Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1776–1809

# Optimal Therapy Scheduling Based on a Pair of Collaterally Sensitive Drugs

• Nara Yoon
• Robert Vander Velde
• Andriy Marusyk
• Jacob G. Scott
Original Article

## 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, $$A_\mathrm{R}$$ and $$B_\mathrm{R}$$, 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 $$A_\mathrm{R}$$ and $$B_\mathrm{R}$$, 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., $$f \Delta t$$-long for DrugA and $$(1-f) \Delta t$$-long for DrugB with $$0 \le f \le 1$$ and $$\Delta t \ge 0$$). 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, $$\mathcal {A/B} = A_\mathrm{R}/B_\mathrm{R}$$, and the effect of each drug. We determine a critical ratio, which we term $$\mathcal {(A/B)}^{*}$$, at which the two drugs are equally effective. As the first stage of the optimal strategy is applied, $$\mathcal {A/B}$$ changes monotonically to $$\mathcal {(A/B)}^{*}$$ and then, during the second stage, remains at $$\mathcal {(A/B)}^{*}$$ 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.

## Keywords

Cancer Evolution of resistance Dynamical systems Optimal control

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© Society for Mathematical Biology 2018

## Authors and Affiliations

• Nara Yoon
• 1
• Robert Vander Velde
• 2
• Andriy Marusyk
• 3
• Jacob G. Scott
• 1
1. 1.Department of Translational Hematology and Oncology ResearchCleveland ClinicClevelandUSA
2. 2.Department of Molecular Medicine, Morsani College of MedicineUniversity of South FloridaTampaUSA
3. 3.Department of Cancer Imaging and MetabolismH. Lee Moffitt Cancer Center and Research InstituteTampaUSA