Annals of Biomedical Engineering

, Volume 44, Issue 2, pp 604–617 | Cite as

Integrated Stent Models Based on Dimension Reduction: Review and Future Perspectives

  • Paolo Zunino
  • Josip Tambača
  • Elena Cutrì
  • Suncica Čanić
  • Luca Formaggia
  • Francesco Migliavacca
Medical Stents: State of the Art and Future Directions


Stent modeling represents a challenging task from both the theoretical and numerical viewpoints, due to its multi-physics nature and to the complex geometrical configuration of these devices. In this light, dimensional model reduction enables a comprehensive geometrical and physical description of stenting at affordable computational costs. In this work, we aim at reviewing dimensional model reduction of stent mechanics and drug release. Firstly, we address model reduction techniques for the description of stent mechanics, aiming to illustrate how a three-dimensional stent model can be transformed into a collection of interconnected one-dimensional rods, called a “stent net”. Secondly, we review available model reduction methods similarly applied to drug release, in which the “stent net” concept is adopted for modeling of drug elution. As a result, drug eluting stents are described as a distribution of concentrated drug release sources located on a graph that fully represents the stent geometry. Lastly, new results about the extension of these model reduction approaches to biodegradable stents are also discussed.


Medical stents Mathematical modeling Dimensional model reduction Stent mechanics Drug eluting stents Biodegradable stents 


  1. 1.
    Antman, S. Nonlinear Problems of Elasticity. New York: Springer, 2005.Google Scholar
  2. 2.
    Babuška, I., and M. Suri. Locking effects in the finite element approximation of elasticity problems. Numerische Mathematik 62(1):439–463, 1992.CrossRefGoogle Scholar
  3. 3.
    Balakrishnan, B., J. F. Dooley, G. Kopia, and E. R. Edelman. Intravascular drug release kinetics dictate arterial drug deposition, retention, and distribution. J. Control. Release 123(2):100–108, 2007.PubMedCentralCrossRefPubMedGoogle Scholar
  4. 4.
    Biscari, P., S. Minisini, D. Pierotti, G. Verzini, and P. Zunino. Controlled release with finite dissolution rate. SIAM J. Appl. Math. 71(3):731–752, 2011.CrossRefGoogle Scholar
  5. 5.
    Bozsak, F., J. M. Chomaz, and A. I. Barakat. Modeling the transport of drugs eluted from stents: physical phenomena driving drug distribution in the arterial wall. Biomech. Model. Mechanobiol. 13(2):327–347, 2014.CrossRefPubMedGoogle Scholar
  6. 6.
    Caiazzo, A., D. Evans, J. L. Falcone, J. Hegewald, E. Lorenz, B. Stahl, D. Wang, J. Bernsdorf, B. Chopard, J. Gunn, R. Hose, M. Krafczyk, P. Lawford, R. Smallwood, D. Walker, and A. Hoekstra. A complex automata approach for in-stent restenosis: two-dimensional multiscale modelling and simulations. J. Comput. Sci. 2(1):9–17, 2011.CrossRefGoogle Scholar
  7. 7.
    Čanić, S., and J. Tambača. Cardiovascular stents as PDE nets: 1D vs. 3D. IMA J. Appl. Math. 77(6):748–770, 2012.CrossRefGoogle Scholar
  8. 8.
    Caputo, M., C. Chiastra, C. Cianciolo, E. Cutrì, G. Dubini, J. Gunn, B. Keller, F. Migliavacca, and P. Zunino. Simulation of oxygen transfer in stented arteries and correlation with in-stent restenosis. Int. J. Numer. Methods Biomed. Eng. 29(12):1373–1387, 2013.CrossRefGoogle Scholar
  9. 9.
    Chiastra, C., S. Morlacchi, D. Gallo, U. Morbiducci, R. Cárdenes, I. Larrabide, and F. Migliavacca. Computational fluid dynamic simulations of image-based stented coronary bifurcation models. J. R. Soc. Interface 10(84):20130193, 2013.PubMedCentralCrossRefPubMedGoogle Scholar
  10. 10.
    Chiastra, C., S. Morlacchi, S. Pereira, G. Dubini, and F. Migliavacca. Computational fluid dynamics of stented coronary bifurcations studied with a hybrid discretization method. Eur. J. Mech. B 35:76–84, 2012.CrossRefGoogle Scholar
  11. 11.
    Cohen, D. S., and T. Erneux. Controlled drug release asymptotics. SIAM J. Appl. Math. 58(4):1193–1204, 1998.CrossRefGoogle Scholar
  12. 12.
    Cutrì, E., P. Zunino, S. Morlacchi, C. Chiastra, and F. Migliavacca. Drug delivery patterns for different stenting techniques in coronary bifurcations: a comparative computational study. Biomech. Model. Mechanobiol. 12(4):657–669, 2013.CrossRefPubMedGoogle Scholar
  13. 13.
    D’Angelo, C., and A. Quarteroni. On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18(8):1481–1504, 2008.CrossRefGoogle Scholar
  14. 14.
    D’Angelo, C., and P. Zunino. Multiscale models of drug delivery by thin implantable devices. Appl. Ind. Math. Italy III Ser. Adv. Math. Appl. Sci. 82:298–310, 2009.Google Scholar
  15. 15.
    D’Angelo, C., P. Zunino, A. Porpora, S. Morlacchi, and F. Migliavacca. Model reduction strategies enable computational analysis of controlled drug release from cardiovascular stents? SIAM J. Appl. Math. 71(6):2312–2333, 2011.CrossRefGoogle Scholar
  16. 16.
    Delfour, M. C., A. Garon, and V. Longo. Modeling and design of coated stents to optimize the effect of the dose. SIAM J. Appl. Math. 65(3):858–881, 2005.CrossRefGoogle Scholar
  17. 17.
    Ellwein, L. M., H. Otake, T. J. Gundert, B. K. Koo, T. Shinke, Y. Honda, J. Shite, and J. F. LaDisa, Jr. Optical coherence tomography for patient-specific 3D artery reconstruction and evaluation of wall shear stress in a left circumflex coronary artery. Cardiovasc. Eng. Technol. 2(3):212–227, 2011.CrossRefGoogle Scholar
  18. 18.
    Foin, N., R. Torii, P. Mortier, M. De Beule, N. Viceconte, P. H. Chan, J. E. Davies, X. Y. Xu, R. Krams, and C. Di Mario. Kissing balloon or sequential dilation of the side branch and main vessel for provisional stenting of bifurcations: lessons from micro-computed tomography and computational simulations. JACC Cardiovasc. Interv. 5(1):47–56, 2012.CrossRefPubMedGoogle Scholar
  19. 19.
    Formaggia, L., S. Minisini, and P. Zunino. Modeling polymeric controlled drug release and transport phenomena in the arterial tissue. Math. Models Methods Appl. Sci. 20(10):1759–1786, 2010.CrossRefGoogle Scholar
  20. 20.
    FreeFem, version 3.12-0 (2d and 3d).
  21. 21.
    Frenning, G. Theoretical investigation of drug release from planar matrix systems: effects of a finite dissolution rate. J. Control. Release 92(3):331–339, 2003.CrossRefPubMedGoogle Scholar
  22. 22.
    Frenning, G. Theoretical analysis of the release of slowly dissolving drugs from spherical matrix systems. J. Control. Release 95(1):109–117, 2004.CrossRefPubMedGoogle Scholar
  23. 23.
    Gastaldi, D., S. Morlacchi, R. Nichetti, C. Capelli, G. Dubini, L. Petrini, and F. Migliavacca. Modelling of the provisional side-branch stenting approach for the treatment of atherosclerotic coronary bifurcations: effects of stent positioning. Biomech. Model. Mechanobiol. 9(5):551–561, 2010.CrossRefPubMedGoogle Scholar
  24. 24.
    Gijsen, F. J. H., F. Migliavacca, S. Schievano, L. Socci, L. Petrini, A. Thury, J. J. Wentzel, A. F. W. van der Steen, P. W. S. Serruys, and G. Dubini. Simulation of stent deployment in a realistic human coronary artery. BioMed. Eng. Online 7:23, 2008.PubMedCentralCrossRefPubMedGoogle Scholar
  25. 25.
    Grassi, M., G. Pontrelli, L. Teresi, G. Grassi, L. Comel, A. Ferluga, and L. Galasso. Novel design of drug delivery in stented arteries: a numerical comparative study. Math. Biosci. Eng. 6(3):493–508, 2009.CrossRefPubMedGoogle Scholar
  26. 26.
    Griso, G. Asymptotic behavior of structures made of curved rods. Anal. Appl. 06(01):11–22, 2008.CrossRefGoogle Scholar
  27. 27.
    Higuchi, T. Rate of release of medicaments from ointment bases containing drugs in suspension. J. Pharm. Sci. 50:874–875, 1961.CrossRefPubMedGoogle Scholar
  28. 28.
    Higuchi, T. Mechanism of sustained-action medication. Theoretical analysis of rate. J. Pharm. Sci. 52:1145–1149, 1963.CrossRefPubMedGoogle Scholar
  29. 29.
    Horner, M., S. Joshi, V. Dhruva, S. Sett, and S. F. C. Stewart. A two-species drug delivery model is required to predict deposition from drug-eluting stents. Cardiovasc. Eng. Technol. 1(3):225–234, 2010.CrossRefGoogle Scholar
  30. 30.
    Hose, D. R., A. J. Narracott, B. Griffiths, S. Mahmood, J. Gunn, D. Sweeney, and P. V. Lawford. A thermal analogy for modelling drug elution from cardiovascular stents. Comput. Methods Biomech. Biomed. Eng. 7(5):257–264, 2004.CrossRefGoogle Scholar
  31. 31.
    Hossainy, S., and S. Prabhu. A mathematical model for predicting drug release from a biodurable drug-eluting stent coating. J. Biomed. Mater. Res. Part A 87(2):487–493, 2008.CrossRefGoogle Scholar
  32. 32.
    Humphrey, J. D., and K. R. Rajagopal. A constrained mixture model for growth and remodeling of soft tissues. Math. Models Methods Appl. Sci. 12(3):407–430, 2002.CrossRefGoogle Scholar
  33. 33.
    Hwang, C. W., D. Wu, and E. R. Edelman. Physiological transport forces govern drug distribution for stent-based delivery. Circulation 104(5):600–605, 2001.CrossRefPubMedGoogle Scholar
  34. 34.
    Jurak, M. T., and J. Tambaca. Linear curved rod model. General curve. Math. Models Methods Appl. Sci. 11(7):1237–1252, 2001.CrossRefGoogle Scholar
  35. 35.
    Karner, G., and K. Perktold. Effect of endothelial injury and increased blood pressure on albumin accumulation in the arterial wall: a numerical study. J. Biomech. 33(6):709–715, 2000.CrossRefPubMedGoogle Scholar
  36. 36.
    Karner, G., K. Perktold, and H. P. Zehentner. Computational modeling of macromolecule transport in the arterial wall. Comput. Methods Biomech. Biomed. Eng. 4(6):491–504, 2001.CrossRefGoogle Scholar
  37. 37.
    Kolandaivelu, K., B. B. Leiden, and E. R. Edelman. Predicting response to endovascular therapies: dissecting the roles of local lesion complexity, systemic comorbidity, and clinical uncertainty. J. Biomech. 47(4):908–921, 2014.CrossRefPubMedGoogle Scholar
  38. 38.
    Li, J., Q. Luo, Z. Xie, Y. Li, and Y. Zeng. Fatigue life analysis and experimental verification of coronary stent. Heart Vessels 25(4):333–337, 2010.CrossRefPubMedGoogle Scholar
  39. 39.
    Lovich, M. A., and E. R. Edelman. Computational simulations of local vascular heparin deposition and distribution. Am. J. Physiol. Heart Circ. Physiol. 271(5):H2014–H2024, 1996.Google Scholar
  40. 40.
    Marrey, R. V., R. Burgermeister, R. B. Grishaber, and R. O. Ritchie. Fatigue and life prediction for cobalt-chromium stents: a fracture mechanics analysis. Biomaterials 27(9):1988–2000, 2006.CrossRefPubMedGoogle Scholar
  41. 41.
    McGinty, S. A decade of modelling drug release from arterial stents. Math. Biosci. 257:80–90, 2014.CrossRefPubMedGoogle Scholar
  42. 42.
    McGinty, S., S. McKee, R. M. Wadsworth, and C. McCormick. Modelling drug-eluting stents. Math. Med. Biol. 28(1):1–29, 2011.CrossRefPubMedGoogle Scholar
  43. 43.
    McGinty, S., S. McKee, R. M. Wadsworth, and C. McCormick. Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent. SIAM J. Appl. Math. 73(6):2004–2028, 2013.CrossRefGoogle Scholar
  44. 44.
    Menzel, A., and E. Kuhl. Frontiers in growth and remodeling. Mech. Res. Commun. 42:1–14, 2012.PubMedCentralCrossRefPubMedGoogle Scholar
  45. 45.
    Migliavacca, F., F. Gervaso, M. Prosi, P. Zunino, S. Minisini, L. Formaggia, and G. Dubini. Expansion and drug elution model of a coronary stent. Comput. Methods Biomech. Biomed. Eng. 10(1):63–73, 2007.CrossRefGoogle Scholar
  46. 46.
    Morlacchi, S., C. Chiastra, E. Cutrì, P. Zunino, F. Burzotta, L. Formaggia, G. Dubini, and F. Migliavacca. 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.CrossRefPubMedGoogle Scholar
  47. 47.
    Morlacchi, S., B. Keller, P. Arcangeli, M. Balzan, F. Migliavacca, G. Dubini, J. Gunn, N. Arnold, A. Narracott, D. Evans, and P. Lawford. Hemodynamics and in-stent restenosis: micro-CT images, histology, and computer simulations. Ann. Biomed. Eng. 39(10):2615–2626, 2011.CrossRefPubMedGoogle Scholar
  48. 48.
    Morlacchi, S., and F. Migliavacca. Modeling stented coronary arteries: where we are, where to go. Ann. Biomed. Eng. 41(7):1428–1444, 2013.CrossRefPubMedGoogle Scholar
  49. 49.
    Morlacchi, S., G. Pennati, L. Petrini, G. Dubini, and F. Migliavacca. Influence of plaque calcifications on coronary stent fracture: a numerical fatigue life analysis including cardiac wall movement. J. Biomech. 47(4):899–907, 2014.CrossRefPubMedGoogle Scholar
  50. 50.
    Mortier, P., G. A. Holzapfel, M. De Beule, D. Van Loo, Y. Taeymans, P. Segers, P. Verdonck, and B. Verhegghe. A novel simulation strategy for stent insertion and deployment in curved coronary bifurcations: comparison of three drug-eluting stents. Ann. Biomed. Eng. 38(1):88–99, 2010.CrossRefPubMedGoogle Scholar
  51. 51.
    Nakazawa, G., A. V. Finn, M. Vorpahl, E. Ladich, R. Kutys, I. Balazs, F. D. Kolodgie, and R. Virmani. Incidence and predictors of drug-eluting stent fracture in human coronary artery: a pathologic analysis. J. Am. Coll. Cardiol. 54(21):1924–1931, 2009.CrossRefPubMedGoogle Scholar
  52. 52.
    Perktold, K., M. Prosi, and P. Zunino. Mathematical models of mass transfer in the vascular walls. Model. Simulat. Appl. 1:244–278, 2009.Google Scholar
  53. 53.
    Pontrelli, G., and F. de Monte. Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transf. 50(17–18):3658–3669, 2007.CrossRefGoogle Scholar
  54. 54.
    Pontrelli, G., and F. De Monte. Modeling of mass dynamics in arterial drug-eluting stents. J. Porous Media 12(1):19–28, 2009.CrossRefGoogle Scholar
  55. 55.
    Pontrelli, G., and F. de Monte. A multi-layer porous wall model for coronary drug-eluting stents. Int. J. Heat Mass Transf. 53(19–20):3629–3637, 2010.CrossRefGoogle Scholar
  56. 56.
    Pontrelli, G., and F. de Monte. A two-phase two-layer model for transdermal drug delivery and percutaneous absorption. Math. Biosci. 257:96–103, 2014.CrossRefPubMedGoogle Scholar
  57. 57.
    Prabhu, S., and S. Hossainy. Modeling of degradation and drug release from a biodegradable stent coating. J. Biomed. Mater. Res. Part A 80(3):732–741, 2007.CrossRefGoogle Scholar
  58. 58.
    Prosi, M., P. Zunino, K. Perktold, and A. Quarteroni. Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow. J. Biomech. 38(4):903–917, 2005.CrossRefPubMedGoogle Scholar
  59. 59.
    Sakharov, D. V., L. V. Kalachev, and D. C. Rijken. Numerical simulation of local pharmacokinetics of a drug after intravascular delivery with an eluting stent. J. Drug Target. 10(6):507–513, 2002.CrossRefPubMedGoogle Scholar
  60. 60.
    Soares, J. S., J. E. Moore, and K. R. Rajagopal. Constitutive framework for biodegradable polymers with applications to biodegradable stents. ASAIO J. 54(3):295–301, 2008.CrossRefPubMedGoogle Scholar
  61. 61.
    Soares, J. S., J. E. Moore, and K. R. Rajagopal. Modeling of deformation-accelerated breakdown of polylactic acid biodegradable stents. J. Med. Device 4(4):041007, 2010.CrossRefGoogle Scholar
  62. 62.
    Soares, J. S., and P. Zunino. A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks. Biomaterials 31(11):3032–3042, 2010.CrossRefPubMedGoogle Scholar
  63. 63.
    Tambača, J. A model of irregular curved rods. In: Applied Mathematics and Scientific Computing, edited by Z. Drmač, V. Hari, and et al. New York: Springer, 2003, pp. 289–299.Google Scholar
  64. 64.
    Tambača, J., S. Čanić, and D. Paniagua. A comparison between fractured Xience-like and Palmaz-like stents using a novel computational model. In: Proceedings of the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society: Engineering the Future of Biomedicine, EMBC, 2009.Google Scholar
  65. 65.
    Tambača, J., M. Kosor, S. Čanic, and D. Paniagua. Mathematical modeling of vascular stents. SIAM J. Appl. Math. 70(6):1922–1952, 2010.CrossRefGoogle Scholar
  66. 66.
    Tambača, J., and I. Velčić. Derivation of the nonlinear bending-torsion model for a junction of elastic rods. Proc. R. Soc. Edinb. Sect. A 142(3):633–664, 2012.CrossRefGoogle Scholar
  67. 67.
    Tambača, J., and B. Žugec. One-dimensional quasistatic model of biodegradable elastic curved rods. Zeitschrift fur Angewandte Mathematik und Physik. 2015.Google Scholar
  68. 68.
    Tzafriri, A. R., A. Groothuis, G. S. Price, and E. R. Edelman. Stent elution rate determines drug deposition and receptor-mediated effects. J. Control. Release 161(3):918–926, 2012.PubMedCentralCrossRefPubMedGoogle Scholar
  69. 69.
    Tzafriri, A. R., A. D. Levin, and E. R. Edelman. Diffusion-limited binding explains binary dose response for local arterial and tumour drug delivery. Cell Prolif. 42(3):348–363, 2009.PubMedCentralCrossRefPubMedGoogle Scholar
  70. 70.
    Vairo, G., M. Cioffi, R. Cottone, G. Dubini, and F. Migliavacca. Drug release from coronary eluting stents: a multidomain approach. J. Biomech. 43(8):1580–1589, 2010.CrossRefPubMedGoogle Scholar
  71. 71.
    Wang, S., and K. Vafai. Analysis of the effect of stent emplacement on LDL transport within an artery. Int. J. Heat Mass Transf. 64:1031–1040, 2013.CrossRefGoogle Scholar
  72. 72.
    Zhu, X., D. W. Pack, and R. D. Braatz. Modelling intravascular delivery from drug-eluting stents with biodurable coating: investigation of anisotropic vascular drug diffusivity and arterial drug distribution. Comput. Methods Biomech. Biomed. Eng. 17(3):187–198, 2014.CrossRefGoogle Scholar
  73. 73.
    Žugec, B. Biodegradable elastic stent model. Ph.D.University of Zagreb. 2014.Google Scholar
  74. 74.
    Zunino, P. Multidimensional pharmacokinetic models applied to the design of drug-eluting stents. Cardiovasc. Eng. 4(2):181–191, 2004.CrossRefGoogle Scholar
  75. 75.
    Zunino, P., C. D’Angelo, L. Petrini, C. Vergara, C. Capelli, and F. Migliavacca. Numerical simulation of drug eluting coronary stents: mechanics, fluid dynamics and drug release. Comput. Methods Appl. Mech. Eng. 198(45–46):3633–3644, 2009.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2015

Authors and Affiliations

  • Paolo Zunino
    • 1
    • 2
  • Josip Tambača
    • 3
  • Elena Cutrì
    • 4
  • Suncica Čanić
    • 5
  • Luca Formaggia
    • 2
  • Francesco Migliavacca
    • 4
  1. 1.Department of Mechanical Engineering and Materials ScienceUniversity of PittsburghPittsburghUSA
  2. 2.MOX, Modeling and Scientific Computing, Department of MathematicsPolitecnico di MilanoMilanItaly
  3. 3.Department of MathematicsUniversity of ZagrebZagrebCroatia
  4. 4.Laboratory of Biological Structure Mechanics, Department of Chemistry, Materials and Chemical Engineering ‘Giulio Natta’Politecnico di MilanoMilanItaly
  5. 5.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations