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A Mathematical Model of Prostate Tumor Growth Under Hormone Therapy with Mutation Inhibitor

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Abstract

This paper extends Jackson’s model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical mutation inhibitors. The new model not only considers the mutation by which androgen-dependent (AD) tumor cells mutate into androgen-independent (AI) ones but also introduces inhibition which is assumed to change the mutation rate. The tumor consists of two types of cells (AD and AI) whose proliferation and apoptosis rates are functions of androgen concentration. The mathematical model represents a free-boundary problem for a nonlinear system of parabolic equations, which describe the evolution of the populations of the above two types of tumor cells. The tumor surface is a free boundary, whose velocity is equal to the cell’s velocity there. Global existence and uniqueness of solutions of this model is proved. Furthermore, explicit formulae of tumor volume at any time t are found in androgen-deprived environment under the assumption of radial symmetry, and therefore the dynamics of tumor growth under androgen-deprived therapy could be predicted by these formulae. Qualitative analysis and numerical simulation show that controlling the mutation may improve the effect of hormone therapy or delay a tumor relapse.

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References

  • Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12, 737–754 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Bellomo, N., Delitala, M.: From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells. Phys. Life Rev. 5, 183–206 (2008)

    Article  Google Scholar 

  • Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Bresch, D., Colin, T., Grenier, E., Ribba, B., Saut, O.: A viscoelastic model for avascular tumor growth, inria-00267292, Version 3, 1 Sep 2008. Preprint

  • Bruchovsky, N., Klotz, L., Crook, J., Malone, S., Ludgate, C., Morris, W.J., Gleave, M.E., Goldenberg, S.L.: 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 

  • Bruchovsky, N., Klotz, L., Crook, J., Larry, S., Goldenberg, S.L.: Locally advanced prostate cancer—biochemical results from a prospective phase II study of intermittent androgen suppression for men with evidence of psa relapse after radiotherapy. Cancer 109, 858–867 (2007)

    Article  Google Scholar 

  • Byrne, H.M., Chaplain, M.A.J.: Modelling the role of cell-cell adhesion in the growth and development of carcinoma. Math. Comput. Model. 12, 1–17 (1996)

    Article  Google Scholar 

  • Byrne, H.M., Alarcón, T.A., Murphy, J., Maini, P.K.: Modelling the response of vascular tumours to chemotherapy: a multiscale approach. Math. Models Methods Appl. Sci. 16, 1219–1241 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Carducci, M.A., Nelson, J.B., Chan-Tack, K.M., et al.: Phenylbutyrate induces apoptosis in human prostate cancer and is more potent than phenylacetate. Clin. Cancer Res. 2, 379–387 (1996)

    Google Scholar 

  • Chung, L.M.: The role of stromal-epithelial interaction in normal and malignant growth. Cancer Surv. 23, 33–42 (1995)

    Google Scholar 

  • De Angelis, E., Jabin, P.E.: Qualitative analysis of a mean field model of tumor-immune system competition. Math. Models Methods Appl. Sci. 13, 187–206 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Ellis, W.J., Vessela, R.L., Buhler, K.R., Bladou, F., True, L.D., Bigler, S.A., Curtis, D., Lange, P.H.: Characterization of a novel androgen-sensitive, prostate-specific antigen-producing prostatic carcinoma xenograft: LuCap 231. Clin. Cancer Res. 2, 1039–1048 (1996)

    Google Scholar 

  • Ferlay, J., et al.: GLOBOCAN 2002: Cancer incidense, mortality and prevalence worldwide. IARC CnacerBase No. 5, Version 2.0. IARC Press, Lyon (2002)

  • Friedman, A.: Mathematical analysis and challenges arising from models of tumour growth. Math. Models Methods Appl. Sci. 17, 1751–1772 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Greenspan, H.: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–235 (1976)

    Article  MathSciNet  Google Scholar 

  • Guo, Q., Tao, Y., Aihara, K.: Mathematical modelling of prostate tumor growth under intermittent androgen suppression with partial differential equations. Int. J. Bifur. Chaos 18, 3789–3797 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Holland, J.F., Frei, E.: Cancer Medicine, 5th edn. Harcourt Asia Pte Ltd., Singapore (2001)

    Google Scholar 

  • Hsing, A., Tsao, L., Devesa, S.: International trends and patterns of prostate cancer incidence and mortality. Int. J. Cancer (Predat. Oncol.) 85, 60–67 (2000)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Isaacs, J.T., Coffey, D.S.: Adaptation versus selection as mechanism response for the relapse of prostatic cancer to androgen therapy as studied in the dunning R-3327-H adenocarcinoma. Cancer Res. 41, 5070–5075 (1981)

    Google Scholar 

  • Jackson, T.L.: Vascular tumor growth and treatment: consequence of polyclonality, competition and dynamic vascular support. J. Math. Biol. 44, 201–226 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Jackson, T.L., Byrne, H.M.: A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math. Biosci. 164, 17–38 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Kamradt, J.M., Pienta, K.J.: Novel molecular targets for prostate cancer therapy. Semin. Oncol. 26, 234–243 (1999)

    Google Scholar 

  • Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Am. Math. Soc. Transl., vol. 23. Am. Math. Soc., Providence (1968)

    Google Scholar 

  • Liu, A.Y., Corey, E., Bladou, F., Lange, P.H., Vessella, R.L.: Prostatic cell lineage markers: emergence of BCL2+ cells of human prostate cancer xenograft LuCaP 23 following castration. Int. J. Cancer 65, 85–89 (1996)

    Article  Google Scholar 

  • Macri, E., Loda, M.: Role of p27 in prostate carcinogenesis. Cancer Metastasis Rev. 17, 337–344 (1998)

    Article  Google Scholar 

  • Ribba, B., Saut, O., Colin, T., Bresch, D., Grenier, E., Boissel, J.P.: A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J. Theor. Biol. 243, 532–541 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Simon, C., Everitt, H., Birtwistle, J., Stevenson, B.: Oxford Handbook of General Practice. Oxford University Press, Oxford (2002)

    Google Scholar 

  • Tao, Y.: A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids. J. Differ. Equ. 247, 49–68 (2009)

    Article  MATH  Google Scholar 

  • Tao, Y., Chen, M.: An elliptic–hyperbolic free boundary problem modelling cancer therapy. Nonlinearity 19, 419–440 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Tao, Y., Guo, Q.: A free boundary problem modelling cancer radiovirotherapy. Math. Models Methods Appl. Sci. 17, 1241–1259 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Tao, Y., Yoshida, N., Guo, Q.: Nonlinear analysis of a model of vascular tumour growth and treatment. Nonlinearity 17, 867–895 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • 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. (2009). doi:10.1142/S021820250900408X

    Google Scholar 

  • Ward, J.P., King, J.R.: Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures. Math. Biosci. 181, 177–207 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Ware, J.L.: Growth factors and their receptors as determinants in the proliferation and metastasis of human prostate cancer. Cancer Metastasis Rev. 12, 287–301 (1993)

    Article  Google Scholar 

Download references

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Correspondence to Youshan Tao.

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Communicated by P.K. Maini.

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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). https://doi.org/10.1007/s00332-009-9056-z

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  • DOI: https://doi.org/10.1007/s00332-009-9056-z

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