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.