Design optimization of stent and its dilatation balloon using kriging surrogate model
 Hongxia Li^{1},
 Tao Liu^{1},
 Minjie Wang^{1},
 Danyang Zhao^{1},
 Aike Qiao^{2},
 Xue Wang^{3},
 Junfeng Gu^{4},
 Zheng Li^{4} and
 Bao Zhu^{5}Email author
DOI: 10.1186/s1293801603076
© The Author(s) 2017
Received: 8 September 2016
Accepted: 25 December 2016
Published: 11 January 2017
Abstract
Background
Although stents have great success of treating cardiovascular disease, it actually undermined by the instent restenosis and their longterm fatigue failure. The geometry of stent affects its service performance and ultimately affects its fatigue life. Besides, improper length of balloon leads to transient mechanical injury to the vessel wall and instent restenosis. Conventional optimization method of stent and its dilatation balloon by comparing several designs and choosing the best one as the optimal design cannot find the global optimal design in the design space. In this study, an adaptive optimization method based on Kriging surrogate model was proposed to optimize the structure of stent and the length of stent dilatation balloon so as to prolong stent service life and improve the performance of stent.
Methods
A finite element simulation based optimization method combing with Kriging surrogate model is proposed to optimize geometries of stent and length of stent dilatation balloon step by step. Kriging surrogate model coupled with design of experiment method is employed to construct the approximate functional relationship between optimization objectives and design variables. Modified rectangular grid is used to select initial training samples in the design space. Expected improvement function is used to balance the local and global searches to find the global optimal result. Finite element method is adopted to simulate the free expansion of balloonexpandable stent and the expansion of stent in stenotic artery. The wellknown Goodman diagram was used for the fatigue life prediction of stent, while dogboning effect was used for stent expansion performance measurement. As the real design cases, diamondshaped stent and svshaped stent were studied to demonstrate how the proposed method can be harnessed to design and refine stent fatigue life and expansion performance computationally.
Results
The fatigue life and expansion performance of both the diamondshaped stent and svshaped stent are designed and refined, respectively. (a) diamondshaped stent: The shortest distance from the data points to the failure line in the Goodman diagram was increased by 22.39%, which indicated a safer service performance of the optimal stent. The dogboning effect was almost completely eliminated, which implies more uniform expansion of stent along its length. Simultaneously, radial elastic recoil (RR) at the proximal and distal ends was reduced by 40.98 and 35% respectively and foreshortening (FS) was also decreased by 1.75%. (b) svshaped stent: The shortest distance from the data point to the failure line in the Goodman diagram was increased by 15.91%. The dogboning effect was also completely eliminated, RR at the proximal and distal ends was reduced by 82.70 and 97.13%, respectively, and the FS was decreased by 16.81%. Numerical results showed that the fatigue life of both stents was refined and the comprehensive expansion performance of them was improved.
Conclusions
This article presents an adaptive optimization method based on the Kriging surrogate model to optimize the structure of stents and the length of their dilatation balloon to prolong stents fatigue life and decreases the dogboning effect of stents during expansion process. Numerical results show that the adaptive optimization method based on Kriging surrogate model can effectively optimize the design of stents and the dilatation balloon. Further investigations containing more design goals and more effective multidisciplinary design optimization method are warranted.
Keywords
Stent Fatigue life Dogboning effect Finite element method Kriging surrogate model Design optimizationBackground
Cardiovascular and cerebrovascular diseases pose a great threat to human beings. Since 1990s, minimally invasive procedures have been introduced to deal with vascular diseases such as percutaneous transluminal coronary angioplasty (PTCA) with stent, which has been widely used in clinical treatment and become one of the most effective therapies to vascular diseases. Compared to drugs and traditional surgeries, this newly developed minimally invasive treatment enjoys a lot of advantages such as being effective and efficient, being relatively easy to perform, causing only minor trauma to patients, ensuring a low infection rate and leading to relatively low cost [1]. However, the development and clinical application of this technology has been impeded by many factors including longterm safety problem of stents, instent restenosis (ISR) due to mechanical injury caused by the stent to vascular wall and inflammatory response of vessel wall against struts. Obviously, stent longterm safety is related to its fatigue life inservice loading and nonuniform stent expansion will cause mechanical damage to the artery wall which has a significant impact on thrombosis and hyperplasia development [2].
As for percutaneous transluminal coronary angioplasty, stent is placed into the stenosis segment of vessel to provide a mechanical support and then the balloon and catheter are removed away. The stent remains in vessel to support vascular wall to ensure smooth blood flow. It also means that the stent would suffer pulsating load all the time in vessel. According to FDA [3], the service life of stent shall be no less than 10 years which means that it should withstand at least 380 million pulsation cycles. FDA also recommends several methods such as Goodman diagrams to test lifetime of stent. Currently, limited by minute structure of stent and vessel as well as the complexity of hemodynamics in stent, researchers often adopt experiment to study fatigue life of stent. However, it often takes 2–3 months to perform the accelerated life test to analyze stent’s fatigue life [4]. Against such a background, it is practically meaningful to explore how to use numerical simulation method to analyze stent’s fatigue life and then optimize geometries of stent based on numerical simulation so as to prolong the service life of stent.
The expanding of stent is not only affected by its geometries but also influenced by the balloon length. Mortier et al. [5] highlighted that the length of balloon is likely to be related to the expanding of stent’s distal ends. It means that under the influence of balloon length the stent may finally take up the shape of a spindle because the distal ends fail to expand enough or it may take the shape of a dogbone because the distal ends expand too much. Such a nonuniform stent expansion may cause mechanical injury to vessel wall and thus leading to instent restenosis. Therefore, it is practically meaningful to find out the proper length of balloon so as to ensure that the stent achieves uniform expansion along its length and to reduce mechanical injury to vessel wall.
Therefore, it is important in stenting to predict and optimize the fatigue life and expansion performance before manufacturing the stent and its dilatation balloon. However, it is hard for traditional methods such as experiment and clinical tests to find the optimal result in stent optimization since the functional relationship between design objectives and variables is nonlinear, complex and implicit. Currently, the common method to optimize stent is to compare several stent designs and choose the best one among them. For example, Migliavacca et al. [6], De Beule et al. [7] and Wang et al. [8] compared the expanding performance of the same type of stent with different geometrics and gave suggestions on the design of stent. This method is relatively easy to use but the optimal stent is actually the relatively better one among a couple of options rather than the real optimal result in the design space. What’s more, since the dilatation of balloonexpandable stent entails highly nonlinear problems such as large deformation, contact and elastoplasticity [9–12], it is difficult to perform optimization by adopting finite element method. As a matter of fact, comparing and analyzing a large amount of geometries of stent and its balloon are timeconsuming and costly.
Fortunately, surrogate model can solve such tricky problems. It is the use of a black box model to create an approximate functional relationship between design objectives and variables, thereby replacing complex engineering computation so as to greatly reduce computational cost. Timmins et al. [13] adopted Lagrange interpolating polynomials (LIPs) to optimize the stent; Shen et al. [14] improved stent’s resistance against compression and decreased internal pressure in expanding stent by employing the artificial neural networks (ANN). Li et al. [15, 16] proposed an adaptive optimization method based on Kriging surrogate model to optimize stent structure to eliminate the dogboning phenomenon during stent expansion process and optimize stent coating to prolong the effective period of drug release. Kriging surrogate model, a semiparameter interpolation technique, is more precise and flexible compared to Lagrange interpolating polynomials and ANN, and thus widely used in multidisciplinary design optimization (MDO).
In the present paper, both the expansion performance of stent and the fatigue life of stent inservice loading were studied. The stent geometries and its dilatation balloon were optimized step by step to improve stent fatigue life and expansion performance. The Kriging model was used to build the relationship between stent fatigue life and stent geometries and the relationship between stent dogboning ratio and length of balloon, respectively, thereby replacing the expensive FEM reanalysis of the fatigue life and dogboning ratio during the optimization. The optimization iterations are based on the approximate relationships for reducing the high computational cost. A ‘spacefiling’ sampling strategy conceptualized as a rectangular grid was used to generate the initial training sample points. In the adaptive optimization process, EI function was adopted to balance local and global searches and tends to find the global optimal design, even with a small sample size. In the present study, an adaptive optimization method was proposed for stent and its dilatation balloon optimization to prolong stent fatigue life and improve its expansion performance, which is hard and timeconsuming to find the optimal design either by experiment or clinic test. As the real design cases, two typical and representative vascular stents named diamondshaped stent and svshaped stent were studied to demonstrate how the proposed method can be harnessed to design and refine stent fatigue life and expansion performance computationally. The numerical results and design optimization method can provide a reference for the design of stent and its dilatation balloon.
Methods
Finite element analysis
 (1)
FEM simulation for the stent fatigue life prediction (FLP): Numerical simulation of the stent deployment derives from relevant literatures [20], which conducted in three steps: first, deployment of stent inside the stenotic artery by imposing a radial displacement to the balloon. Then, stent recoil upon balloon deflation by removing the deployment radial displacement to the balloon. Finally, cardiac cycle of pulsating load by applying a diastolic/systolic blood pressure to the artery.
 (2)FEM simulation for stent expansion performance: There are many finite element models (FEM) used to investigate expansion process of stent in the published studies [21–23]. Among them, four common finite element models of stent expansion were used for the design optimization based on Kriging surrogate model to reduce the dogboning effect of stent by Li et al. [24]. From the previous study, the finite element model of stentballoon expansion with a loading of a timevarying pressure applied to the inner surface of a cylindrical balloon is suitable for design optimization of stent expansion performance using surrogate model combining with FEM,as shown in Fig. 3.
Optimization problem
 1.
Optimization of stent fatigue life: Goodman Diagram is generally employed to predict fatigue life of stent. Data point above or closer to the failure line on the Goodman Diagram indicates that fatigue failure will occur at the zone where the corresponding node located. While, the data point under and far from the failure line indicates a safe service performance. Therefore, the optimization of stent to prolong its fatigue life can be defined as:
 2.
Optimization of stent expansion performance: For balloonexpandable coronary stent, nonuniform expansion along its length often occurs and leads to dogboning effect. It means that the distal ends of stent begin to expand before the proximal part and thus the stent expands into the shape of dogbone. The dogboning ratio can be defined as:
where, d _{ radial } ^{ distal } and d _{ radial } ^{ proximal } denote the distal and proximal radial displacements of stent respectively.
where d _{ radial } ^{ distal } (L) and d _{ radial } ^{ proximal } (L) denote the distal and proximal radial displacements of stent respectively at 32 ms. f(L) is the absolute value of dogboning ratio during the expanding of stent, L refers to the length of balloon, \({\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{L} }}\) and \({\bar{\mathbf{L}}}\) are the upper limit and lower limit for balloon length. In this study, the design space of L of diamondshaped stent and svshaped stent are 4.6 mm ≤ L ≤ 5.1 mm and 6 mm ≤ L ≤ 6.5 mm. When the balloon with the length of \({\bar{\mathbf{L}}}\), the dogboning ratio is larger than 0 and stent takes up the shape of dogbone, while when the balloon with the length of \({\bar{\mathbf{L}}}\), the dogboning ratio is smaller than 0 and stent takes up the shape of spindle.
Material properties
Structure  Balloon  Stent  Thrombus  Vessel 

Material  Rubber  Stainless steel 304  Calcified thrombus  Calcified vessel 
Element type  4node shell element  8node solid element  8node solid element  8node solid element 
Material model  Superelasticity  Bilinearity, isotropy  Linear, isotropy  Linear, isotropy 
Elastic modulus (GPa)  C _{10} = 0.10688E − 2, C _{01} = 0.0710918E − 2  193  0.00219  0.00175 
Poisson’s ratio  0.495  0.3  0.499  0.499 
Ultimate tensile strength (GPa)  0.58  
Yield stress(GPa)  0.315  
Endurance limit(GPa)  0.115 
Optimization algorithm
Results
Optimization results of stent fatigue life
 (1)
Design case of diamondshaped stent: MRG is adopted to select 18 initial training sample points in the design space and after 17 iterations the optimization terminates.
Stent design optimization results
Stents  Variables (mm)  σ _{ m } (MPa)  σ _{ m } (MPa)  D ^{ shortest } (reduced by)  

w _{1}  w _{2}  t _{1}  
Diamond stent  
Original  0.28  0.249  0.12  304.62  2.02  22.39% 
Optimal  0.2685  0.2128  0.1047  238.49  3.51 
Stents  Variables (mm)  σ _{ m } (MPa)  σ _{ m } (MPa)  D ^{ shortest } (reduced by)  

w _{1}  R  t _{2}  
Sv stent  
Original  0.1  0.09  0.1  328.05  3.09  15.91% 
Optimal  0.1111  0.0922  0.0838  283.48  4.78 
 (2)
Design case of svshaped stent: The optimization of svshaped stent geometries to improve its fatigue life stopped after 10 iterations with 18 initial training sample points generated by MRG.
The optimization results were listed in Table 2. After optimization, the width of the struts was increased by 11.1%, the thickness of the stent was decreased by 7.8% and the chamfer radius was reduced by 6.2%. An increase in the width and thickness of struts results in an increase of the radial stiffness of stent, which eventually results in an decrease of amplitude of the applied stress σ _{ a } and decrease of the mean of the applied stress σ _{ m }. Similarly,there is a optimal combination of the width and thickness of strut, as well as the chamfer radius of svshaped stent geometries corresponding to the optimal fatigue life of it.
Goodman diagrams of the original and the optimal stents were illustrated in Fig. 6b, in which σ _{ a } is a function of σ _{ m }. The distance from the data point to the failure line denotes the risk of fatigue fracture of stent in service. After the structure optimization of svshaped stent, the shortest distance from the data point to the failure line was increased by 15.91%, which means the optimal stent has lower risk of fatigue fracture in service compared with the original design.
Optimization results of stents expanding performance
 (1)
Design case of diamondshaped stent: MRG is employed to select 10 initial training sample points in the design space concerning the balloon’s length of diamondshaped stent. After 5 iterations, the optimization terminated. After optimization, dogboning effect almost disappears and stent expands uniformly along its length.
Performance of diamondshaped stent and svshaped stent deployed with original and optimal balloon
(Unit: mm)  

Stents  L  DR  RR  FS  
t = 32 ms  t = 42 ms  Proximal  Distal  
Diamond stent  
Original  5.1  0.0884  0.0868  0.0205  0.0220  0.2149 
Optimal  4.959  0  0.0016  0.0121  0.0143  0.1974 
Sv stent  
Original  6.5  0.0352  0.0363  0.0185  0.0174  0.0571 
Optimal  6.0911  0  0.0027  0.0032  0.0005  0.0475 
 (2)
Design case of svshaped stent: 5 initial training samples were generated by MRG in the design space of the length of balloon placed inside of svshaped stent. 4 iterations were needed to obtain the optimal design. After optimization, the dogboning effect was completely eliminated.
The expansion performance of svshaped stent dilated by the original balloon and optimal balloon is compared as shown in Table 3. The dogboning effect of svshaped stent was completely eliminated after optimization, which indicates a uniform expansion along stent longitudinal direction. Similarly, as the uniform expansion is a important performance of svshaped stent, radial recoil at proximal and distal ends, foreshortening, as well as the dogboning ratio of stent after deflation of balloon were respectively improved by 82.70, 97.13, 16.81 and 92.56%, although they were not considered in the optimization function. The comprehensive performance of svshaped stent was improved after the optimization.
Discussions
An optimization method based on Kriging surrogate model was adopted to optimize the stent and its expanding balloon to prolong the service life of stent and improve its expanding performance. Numerical result shows that the altered adaptive optimization method based on Kriging surrogate model can effectively optimize the stent and its expanding balloon. The blackbox optimization adopting Kriging surrogate model and finite element method can not only find out the optimal result in the design space but is cheaper and more efficient than experiment and clinic test.
Whilst it is more reliable of the data from experiment, which can give a suggestion for stent design, it is hard to find the global optimal design, especially there is coupling effect between design variables. The ISARSTEREO trials [30] provided a compelling clinical evidence for reduce restenosis with thinner struts. Nakatani et al. [31] reported that wider struts result in greater neointimal hyperplasia and poor stent coverage. Most of them are tend to assess one of the variables by fixing others. However, it is hard to study coupling variables, especially the Multiobjective design with coupling variables by clinical trials and experimental. Moreover, since stents are small scale devices subjected to longterm inservice loading of pulsation which is about 4 × 10^{8} cycles [32], direct experimental testing is difficult and timeconsuming to perform.
Therefore, computational approaches represent an assessment tool for stent expansion performance and fatigue lifetime prediction which also considered in several regulatory bodies [3, 33]. However, the functional relationship between design parameters and design objectives of stents is nonlinear, complex, and implicit. Moreover, the multiobjective design of stents involves a number of potentially conflicting performance criteria. Most of the existing framework just studied stents performance by numerical simulation, compared the performance of different types of stents or the same type of stent with different dimensions, and provided the suggestions of stent design. It is easy to study the mechanical properties and analyze the effective factors, but it is difficult to find the globally optimal design in design space.
Therefore, finite element analysis (FEA) based computationally measurable optimization was employed for design of stent geometry. Among them, surrogate modeling methods, which predominantly involves Kriging surrogate model, was constructed to represent the relationship between design goals and design variables. Harewood et al. [34] focused on radial stiffness of stent adopting finite element analysis of a single ring. Li et al [15] optimized stent dogboning using a threedimensional expansion model of balloon, stent, plaque and artery. Li et al [16] focused on pharmaceutically effective time of drug release in a stented artery. When considering multiple objectives, Pant et al [35] and Tammareddi et al [36] constructed and searched the Pareto fronts generated by treating each objective separately. Bressloff [4] recast the optimization as a constrained problem, wherein design improvement is sought in one objective while other objectives were considered as constraints. Among them, as a semiparametric approach, the Kriging model is much more flexible than approaches based on parametric behavioral models.
However, a desirable stent should possess a number of excellent mechanical properties, such as (1) low metal surface coverage; (2) good flexibility; (3) enough radial strength; (4) long fatigue life; (5) low rate of longitudinal shortening; (6) low radial recoil;(7) a small amount of foreshortening; (8) small dogboning effect; (9) good expansibility; (10) good biocompatibility and so on. Therefore, multiobjective optimization of stent design involves a large number of design goals. It is difficult to find the optimal design to improve the overall performance of stenting just by one of the common methods to solve multiobjective problem, such as combining the design objectives in a single weighted objectives function, searching the Pareto fronts, executing the suboptimizations step by step, and taking same design objectives as constraints. In future work, these methods can be used in combination under the premise of rational planning of design objectives and design variables of stent optimization systems to improve the performance of stenting. The design optimization objectives should include stent auxiliary expansion, instent blood flow, drug release, and biomechanical response of vascular tissue. Meanwhile, not only stent structure but also geometries of balloon, structure of polymer coating, and loading process of stent dilatation should be selected as the design variables.
In terms of optimization algorithm, accuracy of Kriging modeling relate to the distribution of simple points in the design space. Li et al [15] studied the sampling methods including Rectangle Grid (RG), Modified Rectangle Grid (MRG), Latin Hypercube Sampling (LHS), and Optimal Latin Hypercube Sampling (Optimal LHS), and pointed out that both MRG and Optimal LHS have better spacefilling properties comparing to RG and LHS. Obviously, increasing the number of sample points is helpful to improving the accuracy of surrogate model. But, analysis each design on samples costs a lot of computing. Consequently, it is a challenging and opportunistic work for further systematic optimization of stenting to study better sampling strategy with a smaller number of points and more efficient surrogate modeling. Furthermore, parallel computing can be used to improve computational efficiency and save computing time.
Although computerbased method has many advantages in stent design and represents an assessment tool for stent performance prediction, it cannot completely replace the experiment studies and clinical tests. It is meaningful and challenging to bridge the gap between the engineering design optimization method and medical communities.
This study suffers from several limits such as: (a) The chemical corrosion of blood to stent hasn’t been considered when evaluating stent’s fatigue life; (b) Since stent expansion process simulation driven by cylindrical balloon has the similar results as the expansion driven by folded balloon and the simulation with cylindrical balloon can significantly save time, balloon pleating/folding hasn’t been considered during the expanding of stent; (c) The optimized results haven’t been testified by experiment and it is only an exploration of the optimization of stent and its expanding balloon.
Conclusions
In this study, an altered adaptive optimization method based on Kriging surrogate model is proposed to optimize the stent and balloon so as to improve the fatigue life of stent as well as its expanding performance. Numerical result proves that this approach can effectively optimize the structure of stent and its expanding balloon. Multiobjective design optimization for stent and its auxiliary system shall be carried out so as to improve the overall performance of stent.
Abbreviations
 PTCA:

percutaneous transluminal coronary angioplasty
 ISR:

instent restenosis
 FDA:

food and drug administration
 FSI:

fluidstructure interaction
 LIPs:

lagrange interpolating polynomials
 ANN:

artificial neural networks
 MDO:

multidisciplinary design optimization
 FEM:

finite element method
 DOE:

design of experiment
 MRG:

modified rectangular grid
 EI:

expected improvement
 VSMC:

vascular smooth muscle cell
Declarations
Authors’ contributions
HL, BZ, MW, AQ and DZ were responsible for the design, data collection and overall investigation, and established the optimization model. TL was responsible for the numerical simulation. HL, JG, XW and ZL were responsible for the optimization method. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Dr. Wang and Dr. Liu in Dalian University of Technology and Dr. Gao in Dalian Jiaotong University for the useful discussion and support.
Competing interests
The authors declare that they have no competing interest.
Funding
This work is fully supported by the National Natural Science Foundation of China (Grant Nos.11502044, 81171107 and 51401045) and China Postdoctoral Science Foundation (Grant No. 2014M561222).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
 Fischman DL, Leon MB, Baim DS, Schatz RA, Savage MP, Penn I, Detre K, Veltri L, Ricci D, Nobuyoshi M, Cleman M, Heuser R, Almond D, Teirstein PS, Fish RD, Colombo A, Brinker J, Moses J, Shaknovich A, Hirshfeld J, Bailey S, Ellis S, Rake R, Goldberg S. A randomized comparison of coronarystent placement and balloon angioplasty in the treatment of coronary artery disease. N Eng J Med. 1994;331(8):496–501.View ArticleGoogle Scholar
 Chua SND, Donald BJM, Hashmi MSJ. Finite element simulation of stent and balloon interaction. 2003;143–144:591–7.Google Scholar
 Food U. Drug Administration, Nonclinical tests and recommended labeling for intravascular stents and associated delivery systems: guidance for industry and FDA staff. US Department of Health and Human Services. Food and Drug Administration, Center for Devices and Radiological Health January. 2005; 13.
 Bressloff NW. Multiobjective design of a biodegradable coronary artery stent. Stud Mechanobiol Tissue Eng Biomater. 2014;15:1–28.View ArticleGoogle Scholar
 Mortier P, De Beule M, Carlier SG, Van Impe R, Verheqqhe B, Verdonck P. Numerical study of the uniformity of balloonexpandable stent deployment. J Biomech Eng. 2008;130(2):021018.View ArticleGoogle Scholar
 Migliavacca F, Petrini L, Colombo M, Auricchio F, Pietrabissa R. Mechanical behavior of coronary stents investigated through the finite element method. J Biomech. 2002;35(6):803–11.View ArticleGoogle Scholar
 De Beule M, Van Impe R, Verhegghe B, Seqers P, Verdonck P. Finite element analysis and stent design: reduction of dogboning. Technol Health Care. 2006;14(4):233–41.Google Scholar
 Wang W, Liang D, Yang D, Qi M. Analysis of the transient expansion behavior and design optimization of coronary stents by finite element method. J Biomech. 2006;39(1):21–32.View ArticleGoogle Scholar
 Tang D, Yang C, Mondal S, Liu F, Canton G, Hatsukami TS, Yuan C. A negative correlation between human carotid atherosclerotic plaque progression and plaque wall stress: in vivo MRIbased 2D/3D FSI models. J Biomech. 2008;41:727–36.View ArticleGoogle Scholar
 Neugebauer M, Glöckler M, Goubergrits L, Kelm M, Kuehne T, Hennemuth A. Interactive virtual stent planning for the treatment of coarctation of the aorta. Int J CARS. 2016;11:133–44.View ArticleGoogle Scholar
 Zahedmanesh H, Kelly DJ, Lally C. Simulation of a balloon expandable stent in a realistic coronary artery determination of the optimum modelling strategy. J Biomech. 2010;43:2126–32.View ArticleGoogle Scholar
 Qiao A, Liu Y, Guo Z. Wall shear stresses in small and large twoway bypass grafts. Med Eng Phys. 2006;28(3):251–8.View ArticleGoogle Scholar
 Timmins LH, Moreno MR, Meyer CA, Criscione JC, Rachev A, Moore JE Jr. Stented artery biomechanics and device design optimization. Med Biol Eng Comput. 2007;45(5):505–13.View ArticleGoogle Scholar
 Shen X, Yi H, Ni Z. Multiobjective design optimization of coronary stent mechanical properties. Chin J Dial Artif Organs. 2012;23(4):14–9.Google Scholar
 Li H, Wang X. Design optimization of balloonexpandable coronary stent. Struct Multidiscip Optim. 2013;48(4):837–47.MathSciNetView ArticleGoogle Scholar
 Li H, Zhang Y, Zhu B, Wu J, Wang X. Drug release analysis and optimization for drugeluting stents. Sci World J. 2013;2013:827839.Google Scholar
 Chua SND, Donald BJM, Hashmi MSJ. Finiteelement simulation of slotted tube (stent) with the presence of plaque and artery by balloon expansion. J Mater Process Tech. 2004;155–156:1772–9.View ArticleGoogle Scholar
 Wu W, Wang WQ, Yang DZ, Qi M. Stent expansion in curved vessel and their interactions: a finite element analysis. J Biomech. 2007;40(11):2580–5.View ArticleGoogle Scholar
 Chua SND, Donald BJM, Hashmi MSJ. Effects of varying slotted tube (stent) geometry on its expansion behaviour using finite element method. J Mater Process Tech. 2004;155–156:1764–71.View ArticleGoogle Scholar
 Azaouzi A, Makradi A, Petit J, Belouettar S, Polit O. On the numerical investigation of cardiovascular balloonexpandable stent using finite element method. Comput Mater Sci. 2013;79:326–35.View ArticleGoogle Scholar
 Dumoulin C, Cochelin B. Mechanical behaviour modelling of balloonexpandable stents. J Biomech. 2000;33:1461–70.View ArticleGoogle Scholar
 Lally C, Dolan F, Prendergast PJ. Cardiovascular stent design and vessel stresses: a finite flement fnalysis. J Biomech. 2005;38:1574–81.View ArticleGoogle Scholar
 Tan LB, Webb DC, Kormi K, AlHassani STS. A method for investigating the mechanical properties of intracoronary stents using finite element numerical simulation. J Cardiol. 2001;78:51–67.Google Scholar
 Li H, Qiu T, Zhu B, Wu J, Wang X. Design optimization of coronary stent based in finite element models. Sci World J. 2013;2013:630243.Google Scholar
 Krige DG. A statistical approach to some basic mine valuation problems on the Witwatersrand. J Chem Metal Min Soc S Afr. 1951;52(6):119–39.Google Scholar
 Lophaven SN, Nielsen HB, Sondergaard J. ‘DACEa Matlab Kriging toolbox’; version 2. Informatics and mathematical modelling. Denmark: Technical University of Denmark; 2002.Google Scholar
 Jones DR, Schonlau M, Welch WJ. Efficient global optimization of expensive blackbox functions. J Global Optim. 1998;13:445–92.MathSciNetView ArticleMATHGoogle Scholar
 Williams CKI. Prediction with gaussian processes: from linear regression to linear prediction and beyond. Learning in graphical models, vol 89. New York: Springer; 1998. p. 599–621.Google Scholar
 Costa JP, Pronzato L, Thierry E (1999) A comparison between kriging and radial basis function networks for nonlinear prediction. Paper presented at: NSIP 1999. Proceedings of the IEEE–EURASIP Workshop on Nonlinear Signal and Image Processing, Antalya, Turkey.
 Kastrati A, Mehilli J, Dirschinger J, Dotzer F, Schühlen H, Neumann F, Fleckenstein M, Pfafferott C, Seyfarth M, Schömig A. Intracoronary stenting and angiographic results: strut thickness effect on restenosis outcome (ISAR–STEREO) trial. Circulation. 2001;103:2816–21.View ArticleGoogle Scholar
 Nakatani S, Nishino M, Taniike M, Makino N, Kato H, Egami Y, Shutta R, Tanouchi J, Yamada Y. Initial findings of impact of strut width on stent coverage and apposition of sirolimuseluting stents assessed by optical coherence tomography. Catheter Cardiovasc Interv. 2012;81(5):776–81.View ArticleGoogle Scholar
 Li J, Luo Q, Xie Z, Li Y, Zeng Y. Fatigue life analysis and experimental verification of coronary stent. Heart Vessel. 2010;25(4):333–7.View ArticleGoogle Scholar
 International Standard ISO 255392. Cardiovascular implantsendovascular devicesPart 2: Vascular stents. 2012.
 Harewood F, Thornton R, Sharp P. Step change in design: exploring sixty stent design variations overnight. www.altairproductdesign.com. 2011. Accessed 8 Jan 2017.
 Pant S, Limbert N, Curzen N, Bressloff N. Multiobjectives design optimisation of coronary stents. Biomaterials. 2011;32:7755–73.View ArticleGoogle Scholar
 Tammareddi S, Sun G, Li Q. Multiobjective robust optimization of coronary stents. Mater Des. 2016;90:682–92.Google Scholar