The inverse dose-rate effect for radon induced lung cancer: a modified approach for risk modelling

Abstract

One of the features of high LET α-ray exposure due to radon inhalation is the well-known inverse dose-rate effect. The longer a given dose is delivered to the lung, the higher is its carcinogenic effect. This introduces the problem of risk extrapolation from high levels of radon exposure typical for uranium miners, down to low levels of radon exposure typical for the general population. An analytical model is presented that accounts for dose-rate effects over the entire exposure range. In accordance with radiobiological considerations and microdosimetric implications, the model provides an adequate description of the inverse dose-rate effect in the higher dose range. At the same time, a linear slope at low exposures that is independent of the dose-rate of exposure is attained.

Appendix

Implementation of the model into the hybrid likelihood algorithm

The hybrid likelihood algorithm has been utilized in the analyses of radiation risks in several studies [19–21]. The following section repeats briefly what is presented in detail in [18].

Consider n individuals (i=1, 2,..., n) of a cohort, C, with exposures starting at the ages e _i who have been observed up to the ages a _i . Let C′ be the subgroup of those who have incurred the tumour at the end of the observation. The total likelihood of the outcome—in terms of a chosen model—can then be expressed as the product of all estimated probabilities, ${\text{exp}}( - R_{i} (a_{i} ))$, to have remained tumour free before age a _i , and the estimated probability densities, r _i (a _i ), of the ‘cases’ to incur the tumour at the observed ages:

$$ \mathcal{L} = {\prod\limits_{i \in C} {{\text{exp}}( - R_{i} (a_{i} ))} }{\prod\limits_{i \in {C}\ifmmode{'}\else$'$\fi } {r_{i} (a_{i} )} }. $$

(12)

The term R _i (a _i ) gives the cumulative hazard function of individual i from the beginning of observation at age e _i to age a _i :

$$ R_{i} (a_{i} ) = {\int\limits_{e_{i} }^{a_{i} } {r_{i} (a){\text{d}}a} }. $$

(13)

The log-likelihood, $L = {\text{ln}}(\mathcal{L})$, can be separated in two terms:

$$ L = L_{1} + L_{0} . $$

(14)

L ₁ is the contribution of the ‘events’, and with (11) one obtains

$$ L_{1} = {\sum\limits_{i \in {C}\ifmmode{'}\else$'$\fi } {{\left[ {{\text{ln}}(r_{0} (a_{i} )) + {\text{ln}}{\left( {1 + (\alpha + \beta \Theta (a_{i} ))g(a_{i} ){\int\limits_{e_{i} }^{a_{i} } {h(a_{i} - e){\text{d}}D_{i} (e)} }} \right)}} \right]}} }. $$

(15)

L ₀ is the contribution of the ‘event-free’ observation periods:

$$ L_{0} = - {\sum\limits_{i \in C} {{\int\limits_{e_{i} }^{a_{i} } {r_{0} (a){\left( {1 + (\alpha + \beta \Theta (a))g(a){\int\limits_{e_{i} }^{a_{i} } {h(a - e){\text{d}}D(e)} }} \right)}{\text{d}}a} }} }. $$

(16)

Equation (16) contains a summation and a double integration, but the summation over all individuals of the cohort can be moved inside the expression, so that it need not be repeated, when the parameters of the model are varied in the maximum likelihood computations:

$$ L_{0} = - {\int\limits_{e_{{{\text{min}}}} }^{a_{{{\text{max}}}} } {r_{0} (a){\left( {n(a) + {\left( {\alpha {\int\limits_{e_{i} }^{a_{i} } {h(a - e)D(a,e){\text{d}}e} } + \beta {\int\limits_{e_{i} }^{a_{i} } {h(a - e)\Theta (a,e){\text{d}}e} }} \right)}g(a)} \right)}{\text{d}}a} }, $$

(17)

where n(a) is the number of individuals (person years) who have been under observation at age a:

$$ n(a) = {\sum\limits_{i \in C_{a} } 1 }, $$

(18)

while D(a, e) is the ‘collective exposure rate’ at age e of all those individuals who have been under observation at age a:

$$ D(a,e) = {\sum\limits_{i \in C_{a} } {D_{i} (e)} }, $$

(19)

Θ(a, e) is the ‘collective dose-rate correction’, at age e of all those individuals who have been under observation at age a, exposed with an average dose-rate Θ(a):

$$ \Theta (a,e) = {\sum\limits_{i \in C_{a} } {D_{i} (e)\Theta _{i} (a)} }. $$

(20)

The average dose-rate Θ(a) is to be calculated according to (5) or (6); e _min then being the earliest age at exposure in the cohort, and a _max being the highest age under observation.

The terms n(a), D(a, e) and Θ(a, e) are independent of the model parameters, i.e. they can be calculated at the beginning of the optimization procedure, without the need for repeated evaluation.

The time since exposure dependence h(a−e)

The time since exposure dependence, h(a−e), that was utilized in the models is of the form:

$$ h(t) = {\text{exp}}( - ({\text{ln}}\,t/t_{0} )^{2} /2\sigma ^{2} ),\quad {\text{with}}\;t = a - e. $$

(21)

Figure 4 shows the solution obtained with the model proposed.

Fig. 4

Time since exposure dependence h(t) (see (21), with t ₀=16.9 year and σ=0.49). After an estimated lag period of about 5 years, the highest risk is reached 17 years since exposure