Aorta Ascending Aneurysm Analysis Using CFD Models towards Possible Anomalies

15 Pages·2017·14.05 MB·English

fluids
Article
Aorta Ascending Aneurysm Analysis Using CFD
Models towards Possible Anomalies
MarianaSimão1,*,JorgeM.Ferreira2,AntónioC.Tomás3,JoséFragata4andHelenaM.Ramos5
1 DepartmentofCivilEngineering,ArchitectureandGeoresources(DECivil),InstitutoSuperiorTécnico,
UniversidadedeLisboa,1600-276Lisboa,Portugal
2 HospitalStaMaria,FacultyofMedicine,UniversidadedeLisboa,1649-028Lisboa,Portugal;
[email protected]
3 CardiothoracicSurgeryDepartment,SantaMartaHospital,1169-024Lisbon,Portugal;
[email protected]
4 NovaMedicalSchool,chronicdiseasesresearchcenter(CEDOC),1169-056Lisbon,Portugal;
[email protected]
5 DepartmentofCivilEngineering,ArchitectureandGeoresources(DECivil),InstitutoSuperiorTécnico,
UniversidadedeLisboa,1600-276Lisboa,Portugal;[email protected]
* Correspondence:[email protected];Tel.:[phone]
Received:14March2017;Accepted:8June2017;Published:10June2017
Abstract: Computationalfluiddynamics(CFD)canbeseenascomplementarytoolalongsidethe
visualizationcapabilitiesofcardiovascularmagneticresonance(CMR)andcomputedtomography
(CT)imagingfordecision-making. InthisresearchCTimagesofthreecases(i.e., ahealthyheart
pilot project and two patients with complex aortic disease) are used to validate and analyse the
corresponding computational results. Three 3D domains of the thoracic aorta were tested under
hemodynamic conditions. Under normal conditions, the flow inside the thoracic aorta is more
streamlined. In the presence of ascending aortic aneurysm, large areas of blue separation zones
(i.e., lowvelocities)areidentified, aswellasaninternalgeometrydeformationoftheaorticwall,
respectively. Thisflowseparationischaracterizedbythereversalofflowandsuddendropofthe
wallshearstress(WSS)intheaorta. Moreover,theaorticaneurysmsimulationsadverselyaffectthe
flowbyincreasingthepressuredropandflowinefficiency,duetotheanatomicalconfigurationof
theascendingaorta. Alteredhemodynamicsledtoavortexformationandlocallyreversedtheflow
thateventuallyinducedalowflowvelocityandoscillatingWSSinthethoracicaorta. Significant
changesinthehemodynamiccharacteristicsaffectthenormalbloodcirculationwithstrongturbulence
occurrence,damagingtheaorticwall,leadingultimatelytotheneedofsurgicalinterventiontoavoid
fatalevents.
Keywords: aorticaneurysm;computationalfluiddynamic(CFD);hemodynamicanalysis;wallshear
stress(WSS);cardiovascularmagneticresonance(CMR);computedtomography(CT)
1. Introduction
Accordingtoanatomicalregions,thethoracicaortagoesfromthehearttothediaphragmwhereit
enterstheabdominalregion,becomingtheabdominalaorta,whichrunsuntiltheaorticbifurcation.
Theaortabeginsastheascendingaorta,turnsintotheaorticarchandtakesaninferiorcoursewherethe
descendingaortastartsitsverticaltrajectorybeforeendingatthebifurcationofthetwocommoniliac
arteries(Figure1). Theascendingaortaoriginatesintheupperpartoftheleftventricle,fromwhichis
separatedbytheaorticvalve.Thisarea,designatedastheaorticroot,doesnothavetheclassicaltubular
shape yet, but follows a bulb-shaped geometry [1]. This region is quite important as its particular
shapeallowsthebloodthatrecoilstobecollectedandbedirectedtothecoronaryostia. Thetubular
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shape,typicaloftheaorta,beginsatthesinotubularjunction,whichmarksthetransitionbetweenthe
between the root and the ascending aorta. The ascending aorta terminates on the aortic arch, which
rootandtheascendingaorta. Theascendingaortaterminatesontheaorticarch,whichhasasteep
has a steep curvature from where the supra-aortic vessels originate superiorly. It continues its course
curvaturefromwherethesupra-aorticvesselsoriginatesuperiorly. Itcontinuesitscourserunning
running dFlouiwds n20w17,a 2r, d31s to the diaphragm, being called the descending aorta [2]. 2 of 15
downwardstothediaphragm,beingcalledthedescendingaorta[2].
between the root and the ascending aorta. The ascending aorta terminates on the aortic arch, which
has a steep curvature from where the supra-aortic vessels originate superiorly. It continues its course
running downwards to the diaphragm, being called the descending aorta [2].

FigFuigreu r1e. 1P.hPyhsyisoiloolgogicicaal laaoorrttaa rroooott iinnt thheen noormrmalaclo cnodnitdioitni.on.
Figure 1. Physiological aorta root in the normal condition.
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oftheaascoerntaditnhga tpaoffreticotns thofe sthege maeonrttab ethtwate eanfftehctes aothreti csvegamlveenat nbdetthweeebnr acthheio caeoprthica livcatlrvue nkan,dp rothdeu cing
ascending portion of the aorta that affects the segment between the aortic valve and the
anenblarragchediocveepshsaellicd itarumnekt,e prr1o.5duticminegs agnr eeantlearrgtheda nvtehsseeol rdigiaimnaeltedri a1m.5 ettimere(sF gigruearteer2 )th[4a–n6 t]h.eA onriagsicneanl ding
brachiocephalic trunk, producing an enlarged vessel diameter 1.5 times greater than the original
aortad,wiamithetear d(Fiaigmuerete 2r) b[4e–t6w].e Aenn a1s.1ceannddin1g. 5aotritma,e wsitthha at doifatmheeteorr bigeitnwaelesni z1e.1, aisndco 1n.5s itdimereesd thtaot bofe tdhiel ated.
diameter o(rFigiginuarl es iz2e), [i4s –co6n].s iAdenre ads tcoe bned diinlagte ado. Trthae, twypiitchal ap rdoigarmesseitoenr o bf eantw aneeeunr y1s.m1 ainn tdh e1 a.5or ttaim beegsi ntsh at of the
Thetypicalprogressionofananeurysmintheaortabeginswithhighbloodpressure,whichincreases
original swizieth, ihsi gcho bnlosiodde prreedss utore ,b we hdicihla itnecdre.a Tsehs eth tey ppriecsasul rpe rwoagvreess osfi oblno oodf aagnai nasnt ethuer wysamlls oinf ththe ea oarotar.t a begins
thepressurewavesofbloodagainstthewallsoftheaorta. Thepresenceofatheroscleroticdiseasealso
The presence of atherosclerotic disease also causes the wall of the aorta to lose its elasticity which
withc haiugshe sbtlhoeowd aplrleosfstuhreea, owrthaitcohl ionscerietsaseelass tthiceit yprweshsicuhrem weaanvsetsh oaft ibtsloaobdil iatygatoindsitl athteea wndalrlesc oofi ltwheit haorta.
means that its ability to dilate and recoil with each pressure wave of blood induced by the heartbeat
The pearcehsepnrecses uorfe awthaveeroosfcbleloroodticin dduisceeadsbey atlhseo hceaaurtsbeesa tthised wecarella soefd t.hEev eanotrutaal ltyo, thloeseel aisttsi cemlaesmticbirtayn ewshich
is decreased. Eventually, the elastic membranes and smooth muscle within the aortic wall deteriorate
meanasn dthsaamnt doito tsht hae bmvielusistseycll detoiwa mditiehltaientre itn hacenreadaos rretesi cc[7ow]i.l a wllidteht eeraiocrha tperaensdsutrhee wveasvseel odfi abmloeotedr iinncdrueacseeds [b7y]. the heartbeat
is decreased. Eventually, the elastic membranes and smooth muscle within the aortic wall deteriorate
and the vessel diameter increases [7].
(a) (b)
Figure 2. Aneurysm of the ascending thoracic aorta: (a) Patient 1; (b) Patient 2.
Figure2.Aneurysmoftheascendingthoracicaorta:(a)Patient1;(b)Patient2.
Understanding the physiology of patients with aorta disease is crucial not only for
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localihnedmivoidduyanl apmatiicesntisn [(c8ao]).m Cparledxioavnasactuolmari ems,agbnuettaicl sroesroenfiannicneg (CthMeRa)s asnedss cmomenptuo(tbefd)i n tdomivoidguraaplhpya (tCieTn)t s[8].
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complex hemodynamic scenarios, simulating multiple conditions for the same patient is still
Understanding the physiology of patients with aorta disease is crucial not only for
understanding local hemodynamics in complex anatomies, but also refining the assessment of
individual patients [8]. Cardiovascular magnetic resonance (CMR) and computed tomography (CT)
imaging have been very useful in evaluating blood flow characteristics [9–11]. Although this imaging
technique (i.e., CMR) provides a 3D flow map of the blood circulation and can be used to explain
complex hemodynamic scenarios, simulating multiple conditions for the same patient is still
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providesa3Dflowmapofthebloodcirculationandcanbeusedtoexplaincomplexhemodynamic
scernealartiiovse,lys ilmonugl a(1ti5n gmimn)u. lAtidpdleiticoonnaldlyit,i ownitsh floimr ittheed sspaamtieal praestioelnuttioisn sctoimllpraerleadti vtoe lCyTl,o dnegsp(i1t5e tmhei n).
relatively long (15 min). Additionally, with limited spatial resolution compared to CT, despite the
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anomalies.
revaenaolmstaolibese. arelevantpotentialclinicaltooltopredictmalfunctionsorfutureanomalies.
Computational fluid dynamics are capable to predict different scenarios at a patient-specific
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level [13]. Nevertheless, for computational models to be potentially integrated into clinical practice
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or used as clinical data (i.e., for decision-making) they need to be validated. Thus, in the present
useodr uassecdli naisc acllidnaictaal (di.ea.t,af o(ir.ed.,e fcoisri odne-cmisiaokni-nmg)akthinegy) nteheedy tnoebede vtoa libdea vteadli.dTahteuds., Tinhuthse, ipnr ethseen ptrreesseenatr ch
research two computational models were developed: one representative of a complex aortic
twroesceoamrcphu ttawtioo ncaolmmpoudtealtsiownearl emdeovdeellos pwede:reo ndeerveeplroepseedn:t aotinvee orefparceosemnptaletixvea oortfi cap actohmoplolgexy ,waoirthtict he
pathology, with the presence of aneurysms, and another illustrative of a normal heart condition.
prepsaetnhcoeloogfya, nweiuthry tshme ps,reasnedncaen ooft haenreuilrluyssmtras,t iavnedo afnaonthoerrm ilalulshteraatritvceo onfd ai tnioonrm. al heart condition.
2. Materials and Methods
2. M2. Matearteiarliaslasn adndM Metehthooddss
2.1. Patient Data
2.12..1P.a PtiaetnietnDt aDtaata
Patients with aortic pathology underwent CT evaluation of cardiac anatomy and function. The
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computed axial tomography (CT), with a spatial resolution of 0.5–0.625 mm in the z-axis, and
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approximately 0.5 mm in the x to y-axes, showed that, morphologically, two of the patients presented
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aneurysms in the thoracic artery trunk. Figures 3–5 show the CT derived anatomical data.
aneurysms in the thoracic artery trunk. Figures 3–5 show the CT derived anatomical data.
presentedaneurysmsinthethoracicarterytrunk. Figures3–5showtheCTderivedanatomicaldata.

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Figure 4. Patient 2 (76 years old) with an ascending aorta aneurysm.
Figure 5 shows the aortic root with the aortic diameter almost constant along the entire
Figure 5 shows the aortic root with the aortic diameter almost constant along the entire thoracic
thoracicFicgouurres 5e .shows the aortic root with the aortic diameter almost constant along the entire thoracic
course.
course.
Fluids2017,2,31 4of15
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Fluids 2017, 2, 31 4 of 15
Fluids 2017, 2, 31 4 of 15

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Figure 5. Healthy patient (pilot) (62 years old) without aorta disease.
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Figure 5. Healthy patient (pilot) (62 years old) without aorta disease.
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from a healthy patient (i.e., without aortic pathology) was also generated (Figure 8).

(a) (b)

Figure (6a. ) A(as)c ending aorta aneurysm (Patient 1): (a) compute(db )( bto)m ography (CT) data and
Figure6.Ascendingaortaaneurysm(Patient1):(a)computedtomography(CT)dataandsegmentation;
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and(b)mesh.
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(a) (b)

Figure( a7). Ascending aorta aneurysm (Patient 2): (a) CT data an(bd) s egmentation; and (b) mesh.

Fig(au)r e 7. Ascending aorta aneurysm (Patient 2): (a) CT data and( bse)g mentation; and (b) mesh.
Figure7.Ascendingaortaaneurysm(Patient2):(a)CTdataandsegmentation;and(b)mesh.
Figure 7. Ascending aorta aneurysm (Patient 2): (a) CT data and segmentation; and (b) mesh.
Fluids2017,2,31 5of15
Fluids 2017, 2, 31 5 of 15

(a) (b)
Figure 8. Healthy patient (pilot): (a) CT data and segmentation; and (b) mesh.
Figure8.Healthypatient(pilot):(a)CTdataandsegmentation;and(b)mesh.
The provided STL geometry presents some relatively complex surfaces, fillets, and blends,
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these tetrahedral elements intersect, their sizes are adjusted with the objective of keeping the elements
usedtoseedthevolumemeshfromwhichtetrahedralelementsgrowelementsinwards. Asthese
as isotropic (similar edge lengths and included angles) as possible and to have reasonably gradual
tetrahedral elements intersect, their sizes are adjusted with the objective of keeping the elements
transitions between smaller and larger elements [14]. However, when these algorithms attempt to
asisotropic(similaredgelengthsandincludedangles)aspossibleandtohavereasonablygradual
remove topological entities, they often need to modify the underlying NURBS (non-uniform rational
transitionsbetweensmallerandlargerelements[14]. However,whenthesealgorithmsattemptto
basis spline) surfaces and are, therefore, somewhat limited. An alternative in COMSOL Multiphysics
removetopologicalentities,theyoftenneedtomodifytheunderlyingNURBS(non-uniformrational
software is to use Virtual Operations, which can keep the existing geometrical representations as a
basisspline)surfacesandare,therefore,somewhatlimited. AnalternativeinCOMSOLMultiphysics
basis for constructing a new alternative topological structure purely for the purposes of meshing and
softdweafirneinisg tthoeu psheyVsiicrst u[1a4l].O perations, which can keep the existing geometrical representations as
abasisfoFrrocmon tshteru Vcitritnugala Onpeewraatilotenrsn oaptitvioentso, pthoelo Fgoicrmal sCtorumcptuosreitep Fuarecelys ffeoartuthree wpuarsp soesleecsteodf mreseushltiinngg and
defiinni nthge tshaempeh gyesoimcse[t1ry4,] .but ignoring faces that lie between two adjacent domains, resulting in a single
dFomroamin.t hLeasVtliyr,t uthael OMpeeshra Ctioonntsroolp Ptoioinntss,, tEhdegFeos,r FmacCeso,m anpdo sDitoemFaaicness fefeaatuturerse, whiadse speoleincttse,d edregseus,l ting
in tfhaecess,a more dogmeoamines tdryu,ribnug tthige nsoetr-iunpg offa tchees pthhyasticlsi.e Hboewtweveeern, thtwesoe gaedojamceetnritc denotmitiaeisn ws,ilrl esstuilll tbine g in
asinpgrelesednot dmuariinng. Lthaes tmlye,sthhinegM steesph. BCyo unstirnogl Pthoeisnet tsy,pEeds goef so,pFearcaetiso,nasn ddurDinogm thaein csomfepaututarteiso,nhail dmeepsho ints,
of the anatomical models, greater control over the meshing process is done by designating geometric
edges,faces,ordomainsduringtheset-upofthephysics. However,thesegeometricentitieswillstill
entities for the control of the mesh size and distribution [14].
bepresentduringthemeshingstep. Byusingthesetypesofoperationsduringthecomputationalmesh
Then, to assess the performance of each structure, two kinds of analyses were undertaken: mesh
oftheanatomicalmodels,greatercontroloverthemeshingprocessisdonebydesignatinggeometric
convergence and mesh sensitivity analysis. These two analyses are both based on the concept of
entitiesforthecontrolofthemeshsizeanddistribution[14].
establishing the independence of the CFD results over the mesh quality. A mesh sensitive analysis
Then, to assess the performance of each structure, two kinds of analyses were undertaken:
was developed by refining the mesh until a critical result, such as the pressure, converges (i.e., does
meshconvergenceandmeshsensitivityanalysis. Thesetwoanalysesarebothbasedontheconceptof
not change significantly with each refinement) [15]. As shown in Figure 9 when the mesh number
estaibnlcisrehainsegs ttho e20in0,d0e0p0,e tnhdee pnrceessoufreth aet tChFe Dcornedsiutilotsn oovf epreathk esymsteoslhic qcounavlietyrg.eAs tmo eas chonsestnasnitt ivvaeluaen.a Tlhyusiss, was
devae lmopesehd wbyithr e2fi2n5,i0n5g9 tehleemmenestsh fuorn ati lnaorcmriatil caaolrrtea saunldt, 2s,u3c5h8,0a5s5t hanedp 2re4s2s,8u5r9e ,elceomnevnetrsg, eress(pi.eec.t,ivdeolyes, not
chanfogre Psaigtinenifit c1a anntldy 2w, withereea ucshedre ifinn tehmis erneste)a[1rc5h].. A sshowninFigure9whenthemeshnumberincreases
to200,00T0h,et hceonpvreersgseunrcee aatntahleyscios nwdaist idoonnoef bpeegainknsiynsgt owliitchc ao nvveeryrg ceosartosea mceosnhs taanndt ivnaclrueme.eTnhtiunsg, tahme esh
withsm22o5o,t0h5n9esesl eomf tehnet msfeosrh abyn o2r5m%a floar oeartcah afunrdth2e,r3 5st8e,p05, r5eaanchdin2g4 21,5805%9 oefl etmhee anutsg,mreenspteedc tmiveeslhy ,dfeonrsPitayt ient
[16].
1and2,wereusedinthisresearch.
The convergence analysis was done beginning with a very coarse mesh and incrementing
the smoothness of the mesh by 25% for each further step, reaching 150% of the augmented mesh
density[16].
Fluids2017,2,31 6of15
Fluids 2017, 2, 31 6 of 15

Figure 9. Mesh sensitivity analysis.
Figure9.Meshsensitivityanalysis.
2.3. Boundary Conditions
2.3. BoundaryConditions
The FEM involves dividing the problem into subdomains, each represented by a set of element
TheFEMinvolvesdividingtheproblemintosubdomains,eachrepresentedbyasetofelement
equations to the original problem, followed by systematically recombining all sets of element
equationstotheoriginalproblem,followedbysystematicallyrecombiningallsetsofelementequations
equations into a global system of equations for the final calculation. The global system of equations
intoaglobalsystemofequationsforthefinalcalculation. Theglobalsystemofequationshasknown
has known solution techniques and can be calculated from the initial values of the original problem
solutiontechniquesandcanbecalculatedfromtheinitialvaluesoftheoriginalproblemtoobtain
to obtain a numerical answer [7]. The 3D incompressible Navier-Stokes equations were used for blood
anumericalanswer[7]. The3DincompressibleNavier-Stokesequationswereusedforbloodflow
flow simulations, with the CFD software COMSOL-Multiphysics Equation (1):
simulations,withtheCFDsoftwareCOMSOL-MultiphysicsEquation(1):
(cid:2034)(cid:1874)
(cid:2025) +(cid:2025)((cid:1874)∙∇)(cid:1874)−∇(cid:2026) =0 (cid:1861)(cid:1866)Ω(cid:3047), ∀(cid:1872)∈(cid:1835)
ρ∂v(cid:2034)(cid:1872)+ρ(v·∇)v−∇σ =0 inΩt, ∀t ∈ I
∂t (1)
(1)
(cid:2008)∇∙·v(cid:1874)==00 in(cid:1861)(cid:1866)ΩΩt,(cid:3047), ∀∀t(cid:1872)∈∈I(cid:1835)
wwhheerree I(cid:1835)==[[00,,T(cid:1846)]]i sist hteheti mtimeien tienrtveravlaaln adnρdi s(cid:2025)t hies dtheen sditeynosiftyb looof db.loTohde. uTnhken uownknnvoawrina bvlaesriaarbeletsh earbelo tohde
vbeloloocdi tyv,evlo,caintyd, th(cid:1874)e, panreds stuhree pthraestsduerpe etnhdast odneptheendCsa uocnh ythset rCesasu,cσh(yv ,spt)r,edsse,s c(cid:2026)r(i(cid:1874)b,e(cid:1868)d)b, ydeasccorinbsetdit ubtyiv ea
constitutive relation that characterizes the rheology of the fluid. In the case of Newtonian fluids the
relationthatcharacterizestherheologyofthefluid. InthecaseofNewtonianfluidsthetotalstress
tteontaslo srtrisesssi mtepnlsyord iesfi snimedplbyy dσe(fvin,epd) b=y −(cid:2026)P((cid:1874)I,+(cid:1868))2=µD−((cid:1842)v(cid:1835)),+µ2i(cid:2020)s(cid:1830)th((cid:1874)e)d, y(cid:2020)n aism tihcev disycnoasmityic ovfisbcloosoidty, oanf dbloDodis,
tahneds t(cid:1830)ra iins rtahtee tsetrnasionr ,rgaitvee tnenbsyoDr, (gvi)v=en(cid:0) ∇byv +(cid:1830)(∇(cid:1874))v=T(cid:1)(/(cid:2008)2(cid:1874). I+n(cid:2008)th(cid:1874)i(cid:3021)s)r⁄e2se. aIrnc ht,htihs erevsisecaorcshit,y tihsev avriisacbolseitayn ids
variable and depends on the shear rate [17]:
dependsontheshearrate[17]:
(cid:113)
γ(cid:2011)==(cid:3493)22t(cid:1872)r(cid:1870)((D(cid:1830)2(cid:2870))) (2(2) )
TThhuuss,, ffoorr aa nnoonn--NNeewwttoonniiaann flfluuiidd,, tthhee ttoottaall ssttrreessss tteennssoorr iiss ggiivveenn bbyy::
(cid:2026)((cid:1874),(cid:1868))=−(cid:1842)(cid:1835)+2(cid:2020)(cid:0)((cid:2011).(cid:4662)(cid:1))(cid:1830)((cid:1874)) (3)
σ(v,p) = −PI+2µ γ D(v) (3)
Different non-Newtonian models have been proposed in expert literature, describing different
speciDfiicfafetiroennst onfo tnh-eN veiwsctoosnitiya nfumncotdioenls (cid:2020)h(a(cid:2011)v)e. bIne ecnasper oofp bolsoeoddi fnloewxps etrhtel iftoecruastu wreil,ld faelslc orinb itnhge bdoifufenrdeendt
svpiseccoifisictayt iloanwsso, futshinegv Cisacorrseituy’sf ushnecatiro-nthµin(nγi)n.gIn mcoadseelo Efqbuloaotidonfl o(4w) s[1th7]e: focuswillfallonthebounded
viscositylaws,usingCarreu’sshear-thinningmodelEquation(4)[17]:
(cid:3041)(cid:2879)(cid:2869)
(cid:2020) =(cid:2020) +((cid:2020) −(cid:2020) )(1+((cid:2019)(cid:2011))(cid:3028)) (cid:3028) (4)
(cid:2998) (cid:2868) (cid:2998)
where γ is the rate of the deforµm=atiµo∞n +ten(µso0r− gµiv∞e)n (cid:0)b1y+ E(qλuγa)taio(cid:1)nn− a(15) for a 3D Cartesian coordina(4te)
system [18]:
FFlluuiiddss 22001177,, 22,, 3311 77 ooff 1155
where γ is the rate of the(cid:2034)d(cid:1874)e(cid:3051)fo(cid:2870)rmat(cid:2034)io(cid:1874)n(cid:3052) t(cid:2870)ensor(cid:2034)(cid:1874)g(cid:3053)iv(cid:2870)en by(cid:2034)E(cid:1874)q(cid:3051)uat(cid:2034)io(cid:1874)(cid:3052)n ((cid:2870)5) fo(cid:2034)r(cid:1874)a(cid:3051)3D(cid:2034)C(cid:1874)(cid:3053)ar(cid:2870)tesian coordinate
(cid:2011) =(cid:3429)2(cid:3421)(cid:3436) (cid:3440) +(cid:4678) (cid:4679) +(cid:3436) (cid:3440) (cid:3425)+(cid:4678) + (cid:4679) +(cid:3436) + (cid:3440)
system[18]: (cid:2034)(cid:1876) (cid:2034)(cid:1877) (cid:2034)(cid:1878) (cid:2034)(cid:1877) (cid:2034)(cid:1876) (cid:2034)(cid:1878) (cid:2034)(cid:1876)
(cid:2870) (cid:2869)⁄(cid:2870) (5)
γ=(cid:34)2(cid:40)(cid:18)∂vx(cid:19)2+(cid:18)∂vy(cid:19)2+(cid:18)+∂(cid:4678)v(cid:2034)z(cid:2034)(cid:1874)(cid:19)(cid:1878)(cid:3052)2(cid:41)++(cid:2034)(cid:2034)(cid:1874)(cid:1877)(cid:18)(cid:3053)(cid:4679)∂v(cid:3433)x + ∂vy(cid:19)2+(cid:18)∂vx + ∂vz(cid:19)2+(cid:18)∂vy + ∂vz(cid:19)2(cid:35)1/2 (5)
∂x ∂y ∂z ∂y ∂x ∂z ∂x ∂z ∂y
TThhee ccoonnssttaannttss ffoorr tthhee CCaarrrreeaauu mmooddeell wweerree eessttaabblliisshheedd ffrroomm iinn vviivvoo ddaattaa aaccccoorrddiinngg ttoo [[1177]]::
(cid:2020) =0.042(cid:1842)(cid:1853).(cid:1871) (cid:2019)=3.31(cid:1871)
(cid:2868)
µ0 =0.042Pa.s λ =3.31s (6)
(cid:2020) =0.00345(cid:1842)(cid:1853).(cid:1871) (cid:1866)=0.375 (6)
(cid:2998)
µ∞ =0.00345Pa.s n =0.375
in which the coefficients (cid:2020) and (cid:2020) are the asymptotic viscosities, that depend on the lowest shear
(cid:2868) (cid:2998)
rinatwesh, iwchhtihche ccooerfrfiecsipeonntsdµs 0toan tdheµ ∞lowareestth peraessysmurpet odtricopv iisnco as iptiiepse, tfhloatwd eapnedn odno nthteh ehliogwheers tsshheeaarr rraatteess,,
rwehspicehctciovrerleys.p Honerdesinto, ath =e 2lo, wfoers tthper eCsasrurreeud mroopdienl aanpdip λe flaonwd nan adreo npathraemheigtehresr tsoh ebaer ersattiems,arteesdp. eIcf tλiv =e l0y.
oHre nr e=i n1,, ath=e v2,isfcoorsitthye isC caornresutamnto adnedl tahned Nλeawntdonniaanr empoadrealm ise rteecrsovtoerbeed ewsittihm tahtee dv.isIcfoλsi=ty 0(cid:2020)or. Tnh=es1e,
(cid:2868)
pthaeravmisecotesritsy reispcroensesntatn tthaen rdeltahxeaNtioenw ttoimniea n(λm), oi.dee.,l tihseryec doevteerremdiwneit hthteh veavliuseco (cid:2011)si tayt µwh.Tichhe tsheep laorwam sehteearrs
0
rraetper eesnednst [t1h9e]r. eTlahxea ptioownetirm laew(λ i)n,di.eex., (tnh)e yredperteesremnitns eththe eslvoaplue eofγ tahtew rahpicidhltyh-edelocwreasshienagr praotretieonnd osf[ t1h9e].
cTuhrevpe oawnder tlhaew sihnadpeex p(na)raremperteesre n(ats) tihs ethsleo psheaopfet htrearnaspitiidolny -bdeetcwreeaesnin tghep ourptipoenr o(cid:2011)f tahnedc uthrvee raanpdidtlhye-
isnhcarpeeaspinarga pmoertteiorn(a o)fi sththe ecushrvape e(it.rea.,n essittiimonabteedtw beye cnutrhvee ufipttpienrgγ ofa nexdptehreimraepnitdally d-iantcar,e saesein [g20p–o2r2t]i.o nof
thecAur vteim(ie.e-v.,aersytiinmga tfeldowby rcauter vwefiastt iinmgpoofseexdp earti mtheen tianlldeta tbao,usened[a2r0y– 2c2o]n.dition. As for the outlet
condAitiotinm, ea- pvareryssinugrefl wowavrea wteaws asseti m(bpoothse addaatpttheedi nfrloetmb o[2u3n]d). aTrhyec obnoduintidoanr.yA csonfodritthioenosu atrleet dceofnindeitdio inn,
Faipgruerses u10re. wavewasset(bothadaptedfrom[23]). TheboundaryconditionsaredefinedinFigure10.
18
16
400
14
12
300
Q (ml/s) 200 810 P (kPa)
6
100 4
2
0 0
0 0.2 0.4 0.6 0.8 1
t (s)
Q (ml/s) P (Pa)
Figure 10. Boundary conditions used in both models.
Figure10.Boundaryconditionsusedinbothmodels.
The simulations used for WSS assessment typically assume that the arterial walls are rigid,
ThesimulationsusedforWSSassessmenttypicallyassumethatthearterialwallsarerigid,which
which greatly simplifies the simulation process. The effect of wall motion on velocity and WSS fields
greatlysimplifiesthesimulationprocess. TheeffectofwallmotiononvelocityandWSSfieldsisoften
is often assumed to be negligible. This may be reasonable in older subjects and certain patient groups
assumedtobenegligible. Thismaybereasonableinoldersubjectsandcertainpatientgroupswith
with increased vascular stiffness. Thus, for simplicity, and also due to the aging of the patients, the
increasedvascularstiffness. Thus,forsimplicity,andalsoduetotheagingofthepatients,thevessel
vessel walls were assumed to be rigid and the no-slip condition was considered.
wallswereassumedtoberigidandtheno-slipconditionwasconsidered.
3. Results
3. Results
3.1. Blood Velocity
3.1. BloodVelocity
Figure 11 shows normal streamlines velocities in the aorta system (between 0.1 m/s and 0.55 m/s)
Figure11showsnormalstreamlinesvelocitiesintheaortasystem(between0.1m/sand0.55m/s)
with the absence of vortex formation in the arch branches.
withtheabsenceofvortexformationinthearchbranches.
Fluids2017,2,31 8of15
FFlluuiiddss 22001177,, 22,, 3311 88 ooff 1155
Fluids 2017, 2, 31 8 of 15

FFFiiiggguuurrreee 111111... HHHeeeaaalllttthhhyyy pppaaatttiiieeennnttt::: vvveeellloooccciiitttyyy ssstttrrreeeaaammmllliiinnneeesss (((pppeeeaaakkk sssyyyssstttooollleee)))...
Figure 11. Healthy patient: velocity streamlines (peak systole).
FFFiiiggguuurrreee 111222 ssshhhooowwwsss lllooowww vvveeellloooccciiitttyyy vvvaaallluuueeesss iiinnn ttthhheee aaaooorrrtttaaa sssyyysssttteeemmm ( ((<<<000...333555 mmm///sss))) aaannnddd mmmaaarrrkkkeeeddd vvvooorrrttteeexxx fffooorrrmmmaaatttiiiooonnn
lllooocccaaattteeeFdddig aaautttr tttehhh 1eee2 ppp srrrhooooxxxwiiimmms aaalolll wssseee ggvgmmemloeeencnnitttt yooo fffv tttahhhleuee eaaasrrr ttiteenerrr tyyyh (e((ccc aoooodddreeetddad sbbbylllusuuteeee)))m... (<0.35 m/s) and marked vortex formation
located at the proximal segment of the artery (coded blue).

FFiigguurree 1122.. PPaattiieenntt 11:: vveelloocciittyy ssttrreeaammlliinneess ((ppeeaakk ssyyssttoollee))..
FFiigguurree 1122.. PPaattiieenntt 11:: vveelloocciittyy ssttrreeaammlliinneess ((ppeeaakk ssyyssttoollee))..
FFiigguurree 1133 sshhoowwss llooww vveelloocciittyy vvaalluueess iinn tthhee aasscceennddiinngg aaoorrttaa ((<<00..22 mm//ss)) aanndd mmaarrkkeedd vvoorrtteexx
ffoorrmmFaaFtitiigigoouunnrr eell oo11cc33aa ttseeshdhdoo waawtts stt lhhlooeeww bb evevggeeililnononcnciiiitnntyygg v vooaaffl l uuttehhesees iniddnee tsshtccheeeenn addasiiscnncegegnn daadiooninrrgttgaa a ao((cocroortatddaee (dd<(< 00bb.2.ll2uu meem))/.. / ssTT))h haaeenn dttdrr aammnnssaaiirrttkiikooeenndd ddvveeooggrrrrtteeeexeex
bfbfooeerrttmwmwaeeateetiinnoo n ntthh loeeloc ddacatiisestttdeaadlal taaartrthcc hheth baaeenn gddbi en ppgnrriioonnxxngiiimmonfgaat llh oddefee dsstehccseeecnn edddneiidnnsigcgne gaanoodarroittnraatg acc hh(acaaoonnrdtggaeee dss( cbaaolbbudrreueu)dpp. tTtlblyhyl ue mmetr)aaa.k knTiisnnhiggtei o aatnllrmmadnooessgsittrt ei99oe00n bdd edeetgwgegrreeereeeensse
btbbheeeetttwwwdeeieseeetnnna l ttthhahreeecmm hd..i a sTTnthhadlii sspa rrrreecoshsxuu ialmlttnssad iil nnpd drdeoisissxcttieuumnrrdabbleie ndddg e ffslaloocoewwrnt adaannicndhdga s snauuogddreddtsaeea nncbh ffralluoonpwwgtel ayascc maccbeeallreekurripaantttgliiyooa nnmlm ((accookosdditnee9gdd0 arrdleemeddgo)).r.s ete 9s0b detewgreeeens
tbheetmw.eeTnh itshreemsu. lTtshiisn rdeissutultrsb iend dflisotwurbanedd fsluodwd aenndfl souwdadcecne lfelorawti oancc(ecloerdaetdiorne d(c)o.ded red).

FFiigguurree 1133.. PPaattiieenntt 22:: vveelloocciittyy ssttrreeaammlliinneess ((ppeeaakk ssyyssttoollee))..
FFiigguurree 1133.. PPaattiieenntt 22:: vveelloocciittyy ssttrreeaammlliinneess ((ppeeaakk ssyyssttoollee))..
Fluids 2017, 2, 31 9 of 15
Fluids2017,2,31 9of15
Fluids 2017, 2, 31 9 of 15
3.2. Wall Shear Stress
33..22.. WWItaa llills SS hhweeaearrll S Strtrereecssossg nized that a higher or lower WSS (HWSS and LWSS) may cause intimal
dysfuItn citsi own,e wll hrieccho gconuizlded r etshualtt ian thhieg hperro gorre slsoiwone ro fW atShSe r(oHscWleSroS siasn [d2 4L].W ExStSr)e mmealyy hciaguhs ev ailnuteims oafl
ItiswellrecognizedthatahigherorlowerWSS(HWSSandLWSS)maycauseintimaldysfunction,
WdySsSfu anrcet aiosnso, cwiahtiecdh wcoituhl dv ersesseul lrt eimn othdee lplirnogg rreessspioonns oibfl ea tfhoerr aonsceluerryossmis [in24it]i.a Etixotnr eamndel yp rhoiggrhe svsaiolune [s2 o5]f
which could result in the progression of atherosclerosis [24]. Extremely high values of WSS are
[W26S]S. Caroen avsesrosceilayt,e rde gwioitnhs voef slsoewl rWemSSo dcoelrlrienlga tree wspiothn sinibwlea rfodr v aansecuurlayrs mre minoitdiaetliloinng a anndd p trhoeg prreossgiroens s[i2o5n]
associated with vessel remodelling responsible for aneurysm initiation and progression [25,26].
o[2f6 a]t.h Ceoronsvcelresreolsyi,s r [e2g7i]o. nAsl toefr leodw h WemSoSd cyonrraemlaitces wleidth t oin aw vaorrdte vxa fsocrumlaart iroenm aondde llloincagl layn rde vtheer sperso tghree fslsoiown,
Conversely, regions of low WSS correlate with inward vascular remodelling and the progression
wofh aitchhe reovsecnleturoaslilsy [2in7d].u Aceltse rleodw h aenmdo doyscniallmatiicnsg l eWd tSoS ai nv otrhteex t fhoorrmacaitci oano artnad. lIonc aclolmy rpeavreisrosens wthiet hfl othwe,
of atherosclerosis [27]. Altered hemodynamics led to a vortex formation and locally reverses the
swmhoicohth e avoernttau aarltlyer yin (dpuactieesn lto-pwil oatn, dF igouscriel l1a4ti)n, glo wW SWSS iSn vtahleu etsh oinr aPcaicti eanotrst a1. aInnd c 2o mcapna bries oonb swerivthe dt hine
flow,whicheventuallyinduceslowandoscillatingWSSinthethoracicaorta. Incomparisonwiththe
tshmeo aostche nadorintag atrhtoerrayc (ipc aatoierntat -(pFiilgout,r Fesig 1u5r ea n1d4) 1, 6lo).w WSS values in Patients 1 and 2 can be observed in
smoothaortaartery(patient-pilot,Figure14),lowWSSvaluesinPatients1and2canbeobservedin
the ascending thoracic aorta (Figures 15 and 16).
theascendingthoracicaorta(Figures15and16).

Figure 14. Patient-pilot: wall shear stress (WSS) contours (at peak systole).
FFiigguurree 1144.. PPaattiieenntt--ppiilloott:: wwaallll sshheeaarr ssttrreessss ((WWSSSS)) ccoonnttoouurrss ((aatt ppeeaakk ssyyssttoollee))..
Figure 14 shows normal values of WSS located in the smooth curvature of the aorta artery. On
Figure14showsnormalvaluesofWSSlocatedinthesmoothcurvatureoftheaortaartery. Onthe
the coFnigtruarrey 1, P4 asthieonwtss 1n aonrmd 2a lp vraelsueenst voaf sWcuSlSa rl oacreaatesd e xinp othseed s tmo oloowth WcuSrSv oatvuerre t iomf eth, we ahoicrhta c aorntterriyb.u Otens
contrary,Patients1and2presentvascularareasexposedtolowWSSovertime,whichcontributes
ttoh ea c ovnatsrcaurlya,r P raetimenotdse 1l lainngd. 2H perreesienn,t tvhaesrceu lias r dairreeacst eexvpiodseendc eto o lfo wan W aSnSe uovryesrm tim, sei,n wceh ilcohc aclolyn-tarilbteurteeds
to a vascular remodelling. Herein, there is direct evidence of an aneurysm, since locally-altered
htoe ma ovdaysncuamlaric sr eamndod veelslisnegl .w Halelr peainr,a mtheetreer si,s i .de.i,r ethcte eovbisdeervnecde oovf earnt dainffeeurreynscme i, ns ivnacleu elso cbaeltlwy-eaeltne rtehde
hemodynamicsandvesselwallparameters,i.e.,theobservedovertdifferenceinvaluesbetweenthe
ahnetmeroidory naanmdi ctsh aen dp ovsetsesreiol rw valels psealr awmaeltle, ras,f fie.ec.t, tthhee ovbasescruveladr orveemrto ddieflfleirnegn caen idn, vcaoluneses qbueetwntelyen, tthhee
anteriorandtheposteriorvesselwall,affectthevascularremodellingand,consequently,theaneurysm
aanneteurriyosrm a ngdr otwhteh p[o27st]e. rFioigr uvree s1se5l swhoawll,s arfefgeicot ntsh eo f vlaoswcu WlarS Sr ecmorordeelalltiendg waintdh , thcoen pseroqgureenstsliyo,n t hoef
growth[27]. Figure15showsregionsoflowWSScorrelatedwiththeprogressionofatherosclerosis.
aatnheeurroysscmle rogsriosw. Athlt e[2re7d]. hFeimguordey 1n5a mshicosw lesd r teog iao vnos rotefx l foowrm WatSioSn c aonrdre lloatceadll yw rietvhe rthsee thpero fglorwes,s wiohni cohf
Alteredhemodynamicsledtoavortexformationandlocallyreversetheflow,whichinduceslowand
ianthdeurcoessc lloerwo sains.d A olstceirleladt ihnegm WodSSy nina mthiec st hleodr atcoi ca avoorrttae.x F fiogrumrea 1ti6o np raensden ltosc laolwly WrevSeSr vsea ltuhees falot wth,e w inhnicehr
oscillatingWSSinthethoracicaorta. Figure16presentslowWSSvaluesattheinnercurvatureofthe
cinudrvuacteusr leo wof atnhde othsocirlalactiicn ago WrtaS Sa nind t hheig thh oWraScSic a ato trhtea . lFeivgeul roef 1t6h ep raeosretnat sa rlochw, WwhSiSc hv amluaeys eaxt pthlaei inn ntheer
thoracicaortaandhighWSSattheleveloftheaortaarch,whichmayexplaintheprogression/extension
pcurorvgaretussrieo nof/ etxhtee nthsioorna coifc tahoer dtai saenadse hinig thh eW fSuStu arte .t he level of the aorta arch, which may explain the
ofthediseaseinthefuture.
progression/extension of the disease in the future.

FFiigguurree 1155.. PPaattiieenntt 11:: wwaallll sshheeaarr ssttrreessss ((WWSSSS)) ccoonnttoouurrss ((aatt ppeeaakk ssyyssttoollee))..
Figure 15. Patient 1: wall shear stress (WSS) contours (at peak systole).
Fluids2017,2,31 10of15
Fluids 2017, 2, 31 10 of 15

FFiigguurree 1166.. PPaattiieenntt 22:: wwaallll sshheeaarr ssttrreessss ((WWSSSS)) ccoonnttoouurrss ((aatt ppeeaakk ssyyssttoollee))..
44.. DDiissccuussssiioonn
TThhee ccoommppuuttaattiioonnaall ssiimmuullaattiioonnss ccoonncceerrnneedd tthhee pprrooffiilleess ooff vveelloocciittyy aanndd WWSSSS iinn tthhee tthhrreeee ccaasseess..
AAccccoorrddiinngg ttoo [[22]],, toto cchhaararactcetreirziez ethteh estsrutrcutcutruer oefo tfheth TeATAA,A th,et hmeamina gineogmeoemtriectarli cTaAlATA pAarapmareatmerest aerres
mareeamsueraesdu: r(cid:1838)ed: L, the a,ntehueryasnmeu lreynsgmth,l eanngdt hD, and, thDe ma,xitmheumm adxiaimmuetmer doifa tmhee atenreuorfytshme. aTnheeu fraycstmor.
(cid:2904)(cid:2885)(cid:2885) TAA (cid:2904)(cid:2885)(cid:2885) TAA
Twhheicfhac atsosrewssheisc thhae slseensgstehs t(cid:1838)hele nagntdh tLhe diaamndettehre (cid:1830)diam eist ekrnDown aiss tkhneo swacncuaslatrh iensdaecxc u(l(cid:2011)a)r iEnqdueaxti(oγn)
(cid:2904)(cid:2885)(cid:2885) TAA (cid:2904)(cid:2885)(cid:2885) TAA
(E7q) u[a2t8i]o. nIf( 7th)e[2 s8a]c.cIufltahre isnadcceuxl aisr cinlodseex tios c1l,o tsheet oan1e,uthreysamne iusr syascmcuislasra c(scpuhlaerri(csaplh),e rbiucat li)f, bitu itsi fcliotsise ctloo s0e,
tthoe0 ,atnheeuarnyesumr yissm miosrme ofuresiffuosrimfo.r mTh.eT hdeefdoerfmoramtioanti odniadmiaemteert errartea te(cid:2031) χcchhaararaccteterrizizeess tthhee nnoonnddeeffoorrmmeedd
aaoorrttaa ddiiaammeetteerr ((d(cid:1856) )) wwitihtht htheem maaxximimuummd diaiammeetteerr ooff tthhee aanneeuurryyssmm.. AA nnoonnaanneeuurryyssmmaall aaoorrttaa iiss
a(cid:2911)n(cid:2924)t(cid:2930)e(cid:2915)r(cid:2928)i(cid:2919)o(cid:2925)r(cid:2928)
ddeeffiinneedd aass D(cid:1830) ==(cid:1856)d [[22]]::
T(cid:2904)A(cid:2885)A(cid:2885) (cid:2911)a(cid:2924)n(cid:2930)t(cid:2915)e(cid:2928)r(cid:2919)i(cid:2925)o(cid:2928)r
β = r(cid:1870)
R
(cid:2010) =
γ = DT(cid:1844)AA (7)
LTAA
(cid:1830)
χ(cid:2011)==d(cid:1838)Dan(cid:2904)TteA(cid:2885)rAi(cid:2885)or (7)
(cid:2904)(cid:2885)(cid:2885)
Toevaluatetheasymmetryβoftheaneurysm,randRaredefinedastheratiomeasuredatthe
(cid:1830)
midsectionoftheTAAfromthelongitudinalz(cid:2031)-a=xisto(cid:2904)t(cid:2885)h(cid:2885)ep osteriorandanteriorwalls,respectively[29].
(cid:1856)
Thus,β =1yieldsanazimuthalsymmetryandβ(cid:2911)=(cid:2924)(cid:2930)(cid:2915)0(cid:2928).(cid:2919)2(cid:2925)(cid:2928)isaTAAforwhichonlytheanteriorwallis
dilateTdo, wevhaillueatthee tphoe satesyrimormweatrllyi s(cid:2010)n eoaf rtlhyefl aante[3u0r]y.sBma,s e(cid:1870)d aonndt h(cid:1844)e saerien ddiecfeins,eidt iassp tohses irbalteioto mideeansutirfeydt haet
tthype emofidasneecutiroyns mofp rtehsee nTteAdAin feraocmh ptahtei enlotn(Tgaitbuldei1n)a.l z-axis to the posterior and anterior walls,
respectively [29]. Thus, (cid:2010) =1 yields an azimuthal symmetry and (cid:2010) =0.2 is a TAA for which only
the anterior wall is dilatedTa, bwleh1il.eG tehoem peotrsictealricohra rwacatlelr iisst incseaofrltyh eflaante [u3r0y]s.m Bsa.sed on these indices, it is
possible to identify the type of aneurysm presented in each patient (Table 1).
d D L
Patient anterior r(cm) R(cm) TAA TAA χ β γ Type
(cm) (cm) (cm)
Table 1. Geometrical characteristics of the aneurysms.
Pilot 2.80 1.40 1.40 2.80 - 1.00 1.00 - -
(cid:2186) (cid:2200) (cid:2174) (cid:2160) (cid:2168) saccularwithazimuthal
Patien1t (cid:1801)3((cid:1814)c.5(cid:1820)m(cid:1805)4(cid:1818))(cid:1809)(cid:1815) (cid:1818) 2.0(c6m) 2(.c7m0) (c4(cid:1794)m.(cid:1775)76(cid:1775)) (c(cid:1794)m6(cid:1775).(cid:1775)9)2 χ1 .35 0β. 76 0γ.6 9 symmetry(35%Toyfpdee formation)
fusiformwithonlyanteriorwall
Pilot2 22.6.890 0.516.40 31.3.430 23.8.808 1-1 .53 1.010.4 4 1.00.107 0-. 34 dilated(44%ofd- eformation)
saccular with azimuthal
1 3.54 2.06 2.70 4.76 6.92 1.35 0.76 0.69
symmetry (35% of deformation)
Accordingto[1,31–35],basedontheTAAshape,differentcriteriafortheTAAcollapsibilitycan
fusiform with only anterior wall
bea2p plied: 2(i.)69if theT0.A56A di3a.m33e ter3is.884 5%h11ig.5h3e rth1.a4n4 the0.1n7o n-d0.e3f4o rmedaortadiameter; (ii)ifthe
dilated (44% of deformation)
asymmetryindexfactoris β < 0.4; (iii)ifthedeformationdiameterratio χ > 3×d ; and(iv)
anterior
if theAscaccocrudlianrgi tnod [e1x,3γ1–3<5]0, b.6a.seHdo owne tvheer ,TtAhAe TshAaAped, idlaiftfieornenatn cdritpeoritae nfotira tlhreu TpAtuAre caollslaopdsiebpileintyd coann
bthee ahpepmlioedd:y (ni)a mif itchleo aTdAsA.T dhiuasm,aeltteerr nisa 4ti5v%e mhiegthheord tshoafnT tAhAe nruonp-tduerefoarsmseesds maoerntat adrieamneeetdere;d (,iis)u icfh thaes
ansuymmemriectarlya innadleyxs ifsacintovro ilsv i(cid:2010)ng<C0F.4D; (miii)o dife tlhset doedfoertemrmatiionne dthiaemWetSeSr rdaitsitor ib(cid:2031)u>tio3n×s(cid:1856)and flow; anpda t(tievr)n isf
(cid:2911)(cid:2924)(cid:2930)(cid:2915)(cid:2928)(cid:2919)(cid:2925)(cid:2928)
(tThaeb sleac2c)u.lar index (cid:2011) <0.6. However, the TAA dilation and potential rupture also depend on the
hemodynamic loads. Thus, alternative methods of TAA rupture assessment are needed, such as