Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T05:44:55.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic aspects of tumour–immune system interaction under a periodic immunotherapy

Part of: Applications - Dynamical systems and ergodic theory Nonlinear operators and their properties Smooth dynamical systems: general theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  27 May 2021

GLADIS TORRES-ESPINO  and
MANUEL ZAMORA
GLADIS TORRES-ESPINO
Affiliation:
Departamento de Matemática, Universidad del Bío-Bío Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Avda. Collao 1202, Casilla 5-C, Concepción, Chile email: gtorres@ubiobio.cl
MANUEL ZAMORA
Affiliation:
Departamento de Matemáticas, Universidad de Oviedo Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA) C/ Federico García Lorca n°18, Oviedo, Spain email: mzamora@uniovi.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a mathematical model proposed in the literature with the aim of describing the interactions between tumor cells and the immune system, when a periodic treatment of immunotherapy is applied. Combining some techniques from non-linear analysis (degree theory, lower and upper solutions, and theory of free-homeomorphisms in the plane), we give a detailed global analysis of the model. We also observe that for certain therapies, the maximum level of aggressiveness of a cancer, for which the treatment works (or does not work), can be computed explicitly. We discuss some strategies for designing therapies. The mathematical analysis is completed with numerical results and conclusions.

Keywords

Tumourimmune systeminteracting populationsperiodic therapiesglobal dynamics

MSC classification

Primary: 37N25: Dynamical systems in biology 37C70: Attractors and repellers, topological structure 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 47H11: Degree theory 37C25: Fixed points, periodic points, fixed-point index theory
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 4 , August 2022 , pp. 606 - 645
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agarwal, R. P. & Regan, D. O. (1996) Singular boundary value problems for superlinear second order ordinary and delay differential equations. J. Differential Equations 130, 333–355.CrossRefGoogle Scholar
Arabameri, A., Asemani, D. & Hajati, J. (2018) Mathematical modeling of in-vivo tumor-immune interactions for the cancer immunotherapy using matured dendritic cells. J. Biolog. Syst. 26, 167–188.CrossRefGoogle Scholar
Bellomo, N., Bellouquid, A. & Delitala, M. (2004) Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition. Math. Models Methods Appl. Sci. 14, 1683–1733.CrossRefGoogle Scholar
Bellomo, N. & Preziosi, L. (2000) Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32, 413–452.CrossRefGoogle Scholar
De Boer, R. J. & Boerlijst, M. C. (1994) Diversity and virulence thresholds in AIDS. Proc. Natl. Acad. Sci. 94, 544–548.CrossRefGoogle Scholar
Bonadonna, G. & Robustelli della Cuna, G. (editors) (1994) Medicina Oncologica, Masson, Milano, (in Italian), pp. 259–272.Google Scholar
Brown, M. (1985) Homeomorphisms of two-dimensional manifolds. Houston J. Math. 11, 455–469.Google Scholar
Brú, A., Albertos, S., Subiza, J. L., García-Asenjo, J. L. & Brú, I. (2003) The universal dynamics of tumor growth. Biophys. J. 85, 2948–2961.CrossRefGoogle ScholarPubMed
Campos, J. & Torres, P. (1999) On the structure of the set of bounded solutions on a periodic Liénard equation. Proc. Amer. Math. Soc. 127, 1453–1462.CrossRefGoogle Scholar
Campos, J., Ortega, R. & Tineo, A. (1997) Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems. J. Differential Equations 138, 157–170.CrossRefGoogle Scholar
Capietto, A., Mawhin, J. & Zanolin, F. (1990) A continuation approach to superlinear periodic boundary value problems. J. Differential Equations 88, 347–395.CrossRefGoogle Scholar
Castelli, R. & Garrione, M. (2018) Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions. J. Differential Equations 265(6), 1502–1543.CrossRefGoogle Scholar
Chouaib, S., Asselin-Paturel, C., Mami-Chouaib, F., Caignard, A. & Blay, J. (1997) The host-tumor immune conflict: from immunosuppression to resistance and destruction. Immunol. Today 18, 493–497.CrossRefGoogle ScholarPubMed
Cid, J. A., Propst, G. & Tvrdy, M. (2014) On the pumping effect in a pipe\tank flow configuration with friction. Physica D 273–274, 28–33.CrossRefGoogle Scholar
De Coster, C. & Habets, P. (2004) The lower and upper solutions method for boundary value problems. In: Handbook of Differential Equations, ODE, Elsevier, p. 69–160.Google Scholar
DeVita, V. T., Hellman, S. & Rosenberg, S. A. (2001) Cancer: Principles and Practice of Oncology. Lippincott Williams and Wilkins.Google Scholar
Diefenbach, A., Jensen, E., Jamieson, A. & Raulet, D. (2001) Rae1 and H60 ligands of the NKG2D receptor stimulate tumor immunity. Nature 413, 165–171.CrossRefGoogle Scholar
Doban, A. L. & Lazar, M. (2017) A switching control law approach for cancer immunotherapy of an evolutionary tumor growth model. Math. Biosci. 284, 40–50.CrossRefGoogle ScholarPubMed
d’Onofrio, A. (2005) A general framework for modeling tumor-inmune system competition and immunotherapy: mathematical analysis and biomedical inferences. Physica D 208, 220–235.CrossRefGoogle Scholar
d’Onofrio, A. & Gandolfi, A. (2004) Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). Math. Biosci. 191, 159–184.CrossRefGoogle ScholarPubMed
d’Onofrio, A. (2006) Tumour-immune system interaction: modeling the tumour-stimulated proliferation of effectors and immunotherapy. Math. Models Methods Appl. Sci. 16, 1375–1401.CrossRefGoogle Scholar
d’Onofrio, A. (2008) Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy. Math. Comput. Model. 47, 614–637.CrossRefGoogle Scholar
d’Onofrio, A. & Gandolfi, A. (2013) Mathematical Oncology. Birkhauser.Google Scholar
Fabry, C. & Fonda, A. (1998) Nonlinear resonance in asymmetric oscillators. J. Differential Equations 147, 58–78.CrossRefGoogle Scholar
Fonda, A. & Toader, R. (2008) Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach. J. Differential Equations 244, 3235–3264.CrossRefGoogle Scholar
Fonda, A. & Ureña, A. J. (2011) Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Disc. Cont. Dyn. Syst. Sec. A 29, 169–192.CrossRefGoogle Scholar
Dudley, M. E., Wunderlich, J. R., Robbins, P. F. et al. (2002) Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes. Science 298, 850–854.CrossRefGoogle ScholarPubMed
Forys, U., Waniewski, J. & Zhivkov, P. (2006) Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy. J. Biol. Syst. 14, 13–30.CrossRefGoogle Scholar
Galach, M. (2003) Dynamics of the tumor-immune system competition: the effect of time delay. Int. J. Applied Math. Comput. Sci. 13(3), 395–406.Google Scholar
Hakl, R., Torres, P. J. & Zamora, M. (2011) Periodic solutions of singular second order differential equations: upper and lower functions. Nonlin. Anal. TMA 74, 7078–7093.CrossRefGoogle Scholar
Hirsch, M. (1988) Systems of differential equations which are competitive or cooperative. III. Competing species. Nonlinearity 1, 51–71.CrossRefGoogle Scholar
Kirschner, D. & Panetta, J. C. (1998) Modeling immnotherapy of the tumor-immune interaction. J. Math. Biol. 37, 235–252.CrossRefGoogle Scholar
Kuznetsov, V. A., Makalkin, I. A., Taylor, M. & Perelson, A. (1994) Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–321.CrossRefGoogle ScholarPubMed
Lomtatidze, A. (2016) Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations. Memoirs Diff. Equ. Math. Phys. 67, 1–129.Google Scholar
Marras, C., Mendola, C., Legnani, F. G. & DiMeco, F. (2003) Immunotherapy and biological modifiers for the treatment of malignant brain tumors. Curr. Opin. Oncol. 15, 204–208.CrossRefGoogle ScholarPubMed
Mawhin, J. (1993) Topological degree and boundary value problems for nonlinear differential equations. In: M. Furi (editor), Topological Methods for Ordinary Differential Equations. Lect. Notes Math., vol. 1537, Springer, Berlin, p. 73–142.Google Scholar
Michelson, S. & Leith, J. (1993) Growth factors and growth control of heterogeneous populations. Bull. Math. Biol. 55, 993–1011.CrossRefGoogle Scholar
Michelson, S. & Leith, J. T. (1994) Dormancy, regression and recurrence: towards a unifying theory of tumor growth control. J. Teor. Biol. 169, 327–338.CrossRefGoogle ScholarPubMed
Michelson, S., Miller, B., Glicksman, A. & Leith, J. (1987) Tumor micro-ecology and competitive interactions. J. Theor. Biol. 128, 233–246.CrossRefGoogle ScholarPubMed
de Mottoni, P. & Schiaffino, A. (1981) Competition systems with periodic coefficients: a geometric approach. J. Math. Biol. 11, 319–335.CrossRefGoogle Scholar
Murray, J. D. (2001) Mathematical Biology: I. An Introduction, Springer.Google Scholar
Nani, F. & Freedman, H. I. (2000) A mathematical model of cancer treatment by immunotherapy. Math. Biosci. 163, 159–199.CrossRefGoogle ScholarPubMed
Ortega, H. (1999) Un Modelo Logístico para Crecimiento Tumoral en Presencia de Células Asesinas. Revista Mexicana de Ingeniería Biomédica 20 (in spanish), 61–67.Google Scholar
Ortega, R. & Tineo, A. (1995) On the number of positive periodic solutions for planar competing Lotka-Volterra systems. J. Math. Anal. Appl. 193, 975–978.CrossRefGoogle Scholar
de Pillis, L. G., Radunskaya, A. E. & Wiseman, C. L. (2005) A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65, 7950–7958.CrossRefGoogle ScholarPubMed
Prendergast, G. C. & Jaffee, E. M. (2013) Cancer Immunotherapy: Immune Suppression and Tumor Growth, 2nd edn. (2013).Google Scholar
Ranchnková, I., Staněk, S., & Tvrdý, M. (2009) Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Vol.. Hindawi Publishing Corporation.CrossRefGoogle Scholar
Robertson-Tessi, M., El-Kareh, A. & Goriely, A. (2012) A mathematical model of tumor-immune interactions. J. Teor. Biol. 294, 56–73.CrossRefGoogle ScholarPubMed
Schmielau, J. & Finn, O. J. (2001) Activated granulocytes and granulocyte-derived gydrogen peroxide are the underlying mechanism of suppression of T–cell function in advanced cancer patients. Cancer Res. 61, 4756–4760.Google Scholar
Smale, E. (1976) On the differential equations of species in competition. J. Math. Biol. 3, 5–7.CrossRefGoogle ScholarPubMed
Sotolongo-Costa, O., Morales-Molina, L., Rodríguez-Pérez, D. Antonraz, J. C. & Chacón-Reyes, M. (2003) Behaviour of tumors under nonstationary therapy. Physica D 178, 242–253.CrossRefGoogle Scholar
Stepanova, N. V. (1980) Course of the immune reaction during the development of a malignant tumor. Biophysics 24, 917–923.Google Scholar
Szymánska, S. (2003) Analysis of the immunotherapy models in the context of cancer dynamics. Int. J. Appl. Math. Comput. Sci. 13, 407–418.Google Scholar
Talkington, A., Dantoin, C. & Durrett, R. (2018) Ordinary differential equation models for adoptive immunotherapy. Bull. Math. Biol. 80, 1059–1083.CrossRefGoogle ScholarPubMed
Vaydia, V. G. & Alexandro, F. J. (1982) Evaluation of some mathematical models for tumor growth. Int. J. Bio-Med. Comput. 13, 19–35.CrossRefGoogle Scholar
de Vladar, H. P. & González, J. A. (2004) Dynamic response of cancer under the influence of immunological activity and therapy. J. Theor. Biol. 227, 335–348.CrossRefGoogle ScholarPubMed
Zitan, A. & Ortega, R. (1994) Existence of asymptotically stable periodic solutions of a forced equation of Liénard type. Nonlinear Anal. 22, 993–103.CrossRefGoogle Scholar