Geodesic
Mathematical Modelling of Cancer Stem Cells Population Behavior
E. Beretta1 ; V. Capasso1 ; N. Morozova2
1 CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
2 CNRS FRE 3377, Laboratoire Epigenetique et Cancer, CEA Saclay 91191 Gif-sur-Yvette, France
Mathematical modelling of natural phenomena, Tome 7 (2012) no. 1, pp. 279-305

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Recent discovery of cancer stem cells in tumorigenic tissues has raised many questions about their nature, origin, function and their behavior in cell culture. Most of current experiments reporting a dynamics of cancer stem cell populations in culture show the eventual stability of the percentages of these cell populations in the whole population of cancer cells, independently of the starting conditions. In this paper we propose a mathematical model of cancer stem cell population behavior, based on specific features of cancer stem cell divisions and including, as a mathematical formalization of cell-cell communications, an underlying field concept. We compare the qualitative behavior of mathematical models of stem cells evolution, without and with an underlying signal. In absence of an underlying field, we propose a mathematical model described by a system of ordinary differential equations, while in presence of an underlying field it is described by a system of delay differential equations, by admitting a delayed signal originated by existing cells. Under realistic assumptions on the parameters, in both cases (ODE without underlying field, and DDE with underlying field) we show in particular the stability of percentages, provided that the delay is sufficiently small. Further, for the DDE case (in presence of an underlying field) we show the possible existence of, either damped or standing, oscillations in the cell populations, in agreement with some existing mathematical literature. The outcomes of the analysis may offer to experimentalists a tool for addressing the issue regarding the possible non-stem to stem cells transition, by determining conditions under which the stability of cancer stem cells population can be obtained only in the case in which such transition can occur. Further, the provided description of the variable corresponding to an underlying field may stimulate further experiments for elucidating the nature of “instructive" signals for cell divisions, underlying a proper pattern of the biological system.
DOI : 10.1051/mmnp/20127113

E. Beretta 1 ; V. Capasso 1 ; N. Morozova 2

1 CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
2 CNRS FRE 3377, Laboratoire Epigenetique et Cancer, CEA Saclay 91191 Gif-sur-Yvette, France 
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E. Beretta; V. Capasso; N. Morozova. Mathematical Modelling of Cancer Stem Cells Population Behavior. Mathematical modelling of natural phenomena, Tome 7 (2012) no. 1, pp. 279-305. doi: 10.1051/mmnp/20127113

[1] M. Al-Hajj, M.S. Wicha, A. Benito-Hernandez, S.J. Morrison, M.F. Clarke Proc. Natl Acad. Sci. USA 2003 3983 3988

[2] S. Bao, Q. Wu, R.E. Mclendon, Y. Hao, Q. Shi, A.B. Hjelmeland, M.W. Dewhirst, D.D. Bigner, J.N. Rich Nature 2006 756 760

[3] B. Barrilleaux, D.G. Phinney, D.J. Prockop, K.C. O’Connor Tissue Eng. 2006 3007 3019

[4] D. Bonnet, J.E. Dick Nat. Med. 1997 730 737

[5] M.F. Clarke, J.E. Dick, P.B. Dirks, C.J. Eaves, C.H. Jamieson, D.L. Jones, J. Visvader, I.L. Weissman, G.M. Wahl Cancer Res. 2006 9339 9344

[6] M. Dean, T. Fojo, S. Bates Nat. Rev. Cancer 2005 275 284

[7] M. Diehn, M.F. Clarke J. Natl. Cancer Inst. 2006 1755 1757

[8] G. Dontu, W.M. Abdallah, J.M. Foley, K.W. Jackson, M.F. Clarke, M.J. Kawamura, M.S. Wicha Genes Dev. 2003 1253 1270

[9] A. D’Onofrio, I.P.M. Tomlison J. Theor. Biol. 2007 367 374

[10] C.E. Eyler, J.N. Rich J. Clin. Oncol. 2008 2839 2845

[11] H.I. Freedman, Y. Kuang Funkcial. Ekvac. 1991 187 209

[12] R.L. Gardner J. Anat. 2002 277 282

[13] J.M. Gimble, A.J. Katz, B.A. Bunnell Circ. Res. 2007 1249 1260

[14] C. Ginestier, M.S. Wicha Breast Cancer Res. 2007 109

[15] J. Guckenheimer, Ph. Holmes. Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, New York, 1983.

[16] P.B. Gupta, T.T. Onder, G. Jiang, K. Tao, C. Kuperwasser, R.A. Weinberg, E.S. Lander Cell 2009 645 659

[17] M.D. Johnston, C.M. Edwards, W.F. Bodmer, P.K. Maini, S.J. Chapman PNAS 2007 4008 4013

[18] S.H. Lang, F. Frame, A. Collins J. Pathol. 2009 299 306

[19] C. Li, D.G. Heidt, P. Dalerba, C.F. Burant, L. Zhang, V. Adsay, M. Wicha, M.F. Clarke, D.M. Simeone Cancer Res. 2007 1030 1037

[20] X. Li, M.T. Lewis, J. Huang, C. Gutierrez, C.K. Osborne, M.F. Wu, S.G. Hilsenbeck, A. Pavlick, X. Zhang, G.C. Chamness J. Natl. Cancer Inst. 2008 672 679

[21] N.J. Maitland, A.T. Collins J. Clin. Oncol. 2008 2862 2870

[22] S.A. Mani, W. Guo, M.J. Liao, E.N. Eaton, A. Ayyanan, A.Y. Zhou, M. Brooks, F. Reinhard, C.C. Zhang, M. Shipitsin, L.L. Campbell, K. Polyak, C. Brisken, J. Yang, R.A. Weinberg Cell 2008 704 715

[23] F. Michor J. Clin. Oncol. 2008 2854 2861

[24] C.A. O’Brien, A. Pollett, S. Gallinger, J.E. Dick Nature 2007 106 110

[25] M.Z. Ratajczak, B. Machalinski, W. Wojakowski, J. Ratajczak, M. Kucia Leukemia 2007 860 867

[26] T. Reya, S.J. Morrison, M.F. Clarke, I.L. Weissman Nature 2001 105 111

[27] L. Ricci-Vitiani, D.G. Lombardi, E. Pilozzi, M. Biffoni, M. Todaro, C. Peschle, R. De Maria Nature 2007 111 115

[28] I. Roeder, M. Herberg, M. Horn Bull. Math. Biol. 2009 602 626

[29] S.K. Singh, I.D. Clarke, M. Terasaki, V.E. Bonn, C. Hawkins, J. Squire, P.B. Dirks Cancer Res. 2003 5821 5828

[30] H. Smith. An introduction to delay differential equations with applications to the life sciences. Springer, New York, 2010.

[31] F. M. Watt, B. L. Hogan Science 2000 1427 1430

[32] R.A. Weinberg. The biology of cancer. Garland Science, New York, 2007.

[33] A. D. Whetton, G. J. Graham Trends Cell Biol. 1999 233 238

[34] L. Wolpert J. Theor. Biol. 1969 1 47

[35] W.A. Woodward, M.S. Chen, F. Behbod, M.P. Alfaro, T.A. Buchholz, J.M. Rosen Proc. Natl. Acad. Sci. USA 2007 618 623

[36] V.P. Zhdanov Chemical Physics Letters 2007 253 256

[37] S. Zhang, C. Balch, M.W. Chan, H.C. Lai, D. Matei, J.M. Schilder, P.S. Yan, T.H. Huang, K.P. Nephew Cancer Res. 2008 4311 4320

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