1 Introduction

Cardiovascular diseases (CVD) have been the principal cause of death in the United States for the past century, with total direct and indirect costs estimated to reach $403.1 billion USD in 2006 [23]. Atherosclerosis is one of the most serious forms of CVD and is characterized by an accumulation of atheromatous lesions in focal regions of the vasculature, leading to a reduction in blood flow. Treatment options include bypass grafting, balloon angioplasty and stent placement, with the latter two gaining popularity due to their less invasive nature. Vascular stents are essentially scaffolds designed to prop open diseased arteries. Beginning with the first human implantation of a balloon expandable stent in 1986 [20], technological advances in stent design, deliverability, and drug-coatings have expanded the application and success rate of the procedure. Presently, the primary concern with this procedure is restenosis, or the development of a new blockage in the stented artery.

Restenosis is a multi-stage process characterized by thrombus deposition, inflammation, proliferation, and arterial wall remodeling, leading to neointimal thickening [4]. In the early days of stent implantation, approximately one-third of patients suffered from restenosis within 6 months of stent deployment [24]. Subsequently, there has been a significant effort devoted to reducing the clinical failures due to restenosis. With advances in stent design, restenosis rates after stent implantation in coronary arteries fell to around 20%, with some variation depending on the design of the stent [12]. Further success in reducing restenosis rates has been achieved with the advancement of drug-eluting stent technology. This technology involves covering stents with thin biodegradable polymer coatings in which anti-proliferative drugs designed to minimize smooth muscle cell proliferation are embedded. While drug-eluting stents have decreased restenosis rates in coronary arteries to near 10% [11], incomplete endothelization of stent struts, lack of demonstrated success in peripheral arteries, and the lack of extensive long-term follow-up studies still limit this technology [9, 10, 16, 25]. Thus, there is still cause to improve stent design for the minimization of restenosis. Improving the biomechanical interaction of the stent and artery wall is one strategy for accomplishing this goal.

The placement of a stent inside an artery has profound implications on the stresses in the artery wall. The changes in artery wall stress are long-term, lasting far beyond the acute injuries associated with deployment issues such as dog-boning and twisting due to balloon folding [15]. Stent oversizing can lead to strut-imposed vascular injury after stent deployment, which has been shown to dictate the extent of intimal thickening in animal models [18]. The presence of a highly rigid stent can subject the artery to extremely high, non-physiologic stress concentrations [1], and large stress gradients at the artery–stent junction [2]. Arterial straightening is another concern after stent deployment. The rigid stent prevents the artery from undergoing cyclic flexure due to the pulsatile nature of the vascular system, and can lessen the degree to which vascular cells express potentially beneficial substances [26]. Moreover, re-endothelialization can also be hindered by stent-induced reductions in cyclic stretch [21, 22]. In addition, stent implantation affects the flow patterns within an artery as both computational and experimental studies have shown [3, 6, 13]. An important challenge to stent design optimization arises from the fact that design properties mediate these concerns and while a particular design characteristic may minimize the effects of one concern, it may exacerbate another. For example, a stent high in radial stiffness will achieve greater lumen gain, but will also induce higher stresses in the artery wall and perhaps limit the degree of cyclic stretch. Therefore, optimization of stent technology will require compromise, balance between competing interests, and an understanding of the impact that varying a given design characteristic may have on arterial wall reaction and restenosis formation.

The finite element method (FEM) is a widely used computational modeling technique that has been employed to investigate the mechanical implications of vascular stenting. Evidence suggests that restenosis rates vary with stent design [12]. Consequently, numerous studies have employed FEM to look at various stent geometries and determine the patterns of arterial wall stress that they induce. These studies have examined commercially available stents [8, 14] and generic designs [1]. Presently, only one study has attempted to evaluate the influence of specific stent design parameters on the stress field induced in a healthy artery wall [1]. The stents were defined by three geometric parameters: strut spacing (h), radius of curvature (ρ), and axial amplitude (f), illustrated in Fig. 2. This study found that stent designs characterized by large strut spacing, a non-zero radius of curvature, and large amplitude induced lower stresses on the vessel. Conversely, designs employing tight strut spacing, a zero radius of curvature, and low amplitude induced higher stresses on the artery. The stent designs that induced higher stresses also distended the artery to a greater degree, although the differences in final minimum diameter relative to the low stress stents in the healthy artery models were small (within 90 μm).

Herein we aim to extend the work of Bedoya et al. [1] and refine the stent strut configuration, via varying specific design parameters, to optimize stent performance. The results from the aforementioned investigation, demonstrated that differences in circumferential stress, radial displacement (indicative of lumen gain), and cyclic radial deflection (change in radial position from diastolic to systolic pressure) were strongly affected by stent strut geometry. Circumferential stress is the most likely culprit in disrupting artery wall structure, so this measure was chosen over others (e.g., von Mises) as a more direct representation of the deleterious effects of stents. While one would like to minimize circumferential stress to reduce trauma induced on the artery wall, from a clinical standpoint a stent must displace the artery to a diameter that will restore blood flow to distal tissues. It is also desirable to allow the artery wall to experience and respond to pulse pressure with as much natural deflection as possible. Herein, we propose an optimization algorithm that considers these three competing solid mechanical concerns after vascular stenting. The algorithm was designed to refine the strut configuration in the middle portion of the stent and identify a unique set of stent geometric parameters that maximizes lumen gain and cyclic radial deflection, while simultaneously minimizing wall stress. Using FEM, we evaluated the accuracy of our algorithm by testing the optimized designs in a non-diseased, three-dimensional, thick-walled, nonlinear model of a stented artery. Our objective in this investigation was to develop an algorithm to process the data obtained from existing computational models that allowed for the generation of optimized stent designs based on reconciling three competing and physiologically relevant solid mechanical factors (circumferential stress, lumen gain, and cyclic radial deflection) of a stented artery.

2 Methods

With the abundance of competing solid mechanical concerns following stent implantation and the need to reconcile these concerns via some form of systematic compromise in stent design considerations, we have developed an algorithm that refines the strut configuration in the middle portion of the stent to provide an optimal mechanical environment. The accuracy of the optimization method was confirmed by evaluating the biomechanical impact of the optimized geometries in computational models of stented arteries.

2.1 Optimization algorithm

Data obtained from previous computational models that analyzed distinct variations in stent geometrical parameters [1] were fit with an optimization function for which a minimum value was sought. Nodal values of tensile circumferential (hoop) stress at diastolic pressure, the pressure at which the stent induces the highest stresses, and radial displacement, at systolic and diastolic pressures, were taken from the inner surface of the middle quadrants of the stented model, defined as the region of interest herein. Behavior at the ends of the stent, in the region of compliance mismatch between the relatively rigid stent and the artery wall, was not considered. For each of the eight stent designs evaluated in Bedoya et al. [1] three values were extracted from the region of interest: percentage of the artery inner surface subjected to Class II critical hoop stresses (greater than 510 kPa) as defined in Bedoya et al. [1], minimum luminal gain (LG), i.e.,

$$ {\text{LG}} = \frac{{A_{{{\text{post}}}} - A_{{{\text{pre}}}} }} {{A_{{{\text{pre}}}} }}, $$
(1)

where A pre is the cross-sectional area of the lumen before stent deployment, A post is the cross-sectional area after stent deployment, and maximum cyclic radial deflection. The values were normalized from 0 to 1 for the eight FEM models. High radial displacement is preferred following stent implantation, therefore the model corresponding to maximum lumen gain was given a value of 0. Cyclic radial deflection should be maximized following stent deployment, thus the model corresponding to the least deviation from the cyclic radial deflection of a normal, healthy artery was given a value of 0. One could also imagine a definition of cyclic radial deflection based on cyclic circumferential stretch. In these models, the local circumferential stretch is nearly directly proportional to radial deflection, so the results would be approximately the same. Conversely, wall stress should deviate as little as possible from that which is present in a normal healthy artery wall to prevent any adverse biological response following stent deployment; therefore the model corresponding to the smallest deviation in circumferential stress from a healthy artery was given a value of 0. A minimal value for the optimization equation given by

$$\begin{aligned}{} G(h,\rho ,f)\,=&\,\alpha G_{\sigma } (h,\rho ,f)^{2} + \beta G_{{{\text{LG}}}} (h,\rho ,f)^{2} \\ & +(1 - \alpha - \beta )G_{{\text{D}}} (h,\rho ,f)^{2} ,\\ \end{aligned}$$
(2)

where h, ρ, and f are the design parameters—strut spacing, radius of curvature, and axial amplitude, respectively—defined in the aforementioned investigation, was sought. G σ(h, ρ, f) was defined as the normalized fraction of the intimal surface subjected to Class II critical stresses (>510 kPa). This threshold value provides an acceptable degree of sensitivity for our model parameters (diameter, wall thickness, artery wall properties, etc.). G LG(h, ρ, f) and G D(h, ρ, f) were defined as the normalized values of minimum luminal gain and maximum cyclic radial deflection within the region of interest, respectively, after stent deployment. The weighting coefficients, α and β, represent the relative importance of the circumferential wall stress, lumen gain, and cyclic radial deflection.

Multi-dimensional linear Lagrange interpolation was implemented to approximate the values of the specific normalized outcomes (G σ , G LG, G D) in the region bounded by the limits of the stent design parameters, defined as the design space herein, modeled in Bedoya et al. [1], with ranges (h 1.2–2.4, ρ 0–0.3, f 0.6–1.8). This parameter range is representative of the commercially available designs, which must incorporate design criteria such as expandability and fatigue behavior. The interpolation method is precisely analogous to that used by the 8-node 3D element [17]; yet rather than nodes in 3-space, our nodes are the eight simulations with the values of the design coordinates (h, ρ, f) as follows: (1.2, 0, 0.6), (1.2, 0.15, 0.6), (1.2, 0.3, 0.6), (1.2, 0.3, 1.2), (2.4, 0.3, 1.2), (2.4, 0, 1.8), (2.4, 0.15, 1.8), (2.4, 0.3, 1.8). The specific normalized outcome values were determined by

$$ \begin{aligned}{} G_{x}\,=&\,G^{1}_{x} (1 - h)(1 - \rho )(1 - f) + G^{2}_{x} (1 - h)(\rho )(1 - f) \\ & + G^{3}_{x} (h)(\rho )(1 - f) + G^{4}_{x} (h)(1 - \rho )(1 - f) \\ & + G^{5}_{x} (1 - h)(1 - \rho )(f) + G^{6}_{x} (1 - h)(\rho )(1 - f) \\ & + G^{7}_{x} (h)(\rho )(f) + G^{8}_{x} (h)(1 - \rho )(f), \\ \end{aligned} $$
(3)

where x corresponds to one of the specific normalized outcomes, G x n (N = 1...8) corresponds to the value of one of the specific normalized outcomes at each of the eight computational models, and h, ρ, and f are incremental positions within the design space. Similarly, the location of G x n is given by

$$ \begin{aligned}{} L_{x}\,=&\,L^{1}_{x} (1 - h)(1 - \rho )(1 - f) + L^{2}_{x} (1 - h)(\rho )(1 - f) \\ & + L^{3}_{x} (h)(\rho )(1 - f) + L^{4}_{x} (h)(1 - \rho )(1 - f) \\ & + L^{5}_{x} (1 - h)(1 - \rho )(f) + L^{6}_{x} (1 - h)(\rho )(1 - f) \\ & + L^{7}_{x} (h)(\rho )(\xi _{3} ) + L^{8}_{x} (h)(1 - \rho )(f), \\ \end{aligned} $$
(4)

where x corresponds to either the h, ρ, or f location and L x n (N = 1...8) corresponds to the location of one of the eight designs evaluated by Bedoya et al. [1]. Substitution of the values of the specific normalized outcomes at each specific incremental location within the design space into Eq. 2, along with predetermined weighting coefficients returns a value for the composite normalized outcome G(h, ρ, f). Figure 1 is a representation of the interpolated data for specified weighting coefficients (α = 0.25 and β = 0.55) within the design space. The geometric stent design parameters (h, ρ, f) corresponding to the minimum G(h, ρ, f) were determined as the optimum design parameters (i.e., the outputs of the optimization algorithm).

Fig. 1
figure 1

Interpolated data within the design space for specified weighing coefficients (α = 0.25 and β = 0.55). Note that this figure illustrates a case where the radius of curvature (ρ) was held constant and G(h, f) was plotted against the strut spacing (h) and axial amplitude (f). The optimization algorithm locates the global minimum varying all three design parameters

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2.2 Finite element modeling

The methods used to construct and model the stent geometries obtained from the optimization scheme have been previously reported in Bedoya et al. [1]. Briefly, a Matlab (MathWorks, Natick, MA, USA) subroutine was written to generate stent designs based on the parameters identified in the optimization algorithm. A separate program was created to automate the generation of three-dimensional stents in MSC.Patran (MSC Software, Santa Ana, CA, USA). Stent designs had an expanded outer radius of 1.2375 mm, which was 10% larger than the systolic radius measured at the inner surface and is consistent with manufacturers’ recommendations and common stenting practice [7]. Because of the repeating nature of the successive rings in this stent design and the fact that any effects from the edges of the stent have dissipated within the first ring segment, the mechanical environment in the middle section of the stent model (the space between the two inner rings) is equivalent to the mechanical environment of any two inner rings of a full length stent [1]. Thus, it is only necessary to model a portion of the full length stent, as seen in Fig. 2. Stent struts had a constant thickness of 100 μm and the stent material was modeled as 316L stainless steel (E = 200 GPa, ν = 0.3).

Fig. 2
figure 2

Design parameters. Generic stent showing the three parameters identified by the optimization algorithm: strut spacing (h), radius of curvature (ρ) and axial amplitude (f). The resulting stent geometries were then modeled to determine the accuracy of the algorithm

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The artery material properties utilized have been previously described [1]. Briefly, mechanical testing of porcine common carotid arteries was conducted using a modified version of the Computer Aided Vascular Experimentation (CAVE) device described by Humphrey et al. [9]. The acquired data were used to determine the constants for the constitutive model which took the form given by Eq. 5

$$ \begin{aligned}{} W\,=&\,C_{{10}} \cdot (I_{1} - 3) + C_{{01}} \cdot (I_{2} - 3) + C_{{11}} \cdot (I_{1} - 3) \cdot (I_{2} - 3) \\ & + C_{{20}} \cdot (I_{1} - 3)^{2} + C_{{30}} \cdot (I_{1} - 3)^{3} , \\ \end{aligned} $$
(5)

where C 10 = 25,466 Pa, C 01 = −11,577 Pa, C 11 = −506 Pa, C 20 = 1,703 Pa, and C 30 = 1,650 Pa. The artery model employed herein was characterized as a straight homogenous circular cylinder with isotropic nonlinear hyperelastic mechanical properties.

The finite element method was employed using MSC.Patran with MSC.Marc as the nonlinear solver (MSC Software, Santa Ana, CA, USA). The boundary conditions applied were the same as those utilized by Bedoya et al. [1]. Briefly, boundary conditions were applied in multiple steps. The quarter vessel model was stretched 59%, simulating the in vivo axial tethering that was measured during vessel harvesting. The vessel was then inflated by applying a pressure of 225 mm Hg, dilating the vessel to a radius greater than the outer radius of the 10% oversized stent. The stent was then translated in the axial direction from its original position outside of the vessel until the stent and vessel midpoints coincided. The pressure was then reduced to systolic and subsequently diastolic. Symmetry displacement boundary conditions were applied to both the stent and vessel to only allow in-plane deformations. An analytical contact boundary condition was applied restricting the future motion of contacting bodies to be only in the normal direction. Mesh independence was determined by convergence of radial displacement (<<1%) and circumferential wall stress (<4.5%) along the length of the stented model at systolic pressure.

3 Results

Results from our optimization algorithm suggest that unique sets of geometric parameters that correspond to global minima in G(h, ρ, f) can be identified over a wide range of optimization weighting coefficients. Post-optimization FEM results indicate that the algorithm accurately predicted G(h, ρ, f), for all stented artery models, in addition to values for percent area subjected to Class II hoop stresses (>510 kPa), minimal lumen gain, and maximum cyclic radial deflection. For example, the optimization algorithm predicted a value of G(h, ρ, f) = 0.201 for α = 0.33 and β = 0.33, compared to the actual value of 0.211 calculated for the stent with the resulting optimized geometry. Furthermore, the algorithm predicted values for percent area subjected to Class II hoop stresses, minimal lumen gain, and maximum cyclic radial deflection of 24%, 0.87, and 0.027 mm, respectively, whereas evaluation of finite element data returned values of 21%, 0.86, and 0.029 mm. In this case, the resulting optimal geometry had a strut spacing (h) of 2.28 mm, radius of curvature (ρ) of 0.3 mm, and axial amplitude (f) of 1.14 mm (Table 1).

Table 1 Values of the objective function G(h, ρ, f) and its components as predicted by the optimization algorithm, and as actually calculated using FEM data from the resulting geometries (shown at left)
Full size table

The ability of the optimization procedure to identify optimal geometries and their resulting mechanical measures was retained over a wide range of weighting coefficients (Table 1). Emphasizing circumferential stress (α = 0.75, β = 0.125) resulted in predicted and actual G(h, ρ, f) values of 0.091 and 0.084, respectively. Moreover, lumen gain and cyclic radial deflection differed by <1 and <<1%, respectively, between the predicted and FEM modeled values. Increasing the importance of lumen gain (α = 0.125, β = 0.75) exhibited similar findings, with only a 6% difference in G(h, ρ, f) values between the optimization algorithm and post-optimization FEM result, with values of 0.172 and 0.182, respectively. Values of G(h, ρ, f) deviated by <2% when emphasis was placed on cyclic radial deflection (α = 0.125, β = 0.125). The optimization algorithm predicted a G(h, ρ, f) value of 0.107, compared to the actual value of 0.109 calculated from the post-FEM results of the optimized geometry. Table 1 summarizes the aforementioned results from the optimization algorithm and finite element modeling for all stent designs investigated. Figure 3 shows the plots of the circumferential stress distribution for the four optimized geometries from the FE results.

Fig. 3
figure 3

Circumferential stress distribution in the four optimized designs

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Noticeable trends in stent geometric parameters were observed between stent designs when emphasis was placed on any of the three mechanical measures (circumferential stress, lumen gain, cyclic radial deflection). For example, increasing the weight on hoop stress (α = 0.75, β = 0.125) resulted in a trend where strut spacing (h) and axial amplitude (f) increased, while radius of curvature (ρ) decreased. In comparing the geometric parameters resulting from α = 0.125, β = 0.75 (emphasize lumen gain) with the optimized parameters resulting from the emphasis on circumferential stress, it is seen that h increases from 1.44 to 2.28 mm, ρ decreases from 0.192 to 0.135 mm, and f increases from 1.128 to 1.68 mm. Emphasizing cyclic deflection resulted in an identical shift in geometric parameters, i.e., h and f increased, ρ decreased. These adjustments make the stent more compliant, and thus more likely to deflect with the artery wall. Examination of the designs resulting from α = 0.33, β = 0.33 (equal weighting) and α = 0.125, β = 0.125 (emphasize cyclic radial deflection) shows that h increases from 2.28 to 2.4 mm, ρ decreases from 0.3 to 0.135 mm, and f increases from 1.14 to 1.8 mm. Furthermore, emphasizing lumen gain also resulted in an apparent shift in optimized geometric parameters. Consider the comparison between the designs that result from α = 0.75, β = 0.125 (emphasize hoop stress) and α = 0.125, β = 0.75 (emphasize lumen gain). In this comparison, the increase in weight of lumen gain results in a decrease in h from 2.28 to 1.44 mm, an increase in ρ from 0.135 to 0.192 mm, and a decrease in f from 1.68 to 1.128 mm. In general, when either a reduction in circumferential stress of an increase in cyclic radial deflection was emphasized, strut spacing (h) and axial amplitude (f) increased, while radius of curvature (ρ) decreased. Conversely, increasing the weight of lumen gain leads to a decrease in strut spacing (h) and axial amplitude (f), while radius of curvature (ρ) increased.

4 Discussion and conclusions

This study demonstrates that medical device design optimization is feasible under complex circumstances. For this specific case of stent design for improved biomechanics, the models are based on geometrically and materially non-linear modeling techniques and output parameters that depend in complex ways on the input (design) parameters. Still, the optimization techniques produced results that were confirmed by modeling the specific optimized geometries. The differences between the predictions of the optimization and the actual mechanical parameters shown in Table 1 are due to curve-fitting non-linear data, and not significant enough to detract from the success of the optimization scheme. Using these techniques, design guidelines for other medical devices may be achieved that incorporate considerations for long-term clinical failure modes. In this case, it was possible, for example, to arrive at stent designs that minimize artery wall stress, which is presumably important for development of restenosis [1, 2, 8]. Optimized geometries in other cases with more or less emphasis on minimum lumen gain or cyclic radial deflection were also identified, indicating that the optimization scheme is reliable over a wide range of desired outputs. While it is possible to cover the entire parameter space with a large number of computational models, this is not feasible in cases such as stented arteries, each of which takes at least several days to converge. The optimization scheme identifies optimal geometries in a more efficient manner.

The effects of the stent geometric parameters on the artery wall mechanical parameters (circumferential stress, lumen gain, and cyclic radial deflection) are due to a combination of factors. First, the design parameters affect the radial rigidity of the stent, independent of the artery, as determined by the degree of radial displacement. It is evident that the stiffer the stent, the greater the lumen gain and zones of high stress, and the lesser degree of cyclic flexure. Second, the design parameters determine the unique geometry of the “panels” of artery wall material between the struts, which in turn lead to variations in stress values and radial displacements. If stent struts were perfectly rigid, larger panels (high strut spacing) would induce higher stresses in the panels as compared to shorter stent panels. However, larger strut spacing also leads to a more compliant stent, which reduces the distending force on the artery wall, and thus the stress. The degree of cyclic radial deflection experienced by the artery wall is also affected by the additional effective stiffness of the panel provided by the distension due to the stent. A greater degree of artery wall stretching in either the axial or circumferential direction (which depends on strut configuration) will make the artery less likely to deform in response to changing pressure. While our optimization study only calculates the collective effects of all mechanisms and not the particular influences on each mechanism, these individual mechanisms provide insight into the sensitivity of the biomechanical environment after stent implantation. There are, of course, other considerations for stent design beyond the scope of this study, such as optimal radiopacity, fatigue behavior, foreshortening, trackability and deliverability. While these are important issues, stent designers are already familiar with the methods to improve them.

It is important to note the limitations associated with the models on which this optimization was based. As pointed out in Bedoya et al. [1], the artery wall is assumed to be a homogeneous, isotropic, hyperelastic cylinder. Features such as stenoses, curvature, etc., were not included. The inclusion of such complicating factors would greatly increase the number of variables required to perform the optimization scheme, such as the stenosis shape and mechanical properties, and the degree of curvature. More importantly, the inclusion of these features would not change the relative rankings of the stents, and thus the optimization results would be largely unchanged. In other words, a stent that minimizes stress in a healthy artery model would also minimize stress in a diseased or curved artery model. This study is also limited to the solid mechanical interaction of the stent and artery wall, and does not include the effects of flow-induced wall shear stress. The effects of these same design parameters on blood flow patterns has been previously investigated [6], wherein it was found that adjusting the parameters such that the struts are aligned with the flow direction provides the least flow disturbance.

There are additional stent design parameters that could be varied to achieve optimized designs, such as stent material (implying changes in material properties) and strut thickness. The influence of these parameters is not difficult to predict. Increasing either the material stiffness or strut thickness would result in higher artery wall stress, higher lumen gain and less cyclic radial deflection. Since these monotonic effects are likely to be preserved despite changes in the parameters varied here, they were kept constant. There is also the additional concern that increasing the number of variables in this kind of an optimization problem could hinder the ability to arrive at an optimal design. The degree and locations of plastic deformation during expansion of the stent (which certainly are related to material properties and strut thickness) may influence the stent/artery wall interaction, but these effects were not included here. It is likely that the variations in geometric parameters dominate over such effects, although this remains to be proven. The relative rankings of the stents for the purposes of optimization are not likely to be affected.

This study is also limited to the changes in artery wall stress induced by the stent, and does not address cellular responses to the additional harmful effects of acute vascular injuries associated with balloon expansion of the artery [5]. Histological studies implicate the disruption of the internal and external elastic laminae in an increased risk for restenosis [18]. The additional artery wall strain associated with balloon pressurization beyond that predicted in our models (also known as stent recoil) is typically just a few percent [19], but could result in additional laminae disruption. The non-uniform expansion of the stent by the balloon (dog-boning) and stent twisting induced by the expansion of folded balloons are examples of stent deployment phenomena that could induce additional acute injury in the artery wall. Axial strain components induced by dog-boning could add to laminae disruption, as could shearing due to stent twisting. The relative importance of these particular injuries on restenosis is unknown. Further investigation of the importance of these acute phenomena relative to the acute and long-term stress from the stent is warranted. In particular, the potential for additional injury at the ends of the stent, where stresses are already high, should be of interest. The present study only included the middle portions of the stented segments. However, it is important to note that the designs that minimized stress in the middle sections also minimized stress at the end segments due to the higher degree of stent compliance [1].

The variety of stent designs identified in this study (Table 1) provide general guidelines for stent design that can be used clinically to treat different disease states (i.e., lesion specific stenting). Atherosclerotic lesions are comprised of a variety of heterogeneously distributed constituents (e.g., soft lipid pool, fibrotic tissue, calcified plaque, etc.) each with unique mechanical properties. Consequently, the overall mechanical behavior of atherosclerotic lesions varies widely across the patient populations. It would be potentially detrimental to implant a stent that can cause unnecessary trauma to the artery wall or not provide sufficient scaffolding to diseased tissue; with due consideration of the effects of given design parameters these unfavorable outcomes can be avoided. For example, if confronted with a relatively stiff, perhaps calcified atherosclerotic plaque, the clinician can choose a design that optimizes minimum luminal gain (maximizing radial displacement), realizing that the minimum requirement of restoring blood flow should be met at the expense of potentially high stresses on the artery wall. Our results indicate that a smaller strut spacing and smaller amplitude would be more appropriate in these cases. On the other hand, if a target lesion is relatively rich in soft lipid content, a design that minimizes artery wall stress could be preferable. Our results indicate that stents with relatively high strut spacing would be more suitable. It should be noted that strut spacing is the limiting factor for axial spacing, thus there are limitations in parameter configurations to ensure that stent designs can be manufactured in the collapsed configuration. These design guidelines, while arrived at using a generic stent design, are generally applicable to any commercially available stent design. The optimization methodology demonstrated here could also be applied specifically to other design schemes that lend themselves to parameterization. Lastly, the importance of artery wall stress in relation to other phenomena that could be involved in the restenotic process should be determined through carefully designed in vivo studies, or perhaps in vitro studies with stents implanted in isolated, perfused arteries.