2013, 10(1): 263-278. doi: 10.3934/mbe.2013.10.263

On a mathematical model of tumor growth based on cancer stem cells

1. 

Departamento de Matemática Aplicada, EUI Informática, Universidad Politécnica de Madrid, 28031 Madrid, Spain

Received  July 2012 Revised  September 2012 Published  December 2012

We consider a simple mathematical model of tumor growth based on cancer stem cells. The model consists of four hyperbolic equations of first order to describe the evolution of different subpopulations of cells: cancer stem cells, progenitor cells, differentiated cells and dead cells. A fifth equation is introduced to model the evolution of the moving boundary. The system includes non-local terms of integral type in the coefficients. Under some restrictions in the parameters we show that there exists a unique homogeneous steady state which is stable.
Citation: J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263
References:
[1]

M. F. Clarke and M. Fuller, Stem cells and cancer: Two faces of eve,, Cell, 124 (2006), 1111.

[2]

A. T. Collins, P. A. Berry, C. Hyde, M. J. Stower and N. J. Maitland, Prospective identification of tumorigenic prostate cancer stem cells,, Cancer Res, 65 (2005), 10946.

[3]

J. E. Dick, Stem cell concepts renew cancer research,, Blood, 112 (2008), 4793.

[4]

A. Friedman, Cancer models and their mathematical analysis,, Tutorials in mathematical biosciences. III, (1872), 223.

[5]

A. Friedman and Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells,, J. Math. Biol., 47 (2003), 391. doi: 10.1007/s00285-003-0199-5.

[6]

C. Fornari, F. Cordero, D. Manini, R. A. Calogero and G. Balbo, Mathematical approach to predict the drug effects on cancer stem cell models,, Proceedings of the CS2Bio 2nd International Workshop on Interactions between Computer Science and Biology, (2011).

[7]

R. Molina-Peña and M. M. Álvarez, A simple mathematical model based on the cancer stem cell hypothesis suggests kinetic commonalities in solid tumor growth,, PLoS ONE, 7 (2012).

[8]

K. Qu and P. Ortoleva, Understanding stem cell differentiation through self-organization theory,, Journal of Theoretical Biology, 250 (2008), 606.

[9]

S. Bapat, "Cancer Steam Cells, Identification and Targets,", Willey Edt. New Jersey, (2009).

[10]

Z. Szymánska, C. Morales Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interaction,, Mathematical Models and Methods in Applied Sciences, 19 (2009), 257.

[11]

J. I. Tello, Mathematical analysis of a model of Morphogenesis,, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343.

[12]

J. I. Tello, On the existence of solutions of a mathematical model of morphogens,, in, (2011).

show all references

References:
[1]

M. F. Clarke and M. Fuller, Stem cells and cancer: Two faces of eve,, Cell, 124 (2006), 1111.

[2]

A. T. Collins, P. A. Berry, C. Hyde, M. J. Stower and N. J. Maitland, Prospective identification of tumorigenic prostate cancer stem cells,, Cancer Res, 65 (2005), 10946.

[3]

J. E. Dick, Stem cell concepts renew cancer research,, Blood, 112 (2008), 4793.

[4]

A. Friedman, Cancer models and their mathematical analysis,, Tutorials in mathematical biosciences. III, (1872), 223.

[5]

A. Friedman and Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells,, J. Math. Biol., 47 (2003), 391. doi: 10.1007/s00285-003-0199-5.

[6]

C. Fornari, F. Cordero, D. Manini, R. A. Calogero and G. Balbo, Mathematical approach to predict the drug effects on cancer stem cell models,, Proceedings of the CS2Bio 2nd International Workshop on Interactions between Computer Science and Biology, (2011).

[7]

R. Molina-Peña and M. M. Álvarez, A simple mathematical model based on the cancer stem cell hypothesis suggests kinetic commonalities in solid tumor growth,, PLoS ONE, 7 (2012).

[8]

K. Qu and P. Ortoleva, Understanding stem cell differentiation through self-organization theory,, Journal of Theoretical Biology, 250 (2008), 606.

[9]

S. Bapat, "Cancer Steam Cells, Identification and Targets,", Willey Edt. New Jersey, (2009).

[10]

Z. Szymánska, C. Morales Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interaction,, Mathematical Models and Methods in Applied Sciences, 19 (2009), 257.

[11]

J. I. Tello, Mathematical analysis of a model of Morphogenesis,, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343.

[12]

J. I. Tello, On the existence of solutions of a mathematical model of morphogens,, in, (2011).

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