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Abstract
Despite major strides in the treatment of cancer, the development of drug resistance remains a major hurdle. One strategy which has been proposed to address this is the sequential application of drug therapies where resistance to one drug induces sensitivity to another drug, a concept called collateral sensitivity. The optimal timing of drug switching in these situations, however, remains unknown. To study this, we developed a dynamical model of sequential therapy on heterogeneous tumors comprised of resistant and sensitive cells. A pair of drugs (DrugA, DrugB) are utilized and are periodically switched during therapy. Assuming resistant cells to one drug are collaterally sensitive to the opposing drug, we classified cancer cells into two groups, [Formula: see text] and [Formula: see text], each of which is a subpopulation of cells resistant to the indicated drug and concurrently sensitive to the other, and we subsequently explored the resulting population dynamics. Specifically, based on a system of ordinary differential equations for [Formula: see text] and [Formula: see text], we determined that the optimal treatment strategy consists of two stages: an initial stage in which a chosen effective drug is utilized until a specific time point, T, and a second stage in which drugs are switched repeatedly, during which each drug is used for a relative duration (i.e., [Formula: see text]-long for DrugA and [Formula: see text]-long for DrugB with [Formula: see text] and [Formula: see text]). We prove that the optimal duration of the initial stage, in which the first drug is administered, T, is shorter than the period in which it remains effective in decreasing the total population, contrary to current clinical intuition. We further analyzed the relationship between population makeup, [Formula: see text], and the effect of each drug. We determine a critical ratio, which we term [Formula: see text], at which the two drugs are equally effective. As the first stage of the optimal strategy is applied, [Formula: see text] changes monotonically to [Formula: see text] and then, during the second stage, remains at [Formula: see text] thereafter. Beyond our analytic results, we explored an individual-based stochastic model and presented the distribution of extinction times for the classes of solutions found. Taken together, our results suggest opportunities to improve therapy scheduling in clinical oncology.
Highlights
Drug resistance is observed in many patients after exposure to cancer therapy, and is a major hurdle in cancer therapy [1]
Much effort has been put into novel drug discovery to combat this, there is a growing interest in determining the optimal sequences, or cycles of drugs that promote collateral sensitivity
To study this second paradigm, we proposed a simple dynamical systems model of tumor evolution in a heterogeneous tumor composed of two cell phenotypes
Summary
Drug resistance is observed in many patients after exposure to cancer therapy, and is a major hurdle in cancer therapy [1]. The simplest manifestation of this heterogeneity can be represented by considering the existence of both therapy resistant and sensitive cell types co-existing prior to therapy [6], with the future cellular composition shaped by the choice of drugs (illustrated in Figure 1 (b)). Researchers (including [18, 22, 23, 27, 28]) have studied the effect of a pair of collaterally sensitive drugs as we propose here, using the Goldie-Coldman model or its variations [19, 28, 29, 30] These models utilize a population structure consisting of four compartments, each of which represents a subpopulation that is either (i) sensitive to the both drugs, (ii) and (iii) resistant to one drug respectively, or (iv) resistant to both. If CP0 (0) < 0, the drug is effective in reducing tumor burden at the beginning, it will eventually regrow (due to drug resistance; see example in Figure 3 (b))
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