Partial Differential Equations
Circadian rhythm and tumour growth
Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 17-22.

We address the following question: can one sustain, on the basis of mathematical models, that for cancer cells, the loss of control by circadian rhythm favours a faster growth? This question, which comes from the observation that tumour growth in mice is enhanced by experimental disruption of the circadian rhythm, may be tackled by mathematical modelling of the cell cycle. For this purpose we consider an age-structured population model with control of death (apoptosis) rates and phase transitions, and two eigenvalues: one for periodic control coefficients (via a variant of Floquet theory in infinite dimension) and one for constant coefficients (taken as the time average of the periodic case). We show by a direct proof that, surprisingly enough considering the above-mentioned observation, the periodic eigenvalue is always greater than the steady state eigenvalue when the sole apoptosis rate is concerned. We also show by numerical simulations when transition rates between the phases of the cell cycle are concerned, that, without further hypotheses, no natural hierarchy between the two eigenvalues exists. This at least shows that, if such models are to take account of the above-mentioned observation, control of death rates inside phases is not sufficient, and that transition rates between phases are a key target in proliferation control.

L'objet de cette Note est de questionner, sur des bases mathématiques, le fait expérimental que les populations de cellules de souris cancéreuses échappant au contrôle circadien, ont tendance à se développer plus vite. Pour cela nous considérons un modèle du cycle cellulaire avec contrôle des taux de mort (apoptose) et de transition entre phases, et deux valeurs propres. L'une est associée aux coefficients périodiques via la théorie de Floquet (dans une version de dimension infinie), l'autre est associée au problème stationnaire avec des coefficients moyens. Nous montrons par une preuve directe que, de façon inattendue si l'on considère l'observation expérimentale évoquée plus haut, la valeur propre périodique est plus grande que la valeur propre stationnaire dans le cas où le contrôle périodique est effectué sur l'apoptose. Nous montrons aussi, par des tests numériques dans le cas où le contrôle périodique est effectué sur le taux de transition d'une phase à l'autre du cycle cellulaire, qu'il n'existe alors aucune hiérarchie naturelle entre les deux types de valeurs propres. Ceci montre au moins que, pour que de tels modèles puissent rendre compte des observations expérimentales ci-dessus, le seul contrôle des taux de mort dans chaque phase est insuffisant, et que les taux de transition entre phases sont une cible clef pour le contrôle de la prolifération.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.10.029
Clairambault, Jean 1, 2; Michel, Philippe 3, 4; Perthame, Benoît 1, 3

1 INRIA, projet BANG, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
2 INSERM E 0354 « Rythmes biologiques et cancer », hôpital Paul-Brousse, 14, avenue Paul-Vaillant-Couturier, 94807 Villejuif cedex, France
3 Département de mathématiques et applications, UMR 8553, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France
4 CEREMADE, université Paris 9 Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
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Clairambault, Jean; Michel, Philippe; Perthame, Benoît. Circadian rhythm and tumour growth. Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 17-22. doi : 10.1016/j.crma.2005.10.029. http://www.numdam.org/articles/10.1016/j.crma.2005.10.029/

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