Geodesic
Fisher type waves in a~model of invasive tumor growth
Matematičeskoe modelirovanie, Tome 19 (2007) no. 6, pp. 31-42.

Voir la notice de l'article provenant de la source Math-Net.Ru

A mathematical model for the invasive tumor growth which includes nutrient distribution in tissue is developed. This model adequately describes constant rate of the tumor linear size growth with time and the formation of necrotic region in the tumor interior, observed in experiments. The existence of autowave solutions is demonstrated and their properties are investigated. The results are compared with the properties of the Kolmogorov–Petrovskii–Piskunov and Fisher equations. The perspective of further investigation and development of the model for studying glioma is discussed. 
@article{MM_2007_19_6_a2,
     author = {A. V. Kolobov and V. V. Gubernov and A. A. Polezhaev},
     title = {Fisher type waves in a~model of invasive tumor growth},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {31--42},
     publisher = {mathdoc},
     volume = {19},
     number = {6},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2007_19_6_a2/}
}
TY  - JOUR
AU  - A. V. Kolobov
AU  - V. V. Gubernov
AU  - A. A. Polezhaev
TI  - Fisher type waves in a~model of invasive tumor growth
JO  - Matematičeskoe modelirovanie
PY  - 2007
SP  - 31
EP  - 42
VL  - 19
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2007_19_6_a2/
LA  - ru
ID  - MM_2007_19_6_a2
ER  - 
%0 Journal Article
%A A. V. Kolobov
%A V. V. Gubernov
%A A. A. Polezhaev
%T Fisher type waves in a~model of invasive tumor growth
%J Matematičeskoe modelirovanie
%D 2007
%P 31-42
%V 19
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2007_19_6_a2/
%G ru
%F MM_2007_19_6_a2
A. V. Kolobov; V. V. Gubernov; A. A. Polezhaev. Fisher type waves in a~model of invasive tumor growth. Matematičeskoe modelirovanie, Tome 19 (2007) no. 6, pp. 31-42. http://geodesic.mathdoc.fr/item/MM_2007_19_6_a2/

[1] H. M. Byrne, “Using mathematics to study solid tumour growth”, Proceedings of the 9th General Meetings of European Women in Mathematics, 1999, 81–107 | MR

[2] L. A. Kunz-Schughart, M. Kreutz, R. Knuechel, “Multicellular spheroids: a three-dimensional in vitro culture system to study tumour biology”, Int. J. Exp. Pathol., 79 (1998), 1–23 | DOI

[3] A. C. Burton, “Rate of growth of solid tumours as a problem of diffusion”, Growth, 30 (1966), 157–176

[4] H. Greenspan, “Models for the growth of solid tumor by diffusion”, Stud. Appl. Math., 51 (1972), 317–340 | Zbl

[5] J. A. Adam, “A mathematical model of tumor growth by diffusion”, Math. Biosci., 94 (1989), 155–157 | DOI | MR

[6] J. A. Adam, S. A. Maggelakis, “Diffusion regulated characteristics of a spherical prevascular carcinoma”, Bull. Math. Biol., 52 (1990), 549–582 | Zbl

[7] M. A. J. Chaplain, D. L. Benson, P.K. Maini, “Nonlinear diffusion of a growth inhibitory factor in multicell spheroids”, Math. Biosci., 121 (1994), 1–13 | DOI | Zbl

[8] H. M. Byrne, M. A. J. Chaplain, “Growth of nonnecrotic tumours in the presence and absence of inhibitors”, Math. Biosci., 130 (1995), 151–181 | DOI | Zbl

[9] H. M. Byrne, M. A. J. Chaplain, “Growth of necrotic tumors in the presence and absence of inhibitors”, Math. Biosci., 135 (1996), 187–216 | DOI | Zbl

[10] J. P. Ward, J. R. King, “Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures”, Math. Biosci., 181 (2003), 177–207 | DOI | MR | Zbl

[11] B. Bazaliy, A. Friedman, “Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumour growth”, Indiana Univ. Math. J., 52 (2003), 1265–1304 | DOI | MR | Zbl

[12] H. Bueno, G. Ercole, A. Zumpano, “Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours”, Nonlinearity, 18 (2005), 1629–1642 | DOI | MR | Zbl

[13] X. Chen, A. Friedman, “A free boundary problem for elliptic-hyperbolic system: an application to tumor growth”, SIAM J. Math. Anal., 35 (2003), 974–986 | DOI | MR | Zbl

[14] Y. Tao, M. Chen, “An elliptic-hyperbolic free boundary problem modelling cancer therapy”, Nonlinearity, 19 (2006), 419–440 | DOI | MR | Zbl

[15] R. M. Sutherland, “Cell and environment interactions in tumor microregions: the multicell spheroid model”, Science, 240 (1988), 177–184 | DOI

[16] M. Castro, C. Molina-Paris, T. S. Deisboeck, “Tumor growth instability and the onset of invasion”, Phys. Rev. E, 72 (2005), 041907 | DOI

[17] A. V. Gusev, A. A. Polezhaev, “Evolyutsiya kletochnoi populyatsii pri nalichii maksimalnoi dopustimoi summarnoi plotnosti kletok”, Kratkie soobscheniya po fizike FIAN, 1997, no. 11–12, 85–90

[18] A. V. Kolobov, A. A. Polezhaev, G. I. Solyanyk, “The role of cell motility in metastatic cell dominance phenomenon: analysis by a mathematical model”, J. of Theoretical Medicine, 3 (2001), 63–77

[19] A. V. Kolobov, A. A. Polezhaev, G. I. Solyanyk, “Stability of tumour shape in pre-angiogenic stage of growth depends on the migration capacity of cancer cells”, Mathematical Modelling and Computing in Biology and Medicine, ed. V. Capasso, Progetto Leonardo, Bologna, 2003, 603–609 | MR

[20] S. A. Astanin, A. V. Kolobov, A. I. Lobanov, T. P. Pimenova, A. A. Polezhaev, G. I. Solyanik, “Vliyanie prostranstvennoi geterogennosti sredy na rost i invaziyu opukholi. Analiz metodami matematicheskogo modelirovaniya”, 2006 (to appear)

[21] R. A. Fisher, “The wave of advance of advantageneous genes”, Ann. Eugenics, 7 (1937), 355–369 | Zbl

[22] J. A. Sherratt, “Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations”, Proc. R. Soc. Lond. B, 241 (1990), 29–36 | DOI

[23] P. Traqui, “From passive diffusion to active cellular migration in mathematical models of tumor invasion”, Acta Biotheoretical, 43 (1995), 443–464 | DOI

[24] K. R. Swanson, C. Bridge, J. D. Murray, E. C. Alvord Jr., “Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion”, Journal of the Neurological Sciences, 216 (2003), 1–10 | DOI

[25] A. N. Kolmogorov, I. G. Petrovskii, N. S. Piskunov, “Issledovanie uravneniya diffuzii, soedinennoi s vozrastaniem kolichestva veschestva, i ego primenenie k odnoi biologicheskoi probleme”, Byul. Mosk. gos. un-ta. Ser. Mat. mekh., 1 (1937), 1–26

[26] V. V. Gubernov, G. N. Mercer, H. S. Sidhu, R. O. Weber, “Evans function stability of combustion waves”, SIAM J. Appl. Math., 63 (2003), 1259–1275 | DOI | MR | Zbl

[27] U. Ebert, W. Saarloos, “Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts”, Physica D, 146 (2000), 1–99 | DOI | MR | Zbl