Optimal design of vascular stents using a network of 1D slender curved rods. (English) Zbl 1507.74228
Summary: In this manuscript we present a mathematical theory and a computational algorithm to study optimal design of mesh-like structures such as metallic stents by changing the stent strut thickness and width to optimize the overall stent compliance. The mathematical constrained optimization problem is to minimize the “compliance functional” over a closed and bounded set of constraints. The compliance functional is the stent’s overall elastic energy. The constraints are the minimal and maximal strut thickness, and a given fixed volume of the stent material. We prove the existence of a minimizer, thereby proving that the constrained optimization problem has a solution. A numerical scheme based on an iteration procedure is introduced, and implemented within a Finite Element Method framework. Optimal design of three different stent prototypes is considered: (1) a single zig-zag ring, which can be found in many complex stent designs on the US market as a basic cell in the modular stent design, (2) a Palmaz-Schatz type stent consisting of 6 zig-zag rings, and (3) a Cypher(TM) type stent consisting of zig-zag rings with sinusoidal connectors. Several interesting optimization solutions are found, some of which have already been implemented in the design of the currently available stents on the US market. The resulting computational algorithm is compared to a Genetic Algorithm, and it is shown that our computational approach outperforms the Genetic Algorithm in the following three key aspects: (1) computation time, (2) accuracy, and (3) maintaining the symmetry of the solution.
MSC:
| 74L15 | Biomechanical solid mechanics |
| 90C90 | Applications of mathematical programming |
| 74P10 | Optimization of other properties in solid mechanics |
| 90C48 | Programming in abstract spaces |
References:
| [1] | Welt, F. G.P., Inflammation and restenosis in the stent era, Arterioscler. Thromb. Vasc. Biol., 22, 11, 1769-1776 (2002) |
| [2] | Costa, M. A., Molecular basis of restenosis and drug-eluting stents, Circulation, 111, 17, 2257-2273 (2005) |
| [3] | Iqbal, J.; Gunn, J.; Serruys, P. W., Coronary stents: historical development, current status and future directions, Br. Med. Bull., 106, 1, 193-211 (2013) |
| [4] | Kastrati, A.; Dirschinger, J.; Boekstegers, P.; Elezi, S.; Schühlen, H.; Pache, J.; Steinbeck, G.; Schmitt, C.; Ulm, K.; Neumann, F. J.; Schömig, A., Influence of stent design on 1-year outcome after coronary stent placement: A randomized comparison of five stent types in 1, 147 unselected patients, Catheter. Cardiovasc. Interv., 50, 3, 290-297 (2000) |
| [5] | Pache, J.; Kastrati, A.; Mehilli, J.; Schühlen, H.; Dotzer, F.; Hausleiter, J.; Fleckenstein, M.; Neumann, F. J.; Sattelberger, U.; Schmitt, C.; Müller, M.; Dirschinger, J.; Schömig, A., Intracoronary stenting and angiographic results: Strut thickness effect on restenosis outcome (ISAR-STEREO) trial, Circulation, 103, 23, 2816-2821 (2001) |
| [6] | Lau, J. S., A stent is not just a stent: stent construction and design do matter in its clinical performance, Singapore Med. J., 45, 7, 305-312 (2004) |
| [7] | Wang, Y.; Čanić, S.; Bukač, M.; Tambača, J., Fluid-structure interaction between pulsatile blood ow and a curved stented coronary artery on a beating heart: a four stent computational study, Comput. Methods Appl. Mech. Engrg., 35, 15, 679-700 (2019) · Zbl 1441.74062 |
| [8] | Zunino, P.; Tambača, J.; Cutri, E.; Čanić, S.; Formaggia, L.; Migliavacca, F., Integrated stent models based on dimension reduction, review and future perspectives, Ann. Biomed. Eng., 44, 2, 604-617 (2016) |
| [9] | James, K. A.; Waisman, H., Layout design of a bi-stable cardiovascular stent using topology optimization, Comput. Methods Appl. Mech. Engrg., 305, 869-890 (2016) · Zbl 1425.74375 |
| [10] | Caputo, M.; Chiastra, C.; Cianciolo, C.; Cutri, E.; Dubini, G.; Gunn, J.; Keller, B.; Zunino, P., Simulation of oxygen transfer in stented arteries and correlation with in-stent restenosis, Int. J. Numer. Methods Biomed. Eng., 29, 12, 1373-1387 (2013) |
| [11] | Morlacchi, S.; Chiastra, E.; Cutri, E.; Zunino, P.; Burzotta, F.; Formaggia, L.; Dubini, G., Stent deformation, physical stress, and drug elution obtained with provisional stenting, conventional culotte and tryton-based culotte to treat bifurcations: a virtual simulation study, Eurointervention, 9, 12, 1441-1453 (2014) |
| [12] | Auricchio, F.; Conti, M.; Ferraro, M.; Morganti, S.; Reali, A.; Taylor, R. L., Innovative and efficient stent flexibility simulations based on isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 295, 347-361 (2015) · Zbl 1423.76204 |
| [13] | Cutri, E.; Zunino, P.; Morlacchi, S.; Chiastra, C.; Migliavacca, F., Drug delivery patterns for different stenting techniques in coronary bifurcations: a comparative computational study, Biomech. Model Mechanobiol., 12, 4, 657-669 (2013) |
| [14] | D’Angelo, C.; Zunino, P.; Porpora, A.; Morlacchi, S.; Migliavacca, F., Model reduction strategies enable computational analysis of controlled drug release from cardiovascular stents, SIAM J. Appl. Math., 71, 6, 2312-2333 (2011) · Zbl 1246.92014 |
| [15] | Simoes, M.; Martinez-Paneda, E., Phase field modelling of fracture and fatigue in shape memory alloys, Comput. Methods Appl. Mech. Engrg., 373, Article 113504 pp. (2020) · Zbl 1506.74022 |
| [16] | Zunino, P.; D’angelo, C.; Petrini, L.; Vergara, C.; Capelli, C.; Migliavacca, F., Numerical simulation of drug eluting coronary stents: mechanics, fluid dynamics and drug release, Comput. Methods Appl. Mech. Eng., 198, 45-46, 3633-3644 (2009) · Zbl 1229.76122 |
| [17] | Russ, J. B.; Li, R. L.; Herschman, A. R.; Waisman, H.; Vedula, V.; Kysar, J. W.; Kalfa, D., Design optimization of a cardiovascular stent with application to a balloon expandable prosthetic heart valve, Mater. Des., 209, Article 109997 pp. (2021), pp. 20 |
| [18] | Schmidt, T.; Abbott, J. D., Coronary stents: History, design, and construction (review article), J. Clin. Med., 7, 6, 126 (2018) |
| [19] | Berry, J. E.; Moore, J. L.; Roychowdhury, W. D.; Routh, S., Experimental and computational flow evaluation of coronary stents, Ann. Biomed. Eng., 28, 386-398 (2000) |
| [20] | Dumoulin, C.; Cochelin, B., Mechanical behaviour modelling of balloon-expandable stents, J. Biomech., 33, 11, 1461-1470 (2000) |
| [21] | Frank, A. O.; Walsh, P. W.; J. E., Moore Jr., Computational fluid dynamics and stent design, Artificial Organs, 26, 7, 614-621 (2002) |
| [22] | Migliavacca, F.; Petrini, L.; Colombo, M.; Auricchio, F.; Pietrabissa, R., Mechanical behavior of coronary stents investigated through the finite element method, J. Biomech., 35, 6, 803-811 (2002) |
| [23] | Morton, A. C.; Crossman, D.; Gunn, J., The influence of physical stent parameters upon restenosis, Pathol. Biol., 52, 4, 196-205 (2004) |
| [24] | Bressloff, N. W.; Ragkousis, G.; Curzen, N., Design optimisation of coronary artery stent systems, Ann. Biomed. Eng., 44, 2, 357-367 (2016) |
| [25] | Karanasiou, G. S.; Papafaklis, M. I.; Conway, C.; Michalis, L. K.; Tzafriri, R.; Edelman, E. R., Stents: biomechanics, biomaterials, and insights from computational modeling, Ann. Biomed. Eng., 45, 4, 853-872 (2017) |
| [26] | Tambača, J.; Kosor, M.; Čanić, S.; Paniagua, D., Mathematical modeling of endovascular stents, SIAM J. Appl. Math., 70, 6, 1922-1952 (2010) · Zbl 1427.74099 |
| [27] | Griso, G., Asymptotic behavior of structures made of curved rods, Anal. Appl. (Singap.), 6, 11-22 (2008) · Zbl 1210.74121 |
| [28] | Čanić, S.; Tambača, J., Cardiovascular stents as PDE nets: 1D vs 3D, IMA J. Appl. Math., 77, 6, 748-779 (2012) · Zbl 1305.92038 |
| [29] | Jurak, M.; Tambača, J., Derivation and justification of a curved rod model, Math. Models Methods Appl. Sci., 9, 991-1014 (1999) · Zbl 1044.74025 |
| [30] | Jurak, M.; Tambača, J., Linear curved rod model, general curve, Math. Models Methods Appl. Sci., 11, 1237-1252 (2001) · Zbl 1036.74032 |
| [31] | Tambača, J.; Velčić, I., Derivation of the nonlinear bending-torsion model for a junction of elastic rods, Proc. R. Soc. Edinburgh Sec. A Math., 142, 633-664 (2012) · Zbl 1316.74036 |
| [32] | Grubišić, L.; Iveković, J.; Tambača, J.; Žugec, B., Mixed formulation of the one-dimensional equilibrium model for elastic stents, Rad HAZU, 21, 219-240 (2017) · Zbl 1383.74055 |
| [33] | Grubišić, L.; Tambača, J., Direct solution method for the equilibrium problem for elastic stents, Numer. Linear Algebra Appl., 26, 3, Article e2231 pp. (2019) · Zbl 1449.74125 |
| [34] | Čanić, S.; Galović, M.; Ljulj, M.; Tambača, J., A dimension-reduction based coupled model of mesh-reinforced shells, SIAM J. Appl. Math., 77, 2, 744-769 (2017) · Zbl 1362.74004 |
| [35] | Cao, D. Q.; Tucker, R. W., Nonlinear dynamics of elastic rods using the cosserat theory: Modelling and simulation, Int. J. Solids Struct., 45, 2, 460-477 (2007) · Zbl 1167.74467 |
| [36] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · Zbl 0788.73002 |
| [37] | Deng, H.; Cheng, L.; Liang, X.; Hayduke, D.; To, A. C., Topology optimization for energy dissipation design of lattice structures through snap-through behavior, Comput. Methods Appl. Mech. Engrg., 358, Article 112641 pp. (2020), 28 · Zbl 1441.74158 |
| [38] | Deng, H.; Hinnenbush, S.; To, A. C., Topology optimization of stretchable metamaterials with Bezier skeleton explicit density (BSED) representation algorithm, Comput. Methods Appl. Mech. Engrg., 366, Article 113093 pp. (2020), 28 pp · Zbl 1442.74166 |
| [39] | Allaire, G., Conception Optimale de Structures (2007), Springer-Verlag: Springer-Verlag Berlin · Zbl 1132.49033 |
| [40] | Allaire, G., Optimal design of structures, map 562, Chapter 4 Department of Applied Mathematics, Ecole Polytechnique, vol. 14 (2015), http://www.cmap.polytechnique.fr/allaire/course_map562.html |
| [41] | US food and drug administration: Summary of safety and effectiveness of cypher(TM) stent (2004) |
| [42] | Kawamoto, H.; Panoulas, V. F.; Sato, K.; Miyazaki, T.; Naganuma, T.; Sticchi, A.; Figini, F.; Latib, A.; Chieffo, A.; Carlino, M.; Montorfano, M.; Colombo, A., Impact of strut width in periprocedural myocardial infarction, JACC Cardiovasc. Interv., 8, 7, 900-909 (2015) |
| [43] | Antman, S. S., Nonlinear Problems of Elasticity (1995), Springer-Verlag · Zbl 0820.73002 |
| [44] | Scardia, L., The nonlinear bending-torsion theory for curved rods as \(? G a m m a\)-limit of three-dimensional elasticity, Asymptotic Anal., 47, 3-4, 317-343 (2006) · Zbl 1133.74027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
