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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 |
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.
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 |

















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 |
Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |
Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |
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