2013, 10(3): 591-608. doi: 10.3934/mbe.2013.10.591

A partial differential equation model of metastasized prostatic cancer

1. 

Department of Mathematics, Ohio State University, Columbus, OH 43210

2. 

Department of Mathematics, Florida State University, Tallahassee, FL 32308, United States

Received  October 2012 Revised  December 2012 Published  April 2013

Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.
Citation: Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591
References:
[1]

D. B. Agus, C. Cordon-Cardo, W. Fox, M. Drobnjak, A. Koff, D. W. Golde and H. I. Scher, Prostate cancer cell cycle regulators: Response to androgen withdrawal and development of androgen independence,, J. Natl. Cancer. Inst., 91 (1999), 1869.  doi: 10.1093/jnci/91.21.1869.  Google Scholar

[2]

G. L. Andriole, E. D. Crawford, R. L. Grubb III, S. S. Buys, D. Chia, T. R. Church, M. N. Fouad, E. P. Gelmann, P. A. Kvale, D. J. Reding, J. L. Weissfeld, L. A. Yokochi, B. O'Brien, J. D. Clapp, J. M. Rathmell, T. L. Riley, R. B. Hayes, B. S. Kramer, G. Izmirlian, A. B. Miller, P. F. Pinsky, P. C. Prorok, J. K. Gohagan and C. D. Berg, Mortality results from a randomized prostate-cancer screening trial,, N. Engl. J. Med., 360 (2009), 1310.  doi: 10.1056/NEJMoa0810696.  Google Scholar

[3]

R. R. Berges, J. Vukanovic, J. I. Epstein, M. CarMichel, L. Cisek, D. E. Johnson, R. W. Veltri, P. C. Walsh and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer,, Clin. Cancer Res., 1 (1995), 473.   Google Scholar

[4]

G. Birkenmeier, F. Struck and R. Gebhardt, Clearance mechanism of prostate specific antigen and its complexes with alpha2-macroglobulin and alpha1-antichymotrypsin,, J. Urol., 162 (1999), 897.   Google Scholar

[5]

X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic systems: An application to tumor growth,, SIAM J. Math. Anal., 35 (2003), 974.  doi: 10.1137/S0036141002418388.  Google Scholar

[6]

M. L. Cher, G. S. Bova, D. H. Moore, E. J. Small, P. R. Carroll, S. S. Pin, J. I. Epstein, W. B. Isaacs and R. H. Jensen, Genetic alterations in untreated metastases and androgen-independent prostate cancer detected by comparative genomic hybridization and allelotyping,, Cancer Res., 56 (1996), 3091.   Google Scholar

[7]

M. W. Dunn and M. W. Kazer, Prostate cancer overview,, Semin. Oncol. Nurs., 27 (2011), 241.  doi: 10.1016/j.soncn.2011.07.002.  Google Scholar

[8]

S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model,, Biol. Direct, 5 (2010), 24.  doi: 10.1186/1745-6150-5-24.  Google Scholar

[9]

B. J. Feldman and D. Feldman, The development of androgen-independent prostate cancer,, Nat. Rev. Cancer, 1 (2001), 34.  doi: 10.1038/35094009.  Google Scholar

[10]

A. Friedman, A multiscale tumor model,, Interface. Free Bound., 10 (2008), 245.  doi: 10.4171/IFB/188.  Google Scholar

[11]

D. Gillatt, Antiandrogen treatments in locally advanced prostate cancer: are they all the same?,, J. Cancer Res. Clin. Oncol., 132 (2006).  doi: 10.1007/s00432-006-0133-5.  Google Scholar

[12]

R. F. Gittes, Carcinoma of the prostate,, N. Engl. J. Med., 324 (1991), 236.  doi: 10.1056/NEJM199101243240406.  Google Scholar

[13]

M. Gleave, S. L. Goldenberg, N. Bruchovsky and P. Rennie, Intermittent androgen suppression for prostate cancer: Rationale and clinical experience,, Prostate Cancer Prostatic Dis., 1 (1998), 289.  doi: 10.3109/9780203091432-43.  Google Scholar

[14]

S. L. Goldenberg, N. Bruchovsky, M. E. Gleave, L. D. Sullivan and K. Akakura, Intermittent androgen suppression in the treatment of prostate cancer: A preliminary report,, Urology, 45 (1995), 839.   Google Scholar

[15]

M. A. Haider, T. H. van der Kwast, J. Tanguay, A. J. Evans, A. Hashmi, G. Lockwood and J. Trachtenberg, Combined T2-weighted and diffusion-weighted MRI for localization of prostate cancer,, AJR Am. J. Roentgenol., 189 (2007), 323.  doi: 10.2214/AJR.07.2211.  Google Scholar

[16]

Y. Hirata, N. Bruchovsky and K. Aihara, Androgen receptor in prostate cancer,, Endocr. Rev., 25 (2004), 276.   Google Scholar

[17]

C. A. Heinlein and C. Chang, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer,, J. Theor. Biol., 264 (2010), 517.  doi: 10.1016/j.jtbi.2010.02.027.  Google Scholar

[18]

A. M. Ideta, G. Tanaka, T. Takeuchi and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer,, J. Nonlinear Sci., 18 (2008), 593.  doi: 10.1007/s00332-008-9031-0.  Google Scholar

[19]

T. L. Jackson, A mathematical model of prostate tumor growth and androgen-independent relapse,, Discrete Cont. Dyn.-B, 4 (2004), 187.  doi: 10.3934/dcdsb.2004.4.187.  Google Scholar

[20]

T. L. Jackson, A mathematical investigation of the multiple pathways to recurrent prostate cancer: Comparison with experimental data,, Neoplasia, 6 (2004), 697.  doi: 10.1593/neo.04259.  Google Scholar

[21]

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 (2011), 19701.  doi: 10.1073/pnas.1115750108.  Google Scholar

[22]

H. V. Jain and A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy,, Discrete Cont. Dyn.-B, ().  doi: 10.3934/dcdsb.2013.18.945.  Google Scholar

[23]

M. Marcelli, W. D. Tilley, C. M. Wilson, J. E. Griffin, J. D. Wilson and M. J. McPhaul, Definition of the human androgen receptor gene structure permits the identification of mutations that cause androgen resistance: premature termination of the receptor protein at amino acid residue 588 causes complete androgen resistance,, Mol. Endocrinol., 4 (1990), 1105.  doi: 10.1210/mend-4-8-1105.  Google Scholar

[24]

H. C. Monro and E. A Gaffney, Modelling chemotherapy resistance in palliation and failed cure,, J. Theor. Biol., 257 (2009), 292.  doi: 10.1016/j.jtbi.2008.12.006.  Google Scholar

[25]

W. D. Nes, Y. O. Lukyanenko, Z. H. Jia, S. Quideau, W. N. Howald, T. K. Pratum, R. R. West and J. C. Hutson, Identification of the lipophilic factor produced by macrophages that stimulates steroidogenesis,, Endocrinology, 141 (2000), 953.  doi: 10.1210/en.141.3.953.  Google Scholar

[26]

T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy,, AIP Advances, 2 (2012).  doi: 10.1063/1.3697848.  Google Scholar

[27]

L. K. Potter, M. G. Zager and H. A. Barton, Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats,, Am. J. Physiol. Endocrinol. Metab., 291 (2006).  doi: 10.1152/ajpendo.00545.2005.  Google Scholar

[28]

A. S. Wright, L. N. Thomas, R. C. Douglas, C. B. Lazier and R. S. Rittmaster, Relative potency of testosterone and dihydrotestosterone in preventing atrophy and apoptosis in the prostate of the castrated rat,, J. Clin. Invest., 98 (1996), 255.  doi: 10.1172/JCI119074.  Google Scholar

[29]

M. Yang, P. Jiang, F-X. Sun, S. Hasegawa, E. Baranov, T. Chishima, H. Shimada, A. R. Moossa and R. M. Hoffman, A fluorescent orthotopic bone metastasis model of human prostate cancer,, Cancer Res., 59 (1999), 781.   Google Scholar

[30]

C. Y-F. Young, B. T. Montgomery, P. E. Andrews, S. Qiu, D. L. Bilhartz and D. J. Tindall, Hormonal regulation of prostate-specific antigen messenger RNA in human prostatic adenocarcinoma cell line LNCaP,, Cancer Res., 51 (1991), 3748.   Google Scholar

[31]

K. Yörükoglu, S. Aktas, C. Güler, M. Sade and Z. Kirkali, Volume-weighted mean nuclear volume in renal cell carcinoma,, Urology, 52 (1998), 44.   Google Scholar

[32]

H. Y. E. Zhau, S. Chang, B. Chen, Y. Wang, H. Zhang, C. Kao, Q. A. Sang, S. J. Pathak and L. W. K. Chung, Androgen-repressed phenotype in human prostate cancer,, Proc. Natl. Acad. Sci. USA, 93 (1996), 15152.  doi: 10.1073/pnas.93.26.15152.  Google Scholar

show all references

References:
[1]

D. B. Agus, C. Cordon-Cardo, W. Fox, M. Drobnjak, A. Koff, D. W. Golde and H. I. Scher, Prostate cancer cell cycle regulators: Response to androgen withdrawal and development of androgen independence,, J. Natl. Cancer. Inst., 91 (1999), 1869.  doi: 10.1093/jnci/91.21.1869.  Google Scholar

[2]

G. L. Andriole, E. D. Crawford, R. L. Grubb III, S. S. Buys, D. Chia, T. R. Church, M. N. Fouad, E. P. Gelmann, P. A. Kvale, D. J. Reding, J. L. Weissfeld, L. A. Yokochi, B. O'Brien, J. D. Clapp, J. M. Rathmell, T. L. Riley, R. B. Hayes, B. S. Kramer, G. Izmirlian, A. B. Miller, P. F. Pinsky, P. C. Prorok, J. K. Gohagan and C. D. Berg, Mortality results from a randomized prostate-cancer screening trial,, N. Engl. J. Med., 360 (2009), 1310.  doi: 10.1056/NEJMoa0810696.  Google Scholar

[3]

R. R. Berges, J. Vukanovic, J. I. Epstein, M. CarMichel, L. Cisek, D. E. Johnson, R. W. Veltri, P. C. Walsh and J. T. Isaacs, Implication of cell kinetic changes during the progression of human prostatic cancer,, Clin. Cancer Res., 1 (1995), 473.   Google Scholar

[4]

G. Birkenmeier, F. Struck and R. Gebhardt, Clearance mechanism of prostate specific antigen and its complexes with alpha2-macroglobulin and alpha1-antichymotrypsin,, J. Urol., 162 (1999), 897.   Google Scholar

[5]

X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic systems: An application to tumor growth,, SIAM J. Math. Anal., 35 (2003), 974.  doi: 10.1137/S0036141002418388.  Google Scholar

[6]

M. L. Cher, G. S. Bova, D. H. Moore, E. J. Small, P. R. Carroll, S. S. Pin, J. I. Epstein, W. B. Isaacs and R. H. Jensen, Genetic alterations in untreated metastases and androgen-independent prostate cancer detected by comparative genomic hybridization and allelotyping,, Cancer Res., 56 (1996), 3091.   Google Scholar

[7]

M. W. Dunn and M. W. Kazer, Prostate cancer overview,, Semin. Oncol. Nurs., 27 (2011), 241.  doi: 10.1016/j.soncn.2011.07.002.  Google Scholar

[8]

S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model,, Biol. Direct, 5 (2010), 24.  doi: 10.1186/1745-6150-5-24.  Google Scholar

[9]

B. J. Feldman and D. Feldman, The development of androgen-independent prostate cancer,, Nat. Rev. Cancer, 1 (2001), 34.  doi: 10.1038/35094009.  Google Scholar

[10]

A. Friedman, A multiscale tumor model,, Interface. Free Bound., 10 (2008), 245.  doi: 10.4171/IFB/188.  Google Scholar

[11]

D. Gillatt, Antiandrogen treatments in locally advanced prostate cancer: are they all the same?,, J. Cancer Res. Clin. Oncol., 132 (2006).  doi: 10.1007/s00432-006-0133-5.  Google Scholar

[12]

R. F. Gittes, Carcinoma of the prostate,, N. Engl. J. Med., 324 (1991), 236.  doi: 10.1056/NEJM199101243240406.  Google Scholar

[13]

M. Gleave, S. L. Goldenberg, N. Bruchovsky and P. Rennie, Intermittent androgen suppression for prostate cancer: Rationale and clinical experience,, Prostate Cancer Prostatic Dis., 1 (1998), 289.  doi: 10.3109/9780203091432-43.  Google Scholar

[14]

S. L. Goldenberg, N. Bruchovsky, M. E. Gleave, L. D. Sullivan and K. Akakura, Intermittent androgen suppression in the treatment of prostate cancer: A preliminary report,, Urology, 45 (1995), 839.   Google Scholar

[15]

M. A. Haider, T. H. van der Kwast, J. Tanguay, A. J. Evans, A. Hashmi, G. Lockwood and J. Trachtenberg, Combined T2-weighted and diffusion-weighted MRI for localization of prostate cancer,, AJR Am. J. Roentgenol., 189 (2007), 323.  doi: 10.2214/AJR.07.2211.  Google Scholar

[16]

Y. Hirata, N. Bruchovsky and K. Aihara, Androgen receptor in prostate cancer,, Endocr. Rev., 25 (2004), 276.   Google Scholar

[17]

C. A. Heinlein and C. Chang, Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer,, J. Theor. Biol., 264 (2010), 517.  doi: 10.1016/j.jtbi.2010.02.027.  Google Scholar

[18]

A. M. Ideta, G. Tanaka, T. Takeuchi and K. Aihara, A mathematical model of intermittent androgen suppression for prostate cancer,, J. Nonlinear Sci., 18 (2008), 593.  doi: 10.1007/s00332-008-9031-0.  Google Scholar

[19]

T. L. Jackson, A mathematical model of prostate tumor growth and androgen-independent relapse,, Discrete Cont. Dyn.-B, 4 (2004), 187.  doi: 10.3934/dcdsb.2004.4.187.  Google Scholar

[20]

T. L. Jackson, A mathematical investigation of the multiple pathways to recurrent prostate cancer: Comparison with experimental data,, Neoplasia, 6 (2004), 697.  doi: 10.1593/neo.04259.  Google Scholar

[21]

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 (2011), 19701.  doi: 10.1073/pnas.1115750108.  Google Scholar

[22]

H. V. Jain and A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy,, Discrete Cont. Dyn.-B, ().  doi: 10.3934/dcdsb.2013.18.945.  Google Scholar

[23]

M. Marcelli, W. D. Tilley, C. M. Wilson, J. E. Griffin, J. D. Wilson and M. J. McPhaul, Definition of the human androgen receptor gene structure permits the identification of mutations that cause androgen resistance: premature termination of the receptor protein at amino acid residue 588 causes complete androgen resistance,, Mol. Endocrinol., 4 (1990), 1105.  doi: 10.1210/mend-4-8-1105.  Google Scholar

[24]

H. C. Monro and E. A Gaffney, Modelling chemotherapy resistance in palliation and failed cure,, J. Theor. Biol., 257 (2009), 292.  doi: 10.1016/j.jtbi.2008.12.006.  Google Scholar

[25]

W. D. Nes, Y. O. Lukyanenko, Z. H. Jia, S. Quideau, W. N. Howald, T. K. Pratum, R. R. West and J. C. Hutson, Identification of the lipophilic factor produced by macrophages that stimulates steroidogenesis,, Endocrinology, 141 (2000), 953.  doi: 10.1210/en.141.3.953.  Google Scholar

[26]

T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy,, AIP Advances, 2 (2012).  doi: 10.1063/1.3697848.  Google Scholar

[27]

L. K. Potter, M. G. Zager and H. A. Barton, Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats,, Am. J. Physiol. Endocrinol. Metab., 291 (2006).  doi: 10.1152/ajpendo.00545.2005.  Google Scholar

[28]

A. S. Wright, L. N. Thomas, R. C. Douglas, C. B. Lazier and R. S. Rittmaster, Relative potency of testosterone and dihydrotestosterone in preventing atrophy and apoptosis in the prostate of the castrated rat,, J. Clin. Invest., 98 (1996), 255.  doi: 10.1172/JCI119074.  Google Scholar

[29]

M. Yang, P. Jiang, F-X. Sun, S. Hasegawa, E. Baranov, T. Chishima, H. Shimada, A. R. Moossa and R. M. Hoffman, A fluorescent orthotopic bone metastasis model of human prostate cancer,, Cancer Res., 59 (1999), 781.   Google Scholar

[30]

C. Y-F. Young, B. T. Montgomery, P. E. Andrews, S. Qiu, D. L. Bilhartz and D. J. Tindall, Hormonal regulation of prostate-specific antigen messenger RNA in human prostatic adenocarcinoma cell line LNCaP,, Cancer Res., 51 (1991), 3748.   Google Scholar

[31]

K. Yörükoglu, S. Aktas, C. Güler, M. Sade and Z. Kirkali, Volume-weighted mean nuclear volume in renal cell carcinoma,, Urology, 52 (1998), 44.   Google Scholar

[32]

H. Y. E. Zhau, S. Chang, B. Chen, Y. Wang, H. Zhang, C. Kao, Q. A. Sang, S. J. Pathak and L. W. K. Chung, Androgen-repressed phenotype in human prostate cancer,, Proc. Natl. Acad. Sci. USA, 93 (1996), 15152.  doi: 10.1073/pnas.93.26.15152.  Google Scholar

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