Advertisement

Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

  • Federica Bubba
  • Benoît Perthame
  • Camille PoucholEmail author
  • Markus Schmidtchen
Article
  • 45 Downloads

Abstract

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a phase-segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson–Bénilan estimates cannot be established in our context. We are led, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an \(L^1\) version in place of the standard upper bound.

Notes

Acknowledgements

F.B. and B.P. have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 740623). M.S. acknowledges the kind invitation to LJLL funded by the previous grant. Furthermore, M.S. received funding for two research visits from the Doris Chen Mobility Award awarded by Imperial College London. C.P. acknowledges support from the Swedish Foundation of Strategic Research Grant AM13-004.

References

  1. 1.
    Aronson, D.G., Bénilan, P.: Régularité des solutions de l’équation des milieux poreux dans \({\mathbb{R}}^n\). CR Acad. Sci. Paris Sér. AB288(2), A103–A105, 1979zbMATHGoogle Scholar
  2. 2.
    Bertsch, M., Dal Passo, R., Mimura, M.: A free boundary problem arising in a simplified tumour growth model of contact inhibition. Interfaces Free Bound. 12(2), 235–250, 2010CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bertsch, M., Gurtin, M.E., Hilhorst, D.: On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities. Nonlinear Anal. Theory Methods Appl. 11(4), 493–499, 1987CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertsch, M., Hilhorst, D., Izuhara, H., Mimura, M.: A nonlinear parabolic–hyperbolic system for contact inhibition of cell-growth. Differ. Equ. Appl. 4(1), 137–157, 2012MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bresch, D., Colin, T., Grenier, E., Ribba, B., Saut, O.: Computational modeling of solid tumor growth: the avascular stage. SIAM J. Sci. Comput. 32(4), 2321–2344, 2010CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bris, C.L., Lions, P.-L.: Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Commun. Partial Differ. Equ. 33(7), 1272–1317, 2008CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Busenberg, S.N., Travis, C.C.: Epidemic models with spatial spread due to population migration. J. Math. Biol. 16(2), 181–198, 1983CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Byrne, H.M., Drasdo, D.: Individual-based and continuum models of growing cell populations: a comparison. Math. Med. Biol. 58(4–5), 657–687, 2003MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carrillo, J.A., Fagioli, S., Santambrogio, F., Schmidtchen, M.: Splitting schemes and segregation in reaction cross-diffusion systems. SIAM J. Math. Anal. 50(5), 5695–5718, 2018CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Craig, K., Kim, I., Yao, Y.: Congested aggregation via Newtonian interaction. Arch. Ration. Mech. Anal. 227(1), 1–67, 2018CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Dambrine, J., Meunier, N., Maury, B., Roudneff-Chupin, A.: A congestion model for cell migration. Commun. Pure Appl. Anal. 11(1), 243–260, 2012CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Degond, P., Hecht, S., Vauchelet, N.: Incompressible limit of a continuum model of tissue growth for two cell populations. arXiv:1809.05442, 2018
  13. 13.
    Gurtin, M.E., Pipkin, A.C.: A note on interacting populations that disperse to avoid crowding. Quart. Appl. Math. 42, 87–94, 1984 CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Gwiazda, P., Perthame, B., Świerczewska-Gwiazda, A.: A two species hyperbolic–parabolic model of tissue growth. arXiv:1809.01867, 2018
  15. 15.
    Hecht, S., Vauchelet, N.: Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint. Commun. Math. Sci. 15(7), 1913, 2017CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Kim, I., Požár, N.: Porous medium equation to Hele–Shaw flow with general initial density. Trans. Am. Math. Soc. 370(2), 873–909, 2018CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Kim, I., Turanova, O.: Uniform convergence for the incompressible limit of a tumor growth model. Ann. Inst. H. Poincaré Anal. Non Linéaire35(5), 1321–1354, 2018CrossRefADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Kim, I.C., Perthame, B., Souganidis, P.E.: Free boundary problems for tumor growth: a viscosity solutions approach. Nonlinear Anal. 138, 207–228, 2016CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Lorenzi, T., Lorz, A., Perthame, B.: On interfaces between cell populations with different mobilities. Kinet. Relat. Models10(1), 299–311, 2017CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Lowengrub, J.S., Frieboes, H.B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S.M., Cristini, V.: Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity23(1), R1–R91, 2010CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Lu, P., Ni, L., Vázquez, J.-L., Villani, C.: Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. (9)91(1), 1–19, 2009CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821, 2010CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Maury, B., Roudneff-Chupin, A., Santambrogio, F.: Congestion-driven dendritic growth. Discrete Contin. Dyn. Syst. 34(4), 1575–1604, 2014CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Maury, B., Roudneff-Chupin, A., Santambrogio, F., Venel, J.: Handling congestion in crowd motion modeling. Netw. Heterog. Media6(3), 485–519, 2011CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Mellet, A., Perthame, B., Quirós, F.: A Hele–Shaw problem for tumor growth. J. Funct. Anal. 273(10), 3061–3093, 2017CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Mészáros, A.R., Santambrogio, F.: Advection–diffusion equations with densityconstraints. Anal. PDE9(3), 615–644, 2016CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Motsch, S., Peurichard, D.: From short-range repulsion to Hele–Shaw problem in a model of tumor growth. J. Math. Biol. 76(1–2), 205–234, 2018CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Perthame, B., Quirós, F., Tang, M., Vauchelet, N.: Derivation of a Hele–Shaw type system from a cell model with active motion. Interfaces Free Bound. 16, 489–508, 2014CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Perthame, B., Quirós, F., Vázquez, J.L.: The Hele–Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal. 212(1), 93–127, 2014CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Preziosi, L.: A review of mathematical models for the formation of vascular networks. J. Theor. Biol. 333, 174–209, 2012 MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ranft, J., Basana, M., Elgeti, J., Joanny, J.-F., Prost, J., Jülicher, F.: Fluidization of tissues by cell division and apoptosis. Natl. Acad. Sci. USA49, 657–687, 2010Google Scholar
  32. 32.
    Roose, T., Chapman, S.J., Maini, P.K.: Mathematical models of avascular tumour growth: a review. SIAM Rev. 49(2), 179–208, 2007CrossRefADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Vázquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford 2007zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Inria MAMBA Team, Laboratoire Jacques-Louis LionsSorbonne Université, CNRS, Université Paris-Diderot SPCParisFrance
  2. 2.Department of MathematicsKTH - Royal institute of TechnologyStockholmSweden
  3. 3.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations