On the Mathematical Modelling of Complex Biological Systems. A Kinetic Theory Approach
Delitala, Marcello
Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 603-618 / Harvested from Biblioteca Digitale Italiana di Matematica

This paper deals with the mathematical modelling, based on the kinetic theory of active particles, of a complex biological living system constituted by different populations of cells. The modelling refers to the competition between immune and tumor cells. Moreover, a qualitative and quantitative analysis is developed, to show how the models can describe several interesting phenomena related to biological applications. A final section highlights further research perspectives related to the modelling of genetic mutations.

Publié le : 2008-10-01
@article{BUMI_2008_9_1_3_603_0,
     author = {Marcello Delitala},
     title = {On the Mathematical Modelling of Complex Biological Systems. A Kinetic Theory Approach},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1},
     year = {2008},
     pages = {603-618},
     zbl = {1188.92002},
     mrnumber = {2455334},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_3_603_0}
}
Delitala, Marcello. On the Mathematical Modelling of Complex Biological Systems. A Kinetic Theory Approach. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 603-618. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_3_603_0/

[1] Bellomo, N., Modelling complex living systems. A kinetic theory and stocastic game approach, Birkhauser, Boston, (2008). | MR 2359781 | Zbl 1140.91007

[2] Bellomo, N. - Li, N. K. - Maini, P. K., On the foundation of cancer modeling - selected topics, speculations, and perspectives, Math. Mod. Meth. Appl. Sci., 18 (2008), 593-646. | MR 2402885 | Zbl 1151.92014

[3] Bellouquid, A. - Delitala, M., Kinetic (cellular) models of cell progression and competition with the immune system, Z. angew. Math. Phys., 55 (2004), 95-317. | MR 2047290 | Zbl 1047.92022

[4] Bellouquid, A. - Delitala, M., Modelling complex biological systems - A kinetic theory approach, Birkhäuser, Boston, (2006). | MR 2248839 | Zbl 1178.92002

[5] De Angelis, E. - Jabin, P.E., Qualitative Analysis of a mean field model of tumor-immune system competition, Math. Mod. Meth. Appl. Sci., 13 (2003), 187-206. | MR 1961000 | Zbl 1043.92012

[6] Delitala, M. - Forni, G., From the mathematical kinetic theory of active particles to modelling genetic mutations and immune competition, Internal Report, Dept. Mathematics, Politecnico, Torino (2008).

[7] Derbel, L., Analysis of a new model for tumor-immune system competition including long time scale effects, Math. Mod. Meth. Appl. Sci., 14 (2004), 1657-1681. | MR 2103095 | Zbl 1057.92036

[8] D'Onofrio, A., Tumor-immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. Mod. Meth. Appl. Sci., 16 (2006), 1375-1401. | MR 2251457 | Zbl 1094.92040

[9] Gatenby, R. A. - Vincent, T. L. - Gillies, R. J., Evolutionary dynamics in carcinogenesis, Math. Mod. Meth. Appl. Sci., 15 (2005), 1619-1638. | MR 2180711 | Zbl 1077.92031

[10] Komarova, N., Stochastic modeling of loss- and gain-of-function mutation in cancer, Math. Mod. Meth. Appl. Sci., 17 (2007), 1647-1673. | MR 2362759 | Zbl 1135.92017

[11] Hanahan, D. - Weinberg, R. A., The Hallmarks of cancer, Cell, 100, 57-70, (2000).

[12] Hartwell, H. L. - Hopfield, J. J. - Leibner, S. - Murray, A. W., From molecular to modular cell biology, Nature, 402 (1999), c47-c52.

[13] Kolev, M. - Kozlowska, E. - Lachowicz, M., Mathematical model of tumor invasion along linear or tubular structures, Math. Comp. Mod., 41 (2005), 1083-1096. | MR 2148091 | Zbl 1085.92019

[14] Merlo, L. M. F. - Pepper, J. W. - Reid, B. J. - Maley, C. C., Cancer as an evolutionary and ecological process, Nature Reviews Cancer, 6 (2006), 924-935.

[15] Perthame, B. - Ryzhik, L., Esponential decay for the fragmentation or cell division equation, J. Diff. Equations, 12, (2005), 155-177. | MR 2114128 | Zbl 1072.35195