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A numerical assessment of wall shear stress changes after endovascular stenting
Franck Nicoud, Hélène Vernhet, Michel Dauzat
To cite this version:
Franck Nicoud, Hélène Vernhet, Michel Dauzat. A numerical assessment of wall shear stress changes after endovascular stenting. Journal of Biomechanics, Elsevier, 2005, 38 (10), pp.2019-2027.
�10.1016/j.jbiomech.2004.09.011�. �hal-00908281�
A Numerial Assessment of Wall Shear Stress Changes
after Endovasular Stenting
F.Nioud
UniversityMontpellierII -CNRSUMR5149
H. Vernhet&M.Dauzat
UniversityMontpellierI -LaboratoryofCardiovasularPhysiologyEA2992
Manusriptnumber: BIO/2003/003572-2ndrevision
Correspondingauthor: Prof.F.Nioud
UniversityofMontpellierII
CC051
34095MontpellierCedex 5
Frane
tel:+33(0)467144846
fax:+33(0)467149316
E-mail:nioudmath.univ-montp2.fr
Keywords: stenting,
restenosis,
wallshearstress,
ompliane,
numerialmethod
Wordount: 3300
Abstrat
This theoretial/numerial study aims at assessing the hemodynami hanges indued by
endovasular stenting. Byusing the lassialone-dimensional linearpressure wavestheory in
elasti vessels, we rst show that the modulus of the reetion oeÆient indued by an en-
dovasularprosthesisismostlikelysmallsineproportionaltothestent-to-wavelengthratio.As
adiret onsequene,thewallmotionoftheelasti(stented)arteryanbepresribedapriori
and the oupled uid-struture problem does not have to be solved for assessing the hemo-
dynami hanges due to stenting.Several 2Daxisymetri alulationsare performed to solve
theunsteady inompressible Navier-Stokesequationson moving meshes fordierent typesof
(stented)arteries.Thenumerialresultssuggestthatendovasularstentinginreasesthesysto-
diastolivariationsofthewallshearstress(by35%atthemiddleofthestent,byalmost50%in
theproximal transitionregion).Additional alulationsshowthat over-dilatedstentsprodue
lesshemodynamiperturbations.Indeed,theinreaseoftheamplitudeofthewallshearstress
variationsoverthe ardiayle is only10 % when thestentradius is equalto the radiusof
theelastiarteryatsystole(insteadofbeingequaltothemeanarteryradius).
1. Motivation& Objetives
Angioplastywithorwithoutendovasularstentingisanestablishedminimallyinvasiveteh-
nique that is used astreatment of olusive disease in medium to large arteries.It has been
appliedextensivelyintheoronary,renal,andperipheralvasularsystems.Theuseofintravas-
ularstentstendsto lowerthe ompliationratealthough restenosisratesashigh as15-30%
after6monthshavebeenobservedinhumanoronaryarteries(Rauetal.1998;Serruysetal.
2002).Coatedstentselutingantimitotiagentssuhassirolimusandpalitaxelinhibitintimal
proliferation.Preliminaryreportshaveshowedextremelylowratesofrestenosisovertheshort
andmid-term(Morie etal.2002; Grubeet al.2003).However,onsideringthelong-standing
hanges in wall mehanis induedby stentplaement,longterm resultsof oated stents are
still required. Besides, a better understanding of the mehanial eets indued by stenting
ouldleadto improvedprosthesiswithlowerfailureratesandsmalleroatingrequirements(if
any).Thisstudyisaontributiontothisquestforbetterstentdesign.
Onepossibleexplanationforthetrendtorestenosisreliesonthehemodynamimodiations
induedby the prosthesis.Changes in wall shear stress indue endothelial dysfuntion(Caro
etal.1969),ultimatelyleadingtointimalhyperplasiaandrestenosis.Daviesetal.(2001)suggest
thatthemagnitudeoftheshear stressisofseondaryimportanetothespatial andtemporal
utuationsofthisquantity.InvivotestingperformedbyRollandetal.(1999);Vernhetetal.
(2000,2001)showthatendovasularstentinginduesalargemodiationofarterialompliane
and thus may modify thepropagation of arterial wavesby introduing additionalreetions.
Therstobjetiveofthisstudyisthentoassesstheamountofpressurewavereetiondueto
stenting(setion2).Anotherexpetedeetoftheomplianemismathinduedbystentingis
tomodifythedetailsofbloodmotioninthestentedarea.Speially,thetime-averaged(over
theardiayle)wallshearstressmightbehanged,aswellasthelevelofitssystolo-diastoli
variations.Theseondgoalofthispaperistolarifytheamountofhangesinbloodmotionthat
theeetsofstentingat amiro-salelevel,inludingthegeometrialdesriptionofthestruts
shape andspaing (Berry etal. 2000; Benardet al.2003). These studies provideinformation
relevantto the hemodynami hangesimmediately after stenting,before the wires havebeen
integratedwith thesurrounding tissue.Forexample,thenumerialresultsobtainedby Berry
etal.(2000)suggestthattheowmayreattahdownstreamofthewireswhenthestrutspaing
is greater than about six wire diameters. On the ontrary, the present study is dealingwith
theglobaleetsoftheomplianemismath,negletingthedetailsoftheprosthesisstruture.
Insteadofonsideringarigidwallwithamiro-salegeometrialdesriptionofthestrutsshape
(Berryetal.2000;Benardetal.2003),wemodeltheprosthesisasauniformelastitubewithits
ownompliane.Theresultsofthisstudyaremorerelevanttolong-standingstenting,afterthe
wires havebeen integratedwith thesurrounding tissue.Thereason foronsidering thispoint
of view is that intimal hyperplasia is ertainly a long lasting phenomenon ompared to the
integrationofthestentwiresbythevessel:forexample,onlyoneweekafterstenting,Robinson
etal.(1988)observedanearlyontinuousendothelialandpseudo-endothelial elllayeronthe
luminal surfaeof thestentedrabbit aorta while intimal hyperplasiais only ompleted after
six 6to 12 weeks for this model. Therefore, amiro-saleanalysis of theblood ow hanges
inludingthestrutsdetailsisonlyrelevanttoautestentingeets.Inthispaperweonsidera
omplementarypointofviewinordertoproviderelevantinformationaboutthelong-standing
hemodynamieets ofstenting.
2. Pressure waves reetion due to stenting
2.1. Basi equations
Thegeneralone-dimensionalequations desribingthepulsatilebloodowin ompliantarter-
iesare well known sinethe work of Hughes& Lubliner (1973). Theydesribe the evolution
respetively. FollowingReuderink et al.(1989),thenon-linear termsareoften negleted.Fur-
thermore, onsidering asegmentwhose rosssetion area(A) andompliane (A 0
=dA=dP)
do not depend on the spae variable x and letting P =
^
Pexp( j!t) and u = u^exp( j!t),
where j 2
= 1and ! is thewavepulsation, thefollowingHelmholtz equationanbederived
fromthelassiallinearwaveequation:
d 2
^
P=dx 2
+k 2
^
P=0 (2.1)
The omplexwavenumber is k = p
!(!+jf
v )A
0
=A, where stands for the blood density
and f
v
is relatedto the visousdrag and anbederivedfrom theWomersley veloity prole
(Womersley 1955). The omplex wave speed is = !=k and the general solution within a
homogeneoussegmentis:
^
P =P +
e
jk (x x0)
+P e
jk (x x0)
; u^= k
!+jf
v
P +
e jk (x x0)
P e
jk (x x0)
(2.2)
wherex
0
istheaxialpositionoftheleftboundaryofthesegmentandP +
and P orrespond
totheamplitudeoftheforwardandbakwardpressurewaves.Theirvaluesaredeterminedin
ordertosatisfytheboundaryonditionsatx=x
0
andx=x
0
+LwhereListhelengthofthe
segment.
2.2. Modellingthe endovasularstenting
In the purpose of modelling the wave reetion indued by an endovasular stent plaed in
anelasti artery, three suessivehomogeneoussegments areonsidered,eah having itsown
set ofonstantareaandompliane (seegure1). Eahphysialquantityin segmentnumber
i (i =1;2;3) is denoted with indie i. Conservation of the total ow rate and energyat the
interfaes1 2and2 3requires,forj=1;2:
A
j
^ u
j (x
0
j +L
j )=A
j+1
^ u
j+1 (x
0
j+1 ) ;
^
P
j (x
0
j +L
j )=
^
P
j+1 (x
0
j+1
) (2.3)
Two boundary onditions at x = x
01
= 0 and x = x
03 +L
3
are needed in order to lose
Figure 1.Shematiofthethreehomogeneoussegmentsusedtobuildamodelofstentedarteryas
regardspropagationandreexion.
presribedatbothsides,leadingtoP +
1
=1andP
3
=0.Thefourremainingwaveamplitudes,
viz.P +
2
;P +
3
;P
1
;P
2
,aredeterminedbysolvingEqs.(2.3)forj=1;2.TheomplexoeÆient
ofwavereetionduetothestentisthendenedasR
stent
=exp( 2jk
1 L
1 )P
1
=P +
1
.Aftersome
algebraonendsout:
R
stent
= A
2 K
2 (A
1 K
1 A
3 K
3 )os(k
2 L
2 ) j(A
1 K
1 A
3 K
3 (A
2 K
2 )
2
)sin(k
2 L
2 )
A
2 K
2 (A
1 K
1 +A
3 K
3 )os(k
2 L
2 ) j(A
1 K
1 A
3 K
3 +(A
2 K
2 )
2
)sin(k
2 L
2 )
(2.4)
where K
i
=k
i
=(!+jf
v
i
). This theoretialresultwill serveasasupport ofthe wall motion
usedinthenumerialstudy.
3. Wall shear stress hanges due to stenting
Thesimple1Danalysisprovidedinsetion2annotbeusedtogaininsightsaboutthedetails
oftheuidmotionmodiationsrelatedto arterystenting.Indeed,noreasonableassumption
regardingtheshapeoftheveloityprolewithinthetransitionareaanbeformulatedapriori
andthemulti-dimensionalNavier-Stokesequationsmustbesolved(assumingNewtonianblood
rheology).The solverused in this study is basedon the projetion method of Chorin (1967)
with nite element disretization and Arbitrary Lagrangian Eulerian formulation to handle
movingboundaries.Thisodehasbeenextensivelyvalidatedbyomputinglassial(Medi&
Figure2.Shematioftheomputationaldomain.
3.1. Computationaldomain
Sine ourobjetiveis to investigate theglobal eet of the ompliane mismath indued by
stenting, the endovasular prosthesis is modelled as auniform dut (the details of the struts
arenotrepresented)whosewallisnotompliant.Suh"prosthesis"isinsertedwithinanelasti
arterywith ompliantwall, asdepited in gure 2.The omputationaldomain is suÆiently
shorttoassumeonstanthostarteryharateristisandnegletthevisousdampingofwaves.
The ow rate entering the domain is taken as Q(x
inlet
;t) = Q
0 +Q
1
exp(j!(t x
inlet
=)),
where ! is thepulsation,x
inlet
is theaxial position oftheinlet setionand Q
0 and Q
1 stand
for the steady and pulsed parts of the ow rate. The mehanialand geometrial data were
obtained from animal experimentations performed by Vernhet et al. (2001): the pulsation is
! = 8, the mean artery radius is R
0
= 1:5 mm, the distensibility oeÆient of the non-
stented artery is A 0
=A = 20;7:10 6
Pa 1
and the length of the stent is L
stent
= 13 mm.
Afterstenting,autemeasurements(Vernhetetal.2001)showthattheomplianeatthestent
levelisapproximatelythreetimessmallerthaninthenon-stentedvessel.Measurementsmade
three months later (Vernhetet al. 2003) show that long-standing stentingis also responsible
forinreasedomplianevaluesupstreamfromthestent,resultinginalargeromplianeratio
of order 5 to 6. In any ase, the ompliane at the stent level is several times smaller than
that of the host vessel and has been negleted during this numerialstudy (rigid prosthesis
assumption). Note however that there would be nofundamental issues in aounting for the
Atually,themotionofthevesselboundaryresultsfromtheouplingbetweentheuidand
wallmehanisand theloal radiusis mostlyrelatedto the pressureeld. Suh aouplingis
diÆult to handle sine the density of blood and tissues are of the same order and beause
therheologyofthevesselsisfarfrom wellunderstood.Under thelinearelastiityassumption,
Tortoriello &Pedrizzetti(2004)reentlyused aperturbativeapproah in orderto replaethe
oupled uid-vessel problem by a asade of two simpler weakly oupled problems. In this
approah, the rst problem provides the exat solution into a rigid vessel, the seond one
approximatesthebloodowmodiationsduetotheompliantwall.Sinethewallshearstress
isexpetedtobeverysensitivetothewallmotion,wedidfollowadierentapproahwherethe
theexatuidowequations(andnottheirtrunatedlinearexpansion)aresolvedfor.Sinewe
aremostlyinterestedin theresponseof theuidmehanistowallmotionperturbations, the
uid-wallouplingproblemanbeavoidedbypresribingthewallmotionapriori.Inabsene
of reetion and in the ase of an elasti uniform artery whose mean radius is R
0
, the wall
motioninduedbyapropagativepressurewaveofpulsation! maybewrittenas:
R (x;t)=R
0
1+e j(!t k x)
; k=
!
(3.1)
where thewavenumberk is relatedto thespeedof the(forward)pressurewave.Note that
=max(R (x;t) R
0 )=R
0
,therelativeamplitudeoftheradiusvariations,isasmallparameter
ofthe problem:it iszerofor aperfetly rigidarterywhile theanimalexperimentsof Vernhet
etal.(2001)suggest'0:05.Thespeedofpropagationisxedbystatingthatthenon-stented
arterybeinguniformalong thestreamwisediretion,themassowrateat anysetionx =X
shouldbethetime-laggedversionofthemassowrateatx=x
inlet
.Theonservationofmass
appliedtothearterysetorx
inlet
<x<X then impliesthat (Nioud2002):
= Q
1
2A
0
+O(1); A
0
=R
2
0
; (3.2)
whereO(1)isatermoforderunitywhihanbenegletedsineisoforder1=aordingto
were usedfortheowrate:Q
0
'2413mm 3
/sand Q
1
'1761mm 3
/s.Eq.(3.2)thenleadsto
'2492mm/s.With!=8,theorrespondingwavelengthis'623mm.Intheasewhere
thevesselisstentedbetweenx
1 andx
2
(seegure2), thewalldisplaementisvirtuallyzeroed
(fullyrigid stent)for x
1
<x<x
2
and thefollowing expressionis valid forany axialposition
andtime:
R (x;t)=R
0
1+f(x)e j(!t k x)
+(1 f(x))ÆR
stent
(3.3)
where the dampingfuntion is f(x) =[1 tanh(x x
1
)℄=2 for x <(x
1 +x
2
)=2 and f(x) =
[1+tanh(x x
2
)℄exp(jk(x
2 x
1
))=2 for x > (x
1 +x
2
)=2 and ÆR
stent
= R
stent R
0 is the
amountof overdilation.Note that the term "overdilation" refers hereto any ase where the
stentradiusisgreaterthanthemeanradiusofthevesselbeforestentingR
0
.Thisissomewhat
dierent from the linialpratie where the term "overdilation"orrespondsto aseswhere
thestentradius(afterreoil)isgreaterthanthemaximum(systoli)radiusofthevessel lose
to the prosthesis (although thedierene between mean andmaximumradius anhardly be
measuredroutinely).
3.2. Numerial results
Several2Daxisymmetrisimulationshavebeenperformedbasedontheomputationaldomain
andwallmotiondesribedin setion3.1.Inallases,thebulkReynoldsnumberbasedonthe
steadypartoftheowrateQ
0
andthemeanradius R
0
isloseto R
b
=102.TheWomersley
numberis W
0
=3:36.Theveloityprole isimposed at theinlet setionx =x
inlet
following
the Womersley solutionin elasti tubesand azeroonstraint onditionis used at theoutlet
setionx=x
outlet
.Inordertoassesstheeet oftheinlet/outletboundaryonditionsonthe
results,omputationaldomainswithtwodierentlengthshavebeenonsidered.Twodierent
spatialresolutionswerealsousedin ordertoassessthespatialdisretizationerrors.Themain
harateristisofthealulationsperformedaregivenintable 1where xisthegridspaing
in thestreamwisediretion in the areax <x <x and r refersto thegrid spaing in the
radial diretion. Runs R1and R2 orrespond to referene alulations without endovasular
prosthesis,thearterybeingfullyrigid(nowalldisplaement)forR1andelastiforR2.Labels
R3(x) and R4orrespond to runs with stenting,the overdilation being non zeroonly for the
latterwhereÆR
stent
=R
0
(thestentradiusisequaltothearteryradiusatsystole).Reallthat
theterm"overdilation"refersheretoastentwhoseradiusisgreaterthanR
0
(seesetion3.1).
Foralltheasesonsideredintable 1,therelativevariationoftheradiusofthestentedvessel
(ÆR (x;t) R
0 )=R
0
is lessthan5%, whih is onsistentwith thelinearelastiityassumption
usedtoderiveEq.(3.1).Whenpresent,thestentisbetweenx
1
=34mmandx
2
=47mm.Runs
whoselabelontains'a'havebeenperformedwithalongeromputationaldomainthanothers.
Labelsontainingletter'b'orrespondtoruns withnermesh inthestreamwisediretion.In
allases,fourardiayleswererstomputedinordertoreahaperiodistate.Afthyle
wasthenomputedinordertoanalyzetheresultsandomparethedierentphysial/numerial
ongurations.ComparisonsoftherunsR2,R2aandR3,R3a,R3b(notshown)indiate that
thereisnosignianteet ofthenumerisinthestentregion(Nioud2002).Thedierenes
observedbetweenR1,R2,R3andR4arethusrelevanttothestentingeets.
Time evolutions of the ow rate at inlet and outlet setions are displayed in gure 3 for
the ase R2. The onstraint that was introdued in setion 3.1 in order to set the speed of
propagation ofthepressurewaveis fullled satisfatorily. Indeed,theowrateat x =x
outlet
isthe signalat x=x
inlet
with atime laglose to(x
outlet x
inlet
)='80=2492'0:032s. In
abseneofendovasularprosthesis,allthephysialquantitiesareself-similarwiththeonstant
speedofpropagationalongtheomputationaldomain.Duetothewalldisplaement,Eq.(3.1),
thewallshearstress(WSS)isnotuniformoverthestreamwisediretion,asdepitedingure4
atfourdierentinstants.Insteaditisalternativelyinreasinganddereasingalongthedomain
dependingonthephaseonsidered.Intheasewherethevesselisnotompliant,thereshould
be no time lag between shear stress signals at dierent loations sine the exatWomersley
proleisimposed at x=x .Aordingly, theWSSismostlyuniform (not exatlyuniform
Run Wallmotion ÆRstent xinlet xoutlet x r #ofgridpoints
R1 R (x;t)=R
0
0:0 nostent 0 80 0:075 0:286 3528
R2 Eq.(3.1) 0:05 nostent 0 80 0:075 0:286 3528
R3 Eq.(3.3) 0:05 0:0 0 80 0:075 0:286 3528
R4 Eq.(3.3) 0:05 0:075 0 80 0:075 0:286 3528
R2a Eq.(3.1) 0:05 nostent 30 110 0:075 0:286 4368
R3a Eq.(3.3) 0:05 0:0 30 110 0:075 0:286 4368
R3b Eq.(3.3) 0:05 0:0 0 80 0:050 0:286 5208
Table1.Listoftheaxisymmetrialulations performedwiththeirmainnumerialharateristis.
Lengthsandaxialpositionsareinmillimetres.
500 1000 1500 2000 2500 3000 3500 4000 4500
1 1.05 1.1 1.15 1.2 1.25
flow rate (mm3/s)
time (s)
Figure3.Timeevolutionsoftheowrateatsetions :x=x
inlet
and :x=x
outlet for
theaseR2.Notethattheoriginoftimeis1ssinethefthyleisanalysed(seetext).
beauseof smallside eetsdue tothe inlet/outletboundaryonditions) overthestreamwise
distaneintheaseR1(seegure4).
The shape of the omputational domain for asesR2, R3 and R4 is shown in gure 5for
times t = nT (orresponding to systole at the inlet setion) and t = (n+1=2)T (diastole).
