INTRODUCTION
Epidemiological studies have played a prominent role in radiological protection. In particular, research results from the Life Span Study (LSS) of the Hiroshima and Nagasaki atomic bomb survivors (^{Preston et al. 2007} ) have been valuable owing to their high-quality diagnosis and dosimetry, wide range of radiation exposure levels, and large sample size including old and young people. The estimated risks, such as excess relative risk (ERR) and excess absolute risk (EAR), are utilized for nominal risk estimation for the system of radiological protection (^{ICRP 2007} ), together with the dose and dose-rate effectiveness factor (^{Rühm et al. 2016} ). Several other epidemiological studies have been reported (^{Preston et al. 2002} ; ^{Jacob et al. 2009} ; ^{Little et al. 2010} ; ^{Schubauer-Berigan et al. 2015} ; ^{Leuraud et al. 2015} ; ^{Richardson et al. 2015} ; ^{Shore et al. 2017} ) that use pooled analysis and meta-analysis methods to estimate the radiation risk of low doses by increasing the sample size of populations assuming similarity. Since the results of epidemiological studies still play an important role in estimating the radiation risk in the latest recommendations of the International Commission on Radiological Protection (^{ICRP 2007} ), their findings will be crucial to the system of radiological protection in years to come.

On the other hand, the relationship between sample size and radiation dose in epidemiological studies where an increase in either cancer mortality or morbidity is detectable was discussed by ^{ICRP (2005)} . In ICRP Publication 99 (2005), an illustrative calculation indicated that a large sample size may be necessary to detect the effect of a low dose of radiation, acknowledging the unrealistically optimistic assumption that the variation in background risks is composed only of statistical uncertainty and decreases monotonically with increasing population size (see Fig. A1 in the appendix). It is well known that an end point of interest in epidemiological studies, such as cancer mortality or morbidity, will be affected by various modification factors, such as age at exposure, attained age, sex, lifestyle (diet, smoking, etc.; ^{HCCP 1996} ), and genetic factors, which will be independently distributed among the population irrespective of radiation exposure. In the analysis of epidemiological studies, although correction is carried out to identify the pure effect of radiation exposure by developing the model to eliminate the effect of modification factors, it is impossible to adjust for such factors inclusively and completely owing to misspecification and so forth. Moreover, there may be other unknown factors affecting the end point of interest, and residual risks (unadjusted risks) may be a factor contributing to the inevitable uncertainty in background risks for control and exposed populations. Consequently, there may be a minimum limit to the provable increase in risk, i.e., a minimum provable risk (MPR) regardless of the sample size.

Thus, to assess the possible effect of such a residual risk in epidemiological studies, the concept of MPR was examined in detail by applying the concept of minimum provable dose (MPD), which has recently been developed for quantitative assessment of protection prioritization on sex- and age-specific radiation-related cancer risks for different tissues and organs for emergency workers by ^{Ogino et al. (2016)} . In this case study, we used the provisional reference values of background lifetime risk of cancer mortality and its variation, which were calculated from the results of a national survey of vital statistics and population census for 47 prefectures in Japan (^{Ogino and Hattori 2014} ), to estimate typical MPR values as well as possible MPR values in a hypothetical epidemiological study.

Additionally, not only carcinogenesis but also noncancer effects are now the focus of radiological protection (^{ICRP 2012} ). Among noncancer effects, circulatory diseases are so highly lethal that it could be of great concern how such diseases will be treated or managed in the system of radiological protection. For Japanese people, estimations of background lifetime risks of noncancer disease mortality, including circulatory diseases (stroke and heart disease), blood disease, respiratory diseases, and so forth, among the 47 prefectures of Japan for 2000, 2005, and 2010 have already been conducted (^{Sasaki et al. 2016} ) using the same methodology as that used to estimate background lifetime risk of cancer mortality (^{Ogino and Hattori 2014} ). It was revealed that although the background lifetime risk of circulatory disease mortality has tended to decrease, the background lifetime risk of circulatory disease mortality itself and also its variation are likely to be slightly higher than those of cancer mortality.

As ^{Ogino et al. (2016)} demonstrated in the previous study, relative comparisons of MPD values were useful in estimating the prioritization of different tissues and organs for radiation-related cancer risks. Likewise, in this case study the MPR for circulatory diseases due to radiation exposure was also estimated and compared with that for cancer .

METHODS
Examination of the concept of MPR
The graphs in Fig. 1 a–c show typical frequency distributions of the risk for a population. Since such distributions have variance due to statistical uncertainty related to the population size N , the frequency is approximately expressed as a Poisson distribution or normal distribution when N is sufficiently large.

Fig. 1: Illustrative examples of the concept of the minimum provable risk (MPR). In (a), graph (A) indicates the distribution of the background risk for two populations. Assuming a small amount of excess risk is added to one of the populations, then the total risk distribution of this population becomes graph (B). When the excess risk is sufficiently large, the total risk distribution of the population becomes graph (B) in (b), and a statistically significant difference between (A) and (B) can be proven. For the calculation described in ICRP Publication 99 (^{ICRP 2005} ), it is assumed that as the population size N increases, the variance decreases monotonically, meaning that a smaller excess risk can be determined, as indicated by the broken line (A') and solid line (B') in (b). The concept of the MPR is shown in (c); σ indicates standard deviation.

Assuming two populations with the same background risk (A), as drawn by the broken line in Fig. 1 a, a small amount of excess risk is added to one of the populations, and the total risk distribution of this population becomes graph (B), as drawn by the solid line in Fig. 1 a. In this case, a statistically significant difference cannot be concluded between (A) and (B) because the two distributions are too close.

On the other hand, when the excess risk is sufficiently large, the total risk distribution of the population becomes graph (B) in Fig. 1 b, and a statistically significant difference between (A) and (B) can be proven. For the calculation described in ICRP Publication 99 (^{ICRP 2005} ), it is assumed that as the population size N increases, the variance will decrease monotonically, meaning that a smaller excess risk can be determined as indicated by the broken line (A') and solid line (B') in Fig. 1 b, by increasing the population size since the variance related to the statistical uncertainty was simply considered.

In actual epidemiological studies, since the risk of interest (such as cancer mortality) for the population will be obscured by various known and unknown factors other than the target risk factor, there will be an eventual limit to this decrease in the variance owing to the unadjusted risk regardless of the population size. The MPR is defined as the minimum risk for which there is an 80% likelihood (or power) of proving an excess risk at the 5% significance level, as shown in Fig. 1 c, on the basis of the example in ICRP Publication 99 (^{ICRP 2005} ).

Derivation of mathematical formula
For illustrative purposes, we assume that (1) there are two populations, a control population and an exposed population, with the same apparent degree of background risk; (2) the risk factor exists among the populations independently; and (3) this risk factor has not been identified or adjusted. In this case, the degree of the unadjusted risk may contribute to the variation. Let R _{b} be the background risk, CV be the coefficient of variation of the background risk, N be the sample size of the population, and R _{b} R be the excess risk for the exposed population. Here, the excess risk is expressed as a product of the background risk R _{b} and the relative risk R . The standard deviations of the background risk for the control population σ_{background} and the exposed population σ_{Total} are expressed as

by considering the propagation of uncertainty. Referring to eqn (A3) in the appendix, a formula that can be used to calculate the excess risk can be defined as

In eqn (3), R _{b} R corresponds to the lower bound of the provable excess risk R _{L} , which is the boundary determining whether the risk is provable or not. To solve eqn (3) for R , it is first rearranged to obtain the following quadratic equation:

By solving eqn (4) and providing a value for R _{b} , the relationship between the lower bound of the provable excess risk R _{L} and the population size N can be obtained.

RESULTS AND DISCUSSION
Typical examples of MPR
The results of a typical calculation of R _{L} are shown in Fig. 2 by assuming provisional values for the necessary parameters for convenience; the background risk R _{b} is 0.3, 0.1, 0.03, and 0.01, and the coefficient of variation CV is 0.3, 0.1, 0.03, 0.01, and 0.003. As shown in Fig. 2 , there is a lower saturated value of the lower bound of provable excess risk, i.e., the MPR, regardless of the population size N , whereas a monotonic decrease due to statistical uncertainty was only assumed in ICRP Publication 99 (^{ICRP 2005} ).^{2}

Fig. 2: Example of relationship between the lower bound of the provable excess risk R _{L} , which is expressed relative to the background risk as R _{b} R , and the population size N considering the variation due to the presence of an unadjusted risk. (a) Results when the coefficient of variation CV is 0.03 and the background risk R _{b} is 0.3, 0.1, 0.03, and 0.01. (b) Results when R _{b} is 0.3 and CV is 0.3, 0.1, 0.03, 0.01, and 0.003.

Possible effect of unadjusted risks in a hypothetical epidemiological study
According to ^{Ogino and Hattori (2014)} , the background lifetime risk of cancer mortality was calculated to be 25.4% for the Japanese population. Moreover, its variation, which is the relative standard deviation of the background lifetime risk of cancer mortality among the 47 prefectures, was calculated to be 0.0413. By substituting these values, i.e., R _{b} = 0.254 and CV = 0.0413,^{3} into eqn (4) and assuming that N is sufficiently large (e.g., 10^{8} ), R is estimated as 0.106, and the MPR for cancer mortality is obtained as MPR = R _{b} R = 0.254 × 0.106 = 0.027 (2.7%).

Thus, when there is potentially the same level of variation in both control and exposed populations in a hypothetical epidemiological study and the additional lifetime cancer mortality risk is less than 2.7% for the exposed population, it may be impossible to prove a statistically significant increase in risk by comparison. As shown in Fig. 1 , increase of the population size does not improve the lower bound of the provable excess risk. In order to prove a slight increase in risk for a population with a certain background risk, it is essential to decrease the CV value, and appropriate correction of the factors is imperative in the analysis in an epidemiological study.

In this case study, the background lifetime cancer mortality risk and its variation among the 47 prefectures in Japan were substituted as the parameter values for R _{b} and CV in the MPR estimation for hypothetical populations; however, the MPR as well as CV values will be dependent on the target disease, end point (mortality or incidence), population, and analytical approach, such as with or without adjustment for the effect of modification factors (smoking habits, salt intake, etc.), in an epidemiological study. Note that the CV value is neither absolute nor intrinsic since it changes with the averaging unit of the prefectural data (^{Ogino et al. 2016} ). The value of R _{b} is a mere mathematical expectation, and statistical data will be affected by the sampling procedure. Therefore, the MPR value also changes.

At the same time, it is noteworthy that there is little variation in lifetime cancer mortality risk (^{Ogino and Hattori 2014} ) in the Japanese population owing to its relatively homogeneous ethnicity, culture, and dietary habits compared with other nations.

Furthermore, these parameter values are simply estimated based on a life-table calculation with statistical mortality data for only 2010. In actual epidemiological studies using a person-year (PY) method, it is necessary to extend the observation period to observe a sufficient number of occurrences of mortality or incidence. With the exception of breast cancer , the greatest contribution to the lifetime cancer risk usually occurs in the terminal phase of life (^{FPCR 2015} ). In general, human lifestyles cannot be managed; thus, changes in body weight, cholesterol, dietary habits, and exercise will affect cancer risk in different ways. Progress in medical diagnosis and therapy and improved public sanitation may suppress infectious diseases and extend the mean life. At the same time, it is likely that the lifetime risk of cancer and circulatory diseases, which are the main diseases in the terminal phase of life, will also vary over time. Changes in the identification codes of diseases will also occur over time, whereas genetic disorders cannot be controlled. In addition, dosimetry errors and selection bias could also be sources of a difference in background lifetime risk between the exposed and control populations.

Consequently, in actual epidemiological studies, a larger number of observations for an end point of interest may be required owing to the large population size and long follow-up period, while unadjusted factors are expected and they could generate variations in the control and exposed populations. It will be indispensable to adjust for such factors precisely from various viewpoints so that small excess risks can be proved epidemiologically.

Comparative assessment of fatal disease risks in a hypothetical population due to radiation exposure
In our previous work (^{Sasaki et al. 2016} ), the background lifetime mortality risk of circulatory disease and its relative standard deviation among the 47 prefectures were calculated to be 30.2% and 0.0492, respectively, using Japanese statistical information as of 2010. These values are slightly larger than those for cancer . As in the previous section, the MPR for circulatory disease mortality was estimated as MPR = R _{b} R = 0.302 × 0.128 = 0.039 (3.9%).

When considering an increased risk due to radiation exposure, the radiation risk factor is defined as the risk per unit dose. On the basis of the latest research using LSS data from Hiroshima and Nagasaki atomic bomb survivors, ^{Ozasa et al. (2012)} and ^{Shimizu et al. (2010)} reported that the ERR values of mortality for all solid cancer and circulatory diseases were 0.42 Gy^{−1} and 0.11 Gy^{−1} , and the EAR values were 26.4 (per 10^{4} PY Gy) and 5.5 (per 10^{4} PY Gy), respectively. In this study, these values were applied simply and directly to a hypothetical population. Table 1 summarizes the related parameters, and the values of (A)/(B) indicate the MPRs for cancer and circulatory disease mortalities considering the increased risk per unit Gy for ERR and per 10^{4} PY Gy for EAR.

Table 1: Minimum provable risks for cancer and circulatory disease mortalities due to radiation exposure.

Hence, in this case study, by applying the concept of MPR to the unit risk per absorbed dose, it is estimated that circulatory disease mortality has a much greater allowance than cancer mortality [approximately five times larger for the ERR model (1.17 vs. 0.25) and seven times larger for the EAR model (0.007 vs. 0.001)]. This implies that it is more difficult to prove that circulatory disease mortality is a radiation risk than to prove that cancer mortality is a radiation risk when hypothetical populations have background risks and CV values similar to those obtained in our previous work (^{Ogino and Hattori 2014} ; ^{Sasaki et al. 2016} ).

It should also be noted that (1) the ERR and EAR values for cancer are estimated for representative subjects at the attained age of 70 y after exposure at an age of 30 y, taking sex, age at exposure, and attained age into consideration as effect modifications (^{Ozasa et al. 2012} ); and (2) the values for circulatory disease include sex, age at exposure, and attained age in the background rates (^{Shimizu et al. 2010} ). The background risks and their variations are lifetime-risk-related values and are obtained using 2010 statistical data with consideration of variation among the 47 prefectures. Therefore, although the values of (A)/(B) shown in Table 1 roughly indicate the MPR per unit radiation exposure as mentioned before, these values should not be compared with the detection limit for an epidemiological study (e.g., 100 mGy in the atomic bomb study). At the same time, a single MPR value should not be treated as an absolute value as mentioned above.

However, since the MPR is capable of providing an objective index for risks of interest in a comprehensive manner that takes the uncertainty in both background risks and radiation-related risks into account, it might contribute to the discussion of the optimization of radiological protection for choosing appropriate countermeasures to reduce the overall risk in various radiation exposure situations.

From another viewpoint, in the previous study (^{Ogino et al. 2016} ), the Shapiro-Wilk test was performed to test the assumption of a normal distribution of background risk. This test for normality was also performed for the background lifetime risk of circulatory disease mortality (p value = 0.791)^{4} for this study, and this test may be necessary for the practical application of MPR to judge the validity of the data to be used.

CONCLUSION
In this study, we developed and discussed the concept of MPR that takes variation in background risk into consideration.

Since an unadjusted risk may introduce uncertainty in epidemiological studies, there will be a limit to the risk that can be proven due to exposure to a very low level of radiation. Nevertheless, results from epidemiological studies provide a strong foundation for radiation risk estimation in the system of radiological protection. Thus, to increase the reliability of such studies and the identification of factors affected, the development of explicit correction methods and standardization are highly desirable.

Assuming that radiation exposure is a risk factor, it was estimated that the MPR for circulatory disease mortality is much greater (approximately five to seven times larger) than that for cancer mortality for hypothetical populations, where their background risks and the corresponding CV values are consistent with those of our previous studies. This implies that circulatory disease mortality will be more difficult to prove as a radiation risk than cancer mortality. However, this estimation was carried out by simply referring to the background lifetime cancer and circulatory disease mortalities and their variation among the 47 prefectures from Japanese statistical information as of 2010, and a single MPR value should not be treated as an absolute value.

In practical radiological protection, it will be important to discuss management of radiation risk from diverse viewpoints. Mortality data were used throughout this study, but incidence data are also applicable on the basis of the concept of MPR. In such cases, the further application of MPR is expected since it can provide an objective index, including not only radiation-related risk but also its variation in the background risk of the population.

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APPENDIX
Reproduction of the calculation of detectable risk in ICRP Publication 99
ICRP Publication 99 (2005) provided an example calculation of the relationship between the radiation dose and the population size N required for 80% likelihood of (power in) detecting an excess risk at a 5% significance level. In this estimation, it was assumed that “the estimated excess would be distributed approximately as a normal random variable with the mean equal to the background rate, 10%, times the dose D , in Gy, and variance equal to 10% times (1 + D ) divided by the population size, N ” (^{ICRP 2005} ). Usually, for the ERR model, the unit of Background is mortality or incidence, and its increment is calculated as Background × D × ERR. For simplification, ERR was assumed to be 1, whereby it can be omitted. Thus, the standard deviations for the background and total (i.e., background plus excess) risks are, respectively, expressed simply as

where D is the radiation dose in the unit of Gy.

As shown in Fig. 1 c in the main text, if the following condition is satisfied, it is judged that a statistically significant excess risk is present in the population:

Given that the excess risk is a function of D and is expressed as 0.1D , the relationship between the population size N and radiation dose D is defined as:

If the background risk is 0.1 (10%), the following equation is obtained by solving eqn (A4) for N :

In Fig. A1, the solid line shows the relationship between D and N obtained using eqn (A5). The four dotted arrows projected onto the horizontal axis correspond to radiation doses of 1 Gy, 0.1 Gy, 0.01 Gy, and 0.001 Gy. In these cases, the required population sizes are 80; 6,400; 620,000; and 62 million, respectively; these sizes coincide with the values in Table 2.4 of ICRP Publication 99 (2005).

According to this consideration, there is no “minimum” value since the larger the population size, the smaller the detectable dose. This graph, as well as Table 2.4 in ICRP Publication 99, not only indicates that there are difficulties in low-dose radiation epidemiological studies owing to the large population size required for statistical significance but also somewhat misleadingly suggests that a slight excess risk due to low-dose radiation exposure can be proved only by increasing the population size. When the coefficient of variation CV is considered in this calculation, as discussed in the main text, the broken curve can be drawn (CV = 0.003), as shown in Fig. A1.

Fig. A1: Relationship between required population sample size and radiation dose at which an increase in risk can be detected. The solid line indicates the case of the assumption in ICRP Publication 99, while the broken line indicates the case in which the coefficient of variation is considered.

Thus, it should be emphasized again that “at low and very low radiation doses, statistical and other variations in baseline risk tend to be the dominant sources of error in both epidemiological and experimental carcinogenesis studies, and estimates of radiation-related risk tend to be highly uncertain because of a weak signal-to-noise ratio and because it is difficult to recognize or to control for subtle confounding factors, ” as stated in ICRP Publication 99 (2005).