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May 2017, 22(3): 1001-1021. doi: 10.3934/dcdsb.2017050

Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer

School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA

* The corresponding author

Received  October 2015 Revised  April 2016 Published  January 2017

Advanced prostate cancer is often treated by androgen deprivation therapy, which is initially effective but gives rise to fatal treatment-resistant cancer. Intermittent androgen deprivation therapy improves the quality of life of patients and may delay resistance towards treatment. Immunotherapy alters the bodies immune system to help fight cancer and has proven effective in certain types of cancer. We propose a model incorporating androgen deprivation therapy (intermittent and continual) in conjunction with dendritic cell vaccine immunotherapy. Simulations are run to determine the sensitivity of cancer growth to dendritic cell vaccine therapy administration schedule. We consider the limiting case where dendritic cells are administered continuously and perform analysis on the full model and the limiting cases of the model to determine necessary conditions for global stability of cancer eradication.

Citation: Erica M. Rutter, Yang Kuang. Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1001-1021. doi: 10.3934/dcdsb.2017050
References:
[1]

R. R. BergesJ. VukanovicJ. I. EpsteinM. CarMichelL. CisekD. E. JohnsonR. W. VeltriP. C. Walsh and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer, Clinical Cancer Research, 1 (1995), 473-480.

[2]

T. E. A. Botrel, O. Clark, R. B. Dos Reis, A. C. L. Pompeo, U. Ferreira, M. V. Sadi and F. F. H. Bretas, Intermittent versus continuous androgen deprivation for locally advanced, recurrent or metastatic prostate cancer: A systematic review and meta-analysis BMC Urology 14 (2014), p9.

[3]

P. A. BurchG. A. CroghanD. A. GastineauL. A. JonesJ. S. KaurJ. W. KylstraR. L. RichardsonF. H. Valone and S. Vuk-Pavlović, Immunotherapy (apc8015, provenge®) targeting prostatic acid phosphatase can induce durable remission of metastatic androgen-independent prostate cancer: A phase 2 trial, The Prostate, 60 (2004), 197-204.

[4]

M. A. Cheever and C. S. Higano, Provenge (sipuleucel-t) in prostate cancer: The first fda-approved therapeutic cancer vaccine, Clinical Cancer Research, 17 (2011), 3520-3526. doi: 10.1158/1078-0432.CCR-10-3126.

[5]

J. M. CrookC. J. O'CallaghanG. DuncanD. P. DearnaleyC. S. HiganoE. M. HorwitzE. FrymireS. MaloneJ. Chin and A. Nabid, Intermittent androgen suppression for rising psa level after radiotherapy, New England Journal of Medicine, 367 (2012), 895-903.

[6]

J. M. CrookE. SzumacherS. MaloneS. Huan and R. Segal, Intermittent androgen suppression in the management of prostate cancer, Urology, 53 (1999), 530-534. doi: 10.1016/S0090-4295(98)00547-0.

[7]

S. EikenberryJ. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), 24-24. doi: 10.1186/1745-6150-5-24.

[8]

R. EverettA. Packer and Y. Kuang, Can mathematical models predict the outcomes of prostate cancer patients undergoing intermittent androgen deprivation therapy?, Research on the Physics of Cancer, 9 (2016), 139-157. doi: 10.1142/9789814730266_0009.

[9]

L. FongD. BrockstedtC. BenikeJ. K. BreenG. StrangC. L. Ruegg and E. G. Engleman, Dendritic cell-based xenoantigen vaccination for prostate cancer immunotherapy, The Journal of Immunology, 167 (2001), 7150-7156. doi: 10.4049/jimmunol.167.12.7150.

[10]

M. Gleave, L. Klotz and S. S. Taneja, The continued debate: Intermittent vs. continuous hormonal ablation for metastatic prostate cancer, Urologic Oncology: Seminars and Original Investigations, 27 (2009), 81–86, Proceedings: Annual Meeting of the Society of Urologic Oncology/Society for Basic Urologic Research (May 2007).

[11]

C. S. HiganoP. F. SchellhammerE. J. SmallP. A. BurchJ. NemunaitisL. YuhN. Provost and M. W. Frohlich, Integrated data from 2 randomized, double-blind, placebo-controlled, phase 3 trials of active cellular immunotherapy with sipuleucel-t in advanced prostate cancer, Cancer, 115 (2009), 3670-3679. doi: 10.1002/cncr.24429.

[12]

Y. Hirata and K. Aihara, Ability of intermittent androgen suppression to selectively create a non-trivial periodic orbit for a type of prostate cancer patients, Journal of Theoretical Biology, 384 (2015), 147-152. doi: 10.1016/j.jtbi.2015.08.010.

[13]

Y. HirataK. AkakuraC. S. HiganoN. Bruchovsky and K. Aihara, Quantitative mathematical modeling of psa dynamics of prostate cancer patients treated with intermittent androgen suppression, Journal of Molecular Cell Biology, 4 (2012), 127-132. doi: 10.1093/jmcb/mjs020.

[14]

Y. HirataS.-i. Azuma and K. Aihara, Model predictive control for optimally scheduling intermittent androgen suppression of prostate cancer, Methods, 67 (2014), 278-281. doi: 10.1016/j.ymeth.2014.03.018.

[15]

Y. HirataN. Bruchovsky and K. Aihara, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer, Journal of Theoretical Biology, 264 (2010), 517-527. doi: 10.1016/j.jtbi.2010.02.027.

[16]

Y. Hirata, M. di~Bernardo, N. Bruchovsky and K. Aihara, Hybrid optimal scheduling for intermittent androgen suppression of prostate cancer Chaos: An Interdisciplinary Journal of Nonlinear Science 20 (2010), 045125.

[17]

M. HussainC. TangenC. HiganoN. Vogelzang and I. Thompson, Evaluating intermittent androgen-deprivation therapy phase iii clinical trials: The devil is in the details, Journal of Clinical Oncology, 34 (2016), 280-285. doi: 10.1200/JCO.2015.62.8065.

[18]

M. HussainC. M. TangenD. L. BerryC. S. HiganoE. D. CrawfordG. LiuG. WildingS. PrescottS. Kanaga Sundaram and E. J. Small, Intermittent versus continuous androgen deprivation in prostate cancer, New England Journal of Medicine, 368 (2013), 1314-1325. doi: 10.1056/NEJMoa1212299.

[19]

A. M. IdetaG. TanakaT. Takeuchi and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer, Journal of Nonlinear Science, 18 (2008), 593-614. doi: 10.1007/s00332-008-9031-0.

[20]

T. Jackson, A mathematical model of prostate tumor growth and androgen-independent relapse, Discrete and Continuous Dynamical Systems -Series B, 4 (2004), 187-201. doi: 10.3934/dcdsb.2004.4.187.

[21]

H. V. JainS. K. ClintonA. Bhinder 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: 10.1073/pnas.1115750108.

[22]

H. V. Jain and A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy, Discrete and Continuous Dynamical Systems -Series B, 18 (2013), 945-967. doi: 10.3934/dcdsb.2013.18.945.

[23]

A. JemalR. SiegelE. WardY. HaoJ. XuT. Murray and M. J. Thun, Cancer statistics, CA: A Cancer Journal for Clinicians, 58 (2008), 71-96. doi: 10.3322/CA.2007.0010.

[24]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127.

[25]

L. Klotz and P. Toren, Androgen deprivation therapy in advanced prostate cancer: Is intermittent therapy the new standard of care?, Current Oncology, 19 (2012), S13-S21. doi: 10.3747/co.19.1298.

[26]

N. Kronik, Y. Kogan, M. Elishmereni, K. Halevi-Tobias, S. Vuk-Pavlovi{\'c} and Z. Agur, Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models PLoS One 5 (2010), e15482.

[27]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology CRC Press, 2016.

[28]

S. Larry GoldenbergN. BruchovskyM. E. GleaveL. D. Sullivan and K. Akakura, Intermittent androgen suppression in the treatment of prostate cancer: A preliminary report, Urology, 45 (1995), 839-845. doi: 10.1016/S0090-4295(99)80092-2.

[29]

U. LedzewiczM. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, Journal of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6.

[30]

U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications, 114 (2002), 609-637. doi: 10.1023/A:1016027113579.

[31]

M. T. Lotze and A. W. Thomson, Dendritic Cells: Biology and Clinical Applications Access Online via Elsevier, 2001.

[32]

J. D. MorkenA. PackerR. A. EverettJ. D. Nagy and Y. Kuang, Mechanisms of resistance to intermittent androgen deprivation in patients with prostate cancer identified by a novel computational method, Cancer Research, 74 (2014), 3673-3683. doi: 10.1158/0008-5472.CAN-13-3162.

[33]

P. S. Nelson, Molecular states underlying androgen receptor activation: A framework for therapeutics targeting androgen signaling in prostate cancer, Journal of Clinical Oncology, 30 (2012), 644-646. doi: 10.1200/JCO.2011.39.1300.

[34]

J. C. Park and M. A. Eisenberger, Intermittent androgen deprivation in prostate cancer: Are we ready to quit?, Journal of Clinical Oncology, 34 (2016), 211-214. doi: 10.1200/JCO.2015.64.1019.

[35]

H. Peng, W. Zhao, H. Tan, Z. Ji, J. Li, K. Li and X. Zhou, Prediction of treatment efficacy for prostate cancer using a mathematical model Scientific Reports 6 (2016), 21599.

[36]

T. Portz and Y. Kuang, A mathematical model for the immunotherapy of advanced prostate cancer, in BIOMAT 2012, World Scientific, 2013, 70–85.

[37]

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.

[38]

S. A. Rosenberg and M. T. Lotze, Cancer immunotherapy using interleukin-2 and interleukin-2-activated lymphocytes, Annual Review of Immunology, 4 (1986), 681-709. doi: 10.1146/annurev.iy.04.040186.003341.

[39]

A. SciarraP. A. AbrahamssonM. BrausiM. GalskyN. MottetO. SartorT. L. Tammela and F. C. da Silva, Intermittent androgen-deprivation therapy in prostate cancer: A critical review focused on phase 3 trials, European Urology, 64 (2013), 722-730. doi: 10.1016/j.eururo.2013.04.020.

[40]

E. J. SmallP. FratesiD. M. ReeseG. StrangR. LausM. V. Peshwa and F. H. Valone, Immunotherapy of hormone-refractory prostate cancer with antigen-loaded dendritic cells, Journal of Clinical Oncology, 18 (2000), 3894-3903.

[41]

G. TanakaY. HirataS. L. GoldenbergN. Bruchovsky and K. Aihara, Mathematical modelling of prostate cancer growth and its application to hormone therapy, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 5029-5044. doi: 10.1098/rsta.2010.0221.

[42]

H. R. Thieme, Convergence results and a poincar{é}-bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267.

[43]

H.-T. TsaiD. F. PensonK. H. MakambiJ. H. LynchS. K. Van Den~Eeden and A. L. Potosky, Efficacy of intermittent androgen deprivation therapy vs conventional continuous androgen deprivation therapy for advanced prostate cancer: a meta-analysis, Urology, 82 (2013), 327-334. doi: 10.1016/j.urology.2013.01.078.

[44]

J. M. WolffP.-A. AbrahamssonJ. Irani and F. Calais da Silva, Is intermittent androgen-deprivation therapy beneficial for patients with advanced prostate cancer?, BJU International, 114 (2014), 476-483. doi: 10.1111/bju.12626.

show all references

References:
[1]

R. R. BergesJ. VukanovicJ. I. EpsteinM. CarMichelL. CisekD. E. JohnsonR. W. VeltriP. C. Walsh and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer, Clinical Cancer Research, 1 (1995), 473-480.

[2]

T. E. A. Botrel, O. Clark, R. B. Dos Reis, A. C. L. Pompeo, U. Ferreira, M. V. Sadi and F. F. H. Bretas, Intermittent versus continuous androgen deprivation for locally advanced, recurrent or metastatic prostate cancer: A systematic review and meta-analysis BMC Urology 14 (2014), p9.

[3]

P. A. BurchG. A. CroghanD. A. GastineauL. A. JonesJ. S. KaurJ. W. KylstraR. L. RichardsonF. H. Valone and S. Vuk-Pavlović, Immunotherapy (apc8015, provenge®) targeting prostatic acid phosphatase can induce durable remission of metastatic androgen-independent prostate cancer: A phase 2 trial, The Prostate, 60 (2004), 197-204.

[4]

M. A. Cheever and C. S. Higano, Provenge (sipuleucel-t) in prostate cancer: The first fda-approved therapeutic cancer vaccine, Clinical Cancer Research, 17 (2011), 3520-3526. doi: 10.1158/1078-0432.CCR-10-3126.

[5]

J. M. CrookC. J. O'CallaghanG. DuncanD. P. DearnaleyC. S. HiganoE. M. HorwitzE. FrymireS. MaloneJ. Chin and A. Nabid, Intermittent androgen suppression for rising psa level after radiotherapy, New England Journal of Medicine, 367 (2012), 895-903.

[6]

J. M. CrookE. SzumacherS. MaloneS. Huan and R. Segal, Intermittent androgen suppression in the management of prostate cancer, Urology, 53 (1999), 530-534. doi: 10.1016/S0090-4295(98)00547-0.

[7]

S. EikenberryJ. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), 24-24. doi: 10.1186/1745-6150-5-24.

[8]

R. EverettA. Packer and Y. Kuang, Can mathematical models predict the outcomes of prostate cancer patients undergoing intermittent androgen deprivation therapy?, Research on the Physics of Cancer, 9 (2016), 139-157. doi: 10.1142/9789814730266_0009.

[9]

L. FongD. BrockstedtC. BenikeJ. K. BreenG. StrangC. L. Ruegg and E. G. Engleman, Dendritic cell-based xenoantigen vaccination for prostate cancer immunotherapy, The Journal of Immunology, 167 (2001), 7150-7156. doi: 10.4049/jimmunol.167.12.7150.

[10]

M. Gleave, L. Klotz and S. S. Taneja, The continued debate: Intermittent vs. continuous hormonal ablation for metastatic prostate cancer, Urologic Oncology: Seminars and Original Investigations, 27 (2009), 81–86, Proceedings: Annual Meeting of the Society of Urologic Oncology/Society for Basic Urologic Research (May 2007).

[11]

C. S. HiganoP. F. SchellhammerE. J. SmallP. A. BurchJ. NemunaitisL. YuhN. Provost and M. W. Frohlich, Integrated data from 2 randomized, double-blind, placebo-controlled, phase 3 trials of active cellular immunotherapy with sipuleucel-t in advanced prostate cancer, Cancer, 115 (2009), 3670-3679. doi: 10.1002/cncr.24429.

[12]

Y. Hirata and K. Aihara, Ability of intermittent androgen suppression to selectively create a non-trivial periodic orbit for a type of prostate cancer patients, Journal of Theoretical Biology, 384 (2015), 147-152. doi: 10.1016/j.jtbi.2015.08.010.

[13]

Y. HirataK. AkakuraC. S. HiganoN. Bruchovsky and K. Aihara, Quantitative mathematical modeling of psa dynamics of prostate cancer patients treated with intermittent androgen suppression, Journal of Molecular Cell Biology, 4 (2012), 127-132. doi: 10.1093/jmcb/mjs020.

[14]

Y. HirataS.-i. Azuma and K. Aihara, Model predictive control for optimally scheduling intermittent androgen suppression of prostate cancer, Methods, 67 (2014), 278-281. doi: 10.1016/j.ymeth.2014.03.018.

[15]

Y. HirataN. Bruchovsky and K. Aihara, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer, Journal of Theoretical Biology, 264 (2010), 517-527. doi: 10.1016/j.jtbi.2010.02.027.

[16]

Y. Hirata, M. di~Bernardo, N. Bruchovsky and K. Aihara, Hybrid optimal scheduling for intermittent androgen suppression of prostate cancer Chaos: An Interdisciplinary Journal of Nonlinear Science 20 (2010), 045125.

[17]

M. HussainC. TangenC. HiganoN. Vogelzang and I. Thompson, Evaluating intermittent androgen-deprivation therapy phase iii clinical trials: The devil is in the details, Journal of Clinical Oncology, 34 (2016), 280-285. doi: 10.1200/JCO.2015.62.8065.

[18]

M. HussainC. M. TangenD. L. BerryC. S. HiganoE. D. CrawfordG. LiuG. WildingS. PrescottS. Kanaga Sundaram and E. J. Small, Intermittent versus continuous androgen deprivation in prostate cancer, New England Journal of Medicine, 368 (2013), 1314-1325. doi: 10.1056/NEJMoa1212299.

[19]

A. M. IdetaG. TanakaT. Takeuchi and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer, Journal of Nonlinear Science, 18 (2008), 593-614. doi: 10.1007/s00332-008-9031-0.

[20]

T. Jackson, A mathematical model of prostate tumor growth and androgen-independent relapse, Discrete and Continuous Dynamical Systems -Series B, 4 (2004), 187-201. doi: 10.3934/dcdsb.2004.4.187.

[21]

H. V. JainS. K. ClintonA. Bhinder 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: 10.1073/pnas.1115750108.

[22]

H. V. Jain and A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy, Discrete and Continuous Dynamical Systems -Series B, 18 (2013), 945-967. doi: 10.3934/dcdsb.2013.18.945.

[23]

A. JemalR. SiegelE. WardY. HaoJ. XuT. Murray and M. J. Thun, Cancer statistics, CA: A Cancer Journal for Clinicians, 58 (2008), 71-96. doi: 10.3322/CA.2007.0010.

[24]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127.

[25]

L. Klotz and P. Toren, Androgen deprivation therapy in advanced prostate cancer: Is intermittent therapy the new standard of care?, Current Oncology, 19 (2012), S13-S21. doi: 10.3747/co.19.1298.

[26]

N. Kronik, Y. Kogan, M. Elishmereni, K. Halevi-Tobias, S. Vuk-Pavlovi{\'c} and Z. Agur, Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models PLoS One 5 (2010), e15482.

[27]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology CRC Press, 2016.

[28]

S. Larry GoldenbergN. BruchovskyM. E. GleaveL. D. Sullivan and K. Akakura, Intermittent androgen suppression in the treatment of prostate cancer: A preliminary report, Urology, 45 (1995), 839-845. doi: 10.1016/S0090-4295(99)80092-2.

[29]

U. LedzewiczM. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, Journal of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6.

[30]

U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications, 114 (2002), 609-637. doi: 10.1023/A:1016027113579.

[31]

M. T. Lotze and A. W. Thomson, Dendritic Cells: Biology and Clinical Applications Access Online via Elsevier, 2001.

[32]

J. D. MorkenA. PackerR. A. EverettJ. D. Nagy and Y. Kuang, Mechanisms of resistance to intermittent androgen deprivation in patients with prostate cancer identified by a novel computational method, Cancer Research, 74 (2014), 3673-3683. doi: 10.1158/0008-5472.CAN-13-3162.

[33]

P. S. Nelson, Molecular states underlying androgen receptor activation: A framework for therapeutics targeting androgen signaling in prostate cancer, Journal of Clinical Oncology, 30 (2012), 644-646. doi: 10.1200/JCO.2011.39.1300.

[34]

J. C. Park and M. A. Eisenberger, Intermittent androgen deprivation in prostate cancer: Are we ready to quit?, Journal of Clinical Oncology, 34 (2016), 211-214. doi: 10.1200/JCO.2015.64.1019.

[35]

H. Peng, W. Zhao, H. Tan, Z. Ji, J. Li, K. Li and X. Zhou, Prediction of treatment efficacy for prostate cancer using a mathematical model Scientific Reports 6 (2016), 21599.

[36]

T. Portz and Y. Kuang, A mathematical model for the immunotherapy of advanced prostate cancer, in BIOMAT 2012, World Scientific, 2013, 70–85.

[37]

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.

[38]

S. A. Rosenberg and M. T. Lotze, Cancer immunotherapy using interleukin-2 and interleukin-2-activated lymphocytes, Annual Review of Immunology, 4 (1986), 681-709. doi: 10.1146/annurev.iy.04.040186.003341.

[39]

A. SciarraP. A. AbrahamssonM. BrausiM. GalskyN. MottetO. SartorT. L. Tammela and F. C. da Silva, Intermittent androgen-deprivation therapy in prostate cancer: A critical review focused on phase 3 trials, European Urology, 64 (2013), 722-730. doi: 10.1016/j.eururo.2013.04.020.

[40]

E. J. SmallP. FratesiD. M. ReeseG. StrangR. LausM. V. Peshwa and F. H. Valone, Immunotherapy of hormone-refractory prostate cancer with antigen-loaded dendritic cells, Journal of Clinical Oncology, 18 (2000), 3894-3903.

[41]

G. TanakaY. HirataS. L. GoldenbergN. Bruchovsky and K. Aihara, Mathematical modelling of prostate cancer growth and its application to hormone therapy, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 5029-5044. doi: 10.1098/rsta.2010.0221.

[42]

H. R. Thieme, Convergence results and a poincar{é}-bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267.

[43]

H.-T. TsaiD. F. PensonK. H. MakambiJ. H. LynchS. K. Van Den~Eeden and A. L. Potosky, Efficacy of intermittent androgen deprivation therapy vs conventional continuous androgen deprivation therapy for advanced prostate cancer: a meta-analysis, Urology, 82 (2013), 327-334. doi: 10.1016/j.urology.2013.01.078.

[44]

J. M. WolffP.-A. AbrahamssonJ. Irani and F. Calais da Silva, Is intermittent androgen-deprivation therapy beneficial for patients with advanced prostate cancer?, BJU International, 114 (2014), 476-483. doi: 10.1111/bju.12626.

Figure 1.  PSA serum concentration, androgen dependent (AD), and androgen independent (AI) cell concentrations for various dendritic cell vaccine injection times with an $e_1=e_2$ of 0.75. More frequent injections result delay of the rise of fatal androgen independent cancer
Figure 2.  Limit cycle solutions for androgen dependent $X_1$, and androgen independent $X_2$ cancer cells. The minimal value of $e_1$ required to produce limit cycle behavior is noted above each solution. As vaccine timing decreases, minimal $e_1$ necessary to have stable disease state decreases
Figure 3.  PSA serum level, AD cell density, and AI cell density for continual dendritic cell vaccinations, with an injection rate 0.04 billion cells for various values of $e_1$. It is apparent that in this continuous case, a wider range of $e_1$ is able to suppress the growth of cancer and elongate the cycles of IAD
Figure 4.  Birfurcation digram for parameter $e_1$, a measure of cytotoxicity of T-cells. Maximum PSA level in black, minimum PSA level in green. Carrying capacity is stable only for $e_1=0$. Immediately after, we have a Hopf bifurcaton and limit cycles, until we reach $e_1\approx 0.66$, after which the disease-free steady state is stable
Figure 5.  Comparisons of the full system and quasi-steady state system for various values of $e_1$: 0, 0.15, 0.25, and 0.65, assuming androgen deprivation therapy is constantly on. These values of $e_1$ show many differing dynamics. The quasi-steady state system closely approximates the full system in every case, but shows slight differences in the case of $e_1=0.15$
Table 1.  Values of parameters (P), explanations, and cited sources of every parameter used in this mathematical model
P Biological Meaning Value Source
$r_1$ AD cell proliferation rate 0.025/day [ 1 ]
$d_1$ AD cell death rate 0.064/day [ 1 ]
$K$ cancer cell carrying capacity 11 billion
$k_4$ AI to AD mutation half-saturation 1.7
$r_2$ AI net cell growth rate 0.006/day [ 1 ]
$m_1$ maximum mutation rate from AD to AI 0.00005/day [ 19 ]
$m_2$ maximum mutation rate from AI to AD 0.00015/day [ 37 ]
$a_0$ base level androgen concentration 30 ng/ml [ 19 ]
$\gamma$ androgen clearance and production rate 0.08/day [ 19 ]
$\omega$ cytokine clearance rate 10/day [ 38 ]
$\mu$ T cell death rate 0.03//day [ 24 ]
$c$ dendritic cell death rate 0.14/day [ 31 ]
$e_1$ max rate T cells kill AD cancer cells 0-1/day [ 24 ]
$g_1$ AD cancer cell saturation level for T cell kill rate 10 x $10^9$ cells [ 24 ]
$e_2$ max rate T cells kill AI cancer cells 0-1/day [ 24 ]
$g_2$ AI cancer cell saturation levelfor T cell kill rate 10 x $10^9$ cells [ 24 ]
$e$ T cell max activation rate 20 x $10^6$ cells/day [ 24 ]
$g$ DC saturation level for T cell activation 400 x $10^6$ cells [ 40 ]
$e_3$ max clonal expansion rate 0.1245/day [ 24 ]
$g_3$ IL-2 saturation level for T cell clonal expansion 1000 ng/ml [ 24 ]
$e_4$ max rate T cells produce IL-2 5 x $10^{-6}$ ng/ml/cell/day [ 24 ]
$g_4$ cancer cell saturation level for T cell stimulation 10 x $10^9$ cells [ 24 ]
$D_1$ DC vaccine dosage 300 x $10^6$ cells [ 40 ]
$c_1$ AD cell PSA level correlation 1 x $10^{-9}$ ng/ml/cell [ 19 ]
$c_2$ AI cell PSA level correlation 1 x $10^{-9} $ ng/ml/cell [ 19 ]
P Biological Meaning Value Source
$r_1$ AD cell proliferation rate 0.025/day [ 1 ]
$d_1$ AD cell death rate 0.064/day [ 1 ]
$K$ cancer cell carrying capacity 11 billion
$k_4$ AI to AD mutation half-saturation 1.7
$r_2$ AI net cell growth rate 0.006/day [ 1 ]
$m_1$ maximum mutation rate from AD to AI 0.00005/day [ 19 ]
$m_2$ maximum mutation rate from AI to AD 0.00015/day [ 37 ]
$a_0$ base level androgen concentration 30 ng/ml [ 19 ]
$\gamma$ androgen clearance and production rate 0.08/day [ 19 ]
$\omega$ cytokine clearance rate 10/day [ 38 ]
$\mu$ T cell death rate 0.03//day [ 24 ]
$c$ dendritic cell death rate 0.14/day [ 31 ]
$e_1$ max rate T cells kill AD cancer cells 0-1/day [ 24 ]
$g_1$ AD cancer cell saturation level for T cell kill rate 10 x $10^9$ cells [ 24 ]
$e_2$ max rate T cells kill AI cancer cells 0-1/day [ 24 ]
$g_2$ AI cancer cell saturation levelfor T cell kill rate 10 x $10^9$ cells [ 24 ]
$e$ T cell max activation rate 20 x $10^6$ cells/day [ 24 ]
$g$ DC saturation level for T cell activation 400 x $10^6$ cells [ 40 ]
$e_3$ max clonal expansion rate 0.1245/day [ 24 ]
$g_3$ IL-2 saturation level for T cell clonal expansion 1000 ng/ml [ 24 ]
$e_4$ max rate T cells produce IL-2 5 x $10^{-6}$ ng/ml/cell/day [ 24 ]
$g_4$ cancer cell saturation level for T cell stimulation 10 x $10^9$ cells [ 24 ]
$D_1$ DC vaccine dosage 300 x $10^6$ cells [ 40 ]
$c_1$ AD cell PSA level correlation 1 x $10^{-9}$ ng/ml/cell [ 19 ]
$c_2$ AI cell PSA level correlation 1 x $10^{-9} $ ng/ml/cell [ 19 ]
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