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.
Similar content being viewed by others
References
Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12, 737–754 (2002)
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)
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)
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)
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)
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)
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)
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)
Chung, L.M.: The role of stromal-epithelial interaction in normal and malignant growth. Cancer Surv. 23, 33–42 (1995)
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)
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)
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)
Greenspan, H.: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–235 (1976)
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)
Holland, J.F., Frei, E.: Cancer Medicine, 5th edn. Harcourt Asia Pte Ltd., Singapore (2001)
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)
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)
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)
Jackson, T.L.: Vascular tumor growth and treatment: consequence of polyclonality, competition and dynamic vascular support. J. Math. Biol. 44, 201–226 (2002)
Jackson, T.L.: A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete Contin. Dyn. Syst. B 4, 187–201 (2004a)
Jackson, T.L.: A mathematical investigation of multiple pathways to recurrent prostate cancer: comparison with experimental data. Neoplasia 6, 697–704 (2004b)
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)
Kamradt, J.M., Pienta, K.J.: Novel molecular targets for prostate cancer therapy. Semin. Oncol. 26, 234–243 (1999)
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)
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)
Macri, E., Loda, M.: Role of p27 in prostate carcinogenesis. Cancer Metastasis Rev. 17, 337–344 (1998)
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)
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)
Simon, C., Everitt, H., Birtwistle, J., Stevenson, B.: Oxford Handbook of General Practice. Oxford University Press, Oxford (2002)
Tao, Y.: A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids. J. Differ. Equ. 247, 49–68 (2009)
Tao, Y., Chen, M.: An elliptic–hyperbolic free boundary problem modelling cancer therapy. Nonlinearity 19, 419–440 (2006)
Tao, Y., Guo, Q.: A free boundary problem modelling cancer radiovirotherapy. Math. Models Methods Appl. Sci. 17, 1241–1259 (2007)
Tao, Y., Yoshida, N., Guo, Q.: Nonlinear analysis of a model of vascular tumour growth and treatment. Nonlinearity 17, 867–895 (2004)
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
Ward, J.P., King, J.R.: Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures. Math. Biosci. 181, 177–207 (2003)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P.K. Maini.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-009-9056-z