 Research
 Open Access
Multiobjective optimization of coronary stent using Kriging surrogate model
 Hongxia Li^{1},
 Junfeng Gu^{2},
 Minjie Wang^{1},
 Danyang Zhao^{1},
 Zheng Li^{2},
 Aike Qiao^{3} and
 Bao Zhu^{4}Email author
https://doi.org/10.1186/s1293801602689
© The Author(s) 2016
 Published: 28 December 2016
Abstract
Background
In stent design optimization, the functional relationship between design parameters and design goals is nonlinear, complex, and implicit and the multiobjective design of stents involves a number of potentially conflicting performance criteria. Therefore it is hard and timeconsuming to find the optimal design of stent either by experiment or clinic test. Fortunately, computational methods have been developed to the point whereby optimization and simulation tools can be used to systematically design devices in a realistic timescale. The aim of the present study is to propose an adaptive optimization method of stent design to improve its expansion performance.
Methods
Multiobjective optimization method based on Kriging surrogate model was proposed to decrease the dogboning effect and the radial elastic recoil of stents to improve stent expansion properties and thus reduce the risk of vascular instent restenosis injury. Integrating design of experiment methods and Kriging surrogate model were employed to construct the relationship between measures of stent dilation performance and geometric design parameters. Expected improvement, an infilling sampling criterion, was employed to balance local and global search with the aim of finding the global optimal design. A typical diamondshaped coronary stentballoon system was taken as an example to test the effectiveness of the optimization method. Finite element method was used to analyze the stent expansion of each design.
Results
27 iterations were needed to obtain the optimal solution. The absolute values of the dogboning ratio at 32 and 42 ms were reduced by 94.21 and 89.43%, respectively. The dogboning effect was almost eliminated after optimization. The average of elastic recoil was reduced by 15.17%.
Conclusion
This article presents FEM based multiobjective optimization method combining with the Kriging surrogate model to decrease both the dogboning effect and radial elastic recoil of stents. The numerical results prove that the proposed optimization method effectively decreased both the dogboning effect and radial elastic recoil of stent. Further investigations containing more design goals and more effective multidisciplinary design optimization method are warranted.
Keywords
 Stent
 Dogboning
 Radial elastic recoil
 Blackbox techniques
 Kriging surrogate model
 Design optimization
Background
As the leading cause of mortality, cardiovascular disease is often related to atherosclerosis which caused by the progressive formation of plaque and eventually results in an obstruction (stenosis) for blood flow through the artery [1–5]. Compared to traditional treatments such as drugs and surgery for coronary artery diseases (narrowing or blockage of the coronary arteries), percutaneous transluminal coronary stenting with the aid of coronary balloon angioplasty is more widely adopted in clinical practice thanks to its high initial success rate, minimal invasive nature, and improved longterm effectiveness. A stent is a wire metal meshed tube placed in the vessel during coronary balloon angioplasty to offer radial strength and to overcome the acute elastic recoil. The stent is put over a balloon catheter and moved into stenosis segment. Then, it expands as the balloon is inflated to open the blocked vessel. After the balloon and catheter are removed, the stent remains in the vessel to act as a scaffold to help prevent arteries from becoming narrowed or blocked again. Nowadays, intravascular stents are routinely and successfully used in medical treatment, but it still needs to be improved. For example, instent restenosis remains the main obstacle for the development of stent. It is known that instent restenosis is caused by artery injury due to stent expansion and vascular inflammation to the stent struts. Therefore, scholarly efforts to improve stent expansion performance and reduce the injury of blood vessel caused by stent implantations in stent design optimization are of great importance.
A desirable stent should possess a number of excellent mechanical properties, including smaller dogboning ratio and smaller radial elastic recoil. The dogboning phenomenon caused by nonuniform balloonstent expansion has a significant impact on the development of thrombus and intimal hyperplasia [6, 7]. A larger dogboning ratio indicates a more serious warpage at the ending struts, which will cause mechanical damage to the vessel wall and results in instent restenosis [8–10]. Additionally, the radial elastic recoil due to elastic deformation of stent has a significant impact on the mechanical support of stent. It is believed that the stent design may affect stent expansion performance such as the dogboning ratio and radial elastic recoil. Thus, it is important to predict the dogboning effect and radial elastic recoil and to optimize the design before manufacturing the stent.
Computational simulation (e.g., finite element analysis (FEA)) can be a very useful tool to study the stent expansion [11–14]. Dumoulin and Cochelin [11] evaluated and characterized the mechanical properties and behaviors of a balloon expandable stent. Etave et al. [15] compared the mechanical performance of two types of stents. In terms of stent design, Migliavacca et al. [16, 17] and Beule et al. [1] assessed the mechanical properties and behavior of balloon expandable stents to determine how the FEA method could be used to optimize stent designs. It is easy to study the mechanical properties and analyze the effective factors, but it is difficult to find the globally optimal solution since the functional relationship between the geometrical parameters and dilation performance of stent is complex, nonlinear and implicit. For the traditional stent design method, a finite number of different designs are assessed and compared with each other to find an optimal design. This method is often used in industry when commercial demands restrict the time spent on developing a better product. However a limited number of discrete points fail to represent all the information in the design space and thus it is very difficult to find the optimal design through the traditional methods. For traditional methods, the designer has to formulate how many different designs can be tested in the available time and their work is also affected by the design methodology, problem fidelity, parameterization and measures of performance. The focus of the study is on the computational design of stent. However, since the stent is minute and boundary conditions for the expanding of stent in the vessel are complicated, it is relatively hard to apply finite element modelling in the optimization of stent. Actually not only the dependences mentioned above but also the availability and power of computers and software shall be considered when the traditional design methods are adopted.
Consequently, some approximation models are widely used in engineering to construct simplified approximations for analysis codes, especially when the analysis is hard and timeconsuming. Incorporating a shape optimization algorithm based on a proven convergence theory into the design process allows engineers to systematically identify the most favourable designs. An adaptive optimization method based on Kriging surrogate model was already proposed to eliminate the dogboning phenomenon by Li et al. [18]. A derivativefree optimization algorithm coupling computational fluid dynamics (CFD) was used for stent design by Gundert et al. [19]. In this paper, Kriging models were used as alternatives to the method of traditional secondorder polynomial response surfaces for constructing global approximations in stent optimization. As a semiparametric approach, the Kriging model [20, 21] is much more flexible than approaches based on parametric behavioral models.
Taking the consideration above in mind, we adopted the Kriging model to create an approximate functional relationship between the design objective and design parameters to replace the expensive reanalysis of the stent dogboning ratio and radial elastic recoil. The optimization iterations are based on the approximate relationship between the design objective and design parameters to reduce the high computational cost. An adaptive optimization method based on the Kriging surrogate model combing with modified rectangular grid (MRG) approach was proposed to minimize the radial elastic recoil and the dogboning effect of stent during the expansion process. Expected improvement (EI) function is employed in the adaptive process [18], which can balance local and global searches and then find the global optimal design even with a small sample size. The FEA solver of ANSYS was used to analyze the measurements of stent expansion performance.
Methods
Finite element model
Bilinear elastic–plastic and hyperelastic (MooneyRivlin) materials were assumed for slotted tube stents and balloon. Data of the material properties used in this study was from previous studies [23, 22].
Optimization problem
Optimization method
A finite element based multiobjective optimization method combining with Kriging surrogate model [18] was constructed for the stent optimization to improve stent expansion performance. Kriging was used to build the approximate functional relationship between the design objective and design variables. A modified rectangular grid (MRG) approach was adopted to generate the initial sample for Kriging. EI function was employed to balance the local and global search to find the global optimal design.
Kriging approximate method
Predictor
Therefore, the function value \(\hat{y}\left( {{\mathbf{x}}^{ * } } \right)\) at every new point x* can be predicted by using Eq. (9).
Sampling strategy
 1.Narrow the range of variables as$$l_{j} \le x_{j} \le \hat{u}_{j} ,\,\,\hat{u}_{j} = u_{j}  \frac{1}{2}\frac{{u_{j}  l_{j} }}{{q_{j}  1}},\quad j = 1,\, \ldots ,\,m$$(11)
 2.Perform rectangular grid (RG) sampling [25] in the narrowed space as$$x_{j}^{i} = l_{j} + k_{j}^{(i)} \frac{{\hat{u}_{j}  l_{j} }}{{q_{j}  1}},\quad k_{j} = 0,\,\,1,\, \ldots ,\,q_{j}  1 \quad i = 1,2,\, \ldots, \, \prod\limits_{j = 1}^{m} {q_{j} }$$(12)
 3.Add a stochastic movement of each sample point in each dimension as$$\frac{{\alpha_{ij} }}{2}\frac{{u_{j}  l_{j} }}{{q_{j}  1}} \quad j = 1,\,2,\, \ldots ,\,m$$(13)
where \({\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha_{ij} \in [ 0, 1]\), which is assumed to be normally distributed.
Expected improvement (EI)
The first term of Eq. (19) refers to the difference between the current minimum response value Y _{min} and the prediction \(\hat{y}\left( {\mathbf{x}} \right)\) at x. Hence, it is large when \(\hat{y}\left( {\mathbf{x}} \right)\) is small. The second term is the product of the root mean squared error (RMSE) σ(x) and the normal density function \(\phi (u)\). \(\phi (u)\) is large when σ(x) is large and \(\hat{y}\left( {\mathbf{x}} \right)\) is closed to Y _{min}. Thus, the expected improvement will be larger when the predicted value is smaller than Y _{min} and/or there is a lot of uncertainty associated with the prediction.
The convergence criterion
Implementation of optimization procedure
 Step 1:

Get a set of n _{ s } samples using MRG.
 Step 2:

Run ANSYS program with Binarysearch method to dilate stent at sample point i, i = 1,…, n _{ s } to nominal diameter and obtain \(d_{\text{radial}}^{\text{distal}}\) and \(d_{\text{radial}}^{\text{proximal}}\), \(R_{{{\text{POI(No}}.i )}}^{\text{loading}}\) and \(R_{{{\text{POI(No}}.i )}}^{\text{unloading}}\), i = 1, …, 4. Calculate f(x _{ i }) in problem (3) at the sample point i, i = 1, …, n _{ s. }
 Step 3:

Find the sample point with the minimum f(x _{ i }) as the initial point for the optimization.
 Step 4:

Get an approximate functional relationship between the design objective f(x) and design variables using Kriging surrogate model based on the trial samples. Calculate EI(x) based on f(x).
 Step 5:

Select optimization algorithm to implement the optimization design based on max EI and obtain the modified design x _{ k. }
 Setp 6:

Get the predictive value \(\hat{y}_{k}\) of x _{ k } based on Kriging and compute f(x _{ k }) by ANSYS program.
 Step 7:

The optimization iteration was stopped when a suitable level of convergence is reached and/or the available time for the optimization process is exhausted. The process of constructing and maximizing EI does not stop until the Euclidean norm between real value f(x _{ k }) and predictive value \(\hat{y}_{k}\) fall below a given tolerance, the Euclidean norm between current and previous iterates falls below a given tolerance, and the criterion stipulated in “The convergence criterion” section is reached. If not, then add the modified design into the set of samples and go to step 3.
Results
Initial training sample points selected by MRG
Samples  W _{1}  W _{2}  W _{3}  W _{4}  W _{5}  T  L  P  DR  RER  

(mm)  (mm)  (mm)  (mm)  (mm)  (mm)  (mm)  (MPa)  32 ms  42 ms  
1  0.309  0.2587  0.2703  0.2355  0.2935  0.1245  4.6935  2.0988  0.2495  0.2518  0.0177 
2  0.3168  0.3013  0.3323  0.2819  0.2065  0.1323  5.6613  1.8081  0.0002  0.0002  0.0205 
3  0.2781  0.3129  0.2316  0.2239  0.2258  0.1052  5.2742  1.7864  0.1537  0.1568  0.0205 
4  0.3129  0.2858  0.2277  0.2781  0.2806  0.1284  5.9839  1.7864  0.2010  0.2037  0.0162 
5  0.3245  0.3323  0.2897  0.2935  0.2226  0.1258  5.8548  1.8289  0.0624  0.0634  0.0186 
6  0.3052  0.2548  0.309  0.3245  0.2871  0.1168  6.3065  1.9537  0.3058  0.3107  0.0152 
7  0.2239  0.3206  0.2355  0.251  0.271  0.1206  6.2419  1.8800  0.1500  0.1529  0.0158 
8  0.2703  0.2819  0.3129  0.3168  0.2839  0.1026  5.0161  1.8908  0.1663  0.1681  0.0172 
9  0.2587  0.3168  0.3361  0.2394  0.2516  0.1219  5.7258  1.9655  0.0434  0.0431  0.0186 
10  0.2626  0.3361  0.3206  0.2974  0.2742  0.1116  4.8871  1.9850  0.2191  0.2234  0.0191 
11  0.3284  0.2394  0.2548  0.3013  0.2581  0.1335  4.629  1.0199  0.2831  0.2874  0.0211 
12  0.251  0.3052  0.3013  0.2742  0.2903  0.1361  5.9194  1.8888  0.0917  0.0918  0.0185 
13  0.2742  0.2974  0.2394  0.2316  0.2484  0.1348  5.3387  1.9205  0.0656  0.0672  0.0194 
14  0.2548  0.2665  0.2239  0.309  0.2194  0.1194  5.1452  1.8079  0.1861  0.1895  0.0217 
15  0.2897  0.2703  0.2665  0.2897  0.2968  0.1039  6.371  1.9168  0.2538  0.2578  0.0151 
16  0.3361  0.3284  0.2587  0.2703  0.2452  0.1077  5.5323  1.8502  0.0013  0.0022  0.0188 
17  0.2432  0.2781  0.3284  0.2548  0.2129  0.1232  5.5968  1.7848  0.0498  0.0501  0.0194 
18  0.2858  0.309  0.2781  0.3206  0.2097  0.1065  6.1129  1.7848  0.0580  0.0578  0.0198 
19  0.2355  0.2626  0.2742  0.2277  0.2161  0.1155  6.0484  1.7490  0.0114  0.0125  0.0172 
20  0.2471  0.2277  0.2819  0.2665  0.2677  0.1103  5.4677  1.8854  0.0402  0.0398  0.0177 
21  0.2935  0.2316  0.2858  0.2587  0.2032  0.1271  4.8226  1.8120  0.2824  0.2865  0.0233 
22  0.2394  0.2742  0.251  0.3052  0.2774  0.1374  4.9516  2.0552  0.1545  0.1556  0.0135 
23  0.2316  0.251  0.3168  0.3129  0.2355  0.1297  6.1774  1.8380  0.1293  0.1307  0.0187 
24  0.2974  0.3245  0.2626  0.3361  0.2419  0.131  4.5645  1.9760  0.3568  0.3630  0.0219 
25  0.2277  0.2935  0.2471  0.3323  0.2323  0.1142  5.2097  1.8269  0.1689  0.1720  0.0204 
26  0.3323  0.2897  0.3245  0.2626  0.2613  0.1129  4.7581  1.9655  0.2288  0.2322  0.0193 
27  0.2819  0.2471  0.3052  0.2432  0.2548  0.1013  6.4355  1.7955  0.1695  0.1732  0.0166 
28  0.2665  0.2239  0.2974  0.3284  0.2387  0.1181  5.7903  1.8350  0.0500  0.0515  0.0182 
29  0.3013  0.2432  0.2935  0.2471  0.2645  0.1387  5.4032  2.0010  0.0348  0.0352  0.0182 
30  0.3206  0.2355  0.2432  0.2858  0.229  0.109  5.0806  1.8160  0.1674  0.1697  0.0202 
Optimization results in details
Optimization results
Stents  W _{ 1 } (mm)  W _{ 2 } (mm)  W _{ 3 } (mm)  W _{ 4 } (mm)  W _{ 5 } (mm)  T (mm)  L (mm)  P (MPa)  DR (t = 32 ms)  DR (t = 42 ms)  RER 

Original stent  0.28  0.28  0.28  0.28  0.249  0.12  5.8  1.8654  0.0622  0.0634  0.0178 
Optimal stent  0.235  0.22  0.22  0.22  0.3  0.1  5.63  1.9077  0.0036  0.0067  0.0151 
Improvement of stent expansion process
Discussions
A FEM based multiobjective optimization method combining with the Kriging surrogate model is proposed to reduce the dogboning effect and the radial elastic recoil of stent to improve the stent expansion performance. Our results show that the proposed optimization method could be used for stent design optimization effectively and conveniently. This provides a new method of stent design and represents a new direction of research. This optimization method combined with experimental verification can serve as a useful tool for stent design before manufacture.
In contrast to the expansive computational simulations employed in the comparison test studies [11, 12, 13, 15, 16, 17], the surrogate modeling approach in which response surface models (RSMs) were used to represent the relationship between design objectives and design variables [27]. Whilst most studies of stent design relate to multiple objectives, some articles only dealt with a single objective function. Harewood et al. [26] focused on radial stiffness of a single ring. Li et al. [28, 29] optimized stent dogboning and drug release, respectively. Grogan et al. [27] performed a single objective optimization for maximum radial strength. When considering multiple objectives, Pant et al. [30] and Bressloff [31] conducted FEA simulation to generate a range of multidisciplinary objectives. Pant et al. [30] constructed the Pareto fronts generated by treating each objective separately. Bressloff [31] recast the optimization as a constrained problem, wherein design improvement is sought in one objective while other objectives were considered as constraints. Multiobjective optimization of stent design involves a large number of design goals. It is difficult to find the optimal solution to improve all of them just by one of the methods to solve multiobjective problem, such as combining the design objectives in a single weighted objectives function, searching the Pareto fronts, 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, including stent auxiliary expansion, instent blood flow, drug release, and biomechanical response of vascular tissue, to improve the performance of stenting.
Some limitations of this study include: (a) FEA model of stent dilation does not contain blood vessels and thrombosis, (b) Balloon folding is not considered during its expansion process, (c) The results of optimal design have made some improvement of stenting performance, but it’s yet short of enough validation through experiment.
Conclusions
This article presents a FEM based multiobjective optimization method combining with the Kriging surrogate model to decrease both the dogboning effect and radial elastic recoil of stents. The Kriging surrogate model coupled with DOE methods was adopted to construct an approximate functional relationship between the objective function and geometries. The EI function was employed to balance local and global searches with the aim of finding out the global optimal design. The proposed optimization method effectively decreased both the dogboning effect and radial elastic recoil of stent. More issues of stent design should be considered and more effective multidisciplinary design optimization method should be investigated to continue our study.
Declarations
Declarations
Authors’ contributions
HL, BZ, MW, DZ and AQ were responsible for the design, data collection and overall investigation, and established the optimization model. JG and ZL were responsible for the optimization method. All authors contributed to the writing of the paper. All authors read and approved the final manuscript.
Authors’ information
Wang’s group (Wang, Zhao and Li) has been doing research in stent design, mechanical properties analysis, structure optimization and micromolding, see website: http://gs1.dlut.edu.cn/Supervisor/Front/dsxx/new/Default.aspx?WebPageName=WangMJ. http://gs1.dlut.edu.cn/Supervisor/Front/dsxx/new/Default.aspx?WebPageName=zhaodanyang. Zhu has been doing research in computational mechanics, finite element method and materials science: http://gs1.dlut.edu.cn/Supervisor/Front/dsxx/new/Default.aspx?WebPageName=zhubao. Gu and Li have been doing research in optimization algorithms, highperformance computing and largescale scientific: http://denm.dlut.edu.cn/info/1589/2878.htm. Qiao’s group has been playing a leading role in graft, stent and blood flow dynamics of aortic dissection: http://life.bjut.edu.cn/szdw/jcrc/20151231/14515449774055993_1.html.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant no. 11502044, 81171107) and Postdoctoral Science Foundation of China (Grant no. 2014M561222).
Competing interests
The authors declare that they have no competing interests.
About this supplement
This article has been published as part of BioMedical Engineering OnLine Volume 15 Supplement 2, 2016. Computational and experimental methods for biological research: cardiovascular diseases and beyond. The full contents of the supplement are available online http://biomedicalengineeringonline.biomedcentral.com/articles/supplements/volume15supplement2 .
Funding
Publication charges for this article have been funded by Postdoctoral Science Foundation of China (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
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