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Comparison ofPerron and Floquet Eigenvalues in AgeStructured Cell Division Cycle Models

Published online by Cambridge University Press:  05 June 2009

J. Clairambault ,
S. Gaubert  and
Th. Lepoutre
J. Clairambault
Affiliation:
INRIA, projet BANG, Domaine de Voluceau, BP 105, 78156 Le Chesnay Cedex France INSERM U 776, Hôpital Paul-Brousse, 14, Av. Paul-Vaillant-Couturier F94807 Villejuif cedex
S. Gaubert
Affiliation:
INRIA Saclay – Ile-de-France, projet MAXPLUS CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Th. Lepoutre*
Affiliation:
INRIA, projet BANG, Domaine de Voluceau, BP 105, 78156 Le Chesnay Cedex France UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
*
thomas.lepoutre@inria.fr
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Abstract

We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to prove several inequalities and equalities on the growth rates. We also discuss about the necessity to take into account the structure of the cell division cycle for chronotherapy modeling. Numerical simulations illustrate the results.

Keywords

cell cyclecircadian rhythmschronotherapystructured PDEsdelay differential equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 3: Cancer modelling (Part 2) , 2009 , pp. 183 - 209
Copyright
© EDP Sciences, 2009

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References

Arino, O.. A survey of structured cell population dynamics. Acta Biotheor., 43 (1995), 3–25. CrossRef
O. Arino and M. Kimmel. Comparison of approaches to modeling of cell population dynamics. SIAM J. Appl. Math., 53(1993), No. 5, 1480–1504.
Arino, O. and Sanchez, E.. A survey of cell population dynamics. J. Theor. Med., 1 (1997), No. 1, 35–51. CrossRef
S. Bernard and H. Herzel. Why do cells cycle with a 24 hour period? Genome Inform., 17, (2006), No. 1, 72–79.
Bekkal Brikci, F., Clairambault, J., and Perthame, B.. Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle. Math. Comput. Modelling, 47 (2008), No. 7-8, 699–713. CrossRef
Bekkal Brikci, F., Clairambault, J., Ribba, B., and Perthame, B.. An age-and-cyclin-structured cell population model for healthy and tumoral tissues. J. Math. Biol., 57 (2008), No. 1, 91–110. CrossRef
Clairambault, J., Gaubert, S., and Perthame, B.. An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations. C. R. Math. Acad. Sci. Paris, 345 (2007), No. 10, 549–554. CrossRef
Clairambault, J., Michel, P., and Perthame, B.. Circadian rhythm and tumour growth. C. R. Acad. Sci., 342 (2006), No. 1, 17–22. CrossRef
J. Clairambault, P. Michel, and B. Perthame. (2007) A mathematical model of the cell cycle and its circadian control, to appear in Mathematical modeling of Biological Systems, Volume I. A. Deutsch and L. Brusch and H. Byrne and G. de Vries and H.-P. Herzel (eds), Birkhäuser, pp 247–259 proceedings of ECMTB conference, Dresden 2005).
R. Dautray and J.L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. Springer, 1988.
Doumic, M.. Analysis of a population model structured by the cells molecular content. Mathematical Modelling of Natural Phenomena, 2 (2007), No. 3, 121–152. CrossRef
E. Filipski, P.F. Innominato., M. Wu, X.M. Liand S. Iacobelli, L.J. Xian, and F. Levi. Effects of light and food schedules on liver and tumor molecular clocks in mice. Journal of the National Cancer Institute, 97 (April 2005), No. 7, 507–517, .
E. Filipski, Verdun M King, X.M. Li, T. G. Granda, M. Mormont, XuHui Liu, B. Claustrat, M. H. Hastings, and F. Levi. Host circadian clock as a control point in tumor progression. J Natl Cancer Inst, 94( May 2002), No 9, 690–697,.
A. Goldbeter. A minimal cascade for the mitotic oscillator involving cyclin and cdc2 kinase. Proc. Nat. Acad. Sci. USA, 88 (October 1991), 9107–9111.
R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, New York, NY, USA, 1986.
J. Keener and J. Sneyd. Mathematical Physiology, volume 8. Springer, 1998.
Levi, F., Altinok, A., Clairambault, J., and Goldbeter, A.. Implications of circadian clocks for the rhythmic delivery of cancer therapeutics. Phil. Trans. R. Soc. A, 366 (2008), 3575–3598. CrossRef
Levi, F. and Schibler, U.. Circadian rhythms: mechanisms and therapeutic implications. Annu. Rev. Pharmacol. Toxicol., 47 (2007), 593–628. CrossRef
J.A.J. Metz and O. Diekmann. The dynamics of physiologically structured populations, volume 68 of L.N. in biomathematics. Springer, 1986.
P. Michel, S. Mischler, and B. Perthame. General relative entropy inequality: an illustration on growth models. J. Math. Pures et Appl., 84 (May 2005), No. 9, 1235–1260.
D. O Morgan. The Cell Cycle. Primers in Biology. Oxford University Press, 2007.
J.D. Murray. Mathematical Biology, volume 1. Springer, 3rd edition, 2002.
B. Novak. Modeling the cell division cycle. Lund(Sweden), April 15-18 1999. Bioinformatics'99. Available online at: http://cellcycle.mkt.bme.hu/people/bnovak/pdfek/lund/talk.pdf.
B. Perthame. Transport equations in biology. Birkhäuser, 2007.
E. Seijo Solis. A report on the discretization of a one-phase model of the cell cycle. Inria internship report, INRIA, 2006.
Tyson, J.J., Chen, K., and Novak, B.. Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol., 2 (2001), 908–916. CrossRef