# Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Model Description

- 1.
- We implement a version of the adjusted replicator dynamics equations to track the frequency evolution of the three sub-populations of cancer cells that make up the tumor. Our model consists of a chemo-sensitive population (S), a population resistant to drug 1 (${R}_{1}$), and a population resistant to drug 2 (${R}_{2}$). The competition among the three populations is determined by a $3\times 3$ payoff matrix A that builds in a fitness cost of resistance. In the absence of chemotherapy, the sensitive population is most fit and will outcompete the resistant populations, saturating the tumor.
- 2.
- We use two time-dependent control functions ${C}_{1}\left(t\right)$ and ${C}_{2}\left(t\right)$ to model the chemotherapy dosing schedules. These chemotherapy dosing functions control three selection pressure parameters ${w}_{1}\left(t\right)$, ${w}_{2}\left(t\right)$, and ${w}_{3}\left(t\right)$ by altering the relative fitness values of the three sub-populations. A dose of chemotherapy (${C}_{i}\left(t\right)>0;{t}_{0}\le t\le {t}_{1}$) lowers the relative fitness of the targeted cell population by altering the selection pressures on the sub-populations to effectively favor ones that are not targeted. This mechanism allows us to ‘design’ favorable fitness landscapes indirectly by adaptively monitoring the sub-population frequencies and altering our dosing schedule in response. Our goals when we design chemotherapy schedules are to: (i) avoid fixation of the sensitive cell population; (ii) avoid chemoresistance (fixation of either of the resistant populations) by keeping the three populations of cells in competition without allowing any of them to saturate the tumor. We implement this by designing schedules that keep us confined to a closed ‘evolutionary cycle’ which keep the sub-populations in competition forever (ideally) for the deterministic ($N=\infty $) model.
- 3.
- We then test the performance of these designed schedules on a finite cell ($N<\infty $) stochastic Moran process model to track the sub-population frequencies during these cycles, where the limit $N\to \infty $ corresponds to our adjusted replicator model from which the cycle was designed.
- 4.
- Since the Moran process model uses a fixed value of N (hence, it cannot be used directly to determine tumor growth where the total cancer cell population increases), we add a tumor growth equation with growth rate determined as a function of the average fitness of the three cancer cell sub-population frequencies comprising the tumor. When the average fitness of the cancer cell populations is above a fixed microenvironmental average, the tumor grows (exponentially), and when it is below, it shrinks (exponentially).

#### 2.1. Three-Component Two-Drug Adjusted Replicator Dynamics Model

#### 2.2. Three-Component Two-Drug Discrete Stochastic Moran Process

#### 2.3. Continuous Limit $N\to \infty $ Which Relates the Moran Process to the Adjusted Replicator System

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A sequence of adaptive chemoschedules that lock the tumor in a closed evolutionary cycle is difficult to achieve for finite cell (shown for $N=12$) populations since sub-population frequencies fluctuate stochastically and are difficult to measure with precision. Middle inset shows a discrete tri-linear plot of a stochastic realization in the $(S,{R}_{1},{R}_{2})$ plane, with $n=24$ steps, starting at point A, using the chemoschedule designed from the deterministic ($N\to \infty $) model.

**Figure 2.**(

**a**) Tri-linear $(S,{R}_{1},{R}_{2})$ coordinate representation of a deterministic evolutionary cycle $ABCA$, from the adjusted replicator equation, and two stochastic realizations (from the Moran process) with N = 10,000 and N = 1,000,000 cells. Inset shows that the stochastic paths are not closed. Background colors show velocity field, hence instantaneous speed of convergence, from the adjusted replicator system. (

**b**) Adaptive chemotherapy schedule with drugs 1 and 2 for closed cycle $ABCA$.

**Figure 3.**(

**a**) Spread of 10,000 trials starting at initial conditions $S=0.8$, ${R}_{1}=0.1$, and ${R}_{2}=0.1$ with principal axis after 1 loop; (

**b**) same spread of trials from (

**a**) plotted with principal component axes; (

**c**) kernel density estimation (KDE) for trials in (

**a**), with darker areas indicating a higher concentration of data, with means ${\mu}_{S}=0.784$, ${\mu}_{R1}=0.106$; (

**d**) the spread of trials closely resembles a multivariate Gaussian distribution composed of the principal components, with singular values ${\sigma}_{1}=0.040$, ${\sigma}_{2}=0.013$.

**Figure 4.**Kernel density estimation showing distribution after successive loops and singular values ${\sigma}_{1}$, ${\sigma}_{2}$. Gaussian spread starts to break down after loops 3 and 4 as some tumors began to saturate. (

**a**) Loop 2, ${\sigma}_{1}=0.072$, ${\sigma}_{2}=0.021$; (

**b**) Loop 3, ${\sigma}_{1}=0.112$, ${\sigma}_{2}=0.025$; (

**c**) Loop 4, ${\sigma}_{1}=0.184$, ${\sigma}_{2}=0.034$; (

**d**) Loop 5, ${\sigma}_{1}=0.261$, ${\sigma}_{2}=0.043$.

**Figure 5.**Histogrammed distributions of the three sub-populations as the number of loops increases. Note that some tumors began to fill at the $S=1$ and ${R}_{1}=1$ corners, distorting the multivariate Gaussian nature of the distributions. Mean $\mu $ and standard deviations $\sigma $ are shown.

**Figure 6.**Same as Figure 5, but for synergistic $e>0$ and antagonistic $e<0$ drug interactions. (

**a**) $e=0.3$; (

**b**) $e=-0.3$.

**Figure 7.**Variance ${\sigma}_{e}$ of the principal components as a function of the cycle number n plotted on log–linear axes, so ${\sigma}_{e}\sim aexp\left({\alpha}_{e}n\right)$. Note that antagonistic interactions grow the slowest, while synergistic interactions grow the fastest. ${\alpha}_{0}$: 0.936 (PCA 1) and 0.580 (PCA 2). ${\alpha}_{0.3}$: 1.02 and 0.670. ${\alpha}_{-0.3}$: 0.885 and 0.543.

**Figure 8.**Tumor growth curve with adjusted replicator model using $g=1.05$ and the stochastic Moran process model (100 runs). The black curve shows untreated growth, blue and yellow show adaptive therapy results. Error bars indicate $90\%$ confidence level. Tumor recurrence sets in at $t\approx 50$ in dimensionless time units.

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**MDPI and ACS Style**

Dua, R.; Ma, Y.; Newton, P.K.
Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model. *Cancers* **2021**, *13*, 2880.
https://doi.org/10.3390/cancers13122880

**AMA Style**

Dua R, Ma Y, Newton PK.
Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model. *Cancers*. 2021; 13(12):2880.
https://doi.org/10.3390/cancers13122880

**Chicago/Turabian Style**

Dua, Rajvir, Yongqian Ma, and Paul K. Newton.
2021. "Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model" *Cancers* 13, no. 12: 2880.
https://doi.org/10.3390/cancers13122880