Predictive formulas expressing relationship between dose rate and survival time in total body irradiation in mice

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.


INTRODUCTION
The Gompertz model (1825) 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 [1][2][3].
The most commonly used methods of parametric estimation for distributions of survival times are the fittings of exponential, lognormal, Weibull, gamma and Gompertz function of survival time.

The "Probacent" Model
On the basis of experimental observations on animals, clinical applications and mathematical reasoning, the author developed a general mathematical model of "probacent"-probability equation that may be applicable as a general approximation method to possibly calculate the probability of safe survival in humans and other living organisms exposed to any harmful or adverse circumstances in overcoming the risk, and further to predict degrees of risk and/or mortality probability in terms of percent probability.In this way, the "probacent" model might make useful predictions of probable outcomes in a variety of biomedical phenomena in protecting exposed subjects [9][10][11][12].
The model of "probacent"-probability equation expressed by Eq.1 was constructed from experimental studies on animals to express survival probability in mice exposed to g-force in terms of magnitude of acceleration and exposure time [9,13]; and to express a relationship among intensity of stimulus or environmental agent (such as drug [9,10,14], heat [15], pH [16], and electroshock [15,17]), duration of exposure and biological response in animals.
The model has been applied to data in the literature to express a relationship among dose rate, duration of exposure and mortality probability in total body irradiation in humans [12,18]; to express carboxyhemoglobin levels of blood as a function of carbon monoxide concentration in air and duration of exposure [19,20]; to express a relationship among plasma acetaminophen concentration, time after ingestion and occurrence of hepatotoxicity in man [21,22]; to predict survival probability in patients with malignant melanoma [23,24]; to predict survival probability in patients with heart transplantation [25]; to express a relationship among age, height and weight, and percentile in Saudi and US children of 6 -16 years of age [26,27]; to predict the percentile of heart weight by body weight from birth to 19 years of age [28,29]; and to predict the percentile of serum cholesterol levels by age in adults [30,31].
The model was applied to the United States life tables, 1992 and 2001 reported by the National Center for Health Statistics (NCHS) to construct formulas expressing age-specific survival probability, death rate and life expectancy in US adults, men and women [11,[32][33][34].
The formula of survival probability is expressed by the following "probacent"-probability equation, Eq.1.
where T = time after biomedical insult, diagnosis of cancer or age; P = "probacent" (abbreviation of probability percentage) = relative biological amount of "reserve" for survival; "probacent" (P) of 0, 50 and 100 corresponds to mean -5 S.D., mean and mean + 5 S.D., respectively; one "probacent" is equivalent to 0.1 S.D. in a normal distribution.In addition, 0, 50 and 100 "probacent" seem to correspond to 0, 50 and 100 percent probability in mathematical prediction problems in terms of percentage.Therefore, the survival probability could be used to predict probabilities in general biomedical phenomena."probacent" values are obtainable from a list of conversion of percent probability into "probacent" that was published by the author (Table 6 of Ref. [9] and Table 4 of Ref. [10]).γ, A and B are constants; A is an intercept and B a slope; γ represents a curvature (a shape of a curve) and expressed by the following equation: If the value of γ becomes equal to one, Eq.1 represents a log-normal distribution.Eq.1 is considered to be fundamentally based on the Gaussian normal distribution.
where D represents death rate in percentage (mortality probability); T is time or age; c, a and b are constants; c represents a curvature (a shape of curve) like γ in Eq.1a; a is an intercept and b a slope.
If the value of constant c becomes equal to one, Eq.2 is essentially similar to the Weibull distribution [1].
Eq.2 was applied to express death rates in US adults [11,33,34].It was found to better express death rates in US elderly population than the Gompertz, the exponential and the Weibull distributions [11].
Mehta and Joshi [35] successfully applied the "probacent"-probability equation, Eqs.1 and 2 to use modelderived data as an input for radiation risk evaluation of Indian population.
A clear and exact quantitative relationship between dose of radiation and mortality in humans is still not known because of lack of human data that would enable to determine LD 50 for humans in total body irradiation.Analysis of human data has been primarily from radiation accidents, radiotherapy and the atomic bomb victims.Consequently, laboratory animals have been used to investigate the relationship between radiation dose and effect in total body irradiation and further to possibly derive a general predictive formula [9,[36][37][38][39][40].
The author has applied Eq.2 to predict mortality probability in total body irradiation without medical support in humans as a function of dose rate and duration of exposure [12]; the formula of the function is constructed on the basis of animal-model predicted data published by Cerveny, MacVittie and Young [18].There is a remarkable agreement between formula-predicted and published estimated LD 50 and also between both mortality probabilities (p value > 0.995).

The Gompertz Modelod
Helligman and Pollard [2] reported that the Gompertz model applied to the Australian national mortality data required a mathematical modification for a curvature noticed in his graphical analysis of the older age population.
Sacher published a comprehensive experimental data on daily dose rates versus average survival times in total body irradiation in mice [36,37].He stated in his discussion that the Gompertz model seemed to be approximately applicable to the data on dose rates versus death rates in mice.However, he did not present a general formula of the Gompertz model in his articles [36,37].
The purpose of this study is to apply each of both the "probacent" and the Gompertz models to the Sacher's reported data above described with an assumption that each model is applicable to the data, and to test which of the models would better fit the reported data on doses versus survival times in mice on the basis of results of statistical analysis.

MATERIALS AND METHODS
Sacher of Argonne National Laboratory, USA reported his comprehensive experimental data on mean aftersurvival times (MAS) of adult LAF1 mice irradiated daily with various doses for the duration of life, beginning at 100 days of age until deaths occurred [36,37].The total number of mice was 4692; 2348 male and 2344 female mice.The range of dose rate is from 6 to 2500 r/day.The MAS is from 5 to 548 days; 5 -6 days at high dose rates (≥1100 r/day) and 501 -548 days at the lowest dose rate, 6 r/day.
The author used the Sacher's reported data [37] to construct general formulas to express relationships between dose rates and mean after-survival times in total body irradiation of γ-ray in male and female mice.
The data on the relationship between dose rate and mean after-survival time in both male and female mice are shown in Table 1 and plotted on a log-log graph paper as illustrated in Figure 1 to improve the overall fit in mathematical analysis.
A closer look at the lines connecting data points, closed and open circles in Figure 1, respectively suggests that there appear to exist roughly four different periods of the first two weeks after beginning of radiation (range of 2500 -330 r/day) in an acute period, two weeks to one month (range: 330 -125 r/day) in an early subacute period, one to five months (range: 125 -43 r/day) in a late subacute period, and five to 18 months (range: 43 -6 r/day) in a chronic period.Each period would have different values of constants in the "probacent" and Gompertz models.
Both models are applied to the Sacher's data with an assumption that dose rates reflect death rates as shown in the relationship among dose rate, duration of exposure and mortality probability in total body irradiation in humans [12], and that both models would be applicable to the data.

Formulas of Mean After-Survival Time (MAS)
The mathematical method to construct formulas of the "probacent" model is described in Appendix and the author's previous articles [33,41].The best-fitting c value is determined by a statistical method of the least sum of squares of curved regression described in the author's previous publication [42].Appendix also describes how to construct formulas of the Gompertz model.

Formulas of "Probacent" Model
Eqs.3-5 with different values of constants, a, b and c are constructed to express relationships between dose rate and mean after-survival time in acute, early subacute, late subacute and chronic periods in male and female mice, respectively.
where R = daily dose rate in r/day; T = mean after-survival time (MAS) in days; a, b and c are constants.
The "probacent" equations for male mice.Acute period (2500 -330 r/day): where R represents daily dose of total bo r/day; T is mean after-survival time (M -survival tim The Gompertz equations for female mice.Acute period (2500 -330 r/day): Early subacute period (330 -125 r/day):

ogram
The computer programs were written in UBASIC IBM PC microcomputer and compatibles for Eqs. 1  In the author's previous studies, the computer program for Eq.1 used a formula of approximation instead of the integral of Eq.1b because the computer cannot perform integral [10,41].Mathematical transformation of the formula of integral, Eq.1b to the formula of approximation in computer programming is described in the author's book [41].In this study, computer programs are used to calculate each equation of the "probacent" and Gompertz models.

Statistical Analysis
A χ 2 goodness-of-f test the fit of mathematical m after-survival times in mice in the Sacher's article [37].The differences are considered statistically significant when p < 0.05.The least square curved regression method described in the author's previous publication [42] is used to determine the best-fitting c value in Eqs.3-5 of the "probacent" model.

RESULTS
Table 1 shows the results of formu ter-survival times total body irradiation in male and female mice.Table 1 also shows comparison of the formula-derived values with the Sacher's reported data on MAS [37].
Differences between both values of formula-derived and reported MAS are statistically not significant (p > 0.995).A close agreement is seen between both values in Table 1.The maximum difference in MAS of acute period of high dose rate, 2500 -330 r/day is ± 1 day in both male and female mice.
Figure 1 illustrates the relationship between dose rate and mean after-survival time (MAS).It seems to the author that the distribution of the Sacher's reported MAS (closed and open circles) are very close to the formuladerived curved line for male and female mice, respectively.
Table 2 shows comparison of the least sum of squares, Σ(E -O) 2 and the least maximum difference, I(E -O)I in the "probacent" and Gompertz models employed in analysis of the Sacher's data on dose rate versus MAS in male and female mice.Here, E is a formula-derived value; O is a Sacher's reported value.
Figures 2 and 3 illustrate relationships between dose rates and mean after-survival times, and two lines: one solid curved line expressing the "probacent" equations and the dashed straight line expressing the Gompertz equations, in male and female mice, respectively.
The least sum of squares of differences and the least maximum difference of the "probacent" model are much smaller than those of the Gompertz model as shown in Table 2.The differences are very large in the late subacute period in both male and female mice; in the same period, the solid line expressing the "probaccent" equation is remarkably curved in contrast to the dashed straight line of the Gompertz equation.
The χ 2 -test p value is greater than 0.995 for the "probacent" formulas for both male and female mice, respectively.In contrast, p value is less than 0.05 for the Gompertz equations for male and female mice, respectively.These results of statistical analysis seem to indicate that the Gompertz model appears oversll not applicable Relationship between dose rate of daily radiation and mean after-survival time (MAS) in LAF1 male and female mice irradiated daily during the duration of life [37].The abscissa represents mean after-survival time in days (log scale) and the ordinate daily dose rate of radiation in r/day (log scale).Data points of closed and open circles indicate MAS of male and female mice, respectively.The solid and dashed curved lines represent MAS of both male and female mice, predicted by Eqs.3-5 of the "probacent" model, respectively.Data points of the Sacher's reported values appear to fall overall very close to or on the formula-derived predicted lines (see text).
Copyright © 2011 SciRes.JBiSE Figure 2. Relationship between dose rate and mean after-survival time (MAS) in LAF1 male mice irradiated daily during the duration of life [37].The abscissa represents mean after-survival time in days and the ordinate daily dose rate of radiation in r/day (lo scale).Data points of closed circles indicate reported MAS of male mice.The solid, curved line and the dashed, straight lines repre g sent MAS of male mice, predicted by the "probacent" model of Eqs.3-5 and the Gompertz model of Eqs.9-12, respectively.Data points appear to overall fall closer to or on the "probacent"-formula-predicted solid, curved line than the lines predicted by the Gompertz equations.

JBiSE
Table 2. Comparison of the least sum of squares, ∑(E -O) 2  and the least maximum-difference, І(E -O)І, in the "proacent" and Gompertz models employed in analysis of the Sa-b cher's data on daily radiation dose versus mean survival time in male and female mice. .p-valu e mice, re spectively; E: formula-derived value of mean after-survival time for male and female mice, respectively; O: Sacher's reported value of mean after-survival time for male and female mice, respectively.
to the Sacher's data, and that the "probacent" model of eath rate, Eq.2 seems to be applicable and d to the reported data than the Gompertz model.

DISCUSSION
Table 1 and Figure 1 reveal a close agreemen the formula-derived v data on mean after-survival times (MAS) (p > 0.995).Table 2 shows that the "probacent" equations, Eqs.3-5 have noticeably smaller least sum of squares and least maximum difference than the Gompertz equations.This finding indicates that the "probacent" model is statistically better fitting to the Sacher's data.The p-value model appears overall not applicable to the Sacher's data.
Data-points of male and female mice appear overall fall much closer to or on the solid curved line represented by the "probacent" equations than the dashed straight lines represented by the Gompertz equations in Figures 2 and 3, respectively.is >0.995 for the "probacent" equations and <0.05 for the Gompertz equations, suggesting that the Gompertz The author feels that in a variety of biomedical phenomena, if Eqs.1 and 2 are applicable, the values of constants γ and c are generally greater than one or less than one but not one, indicating a curved line when plotted on a X-Y graph paper as seen Figures 1-3.The γ and c values are relatively rarely one, indicating a straight line on the graph or otherwise approximately appearing straight.This phenomenon seems to be analogous in physics to that light path is actually curved when passing through a gravitational field of space but appears straight [44,45].
If the γ value becomes equal to one, Eq.1 represents a log-normal distribution.If the c value is one, Eq.2 that is derivable from Eq.1 [33] becomes essentially similar to the Weibull distribution [1].If the base of logarithm is one, the lognormal distribution becomes a norml distribu n tion (log 1 1 = n) [41,46].If the logarithm of one as its base is taken for X axis of time, the Gompertz distribution might be similar to the Weibull distribution, and so might be a specific form of the "probacent"-probability equation.It seems to the author that the "probacent" model may be applicable as a general approximation method to make useful predictions of probable outcomes in a variety of biomedical phenomenon [41,42].
Hematopoietic cells of bone marrow, intestinal tract and central nervous system are most vulnerable to radiation effects [47][48][49][50][51][52].Body responses to lethal radiation effects reflect status of living body in which pathologic changes, physiologic repair (response and regeneration) and inherent aging process are concurrently occurring [37].Death is caused by multi-organ failure.In case of high dose, infection and hemorrhage are earliest contributing factors to death in lethal total body irradiation, damaging most sensitive hematopoietic cells of bone marrow [37,47,52].
High dose rates (330 -2500 r/day) shorten mean after-survival times (MAS) in acute period, causing death within two weeks (Table 1).Decrease in dose rates increases MAS in mice.Cui and his coworkers demonstrated that a fractionated total body irradiation (FTBI) increased survival rates and therapeutic effects of FTBI in bone marrow transplantation in mice [53].This finding is remarkably illustrated by the curved and prolonged line of MAS in the chronic period of very low dose rates (Figures 1-3).
Biologic responses to lethal radiation effects are dependent on radiation dose rate and duration of exposure, and are reflected in survival times.The values of constants, a, b and c of Eq.2 of death rate seem to be determined by the underlying status of biologic responses.
The "probacent" model and findings in this stu m

CONCLUSIONS
The "probacent" model of death rate equation is applied to the Sacher's experimental data on dose rates versus mean after-survival times (MAS) in mice daily irradiated during the duration of life [37].The Gom also applied to the data.structed by both models an confit to data.
In this comparative study of which model would better fit to data, it is found that the "probacent" equations not only well fit the data but also more closely express the relationship between dose rate and mean after-survival time than the Gompertz equations.The Gompertz model a ronic period (higher and lower doses) but seems to be overall not applicable.
The data-points connecting line is actually curved as seen in Figures 1-3, suggesting better fit of the curvedline expressing "probacent" model rather than the straight-line expressing Gompertz model.
The "probacent" model and findings in this study might-be hopefully hel ce to low dose rates for long durations of exposure in total body irradiation, and further in research in a variety of biomedical phenomena.

Figure 1 .
Figure 1.Relationship between dose rate of daily radiation and mean after-survival time (MAS) in LAF1 male and female mice irradiated daily during the duration of life[37].The abscissa represents mean after-survival time in days (log scale) and the ordinate daily dose rate of radiation in r/day (log scale).Data points of closed and open circles indicate MAS of male and female mice, respectively.The solid and dashed curved lines represent MAS of both male and female mice, predicted by Eqs.3-5 of the "probacent" model, respectively.Data points of the Sacher's reported values appear to fall overall very close to or on the formula-derived predicted lines (see text).

Figure 3 .
Figure 3. Relationship between dose rate and mean after-survival time (MAS) in LAF1 female mice irradiated daily during the duration of life [37].The abscissa represents mean after-survival time (MAS) in days and the ordinate daily dose rate of radiation in r/da (log scale).Data points of closed circles indicate reported MAS of female mice.The solid, curved line and the dashed, straight line y s represent MAS of female mice, predicted by the "probacent" model of Eqs.3-5, and by the Gompertz model of Eqs.13-16, respectively.Data points of the Sacher's reported values appear to overall fall closer to or on the "probacent"-formulas-predicted solid, curved line than the Gompertz-formulas-predicted dashed, straight lines.
pertz model is Predictive formulas are d compared regarding their ppears approximately applicable in acute and ch pful in research in human toleran the author's current study is based.rth for his outstanding personal valuable instr .(2003) Statistical methods for dy ight be hopefully helpful in research to investigate 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.

Table 1 .
Mean after-survival times (MAS) of male and female mice irradiated daily for duration of lif ing at 100 days of age