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Labelling experiments and phase-age distributions for multiphase branching processes

Published online by Cambridge University Press:  14 July 2016

P. J. Brockwell  and
W. H. Kuo
P. J. Brockwell
Affiliation:
Michigan State University
W. H. Kuo
Affiliation:
Michigan State University
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Abstract

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x 1, x 2, x 3, x 4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z (4) (a 0,…, a n , t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y 01,…, Y 04,…, Yi 1,…, Yi 4,…, Yn 4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij , i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

Keywords

AGE-DEPENDENT MULTIPHASE BRANCHING PROCESSASYMPTOTIC AGE DISTRIBUTIONMITOTIC CYCLECELL KINETICSLABELLING EXPERIMENT

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 4 , December 1973 , pp. 739 - 747
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] 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
[2] Barrett, J. C. (1970) Optimized parameters for the mitotic cycle. Cell Tissue Kinet. 3, 349–353.Google Scholar
[3] Brockwell, P. J. and Kuo, W. H. (1972) Generalized asymptotic age distributions for a multiphase branching process. Michigan State University Statistical Laboratory, SLP-38.Google Scholar
[4] 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.Google Scholar
[5] Feller, W. (1971) An Introduction to Probability Theory and its Applications. 2nd ed., Vol. II. John Wiley, New York.Google Scholar
[6] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[7] Macdonald, P. D. M. (1970) Statistical inference from the fraction labelled mitoses curve. Biometrika 57, 489–503.Google Scholar
[8] Mendelsohn, M. L. and Takahashi, M. (1972) A critical evaluation of the fraction of labeled mitoses method as applied to the analysis of tumor and other cycles. Chapter III of The Cell Cycle and Cancer . (Baserga, R., ed.). Marcel Dekker, New York.Google Scholar
[9] Powell, E. P. (1955) Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbiol. 15, 492–511.Google Scholar
[10] Steel, G. G. and Hanes, S. (1970) The technique of labelled mitoses: analysis by automatic curve fittings. Cell Tissue Kinet. 4, 93–105.Google Scholar
[11] Takahashi, M. (1968) Theoretical basis for cell cycle analysis, II. J. Theoret. Biol. 18, 195–209.Google Scholar
[12] Trucco, E. and Brockwell, P. J. (1968) Percentage labeled mitoses curves in exponentially growing cell populations. J. Theoret. Biol. 20, 321–337.Google Scholar