# Fractionated Follow-Up Chemotherapy Delays the Onset of Resistance in Bone Metastatic Prostate Cancer

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## Abstract

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## 1. Introduction

## 2. Model

- The fitness function for OBs is given by$${W}_{OB}=(\frac{{\rho}_{\mathrm{OC}}}{{\rho}_{\mathrm{OC}}+{\rho}_{\mathrm{OB}}}(1-B)+({\rho}_{T}+{\rho}_{{T}_{R}})\delta )(1-2s)$$
- The fitness function for OCs is given by$${W}_{OC}=\frac{{\rho}_{\mathrm{OC}}}{{\rho}_{\mathrm{OC}}+{\rho}_{\mathrm{OB}}}(B-1)(1-2s)$$
- Finally, the fitness functions for chemotherapy-sensitive and -resistant tumors are$${W}_{T}=({\rho}_{OC}\gamma +({\rho}_{T}+{\rho}_{{T}_{R}})\u03f5)(1-2s)$$$${W}_{{T}_{R}}={\rho}_{OC}\gamma +({\rho}_{T}+{\rho}_{{T}_{R}})\u03f5-r$$

#### Chemotherapy

## 3. Results

#### 3.1. Tumor Introduction

#### 3.2. Standard Chemotherapy

#### 3.3. Fractionated Treatment

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Tumor Introduction. (

**a**) Bone size and strategy density; (

**b**) stroma.Homeostatic populations were observed by setting ${\rho}_{\mathrm{OB}}\left(0\right)=0.001$, ${\rho}_{\mathrm{OC}}\left(0\right)=0.01$, ${\rho}_{T}\left(0\right)={\rho}_{{T}_{R}}\left(0\right)=0.0$, and $B\left(0\right)=1$. Subfigure (

**a**) shows the model’s recapitulation of the classic bone fluctuation characteristic of bone remodeling units and no distinct strategies leftover. Furthermore, subfigure (

**b**) shows the decrease in stroma after the bone remodeling event. Note the log-scale for the bottom subplot.

**Figure 2.**Tumor introduction. (

**a**) Bone size and strategy density; (

**b**) stroma. Strategy phenotypes in the bottom of (

**a**) were observed to be dominated by the tumor phenotype. Subfigure (

**b**) and the top of (

**a**) showed the characteristic PCa takeover of the bone remodeling complex and the resultant vicious cycle causing dramatic bone growth.

**Figure 3.**Standard chemotherapy. (

**a**) Bone size and strategy density; (

**b**) stroma. Here we see the impact of standard chemotherapy regimens that decrease the PCa-susceptible population and increase the PCa-resistant population. Assuming initial densities of the different population as follows: ${\rho}_{\mathrm{OB}}\left(0\right)=0.001$, ${\rho}_{\mathrm{OC}}\left(0\right)=0.01$, ${\rho}_{T}\left(0\right)={\rho}_{{T}_{R}}\left(0\right)=\phantom{\rule{3.33333pt}{0ex}}0.0$, and $B\left(0\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$.

**Figure 4.**Treatment and treatment holiday combinations. Better treatments are those that reduce tumor burden and extra pathological bone. (

**A**) All potential treatments over 16 40-timestep periods, plotted with respect to average tumor burden and average bone size. (

**B**,

**C**,

**D**) Treatment regiments showing an example of a fractionated treatment (

**B**), continuous block treatment (

**C**), and a fractionated follow-up regiment (

**D**). Assuming initial densities of the different population as follows: ${\rho}_{\mathrm{OB}}\left(0\right)=0.001$, ${\rho}_{\mathrm{OC}}\left(0\right)=0.01$, ${\rho}_{T}\left(0\right)={\rho}_{{T}_{R}}\left(0\right)=0.0$, and $B\left(0\right)=1$.

**Table 1.**The top five treatments as well as an example of continuous and periodic treatments. In the 1st column, each 0 represents a treatment holiday and each 1 represents the application of chemotherapy. The bone size and tumor burden are dimensionless and rounded to three decimal places. Given the qualitative nature of the model, the goal of this table is to show that different treatment schedules impact bone size and tumor burden differently, but the quantitative differences between treatments might not be conserved in reality.

Treatment Schedule | Bone Size | Average Tumor Burden |
---|---|---|

0000000000000000 | 1.754 | 0.773 |

1111100100010010 | 1.160 | 0.352 |

1111010101000001 | 1.163 | 0.358 |

1111010100100100 | 1.165 | 0.358 |

1111011000010010 | 1.166 | 0.356 |

1110110101000100 | 1.172 | 0.366 |

0001111100000000 | 1.615 | 0.512 |

0100100010010011 | 1.709 | 0.666 |

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**MDPI and ACS Style**

Warman, P.I.; Kaznatcheev, A.; Araujo, A.; Lynch, C.C.; Basanta, D. Fractionated Follow-Up Chemotherapy Delays the Onset of Resistance in Bone Metastatic Prostate Cancer. *Games* **2018**, *9*, 19.
https://doi.org/10.3390/g9020019

**AMA Style**

Warman PI, Kaznatcheev A, Araujo A, Lynch CC, Basanta D. Fractionated Follow-Up Chemotherapy Delays the Onset of Resistance in Bone Metastatic Prostate Cancer. *Games*. 2018; 9(2):19.
https://doi.org/10.3390/g9020019

**Chicago/Turabian Style**

Warman, Pranav I., Artem Kaznatcheev, Arturo Araujo, Conor C. Lynch, and David Basanta. 2018. "Fractionated Follow-Up Chemotherapy Delays the Onset of Resistance in Bone Metastatic Prostate Cancer" *Games* 9, no. 2: 19.
https://doi.org/10.3390/g9020019