To learn more of the phase distributions in unperturbed human tumour cells is a prerequisite prior to understanding of those in the perturbed cells. The work is important in understanding the efficiency of anti-cancer therapy. In this paper we investigate the existence, uniqueness and growth rate of the solution to a mathematical model of unperturbed human tumour cell line. At first, we construct the solution of this mathematical model by the method of continuation of solution, and then show the solution is unique. Finally, we find that the growth rate of the solution with respect to time is faster than exponential function. The basic mathematical techniques used here are variation of parameters and upper and lower solutions for differential equations. These results allowed one to estimate the cells population in each phase at specific time while one does not have cells mitosis DNA distribution data and it can also be used to compare with the perturbed cell lines.
Taiwanese J. Math.
16(1):
47-70
(2012).
DOI: 10.11650/twjm/1500406527
B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. C. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol., 47 (2003), 295-312. 1050.92028 10.1007/s00285-003-0203-0B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. C. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol., 47 (2003), 295-312. 1050.92028 10.1007/s00285-003-0203-0
B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. C. Wake and D. J. N. Wall, Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004a), 329-357. 1060.92035 10.1007/s00285-003-0254-2B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. van Brunt, G. C. Wake and D. J. N. Wall, Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004a), 329-357. 1060.92035 10.1007/s00285-003-0254-2
B. Basse, B. C. Baguley, E. Marshall, G. C. Wake and D. J. N. Wall, Modelling cell population growth with applications to cancer therapy in human tumour cell lines, Prog. Biophys. Mol. Biol., 85 (2004b), 353-368. B. Basse, B. C. Baguley, E. Marshall, G. C. Wake and D. J. N. Wall, Modelling cell population growth with applications to cancer therapy in human tumour cell lines, Prog. Biophys. Mol. Biol., 85 (2004b), 353-368.
B. Basse, G. C. Wake, D. J. N. Wall and B. van Brunt, On a cell growth model for plankton, Math. Med. Biol. J. IMA, 21 (2004c), 49-61. 1062.92065 10.1093/imammb/21.1.49B. Basse, G. C. Wake, D. J. N. Wall and B. van Brunt, On a cell growth model for plankton, Math. Med. Biol. J. IMA, 21 (2004c), 49-61. 1062.92065 10.1093/imammb/21.1.49
B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling the flow of cytometric data obtained from unperturbed human tumor cell lines: parameter fitting and comparison, Bulletin of Math. Biol., 67 (2005), 815-830. 1334.92217 10.1016/j.bulm.2004.10.003B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling the flow of cytometric data obtained from unperturbed human tumor cell lines: parameter fitting and comparison, Bulletin of Math. Biol., 67 (2005), 815-830. 1334.92217 10.1016/j.bulm.2004.10.003
G. Bell and E. Anderson, Cell growth and division. Mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351. G. Bell and E. Anderson, Cell growth and division. Mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.
Y. H. Chang and G. C. Jau, The behavior of the solution to a mathematical model for analysis of the cell cycle, CAPP, Vol. 5, No. 4, December, 2006, pp. 779-792. 1135.35318 10.3934/cpaa.2006.5.779Y. H. Chang and G. C. Jau, The behavior of the solution to a mathematical model for analysis of the cell cycle, CAPP, Vol. 5, No. 4, December, 2006, pp. 779-792. 1135.35318 10.3934/cpaa.2006.5.779
G. Chiorino, J. A. J. Metz, D. Tomasoni and P. Ubezio, Desynchronization rate in cell populations: mathematical modeling and experimental data, J. Theor. Biol., 208 (2001), 185-199. G. Chiorino, J. A. J. Metz, D. Tomasoni and P. Ubezio, Desynchronization rate in cell populations: mathematical modeling and experimental data, J. Theor. Biol., 208 (2001), 185-199.
O. Diekmann, Growth, fission and the stable size distribution, J. Math. Biol., 18 (1983), 135-148. 0533.92016 10.1007/BF00280662O. Diekmann, Growth, fission and the stable size distribution, J. Math. Biol., 18 (1983), 135-148. 0533.92016 10.1007/BF00280662
O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248. 0543.92021 10.1007/BF00277748O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248. 0543.92021 10.1007/BF00277748
M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. MR1072714 0744.92026 10.1007/BF00160231M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. MR1072714 0744.92026 10.1007/BF00160231
K. B. Hannsgen and J. J. Tyson, Stability of the steady-state size distribution in a model of cell growth and division, J. Math. Biol., 22 (1985), 293-301. MR813400 0571.92023 10.1007/BF00276487K. B. Hannsgen and J. J. Tyson, Stability of the steady-state size distribution in a model of cell growth and division, J. Math. Biol., 22 (1985), 293-301. MR813400 0571.92023 10.1007/BF00276487
E. Kreyszig, Introductory Functional Analysis with Applications, Springer-Verlag, New York, 1978. 0368.46014E. Kreyszig, Introductory Functional Analysis with Applications, Springer-Verlag, New York, 1978. 0368.46014
F. Montalenti, G. Sena, P. Cappella and P. Ubezio, Simulating cancer-cell kinetics after drug treatment: Application to cisplatin on ovarian carcinoma, Phys. Rev. E, 57(5) (1999), 5877-5887. F. Montalenti, G. Sena, P. Cappella and P. Ubezio, Simulating cancer-cell kinetics after drug treatment: Application to cisplatin on ovarian carcinoma, Phys. Rev. E, 57(5) (1999), 5877-5887.
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 0777.35001C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 0777.35001
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. 0516.47023A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. 0516.47023
B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment, Carcinogenesis and Cell & Tumor Growth of Arino et al. 1995, Vol. 2. (Chapter 14), 1995, pp. 183-200. 0933.92017B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment, Carcinogenesis and Cell & Tumor Growth of Arino et al. 1995, Vol. 2. (Chapter 14), 1995, pp. 183-200. 0933.92017
G. Sena, C. Onado, P. Cappella, F. Montalenti and P. Ubezio, Measuring the complexity of cell cycle arrest and killing of drugs: Kinetics of phase-specific effects induced by taxol, Cytometry, 37 (1999), 113-124. G. Sena, C. Onado, P. Cappella, F. Montalenti and P. Ubezio, Measuring the complexity of cell cycle arrest and killing of drugs: Kinetics of phase-specific effects induced by taxol, Cytometry, 37 (1999), 113-124.
P. Ubezio, Cell cycle simulation for flow cytometry. Computer methods and programs in biomedicine. Section II, Systems and programs, 31(3697) (1990), 255-266. P. Ubezio, Cell cycle simulation for flow cytometry. Computer methods and programs in biomedicine. Section II, Systems and programs, 31(3697) (1990), 255-266.
P. Ubezio, Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations, Discrete Contin. Dyn. Sys. Ser., B4(1) (2004), 323-335. 1044.92037P. Ubezio, Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations, Discrete Contin. Dyn. Sys. Ser., B4(1) (2004), 323-335. 1044.92037