Details

João Oliveira^[a], Ana Jacinta Soares ^[b] and Romina Travaglini ^[c]

Kinetic models leading to pattern formation in the response of the immune system

Pages: 185-212
Received: 7 April 2023;
Accepted in revised form: 16 October 2023
Mathematics Subject Classification: 82C40, 35C20, 35B36, 92C17.
Keywords: Kinetic theory, multicellular systems, chemotaxis, Turing instability, pattern formation.
Authors address:
[a], [b]: Centre of Mathematics, University of Minho, Campus of Gualtar, 4710-057 Braga, Portugal
[c]: Istituto Nazionale di Alta Matematica "Francesco Severi", c/o Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

This research was partially supported by the Portuguese FCT Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. This work is performed in the frame of activities sponsored by INdAM-GNFM and by the Cost Action CA18232.

Abstract: We consider a multicellular system with spatial structure described by the kinetic theory for active particles such that the microscopic state includes dependence on position and velocity, besides the biological activity of the considered populations. The changes in velocity are described by appropriate integral turning operators that include some effects like a velocity-jump process and a volume-filling effect for one population, and a random motion of particles leading to diffusion for another population. The model describes the migration of T-cells driven by cytokines and the possible lesion of particular tissues or organs resulting from inflammation in the response of the immune system within an autoimmune disease. We then derive the hydrodynamic limit of the kinetic system in a diffusive regime and obtain a diffusion-chemotaxis macroscopic model. The stability analysis of the macroscopic system without diffusion is developed, the Turing instability of the complete system and appearance of spatial patterns are investigated. Some numerical simulations are also performed in view of illustrating our theoretical results.

References

[1]: W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol. 9 (1980), 147-177. DOI
[3]: N. Bellomo and A. Bellouquid, From a class of kinetic models to the macroscopic equations for multicellular systems in biology, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), 59-80. DOI
[4]: N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci. 20 (2010), 1179-1207. DOI
[5]: N. Bellomo, C. Bianca and M. Delitala, Complexity analysis and mathematical tools towards the modelling of living systems, Phys. Life Rev. 6 (2009), 144-175. DOI
[6]: N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Model. Dyn. Syst. 20 (1994), 107-122. DOI
[7]: M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys. 124 (2006), 881-912. DOI
[8]: M. Bisi and R. Travaglini, Reaction-diffusion equations derived from kinetic models and their Turing instability, Commun. Math. Sci. 20 (2022), 763-801. DOI
[9]: B. Bozzini, G. Gambino, D. Lacitignola, S. Lupo, M. Sammartino and I. Sgura, Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth, Comput. Math. Appl. 70 (2015), 1948-1969. DOI
[10]: T. M. Brusko, A. L. Putnam and J. A. Bluestone, Human regulatory T cells: role in autoimmune disease and therapeutic opportunities, Immunol. Rev. 223 (2008), 371-390. DOI
[11]: M. Conte, M. Groppi and G. Spiga, Qualitative analysis of kinetic-based models for tumor-immune system interaction., Discrete Contin. Dyn. Syst. Ser. B 23 (2018), 3663-3684. DOI
[12]: N. A. Danke, D. M. Koelle, C. Yee, S. Beheray and W. W. Kwok, Autoreactive T cells in healthy individuals, J. Immunol. 172 (2004), 5967-5972. DOI
[13]: M. Delitala, U. Dianzani, T. Lorenzi and M. Melensi, A mathematical model for immune and autoimmune response mediated by T-cells, Comput. Math. Appl. 66 (2013), 1010-1023. DOI
[14]: R. Della Marca, M. P. M. Ramos, C. Ribeiro and A. J. Soares, Mathematical modelling of oscillating patterns for chronic autoimmune diseases, Math. Methods Appl. Sci. 45 (2022), 7144-7161. DOI
[15]: M. Duan, L. Chang and Z. Jin, Turing patterns of an SI epidemic model with cross-diffusion on complex networks, Phys. A: Stat. Mech. Appl. 533 (2019), 122023. DOI
[16]: E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. DOI
[17]: E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971), 225-234. DOI
[18]: E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol. 30 (1971), 235-248. DOI
[19]: M. Lachowicz, Asymptotic analysis of nonlinear kinetic equations: The hydrodynamic limit, in Lecture Notes on Mathematical Theory of the Boltzmann Equation, World Scientific, 1995, 65-148. DOI
[20]: M. Lachowicz, From microscopic to macroscopic description for generalized kinetic models, Math. Models Methods Appl. Sci. 12 (2002), 985-1005. DOI
[21]: X. Li, W. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math. 78 (2013), 287-306. DOI
[22]: G. Liu and Y. Wang, Pattern formation of a coupled two-cell Schnakenberg model, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), 1051-1062. DOI
[23]: M. Lombardo, R. Barresi, E. Bilotta, F. Gargano, P. Pantano and M. Sammartino, Demyelination patterns in a mathematical model of multiple sclerosis, J. Math. Biol. 75 (2017), 373-417. DOI
[24]: H. G. Othmer and T. Hillen, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math. 61 (2000), 751-775. DOI
[25]: H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math. 62 (2002), 1222-1250. DOI
[26]: K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart. 10 (2002), 501-543. MR
[27]: C. S. Patlak, Random walk with persistence and external bias, Bull. math. biophys. 15 (1953), 311-338. DOI
[28]: A. Poggi and M. R. Zocchi, NK cell autoreactivity and autoimmune diseases, Front. Immunol. 5 (2014), 27. DOI
[29]: I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, II, J. Chem. Phys. 48 (1968), 1695-1700. DOI
[30]: M. P. M. Ramos, C. Ribeiro and A. J. Soares, A kinetic model of T cell autoreactivity in autoimmune diseases, J. Math. Biol. 79 (2019), 2005-2031. DOI
[31]: F. D. Shi and L. Van Kaer, Reciprocal regulation between natural killer cells and autoreactive T cells, Nat. Rev. Immunol. 6 (2006), 751-760. DOI
[32]: D. D. Taub, M. L. Key, D. Clark and S. M. Turcovski-Corrales, Chemotaxis of T lymphocytes on extracellular matrix proteins analysis of the in vitro method to quantitate chemotaxis of human T cells, J. Immunol. Methods 184 (1995), 187-198. DOI
[33]: Z. Tian, M. E. Gershwin and C. Zhang, Regulatory NK cells in autoimmune disease, J. Autoimmun. 39 (2012), 206-215. DOI
[34]: A. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. London, Ser. B, (1952), 37-72. MR
[35]: Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos 17 (2007), 037108. DOI
[36]: C. O. Zachariae, Chemotactic cytokines and inflammation. Biological properties of the lymphocyte and monocyte chemotactic factors ELCF, MCAF and IL-8., Acta Derm. Venereol. Suppl. 181 (1993), 1-37. Pubmed