Skip to main content
SpringerLink
Log in

The Impact of Fractional Differentiation in Terms of Fitting for a Prostate Cancer Model Under Intermittent Androgen Suppression Therapy

  • Chapter
  • First Online:
Mathematical Modelling in Health, Social and Applied Sciences

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

Abstract

Actually, the main motivation for the contents of this chapter is to introduce fractional calculus as a prospective mathematical tool for cancer dynamics, in particular prostate cancer modeling. In this context, firstly, our main problem on the controversial role of androgens for prostate cancer development is handled, and according to our hypothesis a new mathematical model consisting of conventional logistic growth phenomena is constructed versus another prospective model based on ecological phenomena, cell quota. Then, we compare these two models demonstrating the mean squared error (MSE) values for androgen and prostate-specific antigen (PSA) for the first 1.5 cycles of intermittent androgen suppression (IAS) therapy administered to 62 selected patients from Vancouver Prostate Center (Vancouver, BC, Canada). To reduce MSE values, we also generate the fractional version of the model and verify that fractional differentiation provides nearly better data fitting for mathematical modeling. Moreover, with a discussion part, which hints for future works should be taken into account are pointed out.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. W. Deng, C. Li, Analysis of fractional differential equations with multi- orders. Fractals 15(2), 173–182 (2007)

    Article  MathSciNet  Google Scholar 

  2. Y. Ding, H. Ye, A fractional-order differential equation model of HIV infection of CD4 + T-Cells. Math. Comput. Model. 50, 386–392 (2009)

    Article  MathSciNet  Google Scholar 

  3. W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    Article  MathSciNet  Google Scholar 

  4. Z.M. Odibat, N.T. Shawagfeh, Generalized Taylors formula. Appl. Math. Comput. 186, 286–293 (2007)

    MathSciNet  MATH  Google Scholar 

  5. O.O. Mizrak, N. Ozalp, Fractional analog of a chemical system inspired by Braess’ paradox. Comp. Appl. Math. 37, 2503–2518 (2018). https://doi.org/10.1007/s40314017-0462-9

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics 6, 16 (2018). https://doi.org/10.3390/math6020016

    Article  MATH  Google Scholar 

  7. S.R. Denmeade, J.T. Isaacs, A history of prostate cancer treatment. Nat. Rev. Cancer 2(5), 389–396 (2002)

    Article  Google Scholar 

  8. N. Bruchovsky, L. Klotz, J. Crook, S. Malone, C. Ludgate, W.J. Morris, M.E. Gleave, S.L. Goldenberg, Final results of the Canadian prospective phase II trial of intermittent androgen suppression for men in biochemical recurrence after radiotherapy for locally advanced prostate cancer: Clinical parameters. Cancer 107(2), 389–395 (2006)

    Article  Google Scholar 

  9. T. Nishiyama, Serum testosterone levels after medical or surgical androgen deprivation: A comprehensive review of the literature. Urol. Oncol. 32(1), 38–e17–28 (2013)

    Article  Google Scholar 

  10. J.M. Crook, C.J. O’Callaghan, G. Duncan, D.P. Dearnaley, C.S. Higano, E.M. Horwitz, E. Frymire, S. Malone, J. Chin, A. Nabid, P. Warde, T. Corbett, S. Angyal, S.L. Goldenberg, M.K. Gospodarowicz, F. Saad, J.P. Logue, E. Hall, P.F. Schellhammer, K. Ding, L. Klotz, Intermittent androgen suppression for rising PSA level after radiotherapy. N. Engl. J. Med. 367(10), 895–903 (2012)

    Article  Google Scholar 

  11. A.H. Bryce, E.S. Antonarakis, Androgen receptor splice variant 7 in castration resistant prostate cancer: Clinical considerations. Int. J. Urol. 23(8), 646–653 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. P.C. Deaths, Cancer Statistics, 2011 The impact of eliminating socioeconomic and racial disparities on premature cancer deaths (2011)

    Google Scholar 

  14. T.L. Jackson, A mathematical model of prostate tumor growth and androgen-independent relapse. Discret. Contin. Dyn. Syst.-Ser. B 4, 187–202 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  16. T. Shimada, K. Aihara, 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  Google Scholar 

  17. S.E. Eikenberry, J.D. Nagy, Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model. Biol. Direct 5, 24 (2010)

    Article  Google Scholar 

  18. T. Portz, Y. Kuang, J.D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy. AIP Adv. 2, 011002 (2012)

    Article  Google Scholar 

  19. J. Baez, Y. Kuang, Mathematical models of androgen resistance in prostate cancer patients under intermittent androgen suppression therapy. Appl. Sci. 6, 352 (2016). doi: 10.3390/app6110352

    Article  Google Scholar 

  20. T. Portz, Y. Kuang, J. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy. AIP Adv. 2, 1–14 (2012)

    Article  Google Scholar 

  21. Y. Kuang, J.D. Nagy, S.E. Eikenberry, in Introduction to Mathematical Oncology (Chapman and Hall/CRC Mathematical and Computational Biology, 2016)

    Google Scholar 

  22. M. Droop, Some thoughts on nutrient limitation in algae1. J. Phycol. 9(264), 272 (1973)

    Google Scholar 

  23. E.M. Rutter, Y. Kuang, Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer. DCDS-B 22, 1001–1021 (2017)

    Article  MathSciNet  Google Scholar 

  24. N. Bruchovsky, L. Klotz, J. Crook, S. Malone, C. Ludgate, W.J. Morris, M.E. Gleave, S.L. Goldenberg, Final results of the Canadian prospective phase II trial of intermittent androgen suppression for men in biochemical recurrence after radiotherapy for locally advanced prostate cancer. Cancer 107, 389–395 (2006)

    Article  Google Scholar 

  25. N. Bruchovsky, Clinical Research. 2006. Available online: http://www.nicholasbruchovsky.com/clinicalResearch.html. Accessed on 18 July 2018

  26. H. Vardhan Jain, A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discret. Contin. Dyn. Syst.-Ser. B. 18

    Article  MathSciNet  Google Scholar 

  27. Y. Hirata, N. Bruchovsky, K. Aihara, 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 

  28. Q. Guo, Z. Lu, Y. Hirata, K. Aihara, Parameter estimation and optimal scheduling algorithm for a mathematical model of intermittent androgen suppression therapy for prostate cancer. Chaos 23(4), 43125 (2013)

    Article  Google Scholar 

  29. Y. Tao, Q. Guo, K. Aihara, A partial deferential equation model and its reduction to an ordinary differential equation model for prostate tumor growth under intermittent hormone therapy. J. Math. Biol. 1–22 (2013)

    Google Scholar 

  30. Y. Hirata, K. Akakura, C.S. Higano, N. Bruchovsky, K. Aihara, Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression. J. Mol. Cell Biol. 4(3), 127–132 (2012)

    Article  Google Scholar 

  31. Y. Hirata, S.-I. Azuma, K. Aihara, Model predictive control for optimally scheduling intermittent androgen suppression of prostate cancer. Methods 67(3), 278–281 (2014)

    Article  Google Scholar 

Download references

Acknowledgements:

O.O.M. is supported by the Scientific and Research Council of Turkey within the context of 2214-A-Ph.D. Research Fellowship Abroad and by the Scientific Research Unit of Ankara University with the grant number 17L0430006. We are also grateful to Nicholas Bruchovsky for the clinical data set.

Author information

Authors and Affiliations

  1. Department of Mathematics, Karabuk University, Baliklar Kayasi Mevkii, Demir Celik Kampusu, 78050, Karabuk, Turkey

    Ozlem Ozturk Mizrak

  2. Department of Mechatronics Engineering, Karabuk University, Baliklar Kayasi Mevkii, Demir Celik Kampusu, 78050, Karabuk, Turkey

    Cihan Mizrak

  3. School of Science and Technology, Nazarbayev University, Astana, 010000, Republic of Kazakhstan

    Ardak Kashkynbayev

  4. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Yang Kuang

Authors
  1. Ozlem Ozturk Mizrak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cihan Mizrak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ardak Kashkynbayev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yang Kuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ozlem Ozturk Mizrak .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mizrak, O.O., Mizrak, C., Kashkynbayev, A., Kuang, Y. (2020). The Impact of Fractional Differentiation in Terms of Fitting for a Prostate Cancer Model Under Intermittent Androgen Suppression Therapy. In: Dutta, H. (eds) Mathematical Modelling in Health, Social and Applied Sciences. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-2286-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation