Time-integrated radiation risk metrics and interpopulation variability of survival

Abstract

Task Group 115 of the International Commission on Radiological Protection is focusing on mission-related exposures to space radiation and concomitant health risks for space crew members including, among others, risk of cancer development. Uncertainties in cumulative radiation risk estimates come from the stochastic nature of the considered health outcome (i.e., cancer), uncertainties of statistical inference and model parameters, unknown secular trends used for projections of population statistics and unknown variability of survival properties between individuals or population groups. The variability of survival is usually ignored when dealing with large groups, which can be assumed well represented by the statistical data for the contemporary general population, either in a specific country or world averaged. Space crew members differ in many aspects from individuals represented by the general population, including, for example, their lifestyle and health status, nutrition, medical care, training and education. The individuality of response to radiation and lifespan is explored in this modelling study. Task Group 115 is currently evaluating applicability and robustness of various risk metrics for quantification of radiation-attributed risks of cancer for space crew members. This paper demonstrates the impact of interpopulation variability of survival curves on values and uncertainty of the estimates of the time-integrated radiation risk of cancer.

1. Introduction

In recent years, various space agencies and private enterprises have announced their plans to construct space stations and to prepare for space missions to the Moon or Mars. Private industries are currently paving the way for space tourism allowing broader (though still limited) population groups to join short-term space flights. Most of these activities are at early stages of development. Nevertheless, they highlight the need for developing systematic and harmonized approaches in radiological protection given the fact that such a harmonized approach does not yet exist [1], [2], [3], that the cosmic radiation field is complex [4], that accumulated radiation doses can be high if a Solar Particle Event (SPE) occurs during a mission to the Moon [3] or be as high as 1 Sv (without an SPE) on a long-term mission to Mars.[5] In terms of susceptibility to potential health effects from radiation exposures, individuals joining a space mission represent a heterogenous general population due to differences in genetics, sex, age, health status, education background, etc. Thus, application of the system of radiological protection developed by the International Commission on Radiological Protection (ICRP) for radiation exposures on Earth may require rethinking and, possibly, adjustment, to meet the requirements for radiological protection in space [1].

Individuals participating in space missions typically accept known or modelled risks associated with the missions and anticipated environments to be encountered during the missions, including physiological effects of microgravity, psychological stress due to working and living in an isolated space, and potential occurrence of radiation-related adverse health outcomes. Early radiation-induced health effects that manifest as functional impairment of tissue are critical as they may compromise mission success. However, they occur only when incident radiation doses or dose-rates are relatively high, e.g., due to a strong SPE. Late health effects, including development of malignant neoplasms, may appear years after a mission.

Numerous challenges remain for assessing and communicating radiation-induced health risks related to space missions. Such challenges come from unique exposures in space due to the nature of the environments (mixed fields of sparsely and densely ionizing radiations occurring in microgravity) and temporal factors (mission timing and duration) that determine the radiation fields. Further, chronic and, sometimes, acute exposures can occur within a mission, and multiple missions are being flown by ever younger and older people from diverse ethnic and occupational backgrounds. Various metrics can be used to capture and communicate risk of the deleterious effects that may happen to a crew member during a mission or long afterward. For radiological protection purposes, the effective dose metric [6], [7] is an important element on Earth and it is also adopted by several space agencies for exposures in space [2]. However, it is not designed to reflect risks from exposures in space and falls short of fulfilling the need for communicating specific risks to a range of stakeholders, including astronauts and their families, medical support mission planners and designers, and ‘risk balancing’ decisionmakers, and it is limited by specifics of radiation exposure in space. The stakeholders have strongly overlapping but disparate informational needs, for example, projecting the potential health impact to an individual astronaut for a proposed mission. Exposure and risk projections for a specific mission segment is of acute interest to mission planners and specialists who design and optimize radiation shielding to protect space crews from exposure to a sporadic SPE or during traversal of radiation belts. Mission planning requires appropriate metrics to express concomitant risks to ensure risk-informed decision making and optimization of protective measures.

The present paper focusses on quantification of estimates of time-integrated radiation-related cancer risks, using uncertainty of risk model parameters associated with exposures to cosmic radiation during space missions. Multi-model inference, risk transfer, and health statistics for the target population are used in the present paper to consider the potential impact of survival curves that represent various populations. Techniques for assessment of lifetime radiation risks described in the paper were developed within the ICRP Task Group 115, which is focused on radiation exposures and risks in space. At the same time, these approaches have many commonalities with the ICRP methodology for definition of radiation detriment, which is currently reviewed and developed in the ICRP Task Group 122. Correspondingly, this paper effectively contributes to synergy between the two task groups and demonstrates how improvement of radiological protection in space can help advancing radiological protection on Earth, thus contributing to development of a harmonized framework (see more on this in the companion paper by Rühm et al. [1]).

2. Methods

2.1. Radiation risk computation

The methodology of radiation risk computation used in this study follows that described by Ulanowski et al [8] and Walsh et al [9]. The radiation risk of an outcome of interest (here, a malignant neoplasm) is represented by the excess incidence rate of the outcome due to the radiation exposure. Namely, the incidence rate of the outcome in the exposed population, ${\lambda }^{\ast }$ (PY⁻¹), is expressed as a sum of the spontaneous incidence rate of the outcome in the identical non-exposed population, ${\lambda }_0$ (PY⁻¹), and the excess incidence rate attributable to radiation exposure, $h$ (PY⁻¹):

$${\lambda }^{\ast }\left(D\right)={\lambda }_0+h\left(D\right),$$ (1)

where $D$ is the dose of radiation exposure (Gy or Sv).

The excess incidence rate is quantified using pertinent radiation risk models. Uncertainty associated with the risk models is modelled using the full covariance matrix for all model parameters, describing the model baseline and excess rates. Typically, a radiation risk model, used to calculate h (PY⁻¹), provides statistical inference derived from a specific radioepidemiological cohort; therefore, its predictions had to be modified to represent the target population, to which an astronaut is believed to belong. This procedure is conventionally termed ‘risk transfer’. Here, a weighted combination of additive and multiplicative transfer paths for excess incidence rate is used (see details in Ulanowski et al [8]):

$$h=h_m\left(1-f+fB\right),$$ (2)

where $h$ is the excess incidence rate for the target population ($P{Y}^{-1}$) and $h_m$ is the model-calculated excess incidence rate ($P{Y}^{-1}$); $B={\lambda }_0/{\lambda }_m$ is the ratio of the baseline incidence rates as observed for the target population (${\lambda }_0$) and inferred by the radiation risk model from the radioepidemiological cohort (${\lambda }_m$); $f$ is the relative weight of the multiplicative transfer path.

To avoid possible biases due to a risk model selection and to catch uncertainty due to inter-model variability, the Multi-Model Inference (MMI) technique [10] was used for radiation risk computations. The MMI technique uses penalised likelihood to arrange statistical models according to the quality of fit, thus allowing to effectively reject implausible or statistically insignificant models. In this paper, this is implemented by assigning models with weights based on the Akaike Information Criterion (AIC), which represents model-specific deviance penalised with the number of model parameters, followed by risk projections using all plausible models to produce either model-averaged estimates or combined distribution of estimates for assessment of statistics, percentiles and confidence intervals. Given the purpose of this study, no attempt was made to fit new model sets and the selected risk models are presented below.

2.2. Models and data used for calculating radiation risk

For solid cancers, the risk calculations were done using the phenomenological models derived for an aggregated endpoint for all solid cancers in the Life Span Study (LSS) cohort for the follow-up period 1958–2009 [11] without adjustment for smoking [12]. Walsh et al. [12] derived two phenomenological models: one of excess absolute risk (EAR) type with the AIC-weight of 38.4% and one of excess relative risk (ERR) type with the AIC-weight of 61.6%. These radiation risk models for all solid cancers were derived assuming linear dose response.

The two phenomenological models for all leukaemia, excluding chronic lymphoblastic leukaemia (CLL) and adult T-cell leukaemia (ATL), in the LSS cohort for the follow-up period 1958–2001 were used as derived and reported by Hsu et al. [13]: the ERR-type model with AIC-weight of 65.9% and the EAR-type model with AIC-weight of 34.1%. The radiation risk models for leukaemia were fit with linear-quadratic dose response.

In this paper, calculation of the time-integrated radiation risks was done using a custom software tool, which allows computations for several European countries (Denmark, Finland, Germany, Norway, Sweden, Switzerland). The health and life statistics data for the German population were used here, because of better statistics for incidence of malignant neoplasms in a country with larger population. Specifically, cancer incidence and mortality rates were taken for the period 2010–2014 [14] and demographic data were for the period 2013–2015 [15]. In the present study, the following composite endpoints for outcomes of cancer incidence are considered: (a) all solid cancers (ASC), including ICD:10 codes C00–C80; and (b) all leukaemia (LEU) with ICD10 codes C91–C95, excluding CLL (C91.1, C91.4) and ATL (C91.5).

2.3. Risk metrics applied

In the present study, the following risk metrics were considered: life-time attributable risk (LAR), risk of exposure-induced cancer (REIC), and excess lifetime risk (ELR). Besides these more conventional risk metrics, the radiation-attributed decrease of survival (RADS) as well as the relative reduction of lifetime (RRL)––which is closely related to the conventional years of life lost (YLL)––were used in the present study. The conventional radiation risk metrics (see Eqs. (A.1)–(A.4) in Appendix) are defined via survival functions, mortality rates, and incidence rates for the general population. Astronauts comprise a professional group which differs from the general population, because of pre-selection, medical screening and care, training, follow-up and occupation-specific non-radiation hazards. Competing non-radiation-attributable risks, which are hard to quantify, also affect their survival chances, making it difficult to justify the use of the general population’s demographic statistics for this professional group. Correspondingly, credibility of the radiation risk estimates, which are bound to the general population’s demographic and health statistics, can be questioned when applied to atypical population groups, like astronauts or medical patients. The so-called ‘healthy worker effect’ [21] describes the selection bias of very healthy individuals into the astronaut profession and, absent fatal accidents, their high level of fitness through their careers.

An alternative risk metric, termed the Radiation-Attributed Decrease of Survival (RADS [20]), was also calculated in the present study. RADS relies only on integration of the inferred radiation-attributed excess incidence rate and represents the fraction of the disease-free survival probability that is expected to be lost at a given age due to exposure to ionizing radiation:

$$RADS\left(a|a_x,D\right)=1-\exp \left(-{\int }_{a_x}^a{h}_c\left(t|a_x,D\right)\mathrm{d}t\right),$$ (3)

where $h_c\left(t|a_x,D\right)$ is the radiation-attributed excess incidence rate for the disease c after exposure with dose D at age $a_x$.

To facilitate comparison with survival-based metrics (eqs (A.1), (A.2), (A.3), (A.4), the values of RADS were computed either at the mean age in the unexposed population, i.e., at the age equal to the mean disease-free lifetime of the target population for a given outcome. Below, this metric is denoted as $\mathit{RAD}{S}_M$. Alternatively, as it was recently suggested by Walsh et al [22], the value of RADS estimated for astronauts at the age of completion of their professional career, e.g., 65 years, may appear as more appropriate to communicate radiation risks. In this case, RADS provides the risk of radiation-attributed reduction of the survival chances at the age of retirement, which may vary in various populations, and generically indicated here as $\mathit{RAD}{S}_R$.

Another alternative metric included here for the comparison reflects reduction of lifetime. Given an estimate of the years of disease-free life lost due to the radiation exposure, $\mathit{YLL}$, and the expected remaining disease-free lifetime at age of exposure for a non-exposed individual, $T_{a_x}=\frac{1}{S\left(a_x\right)}{\int }_{a_x}^{\infty }S\left(t\right)\mathrm{d}t$, the relative reduction of lifetime, defined as $\mathit{RRL}=\frac{\mathit{YLL}}{T_{a_x}}$, can be used as a relative indicator of detrimental impact of radiation on the lifetime:

$$\mathit{RRL}=\frac{\mathit{YLL}}{T_{a_x}}=\frac{{\int }_{a_x}^{\infty }S\left(t\right)\left(1-\exp \left(-{\int }_{a_x}^t{h}_c\left(x|a_x,D\right)\mathrm{d}x\right)\right)\mathrm{d}t}{{\int }_{a_x}^{\infty }S\left(t\right)\mathrm{d}t},$$ (4)

or, using eq. (3):

$$\mathit{RRL}=\frac{{\int }_{a_x}^{\infty }S\left(t\right)RADS\left(t|a_x,D\right)\mathrm{d}t}{{\int }_{a_x}^{\infty }S\left(t\right)\mathrm{d}t}.$$ (5)

Apparently, eq. (5) can be interpreted as the survival averaged RADS, conditional on survival at age of exposure.

2.4. Survival variability and scaling the hazard

A fundamental assumption made in lifetime risk calculations is that the individual radiation-attributed excess rate can be represented by probabilities of a specific outcome in the hypothetical identical exposed and non-exposed populations. For this, the survival data for specific populations, to which an individual is believed to belong, are used. In case of larger populations, e.g., continental or world populations, the survival data can be represented proportionally to the size of the respective national populations. However, national composition of space crews is not related to size of the national populations; therefore, survival curves for different populations were used to represent plausible range of uncertainty due to different lifetime expectations.

The main purpose of the present study is to investigate impact of cross-populational variability of survival on the estimates of time-integrated radiation risk. For this, the evaluated data and lifetables from the Human Mortality Database [23] were used to represent global variability of survival curves. The lifetables, found in the Human Mortality Database, contain the evaluated data, derived from national demographic statistics, and are homogeneously represented at various age and time intervals up to age 110. For the present study, the 1×1 period lifetables were used for 41 countries, as listed in Table A.1.

The survival curves for different populations were converted to the hazard scaling factors represented as a country-specific time series of ratios of log-transformed survival and log-transformed median of survival data for a given age:

$$R_i\left(a\right)=\frac{\ln S_i\left(a\right)}{\ln \left(\mathrm{m}\mathrm{e}\mathrm{d}\left(S\left(a\right)\right)\right)}.$$ (6)

The randomized survival function S_r is computed by randomly picking a series of the hazard scaling factors R_r(a) and modifying the survival curve S(a) for the selected target population as follows:

$$S_r\left(a\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(R_r\left(a\right)\mathrm{l}\mathrm{n}S\left(a\right)\right).$$ (7)

Use of the scaling factors provided a non-parametric modelling of the variability of hazard (cumulated mortality) across populations by randomly picking a time series of the hazard scaling factors and modifying the survival curve used for radiation risk computations. The random sampling was done uniformly, i.e., the survival curves for all populations were assumed equally probable. This was regarded as more appropriate than use of population-weighted sampling scheme because composition of multinational space crews does not reflect the population size of the countries, which data were used for simulation of uncertainty.

2.5. Global sensitivity index

The effect of inter-populational variability of survival curves on the estimates of radiation risk was evaluated using a variance-based global sensitivity index [24], [25] for a single variable:

$$S_i=\frac{V_i}{V_T}$$ (8)

where V_T is the total variance of a radiation risk estimate with all input variables randomly sampled and V_i is the variance when the i^th variable is fixed. The global sensitivity index $S_i$ represents change of the full variance of a random function if one random variable from several is fixed, specifically, when the same survival curve is selected for integration of risk in all Monte Carlo iterations. Correspondingly, the complementary quantity $1-S_i$ expresses the fraction of the total variance, which can be attributed to variability of survival data, so that it can serve as a measure of sensitivity of the risk estimates to uncertainty of lifetime expectations for different populations.

3. Results

3.1. Variability of survival curves across populations

The survival curves for both sexes were derived from the period lifetables for 41 countries presented in the Human Mortality Database [23] and the survival data for each country were averaged for 10 most recent years available at the date of query. The resulting survival curves are shown in Fig. 1. As seen from the figure, the variability of the average survival curves between populations is substantial, especially, for males. For example, the survival at age 60 varies from 0.61 to 0.93 for males and from 0.85 to 0.96 for females, at age 70 from 0.39 to 0.84 for males and from 0.72 to 0.91 for females, and at age 80 – from 0.17 to 0.64 for males and from 0.44 to 0.80 for females.

Fig. 1.

Average survival curves for females (left) and males (right) based on the lifetables from Human Mortality Database [23] for 41 countries (see text).

The distributions of the survival values for different countries at the given age are asymmetric and skewed, thus no country exists which survival curve can be regarded as representative for a central or median value for the whole dataset in the full range of ages from zero to the maximum lifetime. Age dependence of the survival curves vary differently; therefore, different countries may appear as a median at a given age.

3.2. Hazard scaling factors

The serial hazard scaling factors for all countries are plotted in Fig. 2 for males and females. As seen on the figures, the data for a few countries indicate essentially higher hazards (mortality rates) than for most of the countries. The hazards scaling factors for males are typically larger than that for females with higher maximal values, as well.

Fig. 2.

Hazard normalization factors reflecting variability of survival curves for females (left) and males (right) in 41 countries.

3.3. Radiation risk per unit dose

Performance of different risk metrics was compared using the values of lifetime (LAR, REIC, ELR, RRL) or time-integrated (RADS_R, RADS_M) radiation risk per unit dose computed with the radiation risk models either linear (ASC) or linear-quadratic (LEU) on dose. These values were calculated for males and females exposed at ages 35 and 60 to different doses in range from 0.01 to 3 Sv. The risk values per unit dose for age at exposure 35 are shown in Fig. 3, Fig. 4 for the endpoint ASC and in Fig. 5, Fig. 6 for the endpoint LEU. Each plot in Fig. 3, Fig. 4, Fig. 5, Fig. 6 consists of two sublots showing compatible risk metrics: LAR, REIC and ELR are shown on the left, while RADS_M, RADS_R, and RRL are shown on the right. As indicated above in sub-section 2.3, RADS_R was calculated for the retirement age of 65 years, while RADS_M – for the mean outcome-free lifetime, which for the selected population data (Germany) for the endpoint ASC was equal to 74.4 and 78.3 years for males and females, respectively, and for the endpoint LEU was equal to 78.0 and 82.9 years for males and females, respectively.

Fig. 3.

Lifetime radiation risk per unit dose for various risk metrics as function of dose for all solid cancers for male exposed at age 35. Uncertainty bands result from the risk models, risk transfer and interpopulation variability of survival data. Left plot – conventional lifetime metrics Lifetime Attributable Risk (LAR), Risk of Exposure-Induced Cancer (REIC) and Excess Lifetime Risk (ELR). Right plot – Radiation-Attributed Decrease of Survival at retirement (RADS_R) and at the mean age (RADS_M) and Relative Reduction of Lifetime (RRL).

Fig. 4.

Lifetime radiation risk per unit dose for various risk metrics as function of dose for all solid cancers for female exposed at age 35. Uncertainty bands result from the risk models, risk transfer and interpopulation variability of survival data. Left plot – conventional lifetime metrics Lifetime Attributable Risk (LAR), Risk of Exposure-Induced Cancer (REIC) and Excess Lifetime Risk (ELR). Right plot – Radiation-Attributed Decrease of Survival at retirement (RADS_R) and at the mean age (RADS_M) and Relative Reduction of Lifetime (RRL).

Fig. 5.

Lifetime radiation risk per unit dose for various risk metrics as function of dose for leukaemia for male exposed at age 35. Uncertainty bands result from the risk models, risk transfer and interpopulation variability of survival data. Left plot – conventional lifetime metrics Lifetime Attributable Risk (LAR), Risk of Exposure-Induced Cancer (REIC) and Excess Lifetime Risk (ELR). Right plot – Radiation-Attributed Decrease of Survival at retirement (RADS_R) and at the mean age (RADS_M) and Relative Reduction of Lifetime (RRL).

Fig. 6.

Lifetime radiation risk per unit dose for various risk metrics as function of dose for leukaemia for female exposed at age 35. Uncertainty bands result from the risk models, risk transfer and interpopulation variability of survival data. Left plot – conventional lifetime metrics Lifetime Attributable Risk (LAR), Risk of Exposure-Induced Cancer (REIC) and Excess Lifetime Risk (ELR). Right plot – Radiation-Attributed Decrease of Survival at retirement (RADS_R) and at the mean age (RADS_M) and Relative Reduction of Lifetime (RRL).

Uncertainty bands on Fig. 3, Fig. 4, Fig. 5, Fig. 6 represent the modelled uncertainties (95% confidence interval) coming from the risk models, multi-model inference, risk transfer, health/demographic data and inter-populational variability. Uncertainty due to radiation dose is not modelled, i.e., the risk values are computed for fixed dose values.

As seen on Fig. 3, Fig. 4, the risk per unit dose values for all metrics but LAR decline when dose grows. Increase of radiation-attributed excess rates for a composite endpoint ASC notably reduces disease-free survival and lifetime for exposed populations, thus resulting in reduced risk per unit dose values for REIC and ELR. For the endpoint ASC, LAR is computed with a risk model linear on dose and uses a survival curve for non-exposed population, so its risk per unit dose values do not depend on dose.

For the endpoint LEU (see Fig. 5, Fig. 6), the performance of the risk metrics is different. The selected risk models [13] are linear-quadratic on dose, and the time-integrated radiation risk per unit dose values grow substantially with dose. The incidence rates of leukaemia are significantly smaller than those for the aggregated ASC endpoint; therefore, LAR, REIC and ELR perform very similarly, showing very little difference at doses exceeding 1 Sv. However, the RADS-based metrics (RADS_R, RADS_M, RRL) are different due to different intervals for integration of the inferred radiation-attributed excess rates.

3.4. Sensitivity of risk metrics to interpopulation variability

The risk metrics were calculated for males and females exposed at ages 35 and 60 to different doses ranging from 0.01 to 3 Sv for endpoints ASC and LEU. Effect of interpopulation variability of survival functions on variance of risk estimates was expressed using the global sensitivity index S_i. The asymmetric and skewed distributions of the survival functions also change mean and median values of the generated distributions of the risk estimates; correspondingly, relative changes of the statistical parameters (%) were used to quantify these effects:

$${\delta }_{\mathrm{A}\mathrm{M}}=100\frac{\mathit{AM}-A{M}_1}{\mathit{AM}},{\delta }_{\mathrm{M}\mathrm{D}}=100\frac{\mathit{MD}-M{D}_1}{\mathit{MD}}$$ (9)

where AM and MD denote the mean and the median, respectively, when all variables are randomly sampled, while AM₁ and MD₁ are the mean and the median when interpopulation variability of survival functions is not simulated. The detailed numerical values of the mean (AM) and median (MD) estimates for various risk metrics, their relative changes (${\mathrm{\delta }}_{\mathrm{A}M}$ and ${\mathrm{\delta }}_{\mathit{MD}}$) and the complement to the global sensitivity index ($1-S_1$) are given in Table A.2, Table A.3, Table A.4, Table A.5.

Results for the endpoint ASC and age at exposure 35 (Table A.2) show that contribution of inter-populational variability to the total variance of the survival-dependent risk estimates, i.e., LAR, REIC, ELR and RRL, is in the range 1 − S₁=0.22…0.5 for both sexes. For age at exposure 60 (Table A.4), the complement to the global sensitivity index, 1 − S₁, varies from −1 to 5% for males and from 10 to 16% for females, which can be explained by a stronger bias of the risk estimates for males, which are demonstrated by ${\mathrm{\delta }}_{\mathrm{A}M}$ from −9.3 to −7%, in comparison to that for females from −3.6 to −2%.

For the endpoint LEU and age at exposure 35, effect of interpopulation variability of survival on variance is almost negligible with values of 1 – S₁ ranging from −4 to 2% for males and from −1 to 1% for females. Such weaker sensitivity to interpopulation variability can be explained by time-dependence of the model excess rates (see Fig. A.2, Fig. A.4), which quickly reduce with time after exposure. Variability of survival functions does not significantly change variance of the risk estimates but results in reduction of mean values by 3–4% for males and by 1% for females (Table A.3). However, for age at exposure 60, the highest values of excess rates appear shortly after exposure up to ages 70–75 (see Fig. A.4 in the Appendix) where the largest variations of survival are seen between the populations (see Fig. 1). Correspondingly, the mean values demonstrate stronger bias, by 5–6% for males and 1.5–2% for females, thus resulting in the negative values of the complement to global sensitivity index ranging from −7 to −1% for males and from −3 to −1% for females (Table A.5).

Fig. A.2.

Incidence rates for the endpoint LEU for male s (left) and females (right) exposed at age 35 to dose of 1 Gy. Blue lines – model baseline incidence rates for Excess Relative Risk (ERR) type (solid line) and Excess Absolute Risk (EAR) type (dashed lines) models. Red lines – model excess incidence rates, similarly indicated. Solid line with dots – incidence rate in the target population [14].

Fig. A.4.

Incidence rates for the endpoint LEU for males (left) and females (right) exposed at age 60 to dose of 1 Gy. Blue lines – model baseline incidence rates for Excess Relative Risk (ERR) type (solid line) and Excess Absolute Risk (EAR) type (dashed lines) models. Red lines – model excess incidence rates, similarly indicated. Solid line with dots – incidence rate in the target population [14].

The RADS values demonstrate no dependence on interpopulation variability of survival curves (Table A.2, Table A.3, Table A.4, Table A.5), because this quantity depends only on the radiation-attributed hazard, i.e., the model radiation-attributable excess incidence rate for a given outcome transferred to the target population and integrated until a certain age. Similar to the survival-dependent risk metrics, calculation of RADS involves modelling risk transfer and simulation of uncertainty of the baseline incidence rate for the selected target population; therefore, health and population statistics for the target population are still required and used for computations of RADS estimates. However, uncertainty related to these data is included in the uncertainty of the resulting risk estimates, while application of the multi-model inference reduces impact of these population-specific data on central values of the risk estimates.

4. Discussion

In the present study, several radiation risk metrics have been compared for two different endpoints for cancer after radiation exposure at age 35––all solid cancers (endpoint ASC) and hematopoietic malignancies (endpoint LEU, leukaemia). The compared risk metrics can be characterized as survival-based ones (LAR, REIC, ELR), RADS-based (RADS_M and RADS_R) and a hybrid one (RRL), because the RRL can be also regarded as survival-averaged RADS (see eq. (5).

The survival-based lifetime risk metrics represent radiation risk integrated over the lifetime, differing only in consideration of influence of radiation-attributed excess rates on survival chances. Among these metrics, LAR is a simple approximation, which is valid only if the radiation-attributed excess rates and, correspondingly, reductions of survival chances are negligible, and the same survival curve can be assumed for non-exposed and exposed populations. LAR is a close approximation for REIC for the relatively low doses that are experienced during most short-duration space missions.

For the endpoint ASC with radiation risk models linear on dose, the LAR values per unit dose are the largest among other estimates and do not depend on dose (cf. Fig. 3, Fig. 4), while REIC and ELR take into account radiation-reduced survival and lifetime and demonstrate non-linear dependence on dose, with smaller values per unit dose at higher doses. The RADS and RRL metrics are non-linear on dose by definition and their values per unit dose also reduce as dose grows.

For the endpoint LEU with radiation risk models linear-quadratic on dose, all risk metrics reflect non-linear dependence on dose, but with growing risk per unit dose values as exposure dose (or, e.g., mission duration) increases.

Impact of radiation-attributed excess rates on lifetime and survival curves is more pronounced for the composite endpoint ASC with higher absolute values of the incidence rates (see Fig. A.1, Fig. A.3), than for the endpoint LEU with its substantially, by 10–100 times, lower incidence rates (see Fig. A.2, Fig. A.4). For ASC, the REIC risk per unit dose values match those of LAR at low doses and, correspondingly, low radiation-attributed excess rates, while at the higher doses they deviate downward towards the ELR level. Contrary to that, for the endpoint LEU, the survival-based metrics LAR, REIC and ELR result in almost coinciding risk per unit dose values, demonstrating negligible competing effect of the radiation-attributed excess rates on the lifetime and survival functions.

Fig. A.1.

Incidence rates for the endpoint ASC for males (left) and females (right) exposed at age 35 to dose of 1 Gy. Blue lines – model baseline incidence rates for Excess Relative Risk (ERR) type (solid line) and Excess Absolute Risk (EAR) type (dashed lines) models. Red lines – model excess incidence rates, similarly indicated. Solid line with dots – incidence rate in the target population [14].

Fig. A.3.

Incidence rates for the endpoint ASC for male s (left) and females (right) exposed at age 60 to dose of 1 Gy. Blue lines – model baseline incidence rates for Excess Relative Risk (ERR) type (solid line) and Excess Absolute Risk (EAR) type (dashed lines) models. Red lines – model excess incidence rates, similarly indicated. Solid line with dots – incidence rate in the target population [14].

The radiation risks were calculated taking into account uncertainties of the risk models, multi-model inference, risk transfer, population-specific health statistics and interpopulation variability of survival. The uncertainties of the risk estimates are, approximately, 30–50% and often the different risk metrics appear compatible within the calculated uncertainty bands (see Fig. 4, Fig. 5, Fig. 6).

The interpopulation variability of survival was modelled via variability of survival in the period lifetables for 41 countries [23]. Survival curves in the period lifetables reflect the total mortality rate in a given population averaged for a short period of time (typically, one year), while integration of survival-based radiation risk metrics depends on unknown future trends of survival and mortality in a cohort born in the same year. Relationship between the cohort and period life expectancy is not straightforward and may depend on social, economic, medical and political factors. Discussion on this relationship and examples of secular changes of cohort survival can be found elsewhere [26], [27], [28], [29]. In the present paper, the period lifetables for different countries were used to model variability of survival curves among different populations with different life expectancy and total mortality rates.

To better characterize effect of additional uncertainty due to interpopulation variability of survival the radiation risk values were calculated for ages at exposure 35 and 60. While modelling this uncertainty, it was shown that asymmetric distributions of the survival data resulted in negative biases (down to −10%) of the mean and median risk per unit dose values for both ASC and LEU endpoints, using survival-based risk metrics (LAR, REIC, ELR, RRL). For males, this effect is stronger than for females due to larger variability of the survival values across the populations (Fig. 1). These biases affected the modelled risk values and resulted in smaller values of the complement to the global sensitivity index, which often dropped to negative values. For example, for the endpoint ASC and age at exposure 60 (Table A.4), biases of AM values are from −9.3 to −7% for males and from −3.6 to −2.1% for females. At the same time, the complement to the global sensitivity index 1 – S₁ is close to 0 for males and ranges from 10 to 16% for females, thus clearly illustrating that stronger negative bias is responsible for the reduced complement for males.

By definition, the RADS metrics are not sensitive to interpopulation variability of survival unlike all others (LAR, REIC, ELR, RRL), for which interpopulation variability is responsible for 20–50% of the total variance in case of the composite ASC endpoint and age at exposure 35 (Table A.2) and up to 16% for female and age at exposure 60 (Table A.4). For the LEU endpoint, the situation is different and, for age at exposure 35 years (Table A.3), the highest values of the excess rates are within 10–20 years after exposure, while variability of survival curves at these ages is small and the effect on lifetime risks is also small. For age at exposure 60, the LEU excess rates fall in range of ages from 60 to 75 (Fig. A.4), where dispersion of the survival values between countries is highest (Fig. 1), so that the mean and the median values of the modelled risk per unit dose values are biased downwards by 1.2–8.6% (Table A.5).

Conventional risk metrics LAR, REIC and ELR are well defined, currently in use and can be conveniently perceived as estimates of future lifetime risks. However, being dependent on a specific survival data, they are bound to assumptions that are specific to a certain time and population and sensitive to variability of survival functions, which is often unknown or assumed. This is also applicable to a hybrid metric RRL, which can be interpreted as the survival averaged RADS.

The RADS metrics, being insensitive to time- and population-specific survival functions and their time projections, have some advantages in expressing statistical inferences on radiation risk, as derived from epidemiological studies. However, being not naturally restricted in time by survival functions, RADS values should be specified at a certain age, e.g., age of retirement or age equal to the mean lifetime in the selected population, thus representing the radiation-attributed reduction of survival chances at that age. These specific features of RADS-based metrics make them either more difficult to communicate to a wider audience, including astronauts and their families, or vague due to flexible choices of the age variable.

Contrary to expectations, variance-based global sensitivity index and its complement appeared less informative for expression of interpopulation variability because of non-linear transformation of hazards to survival and resulting downward biases of the model values, which effectively reduce variances and result in lower, sometimes negative, values of the complement to sensitivity index. The bias of the risk estimates due to interpopulation variability demonstrated that additional factors of uncertainty, specific to multinational space crews, may strongly affect as confidence range of the risk estimates as their absolute values. Among such factors, the competing risks, including radiation-attributed non-cancer endpoints and non-radiation factors, like stress, acceleration, microgravity, habitual and psychological factors, may also affect lifetime expectations for space crew members, thus influencing risks projections for radiation-attributed health effects. Correspondingly, comprehensive consideration of multiple competing health-affecting factors may significantly improve plausibility of radiation risk estimates for specific health outcomes.

5. Conclusions

Characterisation of health risks concomitant to radiation exposures in space is challenged by multiple factors, including properties of space radiation, exposure history, protective properties of space vehicles, individual health status and fitness of astronauts as well as by selection of metrics used to express and quantify integral health risks from radiation exposure. Uncertainty in estimates of time-integrated risk metrics originates not only from unknown secular trends of used data but also from individual health status and relevance of the assumed population-specific survival function to an individual expectation of disease-free lifetime.

In the present study, several radiation risk metrics were considered in order to compare and contrast their applicability to understanding and communicating potential risks from exposure to ionizing radiations during space flight for crew members and passengers, e.g., tourists. Among compared metrics were (a) conventional, survival-based and lifetime integrated, radiation risk metrics LAR, REIC and ELR, (b) RADS based on cumulative hazards and (c) RRL which is linked to YLL and reflects the radiation-attributed relative reduction of lifetime after exposure.

No radiation risk metric from the considered in the present paper can be regarded as universally good and fitting all purposes. Some metrics are more convenient for communicating risk expectations to diverse audiences, others are more robust and less sensitive to various uncertainty factors, specific to statistical models and data. Currently, it appears practical to select radiation risk metrics that fit the purpose, based on existing tasks and considered situations. When applied to assessment of radiation risks for space crew members, this, however, does not preclude that continuing efforts to develop generalized risk metrics that represent risks for health and life from multiple stressors, including radiation-attributed and non-radiation-related ones, may provide better and comprehensive way to quantify and communicate health and life risks for human spaceflight.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A.

Conventional risk metrics

Using common notation [16], [17], [18], [19], [20], the conventional time-integrated risk metrics are expressed, as follows.

The lifetime attributable risk (LAR) is an integral of radiation-attributed excess incidence rate during lifetime of the general, non-exposed, population:

$$LAR\left(a_x,D\right)=\frac{1}{S\left(a_x\right)}\left({\int }_{a_x}^{\infty }{\lambda }_c^{\ast }\left(t|a_x,D\right)S\left(t\right)\mathrm{d}t-{\int }_{a_x}^{\infty }{\lambda }_c\left(t\right)S\left(t\right)\mathrm{d}t\right),$$ (A.1)

where $a_x$ is the individual age at exposure; ${\lambda }_c\left(t\right)$ and ${\lambda }_c^{\ast }\left(t\right)$ are the incidence rates of the disease c in the matching non-exposed and exposed populations, respectively, and $S\left(t\right)$ is the disease-free survival function for the disease c in the non-exposed general population.

The risk of exposure-induced cancer (REIC) integrates radiation-attributed excess incidence rate during lifetime of the exposed population, taking into account reduction of the disease-free lifetime due to radiation-attributed health outcomes:

$$REIC\left(a_x,D\right)=\frac{1}{S\left(a_x\right)}\left({\int }_{a_x}^{\infty }{\lambda }_c^{\ast }\left(t|a_x,D\right)S^{\ast }\left(t\right)\mathrm{d}t-{\int }_{a_x}^{\infty }{\lambda }_c\left(t\right)S^{\ast }\left(t\right)\mathrm{d}t\right),$$ (A.2)

where $S^{\ast }\left(t\right)$ is the disease-free survival function for the disease c in the matching exposed population.

The excess lifetime risk (ELR) expresses the difference between estimated incidence rates in the matching exposed and non-exposed populations:

$$ELR\left(a_x,D\right)=\frac{1}{S\left(a_x\right)}\left({\int }_{a_x}^{\infty }{\lambda }_c^{\ast }\left(t|a_x,D\right)S^{\ast }\left(t\right)\mathrm{d}t-{\int }_{a_x}^{\infty }{\lambda }_c\left(t\right)S\left(t\right)\mathrm{d}t\right).$$ (A.3)

Detrimental effects of radiation exposure can be also expressed via reduction of expected disease-free lifetime, called ‘years of life lost’ (YLL), conditional on disease-free survival until the age at exposure $a_x$:

$$YLL\left(a_x,D\right)=\frac{1}{S\left(a_x\right)}\left({\int }_{a_x}^{\infty }S\left(t\right)\mathrm{d}t-{\int }_{a_x}^{\infty }{S}^{\ast }\left(t\right)\mathrm{d}t\right).$$ (A.4)

Integration of these conventional, survival-based, risk metrics is performed over the full lifetime and the upper integration limit is set to infinity. In practical computations, the upper integration limit is defined by available lifetables and, in the present study, was set to 120 years.

List of countries

The population data [30] and survival statistics [23] used in this study represented 41 countries, shown in Table A.1.

Table A.1.

Countries, their populations in 2019 [30] and the period for which the lifetables from the Human Mortality Database [23] were used in calculations of time-integrated radiation risk.

Country			Population in 2019 (million)		Survival data for the period
ISO code	Name	Full name	Males	Females
AU	Australia	Australia	12.551	12.499	2009–2018
AT	Austria	Republic of Austria	4.409	4.546	2010–2019
BY	Belarus	Republic of Belarus	4.4	5.053	2009–2018
BE	Belgium	Kingdom of Belgium	5.744	5.845	2011–2020
BG	Bulgaria	Republic of Bulgaria	3.452	3.651	2008–2017
CA	Canada	Canada	18.392	18.682	2009–2018
CL	Chile	Republic of Chile	9.097	9.373	2008–2017
HR	Croatia	Republic of Croatia	1.979	2.126	2011–2020
CZ	Czechia	The Czech Republic	5.261	5.429	2010–2019
DK	Denmark	The Kingdom of Denmark	2.879	2.913	2011–2020
EE	Estonia	The Republic of Estonia	0.627	0.699	2010–2019
FI	Finland	The Republic of Finland	2.732	2.808	2021–2020
FR	France	The French Republic	31.466	33.525	2009–2018
DE	Germany	The Federal Republic of Germany	40.738	41.921	2008–2017
GR	Greece	Hellenic Republic	5.141	5.333	2010–2019
HK	Hong Kong	Hong Kong (special region of PRC)	3.418	4.018	2010–2019
HU	Hungary	Hungary	4.598	5.062	2011–2020
IS	Iceland	The Republic of Iceland	0.169	0.168	2009–2018
IE	Ireland	Ireland	2.354	2.399	2008–2017
IL	Israel	The State of Israel	4.024	4.085	2007–2016
IT	Italy	The Italian Republic	29.479	31.148	2009–2018
JP	Japan	Japan	61.95	64.91	2010–2019
LV	Latvia	The Republic of Latvia	0.878	1.029	2010–2019
LT	Lithuania	The Republic of Lithuania	1.276	1.483	2010–2019
LU	Luxembourg	Grand Duchy of Luxembourg	0.311	0.305	2010–2019
NL	Netherlands	Kingdom of the Netherlands	8.515	8.582	2010–2019
NZ	New Zealand	New Zealand	2.221	2.297	2004–2013
NO	Norway	The Kingdom of Norway	2.74	2.681	2011–2020
PL	Poland	The Republic of Poland	18.361	19.527	2010–2019
PT	Portugal	The Portuguese Republic	4.824	5.373	2011–2020
KR	Republic of Korea	The Republic of Korea	25.628	25.543	2009–2018
RU	Russia	The Russian Federation	67.026	77.638	2005–2014
SK	Slovakia	The Slovak Republic	2.657	2.8	2010–2019
SI	Slovenia	The Republic of Slovenia	1.035	1.044	2010–2019
ES	Spain	The Kingdom of Spain	22.928	23.765	2009–2018
SE	Sweden	The Kingdom of Sweden	5.026	5.011	2010–2019
CH	Switzerland	The Swiss Confederation	4.294	4.361	2011–2020
TW	Taiwan	Taiwan (province of China)	11.824	11.95	2020–2019
US	The United States of America	The United States of America	162.826	166.239	2020–2019
UA	Ukraine	Ukraine	20.924	24.363	2004–2013
GB	United Kingdom	The United Kingdom of Great Britain and Northern Ireland	33.144	33.997	2009–2018

Table A.2.

Mean (AM) and median (MD) values of time-integrated radiation risk of solid cancers (endpoint ASC) per unit equivalent dose (%/Sv), their relative changes (${\delta }_{\mathrm{A}\mathrm{M}}$ and ${\delta }_{\mathrm{M}\mathrm{D}}$) due to variability of population survival data and a complement for the global sensitivity index S₁ for males and females exposed at age 35.

Dose (Sv)	Male					Female
	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1
	LAR per dose
0.01–3.0 *	12	−7.7	12.1	−10	0.40	19.4	−2.4	19.4	−1.8	0.40
	REIC per dose
0.01	12	−7.7	12	−10	0.40	19	−2.4	19	−1.8	0.40
0.05	12	−7.7	12	−10	0.40	19	−2.4	19	−1.7	0.40
0.15	12	−7.7	12	−9.9	0.41	19	−2.4	19	−1.7	0.40
0.35	12	−7.6	12	−9.7	0.41	19	−2.4	19	−1.7	0.41
1.0	11	−7.5	11	−9.2	0.43	17	−2.3	17	−1.6	0.42
2.0	10	−7.2	10	−8.4	0.46	15	−2.2	15	−1.4	0.45
3.0	9.5	−6.9	9.6	−7.8	0.50	14	−2.1	14	−1.3	0.47
	ELR per dose
0.01	8.5	−6.3	8.4	−9.3	0.27	15	−2.1	15	−1.7	0.31
0.05	8.5	−6.3	8.4	−9.2	0.27	15	−2.1	15	−1.6	0.32
0.15	8.4	−6.2	8.3	−9.1	0.27	15	−2.1	15	−1.6	0.32
0.35	8.2	−6.2	8.1	−8.9	0.27	14	−2.0	14	−1.6	0.32
1.0	7.7	−5.9	7.7	−8.3	0.29	13	−2.0	13	−1.5	0.33
2.0	7.1	−5.5	7.1	−7.4	0.30	12	−1.8	12	−1.3	0.34
3.0	6.5	−5.1	6.5	−6.8	0.32	10	−1.7	10	−1.2	0.35
	RADSR per dose
0.01	5.8	0	5.8	0	0	9.6	0.0	9.4	0.0	0.0
0.05	5.8	0	5.8	0	0	9.5	0.0	9.4	0.0	0.0
0.15	5.8	0	5.8	0	0	9.5	0.0	9.3	0.0	0.0
0.35	5.8	0	5.7	0	0	9.4	0.0	9.2	0.0	0.0
1.0	5.6	0	5.6	0	0	9.1	0.0	9.0	0.0	0.0
2.0	5.5	0	5.5	0	0	8.7	0.0	8.6	0.0	0.0
3.0	5.3	0	5.3	0	0	8.3	0.0	8.2	0.0	0.0
	RADSM per dose
0.01	11	0	11	0	0	18	0.0	18	0.0	0.0
0.05	11	0	11	0	0	18	0.0	17	0.0	0.0
0.15	11	0	11	0	0	18	0.0	17	0.0	0.0
0.35	11	0	11	0	0	17	0.0	17	0.0	0.0
1.0	10	0	11	0	0	16	0.0	16	0.0	0.0
2.0	9.8	0	9.9	0	0	15	0.0	15	0.0	0.0
3.0	9.3	0	9.4	0	0	14	0.0	14	0.0	0.0
	RRL per dose
0.01	4.7	−7.0	4.7	−7.0	0.42	8.4	−2.2	8.3	−2.0	0.22
0.05	4.7	−6.9	4.7	−7.0	0.42	8.4	−2.2	8.3	−2.0	0.22
0.15	4.7	−6.9	4.7	−7.0	0.42	8.3	−2.2	8.2	−2.0	0.22
0.35	4.6	−6.9	4.6	−6.9	0.43	8.2	−2.2	8.1	−2.0	0.23
1.0	4.4	−6.8	4.8	−6.6	0.44	7.8	−2.2	7.7	−1.9	0.23
2.0	4.2	−6.6	4.2	−6.1	0.47	7.2	−2.1	7.2	−1.8	0.24
3.0	4.0	−6.5	4.0	−5.7	0.49	6.7	−2.0	6.7	−1.7	0.24

^*The radiation risk models used for the endpoint ASC are linear on dose, so LAR per dose is constant for a given age and sex

Table A.3.

Mean (AM) and median (MD) values of time-integrated radiation risk of leukaemia (endpoint LEU) per unit equivalent dose (%/Sv), their relative changes (${\delta }_{\mathrm{A}\mathrm{M}}$ and ${\delta }_{\mathrm{M}\mathrm{D}}$) due to variability of population survival data and a complement for the global sensitivity index S₁ for males and females exposed at age 35.

Dose (Sv)	Male					Female
	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1
	LAR per dose
0.01	1.3	−4.1	1.2	−4.5	−0.04	1.0	−1.2	0.9	−1.2	−0.01
0.05	1.4	−4.1	1.3	−4.5	−0.04	1.0	−1.2	1.0	−1.4	−0.01
0.15	1.5	−4.1	1.4	−4.9	−0.03	1.1	−1.2	1.1	−1.4	−0.01
0.35	1.8	−4.1	1.7	−4.7	−0.02	1.3	−1.2	1.3	−1.6	0.00
1.0	2.8	−4.1	2.7	−5.6	0.01	2.1	−1.2	2.0	−1.0	0.01
2.0	4.3	−4.1	4.2	−5.9	0.01	3.2	−1.2	3.0	−1.4	0.01
3.0	5.8	−4.1	5.6	−5.8	0.0	4.3	−1.2	4.1	−1.4	0.00
	REIC per dose
0.01	1.3	−4.1	1.2	−4.5	−0.04	1.0	−1.2	0.9	−1.2	−0.01
0.05	1.4	−4.1	1.3	−4.5	−0.04	1.0	−1.2	1.0	−1.4	−0.01
0.15	1.5	−4.1	1.4	−4.9	−0.03	1.1	−1.2	1.1	−1.4	−0.01
0.35	1.8	−4.1	1.7	−4.7	−0.02	1.3	−1.2	1.3	−1.6	−0.0
1.0	2.7	−4.0	2.6	−5.6	0.01	2.0	−1.2	1.9	−1.0	0.01
2.0	4.1	−4.0	4.0	−5.7	0.02	3.1	−1.2	2.9	−1.3	0.01
3.0	5.2	−3.8	5.1	−5.4	0.01	4.0	−1.1	3.8	−1.2	0.01
	ELR per dose
0.01	1.3	−4.0	1.2	−4.4	−0.04	1.0	−1.2	0.9	−1.3	−0.01
0.05	1.4	−4.0	1.3	−4.3	−0.04	1.0	−1.2	1.0	−1.3	−0.01
0.15	1.5	−4.0	1.4	−4.8	−0.03	1.1	−1.2	1.1	−1.4	−0.01
0.35	1.8	−4.0	1.7	−4.6	−0.02	1.3	−1.2	1.3	−1.6	0.0
1.0	2.7	−4.0	2.6	−5.5	0.01	2.0	−1.2	1.9	−0.9	0.01
2.0	4.0	−3.9	3.9	−5.6	0.01	3.0	−1.1	2.9	−1.4	0.01
3.0	5.1	−3.8	5.0	−5.3	0.01	3.9	−1.1	3.8	−1.3	0.01
	RADSR per dose
0.01	0.9	0.0	0.8	0.0	0.0	0.5	0.0	0.6	0.0	0.0
0.05	0.9	0.0	0.9	0.0	0.0	0.6	0.0	0.6	0.0	0.0
0.15	1.0	0.0	1.0	0.0	0.0	0.7	0.0	0.6	0.0	0.0
0.35	1.2	0.0	1.1	0.0	0.0	0.8	0.0	0.8	0.0	0.0
1.0	1.8	0.0	1.8	0.0	0.0	1.2	0.0	1.2	0.0	0.0
2.0	2.7	0.0	2.7	0.0	0.0	1.9	0.0	1.8	0.0	0.0
3.0	3.6	0.0	3.5	0.0	0.0	2.5	0.0	2.4	0.0	0.0
	RADSM per dose
0.01	1.3	0.0	1.2	0.0	0.0	1.0	0.0	0.9	0.0	0.0
0.05	1.4	0.0	1.3	0.0	0.0	1.0	0.0	0.9	0.0	0.0
0.15	1.5	0.0	1.4	0.0	0.0	1.1	0.0	1.0	0.0	0.0
0.35	1.8	0.0	1.7	0.0	0.0	1.3	0.0	1.3	0.0	0.0
1.0	2.7	0.0	2.6	0.0	0.0	2.0	0.0	1.9	0.0	0.0
2.0	4.0	0.0	4.0	0.0	0.0	3.0	0.0	2.9	0.0	0.0
3.0	5.2	0.0	5.1	0.0	0.0	3.9	0.0	3.8	0.0	0.0
	RRL per dose
0.01	0.7	−3.3	0.7	−3.6	−0.04	0.5	−1.1	0.5	−0.9	−0.01
0.05	0.7	−3.3	0.7	−4.0	−0.03	0.5	−1.1	0.5	−1.2	−0.01
0.15	0.8	−3.3	0.8	−3.9	−0.03	0.6	−1.1	0. 6	−1.2	−0.01
0.35	1.0	−3.3	0.9	−3.7	−0.02	0.7	−1.1	0.7	−1.3	−0.01
1.0	1.5	−3.3	1.4	−3.7	0.02	1.1	−1.1	1.0	−1.2	0.01
2.0	2.2	−3.3	2.1	−4.0	0.02	1.7	−1.1	1.6	−1.1	0.01
3.0	2.9	−3.2	2.8	−4.0	0.01	2.2	−1.1	2.1	−1.1	0.0

Table A.4.

Mean (AM) and median (MD) values of time-integrated radiation risk of solid cancers (endpoint ASC) per unit equivalent dose (%/Sv), their relative changes (${\delta }_{\mathrm{A}\mathrm{M}}$ and ${\delta }_{\mathrm{M}\mathrm{D}}$) due to variability of population survival data and a complement for the global sensitivity index S₁ for males and females exposed at age 60.

Dose (Sv)	Male					Female
	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1
	LAR per dose
0.01–3.0 *	5.6	−8.3	5.2	−19	0.01	7.1	−2.5	6.8	−3.3	0.13
	REIC per dose
0.01	5.6	−8.3	5.2	−19	0.01	7.1	−2.5	6.8	−3.2	0.13
0.05	5.6	−8.3	5.2	−19	0.01	7.1	−2.5	6.8	−3.2	0.13
0.15	5.6	−8.3	5.2	−19	0.01	7.0	−2.5	6.8	−3.2	0.13
0.35	5.5	−8.3	5.2	−19	0.01	6.9	−2.4	6.7	−3.1	0.13
1.0	5.3	−8.2	5.0	−19	0.02	6.7	−2.4	6.5	−3.0	0.14
2.0	5.0	−8.1	4.8	−19	0.03	6.4	−2.4	6.3	−2.9	0.15
3.0	4.8	−8.0	4.6	−18	0.03	6.1	−2.3	6.0	−2.8	0.15
	ELR per dose
0.01	4.2	−7.6	4.0	−15	−0.01	5.8	−2.3	5.6	−2.8	0.10
0.05	4.2	−7.6	4.0	−15	−0.01	5.8	−2.3	5.6	−2.8	0.10
0.15	4.1	−7.5	3.9	−15	−0.01	5.8	−2.3	5.6	−2.8	0.10
0.35	4.1	−7.5	3.9	−15	−0.01	5.7	−2.2	5.5	−2.8	0.10
1.0	3.9	−7.4	3.8	−15	−0.01	5.5	−2.2	5.4	−2.7	0.11
2.0	3.7	−7.2	3.6	−15	0	5.2	−2.2	5.1	−2.6	0.11
3.0	3.5	−7.0	3.5	−14	0.01	5.0	−2.1	4.9	−2.5	0.12
	RADSR per dose
0.01	0.35	0	0.34	0	0	0.38	0	0.36	0	0
0.05	0.35	0	0.34	0	0	0.38	0	0.36	0	0
0.15	0.35	0	0.34	0	0	0.38	0	0.36	0	0
0.35	0.35	0	0.34	0	0	0.38	0	0.36	0	0
1.0	0.35	0	0.34	0	0	0.38	0	0.36	0	0
2.0	0.35	0	0.34	0	0	0.38	0	0.36	0	0
3.0	0.35	0	0.34	0	0	0.38	0	0.36	0	0
	RADSM per dose
0.01	3.6	0	3.7	0	0	4.87	0	4.8	0	0
0.05	3.6	0	3.7	0	0	4.86	0	4.8	0	0
0.15	3.6	0	3.7	0	0	4.85	0	4.8	0	0
0.35	3.5	0	3.7	0	0	4.82	0	4.7	0	0
1.0	3.5	0	3.7	0	0	4.74	0	4.7	0	0
2.0	3.4	0	3.6	0	0	4.62	0	4.5	0	0
3.0	3.3	0	3.5	0	0	4.5	0	4.4	0	0
	RRL per dose
0.01	2.7	−9.3	2.5	−22	0.03	3.2	−3.6	3.1	−4.4	0.14
0.05	2.7	−9.3	2.5	−22	0.03	3.2	−3.6	3.1	−4.4	0.14
0.15	2.7	−9.3	2.5	−22	0.03	3.2	−3.6	3.1	−4.4	0.14
0.35	2.6	−9.3	2.5	−22	0.03	3.2	−3.6	3.1	−4.3	0.14
1.0	2.6	−9.2	2.4	−22	0.04	3.1	−3.5	3.0	−4.2	0.14
2.0	2.5	−9.2	2.3	−21	0.04	3.0	−3.5	2.9	−4.3	0.15
3.0	2.4	−9.1	2.3	−22	0.05	2.9	−3.5	2.8	−4.3	0.16

^*The radiation risk models used for the endpoint ASC are linear on dose, so LAR per dose is constant for a given age and sex

Table A.5.

Mean (AM) and median (MD) values of time-integrated radiation risk of leukaemia (endpoint LEU) per unit equivalent dose (%/Sv), their relative changes (${\delta }_{\mathrm{A}\mathrm{M}}$ and ${\delta }_{\mathrm{M}\mathrm{D}}$) due to variability of population survival data and a complement for the global sensitivity index S₁ for males and females exposed at age 60.

Dose (Sv)	Male					Female
	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1	AM (%/Sv)	${\delta }_{\mathrm{A}\mathrm{M}}$(%)	MD (%/Sv)	${\delta }_{\mathrm{M}\mathrm{D}}$(%)	1−S1
	LAR per dose
0.01	1.9	−5.1	1.6	−6.8	−0.06	1.5	−1.5	1.2	−1.4	−0.02
0.05	2.0	−5.1	1.6	−6.5	−0.06	1.5	−1.5	1.2	−1.8	−0.02
0.15	2.2	−5.1	1.8	−6.2	−0.06	1.7	−1.5	1.4	−1.6	−0.02
0.35	2.6	−5.1	2.2	−6.8	−0.05	2.0	−1.5	1.7	−1.7	−0.01
1.0	4.0	−5.1	3.5	−6.0	−0.04	3.1	−1.5	2.6	−1.5	−0.01
2.0	6.1	−5.1	5.4	−5.9	−0.04	4.7	−1.5	4.1	−1.3	−0.01
3.0	8.3	−5.1	7.3	−6.5	−0.04	6.4	−1.5	5.4	−1.7	−0.01
	REIC per dose
0.01	1.9	−5.1	1.6	−6.8	−0.06	1.5	−1.5	1.2	−1.4	−0.02
0.05	2.0	−5.1	1.6	−6.5	−0.06	1.5	−1.5	1.2	−1.8	−0.02
0.15	2.2	−5.1	1.8	−6.2	−0.06	1.7	−1.5	1.4	−1.5	−0.02
0.35	2.6	−5.1	2.2	−6.7	−0.05	2.0	−1.5	1.7	−1.7	−0.01
1.0	3.9	−5.0	3.4	−5.9	−0.03	3.0	−1.5	2.6	−1.5	−0.01
2.0	5.6	−4.9	5.1	−5.7	−0.02	4.4	−1.5	3.9	−1.2	−0.01
3.0	6.8	−4.7	6.4	−5.8	−0.01	5.5	−1.4	5.0	−1.5	−0.01
	ELR per dose
0.01	1.9	−5.1	1.5	−6.8	−0.06	1.5	−1.5	1.2	−1.5	−0.02
0.05	1.9	−5.0	1.6	−6.6	−0.06	1.5	−1.5	1.2	−1.8	−0.02
0.15	2.1	−5.0	1.8	−6.2	−0.06	1.7	−1.5	1.4	−1.6	−0.02
0.35	2.5	−5.0	2.2	−6.6	−0.05	2.0	−1.5	1.6	−1.7	−0.01
1.0	3.8	−5.0	3.4	−5.9	−0.03	3.0	−1.5	2.6	−1.5	−0.01
2.0	5.5	−4.8	5.0	−5.7	−0.02	4.4	−1.5	3.8	−1.2	−0.01
3.0	6.8	−4.6	6.3	−5.7	−0.01	5.5	−1.4	4.9	−1.5	−0.01
	RADSR per dose
0.01	0.36	0	0.29	0	0	0.24	0	0.19	0	0
0.05	0.38	0	0.30	0	0	0.25	0	0.20	0	0
0.15	0.42	0	0.34	0	0	0.27	0	0.22	0	0
0.35	0.50	0	0.41	0	0	0.33	0	0.27	0	0
1.0	0.76	0	0.65	0	0	0.50	0	0.42	0	0
2.0	1.2	0	0.99	0	0	0.77	0	0.65	0	0
3.0	1.6	0	1.3	0	0	1.03	0	0.87	0	0
	RADSM per dose
0.01	1.7	0	1.4	0	0	1.4	0	1.1	0	0
0.05	1.8	0	1.4	0	0	1.5	0	1.2	0	0
0.15	2.0	0	1.7	0	0	1.6	0	1.3	0	0
0.35	2.3	0	2.0	0	0	1.9	0	1.6	0	0
1.0	3.5	0	3.1	0	0	2.9	0	2.4	0	0
2.0	5.2	0	4.7	0	0	4.2	0	3.7	0	0
3.0	6.5	0	5.9	0	0	5.4	0	4.8	0	0
	RRL per dose
0.01	1.1	−6.2	0.86	−8.6	−0.07	0.83	−2.0	0.65	−2.1	−0.03
0.05	1.1	−6.2	0.90	−8.3	−0.07	0.86	−2.0	0.68	−2.4	−0.02
0.15	1.2	−6.2	1.0	−7.8	−0.06	0.95	−2.0	0.77	−2.5	−0.02
0.35	1.4	−6.2	1.2	−8.2	−0.05	1.1	−2.0	0.93	−2.1	−0.02
1.0	2.2	−6.2	1.9	−7.2	−0.03	1.7	−2.0	1.5	−2.0	−0.02
2.0	3.2	−6.1	2.9	−7.5	−0.02	2.5	−2.0	2.2	−1.8	−0.01
3.0	4.0	−6.0	3.6	−7.1	−0.01	3.3	−1.9	2.9	−2.3	−0.01

Tables of the risk metrics and their statistical properties.

Model incidence rates

In Fig. A.1, Fig. A.2, Fig. A.3, Fig. A.4, model baseline, model excess and target population baseline incidence rates are shown for hypothetic individuals exposed to dose 1 Gy at age 35 and 60. The model incidence rates are computed using the model of Walsh et al. [9] based on the Life Span Study data as published by Grant et al. [11]. The target population curves are time-interpolated data for the German population in 2014 [14]. The blue lines indicate baseline incidence rates: the solid line from the ERR-type model, the dashed one from the EAR-type model, and the solid one with dots for the target population. The red lines show excess incidence rates, computed with the ERR-type model (solid line) and the EAR-type model (dashed line). Fig. A.1, Fig. A.3 present results for the composite endpoint for all solid cancers (ASC) for ages at exposure 35 and 60 years, respectively, and Fig. A.2, Fig. A.4 for hematopoietic malignancies (LEU), as described in sub-section 2.2 for ages at exposure 35 and 60 years, respectively. Left plots in the figures show results for male and the right plots – for female.