Characterization of transport topology in patient specific abdominal aortic aneurysms models

17 Pages·2012·3.24 MB·English

Characterization of the transport topology in patient-specific abdominal
aortic aneurysm models
Amirhossein Arzani and Shawn C. Shadden

Citation: Phys. Fluids 24, 081901 (2012); doi: 10.1063/1.4744984
View online: http://dx.doi.org/10.1063/1.4744984
View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i8
Published by the American Institute of Physics.

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PHYSICSOFFLUIDS24,081901(2012)
Characterization of the transport topology
in patient-specific abdominal aortic aneurysm models
AmirhosseinArzania) andShawnC.Shaddenb)
DepartmentofMechanical,MaterialsandAerospaceEngineering,IllinoisInstituteof
Technology,10W32ndSt.,Chicago,Illinois60616,USA
(Received17April2012;accepted19July2012;publishedonline10August2012)
Abdominalaorticaneurysm(AAA)ischaracterizedbydisturbedbloodflowpatterns
that are hypothesized to contribute to disease progression. The transport topology
in six patient-specific abdominal aortic aneurysms was studied. Velocity data were
obtained by image-based computational fluid dynamics modeling, with magnetic
resonanceimagingprovidingthenecessarysimulationparameters.Finite-timeLya-
punov exponent (FTLE) fields were computed from the velocity data, and used to
identifyLagrangiancoherentstructures(LCS).ThecombinationofFTLEfieldsand
LCS was used to characterize topological flow features such as separation zones,
vortex transport, mixing regions, and flow impingement. These measures offer a
novel perspective into AAA flow. It was observed that all aneurysms exhibited co-
herentvortexformationattheproximalsegmentoftheaneurysm.Theevolutionof
the systolic vortex strongly influences the flow topology in the aneurysm. It was
difficulttopredictthevortexdynamicsfromtheaneurysmmorphology,motivating
theapplicationofimage-basedflowmodeling.(cid:2)C 2012AmericanInstituteofPhysics.
[http://dx.doi.org/10.1063/1.4744984]
I. INTRODUCTION
Abdominalaorticaneurysm(AAA)isapermanentlocalizedwideningoftheabdominalaorta.It
ischaracterizedbydisturbedflowpatterns,includinglowandoscillatorywallshearstresswithhigh
gradients,1 increased particle residence time,2 mild turbulence,3 and a decrease in wall strength.4
Moreover, most AAAs are accompanied by mural thrombus,5 due in part to flow recirculation in
the aneurysm.6 Aneurysm rupture kills thousands annually, and a reliable predictor for aneurysm
progression remains unknown, though maximum AAA diameter, AAA expansion rate, and peak
AAAwallstresshavebeenusedwithvaryingsuccess.4
Studies into AAA hemodynamics have helped reveal possible biomechanical mechanisms
underlying AAA pathogenesis and progression, and these mechanisms have been reviewed in
Refs. 4,7, and 8. Early studies started from idealized models, e.g., to track the formation and
propagation of vortices,and todetermine regions ofhigh shear stressesattheproximal and distal
ends of an aneurysm.9 Finol et al.10 and Deplanoa et al.11 studied common asymmetry in AAA,
reportingtheappearanceofasymmetricvortices,andarelativehighershearingoftheposteriorwall
oftheaneurysm,whichisthemostprevalentlocationofrupture.7Peattieetal.12andSalsacetal.13
demonstratedchangestotheflowpatternsbasedonbulgediameter,andEgelhoffetal.14considered
theeffectsofexercise.
In vivo flow conditions can be far more complex than flow in simplified aneurysm models.
Patient-specificcomputationalmodelshavebeenusedtopredictpossibleinvivoflowpatternsand
wall shear stress in AAA.15 Patient-specific AAA models have also been used in studies of fluid
structureinteraction,16particleimagevelocimetry,17intraluminalthrombusformation,18,19andval-
idationofimage-basedflowmodeling.20 VesselmorphologyhasawidevariabilityinAAAandthe
a)[email protected].
b)[email protected].
1070-6631/2012/24(8)/081901/16/$30.00 24,081901-1 (cid:2)C 2012AmericanInstituteofPhysics
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081901-2 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
geometryoftheaneurysmandsurroundingvasculatureiscriticalindeterminingthehemodynamics.
Lesetal.3 compareddifferentEulerian-basedflowmeasuresinmultiplepatient-specificcomputa-
tionalmodels,andSuhetal.2,21investigatedAAAflowusingaLagrangian-basedparticleresidence
time(PRT)measure.
The purpose of the study herein is to provide a more detailed description of blood flow in-
side AAA. Specifically, this is the first study to compare the transport topology inside different
patient-specificAAAmodels.PreviousstudiesinvestigatingAAAhemodynamicshavereportedon
Eulerian measures, or generic transport characteristics (e.g., PRT), that are unable to convey the
completeflowtopology.InstantaneousEulerianmeasuresplaceinherentlimitsoninterpretationdue
toflowunsteadiness.Whileitmaybereasonabletoinferthebulkfluidmotionfromvelocitydata,
inferencesintotransportmechanismscaneasilyresultinmisinterpretations.Thisisnoteworthybe-
causetransportprocesses,anddisruptionofsuchprocesses,havesignificantphysiologicrelevance.
Furthermore,thetransportmechanicscanbetightlyandsynergisticallycoupledtothefluidforces
thatmayinfluencecellularfunction.22 Ourobjectiveistoprovidemoreprecisecharacterizationof
theunsteadytransporttopologyobservedoverdifferentpatient-specificabdominalaorticaneurysms
underphysiologicalconditions.WeutilizethecomputationofLagrangiancoherentstructure(LCS)
fromfinite-timeLyapunovexponent(FTLE)fields.Thismethodhasbeenappliedpreviouslytocar-
diovascularapplications,23 includingflownearclots,24 inthecarotidarteries,23,25 andflowthrough
the aortic valve.26,27 The present study is organized as follows: In Sec. II A, the procedures for
image-based flow modeling are discussed. Section II B describes the computation of FTLE and
LCS.RelevantfeaturesoftheflowtopologyofthesixpatientsconsideredarepresentedinSec.III,
anddiscussedinSec.IV.
II. METHODS
SixpatientswithsmallAAA(maximaldiameter<5cm)wereconsideredforthisstudy.These
aneurysms demonstrated wide variability, which enabled comparison of flow topologies resulting
from a range of morphology. Magnetic resonance imaging (MRI) was used to obtain anatomical
andvolumetricflowdataforimage-basedcomputationalfluiddynamics(CFD)simulation,andthe
resultingvelocitydatawereusedforsubsequentFTLE/LCScomputations.
A. Imagebasedcomputationalfluiddynamics
Details on the image-based CFD modeling are provided in Ref. 3, and a brief review of the
process is presented here. Image-based geometric models were constructed using the software
package SimVascular.28 Contrast-enhanced magnetic resonance angiography (MRA) was used to
obtain image stacks of the abdominopelivic cavity, creating a bright-blood image of the aorta and
surroundingvessels.Pathswerecreatedalongeachvesselofinterest,andsegmentsofthevessels
wereobtainedalongeachpath.Thesegmentsweresubsequentlyloftedandblendedintoaunified
geometricmodel,asdescribedinRef.29.Eachmodelnominallyincludedtheaortastartingfromthe
levelofthediaphragm,andtheceliactrunk(withhepaticandsplenicarteries),superiormesenteric
artery, the renal arteries, and continued through the internal and external iliac arteries. The model
geometry represents the fluid domain; i.e., in some regions, the model boundary could be due to
muralthrombusratherthanthenativevesselwall.ThecomputationalmodelsaredisplayedinFig.1.
Each geometric model was discretized into a computational mesh that was used for CFD
simulation of the blood flow. Blood was modeled as an incompressible, Newtonian fluid by the
Navier-Stokesequations
∂v
ρ +ρv·∇v=−∇p+μ∇·∇v, (1)
∂t
where ∇·v=0,
and the velocity v and pressure p are functions of space and time (x,t) and the density ρ and
viscosity μ were assumed constant with values of 1.06 g/cm3 and 0.04 P, respectively. To solve
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081901-3 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
Patient1 Patient2 Patient3 Patient4 Patient 5 Patient6
FIG.1. AAAcomputermodelsderivedfromMRI.
Eq. (1), initial and boundary conditions were specified. The velocity was specified at the inflow
facev(x,t)=g(x,t)usingphasecontrastmagneticresonanceimaging(PCMRI)data,asdescribed
below.Zerovelocity(noslip)wasspecifiedalongthewalls.Attheoutlets,aNeumann-typeboundary
condition was imposed by coupling lower order models as described below using the coupled-
multidomainformulation.30Aninitialvalueforvelocityv(x,t)=v0(x)wasassumedknown,which
was obtained from a steady simulation. The Navier-Stokes equations were discretized using a
streamline-upwind Petrov-Galerkin finite element formulation31 with same order interpolation for
velocityandpressure.Thediscretizationwassemidiscrete,i.e.,thespatialdomainwasdiscretized
byfiniteelements(tetrahedra),leavingasystemofordinarydifferentialequationstobeintegratedin
timeusinganimplicit,second-orderaccurategeneralized-αmethod.Thefinite-elementformulation
andintegrationmethodweredescribedinmoredetailinRef.32.Thesolverhasbeenextensivelyused
andvalidatedforimage-basedbloodflowmodeling,includingvalidationinaorticflowstudies.20,33
Directnumericalsimulationwasusedsincetheflowconditionswereobservedtobepredom-
inantlylaminarortransitional,andturbulencewastransientduetothepulsatilenatureoftheflow.
Vessel wall motion was assumed negligible since the abdominal aorta becomes stiffer with AAA
progressionduetoanincreaseincollagenanddecreaseinelastin,34 andoftenanincreaseofmural
thrombus. Nonetheless, the incorporation of realistic variable wall properties and external tissue
supportremainsanoutstandingchallenge,thoughprogressisbeingmade.35
Planar phase contrast magnetic resonance imaging was used to obtain through-plane blood
flow velocities under resting condition at a cross-section of the supraceliac aorta over time.
The expected error in the PCMRI measurement (≈5%) is less than typical variation of cardiac
output for a given patient over a day. The inlet of each model was chosen to correspond to the
PCMRImeasurementplane.ThePCMRIvelocitywasintegratedoverthecross-sectiontoprovide
a time-dependent supraceliac volumetric flow, shown in Fig. 2 for each patient. The resulting
300
Patient1
Patient2
Patient3
200 Patient4
Patient5
)
s Patient6
/
L
m
( 100
Q
0
0 0.5 1 1.5
Time(s)
FIG.2. Volumetricflowratesprescribedattheinletofthemodels.
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081901-4 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
TABLEI. Fluiddynamicandgeometricparametersofcomputermodels.
Patientno. Remax α Dmax(cm) Dmax/Dinfra Qinfra/Qsupra
Patient1 2002 4.92 3.36 1.71 0.48
Patient2 2241 5.20 3.72 1.84 0.66
Patient3 2256 6.22 3.16 1.49 0.59
Patient4 1667 5.86 3.24 1.49 0.47
Patient5 1791 4.86 3.00 1.76 0.69
Patient6 1419 3.63 3.89 2.39 0.31
waveformwasusedtoprescribeatime-dependentWomersleyvelocityprofileatthemodelinlet.29
Three-elementWindkesselmodelswereusedtoimposeoutflowboundaryconditionsateachoutlet
ofthemodel.Theselowerordermodelsofthedownstreamvascularbedswerecoupledtothecom-
putationaldomainusingmethodsdescribedinRef.30.Thesolverwasrununtilthebloodpressure,
measured asthemaximuminspaceateachtimeinstant,became periodicintime.Thesimulation
was continued for an additional five cardiac cycles to produce converged velocity data that were
used for post processing as described in Sec. II B. Mesh independence tests were performed for
onepatient(Patient1)bycomparingresultsfrom≈2million,≈8million,and≈32millionelement
meshesasdescribedinRef.3.The8Mand32Melementmeshesshowedrelativeagreementinthe
flowfieldandthenominaltetrahedraledgesize(500μm)forthe8Melementmeshwasusedforall
patients.
The peak Reynolds number and the Womersley number based on the infrarenal artery were
computedas
4ρQ
Re = max, (2)
max πDμ
(cid:2)
D ωρ
α = , (3)
2 μ
where D is the mean diameter at the infrarenal section, Q is the peak systolic flow rate at the
max
infrarenal section, μ is the dynamic viscosity, ρ is the density, and ω is the heartbeat frequency.
Table I compares these nondimensional parameters, the maximum diameter of the aneurysm, the
ratioofthemaximumdiametertotheinfrarenaldiameter,andthefractionofthesupraceliacflowto
theinfrarenallevel,betweenthepatientmodels.
B. LCScomputations
ArecenttooltostudyadvectivetransportisthecomputationofLagrangiancoherentstructures.
LCS are material surfaces in the flow that have distinguished attracting or repelling properties.36
While the computation of LCS originated to understand chaotic transport, LCS have been shown
toorganize bothlaminarand turbulentfluidmotion.Thismethodiscompelling forunsteady flow
conditions where fluid transport can be difficult to impossible to understand using traditional Eu-
lerian techniques. Numerous applications have demonstrated the utility of LCS for understanding
unsteady flow, including the ability to delineate boundaries of unsteady and interacting vortices,
flow attachment or separation profiles, and the mechanisms underlying mixing, see Ref. 37 for a
review.Computationally,LCScanbeidentifiedashypersurfacesthatlocallymaximizeafinitetime
Lyapunov exponent measure.38,39 This method is relatively straightforward and robust, and hence
hasbecomewidelyused.
ThecomputationoftheFTLEfieldrequirestheadvectionofadensegridoffluidtracersover
thedomainofinterest.Velocitydata,v(x,t),obtainedbyimage-basedCFD,wereusedtosolvethe
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081901-5 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
advectionequation
x˙(t) =v(x,t),
(4)
x(t )=x ,
0 0
over a grid of initial conditions x to provide final locations x(t +T) after an integration time T.
0 0
TheFTLEwascomputedateachinitiallocationx as
0
(cid:3) (cid:3)
(cid:2)(x ,t ;T)= 1 ln(cid:3)∇ x(t +T)(cid:3) . (5)
0 0 |T| x0 0 2
Advection of the particle grid forward in time (T > 0) revealed repelling LCS in the FTLE field,
whereas advection backward in time (T < 0) revealed attracting LCS in the FTLE field. As
the notation in Eq. (5) indicates, the FTLE field corresponds to time t and likewise the values
0
are plotted at the initial locations x , and the integration time T is considered a parameter. Thus,
0
whilecomputedfromaLagrangianframe,theFTLEvaluesaremappedbacktotheinitiallocations,
and plotted as an Eulerian measure. Locally, high values of FTLE appear along hypersufaces that
aregenerallyidentifiedwithLCS.Thisprocedureisrepeatedforarangeofinitialtimest toobtain
0
thetimeevolutionoftheLCS.
The identification of LCS is relatively robust to changes in the integration length |T| used to
compute FTLE. Nonetheless, the signature of any particular LCS in the FTLE field is dependent
ontheLagrangiantimescalesofthedynamics.Ingeneral,sinceweseektounderstandinherently
transient phenomena, the integration time should be chosen long enough for dominant features to
berevealed,butshortenoughfortheFTLEtoberepresentativeofthetransientdynamics.Herein,
the integration time for each patient was set to the respective cardiac cycle length, which is suit-
able for cardiovascular applications.23 Specifically, the dominant structures become well-defined
in less than one cycle and the field does not change much by further increase in integration time
length since most of the tracers used to compute the FTLE become flushed from the domain.
The FTLE field was sampled every t = 1/25th of the cardiac cycle. In our computations, we
0
dropped the 1/T multiplication factor in the definition of FTLE (Eq. (5)). This helps prevent trac-
ers that leave the computational domain in a short time from erroneously increasing the FTLE
value.37
1.EffectofaperiodicityonLCS
Thepresenceoftransitionalflowormildturbulenceobservedintheaneurysmsleadstocycle
to cycle variation of the flow field, and the relevance of LCS derived from a single cardiac cycle
maynotbeclear.Therefore,threeadditionalmethodsforevaluatingFTLE/LCSwereexplored.We
definedanensembleFTLEfromacycle-basedtemporalaverageoftheFTLEfield,i.e.,
(cid:4)n−1
1
(cid:4)(cid:7)(cid:5) (x,t;T)= (cid:7)(x,t +kT;T), (6)
ens
n
k=0
where t ∈ [0, T), and n is the number of cardiac cycles considered. Second, we defined a max-
ensemble FTLE as the maximum FTLE among the corresponding ensembles of the FTLE data,
i.e.,
(cid:7)max−ens(x,t,T)= max {(cid:7)(x,t +kT,T)}. (7)
0≤k≤n−1
Finally,wedefinedavelocity-ensembleFTLE,(cid:7)vel−ens,astheFTLEobtainedfromtheensembled
averagevelocityfield
(cid:4)n−1
1
(cid:4)v(cid:5) (x,t)= v(x,t +kT). (8)
ens
n
k=0
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081901-6 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
Patient1 Patient2 Patient3 Patient4 Patient5 Patient6
FIG.3. Cross-sectionsusedfordisplayingresults.
We used five cardiac cycles of FTLE data for ensemble FTLE and max-ensemble FTLE, and five
cardiaccyclesofvelocitydataforvelocity-ensembleFTLEcomputations.
III. RESULTS
Sectionsofthe3DFTLEfieldareshownintheresultsbelow.CurvesofhighFTLEcorrespond
tocross-sectionalslicesofembeddedLCShypersurfaces.Nominally,twoplanesareshownforeach
patient.Asagittalplaneapproximatelybisectstheleftandrighthemispheresoftheaneurysm,anda
transverseplanewaschosenslightlydistaltothelocationofmaximalbulge,seeFig.3.ForPatient4,a
cross-sectionalviewoftheflowinaniliacaneurysmwasadded,andforPatient5,additionalsections
of the AAA were chosen due to the complexity of the aneurysm shape. Sections of the backward
andforwardFTLEfieldsarepresentedinFigs.4–11forthesixpatients.TheEulerianvelocityfield
isalsoincludedforcomparison.Thecolorbarsarescaledbasedonthespecificmaximumrangeof
eachcase.
A. Patient1
Figure4showstheFTLEandvelocityfieldsforPatient1.Duringpeaksystole(toprow),there
is a region of low backward FTLE in the proximal segment. This region indicates a penetrating
inflow jet into the aneurysm. The low backward FTLE values indicate low sensitivity to initial
conditionswhenparticlesinthisregionareintegratedbackwardintime.Thisimpliesthattheinflow
jetiscoherent.TheindicatedaLCS1markstheboundaryofthejet.Thejetpenetratesdeepintothe
aneurysmandasitenterstheexpansionregion,itrollsupintoalargevortexring.Thecoresofthis
vortex ring on the sagittal cross-section are clear in the backward FTLE during mid-deceleration.
Thejetimpingesontheanteriorwalljustdistaltotheregionofmaximalbulgeasindicatedbythe
arrowintheforwardFTLEfieldduringmid-deceleration.Specifically,intheforwardFTLEfield,
there is an accumulation of repelling LCS orthogonal to the anterior wall, indicating the flow is
directlyimpingingandrollinguponeithersideofthislocation.
As the jet penetrates into the abdominal aorta, it separates from the anterior aortic wall, as
indicated by the separation profile marked aLCS2. This separation induces vortical motion along
the proximal anterior aspect of the bulge and, along with the vortical motion induced from the
roll up of the inflow jet, contributes to a persistent region of recirculation in the proximal an-
terior bulge. The aLCS3 and aLCS4 are separation surfaces from the posterior wall that are
formed by the jet and roll-up. The distal separation from the posterior wall persists during much
of the cardiac cycle. In both anterior and posterior regions of separation, there are dense sets
of LCS, indicating well mixed separation regions, as opposed to stagnant. During the diastolic
phase, both the forward and backward FTLE indicate several coherent vortices spanning the en-
tire aneurysm, which indicates strong stirring/mixing during diastole from mechanisms described
inRef.40.
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081901-7 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
Backward Forward Transverse plane Velocity
Forward (top)
jet
e
ol FTLE
st FTLE FTLE 6.91 Vel(cm/s)
sy 7.33 7.11 56.69
6
ak 6 6
Pe 4 4 4 40
2 20
2 2
0
0 0 0
n aLCS4
tio aLCS2
a
r
e aLCS3
el FTLE
c 7.43 Vel(cm/s)
de FTLE FTLE 34.73
d 7.42 7.49 6
Mi 6 6 4 20
4 4 2 10
aLCS1
2 2 0
0
0 0
e
ol
t
as FTLE
di FTLE FTLE 7.6 Vel(cm/s)
rly 7.6 7.6 6 28.18
a 6 6
E 4 20
4 4
2 10
2 2
0
0 0 0
FIG.4. SagittalandtransversesectionsoftheforwardandbackwardFTLEfields,andsagittalsectionofvelocityfieldfor
Patient1atdifferenttimesofthecardiaccycle.Thetwopointsinthetoptransverseplaneshowwherethesagittalplane
intersects.
B. Patient2
TheresultsforPatient2areshowninFig.5.Forthispatient,theinflowjetdoesnotpenetrate
intothemainaneurysmbulge.Whileavortexringcanbeseenintheproximalabdominalaortain
thebackwardFTLEfieldduringsystole(topleft),thevortexbreaksup,andresultsinmixingand
dissipationintheproximalsegment,andrelativelylowflowinthemainbulge.Thisappearstobea
self-sustainingprocess;thebreakupofthevortexandsubsequentdissipationduringdiastoleleads
todenseLCSintheproximalregionthatcausestheearlybreak-upofthesubsequentsystolicinflow
jet/vortex.Thisleadstostrongermixingproximaltotheaneurysmthanintheaneurysm.Largeflow
separationisobservedfromtheposteriorwall(aLCS1).TheindicatedattractingandrepellingLCS
(aLCS1andrLCS1)togetherformaboundarytoalargevortexformedintheseparationregion,cf.
Ref.41foradiscussionontherelationshipofLCStovortexrings.However,thisseparatedregion
iswell-mixed,asindicatedbyhighFTLEinsidethevortex.Itisthesheddingofthisposteriorwall
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081901-8 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
Backward Forward Transverse plane Velocity
Forward (top)
vortex
aLCS1
FTLE
ole FTLE FTLE 7.42 Vel(cm/s)
st 6.86 6.14 6 61.88
y
s 6 6 4
Peak 4 4 2 40
20
2 2
0
0 0 0
on aLCS1 rLCS1
ti
a
r
cele 6F.9T5LE 6F.T34LE 7F.T5L3E Ve5l5(.c5m9/s)
e
d
6 6 6
d 40
Mi 4 4 4
20
2 2 2
0 0 0 0
e
ol
diast 6F.9TLE 6F.4T6LE 7F.5T5LE Ve2l6(.c0m5/s)
y 6
arl 6 6 4 20
E
4 4
2 10
2 2
0
0 0 0
FIG.5. SagittalandtransversesectionsoftheforwardandbackwardFTLEfields,andsagittalsectionofvelocityfieldfor
Patient2atdifferenttimesofthecardiaccycle.Thetwopointsinthetoptransverseplaneshowwherethesagittalplane
intersects.
vortexduringdiastolethatleadstothemajorityofthemixingobservedinthemainbulge,i.e.,the
mixingappearslessduetothebreakupoftheinflowjet,andmoreduetothedistalvortexformation
anddissipation.
C. Patient3
Figure6showstheresultsforPatient3.ThereisalackofcoherentstructuresintheforwardFTLE
fieldduringmostofsystole,whereasthereareseveralLCSindicatedinthebackwardFTLE.This
indicates relatively low mixing in the aneurysm itself but high mixing proximal to the aneurysm.
Indeed, at the end of systole, there is significant retrograde flow in the abdominal aorta for this
patient and a vortex forms at the proximal segment and propagates upstream resulting in vortex
pinchoff42asindicated.Thisvortexpinchoffeventandsubsequentdissipationintoseveralvortices
duringthediastolicphasecontributestothesubsequenthighbackwardFTLEvaluesandprevalent
coherentstructuresobservedduringsystoleinthebackwardFTLEplots.Specifically,thevortices
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081901-9 A.ArzaniandS.C.Shadden Phys.Fluids24,081901(2012)
Backward Forward Transverseplane Velocity
Forward(top)
stole 6F.T65LE 6F.T3L2E F6T.7L3E Ve5l0(c.5m6/s)
sy 6 6 6
ak 4 4 4 40
e
P
2 2 2 20
0 0 0 0
n
o
ti
a
r FTLE
ele FTLE FTLE 6.39 Vel(cm/s)
c 6.6 6.23 36.72
e 6
d
d 6 6 4
Mi 4 4 20
2
2 2 10
0
0 0 0
rLCS1
pinchoff
e
ol
t FTLE
dias 6F.T5L8E 6F.T1L8E 6.66 Ve3l4(c.m28/s)
y
arl 6 6 4
E 4 4 20
2
10
2 2
0
0 0 0
FIG.6. SagittalandtransversesectionsoftheforwardandbackwardFTLEfields,andsagittalsectionofvelocityfieldfor
Patient3atdifferenttimesofthecardiaccycle.Thetwopointsinthetoptransverseplaneshowwherethesagittalplane
intersects.
formed in the proximal segment due to the retrograde flow get pushed into the aneurysm during
systole.TherearefewandsparseLCSintheforwardFTLEfield,whichindicatesthatthevortices
originating from the proximal segment rapidly dissipate as they are advected into the main bulge.
Therefore, relatively low mixing occurs in the distal segment. For example, the indicated rLCS1,
whichpartitionsfluidflushedfromtheaneurysm,showedveryslowrateofchangeinitsstructureor
movement.
D. Patient4
Figure 7 shows the results for Patient 4. The aLCS1 is the boundary of a proximal inflow jet
totheaneurysm,whichformsawell-definedvortexring.Thevortexringimpingesontheproximal
anteriorwalloftheaneurysmattheendofsystole.Thevortexsubsequentlybreaksupanddissipates
during diastole, but maintains a relatively coherent large scale vortical structure over much of the
cardiac cycle. As with Patient 3, because of the strong dissipation in the proximal segment, and
relatively smooth aneurysm geometry, there is a lack of coherent structures in the forward time
FTLEfieldinthemajorityofthedistalsegment,leadingtolowmixing.Thispatientdemonstrated
verylittlechangeintheFTLEfieldwithtimeinbothplanes(cf.rLCS1);infact,theforwardFTLE
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