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Dynamical Aspects of a Hybrid System Describing Intermittent Androgen Suppression Therapy of Prostate Cancer

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Geometric Properties for Parabolic and Elliptic PDE's (GPPEPDEs 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 176))

Abstract

We consider a mathematical model describing Intermittent Androgen Suppression therapy (IAS therapy) of prostate cancer. The system has a hybrid structure, i.e., the system consists of two different systems by the medium of an unknown binary function denoting the treatment state. In this paper, we shall prove that the hybrid system has a unique solution with the property that the binary function keeps on changing its value. In the clinical point of view, the result asserts that one can plan the IAS therapy for each prostate cancer patient, provided that the tumor satisfies a certain condition.

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References

  1. Akakura, K., Bruchovsky, N., Goldenberg, S.L., Rennie, P.S., Buckley, A.R., Sullivan, L.D.: Effects of intermittent androgen suppression on androgen-dependent tumors: apoptosis and serum prostate specific antigen. Cancer 71, 2782–2790 (1993)

    Article  Google Scholar 

  2. Bladow, F., Vessella, R.L., Buhler, K.R., Ellis, W.J., True, L.D., Lange, P.H.: Cell proliferation and apoptosis during prostatic tumor xenograft involution and regrowth after castration. Int. J. Cancer 67, 785–790 (1996)

    Article  Google Scholar 

  3. Bruchovsky, N., Rennie, P.S., Coldman, A.J., Goldenberg, S.L., Lawson, D.: Effects of androgen withdrawal on the stem cell composition of the Shionogi carcinoma. Cancer Res. 50, 2275–2282 (1990)

    Google Scholar 

  4. Daners, D., Medina, P.K.: Abstract Evolution Equations, Periodic Problems and Appications, Pitman Research Notes in Mathematics Series, vol. 279. Longman Scientific & Technical (1992)

    Google Scholar 

  5. Feldman, B.J., Feldman, D.: The development of androgen-independent prostate cancer. Nat. Rev. Cancer 1, 34–45 (2001)

    Article  Google Scholar 

  6. Friedman, A., Lolas, G.: Analysis of a mathematical model of tumor lymphangiogenesis. Math. Models Methods Appl. Sci. 1, 95–107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gleave, M., Goldenberg, S., Bruchovsky, N., Rennie, P.: Intermittent androgen suppression for protate cancer: rationale and clinical experience. Prostate cancer Prostatic Dis. 1, 289–296 (1998)

    Article  Google Scholar 

  8. Guo, Q., Tao, Y., Aihara, K.: Mathematical modeling of prostate tumor growth under intermittent androgen suppression with partial differential equations. Int. J. Bifurc. Chaos Appl. Sci. Eng. 18, 3789–3797 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hirata, Y., Bruchovsky, N., Aihara, K.: Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer. J. Theor. Biol. 264, 517–527 (2010)

    Article  MathSciNet  Google Scholar 

  10. Huggins, G., Hodges, C.: Studies of prostate cancer: I. The effects of castration, oestrogen and androgen injections on serum phosphates in metastatic carcinoma of the prostate, Cancer Res. 1, 207–293 (1941)

    Google Scholar 

  11. Hatano, T., Hirata, Y., Suzuki, H., Aihara, K.: Comparison between mathematical models of intermittent androgen suppression for prostate cancer. J. Theor. Biol. 366, 33–45 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ideta, A., Tanaka, G., Aihara, K.: A mathematical model of intermittent androgen suppression for prostate cancer. J. Nonlinear Sci. 18, 593–614 (2008)

    Article  MATH  Google Scholar 

  13. Jackson, T.L.: A mathematical model of prostate tumor growth and androgen-independent relapse. Discret. Contin. Dyn. Syst. B 4, 187–201 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jackson, T.L.: A mathematical investigation of multiple pathways to recurrent prostate cancer: comparison with experimental data. Neoplasia 6, 697–704 (2004)

    Article  Google Scholar 

  15. Jain, H.V., Friedman, A.: Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discret. Contin. Dyn. Syst. B 18, 945–967 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23. American Mathematical Society (1967)

    Google Scholar 

  17. Quittner, P., Souplet, P.: Superlinear Parabolic Problems Blow-up, Global Existence and Steady States. Birkhäuser Advanced Texts Basler LehrBücher (2007)

    Google Scholar 

  18. Rennie, P.S., Bruchovsky, N., Coldman, A.J.: Loss of androgen dependence is associated with an increase in tumorigenic stem cells and resistance to cell-death genes. J. Steroid Biochem. Mol. Biol. 37, 843–847 (1990)

    Article  Google Scholar 

  19. Sato, N., Gleave, M.E., Bruchovsky, N., Rennie, P.S., Goldenberg, S.L., Lange, P.H., Sullivan, L.D.: Intermittent androgen suppression delays progression to androgen-independent regulation of prostate-specific antigen gene in the LNCaP prostate tumor model. Biochem. Mol. Biol. 58, 139–146 (1996)

    Google Scholar 

  20. Shimada, T., Aihara, K.: A nonlinear model with competition between prostate tumor cells and its application to intermittent androgen suppression therapy of prostate cancer. Math. Biosci. 214, 134–139 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stamey, T.A., Yang, N., Hay, A.R., McNeal, J.E., Freiha, F.S., Redwine, E.: Prostate-specific antigen as a serum marker for adenocarcinoma of the prostate. N. Engl. J. Med. 317, 909–916 (1987)

    Article  Google Scholar 

  22. Tolis, G., Ackman, D., Stellos, A., Metha, A., Labrie, F., Fazekas, A.T.A., Comaru-Schally, A.M., Schally, A.V.: Tumor growth inhibition in patients with prostatic carcinoma treated with luteinizing hormone-releasing hormone agonists. Proc. Natl. Acad. Sci. USA 79, 1658–1662 (1982)

    Article  Google Scholar 

  23. Tao, Y., Guo, Q., Aihara, K.: A model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression. Math. Models Methods Appl. Sci. 12, 2177–2201 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tao, Y., Guo, Q., Aihara, K.: A mathematical model of prostate tumor growth under hormone therapy with mutation inhibitor. J. Nonlinear Sci. 20, 219–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tao, Y., Guo, Q., Aihara, K.: A partial differential equation model and its reduction to an ordinary differential model for prostate tumor growth under intermittent androgen suppression therapy. J. Math. Biol. 69, 817–838 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The second author was partially supported by Grant-in-Aid for Young Scientists (B), No. 24740097, and by Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (JSPS). The authors would like to express their gratitude to Professor Izumi Takagi of Tohoku University who gave them many useful comments.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Kurumi Hiruko & Shinya Okabe

Authors
  1. Kurumi Hiruko
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  2. Shinya Okabe
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Correspondence to Shinya Okabe .

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Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Milano, Italy

    Filippo Gazzola

  2. Mathematical Institute, Tohoku University, Sendai, Japan

    Kazuhiro Ishige

  3. Dip. di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Monte S. Angelo, Napoli, Italy

    Carlo Nitsch

  4. Dipartimento di Matematica Ulisse Dini, Università degli Studi di Firenze, Firenze, Firenze, Italy

    Paolo Salani

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Hiruko, K., Okabe, S. (2016). Dynamical Aspects of a Hybrid System Describing Intermittent Androgen Suppression Therapy of Prostate Cancer. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_12

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