Improving the Efficiency of Abdominal Aortic Aneurysm Wall Stress Computations

18 Pages·2014·1.89 MB·English

Improving the Efficiency of Abdominal Aortic Aneurysm
Wall Stress Computations
Jaime E. Zelaya1, Sevan Goenezen2, Phong T. Dargon3, Amir-Farzin Azarbal3, Sandra Rugonyi1*
1DepartmentofBiomedicalEngineering,OregonHealth&ScienceUniversity,Portland,Oregon,UnitedStatesofAmerica,2DepartmentofMechanicalEngineering,
TexasA&MUniversity,CollegeStation,Texas,UnitedStatesofAmerica,3DepartmentofSurgery,DivisionofVascularSurgery,OregonHealth&ScienceUniversity,
Portland,Oregon,UnitedStatesofAmerica
Abstract
An abdominal aortic aneurysm is a pathological dilation of the abdominal aorta, which carries a high mortality rate if
ruptured.Themostcommonlyusedsurrogatemarkerofruptureriskisthemaximaltransversediameteroftheaneurysm.
Morerecentstudiessuggestthatwallstressfrommodelsofpatient-specificaneurysmgeometriesextracted,forinstance,
fromcomputedtomographyimagesmaybeamoreaccuratepredictorofruptureriskandanimportantfactorinAAAsize
progression.However,quantificationofwallstressistypicallycomputationallyintensiveandtime-consuming,mainlydueto
thenonlinearmechanicalbehavioroftheabdominalaorticaneurysmwalls.Thesedifficultieshavelimitedthepotentialof
computationalmodelsinclinicalpractice.Tofacilitatecomputationofwall stresses,wepropose to usealinearapproach
thatensuresequilibriumofwallstressesintheaneurysms.Thisproposedlinearmodelapproachiseasytoimplementand
eliminates the burden of nonlinear computations. To assess the accuracy of our proposed approach to compute wall
stresses,resultsfromidealizedandpatient-specificmodelsimulationswerecomparedtothoseobtainedusingconventional
approaches and to those of a hypothetical, reference abdominal aortic aneurysm model. For the reference model, wall
mechanicalpropertiesandtheinitialunloadedandunstressedconfigurationwereassumedtobeknown,andtheresulting
wallstresseswereusedasreferenceforcomparison.Ourproposedlinearapproachaccuratelyapproximateswallstressesfor
varyingmodelgeometriesandwallmaterialproperties.Ourfindingssuggestthattheproposedlinearapproachcouldbe
used asaneffective, efficient,easy-to-use clinical tooltoestimate patient-specific wall stresses.
Citation:ZelayaJE,GoenezenS,DargonPT,AzarbalA-F,RugonyiS(2014)ImprovingtheEfficiencyofAbdominalAorticAneurysmWallStressComputations.PLoS
ONE9(7):e101353.doi:10.1371/journal.pone.0101353
Editor:MohammadR.K.Mofrad,UniversityofCaliforniaBerkeley,UnitedStatesofAmerica
ReceivedDecember2,2013;AcceptedJune5,2014;PublishedJuly9,2014
Copyright: (cid:2) 2014 Zelaya et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.
Funding:ThisworkwassupportedbytheNSFDBI-1052688grantandagrantfromtheOregonClinicalandTranslationalResearchInstitute(OCTRI),National
CenterforAdvancingTranslationalSciences(NCATS)NIHTL1TR000129.Thecontentissolelytheresponsibilityoftheauthorsanddoesnotnecessarilyrepresent
theofficial viewsofgrant givingbodies.Thefundershad norolein studydesign, data collection and analysis,decisionto publish, orpreparationof the
manuscript.
CompetingInterests:Theauthorshavedeclaredthatnocompetinginterestsexist.
*Email:[email protected]
Introduction stresswillresultintissuegrowth(e.g.,increasedwallthickness)
and remodeling (e.g., increased collagen deposition) that
Abdominal aortic aneurysms (AAAs) are pathological dilations
lowers wall stress to homeostatic levels; likewise, an increase
of the abdominal aorta of at least 3 cm in diameter [1][2]. If
in wall shear stress will result in an increase in vascular
ruptured, AAAs carry a mortality rate of 90% [3], claiming
diameter that will lower shear stresses to homeostatic values.
approximately 15,000 American lives annually [2]. Generally,
These G&R mechanisms are postulated to act during AAA
reparative surgery is performed if the AAA maximum transverse
expansion, explaining the possible relationship between wall
diameter measured on computed tomography (CT) scan or
stress and AAA progression. Thus, wall stress has been the
ultrasound imaging exceeds 5.5cm or expands at a rate of
subject of extensive AAA biomechanical research [13] and is
1 cm/year or greater [1]. But use of these criteria is concerning
typically obtained using finite element analysis (FEA)
because AAAs smaller than 5.5 cm can rupture, and larger but
[8][14][15][16]. To compute wall stress, average or systolic
stable AAAs may receive unnecessary surgery [3][4][5][6][7].
intraluminal pressures are conventionally applied to image-
Thus, there is a need to more reliably assess the risks for AAA
derived, patient-specific geometries that are assumed to be
rupture andexpansion[8].
unloaded and unstressed [17][18]. In these models, the AAA
Wall stress has been shown to be a more accurate predictor wallsareassumedtobenonlinearhyperelasticwithmechanical
of AAA rupture and expansion than the maximum transverse properties measured from cadaver tissues or tissues from
diameter [5][9][10]. This is because tissues tear apart when patients undergoing elective repair [13][19][20][21]. Incor-
wall stress exceeds a threshold stress for rupture, which rectly assuming that imaged geometries are unloaded, howev-
depends on the tissue strength. Recent theories of growth er, implies that application of intraluminal pressures to the
and remodeling (G&R) [11][12] postulate that vascular tissues walls will result in overly distorted AAA geometries, typically
grow and remodel so that homeostatic wall stresses are with overestimated wall stress distributions [21][22][23][24].
conserved. According to G&R theories, an increase in wall To resolve this problem, algorithms have been developed for
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boundary conditions, including the effects of internal and
external structures (thrombus and external organs), are
unknown and frequently neglected or approximated. For
AAA wall stresses to become a useful clinical indicator and
be applicable in a clinical environment, a more efficient and
robustmethodologyisneededforestimatingwallstressesfrom
patient-specific geometries.
In this study, we propose modeling AAAs using FEA linear
models as a means of obtaining equilibrium stresses in a more
robust and computationally efficient way. Because linear
models assume infinitesimally small displacements and strains,
the approach preserves the integrity of the imaged geometry,
and the application of intraluminal pressures achieves equilib-
rium of forces and wall stresses directly in the patient-specific
geometry. We assess the accuracy and effectiveness of our
proposedapproachusingidealizedmodelsandpatient-specific
models of AAAs. We also explore the effect of employing
different nonlinear wall material properties and residual
stresses on wall stress computations in idealized models and
compare results with those obtained from linear models.
Problem Formulation and Equations Employed
Todeterminetherelativeaccuracyofthelinearapproachto
Figure1.SchematicshowingtheAAAmodelsemployed.Inall compute AAA wall stresses, we employed three models: a
models, an internal pressure is applied to an initially unloaded, reference model, a conventional model, and our proposed
undeformed configuration. Differences between models are in the
linear model (see Fig. 1). The reference model (Fig. 1A) was
choice of the wall material properties and initial configuration
used as a reference for wall stresses (see below for a more
employed. (A) Reference model: the walls are characterized by
hyperelastic nonlinear material properties; the initial configuration, detailed description of the model). In the reference model,
which is assumed to be known, represents the clinically unknown initial conditions and tissue properties are assumed to be
unloadedandunstressedwallconfiguration,andthedeformed(loaded) known. The conventional model (Fig. 1B) represents the most
configurationrepresentsthedeformedgeometrythatisimagedfrom
commonly used approach to computing wall stress, in which
the patient. (B) Conventional model: the walls are assumed to have
the patient deformed configuration is used as an unloaded,
hyperelasticnonlinearproperties;theinitialconfigurationischosenas
the deformed configuration obtained from the reference model (but unstressed initial configuration and walls are assumed to have
this configuration is assumed to be unloaded and unstressed). After nonlinear, hyperelastic material properties. The linear model
applicationofaninternalpressureintheconventionalmodel,theinitial (Fig. 1C) also uses the patient deformed configuration as an
configuration further deforms into a loaded configuration. (C) Linear
unloaded, unstressed initial configuration, but solution of the
model: the walls are assumed to have linear elastic properties, with
model does not change the wall geometry, and equilibrium
infinitesimallysmalldeformationsandstrains;theinitialconfigurationis
chosen as the deformed configuration obtained from the reference stresses are obtained in the patient geometry. Wall stresses
model (as in the conventional model). Because of the assumptions obtained using the linear and conventional models were
made in the linear model, the initial configuration barely deforms, compared to stresses obtained using the reference model.
preserving the geometrical characteristics of imaged patient geome- Comparisonswereperformedfirstusingidealizedmodels,such
tries.
as straight tubular models representing the arterial wall and
doi:10.1371/journal.pone.0101353.g001
idealized curved axisymmetric models of the AAA. Compar-
isons were then extended to a subject-specific AAA geometry.
Forthetubularmodels,wefurtherexploredtheeffectsofusing
approximating the tissue unloaded configuration from avail-
different reported nonlinear tissue mechanical properties and
able loaded CT scan or magnetic resonance imaging (MRI)
theeffectsofresidualstressesonwallstress.Wethencompared
geometries [18][22][23][25]. Applying intraluminal pressures
wall stresses obtained with nonlinear models to thoseobtained
to these computed unloaded geometries results in wall
using the linear model approach.
deformations that closely approximate the original loaded
AAA geometry and more accurately predict wall stress.
Model Formulation
Although some of these promising methods have been
For all models considered, we solved the equations of
validated, they are difficult to implement and are computa-
equilibrium
tionally intensive [18]. This is because computation of the
undeformedunloadedconfigurationinvolvesthesolutionofan
inverse nonlinear problem. In fact, given the nonlinear +:s~0 inV ð1Þ
properties of AAA walls, even calculation of wall stresses from
a known unloaded, unstressed configuration is involved and
requires extensive computations. Even when using methods to with boundary conditions
recover the unloaded geometry, limitations of current models
include: residual stresses, which are characteristic of vascular
tissues, are neglected; spatial changes in aneurysmal tissue
s:^n~{p^n onC, s:^n~0 onC ð2Þ
properties along and across the wall are neglected; and ‘‘true’’ l o
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where s is the Cauchy stress tensor; p is the intraluminal blood We extensively employed the population average mechanical
pressure and was chosen here as the systolic pressure (0.016 N/ propertiesfoundbyRaghavanandVorp(RVinTable1)[19],
mm2=120mmHg) unless otherwise stated; ^nn is a unit vector as these properties are widely used. To assess the effects of
normaltothewallsurface;Visthebodydomaininthedeformed patient-specific tissue mechanical properties, which are not
configuration;andC andC refertothelumenandoutersurfaces known in clinical practice, we varied the values of a and b
l o
of the deformed configurations of the AAA models, respectively. (c=0)andalsousedcoefficientsobtainedbyPolzeretal.from
Notethatthechoiceofusingsystolicpressureisarbitraryanddoes twodifferentpatient-tissuesamples(P1andP2;seeTable1)in
not affect the results presented, since the hypothetical reference the reference models.
models are alsoassumed tobesubjected tosystolic pressure. In the linear model, the wall was assumed to be an almost
The reference and conventional models assumed nonlinear, incompressible, linear elastic material, characterized by an
hyperelasticwallmaterialproperties.Specifically,AAAwallswere arbitrary, albeit high, Young’s modulus E (which ensures
assumedtobealmostincompressible,homogeneous,andisotropic infinitesimal deformations without compromising wall stress
with an energydensity function Wof theform[19][21] values,seeResults).Linearconstitutiverelationsweregivenby
W~aðI {3ÞzbðI {3Þ2zcðI {3Þ3 ð3Þ
B B B
s~Ce ð5Þ
wherea,bandcarecoefficientsthatdenotethepropertiesofthe
tissue;I isthefirstinvariantoftheLeftCauchy-Green tensorB whereCisthestiffnesstensor,which,foranisotropicmaterial,
B
(I =trB)withB=FFT;andFisthedeformationgradienttensor. depends on E and the Poisson’s ratio n, and e is the
B
Theconstitutiverelationscorrespondingtothenonlinearmaterial infinitesimal strain tensor. Unless otherwise stated, we used
represented inEq.3 are describedby E=8.46109 N/mm2andn=0.4999incomputationsusingthe
linear model.
Differences among the reference, conventional, and linear
models were in the choice of material properties and initial
LW
s~{HIz2 B ð4Þ configurations (see Fig. 1). In all models, the initial configu-
LI
B ration was assumed to be unloaded and unstressed. For the
where Histhehydrostatic pressureandIistheidentity tensor. reference model, the initial configuration was chosen arbi-
The values of coefficients in Eq. 3 were determined from trarily and represented the unstressed and unloaded configu-
human tissue samples subjected to tensile tests. Material ration of the tissue that was assumed to be known in our
properties of AAA tissues have been measured in different models (but which is unknown in clinical practice). For the
studies with varying degrees of accuracy [19][21][26]. One of conventional and linear models, however, the initial configu-
the pioneering studies, by Raghavan and Vorp [19], assumed ration was taken as the deformed configuration of the
that W (Eq. 3) had two terms (c=0) and fitted stress-strain reference model. This choice of initial configuration intended
resultsfromuniaxialtissuetensileteststofindthecoefficientsa to simulate the use of the loaded geometry obtained from CT
and b for each tissue sample. While the study found that the scans or other imaging techniques as an unstressed, unloaded
coefficients vary from sample to sample, it provided a configuration,bothintheconventionalandlinearapproaches.
population average, which is frequently used in the AAA The reference model, conversely, simulates the loading of
literature to represent the material properties of AAA tissues. tissues from the unknown unstressed configuration, and thus
More recently, Polzer et al. [21] measured AAA patient tissue represents a more accurate model of wall stress.
samplesusingbiaxialtensiletestsandfittedtheresultingstress- Once the stress distributions were computed for all models,
strain curves to an energy-density function similar to that of the wall stresses obtained from the conventional and linear
Eq.3butconsistingof5terms,i.e.,W=a(I 23)+b(I 23)2+ modelswerecomparedtothestressesfromthereferencemodel
B B
c(I 2 3)3+f(I 2 3)4+g(I 2 3)5, where f and g are also in order to determine the degree to which conventional and
B B B
coefficients that denote the properties of the tissue. The study linearapproachesapproximatedreferencestresses.Theanalysis
found that even though W was assumed to be isotropic, it was performed on both idealized and patient-specific models of
approximated the mechanical behavior of the tissue well. The AAAs, which are described further below. For idealized thick-
study also found striking variations in mechanical properties wall tubular AAA models, analytical expressions exist for linear
among sampled tissues. Here, for comparison, we used models and were derived in what follows for the hyperelastic
mechanical properties obtained in the Raghavan-Vorp and tissue models.
Polzer studies (see Table 1, f=g=0 for the two patient-
specific tissue material properties selected from Polzer et al.).
Table1. CoefficientsofAAAtissue materialproperties used inthisstudy (see Eq.3).
MaterialModel a(N/mm2) b(N/mm2) c(N/mm2) Reference
Raghavan-Vorp(RV) 0.174 1.881 0 [19]
Polzeretal.sample(P1) 0.0145 0 2.259 [21]
Polzeretal.sample(P2) 0.022 1.461 1.0 [21]
doi:10.1371/journal.pone.0101353.t001
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whererandRaretheradiiofthedeformedandinitial(unstressed)
configurations, respectively (see Fig.2).
Thus, for the thick-wall tube model considered here, F and B
are
0 1
l 0 0
hh
F~BB 0 lrr 0 CC,
@ A
0 0 l
zz
ð10Þ
0l2 0 0 1
hh
B~FFT~BB 0 l2 0 CC
@ rr A
Figure 2. Thick-wall cylindrical model with applied internal 0 0 l2
pressure used for the derivation of analytical solutions. The zz
undeformedconfigurationisassumedtobeunstressedandunloaded;
the deformed configuration is obtained after applying an internal
pressurep.A,BandRrepresenttheinternalwallradius,externalwall 0s 0 0 1
hh
radius and radial coordinate, respectively, in the undeformed config-
ands~B 0 s 0 C~2Bðaz2bðI {3ÞÞ{HI ð11Þ
uration; a, b, and r, represent the internal wall radius, external wall @ rr A B
radius and radial coordinate, respectively, in the deformed configura- 0 0 s
zz
tion.
doi:10.1371/journal.pone.0101353.g002
Replacing intoEq. 6 resultsin
Analytical Expressions for an Axisymmetric Thick-Wall
Tube under Internal Pressure
Consider an axisymmetric thick-wall tube as a simple Lsrr~1(cid:4)2(cid:2)l2 {l2(cid:3)(cid:2)az2b(cid:2)l2 zl2{2(cid:3)(cid:3)(cid:5): ð12Þ
Lr r hh rr hh rr
representation of a blood vessel (see Fig. 2). Initially, the
vessel is assumed to be undeformed, unloaded and unstressed
with an inner radius, A, and outer radius, B. In addition, the Further, foran incompressible material [12],
vessel is assumed to be constrained at both ends in the
longitudinal direction, and a uniform internal pressure p is
l l l ~1 ð13Þ
applied on the luminal surface. Application of the internal rr hh zz
pressure results in a deformed geometry with inner and outer
and, therefore, usingEqs. 9and 13and solvingforr,
radii a and b, respectively.
Employing cylindrical coordinates, the equilibrium equation
r Lr
(Eq. 1)for thiscasereduces to ~1 ð14Þ
RLR
Ls s {s
rrz rr hh~0, ð6Þ
Lr r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r~ R2zb2{B2, ð15Þ
whereristheradiusofthedeformedconfigurationands ands
rr hh
are the Cauchy radial and circumferential stresses, respectively
[12].
r 1
The boundary conditionsfor theproblem (Eq.2) become l ~p , l ~ : ð16Þ
hh ffiBffiffiffi2ffiffi{ffiffiffiffibffiffiffi2ffiffizffiffiffiffiffirffi2ffiffi rr lhh
s ðr~aÞ~{p, ð7Þ
rr
Using Eqs.8, 12and16andsolvingfors thenyields
rr
s ðr~bÞ~0: ð8Þ
rr
s ðrÞ~ð2b{aÞln(cid:7) B2 (cid:8)za(cid:2)B2{b2(cid:3)(cid:7)1{ 1(cid:8)
A.AnalyticalSolutionsforNonlinearHyperelasticTissue rr B2{b2zr2 r2 b2
Model. Weassumethewallpropertiestobeincompressibleand (cid:7)b(cid:8)
nonlinear hyperelastic with a parabolic strain energy density z2ða{2bÞln ð17Þ
r
function W as proposed by Raghavan et al. [19], see Eq. 3 (with
c=0)andconstitutive equations givenby Eq.4. "b2{B2 b2{B2 (cid:2)B2{b2(cid:3)2(cid:7)1 1(cid:8)#
If lhh, lrr, and lzz represent the stretch ratios in the z4b 2B2 {2ðB2{b2zr2Þz 4 r4{b4 ,
circumferential, radial, and longitudinal directions, respectively,
thenfor atube underinternal pressure,
andusing Eq.7,
r Lr
l ~ ,l ~ ,l ~1 ð9Þ
hh R rr LR zz
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weincludedtheequationstogetherwithsomeofthestepsrequired
in the derivation of equations. The constitutive relations for a
{p~ð2b{aÞln(cid:7) B2 (cid:8)za(cid:2)B2{b2(cid:3)(cid:7)1 { 1(cid:8) lmineenatrs ienlapstoilcarmcaotoerrdiainl,ataesss,uamrein[g27i]nfinitesimally small displace-
B2{b2za2 a2 b2
(cid:7)b(cid:8)
z2ða{2bÞln ð18Þ
a E
s ~ ½ne zð1{nÞe (cid:2),
"b2{B2 b2{B2 (B2{b2)2(cid:7)1 1(cid:8)# rr ð1znÞð1{2nÞ hh rr
z4b { z { :
2B2 2ðB2{b2za2Þ 4 a4 b4 s ~ E ½ne zð1{nÞe (cid:2), ð23Þ
hh ð1znÞð1{2nÞ rr hh
s ~nðs zs Þ,
In order to solve Eq. 18 for a and b, an additional equation is zz rr hh
needed. Since the cross-sectional areas of the undeformed and whereEistheYoung’smodulusandnisthePoissonratio,ande
rr
deformed configurations areequal due toincompressibility, and e are the radial and circumferential strains, respectively,
hh
with e =0.
zz
Foralinearaxisymmetrictubewithinternalpressure,thestrain-
p(cid:2)b2{a2(cid:3)~p(cid:2)B2{A2(cid:3), or a2~b2{B2zA2: ð19Þ displacement relations inpolar coordinates are
Lu u
By substituting the equation for a in Eq. 19 into Eq. 18 and err~ Lrr, ehh~ rr, ð24Þ
numericallysolvingforb(givena,b,AandB),thevaluesforband
a that satisfy the boundary conditions (Eqs. 7 and 8) can be where u istheradialdisplacement.
r
obtained. Substituting Eqs. 23 and 24 into Eq. 6 results in the following
Onceaandbarecomputed,Eqs.6and17areusedtoobtainan differential equation:
expression forthecircumferential stress,
" #
Eð1{nÞ L2u 1Lu u
rz r{ r ~0: ð25Þ
{2ð2b{aÞr2 4b(cid:2)b2{B2(cid:3)r2 a(cid:2)B2{b2(cid:3) ð1znÞð1{2nÞ Lr2 r Lr r2
s ðrÞ~ z {
hh ðB2{b2zr2Þ ðB2{b2zr2Þ2 r2
zb(cid:2)B2{b2(cid:3)2(cid:9){3{ 1(cid:10){2ða{2bÞ The solutionof Eq.25is
r4 b4
ð20Þ 1
zð2b{aÞln(cid:7) B2 (cid:8){a(cid:2)B2{b2(cid:3) ur~c1rzc2r, ð26Þ
B2{b2zr2 b2
wherec andc areconstants.Usingtheboundaryconditions(Eqs.
z2ða{2bÞln(cid:7)b(cid:8)z2b(cid:2)b2{B2(cid:3){2b(cid:2)b2{B2(cid:3): 7 and81),wall2stresses areobtained:
r B2 B2{b2zr2
{pa2b2 (cid:9)1 1(cid:10)
s ðrÞ~ { ,
rr ða2{b2Þ b2 r2
ð27Þ
To solveforszz,wefirstfind an equationforH usingEq. 11, {pa2b2 (cid:9)1 1(cid:10) {2npa2
s ðrÞ~ z , s ~ :
hh ða2{b2Þ b2 r2 zz ða2{b2Þ
H~2al2z4bl2(cid:2)l2 zl2{2(cid:3){s : ð21Þ
rr rr hh rr rr
Notethat,duetotheassumptionofsmalldisplacements,while
displacements are computed, the geometry is assumed to remain
SubstitutingEq.21intotheexpressionfors fromEq.11then unchanged(sothata=A,b=B,andr=Rindependentofp).Thus,
zz
yields, stresses depend on pbut not E.
s ~2az4b(cid:2)l2 zl2{2(cid:3){2al2{4bl2(cid:2)l2 zl2{2(cid:3)zs :ð22Þ Effective Stress
zz hh rr rr rr hh rr rr
TheeffectiveorvonMisesstressisameasureoflocalmaximum
stressesthattakesintoaccountthecontributionofnormalstresses
Thustheanalyticalsolutionsforwallstressesinatubularmodel in addition to shear stresses and is extensively used to report
when tissue properties are assumed to be nonlinear hyperelastic stresses in the AAA literature [16][23][28]. However, it is worth
(Eq. 3 with c=0) are given by Eqs. 17, 20 and 22, once the mentioningthatothermeasuresofstressorperhapsstretchmight
deformedinternalandexternaltuberadii,aandb,arecalculated be more relevant in determining AAA risks of expansion and of
using Eqs.18and19. rupture [13]. Because of their wide use, however, we chose to
B. Analytical Solutions for the Linear Tissue reporteffectivestressesinthismanuscript,andforidealizedAAA
Model. Analytical solutions for the case of a thick-wall tube geometries circumferential and radial stresses were also consid-
under internal pressure with linear wall properties under small ered.Notefurtherthat,forthepurposeofthismanuscript,weare
deformationscanbefoundelsewhere(e.g.[27]).Forcompleteness notemployingeffectivestressesasarupturecriterion,butonlyasa
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convenient way of reporting wall stresses. In cylindrical coordi- Wefurtherexploredtheeffectofresidualstressesonwallstress
nates theeffective stress iscalculated as follows: distributions. Residual stresses are the stresses that remain on a
vascularwallafterloadsimposedonthetissuehavebeenremoved.
They manifest in blood vessels as a shrinkage in the axial length
s ~
eff when vessel segments are cut longitudinally (axial stresses) and as
sffiðffisffiffiffiffiffiffi{ffiffiffiffiffisffiffiffiffiffiffiÞffiffi2ffiffizffiffiffiffiffiðffisffiffiffiffiffiffi{ffiffiffiffiffisffiffiffiffiffiffiÞffi2ffiffizffiffiffiffiffiðffiffisffiffiffiffiffiffi{ffiffiffiffiffisffiffiffiffiffiffiÞffiffi2ffiffizffiffiffiffiffi6ffiffi(cid:2)ffiffisffiffiffi2ffiffiffizffiffiffiffiffisffiffi2ffiffiffizffiffiffiffiffisffiffiffi2ffiffiffi(cid:3)ffiffið28Þ anopeningoftheunloadedcircularcross-section,characterizedby
rr hh rr zz hh zz rh rz hz an opening angle [30], when vessel segments are cut radially
2 (circumferential stresses). To model circumferential residual
stresses in tubular models of AAA, we started from a 2D open
sectorintheinitialconfiguration(seeFig.3A).Thedimensionsof
the sector were determined so that the closed unloaded
Models of AAA
configuration was the same, independent of opening angle. The
opensectorwasmodeledasaplanestrain2DprobleminADINA,
The specific geometrical models of AAA considered and
andsymmetry was consideredbymodeling halfof thesector (see
strategies employed to solve for the model wall stresses are
Fig. 3B). The open sector was then closed by imposing a
described below.
displacementinthedirectionofclosureononeendofthesector.
Note that this way of modeling residual stresses cannot be
Axisymmetric Thick-Wall Tubular Model of AAA
implementedinpatient-specificmodelsoftheAAA.Oncethe2D
The arterial wall was first modeled as an axisymmetric, thick- segment was closed, an internal pressure (p=0.016N/mm2) was
wall, straight circular tube with applied internal pressure and no
imposed to obtain the distribution of wall stresses. The obtained
longitudinal strain. To determine how geometry affects stress
deformedconfigurationservedastheinitial,unloaded,unstressed
distributions, wall stresses were computed from the analytical
geometryofthelinear model,andwall stressesobtainedwiththe
solutions using different initial configurations (i.e. initial tube
linear model and nonlinear models with varying residual stresses
dimensions,seeFig.2).Theseinitialconfigurationgeometrieswere
were then compared. Different hyperelastic material properties
employed in the reference model, and the resulting deformed (RV, P1 and P2; see Table 1) were employed to determine the
configuration of the reference model was used as the initial effectof tissuemechanical propertieson residualandloaded wall
unloadedconfigurationintheconventionalandlinearmodels(see stresses.Thegeometrywasdiscretizedusingmixed9/3elements,
Fig. 1). Additionally, an array of material property values was andconvergenceofresultswasachievedusing180elements,with
testedtoassesstheeffectoftissuemechanicalpropertiesonstress 3 elements spanningthewall thickness.
distributions.Tosimulatetheclinicalsituationinwhichwalltissue
propertiesarenotknown,weallowedthevaluesofaandbtovary Idealized AAA Models with Non-Uniform Wall Thickness
inthereferencemodel(c=0),whileusingRVpopulationaverage
We implemented idealized geometrical models of AAAs with
values in the conventional model (see Table 1) and a constant
non-uniform wall thickness to explore the effect of varying
elasticity (E=8.46109N/mm2)inthelinear model. Wallstresses
thickness on wall stress computations. To this end, we started
from the linear and conventional models were then compared to
using a thick-wall tubular model of the AAA with no residual
corresponding reference stresses. stresses, in which wall thickness varied longitudinally. We also
We also studied the effect of using different AAA tissue simulated a model in which the wall thickness varied circumfer-
properties (RV, P1, and P2, see Table 1) on wall stresses. Since entially, assuming plane strain conditions. Analytical solutions
analytical expressions were not available for all material proper- were not available for these cases; therefore, the analysis was
ties, we employed FEA implemented in ADINA (v8.8.3, ADINA performedusingFEAinADINA.Mixed9/3elementswereused
R & D, Inc., Watertown, MA) to solve for wall stresses in an to discretize the geometries. RV material properties were used
idealized 2D axisymmetric tubular model. To discretize the 2D here for the reference and conventional models. Reference,
geometry,weusedoptimal9/3axisymmetricelements,whichare conventional,andlinearmodelsweresimulatedwiththedescribed
quadrilateralmixeddisplacement/pressurebasedelements(with9 non-uniform wall thickness. Wall stresses were then compared to
displacement degrees of freedom and 3 pressure degrees of establish the accuracy of the linear model in accounting for
freedom) that satisfy the inf-sup condition, ensuring numerical changes inwallthickness.
stabilitywhensolvingproblemsinvolvingincompressibleoralmost
incompressible media such as the AAA wall tissue [29]. In our Idealized Curved Axisymmetric Model of AAA
models, we used six elements spanning the thickness of the wall. Toassesstheeffectofwallcurvatureonthestresses,thearterial
Simulationswereperformedsuchthatthedeformedconfiguration wall was modeled as a curved axisymmetric structure. The outer
ofthehyperelasticmodels,usedhereasreferencemodels,wasthe wall radius B of the 2D axisymmetric initial configuration was
same regardless of the specific nonlinear material property specified by thefollowing equation:
employed. This deformed configuration, further, was used as the
initial unloaded geometry for the linear model. Wall stresses
obtained from the reference models were then compared to
pZ
stresses fromthelinear model. B~7:5cos( )z17:5, Z[½{65,65(cid:2), ð29Þ
65
For the nonlinear FEA models presented here and throughout
thestudy,theconvergencecriterionforequilibriumiterationswas whereZis the height, and B andZare in mm. The height was
specified by energy. The convergence ratio for out-of-balance chosentobe130mm[31].Themaximumdiameterwas50mm,
energy was set to a tolerance value of 0.001. The nonlinear andthewallthicknesswas1.5mm,thereportedmedianthickness
iteration scheme used was the full Newton method, and the [14][19]. The model was constrained at both ends in the
maximum number of iterations implemented for every time step longitudinaldirectionbutwasallowedtomoveanddeformfreely
was set to 15. Convergence was achieved for non-linear models intheradialdirection.Theanalysiswasperformed usingFEAin
using 15to60time steps. ADINA,withtheAAAwalldiscretizedusing9/3mixedelements.
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For these models, we further incorporated an intraluminal the MPR determines differences in stresses between the wall and
thrombus(ILT)inourFEAsimulationsandcomparedresultswith ILT, the linear models that included a thrombus were imple-
andwithoutthethrombus.WhentheILTwasmodeled,thelumen mentedassuminganMPRbetweentheelasticmoduliEofthewall
radius, L,was specified by, and ILT. MPR was first set at 6.7, which is the ratio of the
populationaveragevaluesofthewallandILT,i.e.,a=0.174 N/
mm2/D =0.026 N/mm2. In order to determine how implemen-
1
tation of different MPRs affected the distribution of stresses, we
pZ
L~1:25cos( )z8:75, Z[½{65,65(cid:2), ð30Þ varied MPR in our computations and compared computed wall
65
stresses.TheAAAwallmaterialpropertiesweremodeledusingthe
and the ends of the thrombus were fixed in the longitudinal energy-densityfunctionWproposedbyRaghavanandVorp[19],
directiononly.Thethrombuswasalsodiscretizedusingmixed9/3 Eq. 3 with c=0. Varying MPRs were obtained by changing the
elements. coefficient of the thrombus D1 and the coefficient a, for the wall.
Like the AAA wall, the thrombus was treated as a nonlinear, ValuesofMPRconsideredwere4,6.7,and10.25.AnMPRof4
homogeneous, isotropic, incompressible material but with the wasachievedbymodelingaweakwallstiffness(a=0.144N/mm2,
following energy-density function: b=1.152N/mm2) and relatively stiff thrombus (D1=0.036 N/
mm2, D =0.0356N/mm2). Conversely, an MPR of 10.25 was
2
achieved by modeling a relatively stiff wall (a=0.204N/mm2,
W~D1ðIIB{3ÞzD2ðIIB{3Þ2, ð31Þ b=2.61N/mm2) and weak thrombus (D1=0.0199 N/mm2,
D =0.0216N/mm2) [19][32]. For the linear model, the wall
2
where D1 and D2 are coefficients [32], and IIB is the second elasticitymoduluswassetatavalueofE=8.46109N/mm2,and
invariant of the Left Cauchy-Green deformation tensor B
different MPR values were generated by varying the elasticity
(II =0.5 [(trB)2-tr(B)2]). The ranges of measured values for D
B 1 modulusofthethrombus.Forthemodelsexcludingandincluding
and D2 (95th percentile confidence intervals) obtained from theILT,convergencewasachievedwith390and1,690elements,
patients undergoing elective repair were as follows: D1=0.0199– respectively,with3and10elementsspanningthethicknessofthe
0.036 N/mm2andD2=0.0216–0.0356N/mm2[32].Ingeneral, wall andthrombus,respectively.
thethrombusismorecompliantthanthetissuewall.Thestiffness
ratio between the wall and the ILT, which we refer to as the Subject-Specific Model of AAA
materialpropertyratio(MPR),wascomputedfromthenonlinear
To assess the effect of AAA geometry on wall stress
models using the ratio of the wall coefficient a (Eq. 3) and the
distributions and the degree to which the conventional and
intraluminalthrombuscoefficientD (Eq.31),i.e.,(a/D ).Because
1 1 linear models correctly capture these distributions, a patient-
specific model was implemented. The initial configuration of
the patient-specific AAA had no ILT and was extracted from
contrast-enhanced spiral CT scan images of a de-identified
patient. The images for this retrospective study were provided
by the Oregon Health & Science University Department of
VascularSurgeryfollowingOHSUInstitutionalReviewBoard
(IRB) protocols. Patient consent has been waived by the
OHSU IRB, as this retrospective study constituted a minimal
riskchartreview.ThisstudywasapprovedbytheOHSUIRB.
Extraction of the AAA geometry from CT scan images was
achieved by using a semi-automated custom-made segmenta-
tion program. The segmentation of the vessel lumen com-
menced below the aortic-renal intersection and ended at the
aortic-iliac bifurcation. The first contour was manually traced
aroundthecontrast-enhancedlumenoftheimage.Subsequent
contours were automatically obtained along the AAA midline
and around the contrasted lumen. Automatically segmented
contourswereexaminedandcorrectedmanuallywhenneeded.
Smoothing algorithms were applied to reduce surface-extrac-
tion noise. Contours were then ‘‘stacked’’ to generate the
Figure3.Modelingofresidualstressesinatubularvessel.(A) three-dimensional AAA lumen geometry in the form of a
Schematics of the physical tissue configurations: an initial unloaded, surface mesh. The mesh coordinates were imported into a
unstressed and undeformed circular section (with radii A and B) is custom-made MATLAB (vR2010b, MathWorks, Inc., Natick,
closed,whichgeneratesresidualstressesintheunloadedconfiguration. MA) program written to uniformly displace each vertex
Thisclosedconfiguration(withradiia andb)isloadedtogeneratethe
0 0 1.5 mm radially outward to generate the AAA outer wall
finaldeformedandloadedconfiguration(radiiaandb)thatrepresents
geometry. This geometry was used as the unloaded configu-
arterialtissuesunderload.Theopensectorschematicsalsoshowthe
definition of the opening angle, Q, in relation to the angle h. (B) ration of the reference model. The deformed configuration
Schematics of the FEA model implemented to compute residual obtained from the reference model, assuming RV material
stressesandtheireffectsontheloadedconfiguration.Wefirstmodeled properties, was subsequently employed as the unloaded
an unstressed, unloaded and undeformed open sector. We then configuration for the conventional and linear models, as done
imposedahorizontaldisplacement ontheopenendofthesectorto
with theothergeometrical AAA models describedbefore.The
closethesegmentandgenerateresidualstresses.Theclosedsectorwas
patient-specificAAA geometrywasimportedinto ADINAand
then loaded with an internal pressure to compute the loaded
configurationandresultingwallstresses. discretized using 3D, 27/4 hexahedral mixed elements with 3
doi:10.1371/journal.pone.0101353.g003 elements spanning the wall thickness. The 27/4 mixed
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LinearModelofWallStressComputations
displacement/pressureelementsarethe3Dcounterpartsofthe of elasticity modulus, the wall displacements computed for
9/3 mixed elements (with 27 displacement degrees of freedom the idealized and patient-specific AAA models were negligible
and 4 pressure degrees of freedom) and also satisfy the inf-sup (,1.361029 mm).
condition[29].Theendsofthemodelwereconstrainedin the
longitudinal direction, and selected end-nodes were con- Axisymmetric Tubular Model of an AAA with Parabolic
strained in all directions to prevent rigid body motion. Energy-Density Function
Convergence of results was achieved using 4,800 elements.
Wall stress versus normalized wall thickness plots were initially
generated for the axisymmetric tubular model to determine how
Wall Stress Comparisons
the wall stresses of thelinear and conventional models compared
Wallstressesobtainedfromtheconventionalandlinearmodels tothoseofthereferencemodel(seeFig.4).Here,conventionaland
werecomparedtostressesobtainedfromreferencemodels.Point- reference models used the RV material properties (see Table 1).
by-point differences instresses were computedas follows: Compared to the radial (s ) and axial stresses (s ), the
rr zz
circumferential stresses (s ) had larger magnitude values,
hh
Ds{s(cid:3)D contributing the greatest weight to the calculation of effective
i i ð32Þ
Ds(cid:3)iD stresses (see Eq. 28). Values of shh computed using the linear
model were closer to those obtained from the reference model
where s is the stress of interest (conventional or linear model; than values obtained using the conventional model. A similar
i
circumferential, radial or effective stress) and s*i is the findingwasobservedfortheeffectivestress.Ontheotherhand,srr
corresponding stress in the reference model. Eq. 32 was also wasalmostthesamethroughoutthewallthicknessforallmodels,
usedtocalculatedifferencesinmaximumeffectivestresseswith representing the effect of the pressure boundary conditions on
respect to those in the reference model. For models solved radial stresses.
using FEA, stress differences with respect to the reference
model were plotted for the whole model and over the wall
thickness. These analyses allowed for an objective comparison
of wall stress.
To facilitate comparisons of solutions over a range of tissue
mechanicalproperties(characterizedbyEq.3withaandb;c=0),
differencesinstresswereintegratedoverthenormalizedthickness
and normalizedtothereference stress integral,
(cid:7)ð1 (cid:8)
(cid:11)(cid:11)si{s(cid:3)i(cid:11)(cid:11)dr0
0 ð33Þ
ð1
(cid:11)(cid:11)si{s(cid:3)i(cid:11)(cid:11)dr0
0
wheredr’isthenormalizedthicknessdifferential.Forconvenience,
this calculation was employed only for the axisymmetric tubular
models usinganalytical solutionsof stresses.
Results
Convergence of Linear Model to Equilibrium Stresses
To ensure that the linear model with applied internal
pressures achieved the same equilibrium stresses independent
oftheYoung’smodulus,E,employed,aconvergencestudywas
first performed. Analytically, for the axisymmetric tubular
model,wallstressesdependedontheradiusandwallthickness
of the initial, undeformed configuration and the applied
internal pressure. Further, wall stresses depended on wall
material properties in the case of hyperelastic tissues (see Eqs.
17, 20 and 22) but were independent of wall mechanical
properties when tissues were assumed to be linear and elastic
with infinitesimally small deformations and strains (see Eqs.
27). As expected, when the linear axisymmetric models were
Figure 4. Stress comparisons among the reference, conven-
implemented using FEA (assuming small displacements and tional,andlinearmodelsofanaxisymmetricthick-walltubular
strains), wall stresses did not vary significantly (,0.1%) as E geometry.(A)Circumferentialwallstress;(B)effectivewallstress;and
was increased from 1 to 1010 N/mm2. Similarly, wall stresses (C)radialwallstressdistributionsareplottedacrossthenormalizedwall
did not vary significantly with varying values of E (,0.2%) in thickness. For the reference model, the inner and outer radii in the
deformedconfigurationwere14.8mmand16.1mm,respectively;for
linear idealized and patient-specific models of an AAA.
the conventional model, the inner and outer radii in the deformed
Estimation of equilibrium stresses using the linear model configuration were 16.27mm and 17.53mm, respectively; applied
was therefore effectively independent of the E employed. internalpressure,p=0.016N/mm2(120mmHg);RVmaterialproperties
WechoseanarbitraryhighYoung’smodulus(E=8.46109 N/ wereusedforboththereferenceandconventionalmodels.
mm2) to use in our linear models. In applying this choice doi:10.1371/journal.pone.0101353.g004
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Figure5.Relativedifferencesineffectivewallstressdistributionsforthecaseofatubulararterialmodel.Thefigureshowsdifferences
ofeffectivewallstressdistributionsobtainedfromtheconventionalandlinearmodels,withrespecttothestressdistributionsfromthereference
model,computedusingEq.33.TosimulatetheclinicalsituationwherethematerialpropertiesoftheAAAareunknown,thematerialconstantsaand
b(c=0)werevariedinthereferencemodel(a andb reportedvalues),whereaspopulationaverageaandb(a=0.174N/mm2,b=1.881N/mm2;
ref ref
RVmaterialproperties)wereusedintheconventionalmodel,andconstantelasticity(E=8.46109N/mm2)wasusedinthelinearmodel.Further,the
initial geometry was varied to represent different aneurysm sizes and wall thicknesses. In all cases, applied internal pressure was 0.016N/mm2
(120mmHg).Reportedgeometricalmodelexternalradius,B,andwallthickness,h,correspondtotheinitialconfigurationofthereferencemodel.
0
Thedashedlinesindicatethephysiologicalrangeofthematerialpropertyvaluesfora .
ref
doi:10.1371/journal.pone.0101353.g005
Forathick-walltubularmodelunderequilibrium,thefollowing conditions were the same. The integral, however, was larger for
condition issatisfied, the conventional model (0.26N/mm), reflecting the additional
radial expansionof thewallunder theconventionalapproach.
ð1 To assess how closely the linear and conventional approaches
h shhdr0~pa, ð34Þ approximated reference stresses under different conditions, com-
0 putationswereperformedforarangeofaandbvalues(c=0)and
where h and a are the wall thickness and lumen radius of the differentinitialgeometries(seeFig.5).Toeffectivelycompareand
deformed configuration, respectively; dr’ is the normalized visualize stress differences (with respect to reference stresses) as a
thickness differential; and p the applied internal pressure (see functionoftheparametersaandbinthereferencemodel,weused
Fig.2).Theintegralofs overthethickness(lefthandsideofEq. Eq.33sothateachcase(linear,conventional)wasrepresentedby
hh
34), was about 0.24N/mm for both the linear and reference onevalue,whichwechosetoreportasapercentstressdifference.
models.Thiswas expectedsincethedeformed wallconfiguration We found that, irrespective of the tissue properties used in the
was practically identical for both models, and the boundary reference model, stresses obtained using the linear model were
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LinearModelofWallStressComputations
Figure6.Relativedifferencesinmaximumeffectivewallstressdistributionsforatubulararterialmodel.Modelssimulatedarethe
sameasforFig.5,butdifferencesinmaximalwallstresseswithrespecttoreferencestresses,computedusingEq.32,arereportedinstead.The
reportedmaterialcoefficientsa andb correspondtothoseofthereferencemodel.RVmaterialproperties(a=0.174N/mm2,b=1.881N/mm2)
ref ref
wereusedintheconventionalmodel,andaconstantelasticity(E=8.46109N/mm2)wasusedinthelinearmodel.Appliedinternalpressurewas
0.016N/mm2(120mmHg).Theinitialgeometrywasvariedtorepresentdifferentaneurysmsizesandwallthicknesses.Thedashedlinesindicatethe
physiologicalrangeofthematerialpropertyvaluesfora
ref.
doi:10.1371/journal.pone.0101353.g006
closertothestressesinthereferencemodelthanwerethestresses 5D; 5E and 5F). In general, however, the linear model
obtained using the conventional model (see Fig. 5). Differences approximated reference stresses better than the conventional
withrespecttoreferencestressesdecreasedforboththelinearand model.
conventional models as a increased. Increasing b, however, had To determine how well the maximum effective stress is
only a small effect on stress differences. Increasing the model approximated by the linear and conventional approaches, we
external radius, B, while keeping a constant wall thickness, h , examined our previous results (Fig. 5) but reported differences in
0
resulted in increased stress differences between the conventional maximum effective stress (see Fig. 6). Maximum stresses were
and reference models and decreased differences between linear generally overestimated in the conventional model and underes-
andreferencemodels(compareFigs.5A,5Cand5E;and5B,5D timatedinthelinearmodel(seeFig.4)whentissuepropertiesfor
and 5F). Both B and h0 correspond to the initial configuration of the reference model were within physiological range. We found
the reference model. Increasing h0 while keeping B constant that in most cases, the linear model approximated the maximum
resulted in decreased stress differences between the conventional stressbetterthantheconventionalmodelwithinthephysiological
and reference models but an increased difference between the range of a and b, and the difference gap between linear and
linear and reference models (compare Figs. 5A and 5B; 5C and
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