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Journal of Mathematical Biology

, Volume 72, Issue 4, pp 973–996 | Cite as

Mathematical modeling and simulation of the evolution of plaques in blood vessels

  • Yifan Yang
  • Willi Jäger
  • Maria Neuss-RaduEmail author
  • Thomas Richter
Article

Abstract

In this paper, a model is developed for the evolution of plaques in arteries, which is one of the main causes for the blockage of blood flow. Plaque rupture and spread of torn-off material may cause closures in the down-stream vessel system and lead to ischemic brain or myocardial infarctions. The model covers the flow of blood and its interaction with the vessel wall. It is based on the assumption that the penetration of monocytes from the blood flow into the vessel wall, and the accumulation of foam cells increasing the volume, are main factors for the growth of plaques. The dynamics of the vessel wall is governed by a deformation gradient, which is given as composition of a purely elastic tensor, and a tensor modeling the biologically caused volume growth. An equation for the evolution of the metric is derived quantifying the changing geometry of the vessel wall. To calculate numerically the solutions of the arising free boundary problem, the model system of partial differential equations is transformed to an ALE (Arbitrary Lagrangian-Eulerian) formulation, where all equations are given in fixed domains. The numerical calculations are using newly developed algorithms for ALE systems. The results of the simulations, obtained for realistic system parameters, are in good qualitative agreement with observations. They demonstrate that the basic modeling assumption can be justified. The increase of stresses in the vessel wall can be computed. Medical treatment tries to prevent critical stress values, which may cause plaque rupture and its consequences.

Keywords

Atherosclerotic plaque formation Fluid-structure interaction Coupling biochemical reactions and biomechanics  Modeling tissue growth Computing wall stresses 

Mathematics Subject Classification

35Q30 74L15 92C10 92C50 

Notes

Acknowledgments

The work of the first author was supported in the framework the Pioneering Projects of IWR, University of Heidelberg.

References

  1. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40(12):1297-1316CrossRefMathSciNetzbMATHGoogle Scholar
  2. Barrett KE, Boitano S, Barman SM, Brooks HL (2010) Ganongs review of medical physiology, 23rd edn. McGraw Hill Professional, USAGoogle Scholar
  3. Boyd J, Buick JM, Green S (2007) Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phy Fluids 19(9):093103CrossRefGoogle Scholar
  4. Ciarlet PG (1988) Mathematical Elasticity, vol.I: Three-Dimensional Elasticity. North-Holland, AmsterdamGoogle Scholar
  5. Doktorski I (2007) Mechanical model for biofilm growth phase. PhD thesis, University of HeidelbergGoogle Scholar
  6. Dunne T, Rannacher R, Richter T (2010) Numerical simulation of fluid-structure interaction based on monolithic variational formulations. Fundamental Trends in Fluid-Structure Interaction., vol 1 of Contemporary Challenges in Mathematical Fluid Dynamics and Its ApplicationsWorld Scientific, Singapore, pp 1-75Google Scholar
  7. El Khatib N, Génieys S, Volpert V (2007) Atherosclerosis Initiation Modeled as an Inflammatory Process. Math Model Nat Phenom 2:126-141CrossRefMathSciNetGoogle Scholar
  8. Fasano A, Santos RF, Sequeira A (2011) Blood coagulation: a puzzle for biologists, a maze for mathematicians. In: Ambrosi D, Quarteroni A, Rozza G (eds) Modelling of physiological flows. Springer-Verlag, Italia, pp 41-75Google Scholar
  9. Fernández MA, Formaggia L, Gerbeau J-F, Quarteroni A (2009) The derivation of the equations for fluids and structure. Cardiovascular Mathematics., vol 1, Springer, Milan, pp 77-121Google Scholar
  10. Fogelson AL (1992) Continuum models of platelet aggregation: formulation and mechanical properties. SIAM J Appl Math 52(4):1089-1110CrossRefMathSciNetzbMATHGoogle Scholar
  11. Formaggia L, Moura A, Nobile F (2007) On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM. Math Model Num Anal 41(04):743-769CrossRefMathSciNetzbMATHGoogle Scholar
  12. Fung YC (1984) Biodynam Circ. Springer-Verlag, New YorkGoogle Scholar
  13. Hahn C, Schwartz MA (2009) Mechanotransduction in vascular physiology and atherogenesis. Nat Rev Mol Cell Biol 10:53-62CrossRefGoogle Scholar
  14. Holzapfel G (2000) Nonlinear solid mechanics, a continuum approach for engineering. John Wiley and Sons, ChichesterzbMATHGoogle Scholar
  15. Holzapfel GA, Stadler M, Schulze-Bauer CAJ (2002) A layer-specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing. Ann Biomed Eng 30:753-767CrossRefGoogle Scholar
  16. Hron J, Madlik M (2007) Fluid-structure interaction with applications in biomechanics. Nonlinear Anal Real World Appl 8:1431-1458CrossRefMathSciNetzbMATHGoogle Scholar
  17. Humphrey JD (2002) Cardiovascular solid mechanics, cells, tissues, and organs. Springer, NewYorkCrossRefGoogle Scholar
  18. Ibragimov AI, McNeal CJ, Ritter LR, Walton JR (2005) A mathematical model of atherogenesis as an inflammatory response. Math Med Biol 22(4):305-333CrossRefzbMATHGoogle Scholar
  19. Janela J, Moura A, Sequeira A (2010) A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. J Comp Appl Math 234(9):2783-2791CrossRefMathSciNetzbMATHGoogle Scholar
  20. Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, CambridgezbMATHGoogle Scholar
  21. Jones GW, Chapman SJ (2012) Modeling growth in biological materials. SIAM Rev 54(1):52-118CrossRefMathSciNetzbMATHGoogle Scholar
  22. Kalita P, Schaefer R (2008) Mechanical models of artery walls. Arch Comp Methods Eng 15:1-36CrossRefMathSciNetzbMATHGoogle Scholar
  23. Li ZY, Howarth SPS, Tang T, Gillard JH (2006) How critical is fibrous cap thickness to carotid plaque stability? Stroke 37(5):1195-1199CrossRefGoogle Scholar
  24. Ougrinovskaia A, Thompson R, Myerscough M (2010) An ODE model of early stages of atherosclerosis: mechanisms of theinflammatory response. Bull Math Biol 72:1534-1561CrossRefMathSciNetzbMATHGoogle Scholar
  25. Pasterkamp G, Falk E (2000) Atherosclerotic plaque rupture: an overview. J Clin Basic Cardiol 3:81-86Google Scholar
  26. Quarteroni A, Formaggia L (2004) Mathematical modelling and numerical simulation of the cardiovascular system. In: Handbook of numerical analysis 12. Elsevier, Amsterda, pp 3-127Google Scholar
  27. Quarteroni A, Tuveri M, Veneziani A (2000) Computational vascular fluid dynamics: problems, models and methods. Comp Visual Sci 2:163-197CrossRefzbMATHGoogle Scholar
  28. Quarteroni A, Veneziani A, Zunino P (2001) Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls. SIAM J Numer Anal 39(5):1488-1511CrossRefMathSciNetzbMATHGoogle Scholar
  29. Rajagopal KR, Srinivasa AR (2004) On thermomechanical restrictions of continua. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 460, The Royal Society, pp 631-651Google Scholar
  30. Richter T (2011) Gascoigne. Lecture Notes, University of Heidelberg, http://numerik.uni-hd.de/ richter/SS11/gascoigne/index.html
  31. Robertson AM (2008) Review of relevant continuum mechanics. In: Hemodynamical flows: modeling, analysis and simulation. Springer, pp 1-62Google Scholar
  32. Robertson AM, Sequeira A, Kameneva MV (2008) Hemorheology. In: Hemodynamical flows: modeling, analysis and simulation. Springer, pp 63-120Google Scholar
  33. Tang D, Yang C, Kobayashi S, Zheng J, Woodard PK, Teng Z, Billiar K, Bach R, Ku DN (2009) 3D MRI-based anisotropic FSI models with cyclic bending for human coronary atherosclerotic plaque mechanical analysis. J. Biomech. Eng. 131(6):061010CrossRefGoogle Scholar
  34. Tang D, Yang C, Mondal S, Liu F, Canton G, Hatsukami TS, Yuan C (2008) A negative correlation between human carotid atherosclerotic plaque progression and plaque wall stress: In vivo MRI-based 2D/3D FSI models. J Biomech 41(4):727-736CrossRefGoogle Scholar
  35. Tang D, Yang C, Zheng J, Woodard PK, Sicard GA, Saffitz JE, Yuan C (2004) 3D MRI-based multicomponent FSI models for atherosclerotic plaques. Ann Biomed Eng 32:947-960CrossRefGoogle Scholar
  36. Turek S, Hron J, Madlik M, Razzaq M, Wobker H, Acker JF (2010) Numerical Simulation and Benchmarking of a Monolithic Multigrid Solver for Fluid-Structure Interaction Problems with Application to Hemodynamics. Fluid Structure Interaction II, Springer-Berlin-Heidelberg, pp 193-220Google Scholar
  37. VanEpps JS, Vorp DA (2007) Mechanopathobiology of atherogenesis: a review. J Surg Res 142:202-217CrossRefGoogle Scholar
  38. Weller F (2008) Platelet deposition in non-parallel flow. J Math Biol 57:333-359CrossRefMathSciNetzbMATHGoogle Scholar
  39. Weller F, Neuss-Radu M, Jäger W (2013) Analysis of a free boundary problem modeling thrombus growth. SIAM J Math Anal 45:809-833CrossRefMathSciNetzbMATHGoogle Scholar
  40. Wick T (2011) Fluid-structure interactions using different mesh motion techniques. Comp Struct 89:1456-1467CrossRefGoogle Scholar
  41. Yang Y, Richter T, Jäger W, Neuss-Radu M. An ALE approach to mechano-chemical processes in fluid-structure interactions (in preparation)Google Scholar
  42. Zamir M (2005) The physics of coronary blood flow, series: biological and medical physics, biomedical engineering. Springer, New YorkGoogle Scholar
  43. Zohdi TI, Holzapfel GA, Berger SA (2004) A phenomenological model for atherosclerotic plaque growth and rupture. J Theor Biol 227:437-443CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yifan Yang
    • 1
  • Willi Jäger
    • 1
  • Maria Neuss-Radu
    • 2
    Email author
  • Thomas Richter
    • 3
  1. 1.Interdisciplinary Center for Scientific ComputingHeidelberg UniversityHeidelbergGermany
  2. 2.Mathematics DepartmentUniversity of Erlangen-NurembergErlangenGermany
  3. 3.Institute for Applied MathematicsHeidelberg UniversityHeidelbergGermany

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