Bayesian approaches for radiation dose-response estimation
Yeongwoo Park 1 · Young Min Kim 2 · Yongku Kim 3
1 Health Insurance Research Institude, National Health Insurance Service
23 Department of Statistics, Kyungpook National University
Received 22 June 2020, revised 7 July 2020, accepted 8 July 2020
Abstract
Many people are exposed to radiation from frequently receiving X-rays or CT scans at the hospital. Exposure at low dose does not give an immediate response to the body. However, it is well known that the risk of cancer incidence rate is increasing over time. Therefore, in order to accurately quantify the risk of radiation exposure, many studies have been actively conducted. However, the Bayesian approach has not been rarely applied as a way to quantify radiation exposure risk. In this paper, the Bayesian approaches were introduced for several dose-response models to estimate the risk of radiation exposure, and the proposed model was applied to Life Span Study (LSS) solid cancer incidence data obtained from atomic bomb survivors (1958-2009). It was confirmed that the dose-response model of Linear nonthreshold (LNT), Quadratic, Linear-Quadratic, and Threshold showed similar results to those without the Bayesian analysis method. In the case of the piecewise linear dose-response model, the changes in slope associated with radiation exposure were found to vary with the dose ranges.
Keywords: Bayesian analysis, dose-response model, excess relative risk, piecewise linear model, radiation exposure.
1. Introduction
Nowadays, many people are exposed to various radiation from frequently receiving X-ray or CT scans at the hospital or radon emitted from soil, rocks, and construction materials.
Exposure at low dose does not give an immediate response to the body. However, it is well known that the risk of cancer incidence rate is increasing over time. Therefore, in order to accurately quantify the risk of radiation exposure, many studies have been actively conducted with persistence (BEIR VII, 2006; Grant et al., 2017; UNSCEAR, 2019).
In particular, risks associated with radiation exposure are estimated by using various methods of statistics. Expectile regression and percentile regression are applied as methods
1
Researcher, Health Insurance Research Institude, National Health Insurance Service, Wonju 26464, Korea
2
Assistant professor, Department of Statistics, Kyungpook National University, Daegu 41566, Korea
3
Corresponding auther: Associate professor, Department of Statistics, Kyungpook National University,
Daegu 41566, Korea. E-mail: [email protected]
for estimating trends in extreme regions, such as the tail of response variables (Kim and Kang, 2018). These techniques have been actively studied in linear regression analysis, but rarely proceeded in the nonlinear Poisson regression analysis, such as excess relative risk (ERR). An (2018) presented the estimates of the expectile regression for the approximate linear model of the non-linear model, and Lee (2020) presented the results using percentile regression. Dropkin (2016) analyzed by applying the Generalized Additive Model (GAM) for radiation risk of women surviving the a-bombs at low dose. To solve the ambiguity of cancer risk to the quantitative association at low doses, Sasaki et al. (2014) conducted a cancer risk assessment in A-bomb survivors through a nonparametric statistical procedure based on the noise cancellation process of the artificial neural network (ANN) theorem. Furukawa et al. (2016) considered a Bayesian semiparametric model using piecewise-linear dose-response model to reduce the uncertainty of low dose. As such studies, the Bayesian approach has not been rarely applied as a way to quantify radiation exposure risk.
The risk assessment method for various radiation exposures is to quantify the potential cancer incidence rate. In the Life Span Study of atomic bomb survivors in Hiroshima and Nagasaki, radiation related cancers are assessed using excess relative risk (Grant et al., 2017).
ERR means how much the person’s risk has been exceeded compared to those who have not been exposed to radiation. In general a Poisson nonlinear model, mixed both log-linear and linear term, is used as a statistical model for estimating the parameter of excess risk. The important thing is to quantitatively express the degree of risk associated with exposure, and it is necessary to understand the exact radiation exposure dose and response relationship.
The linear nonthreshold (LNT) model is used as the most common approach to estimate the ERR model in radiation studies (BEIR VII, 2006; Grant et al., 2017). The LNT model is advantageous for interpretation because it assumes that the risk increases with increasing exposure. However, when estimating the model, the slope of the LNT has the disadvantage that it is usually determined by observations at higher dose. In addition, when estimating risk by dividing the dose range, it was found that the risk estimate at lower dose is outside the confidence interval of the risk estimate including the high dose range. This shows that the uncertainty of the risk estimate at lower dose is underestimated (Furukawa et al., 2016).
Therefore, we have considered models that are frequently used as dose-response models and piecewise linear model with different slopes in each range using cutpoints.
Through previous studies, we realized that few studies have been conducted to estimate radiation risk using the Bayesian approach that can use prior information of parameters.
In addition, we were examined the limitations of the LNT model widely used in radiation epidemiology. In this paper, to estimate the exact risk of radiation exposure, we tried to es- timate several dose-response models using the Bayesian approaches and compare the models based on the results (Kim, 2019; Park et al., 2017).
2. Materials and methods
2.1. Life span study data
We applied the Life Span Study solid cancer incidence data (1958-2009) provided by Radi-
ation Effects Research Foundation (RERF) to estimate the excess relative risk for radiation
exposure. LSS incidence data is a cohort data constructed by sampling an atomic bomb
survivors in Hiroshima and Nagasaki and it is still being followed up. Eligible members of
the LSS incidence data were 111,917, of which 6,473 who could not estimate radiation dose were excluded from the analysis. Therefore, 105,444 people were used for this analysis, and 22,538 were diagnosed as first primary solid cancer between 1958 and 2009. The data is consisted of a person-years table stratified by city, gender, radiation dose, exposure age, and attained age. More details about the LSS data are available at the site of the RERF (https://www.rerf.or.jp/en/library/data-en/lssinc17e/).
The characteristics of subjects in the LSS solid cancer incidence data are shown at table 2.1. There are several methods for estimating radiation dose, but in this study, we used DS02R1 weighted absorbed colon dose (Gy). The range of 0.005-0.5Gy was 39,031 persons, which was the most frequent, and more than half of the atomic bomb survivors were not in the city at the time of the bombing or were exposed to very low doses.
Table 2.1 Distribution of colon dose by sex, city and age at exposure DS02R1 weigthed absorbed colon dose (Gy)
subject NIC
∗0-0.005 0.005-0.5 0.5-1 1≤
Total 105,444(100) 25,239(23.9) 35,978(34.1) 39,031(37.0) 3,136(3.0) 2,060(2.0) Sex
male 42,910(100) 10,488(24.4) 14,574(34.0) 15,608(36.4) 1,282(3.0) 958(2.2) female 62,534(100) 14,751(23.6) 21,404(34.2) 23,423(37.5) 1,854(3.0) 1,102(1.8) City
Hiroshima 73,401(100) 19,249(26.2) 20,087(27.4) 30,556(41.6) 2,100(2.9) 1,409(1.9) Nagasaki 32,043(100) 5,990(18.7) 15,891(49.6) 8,475(26.4) 1,036(3.2) 651(2.0) Age at exposure
<10 22,708(100) 4,995(22.0) 7,928(34.9) 8,909(39.2) 505(2.2) 371(1.6) 10-19 23,079(100) 5,878(18.7) 7,973(34.5) 7,750(33.6) 892(3.9) 586(2.5) 20-29 14,251(100) 3,675(25.8) 4,718(33.1) 5,070(35.6) 478(3.4) 310(2.2) 30-39 15,838(100) 4,034(25.5) 5,127(32.4) 5,953(37.6) 418(2.6) 306(1.9) 40-49 16,074(100) 3,727(23.2) 5,472(34.0) 6,067(37.7) 504(3.1) 304(1.9) 50-59 9,379(100) 1,996(21.3) 3,306(35.2) 3,678(39.2) 258(2.8) 141(1.5) 60≤ 4,115(100) 934(22.7) 1,454(35.3) 1,604(39.0) 81(2.0) 42(1.0)
∗
NIC: Not in the city
2.2. Excess relative risk model
The excess relative risk (ERR) is a risk measurement method that quantifies how much the level of risk among persons with a given level of exposure exceeds the risk of non-exposed persons (WHO, 2009). To quantify the risks at exposure, relative risk (RR) is generally used, but the excess relative risk (ERR=RR-1) is used more commonly in radiation epidemiology.
It can be expressed as follows.
λ d (x) = λ 0 (x)(1 + ERR).
λ d (x) is the incidence rate with the characteristic of x when exposed to radiation and λ 0 (x) is the cancer incidence rate with the characteristic of x when not exposed to radiation.
To calculate the risk of radiation exposure, it is necessary to estimate the incidence of solid cancer from exposure and non-exposure, which is explained using the poisson relative risk models and can be expressed as:
Y i ∼ P ois(P Y i e η
i(1 + ERR(d i , s i , e i , a i ))), i = 1, · · · , n,
where Y i , P Y i , d i , and e η
iare the number of solid cancer, the number of person-years of
follow up, the mean absorbed colon dose and the background risk for stratum i respectively.
The linear predictor for the background risk has the form η i = α 0 + α 1 x 1i + · · · + α 11 x 11i , where x is constructed by multiplying city(c), sex(s), attained age(a), birth year(b) and location at the time of bombing(l). In more detail, the background risk describes the effects of sex-specific quadratic splines log-attained age, sex-specific quadratic log-attained age over 70, year of birth, city and NIC (not in city). The form of poisson relative risk is not a typical log-linear poisson model that is generally known (McCullagh and Nelder, 1989). Because the log-linear term and the linear term are mixed in the model, it is described as a generalized nonlinear model.
The ERR model related to radiation risk can be divided into two effects, the dose effect and the modification effect, and can be expressed as follows(BEIR VII, 2006; Furukawa et al., 2016).
ERR(d, s, e, a) = ρ(d)(s, e, a)
= ρ(d)θ s exp{γ (e − 30)
10 + φlog(a/70)},
where ρ(d) is dose-response function which represents the main effect of radiation and
(s, e, a) is modification effect function which represents how the excess risk associated with radiation exposure depended on exposure age(e), sex(s), attaind age(a).
2.3. Dose-response model
For applying the Bayesian method, we have considered several dose-response functions that are widely used to estimate excess relative risk for radiation exposure. The following dose-response models were considered:
Linear nonthreshold: ρ(d) = β 1 d Quadratic: ρ(d) = β 1 d 2 Linear-Quadratic: ρ(d) = β 1 d + β 2 d 2
Threshold: ρ(d) = β 1 (d − δ 1 ) + ,
where (d − δ 1 ) + =(d − δ 1 ) if d > δ 1 and 0 otherwise. The linear nonthreshold (LNT) model is a simple model that assumes that a certain risk increases in proportion to the dose level and is the most considered model in radiation epidemiology studies. Quadratic and Linear- Quadratic models are for considering curvature. The quadratic is a model that assumes that small risks occur at low doses and that the risk increases rapidly at high doses. The Linear-Quadratic is a model that assumes that the risk increase will have a different form from the LNT model at a high dose. The threshold is a model that focuses on the nature of the dose-response for a limited dose range from 0 to D lim , explaining that it has no effect at a specific low dose level.
As described above, the LNT model has a problem of underestimating uncertainty at low doses. Reflecting on this problem, we also considered the following piecewise linear dose- response model for excess relative risk.
ρ(d) =
C
X
k=1
β k h k (d),
h k (d) = {min(d, δ k ) − δ (k−1) } I(d > δ k−1 ), k = 1, · · · , C,
where I(.) is the indicator function such that I(D > D lim )=1 if D > D lim is true and 0 otherwise. The piecewise linear dose-response model divides the dose into specific range and estimates the degree of risk for each range. Even if different slopes are given for each range, it is set to have a continuous form using h k (d) functions. It is reasonable that the risks located in proximity are similar, rather than far away. Therefore, we assumed that the slopes of adjacent ranges are affected by each other, and the following conditional distribution is considered to reflect this assumption.
[β j |β j−1 ] ∼ N (β j−1 , σ 2 ), j = 1, · · · , C.
Because the risk of the next range is based on the risk of previous range, this type of assumption can be said to have an autocorrelation structure for the slope of the range. σ is responsible for determining the degree of smoothness of the dose-response curve. If σ → 0, the dose-response model has a linear form. If σ has a large value, it will have a complicated form because of the change in risk.
3. Application
To estimate the risk of radiation exposure, several dose-response models were applied to LSS solid cancer incidence data obtained from atomic bomb survivors (1958-2009). First of all, Y i is called solid cancer incidence at i stratum, i = 1, ..., n, and the distribution is assumed as follows.
Y i ∼ P ois(P Y i e η
i(1 + ERR(d i , s i , e i , a i ))).
Since the parameters don’t have information about the dose-response models of LNT, Quadratic, Linear-Quadratic, and Threshold , the independent non-informative prior is con- sidered. On the other hand, in the piecewise-linear model, the slopes of the adjacent ranges are judged to be similar to each other, so it is considered as a prior model.
[β j |β j−1 ] ∼ N (β j−1 , σ 2 ), j = 1, · · · , C.
To apply the piecewise linear model, we set the cutpoint using the 22 locations: 0, 0.005, 0.02, 0.04, 0.06, 0.08, 0.1, 0.125, 0.15, 0.175, 0.2, 0.25, 0.3, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.5 and 3, which are a mixture of equally spaced percentiles and intervals and are a conventional stratification criteria of person year in LSS data.
The smoothness parameter, σ, is considered as a hyper parameter and a non-information prior is considered to estimate the values well explained by the data. We thought that the values would change depending on the interval of the range, so we estimated it by splitting the interval at 0.2Gy.
The joint posterior distribution for the most complex dose-response model, piecewise linear
model, is as follows.
π(α, β, γ, φ|x) ∝
n
Y
i=1
[P Y i e η
i(1 +
C
X
k=1
β k h k (d))exp(γe ∗ i + φa ∗ i )] Y
i× exp[−
n
X
i=1
P Y i e η
i(1 +
C
X
k=1
β k h k (d))exp(γe ∗ i + φa ∗ i ))]
×
n
Y
i=1
1
Y i ! × π(α 0 ) × π(α 1 ) × π(α 2 ) × π(α 3 ) × · · · × π(α 11 )
× π(β 1 ) × π(β 2 |β 1 ) × π(β 3 |β 2 ) × · · · × π(β 22 |β 21 ) × π(γ) × π(φ).
Since it is difficult to find the joint posterior distribution of the proposed model, the Gibbs sampler was used to generate a random sample from the conditional posterior distribution.
To operate the Gibbs sampler, the conditional posterior distribution of each random variable was calculated, and most of the conditional posterior distribution does not belong to well- known family of distribution, so a Metropolis-Hastings algorithm is inserted to estimate the parameters. In order to confirm the efficiency of the generated Gibbs sampler, chains with a parallel structure were formed, and the rejection rate and the Gelman-Rubin statistics (G-R statistics) of the Metropolis-Hastings algorithm were confirmed.
When the conditional posterior distribution of each parameter is obtained based on the joint posterior distribution, most of them do not belong to the generally well-known family of distribution. Therefore, a random walk chain was used as a candidate-generating density function for Metropolis-Hastings to generate samples. Because we considered the condition that the support of random variables is greater than 0 for some interest parameters (β), the truncated normal distribution was set as the candidate-generating density function.
q 1 (y|x) = φ(y|x, σ 2 ) 1 − Φ(0|x, σ 2 ) , where φ(y|x, σ 2 ) = (1/ √
2πσ 2 exp{−(1/2σ 2 )(y − x) 2 } and Φ(0|x, σ 2 ) = R 0
∞ φ(y|x, σ 2 ).
In order to confirm the efficiency and convergence of Gibbs sampling, the rejection rate and the Gelman-Rubin Statistics were checked (Gelman et al., 2013). In this study, the rejection rates of the Metropolis-Hastings algorithm were found to be about 20∼50%, and the G-R statistics were found to have a value of 1.00-1.23. In the Bayesian approach, two parallel chains were used to check the convergence of the chain. After generating 20,000 samples, the first 5,000 samples were removed to eliminate the influence on the initial value, and every 30th samples were selected to eliminate the autocorrelation.
The posterior mean was used for parameter estimator and risk estimates are gender-
averaged excess relative risks at age 70 after exposure at age 30. Table 3.1 shows excess
relative risks for all solid cancer in relation to several radiation dose using the obtained
parameters, after fitting the several dose-response models. In more detail, Figure 3.1 shows
the excess relative risk for all solid cancer estimated over the entire dose range (left) and
at 0-0.5Gy (right). At the entire dose range, it can be seen that the models other than the
quadratic have a roughly similar form, and the risk can be seen to increase upwardly at the
quadratic term models. The piecewise linear model was estimated to be higher than the risk of the LNT model at 0.5Gy and lowered at 1.75Gy. At 0-0.5Gy, the LNT model was found to have the highest risk at low doses, and it is considered reasonable to assume that the LNT model is applied as the radiation protection guidelines (ICRP, 1990). The Linear-Quadratic model and the piecewise linear model showed a similar shape at low doses, and the threshold model showed that the radiation effect was not increased until a specific dose.
Table 3.1 Exess relative risk for all solid cancer in relation to several radiation dose
Dose LNT Q LQ Threshold Piecewise
0.005 0.00242 0.00001 0.00197 0.00000 0.00196 0.01 0.00485 0.00003 0.00394 0.00000 0.00382 0.1 0.04846 0.00278 0.04003 0.02078 0.03921 0.5 0.24230 0.06950 0.21316 0.21630 0.21596 1.0 0.48460 0.27799 0.45878 0.46069 0.50265 2.0 0.96920 1.11195 1.01745 0.94948 1.00469 3.0 1.45380 2.50190 1.76600 1.43827 1.36336
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00.51.01.5
Weighted absorbed colon dose(Gy)
Excess Relative Risk
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00.51.01.5
Linear nonthreshold Quadratic Linear−Quadratic Threshold Piecewise linear
0.0 0.1 0.2 0.3 0.4 0.5
0.000.050.100.150.200.25
Weighted absorbed colon dose(Gy)
Excess Relative Risk
0.0 0.1 0.2 0.3 0.4 0.5
0.000.050.100.150.200.25
Linear nonthreshold Quadratic Linear−Quadratic Threshold Piecewise linear
