The Gompertz model is the long-time well-known mathematical model of exponential expression among mortality models in the literature that are used to describe mortality and survival data of a population. The death rate of the “probacent” model developed by the author based on animal experiments, clinical applications and mathematical reasoning was applied to predict age-specific death rates in the US elderly population, 2001, and to express a relationship among dose rate, duration of exposure and mortality probability in total body irradiation in humans. The results of both studies revealed a remarkable agreement between “probacent”-formula-predicted and published-reported values of death rates in the US elderly population or mortality probabilities in total body irradiation in humans (p - value > 0.995 in χ² test in each study). In this study, both the Gompertz and “probacent” models are applied to the Sacher’s comprehensive experimental data on survival times of mice daily exposed to various doses of total body irradiation until death occurs with an assumption that each of both models is applicable to the data. The purpose of this study is to construct general formulas expressing relationship between dose rate and survival time in total body irradiation in mice. In addition, it is attempted to test which model better fits the reported data. The results of the comparative study revealed that the “probacent” model not only fit the Sacher’s reported data but also remarkably better fit the reported data than the Gompertz model. The “probacent” model might be hopefully helpful in research in human tolerance to low dose rates for long durations of exposure in total body irradiation, and further in research in a variety of biomedical phenomena.
KeywordsLethal Radiation DoseTotal Body IrradiationFormula of Survival Time in MiceDose-Survival Curve“Probacent” ModelGompertz Model
Lee, E.T. and Wang. J.W. (2003) Statistical methods for survival data. John Wiley & Sons, Hoboken, pp. 8-197. doi:10.1002/0471458546.ch2
Heligman, L. and Pollard, J.H. (1980) The age pattern of mortality. Journal of Inst. Actuaries, 107, 49-80.
Gompertz, B. (1994) Parametric models. In: Statistical methods in medical research, Blackwell Science, Cambridge, pp. 482-483.
Kaplan, E.L. and Meier, P. (1958) Nonparametric estimation for incomplete observations. Journal of American Statistical Association, 53, 457-481. doi:10.2307/2281868
Gordis, L. (2004) The Kaplan-Meier method. In: Epidemiology, Elsevier Saunders, Philadelphia, pp. 104-106.
Mould, R.F. (1976) Calculation of survival rates by the life Table and other methods. Clinical Radiology, 27, 33-38. doi:10.1016/S0009-9260(76)80011-6
American Joint Commission (1983) Reporting of cancer survival and results. In: Manual for Staging of Cancer. Lippincott, New York, pp. 11-21.
Cox, D.R. and Oaks, D. (1985) Distribution of failure times; parametric Statistical analysis: Simple sample; single-sample nonparametric methods. In: Analysis of survival data, Chapman and Hall, London, pp. 13-61.
Chung, S.J. (1960) Studies on a mathematical relationship between stress an response in biological phenomena. Republic of Korea Journal of the National Academy of Sciences, 2, 115-162.
Chung, S.J. (1986) Computer-assisted predictive mathematical relationship among metrazol dose and time and mortality in mice. Computer Methods and Programs in Biomedicine, 22, 275-284. doi:10.1016/0169-2607(86)90004-0
Chung, S.J. (2007) Computer-assisted predictive formulas expressing survival probability and life expectancy in US adults, men and women, 2001. Computer Methods and Programs in Biomedicine, 86, 197-209. doi:10.1016/j.cmpb.2007.02.009
Chung, S.J. (2011) Predictive formulas expressing relationship among dose rate, duration of exposure and mortality probability in total body irradiation in humans. Journal of Biomedical Science and Engineering, 4, 497- 505. doi:10.4236/jbise.2011.47063
Chung, S.J. (1959) Studies of positive radial acceleration on mice. Journal of Applied Physiology, 14, 52-54.
Boak, H. and Chung, S.J. (1962) Studies on a relationship between dose, time and percentage of occurrence of response and a method of evaluation of combined action in drugs. The New Medical Journal, 5, 35-82.
Kim, C.C. and Chung, S.J. (1962) Studies on a relationship between stress, duration of exposure and percentage of response in goldfish to single, double, and triple stresses of acceleration, electroshock, heat, chemical and osmotic stimuli. Republic of Korea Theses of Catholic Medical College, 5, 257-336.
Cho, D.W. and Chung, S.J. (1961) Studies of tolerance of Paramecium caudatum to hydroxyl ions. Bulletin of Yamaguchi Medical School, 8, 151-160.
Chung, S.J. (1989) Computer-assisted mathematical relationship among electroshock voltage and duration and occurrence of convulsion in mice. Computer Methods and Programs in Biomedicine, 28, 23-30. doi:10.1016/0169-2607(89)90177-6
Cerveny, T.J., MacVittie, T.J. and Young, R.W. (1989) Acute radiation syndrome in humans. In: R.J. Walker and T.J. Cerveny, Ed., Medical consequences of nuclear warfare, TMM Publishers, Office of the Surgeon General, Falls Church, Virginia, 15-36.
Forbes, W.H., Sergent, F. and Roughton, F.J.W. (1988) The risk of carbon monoxide uptake by normal men. American Journal of Physiology, 143, 594-608.
Chung, S.J. (1988) Formula predicting carboxyhemoglobin resulting from carbon monoxide exposure. Veterinary and Human Toxicology, 30, 528-532.
Prescott, L.F. Roscoe, P., Wright, N. and Brown, S.S. (1991) Plasma paracetamol half-life and hepatic necrosis in patients with paracetamol overdosage. Lancet, 1, 519- 522.
Chung, S.J. (1989) Computer-assisted predictive mathematical relationship among plasma acetaminophen concentration, time after ingestion and occurrence of hepatotoxicity in man. Computer Methods and Programs in Biomedicine, 28, 37-43. doi:10.1016/0169-2607(89)90179-X
Chung S.J. (1991) Formula predicting survival in patients with invasive malignant melanoma. International Journal of Biomedical Computing, 28, 151-159. doi:10.1016/0020-7101(91)90051-F
Chung, S.J. (1994) Formula expressing relationship among lesion thickness, time after diagnosis and survival probability in patients with malignant melanoma. International Journal of Biomedical Computing, 37, 171-180. doi:10.1016/0020-7101(94)90139-2
Chung, S.J. (1993) Formula predicting survival probability in patients with heart transplantation. International Journal of Biomedical Computing, 32, 211-221. doi:10.1016/0020-7101(93)90015-X
Magbool, G., Kaul, K.K., Corea, J.R., Osman, M. and Atfaj, A. (1993) weight and height in Saudi children six to 16 years from the Eastern Province. Annals of Saudi Medicine, 13, 344-349.
Chung, S.J. (1994) Formulas expressing relationship among age, height and weight, and percentile in Saudi and US children of ages of 6-16 years. International Journal of Biomedical Computing, 37, 258-272. doi:10.1016/0020-7101(94)90124-4
Sholz, D.C., Kitzman, D.W., Hagen, P.T., Ilstrup, D.H. and Edwards. W. D. (1998) Age-related changes in normal heart during the first 10 decades of life, Part I. (Growth): A quantitative anatomic study of 200 specimens from subjects from birth to 19 years old. Mayo Clinic Proceedings, 13, 126-136, 637.
Chung, S.J. (1990) Formulas predicting the percentiles of heart weight by body weight in subjects from birth to 19 years of age. International Journal of Biomedical Computing, 26, 257-269. doi:10.1016/0020-7101(90)90049-Z
Chung, S.J. (1990) Formulas predicting the percentile of serum cholesterol levels by age in adults. Archives of Pathology and Laboratory Medicine, 114, 869-895.
Chung, S.J. (1992) Relationship among age, serum cholesterol level and population percentile in adults. International Journal of Biomedical Computing, 31, 99-116. doi:10.1016/0020-7101(92)90066-2
Arias, E. (2004) United States life Tables, 2001. National Vital Statistics Report, 52, 1-40.
Chung, S.J. (1995) Formulas expressing life expectancy, survival probability and death rate in life tables at various ages in US adults. International Journal of Biomedical Computing, 39, 209-217. doi:10.1016/0020-7101(94)01068-C
Chung, S.J. (1997) Comprehensive life tables of computer-assisted predictive mathematical relationship among age and life expectancy, survival probability or death rate in US adults. Computer Methods and Programs, 52, 67-73. doi:10.1016/S0169-2607(96)01778-6
Mehta, S.C. and Joshi, H.C. (2004) Model based point estimates of survival/death rate: An input for radiation risk evaluation in Indian context. Indian Journal of Nuclear Medicine, 19, 16-18.
Sacher, G.A. (1956) On the statistical nature of mortality, with especial reference to chronic radiation mortality, Radiology, 67, 250-257.
Sacher, G.A. (1959) On the radiation of lethality to radiation injury, and its relevance for the prediction problem. IXth International Congress of Radiology, München, 23- 30 December 1959.
Travis, E.L., Peters, L.J., Thames, H.D., et al. (1985) Effect of dose-rate on total body irradiation: Lethality and pathologic findings. Radiology and Oncology, 4, 341-351. doi:10.1016/S0167-8140(85)80122-5
Ellington, F. (1947) Influence of dose fractionation on the lethal X-ray effect produced by total body irradiation in mice. Radiology, 47, 238-241.
Grahn, D. (1958) Acute radiation response of mice from a cross between radio-sensitive and radio-resistant strains. Genetics, 43, 835-843.
Chung, S.J. (2009) Seeking a New World: A New Philosophy of Confucius and Kim Hang. iUniverse, Bloomington, pp. 68-76,153.
Chung, S.J. (2011) Computer program of nonlinear, curved regression for “probacent”-probability equation in biomedicine. Journal of Biomedical Science and Engineering, 4, 620-630. doi:10.4236/jbise.2011.49078
Dixon, W.J. and Massey Jr., F.J. (1957) Introduction to Statistical; Analysis, McGraw-Hill, New York, pp. 191- 227.
Einstein, A. (1905) The special theory of relativity. In: S. W. Hawking, A Brief History of Time, Bantam Books, New York, pp. 1-61.
Suplee, C (1999) Physics in the 20th Century, Hany N. Abrahams, Inc., New York, pp. 82-180.
Chung, S.J. (2010) The book of right change: A New Philosophy of Asia. iUniverse, Bloomington, p. 10.
Warren, S. (1961) The pathology of ionizing radiation, Charles C. Thomas Publisher, Springfield.
Li, X.H., Fu, D., Latif, N.H., et al. (2010) δ-tocotrienol protects mouse and human hematopoietic progenitors from γ-irradiation through extracellular signal-regulated kinase/mammalian target of rapamycin signaling. Hae- matologica, 95, 1996-2004. doi:10.3324/haematol.2010.026492
Komarova, E.A., Kondratov, R.V., Wang, K., et al. (2004) Dual effect of p53 on radiation sensitivity in vivo: p53 promotes hematopoietic injury, but protects from gastro-intestinal syndrome in mice. Oncogene, 23, 3265- 3271. doi:10.1038/sj.onc.1207494
Department of Radiology, University of Illinois (1998) Whole body irradiation: Lesson from Chernobyl. www.uic.edu/com/uhrd/manuscript/section4/section4.html.
Jones, T.D., Morris. M.D., Wells, S.M. and Young, R.W. (1986) Animal mortality resulting from uniform exposures to photon radiations: calculated ld50 and a compilation of experimental data, Oak Ridge National Laboratory, Oak Ridge.
Donnelly, E.H., Nembauser, J.B., Smith, J.M., et al. (2010) Acute radiation syndrome: Assessment and management. Southern Medical Journal, 103, 541-544. doi:10.1097/SMJ.0b013e3181ddd571
Cui, Y. Yang, G., Fan, T., et al. (2002) Optimal protocol for total body irradiation for allogeneic bone marrow transplantation in mice. Bone Marrow Transplantation, 30, 843-849. doi:10.1038/sj.bmt.1703766