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Autowaves in the Model of Infiltrative Tumour Growth with Migration-Proliferation Dichotomy

Published online by Cambridge University Press:  15 June 2011

A.V. Kolobov ,
V.V. Gubernov  and
A.A. Polezhaev
A.V. Kolobov
Affiliation:
Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninskij prosp. 53, Moscow 119991, Russia
V.V. Gubernov
Affiliation:
Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninskij prosp. 53, Moscow 119991, Russia
A.A. Polezhaev*
Affiliation:
Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninskij prosp. 53, Moscow 119991, Russia
*
⋆Corresponding author. E-mail: apol@lpi.ru
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Abstract

A mathematical model of infiltrative tumour growth is investigated taking into account transitions between two possible states of malignant cells: proliferation and migration. These transitions are considered to depend on oxygen level in a threshold manner where high oxygen concentration allows cell proliferation, while concentration below a certain critical value induces cell migration. The infiltrative tumour spreading rate dependence on model parameters is obtained. It is shown that the tumour growth rate depends on the tissue oxygen level in a threshold manner.

Keywords

tumor growthproliferation and migration dichotomyoxygen
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 7: Mathematical modeling in biomedical applications , 2011 , pp. 27 - 38
Copyright
© EDP Sciences, 2011

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References

Burton, A.C.. Rate of growth of solid tumours as a problem of diffusion. Growth, 30 (1966), 157–176. Google ScholarPubMed
Bueno, H., Ercole, G., Zumpano, A.. Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours. Nonlinearity, 18 (2005), 1629–1642. CrossRefGoogle Scholar
Byrne, H.M., Chaplain, M.A.J.. Growth of necrotic tumours in the presence and absence of inhibitors. Math. Biosci., 135 (1996), 187–216. CrossRefGoogle ScholarPubMed
Chintala, S.K., Rao, J.R.. Invasion of human glioma: role of extracellular matrix proteins. Frontiers in Bioscience, 1 (1996), 324–339. Google ScholarPubMed
S., Fedotov, V., Mendez. Continuous-time random walks and traveling fronts. Phys. Rev. E, 66 (2002), 030102. Google Scholar
S., Fedotov, A., Iomin. Migration and Proliferation Dichotomy in Tumor-Cell Invasion. Phys. Rev. Lett., 98 (2007), 118101. Google Scholar
Giese, A., Bjerkvig, R., Berens, M.E., Westphal, M.. Cost of migration: invasion of malignant gliomas and implications for treatment. J. Clinical Oncology, 21 (2003), 1624–1636. CrossRefGoogle ScholarPubMed
Greenspan, H.P.. On the growth and stability of cell cultures and solid tumors. J. Theor. Biol., 56 (1976), 229–242. CrossRefGoogle ScholarPubMed
Gusev, A., Polezhaev, A.. Modelling of a cell population evolution for the case existence of maximal possible total cell density. Kratkie soobscheniya po fizike FIAN, 11-12 (1997), 85–90. Google Scholar
Hanahan, D., Weinberg, R.A.. The hallmarks of cancer. Cell, 100 (2000), 57–70. CrossRefGoogle ScholarPubMed
Harris, A.L.. Hypoxia – a key regulatory factor in tumour growth. Nat. Rev. Cancer, 2 (2002), 38–47. CrossRefGoogle ScholarPubMed
H., Hatzikirou, D., Basanta, M., Simon, K., Schaller, A., Deutsch. ’Go or Grow’: the key to the emergence of invasion in tumour progression? Math. Med. Biol., 7 (2010), 1–17. Google Scholar
A., Iomin. Toy model of fractional transport of cancer cells due to self-entrapping. Phys. Rev. E, 73 (2006), 061918. Google Scholar
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 & Computing in Biology and Medicine (Ed. V. Capasso), 2003, 603–609.
Laird, A.K.. Dynamics of tumor growth. Br. J. Cancer, 18 (1964), 490–502. CrossRefGoogle Scholar
J.S., Lowengrub, H.B., Frieboes, F., Jin, Chuang, Y-L., Li, X., Macklin, P., Wise, S.M., Cristini, V.. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity, 23 (2010), 1–91. Google Scholar
van Saarloos, W.. Front propagation into unstable states. Physics Reports, 386 (2003), 29–222. CrossRefGoogle Scholar
Sherratt, J.A., Chaplain, M.A.J.. A new mathematical model for avascular tumour growth. J. Math. Biol., 43 (2001), 291–312. CrossRefGoogle ScholarPubMed
K.R., Swansona, C., Bridge, J.D., Murray, Alvord, E.C. Jr. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurolog. Sci., 216 (2003), 1–10. Google Scholar
Tao, Y., Chen, M.. An elliptic-hyperbolic free boundary problem modelling cancer therapy. Nonlinearity, 19 (2006), 419–440. CrossRefGoogle Scholar
Thomlinson, R.H., Gray, L.H.. The histological structure of some human lung cancers and the possible implications for radiotherapy. Br. J. Cancer, 9 (1955), 539–549. CrossRefGoogle ScholarPubMed
Ward, J.P., King, J.R.. Mathematical modelling of avascular-tumour growth. IMA J. Math. Appl. Med. Biol., 14 (1997), 39–69. CrossRefGoogle Scholar