  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.
