Evolutionary of Prostate Cancer Stem Cells in Prostate Duct
Authors: Zachariah Sinkala
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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328566Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1468
 M. Abercrombie, Contact inhibition in tissue. In vitro 6, 128-140, 1970.
 R. Agami, All Roads Lead toIKKÐö, Cell 129, 1043-1045, 2007.
 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.
 M. Bertsch, M. E. Gurtin, D. Hilhorst, L. S. Peletier, On interacting populations that disperse to avoid crowding: perservation of segregation, 1985
 J. Canosa, On a nonlinear diffusion equation describing population growth. IBM J. Res. Dev. 17, 307-313, 1973.
 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.
 T. C. Collins, N. J. Maitland, Prostate cancer stem cells. European J. of Cancer 42, 1213-1218, 2006.
 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.
 F. F. Costa K. Le Blanc, B. Brodin, Cancer/Testis antigens, stem cells and cancer. Stem Cells 12(2), 398-404, 2006.
 D. Dingli, F. Michor, Successful therapy must eradicate cancer stem cells. Stem Cells 24, 2603-2610, 2006.
 G. P. Dunn, et al., The three Es of cancer immunoediting. Annu. Rev. Immunol. 22, 329360, 2004.
 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.
 R. A. Fisher, The wave of advance of advantageneous genes. Ann. Eugenics 7, 353-369, 1937.
 R. Ganguly, and I. K. Puri, Mathematical model for the cancer stem cell hypothesis. Cell Proliferation 39, 3-14, 2006.
 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.
 S. Gakkhar, S. Brahampal, R. K. Naji, Dynamical behavior of two predators competing over a single prey. Biosystems, 90, 808-817, 2007.
 M. C. Garnick, M. C. Fair, Combating Prostate Cancer. Scientific American 279(6), 74-83, 1998.
 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.
 M. Greaves, M., 2000. Cancer: The Evolutionary Legancy, Oxford University Press.
 L. Han, A. Publiese, Epidemics in two competing species. Nonlinear Analysis: Real World Applications, 10(2), 723-744, 2009.
 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.
 M. Jakobisiak, et al. Natural mechanisms protecting against cancer. Immunol. Lett. 90, 103122, 2003.
 B. T. Kawasaki, and W. L. Farrar Cancer stem cells, CD200 and immunoevasion, Trends in Immunology, 29(10), Issue 10, 464-468, 2008.
 E. F. Keller, L. A. Segel, Model for chemotaxis. J. Theor. Biol. 30, 225-234, 1971.
 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.
 H. Kitano, Cancer therapy as a robust system: implications for anticancer therapy, Nature Reviews - Cancer, 4, 227-235, 2004.
 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.
 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.
 I. C. Mackenzie, Stem cell properties and epithelial malignancies European Journal of Cancer. 42(9), 1204-1212, 2006.
 N. J. Maitland, A. T. Collins, prostate cancer stem cells: new therapeutic targets? European Journal of Cancer Supplements, 5(4), 2007.
 J. Michalowski, Common molecule notifies immune system of prostate cancer. www.eurekalert.org, Publication release, January 10, 2008.
 F. Michor, M. A. Nowak, S. A. Frank, Y. Iwasa, Stochastic elimination of cancer cells. Proc. R. Soc. Lond. B 270, 2017-2024, 2003.
 S. J. Morrison, A. C. Spradling, Stem Cells and Niches: Mechanisms that promote stem cell maintenance throughout life, Cell 132, 598-611, 2008.
 D. Mukherjee, Uniform persistence in a generalized prey-predator system with parasitic infection. Biosystems, 47, 149-155, 1998.
 J. A. Nemeth, C. Lee, Prostatic ductal system in rats: regional variation in stromal organization. Prostate 28, 124128, 1996.
 P. C. Nowell, The clonal expansion of tumor cell populations. Science 194, 23-28, 1996.
 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.
 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.
 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.
 J. A. Schalken, G. Van Leenders, Cellular and molecular biology of the prostate: stem cell biology, Urology, November 62(Supplement 5A), 11-20, 2003.
 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.
 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.
 J. A. Sherratt, M. A. J. Chaplain, A new mathematical model for avascular tumour growth. J. Math. Biol. 43, 291-312, 2001.
 T. Quinn, and Z. Sinkala, Dynamics of prostate cancer stem cells with diffusion and organism response, Biosystems, 96(1), 69-79, 2009.
 T. L. Soon, A. K. Cheng, A numerical simulation of avascular tumour growth. Anziam. J. 46, 902-917, 2005.
 Y. Sugimura, G. R. Cunha, A. A. Donjacour, Morphogenesis of ductal networks in the mouse prostate. Biol. Reprod. 34, 961971, 1986.
 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.
 D. Wodarz, N. Komarov, Computational Biology of Cancer. World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
 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
 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.