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## Androgen driven evolutionary population dynamics in prostate cancer growth

 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia 2 Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received  October 2019 Revised  January 2020 Published  September 2020

Prostate cancer worldwide is regarded the second most frequent diagnosed cancer in men with (899,000 new cases) while in common cancer it is the fifth. Regarding the treatment of progressive prostate cancer the most common and effective is the intermittent androgen deprivation therapy. Usually this treatment is effective initially at regressing tumorigenesis, mostly a resistance to treatment can been seen from patients and is known as the castration-resistant prostate cancer (CRPC), so there is no any treatment and becomes fatal. Therefore, we proposed a new mathematical model for the prostate cancer growth with fractional derivative. Initially, we present the model formulation in detail and then apply the fractional operator Atangana-Baleanu to the model. The fractional model will be studied further to analyze and show its existence of solution. Then, we provide a new iterative scheme for the numerical solution of the prostate cancer growth model. The analytical results are validated by considering various values assigned to the fractional order parameter $\alpha.$

Citation: Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020426
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##### References:
The dynamics of the prostate cancer growth model with $\alpha = 1$
The dynamics of the prostate cancer growth model with $\alpha=0.98$
The dynamics of the prostate cancer growth model with $\alpha=0.96$
The dynamics of the prostate cancer growth model with $\alpha=0.9$
The dynamics of the prostate cancer growth model with $\alpha=0.8$
The dynamics of the prostate cancer growth model with $\alpha=0.7$
The dynamics of the prostate cancer growth model with $\alpha = 0.5$
Phase portraits of the model considering different variables with $\alpha = 1$
Phase portraits of the model considering different variables with $\alpha=0.98$
Phase portraits of the model considering different variables with $\alpha=0.96$
Phase portraits of the model considering different variables with $\alpha = 0.94$
Parameters and their values used in the model simulation.
 Parameter Description value Reference $\mu_m$ Maximum proliferation rate 0.025 day$^{-1}$ [5] $q_1, q_2$ Minimum AD and AI cell quota 0.175, 0.1 day$^{-1}$ [6] $\delta_1, \delta_2$ Androgen-independent rate 0.02 day$^{-1}$ [12] $d_1, d_2$ AD and AI cell apoptosis rate 0.015, 0.015 day$^{-1}$ [5] $c_1$ Mutation rate from AD to AI 0.00015 day$^{-1}$ [12] $c_2$ Mutation rate from AI to AD 0.0001 day$^{-1}$ [12] $q_m$ Cell maximum quota 5 day$^{-1}$ Assumed $\nu_m$ Uptake rate of the maximum cell quota 0.275 nM day$^{-1}$ Assumed $\nu_h$ Uptake rate half-saturation level 4 nM d Assumed $b$ Degradation rate of cell quota 0.09 day$^{-1}$ [12] $K_1$ Half-saturation level from AD to AI mutation 0.08 nM [12] $K_2$ Half-saturation level from AI to AD 1.7 nM [12] $R_1, R_2$ Androgen dependent rates 1.3, 0.8 Assumed $\gamma_1$ Androgen clearance rate 0.08 [12] $a_0$ Normal androgen concentration 10 [12] $\sigma_0$ Production rate of PSA 0.004 [12] $\sigma_1, \sigma_2$ AD and AI Production rate of PSA 0.05, 0.05 [12] $\varpi_1, \varpi_2$ Half saturation level of AD and AI PSA 1.3, 1.1 [12] $\delta_3$ PSA clearance rate 0.08 [12]
 Parameter Description value Reference $\mu_m$ Maximum proliferation rate 0.025 day$^{-1}$ [5] $q_1, q_2$ Minimum AD and AI cell quota 0.175, 0.1 day$^{-1}$ [6] $\delta_1, \delta_2$ Androgen-independent rate 0.02 day$^{-1}$ [12] $d_1, d_2$ AD and AI cell apoptosis rate 0.015, 0.015 day$^{-1}$ [5] $c_1$ Mutation rate from AD to AI 0.00015 day$^{-1}$ [12] $c_2$ Mutation rate from AI to AD 0.0001 day$^{-1}$ [12] $q_m$ Cell maximum quota 5 day$^{-1}$ Assumed $\nu_m$ Uptake rate of the maximum cell quota 0.275 nM day$^{-1}$ Assumed $\nu_h$ Uptake rate half-saturation level 4 nM d Assumed $b$ Degradation rate of cell quota 0.09 day$^{-1}$ [12] $K_1$ Half-saturation level from AD to AI mutation 0.08 nM [12] $K_2$ Half-saturation level from AI to AD 1.7 nM [12] $R_1, R_2$ Androgen dependent rates 1.3, 0.8 Assumed $\gamma_1$ Androgen clearance rate 0.08 [12] $a_0$ Normal androgen concentration 10 [12] $\sigma_0$ Production rate of PSA 0.004 [12] $\sigma_1, \sigma_2$ AD and AI Production rate of PSA 0.05, 0.05 [12] $\varpi_1, \varpi_2$ Half saturation level of AD and AI PSA 1.3, 1.1 [12] $\delta_3$ PSA clearance rate 0.08 [12]
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