Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T18:26:27.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical Modelling of Cancer Stem Cells PopulationBehavior

Published online by Cambridge University Press:  25 January 2012

E. Beretta ,
V. Capasso  and
N. Morozova
E. Beretta
Affiliation:
CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
V. Capasso*
Affiliation:
CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
N. Morozova
Affiliation:
CNRS FRE 3377, Laboratoire Epigenetique et Cancer, CEA Saclay 91191 Gif-sur-Yvette, France
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questionsabout their nature, origin, function and their behavior in cell culture. Most of currentexperiments reporting a dynamics of cancer stem cell populations in culture show theeventual stability of the percentages of these cell populations in the whole population ofcancer cells, independently of the starting conditions. In this paper we propose amathematical model of cancer stem cell population behavior, based on specific features ofcancer stem cell divisions and including, as a mathematical formalization of cell-cellcommunications, an underlying field concept. We compare the qualitative behavior ofmathematical models of stem cells evolution, without and with an underlying signal. Inabsence of an underlying field, we propose a mathematical model described by a system ofordinary differential equations, while in presence of an underlying field it is describedby a system of delay differential equations, by admitting a delayed signal originated byexisting cells. Under realistic assumptions on the parameters, in both cases (ODE withoutunderlying field, and DDE with underlying field) we show in particular the stability ofpercentages, provided that the delay is sufficiently small. Further, for the DDE case (inpresence of an underlying field) we show the possible existence of, either damped orstanding, oscillations in the cell populations, in agreement with some existingmathematical literature. The outcomes of the analysis may offer to experimentalists a toolfor addressing the issue regarding the possible non-stem to stem cells transition, bydetermining conditions under which the stability of cancer stem cells population can beobtained only in the case in which such transition can occur. Further, the provideddescription of the variable corresponding to an underlying field may stimulate furtherexperiments for elucidating the nature of “instructive" signals for cell divisions,underlying a proper pattern of the biological system.

Keywords

cancer stem cellsdelay differential equationsqualitative behaviorstabilityoscillations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 279 - 305
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al-Hajj, M., Wicha, M.S., Benito-Hernandez, A., Morrison, S.J., Clarke, M.F.. Prospective identification of tumorigenic breast cancer cells. Proc. Natl Acad. Sci. USA, 100 (2003), 3983–3988. CrossRefGoogle ScholarPubMed
Bao, S., Wu, Q., McLendon, R.E., Hao, Y., Shi, Q., Hjelmeland, A.B., Dewhirst, M.W., Bigner, D.D., Rich, J.N.. Glioma stem cells promote radioresistance by preferential activation of the DNA damage response. Nature, 444 (2006), 756–760. CrossRefGoogle ScholarPubMed
Barrilleaux, B., Phinney, D.G., Prockop, D.J., O’Connor, K.C.. Review : ex vivo engineering of living tissues with adult stem cells. Tissue Eng., 12 (2006), 3007–3019. CrossRefGoogle ScholarPubMed
Bonnet, D., Dick, J.E.. Human acute myeloid leukemia is organized as a hierarchy that originates from a primitive hematopoietic cell. Nat. Med., 3 (1997), 730–737. CrossRefGoogle ScholarPubMed
Clarke, M.F., Dick, J.E., Dirks, P.B., Eaves, C.J., Jamieson, C.H., Jones, D.L., Visvader, J., Weissman, I.L., Wahl, G.M.. Cancer stem cells-Perspectives on current status and future directions : AACR workshop on cancer stem cells. Cancer Res., 66 (2006), 9339–9344. CrossRefGoogle Scholar
Dean, M., Fojo, T., Bates, S.. Tumour stem cells and drug resistance. Nat. Rev. Cancer, 5 (2005), 275–284. CrossRefGoogle ScholarPubMed
Diehn, M., Clarke, M.F.. Cancer stem cells and radiotherapy : new insights into tumor radioresistance. J. Natl. Cancer Inst., 98 (2006), 1755–1757. CrossRefGoogle ScholarPubMed
Dontu, G., Abdallah, W.M., Foley, J.M., Jackson, K.W., Clarke, M.F., Kawamura, M.J., Wicha, M.S.. In vitro propagation and transcriptionalprofiling of human mammary stem/progenitor cells. Genes Dev., 17 (2003), 1253–1270. CrossRefGoogle ScholarPubMed
D’Onofrio, A., Tomlison, I.P.M.. A nonlinear mathematical model of cell renewal, turnover and tumorigenesys in colon crypts. J. Theor. Biol., 244 (2007), 367–374. CrossRefGoogle Scholar
Eyler, C.E., Rich, J.N.. Survival of the fittest : cancer stem cells in therapeutic resistance and angiogenesis. J. Clin. Oncol., 26 (2008), 2839–2845. CrossRefGoogle ScholarPubMed
Freedman, H.I., Kuang, Y.. Stability switches in linear scalar neutral delay equations. Funkcial. Ekvac., 34 (1991), 187–209. Google Scholar
Gardner, R.L.. Stem cells : potency, plasticity and public perception. J. Anat., 200 (2002), 277–282. CrossRefGoogle ScholarPubMed
Gimble, J.M., Katz, A.J., Bunnell, B.A.. Adipose-derived stem cells for regenerative medicine. Circ. Res., 100 (2007), 1249–1260. CrossRefGoogle ScholarPubMed
Ginestier, C., Wicha, M.S.. Mammary stem cell number as a determinate of breast cancer risk. Breast Cancer Res., 9 (2007), 109. CrossRefGoogle ScholarPubMed
J. Guckenheimer, Ph. Holmes. Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, New York, 1983.
Gupta, P.B., Onder, T.T., Jiang, G., Tao, K., Kuperwasser, C., Weinberg, R.A., Lander, E.S.. Identification of selective inhibitors of cancer stem cells by high-throughput screening. Cell, 138 (2009), 645–659. CrossRefGoogle ScholarPubMed
Johnston, M.D., Edwards, C.M., Bodmer, W.F., Maini, P.K., Chapman, S.J.. Mathematical modelling of cell population dynamics in the colonic crypt and in colorectal cancer. PNAS, 104 (2007), 4008–4013. CrossRefGoogle ScholarPubMed
Lang, S.H., Frame, F., Collins, A., Prostate cancer stem cells. J. Pathol., 217 (2009), 299–306. CrossRefGoogle ScholarPubMed
Li, C., Heidt, D.G., Dalerba, P., Burant, C.F., Zhang, L., Adsay, V., Wicha, M., Clarke, M.F., Simeone, D.M.. Identification of pancreatic cancer stem cells. Cancer Res., 67 (2007), 1030–1037. CrossRefGoogle ScholarPubMed
Li, X., Lewis, M.T., Huang, J., Gutierrez, C., Osborne, C.K., Wu, M.F., Hilsenbeck, S.G., Pavlick, A., Zhang, X., Chamness, G.C., et al. Intrinsic resistance of tumorigenic breast cancer cells to chemotherapy. J. Natl. Cancer Inst., 100 (2008), 672–679. CrossRefGoogle Scholar
Maitland, N.J., Collins, A.T.. Prostate cancer stem cells : a new target for therapy. J. Clin. Oncol., 26 (2008), 2862–2870. CrossRefGoogle ScholarPubMed
Mani, S.A., Guo, W., Liao, M.J., Eaton, E.N., Ayyanan, A., Zhou, A.Y., Brooks, M., Reinhard, F., Zhang, C.C., Shipitsin, M., Campbell, L.L., Polyak, K., Brisken, C., Yang, J., Weinberg, R.A.. The epithelial-mesenchymal transition generates cells with properties of stem cells. Cell, 133 (2008), 704–715. CrossRefGoogle ScholarPubMed
Michor, F.. Mathematical models of cancer stem cells. J. Clin. Oncol., 26 (2008), 2854–2861. CrossRefGoogle ScholarPubMed
O’Brien, C.A., Pollett, A., Gallinger, S., Dick, J.E.. A human colon cancer cell capable of initiating tumour growth in immunodeficient mice. Nature, 445 (2007), 106–110. CrossRefGoogle ScholarPubMed
Ratajczak, M.Z., Machalinski, B., Wojakowski, W., Ratajczak, J., Kucia, M.. A hypothesis for an embryonic origin of pluripotent Oct-4(+) stem cells in adult bone marrow and other tissues. Leukemia, 21 (2007), 860–867. Google ScholarPubMed
Reya, T., Morrison, S.J., Clarke, M.F., Weissman, I.L.. Stem cells, cancer, and cancer stem cells. Nature, 414 (2001), 105–111. CrossRefGoogle ScholarPubMed
Ricci-Vitiani, L., Lombardi, D.G., Pilozzi, E., Biffoni, M., Todaro, M., Peschle, C., De Maria, R.. Identification and expansion of human colon-cancer-initiating cells. Nature, 445 (2007), 111–115. CrossRefGoogle ScholarPubMed
Roeder, I., Herberg, M., Horn, M.. An "Age" structured model of hemapoietic stem cell organization with application to chronic myeloid leukemia. Bull. Math. Biol., 71 (2009), 602–626. CrossRefGoogle Scholar
Singh, S.K., Clarke, I.D., Terasaki, M., Bonn, V.E., Hawkins, C., Squire, J., Dirks, P.B.. Identification of a cancer stem cell in human brain tumors. Cancer Res. 63 (2003), 5821–5828. Google ScholarPubMed
H. Smith. An introduction to delay differential equations with applications to the life sciences. Springer, New York, 2010.
Watt, F. M., Hogan, B. L.. Out of Eden : stem cells and their niches. Science, 287 (2000), 1427–1430 CrossRefGoogle ScholarPubMed
R.A. Weinberg. The biology of cancer. Garland Science, New York, 2007.
Whetton, A. D., Graham, G. J.. Homing and mobilization in the stem cell niche. Trends Cell Biol., 9 (1999), 233–238 CrossRefGoogle ScholarPubMed
Wolpert, L.. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol., 25 (1969), 1–47 CrossRefGoogle ScholarPubMed
Woodward, W.A., Chen, M.S., Behbod, F., Alfaro, M.P., Buchholz, T.A., Rosen, J.M.. WNT/beta-catenin mediates radiation resistance of mouse mammary progenitor cells. Proc. Natl. Acad. Sci. USA 104 (2007), 618–623. CrossRefGoogle ScholarPubMed
Zhdanov, V.P.. Effect of cell-cell communication on the kinetics of proliferation and differentiation of stem cells. Chemical Physics Letters, 437 (2007), 253–256. CrossRefGoogle Scholar
Zhang, S., Balch, C., Chan, M.W., Lai, H.C., Matei, D., Schilder, J.M., Yan, P.S., Huang, T.H., Nephew, K.P.. Identification and characterization of ovarian cancer-initiating cells from primary human tumors. Cancer Res., 68 (2008), 4311–4320. CrossRefGoogle ScholarPubMed