-
Mechanical Behavior of fully expanded
endovascular stents
J. Tambaca S. Canic D. Paniagua, M.D., M. Kosor
December 21, 2008
1 Introduction
Endoluminal stents are used in the cardiovascular system
(coronary arteries, pulmonaryarteries, aorta, large systemic veins
and arteries, etc.) as well as in the tracheobronchial,biliary and
urogenital systems. They play a crucial role in the treatment of
coronary arterydisease (CAD). One of the complications following
the treatment of CAD using stents isre-stenosis. Clinical studies
correlated re-stenosis with geometric properties of stents, suchas
the number of stent struts, the strut width and thickness, and the
geometry of the cross-section of each stent strut, [9], [14], [16],
[12]. At the same time these geometric propertiesplay a key role in
the overall mechanical properties of a stent.
There is a large number of stents with different geometric and
mechanical featuresavailable on the market. Knowing the mechanical
properties of a stent is important indetermining what pressure
loads can a stent sustain when inserted into a native
artery.Different medical applications require stents with different
mechanical properties. Thetherapeutic efficacy of stents depends
among other things on their mechanical properties [8].
Department of Mathematics, University of Zagreb, Bijenicka 30,
10000 Zagreb, Croatia([email protected]). Research supported in part
by MZOS-Croatia under grant 037-0693014-2765 and bythe NSF/NIH
under grant DMS-0443826
Department of Mathematics, University of Houston, 4800 Calhoun
Rd., Houston TX 77204-3476, USA([email protected]). Research
supported in part by the Texas Higher Education Board ARP
003652-0051-2006, NSF/NIH under grant DMS-0443826, UH GEAR-2007
grant, and by the NSF under grantDMS-0806941.
Co-Director, Cardiac Catheterization Laboratories Michael E.
DeBakey Veterans Medical Center, Assis-tant Professor of Medicine
Baylor College of Medicine, St Lukes Episcopal Hospital/Texas Heart
Institute
Department of Mathematics, University of Houston, Houston, 4800
Calhoun Rd., Houston TX 77204-3476, USA ([email protected]).
Graduate student support through the NSF/NIH under grant
DMS-0443826
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Figure 1: Deployment of a coronary stent.
Computational studies of mechanical properties of vascular
stents are a way to improvetheir design and performance. Even
though a lot of attention has been devoted in car-diovascular
literature to the use of endovascular prostheses over the past
10-15 years, theengineering and mathematical literature on the
computational studies of the mechanicalproperties of stents is not
nearly as rich.
Various issues in stent design and performance are important.
They range from thestudy of large deformations that a stent
undergoes during balloon expansion, for which non-linear elasticity
and plasticity need to be considered, all the way to the small
deformationsexhibited by an already expanded stent inserted into an
artery, for which linear elasticitymight be adequate. A range of
issues has been studied in [5, 15] and the references
therein,involving several different approaches. Most approaches use
commercial software packagesbased on the three-dimensional Finite
Element Method (FEM) structure approximations[4]. Finite Element
Method is nowadays a methodology well known and widely used inmany
engineering fields. However, it is worth remembering that the
reliability of the resultsclearly depends on the assumptions and
hypotheses adopted in the analysis. Often timesthe assumptions and
hypotheses are not clearly specified in a published study, making
theinterpretation of the results difficult [15].
In this manuscript we study how overall mechanical properties of
stents depend on thegeometry of a stent and on the mechanical
properties of alloys used in the stent production.This is the first
manuscript in which several stents and pressure loads are
considered andcompared for a stent in its expanded state. The
geometric parameters considered are thenumber of stent struts,
their geometric distribution, the thickness, width and length of
eachstrut, and the radius and total length of the stent. The
mechanical parameters of the stentstruts are the elastic (Youngs)
modulus and the Poissons ratio of the strut material.
What facilitated this research is a novel mathematical and
computational algorithmthat allows calculation of mechanical stent
properties 1000 times faster than the standardapproaches. The
mathematical model, recently developed by the authors in [17]
describesa stent as a mesh of one-dimensional, linearly elastic
curved rods [10, 11, 18]. The com-
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putational implementation of the new mathematical model was
performed using a FiniteElement Method. This novel algorithm is
simple and it requires much less computer mem-ory and computational
time than the classical three-dimensional FEM approaches using
theblack-box software such as ANSYS [4]. Our FEM algorithm can be
run on a standardlaptop configuration and can simulate the
mechanical response of a stent with any givengeometry within one
minute.
2 The Mathematical Model
A stent is a three-dimensional body which can be defined as a
union of three-dimensionalstruts made from a metalic alloy (see
Figure 2). The mechanical behavior of stents is usually
Figure 2: A stent with nC = 6 and nL = 9
described by the theory of elasto-plasticity [1, 13].
Elasto-plasticity describes deformation ofmaterials as a function
of the applied load. For relatively small loads a stent behaves as
anelastic body, which means that after the load is removed, the
stent will assume its originalconfiguration before the application
of the load. If the magnitude of the load is increased,a plastic
deformation takes place, which means that the stent deforms
permanently, i.e.,the deformation is irreversible. This corresponds
to, for example, a deformation that leadsto a fully expanded stent
from its undeformed, initial state using balloon expansion.
Theresponse of a material to an applied load is usually described
by a stress-strain relationship,see Figure 3. Stress is a measure
of the average amount of force F exerted per unit areaA: = F/A.
Strain e is the geometric measure of deformation representing the
relative
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Str
ess
N/m
2
Plastic deformation
Elastic deformation
Yield stress Fracture
Strain %
Figure 3: Stress-strain relationship
displacement between particles in the material body. For
example, extensional strain e ofa material line element or fiber
axially loaded is expressed as the change in length L perunit of
the original length L of the line element or fibers: e = L/L.
The stress (load) beyond which a material will undergo a plastic
deformation is calledthe Yield Stress. See Figure 3. Different
materials and different stent configurations havedifferent yield
stress.
If a relatively small load is applied to the structure in this
new configuration aftera plastic deformation has taken place, the
structure will again behave as an elastic body.This corresponds to,
for example, the cyclic radial and longitudinal deformations that a
fullyexpanded stent undergoes after the insertion into the lumen of
a native artery. This time-dependent elastic deformation is a
result of the application of the exterior forces exerted bythe
arterial tissue combined with the interior forces exerted by the
blood flow stress. It is thiselastic regime that we are interested
in this manuscript. Which geometric and mechanicalcharacteristics
will make a given stent stiffer under uniform compression? Which
geometricand mechanical characteristics will make a given stent
stiffer under bending loads? Whatdoes a stent deformation look like
if it is inserted in a vessel lumen that is not
uniformlycylindrical due to the plaque deposits? This manuscript
gives answers to these questions ina particularly simple
fashion.
First we recall some basic concepts from structure mechanics. As
mentioned earlier, theresponse of a material to an applied load is
described by a stress-strain relationship, seeFigure 3. Stress
corresponds to the loads applied, while deformation is measured by
straine. In this manuscript we are interested only in the elastic
deformations that result after the
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delivery of an endoluminal stent into the diseased region. In
this case, there is a function F ,which is in general nonlinear,
such that the stress can be expresses as a nonlinear functionof
strain
= F (e).
If we additionally assume that stress is small then this
relation can be approximated bya linear relation
= Ae,
with A = F (0) where F denotes the derivative of F with respect
to e. See Appendix formore details. This is known as the Hookes Law
which represents a constitutive law for alinearly elastic solid.
Strain e of an elastic body can be expressed in terms of the
gradient of the displacement u. If the displacement is small, the
strain e and displacement u satisfythe following linear
relationship:
e(u) =1
2
(
uT + u)
,
which is known as the infinitesimal strain tensor. Here uT
denotes the transpose of u.See Appendix for more details. This
leads us to the linear stress-displacement relationshipof the
following form:
=1
2A
(
uT + u)
. (1)
This relationship says, among other things, that if one doubles
the load, the displacementwill double as well. The coefficients in
the matrix A describe the elastic properties of agiven elastic
solid. For an isotropic material they can be given in terms of the
Youngsmodulus E, describing the stiffness of an elastic solid, and
the Poissons ratio , describingthe compressibility of an elastic
solid. Equations in the Appendix show how E and appear as
coefficients in the matrix A. We will use equation (1) to model the
mechanicalproperties of stent struts.
The geometric properties of a stent are described by the number
of vertices in thecircumferential and in the longitudinal
direction, the thickness, width and length of eachstent strut (we
assume that the cross-section of a strut is rectangular), the
overall refer-ence radius of the expanded stent, and the total
(reference) length of the expanded stent.Notice that the maximum
expanded radius of a stent and the total length of the
expandedstent depend on the number and length of stent struts in
circumferential and longitudinaldirection. See Table 1 for the list
of all the parameters.
In [17] we developed a new mathematical model that approximates
the stent frame asa mesh of one-dimensional curved rods (struts). A
Finite Element Method (FEM) wasdeveloped to calculate the solution
to the mathematical model [17]. This new approach canbe applied to
stents with arbitrary geometries. The one-dimensional approximation
ofstent struts as curved rods makes the FEM method simulation
incomparably simpler and
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E Youngs modulus of stent struts
Poissons ratio of stent struts
nL number of vertices in the axial (longitudinal) direction
nC number of vertices in the circumferential direction
t thickness of each stent strut
w width of each stent strut
l length of each stent strut
R reference radius of an expanded stent
L total (reference) length of an expanded stent
Table 1: Stent parameters.
faster than the standard approaches using a black-box software
such as ANSYS, whichapproximate stent struts as three-dimensional
bodies. We developed a code in C++ toimplement our approach. We
have been working with frames consisting of 50-300 vertices.The
time to solve the problem numerically varies from 0.3 to 5 seconds
on a server withone 3.00 GHz processor and 2GB of RAM. This is in
contrast with standard approachesusing 3D approximations of stent
struts that take several hours to a day for a simulationof one
stent configuration. In addition, often times the number of nodes
that it takesto approximate each 3D stent strut with suffucient
accuracy exceeds the computationalcapabilities (memory
requirements) of standard machines such as those described above.If
one is interested in a patient-specific calculations that are
performed in real time, thealgorithm we developed in [17] is the
one to be used.
3 Results
Through a series of examples we will show the response of
several different stent configu-rations to the pressure load
exerted on a stent in its expanded state. We will consider twotypes
of loading: uniform compression and loading causing bending of
stents.
Uniform compression: In all the examples below a uniformly
distributed force inradial direction will be applied to stents
causing compression. Radial displacement fromthe expanded
configuration will be measured. The compression force will
correspond tothe pressure load of 0.5 atmospheres. The force is
calculated by considering the 0.5 atmpressue load of a cylinder
(e.g., blood vessel) of length L acting on a stent of length L.This
pressure load is physiologically reasonable. Namely, we can use the
Law of Laplace toestimate exterior pressure loads to an inserted
stent. Recall that the Law of Laplace whichrelates the displacement
u of the arterial wall with the transmurral pressure p p0 reads
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[6, 19]:
p p0 =Eh
(1 2)R2u, (2)
where E is the Youngs modulus of the vessel wall, h is the
vessel wall thickness, R the vessel(reference) radius and the
Poisson ratio. For incompressible materials such as arterialwalls,
= 1/2. The Youngs modulus of a coronary artery is between 105Pa and
106Pa, seee.g., [2] and the references therein. For our calculation
let us take the intermediate value ofE = 5 105 Pa, and let us take
the reference coronary artery radius to be around 1.3mmwith the
vessel wall thickness h = 1mm. Stents are typically oversized by
10% of the nativevessel radius to provide reasonable fixation.
Thus, 10% displacement of a coronary arteryof radius 1.3mm is
0.13mm. This gives u = 0.13mm. By plugging these values into
formula(2) one gets p p0 5 10
4Pa which equals 0.5 atm. Thus, a pressure load of 0.5 atm
isnecessary to expand a coronary artery by 10% of its reference
radius.
As we shall see below, it will be useful to calculate the total
force, corresponding to thepressure of 0.5 atm, by which an artery
acts on a stent of length L and radius R. This forceis equal in
magnitude, but of opposite sign, to the total force that is
necessary to expandthe section of the vessel of length L and radius
R by 10%. Since pressure equals force perunit area, the
corresponding total force F that is needed to expand a section of
an arteryof length L and radius R by 10% is
F = (p p0) 2RL = 0.5atm 2RL,
where 2RL is the luminal area of the arterial section of length
L and radius R. We willuse this expression in the examples below to
calculate the total force by which an expandedartery acts onto a
stent with given geometric characteristics that depend on the
expandedstent radius R and length L.
Bending: In the examples below we will be calculating stent
deformation to forcescausing bending. These forces will be applied
pointwise to the center of a given stent (at 2-4 points in the
center) and to the end points (at 1 point near each end of a
stent). The forceat the end points is applied in the opposite
direction from the force applied to the centerof the stent. The
magnitude of the total applied force is calculated for each stents
to beequal to the force that a curved vessel, with the radius of
curvature Rc = 2.5cm, exerts ona straight stent that is inserted
into the curved vessel. Stents with higher bending rigiditywill
deform less, while stents with low bending rigidity will deform
more. Graphs showingthe curvature of deformed stents considered in
this manuscript are shown in Figures 14(right), 22 and 30. The
curvature of each stent is calculated as the reciprocal of the
radiusof curvature for each deformed stent 1/Rc.
The first example shows some basic properties of stent
deformation for a stent withuniform geometry (Palmaz-like). When a
uniformly distributed pressure is applied tothe surface of a
uniform stent, maximal radial and longitudinal displacement occur
at the
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Figure 4: A photograph of a stent with uniform geometry (Palmaz
stent by Cordis).
Figure 5: A stent with nC = 6 and nL = 6
end points of a stent. At the same time the maximum rotation of
the cross-section of eachstrut occurs in the middle of the strut
with maximum rotations in the middle of a strutoccurring for the
end struts.
Example 1 A Palmaz-like stainless steel stent (316L) such as the
one shown in Figure 4,with uniform geometry containing 8 vertices
in the circumferential direction and 7 verticesin the longitudinal
(axial) direction is considered. The length of each strut is 2mm.
Thestent has been expanded to the radius of 1.5mm into its
reference configuration. The stent issubject to the uniformly
distributed compression forces corresponding to the pressure load
of0.5 atmosphere. As a result, the stent deforms and exhibits both
the radial and longitudinaldisplacement. Our simulations show that
the maximum radial and longitudinal displacementfor this
Palmaz-like stent is assumed at the end-points of the stent. For
the exterior pres-sure load the maximum radial displacement is
equal to 0.166951mm (11% of the referenceconfiguration). See Figure
6. For the interior pressure load the maximum displacementis
0.121936 mm (8% of the reference configuration). See Figure 7.
Notice the well-known
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dogboning effect for an inflated stent shown in Figure 7. Figure
6 shows the magnitude ofthe radial and longitudinal displacement,
and the magnitude of rotation of the cross-section.Negative
displacement in Figure 6 corresponds to compression, while positive
displacementto expansion. Notice that the maximum radial and
longitudinal displacements occur at theend points of a stent, and
that the largest cross-section rotation occurs at the middle of
thestrut with the maximum cross-section rotation occuring for the
end-struts.
-0.000166951
-0.000121936 -0.002-0.0010.0000.001
0.002
0.000 0.005 0.010
-0.002
-0.001
0.000
0.001
0.002
0
0.0000477377 -0.002-0.0010.0000.001
0.002
0.000 0.005 0.010
-0.002
-0.001
0.000
0.001
0.002
0
0.0481664 -0.002-0.0010.0000.001
0.002
0.000 0.005 0.010
-0.002
-0.001
0.000
0.001
0.002
Figure 6: The magnitude of the radial (left) and longitudinal
(center) displacement, and the rotationof the cross-sections
(right) for a Palmaz-like stent in Example 1.
0.000121936
0.000166951 -0.002-0.0010.0000.001
0.002
0.000 0.005 0.010
-0.002
-0.001
0.000
0.001
0.002
Figure 7: Palmaz-like stent from Example 1 exposed to the
uniform pressure of 0.5 atm applied tothe interior surface of the
stent causing expansion. The figure shows the dogboning effect
(flaring ofthe proximal and distal ends of the stent).
The next example shows that fully expanded stents are stiffer
than those that are notfully expanded.
Example 2 All the stent parameters in this example as well as
the magnitude of the pres-sure load are the same as those for
Palmaz-like stent in Example 1. The only difference isthe reference
configuration to which the stent has been expanded: the stent is
now expandedto 0.66 of the the reference configuration in the
previous example, i.e., to 1mm. Our resultsshow that the maximal
radial displacement is now 0.175536mm which is considerably
largerthat that in the previous example, giving now a displacement
of 17.5% of the reference con-figuration. Thus, we conclude that
fully expanded stents are stiffer than those not expandedto the
maximal diameter. See Figure 8.
The following example shows that non-uniform pressure loads
cause higher stent defor-mations. This corresponds to, for example,
a situation when a stent is inserted in a vessel
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-0.000175536
-0.000128154
-0.0010.0000.001
0.000 0.005 0.010
-0.001
0.000
0.001
0
0.0000326297
-0.0010.0000.001
0.000 0.005 0.010
-0.001
0.000
0.001
Figure 8: The magnitude of the radial (left) and longitudinal
(right) displacement for a Palmaz-likestent in Example 2.
lumen with either high diameter gradients or a non-axially
symmetric geometry which canoccur due to, for example, plaque
deposits that have not been uniformly pushed againstthe wall of a
diseased artery during balloon angioplasty. We will show that in
this case theload that a stent can support is much smaller than in
the case when uniform pressure loadis applied.
-0.000308724
0.000057008
-0.002-0.0010.0000.0010.002
0.000 0.005 0.010
-0.002
-0.001
0.000
0.001
0.002
Figure 9: The figure on the left shows deformation of a
Palmaz-like stent under the load appliedto the middle ring of the
stent. The stent struts are colored based on the magnitude of the
radialdisplacement. They are superimposed over the reference
configuration shown in grey. The figure onthe right shows a
possible scenario in which higher loads in the middle of a stent
may appear dueto the plaque deposits.
Example 3 A Palmaz-like stent from Example 1 is considered with
the reference radius of1.5mm. Radial force is applied to the eight
points in the middle of the stent. The forceapplied at each of the
eight points is equal to 1/8 of the total force that corresponds
tothe uniform pressure load of 0.5 atm. Figure 9 shows that the
radial displacement of thestent with the maximum displacement of
0.308724mm (20.5 %) occuring in the middle ofthe stent. This is
considerable larger than the maximal displacement of the same
stentunder uniform compression (11% displacement), studied in
Example 1. Thus, stents thatare subject to a non-uniformly
distributed radial force deform more than those subject to
auniformly distributed radial force.
The next example shows the behavior of a non-uniform
Express-like stent (Express stentby Boston Scientific) which is
believed to have superior flexibility properties. We comparethe
response of an Express-like stent with a uniform stent
(Palmaz-like) from Example 1 tothe uniform compression forces and
to the forces causing bending.
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Figure 10: The figure on the left shows a close-up of the
geometry of a non-uniform stent (Express,Boston Scientific) while
the figure on the right shows our computationally generated
geometry of anExpress-like stent.
Example 4 A non-uniform Express-like stent (Express by Boston
Scientific), with the ge-ometric structure consisting of
alternating zig-zag rings with n1C = 6 and n
2
C = 9 verticesin the circumferential direction and nL = 30
vertices in the longitudinal direction, and withstraight
longitudinal struts connecting the rings, is considered. The
expanded stent radiuswas R = 1.5mm and expanded length was L =
17mm. A photograph of an expanded Expressstent with these geometric
characteristics is shown in Figure 11 (left). The stent is sub-ject
to the uniformly distributed compression force and to bending
forces. The mechanicalresponse of the Express-like stent is
compared with that of a uniform stent (Palmaz-like)with the
eqivalent geometric characteristics: nC = 7, nL = 30, and the same
fully expandedradius R = 1.5mm and length L = 17mm. The following
conclusions can be drawn.
-6
-3.8552 10
-6
-1.29332 10
-0.002-0.0010.0000.0010.002
0.00 0.01 0.02
-0.002
-0.001
0.000
0.001
0.002
Figure 11: Left: Express stent (Boston Scientific) with n1C
= 6, n2C
= 8 and nL = 30 expandedto R = 1.5mm; Right: Computationally
generated Express-like stent with the stent struts coloredwith
respect to the radial displacement under uniform compression.
Uniform compression: The stents were exposed to the uniformly
distributed com-pression force of the same magnitude. Figure 11
(right) and Figure 12 show the deformedstents with stent struts
colored by the magnitude of the radial displacement. The
followingobservations can be made:
Express-like stent is softer than the equivalent uniform stent
(Palmaz-like) (comparethe maximal displacement values shown in the
scale bar on the left of each figure).
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-7
-6.81503 10
-7
-4.83189 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 12: Uniform stent (Palmaz-like) under compression. Stent
struts are colored based on themagnitude of the radial displacment
(compression). Blue denotes large deformation, and red denotessmall
deformation.
Express-like stent is stiffest at the zig-zag rings consisting
of shorter stent struts(colored in red in Figure 11 (right)).
Longitudinal extension of Express-like stent under compression
is smaller than that ofthe uniform (Palmaz-like) stent. Figure 14
(left) shows a comparison in longitudinalextension for the two
stents.
-0.000448836
0.000414923
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-0.0000216032
0.000014107
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 13: Left: Express-like stent exposed to bending forces.
Right: Uniform (Palmaz-like) stentexposed to bending foces. Stent
struts are colored based on the magnitude of radial
displacement.Express-like stent bends more than an equivalent
uniform stent.
Bending: Express-like stent and uniform stent were exposed to
the same forcescausing bending. Figure 13 shows that the
Express-like stent is more flexible than isthe uniform stent. A
comparison in bending curvatures of the two stents is shown
inFigure 14 (right). A measure of curvature is calculated as a
reciprocal of the radiusof curvature for a middle curve of each
stent and reported in the graph in Figure 14(right).
Example 5 (Cypher-like stents). Several non-uniform Cypher-like
stents are consid-ered. Figure 15(left) shows Cypher stent by
Cordis, and Figure 15(right) shows the compu-tationally generated
geometry of a Cypher-like stent.
The stent geometry is that of the closed-cell FLEXSEGMENT. The
stent struts are madeof 316L stainless steel. The struts are
organized in rings (alternate, reflected rings), which
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Longitudinal extension under uniform compression
(percentage)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Stent E Stent UE
Curvature
0
2
4
6
8
10
12
14
16
18
20
Stent E Stent UE
Figure 14: Left: Longitudinal extension under compression;
Right: Magnitude of curvature underbending forces. The graphs show
a comparison between an Express-like stent and a uniform
Palmaz-like stent.
strut thickness 1403 m
Figure 15: Left: Cypher stent by Cordis; Right: Computationally
generated Cypher-like stent Cwith nC = 6, nL = 16, main strut
thickness 140m, connecting strut thickness 140/3m.
are connected with a sinusoidally-shaped struts. The number of
vertices in the circumferen-tial direction nC = 6, and in the
longitudinal direction nL = 16. The rings strut thicknessis 140 m.
In this example We consider the following Cypher-like stents:1.
Stent C, most closely resembles Cypher stent; has the
sinusoidally-shaped connecting
strut thickness 140 m strut thickness 1403m
Figure 16: Left: Cypher-like stent C-h with stent strut
thickness 140m; Right: Cypher-like stentC2 with main strut
thickness 140m, connecting strut thickness 140/3m (just as stent
C), but withopen cell desing (every other connecting strut
missing).
struts of width 140/3m.2. Stent C-h has the sinusoidally-shaped
connecting struts of width 140m.3. Stent C2 has the
sinusoidally-shaped connecting struts of width 140/3m, just as
theCypher stent, but the longitudinal connecting struts connect
every other vertex in the cir-cumpherential direction giving rise
to the total amount of connecting struts which is half thenumber of
the connecting struts in stents C and C2 (open-cell design).See
Figures 15 (right) and 16 for the corresponding geometries.
The performance of the three stents will be compared with the
uniform stent (Palmaz-
13
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-0.0000209791
-6
-8.07087 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-6
-6.90517 10
-6
-1.98935 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 17: Left: Cypher-like stent C with the main struts of
thickness t1 = 0.14mm and with thesinusoidal connecting struts of
thickness t2 = 0.14/3mm. Right: Cypher-like stent C-h with the
mainstruts of thickness t1 = 0.14mm and with sinusoidal connecting
struts of thickness t2 = 0.14mm.Stent struts are colored by radial
displacement under the application of a uniform radial force
causingcompression.
-0.0000245928
-6
-3.0573 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-6
-4.84834 10
-6
-3.22986 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 18: Left: Cypher-like stent C2 with open cell design and
main struts of thickness t1 =0.14mm and with the sinusoidal
connecting struts of thickness t2 = 0.14/3mm. Right: Uniformstent
UC (Palmaz-like) with the struts of thickness t1 = 0.14mm. Stent
struts are colored by radialdisplacement under the application of a
uniform radial force causing compression.
like), named Stent UC, which has equivalent geometric
characteristics (nC = 6, nL = 16with expanded radius R = 1.5mm).
The stents were exposed to the unformly distributedradial force and
to the force causing bending. Results showing the stents response
are shownnext.
Uniform compression: The stents were exposed to the uniformly
distributed com-pression force that corresponds to the pressure of
0.5 atm. Figure 17 and Figure 18 left showthe response of each
stent to the uniformly distributed radial force. This is compared
withthe response of a uniform stent UC (Palmaz-like) with the
equivalent geometric character-istics, shown in Figure 18 right.
Maximum radial displacement is colored in blue, minimumradial
displacement is colored in red.
Several conclusions can be drawn:
1. The response of the stents considered in this example from
hardest to softest is thefollowing: UC (hardest), C-h, C, C2
(softest).
2. Cypher-like stents C and C2 have similar response to
compression: the lowest defor-mation occurs at the main (zig-zag)
struts, while the largest deformation occurs at
14
-
the soft (sinusoidal) connecting struts.
3. Cypher-like stent C-h deforms more in the middle, and less at
the end struts. Thisis in contrast with the uniform stent UC
(Palmaz-like) which deformes more at theends than in the
middle.
4. Stent C and stent C2, which are the two stents with thinner
connecting struts (thick-ness of connecting struts = 140/3m) have
largest longitudinal extension. This isshown by the diagram in
Figure 19 where longitudinal extension is compared for thestents C,
C-h, C2 and UC.
Longitudinal extension under uniform compression
(percentage)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Stent C Stent C-h Stent C2 Stent UC
Figure 19: Comparison of longitudinal extension under uniform
radial compression for stents C,C-h, C2, and UC.
Bending: Cypher-like stents C, C-h, C2 and uniform stent UC
(Plamaz-type) wereexposed to bending forces. The response of the
four stents is presented in Figures 20 and21. The following
conclusions can be drawn:
-0.000412592
0.000401004
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-0.0000335185
0.0000246116
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 20: Cypher-like stents: Stent C (left) and Stent C-h
(right) deformed due to bending forces.The pictures are colored by
the magnitude of radial displacement.
1. Cypher-like stent C-h with thick connecting struts is
minimally flexible. This stentresists bending to the same order of
magnitude as does the uniform (Palmaz-like)stent.
2. The two stents with thin connecting struts, C and C2, are the
most flexible.
15
-
-0.00104389
0.00103808
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-0.0000290821
0.0000118703
-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
Figure 21: Cypher-like stents: Stent C2 (left) and Stent UC
(right) deformed due to bending forces.The picutures are colored by
the magnitude of radial displacement.
3. Stent C2 with half the number of connecting thin struts (an
open-cell design) is byfar the most flexible of the four stents
considered.
Curvature
0
5
10
15
20
25
30
35
40
45
50
Stent C Stent C-h Stent C2 Stent UC
Figure 22: Comparison between stents C, C-h, C2, and UC to
bending. The figure shows a measureof curvature (the reciprocal of
the radius of curvature) for the four stents. Stent C2 is by far
themost flexible of the four.
Figure 22 shows the curvature of the four stents considered in
this example. The cur-vature is calculated as the reciprocal of the
radius of curvature of the middle curve foreach stent. The table
shown in Figure 22 indicates that Cypher-like stent C2 with
opencell design is by far the most flexible of the four stents.
This stent has comparable radialstiffness to Cypher-like stent C.
Thus, Cypher-like stent C2 with open cell design appearsto hold
compression as well as does Cypher-like stent C, but it is more
flexible than any ofthe stents considered in this example.
Example 6 Xience-like stents. Several non-uniform Xience-like
stents are considered.Figure 23 (left) shows Xience stent by Abbott
and Figure 23 (right) shows a computationallygenerated geometry of
Xience-like stent. The stent geometry is that of Multi-Link
MiniVision. The stent struts are made of Cobalt Chromium (CoCr)
(L-605) with thickness0.08mm. The stent struts are organized in
rings connected with struts which are eitherstraight, as in Figure
24 (right), or the connecting struts have a small curved deviation
asshown in Figure 24 (left). We name these two Xience-like stents
as follows:1. Stent X, the stent shown in Figure 24 (left) and in
Figure 23 (right);2. Stent X-s, the stent with straight connecting
struts shown in Figure 24 (right).
16
-
Figure 23: Xience stent by Abbott (left); Computationally
generated Xience-like stent (right)showing half of the mesh.
Figure 24: Xience-like stent X (left) and Xience-like stent X-s
(right) with nC = 8 and nL = 24
Each of the stents has nC = 6 vertices in the circumferential
direction and nL = 24 verticesin the longitudinal direction. We
will be comparing the performance of these stents witha uniform
stent (Palmaz-like), which we call Stent UX. For the comparison
reasons, wewill be assuming that Stent UX is made of the same alloy
as stents X and X-s (CoCr,Youngs modulus E = 2.43 1011Pa), and has
an equivalent geometry consisting of nC = 6vertices in the
circumferential direction, nL = 24 in the longitudinal direction.
Stent UXwill be expanded to a diameter of 3mm (reference radius R =
1.5mm), just as the other twoXience-like stents. The stents were
exposed to the unformly distributed radial force and tothe force
causing bending. Results showing the stents response are shown
next.
Uniform Compression: The stents were exposed to the uniformly
distributed forcecausing compression. Figures 25 and 26 show the
response of stents X, X-s and UX. Thefigures are colored based on
the magnitude of radial displacement.
-0.0000259656
-6
-8.34041 10-0.0020.0000.002
0.00 0.01 0.02
-0.002
0.000
0.002
-0.000024184
-0.0000103393
-0.002-0.0010.0000.0010.002
0.00 0.01 0.02
-0.002
-0.001
0.000
0.001
0.002
Figure 25: Left: Xience-like stent X ; Right: Xience-like stent
X-s. Stent struts are colored by themagnitude of radial
displacement under uniformly distributed compression force.
The following conclusion can be drawn:
1. Both Xience-like stents X and X-s are slightly softer in the
middle and harder at theends. This is in contrast with the uniform
stent UX which is softer at the end pointsand harder in the
middle.
17
-
-6
-8.73161 10
-6
-5.82126 10
-0.002-0.0010.0000.0010.002
0.00 0.01 0.02
-0.002
-0.001
0.000
0.001
0.002
Figure 26: Palmaz-like uniform stent UX. Stent struts are
colored by the magnitude of radialdisplacement under uniformly
distributed compression force.
2. Xience-like stent X experiences the largest radial
displacement at the connectingstruts, shown in light blue in Figure
25 (left), and the smallest radial displacement atthe end struts,
shown in red in Figure 25 (left).
3. Xience-like stent X-s experiences the largest radial
displacement at the points whereconnecting struts meet the main
zig-zag struts at the interior angle, shown in lightblue in Figure
25 (right), and the smallest radial displacement at the end struts,
shownin red in Figure 25 (right).
4. The radial deformation for Stents X and X-s is of the same
order of magnitude.
5. Radial stiffness of the uniform stent UX is larger than that
for the two Xience-likestents X and X-s.
6. A digram shown in Figure 27 indicates that Xience-like stents
are stiffer in the lon-gitudinal direction than the uniform stent
(Palmaz-like) since longitudinal elongationafter uniform
compression for Stents X and X-s is smaller than that for Stent
UX.We attribute this behavior to the fact that the zig-zag rings
are in phase, namely,they are not alternating with the reflected
rings as is the case with the Cypher-likestents.
Bending: The response of the three stents to bending forces is
presented in Figures 28and 29. The following conclusions can be
drawn:
1. Xience-like stents X and X-s are considerably softer when
exposed to bending forcesthan the uniform stent UX. See diagram in
Figure 30 which shows the bending cur-vature for the three stents
considered in this example.
2. Stent X is softer to bending loads than Stent X-s.
18
-
Longitudinal extension under uniform compression
(percentage)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Stent UX Stent X Stent X-s
Figure 27: Longitudinal extension under uniform compression for
the Xience-like stents X, X-s andfor the Palmaz-like uniform stent
UX.
-0.00356515
0.00354223
-0.004-0.0020.0000.0020.004
0.00 0.01 0.02
-0.004
-0.002
0.000
0.002
0.004
-0.00270635
0.0026686
-0.004-0.0020.0000.0020.004
0.00 0.01 0.02
-0.004
-0.002
0.000
0.002
0.004
Figure 28: Left: Xience-like stent X; Right: Xcience-like stent
X-s. Stent struts are colored by themagnitude of radial
displacement under forces causing bending.
4 Appendix
We list here some basic formulas that characterize mechanics of
isotropic, linearly elasticmaterials. As before, we denote by u the
displacement vector in the three-dimensional space
u =
u1u2u3
.
Deformation is measured by strain e. For small deformations,
e(u) =1
2
(
u + (u)T)
,
which is known as the inifinitesimal strain tensor. Here u
denotes the gradient of u, whichis the matrix of partial
derivatives of u denoting the change (derivative) of each of the
threecoordinates of u in each of the three independent spatial
directions. Thus, such a matrix
19
-
-0.000199815
0.000133202
-0.002-0.0010.0000.0010.002
0.00 0.01 0.02
-0.002
-0.001
0.000
0.001
0.002
Figure 29: Palmaz-like stent UX. Stent struts are colored by the
magnitude of radial displacementunder forces causing bending.
Curvature
0
20
40
60
80
100
120
Stent UX Stent X Stent X-s
Figure 30: Comparison between stents X, X-s, and UX to bending.
The figure shows a measure ofcurvature (the reciprocal of the
radius of curvature) for the three stents. Stent X is the most
flexibleof the three.
has to have 9 entries. Indeed, the gradient of u is defined to
be
u =
u1x1
u1x2
u1x3
u2x1
u2x2
u2x3
u3x1
u3x2
u3x3
.
The transpose of u, denoted by (u)T , is the matrix which is
obtained from u byswitching its rows and columns to obtain
(u)T =
u1x1
u2x1
u3x1
u1x2
u2x2
u3x2
u1x3
u2x3
u3x3
.
Thus, inifinitesimal strain e(u) can be written in components as
a matrix of the followingform:
e(u) =1
2
(
u + (u)T)
=
u1x1
1
2
(
u1x2
+ u2x1
)
1
2
(
u1x3
+ u3x1
)
1
2
(
u2x1
+ u1x2
)
u2x2
1
2
(
u2x3
+ u3x2
)
1
2
(
u3x1
+ u1x3
)
1
2
(
u3x2
+ u2x3
)
u3x3
(3)
20
-
We denote by eij for i, j = 1, 2, 3 the entries of the strain
matrix (3). Thus,
e11 =u1x1
, e12 =1
2
(
u1x2
+u2x1
)
, e13 =1
2
(
u1x3
+u3x1
)
, ...
and so on. For more details about the relationship between
infinitesimal strain given by (3)and strain defined via e = L/L,
please see Section 2.1.5 in [1].
Keeping this notation in mind, we are now in a position to write
the consitutive law foran isotropic, linearly elastic solid. The
consitutive law describes how a solid deforms afteran application
of a force. Since strain measures the deformation and stress
corresponds tothe force, the constitutive law is given in terms of
the stress-strain relationship. For anisotropic, linearly elastic
solid the stress-strain relationship is given by = Ae which canbe
written in components as:
112233231312
=E
(1 + )(1 2)
1 0 0 0 1 0 0 0 1 0 0 00 0 0 12
20 0
0 0 0 0 122
00 0 0 0 0 12
2
e11e22e33
2e232e132e12
.
(4)Matrix A contains the parameters that define the behavior of
the underlying material. Foran isotropic, linarly elastic solid
only two parameters are necessary for a complete descriptionof the
solid behavior: the Youngs modulus of elasticity E and the Poisson
ratio .
The Youngs modulus of elasticity E is the slope of the
stress-strain curve in uniaxialtension. It has dimensions of stress
(N/m2 = Pa) and is usually large - for steel, E =210 109Pa. One can
think of E as the measure of the stiffness of a solid. The larger
thevalue of E, the stiffer the solid.
Poissons ration is the ratio of lateral to longitudinal strain
in uniaxial tensile stress.It is dimensionless and typically ranges
from 0.2 - 0.49, and is around 0.3 for most metals.One can think of
as the measure of the compressibility of the solid. If = 0.5, the
solidis incompressible - its volume remains constant no matter how
it is deformed.
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21
-
[3] P.G. Ciarlet, Mathematical Elasticity. Volume I:
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[17] J. Tambaca, M. Kosor, S. Canic, D. Paniagua. Mathematical
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