Simple Linear Model of Tumor Growth in a Changing Environment

DOI: 10.4236/am.2015.67104   PDF   HTML   XML   3,035 Downloads   3,656 Views  


In an environment that is neither static nor in equilibrium, but is dynamic and changing, the kinetics of the reactions that cause the growth of a tumor, which depend on the state of the evolving environment, cannot be parametrized in terms of constant rates. We propose a simple model for describing the growth on an untreated tumor in such environments, which is characterized by a minimal number of parameters and is generalizable to include the effects of various types of therapies. In the simplest version that we consider here, it consists of a linear equation with a time-dependent growth rate, which we interpret as the coupling of the system with a dynamic environment. A complete solution is given in terms of the integral of the growth rate. The essential features of the general solution are illustrated with a few examples, and comparison is made with the models that have been proposed to describe recent data.

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Nieves, J. and Ubriaco, M. (2015) Simple Linear Model of Tumor Growth in a Changing Environment. Applied Mathematics, 6, 1139-1147. doi: 10.4236/am.2015.67104.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Laird, A.K. (1965) Dynamics of Tumor Growth: Comparison of Growth Rates. British Journal of Cancer, 19, 278-291.
[2] Norton, L., Simon, R. and Brereton, H. (1976) Predicting the Course of Gompertzian Growth. Nature, 264, 542-545.
[3] Norton, L. (1988) A Gompertzian Model of Human Breast Cancer. Cancer Research, 48, 7067.
[4] Calderon, C. and Kwembe, T. (1991) Modeling Tumor Growth. Mathematical Biosciences, 103, 97-114.
[5] Hart, D., Schochat, E. and Agur, Z. (1998) The Law Growth of Primary Breast Cancer as Inferred from Mammography Screening Trials Data. British Journal of Cancer, 78, 382-387.
[6] Spratt, J.A., von Fournier, D. and Spratt, J.S. (1993) Decelerating Growth and Human Breast Cancer. Cancer, 71, 2013-2019, and references therein.<2013::AID-CNCR2820710615>3.0.CO;2-V
[7] Dingli, D., Cascino, M.D., Josic, K., Rusell, J. and Bajzer, Z. (2006) Mathematical Modeling of Cancer Radiovirotherapy. Mathematical Biosciences, 199, 55-78, and references therein.
[8] Araujo, R.P. and McElwain, D.L.S. (2004) A History of the Study of Solid Tumor Growth: The Contribution of Mathematical Modeling. Bulletin of Mathematical Biology, 66, 1039-1091.
[9] Byrne, H.M., Alarcon, T., Owen, M.R., Web, S.D. and Maini, O.K. (2006) Modelling Aspects of Cancer Dynamics: A Review. Philosophical Transactions of the Royal Society A, 364, 1563-1578.
[10] Delsanto, P.P., Guiot, C. and Degiorgis, P.G. (2004) Growth Model for Multicelullar Spheroids. Applied Physics Letters, 85, 4225.
[11] Zhong, W.R., Shao, Y.Z. and He, Z.H. (2006) Pure Multiplicative Stochastic Resonance of a Theoretical Antitumor Model with Seasonal Modulability. Physical Review E, 73, 060902(R).
[12] Zhong, W.R., Shao, Y.Z. and He, Z.H. (2006) Temporal Fluctuation-Induced Transition in a Tumor Model. Physical Review E, 74, 011916.
[13] Breward, C.J.W., Byrne, H.M. and Leweis, C.E. (2003) A Multiphase Model Describing Vascular Tumor Growth. Bulletin of Mathematical Biology, 65, 609-640.
[14] Durrett, R. (2013) Cancel Modeling: A Personal Perspective. Notices of the AMS, 60, 304-309.
[15] Berstein, J. (1988) Kinetic Theory in the Expanding Universe. Cambridge University, New York.

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