- Email: [email protected]
The risk of leukemia from low doses of low-LET radiation
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and Computer
Modelling
33 (2001)
1307-1313 www.elsevier.nl/locate/mcm
The Risk of Leukemia from Low Doses of Low-LET Radiation M. ZAIDER Department Memorial 1275 York
of Medical
Sloan-Kettering
Avenue,
Physics Cancer
New York,
NY
Center
10021,
U.S.A.
[email protected]
Abstract-h this communication, we examine the evidence (if any) for a nonlinear dose response in relation to leukemia mortality in the Japanese A-bomb population. Specifically, we seek an estimate of the probability that, at low doses of radiation, the relative risk (RR) is smaller than one. Using Bayesian bootstrap techniques, we find that there is a 90% probability that-for at least One dose group at or below 200 mSv-there is a reduction of effect relative to the control group. We take this as evidence for radiation hormesis (end point: leukemia mortality) at low doses of radiation. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Leukemia
risk, A-bomb survivors,
Low doses of radiation,
Hormesis.
INTRODUCTION Leukemia
to ionizing radiation and it has been shown to be a such as the Japanese A-bomb survivors [l]. However, while the risk from substantial doses is quite generally recognized, there is uncertainty at lower doses that are of the order of several hundred milligray. principal
was the
hazard
In formulating
first
malignancy
in irradiated
radiation
attributed
populations
risks, the incidence
of deleterious
changes
is generally
related
to the
average dose (equivalent) in an irradiated organ. The “risk factor” is defined as the probability of an effect per unit of the average dose with the implication that, in the range of concern in it is irrelevant how radiation protection, the dose-effect curve is linear. With this assumption, the dose varies throughout the organ because each portion is affected in proportion to the dose. However, when the dose-effect relation is not linear, the response to a given average dose must of depend on the distribution of the dose within the organ. In the case of positive curvature t,he effect vs. dose plot, increasing nonuniformity must result in a higher effect probability. In t,he caSe where low doses of radiation reduce the spontaneous incidence, the dose response curve displays a quasithreshold; exposure to an average dose that is smaller than the threshold value can nevertheless result in a positive effect when the dose distribution is broad enough to extend sufficiently beyond the threshold value. This could lead to erroneous conclusions concerning the shape
of the response
curve at low doses.
I am indebted to Prof. H. H. Rossi and Prof. A. Yakovlev for their interest in this work and for liberal suggestions. This paper makes use of data obtained from the Radiation Effects Research Foundation (RERF) in Hiroshima, Japan. RERF is a private foundation funded equally by the Japanese Ministry of Health and Welfare and the U.S. Department of Energy through the U.S. Academy of Sciences. The conclusions in this paper
@ 2001 Elsevier
Science
Ltd. All rights reserved.
Typeset
by AM-m
1308
hf. ZAIDER
There is substantial evidence that the risk of leukemia increases more than proportionately with dose. The shape of the curve must depend on the irradiation pattern, but in the case of the Japanese A-bomb survivors, risks at 0.1 Sv and 1 Sv are said to differ by a factor of 20 [l]. In fact, an analysis by Shimizu et al. [2] appears to show a reduction of leukemia mortality at doses less than 0.2 Sv. A linear-quadratic fit yields a relative risk (RR) of about 0.75 at 100 mSv. They state that “it can not be said that there is a downward dose response in this dose range” because the “linear coefficient with minus sign in the linear-quadratic model is not statistically significant”. In this paper, we reexamine the Japanese A-bomb data for evidence of nonlinear response (here, leukemia mortality) at low doses of radiation. We do this by estimating the probability t,hat, for at least one dose group, the relative risk is less than one. Should this be indeed the case, we shall take it as an indication of hermetic response. The analysis presented here attempts to circumvent two common difficulties: (a) the fact that in traditional statistical analysis the probability distribution function (pdf) of an estimator obtained from a finite sample is usually not available; here we are interested in the probability that RR < 1, and (b) the limited Japanese data which-even with the first complication mentioned above removed-may not result in a conclusive answer for specific dose bins. To answer the first problem, we shall analyze the data using a method known as Bayesian bootstrap [3]; this approach, which is nonparametric, uses repeated sampling from the data to find the pdf of an estimator. If there are no dose points for which the probability that RR < 1 is significantly closer to 100% (e.g., Pr{RR < 1) > O.OO),we shall combine several dose points to calculate the probability that at least at one of them the relative risk is less than one. Thus, Pr{at least one RRi < 1; i 5 n} = 1 - Pr{RRi
> 1}Pr{RR2
2 1). . . Pr{RR,
2 1).
(1)
This would be taken as proof, or lack thereof, of a nonlinear response at low doses. Although possible mechanisms responsible for a red&ion of the cancer mortality at low doses have been discussed in the literature [4-81, its immediate cause (if indeed present) in human populations remains a matter of speculation. As far as the initiation stage is concerned, microdosimetric arguments show that if a tumor is monoclonal in origin, then at low doses the probability of the formation of a malignant cell must increase linearly with dose. However, inhibitory factors may prevent most of the transformed (or pretransformed) cells from developing into a tumor. A decrease in cancer mortality may occur if, for instance, (a) some of these factors are radiation dependent, and (b) they act selectively on transformed cells compared to normal cells. This will be the case if, for example, at low doses the number of preexisting malignant cells killed exceeds the number of normal cells transformed. Yakovlev and Polig [6] have developed a stochastic model that describes the competition between cell killing and tumor promotion. Covelli et al. [8] observed a striking case of reduction of the incidence of malignant lymphoma in mice irradiated with x-rays and neutrons. They interpreted their results in terms of a combination of cell transformation and cell killing. The x-ray data were fitted by M =
(~0+ azD”> emAD,
(2)
where A,4 is the mortality at dose D, a0 is the fractional control incidence, and u2 and X are constants. In this analysis, cell killing influences the dose-effect relation not only at large doses but also at small doses. This is especially evident in this case because a0 is large (0.56). Equation (2) provides a good fit to a curve which initially decreases between 0 and 1 Gy to about 80% of the control incidence, rises barely above it near 2 Gy, and then declines.
Risk of Leukemia
1309
DATA ANALYSIS The data used in this analysis were taken from the Life Span Study (LSS) Report 12, Detailed Cancer Mortality Data and Supplementary Tables (see [9], 1rereafter referred to as Report 12). Of the various factors used in this report to cross-classify the Japanese cohort, we retain here only city (Hiroshima or Nagasaki), follow-up interval (ten periods of between two and five years each), organ dose equivalent-the calculation of which is explained below, and cause of death. For each such group (cell) i, the information available is: number of deaths (di), gamma and neutron organ doses (D,,i, D,+),
person-years for all people ever at risk in the cell (PY,),
and the time
interval (Ati). For any particular cause of death, the probability of survival up to time tk (the end of the period Atk = tk - tk_i) was estimated using the actuarial method [lo]:
S(tk,D)
[I- &A,i]
= fi i=l
.
Here the quantity in square brackets represents the probability that a person who survived up to time ti_1 all risks will not succumb to a particular site-specific cancer death (here, leukemia) during Ati; and PY.i/A& is the average number of persons at risk during the interval At,. With this, the relative risk (RR) for the entire period [0, tk] is RR(tk,D)
=
1 - S(tk, D) = 0)’
1 - S(tk,D
In the following, RR will refer to the end of the last interval recorded in Report 12 (December 31. 1990), and tk will be omitted. The control group (labeled D = 0) includes all persons with D 5 0.005 Gy. In equations (3) and (4), D refers to the total gamma dose equivalent. converting the neutron dose to gamma dose equivalent as follows [ll]:
D = D,
It is calculated by
(5)
-I- wDnr
where w = J(oy
+ 2PD,)2 + 4o,PD,
- (o, + 2PD,)
2PD,
(6)
Here we have used fll]: a-, = 1.G x 10m2Gy-‘, o’, = 83.5 x lop2 Gy-l, and /? = 5.0 x lo-” Gy-“. We further group the data in dose bins with limiting points identical to those used in Report 12 (specifically: 0.005, 0.02, 0.05, 0.1, 0.2, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0Gy). This particular definition of the dose bins is, of course, quite arbitrary but not essential for the present analysis. Figure 1 shows the results of one such analysis (leukemia mortality at Nagasaki); the error bars were calculated with Greenwood’s formula [lo].
BOOTSTRAP CONFIDENCE FOR THE RELATIVE
INTERVALS RISK
The bootstrap method [3] consists of using the available data (here, the number of deaths in each cell) for generating new (bootstrapped) “measurements” similar to those described in Report 12. The analysis described above is repeated for a large number of bootstrapped data sets, and the probability distribution of RR for each dose group is thus calculated. Consider a particular cell (to simplify the notation, we suppress here the subscript i). The number of observed leukemia deaths, d, is one particular realization of a Bernoulli variable which is distributed according to a binomial process: f(d ) 6) = (;)&(l
- 9)+
1310
hf. ZAIDER
20l-
16
Leukemia (mortality) Nagasaki 1950-90
12
6
4
0
Figure 1. Relative risk as a function of equivalent dose for leukemia mortality at Nagasaki. The results shown were obtained using the actuarial method, equation (3).
Here n is the average number of persons at risk in the cell (PY/At). To bootstrap the data {d}, we need estimators for the probability of death, 8, in each cell. The parameter 0 is obviously an unobservable quantity and its possible numerical values are thus describable in terms of a probability distribution, h(0) de, which quantifies our judgment as to the likelihood that 0 will take a value in any particular interval de; this can be based on prior information we have about .9 (for instance, that it must take values in the interval [0, l]), and/or on available data. The question
w
I 4 4 = p(e)I’(d
l?(n + 2) ed(i + l)P(n - d + 1)
- ey(
(8)
which reads: the probability distribution of 8, given that in a cohort of n persons d deaths were observed, is the product of p(B), the prior distribution of 0, and the likelihood of observing d given 8. The key problem in using equation (8) is deciding what prior p(B) to use. It would appear that for lack of any a priori knowledge about 0, one should take p(0) = constant in the interval [O,11 (that is, all values of 8 are equiprobable). However, with this, equation (8) yields
gEd+l
(9)
12’
a result which seems at variance with the classical (and certainly more intuitive) estimator of 0, namely d/n. This difficulty has been addressed by Jeffreys [13] and Jaynes [14] who give groupinvariance arguments for using a nonuniform prior where more weight is given to the end points of the interval [0,11,specifically, p(‘)
1 = e(i _ 0).
With this, the renormalized equation (8) becomes
r(n) _ djed-l(i h(e 1d,n) = r(d)r(n
- e)n+l,
(11)
valid for d 2 1 and (n - d) 2 1. In particular,
&d
12’
g; = (dln)(l - d/n) n+l
.
(12)
Risk of Leukemia
1311
(11) is not valid for d = 0 because then the distribution h(B) is not normalizable. To avoid this problem, for those cells where d = 0 we use the value d = E instead, where 0 < E < 1.
Equation
This is not a difficulty because at any value of E the calculated probability (that RR < 1) will be a lower limit of the “true” probability; this was verified by taking the limit E -+ 0. To recapitulate, bootstrap samples of the data in Report 12 are generated as follows. (a) For each cell, a value 8 is selected from the distribution, equation (11). (b) For each cell a new number of deaths, d, is chosen from the binomial distribution, equation (7). (c) The data are re-analyzed and relative risks, RR(Di), are generated. In particular, at each dose we record the fraction of samples with RR < 1.
RESULTS
AND DISCUSSION
Analyses of the Japanese A-bomb data have used the DS86 dosimetry system [15]. It is now apparent that the DS86 substantially underestimates the neutron component of the absorbed dose at Hiroshima (for a review, see [16]). Th is is indicated by physical measurements of larger slow neutron fluence than is to be expected on the basis of the DS86 [17], and by biological studies that were based on the DS86 and resulted in different dose-effect curves for the chromosome aberrations in the two cities [18]. To avoid these problems we limit this analysis to the Nagasaki data only. Figure 2 shows estimators for the average relative risk of leukemia mortality as a function of dose equivalent,. In this calculation, E = 0.1. The error bars shown correspond to one standard error, and were obtained by bootstrapping 1000 samples. For individual dose points, the probability that RR < 1 is never larger than 53%. Nevertheless, with the aid of equation (l), we can calculate the probability that at least one RR is less than 1 in the range of doses [0, D]. The result is plotted in Figure 3 as a function of dose. Thus, for instance, for doses less than 200mSv the probability that at least one dose group shows a reduction in relative risk (< 1) is larger than 90%. This probability increases further (close to 100%) at larger doses. 20 Leukemia
(mortality)
-1
lo-”
10"
Dose
‘cfquivalent
1
Figure 2. Same as Figure 1, however, using the bootstrapped represent one standard deviation.
data.
The error bars
Based on this analysis, one must conclude that the relationship between absorbed doses of photons and leukemia induction is not linear. Moreover, it is quite likely that a hermetic response
1312
hf. ZAIDER
l-
0
0.9 z? 2
0.6 -
9
0.7 -
1 2
0.6 -
3
0.5 -
:
0.4 -
tf
0.3 f’
1 c) ‘i:lu
0.2 -
a
00000
00
0
0
Leukemia (mortality) Nagasaki 1950-90
0.1 0
I
16” Figure 3. The probability hasRR< 1.
0
0
I I I lllll
-1
I
I I I IIII[
Dose l&.Gval~~t
0
I
I I I IIIg
1
/ do
that, for at least one dose group in the range [O,D], one
exists at a dose at or below approximately 200 mSv. Although dosimetric uncertainties remain, one must note that among epidemiological studies, that of the A-bomb survivors from Nagasaki involves cohorts what were subject to the least nonuniformity of gamma ray dose and only a small neutron component. This analysis has obvious implications for radiation protection. Risk estimates for leukemia mortality from studies where the dose in the bone marrow was not uniform valid only for that particular dose distribution. Because the average dose in cannot be used to evaluate leukemia risk, arguments concerning the linearity of curve at low doses that make use of the average dose in the organ would most
must be assumed the bone marrow the dose-response likely be in error.
REFERENCES 1. D.A. Pierce, Y. Shimizu, D.L. Preston, M. Vaeth and K. Mabuchi, Studies of the mortality of atomic bomb survivors. Report 12, Part I. Cancer: 1950-1990, R&at. Res. 146, l-27 (1996). 2. Y. Shimizu, H. Kato, W.J. Schull and J. Mabuchi, Dose response analysis among atomic bomb survivors exposed to low-level radiation, In Low Dose Irradiation and Biological Defense Mechanisms, (Edited by T. Sughara et, al.), pp. 71-74, Elsevier Science, New York, (1992). 3. B. Efron and R. Tibshibani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistical Science 1, 54-77 (1986). 4. T.D. Luckey, Radiation Hormesis;CRC Press, Boca Raton, FL, (1991). 5. A.Y. Yakovlev, A.D. Tsodikov and L. Bass, A stochastic model of hormesis, Math. Biosciences 116, 197-219 (1993). 6. A. Yakovlev and E. Polig, A diversity of responses displayed by a stochastic model of radiation carcinogenesis allowing for cell death, Math. Biosciences 132, l-33 (1996). 7. A. Yakovlev, W. Mueller, L. Pavlova and E. Polig, Do cells repair precancerous lesions induced by radiation, A4uth. Biosciences 142, 107-117 (1997). 8. V. Covelli, V. DiMajo, M. Coppola and S. Rebeasi, The dose response relationship for myeloid leukemia and malignant lymphoma in BC3F mice, Radiat. Res. 119, 553-561 (1989). 9. RERF, Radiation Effects Research Foundation, Life Span Study Report 12: Detailed Cancer Mortality Data and Supplementary Tables, Hiroshima, Japan, (1996). 10. R.G. Miller, Survival Analysis. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley and Sons, New York, (1981). 11. H.H. Rossi and M. Zaider, Contribution of neutrons to the biological effects in Hiroshima, Health Phys. 58, 645-647 (1990). 12. H. Jeffreys. Theory of Probability, Clarendon Press, Oxford. (1985). 13. E.T. Jaynes, Prior probabilities, IEEE Tkans. Syst. Sci. Cybern. SSC-4, 227-241 (1968). 14. S.J. Press, Bayesian Statistics, John Wiley and Sons, New York, (1989).
Risk of Leukemia 15.
1313
D.C. Kaul, S.D. Egbert, M.D. Otis, T. Kuhn, G.D. Kerr, K.F. Eckerman, M. Cristy, T. hlaruyama, J.C. Ryman and J.S. Tang, Organ dosimetry, In U.S.-Japan Joint Reassessment of Atomic Bomb Radiation Dosimetry in Hiroshima and Nagasaki, Final Report, (Edited by W.C. Roesch), pp. 306-404, The Radiation Effects Research Foundation, Hiroshima, (1987). 16. W. Riihm, A.M. Kellerer, G. Korschinek, T. Faesterman, K. Knie, G. Rugel, K. Kato and E. Nolte, The dosimetry system DS86 and the neutron discrepancy in Hiroshima-Historical review, present status, and future options, Radiat. Environ. Biophys. 37, OO&OOO(1999). 17. T. Ytraume, S.D. Egbert, W.A. Woolson, R.C. Finkel, R.C. Kubik, H.E. Gove, P. Sharma aud P. Ho&i, Neutron discrepancies in the DS86 Hiroshima dosimetry system, Health Phys. 63, 412-426 (1992). 18. D.O. Stram, R. Sposto, D. Preston, S. Abrahamson, T. Honda and A.A. Awa, Stable chromosome aberrations among A-bomb survivors: An update, Radiat. Res. 136, 29-36 (1993).
Mathematical
and Computer
Modelling
33 (2001)
1307-1313 www.elsevier.nl/locate/mcm
The Risk of Leukemia from Low Doses of Low-LET Radiation M. ZAIDER Department Memorial 1275 York
of Medical
Sloan-Kettering
Avenue,
Physics Cancer
New York,
NY
Center
10021,
U.S.A.
[email protected]
Abstract-h this communication, we examine the evidence (if any) for a nonlinear dose response in relation to leukemia mortality in the Japanese A-bomb population. Specifically, we seek an estimate of the probability that, at low doses of radiation, the relative risk (RR) is smaller than one. Using Bayesian bootstrap techniques, we find that there is a 90% probability that-for at least One dose group at or below 200 mSv-there is a reduction of effect relative to the control group. We take this as evidence for radiation hormesis (end point: leukemia mortality) at low doses of radiation. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Leukemia
risk, A-bomb survivors,
Low doses of radiation,
Hormesis.
INTRODUCTION Leukemia
to ionizing radiation and it has been shown to be a such as the Japanese A-bomb survivors [l]. However, while the risk from substantial doses is quite generally recognized, there is uncertainty at lower doses that are of the order of several hundred milligray. principal
was the
hazard
In formulating
first
malignancy
in irradiated
radiation
attributed
populations
risks, the incidence
of deleterious
changes
is generally
related
to the
average dose (equivalent) in an irradiated organ. The “risk factor” is defined as the probability of an effect per unit of the average dose with the implication that, in the range of concern in it is irrelevant how radiation protection, the dose-effect curve is linear. With this assumption, the dose varies throughout the organ because each portion is affected in proportion to the dose. However, when the dose-effect relation is not linear, the response to a given average dose must of depend on the distribution of the dose within the organ. In the case of positive curvature t,he effect vs. dose plot, increasing nonuniformity must result in a higher effect probability. In t,he caSe where low doses of radiation reduce the spontaneous incidence, the dose response curve displays a quasithreshold; exposure to an average dose that is smaller than the threshold value can nevertheless result in a positive effect when the dose distribution is broad enough to extend sufficiently beyond the threshold value. This could lead to erroneous conclusions concerning the shape
of the response
curve at low doses.
I am indebted to Prof. H. H. Rossi and Prof. A. Yakovlev for their interest in this work and for liberal suggestions. This paper makes use of data obtained from the Radiation Effects Research Foundation (RERF) in Hiroshima, Japan. RERF is a private foundation funded equally by the Japanese Ministry of Health and Welfare and the U.S. Department of Energy through the U.S. Academy of Sciences. The conclusions in this paper
@ 2001 Elsevier
Science
Ltd. All rights reserved.
Typeset
by AM-m
1308
hf. ZAIDER
There is substantial evidence that the risk of leukemia increases more than proportionately with dose. The shape of the curve must depend on the irradiation pattern, but in the case of the Japanese A-bomb survivors, risks at 0.1 Sv and 1 Sv are said to differ by a factor of 20 [l]. In fact, an analysis by Shimizu et al. [2] appears to show a reduction of leukemia mortality at doses less than 0.2 Sv. A linear-quadratic fit yields a relative risk (RR) of about 0.75 at 100 mSv. They state that “it can not be said that there is a downward dose response in this dose range” because the “linear coefficient with minus sign in the linear-quadratic model is not statistically significant”. In this paper, we reexamine the Japanese A-bomb data for evidence of nonlinear response (here, leukemia mortality) at low doses of radiation. We do this by estimating the probability t,hat, for at least one dose group, the relative risk is less than one. Should this be indeed the case, we shall take it as an indication of hermetic response. The analysis presented here attempts to circumvent two common difficulties: (a) the fact that in traditional statistical analysis the probability distribution function (pdf) of an estimator obtained from a finite sample is usually not available; here we are interested in the probability that RR < 1, and (b) the limited Japanese data which-even with the first complication mentioned above removed-may not result in a conclusive answer for specific dose bins. To answer the first problem, we shall analyze the data using a method known as Bayesian bootstrap [3]; this approach, which is nonparametric, uses repeated sampling from the data to find the pdf of an estimator. If there are no dose points for which the probability that RR < 1 is significantly closer to 100% (e.g., Pr{RR < 1) > O.OO),we shall combine several dose points to calculate the probability that at least at one of them the relative risk is less than one. Thus, Pr{at least one RRi < 1; i 5 n} = 1 - Pr{RRi
> 1}Pr{RR2
2 1). . . Pr{RR,
2 1).
(1)
This would be taken as proof, or lack thereof, of a nonlinear response at low doses. Although possible mechanisms responsible for a red&ion of the cancer mortality at low doses have been discussed in the literature [4-81, its immediate cause (if indeed present) in human populations remains a matter of speculation. As far as the initiation stage is concerned, microdosimetric arguments show that if a tumor is monoclonal in origin, then at low doses the probability of the formation of a malignant cell must increase linearly with dose. However, inhibitory factors may prevent most of the transformed (or pretransformed) cells from developing into a tumor. A decrease in cancer mortality may occur if, for instance, (a) some of these factors are radiation dependent, and (b) they act selectively on transformed cells compared to normal cells. This will be the case if, for example, at low doses the number of preexisting malignant cells killed exceeds the number of normal cells transformed. Yakovlev and Polig [6] have developed a stochastic model that describes the competition between cell killing and tumor promotion. Covelli et al. [8] observed a striking case of reduction of the incidence of malignant lymphoma in mice irradiated with x-rays and neutrons. They interpreted their results in terms of a combination of cell transformation and cell killing. The x-ray data were fitted by M =
(~0+ azD”> emAD,
(2)
where A,4 is the mortality at dose D, a0 is the fractional control incidence, and u2 and X are constants. In this analysis, cell killing influences the dose-effect relation not only at large doses but also at small doses. This is especially evident in this case because a0 is large (0.56). Equation (2) provides a good fit to a curve which initially decreases between 0 and 1 Gy to about 80% of the control incidence, rises barely above it near 2 Gy, and then declines.
Risk of Leukemia
1309
DATA ANALYSIS The data used in this analysis were taken from the Life Span Study (LSS) Report 12, Detailed Cancer Mortality Data and Supplementary Tables (see [9], 1rereafter referred to as Report 12). Of the various factors used in this report to cross-classify the Japanese cohort, we retain here only city (Hiroshima or Nagasaki), follow-up interval (ten periods of between two and five years each), organ dose equivalent-the calculation of which is explained below, and cause of death. For each such group (cell) i, the information available is: number of deaths (di), gamma and neutron organ doses (D,,i, D,+),
person-years for all people ever at risk in the cell (PY,),
and the time
interval (Ati). For any particular cause of death, the probability of survival up to time tk (the end of the period Atk = tk - tk_i) was estimated using the actuarial method [lo]:
S(tk,D)
[I- &A,i]
= fi i=l
.
Here the quantity in square brackets represents the probability that a person who survived up to time ti_1 all risks will not succumb to a particular site-specific cancer death (here, leukemia) during Ati; and PY.i/A& is the average number of persons at risk during the interval At,. With this, the relative risk (RR) for the entire period [0, tk] is RR(tk,D)
=
1 - S(tk, D) = 0)’
1 - S(tk,D
In the following, RR will refer to the end of the last interval recorded in Report 12 (December 31. 1990), and tk will be omitted. The control group (labeled D = 0) includes all persons with D 5 0.005 Gy. In equations (3) and (4), D refers to the total gamma dose equivalent. converting the neutron dose to gamma dose equivalent as follows [ll]:
D = D,
It is calculated by
(5)
-I- wDnr
where w = J(oy
+ 2PD,)2 + 4o,PD,
- (o, + 2PD,)
2PD,
(6)
Here we have used fll]: a-, = 1.G x 10m2Gy-‘, o’, = 83.5 x lop2 Gy-l, and /? = 5.0 x lo-” Gy-“. We further group the data in dose bins with limiting points identical to those used in Report 12 (specifically: 0.005, 0.02, 0.05, 0.1, 0.2, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0Gy). This particular definition of the dose bins is, of course, quite arbitrary but not essential for the present analysis. Figure 1 shows the results of one such analysis (leukemia mortality at Nagasaki); the error bars were calculated with Greenwood’s formula [lo].
BOOTSTRAP CONFIDENCE FOR THE RELATIVE
INTERVALS RISK
The bootstrap method [3] consists of using the available data (here, the number of deaths in each cell) for generating new (bootstrapped) “measurements” similar to those described in Report 12. The analysis described above is repeated for a large number of bootstrapped data sets, and the probability distribution of RR for each dose group is thus calculated. Consider a particular cell (to simplify the notation, we suppress here the subscript i). The number of observed leukemia deaths, d, is one particular realization of a Bernoulli variable which is distributed according to a binomial process: f(d ) 6) = (;)&(l
- 9)+
1310
hf. ZAIDER
20l-
16
Leukemia (mortality) Nagasaki 1950-90
12
6
4
0
Figure 1. Relative risk as a function of equivalent dose for leukemia mortality at Nagasaki. The results shown were obtained using the actuarial method, equation (3).
Here n is the average number of persons at risk in the cell (PY/At). To bootstrap the data {d}, we need estimators for the probability of death, 8, in each cell. The parameter 0 is obviously an unobservable quantity and its possible numerical values are thus describable in terms of a probability distribution, h(0) de, which quantifies our judgment as to the likelihood that 0 will take a value in any particular interval de; this can be based on prior information we have about .9 (for instance, that it must take values in the interval [0, l]), and/or on available data. The question
w
I 4 4 = p(e)I’(d
l?(n + 2) ed(i + l)P(n - d + 1)
- ey(
(8)
which reads: the probability distribution of 8, given that in a cohort of n persons d deaths were observed, is the product of p(B), the prior distribution of 0, and the likelihood of observing d given 8. The key problem in using equation (8) is deciding what prior p(B) to use. It would appear that for lack of any a priori knowledge about 0, one should take p(0) = constant in the interval [O,11 (that is, all values of 8 are equiprobable). However, with this, equation (8) yields
gEd+l
(9)
12’
a result which seems at variance with the classical (and certainly more intuitive) estimator of 0, namely d/n. This difficulty has been addressed by Jeffreys [13] and Jaynes [14] who give groupinvariance arguments for using a nonuniform prior where more weight is given to the end points of the interval [0,11,specifically, p(‘)
1 = e(i _ 0).
With this, the renormalized equation (8) becomes
r(n) _ djed-l(i h(e 1d,n) = r(d)r(n
- e)n+l,
(11)
valid for d 2 1 and (n - d) 2 1. In particular,
&d
12’
g; = (dln)(l - d/n) n+l
.
(12)
Risk of Leukemia
1311
(11) is not valid for d = 0 because then the distribution h(B) is not normalizable. To avoid this problem, for those cells where d = 0 we use the value d = E instead, where 0 < E < 1.
Equation
This is not a difficulty because at any value of E the calculated probability (that RR < 1) will be a lower limit of the “true” probability; this was verified by taking the limit E -+ 0. To recapitulate, bootstrap samples of the data in Report 12 are generated as follows. (a) For each cell, a value 8 is selected from the distribution, equation (11). (b) For each cell a new number of deaths, d, is chosen from the binomial distribution, equation (7). (c) The data are re-analyzed and relative risks, RR(Di), are generated. In particular, at each dose we record the fraction of samples with RR < 1.
RESULTS
AND DISCUSSION
Analyses of the Japanese A-bomb data have used the DS86 dosimetry system [15]. It is now apparent that the DS86 substantially underestimates the neutron component of the absorbed dose at Hiroshima (for a review, see [16]). Th is is indicated by physical measurements of larger slow neutron fluence than is to be expected on the basis of the DS86 [17], and by biological studies that were based on the DS86 and resulted in different dose-effect curves for the chromosome aberrations in the two cities [18]. To avoid these problems we limit this analysis to the Nagasaki data only. Figure 2 shows estimators for the average relative risk of leukemia mortality as a function of dose equivalent,. In this calculation, E = 0.1. The error bars shown correspond to one standard error, and were obtained by bootstrapping 1000 samples. For individual dose points, the probability that RR < 1 is never larger than 53%. Nevertheless, with the aid of equation (l), we can calculate the probability that at least one RR is less than 1 in the range of doses [0, D]. The result is plotted in Figure 3 as a function of dose. Thus, for instance, for doses less than 200mSv the probability that at least one dose group shows a reduction in relative risk (< 1) is larger than 90%. This probability increases further (close to 100%) at larger doses. 20 Leukemia
(mortality)
-1
lo-”
10"
Dose
‘cfquivalent
1
Figure 2. Same as Figure 1, however, using the bootstrapped represent one standard deviation.
data.
The error bars
Based on this analysis, one must conclude that the relationship between absorbed doses of photons and leukemia induction is not linear. Moreover, it is quite likely that a hermetic response
1312
hf. ZAIDER
l-
0
0.9 z? 2
0.6 -
9
0.7 -
1 2
0.6 -
3
0.5 -
:
0.4 -
tf
0.3 f’
1 c) ‘i:lu
0.2 -
a
00000
00
0
0
Leukemia (mortality) Nagasaki 1950-90
0.1 0
I
16” Figure 3. The probability hasRR< 1.
0
0
I I I lllll
-1
I
I I I IIII[
Dose l&.Gval~~t
0
I
I I I IIIg
1
/ do
that, for at least one dose group in the range [O,D], one
exists at a dose at or below approximately 200 mSv. Although dosimetric uncertainties remain, one must note that among epidemiological studies, that of the A-bomb survivors from Nagasaki involves cohorts what were subject to the least nonuniformity of gamma ray dose and only a small neutron component. This analysis has obvious implications for radiation protection. Risk estimates for leukemia mortality from studies where the dose in the bone marrow was not uniform valid only for that particular dose distribution. Because the average dose in cannot be used to evaluate leukemia risk, arguments concerning the linearity of curve at low doses that make use of the average dose in the organ would most
must be assumed the bone marrow the dose-response likely be in error.
REFERENCES 1. D.A. Pierce, Y. Shimizu, D.L. Preston, M. Vaeth and K. Mabuchi, Studies of the mortality of atomic bomb survivors. Report 12, Part I. Cancer: 1950-1990, R&at. Res. 146, l-27 (1996). 2. Y. Shimizu, H. Kato, W.J. Schull and J. Mabuchi, Dose response analysis among atomic bomb survivors exposed to low-level radiation, In Low Dose Irradiation and Biological Defense Mechanisms, (Edited by T. Sughara et, al.), pp. 71-74, Elsevier Science, New York, (1992). 3. B. Efron and R. Tibshibani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistical Science 1, 54-77 (1986). 4. T.D. Luckey, Radiation Hormesis;CRC Press, Boca Raton, FL, (1991). 5. A.Y. Yakovlev, A.D. Tsodikov and L. Bass, A stochastic model of hormesis, Math. Biosciences 116, 197-219 (1993). 6. A. Yakovlev and E. Polig, A diversity of responses displayed by a stochastic model of radiation carcinogenesis allowing for cell death, Math. Biosciences 132, l-33 (1996). 7. A. Yakovlev, W. Mueller, L. Pavlova and E. Polig, Do cells repair precancerous lesions induced by radiation, A4uth. Biosciences 142, 107-117 (1997). 8. V. Covelli, V. DiMajo, M. Coppola and S. Rebeasi, The dose response relationship for myeloid leukemia and malignant lymphoma in BC3F mice, Radiat. Res. 119, 553-561 (1989). 9. RERF, Radiation Effects Research Foundation, Life Span Study Report 12: Detailed Cancer Mortality Data and Supplementary Tables, Hiroshima, Japan, (1996). 10. R.G. Miller, Survival Analysis. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley and Sons, New York, (1981). 11. H.H. Rossi and M. Zaider, Contribution of neutrons to the biological effects in Hiroshima, Health Phys. 58, 645-647 (1990). 12. H. Jeffreys. Theory of Probability, Clarendon Press, Oxford. (1985). 13. E.T. Jaynes, Prior probabilities, IEEE Tkans. Syst. Sci. Cybern. SSC-4, 227-241 (1968). 14. S.J. Press, Bayesian Statistics, John Wiley and Sons, New York, (1989).
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D.C. Kaul, S.D. Egbert, M.D. Otis, T. Kuhn, G.D. Kerr, K.F. Eckerman, M. Cristy, T. hlaruyama, J.C. Ryman and J.S. Tang, Organ dosimetry, In U.S.-Japan Joint Reassessment of Atomic Bomb Radiation Dosimetry in Hiroshima and Nagasaki, Final Report, (Edited by W.C. Roesch), pp. 306-404, The Radiation Effects Research Foundation, Hiroshima, (1987). 16. W. Riihm, A.M. Kellerer, G. Korschinek, T. Faesterman, K. Knie, G. Rugel, K. Kato and E. Nolte, The dosimetry system DS86 and the neutron discrepancy in Hiroshima-Historical review, present status, and future options, Radiat. Environ. Biophys. 37, OO&OOO(1999). 17. T. Ytraume, S.D. Egbert, W.A. Woolson, R.C. Finkel, R.C. Kubik, H.E. Gove, P. Sharma aud P. Ho&i, Neutron discrepancies in the DS86 Hiroshima dosimetry system, Health Phys. 63, 412-426 (1992). 18. D.O. Stram, R. Sposto, D. Preston, S. Abrahamson, T. Honda and A.A. Awa, Stable chromosome aberrations among A-bomb survivors: An update, Radiat. Res. 136, 29-36 (1993).