A Rigorous Sharp Interface Limit of a Diffuse Interface Model Related to Tumor Growth

Abstract

In this paper, we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter \(\varepsilon \), representing the interface thickness between the tumorous and non-tumorous cells, tends to zero. More in particular, we analyze here a gradient-flow-type model arising from a modification of the recently introduced model for tumor growth dynamics in Hawkins-Daruud et al. (Int J Numer Math Biomed Eng 28:3–24, 2011) (cf. also Hilhorst et al. Math Models Methods Appl Sci 25:1011–1043, 2015). Exploiting the techniques related to both gradient flows and gamma convergence, we recover a condition on the interface \(\Gamma \) relating the chemical and double-well potentials, the mean curvature, and the normal velocity.

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References

  1. Abels, H., Lengeler, D.: On sharp interface limits for diffuse interface models for two-phase flows. Interfaces Free Bound. 16, 395–418 (2014)

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variations and Free Discontinuity Problems. Clarendon Press, Oxford (2000)

    Google Scholar 

  3. Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modeling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18(4), 593–646 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  4. Chen, X.: Global asymptotic limit of solutions of the Cahn–Hilliard equation. J. Differ. Geom. 44, 262–311 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  5. Chen, Y., Wise, S.M., Shenoy, V.B., Lowengrub, J.S.: A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane. Int. J. Numer. Methods Biomed. Eng. 30, 726–754 (2014)

    MathSciNet  Article  Google Scholar 

  6. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth. Nonlinear Anal. Real World Appl. 26, 93–108 (2015a)

    MathSciNet  Article  MATH  Google Scholar 

  7. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discrete Contin. Dyn. Syst. Ser. S 1–19. Preprint arXiv:1503.00927 [math.AP] (2015b) (To appear)

  8. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal Distributed Control of a Diffuse Interface Model of Tumor Growth, pp. 1–32. Preprint arXiv:1601.04567v1 (2016)

  9. Cristini, V., Frieboes, H.B., Li, X., Lowengrub, J.S., Macklin, P., Sanga, S., Wise, S.M., Zheng, X.: Nonlinear modeling and simulation of tumor growth. In: Bellomo, N., Chaplain, M., de Angelis, E. (eds.) Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy. Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Boston (2008)

    Google Scholar 

  10. Cristini, V., Li, X., Lowengrub, J.S., Wise, S.M.: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58, 723–763 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  11. Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer. An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  12. Dai, M., Feireisl, E., Rocca, E., Schimperna, G., Schonbek, M.: Analysis of a Diffuse Interface Model of Multispecies Tumor Growth, pp. 1–18. Preprint arXiv:1507.07683 (2015)

  13. Fasano, A., Bertuzzi, A., Gandolfi, A.: Mathematical modelling of tumour growth and treatment. In: Quarteroni, A., Formaggia, L., Veneziani, A. (eds.) Complex Systems in Biomedicine. Biomedical and Life Science. Springer, Berlin (2006)

  14. Friedman, A., Bellomo, N., Maini, P.K.: Mathematical analysis and challenges arising from models of tumor growth. Math. Models Methods Appl. Sci. 17, 1751–1772 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  15. Frigeri, S., Grasselli, M., Rocca, E.: On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26, 215–243 (2015)

    Article  Google Scholar 

  16. Garcke, H., Lam, K.F.: Well-posedness of a Cahn–Hilliard System Modelling Tumour Growth with Chemotaxis and Active Transport, pp. 1–28. Preprint arXiv:1511.06143 (2015)

  17. Garcke, H., Lam, K.F., Sitka, E., Styles, V.: A Cahn–Hilliard–Darcy Model for Tumour Growth with Chemotaxis and Active Transport, pp. 1–45. Preprint arXiv:1508.00437 (2015)

  18. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  19. Giacomin, G., Lebowitz, J.L.: Phase Segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  20. Graziano, L., Preziosi, L.: Mechanics in tumor growth. In: Mollica, F., Preziosi, L., Rajagopal, K.R. (eds.) Modeling of Biological Materials. Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Boston (2007)

    Google Scholar 

  21. Hawkins-Daruud, A., van der Zee, K.G., Oden, J.T.: Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Math. Biomed. Eng. 28, 3–24 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  22. Hilhorst, D., Kampmann, J., Nguyen, T.N., van der Zee, K.G.: Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25, 1011–1043 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  23. Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259, 3032–3077 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  24. Le, N.Q.: A gamma-convergence approach to the Cahn–Hilliard equation. Calc. Var. Partial Differ. Equ. 32, 499–522 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  25. Lowengrub, J.S., Frieboes, H.B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S.M., Cristini, V.: Nonlinear modeling of cancer: bridging the gap between cells and tumors. Nonlinearity 23, R1–R9 (2010)

    Article  MATH  Google Scholar 

  26. Lowengrub, J.S., Titi, E., Zhao, K.: Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24, 1–44 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. Luckhaus, S., Modica, L.: The Gibbs–Thompson relation whitin the gradient theory of phase transitions. Arch. Ration. Mech. Anal. 107(1), 71–83 (1989)

    Article  MATH  Google Scholar 

  28. Modica, L., Mortola, S.: Un esempio di \(\Gamma \)-convergenza, (Italian). Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  29. Roger, M., Tonegawa, Y.: Convergence of phase-field approximations to the Gibbs–Thomson law. Calc. Var. Partial Differ. Equ. 32, 111–136 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  30. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg–Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  31. Taylor, M.E.: Partial Differential Equations, Basic Theory. Springer, Berlin (1999)

    Google Scholar 

  32. Tonegawa, Y.: Phase field model with a variable chemical potential. Proc. R. Soc. Edinb. Sect. A 134(4), 993–1019 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  33. Tonegawa, Y.: A diffuse interface whose chemical potential lies in Sobolev spaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4(3), 487–510 (2005)

    MATH  Google Scholar 

  34. Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth—I: model and numerical method. J. Theor. Biol. 253, 524–543 (2008)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and their precious suggestions. The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) is gratefully acknowledged by the authors. The present paper also benefits from the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM) and the IMATI-C.N.R. Pavia.

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Correspondence to Riccardo Scala.

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Communicated by Irene Fonseca.

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Rocca, E., Scala, R. A Rigorous Sharp Interface Limit of a Diffuse Interface Model Related to Tumor Growth. J Nonlinear Sci 27, 847–872 (2017). https://doi.org/10.1007/s00332-016-9352-3

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Keywords

  • Sharp interface limit
  • Gamma convergence
  • Gradient-flow
  • Diffuse interface models
  • Cahn–Hilliard equation
  • Reaction–diffusion equation
  • Nonlocal operators
  • Tumor growth

Mathematics Subject Classification

  • 49J40
  • 82B24
  • 35K46
  • 35K57
  • 35R11
  • 92B05