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

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.

Appendices

Appendix A: Further details on fitted models

Leukaemia

The baseline leukaemia incidence in the stratum with city c=0=Hiroshima, c=1=Nagasaki), sex s (s=0=male, s=1=female), and average age at exposure a and average time since exposure t is given by:

$$ \lambda {\left( {c,s,a,t} \right)} = \exp {\left[ {\delta _{1} + \delta _{2} \cdot s + \delta _{3} \cdot c + \delta _{4} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]} + \delta _{5} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]}^{2} + \delta _{6} \cdot \ln {\left[ {t/25} \right]}} \right]} $$

(8)

The function G(s,a,t) modifying the radiation-induced excess leukaemia risk is given by:

$$ G{\left( {s,a,t} \right)} = \exp {\left[ {\pi _{1} \cdot s + \pi _{2} \cdot \ln {\left[ {t/25} \right]}} \right]} $$

(9)

The knotpoints used for the ten-knotpoint model are 0.00176, 0.04011, 0.15287, 0.33687, 0.71207, 1.21019, 1.66890, 2.31870, 3.05143 and 4.0 Sv, while for the three-knotpoint model they are 0.00176, 0.04011 and 0.055 Sv. With the exception of the largest points, these represent the person-year weighted posterior means from the first stage of the MCMC fits, by “nominal” (DS86) bone-marrow dose group.

Solid cancer

The baseline solid cancer incidence in the stratum with city c, sex s, average age at exposure a and average time since exposure t is given by:

$$ \lambda {\left( {c,s,a,t} \right)} = \exp {\left[ \begin{aligned} & \delta _{1} + \delta _{2} \cdot s + \delta _{3} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]} + \delta _{4} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]} \cdot s \\ & + \delta _{5} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]}^{2} + \delta _{6} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]}^{2} \cdot s + \delta _{7} \cdot \ln {\left[ {t/25} \right]} \\ & + \delta _{8} \cdot \ln {\left[ {t/25} \right]} \cdot s \\ \end{aligned} \right]} $$

(10)

The function G(s,a,t) modifying the radiation-induced excess solid cancer risk is given by:

$$ G{\left( {s,a,t} \right)} = \exp {\left[ {\pi _{1} \cdot s + \pi _{2} \cdot \ln {\left[ {t/25} \right]} + \pi _{3} \cdot \ln {\left[ {{\left( {t + a} \right)}/50} \right]}} \right]} $$

(11)

The knotpoints used for the ten-knotpoint model are 0.00176, 0.03997, 0.14694, 0.32338, 0.67825, 1.15012, 1.58285, 2.09054, 2.58584 and 4.0 Sv, while for the three-knotpoint model they are 0.00176, 0.03997 and 0.055 Sv. With the exception of the largest points, these represent the person-year weighted posterior means from the first stage of the MCMC fits, by “nominal” (DS86) colon dose group.

Appendix B: Risk summaries from other studies

The cancer risks derived from other studies are shown in Table 5, and in combination with those calculated here (from Tables 3 and 4), in Figs. 7 and 8. With the exception of the studies of Little et al. [31] and BEIR V [1] these are straightforwardly derived from the publications.

For the study of Little et al. [31] the leukaemia risk was given as 0.91×10⁻² Sv⁻¹ (90% CI 0.026, 2.33); the 95% CI were estimated by inflating the upper and lower parts of the CI by the factor N_0.975/N_0.95=1.96/1.64=1.19, where N_0.95=1.64 and N_0.975=1.96 are the 95% and 97.5% points of the standard normal distribution. The all cancer risk was given by Little et al. [31] as 10.2×10⁻² Sv⁻¹ (90% CI 3.478, 28.5). The mean solid cancer (=all cancers−leukaemia) risk can therefore estimated to be 10.2–0.91×10⁻² Sv⁻¹=9.29×10⁻² Sv⁻¹. The 97.5% point for solid cancers is estimated by approximating the upper tail of the all cancer risk distribution by a normal random variable, with standard deviation (SD) equal to $\frac{{28.5 - 10.2}} {{N_{{0.95}} }}\;{\text{x}}\;10^{{ - 2}} \;{\text{Sv}}^{{ - 1}} = 11.13\;{\text{x}}\;10^{{ - 2}} \;{\text{Sv}}^{{ - 1}}$; similarly for leukaemia the corresponding upper SD is estimated by $\frac{{2.33 - 0.91}} {{N_{{0.95}} }}\;{\text{x}}\;10^{{ - 2}} \;{\text{Sv}}^{{ - 1}} = 0.86\;{\text{x}}\;10^{{ - 2}} \;{\text{Sv}}^{{ - 1}}$. From this can be estimated the solid cancer upper SD by [11.13²−0.86²]^0.5×10⁻² Sv⁻¹=11.09×10⁻²Sv⁻¹, and therefore the 97.5% point for solid cancer risk by 10.2−0.91+11.09⋅N_0.975×10⁻²Sv⁻¹=31.03×10⁻²Sv⁻¹. The lower confidence limit is similarly derived.

BEIR V [1] estimated the risks for leukaemia to be 1.1×10⁻² Sv⁻¹ (90% CI 0.5, 2.8) for males and 0.8×10⁻² Sv⁻¹ (90% CI 0.3, 1.9) for females. The mean risk for the total population (males+females) can therefore be estimated as 0.5×[1.1+0.8]×10⁻² Sv⁻¹=0.95×10⁻² Sv⁻¹. The 97.5% point is estimated by approximating the upper tail of the male risk distribution by a normal random variable, with SD equal to $\frac{{2.8 - 1.1}} {{N_{{0.95}} }} \times 10^{{ - 2}} \;{\text{Sv}}^{{ - 1}} = 1.03 \times 10^{{ - 2}} \;{\text{Sv}}^{{ - 1}}$. For females the analogous quantity is estimated as $\frac{{1.9 - 0.8}} {{N_{{0.95}} }} \times 10^{{ - 2}} \;{\text{Sv}}^{{ - 1}} = 0.67 \times 10^{{ - 2}} \;{\text{Sv}}^{{ - 1}}$. Therefore the 97.5% point of the risk for the total population can be estimated by X_0.975, satisfying:

$$ 0.975 = 0.5 \cdot {\left[ {{\text{erf}}{\left( {\frac{{X_{{0.975}} - 0.011}} {{0.0103}}} \right)} + {\text{erf}}{\left( {\frac{{X_{{0.975}} - 0.008}} {{0.0067}}} \right)}} \right]} $$

(12)

where erf(x) is the error function. This can be calculated by numerical means to be X_0.975=2.81×10⁻²Sv⁻¹. The lower confidence limit is similarly derived, as also the analogous estimates for solid cancers.