Bulletin of Mathematical Biology

, Volume 56, Issue 6, pp 1121–1141 | Cite as

Intrinsic time scaling in survival analysis: Application to biological populations

  • T. Eakin
Article
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Revised:

Abstract

A method of dimensionless time-scaling based on extrinsic expectation of life at birth but intrinsic to a system generating a survival distribution is introduced. Such scaling allows the survival fraction function and its associated mortality function to serve as Green's functions for their generalized equivalents. i.e. a “population” function and a “death” function. The analytical mechanics of utilizing these concepts are formulated, applied to the classical Gompertz and Weibull survival models, and discussed with respect to biological relevance.

Keywords

Nematode Population Survival Distribution Gompertz Model Birth Function Biological Population 
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Literature

  1. Carey, J. R., P. Liedo, D. Orozco and J. W. Vaupel. 1992. Slowing of mortality rates at older ages in large medfly cohorts.Science 258, 457–461.Google Scholar
  2. Curtsinger, J. W., H. H. Fukui, D. R. Townsend and J. W. Vaupel. 1992. Demography of genotypes: Failure of the limited life-span paradigm inDrosophila melanogaster.Science 258, 461–463.Google Scholar
  3. Davis, P. J. 1965. Gamma function and related functions. InHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun (Eds), pp. 253–293. New York: Dover Publications, Inc.Google Scholar
  4. Eakin, T. 1992. An intrinsic time scaling for survival analyses of biological populations. UT-CHPC Technical Report Series, University of Texas, Austin # CHPC TR-1992-0009.Google Scholar
  5. Eakin, T., R. Shouman, Y. Qi, G. Liu and M. Witten. 1992. Issues of parametric estimation in multiparametric gerontologic survival models. UT-CHPC Technical Report Series, University of Texas, Austin # CHPC TR-1992-0002.Google Scholar
  6. Eakin, T. and M. Witten. 1993. Intrinsic prolate rectangularity in transition regions of survival curves: How square is the survival curve of a given species? UT-CHPC Technical Report Series, University of Texas, Austin # CHPC TR-1993-0006.Google Scholar
  7. Gompertz, B. 1820. A sketch on the analysis and the notation applicable to the value of life contingencies.Phil. Trans. Roy. Soc. 110, 214–294.Google Scholar
  8. Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality and on a new mode of determining the value of life contingencies.Phil. Trans. Roy. Soc. 115, 513–585.Google Scholar
  9. Gompertz, B. 1862. A supplement to two papers published in the transactions of the Royal Society, “On the science connected with human mortality”, the one published in 1820 and the other in 1825.Phil. Trans. Roy. Soc. 52, 511–559.Google Scholar
  10. Iwasaki, K., C. A. Gleiser, E. J. Masoro, C. A. McMahan, E. J. Seo and B. P. Yu. 1988a. The influence of dietary protein source in longevity and age-related disease processes of Fischer rats.J. Gerontol. Biol. Sci. 43, B5-B12.Google Scholar
  11. Iwasaki, K., C. A. Gleiser, E. J. Masoro, C. A. McMahan, E. J. Seo and B. P. Yu. 1988b. Influence of the restriction of individual dietary components on longevity and age related disease of Fischer rats: The fat component and the mineral component.J. Gerontol. Biol. Sci. 43, B13-B21.Google Scholar
  12. Izmaylov, D. M., L. K. Obukhova, O. V. Okladnova and A. P. Akifyev. 1993a. Phenomenon of life span instability inDrosophila melanogaster: I. Nonrandom origin of life span variations in successive generations.Expl. Gerontol. 28, 169–180.CrossRefGoogle Scholar
  13. Izmaylov, D. M., L. K. Obukhova, O. V. Okladnova and A. P. Akifyev. 1993b. Phenomenon of life span instability inDrosophila melanogaster: II. Change in rhythm of natural variations of life span after single exposure to γ-irradiation.Expl. Gerontol. 28, 180–194.Google Scholar
  14. Johnson, T. E. 1990. Increased life-span ofage-1 mutants inCaenorhabditis elegans and lower Gompertz rate of aging.Science 249, 908–912.Google Scholar
  15. Kenyon, C., J. Chang, G. Gensch, A. Rudner and R. Tabtiang. 1993. AC. elegans mutant that lives twice as long as wild type.Nature 366, 461–464.CrossRefGoogle Scholar
  16. Lin, C. C. and L. A. Segel. 1988.Mathematics Applied to Deterministic Problems in the Natural Sciences. Philadelphia: Society for Industrial and Applied Mathematics.MATHGoogle Scholar
  17. Liu, G., R. Shouman, Y. Qi, T. Eakin and M. Witten. 1992. GAIA Project. Access and submission to the multispecies survival database. UT-CHPC Technical Report Series. University of Texas Austin # CHPC TR-1992-0012.Google Scholar
  18. McKendrick, A. G. 1926. Application of mathematics to medical problems.Proc. Edin. Math. Soc. 44, 98–130.CrossRefGoogle Scholar
  19. Metz, J. A. J. and O. Diekmann. 1986. Age dependence. InThe Dynamics of Physiologically Structured Populations, J. A. J. Metz and O. Diekmann (Eds), pp. 136–184. New York: Springer-Verlag.Google Scholar
  20. Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery. 1992.Numerical Recipes in C. The Art of Scientific Computing, 2nd edn. Cambridge: Cambridge University Press.MATHGoogle Scholar
  21. Shimokawa, I., Y. Higami, G. B. Hubbard, C. A. McMahan, E. J. Masoro and B. P. Yu. 1993. Diet and the suitability of male Fischer 344 rat as a model for aging research.J. Gerontol. Biol. Sci. 48, B27-B32.Google Scholar
  22. Shouman, R. and M. Witten. 1993. Survival estimates and sample size: What can we conclude? UT-CHPC Technical Report Series, University of Texas, Austin # CHPC TR-1993-0005.Google Scholar
  23. Slater, L. J. 1965. Confluent hypergeometric functions. InHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun (Eds), pp. 503–535. New York: Dover Publications, Inc.Google Scholar
  24. van der Laan, C. G. and N. M. Temme. 1984.Calculations of Special Functions: The gamma function, the exponential integrals and error-like functions. Amsterdam: Mathematisch Centrum.Google Scholar
  25. Weibull, W. 1951. A statistical distribution of wide applicability.J. Appl. Mechanics 18, 293–297.MATHGoogle Scholar
  26. Witten, M. 1985. Predicting maximum lifespan and other lifespan variables in lifespan studies: Some useable reslts and formulae.The Gerontologist 25, 215.Google Scholar
  27. Witten, M. 1988. A return to time, cells, systems, and aging: V. Further thoughts on Gompertzian survival dynamics—the geriatric years.Mechanism of Ageing and Development 46, 175–200.CrossRefGoogle Scholar
  28. Witten, M. 1989. Quantifying the concepts of rate and acceleration/deceleration of aging.Growth, Development and Aging 53, 7–16.Google Scholar
  29. Witten, M. 1992. Might stochasticity and sampling variation be a possible explanation for variation in clonal population survival curves. UT-CHPC Technical Report Series. University of Texas, Austin # CHPC TR-1992-0006.Google Scholar
  30. Yu, B. P., E. J. Masoro, I. Murata, H. A. Bertrand and F. T. Lynd. 1982. Lifespan study of SPF Fischer 344 male rats fedad libitum or restricted diets: longevity, growth, lean body mass and disease.J. Gerontol. 37, 130–141.Google Scholar
  31. Yu, B. P., E. J. Masoro and C. A. McMahan. 1985. Nutritional influences on aging of Fischer 344 rats: I. Physical, metabolic, and longevity characteristics.J. Gerontol. 40, 657–670.Google Scholar

Copyright information

© Society for Mathematical Biology 1994

Authors and Affiliations

  • T. Eakin
    • 1
  1. 1.Balcones Research CenterUniversity of Texas System, Center for High Performance Computing, Department of Applications Research and DevelopmentAustinU.S.A.

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