# Flexible dose-response models for Japanese atomic bomb survivor data: Bayesian estimation and prediction of cancer risk

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DOI: 10.1007/s00411-004-0258-3

- Cite this article as:
- Bennett, J., Little, M.P. & Richardson, S. Radiat Environ Biophys (2004) 43: 233. doi:10.1007/s00411-004-0258-3

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## Abstract

Generalised absolute risk models were fitted to the latest Japanese atomic bomb survivor cancer incidence data using Bayesian Markov Chain Monte Carlo methods, taking account of random errors in the DS86 dose estimates. The resulting uncertainty distributions in the relative risk model parameters were used to derive uncertainties in population cancer risks for a current UK population. Because of evidence for irregularities in the low-dose dose response, flexible dose-response models were used, consisting of a linear-quadratic-exponential model, used to model the high-dose part of the dose response, together with piecewise-linear adjustments for the two lowest dose groups. Following an assumed administered dose of 0.001 Sv, lifetime leukaemia radiation-induced incidence risks were estimated to be 1.11×10^{−2} Sv^{−1} (95% Bayesian CI −0.61, 2.38) using this model. Following an assumed administered dose of 0.001 Sv, lifetime solid cancer radiation-induced incidence risks were calculated to be 7.28×10^{−2} Sv^{−1} (95% Bayesian CI −10.63, 22.10) using this model. Overall, cancer incidence risks predicted by Bayesian Markov Chain Monte Carlo methods are similar to those derived by classical likelihood-based methods and which form the basis of established estimates of radiation-induced cancer risk.

## Introduction

The Japanese atomic bomb survivor Life Span Study (LSS) cohort, even though exposed at high doses and dose-rates, is undoubtedly the most important group for the purposes of assessing the risks arising from exposure to ionising radiation, and indeed this body of data has been used by many scientific committees for the purposes of estimating cancer risks [1, 2, 3]. Calculation of cancer risks in populations other than this Japanese cohort is based on the assumption that risk coefficients derived from the LSS can be used to estimate risk in another population, an assumption generally made by various scientific committees that have used the LSS for this purpose [1, 2, 3]. In this paper the risk estimates derived from the LSS cohort will be used to predict the cancer risks for a current UK population. The main aim of the analysis is to reflect in the risk projection the impact of some of the sources of uncertainty on the risk model parameters.

A major source of uncertainty concerns the extrapolation of risks at high dose and high dose-rates to those at low doses and low dose-rates. Crucial to the resolution of this area of uncertainty is the flexible modelling of the dose-response relationship and the importance of both systematic and random dosimetric errors for analyses of the Japanese atomic bomb survivor data or other exposed groups. The problem of allowing for errors in dose assessments when estimating dose-response relationships has recently been the subject of much interest in epidemiology. It is well recognised that measurement error can substantially alter the shape of this relationship and hence the derived population risk estimates [4]. The problem of dosimetric error for the Radiation Effects Research Foundation (RERF) data has been investigated by Jablon [5], Gilbert [6] and subsequently in a series of papers by Pierce et al. [7, 8, 9]. Pierce and colleagues carried out a dose adjustment prior to the model fitting, allowing for random dosimetric errors. A similar procedure was followed by Little and Muirhead [10, 11, 12, 13, 14] and Little [15], who also took account of certain systematic errors in dose [13], which was not done in previous analyses. This dose adjustment entails the substitution of the “estimated dose” (“nominal” DS86 dose estimate) by the expectation of the “true dose” given the estimated one. This approach to measurement error correction is an example of “regression calibration” which, as emphasised by Carroll et al. [16], is an approximate method in non-linear dose-effect relationships. It leads to reasonable adjusted point estimates of the model parameters but does not fully take account of all the variability induced by the measurement errors. Within the context of the LSS cancer mortality or incidence data the extra variability not taken into account is relatively small [7, 9] and this regression-calibration method leads to adjusted risk model parameters that can be used in risk projection.

A Bayesian approach to the measurement error problem has recently been developed [17, 18, 19] which rests on the formulation of conditional independence relationships between different model components, following the general structure outlined by Clayton [20]. In this approach three basic sub-models are distinguished and linked: the disease model, the measurement model and the exposure model. The power of this Bayesian approach is that the dosimetric uncertainty is reflected in the variability of the model parameters. An adapted Bayesian method of correction for measurement error—the two-stage Bayesian method—has already been applied to the fitting of generalised relative risk models to the LSS Report 12 Japanese atomic bomb survivor cancer mortality data [21, 22, 23]. The posterior distribution of the risk parameters was obtained by samples from Markov Chain Monte Carlo (MCMC) algorithms.

Both regression-calibration and Bayesian MCMC methods for taking measurement error into account use a number of assumptions on the size of the dosimetric error and its distribution and an approximation argument concerning the exposure distribution at the group level. These baseline assumptions are similar though not identical in both methods, hence the resulting risk estimates and their variability can differ.

In this paper the posterior distribution of risk parameters obtained from a Bayesian MCMC algorithm will be applied to estimate uncertainties in population cancer incidence risks for a current UK population. Exploiting the flexibility of the Bayesian framework, a number of dose-response models will be investigated, that include both semi-parametric (spline) and parametric components, to compute cancer risks.

## Materials and methods

### Data

The latest versions of the leukaemia [24] and solid cancer [25] incidence datasets for the LSS cohort of A-bomb survivors were employed. The leukaemia incidence data of Preston et al. [24] involved the follow-up of 86,332 survivors of the atomic bombings of Hiroshima and Nagasaki with shielded kerma dose <4 Gy from 1 October 1950 to 31 December 1987. By the end of 1987 there had been 192 cases of the 3 main radiogenic leukaemia subtypes, i.e. acute myeloid leukaemia (AML), chronic myeloid leukaemia (CML), acute lymphocytic leukaemia (AML). (Note: the number of survivors given by Preston et al. in the open literature report [24] (86,293) is slightly smaller than that estimated here, a result of the analysis dataset used by Preston et al. being subsequently amended before being made publicly available.) The solid cancer incidence data of Thompson et al. [25] involved the follow-up of a slightly smaller cohort of 79,972 survivors with shielded kerma dose <4 Gy from 1 January 1958 (when the cancer registries were established in Hiroshima and Nagasaki) to 31 December 1987. By the end of 1987 there had been 8,613 solid cancer cases. Following the example of Preston et al. [24] and Thompson et al. [25] in their analysis of these datasets, all survivors with shielded kerma dose >4 Gy were omitted from the modelling of cancer (the second stage in the Bayesian MCMC model fitting: see Bayesian modelling at the strata level section), because of possible systematic errors in the highest dose estimates and possible cell sterilisation effects. However, the full dataset, including the survivors with shielded kerma dose >4 Gy, was used to assess the dosimetric error structure (the first stage in the Bayesian MCMC model fitting: see Bayesian modelling at the strata level section).

The organ dose used for the solid cancer data was that to the colon, whereas bone marrow dose was used for leukaemia; in both cases a neutron relative biological effectiveness (RBE) of 10 was assumed, as used in previous analyses [23, 24, 25]. All “nominal” organ doses were calculated using the latest publicly available dosimetry, the so-called DS86 dosimetry [26]. (Note: this is not the most current set of dose estimates for the atomic bomb survivors: see Discussion section.) Individual data were not available, so that all analyses used the stratified data. The stratification employed is very similar to that previously used [24, 25], and is defined by time since exposure, age at exposure, city, sex and (“nominal”) dose.

### Models used

*c*, sex

*s*, age at exposure

*a*, time since exposure

*t*, and “true” average organ dose \( \bar{D}, \)can be partitioned as:

*EAR*(

*s*,

*a*,

*t*,\( \bar{D} \)) describes the radiation-induced excess attributable risk (EAR), and is assumed to be zero for zero average organ dose;

*λ*(

*c*,

*s*,

*a*,

*t*) describes the “baseline” (zero dose) cancer risk. Therefore the number of cancer cases in the stratum is a Poisson random variable with expectation:

*PY*is the number of (migration-adjusted) person years of follow-up in the stratum. Previous dose-response relationships considered for these data include linear and quadratic models, and certain more general forms [7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23]. In order to explore the form of model best suited to the data a piecewise-linear model was initially considered. This is an extremely flexible class of models chosen primarily for the information that might be gained from their fitting rather than as candidates for the final models. The knotpoints were initially fixed at the average posterior dose of each “nominal” DS86 dose group (see Appendix A for further details). In order to make further inferences, some of these were later altered. Therefore, for the purposes of exploratory dose-response analysis (as shown in Figs. 1 and 3), a model was fitted in which the radiation-induced excess risk was given by a piecewise-linear function of the true mean dose with 10 knotpoints, 0=

*K*

_{0}<

*K*

_{1}<...<

*K*

_{10}:

*θ*

_{0}for mean dose \( \bar{D} \)=0, and thereafter to be

*θ*

_{i}for mean dose \( \bar{D} \)=

*K*

_{i}, linear in between these points and linear for \( \bar{D} \)>

*K*

_{9}. Further details on the function,

*G*(

*s*,

*a*,

*t*), modifying the excess risk, also on the baseline risk,

*λ*(

*c*,

*s*,

*a*,

*t*), are given in Appendix A. In a second step, a more parsimonious dose-response model was also fitted, employing a combination of a piecewise-linear function with three knotpoints describing the low-dose data, and a linear-quadratic-exponential function describing the high-dose data:

*θ*

_{0}for mean dose \( \bar{D} \)=0, and thereafter to be

*θ*

_{1},

*θ*

_{2},

*θ*

_{3}for mean dose \( \bar{D} \)=

*K*

_{1},

*K*

_{2},

*K*

_{3}, respectively, linear in between these points and constant, =

*θ*

_{3}, for \( \bar{D} \)>

*K*

_{3}. The second part of this function inside the braces {}, \( {\left( {\beta _{1} \cdot {\left[ {\ifmmode\expandafter\bar\else\expandafter\=\fi{D} - K_{3} } \right]} + \beta _{2} \cdot {\left[ {\ifmmode\expandafter\bar\else\expandafter\=\fi{D} - K_{3} } \right]}^{2} } \right)} \cdot \exp {\left[ {\beta _{3} \cdot {\left[ {\ifmmode\expandafter\bar\else\expandafter\=\fi{D} - K_{3} } \right]}} \right]} \cdot 1_{{K_{3} < \ifmmode\expandafter\bar\else\expandafter\=\fi{D}}} \), is zero for mean dose \( \bar{D} \)<

*K*

_{3}, and for mean dose \( \bar{D} \)≥

*K*

_{3}is given by a product of a linear-quadratic initiation term,

*β*

_{1}⋅[\( \bar{D} \)–

*K*

_{3}]+

*β*

_{2}⋅[\( \bar{D}\)–

*K*

_{3}]

^{2}, and an exponential cell-sterilisation term, exp[

*β*

_{3}⋅[\( \bar{D} \)–

*K*

_{3}]]. For purposes of comparison, an otherwise similar model without knotpoints was also fitted:

It should be emphasised that the “true” stratum-average organ dose, \( \bar{D}, \)is not known; the only recorded dosimetric quantity in any stratum is the “nominal” stratum-average (DS86) organ dose, \( \bar{d}. \)

### Model fitting

#### Measurement error at the individual level

The natural modelling of measurement error in Bayesian MCMC methods is at the individual level. The stratification creates groups of subjects, and so requires transfer of the modelling of measurement error on the* individual dose* to the measurement error on the* mean dose* over the stratum. At an individual level, the “true” dose distribution in each of the two cities (Hiroshima, Nagasaki) is modelled by a Weibull distribution, in which the probability of the “true” dose being greater than* D* is given by exp(−*θ*⋅*D*^{ϕ}), as assumed by Pierce et al. [7, 8, 9]. Jablon [5] investigated the errors in the Japanese atomic bomb dosimetry and found that these errors were most likely to be log-normal, with a geometric standard deviation (GSD) of about 30%. Furthermore, a “classical” measurement error model is employed since the main component of the measurement error comes from the declaration by the survivor of their location and orientation with respect to the bomb at the time of explosion [26]. Therefore, in this paper the distribution of the “nominal” dose,* d*, given the “true” individual dose,* D*, is assumed to be log-normal with median* D*. Following the example of Pierce et al. [7], in this paper the “nominal” dose is assumed to be log-normally distributed with 35% GSD errors.

#### Bayesian modelling at the strata level

A two-stage method is used for modelling the stratum-specific dosimetric uncertainties, very similar to that used by Little et al. [23], and described in more detail there. In the first stage, for each stratum* i* (defined by city, sex and age at exposure group) and dose group* j*, the distribution of the “true” mean dose \( \bar{D}_{{ij}} \) is computed by Monte Carlo integration according to an iterative procedure [23]. The FORTRAN program used to perform these calculations was written by the second author, and is available on request. This procedure was necessitated by the grouped nature of the data, in particular by the fact that individual “nominal” doses were not available. In the second stage, the derived distribution of all the \( \bar{D}_{{ij}} \) is then used together with the EAR disease models (1, 3, 4, 5) to derive the posterior distribution of the parameters of these EAR models. The Bayesian sampling was performed using WinBUGS [27], and the WinBUGS scripts for this are available from the second author on request.

Leukaemia incidence risk model parameters for models (4) and (5)

Parameter | Mean | Centile | |||||
---|---|---|---|---|---|---|---|

2.5% | 5% | 50% | 95% | 97.5% | |||

Linear-quadratic-exponential + three knotpoints model (4) | |||||||

| 2.5525 | −1.4340 | −0.4586 | 2.6650 | 5.1600 | 5.6880 | |

| 2.9528 | −0.7245 | −0.5683 | 2.0350 | 9.3530 | 11.6900 | |

| −0.1908 | −0.8662 | −0.7648 | −0.2120 | 0.4711 | 0.6024 | |

| −0.5625 | −0.9974 | −0.9278 | −0.5593 | −0.2018 | −0.1367 | |

| −0.6928 | −1.1050 | −1.0390 | −0.6893 | −0.3600 | −0.3006 | |

Linear-quadratic-exponential model (5) | |||||||

| 0.5932 | −1.3330 | −1.0150 | 0.5924 | 2.1700 | 2.4930 | |

| 4.5320 | −0.0745 | 0.2544 | 4.0700 | 10.4000 | 11.8200 | |

| −0.3654 | −0.9595 | −0.8797 | −0.4029 | 0.3081 | 0.4724 | |

| −0.6699 | −1.2020 | −1.1100 | −0.6669 | −0.2341 | −0.1557 | |

| −1.0070 | −1.4820 | −1.4070 | −1.0010 | −0.6298 | −0.5512 |

Solid cancer incidence risk model parameters for models (4) and (5)

Parameter | Mean | Centile | |||||
---|---|---|---|---|---|---|---|

2.5% | 5% | 50% | 95% | 97.5% | |||

Linear-quadratic-exponential + three knotpoints model (4) | |||||||

| 10.4640 | −14.5800 | −10.2800 | 11.3100 | 29.1500 | 32.1200 | |

| 36.3070 | −6.8230 | −3.3060 | 36.4600 | 73.8200 | 81.2100 | |

| −0.6430 | −1.1610 | −1.0920 | −0.7140 | 0.1183 | 0.5212 | |

| 0.4461 | 0.0980 | 0.1473 | 0.4407 | 0.7691 | 0.8427 | |

| 0.8322 | 0.2905 | 0.3765 | 0.8262 | 1.2990 | 1.4000 | |

| 1.2127 | 0.6267 | 0.7209 | 1.2180 | 1.7040 | 1.7870 | |

Linear-quadratic-exponential model (5) | |||||||

| 12.4036 | −0.5808 | 1.7061 | 12.5800 | 22.4100 | 24.1898 | |

| 21.5593 | −6.6020 | −5.8900 | 21.3500 | 52.4900 | 57.7900 | |

| −0.4292 | −1.0390 | −0.9731 | −0.5540 | 0.6977 | 0.8555 | |

| 0.6020 | 0.2011 | 0.2595 | 0.5866 | 0.9904 | 1.0760 | |

| 0.8221 | 0.2206 | 0.3212 | 0.8153 | 1.3610 | 1.4610 | |

| 0.9692 | 0.3581 | 0.4573 | 0.9734 | 1.4690 | 1.5620 |

*θ*and

*ϕ*, were assumed to have gamma prior distributions with mean 0.001 and coefficient of variation 100; most other parameters have normal prior distributions with mean 0 and variance 100. Care was taken to check stability and convergence of the posterior MCMC samples. A total of 70,000 samples were taken for leukaemia and solid cancer, after 10,000 samples were discarded in each case to allow the Markov chains to reach their stationary equilibrium distributions. Each set of model parameter values from this sample was used to calculate a measure of population cancer risk for a current UK population, as detailed in the section “Risk Projection”. This sample of parameter values is therefore associated with a sample of population cancer risks for a current UK population. The mean and various centiles of population risk for this sample are given in Tables 3 and 4; the distribution of the risk is illustrated graphically in Figs. 5 and 6.

Lifetime leukaemia radiation exposure-induced incidence risks (per Sv) predicted by Bayesian Markov Chain Monte Carlo method for a population in equilibrium having the mortality rates of the 1998 England and Wales population

Dose (Sv) | Mean | Centile | |||||
---|---|---|---|---|---|---|---|

2.5% | 5% | 50% | 95% | 97.5% | |||

Linear-quadratic-exponential + three knotpoints model (4) | |||||||

0.001 | 0.0111 | −0.0061 | −0.0020 | 0.0117 | 0.0217 | 0.0238 | |

0.01 | 0.0112 | −0.0057 | −0.0016 | 0.0118 | 0.0217 | 0.0238 | |

0.1 | 0.0122 | −0.0019 | 0.0013 | 0.0125 | 0.0218 | 0.0238 | |

1.0 | 0.0180 | 0.0122 | 0.0131 | 0.0178 | 0.0234 | 0.0247 | |

Linear-quadratic-exponential model (5) | |||||||

0.001 | 0.0033 | −0.0077 | −0.0059 | 0.0035 | 0.0116 | 0.0130 | |

0.01 | 0.0035 | −0.0071 | −0.0054 | 0.0037 | 0.0117 | 0.0130 | |

0.1 | 0.0057 | −0.0018 | −0.0007 | 0.0057 | 0.0124 | 0.0138 | |

1.0 | 0.0187 | 0.0124 | 0.0131 | 0.0182 | 0.0257 | 0.0276 |

Lifetime solid cancer radiation exposure-induced incidence risks (per Sv) predicted by Bayesian Markov Chain Monte Carlo method for a population in equilibrium having the mortality rates of the 1998 England and Wales population

Dose (Sv) | Mean | Centile | |||||
---|---|---|---|---|---|---|---|

2.5% | 5% | 50% | 95% | 97.5% | |||

Linear-quadratic-exponential + three knotpoints model (4) | |||||||

0.001 | 0.0728 | −0.1063 | −0.0758 | 0.0803 | 0.2005 | 0.2210 | |

0.01 | 0.0748 | −0.1006 | −0.0710 | 0.0820 | 0.2006 | 0.2210 | |

0.1 | 0.0936 | −0.0513 | −0.0263 | 0.0975 | 0.2020 | 0.2206 | |

1.0 | 0.1647 | 0.1189 | 0.1251 | 0.1631 | 0.2096 | 0.2196 | |

Linear-quadratic-exponential model (5) | |||||||

0.001 | 0.0900 | −0.0040 | 0.0134 | 0.0935 | 0.1562 | 0.1685 | |

0.01 | 0.0912 | −0.0006 | 0.0163 | 0.0944 | 0.1564 | 0.1686 | |

0.1 | 0.1020 | 0.0297 | 0.0423 | 0.1030 | 0.1579 | 0.1697 | |

1.0 | 0.1471 | 0.1100 | 0.1153 | 0.1459 | 0.1835 | 0.1916 |

## Results

The results of fitting model (3) to the leukaemia and solid cancer data are shown in Figs. 1 and 3. The exploratory fits revealed that while a linear-quadratic-exponential model, \( F{\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{D}} \right)} = {\left( {\beta _{1} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{D} + \beta _{2} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{D}^{2} } \right)} \cdot \exp {\left[ {\beta _{3} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{D}} \right]} \) would be suitable to describe the high-dose part of the dose-response relationship, there were anomalies in the dose response at low doses. In particular, there were indications that for leukaemia the dose-response initially sharply increased near 0 Sv, then decreased with dose before increasing again at higher dose (Fig. 1). In contrast, for solid cancer, after the initial sharp increase in risk near 0 Sv, there was a more progressive increase thereafter (Fig. 3). This exploratory analysis therefore indicated that the more parsimonious dose-response model (4) could be fitted.

The results of fitting model (4) to the leukaemia and solid cancer data are shown in Figs. 2 and 4. The parameters for the model fits are shown in Tables 1 and 2. Data at the very low doses are clustered around 2 points, 0.0018 Sv and 0.04 Sv. These 2 clusters, representing the lowest 2 “nominal” dose groups, respectively account for 79.4% (68,507/86,332) of the individuals and 51.0% (98/192) of cases in the leukaemia dataset and 80.3% (64,181/79,972) of individuals and 75.6% (6,509/8,613) of cases in the solid cancer dataset. For the leukaemia dataset the mean height of the piecewise part of the dose-response model at 0.0018 Sv is above that for the second cluster at 0.04 Sv, while for solid cancer it is slightly below that point (Figs. 2, 4). The improvement in fit obtained by including three knotpoints to cover this anomaly at low doses did not notably decrease the deviance, either for leukaemia or for solid cancers. However, inference was made from this linear-quadratic-exponential model with knotpoints (Eq. 4) for the lowest doses, in order to ensure that the parameters of the dose-effect relationship for higher doses were not artificially modified by the constraints imposed at low doses.

From Table 1 it is seen that the linear parameter,* β*_{1}, which dominates the low-dose leukaemia dose response, is much larger, and more widely dispersed in the model with piecewise-linear adjustments for the two lowest dose groups than for the one without these adjustments. For solid cancer, Table 2 shows that the linear parameter,* β*_{1}, is slightly lower, and more widely dispersed, in the model with piecewise-linear adjustments for the two lowest dose groups than for the one without these adjustments.

### Risk projection

For the purposes of calculating cancer incidence risks in this UK population, high dose-rate risks are evaluated from models (4 and 5) fitted to the Japanese solid cancer and leukaemia data separately.

### Formulation

*D*

_{t}, will be assessed, that is assumed to be administered at age

*a*for sex

*s*so that the test dose only affects the rates of a specific cancer type

*c*(

*c*=solid cancers, leukaemia). The risk measure used is

*radiation exposure-induced cancer incidence*(RECI) per unit dose:

*EAR*

_{c}(

*s*,

*a*,

*t*,

*D*) is the excess absolute incidence rate for cancer type

*c*associated with the linear-quadratic-exponential part of models (4) and (5),

*μ*(

*s*,

*a*) is the all-cause (cancer and non-cancer) mortality rate, and

*s*surviving to age

*t*given that they have survived to age

*a*(≤

*t*). Persons are assumed capable of surviving in principle up to the age of

*y*

_{T}(=121 years here), at which point they are assumed to die instantaneously (i.e. the population is truncated at that age). This risk measure was employed by the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) [3] and by Little et al. [28]. It is closely related to the analogous measure for cancer mortality, the risk of exposure-induced death (REID). These and other measures of risk are discussed in more detail by Thomas et al. [29]. The non-constancy of the measure of risk as a function of the test dose,

*D*

_{t}, should be noted, a result of the non-linearity of

*EAR*

_{c}(

*s*,

*a*,

*t*,

*D*

_{t}). In calculating the risks, the age-specific and sex-specific mortality rates of the current (1998) UK population [30] are used as the baseline mortality rates,

*μ*(

*s*,

*a*). Tables 3 and 4 display age-averaged risks for a hypothetical population which is assumed to be in equilibrium given these mortality rates and which has the male:female ratio for those less than 1 year of age of the current (1998) UK population [30].

### Leukaemia risks

Table 3 shows that following an assumed administered dose of 1 Sv, leukaemia incidence risks are 1.80×10^{−2} Sv^{−1} (95% Bayesian CI, BCI, 1.22, 2.47) using a linear-quadratic-exponential model with piecewise-linear adjustments for the two lowest dose groups, or 1.87×10^{−2} Sv^{−1} (95% BCI 1.24, 2.76) using a linear-quadratic-exponential model; following an assumed administered dose of 0.001 Sv, leukaemia incidence risks using these two models were 1.11×10^{−2} Sv^{−1} (95% BCI −0.61, 2.38) and 0.33×10^{−2} Sv^{−1} (95% BCI −0.77, 1.30), respectively. The reason for low-dose leukaemia risks being larger and more widely dispersed for the linear-quadratic-exponential model with piecewise-linear adjustments is because, as noted above, the linear parameter,* β*_{1}, which dominates the low-dose dose response, is much larger and more widely dispersed in this model than for the one without piecewise-linear adjustments (Table 1). Figure 5 illustrates the shapes of the cancer risk distributions, which are somewhat left-skewed, particularly for the model with low-dose adjustment.

Table 3 shows that as the test dose increases the leukaemia risk generally increases, as would be expected given the linear-quadratic-exponential nature of the dose response.

### Solid tumour risks

Table 4 shows that following an assumed administered dose of 1 Sv, solid cancer incidence risks were calculated to be 16.47×10^{−2} Sv^{−1} (95% BCI 11.89, 21.96) using a linear-quadratic-exponential model with piecewise-linear adjustments for the two lowest dose groups, or 14.71×10^{−2} Sv^{−1} (95% BCI 11.00, 19.16) using a linear-quadratic-exponential model; following an assumed administered dose of 0.001 Sv, solid cancer incidence risks using these two models were 7.28×10^{−2} Sv^{−1} (95% BCI −10.63, 22.10) and 9.00×10^{−2} Sv^{−1} (95% BCI −0.40, 16.85), respectively. The reason for solid cancer risks being more widely dispersed for the linear-quadratic-exponential model with piecewise-linear adjustments is, as noted above, that the linear parameter,* β*_{1}, is more widely dispersed for the model with piecewise-linear adjustments (Table 2). Figure 6 illustrates the shapes of the cancer risk distributions, which are somewhat left-skewed, particularly for the model with piecewise-linear low-dose adjustment. Figure 6 also demonstrates the markedly wider distribution of risks predicted by the model with piecewise-linear low-dose adjustment.

## Discussion

In this paper a class of risk models with a high degree of flexibility in the dose response has been fitted to the latest Japanese atomic bomb survivor cancer incidence data. The resulting uncertainty distributions in the model parameters are used to derive uncertainties in population cancer risks for a current UK population. The Bayesian MCMC methodology that is employed in fitting the risk models is unusual, at least as applied to the Japanese LSS dataset, and explicitly takes account of random errors in the DS86 dose estimates, as well as uncertainties in the model parameters determining the shape of the dose response and the variation of risk as a function of time after exposure and age at exposure. As would be expected, the resulting central estimates of risk that it gives are similar to those obtained using likelihood-based methods without adjustment for dosimetric error, as is discussed below. However, the uncertainties in risk that are estimated here tend to be wider than those previously given.

Risk estimates have been derived using two forms of model for the dose response, a linear-quadratic-exponential model with piecewise-linear adjustments for the two lowest dose groups, or a simple linear-quadratic-exponential model. The reason the first of these models was fitted is because of indications that the population in the lowest dose groups in the atomic bomb survivors may be unrepresentative of that in the higher-dose groups, so that the cancer risks do not belong to the same curve. Certainly for leukaemia there are grounds for believing that by anchoring the risk at that predicted by the lowest dose group one may be biasing the low-dose radiation-induced excess risk downwards by a factor of about 3 (Tables 1, 3). Even for solid cancers, where there are few indications of such bias, by using the more flexible class of models a better envelope of the uncertainty in low-dose risk may be obtained by using these models. For this reason, from now on only this more flexible dose-response model will be considered in all comparisons with cancer risks derived in other publications.

### Comparison of risks with those previously estimated

^{−2}radiation exposure-induced leukaemia cases Sv

^{−1}, is encompassed by the larger uncertainty range derived from Little et al. [31], −0.14–2.60×10

^{−2}radiation-induced leukaemia deaths Sv

^{−1}. Alone among the studies presented in Table 5, the risks in the study of Little et al. [31] are not directly based on models fitted to the Japanese atomic bomb survivors. Little et al. [31] assessed risks and uncertainties using a novel expert-elicitation methodology within a Bayesian framework which involves elicitation of uncertainties in cancer risk from a panel of experts.

Leukaemia and solid cancer risks in various other studies

Cancer type | Reference | Population | Test dose | Cancer risk Sv |
---|---|---|---|---|

Leukaemia | Little et al. [23] | UK | 0.001 | 0.84 (0.02, 2.04) |

US NRC [1] | US | 0.1 | 0.95 (0.26, 2.81) | |

UNSCEAR [3] | UK | 0.1 | 0.5 | |

UNSCEAR [3] | Japanese | 0.1 | 0.4 | |

ICRP [2] | UK | 1.0 | 0.75 | |

UNSCEAR [3] | UK | 1.0 | 1.08 | |

UNSCEAR [3] | Japanese | 1.0 | 0.87 | |

Little et al. [31] | EU/US | 1.0 | 0.91 (−0.14, 2.60) | |

Little et al. [23] | UK | 1.0 | 1.93 (1.14, 3.38) | |

Solid | Little et al. [28] | UK | 0.001 | 12.13–21.98 |

Little et al. [23] | UK | 0.001 | 12.10 (9.46, 15.05) | |

US NRC [1] | US | 0.1 | 6.95 (4.17, 10.93) | |

UNSCEAR [3] | UK | 0.1 | 35.0 | |

UNSCEAR [3] | Japanese | 0.1 | 25.0 | |

ICRP [2] | UK | 1.0 | 8.95 | |

UNSCEAR [3] | UK | 1.0 | 26.0 | |

UNSCEAR [3] | Japanese | 1.0 | 19.8 | |

Little et al. [31] | EU/US | 1.0 | 9.29 (1.35, 31.03) | |

Little et al. [23] | UK | 1.0 | 10.36 (8.41, 12.42) |

^{−2}radiation exposure-induced cancer cases Sv

^{−1}(95% BCI −5.13, 22.06), and the incidence risks calculated (for a UK population) by UNSCEAR [3], which range between 18.0–35.0×10

^{−2}radiation exposure-induced cancer cases Sv

^{−1}. UNSCEAR employ two methods of transfer of risk between the atomic bomb survivors and the current UK population, relative and absolute risk transfer. As can be seen from Table 5, models with absolute transfer of risks, which correspond to what is done in this paper, produce lower risks, in the range 18.0–29.0×10

^{−2}radiation exposure-induced cancer cases Sv

^{−1}. The reason for this is that cancer incidence rates are generally lower in Japan than in the UK [32]. UNSCEAR also use two different risk projection models, one in which relative risk varies with age at exposure, the other in which it varies proportional to a power of attained age. The age-at-exposure model generally produces higher risks. These should be contrasted with the excess absolute risk model employed here.

In comparing risks between the various studies a number of points should be considered. As shown above, the largest source of uncertainty for cancer risk evaluations is model uncertainty, and arguably this is the most serious shortcoming of this and previous attempts at determining uncertainties in late health effects, as discussed at greater length by Little et al. [23]. Approximately 52 years after exposure about half the LSS cohort is still alive [33], so that population risk calculations crucially depend on extrapolating the current mortality and incidence follow-up of this group to the end of life. As such, risk models with equivalent statistical fit to the LSS dataset can yield divergent risk estimates. Uncertainties due to risk projection are greatest for solid cancers, because the radiation-associated excess risk is still increasing in this group. For leukaemia the excess risk is reducing over time (Table 1) and most models used in other studies predict, as here, very few radiation-associated leukaemia deaths or cases from the current follow-up point to extinction. Related to the uncertainties resulting from the method of risk projection to the end of life, are those due to the method of extrapolation from the Japanese atomic bomb survivors to the population of interest (e.g. a current UK population, as used here); that these are also important is demonstrated by the difference between the risks calculated by UNSCEAR [3] for the UK population using two different methods of risk transfer (Table 5). The test dose used also makes some difference to risk estimates, as shown by Tables 3, 4 and 5. The different measures of risk, in particular the excess cancer death measure used by BEIR V [1] and the radiation exposure-induced cancer incidence/death measure used in most other studies, and the distinction between mortality and incidence risks should also be noted. These are discussed at greater length by Little et al. [23] and Thomas et al. [29].

### Other considerations

A distinctive feature of the models considered in this paper is their attempt to decouple the very low-dose part of the dose response in the LSS data from the higher-dose part. As noted above, this is because of indications that the population represented by the low-dose groups may be unrepresentative of that in the high-dose groups. For similar reasons the so-called not-in-city (NIC) group of survivors has been excluded from most recent analysis of the LSS [24, 25, 33, 34] because of doubts as to how socio-economically similar this group is to those in either city (defined as <10 km from the hypocentre) at the time of the bombings. There is evidence of lower mortality in this group compared with those in the city, but exposed at low dose [35]. There is also evidence that cancer rates in the so-called “distal” survivors (those exposed >3000 m from the hypocentre) have increased from being slightly less than those of the low-dose “proximal” group (those exposed <3000 m from the hypocentre) to being about 10% greater, by 1990 [36]. The analyses of this paper suggest that there are irregularities in the low-dose (<0.1 Sv) part of the dose response, particularly for leukaemia (Figs. 1, 2), which are unlikely to be due to the radiation dose received. Assuming that they are not due to chance, it is possible that they may reflect residual socio-economic heterogeneity within this low-dose part of the LSS cohort [35, 36]. Taking account of this irregularity in the dose response results in a substantial increase in the low-dose leukaemia risk (Tables 3, 5). It is arguable that by taking account of this feature of the dose response a better central estimate of risk, as well as of its uncertainty, is derived. It would be desirable to take account of the variation of cancer rate by distal and proximal category as well as dose, which is acting as a proxy for the proportion of survivors in these categories in the very low-dose group, using the data recently analysed by Pierce and Preston [36], but this data is not publicly available.

Likewise, by allowing for possible high-dose reductions in risk, using an exponential modifier of the dose response as in Eq. 4, it is less necessary to consider restricting the data, for example by omitting survivors with doses >2 Gy, as others did [13, 37, 38], in order to minimise the influence of the high-dose part of the dose response on assessments of low-dose curvature. In contrast to the analyses employed here, the analyses of Little and Muirhead [13], Kellerer et al. [37] and Walsh et al. [38] used forms of dose response that were linear-quadratic in gamma dose and linear in neutron dose, without the exponential modifier of dose used here to adjust for high-dose curvature.

In principle, the modelling conducted in this paper could be extended to take account of model uncertainties, and in particular the use of absolute as well as relative risk models, or indeed hybrid relative/absolute models of the sort employed by Muirhead and Darby [39] and by Little et al. [40]. It would also be possible to extend the models to take account of other sorts of errors in the DS86 dosimetry than those considered here. For some time it was thought that the neutron dose estimates for the Hiroshima atomic bomb survivors using the most recently published (DS86) dosimetry were systematic underestimates, particularly for survivors beyond 1000 m from the hypocentre [26, 41]. The DS86 gamma dose estimates were thought to be more reliable [26], as also the Nagasaki neutron dose estimates [42]. Analysis has been performed attempting to take account both of random errors in DS86 dose estimates as well as such systematic discrepancies [13]. However, recent analysis of the totality of data, including fast-neutron activation products, suggests that there are no appreciable systematic errors in the DS86 Hiroshima neutron dose estimates [43, 44]. The most current set of atomic bomb survivor dose estimates, the so-called DS02 dosimetry, differ slightly from the DS86 system, both for neutron and gamma doses, by amounts generally no more than 20% in the range up to 1500 m from the two hypocentres where survivors received the most appreciable doses [44]. Analyses of the RERF epidemiological data using the new dosimetry indicate that cancer risks might decrease by about 8% as a result, with no appreciable change in the shape of the dose response or in the age-time patterns of excess risk [45].

The analyses of this paper for all solid tumours are based on weighted colon dose, evaluated using a neutron RBE of 10, with a model that is linear-quadratic-exponential in this variable. Radiobiological data suggests that the dose response should be linear for neutrons and linear-quadratic for gamma rays, as proposed by Little and Muirhead [13], Kellerer et al. [37] and Walsh et al. [38]. In addition, for organs near the surface of the body RBE-10-weighted colon dose will underestimate the true organ dose; in particular the neutron component of dose will be underestimated [37, 38]. Walsh et al. [38] demonstrated that, depending on the assumed neutron weighting factor, using organ-specific doses would result in values of the linear relative risk parameter for solid cancer mortality being slightly higher, by up to about 10%, than that calculated using organ-averaged dose. However, employing organ-specific doses yielded estimates of curvature in the dose response that were both larger and more uncertain than those obtained using organ-averaged dose [38].

The analysis given here is based on the follow-up of the incidence data up to the end of 1987 using the previous (DS86) dosimetry [24, 25]. At least for solid cancers, incidence follow-up has now been extended to the end of 1994 [36, 46]. However this dataset is not yet publicly available. It would clearly be desirable to repeat the analysis using more complete follow-up, also using the most current (DS02) dosimetry. It is hoped that these data will become available shortly.

The Bayesian MCMC methodology that has been employed in model fitting here and in a previous analysis of the Japanese mortality data [23] is unusual, at least as applied to the LSS dataset, and explicitly takes account of random errors in the DS86 dose estimates, as well as uncertainties in the model parameters determining the shape of the dose response and the variation of risk as a function of time after exposure and age at exposure. As previously discussed [23], despite the novelty of this technique, the results that it gives are similar to those obtained using regression-calibration methods used by other researchers [7, 8, 9, 10, 11, 12, 13, 14, 15], as well as those obtained by others using likelihood-based methods without adjustment for dosimetric error [1, 2, 3]. The reasons for this are not surprising. When uninformative priors are assumed, the difference between the Bayesian MCMC approach and the first-order regression-calibration approach is between using the exact likelihood, and using the regression-calibration approximation, obtained by substituting in the likelihood for the true dose,* D*, with the conditional expectation, \( E{\left[ {D|d} \right]} \), of the true dose given the nominal dose,* d*. It is well known that when dosimetric errors are not too large, as is the case here, the first-order regression-calibration parameter estimates are a good approximation to the full likelihood-based estimates [16, 47, 48].

However, there is an important practical advantage arising from use of the Bayesian MCMC approach to calculate population risks. In this paper assessment is made of uncertainties in a non-linear population risk function* RECI*_{c}(*s*,*a*,*D*_{t})=*RECI*_{c}(*s*,*a*,*D*_{t},*θ*) of the model parameters* θ*. The Bayesian MCMC approach produces an arbitrarily large collection of realisations (*θ*_{j})_{j} sampled from the posterior parameter distribution. The uncertainty in the function* RECI*_{c}(*s*,*a*,*D*_{t},*θ*) can be directly evaluated by describing the distribution of values (*RECI*_{c}(*s*,*a*,*D*_{t},*θ*_{j}))_{j}.

It should be noted that in the RERF analyses of Pierce et al. [7, 8, 9], regression-calibration adjustments are restricted to relatively high-dose exposure groups (those with shielded kerma >0.1 Gy); in contrast, most other researchers [10, 11, 12, 13, 14, 15, 21, 22, 23] assumed uniform dose error in all dose groups, which in the low-dose groups tends to result in posterior “true” dose estimates being greater than the “nominal” doses. Arguably, the exclusion from adjustment of the <0.1 Gy survivors by Pierce et al. [7, 8, 9] is arbitrary. However, in practice the degree of adjustment is slight in absolute terms, and would not be expected to make a substantive difference to the results [15].

Finally, comment should be made on the two-stage modelling of dosimetric error adopted here and in a previous paper [23]. This is only necessary because of the grouped nature of the RERF data in its publicly available form. Availability of individual data would certainly facilitate a simpler analysis of this data, although there is no reason to suppose that the results that might be obtained would materially differ from those presented. Subject to individual privacy being maintained (for example by releasing records without name, and year of birth only), it is difficult to see why this individual data should not be made available in the public domain.

## Acknowledgements

The authors are grateful for the detailed and helpful comments of the two referees. This report makes use of data obtained from the Radiation Effects Research Foundation (RERF), Hiroshima and Nagasaki, Japan. RERF is a private, non-profit foundation funded by the Japanese Ministry of Health, Labour and Welfare and the US Department of Energy, the latter through the National Academy of Sciences. The conclusions in this report are those of the authors and do not necessarily reflect the scientific judgment of RERF or its funding agencies. This work was funded partially by the European Commission under contract FIGD-CT-2000–0079.