Department of Solid Mechanics, Materials andSystems, Faculty of Engineering, Tel Aviv
University, Ramat Aviv 69978, Israel
Shmuel EinavDepartment of Biomedical Engineering, Faculty of
Engineering, Tel Aviv University, Ramat Aviv69978, Israel
Leonid SlepyanDepartment of Solid Mechanics, Materials and
Systems, Faculty of Engineering, Tel AvivUniversity, Ramat Aviv 69978, Israel
The Cardiocoil Stent-ArteryInteractionAn analytical approach for the mechanical interaction of the self-expanding Cardstent with the stenosed artery is presented. The damage factor as the contact strestent-artery interface is determined. The stent is considered as an elastic helichaving a nonlinear pressure-displacement dependence, while the artery is modeleelastic cylindrical shell. An influence of a moderate relative thickness of the shestimated. The equations for both the stent and the artery are presented in theassociated helical coordinates. The computational efficiency of the model enabcarry out a parametric study of the damage factor. Comparative examinations arducted for the stents made of the helical rods with circular and rectangular crostions. It was found, in particular, that, under same other conditions, the damagefor the stent with a circular cross section may be two times larger than thatrectangular one.fDOI: 10.1115/1.1871194g
Keywords: Self-Expanding Stent, Damage Factor, Analytic Solution, Parametric S
-staf thomble
loa
ingimadm
ion
ensisntcoth
y rme
pe
abr, hlthth
ropf adifitre
icaith
urin
xten--
suit-led ust force of
con-pen-t foreachvidein theed nu-
tingstentTheandec. 5,
a ho-braneolza-rgeel,
asl the
studyy isoge-er isells
or aanu
1 IntroductionThe unacceptably high restenosis ratef1g after the stenting pro
cedure stimulated many studies of the topic in order to underthe mechanics involved in the process. At the final stage oprocedure, the stent interacts with the artery by applying spressure at the contact area. This pressure has a considerafluence on the stress state within the artery wallsf2g. When thestent is unsuitable and the pressure is too high it may causeinjury of the artery as well as high stresses in the arterial wBoth these factors increase the risk for a restenosis, includnarrowing of the lumen. A correlation between the suboptstent dilatation with the occurrence of restenosis was reporteAkiyama f3g. Injury of the artery was also shown to cause inflamation resulting in rapid multiplication of cells and the formatof a layer sneointimad, producing a narrowing of the lumenf4g.Rachevf5g and Rachev et al.f6g investigated the stress dependremodeling of the vessel wall as possible causes of resteno
The stress state in the artery that had been injured after stedepends upon its geometry and the mechanical properties inbination with the properties of the stent. The stiffness and omechanical properties of stents were investigated by mansearchers. Most of the studies are based either on the experiapproachf7–10g or on the numerical modelingf11,12g. Analyticalmodels were also used in a number of studies to verify the exmentsf13,14g.
Stent-artery interaction which requires a model that is capof simultaneously evaluating the stent and the artery, howevebeen less investigated. Holzapfel et al.f15g used a numericamodel for the balloon-expended Palmaz-Schatz stent: the auconsidered the problem in a framework of finite strains andarterial wall being composed of several layers with different perties. Auriccio et al.f16g also investigated the interaction oballoon expanded stent with an artery and suggested a modesign in order to reduce the nonuniformity of the contact sdistribution.
The goal of the present research is to investigate theoretthe main features of the mechanical interaction of an artery wself-expanding stent undergoing only elastic deformations d
1To whom correspondence should be addressed.Contributed by the Bioengineering Division for publication In the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript receieved: January 27, 2004 Final m
script received: September 10, 2004. Associate Editor: Fumihiko Kajiya.
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
ndeein-
calll.
al
by-
t.ingm-ere-ntal
ri-
leas
orse-
edss
llyag
deployment. We concentrate on Cardiocoil nitinol stentf17g. Theexperimental and clinical performance of this stent was esively studied by several authorsf17–21g. The crucial role of appropriate stent sizing was emphasized by Hong et al.f20g. TheCardiocoil stent has a relatively simple geometric structureable for the purposes of mathematical description. This enabto develop an efficient analytical interaction model convenienthe analysis. Specifically, our model demonstrates the influenthe geometric and elastic parameters of the problem on thedam-age factordefined by the stresses appearing at the interfacetact zone. Knowledge of this factor and, in particular, its dedence upon the stent-artery radial mismatch is importanchoosing the correct size and type of Cardiocoil stent forpatient. In addition, the obtained analytic solution could proconvenient benchmark problems to be used as guidelinessearch for more accurate and, consequently, more advancmerical approaches.
The analytical model of an elastic circular cylinder representhe artery is given in the next section. Section 3 describes awhich is considered to represent an elastic curvilinear rod.nonlinear stent-artery interaction problem is formulatedsolved in Sec. 4. The numerical results are presented in Sfollowed by the concluding remarks.
2 Artery ModelThe assumption that an arterial wall can be modeled as
mogeneous elastic layer and an artery as a cylindrical memor a hollow elastic cylinder was accepted by many authors. Hpfel et al.f22g considered this model in the framework of a lastrain analysis. Rachevf5g used a different two-layered modwhile Auriccio et al.f16g considered the artery and the plaquedissimilar homogeneous isotropic materials. In order to reveamain features of the stent-artery interaction in the presentwe employed a relatively simple model in which the arterconsidered as being a circular hollow cylinder made of homneous elastic material. As such, the geometry of the cylindcompletely defined by the wall thicknessh and the radius of thmiddle surfaceR sFig. 1d. The elastic properties of the artery waare described by the Young modulusE and the Poisson ration.
Note that the analytical approach used here is still valid f-
more sophisticated artery model, for example, for a three-layer
APRIL 2005, Vol. 127 / 3375 by ASME
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
rthe
s osynnt, tadthb
l be
div-sursua,
coel
n is
eteem
efohatermrte
sescr
th
nearordi-yof
thehhelliny the
eriodicyny
ess
c
tnding
e
d
e,pe ofenial to, into
Downloa
model corresponding to Intima, Media, and Advestitia or fomultilayer model, while the simplest formulation adopted inpresent paper gives us the first approximation.
The solution of the stent-artery interaction problem hingethe knowledge of the compliance of both constituents of thetem. The radial compliance of the cylinder subjected to the ipressure along the helical strip is calculated in two steps. Firsproblem on the corresponding thin-walled shell with helical loing is solved analytically. Then, in order to take into accountfinite relative thickness of the shell, this solution is correctedthe use of numerical results of an auxiliary problem which wilspecified later.
The system of curvilinear cylindrical coordinatesx,y,z with y=Ru is defined as shown in Fig. 1, and the correspondingplacements are denoted asu,v ,w, respectively. The elastic behaior of the thin-walled shell subjected to normal internal prespsx,yd sthe stent-artery contact is assumed to be frictionlesd isdefined by a system of three partial differential equilibrium eqtions with respect tousx,yd ,vsx,yd andwsx,yd ssee, for exampleRef. f23gd.
The pressure is applied to the artery by the stent which issidered as a helical rod. Consequently, the contact zone is a hstrip of width b defined by the rod’s cross section. Its locatiocompletely determined by the anglea between thex axis and thevector tangential to the helixssee Fig. 2sadd. Note, that the valueacorresponds to the deformed state of the stent, it will be dmined during the solution of stent-artery interaction problClearly
a = arctan2pR
Hs1d
whereH is the pitch of the helixssee Fig. 2sadd.In the present work we concentrated upon studying the d
mations appearing in the middle part of the stent-artery meccal system and did not consider the regions where the stentnates. Consequently, the distributed loading applied to the ais assumed to be constant along the contact strip. In this castress-strain state within the artery will possess the samesymmetry as the contact helix zone itself. This fact dictates
Fig. 1 Artery segment and cylindrical coordinate system
Fig. 2 Cylindrical shell with helical coordinate system „a…; the
shell deployed to a strip „b…
338 / Vol. 127, APRIL 2005
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
a
ns-erhe-ey
s-
e
-
n-ical
r-.
r-ni-
i-rytheewe
need for employing a nonstandard system of helical curvilicoordinates for the analysis. In such a system, the radial conatez remains the same while the coordinatesx,y are replaced bthe helical coordinatesq,s as two mutually orthogonal systemshelices ssee Fig. 2sadd. On the cylindrical surfacez=const=R,these coordinates vary within the following limits −a,q,a,−`,s,`. The cylindrical and the helical coordinate lines onshell surface are presented in Fig. 2sbd where the strip of the widt2a=Hsina is depicted. The strip is obtained by cutting the salong the coordinate lineq=const and deploying it to a plafigure. The relations between the coordinates are given brotation matrix as it is seen from Fig. 2sbd:
Hs
qJ = H sina cosa
− cosa sinaJHy
xJ s2d
For the considered symmetry the stresses and strains are pin x and y with the periodsH and 2pR, respectively, while theare independent ofs. This allows us to consider the fields at agivens for arbitrary extendedq. In such representation, the strfields are periodic in theq direction with the period equal to 2a.The radial displacements being independent ofs are also periodiin q:
w = wsqd s3d
Regarding the remaining two displacementsu and v, one mustake into account that they have components linearly depeon x, i.e.,
u = usqd + «0x s4d
v = vsqd + c0x s5d
where«0 andc0 are the averaged strain along thex axis and thangle of torsion about this axis, respectively.
Substituting the expressions forx,y from s2d into the mentionesystem of partial differential equations and usings3d–s5d, aftersome manipulation, we obtain
t1]2u
]q2 −n+
2
]2v]q2sin 2a +
n
R
]w
]qsina = 0 s6d
s1 + k2dt2]2v]q2 −
cosa
R
]w
]q−
n+
2
]2u
]q2sin 2a + k2R]3w
]q3 cosa = 0
s7d
]4w
]q4 +cosa
R
]3v]q3 +
1
k2R3Fw
R−
] v]q
cosa + n«0 + n]u
]qsinaG =
psqdDs
s8d
wheren± =1 ± n
2, k2 =
h2
12R2, Ds =Eh3
12s1 − n2d,
t1 = sin2 a + n−cos2 a, t2 = n−sin2 a + cos2 a
It is assumed that the coordinate lineq=0 corresponds to thmiddle of the helical contact zone of widthb ssee Fig. 2sbdd. Thenin accordance with the helical symmetry of the deformed shathe shell, the radial displacementswsqd are represented by an evfunction and the nontrivial parts of the displacements tangentthe shell surfaceusqd andvsqd are odd functions. Consequentlyview of the 2a periodicity of these functions, it is suitablepresent them by the following Fourier series:
hu,vj = o`
hun,vnjsinpnq
as9d
n=1
Transactions of the ASME
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
rste
n-effi-
arily
Downloa
w = on=0
`
wncospnq
as10d
The unknown coefficients are defined after substitution ofs9d ands10d into s6d–s8d in a standard manner. Multiplication of the fitwo equations by the sinsnpq/ad and the third one by th
sby
oe
athe
ththereeq
alasell
la
calth
ut
etion
be
Journal of Biomechanical Engineering
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
cossnpq/ad and integration overq yields the system of three liear algebraic equations for deriving the triple of unknown cocients un,vn, and wn for each n=1,2,…. For formulating theproblem on stent artery compatibility, we are interested primin radial displacement. The coefficientswn, n=1,2,… are foundto be
The remaining coefficientw0 appearing ins10d, which representthe average radial displacement of the artery, is obtainedsimple integration of Eq.s8d:
w0 =R4k2Q0
2Dsa− n«0R s12d
The valuesQn in the earlier equations represent the Fourier cficients in the expansion of an external pressure
Qn =E−a
a
psqdcossnpq/addq s13d
Since our intention is to investigate the problem of pressureplied to an artery by a stent, it is worthwhile to considerspecific case of distributed line loading
psqd = Qds0d s14d
wheredsqd denotes the Dirac-delta function. For this case
Qn = Q n= 0,1,2,… s15dThe magnitude of the average elongation per unit length in
x direction«0 in Eq. s12d together with the rotation per unit lengc0 are derived from the boundary conditions. In the considproblem these conditions are expressed by the conditions oflibrium in a cross section of the shell perpendicular to thex axis.In accordance with the assumption that there is no tangentiteraction between the stent and the artery, we consider the cpure normal loading applied to the internal surface of the shethe radial direction. Therefore, the resultant axial forceFx and thetorque momentMT applied to the cross section, which is a circuring with the radiusR, are equal to zero
Fx = 0, MT = 0 s16d
The resultants are expressed in terms of the internal forceslated per unit length of the ring. Deriving these forces fromobtained expressions for the displacements and carrying ointegration one obtains
Fx = 2pksR«0 + nw0ds17d
MT = pkR2sn − 1dc0
Hence, from the assumptionss16d, it follows that the averaglongitudinal and rotational components of the shell deformaare given by
«0 = − nw0
Rand c0 = 0 s18d
Consequently, in accordance withs12d, the value ofw0 is found to
a
f-
p-
e
dui-
in-of
in
r
cu-ethe
s
w0 =R4k2Q0
2Dsas1 − n2d=
Q0R2
2aEhs19d
Finally, the radial displacement of the arterywa along the helixq=0 in accordance withs10d is
wa = on=0
`
wn s20d
where the coefficientswn are defined froms11d and s19d and depend upon the applied loading. This loading will be found lfrom the compatibility conditions for the deformations of a sand an artery.
Let us now verify the obtained solution by examining a limitcase. Consider the case of the line loadings14d and s15d andassume that the anglea increases. Then the pitch of the helixHdefining the distance between the loading lines, as it is seens1d, will decrease and in the limiting situation, whena approache90 deg, the shell will be subjected to the uniform radial presp0 which is the average of the external forces
p0 =Q
2as21d
Substituting the latter relation tos19d srecall, that for the linloading Q0;Qd one obtains the radial displacement of the scoinciding with the average value
w0 =p0R
2
Ehs22d
The last formula represents the well-known solution for the plem of an elastic ring subjected to the internal pressure. Theculations based on the suggested algorithm were carried outaclose to 90 deg, and the difference from the limiting analytic vs22d was found to be less than 0.1%.
The ratio of the wall thickness to diameter in normal arterie0.1. Consequently, the radial compliance determined by meathe thin-walled shell model represents a rather rough approtion. In order to improve the model we will compare the comances for the thin-walled shell and for the thick hollow cylindean auxiliary problem. The obtained difference will be expreby a coefficient which will correct the solution for the thin-walshell under helical loading.
In the auxiliary problem the loading to the shell/cylinder insurface acts not along the helical line but along the periodictem of circular rings perpendicular to the longitudinal axis.distance between the rings is equal to the pitchH of the helix.This problem, in contrast to the original one, possesses the
symmetry and very convenient for a solution.
APRIL 2005, Vol. 127 / 339
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
. Tineets
emoofybyh
on
rm
tingth
the
thintdin
s
thed
tions
tentssia
eing itsimepli-
ere-
de-
, in
an
red.sr-ionre-ralssuresical
deliv-
is
del
studylen-
m, a.09nablyeffectch arelyticalent to
dis-ue inng,rtion
tolthy
Downloa
The calculations were based on the finite element methodradial displacement of the thick-walled artery was then obtafrom that for a thin-walled shell models20d using the correctivmultiplier equal to the ratio of the corresponding displacementhe auxiliary problem.
3 The Stent ModelThe second constituent of the considered mechanical syst
the Cardiocoil stentf17g. It is depicted in Fig. 3. In order tdescribe its mechanical behavior we will employ the modelhelical rod sas an inclusion in an elastic matrixd developed bSlepyan et al.f24g. The spatial location of the wire is definedthe parameters of the corresponding helixr0,a0 passing througthe center of the wire cross sections. The curvaturek0 and torsiont0 of the helix as defined by the use of the Frenet orthogcoordinatest ,n,b shown in Fig. 4 are
k0 =sin2 a0
r0s23d
t0 =sina0cosa0
r0s24d
We will further denote the parameters related to the undefosinitiald position of the wire by the superscripti, namelyr0
i ,a0i ,k0
i ,t0i . The principal vector and the principal moment ac
at the wire cross section can be presented by the use ofs-independent componentssthe s coordinate is the same as inprevious sectiond
Q = Tt + Nn + Bb s25d
M = Mtt + Mnn + Mbb s26dOne of the initial assumptions of the considered model ofstent-artery interaction is that the tangential stresses at theface are negligibly small. Consequently, the only external loaapplied to the stent is the distributed normal forceqn directedalong the vectorn. Then from the equilibrium equations it followthat N=0, Mn=0 and
k0T − t0B = − qn s27d
Fig. 3 Curvilinear helical wire subjected to radial loading qn
Fig. 4 Frenet orthogonal coordinates used for the stent
description
340 / Vol. 127, APRIL 2005
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
hed
in
is
a
al
ed
eir
eer-g
k0Mt − t0Mb − B = 0 s28d
In addition, since the stent is supposed to be unstressed inxdirection and untwisted around thex axis, it may be concludethat Qx=0 andMx=0 or
Tcosa0 + Bsina0 = 0 s29d
Mbsina0 + Mtcosa0 + sTsina0 − Bcosa0dr0 = 0 s30dFinally, the elasticity equations relating the stent deformawith the internal moments are given by
Mb = EIsk0 − k0i d s31d
Mt = GIpst0 − t0i d s32d
where E, G are the Young and the shear modulus of the smaterial, respectively,I is the moment of inertia of the wire crosection about theb direction, andIp is the polar moment of inertof the section.
The nonlinear system of Eqs.s23d–s32d completely defined thstress-strain state of the stent and can be employed for derivcompliance in the radial direction, a property which is of primportance for the study of stent-artery interaction. This comance expressed by the dependencer0sqnd is determined in thfollowing way. First, the couplesr0,a0 providing the solution foEqs.s23d–s26d and s28d–s32d are found by an incremental procdure with the r0
i ,a0i starting point corresponding to the un
formed position. The obtained values are then substituted ins27dand the magnitude ofqn corresponding to the givenr0 is calcu-lated. An illustrative numerical result is presented in Fig. 5which the dependence of the normalized stent diameterd0 uponthe external line pressureqn is depicted. Here, a stent withouter diameterd0
i =5 mm and with a helix anglea0i =81.2 deg
made of a circular wire with a diameter 0.25 mm is consideThe material of the stent is NiTinol with an elastic moduluE=46.6 GPa and a Poisson ration=0.3. The type of the nonlineaity with a positive second derivative in the working reg3 mm,d0,5 mm conforms with the experimental results psented by Jedwab and Clercf14g for a stent composed of sevehelices. The information on the decreasing of the external prefor a very large stent deformations is understood from the phyconsiderations and probably may be used in studying stentery problem. The longitudinal stent deformation in thex directionis defined by the deviation of the helix anglea0 from its initialvalue a0
i . The dependence ofa0 upon the external pressurefound to be similar to the behavior ofd0.
Additional support for the validity of the developed stent mocomes from experimental results of Schrader and Beyarf9g. Thecompliance of stents was experimentally investigated in thatand expressed in terms of the external pressurepr and the radiadeformation«r. A comparison of the analytical and the experimtal results for the NiTinol stent with an external diameter 3 mhelix angle a0
i =81.2 deg and a wire cross section 030.12 mm is presented in Fig. 6. The agreement is reasogood, the difference that does exist may be explained by theof the shear stresses at the loaded surface of the stent whipresent in the experimental device and are absent in the anamodel. Consequently, the pressure required in the experimreach the specific stent radial contraction is higher.
4 Stent-Artery InteractionIn developing a suitable model for stent interaction with a
eased artery we have to take into account the role of plaqdiminishing the width of arterial lumen. In the clinical settiballoon angioplasty is usually carried out before the stent inse
in order to bring the inner diameterddisin of the diseased artery
the sizeddisin which is equal to the inner diameter of a hea
in
artery dheal. As a result, the preinflation situation in which the
Transactions of the ASME
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
eqnsen
surte
olzomtakecos
s aonarst
ofc
oth
eomet-el
of theandl the, use
sk in
the
theofh are
esults.
Downloa
outer diameters of the diseased and the healthy arteries arebut the inner diameters are different reverts to the situatiowhich the inner diameters are the same but the outer onedifferent. The outer diameterddis
out of the diseased artery is thdetermined from mass conservation
sddisoutd2 − sddis
in d2 = sddisoutd2 − sddis
in d2 s33dThe tube obtained after inflation will include the arterial tis
and the existing plaque. The elastic properties of the sick aare significantly influenced by the desease. According to Hphel f15g the increase in the isotropic elastic responce of the sparts of the Intima may approach 150%, a certain changeplace even in Media and only Adventitia remains unaltered. Nertheless we will assume here that the tube material may besidered as isotropic elastic with the same elastic propertiethose of a healthy artery. The earlier simplifying assumptionsomewhat presumptuous, but we contend that they are reaswithin the framework of the present study to obtain the fiapproximation analytical results.
Stent-artery interaction lends itself well to study by meansmodel. The self-expanding stent and the postinflation arterybe considered as two elastic springs. Inserting one into an
Fig. 5 The influence of the external raized outer stent diameter d0
Fig. 6 The external radial pressure otraction «r for the 3 mm stent with
Ã0.12 mm
Journal of Biomechanical Engineering
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
ualinare
erya-ees
v-n-asreble
-
aaner
leads to a stress-strain state that appears as the result of a gric mismatch between the springssi.e., the outer diameter of thstent exceeds the inner diameter of the arteryd and the externaforce si.e., the blood pressured. Note that one of the springssthestentd is assumed nonlinear. The nonlinear elastic responceNiTinol coil stent is caused by the geometrical nonlinearity,we take this into account. In contrast, for the artery modephysical nonlinearity may have an importance; we, howeverthe linearly elastic model. The determinationsand the accountd ofthe artery physical nonlinearity seems to be an important tathis field.
The compatibility condition that needs to be fulfilled atstent-artery interface has the form
ddisin + 2wa = d0
i + 2ws s34d
where wa and ws are the respective radial displacements ofartery and the stent at the interface, andd0
i is the outer diameterthe stent in the unloaded position. The displacements whiccaused by the average blood pressurepb and by the pressurepsaatthe stent-artery interface, can be easily calculated using the rfor the radial compliances obtained in the previous sections
l line pressure qn on the normal-
e stent pr vs relative radial con-rectangular cross section 0.09
dia
n ththe
APRIL 2005, Vol. 127 / 341
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
The unknown interface pressurepsa is derived by a simple triaand-error procedure in order to fulfills34d. A damage factor chaacterizing the level of forces acting between the stent and arsurfaces is defined as the normalized interface pressure
D =psa
pbs35d
A parametric study of this factor will be presented in the folloing section.
5 Numerical ResultsThe manufacturer’s recommendation for choosing the sten
ameter for a specific patient is based mainly on the mismbetween the outer radius of the stent and the inner radius oartery. On the other hand, the same radial mismatch may levery different contact pressuressand, consequently, damage ftors that defines arterial injuryd, depending upon the complianof the stent, the diameter of the artery and the percent of steThis is important for choosing the correct therapeutic soluThe numerical results presented later illustrate the earlier ddencies for the Cardiocoil stent.
Table 1 Interaction of arteries having 75% area
sad
Inner diameter of inflated artery before stentingsmmdDifference between the stent and the artery diamCircular Artery diameter after stent imcross section Damage factor-DRectangular Artery diameter after stent imcross section Damage factor-D
sbd
Inner diameter of inflated artery before stentingsmmdDifference between the stent and the artery diamCircular Artery diameter after stent imcross section Damage factor-DRectangular Artery diameter after stent imcross section Damage factor-D
Fig. 7 Dependence of the damage fmismatch Dd for the case of a circulaand the stent diameter is varied for e
sponds to the healthy 4.75 mm artery.
342 / Vol. 127, APRIL 2005
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
ial
-
i-chheto
is..n-
The damage factors for stents with rectangular and circross sections are given in Table 1. Two standard outer steameters of 3 mmsTable 1ad and 5 mmsTable 1bd are consideredAccordingly, circular wires have diameters of 0.15 and 0.25and the dimensions of the rectangular cross section are30.12 mm and 0.13530.18 mm. The results are calculatedthree radial stent-artery mismatchesDd=0.1,0.3,0.5 mm whiccorrespond to the healthy arteries with diameters 2.9, 2.7, 2.5respectively, for Table 1a, and 4.9, 4.7, 4.5 mm for Table 1b.area stenosis is assumed to be 75% for all the cases. Theshow that the increase of the lumen after stent implantation gmonotonically with the increase of the radial mismatch andproaches the maximum value of 0.21 mm for the circular wire0.1 mm for the rectangular onesTable 1ad. The smaller extent oincrease in the latter case is understandable since the rectawire stent is more compliant. Consequently, the damage factthis stent is found to be significantly less than that for thewith a circular cross section. Specifically, for the maximal ramismatch 0.5 mm the difference between the damage factabout 240% for the 5 mm stents and 180% for the 3 mm on
Moreover, it can be noted that the damage factor seemsproportional to the radial stent-artery mismatch. This fact is i
or D upon the radial stent-arteryire. The arterial diameter is fixed
h curve. The dashed line corre-
ste
eteDdpla
pla
eteDdpla
pla
actr wac
Transactions of the ASME
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
raen
cloofntapr
nd,l to
r fr,d
tref t
t th
adialomes
mayor. ThemageTheg ofn themage
r thethe
Downloa
trated by the data presented in Fig. 7 which presentsD for injuredarteries of 2.75, 3.75, and 4.75 mm with 75% stenosis. The gfor the healthy artery is also given for reference. The dependof the damage factor upon the mismatch is found to be veryto linear. For the limiting valueDd=0 when the outer diameterthe stent is equal to the inner diameter of the artery, the copressure under the stent is, in fact, equal to average bloodsure. Consequently, the stent-artery interaction vanishes aaccordance withs35d, the damage factor in this case is equaunity.
The influence of the stent geometry on the damage facto75% area stenosis is presented in Figs. 8 and 9. In the formediameter of the circular wire varies. The results for the 3 anmm stents with a 0.25 mm radial mismatch reveal a generalof moderate increasing of the damage factor with increasing owire diameter. This phenomenon is in agreement with the fac
Fig. 8 Dependence of the damage facmm diameter and 5 mm diameter stentsis 0.25 mm
Fig. 9 Dependence of the damage faundeformed position … for the 3 mm
stents. The radial mismatch with the artery
Journal of Biomechanical Engineering
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
phcese
ctes-in
orthe5ndheat
enlarging of the wire diameter leads to the increasing of the rstiffness of the stent. On the other hand, when the wire becthicker, the stent-artery contact area increases as well, whichdecrease the contact stresses that define the damage factlatter observation helps to understand the reduction of the dafactor for the 3 mm stent observed for large wire diameters.two mentioned effects of stent radial stiffening and increasinthe interface contact area having the opposite influence ocontact stresses explain the nonmonotonic behavior of the dafactor as a function of the helix anglea sFig. 9d. It is interesting tonote that for the circular wire stent, the maximum is seen foangles in the vicinity of 80 deg, and this is the value forCardiocoil stents currently employed in clinical practice.
upon the wire’s diameter for the 3e radial mismatch with the artery
r upon the helix angle a „in theshed lines … and 5 mm „solid lines …
tor, th
cto„da
is 0.25 mm for all the cases.
APRIL 2005, Vol. 127 / 343
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
comredi
aropuariceffy f
t.prohacof tred. Tte
merin
anntionesialte oedn tf ththdic
pliepetiene i
et
i, Mroniolla
rea
, S.,nduce
S. H.,87 to
t Re-ech.,
el ofent to
, ands of
nicalsis,”
Me-iagn.,
nicalhoice.001,Ele-
of
om-Inf.,
andEx-
yer-astymed.
is ofeth-
96,
r, M.,nine
ber-ighlyeart
on,s in
999,gio-
996,nite-
om-
an
Downloa
6 Concluding RemarksThe study of the stent-artery interaction phenomenon is a
plicated mechanical problem. It requires developing a thdimensional model for a system including the constituents offerent origin ssteel, tissue, plaqued characterizing by a nonlinebehavior and having a large gap in geometric and elastic prties. In addition, the data defining these properties are usknown with a limited accuracy. Consequently, a pure numeapproach to the problem requires tremendous computationaland the results in the literature are usually presented onlspecific parameter combinations.
The approach suggested in the present study is differenorder to understand the basic mechanics of the interactionlem several rather presumptuous simplifying assumptionsbeen made. In particular, the material of the injured artery issidered as linear isotropic elastic one, the prestretching oartery in the circumferential and longitudinal directions is ignothe influence of stent ends on the stress state is neglectedenabled to develop a very efficient semi-analytical model of inaction of the artery with Cardiocoil stent and carry out a pararic study. The interaction is characterized by adamage factodefined as the ratio of the contact stresses at the stent-arteryface to average blood pressure. It is obvious that this factor cbe either too small or too large: if it is too small, the interacbetween the stent and the artery is absent and the stent dofulfill its supporting function, and if it is too large, the arterinjury may lead to growth of neointima and restenosis. In spithe simplifying assumptions the obtained results are believgive an acceptably accurate description of the differences idamage factor resulting from the variation in the properties ostent and of the artery. Establishing of an optimal value forfactor based on an analysis of known data collected from meexperience is an important topic for future studies.
The results we obtained for the Cardicoil stent can be apby the designers of this type of stent as well as by medicalsonnel for choosing the most suitable stent for a specific paFuture research should include analytical/numerical derivatiodamage factors for other stent types and investigation of thfluence of plaque geometry on the damage factor.
AcknowledgmentThis work was partly supported by Nicholas and Elisab
and Figulla, H. R., 2003, “Determinants of Target Vessel Failure in ChTotal Coronary Occlusions after Stent Implantation—The Influence of Ceral Function and Coronary Hemodynamics,” J. Am. Coll. Cardiol.,42s2d, pp.219–225.
f2g De Belder, A., and Thomas, M. R., 1998, “The Pathophysiology and Tment of In-stent Restenosis,” Stent,1s3d, pp. 74–82.
344 / Vol. 127, APRIL 2005
ded 13 Apr 2010 to 132.66.235.242. Redistribution subject to ASME lic
-e-f-
er-llyalortor
Inb-ven-he,hisr-t-
ter-ot
not
ftohee
isal
dr-
nt.ofn-
h
.,ct-
t-
f3g Akiyama, T., Di Mario, C., Reimers, B., Ferraro, M., Moussa, I., Blenginoand Colombo, A., 1997, “Does the High-Pressure Stent Expansion IMore Restenosis?,” J. Am. Coll. Cardiol.,29, p. 368A.
f4g Oesterle, S. N., Whitbourn, R., Fitzgerald, P. J., Yeung, A. C., Stertzer,Dake, M. D., Yock, P. G., and Virmani, R., 1997, “The Stent Decade: 191997,” Am. Heart J.,136, pp. 578–599.
f5g Rachev, A., 1997, “Theoretical Study of the Effect of Stress-Dependenmodeling on Arterial Geometry under Hypertensive Conditions,” J. Biom30, pp. 819–827.
f6g Rachev, A., Manoach, E., Berry, J., and Moore, J. E., Jr., 2000, “A ModStress-Induced Geometrical Remodeling of Vessel Segments AdjacStents and Artery/Graft Anastomoses,” J. Theor. Biol.,206, pp. 429–443.
f7g Fluecker, F., Sternthal, H., and Klein, G. E., 1994, “Strength, ElasticityPlasticity of Expandable Metal Stents: In Vitro Studies with Three TypeStress,” J. Vasc. Interv Radiol.,5, pp. 745–750.
f8g Lossef, S. V., Luts, R. J., and Mandorf, J., 1994, “Comparison of MechaDeformation Properties of Metallic Stents with Use of Stress-Strain AnalyJ. Vasc. Interv Radiol.,5, pp. 341–349.
f9g Schrader, C. S., and Beyar, R., 1998, “Evaluation of the Compressivechanical Properties of Endoluminal Metal Stents,” Cathet Cardiovasc. D44, pp. 179–187.
f10g Dyet, J. F., Watts, G., Ettles, D. F., and Nicholson, A. A., 2000, “MechaProperties of Metallic Stents: How Do These Properties Influence the Cof Stent for Specific Lesions?,” Cardiovasc. Intervent Radiol.,23, pp. 47–54
f11g Etave, F., Finet, G., Boivin, M., Boyer, J., Rioufol, G., and Thollet, G., 2“Mechanical Properties of Coronary Stents Determined by Using Finitement Analysis,” J. Biomech.,34, pp. 1065–1075.
f12g Dumoulin, C., and Cochelin B., 2000, “Mechanical Behavior ModelingBalloon-Expandable Stents,” J. Biomech.,33, pp. 1461–1470.
f13g Loshakove, A., and Azhari, H., 1997, “Mathematical Formulation for Cputing the Performance of Self Expanding Helical Stents,” Int. J. Med.44, pp. 127–133.
f14g Jedwab, M. R., and Clerc, C. O., 1993, “A Study of the GeometricalMechanical Properties of a Self-Expanding Metallic Stent—Theory andperiment,” J. Appl. Biomater,4, pp. 77–85.
f15g Holzapfel, G. A., Stadler, M., and Schulze-Bauer, C. A.J., 2002, “A LaSpecific Three-Dimensional Model for the Simulation of Balloon Angioplusing Magnetic Resonance Imaging and Mechanical Testing,” Ann. BioEng., 30, pp. 753–767.
f16g Auriccio, F., Di Loreto, M., and Sacco, E., 2001, “Finite Element Analysa Stenotic Artery Revascularization through a Stent Insertion,” Comput. Mods Biomechanics and Biomechanical Engineering,4, pp. 249–264.
f17g Handbook of Coronary Stents, P. Serruys, ed., Martin Dunitz, London, 19Chap. 10.1.
f18g Beyar, R., Henry, M., Shofti, R., Grenadier, E., Globerman, E., and Beya1994, “Self Expandable Nitinol Stent for Cardiovascular Applications: Caand Human Experience,” Cathet Cardiovasc. Diagn.,32, pp. 162–170.
f19g Grenadier, E., Shofti, S., Beyar, M., Lichtig, H., Mordechovitz, D., Gloman, O., Markiewics, W., and Beyar, R., 1994, “Self Expandable and HFlexible Nitinol Stent: Immediate and Long Term Results in Dogs, Am. HJ., 128, pp. 870–878.
f20g Hong, M. K., Beyar, R., Kornowski, R., Tio, F. O., Bramwell, O., and LeM. B., 1997, “Acute and Chronic Effects of Self-Expanding Nitinol StentPorcine Coronary Arteries,” Coron. Artery Dis.,8s1d, pp. 45–48.
f21g Roguin, A., Grenadier, E., Linn, S., Markiewicz, W., and Beyar, R., 1“Continued Expansion of the Nitinol Self-Expanding Coronary Stent: Angraphic Analysis and One-Year Clinical Follow-Up,” Am. Heart J.,138s2d,pp. 326–333.
f22g Holzapfel, G. A., Eberlein, R., Wriggers, P., and Wezsäcker, H. W., 1“Large Strain Analysis of Soft Biological Membranes: Formulation and FiElement Analysis,” Comput. Methods Appl. Mech. Eng.,132, pp. 45–61.
f23g Vinson, J. R., 1993,The Behavior of Shells Composed of Isotropic and Cposite Materials, Kluwer Academic, the Netherlands.
f24g Slepyan, L. I., Krylov, V. I., and Parnes, R., 2000, “Helical Inclusion inElastic Matrix,” J. Mech. Phys. Solids,48, pp. 827–865.
Transactions of the ASME
ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm
![2[125I]Iodomelatonin binding and interaction with beta-adrenergic signaling in chick heart/coronary artery physiology](https://static.documents.pub/doc/80x56/63556deef4b7d3d11c0ca4a8/2125iiodomelatonin-binding-and-interaction-with-beta-adrenergic-signaling-in-chick.jpg)