This paper begins from the publicly available data, and I do not know if a comparable analysis of the anonymous individual data would show similar results.

In re-analysing portions of the 1950–90 grouped data, the approach here has four features.

- 1)
To predict risks at 10 mSv, the 0 – 20 mSv data is analysed directly.

- 2)
A variable lag period is used to analyse latency.

- 3)
Dosimetry data is not reduced to categories before modelling. Dose is taken as a numerical variable, defined on the grouped data cells, but results are also tested in a 4-category model with baseline defined by D_{φ} = 0 and cutpoints which roughly equipartition the p-y in non-baseline categories.

- 4)
Linearity of the dose response is tested by nesting within more complex models.

With a fixed 5 year lag, none of the cancers considered here show significant effects in the 0 – 20 mSv dose range using a linear model. Allowing latency to vary in this model gives significant positive responses for the liver and all-solid cancers.

Latency reflects biochemical changes required after initial radiation if mutant cells are to progress and form a tumour eventually identified as cause of death, and historical changes in environmental factors which interact with radiation for a particular cancer. Thus latency may be organ and gender specific.

Rothman [13] illustrates how ignoring latency may mask important effects, whether or not the original exposure was brief. Analysis using the lagged dose D_{φ} is a simple approach depending on only one parameter. The response might be clearer by modelling the effect of D*w, with w some more general function of Time-Since-Exposure. Such models have been applied to lung cancer mortality in uranium miner cohorts [14, 15].

If the linear model were appropriate throughout the low dose region, we might expect ERR_{0.025,φ} ~ 0.025(ERR_{1,φ}) whatever data were used to estimate each ERR. In fact, the ERR values are often comparable. Two non-linear models give significant improvements in the 0 – 20 mSv dose range for the stomach, liver, lung, uterus and all-solid cancers, and for various gender specific sites. These improvements are strong, for example p < 0.001 when comparing the two-phase and linear models (M/F) for stomach, liver, lung, and all-solid; and p < 0.000001 for the liver.

Unlike the linear and control models, the transient and two-phase models require extensive computation as the Deviance may have multiple local minima at any choice of latency φ (fixed when fitting the model). Computation involves a search for local minima, selection of the minimum Deviance at φ, and then a comparison amongst these minima for different φ values. The optimal φ is chosen to give the absolute minimum Deviance, with or without the constraint that ERR_{0.025} be non-negative. The search is streamlined by restricting φ to 5, 6, ... 44 and later refined to consider all φ (to 2 decimal places) in a range which appears likely to contain the optimal value.

I do not know of any general analytical method which might limit the total number of local minima at fixed φ in this data. Instead, the τ axis is partitioned (see Methods). Fitting the model with τ constrained to an interval typically yields a τ value at either endpoint, reflecting the constraint, except for those intervals which contain τ values at which the Deviance attains a local minimum.

Searches begin from the control parameter values which optimise the control model, while τ is confined to the relevant interval and β,σ are initially set to 0. Conceivably, this choice of initial conditions may cause the Newton-Raphson iteration to miss some local minima, though testing other initial conditions did not detect any other solutions. In any case the minimum Deviance at any particular φ can be no higher than the values found here, so any missing minima could only strengthen the evidence of non-linearity.

Fitting the two-phase model to the lung data illustrates these issues. When φ = 13 three local minima are detected. At τ = 4.51, Dev = 1040.35. At τ = 13.86, Dev = 1040.87. At τ = 175.63, Dev = 1050.25. The minimum Dev at φ = 13 is thus 1040.35. The linear model has Dev = 1053.64. Thus LRT_{2p-lin} for comparing the two-phase and linear models is 1053.64 – 1040.35 = 13.29, and it is this value which is displayed in Figure 6 when φ = 13. Likewise the ERR value computed at this minimum Dev is displayed in Figure 7 when φ = 13. The resulting graphs indicate the region to be searched for an optimal choice of φ, subject to the constraint ERR ≥ 0. This optimum is φ = 13.6, the value shown in Table 3. At this latency there are again 3 local minima, two of which have similar Dev. At φ = 13.6 the minimum at τ = 4.45 with Dev = 1038.58 has ERR_{1} = 0.88 while the local minimum at τ = 13.54 with Dev = 1039.26 has ERR_{1} = 1.10. Note that while τ varies widely without appreciable change in Dev, ERR_{1} is much more stable. In this example, the 95%CI is (0.12, 3.36). For this reason, ERR is a much better focus for analysis than the model parameters themselves.

Confidence intervals for ERR are often somewhat wider in the two-phase model than in the simpler transient model which is sufficient to describe the lung and all-solid (M/F, F) data. However, as well as improving the fit for the stomach, liver and all-solid (M) the two-phase model gives a more coherent account of the latency regions of significant positive or negative dose response.

Linear extrapolation of the LSS12 results shows almost no response at the doses considered here. Whilst LSS12 uses organ doses and a different system of controls, these factors do not account for the large discrepancy in risk estimates. Alternative controls affect the estimates by a factor of 2 or less, and the use of organ doses has even less impact. Significant discrepancies arise when the dose range is restricted to 0 – 20 mSv and latency is included in the analysis.

Stewart and Kneale [16] found evidence of selection bias in the LSS 1950 – 1985 cohort, thought to reflect the fact that only those victims able to survive from 1945 – 1950 were eligible to enter the cohort. The test group used by Stewart and Kneale to detect bias included less than 4% of the total cohort but had nearly 30% of high doses (> 1000 mSv). It is plausible that such bias would be reduced in the 0 – 20 mSv subcohort, but I cannot test this from the publicly available RERF data.

Pierce and Preston [17] analysed all-solid cancer incidence in Japanese survivors from the 1958 – 1994 tumour registry data for the range 0 – 500 mSv, using a linear ERR model based on colon dose and a categorical model with cutpoints 0, 5, 20, 100, 200, 250, 300, 400 mSv (colon dose). Estimates correspond to ERR ~ 0.006 at 10 mSv. Likewise, if the linear model here is applied to all-solid cancer mortality (1950–90) in the 0 – 500 mSv dose range with latency 5 years, ERR_{1} = 0.004 with LRT = 8.58.

Pierce and Preston focus on survivors who were exposed relatively near the hypocentres of the A-bombs and exclude distal survivors ( ≥ 3 km distant) on the grounds that they had higher baseline cancer rates and that some lifestyle cancer risk factors correlate with urban-rural distinctions, though cigarette smoking had almost no correlation with estimated dose or distance from the hypocentre. Excluding the distal group lowered the baseline by about 5% in their data. Although that is significant in relation to the estimates of ERR in the RERF studies, it is marginal compared to the ERR values found here with the latency models.

The 0 – 20 mSv subcohort contains many proximal as well as distal survivors (10,159 proximal survivors in the incidence dataset had doses below 5 mSv). The results here may of course reflect other risk factors which may correlate and/or interact with radiation dose, but which could only be approached through the individual data. Investigation of possible confounders should also consider latency and non-linear models such as those analysed here, given their clear superiority to the linear model with fixed 5 year lag, for the low-dose grouped data.

This paper is based on the DS86 dosimetry and the results could reflect dosimetry errors, arising from incorrect estimation of the flash dose or by omitting other radiation sources. The doses received in Hiroshima and Nagasaki include the flash dose (used here), induced radioactivity in building materials or soil which persisted for several weeks, "black rain" which fell in the immediate aftermath of the bombings, natural background radiation, global fallout from atmospheric weapons tests, occupational and medical exposures. The public data does not include any individual occupational or medical exposures. Natural background and global fallout should not be correlated with the exposures arising directly from the bombs in 1945, and would be expected to bias results towards the null. Doses from induced radioactivity and "black rain" could be relevant, but currently available RERF public datasets do not include either of these two additional sources.

Errors in the flash dose itself are unlikely to explain the results. Non-linearity and large values of ERR at 10 mSv persist when the zero-dose data is deleted from the 0 – 20 mSv subcohort and likewise when ten intervals spanning 0 – 20 mSv are used to delete dose ranges from the data.

The DS02 dataset shows that there is virtually no misclassification between the DS86 categories "below 5 mSv" and "above 5 mSv". For the liver at latency 36.9 years, it makes little difference whether dose is taken as a categorical variable defined by the 5 mSv cutpoint in the data stratification, or as a numerical value analysed with the linear model.

DS02 and DS86 are in reasonable agreement above 5 mSv. If datacells with DS86 dose below 5 mSv are excluded from the 0 – 20 mSv dose range, the two-phase model is no longer a significant improvement on the linear model for the liver. However, very similar evidence of non-linearity for the liver arises in the 5 mSv – 500 mSv dose range where DS86 is a reasonable approximation to DS02, and in the 0 – 500 mSv dose range considered by Pierce and Preston [17].

Analysis of solid cancers using DS02 in the 0 – 20 mSv dose range gives estimates of ERR comparable to those derived from DS86.

These various arguments suggest that the results are unlikely to be explained by errors in dosimetry of the flash dose, although DS02 like DS86 is subject to some uncertainty due to random errors in specifying individual location and shielding. Separately, induced radioactivity and "black rain" represent additional doses not reported by DS86 or DS02.

Induced radioactivity appears unlikely to fully explain the results. According to the RERF website [6] "The closer to the hypocentre, the higher was the dose [from induced radioactivity]. Past investigations suggested that the maximum cumulative dose at the hypocentre from immediately after the bombing until today is 0.8 Gy in Hiroshima and 0.3–0.4 Gy in Nagasaki. When the distance is 0.5 km or 1.0 km from the hypocentre, the estimates are about 1/10 and 1/100 of the value at the hypocentre, respectively." The issue was examined in detail in the DS86 Final Report Chapter 6 and an Appendix to this Chapter [8]. The cumulative dose from induced radioactivity decreases exponentially with distance from the hypocentre.

From the DS02 dataset, the minimum distance from the hypocentre amongst cells with DS86 doses below 20 mSv is 2.081 km while for cells with DS86 doses below 500 mSv it is 1.210 km. For the 5 – 500 mSv range the total cumulative impact of induced radioactivity would be below 8 mSv in Hiroshima and below 4 mSv in Nagasaki. Consider the unlikely possibility that induced radioactivity adds 8 mSv to those cells which contain Hiroshima liver cancer deaths, and 4 mSv to those cells which contain Nagasaki liver cancer deaths, while leaving all other cells unaffected. Under this extreme assumption, the two-phase model at latency φ = 36.9 years has solutions with ERR_{1} < 0 but the linear model gives LRT_{lin-con} = 11.03 (1 df) and ERR_{1} = 0.029 (0.010, 0.054). Conversely, suppose that induced radioactivity adds 8 mSv to those Hiroshima cells which do not contain liver cancer deaths, and 4 mSv to those Nagasaki cells which do not contain liver cancer deaths, while leaving all other cells unaffected. Then the linear model at latency φ = 36.9 years gives LRT_{lin-con} = 4.63 (1 df) and ERR_{1} = 0.016 (0.001, 0.036), which is still 4.3 times higher than the LSS12 estimate. These two extreme assumptions may perhaps provide limits on the scope for induced radioactivity to affect the linear model in the 5 – 500 mSv subcohort, where the corresponding results without the addition of induced radioactivity are LRT_{lin-con} = 7.25 (1 df) and ERR_{1} = 0.021 (0.005, 0.044). For the 0 – 20 mSv dose range the total cumulative impact of induced radioactivity would be below 0.08 mSv in Hiroshima and below 0.04 mSv in Nagasaki. Whatever consequence this may have for the two-phase model, its effect on the linear model is negligible.

"Black rain" fell primarily at some distance from the hypocentres. According to the RERF website, "Because of wind, the rain mainly fell in northwestern Hiroshima (Koi-Takasu area) and in eastern Nagasaki (Nishiyama area). The maximum estimates of dose due to fallout are 0.01–0.03 Gy in Hiroshima and 0.2–0.4 Gy in Nagasaki. The corresponding doses at the hypocentres are believed to be only about 1/10 of these values." According to Chapter 6 of the DS86 Final Report [8] the rainfall was concentrated around 3000 m from the hypocentres in both cities. From the DS02 dataset, the maximum distance from the hypocentre amongst cells with DS86 doses above 5 mSv is 2.683 km, somewhat closer to the hypocentres than the main rainfall areas. Maximum doses from rainfall were much lower in Hiroshima, though RERF do not give enough detail to estimate an upper bound for the dose from "black rain" in Hiroshima at distances below 2.683 km. However, if the Hiroshima data for the liver is analysed separately for the 5 – 500 mSv subcohort, the two-phase model remains a significant improvement over the linear model at latency φ = 36.9. LRT_{2p-lin} = 5.718 (1 df) and ERR_{1} = 0.674 (-0.06, 1.92) while for the linear model LRT_{lin-con} = 7.946 and ERR_{1} = 0.027 (0.007, 0.055).

Compounding the uncertainty over doses from "black rain", individuals may have travelled into a rainfall area even if they were outside it when the bombs exploded. RERF's estimates refer to external doses, omitting any inhaled or ingested radiation. In any case the conclusions are provisional until a dataset for specific solid cancers showing DS02 flash doses, induced radioactivity and "black rain" becomes available. Potential confounding by other risk factors cannot be excluded. The neutron RBE of 10 used here, as in LSS12 and elsewhere, may be inappropriate. Despite all these reservations, which also affect the LSS studies, it is striking that both the 0 – 20 mSv and 5 – 500 mSv subcohorts show non-linear dose response for the liver and the two-phase model gives comparable estimates from both subcohorts for Excess Relative Risk at 10 mSv with latency 36.9 years. "Black rain" and induced radioactivity have quite different impacts within these two subcohorts.

I do not know whether these results and the optimal latencies are specific to the A-bomb cohort. In their study of gamma radiation and mortality in the Oak Ridge workforce, Frome et al. [18] find a 'Low-dose β' value of 2.9 with LRT = 3.12 (p = 0.08) for the combined digestive category using a multiplicative model (without latency) with dose restricted to 0 – 640 mSv. Since their unit dose is 1 Sv this corresponds to ERR_{1} = 0.029.

Non-linear models are applied here without assuming any cellular mechanism, and their success in fitting the cohort data does not prove that any particular cell mechanism operated there. However, the two-phase model adapts and simplifies a model derived by Brenner et al. [19] to explain the 'oncogenic transformation frequency' of cells exposed to broad beam irradiation by α particles:

TF = νq<N> + σ [1-e^{(-k<N>)}] [e^{(-q<N>)}]

TF is the number of transformed cells per surviving cell (Excess Relative Risk of transformation); ν is the transformation frequency for cells struck directly by exactly one α particle, q the surviving fraction of cells struck directly by exactly one α particle, <N> the mean number of α particles striking each cell in the broad beam irradiation (a dose variable), k the number of cells receiving the bystander signal emitted from a cell struck directly by one or more α particles, and σ the (presumed) hypersensitive fraction of bystander cells which are transformed on receipt of any bystander signal. Note that the first term in the TF expression refers to direct effects, whilst the second term refers to bystander effects. As a function of <N>, TF is approximately linear with slope ν q + σ k at very low doses and approximately linear with slope ν q at higher doses.

The two-phase model is asymptotically linear with slope σ + β at very low doses and asymptotically linear with slope β at higher doses. It generates curves of the same qualitative shape as the TF cell model.

In the cell model, the ratio of asymptotic slopes is **R** = 1+(σ k/ν q). From targetted microbeam experiments on C3H 10T^{1}/2 mouse fibroblast cells, Brenner et al. estimate ν = 1.3 × 10^{-4}, σ = 6.4 × 10^{-4}, and q = 0.8. Thus **R** ~ 1 + 6.2 k. In a subsequent paper Brenner and Sachs [20] suggest that k ~ 50 (so **R** ~ 311) for the human lung, from modelling dose-rate effects in (male) uranium miners exposed to radon.

In the two-phase model, the ratio of asymptotic slopes is **R** = 1 + σ/β. For the male lung, with optimal latency 13.55 years, **R** = 202.22. For the lung, the two-phase model is not significant against the transient and the confidence region for **R** is infinite and includes negative values. But for the stomach (M/F) **R** = 262.62 with 95% CI (102.41, 975.09), and for the liver (M/F) **R** = 204.11 (58.29, 792.85). Thus the low dose A-bomb data is compatible with a bystander model using roughly comparable values of k, the number of cells receiving the bystander signal.

The flash dose comprised gamma and neutron doses. An appropriate cell model for bystander effects following gamma radiation may differ from that developed for α particles, where the impact of a single track on one cell is quite large. However fission neutrons, like α particles, are high LET (linear energy transfer) and might elicit a similar bystander signal. Induction of genomic instability in unirradiated bystander cells has been demonstrated for neutron irradiation in mice [21].

Non-linear dose response curves can also arise from a hypersensitive population subset [22]. Sharp et al. [23] analysed primary liver cancer mortality in relation to Hepatitis B and C in Japanese A-bomb survivors and found a very strong supermultiplicative interaction between HVC and radiation dose, but only for the high and medium dose ranges. For doses below 18 mSv, there was no significant interaction. However, Sharp et al. did not include latency and truncated all liver doses below 3 mSv. I could not analyse primary liver cancer because the data in r12canc.dat shows all liver cancers. Persistent inflammation including chronic liver disease has been detected in Japanese survivors 40 years after the bombing and correlated with radiation dose [24].