Chinese Annals of Mathematics, Series B

, Volume 34, Issue 2, pp 295–318 | Cite as

Composite waves for a cell population system modeling tumor growth and invasion

  • Min Tang
  • Nicolas Vauchelet
  • Ibrahim Cheddadi
  • Irene Vignon-Clementel
  • Dirk Drasdo
  • Benoît Perthame
Article

Abstract

In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or/and on the mechanical stress in the tumor.

For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, the authors prove that there are always traveling waves above a minimal speed, and analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions, and in particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.

Keywords

Traveling waves Reaction-diffusion Tumor growth Elastic material 

2000 MR Subject Classification

35J60 35K57 74J30 92C10 

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References

  1. [1]
    Adam, J. and Bellomo, N., A Survey of Models for Tumor-Immune System Dynamics, Birkhäuser, Boston, 1997.MATHCrossRefGoogle Scholar
  2. [2]
    Ambrosi, D. and Preziosi, L., On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sci., 12(5), 2002, 737–754.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Anderson, A., Chaplain, M. A. J. and Rejniak, K., Single-Cell-Based Models in Biology and Medicine, Birkhauser, Basel, 2007.MATHCrossRefGoogle Scholar
  4. [4]
    Araujo, R. and McElwain, D., A history of the study of solid tumour growth: the contribution of mathematical models, Bull Math. Biol., 66, 2004, 1039–1091.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Bellomo, N., Li, N. K. and Maini, P. K., On the foundations of cancer modelling: selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 4, 2008, 593–646.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Bellomo, N. and Preziosi, L., Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Model., 32, 2000, 413–452.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Berestycki, H. and Hamel, F., Reaction-Diffusion Equations and Popagation Phenomena, Springer-Verlag, New York, 2012.Google Scholar
  8. [8]
    Breward, C. J. W., Byrne, H. M. and Lewis, C. E., The role of cell-cell interactions in a two-phase model for avascular tumour growth, J. Math. Biol., 45(2), 2002, 125–152.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Byrne, H. and Drasdo, D., Individual-based and continuum models of growing cell populations: a comparison, J. Math. Biol., 58, 2009, 657–687.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Byrne, H. M., King, J. R., McElwain, D. L. S. and Preziosi, L., A two-phase model of solid tumor growth, Appl. Math. Lett., 16, 2003, 567–573.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Byrne, H. and Preziosi, L., Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol., 20, 2003, 341–366.MATHCrossRefGoogle Scholar
  12. [12]
    Chaplain, M. A. J., Graziano, L. and Preziosi, L., Mathematical modeling of the loss of tissue compression responsiveness and its role in solid tumor development, Math. Med. Biol., 23, 2006, 197–229.MATHCrossRefGoogle Scholar
  13. [13]
    Chatelain, C., Balois, T., Ciarletta, P. and Amar, M., Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture, New Journal of Physics, 13, 2011, 115013+21.CrossRefGoogle Scholar
  14. [14]
    Chedaddi, I., Vignon-Clementel, I. E., Hoehme, S., et al., On constructing discrete and continuous models for cell population growth with quantitatively equal dynamics, in preparation.Google Scholar
  15. [15]
    Ciarletta, P., Foret, L. and Amar, M. B., The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis, J. R. Soc. Interface, 8(56), 2011, 345–368.CrossRefGoogle Scholar
  16. [16]
    Colin, T., Bresch, D., Grenier, E., et al., Computational modeling of solid tumor growth: the avascular stage, SIAM J. Sci. Comput., 32(4), 2010, 2321–2344.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Cristini, V., Lowengrub, J. and Nie, Q., Nonlinear simulations of tumor growth, J. Math. Biol., 46, 2003, 191–224.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    De Angelis, E. and Preziosi, L., Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem, Math. Models Methods Appl. Sci., 10(3), 2000, 379–407.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Drasdo, D., On selected individual-based approaches to the dynamics of multicellular systems, Multiscale Modeling, W. Alt, M. Chaplain and M. Griebel (eds.), Birkhauser, Basel, 2003.Google Scholar
  20. [20]
    Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, A. M. S., Providence, RI, 1998.MATHGoogle Scholar
  21. [21]
    Friedman, A., A hierarchy of cancer models and their mathematical challenges, DCDS(B), 4(1), 2004, 147–159.MATHGoogle Scholar
  22. [22]
    Funaki, M., Mimura, M. and Tsujikawa, A., Traveling front solutions in a chemotaxis-growth model, Interfaces and Free Boundaries, 8, 2006, 223–245.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Gardner, R. A., Existence of travelling wave solution of predator-prey systems via the connection index, SIAM J. Appl. Math., 44, 1984, 56–76.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Hoehme, S. and Drasdo, D., A cell-based simulation software for multi-cellular systems, Bioinformatics, 26(20), 2010, 2641–2642.CrossRefGoogle Scholar
  25. [25]
    Lowengrub, J. S., Frieboes, H. B., Jin, F., et al., Nonlinear modelling of cancer: bridging the gap between cells and tumours, Nonlinearity, 23, 2010, R1–R91.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Murray, J. D., Mathematical biology, Springer-Verlag, New York, 1989.MATHCrossRefGoogle Scholar
  27. [27]
    Nadin, G., Perthame, B. and Ryzhik, L., Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces and Free Boundaries, 10, 2008, 517–538.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Perthame, B., Quirós, F. and Vázquez, J. L., The Hele-Shaw asymptotics for mechanical models of tumor growth, in preparation.Google Scholar
  29. [29]
    Preziosi, L. and Tosin, A., Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications, J. Math. Biol., 58, 2009, 625–656.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Radszuweit, M., Block, M., Hengstler, J. G., et al., Comparing the growth kinetics of cell populations in two and three dimensions, Phys. Rev. E, 79, 2009, 051907-1–12.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Ranft, J., Basan, M., Elgeti, J., et al., Fluidization of tissues by cell division and apaptosis, PNAS, 107(49), 2010, 20863–20868.CrossRefGoogle Scholar
  32. [32]
    Roose, T., Chapman, S. and Maini, P., Mathematical models of avascular tumour growth: a review, SIAM Rev., 49(2), 2007, 179–208.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Sánchez-Garduño, F. and Maini, P. K., Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Diff. Eq., 117(2), 1995, 281–319.MATHCrossRefGoogle Scholar
  34. [34]
    Weinberger, H. F., Lewis, M. A. and Li, B., Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45, 2002, 183–218.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Min Tang
    • 1
    • 2
  • Nicolas Vauchelet
    • 3
    • 4
  • Ibrahim Cheddadi
    • 3
    • 4
  • Irene Vignon-Clementel
    • 3
    • 4
  • Dirk Drasdo
    • 3
    • 4
  • Benoît Perthame
    • 3
    • 4
  1. 1.Department of mathematics, Institute of Natural Sciences and MOE-LSCShanghai Jiao Tong UniversityShanghaiChina
  2. 2.INRIA Paris RocquencourtParisFrance
  3. 3.Laboratoire Jacques-Louis LionsUPMC Univ Paris 06 and CNRS UMR 7598ParisFrance
  4. 4.INRIA Paris RocquencourtParisFrance

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