The generalization of the model of Rescigno and De Lisi immune system reactions in case of multiple tumors of the cylindrical form
Matematičeskoe modelirovanie, Tome 28 (2016) no. 10, pp. 65-79.

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The article is devoted to the description of several models of the immune response to the appearance of neoplastic process. The models are built on the basis of mathematical model of Resigno and De Lisi, and they are an improvement over the classical model due to the consideration of new geometrical forms of tumors, and by increasing their number. This is important from a practical point of view, because in real life, as a rule, the number of tumors is not a single, and shape of tumors is far from spherical. In addition, the new models take into account that the development of a malignant tumor, there is suppression of the General condition of the immune system, and it also assumes the possibility of receipt of lymphocytes from the outside that allows you to simulate the immune response and gives greater opportunity for neutralization of tumor process and reduce the likelihood of developing the disease.
Keywords: mathematical model, tumor.
Mots-clés : immune response
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N. I. Eremeeva. The generalization of the model of Rescigno and De Lisi immune system reactions in case of multiple tumors of the cylindrical form. Matematičeskoe modelirovanie, Tome 28 (2016) no. 10, pp. 65-79. http://geodesic.mathdoc.fr/item/MM_2016_28_10_a4/

[1] M. I. Davydov, E. M. Aksel, “Statistika zlokachestvennykh novoobrazovanii v Rossii i stranakh SNG v 2007 g.”, Vestnik RONTH im. N. N. Blokhina RAMN, 20:3 (2009), 8–156

[2] Statistika zabolevaemosti rakom, Meditsinskaia entsiklopediia «Kniga medika», (data obrashcheniia: 2.10.2013) http://www.knigamedika.ru

[3] Iu. M. Romanovskii, N. V. Stepanova, D. S. Chernavskii, Matematicheskaia Biofizika, Nauka, M., 1984, 304 pp.

[4] V. G. Knigavko, M. A. Bondarenko, “Mathematical modeling of oxygen diffusion and consumption in malignant tumor”, Biophysics, 50:3 (2005), 579–583

[5] S. A. Astanin, A. I. Lobanov, “Trekhmernaia model rosta nevaskuliarizirovannoi opukholi v tkani”, Matematika. Kompiuter. Obrazovanie, v. 1, 2005, 759–769

[6] V. A. Slepkov, V. G. Sukhovolsky, Z. G. Khlebopros, “Population dynamics in modeling tumor growth”, Biophysics, 52:4 (2007), 426–431

[7] A. V. Kolobov, A. A. Anashkina, V. V. Gubernov, A. A. Polezhaev, “Matematicheskaia model rosta opukholi s uchetom dikhotomii migrazii i proliferathii”, Compiuternye issledovaniia i modelirovanie, 1:4 (2009), 415–422 | Zbl

[8] I. V. Zhukova, E. P. Kolpak, “Mathematiceskie modeli zlokachestvennoi opukholi”, Vestnik S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform., 2014, no. 3, 5–18

[9] D. A. Bratsun, A. P. Zakharov, L. M. Pismen, “Mnogourovnevoe matematicheskoe modelirovanie vozniknoveniia I rosta opukholi v tkani epiteliia”, Compiuternye issledovaniia i modelirovanie, 6:4 (2014), 585–604 | Zbl

[10] M. Gyllenberg, G. Webb, “A nonlinear structured population model of tumour growth with quiescence”, J. Math. Biol., 28 (1990), 671–684 | DOI | MR

[11] M. A. J. Chaplain, J. A. Sherratt, “A new mathematical model for avascular tumor growth”, Journal of Mathematical Biology, 43:4 (2000), 291–312 | MR

[12] H. M. Byrne, C. J. W. Breward, C. E. Lewis, “The role of cell-cell interactions in a two-phase model for avascular tumour growth”, Journal of Mathematical Biology, 45:2 (2001), 125–131 | MR

[13] T. Alarcon, “A cellular automaton model for tumour growth in inhomogeneous environment”, J. Theor. Biol., 225 (2003), 257–274 | DOI | MR

[14] H. M. Byrne, J. R. King, D. L. S. McElwain, “A two-phase model of solid tumor growth”, Appl. Math. Lett., 16 (2003), 567–573 | DOI | MR | Zbl

[15] Y. Kim, M. A. Stolarska, H. G. Othmer, “A hybrid model for tumour spheroid growth in vitro”, Math. Mod. Meth. Appl. Sci., 17 (2007), 1773–1798 | DOI | MR | Zbl

[16] V. S. Shapot, “O proiavleniiakh i mekhanizmakh sistemnogo deistviia na organizm”, Patofiziologiia i eksperimentalnaia terapiia, 1974, no. 3, 3

[17] D. K. Arrowsmith, C. M. Place, Ordinary differential equations: a qualitative approach with applications, Chapman and Hall, London–New York, 1982, 252 pp. | MR | Zbl

[18] G. W. Swan, Some Current Mathematical Topics in Cancer Research, University Microfilms International, MI, 1977

[19] D. Liu, S. Ruan, D. Zhu, “Bifurcation analysis in models of tumor and Immune system interactions”, Discrete and Continuous Dynamical Systems. Series B, 12 (2009), 151–168 | DOI | MR | Zbl