# Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Clinical Trial Data

## 3. Formulation of Mathematical Models

#### 3.1. Model 1: Single Population Model

#### 3.2. Model 2: Two Population Model

#### 3.3. Derivation of $dQ/dt$

#### 3.4. Portz, Kuang, and Nagy (PKN) Model

## 4. Model Dynamics

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 5. Parameter Estimation

#### 5.1. Sensitivity Analysis

## 6. Comparison of Models

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Heinlein, C.; Chang, C. Androgen receptor in prostate cancer. Endocr. Rev.
**2004**, 25, 276–308. [Google Scholar] [CrossRef] [PubMed] - Shafi, A.; Yen, A.; Weigel, N. Androgen receptors in hormone-dependent and castration-resistant prostate cancer. Pharmacol. Ther.
**2013**, 140, 223–238. [Google Scholar] [CrossRef] [PubMed] - Tsao, C.K.; Small, A.; Galsky, M.; Oh, W. Overcoming castration resistance in prostate cancer. Curr. Opin. Urol.
**2012**, 22, 167–174. [Google Scholar] [CrossRef] [PubMed] - Bruchovsky, N.; Klotz, L.; Crook, J.; Phillips, N.; Abersbach, J.; Goldenberg, S. Quality of life, morbidity, and mortality results of a prospective phase II study of intermittent androgen suppression for men with evidence of prostate-specific antigen relapse after radiation therapy for locally advanced prostate cancer. Clin. Genitourin. Cancer
**2008**, 6, 46–52. [Google Scholar] [CrossRef] [PubMed] - Feldman, B.; Feldman, D. The development of androgen-independent prostate cancer. Nat. Rev. Cancer
**2001**, 1, 34–45. [Google Scholar] [CrossRef] [PubMed] - Hussain, M.; Tangen, C.; Berry, D.; Higano, C.; Crawford, E.; Liu, G.; Wilding, G.; Prescott, S.; Sundaram, S.K.; Small, E.J.; et al. Intermittent versus continuous androgen deprivation in prostate cancer. N. Engl. J. Med.
**2013**, 368, 1314–1325. [Google Scholar] [CrossRef] [PubMed] - Karantanos, T.; Evans, C.P.; Tombal, B.; Thompson, T.C.; Montironi, R.; Isaacs, W.B. Understanding the mechanisms of androgen deprivation resistance in prostate cancer at the molecular level. Eur. Urol.
**2015**, 67, 470–479. [Google Scholar] [CrossRef] [PubMed] - Klotz, L.; Toren, P. Androgen deprivation therapy in advanced prostate cancer: Is intermittent therapy the new standard of care? Curr. Oncol.
**2012**, 19, S13–S21. [Google Scholar] [CrossRef] [PubMed] - Bruchovsky, N.; Klotz, L.; Crook, J.; Malone, S.; Ludgate, C.; Morris, W.J.; Gleave, M.E.; Goldenberg, S.L. Final results of the Canadian prospective phase II trial of intermittent androgen suppression for men in biochemical recurrence after radiotherapy for locally advanced prostate cancer. Cancer
**2006**, 107, 389–395. [Google Scholar] [CrossRef] [PubMed] - Gleave, M. Prime time for intermittent androgen suppression. Eur. Urol.
**2014**, 66, 240–242. [Google Scholar] [CrossRef] [PubMed] - Jackson, T. A Mathematical Investigation of the Multiple Pathways to Recurrent Prostate Cancer: Comparison with Experimental Data. Neoplasia
**2004**, 6, 697–704. [Google Scholar] [CrossRef] [PubMed] - Ideta, A.; Tanaka, G.; Takeuchi, T.; Aihara, K. A mathematical model of intermittent androgen suppression for prostate cancer. J. Nonlinear Sci.
**2008**, 18, 593–614. [Google Scholar] [CrossRef] - Portz, T.; Kuang, Y.; Nagy, J. A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy. AIP Adv.
**2012**, 2, 1–14. [Google Scholar] [CrossRef] - Hirata, Y.; Bruchovsky, N.; Aihara, K. Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer. J. Theor. Biol.
**2010**, 264, 517–527. [Google Scholar] [CrossRef] [PubMed] - Swanson, K.; True, L.; Lin, D.; Buhler, K.; Vessella, R.; Murray, J. A quantitative model for the dynamics of serum prostate-specific antigen as a marker for cancerous growth: An explanation for a medical anomaly. Am. J. Pathol.
**2001**, 158, 2195–2199. [Google Scholar] [CrossRef] - Jain, H.; Clinton, S.; Bhinder, A.; Friedman, A. Mathematical modeling of prostate cancer progression in response to androgen ablation therapy. Proc. Natl. Acad. Sci. USA
**2011**, 108, 19701–19706. [Google Scholar] [CrossRef] [PubMed] - Jain, H.; Friedman, A. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discret. Contin. Dyn. Syst. B
**2013**, 18, 945–967. [Google Scholar] [CrossRef] - Jain, H.; Friedman, A. A partial differential equation model of metastasized prostatic cancer. Math. Biosci. Eng.
**2013**, 10, 591–608. [Google Scholar] [CrossRef] [PubMed] - Kuang, Y.; Nagy, J.; Eikenberry, S. Introduction to Mathematical Oncology; CRC Press: Boca Raton, FC, USA, 2016. [Google Scholar]
- Guo, Q.; Lu, Z.; Hirata, Y.; Aihara, K. Parameter estimation and optimal scheduling algorithm for a mathematical model of intermittent androgen suppression therapy for prostate cancer. Chaos
**2013**, 23, 43125. [Google Scholar] [CrossRef] [PubMed] - Tao, Y.; Guo, Q.; Aihara, K. A partial differential equation model and its reduction to an ordinary differential equation model for prostate tumor growth under intermittent hormone therapy. J. Math. Biol.
**2014**, 69, 817–838. [Google Scholar] [CrossRef] [PubMed] - Suzuki, Y.; Sakai, D.; Nomura, T.; Hirata, Y.; Aihara, K. A new protocol for intermittent androgen suppression therapy of prostate cancer with unstable saddle-point dynamics. J. Theor. Biol.
**2014**, 350, 1–16. [Google Scholar] [CrossRef] [PubMed] - Hirata, Y.; Akakura, K.; Higano, C.; Bruchovsky, N.; Aihara, K. Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression. J. Mol. Cell Biol.
**2012**, 4, 127–132. [Google Scholar] [CrossRef] [PubMed] - Hirata, Y.; Tanaka, G.; Bruchovsky, N.; Aihara, K. Mathematically modelling and controlling prostate cancer under intermittent hormone therapy. Asian J. Androl.
**2012**, 14, 270–277. [Google Scholar] [CrossRef] [PubMed] - Hirata, Y.; Azuma, S.; Aihara, K. Model predictive control for optimally scheduling intermittent androgen suppression of prostate cancer. Methods
**2014**, 67, 278–281. [Google Scholar] [CrossRef] [PubMed] - Jackson, T. A mathematical model of prostate tumor growth and androgen-independent relapse. Discret. Contin. Dyn. Syst. B
**2004**, 4, 187–202. [Google Scholar] [CrossRef] - Droop, M. Some thoughts on nutrient limitation in algae1. J. Phycol.
**1973**, 9, 264–272. [Google Scholar] [CrossRef] - Everett, R.; Packer, A.; Kuang, Y. Can Mathematical Models Predict the Outcomes of Prostate Cancer Patients Undergoing Intermittent Androgen Deprivation Therapy? Biophys. Rev. Lett.
**2014**, 9, 173–191. [Google Scholar] [CrossRef] - Roy, A.; Chatterjee, B. Androgen action. Crit. Rev. Eukaryot. Gene Expr.
**1995**, 5, 157–176. [Google Scholar] [CrossRef] [PubMed] - Bruchovsky, N. Clinical Research. 2006. Available online: http://www.nicholasbruchovsky.com/clinicalResearch.html (accessed on 11 November 2016).
- Vollmer, R. Tumor Length in Prostate Cancer. Am. J. Clin. Pathol.
**2008**, 130, 77–82. [Google Scholar] [CrossRef] [PubMed] - Morken, J.; Packer, A.; Everett, R.; Nagy, J.; Kuang, Y. Mechanisms of resistance to intermittent androgen deprivation in patients with prostate cancer identified by a novel computational method. Cancer Res.
**2014**, 74, 3673–3683. [Google Scholar] [CrossRef] [PubMed] - Berges, R.R.; Vukanovic, J.; Epstein, J.I.; CarMichel, M.; Cisek, L.; Johnson, D.E.; Veltri, R.W.; Walsh, P.C.; Isaacs, J.T. Implication of cell kinetic changes during the progression of human prostatic cancer. Clin. Cancer Res.
**1995**, 1, 473–480. [Google Scholar] [PubMed] - Nishiyama, T. Serum testosterone levels after medical or surgical androgen deprivation: A comprehensive review of the literature. Urol. Oncol.
**2013**, 32, 38.e17–38.e28. [Google Scholar] [CrossRef] [PubMed] - Pell, B.; Baez, J.; Phan, T.; Gao, D.; C, G.; Kuang, Y. Patch Models of EVD Transmission Dynamics; Springler: Cham, Switzerland, 2016. [Google Scholar]
- Chowell, G.; Simonsen, L.; Kuang, Y.; Sciences, S. Is West Africa Approaching a Catastrophic Phase or is the 2014 Ebola Epidemic Slowing Down? Different Models Yield Different Answers for Liberia. PLOS Curr. Outbreaks
**2014**. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Sample data for prostate-specific antigen (PSA) and androgen data for a patient in a clinical trial.

**Figure 2.**Bifurcation diagram displaying ${x}_{1}$ cell population vs. parameters μ and γ (

**left**) and γ and ${d}_{1}$ (

**right**). This figure depicts the regions in which ${x}_{1}$ can go extinct. This happens when androgen levels γ are very low, or cancer cells’ proliferation rate μ is very low, or cancer cells’ death rate ${d}_{1}$ is very high.

**Figure 5.**Simulations of fittings for every model for 1.5 cycles of treatment (

**left**of gray line), and one cycle of forecast (

**right**of gray line). For these four patients, we can see that models fit data at comparable accuracy but Model 2 perform much better in PSA forecasting.

**Figure 6.**Simulations of fittings of androgen levels for Models 1 and 2. These two models have comparable goodness in fitting androgen data as their derivations are very similar.

**Figure 7.**Cancer cells in resistant and non-resistant patient for Model 1. For the non-resistant patient we see a slight increase in volume over the course of several cycles. In the resistant patient we see that cancer volume has grown substantially.

**Figure 8.**Cancer cells in resistant and non-resistant patients for Model 2. For the non-resistant patient, we see an increase in the volume of CR cells, but the original volume is about the same. In the resistant patient, we see that cancer volume has grown to double the volume compared to the non resistant patient.

Parameters | Definition | Range | Units | Source |
---|---|---|---|---|

μ | Maximum proliferation rate | 0.001–0.09 | day${}^{-1}$ | [33] |

q | Minimum cell quota | 0.1–0.5 | nM | [34] |

${q}_{1}$ | Minimum CS cell quota | 0.1–0.5 | nM | [34] |

${q}_{2}$ | Minimum CR cell quota | 0.1–0.3 | nM | [34] |

b | Prostate baseline PSA | 0.1–2.5 | 10${}^{-3}$μg/L/nM/day | [9] |

σ | Tumor PSA production rate | 0.001–0.9 | μg/L/nM/mm${}^{3}$/day | [28] |

ϵ | PSA clearance rate | 0.001–0.01 | day${}^{-1}$ | [28] |

d | Maximum cell death rate | 0.0001–0.09 | day${}^{-1}$ | [33] |

${d}_{1}$ | Maximum CS cell death rate | 0.001–0.09 | day${}^{-1}$ | [33] |

${d}_{2}$ | Maximum CR cell death rate | 0.0001–0.001 | day${}^{-1}$ | [33] |

${\delta}_{1}$ | Density death rate | 0.1–9 × ${10}^{-5}$ | 1/day/mm${}^{3}$ | [31] |

${\delta}_{2}$ | Density death rate | 0.01–4.5 × ${10}^{-4}$ | 1/day/mm${}^{3}$ | [31] |

R | Cell death rate half-saturation level | 0–3 | nM | [28] |

${R}_{1}$ | CS cell death rate half-saturation level | 0–3 | nM | [28] |

${R}_{2}$ | CR cell death rate half-saturation level | 0–3 | nM | [28] |

${c}_{1}$ | Maximum CS to CR rate | ${10}^{-5}$–${10}^{-4}$ | day^{−1} | [12] |

K | CS to CR half-saturation level | 0–1 | nM | [28] |

${\gamma}_{1}$ | Testes androgen production | 20 | day${}^{-1}$ | ad hoc |

${\gamma}_{2}$ | Secondary androgen production | 0.001–0.01 | day${}^{-1}$ | ad hoc |

${Q}_{m}$ | Maximum androgen | 15–30 | nM | [9] |

ν | death rate decay rate | 0.01 | unitless | ad hoc |

**Table 2.**Comparison of Mean Squared Error (MSE) for Androgen and prostate-specific antigen (PSA) for the first 1.5 cycles.

Model | PSA | Androgen | ||||
---|---|---|---|---|---|---|

Min | Mean | Max | Min | Mean | Max | |

PKN Model | 0.5119 | 9.4463 | 93.1587 | N/A | N/A | N/A |

Model 1 | 0.9735 | 8.6763 | 71.8471 | 5.0351 | 100.1071 | 710.2604 |

Model 2 | 0.2461 | 10.3993 | 137.4345 | 5.1283 | 101.4763 | 710.4412 |

Model | Min | Mean | Max |
---|---|---|---|

PKN Model | 12.234 | 162.5494 | 1868.6394 |

Model 1 | 11.3935 | 141.9280 | 1663.0218 |

Model 2 | 2.2727 | 56.3478 | 278.4050 |

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**MDPI and ACS Style**

Baez, J.; Kuang, Y.
Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy. *Appl. Sci.* **2016**, *6*, 352.
https://doi.org/10.3390/app6110352

**AMA Style**

Baez J, Kuang Y.
Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy. *Applied Sciences*. 2016; 6(11):352.
https://doi.org/10.3390/app6110352

**Chicago/Turabian Style**

Baez, Javier, and Yang Kuang.
2016. "Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy" *Applied Sciences* 6, no. 11: 352.
https://doi.org/10.3390/app6110352