Control Structures of Drug Resistance in Cancer Chemotherapy

The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. Resistance manifests through a diverse set of molecular mechanisms, such as the upregulation of efflux transporters on the cell membrane, enhanced DNA damage repair mechanisms, and/or the presence of cancer stem cells. Classically, these mechanisms are understood as conferred to the cell by random genetic mutations, from which clonal expansion occurs via Darwian evolution. However, the recent experimental discovery of epigenetics and phenotype plasticity complicates this hypothesis. It is now believed that chemotherapy can produce drug-resistant clones. In this work, we study a previously introduced framework of drug-induced resistance, which incorporates both random and drug induced effects. A time-optimal control problem is then presented and analyzed utilizing differential-geometric techniques. Specifically, we seek the treatment protocol which prolongs patients life by maximizing the time of treatment until a critical tumor size is reached. The general optimal control structure is determined as a combination of both bang-bang and path-constrained arcs. Numerical results are presented which demonstrate decreasing treatment efficacy as a function of the ability of the drug to induce resistance. Thus, drug-induced resistance may dramatically effect the outcome of chemotherapy, implying that factors besides cytotoxicity should be considered when designing treatment regimens.