×

Metamodeling the learning-hiding competition between tumours and the immune system: A kinematic approach. (English) Zbl 1201.34071

Summary: The competitive interaction between the Immune System and tumours is very complex, being non-linear and, to some extent, evolutionary. A fundamental aspect of this evolution is the asynchronous process of mutual learning of the two populations involved - the tumoural and the immune cells. In this work, to describe them, we propose a simple non-linear family of super-macroscopic models with non-monotonically time-varying coefficients. Numerical simulations of transitory phases complement the theoretical analysis.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92C60 Medical epidemiology
92C50 Medical applications (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] de Vito, V. T.; Hellman, J.; Rosenberg, S. A., Cancer: Principles and Practice of Oncology (2005), J.P. Lippincott: J.P. Lippincott Philadelphia
[2] Boyle, P.; d’Onofrio, A.; Maisonneuve, P.; Severi, G.; Robertson, C.; Tubiana, M.; Veronesi, U., Measuring progress against cancer in Europe: Has the 15
[3] Pardoll, D., Does the immune system see tumors as foreign or self?, Annual Review of Immunology, 21, 807-839 (2003)
[4] Ehrlich, P., Ueber den jetzigen stand der Karzinomforschung, Vortrag gehalten vor den studenten der Amsterdamer Universitaet, Vereinigung fuer wissenschaftliche Arbeit 1 June 1908, (Ehrlich, P., Beitraege zur Experimentellen Pathologie und Chemotherapie (1909), Akademische Verlagsgesellschaft: Akademische Verlagsgesellschaft Leipzig), 118-164, (Reprinted in: F. Himmelweit (Ed.), The Collected Papers of Paul Ehrlich, Pergamon Press, London, 1957)
[5] Dunn, G. P.; Old, L. J.; Schreiber, R. D., The three ES of cancer immunoediting, Annual Review of Immunology, 22, 322-360 (2004)
[6] Stepanova, N. V., Course of the immune reaction during the development of a malignant tumor, Biophysica, 24, 917-923 (1980)
[7] Kuznetsov, V. A.; Makalkin, I. A.; Taylor, M. A.; Perelson, A. S., Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56, 295-321 (1994) · Zbl 0789.92019
[8] Kuznetsov, V. A.; Knott, G. D., Modeling tumor regrowth and immunotherapy, Mathematical and Computer Modelling, 33, 1275-1287 (2001) · Zbl 1004.92021
[9] de Pillis, L. G.; Radunskaya, A. E.; Wiseman, C. L., A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65, 7950-7958 (2005)
[10] d’Onofrio, A., A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208, 220-235 (2005) · Zbl 1087.34028
[11] d’Onofrio, A., Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Sciences, 16, 1375-1401 (2006) · Zbl 1094.92040
[12] d’Onofrio, A., Tumor evasion from immune system control: Strategies of a MISS to become a MASS, Chaos, Solitons and Fractals, 31, 261-268 (2007) · Zbl 1133.92016
[13] d’Onofrio, A., Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Mathematical and Computer Modelling, 47, 614-637 (2008) · Zbl 1148.92026
[14] Bellomo, N.; Bellouquid, A.; Delitala, M., Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition, Mathematical Models and Methods in Applied Sciences, 14, 1683-1733 (2004) · Zbl 1060.92029
[15] Bellomo, N.; Delitala, M., From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Review, 5, 183-206 (2008)
[16] Bellomo, N., Modeling Complex Living Systems—Kinetic Theory and Stochastic Game Approach (2008), Birkhäeuser, Springer: Birkhäeuser, Springer Boston · Zbl 1140.91007
[17] Bellouquid, A.; Delitala, M., Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach (2006), Birkhäeuser, Springer: Birkhäeuser, Springer Boston · Zbl 1178.92002
[18] Cattani, C.; Ciancio, A., Qualitative analysis of second order models of tumor-immune system competition, Mathematical and Computer Modelling, 47, 1339-1355 (2008) · Zbl 1145.34303
[19] Cattani, C.; Ciancio, A., Third order model for tumor-immune system competition, (Proc. 4-th International Colloquium Mathematics in Engineering and Numerical Physics. Proc. 4-th International Colloquium Mathematics in Engineering and Numerical Physics, October 6-8, 2006, Bucharest, Ro (2007), Geometry Balkan Press: Geometry Balkan Press Bucharest), 30-37 · Zbl 1132.34319
[20] Melief, C. J.M., Cancer immunology: Cat and mouse games, Nature, 437, 41-42 (2005)
[21] Abrams, D., The evolution of predator-prey systems: Theory and evidence, Annual Review of Ecological Systems, 31, 79-105 (2000)
[22] Cheon, T., Evolutionary stability of ecological hierarchy, Physical Review Letters, 90 (2003), Art. no. 258105
[23] Bobryk, R. B.; Chrzeszczyk, A., Transitions induced by bounded noise, Physica A, 358, 263-272 (2005)
[24] Thieme, H. R., Mathematics in Population Biology (2003), Princeton University Press: Princeton University Press Princeton · Zbl 1054.92042
[25] Hale, J. K.; Kocack, H., Dynamics and Bifurcations (2003), Springer: Springer Heidelberg, New York
[26] Cucker, F.; Smale, S., On the mathematical foundations of learning, Bulletin of the American Mathematical Society (New Series), 39, 1-49 (2002) · Zbl 0983.68162
[27] Cattani, C.; Ciancio, A., Hybrid two scales mathematical tools for active particles modelling complex systems with learning hiding dynamics, Mathematical Modelling and Methods in Applied Sciences, 17, 171-187 (2007) · Zbl 1142.82019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.