidentifiedfromMillsetal.(1970).Two
ds ¼0:05*L ðx;y;zÞ[GSinlet<GSoutlet ð9Þ additional static CSS analyses applied the maximum
fluidpressure(P )fromtheFSIapproach.Aspatially
max
As the AAA is situated in vivo along the spinal
uniformmaximumpressureconditionwasutilisedinthe
column, which provides support and resistance in the
CSS analysis, whereas the non-uniform pressure
U,mx
posteriordirection,theadditionofcontactwithavirtual
distribution was mapped from the fluid to the solid
spinewasincludedinthecomputationaldomainbyfixing
domainfor the CSS analysis.
N,mx
thedisplacementalongtheposteriorsurfaceofthemodel.
The aforementioned external pressure of 12mmHg was
appliedtotheremainingoutersurfaceofthearterialwall
2.5 Numerical discretization of the fluid–structure
aswell.Therefore,sixsetsofboundaryconditionswere
comparedwiththeb¼0.6(UW)model:fixed(F),axial interaction problem
stretch (AX), external pressure with fixed ends (F/EX) The fluid and solid domains are coupled in the FSI
andunderaxialstretch(AX/EX),spinalcontact(SP)with approachthroughtractionequilibriumanddisplacement
306 C.M. Scotti et al.
compatibility (Bathe et al. 1999), shown in Equations
(10) and (11).
d ¼d ðx;y;zÞ[GS >GF ð10Þ
s f wall wall
t ·n^ ¼t ·n^ ðx;y;zÞ[GS >GF ð11Þ
s s f f wall wall
Neither of these conditions requires identical, matching
meshes between the two domains and instead supports
the use of solution mapping to establish equilibrium,
which is describedindetail byZhanget al. (2003).
The finite element method (FEM) within ADINA is
usedtoestablishthenumericalsolutionfortheFSIandCSS
analyses.Linear,hexahedral,eight-nodedelementsgener-
atedbyGridgen(Pointwise,Inc.,FortWorth,TX)areused
todiscretisethefluidandsolid.The10aneurysmmodels
andthenon-aneurysmalcontrolareallcomposedof17,280
hexahedralelements(19,093nodes)forthefluidand5760
hexahedral elements (8784 nodes) for the solid domains.
Figure3illustratesthefluidandsolidmeshesforb¼0.6
whileFigure4showsthetime-periodicconvergenceofthe
z-componentofthevelocityanddisplacementfortheFSI
approach. Time-periodic convergence is said to be
achieved because the change between cardiac cycles is
minimisedforboththez-componentofthevelocityandthe
displacement. Since it follows nearly the same dynamic Figure3. Computationaldomainforb¼0.6models:(a)fluid
pattern between periods it is said to be converged; the mesh(b)solidmeshwithvariablewallthickness.Availablein
results will be nearly the same regardless of how many colouronline.
additional time cycles are simulated. Mesh sensitivity
analyses were conducted as discussed in Scotti et al. For this study, the two domains are coupled directly
(2005a),howeverthemeshusedinthepresentstudywas within ADINA, rather than employing the more time-
chosenduetoitsadequatecompromisebetweenreasonable consuming iterative approach. The direct coupling
CPU simulation times (40 CPU-hours/cardiac cycle on combines the fluid and solid matrices in its solution
average)andsmallrelativeerrorsoftheprimaryvariables process, using the Newton–Raphson method shown in
atrandomlyselectednodalpoints. Equation(14),withkastheequilibriumiterationnumber
The governing equations are applied to the finite (Zhang et al. 2003).
elementsofthecomputationaldomainintheirweakformto
generateamatrixofequations.Thefluiddomainusesthe (cid:3)›FðXkÞ(cid:4)21
Petrov–Galerkinformulation (Batheand Zhang2002)to Xkþ1 ¼Xk2 FðXkÞ ð14Þ
›X
interpolate the flow conditions over the elements. For a
givendomain,V,thefiniteelementsolutionforuhandvh
The general method for the FSI is shown in Zhang
is generated from subspaces which satisfy mass and
etal.(2003)andstartswiththefluidnodaldisplacements
momentum conservation as shown in Equations (12)
obtained from the ALE procedure. These displacements
and(13).
are then correlated with the velocity along the FSI
ð boundary. The fluid stresses found by the interpolation
q7·ðruhÞdV ð12Þ scheme are then used to obtain the load on the solid
V
domain. The fluid and solid domains are solved until
ð (cid:3)›ruh (cid:4) equilibrium in the fluidand solid is established.
w þ7·ðruhvh2tðuh;pÞÞ dV ð13Þ
›t Time integration is completed using the Euler back-
V
ward method for velocity and acceleration (Zhang et al.
The solid domain employs mixed-interpolation 2003),whichisbasedontheEuler-amethodanda¼1for
hexahedral elements, preferred in modelling nearly numericalstability.Convergenceateachtimestepisfound
incompressible media, which use constant functions to when the stress and displacement residuals are less than
interpolate pressure and bilinear functionstointerpolate theirrespectivetolerancesof0.01.Pulsatilebloodflowis
displacements on each solid element. simulatedovereighttoelevencycleswithatimestepsize
Computer Methods in Biomechanics and Biomedical Engineering 307
Figure4. Timeperiodicconvergenceplotforb¼0.6withuniformwallthickness:(a)velocityinthez-directionand(b)displacementfor
theFSIanalysis.Theinsetsshowaschematicofthenodalpointlocationsusedforthetimeconvergencestudies.Availableincolouronline.
ofDt ¼1x1023suntilperiodicconvergenceisachieved. 3. Results and discussion
FSI
ThesimulationswereperformedonaTru64Unixoperating 3.1 Solid mechanicsvs. fluid–solid dynamics
systemusinguptoeight1.15GHzEV7processorsandin-
PreviousstudiesofAAAs,havereliedonthewallstress
memorycomputing.TheCSSapproachesonlyutilisethe
predicted by quasi-static CSS techniques (Vorp et al.
soliddomain;hence,thefinalmatrixassemblyconsistsof 1998;Raghavanetal.2000;Fillingeretal.2002,2003).
onlysolidelementequationsandthecomputationaltimes This is in part due to the reduced computational time
aresignificantlylesswhencomparedwiththeFSI. requiredfortheseanalyseswhichmakesitadvantageous
308 C.M. Scotti et al.
foritspotentialuseasatoolinaclinicalsetting.However, mechanics (Fillinger et al. 2002). The CSS and
U,mx
the accuracy of CSS-based rupture potential prediction CSS approach applied the maximum pressure found
N,mx
with an assumed uniform intraluminal pressure remains from the FSI analysis as either a uniform (U) or non-
unclear, which has led to the investigation of FSI uniform (N) load in a CSS analysis. Consequently the
techniques. The primary argument for utilizing CSS stressesaresimilar,withhighlyasymmetricmodelswitha
methods is the small pressure drop across the AAA, uniform wall and the VW models showing a small
estimated in a previous study at 0.1kPa (0.075mmHg) (,0.5%)effectontheresultingwallstress.
(Wolters et al. 2005). However, we have shown that Figure 6(a) shows the predicted maximum stress of
pressurewithintheAAAsacmayexceedthepeaksystolic the b¼0.6 UW model using the four computational
pressure in the iliac arteries, while the intra-aneurysmal approaches and compares them with the transient aortic
pressuregradientrangesfrom1.8–4.6mmHg(Scottiand pressurewaveform reproduced from Mills et al. (1970).
Finol 2007). Consequently, the assumptions that maxi- For b¼0.6, the nearly identical maximum wall stress
mumstressoccursatpeaksystoleandthatpeaksystolic predicted by the CSS , CSS and FSI methods is
U,mx N,mx
pressureistheonlyfluidvariableresponsibleforthewall noticeably larger than that predicted when peak systolic
mechanicsneedstobere-examined. pressure is applied in the CSS approach. With the
U,pk
Table1andFigure5demonstratetheunderestimation pressurewaveformsuperimposedtothewallstresses,the
ofmaximumwallstressresultingfromapplyingaCSS FSI results show the maximum wall stress occurring
U,pk
method with respect to the baseline FSI method. A earlierinthecardiaccycle,att¼0.46s,whichcoincides
previous comparison between the static and transient withthemaximumluminalvolumeshowninFigure6(b).
(CSS and CSS , respectively) solid stress analyses has Comparing the wall stress with lumen volume also
S T
yielded markedly similar results, with less than 1% demonstratestheeffectofthemomentumcarriedbyfluid
difference in wall stress (Scotti et al. 2005a; Scotti and flow and the outlet boundary condition. Once peak
Finol2007).However,themaximumwallstresspredicted systolic flow is reached, the complex flow dynamics
byCSS underestimatestheFSI-predictedmaximumby within the AAA is characterised by several instances of
U,pk
12.1–19.8%fortheUWmodels,and17.8–20.8%forthe retrogradeflowwithvortexshedding,resultinginphases
VWmodels.Thisisdue,inpart,totheappliedboundary ofsacexpansionandejectionofvorticesfromtheAAA
conditions for the CSS and FSI. The CSS applied a sac. This sequence of expansion and contraction of the
U,pk
peak systolic pressure of 116.8mmHg, derived from the arterial wall is reflected in Figure 6(b) as oscillations in
waveform shown in Figure 2(b). This waveform was the 0.55s , t,1.1srange.
applied as the outlet condition for the FSI analysis, Furthermore, these fluctuations of volume and wall
resultinginahigherpressureactingintheAAAsac.This stress are not exclusive to the FSI of an AAA. Figure 7
pressure induces a proportionally higher wall stress, shows a comparison of the biomechanics between a
resulting in comparable wall stress distributions and control(non-aneurysmal)infrarenalaortamodelandthe
magnitudesfoundinFigure5(b)–(d)andTable1forthe b¼1.0 aneurysm model. The wall stress distributions
FSI, CSS and CSS techniques. These elevated along the inner anterior wall are shown in Figure 7(a)
U,mx N,mx
pressures are the effectofthe downstreamdynamicsand scaled to their respective maxima, which occurs when
pressure condition rather than hypertension, which has the change in volume of the fluid domain is at its
been shown to have a significant effect on AAA wall maximum.Theeffectsofcurvatureandthephysiological
Table1. MaximumwallstresspredictedbyFSIandCSSmethods1.
b 0.2 0.4 0.6 0.8 1.0
UW
FSI 37.6 35.4 35.0 33.3 31.3
CSS 31.4(216.4%) 28.5(219.6%) 28.1(219.8%) 27.8(216.7%) 27.5(212.1%)
U,pk
CSS 37.7(þ0.3%) 35.5(þ0.2%) 35.0 33.3 31.3
U,mx
CSS 37.6 35.4 35.0 33.3 31.3
N,mx
VW
FSI 124.5 113.4 106.2 97.0 93.2
CSS 99.6(220.0%) 90.4(220.3%) 84.1(220.8%) 79.8(217.8%) 76.6(217.8%)
U,pk
CSS 125.1(þ0.4%) 113.8(þ0.3%) 106.4(þ0.2%) 97.1(þ0.2%) 93.4(þ0.2%)
U,mx
CSS 124.5 113.4 106.2 97.0 93.2
N,mx
1Maximumwallstress(inN/cm2)andcomparisonbetweenthetwonumericalmethods.Theparenthesesshowthe%differenceofthestressobtainedwiththeCSS methods
S
withrespecttothebaselineFSImethod.CSSU,pkindicatesstaticstressanalysesconductedwithauniformpressureof116.8mmHg,correspondingtopeaksystolefortheFSI
outflowboundarycondition;CSSU,mxwereconductedwiththemaximumFSIpredictedwallpressureapplieduniformlyontheinnersurfaceoftheAAAwall;andCSSN,mx
wereconductedwiththemaximumFSIpredictedwallpressuredistributionappliednon-uniformlyontheinnersurfaceoftheAAAwall.
Computer Methods in Biomechanics and Biomedical Engineering 309
Figure5. Comparisonofmaximumwallstressdistributionsforb¼0.6UWmodelpredictedby:staticCSS,under(a)peaksystolic
pressure {CSS } (b) maximum, uniform FSI-predicted pressure {CSS }, (c) maximum, non-uniform FSI-predicted pressure
U,pk U,mx
{CSS },and(d)fluid–structureinteraction{FSI}.ThesymbolDindicatesthelocationofthemaximumwallstress.Availablein
N,mx
colouronline.
significance of aneurysm development are underscored conditions at t¼0 (see Figure 2) yield a change in
by the increase in wall stress between the non- volume for both the non-aneurysmal and AAA models
aneurysmal aorta and the AAA model. Since both greaterthanzeroattheonsetofthelastpulsatilecyclein
modelsaresymmetricwithrespecttothecentreline,the Figure 7(b) once the periodic steady state is achieved.
curvature of the AAA model causes the wall stress to Furthermore, the complex fluid dynamics within a
localise at the proximal and distal ends, more than dilated aorta results in a greater momentum change and
doublingthestressinthenon-aneurysmalaorta.Invivo, flow-induced wall pressure, and thus increases the
this elevated stress is believed to act on a weaker, differential volume in comparison with the non-
potentially thinner aortic wall, placing the AAA at an aneurysmal aorta. These fluctuations can be dampened
increased risk for deleterious event of rupture. Recent or removed altogether with additional boundary con-
studies have shown that hemodynamic forces may ditions such as external pressure or alternative outlet
modulate AAA inflammation and diameter enlargement boundary conditions on the fluid domain, such as
via wall shear or strain-related reductions in oxidative impedance orresistance.
stress at these localised regions of change in vessel
curvature, making them more susceptible to weakening 3.2 Geometry effectsin FSI (I): sagittal asymmetry
(Shoetal.2004,2002).Forthenon-aneurysmalaortathe Flow dynamics within the arterial tree can have a
maximumstressisachievedearlierthantheAAAmodel, significant effect on disease progression and cellular
at t¼0.43s rather than t¼0.46s, as shown in Figure mechanics. Among other findings, wall shear
7(b). Also present are subtle fluctuations in luminal stress gradients, the development of vortices, and
volume in the non-aneurysmal aorta after peak systolic recirculating blood flow have been shown to correlate
pressure. The non-zero velocity and pressure boundary with atherosclerosis, intimal thickening, and potential
310 C.M. Scotti et al.
Figure6. (a)Comparisonofthetemporaldistributionofwallstressfortheb¼0.6UWmodelforstaticCSSandFSImethods.The
aorticpressurewaveformisshownonthesecondaryy-axis.Thevaluesshownareforthemeshelementwherethepeakwallstress
occurs for each computational technique. (b) Temporal variation of luminal volume and wall stress for the b¼0.6 UW model
(; ¼206.6cm3).
0
arterialexpansion(Adolphetal.1997;Tayloretal.1998; pressure changes. Since the pressuregradient across the
Bonert et al. 2003; Sho et al. 2002, 2004). As shown in aneurysm models is similar regardless of asymmetry, as
Figure 8, the compliant arterial wall prevents vortex shown in Figure 8, the largervolumeof the asymmetric
formation in the aneurysm sac, unlike previously model(b¼0.2)atmaximumdeformationyieldsalarger
observedflowpatternsinrigidwallstudiesofasymmetric compliance.Thechangesinluminalvolumeareshownin
AAAmodelsundersystolicflowdecelerationatt¼0.5s Table2,withthemaximumoccurringafterpeaksystolic
(peak systolic pressure in reference to the outlet normal flow and prior to peak systolic pressure conditions.
traction boundary condition) (Finol et al. 2003b). This Thetimelagbetweenthemaximuminletvelocityandthe
streamlined profile along the mid-sagittal plane of the maximumoutletpressureaffectsthedynamicswithinthe
AAAisverysimilartothatshowninScottietal.(2005)for AAAsac,contributingtothephaseshift.Withtheinflux
whichlinearlyelasticwallpropertiesweremodelledanda ofbloodflowingintotheAAAsacandtheresistanceofthe
different inlet velocity waveform was used. In Figure 8, opposing pressure boundary condition, the compliant
themaximumvelocityisabout20%largerfortheb¼0.2 aneurysmal segment of the idealised aorta expands by
model,dueinparttothephaseshiftandlargercompliance 15.2% for the most asymmetric model vs. 12.8% for
of the arterial wall. Volumetric compliance of a blood the axisymmetric model. The larger expansion of the
vessel is based on the relationship betweenvolume and b¼0.2modelresultsinanincreaseintheelasticenergy