MATHEMATICAL MODELING AND NUMERICALSIMULATION FOR CELLULAR SYSTEMS
Keywords:
Mathematical Modeling, Well-Posedness, Analysis & SimulationAbstract
In this work, we are interested in modeling biological cell migration and proliferation. These models provide a mathematical framework to analyze and simulate the intricate dynamics of tumor growth, considering factors such as cell proliferation, angiogenesis, and interactions with the surrounding microenvironment. This can help to develop new therapies, design biomaterials, and engineer tissues for regenerative medicine. We are going to derive the “simplest” mathematical model able to fulfill some requirements, such as travelling waves and sharp cell fronts. When the cell density is large enough, the continuous medium assumption is a good approximation and partial differential equations can be written. The well-posedness of the derived model is proved using semi-group theory and the numerical study has been carried out by the use of finite element methods. The obtained numerical results are shown to be efficient and qualitatively significant.
Downloads
References
[1] Adams, A.R., Fournier, J.J.F. (2003). Sobolev Spaces, 2nd ed. Elsevier Science Ltd. 3
[2] Allaire, G. (2005). Analyse numerique et optimisation. Une introduction `a la mod´elisation math´ematique et `a la simulation num´erique. Paris. 4
[3] Blanchet, A. (2010). A Gradient flow approach to Keller-Segel Systems. https://www.tse−fr.eu/sites/default/files/medias/doc/by/blanchet/dec2010/b4.pdf. 1, 2.2
[4] Brady, R., Enderling, H. (2019). Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to. Bull Math Biol, 81(12), 3722–3731. DOI: https://doi.org/10.1007/s11538-019-00640-x. 1
[5] Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. DOI : 10.1007/978-0-387-70914-7. 2.2, 3.1, 3.3, 3.2, 3.3, 4
[6] Meriem Boussebha and Fatma Zohra Nouri (2025), A Mathematical and Numerical Study of a Model for an Avascular Tumor Evolution, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 32, 247-267. 1, 2.1, 3.3
[7] N. Djedaidi, S. Gasmi and F.Z. Nouri (2023), Interface dynamics for a bi-phasic problem in heterogeneous porous media, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms 30 (2023), pp. 21-33. 3.2, 3.3, 3.4
[8] https://freefem.org/Edition 2024 5
[9] Hadamard, J. (1952). Lectures on Cauchy’s problem in linear partial differential equations. Dover, reprint. 2.2
[10] A. Hadji and F.Z. Nouri (2022), Mathematical and numerical study for a bioglass bioactivity degradation, International Journal of Advanced Science and Research , Vol 7, Issue 2, 2022, pp. 15-22. 1, 3.2
[11] Jones, D., Plank, M., Sleeman, B. (2009). Differential Equations and Mathematical Biology. New York: Chapman and Hall/CRC. DOI : http : //doi.org/10.1201/9781420083583. 1, 2.1
[12] Kolobov, A.V., Gubernov, V.V., Polezhaev, A.A. (2009). Autowaves in a model of invasive tumor growth. IOPHYSICS, 54, 232–237. DOI: https : //doi.org/10.1134/S0006350909020195. 1
[13] Macias-Diaz, J.E., Gallegos, A. (2018). A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy. J. Math Chem, 56, 1985–2000. DOI: https://doi.org/10.1007/s10910-017-0818-9. 1, 2.1
[14] F.Z. Nouri (2019), A New Approach for a Multiphase Flow Problem, Int. J. of Maths and Computation vol. 30 issue No 3, pp.32-42. 2.1, 3.2, 3.4, 5
[15] Pazy, A. (1992). Semi-groups of Linear Operators and Applications To Partial Differential Equations, 2nd ed. Springer Verlag, Berlin-Heidelberg New York-Tokyo. 2.2, 3.1, 3.1
[16] Raymond, J.P. (n.d.). Evolution equation, University of Paul Sabatier. www.math.univ-toulouse.fr/ raymond/book evolution.pdf. 3.4, 3.5
[17] Redouane, F., Jamshed, W., Devi, S.S.U. et al. (2022). Heat flow saturate of Ag/MgO-water hybrid nanofluid in heated trigonal enclosure with rotate cylindrical cavity by using Galerkin finite element. Sci Rep 12, 2302. https://doi.org/10.1038/s41598-022-06134-6 1
[18] Rejniak, K.A. (2007). An immersed boundary framework for modeling the growth of individual cells: an application to the early tumor development. J. Theor. Biol., 247(1), 186–204. 1, 2.1
[19] Rejniak, K.A. (2011). Hybrid models of tumor growth. Wiley Interdiscip. Rev. Syst. Biol. Med., 3(1), 115–125. 1
[20] Sabir, M., et al. (2017). A mathematical model of tumor hypoxia targeting in cancer treatment and its numerical simulation. Computers and Mathematics with Applications. DOI: http://dx.doi.org/10.1016/j.camwa.2017.08.019. 1
[21] Xu, J., Ding, L., Mei, J. et al. (2025) Dual roles and therapeutic targeting of tumor-associated macrophages in tumor microenvironments. Sig Transduct Target Ther 10, 268, 1-29. https://doi.org/10.1038/s41392-025-02325-5 1, 2.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Aymen Hadji, Ibtissem Hadji, Fatma Zohra Nouri
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
