Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

¹ UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu 75252 Paris Cedex 05, France.
lorz@ann.jussieu.fr
² INRIA-Rocquencourt, EPI BANG, France.
jean.clairambault@inria.fr
³ Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, I10129 Torino, Italy.
tommaso.lorenzi@polito.it
⁴ Institut des Sciences de l’Evolution, CNRS, Université Montpellier 2, Place Eugene Bataillon, 34095 Montpellier, France.
⁵ Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, New Mexico, USA.
mhochber@univ-montp2.fr
⁶ Institut Universitaire de France, France.
benoit.perthame@upmc.fr

Received: 22 November 2011
Revised: 30 May 2012

Abstract

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton–Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.