Resistance to anticancer drugs is a complex process that results from alterations in drug targets; development of alternative pathways for growth activation; changes in cellular pharmacology, including increased drug efflux; regulatory changes that alter differentiation pathways or pathways for response to environmental adversity; and/or changes in the local physiology of the cancer, such as blood supply, tissue hydrodynamics, behavior of neighboring cells, and immune system response. All of these specific mechanisms are facilitated by the intrinsic hallmarks of cancer, such as tumor cell heterogeneity, redundancy of growth-promoting pathways, increased mutation rate and/or epigenetic alterations, and the dynamic variation of tumor behavior in time and space. Understanding the relative contribution of each of these factors is further complicated by the lack of adequate in vitro models that mimic clinical cancers. Several strategies to use current knowledge of drug resistance to improve treatment of cancer are suggested.

Resistance to chemotherapy is a key impediment to successful cancer treatment that has been intensively studied for the last three decades. Several central mechanisms have been identified as contributing to the resistance. In the case of multidrug resistance (MDR), the cell becomes resistant to a variety of structurally and mechanistically unrelated drugs in addition to the drug initially administered. Mathematical models of drug resistance have dealt with many of the known aspects of this field, such as pharmacologic sanctuary and location/diffusion resistance, intrinsic resistance that is therapy independent, therapy-dependent cellular alterations including induced resistance (dose-dependent) and acquired resistance (dose-independent). In addition, there are mathematical models that take into account the kinetic/phase resistance, and models that investigate intra-cellular mechanisms based on specific biological functions (such as ABC transporters, apoptosis and repair mechanisms). This review covers aspects of MDR that have been mathematically studied, and explains how, from a methodological perspective, mathematics can be used to study drug resistance. We discuss quantitative approaches of mathematical analysis, and demonstrate how mathematics can be used in combination with other experimental and clinical tools. We emphasize the potential benefits of integrating analytical and mathematical methods into future clinical and experimental studies of drug resistance.