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Journal of Biomechanics 38 (2005) 1574–1581
Cardiovascular stent design and vessel stresses: a finite
element analysis
C. Lallya, F. Dolanb, P.J. Prendergasta,
Centre for Bioengineering, Department of Mechanical Engineering, Trinity College, Dublin 2, Ireland
Medtronic Vascular, Parkmore Industrial Estate, Galway, Ireland
Accepted 30 July 2004
Intravascular stents of various designs are currently in use to restore patency in atherosclerotic coronary arteries and it has been
found that different stents have different in-stent restenosis rates. It has been hypothesized that the level of vascular injury caused to
a vessel by a stent determines the level of restenosis. Computational studies may be used to investigate the mechanical behaviour of
stents and to determine the biomechanical interaction between the stent and the artery in a stenting procedure. In this paper, we test
the hypothesis that two different stent designs will provoke different levels of stress within an atherosclerotic artery and hence cause
different levels of vascular injury. The stents analysed using the finite-element method were the S7 (Medtronic AVE) and the NIR
(Boston Scientific) stent designs. An analysis of the arterial wall stresses in the stented arteries indicates that the modular S7 stent
design causes lower stress to an atherosclerotic vessel with a localized stenotic lesion compared to the slotted tube NIR design. These
results correlate with observed clinical restenosis rates, which have found higher restenosis rates in the NIR compared with the S7
stent design. Therefore, the testing methodology outlined here is proposed as a pre-clinical testing tool, which could be used to
compare and contrast existing stent designs and to develop novel stent designs.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Coronary stent; Finite-element method; Restenosis; Vascular injury; Arterial-wall mechanics
1. Introduction
Many clinical studies have been carried out to
investigate the performance of cardiovascular stents
and it has been found that different stents have different
in-stent restenosis rates (McClean and Eigler, 2002;
Colombo et al., 2002) with in-stent restenosis found to
occur in as many as 20–50% of stented vessels (Grewe et
al., 2000). Numerous computational studies have been
carried out to investigate the expansion and mechanical
behaviour of different stent designs; both balloonexpanding stent designs (Dumoulin and Cochelin,
2000; Etave et al., 2001; Tan et al., 2001; Migliavacca
Corresponding author. Tel.: +353-1-608-1383; fax: +353-1-6795554.
E-mail address: [email protected] (P.J. Prendergast).
URL: http://www.biomechanics.ie.
0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
et al., 2002; Chua et al., 2003; Petrini et al., 2004) and
nitinol self-expanding stents (Whitcher, 1997). However,
very few analyses have been performed on the interaction between the stent and the artery, even though
vascular injury has been hypothesized as the stimulus
for the formation of occlusive intimal hyperplasia and
eventual restenosis (Edelman and Rogers, 1998). In fact,
vascular injury caused to a vessel by the implantation of
a stent, whether defined by the depth of penetration of
the stent wires or by an aggressiveness score, has
consistently been found to determine the degree of
restenosis (Schwartz and Holmes, 1994; Arakawa et al.,
1998; Hoffman et al., 1999) and restenosis is therefore
strongly linked to stent design (Rogers and Edelman,
1995; Kastrati et al., 2000).
In recent years, drug-eluting stents have emerged as
an alternative to bare metal stents and are coated in an
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
anti-proliferative drug to prevent in-stent restenosis.
Although, these stents have shown much promise it is
still generally accepted that the optimal approach to
coronary revascularization still lies in further developments in both stent design and drug-eluting stents
(Lepor et al., 2002). Rogers et al. (1999) used the finiteelement method to carry out a 2D analysis to investigate
balloon–artery interactions during stent placement and
showed that factors such as the balloon-inflation
pressures, stent–strut openings and balloon compliance
can influence the contact stresses between the balloon
and the arterial tissue within the stent–struts and hence
vessel injury. However, their study was limited to a 2D
analysis and furthermore used linear elastic material
properties to model the arterial tissue. A study of the
arterial tissue within repeating units of different
commercially available stent designs has been reported
by Prendergast et al. (2003); they reported tissue
prolapse and arterial wall stresses; however stent–artery
contact was not simulated in that analysis nor were the
models fully three dimensional (3D) in so far as only one
repeating unit of each stent design was modelled. A fully
3D model has been developed by Auricchio et al. (2001),
to look at an improvement to a stent design; however,
the arterial tissue stresses were not reported. The most
detailed numerical model to-date of a cardiovascular
intervention is that by Holzapfel et al. (2002). They have
developed a 3D model of a harvested cadaveric vessel
using the finite-element method. The stresses induced
within the vessel for a balloon angioplasty and a
Palmaz–Schatz stenting procedure were determined.
However, the localized stresses around the stent struts,
i.e., the stresses most likely to provoke in-stent restenosis, were not computed and consequently it would
not be possible to determine differences in the level of
vascular injury as a function of stent design from their
In this paper, we test the hypothesis that two different
stent designs will provoke different levels of stress in the
vascular wall. The stents analysed were the S7 (Medtronic AVE, Minnesota, USA) and the NIR (Boston
Scientific, Massachusetts, USA) stent designs. If this
hypothesis is confirmed, and if the two stents also have
significantly different restenosis rates, the study would
support the use of computer-based finite-element
analysis as a pre-clinical testing methodology to analyse
the biomechanical attributes of cardiovascular stents.
2. Materials and methods
A finite-element analysis requires the geometry and
material properties of the stent and blood vessel and
appropriate loading conditions to simulate the stenting
procedure, as described below. The finite-element soft-
ware used was MSC Marc/Mentat (Santa Ana, CA,
2.1. Model geometry
The 3D geometry of the repeatable units of the fully
expanded 3.5 mm diameter NIR and S7 stents was
determined using a coordinate measurement technique
reported earlier (Prendergast et al., 2003). The thickness
of the struts was 0.1 mm. Using the repeating unit
geometry of each stent design, solid models of the full
stents were generated. The solid models generated were
of the stents in a planar state, i.e., the geometry of the
stents if they were cut open longitudinally and flattened
out. The stent solid models was then meshed and
‘wrapped’ into a cylindrical shape by transferring the
nodal coordinates from a Cartesian coordinate system
into a cylindrical coordinate system. In this way FE
meshes were generated for each stent design as shown in
Fig. 1. The two stent designs differed greatly; the NIR is
a slotted tube laser-cut stent with 7 crowns whilst the S7
is a modular stent with a circular cross-section, 10
crowns and welded joints.
The atherosclerotic coronary artery was modelled as
an idealized vessel and represented by a cylinder with
outside diameter of 4 mm and had a localized crescentshaped axisymmetric stenosis with minimum lumen
diameter of 2 mm, see Fig. 2. The plaque corresponds
to a maximum stenosis of 56% of the proximal and
distal lumen cross-sectional area.
The adaptive meshing capability within Marc/Mentat
was used in the models for the arterial tissue to allow the
mesh to adapt and refine at contact areas. This enabled
the elements in the region of highest stress gradients to
subdivide and the finer mesh allowed for more accurate
stress and strain evaluation in critical regions.
2.2. Material properties
The two materials of the artery wall, arterial tissue
and stenotic plaque, were modelled using a 5-parameter
third-order Mooney–Rivlin hyperelastic constitutive
equation. This has been found to adequately describe
the non-linear stress-strain relationship of elastic arterial
tissue (Lally and Prendergast, 2003). The general
polynomial form of the strain energy density function
in terms of the strain invariants, given by Maurel et al.
(1998) for an isotropic hyperelastic material is
W ðI 1 ; I 2 ; I 3 Þ ¼
aijk ðI 1 3Þm ðI 2 3Þn ðI 3 3Þo ;
a000 ¼ 0;
where W is the strain-energy density function of the
hyperelastic material, I1, I2 and I3 are the strain
invariants and aijk are the hyperelastic constants. If the
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
Fig. 1. Finite-element meshes of the fully expanded stents and exploded views of one repeating unit; (a) NIR stent and (b) S7.
used to model the arterial tissue in this study is a specific
form of Eq. (1) whereby the strain-energy density
function is a third-order hyperelastic model suitable
for an incompressible isotropic material and has the
form given in Eq. (3) (Mooney, 1940).
W ¼ a10 ðI 1 3Þ þ a01 ðI 2 3Þ þ a20 ðI 1 3Þ2
þ a11 ðI 1 3ÞðI 2 3Þ þ a30 ðI 1 3Þ3 :
Fig. 2. Atherosclerotic coronary vessel geometry; artery outer radius,
R0=2 mm, Non-stenosed artery inner radius, RI=1.5 mm, stenosis
inner radius, RP=1 mm, stent radius, RS=1.75 mm.
principal stretches of the material are denoted l1, l2 and
l3, then the strain invariants for the material may be
defined as
I 1 ¼ l21 þ l22 þ l23 ;
I 2 ¼ l21 l22 þ l21 l23 þ l22 l23 ;
I 3 ¼ l21 l22 l23 :
Arterial tissue may be taken as incompressible based on
the results of previous studies (Carew et al.,1968;
Dobrin and Rovick, 1969). I3=1 for an incompressible
material. The specific hyperelastic-constitutive model
Using Eq (3) the stress components can be obtained by
differentiating the strain-energy density function, W,
with respect to the corresponding strain components
(Humphrey, 2002).
The arterial tissue material model was determined by
fitting to data from uniaxial and equibiaxial tension tests
of human femoral arterial tissue. The uniaxial and
equibiaxial experimental and hyperelastic material
model data are shown in Fig. 3. More details of the
determination of the experimental data and this
hyperelastic-material model are given in Prendergast et
al. (2003). The hyperelastic constitutive model used to
represent the plaque tissue in the vessel with a localized
stenotic lesion was determined by fitting to published
data for human calcified plaques (Loree et al., 1994), see
Fig. 4. Table 1 summarizes the constants used for the
hyperelastic constitutive equations to define the two
material models. The stent material was modelled as
linear elastic 316L stainless steel (E ¼ 200 GPa; n ¼ 0:3).
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
2.3. Boundary conditions
Stress (MPa)
Extension Ratio
Fig. 3. Human femoral arterial tissue properties (solid lines) and the
mechanical properties of the hyperelastic-material model for arterial
tissue (dashed lines) (Prendergast et al., 2003).
Fig. 4. Human calcified plaque properties (data points), adapted from
Loree et al. (1994) (with permission), and the mechanical properties of
the hyperelastic material model for plaque tissue (solid lines).
Table 1
Hyperelastic constants to describe the arterial tissue (Prendergast et al.,
2003) and stenotic plaque non-linear elastic behaviour
Arterial wall tissue (kPa)
Stenotic plaque tissue (kPa)
The parameters describe a Mooney–Rivlin model of the form given in
Eq. (1).
The loading and restraint conditions were applied to
the stent/artery construct in two steps. It involved use of
the feature that exists in Marc/Mentat (and many other
finite-element codes as well) that allows elements to be
activated and deactivated during an analysis (called
‘element birth and death’). In the first step, the stent
elements were deactivated and the vessel was expanded
to a diameter greater than that of the expanded stent by
applying a sufficient internal pressure to the vessel
(13 MPa). In the second step, the elements of the stent
were activated and the pressure on the inner lumen of
the artery was gradually reduced to a value of 13.3 kPa,
corresponding to mean blood pressure of 100 mmHg.
Due to the elastic nature of the hyperelastic arterial
tissue the vessel contracted around the stent with the
stent behaving as a scaffold within the vessel. Frictionless contact between the stent and the artery was
The contact algorithm implemented in marc/mentat
was the direct constraint method. In this procedure, the
motion of the bodies are tracked, and when contact
occurs, direct constraints are placed on the motion using
boundary conditions—both kinematic constraints on
transformed degrees of freedom and nodal forces.
Deformable–deformable contact was used to describe
the contact between the two contact bodies, i.e., the
stent and the artery. Both contact bodies were mathematically defined as analytical (NURB) surfaces. This
leads to a more accurate solution because the normal to
the contact surface is recalculated each iteration based
upon the current surface position.
A longitudinal stretch of 1.2 was applied to the artery
in an attempt to simulate the longitudinal tethering
observed on coronary arteries in vivo (Weizsacker et al.,
1983; Ogden and Schulze-Bauer, 2000). Axial restraints
were applied to one end of the vessel. During the
analysis the stent was restrained at one node in the
circumferential direction to prevent rigid body rotations. Because of cyclic symmetry it was not necessary to
model the entire stented vessel but rather only segments
of the stented artery had to be modelled. Using the
cyclic symmetry capability in Marc/Mentat, the nodes
on the two cyclically symmetrical faces were coupled or
tied. This allows out of plane motion (unlike planar
symmetry constraints) and was therefore valid for
representing a cyclically repeating segment of the stented
vessel. Planar symmetry constraints cannot accurately
model a cyclically repeating segment of a stenting
procedure since the arterial geometry and the stent
geometry in contact are not uniform in the axial
direction. The NIR stent was represented by a cyclically
repeating one-seventh segment whilst the S7 could only
be represented by half-cyclic symmetry due to the
position of the welds on the stent.
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
Fig. 5. The deformation of the artery, stenotic material and stent for (a) the NIR stent and (b) the S7 stent (one quarter of the artery and stenosis
removed for viewing).
The element and node number used to represent the
geometry of the two stenting simulations differed greatly
as a result of the cyclic symmetry constraint applied.
The one-seventh segment of the NIR stent was
adequately represented by 125 elements and 470 nodes.
For the half model of the S7, 5040 elements and 9021
nodes were necessary to model the circular cross-section
of the stent. Although the mesh density used for the
plaque and artery were the same in each model the
number of elements differed since only one-seventh of
the artery was represented in the NIR simulations whilst
half of the artery had to be defined in the simulation of
the S7 stenting procedure. This corresponded to 11,460
elements and 14,154 nodes for the NIR stenting
procedure simulation and 40,040 elements and 47,783
nodes for the S7.
3. Results
The adaptive meshing incorporated into the simulations resulted in an increase in the number of elements
and nodes defining the hyperelastic artery and plaque
materials in both stenting simulations. The final number
of elements that defined the artery was 22,072 (30,547
nodes) and 60,354 elements (80,293 nodes) in the NIR
and S7 simulations, respectively.
The finite-element models predicted that both stents
restored patency to the stenosed vessels, see Fig. 5a
and b. As shown in Fig. 5a and b, the tissue drapes
between the repeating units of the stents. Quantifying
this, we find that the maximum tissue prolapse between
the stent–struts to be 0.056 mm in the S7 stent compared
with 0.124 mm for the NIR stent in the stenotic vessel.
The contact area between the stent and the artery was
13.9 mm2 for the NIR stent and 11.3 mm2 the S7 stent.
The stresses induced within each stented vessel
differed with a larger volume of highly stressed vascular
wall predicted for the NIR stent, see Figs. 6 and 7. By
analysing the volumes of the material stressed at
different stress levels it was found that very high tensile
stresses (44 MPa) occurred in 21% of the artery stented
Fig. 6. Maximum principal stresses in atherosclerotic vessels stented
with NIR and S7 stent. For symmetry reasons only 1/7 of the NIR and
1/2 of the S7 needs to be modelled.
with the NIR stent as compared to only 4% with the S7,
see Fig. 7 for detailed results.
Radial retraction was observed in the two stents as a
result of the radial compressive forces exerted by the
artery on the stents. This is to be expected due to
bending of the stent struts and continued until
equilibrium was reached between the radial strength of
the stents and the radial compressive forces exerted by
the artery. The retraction was found to be greater in the
S7 compared with the NIR, see Fig. 8. The final lumen
diameter corresponded to a final stenosis of 14% for the
Volume of Arterial
Tissue (mm 3)
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
11 < 2
0< σ
2 < 1σ1 < 4
σ11 > 4
Maximum Principal Stress (σ1), M
Fig. 7. Maximum principal stress volumes for the arterial tissue in the
atherosclerotic vessel stented with an S7 and NIR stent.
Fig. 8. Stent radial retraction in the models of the stented vessels.
NIR and 23% for the S7 stent, relative to the initial
proximal and distal lumen area (i.e. the lumen area with
a radius of 1.5 mm). However, the lumen cross-section
was altered as a result of the axial stretch applied. As a
result, the final lumen area at the stenosis corresponded
to 8% increase in the lumen area, relative to the final
proximal and distal lumen cross-sectional area in the
NIR stenting simulation. It corresponded to 4%
reduction (stenosis) in the final lumen cross-sectional
area proximal and distal to the stenosis in the vessel
stented with the S7 stent.
4. Discussion and conclusions
For a vessel with a localized stenotic lesion, based on
the stresses induced in the vessel wall in these simulations, it is expected that the S7 would cause less vascular
injury than the NIR. The lower volume of tissue stressed
to high levels with the S7 stent compared to the same
idealized geometry stented with the NIR made from the
same material, can be attributed to the greater
conformability of the S7 stent to the inner lumen
geometry of the vessel. This includes the greater radial
retraction observed in the S7 at the site of the vessel
stenosis, see Fig. 8. By retracting, the stent maintains the
lower stresses on the vessel; however the S7 maintains
sufficient patency and superior scaffolding properties
(0.056 mm maximum tissue prolapse compared with
0.124 mm for the NIR stent). In fact this value of
maximum tissue prolapse is also very low compared
with other stents designs (BeStent 2, Medtronic AVE;
VELOCITY, Cordis; TETRA, Guidant), in which the
maximum tissue prolapse within a repeatable stent unit
of each of these designs has been determined in an
earlier study (Prendergast et al., 2003).
Stent retraction is measured by the radial displacement of the stent; however, the tissue prolapse is tissue
protrusion between the stent–struts and it defines the
scaffolding properties of the stent. These results
illustrate the coupling that exists between the stresses
imposed on the vessel wall and the radial retraction of
the stent. The radial retraction observed here is
exaggerated as a result of the method used to simulate
the stenting procedure, i.e., an expansion of the artery
past the stent and the subsequent collapse of the artery
around the stent. However, the simulations illustrate
that if the two stent designs analysed were deployed
using the same pressure in the same vessel that the NIR
stent would induce higher stresses in the vessel wall.
Clearly, an optimum stent design should retract
sufficiently to prevent overstressing the vessel wall but
still maintain patency of the vessel and plaque scaffolding. In fact, taking account of the differences in
magnitude of radial retraction and tissue prolapse for
the two stent designs, the minimum lumen diameter is
predicted to be 2.63 and 2.79 mm at the vessel stenosis
for the S7 and NIR stent, respectively. These correspond
to 8% increase in the stenosis lumen area for the NIR
stent and 4% reduction in the stenosis lumen area for
the S7, relative to the final proximal and distal lumen
area of the stented vessel. It is clear that the marginally
higher lumen diameter achieved by the NIR stent is at
the expense of large areas of considerably higher stress.
These stresses may provoke a greater response to injury
by the vessel wall and ultimately restenosis. This
demonstrates the high risks associated with choosing
an oversized rigid stent design as compared to the use of
the more flexible S7 stent.
The clinical angiographic restenosis rate reported for
the NIR stent is 19% at 6-month follow-up (Rutsch et
al., 2000), and 19.3% at 9-month follow-up (Baim et al.,
2001). The angiographic restenosis rate reported for the
S7 stent at 6-month follow-up is 10.1% (Medtronic
DISTANCE trial). The results of this study offer an
explanation for this lower restenosis rate when compared with the NIR stent; the S7 stent would cause less
vascular injury to the stented vessel and therefore would
be expected to have a lower restenosis rate than the NIR
stent whilst maintaining superior scaffolding properties
than the NIR stent. It should also be noted that the S7
achieves high patency (only 4% stenosis) and excellent
scaffolding whilst also having a lower stent/artery
contact area (11.3 mm2 as compared to 13.9 mm2 for
C. Lally et al. / Journal of Biomechanics 38 (2005) 1574–1581
the NIR). Maintaining the metal/tissue contact area low
also reduces the thrombotic response of the vessel to the
presence of a foreign material (Rogers and Edelman,
The main limitation of this study is that the arteries in
which the stents are implanted are an idealized
representation of stenosed coronary arteries. It is
hypothesized, however, that the tortuosity of a realistic
coronary artery model would only serve to further
demonstrate the lower stresses generated by a flexible
stent that can conform well to the vessel curvature
relative to a stiffer stent. A more tortuous arterial
geometry might also show that a rigid stent would cause
even further high-stress concentrations and hence
vascular injury at the ends of the stent, which could
embed in the artery wall due to the relative nonconformability of the stent.
No rupture or damage mechanism has been incorporated into our plaque model. The high stresses observed
in the calcified plaque are high enough that fracture
could occur in this relatively brittle material. An analysis
of the fracture process of the plaque is, however, beyond
the scope of the present study. This study is currently
being extended to develop a computational model of the
restenosis mechanism using the stresses induced within
the artery wall as a damage stimulus for the growth of
restenotic tissue (Lally et al., 2004).
Finally, the process of stent expansion is not modelled
in our simulations. This limits the study since it cannot
be used to analyse the shearing force that an expanding
stent could impose on an artery. In this respect, the
approach used in this paper is suitable for studies that
aim to look at the influence of stent placement on vessel
wall stresses and stent-induced vascular injury after stent
deployment since the stent geometry has been obtained
from a stent in its expanded state.
The methodology described in this paper is proposed
as a method to compare and analyse existing stent
designs and can now also be used to develop new stent
designs. Since this study already shows that stress
predictions can be correlated with in-stent restenosis, it
would be worthwhile to develop finite-element models of
realistic vessel geometries, which can be obtained from
Intravascular Ultrasound (IVUS) imaging. By determining full 3D patient specific geometries pre-stenting it
may be possible to choose a stent design based on the
patient’s specific coronary artery stenosis geometry and
thereby optimize the outcome of stenting procedures.
Project funded by an Applied Research Grant
awarded by Enterprise Ireland to Medtronic Vascular,
Galway, Ireland, and the Centre for Bioengineering,
Trinity College, Dublin, Ireland, under the Programme
for Research in Third Level Institutions, administered
by the HEA.
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