# High Accuracy Indicators of Androgen Suppression Therapy Failure for Prostate Cancer—A Modeling Study

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### Model Quick-Guide Box

_{2}as the model calibrates its value over the course of treatment (Figure 1b). This means ${x}_{2}$ represents the currently dominant resistant clone.

_{1}and q

_{2}parameters. Additionally, the model should reflect the divergence between PSA and androgen levels observed in later cycles of resistant patients.

## 3. Results

_{2}ratio SVM threshold was 79%. The accuracy for the androgen/PSA SVM threshold was 85%. For the Max thresholds, we randomized the data, performed the five-fold cross-validation, and then replicated the procedure 100 times. The mean accuracies of the q

_{2}ratio SVM thresholds were 87% for the training sample and 84% for the holdout sample. The mean accuracies for the androgen/PSA SVM thresholds were 89% for the training sample and 85% for the holdout sample.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematics of model foundation and evolutionary framework. (

**a**) Androgen (testosterone) enters the cancer cells. Some is converted to the potent dihydrotestosterone (DHT) with the help of 5-$\alpha $ reductase. Both then bind to the androgen receptors (AR). The bound androgen receptors send proliferative signals for cancer to grow and produce PSA (P). PSA then leaks into the bloodstream. (

**b**) The distribution of the minimum cell quota (q) prior to treatment, q profile skews toward higher values of q. This means most cancer cells are initially sensitive to treatment. During each treatment, this evolutionary landscape shifts toward a lower average q, meaning an increasing number of cells become less dependent on exogenous androgen.

**Figure 2.**Model validation: Best-fit model solutions to the dynamics of serum androgen and PSA levels. Circles represent patient measurements, and the solid lines are solutions of model (model equation number). ‘CS’ = castration susceptible tumor cell population; ‘CR’ = castration-resistant population. Panel (

**a**) was produced by a short dataset 1.5 cycles long, and panel (

**b**) by a dataset 2.5 cycles long.

**Figure 3.**The predictive potential of the q

_{2}ratio: The scatterplot (

**a**) indicates the value of the q

_{2}ratio for every patient in the dataset. The ratio is between the initial and final values of the q

_{2}parameter calculated by the mathematical model. Max (dotted line) and SVM (solid line) threshold values are shown. The confusion matrix (

**b**) compares actual patient outcomes with outcomes predicted by q

_{2}ratio with respect to the thresholds.

**Figure 4.**The predictive potential of the Androgen/PSA ratio: Scatterplot (

**a**) shows the value of the androgen/PSA ratio for every patient when calculated using mean values of androgen and PSA from the first 200 days of treatment. Scatterplot (

**a**) demonstrates that there is little correlation between the value of the ratio and treatment outcome when calculated in this manner. Scatterplot (

**b**) shows the same ratio calculated using mean androgen and PSA values from the patient’s final on-treatment cycle, not exceeding 200 days. For the purposes of this figure, all ratio values greater than five are represented as five. Scatterplot (

**b**) shows two thresholds below which values of the androgen/PSA ratio indicate impending treatment failure. The confusion matrix (

**c**) compares actual patient outcomes to outcomes predicted by the ratio with respect to the two thresholds.

**Table 1.**Parameter definitions and boundaries: This table describes the physiological interpretations of the fifteen parameters used in this model [5,26]. The range column indicates the upper and lower bounds within which an error-minimizing function may establish an optimal value with respect to a concrete set of patient data. The * in place of upper and lower bounds of ${A}_{0}$ is because the range of ${A}_{0}$ is patient specific and is set to the patient’s maximum recorded androgen data$\pm 10$.

Parameter | Description | Range | Unit |
---|---|---|---|

$\mathsf{{\rm M}}$ | max proliferation Rate | 0.001–0.09 | [day]^{−1} |

q_{1} | $\mathrm{minimum}\mathrm{cell}\mathrm{quota}\mathrm{for}{x}_{1}$ to proliferate | 0.41–1.73 | [nmol][day]^{−1} |

q_{2} | $\mathrm{minimum}\mathrm{cell}\mathrm{quota}\mathrm{for}{x}_{2}$ to proliferate | 0.01–0.41 | [nmole][day]^{−1} |

d | density death rate | 0.001–0.30 | [L]^{−1}[day]^{−1} |

c | maximum mutation rate | 0.00015–0.00015 | [day]^{−1} |

K | half-saturation constant for mutation | 1–1 | [nmole][day]^{−1} |

${\mathsf{\gamma}}_{1}$ | androgen production by testes | 0.008–0.8 | [nmol][day]^{−1} |

${\mathsf{\gamma}}_{2}$ | androgen production rate by adrenal gland | 0.005–0.005 | [nmol][day]^{−1} |

${A}_{0}$ | homeostasis serum androgen level | * | [nmol] |

$\mathsf{\Delta}$ | androgen degradation rate | 0.03–0.15 | [day]^{−1} |

b | baseline PSA production rate | 0.0001–0.1 | $[$g][nmol]^{−1}[day]^{−1} |

${\mathsf{\sigma}}_{1}$ | $\mathrm{maximum}\mathrm{PSA}\mathrm{production}\mathrm{rate}\mathrm{by}{x}_{1}$ | 0.001–1 | $[$g][nmol]^{−1}[L]^{−1}[day]^{−1} |

${\mathsf{\sigma}}_{2}$ | maximum PSA production rate by x_{2} | 0.001–1 | $[$g][nmol]^{−1}[L]^{−1}[day]^{−1} |

$\u03f5$ | PSA clearance rate | 0.0001–0.1 | [day]^{−1} |

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**MDPI and ACS Style**

Meade, W.; Weber, A.; Phan, T.; Hampston, E.; Resa, L.F.; Nagy, J.; Kuang, Y. High Accuracy Indicators of Androgen Suppression Therapy Failure for Prostate Cancer—A Modeling Study. *Cancers* **2022**, *14*, 4033.
https://doi.org/10.3390/cancers14164033

**AMA Style**

Meade W, Weber A, Phan T, Hampston E, Resa LF, Nagy J, Kuang Y. High Accuracy Indicators of Androgen Suppression Therapy Failure for Prostate Cancer—A Modeling Study. *Cancers*. 2022; 14(16):4033.
https://doi.org/10.3390/cancers14164033

**Chicago/Turabian Style**

Meade, William, Allison Weber, Tin Phan, Emily Hampston, Laura Figueroa Resa, John Nagy, and Yang Kuang. 2022. "High Accuracy Indicators of Androgen Suppression Therapy Failure for Prostate Cancer—A Modeling Study" *Cancers* 14, no. 16: 4033.
https://doi.org/10.3390/cancers14164033