Evolutionary of Prostate Cancer Stem Cells in Prostate Duct
A systems approach model for prostate cancer in prostate duct, as a sub-system of the organism is developed. It is accomplished in two steps. First this research work starts with a nonlinear system of coupled Fokker-Plank equations which models continuous process of the system like motion of cells. Then extended to PDEs that include discontinuous processes like cell mutations, proliferation and deaths. The discontinuous processes is modeled by using intensity poisson processes. The model incorporates the features of the prostate duct. The system of PDEs spatial coordinate is along the proximal distal axis. Its parameters depend on features of the prostate duct. The movement of cells is biased towards distal region and mutations of prostate cancer cells is localized in the proximal region. Numerical solutions of the full system of equations are provided, and are exhibit traveling wave fronts phenomena. This motivates the use of the standard transformation to derive a canonically related system of ODEs for traveling wave solutions. The results obtained show persistence of prostate cancer by showing that the non-negative cone for the traveling wave system is time invariant. The traveling waves have a unique global attractor is proved also. Biologically, the global attractor verifies that evolution of prostate cancer stem cells exhibit the avascular tumor growth. These numerical solutions show that altering prostate stem cell movement or mutation of prostate cancer cells lead to avascular tumor. Conclusion with comments on clinical implications of the model is discussed.
[1] M. Abercrombie, Contact inhibition in tissue. In vitro 6, 128-140, 1970.
[2] R. Agami, All Roads Lead toIKKÐö, Cell 129, 1043-1045, 2007.
[3] Anderson Cancer Center, University of Texas, 2008. Drug therapy to
bolster immune system cells found effective toward childhood cancer.
Article on work of Lee, D. (Ph.D.), Science Daily, May, 2008.
[4] M. Bertsch, M. E. Gurtin, D. Hilhorst, L. S. Peletier, On interacting
populations that disperse to avoid crowding: perservation of segregation,
1985
[5] J. Canosa, On a nonlinear diffusion equation describing population
growth. IBM J. Res. Dev. 17, 307-313, 1973.
[6] L. G. Charles, C. X. Yilin, , N. P. Restifo, B. Roessler, M. G. Sanda,
Antitumor efficacy of tumor-antigen-encoding recombinant poxvirus immunization
in Dunning rat prostate cancer: implications for clinical
genetic vaccine development. World Journal of Urology, April, 18 (2),
136-142, 2000.
[7] T. C. Collins, N. J. Maitland, Prostate cancer stem cells. European J. of
Cancer 42, 1213-1218, 2006.
[8] E. Conway, D. Huff, J. Smoller, Large time behavior of solutions of
systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math.,
35 (1), July, 1-16, 1978.
[9] F. F. Costa K. Le Blanc, B. Brodin, Cancer/Testis antigens, stem cells
and cancer. Stem Cells 12(2), 398-404, 2006.
[10] D. Dingli, F. Michor, Successful therapy must eradicate cancer stem
cells. Stem Cells 24, 2603-2610, 2006.
[11] G. P. Dunn, et al., The three Es of cancer immunoediting. Annu. Rev.
Immunol. 22, 329360, 2004.
[12] M. Fasso, R. Waitz, H. Yafei, R. Tae, N. M. Greenberg, N. Shastri,
L. Fong, J. P. Allison, SPAS-1 (stimulator of prostatic adenocarcinomaspecific
T cells)/SH3GLB2: A prostate tumor antigen identified by CTLA-
4 blockade. Proceedings of the National Academy of Sciences of the
United States of America (0027-8424). 105(9), 3509-3514, 2008.
[13] R. A. Fisher, The wave of advance of advantageneous genes. Ann.
Eugenics 7, 353-369, 1937.
[14] R. Ganguly, and I. K. Puri, Mathematical model for the cancer stem cell
hypothesis. Cell Proliferation 39, 3-14, 2006.
[15] A. L. Garner, Y. Y. Lau, S. W. Jordan, M. D. Uhler, R. M. Gilgenbach,
Implications of a simple mathematical model to cancer cell population
dynamics. Cell Prolif., 39, 15-28, 2006.
[16] S. Gakkhar, S. Brahampal, R. K. Naji, Dynamical behavior of two
predators competing over a single prey. Biosystems, 90, 808-817, 2007.
[17] M. C. Garnick, M. C. Fair, Combating Prostate Cancer. Scientific
American 279(6), 74-83, 1998.
[18] M. C. Gong., J. B. Latouche, A. Krause, W. D. W. Heston, N. H.
Bander, M. Sadelain, Cancer patient T cells genetically targeted to
prostate-specific membrane antigen specifically lyse prostate cancer cells
and release cytokines in response to prostate-specific membrane antigen.
Neoplasia, 1(2), June. 123-127, 1999.
[19] M. Greaves, M., 2000. Cancer: The Evolutionary Legancy, Oxford
University Press.
[20] L. Han, A. Publiese, Epidemics in two competing species. Nonlinear
Analysis: Real World Applications, 10(2), 723-744, 2009.
[21] A. L. Harzstark, E. J. Small, Immunotherapy for prostate cancer using
antigen-loaded antigen-presenting cells: APC8015 (provenge). Expert
Opinion on Biological Therapy, August, 7(8), 1275-1280, 2007.
[22] M. Jakobisiak, et al. Natural mechanisms protecting against cancer.
Immunol. Lett. 90, 103122, 2003.
[23] B. T. Kawasaki, and W. L. Farrar Cancer stem cells, CD200 and
immunoevasion, Trends in Immunology, 29(10), Issue 10, 464-468, 2008.
[24] E. F. Keller, L. A. Segel, Model for chemotaxis. J. Theor. Biol. 30,
225-234, 1971.
[25] Y. Kiniwa, Y. Miyahara, H. Y. Wang, W. Peng, G. Peng, T.M. Wheeler,
T. C. Thompson, J. Lloyd, CD8+ Foxp3+ T cells mediate immunosuppression
in prostate cancer. Clinical Cancer Research, December 1(13),
6947-6958, 2007.
[26] H. Kitano, Cancer therapy as a robust system: implications for anticancer
therapy, Nature Reviews - Cancer, 4, 227-235, 2004.
[27] J. L. Lao, and D. T. Kamei, Investigation of cellular movement in the
prostate epithelium using an Agent-based model, J. Theor. biol. 250, 642-
654, 2008.
[28] C. Lee, J. A. Sensibar, S. M. Dudek, R. A. Hiipakka, S. T. Liao, Prostatic
ductal system in rats: regional variation in morphological and functional
activities. Biol. Reprod. 43, 10791086, 1990.
[29] I. C. Mackenzie, Stem cell properties and epithelial malignancies European
Journal of Cancer. 42(9), 1204-1212, 2006.
[30] N. J. Maitland, A. T. Collins, prostate cancer stem cells: new therapeutic
targets? European Journal of Cancer Supplements, 5(4), 2007.
[31] J. Michalowski, Common molecule notifies immune system of prostate
cancer. www.eurekalert.org, Publication release, January 10, 2008.
[32] F. Michor, M. A. Nowak, S. A. Frank, Y. Iwasa, Stochastic elimination
of cancer cells. Proc. R. Soc. Lond. B 270, 2017-2024, 2003.
[33] S. J. Morrison, A. C. Spradling, Stem Cells and Niches: Mechanisms
that promote stem cell maintenance throughout life, Cell 132, 598-611,
2008.
[34] D. Mukherjee, Uniform persistence in a generalized prey-predator
system with parasitic infection. Biosystems, 47, 149-155, 1998.
[35] J. A. Nemeth, C. Lee, Prostatic ductal system in rats: regional variation
in stromal organization. Prostate 28, 124128, 1996.
[36] P. C. Nowell, The clonal expansion of tumor cell populations. Science
194, 23-28, 1996.
[37] B. I. Rini., V. Weinberg, L. Fong, S. Conry, R. M. Hershberg, E. J. Small,
Combination immunotherapy with prostatic acid phosphatase pulsed
antigen-presenting cells (provenge) plus bevacizumab in patients with
serologic progression of prostate cancer after definitive local therapy.
Cancer, July, 107(1), 67-74, 2006.
[38] M. Rouleau, J. Leger, M. Tenniswood, M., 1990. Ductal heterogeneity
of cytokeratins, gene expression, and cell death in the rat ventral prostate.
Mol. Endocrinol. 4, 20032013, 1990.
[39] Savage, P., Vosseller, K., Kang, C., Larimore, K., Riedel, E., Wojnoonski,
K., Jungbluth, A.A., Allison, J.P., 2008. Recognition of a ubiquitous
self antigen by prostate cancer infiltrating CD8+ T lymphocytes. Science,
January 319 (5860), 215- 220.
[40] J. A. Schalken, G. Van Leenders, Cellular and molecular biology of
the prostate: stem cell biology, Urology, November 62(Supplement 5A),
11-20, 2003.
[41] Schreiber, H. and Rowley, D.A., Mutations in cancer cells can give rise
to tumor-specific antigens, but abnormal processing of normal molecules
in these cells can also elicit an immune response. Science 319 (5860),
164-165, 2008.
[42] J. A. Sherratt, Wave front propagation in a competition equation with a
new motility term modeling contact inhibition between cell populations.
Proc. R. Soc. Lond. A 456, 2365-2387, 2000.
[43] J. A. Sherratt, M. A. J. Chaplain, A new mathematical model for
avascular tumour growth. J. Math. Biol. 43, 291-312, 2001.
[44] T. Quinn, and Z. Sinkala, Dynamics of prostate cancer stem cells with
diffusion and organism response, Biosystems, 96(1), 69-79, 2009.
[45] T. L. Soon, A. K. Cheng, A numerical simulation of avascular tumour
growth. Anziam. J. 46, 902-917, 2005.
[46] Y. Sugimura, G. R. Cunha, A. A. Donjacour, Morphogenesis of ductal
networks in the mouse prostate. Biol. Reprod. 34, 961971, 1986.
[47] A. R. Uzgare, J. T. Isaacs, Prostate cancer: potential targets of antiproliferating
and apoptotic signaling pathways, International J. of Biochemistry
and Cell Biology, 37, 707-714, 2005.
[48] D. Wodarz, N. Komarov, Computational Biology of Cancer. World
Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
[49] G. Wang, B. Kovalenko, E. L. Wilson, and D. Moscatelli, Vascular
Density is Highest in the Proximal Region of the Mouse Prostate. Prostate.
67(9): 968975, 2007
[50] X. Zang, T. Houston, H. A. Al-Ahmadie, A. M. Serio, V. E. Reuter,
J. A. Eastham, P. T. Scardino, P. Sharma, J. P. Allison, B7-H3 and B7x
are highly expressed in human prostate cancer and associated with disease
spread and poor outcome. PNAS, Biological Sciences/Immunology,
December, 104(49), 19458-19463, 2007.
[1] M. Abercrombie, Contact inhibition in tissue. In vitro 6, 128-140, 1970.
[2] R. Agami, All Roads Lead toIKKÐö, Cell 129, 1043-1045, 2007.
[3] Anderson Cancer Center, University of Texas, 2008. Drug therapy to
bolster immune system cells found effective toward childhood cancer.
Article on work of Lee, D. (Ph.D.), Science Daily, May, 2008.
[4] M. Bertsch, M. E. Gurtin, D. Hilhorst, L. S. Peletier, On interacting
populations that disperse to avoid crowding: perservation of segregation,
1985
[5] J. Canosa, On a nonlinear diffusion equation describing population
growth. IBM J. Res. Dev. 17, 307-313, 1973.
[6] L. G. Charles, C. X. Yilin, , N. P. Restifo, B. Roessler, M. G. Sanda,
Antitumor efficacy of tumor-antigen-encoding recombinant poxvirus immunization
in Dunning rat prostate cancer: implications for clinical
genetic vaccine development. World Journal of Urology, April, 18 (2),
136-142, 2000.
[7] T. C. Collins, N. J. Maitland, Prostate cancer stem cells. European J. of
Cancer 42, 1213-1218, 2006.
[8] E. Conway, D. Huff, J. Smoller, Large time behavior of solutions of
systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math.,
35 (1), July, 1-16, 1978.
[9] F. F. Costa K. Le Blanc, B. Brodin, Cancer/Testis antigens, stem cells
and cancer. Stem Cells 12(2), 398-404, 2006.
[10] D. Dingli, F. Michor, Successful therapy must eradicate cancer stem
cells. Stem Cells 24, 2603-2610, 2006.
[11] G. P. Dunn, et al., The three Es of cancer immunoediting. Annu. Rev.
Immunol. 22, 329360, 2004.
[12] M. Fasso, R. Waitz, H. Yafei, R. Tae, N. M. Greenberg, N. Shastri,
L. Fong, J. P. Allison, SPAS-1 (stimulator of prostatic adenocarcinomaspecific
T cells)/SH3GLB2: A prostate tumor antigen identified by CTLA-
4 blockade. Proceedings of the National Academy of Sciences of the
United States of America (0027-8424). 105(9), 3509-3514, 2008.
[13] R. A. Fisher, The wave of advance of advantageneous genes. Ann.
Eugenics 7, 353-369, 1937.
[14] R. Ganguly, and I. K. Puri, Mathematical model for the cancer stem cell
hypothesis. Cell Proliferation 39, 3-14, 2006.
[15] A. L. Garner, Y. Y. Lau, S. W. Jordan, M. D. Uhler, R. M. Gilgenbach,
Implications of a simple mathematical model to cancer cell population
dynamics. Cell Prolif., 39, 15-28, 2006.
[16] S. Gakkhar, S. Brahampal, R. K. Naji, Dynamical behavior of two
predators competing over a single prey. Biosystems, 90, 808-817, 2007.
[17] M. C. Garnick, M. C. Fair, Combating Prostate Cancer. Scientific
American 279(6), 74-83, 1998.
[18] M. C. Gong., J. B. Latouche, A. Krause, W. D. W. Heston, N. H.
Bander, M. Sadelain, Cancer patient T cells genetically targeted to
prostate-specific membrane antigen specifically lyse prostate cancer cells
and release cytokines in response to prostate-specific membrane antigen.
Neoplasia, 1(2), June. 123-127, 1999.
[19] M. Greaves, M., 2000. Cancer: The Evolutionary Legancy, Oxford
University Press.
[20] L. Han, A. Publiese, Epidemics in two competing species. Nonlinear
Analysis: Real World Applications, 10(2), 723-744, 2009.
[21] A. L. Harzstark, E. J. Small, Immunotherapy for prostate cancer using
antigen-loaded antigen-presenting cells: APC8015 (provenge). Expert
Opinion on Biological Therapy, August, 7(8), 1275-1280, 2007.
[22] M. Jakobisiak, et al. Natural mechanisms protecting against cancer.
Immunol. Lett. 90, 103122, 2003.
[23] B. T. Kawasaki, and W. L. Farrar Cancer stem cells, CD200 and
immunoevasion, Trends in Immunology, 29(10), Issue 10, 464-468, 2008.
[24] E. F. Keller, L. A. Segel, Model for chemotaxis. J. Theor. Biol. 30,
225-234, 1971.
[25] Y. Kiniwa, Y. Miyahara, H. Y. Wang, W. Peng, G. Peng, T.M. Wheeler,
T. C. Thompson, J. Lloyd, CD8+ Foxp3+ T cells mediate immunosuppression
in prostate cancer. Clinical Cancer Research, December 1(13),
6947-6958, 2007.
[26] H. Kitano, Cancer therapy as a robust system: implications for anticancer
therapy, Nature Reviews - Cancer, 4, 227-235, 2004.
[27] J. L. Lao, and D. T. Kamei, Investigation of cellular movement in the
prostate epithelium using an Agent-based model, J. Theor. biol. 250, 642-
654, 2008.
[28] C. Lee, J. A. Sensibar, S. M. Dudek, R. A. Hiipakka, S. T. Liao, Prostatic
ductal system in rats: regional variation in morphological and functional
activities. Biol. Reprod. 43, 10791086, 1990.
[29] I. C. Mackenzie, Stem cell properties and epithelial malignancies European
Journal of Cancer. 42(9), 1204-1212, 2006.
[30] N. J. Maitland, A. T. Collins, prostate cancer stem cells: new therapeutic
targets? European Journal of Cancer Supplements, 5(4), 2007.
[31] J. Michalowski, Common molecule notifies immune system of prostate
cancer. www.eurekalert.org, Publication release, January 10, 2008.
[32] F. Michor, M. A. Nowak, S. A. Frank, Y. Iwasa, Stochastic elimination
of cancer cells. Proc. R. Soc. Lond. B 270, 2017-2024, 2003.
[33] S. J. Morrison, A. C. Spradling, Stem Cells and Niches: Mechanisms
that promote stem cell maintenance throughout life, Cell 132, 598-611,
2008.
[34] D. Mukherjee, Uniform persistence in a generalized prey-predator
system with parasitic infection. Biosystems, 47, 149-155, 1998.
[35] J. A. Nemeth, C. Lee, Prostatic ductal system in rats: regional variation
in stromal organization. Prostate 28, 124128, 1996.
[36] P. C. Nowell, The clonal expansion of tumor cell populations. Science
194, 23-28, 1996.
[37] B. I. Rini., V. Weinberg, L. Fong, S. Conry, R. M. Hershberg, E. J. Small,
Combination immunotherapy with prostatic acid phosphatase pulsed
antigen-presenting cells (provenge) plus bevacizumab in patients with
serologic progression of prostate cancer after definitive local therapy.
Cancer, July, 107(1), 67-74, 2006.
[38] M. Rouleau, J. Leger, M. Tenniswood, M., 1990. Ductal heterogeneity
of cytokeratins, gene expression, and cell death in the rat ventral prostate.
Mol. Endocrinol. 4, 20032013, 1990.
[39] Savage, P., Vosseller, K., Kang, C., Larimore, K., Riedel, E., Wojnoonski,
K., Jungbluth, A.A., Allison, J.P., 2008. Recognition of a ubiquitous
self antigen by prostate cancer infiltrating CD8+ T lymphocytes. Science,
January 319 (5860), 215- 220.
[40] J. A. Schalken, G. Van Leenders, Cellular and molecular biology of
the prostate: stem cell biology, Urology, November 62(Supplement 5A),
11-20, 2003.
[41] Schreiber, H. and Rowley, D.A., Mutations in cancer cells can give rise
to tumor-specific antigens, but abnormal processing of normal molecules
in these cells can also elicit an immune response. Science 319 (5860),
164-165, 2008.
[42] J. A. Sherratt, Wave front propagation in a competition equation with a
new motility term modeling contact inhibition between cell populations.
Proc. R. Soc. Lond. A 456, 2365-2387, 2000.
[43] J. A. Sherratt, M. A. J. Chaplain, A new mathematical model for
avascular tumour growth. J. Math. Biol. 43, 291-312, 2001.
[44] T. Quinn, and Z. Sinkala, Dynamics of prostate cancer stem cells with
diffusion and organism response, Biosystems, 96(1), 69-79, 2009.
[45] T. L. Soon, A. K. Cheng, A numerical simulation of avascular tumour
growth. Anziam. J. 46, 902-917, 2005.
[46] Y. Sugimura, G. R. Cunha, A. A. Donjacour, Morphogenesis of ductal
networks in the mouse prostate. Biol. Reprod. 34, 961971, 1986.
[47] A. R. Uzgare, J. T. Isaacs, Prostate cancer: potential targets of antiproliferating
and apoptotic signaling pathways, International J. of Biochemistry
and Cell Biology, 37, 707-714, 2005.
[48] D. Wodarz, N. Komarov, Computational Biology of Cancer. World
Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
[49] G. Wang, B. Kovalenko, E. L. Wilson, and D. Moscatelli, Vascular
Density is Highest in the Proximal Region of the Mouse Prostate. Prostate.
67(9): 968975, 2007
[50] X. Zang, T. Houston, H. A. Al-Ahmadie, A. M. Serio, V. E. Reuter,
J. A. Eastham, P. T. Scardino, P. Sharma, J. P. Allison, B7-H3 and B7x
are highly expressed in human prostate cancer and associated with disease
spread and poor outcome. PNAS, Biological Sciences/Immunology,
December, 104(49), 19458-19463, 2007.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49652", author = "Zachariah Sinkala", title = "Evolutionary of Prostate Cancer Stem Cells in Prostate Duct", abstract = "A systems approach model for prostate cancer in prostate duct, as a sub-system of the organism is developed. It is accomplished in two steps. First this research work starts with a nonlinear system of coupled Fokker-Plank equations which models continuous process of the system like motion of cells. Then extended to PDEs that include discontinuous processes like cell mutations, proliferation and deaths. The discontinuous processes is modeled by using intensity poisson processes. The model incorporates the features of the prostate duct. The system of PDEs spatial coordinate is along the proximal distal axis. Its parameters depend on features of the prostate duct. The movement of cells is biased towards distal region and mutations of prostate cancer cells is localized in the proximal region. Numerical solutions of the full system of equations are provided, and are exhibit traveling wave fronts phenomena. This motivates the use of the standard transformation to derive a canonically related system of ODEs for traveling wave solutions. The results obtained show persistence of prostate cancer by showing that the non-negative cone for the traveling wave system is time invariant. The traveling waves have a unique global attractor is proved also. Biologically, the global attractor verifies that evolution of prostate cancer stem cells exhibit the avascular tumor growth. These numerical solutions show that altering prostate stem cell movement or mutation of prostate cancer cells lead to avascular tumor. Conclusion with comments on clinical implications of the model is discussed.
", keywords = "Fokker-Plank equations, global attractor, stem cell.", volume = "4", number = "5", pages = "501-11", }
