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

Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models

Jean Clairambault (1) , Stéphane Gaubert (2, 3) , Thomas Lepoutre (1)

1 BANG - Nonlinear Analysis for Biology and Geophysical flows
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 MAXPLUS - Max-plus algebras and mathematics of decision

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.

Mots clés: cell cycle; circadian rhythms; chronotherapy; structured PDEs; delay differential equations.; delay differential equations

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Systèmes dynamiques [math.DS]

Jean Clairambault, Stéphane Gaubert, Thomas Lepoutre. Comparison of Perron and Floquet Eigenvalues in Age Structured Cell Division Cycle Models. Mathematical Modelling of Natural Phenomena, 2009, 4 (3), pp.183-209. ⟨10.1051/mmnp/20094308⟩. ⟨hal-00344039⟩