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Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression, "Beyond the Abstract," by Kazuyuki Aihara

BERKELEY, CA (UroToday.com) - This review paper summarizes the development of mathematical models for prostate cancer under intermittent androgen suppression (IAS), especially featuring the recent model of Hirata et al.[1] Our modeling of prostate cancer under IAS started with the Ideta model[2] which is the first model for IAS, and currently a standard model for IAS in the field of mathematical biology. The Ideta model includes three variables. The first variable is related to androgen dynamics, while the second and the third variables are related to androgen-dependent and androgen-independent cancer cells, respectively. Based on this model, several subsequent models have been developed. Such models include a model considering the effects of competitions among cancer cell[3] and models taking into account effects of spatial distribution.[4, 5, 6, 7, 8 ] These models consider more-detailed biochemical dynamics in terms of androgen and androgen receptors.

Among these extended models, the model of Hirata et al.[1] is most practical because the model is a piecewise linear model and easy to handle mathematically, as well as clinically. This model contains three variables: the first variable is related to androgen-dependent cancer cells, the other two variables, related to two different types of androgen-independent cancer cells, namely the adapted and mutated types. The effect of androgen dynamics is expressed by the switching between on-treatment and off-treatment periods. Namely, in on-treatment periods, androgen-dependent cancer cells may change to the two types of androgen-independent cancer cells. In off-treatment periods, the first type of adapted androgen-independent cancer cells may change back to androgen-dependent cancer cells, while the second type of mutated androgen-independent cancer cells go back to neither androgen-dependent cancer cells nor the first type of adapted androgen-independent cancer cells because the genes are mutated. Prostate specific antigen (PSA) is represented by the linear sum of the three variables, for simplicity. We have checked the assumptions of this model with clinical datasets: the piecewise linearity is enough to model the nonlinear dynamics of prostate cancer under IAS; the switching between on- and off-treatment periods is enough to describe the androgen-related dynamics. The model can fit clinical datasets quantitatively well.[1, 9] Due to the simplicity of the model, we can derive its solution analytically.[14] In addition, we can obtain the inequality conditions of parameters, fitted in a personalized way, for classifying each patient to three types.[9] The classification, by the model, correlates well with the diagnoses made by medical doctors.[1, 9, 12]

We are very close to the truly clinical application of our model. We are developing mathematical tools making the analysis more practical, such as inference of patient's state and parameters from very short observations of PSA data,[10] to overcome the uncertainty due to the observations. Clinical trials might be necessary in which the scheduling of on- and off-treatment periods is robustly optimized by the Hirata model and another model[11] to personalize IAS to each individual patient. Our mathematical modeling of IAS not only provides personalized treatment scheduling to each prostate cancer patient, but also shows new fundamental mathematical problems which commonly appear when we try to apply mathematics to medical problems.[14]

References:

  1. Y. Hirata, N. Bruchovsky, and K. Aihara, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer. J. Theor. Boil. 264, 517-527 (2010).
  2. A. M. Ideta, G. Tanaka, T. Tkeuchi, and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer. J. nonlinear Sci. 18, 593-614 (2008).
  3. T. Shimada and 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).
  4. Q. Guo, Y. Tao, and K. Aihara, Mathematical modelling of prostate toumor growth under intermittent androgen suppression with partial differential equations. Int. J. Bifurcat. Chaos 18, 3789-3797 (2008).
  5. Y. Tao, Q. Guo, and K. Aihara, A model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression. Math. Models Meth. Appl. Sci. 19, 2177-2201 (2009).
  6. Y. Tao, Q. Guo, and K. Aihara, A mathematical model of prostate tumor growth under hormone therapy with mutation inhibitor. J. Nonlinear Sci. 20, 219-240 (2010).
  7. H. V. Jain, S. K. Clinton, A Bhinder, and A. Friedman, Mathematical modeling of prostate cancer progression in response to androgen ablation therapy. Proc. Natl Acad. Sci. USA 108, 19701-19706 (2011).
  8. T. Portz, Y. Kuang, and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy. AIP Adv. 2, 011002 (2012).
  9. Y. Hirata, K. Akakura, C. S. Higano, N. Bruchovsky, and K. Aihara, Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression. J. Mol. Cell. Biol. 4, 127-132 (2012).
  10. H. Kuramae, Y. Hirata, N. Bruchovsky, K. Aihara, and H. Suzuki, Nonlinear sysmtes identification by combining regression with bootstrap resampling. Chaos 21, 043121 (2011).
  11. T. Suzuki, N. Bruchovsky, and K. Aihara, Piecewise affine systems modelling for optimizing hormone therapy of prostate cancer. Phil. Trans. R. Soc. A, 368, 5045-5059 (2010).
  12. G. Tanaka, Y. Hirata, S. L. Goldenberg, N. Bruchovsky, and K. Aihara, Mathematical modelling of prostate cancer growth and its application to hormone therapy. Phil. Trans. R. Soc. A 368, 5029-5044 (2010); correction, Phil. Trans. R. Soc. A 369, 496 (2011).
  13. K. Aihara(Ed.), Theory of Hybrid Dynamical Systems and its Applications to Biological and Medical Systems. A Theme Issue of Phil. Trans. R. Soc. A, 368, 1930 (2010).
  14. Y. Hirata, M. di Bernardo, N. Bruchovsky, and K. Aihara, Hybrid optimal scheduling for intermittent androgen suppression of prostate cancer. Chaos 20, 045125 (2010).

 

Written by:
Kazuyuki Aihara as part of Beyond the Abstract on UroToday.com. This initiative offers a method of publishing for the professional urology community. Authors are given an opportunity to expand on the circumstances, limitations etc... of their research by referencing the published abstract.

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

 

Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression - Abstract

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