Nara Yoon, Nikhil Krishnan, Jacob Scott

Read the manuscript**Figure 1. (a) A collateral sensitivity map**: This map represents the results of a hypothetical experiment in which tumors are exposed to one drug (row), and after resistance to this drug develops are tested against another drug (column). The color is then the change in sensitivity from the wild type to the evolved strain. **(b) A collateral sensitivity network** showing every drug pair and their collateral sensitivity relationship. Nodes represent drugs, and directed edges point from drugs, which when resistance develops, end with sensitivity to another (edge terminus). The three drugs connected by the red arrows is an example of collaterally sensitive drug cycle.

**Figure 2: Population dynamics under $Drug \ i$ therapy.**

\begin{equation} \frac{d\mathcal{R}}{dt}=\mathcal{D}\left(\underset{1 \leq i \leq N}{\operatorname{argmax}} \ \text{ effect of }Drug\ i\text{ at } t \right) \ \mathcal{R}. \end{equation}

This is an abstract strategy of treatment, as it involves unrealistic instantaneous drug change. However, we can approximate the behavior under the strategy by a piece-wise continuous differential system with drug selection on discrete time points. An example simulation of the approximated optimal therapy with a cycle of 4 hypothetical drugs ($N=4$) is show on Figure 3. Expectedly, the cell subpopulations, effectiveness levels of each drug, and utilized drug sequences are dynamically changing over the therapy period.

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**Figure 3:** Optimal therapeutic dynamics with a cycle of four symmetric drugs. (a) Histories of subpopulations and total population with gray lines/curves indicating the ends of stages and exponential curves compatible to the last stage. (b) Effects of the drugs (rate of total population change under the drugs) along the optimal population dynamics. (c) Stage-wise relative frequencies of the drugs chosen in the optimal histories simulation.

The optimal dynamics of cell population (Equation (1)) is equivalent to the following differential system with a finite number ($\leq N-1$) of jump discontinuity that occurs at a "Stage" change.

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**Figure 4:** Algorithm and example simulation for the administration of the optimal drug schedules when total population data is available, but not subpopulation nor drug pa- rameters. (a) Diagram of the classification of “effective” and “ineffective” drugs, and the threshold of drug effectiveness (ef∗). The effectiveness levels of the most effective drug (efm), effective drugs and ineffective drugs are indicated by blue-filled circle, black-filled circles and empty circles, respectively. (b) Flow chart of the optimal therapy algorithm based on the “effective” drugs from (a). (c) Comparison between the realistic approximation the optimal therapy generated by the algorithm (b) – solid curves, and the actual one by Equation (1) – dashed curves. ε = 0.03 and η = 0.5 and the other used parameters and initial tumor status are the same as in the example in Figure 3. (d) Errors between the ap- proximated and actual optimal histories, over the range of the parameter of effective drug window.

- A. Dhawan, D. Nichol, F. Kinose, M. E. Abazeed, A. Marusyk, E. B. Haura, and J. G.Scott, “Collateral sensitivity networks reveal evolutionary instability and novel treat-ment strategies in alk mutated non-small cell lung cancer,”Scientific reports, vol.7, no.1,p.1232,2017.
- N. Yoon, R. Vander Velde, A. Marusyk, and J. G. Scott, “Optimal therapy schedulingbased on a pair of collaterally sensitive drugs,”Bulletin of mathematical biology, vol.80,no.7, pp.1776–1809,2018.
- N. Yoon, N. P. Krishnan, and J. G. Scott, “Modeling of collaterally sensitive drug cycles,and optimization of the drug effect in the spirit of adaptive therapy,”bioRxiv,2020.

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