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Numerical analysis of a chemotaxis model for tumor invasion

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Abstract

This paper is devoted to the study of a time-discrete scheme and its corresponding fully discretization approximating a d-dimensional chemotaxis model describing tumor invasion, d ≤ 3. This model describes the chemotactic attraction experienced by the tumor cells and induced by a so-called active extracellular matrix, which is a chemical signal produced by a biological reaction between the extracellular matrix and a matrix-degrading enzyme. In order to construct the numerical approximations and to control the chemo-attraction term in the tumor cells equation, we introduce an equivalent model with a new variable given by the gradient of the active extracellular matrix and use an inductive strategy. Then, we consider a first-order and non-linear time-discrete scheme which is mass-conservative and possesses the property of positivity for all the biological variables. After, we study the corresponding fully discrete finite element with “mass-lumping” approximation proving well-posedness, mass-conservation and the non-negativity of the extracellular matrix, the degrading enzyme, and the active extracellular matrix. In addition, we obtain uniform strong estimates required in the convergence analysis, and we prove optimal error estimates and convergence towards regular solutions. Finally, we provide some numerical results in agreement with our theoretical analysis with respect to the positivity and the error estimates.

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References

  1. Amrouche, C., Seloula, N.E.H.: lp-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23(1), 37–92 (2013)

    Article  MathSciNet  Google Scholar 

  2. Anderson, A.R.A., Chaplain, M.A.J., Newman, E.L., Steele, R.J.C., Thompson, A.M.: Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129–254 (2000)

    Article  Google Scholar 

  3. Barrett, J., Blowey, J.: Finite element approximation of a nonlinear cross-diffusion population model. Numer. Math. 98(2), 195–221 (2004)

    Article  MathSciNet  Google Scholar 

  4. Becker, R., Feng, X., Prohl, A.: Finite element approximations of the Ericksen- Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46, 1704–1731 (2008)

    Article  MathSciNet  Google Scholar 

  5. Bessemoulin-Chatard, M., Jüngel, A.: A finite volume scheme for a Keller-Segel model with additional cross-diffusion. IMA J. Numer. Anal. 34(1), 96–122 (2014)

    Article  MathSciNet  Google Scholar 

  6. Chamoun, G., Saad, M., Talhouk, R.: Monotone combined edge finite volume-finite element scheme for anisotropic Keller-Segel model. Numer. Methods Partial Differ. Equ. 30(3), 1030–1065 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chaplain, M.A.J., Anderson, A.R.A.: Mathematical modelling of tissue invasion. In: Preziosi, L. (ed.) Cancer Modelling and Simulation, pp 267–297. Chapman & Hall/CRT (2003)

  8. Chaplain, M.A.J., Lachowicz, M., Szymańska, Z., Wrzosek, D.: Mathematical modelling of cancer invasion: the importance of cell-cell adhesion and cell-matrix adhesion. Math. Models Methods Appl. Sci. 21, 719–743 (2011)

    Article  MathSciNet  Google Scholar 

  9. Chertock, A., Kurganov, A.: A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111(2), 169–205 (2008)

    Article  MathSciNet  Google Scholar 

  10. Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004)

    Article  MathSciNet  Google Scholar 

  11. Duarte-Rodríguez, A., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A., Villamizar-Roa, E.J.: Numerical analysis for a chemotaxis-Navier-Stokes system. ESAIM Math. Model. Numer. Anal. 55(suppl.), S417–S445 (2021)

    Article  MathSciNet  Google Scholar 

  12. Duczek, S., Gravenkamp, H.: Critical assessment of different mass lumping schemes for higher order serendipity finite elements. Comput. Methods Appl. Mech. Eng. 350, 836–897 (2019)

    Article  MathSciNet  Google Scholar 

  13. Fernández-Romero, A., Guillén-González, F., Suárez, A.: A Glioblastoma PDE-ODE model including chemotaxis and vasculature. ESAIM Math. Model. Numer. Anal. 56(2), 407–431 (2022)

    Article  MathSciNet  Google Scholar 

  14. Fujie, K., Ito, A., Yokota, T.: Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type. Adv. Math. Sci. Appl. 24, 67–84 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Fujie, K., Ito, A., Winkler, M., Yokota, T.: Stabilization in a chemotaxis model for tumor invasion. Discrete Contin. Dyn. Syst. 36(1), 151–169 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Guillén-González, F., Gutiérrez-Santacreu, J. V.: From a cell model with active motion to a Hele-Shaw-like system: a numerical approach. Numer. Math. 143(1), 107–137 (2019)

    Article  MathSciNet  Google Scholar 

  17. Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A.: Unconditionally energy stable fully discrete schemes for a chemo-repulsion model. Math. Comp. 88, 2069–2099 (2019)

    Article  MathSciNet  Google Scholar 

  18. Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A.: Study of a chemo-repulsion model with quadratic production. Part II: analysis of an unconditionally energy-stable fully discrete scheme. Comput. Math. Appl. 80, 636–652 (2020)

    Article  MathSciNet  Google Scholar 

  19. Gutiérrez-Santacreu, J. V., Rodríguez-Galván, J. R.: Analysis of a fully discrete approximation for the classical Keller-Segel model: lower and a priori bounds. Comput. Math. Appl. 85, 69–81 (2021)

    Article  MathSciNet  Google Scholar 

  20. Hecht, F.: New development in freefem++. J. Numer. Math. 20 (3-4), 251–265 (2012)

    Article  MathSciNet  Google Scholar 

  21. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  Google Scholar 

  22. Hillen, T., Painter, K., Winkler, M.: Convergence of a cancer invasion model to a logistic chemotaxis model. Math. Models Methods Appl. Sci. 23, 165–198 (2013)

    Article  MathSciNet  Google Scholar 

  23. Jin, H.-Y., Xiang, T.: Boundedness and exponential convergence in a chemotaxis model for tumor invasion. Nonlinearity 29(12), 3579–3596 (2016)

    Article  MathSciNet  Google Scholar 

  24. Marciniak-Czochra, A., Ptashnyk, M.: Boundedness of solutions of a haptotaxis model. Math. Models Methods Appl. Sci. 20, 449–476 (2009)

    Article  MathSciNet  Google Scholar 

  25. Morales-Rodrigo, C.: Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours. Math Comput. Model. 47, 604–613 (2008)

    Article  MathSciNet  Google Scholar 

  26. Necas, J.: Les Methodes Directes En Theorie Des Equations Elliptiques. Editeurs Academia, Prague (1967)

    MATH  Google Scholar 

  27. Niño-Celis, V., Rueda-Gómez, D. A., Villamizar-Roa, E. J.: Convergence and positivity of finite element methods for a haptotaxis model of tumoral invasion. Comput. Math. Appl. 89, 20–33 (2021)

    Article  MathSciNet  Google Scholar 

  28. Saito, N.: Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Commun. Pure Appl. Anal. 11, 339–364 (2012)

    Article  MathSciNet  Google Scholar 

  29. Stinner, C., Surulescu, C., Winkler, M.: Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007 (2014)

    Article  MathSciNet  Google Scholar 

  30. Szymańska, Z., Morales-Rodrigo, C., Lachowicz, M., Chaplain, M.: Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 19, 257–281 (2009)

    Article  MathSciNet  Google Scholar 

  31. Tao, Y.: Global existence for a haptotaxis model of cancer invasion with tissue remodeling. Nonlinear Anal. Real World Appl. 12, 418–435 (2011)

    Article  MathSciNet  Google Scholar 

  32. Zhang, J., Zhu, J., Zhang, R.: Characteristic splitting mixed finite element analysis of Keller-Segel chemotaxis models. Appl. Math. Comput. 278, 33–44 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Zhigun, A., Surulescu, C., Uatay, A.: Global existence for a degenerate haptotaxis model of cancer invasion. Z. Angew. Math. Phys. 67(29), Art. 146 (2016)

    Article  MathSciNet  Google Scholar 

  34. Zhou, G., Saito, N.: Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis. Numer. Math. 135(1), 265–311 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments.

Funding

The authors have been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, Colombia.

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Authors and Affiliations

  1. Escuela de Matemáticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia

    Jhean E. Pérez-López, Diego A. Rueda-Gómez & Élder J. Villamizar-Roa

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  1. Jhean E. Pérez-López
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  2. Diego A. Rueda-Gómez
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  3. Élder J. Villamizar-Roa
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Correspondence to Diego A. Rueda-Gómez.

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Communicated by: Siddhartha Mishra

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Pérez-López, J.E., Rueda-Gómez, D.A. & Villamizar-Roa, É.J. Numerical analysis of a chemotaxis model for tumor invasion. Adv Comput Math 48, 26 (2022). https://doi.org/10.1007/s10444-022-09949-z

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  • DOI: https://doi.org/10.1007/s10444-022-09949-z

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