Advertisement

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

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 

Mathematics Subject Classification

35J60 35K57 74J30 92C10 

References

  1. 1.
    Adam, J., Bellomo, N.: A Survey of Models for Tumor-Immune System Dynamics. Birkhäuser, Boston (1997) CrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12(5), 737–754 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, A., Chaplain, M.A.J., Rejniak, K.: Single-Cell-Based Models in Biology and Medicine. Birkhauser, Basel (2007) CrossRefzbMATHGoogle Scholar
  4. 4.
    Araujo, R., McElwain, D.: A history of the study of solid tumor growth: the contribution of mathematical models. Bull. Math. Biol. 66, 1039–1091 (2004) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 4, 593–646 (2008) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bellomo, N., Preziosi, L.: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32, 413–452 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berestycki, H., Hamel, F.: Reaction-Diffusion Equations and Propagation Phenomena. Springer, New York (2012) Google Scholar
  8. 8.
    Breward, C.J.W., Byrne, H.M., Lewis, C.E.: The role of cell-cell interactions in a two-phase model for avascular tumor growth. J. Math. Biol. 45(2), 125–152 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Byrne, H., Drasdo, D.: Individual-based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58, 657–687 (2009) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Byrne, H.M., King, J.R., McElwain, D.L.S., Preziosi, L.: A two-phase model of solid tumor growth. Appl. Math. Lett. 16, 567–573 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Byrne, H., Preziosi, L.: Modelling solid tumor growth using the theory of mixtures. Math. Med. Biol. 20, 341–366 (2003) CrossRefzbMATHGoogle Scholar
  12. 12.
    Chaplain, M.A.J., Graziano, L., Preziosi, L.: Mathematical modeling of the loss of tissue compression responsiveness and its role in solid tumor development. Math. Med. Biol. 23, 197–229 (2006) CrossRefzbMATHGoogle Scholar
  13. 13.
    Chatelain, C., Balois, T., Ciarletta, P., Amar, M.: Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture. New J. Phys. 13, 115013+21 (2011) 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. (2013, in preparation) Google Scholar
  15. 15.
    Ciarletta, P., Foret, L., 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), 345–368 (2011) 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), 2321–2344 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cristini, V., Lowengrub, J., Nie, Q.: Nonlinear simulations of tumor growth. J. Math. Biol. 46, 191–224 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    De Angelis, E., Preziosi, L.: Advection-diffusion models for solid tumor evolution in vivo and related free boundary problem. Math. Models Methods Appl. Sci. 10(3), 379–407 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Drasdo, D.: In: Alt, W., Chaplain, M., Griebel, M. (eds.) On Selected Individual-Based Approaches to the Dynamics of Multicellular Systems, Multiscale Modeling. Birkhauser, Basel (2003) Google Scholar
  20. 20.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. AMS, Providence (1998) zbMATHGoogle Scholar
  21. 21.
    Friedman, A.: A hierarchy of cancer models and their mathematical challenges. Discrete Contin. Dyn. Syst., Ser. B 4(1), 147–159 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Funaki, M., Mimura, M., Tsujikawa, A.: Traveling front solutions in a chemotaxis-growth model. Interfaces Free Bound. 8, 223–245 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gardner, R.A.: Existence of travelling wave solution of predator-prey systems via the connection index. SIAM J. Appl. Math. 44, 56–76 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hoehme, S., Drasdo, D.: A cell-based simulation software for multi-cellular systems. Bioinformatics 26(20), 2641–2642 (2010) CrossRefGoogle Scholar
  25. 25.
    Lowengrub, J.S., Frieboes, H.B., Jin, F., et al.: Nonlinear modelling of cancer: bridging the gap between cells and tumors. Nonlinearity 23, R1–R91 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Murray, J.D.: Mathematical Biology. Springer, New York (1989) CrossRefzbMATHGoogle Scholar
  27. 27.
    Nadin, G., Perthame, B., Ryzhik, L.: Traveling waves for the Keller-Segel system with Fisher birth terms. Interfaces Free Bound. 10, 517–538 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Perthame, B., Quirós, F., Vázquez, J.L.: The Hele-Shaw asymptotics for mechanical models of tumor growth. (2013, in preparation) Google Scholar
  29. 29.
    Preziosi, L., Tosin, A.: Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58, 625–656 (2009) 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, 051907 (2009) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ranft, J., Basan, M., Elgeti, J., et al.: Fluidization of tissues by cell division and apoptosis. Proc. Natl. Acad. Sci. USA 107(49), 20863–20868 (2010) CrossRefGoogle Scholar
  32. 32.
    Roose, T., Chapman, S., Maini, P.: Mathematical models of avascular tumor growth: a review. SIAM Rev. 49(2), 179–208 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sánchez-Garduño, F., Maini, P.K.: Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Differ. Equ. 117(2), 281–319 (1995) CrossRefzbMATHGoogle Scholar
  34. 34.
    Weinberger, H.F., Lewis, M.A., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Min Tang
    • 1
    • 2
  • Nicolas Vauchelet
    • 2
    • 3
  • Ibrahim Cheddadi
    • 2
    • 3
  • Irene Vignon-Clementel
    • 2
    • 3
  • Dirk Drasdo
    • 2
    • 3
  • Benoît Perthame
    • 2
    • 3
  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

Personalised recommendations