Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T18:02:27.091Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Model of Carcinogenesis and Tumor Size at Detection

Part of: Distribution theory - Probability Applications

Published online by Cambridge University Press:  01 July 2016

L. G. Hanin ,
S. T. Rachev ,
A. D. Tsodikov  and
A. Yu. Yakovlev
L. G. Hanin*
Affiliation:
Wayne State University
S. T. Rachev*
Affiliation:
University of California at Santa Barbara
A. D. Tsodikov*
Affiliation:
Universität Leipzig
A. Yu. Yakovlev*
Affiliation:
University of Utah
*
∗ Postal address: Department of Mathematics, Wayne State University, Detroit, Ml 48202, USA.
∗∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.
∗∗∗ Postal address: IMISE, Universität Leipzig, Liebigstr. 27, 04103 Leipzig, Germany.
∗∗∗∗ Postal address: Huntsman Cancer Institute, Division of Public Health Science, University of Utah, 546 Chipeta Way, Suite 1100, Salt Lake City, Utah 84108, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.

Keywords

STOCHASTIC MODELASYMPTOTIC PROPERTIESCARCINOGENESISCONDITIONAL DISTRIBUTIONTUMOR SIZESTABILITY TO PRIORSBAYES THEOREMPROBABILITY METRICS

MSC classification

Primary: 60E05: Distributions
Secondary: 62P10: Applications to biology and medical sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 607 - 628
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ainsworth, E. J. (1982) Radiation carcinogenesis-perspectives. In Probability Models and Cancer. ed. Le Cam, L. and Neyman, L.. North-Holland, Amsterdam. pp. 99–169.Google Scholar
[2] Atkinson, N. E., Bartoszynski, R., Brown, B. W. and Thompson, J. R. (1983) On estimating the growth function of tumors. Math. Biosci. 67, 145–166.CrossRefGoogle Scholar
[3] Atkinson, N. E., Brown, B. W. and Thompson, J. R. (1987) On the lack of concordance between primary and secondary tumor growth rates. J. Nat. Cancer Inst. 78, 425–435.Google Scholar
[4] Bartoszynski, R. (1987) A modeling approach to metastatic progression of cancer. In Cancer Modeling. ed. Thompson, J. R. and Brown, B. W.. Marcel Dekker, New York. pp. 237–267.Google Scholar
[5] Bartoszynski, R., Jones, B. F. and Klein, J. P. (1985) Some stochastic models of cancer metastases. Commun. Statist. Stoch. Models 1, 317–339.CrossRefGoogle Scholar
[6] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[7] Brown, B. W., Atkinson, N. E., Bartozynski, R. and Montague, E. D. (1984) Estimation of human tumor growth rate from distribution of tumor size at detection. J. Nat. Cancer Inst. 72, 31–38.CrossRefGoogle ScholarPubMed
[8] Clifton, K. H., Tanner, M. A. and Gould, M. N. (1986) Assessment of radiogenic cancer initiation frequency per clonogenic rat mammary cell in vivo. Cancer Res. 46, 2390–2395.Google ScholarPubMed
[9] Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
[10] Frankenberg-Schwager, M. (1989) Review of repair kinetics for DNA damage induced in eukaryotic cells in vitro by ionizing radiation. Radiother. Onc. 14, 307–320.CrossRefGoogle ScholarPubMed
[11] Kennedy, A. R. (1985) Evidence that the first step leading to carcinogen-induced malignant transformation is a high frequency, common event. In Carcinogenesis: A Comprehensive Survey. Vol. 9: Mammalian Cell Transformation: Mechanism of Carcinogenesis and Assays for Carcinogenesis. ed. Barrett, J. C. and Tennant, R. W.. Raven, New York. pp. 455–464.Google Scholar
[12] Kimmel, M. and Flehinger, B. J. (1991) Nonparametric estimation of the size-metastasis relationship in solid cancers. Biometrics 47, 987–1004.CrossRefGoogle ScholarPubMed
[13] Klebanov, L. B., Rachev, S. T. and Yakovlev, A. Yu. (1993) A stochastic model of radiation carcinogenesis: Latent time distributions and their properties. Math. Biosci. 113, 51–75.CrossRefGoogle ScholarPubMed
[14] Klebanov, L. B., Rachev, S. T. and Yakovlev, A. Yu. (1993) On the parametric estimation of survival functions. Statist. Decisions 3, 83–102.Google Scholar
[15] Klein, J. P. and Bartoszynski, R. (1991) Estimation of growth and metastatic rates of primary breast cancer. In Mathematical Population Dynamics. ed. Arino, O., Axelrod, D. E. and Kimmel, M.. Marcel Dekker, New York, pp. 397–412.Google Scholar
[16] Kopp-Schneider, A., Portier, C. J., and Rippmann, F. (1991) The application of a multistage model that incorporates DNA damage and repair to the analysis of initiation/promotion experiments. Math. Biosci. 105, 139–166.CrossRefGoogle Scholar
[17] Myasnikova, E. M., Rachev, S. T. and Yakovlev, A. Yu. (1995) Queueing models of potentially lethal damage repair in irradiated cells. Math. Biosci. To appear.CrossRefGoogle Scholar
[18] Pericchi, L. R., Sansó, B. and Smith, A. F. M. (1993) Posterior cumulant relationships in Bayesian inference involving the exponential family. J. Amer. Statist. Assoc. 88, 1419–1426.CrossRefGoogle Scholar
[19] Rachev, S. T. (1991) Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester.Google Scholar
[20] Rochefordiere, A., Asselain, B., Campana, F., Scholl, S. M., Fenton, J., Vilcoq, J. R., Durand, J.-C., Pouillart, P., Magdelenat, H. and Fourquet, A. (1993) Age as prognostic factor in premenopausal breast carcinoma. Lancet 341, 1039–1043.CrossRefGoogle ScholarPubMed
[21] Tobias, C. A., Blakely, E. A., Ngo, F. Q. H., and Yang, T. C. Y. (1980) The repair-misrepair model of cell survival. In Radiation Biology in Cancer Research. ed. Meyn, R. E. and Withers, H. R.. Raven, New York. pp. 195–229.Google Scholar
[22] Tsodikov, A. D., Asselain, B., and Yakovlev, A. Y. (1997) A distribution of tumor size at detection: An application to breast cancer data. Biometrics. In press.CrossRefGoogle Scholar
[23] Tyurin, Yu. N., Yakovlev, A. Yu., Shi, J., and Bass, L. (1995) Testing a model of aging in animal experiments. Biometrics 51, 363–372.CrossRefGoogle Scholar
[24] Yakovlev, A. Yu and Polig, E. (1996) A diversity of responses displayed by a stochastic model of radiation carcinogenesis allowing for cell death. Math. Biosci. 132, 1–33.CrossRefGoogle ScholarPubMed
[25] Yakovlev, A. and Tsodikov, A. Yu. (1996) Stochastic Models of Tumor Latency and Their Biostatistical Applications. World Scientific, Singapore.Google Scholar
[26] Yakovlev, A. Yu., Tsodikov, A. D., and Anisimov, V. N. (1995) A new model of aging: Specific versions and their application. Biom. J. 37, 435–448.CrossRefGoogle Scholar
[27] Yakovlev, A. Yu., Tsodikov, A. D. and Bass, L. (1993) A stochastic model of hormesis. Math. Biosci. 116, 197–219.CrossRefGoogle Scholar
[28] Yakovlev, A. Yu. and Zorin, A. V. (1988) Computer Simulation in Cell Radiobiology. (Lecture Notes in Biomathematics 74.) Springer, Berlin.CrossRefGoogle Scholar
[29] Yang, G. L. and Chen, W. (1991) A stochastic two-stage carcinogenesis model: A new approach to computing the probability of observing tumor in animal bioassays. Math. Biosci. 104, 247–258.CrossRefGoogle Scholar
[30] Zelen, M. and Feinleib, M. (1969) On the theory of screening for chronic disease. Biometrika 56, 601–614.CrossRefGoogle Scholar
[31] Zhu, L. X. and Hill, C. K. (1991) Repair of potentially mutagenic damage and radiation quality. Radiat. Res. 127, 184–189.CrossRefGoogle ScholarPubMed
[32] Zolotarev, V. M. (1986) Modern Summation Theory for Independent Random Variables. Nauka, Moscow. (In Russian.) Google Scholar