Research Article

Comparison of Perron and Floquet Eigenvalues in Age Structured Cell Division Cycle Models

J. Clairambault^a1a2, S. Gaubert^a3a4 and Th. Lepoutre^a1a5

^a1 INRIA, projet BANG, Domaine de Voluceau, BP 105, 78156 Le Chesnay Cedex France

^a2 INSERM U 776, Hôpital Paul-Brousse, 14, Av. Paul-Vaillant-Couturier F94807 Villejuif cedex

^a3 INRIA Saclay – Ile-de-France, projet MAXPLUS

^a4 CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France

^a5 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

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.

(Online publication June 5 2009)

Key Words:

• cell cycle;
• circadian rhythms;
• chronotherapy;
• structured PDEs;
• delay differential equations

Mathematics Subject Classification:

• 35F05;
• 35P05;
• 35P15;
• 92B05;
• 92D25

Correspondence:

^c1 thomas.lepoutre@inria.fr