# American Institute of Mathematical Sciences

October  2021, 14(10): 3419-3440. doi: 10.3934/dcdss.2020426

## 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  October 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3419-3440. doi: 10.3934/dcdss.2020426
##### References:
 [1] K. Akakura, N. Bruchovsky, S. L. Goldenberg, P. S. Rennie, A. R. Buckley and L. D. Sullivan, Effects of intermittent androgen suppression on androgen-dependent tumors. Apoptosis and serum prostate-specific antigen, Cancer, 71 (1993), 2782-2790.  doi: 10.1002/1097-0142(19930501)71:9<2782::AID-CNCR2820710916>3.0.CO;2-Z. [2] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, preprint, (2016), arXiv: 1602.03408. [3] A. Atangana and T. Mekkaoui, Capturing complexities with composite operator and differential operators with non-singular kernel, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 023103, 15 pp. doi: 10.1063/1.5085927. [4] A. Atangana and S. Jain, The role of power decay, exponential decay and Mittag–Leffler functions waiting time distribution: Application of cancer spread, Physica A: Statistical Mechanics and its Applications, 512 (2018), 330-351.  doi: 10.1016/j.physa.2018.08.033. [5] R. R. Berges, J. Vukanovic, J. I. Epstein, M. CarMichel, L. Cisek, D. E. Johnson, R. W. Veltri, P. C. Walsh and and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer, Clinical Cancer Research, 1 (1995), 473-480. [6] N. Bruchovsky, L. Klotz, J. Crook, N. Phillips, J. Abersbach and S. L. Goldenberg, 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, Clinical Genitourinary Cancer, 6 (2008), 46-52. [7] S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), Art. No. 24. doi: 10.1186/1745-6150-5-24. [8] M. M. El-Dessoky and M. A. Khan, Application of fractional calculus to combined modified function projective synchronization of different systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013107. doi: 10.1063/1.5079955. [9] B. J. Feldman and D. Feldman, The development of androgen-independent prostate cancer, Nature Reviews Cancer, 1 (2001), 34-45.  doi: 10.1038/35094009. [10] C. A. Heinlein and C. Chang, Androgen receptor in prostate cancer, Endocrine Reviews, 25 (2004), 276-308.  doi: 10.1210/er.2002-0032. [11] J. Holzbeierlein, P. Lal, E. LaTulippe, A. Smith, J. Satagopan and et al., Gene expression analysis of human prostate carcinoma during hormonal therapy identifies androgen-responsive genes and mechanisms of therapy resistance, The American Journal of Pathology, 164 (2004), 217-227.  doi: 10.1016/S0002-9440(10)63112-4. [12] A. M. Ideta, G. Tanaka, T. Takeuchi, and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer, Journal of Nonlinear Science, 18 (2008), Art. No. 593. doi: 10.1007/s00332-008-9031-0. [13] H. V. Jain, S. K. Clinton, A. Bhinder and and A. Friedman, Mathematical modeling of prostate cancer progression in response to androgen ablation therapy, Proceedings of the National Academy of Sciences, 108 (2011), 19701-19706.  doi: 10.3934/dcdsb.2013.18.945. [14] M. A. Khan, S. Ullah, K. O. Okosun and K. Shah, A fractional order pine wilt disease model with Caputo–Fabrizio derivative, Advances in Difference Equations, (2018), Paper No. 410, 18 pp. doi: 10.1186/s13662-018-1868-4. [15] M. A. Khan, S. Ullah and and M. Farooq, A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative, Chaos, Solitons & Fractals, 116 (2018), 227-238.  doi: 10.1016/j.chaos.2018.09.039. [16] G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova, and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences, 113 (2016), E7663–E7671. doi: 10.1073/pnas.1615791113. [17] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan and and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A: Statistical Mechanics and its Applications, 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018. [18] P. S. Nelson, Molecular states underlying androgen receptor activation: A framework for therapeutics targeting androgen signaling in prostate cancer, Journal of Clinical Oncology, 30 (2011), 644-646.  doi: 10.1200/JCO.2011.39.1300. [19] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, in Mathematics in Science and Engineering, Vol. 198, Academic Press Inc., San Diego, CA, 1999. [20] T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy, AIP Advances, 2 (2012), 011002. doi: 10.1063/1.3697848. [21] M. H. Rashid and U. B. Chaudhary, Intermittent androgen deprivation therapy for prostate cancer, The Oncologist, 9 (2004), 295-301.  doi: 10.1634/theoncologist.9-3-295. [22] G. Tanaka, Y. Hirata, S. L. Goldenberg, N. Bruchovsky and K. Aihara, Mathematical modelling of prostate cancer growth and its application to hormone therapy, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 5029-5044.  doi: 10.1098/rsta.2010.0221. [23] M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), Art. No. 444. doi: 10.1140/epjp/i2017-11717-0. [24] S. Ullah, M. A. Khan and and M. Farooq, A fractional model for the dynamics of TB virus, Chaos, Solitons & Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001. [25] S. Ullah, M. A. Khan, and M. Farooq, A new fractional model for the dynamics of the hepatitis b virus using the caputo-fabrizio derivative, The European Physical Journal Plus, 133 (2018), Art. No. 237. doi: 10.1140/epjp/i2018-12072-4.

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##### References:
 [1] K. Akakura, N. Bruchovsky, S. L. Goldenberg, P. S. Rennie, A. R. Buckley and L. D. Sullivan, Effects of intermittent androgen suppression on androgen-dependent tumors. Apoptosis and serum prostate-specific antigen, Cancer, 71 (1993), 2782-2790.  doi: 10.1002/1097-0142(19930501)71:9<2782::AID-CNCR2820710916>3.0.CO;2-Z. [2] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, preprint, (2016), arXiv: 1602.03408. [3] A. Atangana and T. Mekkaoui, Capturing complexities with composite operator and differential operators with non-singular kernel, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 023103, 15 pp. doi: 10.1063/1.5085927. [4] A. Atangana and S. Jain, The role of power decay, exponential decay and Mittag–Leffler functions waiting time distribution: Application of cancer spread, Physica A: Statistical Mechanics and its Applications, 512 (2018), 330-351.  doi: 10.1016/j.physa.2018.08.033. [5] R. R. Berges, J. Vukanovic, J. I. Epstein, M. CarMichel, L. Cisek, D. E. Johnson, R. W. Veltri, P. C. Walsh and and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer, Clinical Cancer Research, 1 (1995), 473-480. [6] N. Bruchovsky, L. Klotz, J. Crook, N. Phillips, J. Abersbach and S. L. Goldenberg, 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, Clinical Genitourinary Cancer, 6 (2008), 46-52. [7] S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), Art. No. 24. doi: 10.1186/1745-6150-5-24. [8] M. M. El-Dessoky and M. A. Khan, Application of fractional calculus to combined modified function projective synchronization of different systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013107. doi: 10.1063/1.5079955. [9] B. J. Feldman and D. Feldman, The development of androgen-independent prostate cancer, Nature Reviews Cancer, 1 (2001), 34-45.  doi: 10.1038/35094009. [10] C. A. Heinlein and C. Chang, Androgen receptor in prostate cancer, Endocrine Reviews, 25 (2004), 276-308.  doi: 10.1210/er.2002-0032. [11] J. Holzbeierlein, P. Lal, E. LaTulippe, A. Smith, J. Satagopan and et al., Gene expression analysis of human prostate carcinoma during hormonal therapy identifies androgen-responsive genes and mechanisms of therapy resistance, The American Journal of Pathology, 164 (2004), 217-227.  doi: 10.1016/S0002-9440(10)63112-4. [12] A. M. Ideta, G. Tanaka, T. Takeuchi, and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer, Journal of Nonlinear Science, 18 (2008), Art. No. 593. doi: 10.1007/s00332-008-9031-0. [13] H. V. Jain, S. K. Clinton, A. Bhinder and and A. Friedman, Mathematical modeling of prostate cancer progression in response to androgen ablation therapy, Proceedings of the National Academy of Sciences, 108 (2011), 19701-19706.  doi: 10.3934/dcdsb.2013.18.945. [14] M. A. Khan, S. Ullah, K. O. Okosun and K. Shah, A fractional order pine wilt disease model with Caputo–Fabrizio derivative, Advances in Difference Equations, (2018), Paper No. 410, 18 pp. doi: 10.1186/s13662-018-1868-4. [15] M. A. Khan, S. Ullah and and M. Farooq, A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative, Chaos, Solitons & Fractals, 116 (2018), 227-238.  doi: 10.1016/j.chaos.2018.09.039. [16] G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova, and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences, 113 (2016), E7663–E7671. doi: 10.1073/pnas.1615791113. [17] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan and and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A: Statistical Mechanics and its Applications, 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018. [18] P. S. Nelson, Molecular states underlying androgen receptor activation: A framework for therapeutics targeting androgen signaling in prostate cancer, Journal of Clinical Oncology, 30 (2011), 644-646.  doi: 10.1200/JCO.2011.39.1300. [19] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, in Mathematics in Science and Engineering, Vol. 198, Academic Press Inc., San Diego, CA, 1999. [20] T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy, AIP Advances, 2 (2012), 011002. doi: 10.1063/1.3697848. [21] M. H. Rashid and U. B. Chaudhary, Intermittent androgen deprivation therapy for prostate cancer, The Oncologist, 9 (2004), 295-301.  doi: 10.1634/theoncologist.9-3-295. [22] G. Tanaka, Y. Hirata, S. L. Goldenberg, N. Bruchovsky and K. Aihara, Mathematical modelling of prostate cancer growth and its application to hormone therapy, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 5029-5044.  doi: 10.1098/rsta.2010.0221. [23] M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), Art. No. 444. doi: 10.1140/epjp/i2017-11717-0. [24] S. Ullah, M. A. Khan and and M. Farooq, A fractional model for the dynamics of TB virus, Chaos, Solitons & Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001. [25] S. Ullah, M. A. Khan, and M. Farooq, A new fractional model for the dynamics of the hepatitis b virus using the caputo-fabrizio derivative, The European Physical Journal Plus, 133 (2018), Art. No. 237. doi: 10.1140/epjp/i2018-12072-4.
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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