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Stochastic models in cell kinetics

Published online by Cambridge University Press:  14 July 2016

Peter J. Brockwell
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Abstract

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

Keywords

CELL KINETICSAGE-DEPENDENT BRANCHING PROCESSESPERIODIC EXPONENTIAL GROWTH
Type
Part 3 - Stochastic Models in Biology and Field Trials
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 91 - 111
Copyright
Copyright © Applied Probability Trust 1988 

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References

Barrett, J. C. (1966) A mathematical model of the mitotic cycle and its application to the interpretation of percentage labelled mitoses data. J. Nat. Cancer Inst. 37, 443–450.Google Scholar
Barrett, J. C. (1970) Optimized parameters for the mitotic cycle. Cell Tissue Kinet. 3, 349–353.Google ScholarPubMed
Bell, G. I. and Anderson, E. C. (1967) Cell growth and division I. Biophys. J. 7, 329–351.CrossRefGoogle ScholarPubMed
Brockwell, P. J. (1975) Random walks, diffusion models and diurnal rhythm in cell kinetics. In Mathematical Models in Cell Kinetics, ed. Valleron, A.-J. European Press, Ghent.Google Scholar
Brockwell, P. J. and Kuo, W. H. (1973) Labelling experiments and phase-age distributions for multiphase branching processes. J. Appl. Prob. 10, 739–747.CrossRefGoogle Scholar
Brockwell, P. J. and Trucco, E. (1970) On the decomposition by generations of the FLM function. J. Theoret. Biol. 26, 149–179.CrossRefGoogle Scholar
Brockwell, P. J., Trucco, E. and Fry, R. J. M. (1972) The determination of cell-cycle parameters from measurements of the fraction of labeled mitoses. Bull. Math. Biophys. 34, 1–12.CrossRefGoogle ScholarPubMed
Brockwell, P. J., Resnick, S. I. and Pacheco-Santiago, N. (1982) Extreme values, range and weak convergence of integrals of Markov chains. J. Appl. Prob. 19, 272–288.CrossRefGoogle Scholar
Brown, J. M. and Berry, R. J. (1968) The relationship between diurnal variation of the number of cells in mitosis and the number of cells synthesizing DNA in the epithelium of the hamster cheek pouch. Cell Tissue Kinet. 1, 23.Google Scholar
Cleaver, J. E. (1967) Thymidine Metabolism and Cell Kinetics. North-Holland, Amsterdam.Google Scholar
Clifford, P. and Sudbury, A. (1972) The linear cell-size-dependent branching process. J. Appl. Prob. 9, 687–696.CrossRefGoogle Scholar
Cowan, R. and Staudte, R. (1986) The bifurcating autoregression model in cell lineage studies. Biometrics 42, 769–783.CrossRefGoogle ScholarPubMed
Hopper, J. L. and Brockwell, P. J. (1978) A stochastic model for cell populations with diurnal rhythm. Cell Tissue Kinet. 11, 205–225.Google Scholar
Izquierdo, J. N. and Gibbs, S. J. (1974) Turnover of cell-renewing populations undergoing circadian rhythms in cell proliferation. Cell Tissue Kinet. 7, 99.Google ScholarPubMed
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Jagers, P. (1983) Stochastic models for cell kinetics. Bull. Math. Biol. 45, 507–519.CrossRefGoogle ScholarPubMed
Jagers, P. and Nerman, O. (1985) Branching processes in periodically varying environment. Ann. Prob. 13, 254–268.CrossRefGoogle Scholar
Kendall, D. G. (1948) On the role of variable generation time in the development of a stochastic birth process. Biometrika 35, 316–330.CrossRefGoogle Scholar
Klein, B. and Valleron, A.-J. (1978) A compartmental model for the study of diurnal rhythms in cell proliferation. J. Theoret. Biol. 64, 27–42.CrossRefGoogle Scholar
Mckendrick, A. G. (1926) Application of mathematics to medical problems. Proc. Edinburgh Math. Soc. 44, 98–130.CrossRefGoogle Scholar
Macdonald, P. D. M. (1970) Statistical inference from the fraction labelled mitoses curve. Biometrika 57, 489–503.CrossRefGoogle Scholar
Macdonald, P. D. M. (1973) On the statistics of cell proliferation. In The Mathematical Theory of the Dynamics of Biological Populations, ed. Bartlett, M. S. and Hiorns, R. W., Academic Press, London.Google Scholar
Macdonald, P. D. M. (1981) Measuring circadian rhythms in cell populations. On the statistics of cell proliferation. In The Mathematical Theory of the Dynamics of Biological Populations II, ed. Hiorns, R. W. and Cooke, D., Academic Press, London.Google Scholar
Møller, U., Larsen, J. K. and Faber, M. (1974) The influence of injected tritiated thymidine on the mitotic circadian rhythm in the epithelium of the hamster cheek pouch. Cell Tissue Kinet. 7, 231–239.Google ScholarPubMed
Quastler, H. and Sherman, F. G. (1959) Cell population kinetics in the intestinal epithelium of the mouse. Exp. Cell Res. 16, 420–438.CrossRefGoogle Scholar
Stuart, R. N. and Merkle, T. C. (1959). The calculation of treatment schedules for cancer chemotherapy, Part II. UCRL-14505–2, Lawrence Radiation Laboratory, Livermore, California.Google Scholar
Sudbury, A. (1981) The expected population size in a cell-size-dependent branching process. J. Appl. Prob. 18, 65–75.CrossRefGoogle Scholar
Takahashi, M. (1966) Theoretical basis for cell-cycle analysis. I. Labelled mitosis wave method. J. Theoret. Biol. 13, 202–211.CrossRefGoogle Scholar
Takahashi, M. (1968) Theoretical basis for cell-cycle analysis. II. Further studies on labelled mitosis wave method. J. Theoret. Biol. 18, 195–209.CrossRefGoogle ScholarPubMed
Trucco, E. and Brockwell, P. J. (1968) Percentage labeled mitoses curves in exponentially growing cell populations. J. Theoret. Biol. 20, 321–337.CrossRefGoogle ScholarPubMed
Valleron, A.-J. and Macdonald, P. D. M., (eds) (1978) Biomathematics and Cell Kinetics. North-Holland, Amsterdam.Google Scholar
Wimber, D. E. (1963) Methods for studying cell proliferation with emphasis on DNA labels. In Cell Proliferation, ed. Lamerton, L. F. and Fry, R. J. M. Blackwell, Oxford.Google Scholar