Journal of Mathematical Biology

, Volume 72, Issue 4, pp 973–996

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

• Yifan Yang
• Willi Jäger
• 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):1297-1316
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):093103
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 2:126-141
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):1089-1110
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):743-769
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 10:53-62
14. Holzapfel G (2000) Nonlinear solid mechanics, a continuum approach for engineering. John Wiley and Sons, Chichester
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 30:753-767
16. Hron J, Madlik M (2007) Fluid-structure interaction with applications in biomechanics. Nonlinear Anal Real World Appl 8:1431-1458
17. Humphrey JD (2002) Cardiovascular solid mechanics, cells, tissues, and organs. Springer, NewYork
18. Ibragimov AI, McNeal CJ, Ritter LR, Walton JR (2005) A mathematical model of atherogenesis as an inflammatory response. Math Med Biol 22(4):305-333
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):2783-2791
20. Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, Cambridge
21. Jones GW, Chapman SJ (2012) Modeling growth in biological materials. SIAM Rev 54(1):52-118
22. Kalita P, Schaefer R (2008) Mechanical models of artery walls. Arch Comp Methods Eng 15:1-36
23. Li ZY, Howarth SPS, Tang T, Gillard JH (2006) How critical is fibrous cap thickness to carotid plaque stability? Stroke 37(5):1195-1199
24. Ougrinovskaia A, Thompson R, Myerscough M (2010) An ODE model of early stages of atherosclerosis: mechanisms of theinflammatory response. Bull Math Biol 72:1534-1561
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 2:163-197
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):1488-1511
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):061010
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):727-736
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 32:947-960
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 142:202-217
38. Weller F (2008) Platelet deposition in non-parallel flow. J Math Biol 57:333-359
39. Weller F, Neuss-Radu M, Jäger W (2013) Analysis of a free boundary problem modeling thrombus growth. SIAM J Math Anal 45:809-833
40. Wick T (2011) Fluid-structure interactions using different mesh motion techniques. Comp Struct 89:1456-1467
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 227:437-443

• Yifan Yang
• 1
• Willi Jäger
• 1