Integrating stochastic chemotaxis–haptotaxis mechanisms in cancer invasion: A multiscale derivation and computational perspective

Abdelghafour Atlas; Mostafa Bendahmane; Fahd Karami; Jacques Tagoudjeu; Mohamed Zagour; Abdelghafour Atlas; Mostafa Bendahmane; Fahd Karami; Jacques Tagoudjeu; Mohamed Zagour

doi:10.3934/mbe.2025097

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 10: 2641-2671. doi: 10.3934/mbe.2025097

Previous Article Next Article

Research article Special Issues

Integrating stochastic chemotaxis–haptotaxis mechanisms in cancer invasion: A multiscale derivation and computational perspective

1.
Ecole Nationale des Sciences Appliquées de Marrakech, Université Cadi Ayyad, B.P. 575 Marrakech, Maroc
2.
Institut de Mathématiques de Bordeaux, Université de Bordeaux, Bordeaux Cedex, France
3.
École Supérieure de Technologie d'Essaouira, Université Cadi Ayyad, Essaouira El Jadida, Essaouira, Maroc
4.
École Nationale Supérieure Polytechnique de Yaoundé, Universite de Yaoundé I, Yaoundé, Cameroun
5.
Euromed University of Fes, UEMF, Morocco

Received: 24 May 2025 Revised: 05 August 2025 Accepted: 05 August 2025 Published: 25 August 2025

This paper deals with the multiscale derivation of a nonlinear stochastic chemotaxis–haptotaxis system of cancerous tissue invasion from a new stochastic kinetic theory model based on the micro–macro decomposition technique. We show that this approach technically can lead to some systems known in the literature, such as the filling volume effect, and a new system by taking the stochasticity effect and nonlocal diffusion into account. We develop an asymptotic–preserving numerical scheme to solve the obtained equivalent micro–macro formulation numerically. The objective is to provide a uniformly stable scheme regarding the small parameters and consistency with the diffusion limit. Various numerical examples validate the proposed approach. Finally, we provide numerical simulations in the two-dimensional setting obtained by the macroscopic stochastic model.
- disease prevention,
- public health,
- kinetic theory,
- stochastic chemotaxis–haptotaxis system,
- cell biology,
- asymptotic–preserving scheme,
- numerical method
Citation: Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Jacques Tagoudjeu, Mohamed Zagour. Integrating stochastic chemotaxis–haptotaxis mechanisms in cancer invasion: A multiscale derivation and computational perspective[J]. Mathematical Biosciences and Engineering, 2025, 22(10): 2641-2671. doi: 10.3934/mbe.2025097

Related Papers:

Abstract

This paper deals with the multiscale derivation of a nonlinear stochastic chemotaxis–haptotaxis system of cancerous tissue invasion from a new stochastic kinetic theory model based on the micro–macro decomposition technique. We show that this approach technically can lead to some systems known in the literature, such as the filling volume effect, and a new system by taking the stochasticity effect and nonlocal diffusion into account. We develop an asymptotic–preserving numerical scheme to solve the obtained equivalent micro–macro formulation numerically. The objective is to provide a uniformly stable scheme regarding the small parameters and consistency with the diffusion limit. Various numerical examples validate the proposed approach. Finally, we provide numerical simulations in the two-dimensional setting obtained by the macroscopic stochastic model.

References

[1]	A. Gerisch, M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theor. Biol., 250 (2008), 684–704. https://doi.org/10.1016/j.jtbi.2007.10.026 doi: 10.1016/j.jtbi.2007.10.026
[2]	H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nat. Rev. Cancer, 10 (2010), 221–230. https://doi.org/10.1038/nrc2808 doi: 10.1038/nrc2808
[3]	S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann, L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427–1438. https://doi.org/10.1083/jcb.110.4.1427 doi: 10.1083/jcb.110.4.1427
[4]	M. A. J. Chaplain, G. Lolas, Mathematical modeling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685–1734. https://doi.org/10.1142/S0218202505000947 doi: 10.1142/S0218202505000947
[5]	M. A. J. Chaplain, G. Lolas, Mathematical modeling of cancer cell invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399–439. https://doi.org/10.3934/nhm.2006.1.399 doi: 10.3934/nhm.2006.1.399
[6]	A. Friedman, G. Lolas, Analysis of a mathematical model of tumor lymph-angiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95–107. https://doi.org/10.1142/S0218202505003915 doi: 10.1142/S0218202505003915
[7]	R. A. Gatenby, E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745–5753.
[8]	G. Meral, C. Stinner, C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, IMA J. Appl. Math., 80 (2015), 1300–1321. https://doi.org/10.1093/imamat/hxu055 doi: 10.1093/imamat/hxu055
[9]	C. Stinner, C. Surulescu, M. Winkler, Global weak solutions in a pde-ode system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969–2007. https://doi.org/10.1137/13094058X doi: 10.1137/13094058X
[10]	J. A. Adam, N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, Birkhäuser, Boston, 1996. https://doi.org/10.1007/978-0-8176-8119-7
[11]	N. Bellomo, A. Bellouquid, E. D. Angelis, The modeling of the immune competition by generalized kinetic (boltzmann) models: Review and research perspectives, Math. Comput. Model., 37 (2003), 1131–1142. https://doi.org/10.1016/S0895-7177(03)80007-9 doi: 10.1016/S0895-7177(03)80007-9
[12]	N. V. Mantzaris, S. Webb, H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111–187. https://doi.org/10.1007/s00285-003-0262-2 doi: 10.1007/s00285-003-0262-2
[13]	Z. Szymańska, Analysis of immunotherapy models in the context of cancer dynamics, Appl. Math. Comput. Sci., 13 (2003), 407–418.
[14]	N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of keller-segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663–1763. https://doi.org/10.1142/S021820251550044X doi: 10.1142/S021820251550044X
[15]	K. Choi, M. J. Kang, Y. S. Kwon, A. F. Vasseur, Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, Math. Models Methods Appl. Sci., 30 (2020), 387–437. https://doi.org/10.1142/S0218202520500104 doi: 10.1142/S0218202520500104
[16]	C. Jin, Global classical solution and boundedness to a chemotaxis–haptotaxis model with re-establishment mechanisms, Bull. London Math. Soc., 50 (2018), 598–618. https://doi.org/10.1112/blms.12160 doi: 10.1112/blms.12160
[17]	P. Y. Pang, Y. Wang, Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28 (2018), 2211–2235. https://doi.org/10.1142/S0218202518400134 doi: 10.1142/S0218202518400134
[18]	Y. Tao, M. Winkler, A chemotaxis–haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047–2067. https://doi.org/10.3934/cpaa.2019092 doi: 10.3934/cpaa.2019092
[19]	C. Engwer, C. Stinner, C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017), 1355–1390. https://doi.org/10.1142/S0218202517400188 doi: 10.1142/S0218202517400188
[20]	T. Xiang, J. Zheng, A new result for 2d boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source, Nonlinearity, 32 (2019), 4890–4911. https://doi.org/10.1088/1361-6544/ab41d5 doi: 10.1088/1361-6544/ab41d5
[21]	N. Kolbe, J. Katuchová, N. Sfakianakis, N. Hellmann, M. Lukáčová-Medviďová, Numerical study of cancer cell invasion dynamics using adaptive mesh refinement: The urokinase model, preprint, arXiv: 1408.0642.
[22]	Y. Epshteyn, Discontinuous galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168–181. https://doi.org/10.1016/j.cam.2008.04.030 doi: 10.1016/j.cam.2008.04.030
[23]	S. Ganesan, S. Lingeshwaran, Galerkin finite element method for cancer invasion mathematical model, Comput. Math. Appl., 73 (2017), 2603–2617. https://doi.org/10.1016/j.camwa.2017.04.006 doi: 10.1016/j.camwa.2017.04.006
[24]	J. Manimaran, L. Shangerganesh, Solvability and numerical simulations for tumor invasion model with nonlinear diffusion, Comput. Math. Methods, 2 (2020), e1068. https://doi.org/10.1002/cmm4.1068 doi: 10.1002/cmm4.1068
[25]	D. Balding, D. McElwain, A mathematical model of tumor-induced capillary growth, J. Theoret. Biol., 114 (1985), 53–73. https://doi.org/10.1016/S0022-5193(85)80255-1 doi: 10.1016/S0022-5193(85)80255-1
[26]	H. Joshi, M. Yavuz, O. Taylan, A. Alkabaa, Dynamic analysis of fractal-fractional cancer model under chemotherapy drug with generalized mittag-leffler kernel, Comput. Methods Programs Biomed., 260 (2024), 108565. https://doi.org/10.1016/j.cmpb.2024.108565 doi: 10.1016/j.cmpb.2024.108565
[27]	U. Dobramysl, M. Mobilia, M. Pleimling, U. C. Täuber, Stochastic population dynamics in spatially extended predator-prey systems, J. Phys. A, 51 (2018), 063001. https://doi.org/10.1088/1751-8121/aa95c7 doi: 10.1088/1751-8121/aa95c7
[28]	M. Bandyopadhyay, J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913–936. https://doi.org/10.1088/0951-7715/18/2/022 doi: 10.1088/0951-7715/18/2/022
[29]	M. Bendahmane, H. N. Nzeti, J. Tagoudjeu, M. Zagour, Stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises, Chaos, 33 (2023), 073103. https://doi.org/10.1063/5.0140102 doi: 10.1063/5.0140102
[30]	G. D. Prato, A. Ichikawa, Optimal control of linear systems with almost periodic inputs, SIAM J. Control Optim., 25 (1987), 1007–1019. https://doi.org/10.1137/0325055 doi: 10.1137/0325055
[31]	C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.
[32]	T. Bose, S. Trimper, Stochastic model for tumor growth with immunization, Phys. Rev. E, 79 (2009), 051903. https://doi.org/10.1103/PhysRevE.79.051903 doi: 10.1103/PhysRevE.79.051903
[33]	A. Bellouquid, J. Tagoudjeu, An asymptotic–preserving scheme for kinetic models for chemotaxis phenomena, Commun. Appl. Ind. Math., 9 (2018), 61–75. https://doi.org/10.2478/caim-2018-0010 doi: 10.2478/caim-2018-0010
[34]	M. Bendahmane, J. Tagoudjeu, M. Zagour, Odd-even based asymptotic–preserving scheme for a 2d stochastic kinetic-fluid model, J. Comput. Phys., 471 (2022), 111649. https://doi.org/10.1016/j.jcp.2022.111649 doi: 10.1016/j.jcp.2022.111649
[35]	F. Filbet, P. Laurençot, B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189–207. https://doi.org/10.1007/s00285-004-0286-2 doi: 10.1007/s00285-004-0286-2
[36]	A. Atlas, M. Bendahmane, F. Karami, D. Meskine, M. Zagour, Kinetic-fluid derivation and mathematical analysis of nonlocal cross-diffusion-fluid system, Appl. Math. Model., 82 (2020), 379–408. https://doi.org/10.1016/j.apm.2019.11.036 doi: 10.1016/j.apm.2019.11.036
[37]	M. Bendahmane, F. Karami, M. Zagour, Kinetic-fluid derivation and mathematical analysis of the cross-diffusion-brinkman system, Math. Meth. Appl. Sci., 41 (2018), 6288–6311. https://doi.org/10.1002/mma.5139 doi: 10.1002/mma.5139
[38]	N. Bellomo, A. Bellouquid, On the derivation of angiogenesis tissue models: From the micro-scale to the macro-scale, Math. Mech. Solids, 20 (2014), 1–12. https://doi.org/10.1177/1081286514544855 doi: 10.1177/1081286514544855
[39]	M. Bendahmane, F. Karami, M. Zagour, Multiscale derivation of deterministic and stochastic cross-diffusion models in a fluid: A review, Chaos, 34 (2024), 122101. https://doi.org/10.1063/5.0238999 doi: 10.1063/5.0238999
[40]	S. Jin, Efficient asymptotic–preserving (ap) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441–454. https://doi.org/10.1137/S1064827598334599 doi: 10.1137/S1064827598334599
[41]	A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J. Sci. Comput., 19 (1998), 2032–2050. https://doi.org/10.1137/S1064827595286177 doi: 10.1137/S1064827595286177
[42]	N. Crouseilles, M. Lemou, An asymptotic–preserving scheme based on a micro–macro decomposition for collisional vlasov equations: Diffusion and high-field scaling limits, Kinet. Relat. Models, 4 (2011), 441–477. https://doi.org/10.3934/krm.2011.4.441 doi: 10.3934/krm.2011.4.441
[43]	G. Dimarco, L. Pareschi, Asymptotic–preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51 (2013), 1064–1087. https://doi.org/10.1137/12087606X doi: 10.1137/12087606X
[44]	J. Jang, F. Li, J. M. Qiu, T. Xiong, Analysis of asymptotic–preserving dg-imex schemes for linear kinetic transport equations in a diffusive scaling, SIAM J. Numer. Anal., 52 (2014), 2048–2072. https://doi.org/10.1137/130938955 doi: 10.1137/130938955
[45]	M. Bennoune, M. Lemou, L. Mieussens, An asymptotic–preserving scheme for the kac model of the boltzmann equation in the diffusion limit, Continuum Mech. Thermodyn., 21 (2009), 401–421. https://doi.org/10.1007/s00161-009-0116-2 doi: 10.1007/s00161-009-0116-2
[46]	A. Dietrich, N. Kolbe, N. Sfakianakis, C. Surulescu, Multiscale modeling of glioma invasion: From receptor binding to flux-limited macroscopic pdes, Multiscale Model. Simul., 20 (2022), 685–713. https://doi.org/10.1137/21M1412104 doi: 10.1137/21M1412104
[47]	J. A. Carrillo, B. Yan, An asymptotic–preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336–361. https://doi.org/10.1137/110851687 doi: 10.1137/110851687
[48]	V. Andasari, A. Gerisch, G. Lolas, A. P. South, M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141–171. https://doi.org/10.1007/s00285-010-0369-1 doi: 10.1007/s00285-010-0369-1
[49]	P. Zheng, C. Mu, X. Song, On the boundedness and decay of solutions for a chemotaxis–haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. A, 36 (2016), 1737–1757. https://doi.org/10.3934/dcds.2016.36.1737 doi: 10.3934/dcds.2016.36.1737
[50]	N. Bellomo, K. J. Painter, Y. Tao, M. Winkler, Occurrence vs. absence of taxis-driven instabilities in may-nowak model for virus infection, SIAM J. Appl. Math., 79 (2019), 1990–2010. https://doi.org/10.1137/19M1250261 doi: 10.1137/19M1250261
[51]	M. Winkler, Boundedness in a chemotaxis-may-nowak model for virus dynamics with mildly saturated chemotactic sensitivity, Acta Appl. Math., 163 (2019), 1–17. https://doi.org/10.1007/s10440-018-0211-0 doi: 10.1007/s10440-018-0211-0

Reader Comments

Your name:*

Email:*
© 2025 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)