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A model for asymmetrical cell division
1. | Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand, New Zealand |
2. | Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand |
References:
[1] |
B. Basse, B. Baguley, E. Marshell, W. Joseph, B. Van-Brunt, G. C. Wake and D. Wall, Modelling cell death in human tumor cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004), 329-357.
doi: 10.1007/s00285-003-0254-2. |
[2] |
Basse, G. C. Wake, D. J. N. Wall and B. Van-Brunt, On a cell-growth model for plankton, Mathematical medicine and biology, 21 (2004), 49-61. |
[3] |
R. Begg, Cell-population Growth Modeling and Functional Differential Equations, Ph.D thesis, University of Canterbury, New Zealand, 2007. |
[4] |
R. Begg, D. J. N. Wall and G. C. Wake, On a functional equation model of transient cell growth, Mathematical medicine and biology, 22 (2005), 371-390.
doi: 10.1093/imammb/dqi015. |
[5] |
M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to self similarity for the fragmentation equation in $L^1$ spaces, Communications in Applied and Industrial Mathematics, 1 (2010), 299-308. |
[6] |
M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathémathiques Pures et Appliquée, 96 (2011), 334-362.
doi: 10.1016/j.matpur.2011.01.003. |
[7] |
F. P. Da Costa, M. Grinfeld and J. B. Mcleod, Unimodality of steady size distributions of growing cell populations, J.evol.equ., 1 (2001), 405-409.
doi: 10.1007/PL00001379. |
[8] |
O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, Jour. Math. Biol., 19 (1984), 227-248.
doi: 10.1007/BF00277748. |
[9] |
A. J. Hall and G. C. Wake, A functional differential equation arising in modelling of cell growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424-435.
doi: 10.1017/S0334270000006366. |
[10] |
A. J. Hall, G. C. Wake and P. W. Gandar, Steady size distributions for cells in one dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123.
doi: 10.1007/BF00160330. |
[11] |
H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72 (1984), 19-50.
doi: 10.1016/0025-5564(84)90059-2. |
[12] |
P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[13] |
T. R. Malthus, An Essay on the Principle of Population, St. Paul's London, 1798. |
[14] |
A. G. Mckendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130. |
[15] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-662-13159-6. |
[16] |
P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697-702.
doi: 10.1016/j.crma.2004.03.006. |
[17] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[18] |
R. A. Neumïler and J. A. Knoblich, Dividing cellular asymmetry: Asymmetric cell division and its implications for stem cells and cancer, Genes Dev., 23 (2009), 2675-2699. |
[19] |
B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, Journal of Differential Equations, 210 (2005), 155-177.
doi: 10.1016/j.jde.2004.10.018. |
[20] |
T. Suebcharoen, B. Van-Brunt and G. C. Wake, Asymmetric cell division in a size-structured growth model, Differential and Integral Equations, 24 (2011), 787-799. |
[21] |
B. Van-Brunt, G. C. Wake and H. K. Kim, A singular Sturm-Liouville problem involving an advanced functional differential equation, European Journal of Applied Mathematics, 12 (2001), 625-644.
doi: 10.1017/S0956792501004624. |
[22] |
B. Van-Brunt and M. Vlieg-Hulstman, An eigenvalue problem involving a functional differential equation arising in a cell growth model, ANZIAM J., 51 (2010), 383-393.
doi: 10.1017/S1446181110000866. |
show all references
References:
[1] |
B. Basse, B. Baguley, E. Marshell, W. Joseph, B. Van-Brunt, G. C. Wake and D. Wall, Modelling cell death in human tumor cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004), 329-357.
doi: 10.1007/s00285-003-0254-2. |
[2] |
Basse, G. C. Wake, D. J. N. Wall and B. Van-Brunt, On a cell-growth model for plankton, Mathematical medicine and biology, 21 (2004), 49-61. |
[3] |
R. Begg, Cell-population Growth Modeling and Functional Differential Equations, Ph.D thesis, University of Canterbury, New Zealand, 2007. |
[4] |
R. Begg, D. J. N. Wall and G. C. Wake, On a functional equation model of transient cell growth, Mathematical medicine and biology, 22 (2005), 371-390.
doi: 10.1093/imammb/dqi015. |
[5] |
M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to self similarity for the fragmentation equation in $L^1$ spaces, Communications in Applied and Industrial Mathematics, 1 (2010), 299-308. |
[6] |
M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathémathiques Pures et Appliquée, 96 (2011), 334-362.
doi: 10.1016/j.matpur.2011.01.003. |
[7] |
F. P. Da Costa, M. Grinfeld and J. B. Mcleod, Unimodality of steady size distributions of growing cell populations, J.evol.equ., 1 (2001), 405-409.
doi: 10.1007/PL00001379. |
[8] |
O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, Jour. Math. Biol., 19 (1984), 227-248.
doi: 10.1007/BF00277748. |
[9] |
A. J. Hall and G. C. Wake, A functional differential equation arising in modelling of cell growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424-435.
doi: 10.1017/S0334270000006366. |
[10] |
A. J. Hall, G. C. Wake and P. W. Gandar, Steady size distributions for cells in one dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123.
doi: 10.1007/BF00160330. |
[11] |
H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72 (1984), 19-50.
doi: 10.1016/0025-5564(84)90059-2. |
[12] |
P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[13] |
T. R. Malthus, An Essay on the Principle of Population, St. Paul's London, 1798. |
[14] |
A. G. Mckendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130. |
[15] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-662-13159-6. |
[16] |
P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697-702.
doi: 10.1016/j.crma.2004.03.006. |
[17] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[18] |
R. A. Neumïler and J. A. Knoblich, Dividing cellular asymmetry: Asymmetric cell division and its implications for stem cells and cancer, Genes Dev., 23 (2009), 2675-2699. |
[19] |
B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, Journal of Differential Equations, 210 (2005), 155-177.
doi: 10.1016/j.jde.2004.10.018. |
[20] |
T. Suebcharoen, B. Van-Brunt and G. C. Wake, Asymmetric cell division in a size-structured growth model, Differential and Integral Equations, 24 (2011), 787-799. |
[21] |
B. Van-Brunt, G. C. Wake and H. K. Kim, A singular Sturm-Liouville problem involving an advanced functional differential equation, European Journal of Applied Mathematics, 12 (2001), 625-644.
doi: 10.1017/S0956792501004624. |
[22] |
B. Van-Brunt and M. Vlieg-Hulstman, An eigenvalue problem involving a functional differential equation arising in a cell growth model, ANZIAM J., 51 (2010), 383-393.
doi: 10.1017/S1446181110000866. |
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