Versal deformation and static bifurcation diagrams for the cancer cell population model
Authors:
Vladimir Balan and Ileana Rodica Nicola
Journal:
Quart. Appl. Math. 67 (2009), 755-770
MSC (2000):
Primary 37G10, 37G35, 70K50
DOI:
https://doi.org/10.1090/S0033-569X-09-01169-X
Published electronically:
May 14, 2009
MathSciNet review:
2588235
Full-text PDF Free Access
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Additional Information
Abstract: The paper studies the existence of rest-points and the static bifurcation diagrams of a given nonlinear differential system modeling the cancer cell population evolution from biology. To this aim, the nullclines, the equilibrium points, the transient set, the static bifurcation equation and the associated versal deformation are investigated. The results are further discussed in view of potential applications to cancer therapy.
References
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- G.I. Solyanik, N.M. Berezetskaya, R.I. Bulkiewicz, G.I. Kulik, Different growth patterns of a cancer cell population as a function of its starting growth characteristics: Analysis by mathematical modelling, Cell Prolif., vol. 28, No. 5, (1995), 263-278.
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References
- V. Balakotaiah, D. Luss, Analysis of the multiplicity patterns of a CSTR, Chem. Eng. Commun., vol. 13, (1981), 111-132.
- S.J. Beebe, P.M. Fox, L.J. Rec, E.L. Willis, K.H. Schoenbach, Nanosecond, high-intensity pulses electric fields induce apoptosis in human cells, Fed. Am. Soc. Exper. Biol. J., vol. 17, (2003), 1493-1495.
- F. Behbod, J.M. Rosen, Will cancer stem provide new therapeutic targets?, Carcinogenesis, vol. 26, (2004), 703-711.
- R. Curtu, Dynamics and bifurcations of the Gray-Scott model in the presence of non-catalyzed conversion (in Romanian), Ed. Univ. Transilvania, Braşov, 2002. MR 1993249 (2004g:37117)
- R. Curtu, The static bifurcation diagram for the Gray-Scott model, Int. Jour. of Bifurcation and Chaos, vol. 11, 9 (2001), 2483-2491. MR 1862635 (2002h:37172)
- J.P. Freyer, R.M. Sutherland, Regulation of growth saturation and development of necrosis in EMT6/R0 multicellular spheroids by the glucose and oxygen supply, Cancer Res., vol. 46, (1986), 3504-3512.
- A.L. Garner, Y.Y. Lau, D.W. Jordan, M.D. Uhler, R.M. Gilgenbach, Implication of a simple mathematical model to cancer cell population dynamics, Cell Prolif., vol. 39, (2006), 15-28.
- M. Golubitsky, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1985. MR 771477 (86e:58014)
- A.M. Luciani, A. Rosi, P. Matarrese, G. Arancia, L. Guidoni, V. Viti, Changes in cell volume and internal sodium concentration in HrLa cells during exponential growth and following Ionidamine treatment, Eur. J. Cell Biol., vol. 80, (2001), 187-195.
- T. Reya, S.J. Morrison, M.F. Clarke, I.L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, vol. 414, (2001), 105-111.
- K.H. Schoenbach, R.P. Joshi, J.F. Kolb, N. Chen, M. Stacey, P.F. Blackmore, E.S. Buescher, S.J. Beebe, Ultrashort electrical pulses open a new gateway into biological cells, Proc. IEEE, vol. 92, No. 7, (2004), 1122-1137.
- G.I. Solyanik, N.M. Berezetskaya, R.I. Bulkiewicz, G.I. Kulik, Different growth patterns of a cancer cell population as a function of its starting growth characteristics: Analysis by mathematical modelling, Cell Prolif., vol. 28, No. 5, (1995), 263-278.
- M. Stacey, J. Stickley, P. Fox, V. Statler, K. Schoenbach, S.J. Beebe, S. Buescher, Differential effects in cells exposed to ultra-short high intensity electric fields: Cell survival, DNA damage, and cell-cycle analysis, Mutat. Res., vol. 542, (2003), 65-75.
- C.A. Wallen, R. Higashicubo, L.A. Dethlefsen, Murine mammary tumour cells in vitro. I. The development of a quiescent state, Cell Tissue Kinet., vol. 17, (1984), 65-78.
- C.A. Wallen, R. Higashicubo, L.A. Dethlefsen, Murine mammary tumour cells in vitro. II. Recruitment of quiescent state, Cell Tissue Kinet., vol. 17, (1984), 79-89.
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Additional Information
Vladimir Balan
Affiliation:
University “Politehnica” of Bucharest, Faculty of Applied Sciences, Dept. Mathematics-Informatics I, Splaiul Indpendentei 313, RO-060042 Bucharest, Romania
ORCID:
[object Object]
Email:
vbalan@mathem.pub.ro
Ileana Rodica Nicola
Affiliation:
University “Spiru Haret” of Bucharest, Faculty of Mathematics and Informatics, Ion Ghica Str. 13, RO-030045 Bucharest, Romania
Email:
nicola_rodica@yahoo.com
Keywords:
Static bifurcation diagram,
versal deformation,
transient set,
hysteresis set,
nullclines,
normal form,
quiescent cell population,
proliferating cell population
Received by editor(s):
August 25, 2008
Published electronically:
May 14, 2009
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.
