Geodesic
Comparison of Perron and Floquet Eigenvalues in Age Structured Cell Division Cycle Models
J. Clairambault1, 2 ; S. Gaubert3, 4 ; Th. Lepoutre1, 5
1 INRIA, projet BANG, Domaine de Voluceau, BP 105, 78156 Le Chesnay Cedex France
2 INSERM U 776, Hôpital Paul-Brousse, 14, Av. Paul-Vaillant-Couturier F94807 Villejuif cedex
3 INRIA Saclay – Ile-de-France, projet MAXPLUS
4 CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France
5 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
Mathematical modelling of natural phenomena, Tome 4 (2009) no. 3, pp. 183-209.

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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.
DOI : 10.1051/mmnp/20094308

J. Clairambault 1, 2 ; S. Gaubert 3, 4 ; Th. Lepoutre 1, 5

1 INRIA, projet BANG, Domaine de Voluceau, BP 105, 78156 Le Chesnay Cedex France
2 INSERM U 776, Hôpital Paul-Brousse, 14, Av. Paul-Vaillant-Couturier F94807 Villejuif cedex
3 INRIA Saclay – Ile-de-France, projet MAXPLUS
4 CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France
5 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France 
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J. Clairambault; S. Gaubert; Th. Lepoutre. Comparison of Perron and Floquet Eigenvalues in Age Structured Cell Division Cycle Models. Mathematical modelling of natural phenomena, Tome 4 (2009) no. 3, pp. 183-209. doi : 10.1051/mmnp/20094308. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20094308/

[1] O. Arino Acta Biotheor. 1995 3 25

[2] O. Arino and M. Kimmel. Comparison of approaches to modeling of cell population dynamics. SIAM J. Appl. Math., 53(1993), No. 5, 1480–1504.

[3] O. Arino, E. Sanchez J. Theor. Med. 1997 35 51

[4] S. Bernard and H. Herzel. Why do cells cycle with a 24 hour period? Genome Inform., 17, (2006), No. 1, 72–79.

[5] F. Bekkal Brikci, J. Clairambault, B. Perthame Math. Comput. Modelling 2008 699 713

[6] F. Bekkal Brikci, J. Clairambault, B. Ribba, B. Perthame J. Math. Biol. 2008 91 110

[7] J. Clairambault, S. Gaubert, B. Perthame C. R. Math. Acad. Sci. Paris 2007 549 554

[8] J. Clairambault, P. Michel, B. Perthame C. R. Acad. Sci. 2006 17 22

[9] 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).

[10] R. Dautray and J.L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. Springer, 1988.

[11] M. Doumic Mathematical Modelling of Natural Phenomena 2007 121 152

[12] 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, .

[13] 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,.

[14] A. Goldbeter. A minimal cascade for the mitotic oscillator involving cyclin and cdc2 kinase. Proc. Nat. Acad. Sci. USA, 88 (October 1991), 9107–9111.

[15] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, New York, NY, USA, 1986.

[16] J. Keener and J. Sneyd. Mathematical Physiology, volume 8. Springer, 1998.

[17] F. Levi, A. Altinok, J. Clairambault, A. Goldbeter Phil. Trans. R. Soc. A 2008 3575 3598

[18] F. Levi, U. Schibler Annu. Rev. Pharmacol. Toxicol. 2007 593 628

[19] J.A.J. Metz and O. Diekmann. The dynamics of physiologically structured populations, volume 68 of L.N. in biomathematics. Springer, 1986.

[20] 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.

[21] D. O Morgan. The Cell Cycle. Primers in Biology. Oxford University Press, 2007.

[22] J.D. Murray. Mathematical Biology, volume 1. Springer, 3rd edition, 2002.

[23] 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.

[24] B. Perthame. Transport equations in biology. Birkhäuser, 2007.

[25] E. Seijo Solis. A report on the discretization of a one-phase model of the cell cycle. Inria internship report, INRIA, 2006.

[26] J.J. Tyson, K. Chen, B. Novak Nat. Rev. Mol. Cell Biol. 2001 908 916

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