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June
2017, 14(3): 777-804.
doi: 10.3934/mbe.2017043
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 |
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.
Keywords:
Prostate cancer,
androgen suppression,
differential equations,
sensitivity functions,
relapse.
Mathematics Subject Classification:
Primary: 92C50, 93A30; Secondary: 34C60.
Citation:
Alacia M. Voth, John G. Alford, Edward W. Swim. Mathematical modeling of continuous and intermittent androgen suppression for the treatment of advanced prostate cancer. Mathematical Biosciences & Engineering,
2017, 14
(3)
: 777-804.
doi: 10.3934/mbe.2017043
![]()
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, 2nd 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. Google Scholar |
| [11] |
Prostate cancer treatment (PDQ) -Patient Version National Cancer Institute, 2016. Available from: https://www.cancer.gov/types/prostate/patient/prostate-treatment-pdq.Google Scholar |
| [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. Google Scholar |
| [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, 2nd 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.Google Scholar |
show all references
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, 2nd 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. Google Scholar |
| [11] |
Prostate cancer treatment (PDQ) -Patient Version National Cancer Institute, 2016. Available from: https://www.cancer.gov/types/prostate/patient/prostate-treatment-pdq.Google Scholar |
| [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. Google Scholar |
| [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, 2nd 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.Google Scholar |
Figure 2.
CAS sensitivity of androgen concentration, sγa, with respect to the concentration rate γ. The negative sensitivity indicates that as γ is increased, androgen concentration will decrease
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)
Figure 4.
CAS sensitivity of androgen independent tumor cells, sγx2, with respect to the androgen concentration rate γ.
Figure 5.
CAS sensitivity of androgen independent tumor cells, sx2m1, with respect to the maximum mutation rate m1
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).
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.
Figure 8.
Sensitivity analysis for androgen concentration, sγa, androgen dependent tumor cells, sγx1, and androgen independent tumor cells, sγx2, 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.
Figure 9.
Sensitivity analysis for androgen concentration, sam1, androgen dependent tumor cells, sx1m1, and androgen independent tumor cells, sx2m1, 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.
Figure 10.
Sensitivity analysis for androgen concentration, sar0, androgen dependent tumor cells, sx1r0, and androgen independent tumor cells, sx2r0, 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.
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).
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
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).
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.
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).
Figure 16.
This plot depicts the relapse time as it depends on κ for the continuous model. The dashed curve is the relapse time TR (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.
Figure 17.
This figure shows the sensitivity indices of relapse time TR from (67) for the periodic model. The box plots summarize the values of $\Upsilon$ θTR 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 TR was computed from (61). In the bottom row κ = 1 − β2/α2 and the relapse time TR was computed from (64). The parameter values p are displayed along the horizontal axis. Outliers are not displayed.
Figure 18.
This figure shows the sensitivity indices of relapse time TR from (67) for the continuous model. The box plots summarize the values of $\Upsilon$ θTR computed from 100 random draws of κ distributed normally with mean 0.5 and standard deviation 0.165. The relapse time TR was computed from (61) with |ζ1| replaced by (66). The parameter values p are displayed along the horizontal axis. Outliers are not displayed.
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 | 30 nmol/l | 26.25-33.75 nmol/l | |
| Androgen concentration rate | 0.08 days |
0.0425-0.1175 days |
|
| Androgen dependent proliferation rate | 0.0204 days |
0.0129-0.0279 days |
|
| Androgen dependent apoptosis rate | 0.0076 days |
0.00685-0.00835 days |
|
| Androgen independent proliferation rate | 0.0242 days |
0.0216-0.0268 days |
|
| Androgen independent apoptosis rate | 0.0168 days |
0.0130-0.0206 days |
|
| Maximum mutation rate | 0.00005 days |
1.25-8.75 |
|
| AD proliferation half-saturation level | 2 nmol/l | 1.25-2.75 nmol/l | |
| AD androgen free apoptosis constant | 8 | 7.25-8.75 | |
| AD apoptosis rate half-saturation level | 0.5 nmol/l | 0.275-0.725 nmol/l | |
| Minimum PSA concentration | 10 ng/ml | 6.5-13.5 ng/ml | |
| Maximum PSA concentration | 15 ng/ml | 11.5-18.5 ng/ml | |
| Treatment transition rate | 1100 days |
500-1700 days |
| Parameter | Variable | Baseline Value | Range |
| Normal androgen level | 30 nmol/l | 26.25-33.75 nmol/l | |
| Androgen concentration rate | 0.08 days |
0.0425-0.1175 days |
|
| Androgen dependent proliferation rate | 0.0204 days |
0.0129-0.0279 days |
|
| Androgen dependent apoptosis rate | 0.0076 days |
0.00685-0.00835 days |
|
| Androgen independent proliferation rate | 0.0242 days |
0.0216-0.0268 days |
|
| Androgen independent apoptosis rate | 0.0168 days |
0.0130-0.0206 days |
|
| Maximum mutation rate | 0.00005 days |
1.25-8.75 |
|
| AD proliferation half-saturation level | 2 nmol/l | 1.25-2.75 nmol/l | |
| AD androgen free apoptosis constant | 8 | 7.25-8.75 | |
| AD apoptosis rate half-saturation level | 0.5 nmol/l | 0.275-0.725 nmol/l | |
| Minimum PSA concentration | 10 ng/ml | 6.5-13.5 ng/ml | |
| Maximum PSA concentration | 15 ng/ml | 11.5-18.5 ng/ml | |
| Treatment transition rate | 1100 days |
500-1700 days |
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%.
Table 3.
Average sensitivity indices of relapse time TR 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 |
| Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |
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