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A Mathematical Model of Cancer Stem Cell Lineage PopulationDynamics with Mutation Accumulation and Telomere Length Hierarchies

Published online by Cambridge University Press:  25 January 2012

G. Kapitanov
G. Kapitanov*
Affiliation:
Vanderbilt University Department of Mathematics 1326 Stevenson Center, Nashville, TN 37240, USA
*
⋆E-mail: [email protected]
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Abstract

There is evidence that cancer develops when cells acquire a sequence of mutations thatalter normal cell characteristics. This sequence determines a hierarchy among the cells,based on how many more mutations they need to accumulate in order to become cancerous.When cells divide, they exhibit telomere loss and differentiate, which defines anothercell hierarchy, on top of which is the stem cell. We propose a mutation-generation model,which combines the mutation-accumulation hierarchy with the differentiation hierarchy ofthe cells, allowing us to take a step further in examining cancer development and growth.The results of the model support the hypothesis of the cancer stem cell’s role in cancerpathogenesis: a very small fraction of the cancer cell population is responsible for thecancer growth and development. Also, according to the model, the nature of mutationaccumulation is sufficient to explain the faster growth of the cancer cell population.However, numerical results show that in order for a cancer to develop within a reasonabletime frame, cancer cells need to exhibit a higher proliferation rate than normalcells.

Keywords

cancerstem cellscancer stem cellmathematical modelmutationtelomere shortening
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 136 - 165
Copyright
© EDP Sciences, 2012

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References

Ahmed, S., Passos, J.F., Birket, M.J., Beckmann, T., Brings, S., Peters, H., Birch-Machin, M.A., von Zglinicki, T., Saretzki, G.. Telomerase Does Not Counteract Telomere Shortening But Protects Mitochondrial Function Under Oxidative Stress. Journal of Cell Science, 121 (2008), No. 7, 1046–1053. CrossRefGoogle Scholar
Arino, O., Kimmel, M., Webb, G.F.. Mathematical Modeling of the Loss of Telomere Sequences. J. theor. Biol., 177 (1995), No. 1, 45–57. CrossRefGoogle ScholarPubMed
Arino, O., Sánchez, E., Webb, G.F.. Polynomial Growth Dynamics of Telomere Loss in a Heterogeneous Cell Population. Dynamic Control Discrete Impulsive System, 3 (1997), No. 3, 263–282. Google Scholar
Armitage, P., Doll, R.. The age distribution of cancer and a multi-stage theory of carcinogenosis. IJE, 33 (2004), No. 6, 1174–1179. Google Scholar
Bagheri, S., Nosrati, M., Li, S., Fong, S., Torabian, S., Rangel, J., Moore, D.H., Federman, S., LaPosa, R.R., Baehner, F.L., Sagebiel, R.W., Cleaver, J.E., Haqq, C., Debs, R.J., Blackburn, E.H., Kashani-Sabet, M.. Genes and pathways downstream of telomerase in melanoma metastasis. PNAS, 103 (2006), No. 30, 11306–11311. CrossRefGoogle ScholarPubMed
Banks, H.T., Sutton, K.L., Thompson, W.C., Bocharov, G., Roose, D., Schenkel, T., Meyerhans, A.. Estimation of Cell Proliferation Dynamics Using CFSE Data. Bulletin of Mathematical Biology, 73 (2011), 1, 116–150. CrossRefGoogle ScholarPubMed
Bernard, S., Pujo-Menjouet, L., Mackey, M.C.. Analysis of Cell Kinetics Using a Cell Division Marker : Mathematical Modeling of Experimental Data. Biophysical Journal, 84 (2003), No. 5, 3414–3424. CrossRefGoogle ScholarPubMed
D.S. Bernstein. Matrix Mathematics, Second Edition., Princeton University Press, 2009.
Bonnet, D., Dick, J.E.. Human Acute Myeloid Leukemia is Organized as a Hierarchy That Originates From a Primitive Hematopoetic Cell. Nature Medicine, 3 (1997), No. 7, 730–737. CrossRefGoogle Scholar
Brú, A., Albertos, S., Subiza, J.L., García-Asenjo, J.L., Brú, I.. The Universal Dynamics of Tumor Growth. Biophysical Journal, 85 (2003), No. 5, 2948–2961. CrossRefGoogle ScholarPubMed
Cohen, E.D., Tian, Y., Morrisey, E.E.. Wnt signaling : an essential regulator of cardiovascular differentiation, morphogenesis and progenitor self-renewal. Development, 135 (2008), No. 5, 789–798. CrossRefGoogle ScholarPubMed
Collins, A.T., Berry, P.A., Hyde, C., Stower, M.J., Maitland, N.J.. Prospective Identification of Tumorigenic Prostate Cancer Stem Cells. Cancer Res., 65 (2005), No. 23, 10946–10951. CrossRefGoogle Scholar
Dalerba, P., Dylla, S.J., Park, I.-K., Liu, R., Wang, X., Cho, R.W., Hoey, T., Gurney, A., Huang, E.H., Simeone, D.M., Shelton, A.A., Parmiani, G., Castelli, C., Clarke, M.F.. Phenotypic characterization of human colorectal cancer stem cells. PNAS, 104 (2007), No. 24, 10158–10163. CrossRefGoogle ScholarPubMed
de Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.. A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth. Cancer Res., 65 (2005), No. 17, 7950–7958. CrossRefGoogle ScholarPubMed
Deasy, B.M., Jankowski, R.J., Payne, T.R., Cao, B., Goff, J.P., Greenberger, J.S., Huard, J.. Modeling Stem Cell Population Growth : Incorporating Terms for Proliferative Heterogeneity. Stem Cells, 21 (2003), No. 5, 536 – 545. CrossRefGoogle ScholarPubMed
Dick, J.E.. Breast cancer stem cells revealed. PNAS, 100 (2003), No. 7, 3547–3549. CrossRefGoogle ScholarPubMed
Dingli, D., Michor, F.. Successful Therapy Must Eradicate Cancer Stem Cells. Stem Cells, 24 (2006), No. 12, 2603–2610. CrossRefGoogle ScholarPubMed
Dyson, J., Sánchez, E., Villella-Bressan, R., Webb, G.F.. Stabilization of telomeres in nonlinear models of proliferating cell lines. Journal of Theoretical Biology, 244 (2007), No. 3, 400–408. CrossRefGoogle ScholarPubMed
Dyson, J., Villella-Bressan, R., Webb, G.F.. Asymptotic Behaviour Of Solutions To Abstract Logistic Equations. Mathematical Biosciences, 206 (2007), No. 2, 216–232. CrossRefGoogle ScholarPubMed
Enderling, H., Park, D., Hlatky, L., Hahnfeldt, P.. The Importance of Spatial Distribution of Stemness and Proliferation State in Determining Tumor Radioresponse. Math. Model. Nat. Phenom., 4 (2009), No. 3, 117–133. CrossRefGoogle Scholar
Eramo, A., Lotti, F., Sette, G., Pilozzi, E., Biffoni, M., Di Virgilio, A., Conticello, C., Ruco, L., Peschle, C., De Maria, R.. Identification and expansion of the tumorigenic lung cancer stem cell population. Cell Death and Differentiation, 15 (2008), No. 3, 504–514. CrossRefGoogle ScholarPubMed
Fearon, E.R., Vogelstein, B.. A Genetic Model for Colorectal Tumorigenesis. Cell, 61 (1990), 759–767. CrossRefGoogle ScholarPubMed
Frenck, R.W. Jr., Blackburn, E.H, Shannon, K.M.. The rate of telomere sequence loss in human leukocytes varies with age. PNAS, 95 (1998), No. 10, 5607-5610. CrossRefGoogle Scholar
Gentry, S.N., Ashkenazi, R., Jackson, T.L.. A Maturity Structured Mathematical Model of Mutation Acquisition in the Absence of Homeostatic Regulation. Math. Model. Nat. Phenom., 4 (2009), 403–422. CrossRefGoogle Scholar
Hanahan, D., Weinberg, R.A.. The Hallmarks of Cancer : The Next Generation. Cell., 144 (2011), No. 5, 646–674. CrossRefGoogle ScholarPubMed
Huffman, K.E., Levene, S.D., Tesmer, V.M., Shay, J.W., Wright, W.E.. Telomere Shortening Is Proportional to the Size of the G-rich Telomeric 3’-Overhang. The Journal of Biological Chemistry, 275 (2000), No. 26, 19719–19722. CrossRefGoogle ScholarPubMed
Knudson, A. G., Two genetic hits (more or less) to cancer. Nat. Rev. Cancer., 1 (2001), 157–162. CrossRefGoogle Scholar
Lang, S.H., Frame, F.M., Collins, A.T.. Prostate cancer stem cells. Journal of Pathology, 217 (2009), No. 9, 299–306. CrossRefGoogle ScholarPubMed
Levy, M.Z., Allsopp, R.C., Futcher, A.B., Greider, C.W., Harley, C.B.. Telomere End-replication Problem and Cell Aging. J. Mol. Biol., 225 (1992), No. 4, 951–960. CrossRefGoogle ScholarPubMed
Lodish, H., Flygare, J., Chou, S.. From stem cell to erythroblast : Regulation of red cell production at multiple levels by multiple hormones. IUBMB Life, 62 (2010), No. 7, 492–496. CrossRefGoogle ScholarPubMed
Marciniak-Czochra, A.. Mathematical models of stem cells renewal and differentiation. Oberwolfach Reports, 2 (2009), 3414–3424. Google Scholar
Morrison, S.J., Uchida, N., Weissman, I.L.. The biology of hematopoietic stem cells. Annu. Rev. Cell Dev. Biol., 11 (1995), 35–71. CrossRefGoogle Scholar
P. Olofsson. Modeling of the Process of Telomere Shortening : an Overview.
L. Perko. Differential Equations and Dynamical Systems, 3rd edition. Springer, New York, NY, 2001.
Roegiers, F., Jan, Y.N.. Asymmetric cell division. Current Opinion in Cell Biology, 16 (2004), No. 2, 195-205. CrossRefGoogle Scholar
Simon, G.R., Wagner, H.. Small Cell Lung Cancer*. Chest, 123 (2003), No. 1, 259–271. CrossRefGoogle ScholarPubMed
Singh, S.K., Clarke, I.D., Terasaki, M., Bonn, V.E., Hawkins, C., Squire, J., Dirks, P.B.. Identification of a Cancer Stem Cell in Human Brain Tumors. Cancer Res., 63 (2003), No. 18, 5821–5828. Google ScholarPubMed
Skehan, P., Friedman, S.J.. Non-exponential growth by mammalian cells in culture. Cell Tissue Kinet., 17 (1984), No. 4, 335–343. Google ScholarPubMed
Solyanik, G.I., Berezetskaya, N.M., Bulkiewicz, R.I., Kulik, G.I.. Different growth patterns of a cancer cell population as a function of its starting growth characteristics : analysis by mathematical modelling. Cell Prolif, 28 (1995), No. 5, 263–278. CrossRefGoogle ScholarPubMed
Spangrude, G.J., Heimfeld, S., Weissman, I.L.. Purification and characterization of mouse hematopoietic stem cells. Science, 244 (1988), No. 4861, 58–62. CrossRefGoogle Scholar
Speer, J.F., Petrosky, V.E., Retsky, M.W., Wardwell", R.H., A Stochastic Numerical Model of Breast Cancer Growth That Simulates Clinical Data. Cancer Res., 44 (1984), No. 9, 4124–4130. Google ScholarPubMed
Sprouffske, K., Pepper, J.W., Maley, C.C.. Accurate Reconstruction of the Temporal Order of Mutations in Neoplastic Progression. Cancer Prev. Res., 4 (2011), No. 7, 1135–1144. CrossRefGoogle ScholarPubMed
Stewart, S.A., Hahn, W.C., O’Connor, B.F., Banner, E.N., Lundberg, A.S., Modha, P., Mizuno, H., Brooks, M.W., Fleming, M., Zimonjic, D.B., Popescu, N.C., Weinberg, R.A.. Telomerase contributes to tumorigenesis by a telomere length-independent mechanism. PNAS, 99 (2002), No. 20, 12606–12611. CrossRefGoogle ScholarPubMed
Stratton, M.R., Campbell, P.J., Futreal, P.A.. The Cancer Genome. Nature, 458 (2009), No. 7239, 156–182. CrossRefGoogle ScholarPubMed
van der Flier, L.G., Clevers, H.. Stem Cells, Self-Renewal, and Differentiation in the Intestinal Epithelium. Annu. Rev. Physiol., 71 (2009), No. 1, 241–260. CrossRefGoogle ScholarPubMed
von Zglinicki, T.. Oxidative Stress Shortens Telomeres. Trends in Biochemical Scoences, 27 (2002), No. 7, 339–344. CrossRefGoogle ScholarPubMed
Webb, G.F.. Logistic Models Of Structured Population Growth. Comp and Maths. with Appls., 12 (1986), No. 4-5A, 527–539. CrossRefGoogle Scholar
Weinstein, G.D., McCullough, J.L., Ross, P.. Cell Proliferation in Normal Epidermis. The Journal of Investigative Dermatology, 82 (1984), No. 6, 623–628. CrossRefGoogle ScholarPubMed
Wilson, G.D., McNally, N.J., Dische, S., Saunders, M.I., Des Rochers, C., Lewis, A.A., Bennett, M.H.. Measurement of cell kinetics in human tumours in vivo using bromodeoxyuridine incorporation and flow cytometry. Br. J. Cancer, 58 (1988), No. 4, 423–431. CrossRefGoogle Scholar
Ying, Q.-L., Wray, J., Nichols, J., Batlle-Morera, L., Doble, B., Woodgett, J., Cohen, P., Smith, A.. The ground state of embryonic stem cell self-renewal. Nature, 453 (2008), No. 7194, 519–523. CrossRefGoogle ScholarPubMed