Mathematical modeling of continuous and intermittent androgen suppression for the treatment of advanced prostate cancer

Alacia M. Voth; John G. Alford; Edward W. Swim

doi:10.3934/mbe.2017043

Article Contents

2017, Volume 14, Issue 3: 777-804. Doi: 10.3934/mbe.2017043

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Mathematical modeling of continuous and intermittent androgen suppression for the treatment of advanced prostate cancer

1.
United Services Automobile Association, 9800 Fredericksburg Rd, San Antonio, TX 78288, USA
2.
Sam Houston State University, Department of Mathematics and Statistics, Huntsville, TX 77341, USA

* Corresponding author: edward.swim@shsu.edu
* Corresponding author: edward.swim@shsu.edu

Received: August 03, 2016

Published: September 2017

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Abstract

Prostate cancer is one of the most prevalent types of cancer among men. It is stimulated by the androgens, or male sexual hormones, which circulate in the blood and diffuse into the tissue where they stimulate the prostate tumor to grow. One of the most important treatments for advanced prostate cancer has become androgen deprivation therapy (ADT). In this paper we present three different models of ADT for prostate cancer: continuous androgen suppression (CAS), intermittent androgen suppression (IAS), and periodic androgen suppression. Currently, many patients in the U.S. receive CAS therapy of ADT, but many undergo a relapse after several years and experience adverse side effects while receiving treatment. Some clinical studies have introduced various IAS regimens in order to delay the time to relapse, and/or to reduce the economic costs and adverse side effects. We will compute and analyze parameter sensitivity analysis for CAS and IAS which may give insight to plan effective data collection in a future clinical trial. Moreover, a periodic model for IAS is used to develop an analytical formulation for relapse times which then provides information about the sensitivity of relapse to the parameters in our models.

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Mathematics Subject Classification: Primary: 92C50, 93A30; Secondary: 34C60.

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Figure 1. Solutions for all three compartments in the CAS model are presented. Baseline values for all parameters are used, as shown in Table 1.

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Figure 2. CAS sensitivity of androgen concentration, s_γ^a, with respect to the concentration rate γ. The negative sensitivity indicates that as γ is increased, androgen concentration will decrease

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Figure 3. CAS sensitivity of androgen dependent cells (x₁) to the proliferation rate α₁, shown in (a), and the apoptosis rate β₁, shown in (b)

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Figure 4. CAS sensitivity of androgen independent tumor cells, s_γ^x₂, with respect to the androgen concentration rate γ.

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Figure 5. CAS sensitivity of androgen independent tumor cells, s^x₂_m₁, with respect to the maximum mutation rate m₁

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Figure 6. Sample plots of the treatment function u(t), as determined by (19), based on a parameter value of λ = 1100. Here we illustrate the behavior for κ = 0 in (a) and κ = 1 − β₂/α₂ in (b).

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Figure 7. These graphs illustrate the long term behavior of the androgen levels a(t), plotted as a dotted line, and serum PSA concentration y(t), plotted as a solid line, using the model from equations (22)-(25). The plots in (a) correspond to κ = 0 and in (b) correspond to κ = 1 − β₂/α₂.

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Figure 8. Sensitivity analysis for androgen concentration, s_γ^a, androgen dependent tumor cells, s_γ^x₁, and androgen independent tumor cells, s_γ^x₂, with respect to the parameter γ for intermittent treatment. The left column presents results for κ = 0, while the right column presents the sensitivities when κ = 1 − β₂/α₂. Here red depicts on-treatment while black depicts off-treatment.

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Figure 9. Sensitivity analysis for androgen concentration, s^a_m₁, androgen dependent tumor cells, s^x₁_m₁, and androgen independent tumor cells, s^x₂_m₁, with respect to the mutation parameter m₁ for intermittent treatment. The left column presents results for κ = 0, while the right column presents the sensitivities when κ = 1 − β₂/α₂. Here red depicts on-treatment while black depicts off-treatment.

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Figure 10. Sensitivity analysis for androgen concentration, s^a_r₀, androgen dependent tumor cells, s^x₁_r₀, and androgen independent tumor cells, s^x₂_r₀, with respect to the parameter r₀ for intermittent treatment. The left column presents results for κ = 0, while the right column presents the sensitivities when κ = 1 − β₂/α₂. Here red depicts on-treatment while black depicts off-treatment.

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Figure 11. These boxplots illustrate the relative sensitivity of the IAS model to each parameter for the case κ = 0. In subfigure (b) data for the three parameters for which the median relative sensitivity exceeded the others by at least an order of magnitude are summarized. For clarity of the display, boxplots for the remaining parameters are presented in (a).

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Figure 12. These boxplots illustrate the relative sensitivity of the IAS model to each parameter for the case κ = 1 − β₂/α₂. Here we have preserved the same arrangement of parameters as displayed in Figure 11

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Figure 13. These curves show the relapse time from (36) with y = 20 and the on-treatment time from (37) where κ = 0 in (a) and κ = 1 − β₂/α₂ in (b). The solid curves use the ITTA model for IAS with u as in (11) and the dashed curves use our model for IAS with u as in (19).

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Figure 14. The periodic androgen suppression model where y(t) = x₁(t) + x₂(t) is plotted (solid) with x₁(t) and x₂(t) computed as in (41) with κ = 0 on the left and κ = 1 − β₂/α₂ on the right. The androgen level a(t) is also plotted (dash-dot) and oscillates with period T = 200 in both cases. The initial conditions are a(0) = 30, x₁(0) = 15, and x₂(0) = 0.01.

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Figure 15. The solid and dash-dot curves show the relapse time TR and on-treatment time Ton respectively for the PAS model with TR as in (36) where y = 20 and Ton from (37). Here we use κ = 0 in (a) and κ = 1 − β₂/α₂ in (b). The dashed curves show the approximation of relapse time from (61) in (a) and (64) in (b).

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Figure 16. This plot depicts the relapse time as it depends on κ for the continuous model. The dashed curve is the relapse time T_R (where y = 20) as a function of κ estimated from (61) with |ζ₁| as in (66). The solid curve is the relapse time as it depends on κ computed using the model equations (2) and (3) with continuous androgen suppression and solving x₁(t) + x₂(t) = 20 for t.

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Figure 17. This figure shows the sensitivity indices of relapse time T_R from (67) for the periodic model. The box plots summarize the values of $\Upsilon$_θ^T_R computed from 100 random draws of the period T distributed normally with mean 425 days and standard deviation 25 days. In the top row κ = 0 and the relapse time T_R was computed from (61). In the bottom row κ = 1 − β₂/α₂ and the relapse time T_R was computed from (64). The parameter values p are displayed along the horizontal axis. Outliers are not displayed.

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Figure 18. This figure shows the sensitivity indices of relapse time T_R from (67) for the continuous model. The box plots summarize the values of $\Upsilon$_θ^T_R computed from 100 random draws of κ distributed normally with mean 0.5 and standard deviation 0.165. The relapse time T_R was computed from (61) with |ζ₁| replaced by (66). The parameter values p are displayed along the horizontal axis. Outliers are not displayed.

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Table 1. Parameter values used for continuous, intermittent and periodic androgen suppression models are listed with their baseline value, units of measurement, and a range of values used for data simulation

Parameter	Variable	Baseline Value	Range
Normal androgen level	$a_0$	30 nmol/l	26.25-33.75 nmol/l
Androgen concentration rate	$\gamma$	0.08 days$^{-1}$	0.0425-0.1175 days$^{-1}$
Androgen dependent proliferation rate	$\alpha_1$	0.0204 days$^{-1}$	0.0129-0.0279 days$^{-1}$
Androgen dependent apoptosis rate	$\beta_1$	0.0076 days$^{-1}$	0.00685-0.00835 days$^{-1}$
Androgen independent proliferation rate	$\alpha_2$	0.0242 days$^{-1}$	0.0216-0.0268 days$^{-1}$
Androgen independent apoptosis rate	$\beta_2$	0.0168 days$^{-1}$	0.0130-0.0206 days$^{-1}$
Maximum mutation rate	$m_1$	0.00005 days$^{-1}$	1.25-8.75$\times10^{-5}$ days$^{-1}$
AD proliferation half-saturation level	$k_2$	2 nmol/l	1.25-2.75 nmol/l
AD androgen free apoptosis constant	$k_3$	8	7.25-8.75
AD apoptosis rate half-saturation level	$k_4$	0.5 nmol/l	0.275-0.725 nmol/l
Minimum PSA concentration	$r_0$	10 ng/ml	6.5-13.5 ng/ml
Maximum PSA concentration	$r_1$	15 ng/ml	11.5-18.5 ng/ml
Treatment transition rate	$\lambda$	1100 days$^{-1}$	500-1700 days$^{-1}$

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Table 2. Median relative sensitivities of the cost function J to the parameters in the IAS model. These statistics are based on 100 simulated data sets generated using random samples of the parameters from independent normal distributions with means given by the baseline values in Table 1 and standard deviations that limit the sample coefficients of variation to no more than 30%.

$\theta$	$\Upsilon^J_{\theta}$, $\kappa=0$	$\Upsilon^J_{\theta}$, $\kappa=1-\beta_2/\alpha_2$
$r_0$	$-1362$	$-50.19$
$\alpha_2$	$-292.9$	$-9.261$
$\beta_2$	$+189.3$	$+6.416$
$m_1$	$-9.609$	$-0.2761$
$\alpha_1$	$-8.772$	$+0.8986$
$\beta_1$	$+6.459$	$+1.246$
$\gamma$	$+3.463$	$-0.3032$
$k_3$	$+2.658$	$+0.07086$
$k_4$	$+1.807$	$+0.1499$
$k_2$	$+1.433$	$+0.2360$
$a_0$	$+0.3096$	$-0.6929$
$\lambda$	$-0.03408$	$-0.003861$
$r_1$	$-0.002916$	$-0.0003114$

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Table 3. Average sensitivity indices of relapse time T_R for the data depicted in Figures 17 and 18 where Periodic (ⅰ) is κ = 0 and Periodic (ⅱ) is κ = 1 − β₂/α₂

Parameter	Periodic (ⅰ)	Periodic (ⅱ)	Continuous
$a_0$	$-0.017$	$-0.0091$	$-0.0058$
$\gamma$	$+0.069$	$+0.035$	$+0.032$
$\alpha_1$	$-0.075$	$-0.041$	$-0.054$
$\beta_1$	$+0.13$	$+0.070$	$+0.11$
$\alpha_2$	$-3.12$	$-1.69$	$-3.15$
$\beta_2$	$+2.18$	$+1.18$	$+2.20$
$m_1$	$-0.13$	$-0.074$	$-0.13$
$k_2$	$+0.0069$	$+0.0037$	$+0.0034$
$k_3$	$+0.10$	$+0.054$	$+0.091$
$k_4$	$+0.0099$	$+0.0053$	$+0.0024$
$T$	$+0.033$	$+0.47$	$-$

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References

[1]	P.-A. Abrahamsson, Potential benefits of intermittent androgen suppression therapy in the treatment of prostate cancer: A systematic review of the literature, European Urology, 57 (2010), 49-59. doi: 10.1016/j.eururo.2009.07.049.
[2]	H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, J. Inverse Ill-Posed Probl., 15 (2007), 683-708. doi: 10.1515/jiip.2007.038.
[3]	H.T. Banks and D.M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625. doi: 10.1007/s00285-004-0299-x.
[4]	N. C. Buchan and S. L. Goldenberg, Intermittent androgen suppression for prostate cancer, Nature Reviews Urology, 7 (2010), 552-560. doi: 10.1038/nrurol.2010.141.
[5]	N. Chitnis, J.M. Hyman and J.M. Chushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0.
[6]	R.A. Everett, A.M. Packer and Y. Kuang, Can mathematical models predict the outcomes of prostate cancer patients undergoing intermittent androgen deprivation therapy?, Biophys. Rev. Lett., 9 (2014), 139-157. doi: 10.1142/9789814730266_0009.
[7]	J. K. Hale, Ordinary Differential Equations, 2^nd edition, Krieger Publishing, Malabar FL, 1980.
[8]	Y. Hirata, N. Bruchovsky and K. Aihara, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer, J. Theor. Biol., 264 (2010), 517-527. doi: 10.1016/j.jtbi.2010.02.027.
[9]	A.M. Ideta, G. Tanaka, T. Takeuchi and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer, J. Nonlinear Sci., 18 (2008), 593-614. doi: 10.1007/s00332-008-9031-0.
[10]	H. Lepor and N.D. Shore, LHRH agonists for the treatment of prostate cancer: 2012, Reviews in Urology, 14 (2012), 1-12.
[11]	Prostate cancer treatment (PDQ) -Patient Version National Cancer Institute, 2016. Available from: https://www.cancer.gov/types/prostate/patient/prostate-treatment-pdq.
[12]	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), 1-14. doi: 10.1063/1.3697848.
[13]	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.
[14]	F.G. Rick and A.V. Schally, Bench-to-bedside development of agonists and antagonists of luteinizing hormone-releasing hormone for treatment of advanced prostate cancer, Urologic Oncology: Seminars and Original Investigations, 33 (2015), 270-274. doi: 10.1016/j.urolonc.2014.11.006.
[15]	A. Sciarra, P.A. Abrahamsson, M. Brausi, M. Galsky, N. Mottet, O. Sartor, T.L.J. Tammela and F.C. da Silva, Intermittent androgen-depravation therapy in prostate cancer: a critical review focused on phase 3 trials, European Urology, 64 (2013), 722-730.
[16]	L.G. Stanley, Sensitivity equation methods for parameter dependent elliptic equations, Numer. Funct. Anal. Optim., 22 (2001), 721-748. doi: 10.1081/NFA-100105315.
[17]	Y. Suzuki, D. Sakai, T. Nomura, Y. Hirata and K. Aihara, A new protocol for intermittent androgen suppresion therapy of prostate cancer with unstable saddle-point dynamics, J. Theor. Biol., 350 (2014), 1-16. doi: 10.1016/j.jtbi.2014.02.004.
[18]	G. Tanaka, K. Tsumoto, S. Tsuji and K. Aihara, Analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer, Physica D, 237 (2008), 2616-2627. doi: 10.1016/j.physd.2008.03.044.
[19]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2^nd edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.
[20]	L. Voth, The Exploration and Computations of Mathematical Models of Intermittent Treatment for Prostate Cancer, M. S. thesis, Sam Houston University, 2012.