Simple Linear Model of Tumor Growth in a Changing Environment

Abstract

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.

Share and Cite:

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.

References

[1] Laird, A.K. (1965) Dynamics of Tumor Growth: Comparison of Growth Rates. British Journal of Cancer, 19, 278-291.
http://dx.doi.org/10.1038/bjc.1965.32
[2] Norton, L., Simon, R. and Brereton, H. (1976) Predicting the Course of Gompertzian Growth. Nature, 264, 542-545.
http://dx.doi.org/10.1038/264542a0
[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.
http://dx.doi.org/10.1016/0025-5564(91)90093-X
[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.
http://dx.doi.org/10.1038/bjc.1998.503
[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.
http://dx.doi.org/10.1002/1097-0142(19930315)71:6<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.
http://dx.doi.org/10.1016/j.mbs.2005.11.001
[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.
http://dx.doi.org/10.1016/j.bulm.2003.11.002
[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.
http://dx.doi.org/10.1063/1.1812842
[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).
http://dx.doi.org/10.1103/PhysRevE.73.060902
[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.
http://dx.doi.org/10.1103/PhysRevE.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.
http://dx.doi.org/10.1016/S0092-8240(03)00027-2
[14] Durrett, R. (2013) Cancel Modeling: A Personal Perspective. Notices of the AMS, 60, 304-309.
http://dx.doi.org/10.1090/noti953
[15] Berstein, J. (1988) Kinetic Theory in the Expanding Universe. Cambridge University, New York.
http://dx.doi.org/10.1017/CBO9780511564185

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.