Analytical and numerical solution and multi-objective optimization of tetra-star-chiral auxetic stents

Discover Applied Sciences Research Analytical and numerical solution and multi‑objective optimization of tetra‑star‑chiral auxetic stents Parsa Behinfar1 · Amir Nourani1 Received: 18 June 2023 / Accepted: 19 December 2023 © The Author(s) 2024  OPEN Abstract The present study examines the mechanical properties of auxetic stents with the tetra-star-chiral structure. The tetrastar-chiral geometry is parametrically modeled. Then, the design of experiments (DOE) is developed by defining the elastic properties of the stents and using the response surface method (RSM). Finite element (FE) analysis is performed in order to find a polynomial relationship between the geometric parameters as inputs and the elastic parameters as the outputs. Then, the optimal stent is found in terms of elasticity parameters by using RSM and NSGA-II methods and the two-dimensional Pareto front is plotted. The optimal parameters of the stent including flexural stiffness, axial elasticity modulus, radial elasticity modulus and Poisson’s ratio are obtained as 10.66 mPa m4, 5.37 MPa, 33.2 MPa and − 0.41, respectively. Moreover, a method is proposed to find an analytical solution for metal elastic stents in order to verify the FE model results, and also the blood vessel compliance of the optimal stent is examined. Article Highlights • • • • • Axial tensile and bending tests are performed on a number of auxetic stent at a similar strain and moment respectively. Four main elasticity parameters (i.e., E, Er, EI, 𝜈 ) are calculated as a function of geometry parameters. Using castigliano’s theorem, the results of FEM tests are verified analytically Multi-objective optimization is performed and a set of optimal stents in terms of elasticity parameters is obtained. Using the compliance mathematical criteria, the optimized stent is seen to be suited for vessel wall. Keywords Auxetic · Finite element · Optimization · Analytical solution · Compliance List of symbols E ∗ Elasticity modulus of structure (MPa) EI Flexural rigidity of structure (Pa m4) Er Radial elasticity modulus of structure (MPa) F Force (N) L2 Length of smaller link (mm) L1 Distance between 2 horizontal link (mm) Lstent Length of stent (mm) M Moment (N m) n1 No. of circumferential cells (–) * Amir Nourani, | 1Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran. Discover Applied Sciences (2024) 6:39 | https://doi.org/10.1007/s42452-024-05663-1 Vol.:(0123456789) Research Discover Applied Sciences (2024) 6:39 | https://doi.org/10.1007/s42452-024-05663-1 n2 No. of axial cells (–) Ravg Average radius of stent (mm) RPi Stent radius at pressure pi (mm) t Thickness of link (mm) t3 Stent wall thickness (mm) U Strain energy (J) 𝜀axial Axial strain (–) 𝜀radial Radial strain (–) 𝜈 ∗ Poisson’s ratio of structure (–) Abbreviations DOE Design of experiment FE Finite element FEM Finite element method GA Genetic algorithm RSM Response surface method 1 Introduction Auxetic metamaterials have been widely used in recent years. They have different elastic characteristics, e.g., a negative Poisson’s ratio [1–5], with respect to regular materials. These irregularities give them some desirable capabilities in mechanical and medical engineering such as energy absorption, fracture resistance and shear resistance [6–8]. They have also been employed in medical and biomechanical industries, e.g., as knee straps, in-vessel stents, and orthopedic and sports shoes [9–11]. Since auxetic structures can expand simultaneously in the longitudinal and transverse directions, they are widely employed in angiography [12], in which stents and balloons are sent through the blood vessel to widen it. Khoshgoftar and Abbaszadeh [13] evaluated the effects of the geometric parameters of a honeycomb structure, e.g., thickness, aspect ratio, and the angle between cell components, on Poisson’s ratio; it was shown that a smaller Poisson’s ratio can be obtained by decreasing thickness, cell angle and number of cells. Poncin and Proft [14] evaluated different materials and methods in the fabrication of artery stents. They reported some stainless steel, cobalt, titanium, Nitinol, and magnesium alloys to be optimal materials in terms of body compatibility and ease of fabrication. Eshghi et al. [15] conducted the FE analysis of expanded stents on a balloon within blood vessels in the presence of a plaque. They also reported the diameter variation and bending of stent while the applied inlet pressure increased. Bhullar [16] explored the effects of a negative Poisson’s ratio on the performance of esophagus-placed stents. The properties of a number of stents with different geometries were obtained under tensile loads. Ren et al. [17] employed tensile and compressive tests to analyze the effects of geometrical parameters on the elastic parameters (e.g., Poisson’s ratio) of metal stents, it was found that, in a proper value of pattern scale factor (PSF), Poisson’s ratio and force–displacement curve for both compressive and tensile loading conditions are the same. Wu et al. [18] compared chiral structures with circular and oval nodes. The in-plane elastic properties of the structures were obtained using FEM. Tensile tests were performed, and the geometrical parameters (as the inputs) and Poisson’s ratio (as the output) were related. The two structures were found to show promising performances as a stent in blood vessels due to high expansion and axial stability. Ruan et al. [19] examined anti-chiral stents. Poisson’s ratio was plotted versus geometric parameters. It was found that the maximum stress would be smaller when the number of cells in the axial direction was higher. A review of the literature on auxetic stents indicates research gaps. For example, studies on the optimization of stent parameters mostly implemented single-objective optimization. Also, the bending performance and analytical solution for mechanical properties of stents have not been studied. In this study, RSM method is used to find a mathematical relationship between geometrical parameters as the inputs and elasticity parameters as the outputs for some typical biomaterials. In order to carry out designed RSM tests, FE modeling is applied. Also, a comprehensive analytical method using Castigliano’s theorem to evaluate tetra-star-chiral structure is proposed. Moreover, the multi-objective optimization of stents using NSGA-II Vol:.(1234567890) Discover Applied Sciences (2024) 6:39 Research | https://doi.org/10.1007/s42452-024-05663-1 and RSM methods is performed. Finally the compliance of the optimized stents is measured as a criterion to evaluate the compatibility of the stent to be used in the blood vessels. In the next section, we consider to introduce our method of geometric modeling using Catia software, design of FEM experiments using RSM method, analytical solution, multi-objective optimization and compliance analysis of auxetic tetrastar-chi (...truncated)