Research article Special Issues

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

  • Received: 27 February 2019 Accepted: 17 June 2019 Published: 22 June 2019
  • This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.

    Citation: Alina Chertock, Pierre Degond, Sophie Hecht, Jean-Paul Vincent. Incompressible limit of a continuum model of tissue growth with segregation for two cell populations[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5804-5835. doi: 10.3934/mbe.2019290

    Related Papers:

  • This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.


    加载中


    [1] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Part. Diff. Eq., 26(2001), 101–174, URL https://doi.org/10.1081/PDE-100002243.
    [2] D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasser-stein metric, ESAIM: M2AN, 33(1999), 837–852, URL https://doi.org/10.1051/m2an: 1999166.
    [3] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29(1998), 1–17, URL https://doi.org/10.1137/S0036141096303359.
    [4] D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Phys. Biol., 2(2015), 133–147.
    [5] J. Galle, M. Loeffler and D. Drasdo, Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro, Biophys. J., 88(2005), 62–75, URL http://www.sciencedirect.com/science/article/pii/S0006349505730873.
    [6] R. Araujo and D. McElwain, A history of the study of solid tumour growth: the contribution of mathematical modelling, D.L.S. Bull. Math. Biol., 66(2004), 1039.
    [7] D. Bresch, T. Colin, E. Grenier, et al., Computational modeling of solid tumor growth: the avascular stage, SIAM J. Sci. Comput., 32(2010), 2321–2344, URL https://doi.org/10.1137/070708895.
    [8] H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135(1996), 187–216, URL http://www.sciencedirect.com/science/article/pii/0025556496000235.
    [9] P. Ciarletta, L. Foret and M. Ben Amar, The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis, J. Royal Soc. Interface, 8(2011), 345–368, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3030817/.
    [10] J. Ranft, M. Basan, J. Elgeti, et al., Fluidization of tissues by cell division and apoptosis, Proc. Natl. Acad. Sci., 107(2010), 20863–20868.
    [11] S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Part. Diff. Eq., 33(2008), 636–655, URL https://doi.org/10.1080/03605300701743848.
    [12] A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Am. Math. Soc., 360(2008), 5291–5342.
    [13] H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51(1972), 317–340, URL http://dx.doi.org/10.1002/sapm1972514317.
    [14] J. S. Lowengrub, H. B. Frieboes, F. Jin, et al., Nonlinear modelling of cancer: bridging the gap between cells and tumours, Nonlinearity, 23(2010), R1–R9, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2929802/.
    [15] D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem aris-ing in biology, Nonlinear Anal. Real, 4(2003), 261 – 285, URL http://www.sciencedirect.com/science/article/pii/S1468121802000093.
    [16] B. Perthame, F. Quirós and J. L. Vázquez, The Hele–Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., 212(2014), 93–127, URL https://doi.org/10.1007/s00205-013-0704-y.
    [17] B. Perthame, F. Quiròs, M. Tang et al., Derivation of a Hele-Shaw type system from a cell model with active motion, Interfaces Free Bound., 14(2014), 489–508.
    [18] B. Perthame and N. Vauchelet, Incompressible limit of a mechanical model of tumour growth with viscosity, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 373(2015), 20140283, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4535270/.
    [19] H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell popu-lations: a comparison, J. Math. Biol., 58(2008), 657, URL https://doi.org/10.1007/s00285-008-0212-0.
    [20] J. L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, 2007.
    [21] S. Hecht and N. Vauchelet, Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint, Commun. Math. Sci., 15(2017), 1913–1932, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5669502/.
    [22] A. Friedman and S. Y. Huang, Asymptotic behavior of solutions of ut = ∆φm(u) as m → ∞ with inconsistent initial values, Analyse Math. Appl., (1988), 165–180.
    [23] P. Bénilan, L. Boccardo and M. Herrero, On the limit of solutions of ut = ∆um as m → ∞., Interfaces Free Bound., 12(1989).
    [24] P. Bénilan and N. Igbida, La limite de la solution de ut = ∆pum lorsque m → ∞., C. R. Acad. Sci. Paris Sier, 321(1995), 1323–1328.
    [25] C. Elliott, M. Herrero, J. King, et al., The mesa problem: diffusion patterns for u t = ∇(u m ∇u) as m → ∞, IMA J. Appl. Math., 2(1986), 147–154.
    [26] O. Gil and F. Quirós, Convergence of the porous media equation to Hele-Shaw, Nonlinear Anal., 44(2001), 1111–1131, URL http://dx.doi.org/10.1016/S0362-546X(99)00325-9.
    [27] A. K. Dutt, Turing pattern amplitude equation for a model glycolytic reaction-diffusion system, J. Math. Chem., 48(2010), 841–855, URL https://doi.org/10.1007/s10910-010-9699-x.
    [28] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. The-oret. Biol., 26(1970), 399–415, URL http://www.sciencedirect.com/science/article/pii/0022519370900925.
    [29] A. J. Lotka, Contribution to the theory of periodic reactions, J. Phys. Chem., 14(1909), 271–274, URL http://dx.doi.org/10.1021/j150111a004.
    [30] M. Bertsch, M. E. Gurtin and L. A. P. D. Hilhorst, On interacting populations that disperse to avoid crowding: the effect of a sedentary colony, Q. Appl. Math., 19(1984), 1–12.
    [31] S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migra-tion, J. Math. Biol., 16(1983), 181–198, URL https://doi.org/10.1007/BF00276056.
    [32] M. Bertsch, R. D. Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces Free Bound., 12(2010), 235–250.
    [33] M. Bertsch, D. Hilhorst, H. Izuhara, et al., A non linear parabolic-hyperbolic system for contact inhibition of cell growth, Differ. Equ. Appl., 4(2010), 137–157.
    [34] A. J. Sherratt and M. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43(2001), 291–312.
    [35] G. Galiano, S. Shmarev and J. Velasco, Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem, Discrete Contin. Dyn. Syst., 96(2015), 339–357, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10564.
    [36] C. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Ration. Mech. Anal., 35(1986), 1479–1501.
    [37] J. A. Carrillo, S. Fagioli, F. Santambrogio, et al., Splitting schemes and segregation in reaction cross-diffusion systems, J. Math. Anal., 50(2017),5695–5718.
    [38] I. Kim and A. R. Mészáros, On nonlinear cross-diffusion systems: an optimal transport approach, Calc. Var. Partial Diff. Eq., 57 (2018), 79, URL https://doi.org/10.1007/s00526-018-1351-9.
    [39] A. Chertock, P. Degond and J. Neusser, An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations, J. Comput. Phys., 335(2017), 387 – 403, URL http://www.sciencedirect.com/science/article/pii/S0021999117300463.
    [40] C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase tran-sitions, ZAMM Z. Angew. Math. Mech, 85(2005), 839–857.
    [41] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17(2015), 233–258.
    [42] S. Gottlieb, D. Ketcheson and C. W. Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd. (2011).
    [43] S. Gottlieb, C. W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43(2001), 89–112.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3133) PDF downloads(652) Cited by(3)

Article outline

Figures and Tables

Figures(6)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog