Quantitative assessment of provability of radiation-related cancers considering unavoidable existence of unadjusted risk factors

The attribution of stochastic effects to exposure to ionizing radiation has been qualitatively discussed by introducing two distinct concepts of provability and probability. This study aims to develop a method of quantitatively assessing the provability of radiation-related cancers. To this end, the ‘minimum provable dose’ (MPD) was developed and applied to actual cancer mortality in Japan. The background lifetime risk of cancer mortality was calculated for the esophagus, stomach, colon, liver, lungs, skin, breasts, ovaries, bladder, and bone marrow as well as the age-specific risk coefficients reproducing those given in the 2007 Recommendations of the International Commission on Radiological Protection (ICRP). Comparing the relative ratio of MPDs, which was defined herein as the ‘provability index’ (PI), we quantitatively ranked radiation-related cancers for different tissues and organs predicated on provability for ages of 10, 30, 50, and 0–85+  years at exposure. We discuss the radiological protection of male emergency workers focusing on cancers highly prioritized according to the ranking (i.e. colon, bone marrow, and bladder). The present study proposed the system to quantitatively evaluate the level of radiological protection taking into account the variations of the background cancer risk on the provability of radiation-related cancers.

Keywords: radiation exposure, stochastic effect, cancer, provability, lifetime risk, risk coefficient (Some figures may appear in colour only in the online journal)

Introduction
The United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) recently published a report on the attribution of health effects to exposure to ionizing radiation and on inferring risks [1]. A key conclusion of the report is that stochastic effects cannot be unequivocally attributed to radiation exposure for any individual at present owing to other possible causes, but an increased frequency of occurrence of the stochastic effects can be attributed to radiation exposure for a population through epidemiological analysis. Thyroid cancers and leukemia are well-known examples of such stochastic effects following exposure in early childhood [2]. This is because of low spontaneous incidence and high radiosensitivity for these diseases, making an increased occurrence of radiation-related cancers sufficient to be revealed as actual health effects.
There have also been discussions on the attribution of stochastic effects to radiation exposure by introducing two distinct concepts. One is 'provability' for the attribution of radiation effects as a retrospective notion, which is defined as an ability to reveal by evidence the actual occurrence of radiation health effects that have actually been incurred in past exposure. The other is 'probability' for the inference of radiation risk as a prospective notion, which is defined as an ability to estimate by inference the prospective plausibility of hazardous outcomes [3,4]. These clear distinctions have been identified as one of the most critical challenges to be resolved by the radiological protection community particularly after the Fukushima Daiichi nuclear power plant accident [5], but this important concept of provability has not yet been sufficiently explored in a quantitative manner.
Here, we set out to quantitatively assess the provability concept. First, we developed a method of examining the relationship between a spontaneous occurrence of cancers in an unexposed population and the total occurrence of cancers in an exposed population. We represented the concept of provability in terms of a 'minimum provable dose' (MPD), above which a significant increase in the occurrence of radiation-related cancers in a population can be proved by evidence as an actual occurrence of health effects, considering the deviation of the spontaneous occurrence of cancers in the population. Second, we applied this approach to actual cancer mortality in Japan. We calculated the background lifetime risk of cancer mortality for the Japanese population, which is the estimate of the cumulative spontaneous occurrence of cancer mortality for each gender independent of radiation exposure. Third, we calculated age-and gender-specific cancer risk coefficients for the Japanese population, which are estimates of cumulative radiation-related cancers that develop in their lifetime after radiation exposure, reproducing those given in the 2007 Recommendations of the International Commission on Radiological Protection (ICRP) [6]. Last, we calculated age-and genderspecific MPDs for each tissue and organ subject to radiological protection (i.e. the esophagus, stomach, colon, liver, lungs, breasts, ovaries, bladder, and bone marrow) using the lifetime background risk and the risk coefficients of cancer mortality. The relative ratio of the MPDs was subsequently defined as the 'provability index' (PI), which allows quantitative ranking of radiation-related cancers for different tissues and organs in a consistent manner from the viewpoint of provability. These developments and calculations are described in detail in sections 2-4.

Homogeneity of the population
Since it is impossible at present to distinguish radiation-related cancers from those owing to other possible causes, radioepidemiological data are susceptible to statistical uncertainties [7]. Namely, with decreasing level of radiation exposure, the detection of a significant increase in radiation-related cancers becomes more difficult and requires a larger sample size. In this respect, ICRP has provided an illustrative example, which estimated that sample sizes of 6 390, 620 000, and 61.8 million persons are required for radiation exposure at absorbed doses of 100 mGy, 10 mGy, and 1 mGy to the tissues and organs, respectively [8]. Note that it considered the standard deviation of the sample size calculated by assuming an absolutely homogeneous population in terms of the spontaneous incidences of cancers (i.e. a constant background risk of 10%) for heuristic purposes, and hence the relationship between the sample size and radiation dose was given as a simple linear plot, acknowledging this unrealistically optimistic assumption.
In radioepidemiological analysis, risk factors such as age and gender are generally adjusted to study such a homogeneous population, but other risk factors such as diet, genetic susceptibility, other environmental exposures, and demographic and lifestyle factors are not fully adjusted because of lack of knowledge [1]. Therefore, the unavoidable existence of such unadjusted risk factors should affect the provability of radiation-related cancers in an inhomogeneous population. In such realistic cases, no matter how large the samples were collected, the radiation dose is no longer inversely proportional to the square of sample size as shown in the above example, and the level of unavoidable existence of unadjusted risk factors becomes dominant with respect to the provability of radiation-related cancers. To this end, we developed a method of estimating the effect of the inhomogeneous population in terms of the background cancer risk on the provability of radiation-related cancers by setting expedient assumptions as described in sections 2.2 and 2.3. Figure 1 describes the relationship between the frequencies of the spontaneous occurrence of cancers in an unexposed population and the total occurrence of cancers in an exposed population in terms of the lifetime cancer risk. An inference of spontaneous occurrence is shown as a background cancer risk (dashed line), and that of spontaneous and radiation-related occurrence is shown as a total cancer risk (solid line). Here we assumed that the number of cancer occurrences is sufficiently large for the Poisson distribution to be satisfactorily approximated by a normal distribution. The standard deviations of the background and total cancer risks are denoted as σ Background and σ Total , respectively.

Minimum provable risk
When radiation dose is generally low, the distributions of the frequencies of the background and total cancer risks shown in figure 1 are located close to each other, and the incremental risk posed by the radiation exposure cannot be revealed by evidence. Conversely, when radiation dose is sufficiently high for the distribution of the total cancer risk to be located apart from that of the background cancer risk, the incremental risk can be revealed with a given statistical power. To quantitatively determine this border condition, the minimum provable risk was defined as the incremental risk needed to reveal a significant increase in the occurrence of radiation-related cancers. The statistical condition set in this definition was a 80% power to detect an excess risk at the 5% significance level, following the illustrative example of the population size and the radiation dose provided by ICRP [8].
Considering the characteristics of the normal distributions for the background and total cancer risks, the 95th percentile of the frequency from the lower bound in the background cancer risk distribution corresponds to 1.645σ Background from the median, and the 80th percentile of the frequency from the upper bound in the total cancer risk distribution corresponds to 0.842σ Total from the median. Accordingly, the minimum provable risk is calculated as the sum of 1.645σ Background and 0.842σ Total as shown in figure 1. This means that when the radiation exposure imposes a higher risk than the minimum provable risk (i.e. 1.645σ Background +0.842 σ Total ), a significant increase in the occurrence of radiation-related cancers in a population can be proved by evidence as an actual occurrence of health effects, but this is not the case when the radiation exposure imposes less risk than the minimum provable risk.

Minimum provable dose (MPD)
To quantitatively incorporate the provability concept in terms of the radiation dose, the minimum provable risk was converted to MPD, which allows ranking of radiation-related cancers in different tissues and organs in a consistent manner from the viewpoint of provability. In the conversion, we used cancer risk coefficients, which are estimates of cumulative radiationrelated cancers that develop in a person's lifetime after each age at exposure. ICRP provides gender-averaged nominal risk coefficients of cancer incidence in  [6]. However, these risk coefficients are ageat-exposure-averaged values intended for the derivation of the relative detriment in a representative population for the purpose of radiological protection. Since the risk factors of age and gender are important determinants in cancer risk estimates, these nominal approaches are not suitable for the calculation of the MPD and application to the PI, although we recognize the importance of the nominal approach for the purpose of radiological protection. For this reason, we calculated age-and gender-specific cancer risk coefficients using the basic methodology developed in the 2007 Recommendations [6]. Details of the calculations are given in section 4. The MPD was calculated by dividing the minimum provable risk (i.e. 1.645σ Background + 0.842σ Total ) by the age-and gender-specific cancer risk coefficients (i.e. cancer cases per 10 000 persons per sievert of the equivalent dose). Therefore, the MPD was also given as age-and gender-specific numerical values in the unit of sievert for the equivalent dose for each tissue and organ. For this calculation, we set several expedient assumptions. First, there is effectively some lower bound on the radiation dose (i.e. MPD) for which a significant increase in radiation-related cancers can be revealed with the given statistical power as stated for the minimum provable risk, regardless of how large the sample is. Namely, the MPD was calculated with the assumption of a sufficient population size. Second, we assumed a lifetime follow up for the cancer occurrence. Third, the number of cancer cases is sufficiently large for the Poisson distribution to be approximated by the normal distribution. Moreover, the standard deviations of the background risk and total risk were assumed to be equal (i.e. σ Background = σ Total ) and calculated from regional variations of the background cancer risk in Japan, whose ethnicity can in general be considered to be relatively uniform. Details of the calculations are given in section 3.
Note that the regional variations of the background cancer risk are no more than the substitute index for assuming the level of the unavoidable existence of the unadjusted risk factors in this study. Furthermore, the MPD should not be considered as an absolute value because such a calculation of the standard deviation of the background cancer risk is dependent on the regional variations considered such as the district, prefecture, and municipality. Given the background risk of cancer mortality in the 47 prefectures of Japan, the standard deviation could be calculated by assuming a normal distribution. However, the larger the averaging unit of prefectures used in the calculation of the background cancer risk, the smaller the calculated standard deviation. For instance, the calculated standard deviation when cancer cases are combined in 23 groups consisting of 2 prefectures and a residual prefecture will be much smaller than the standard deviation calculated among the 47 prefectures. In this way, the standard deviation calculated using prefectural variations is not an absolute value, and the same is true for the MPD. Therefore, the MPD should be incorporated as relative values to compare cancer provability among different tissues and organs. In consequence, the concept of PI was developed in this study as described in section 2.4. The effects of the standard deviation calculated with different averaging units of prefectures on the calculations of the MPD and PI are further examined in the discussion.

Provability index (PI)
Considering the characteristics of the above-mentioned MPD, the PI was defined as a quantitative measure that can specify the relative provability of radiation-related cancers, and was calculated as follows. First, MPDs were obtained as age-and gender-specific values for tissues and organs, i.e. the esophagus, stomach, colon, liver, lungs, breasts, ovaries, bladder, and bone marrow, subject to the radiological protection in the 2007 Recommendations [6]. Breast cancer and ovarian cancer were only analyzed for females, and skin was omitted from the analysis due to the low lethality fraction (i.e. 0.002) [6]. Second, the lowest MPD was identified among all tissues and organs, ages at exposure, and genders. Last, the lowest MPD was divided by each MPD. Namely, the lowest MPD was given a value of one, and the other MPDs were converted to values relative to the lowest MPD. Comparing the values of the PI among different tissues and organs, we could quantitatively rank radiation-related cancers in a consistent manner from the viewpoint of provability. Note that one might define the PI using the lowest MPDs for each gender and age range as the numerator, which results in a set of PIs that are gender and age-range specific. However, we defined the PI here using the lowest MPD among all tissues and organs, ages, and genders, so that we were able to compare the provability of radiation-related cancers among different genders and ages in a consistent manner.

Background cancer risk
As mentioned above, we applied the MPD approach to actual cancer mortality in Japan. In the previous study, we found that the gender-averaged background lifetime risk of all cancer mortalities ranges from 23.7% to 28.3% among the 47 prefectures of Japan, and the arithmetic mean was 25.4% [9]. Here, gender-specific background lifetime risks of cancer mortality were similarly calculated for each tissue and organ. The calculations are described in detail in the following.
The background risk of cancer mortality was defined as the estimated probability that a newborn child dies of cancer in his or her lifetime regardless of radiation exposure, calculated under the assumption that current mortality rates at all age intervals remain constant during the lifetime. The mathematical derivation of the background cancer risk used in this paper was the same as that by Wun et al [10], which has been used in reports by the National Cancer Center of Japan [11] as well as the Surveillance, Epidemiology, and End Results Program of the US National Cancer Institute [12] and the American Cancer Society [13].
To calculate the background risk of cancer mortality by constructing a life table, statistical data on the total mortality rate, cancer-specific mortality rate, and population for 5 year age intervals are necessary. Here, the population data originated from the latest 19th population census in 2010, which has been conducted almost every 5 years by the Ministry of Internal Affairs and Communications of Japan [14], and the mortality data originated from the vital statistics in 2010, which have been compiled yearly by the Ministry of Health, Labour and Welfare (MHLW) of Japan [15].
The mortality and population data obtained from these national surveys are stratified into 5 year age intervals of 0-4, 5-9, …, 80-84, with a final open interval of 85+ for use in the life table. A hypothetical cohort of 1 million live births was set, and the decrease in population in each age interval was calculated by converting the total mortality rate (i.e. cancer and noncancer mortality rates) in each age interval to probabilities using an exponential model. The initial population in the age interval of 5-9 (q 5 ) was given by equation (1), where α 0 represents the total mortality rate for the previous 5 year age interval of 0-4 and q 0 represents the 1 million live births. The equations shown hereinafter are the same as those used in the reports of the National Cancer Center of Japan [11]. Subsequently, the initial population in the next age interval of 10-14 (q 10 ) was given by equation (2), and the initial population in each age interval (q i ) was generally given by equation (3), where i represents the initial age of the interval (i.e. i = 5, 10, …, 85), q i represents the initial population in the age interval with initial age i, and α i represents the total mortality rate for the age interval with initial age i.
The cancer mortality (m i ) for the age interval with the initial age i was given by equation (4), where β i represents the corresponding cancer mortality rate.
The cancer mortality for the final open interval of 85+ (m 85+ ) was given by equation (5). Finally, the background risk of cancer mortality was calculated by dividing the cumulative cancer mortality for all age intervals (i.e. m 0 , m 5 , …, m 85 , and m 85+ ) by the hypothetical cohort of 1 million live births.

Reproduction of nominal risk coefficients in the ICRP 2007 Recommendations
To verify the validity of the method of calculating age-and gender-specific cancer risk coefficients for the Japanese population for use in the MPD calculation, we first reproduced the age-and gender-averaged nominal risk coefficients given in table A.4.1 of the 2007 Recommendations [6]. The tissues and organs selected were the esophagus, stomach, colon, liver, lungs, breasts, ovaries, bladder, thyroid, and bone marrow. Skin was excluded here because the information on the parameters necessary for calculating the risk coefficients, such as the background incidence rate and excess relative risk, is not available in the 2007 Recommendations [6].
First, we calculated the exposed-and attained-age specific excess relative risk (ERR per gray (Gy)) and the exposed-and age-specific excess absolute risk (EAR per 10 000 persons per Gy) using equations (6)-(8), where n ERR x represents ERR at attained age n after radiation exposure at age x. In equation (6), the n ERR 30 was estimated to increase in proportion to the power θ of the attained age n with which the ERR varies. The risk estimates were adjusted downward by a factor of two except for the bone marrow to account for the dose and doserate effectiveness factor (DDREF) following the 2007 Recommendations [6]. The values of 70 ERR 30 for each gender and for each tissue and organ were obtained from table A.4.6 of the 2007 Recommendations [6]. The n ERR x was calculated using equations (7) and (8), depending on the value of n (i.e. equation (7) for n < 30 and equation (8) for n > 30), where δ represents the change in ERR per decade increase. The age at exposure x and the attained age n were considered for 5 year age groups and for every age, respectively. Similarly, the exposedand attained-age-specific EAR was calculated using the same equations as used for the ERR calculations. ( ) Second, life tables [16] for each exposure at each age were developed to calculate the risk coefficients of cancer incidence. To calculate the number of survivors exposed at age x, we used equations (9)-(12), where n l x is the surviving number at exposed age x (age 0-90) and attained age n (age 0-100), n d x is the background mortality rate given in tables A. 4 When n x 1 10 When + ≥ + n x 1 10, 10 EAR 100 000 .
x n x n x n x n 1 (12) Third, the ERR coefficient ( all ERRC x ) and EAR coefficient ( all EARC x ), which are the estimated numbers of excess cases of cancer incidence per 10 000 persons in the relative and absolute risk models, respectively, were calculated by summing the cases of cancer incidence at each attained age using equations (13) and (14). The value of 0.1 in equation (13)   Last, the nominal risk coefficient was calculated using equation (15), where x C 0

85
ERR 18 x all EAR 18 x all represent the means of the risk coefficients per 10 000 persons per Gy for each exposure at each age (i.e. 18 age groups) by adjusting differences in the age structures owing to the initial setting of the population as 100 000 persons for each age at exposure.
To transfer risk estimates across populations, ERR:EAR weights of 0:100% were assigned for the breasts and bone marrow (i.e. y = 0 in equation (15)), 100:0% for the thyroid (i.e. y = 1), 30:70% for the lungs (i.e. y = 0.3), and 50:50% for all other tissues and organs (i.e. y = 0.5), following Box A.1 (c) in the 2007 Recommendations [6]. The latency was assumed to be 10 years for solid cancers (i.e. esophagus, stomach, colon, liver, lungs, breasts, ovaries, bladder, thyroid) and 2 years for leukemia (i.e. bone marrow). For leukemia, the onset period for the risk calculation was assumed to be from x + 4 years to x + 34 years, and the EARC at 1 mGy was firstly calculated and then adjusted for the risk at 1 Gy. We used the EAR of leukemia obtained by Preston [17]. The value of the EAR 9 years after exposure was used as a constant value for the period of 7 years (i.e. 2-9 years) to provide a realistic value of the EAR. Table 1 shows a comparison between the nominal risk coefficients obtained here and those given in the 2007 Recommendations [6]. It was found that our results for all tissues and organs except for the bone marrow concur with the values in the 2007 Recommendations within 30%. For the bone marrow, the reproduced risk coefficients were 119% and 141% larger than the ICRP values. This was because details of the calculation method for leukemia are not available in ICRP Publications, and thus we calculated the risk coefficients by the original method as stated above.

Gender-and age-specific risk coefficients for the Japanese
After verifying the method of calculating the nominal risk coefficients of cancer incidence, as shown in table 1, we calculated gender-and age-specific risk coefficients of cancer incidence for the Japanese population using the method in section 4.1 and specific data for Japan as follows. The background cases of cancer incidence per 100 000 at age n (C n ) for the Japanese population were obtained from the annual report of 'Monitoring of Cancer Incidence in Japan (MCIJ) in 2010' [18], and the background mortality rate ( n d x ) was obtained from a complete life table for 2010 compiled by MHLW [19]. The year 2010 was selected for consistency with the calculation of the background lifetime risk of cancer mortality as described in section 3. For skin cancer, the parameters necessary for calculating the cancer risk coefficients were not available in the 2007 Recommendation [6], and thus we applied the values of 30 EAR 70 and 30 ERR 70 obtained by Preston [20].  Table 2 shows the result of the background lifetime risk of cancer mortality for each gender, tissue and organ, and prefecture in Japan. The 10th revision of the International Classification of Diseases (ICD) [21] was shown for the tissues and organs, namely the esophagus (C15), stomach (C16), colon (C18), liver (C22), lung (C33-34), skin (C43-44), breast (C50), ovary (C56), bladder (C67), and bone marrow (C91-95) as well as all types of cancer (C00-C97). The background lifetime risk of cancer mortality for the thyroid could not be calculated because the thyroid-specific mortality data are not available from the vital statistics in 2010 [15]. For reference, arithmetic means of the prefectural data are shown in the bottom of table 2. These prefectural variations in the lifetime background risk of cancer mortality were used to calculate the standard deviations of the background cancer risk for the Japanese as shown in figure 1. Table 3 shows the obtained age-specific risk coefficients of cancer mortality for the Japanese population. The risk coefficients of cancer incidence calculated by the method described in the section 4.2 were converted to those of cancer mortality by multiplying by the tissue-and organspecific lethality fraction given by the ICRP 2007 Recommendations [6] as shown in the second column of table 3. This conversion was employed because we here applied the provability concept to the actual cancer mortality, not cancer incidence, in this study. For tissues and organs with high lethality fractions such as the esophagus, stomach, liver, and lungs, the risk coefficients of cancer mortality are similar to those of cancer incidence. However, for tissues and organs with low lethality fractions (e.g. 0.002 for skin), the risk coefficients of cancer mortality are extremely low compared with those for cancer incidence. Considering the high calculated PI for the skin and little practical concern, skin was excluded in the following analysis. Table 4 shows the result of the MPD calculation for cancer mortality in the case of using the standard deviation obtained from the prefectural variation of the background lifetime risk of cancer mortality among the 47 prefectures. For the sake of simplicity, we set four age groups at exposure of 10 (10-14), 30 (30)(31)(32)(33)(34), 50 (50-54), and 0-85+ years. The smallest MPD among all tissues and organs and age groups at exposure was found to be an equivalent dose of 250 mSv for the breast in females in the 10 year-old group. It should be emphasized throughout this paper that numerical values of the MPD are not absolute and should not be compared with any protection quantities. This is because the concepts of the MPD are developed here just in order to compare the PI for radiation-related cancer, defined as the relative ratio of the MPDs among different tissues and organs. Figure 2 shows the obtained PI calculation. The solid and dashed lines show the results for males and females, respectively. On the basis of the results of the MPD calculation, the PI for the breast in females in the 10 year-old group at exposure, for which the MPD was the minimum among all tissues and organs and age groups, was set as one, and the PIs for all other tissues and organs were calculated by dividing the smallest MPD for the breast by each MPD. The exclusion of the thyroid in the MPD calculation of cancer mortality does not affect the result of the top ranking in the PI calculation. Table 5 shows the quantitative ranking of radiation-related cancers obtained using the results of the PI calculation. The arrows and numbers beside the tissues and organs represent changes in the ranking as compared with the ranking based on the risk coefficient of cancer mortality. Namely, the up arrow represents an increase in the ranking and the down arrow represents a decrease in the ranking. For instance, for the case of 10 years of age at exposure, female breast is ranked as fourth place on the basis of the risk coefficient of cancer mortality (table 3), whereas it is ranked as first place on the basis of the PI. Thus, the number beside 'Breast, female' is ' 3'. By comparing the numbers among the 16 tissues and organs for each age group at exposure, it was found that the tissues and organs whose ranking based on the PI increased by more than 6 compared with the ranking based on the risk coefficient were the  Age-specific risk coefficients of cancer mortality for the Japanese population.  bladder in males and females, the ovary in females, and the esophagus in females. On the other hand, the tissues and organs whose ranking decreased by more than 6 were the liver in males and females, the stomach in males and the lung in males. In addition, female breast, female lung, male colon, female esophagus, and female bone marrow were ranked in the top three on the basis of the PI, while these tissues and organs except for female lung had lower rankings in terms  of the risk coefficient. By considering these changes in the ranking, we discuss the radiological protection for male emergency workers in section 6.2.

Variations in background cancer risk
Here, the prefectural data on the background risk of cancer mortality in Japan shown in table 2 were used to calculate the standard deviations for the background and the total cancer risks shown in figure 1. In the calculation, the Shapiro-Wilk test [22] was performed with respect to distributional assumptions of the normality. The null hypothesis that the data are normally distributed was accepted (i.e. p-value is less than 0.05) for esophagus, colon, liver, lung, skin, breasts, ovaries, bladder and all cancer in both genders, and the stomach in females. The same hypothesis was rejected for bone marrow in both genders and stomach in males. It is beyond the scope of this paper to discuss details of the cause of each cancer, but it has been reported that Japan has a high prevalence of adult T-cell leukemia [23], especially in Okinawa prefecture and the Kyushu region which consists of Fukuoka, Saga, Nagasaki, Kumamoto, Oita, Miyazaki and Kagoshima prefectures. The background cancer risk of bone marrow was also calculated to be higher when compared to other prefectures as shown in table 2. An association between Helicobacter pylori virulence factors and gastrointestinal diseases has also been discussed for the situation in Okinawa prefecture where the incidence of gastric cancer is the lowest in Japan [24], and the background risk of stomach cancer in males was calculated to be the lowest in this study. In fact, the null hypothesis for normality in bone marrow and stomach data was accepted when the regional data were not included in the normality test. In this regard, the standard deviations of the background risk of cancer mortality of bone marrow and stomach would be smaller than those calculated in this study, which leads to smaller values of the MPD and accordingly increased degree of the provability. Another noteworthy point is the averaging unit of the prefectural data on the background risk. It is generally known in the multiple data analysis that the higher the averaging unit of the data, the lower the standard deviation of data following the normal distribution, and the same is true here for the MPD calculation. Therefore, we quantitatively examined the dependence of the MPD calculation on the averaging unit of prefectural data by randomly selecting prefectural data. Figure 3 shows the dependence of the calculated median of the standard deviation of the background lifetime risk of cancer mortality per 1 million hypothetical persons on the averaging unit of prefectural data. Here, the esophagus, stomach, and colon in males were chosen as examples. When the standard deviation was calculated using the data from all 47 prefectures, there was only one combination of the prefectural data, and thus the standard deviation was calculated to be a constant value, for instance, 2572 for the esophagus. Next, when the 47 prefectures were randomly combined into 23 groups of two prefectures and a residual prefecture in 10 000 different ways, the median standard deviation was calculated to be 1820. Although the total number of possible combinations in this case is 1.2 × 10 30 , we chose 10 000 as the number of combinations because several calculations were conducted for 100, 300, 1000, 3000, and 10 000 combinations, and the median of the standard deviation was found to converge when more than 3000 combinations were considered. Furthermore, we examined the standard deviation for different averaging units of prefectural data (i.e. 3 prefectures × 15 groups, 4 prefectures × 11 groups, …, 10 prefectures × 4 groups). From these calculations, it was verified that the standard deviation of the background cancer risk generally decreases with increasing the averaging unit of prefectural data. This is why the calculated MPD values are not absolute values, and why we developed the concept of the PI defined as the relative ratio of the MPD for tissues and organs. We conducted the same analysis for all tissues and organs in addition to the esophagus, stomach, and colon. Interestingly, the median of the standard deviation similarly decreased with increasing averaging unit of the prefectural data, and the relative decrease was almost constant for all tissues and organs. This means that the PI remains almost constant even if the averaging unit of prefectural data changes. For this reason, the results of the PI calculation shown in figure 2 can be considered as valid despite the dependence on the averaging unit of prefectural data.
The above approach using the regional variations was applied here to expediently embody the inhomogeneous population in terms of the background cancer risk due to the unavoidable existence of the unadjusted risk factors mentioned by UNSCEAR such as diet, genetic susceptibility, environmental exposures, and demographic and lifestyle factors [1]. Note that authors recognize that the regional variations are no more than the substitute index for assuming the level of the unavoidable existence of the unadjusted risk factors, and that the numerical values of the MPD should not be compared with any protection quantities. Furthermore, the present study does not show that the current situation regarding the background lifetime risk of cancer mortality in Japan is acceptable nor that there is no radiation-related cancer risk below a certain radiation dose, and should not be interpreted in such ways. Cancer control programs should be promoted in municipalities on the basis of the Cancer Control Act approved in 2006 in Japan [25] to reduce the background cancer risk itself. Moreover, the background risk of cancer mortality calculated here is based on vital statistics data for 2010 in Japan and can change over time. To determine whether the present findings are universal, the validity of the above approach should be verified in the future along with the update of the cancer risk coefficients.

Radiological protection for male workers in emergency situation
The UNSCEAR report on the attribution of health effects to radiation exposure mentions that, in general, increases in the incidence of health effects in populations cannot be reliably attributed to chronic exposure to radiation at levels that are typical of global average background levels of radiation [1]. On the other hand, compared with natural background radiation, levels of radiation exposure can be much higher in an emergency exposure situation, defined in the system of radiological protection as an unexpected situation that occurs during a planned operation or from a malicious act or any other situation that requires urgent protective action to avoid or reduce undesirable consequences [6]. After such emergency situations, a discernible increase in radiation-related cancers is one of the main concerns of society including those who are exposed to the higher levels of radiation. From this viewpoint, we discuss radiological protection for emergency workers using the results of the PI calculation. Female workers are not considered here because most of the workers involved in the emergency response work at nuclear or radiological accidents are generally males, such as the emergency response workers after the Fukushima nuclear accident [26].
The maximum PI for males was found in the colon throughout all age groups (table 5). For the 10 year-old group, the colon was followed by the stomach, bone marrow, and bladder, where the PI for the stomach was less than half for the colon and that for the bone marrow and bladder was less than one third for the colon as shown in figure 2. Similarly, the lung, stomach, and bone marrow followed the colon for the 30 year-old group with differences in the PI of more than a factor of 2. For the 50 year-old group, the lung and bone marrow followed the colon and had very similar PI, and the bladder followed these three tissues and organs. Considering the position of the colon, bone marrow, and bladder in the body, these results imply the importance of local protection of the abdomen for male emergency workers, in which the colon, bone marrow, and bladder are located. Furthermore, it can be considered from the result of the PI calculation that by reducing the radiation dose received at the colon by a factor of two for young workers, the PI for the colon becomes close to that for other tissues and organs, and thus a prominent increase in radiation-related colon cancers can be avoided by providing the same protection levels as those for other tissues and organs. For such local protection, a personal shielding device intended to prevent acute radiation syndrome (ARS) by reducing the marrow dose after an accident [27] may also be effective for preventing a discernible increase in stochastic effects.

Cancer mortality and incidence
The concept of the MPD can be universally applied not only to mortality but also to the incidence of cancers, given the background cancer risk and the cancer risk coefficient. The reason why we presently applied the MPD concept to actual cancer mortality in Japan was that the registration accuracy for cancer mortality is relatively matured compared with that for cancer incidence in Japan. Population-based registration of mortality has been nationally developed by the central government based on the Family Registration Law since 1898. All mortality data must be submitted to the central government via municipalities, and the data collected each year are made publically available in September of the following year. In contrast, population-based registration of cancer incidence has been conducted by local governments but not by the central government, and the reporting of cancer incidence has not been a mandatory task for medical institutions. However, the Act on the Promotion of Cancer Registration enacted in 2013 will change this situation. Each hospital manager has to report information on any primary cancer first diagnosed in their institutions to the prefectural governors from January 1, 2016. The cancer statistics in Japan will thus be more stable, reliable, and complete at least 3 years after the first report on cancer statistics from the National Cancer Registry in 2018 [28].
We have already observed such a transition period in terms of improved population-based registration of cancer incidence in national reports. The National Cancer Center has published MCIJ reports annually since 2003, which are based on the data collected by local government 3 years earlier. For instance, cancer incidence data in the year of 2011 became available in March 2015 through the MCIJ 2011 report [29]. The series of MCIJ annual reports set consistent qualification criteria for cancer registration (i.e. death certificate only (DCO) rates <25% or death certificate notification (DCN) rates <30%, and incidence/mortality (IM) ratio ≥1.5), and prefectural data that did not satisfy these criteria were excluded to maintain accuracy in their estimation of cancer incidence rates in Japan. Although only 12 prefectures satisfied these criteria in the MCIJ 2005 report [30], reflecting recent action towards complying with the Act on the Promotion of Cancer Registration enacted in 2013, 28 and 39 prefectures satisfied the criteria in the MCIJ 2010 [18] and 2011 reports [29], respectively. Thus, the population-based registration of cancer incidence will further be improved, making it possible to similarly deal with the cancer incidence data for each prefecture and that for cancer mortality and to quantitatively analyze the MPD for cancer incidence with the same approach as that for cancer mortality.

Conclusions
Here we have presented a method of quantitatively developing the concept of provability of radiation-related cancers. The general ideas of MPD and PI were developed and applied to actual cancer mortality in Japan. The background lifetime risk of cancer mortality for each gender in the 47 prefectures was calculated for the esophagus, stomach, colon, liver, lungs, skin, breasts, ovaries, bladder, and bone marrow. The regional variations of the background cancer risk approximated by the normal distribution were used as the substitute index for assuming the level of the unavoidable existence of the unadjusted risk factors, which dominantly affects the provability of radiation-related cancers in the population. The nominal risk coefficients for these tissues and organs except for the bone marrow closely reproduced those given in the 2007 Recommendations. The age-specific risk coefficients of cancer mortality in Japan were calculated and used for the MPD calculation. The values of the MPD were not absolute and developed only for the purpose of comparison of the PI, and therefore cannot be compared with any protection quantities. From the PI calculation, the importance of local protection of the abdomen for male emergency workers was proposed. Finally, it should be noted that the lifetime background risk of cancer mortality calculated here was based on vital statistics data as of 2010 in Japan and can change over time. To determine whether the present findings are universal, the validity of our approach should be verified in the future along with the update of the cancer risk coefficients. Furthermore, it can be expected that the population-based registration of cancer incidence in Japan will be further improved, allowing more quanti tative analysis of the MPD for cancer incidence.