Androgen driven evolutionary population dynamics in prostate cancer growth

Ebraheem O. Alzahrani; Muhammad Altaf Khan

doi:10.3934/dcdss.2020426

Article Contents

2021, Volume 14, Issue 10: 3419-3440. Doi: 10.3934/dcdss.2020426

This issue Previous Article Analysis and new applications of fractal fractional differential equations with power law kernel Next Article On solutions of fractal fractional differential equations

Androgen driven evolutionary population dynamics in prostate cancer growth

Ebraheem O. Alzahrani^1, and
Muhammad Altaf Khan^2,3, ,

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

^* Corresponding author: Muhammad Altaf Khan (muhammad.altaf.khan@tdtu.edu.vn)
^* Corresponding author: Muhammad Altaf Khan (muhammad.altaf.khan@tdtu.edu.vn)

Received: September 30, 2019

Revised: December 31, 2019

Early access: September 2020

Published: October 2021

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Abstract

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. $

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Mathematics Subject Classification: Primary: 34A08, 92B05; Secondary: 65R99.

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Figure 1. The dynamics of the prostate cancer growth model with $ \alpha = 1 $

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Figure 2. The dynamics of the prostate cancer growth model with $ \alpha=0.98 $

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Figure 3. The dynamics of the prostate cancer growth model with $ \alpha=0.96 $

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Figure 4. The dynamics of the prostate cancer growth model with $ \alpha=0.9 $

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Figure 5. The dynamics of the prostate cancer growth model with $ \alpha=0.8 $

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Figure 6. The dynamics of the prostate cancer growth model with $ \alpha=0.7 $

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Figure 7. The dynamics of the prostate cancer growth model with $ \alpha = 0.5 $

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Figure 8. Phase portraits of the model considering different variables with $ \alpha = 1 $

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Figure 9. Phase portraits of the model considering different variables with $ \alpha=0.98 $

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Figure 10. Phase portraits of the model considering different variables with $ \alpha=0.96 $

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Figure 11. Phase portraits of the model considering different variables with $ \alpha = 0.94 $

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Table 1. 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]

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