Modelling tumour-immunity interactions with different stimulation functions

Petar Zhivkov; Jacek Waniewski

International Journal of Applied Mathematics and Computer Science (2003)

  • Volume: 13, Issue: 3, page 307-315
  • ISSN: 1641-876X

Abstract

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Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.

How to cite

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Zhivkov, Petar, and Waniewski, Jacek. "Modelling tumour-immunity interactions with different stimulation functions." International Journal of Applied Mathematics and Computer Science 13.3 (2003): 307-315. <http://eudml.org/doc/207645>.

@article{Zhivkov2003,
abstract = {Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.},
author = {Zhivkov, Petar, Waniewski, Jacek},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {stability analysis; immunotherapy; critical points; ordinary differential equations},
language = {eng},
number = {3},
pages = {307-315},
title = {Modelling tumour-immunity interactions with different stimulation functions},
url = {http://eudml.org/doc/207645},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Zhivkov, Petar
AU - Waniewski, Jacek
TI - Modelling tumour-immunity interactions with different stimulation functions
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 3
SP - 307
EP - 315
AB - Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.
LA - eng
KW - stability analysis; immunotherapy; critical points; ordinary differential equations
UR - http://eudml.org/doc/207645
ER -

References

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  1. Bell G. (1973): Predator-prey equation simulating an immune response. - Math. Biosci., Vol. 16, pp. 291-314. Zbl0253.92003
  2. Chen Ch-H. and Wu T.C. (1998): Experimental vaccines strategies for cancer immunotherapy. - J. Biomed. Sci., Vol. 5, No. 5, pp. 231-252. 
  3. de Pillis L.G. and Radunskaya A.E. (2001): A mathematical tumor model with immune resistance and drug therapy: An optimal control approach. -J. Theor. Med., Vol. 3, pp. 79-100. Zbl0985.92023
  4. Fong L. and Engleman E. (2000): Dendritic cells in cancer immunology. - Ann. Rev. Immunol., Vol. 18, pp. 245-273. 
  5. Ginzburg L.R. and Akcakaya H.R. (1992): Consequences of ratio-dependent predation for steady-state properties of ecosystems. - Ecology, Vol. 73, pp. 1536-1543. 
  6. Kuby J. (1998): Immunology. - New York: Freeman Co. 
  7. Mayer H., Zaenker K.S. and an der Heiden U. (1995): A basic mathematical model of the immune response. - Chaos, Vol. 5, No. 1, pp. 155-161. 
  8. Moingeon P. (2001): Cancer vaccines. - Vaccines, Vol. 19, No. 11-12, pp. 1305-1326. 
  9. Prikrylova D., Jilek M. and Waniewski J. (1992): Mathematical Modelling of the Immune Response. - Boca Raton: CRC Press. Zbl0900.92082
  10. Romanovski I., Stepanova N. and Chernavski D. (1975): Mathematical Modelling in Biophysics. - Moscow: Nauka, (in Russian). 
  11. Rosenberg St. (2001): Progress in human tumour immunology and immunotherapy. - Nature, Vol. 411, No. 6835, pp. 380-385. 
  12. William E. (Ed.) (1984): Fundamental Immunology. - New York: Raven Press. 

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