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(tambaca@math.hr). 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(canic@math.uh.edu). 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 (mate.kosor@gmail.com). 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

3

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

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0.002

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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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-0.0010.0000.001

0.000 0.005 0.010

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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.

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0.000057008

-0.002-0.0010.0000.0010.002

0.000 0.005 0.010

-0.002

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0.000

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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

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-6

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-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).

11

-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

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0.000014107

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0.00 0.01 0.02

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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

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0.045

Stent E Stent UE

Curvature

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2

4

6

8

10

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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-

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-6

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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.

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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:

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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.

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0.0000118703

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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.

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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)

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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.

References

[1] A. F. Bower. Applied Mechanics of Solids. http://solidmechanics.org/ 2008.

[2] S. Canic, C.J. Hartley, D. Rosenstrauch, J. Tambaca, G. Guidoboni, A. Mikelic. BloodFlow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics andExperimental Validation. Annals of Biomedical Engineering. 34 (2006), pp.575 - 592.

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[3] P.G. Ciarlet, Mathematical Elasticity. Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988.

[4] Computational software ANSYS. http://www.ansys.com/ Ansys Inc. 2008.

[5] C. Dumoulin and B. Cochelin. Mechanical behaviour modelling of balloon-expandablestents Journal of Biomechanics 33(11) (2000) 14611470.

[6] Y.C. Fung Biomechanics: Circulation. Second Edition. Springer. New York. 1993.

[7] Y.C. Fung Biomechanics: Mechanical Properties of Living Tissues. Second Edition.Springer 1993.

[8] G. Hausdorf. Mechanical and Biophysical Aspects of Stents,in Catheter Based De-vices for the Treatment of Non-coronary Cardiovascular Diseases in Adults and Chil-dren. Ed. P. Syamasundar Rao and Morton J. Kern. Lippincott Williams & Wilkins,Philadelphia 2003.

[9] V. Hoang, Stent Design and Engineer Coating Over Flow Removal Tool, Team #3(Vimage), 10/29/04.

[10] M. Jurak, J. Tambaca, Derivation and justification of a curved rod model, Mathemat-ical Models and Methods in Applied Sciences 9 (1999) 7, 9911014.

[11] M. Jurak, J. Tambaca, Linear curved rod model. General curve, Mathematical Modelsand Methods in Applied Sciences 11 (2001) 7, 12371252.

[12] KW Lau, A. Johan, U. Sigwart, JS Hung, A stent is not just a stent: stent constructionand design do matter in its clinical performance, Singapore Medical Journal 45 (2004)7, 305312.

[13] V. A. Lumbarda. Elastoplasticity Theory, CRC Press Mechanical Engineering Series2001.

[14] McClean Dougal R, MD, Eigler N, MD, Stent Design: Implications for Restenosis,MedReviews, LLC 2002.

[15] F. Migliavacca, L. Petrini, M. Colombo, F. Auricchio, R. Pietrabissa. Mechanicalbehavior of coronary stents investigated through the finite element method Jurnal ofBiomechanics 35 (2002) 803811.

[16] Allison C. Morton, D. Crossman, J. Gunn., The influence of physical stent parametersupon restenosis, Pathologie Biologie 52 (2004), (196-205).

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[17] J. Tambaca, M. Kosor, S. Canic, D. Paniagua. Mathematical Modeling of VascularStents. Submitted to SIAM J. Appl. Math. (2008).

[18] J. Tambaca, A numerical method for solving the curved rod model, ZAMM 86 (2006),210221.

[19] N. Westerhof, F. Bosman, C.D. Vries, and A. Noordergraaf. Analog studies of thehuman systemic arterial tree. J. of Biomech., 2 (1969), 121143.

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