A hybrid likelihood algorithm for risk modelling

The risk of radiation-induced cancer is assessed through the follow-up of large cohorts, such as atomic bomb survivors or underground miners who have been occupationally exposed to radon and its decay products. The models relate to the dose, age and time dependence of the excess tumour rates, and they contain parameters that are estimated in terms of maximum likelihood computations. The computations are performed with the software package EPI-CURE, which contains the two main options of person-by person regression or of Poisson regression with grouped data. The Poisson regression is most frequently employed, but there are certain models that require an excessive number of cells when grouped data are used. One example involves computations that account explicitly for the temporal distribution of continuous exposures, as they occur with underground miners. In past work such models had to be approximated, but it is shown here that they can be treated explicitly in a suitably reformulated person-by person computation of the likelihood. The algorithm uses the familiar partitioning of the log-likelihood into two terms, L1 and L0. The first term, L1, represents the contribution of the 'events' (tumours). It needs to be evaluated in the usual way, but constitutes no computational problem. The second term, L0, represents the event-free periods of observation. It is, in its usual form, unmanageable for large cohorts. However, it can be reduced to a simple form, in which the number of computational steps is independent of cohort size. The method requires less computing time and computer memory, but more importantly it leads to more stable numerical results by obviating the need for grouping the data. The algorithm may be most relevant to radiation risk modelling, but it can facilitate the modelling of failure-time data in general.

Probability of causation for lung cancer after exposure to radon progeny: a comparison of models and data

The estimates of lung cancer risk due to the exposure to radon decay products are based on different data sets from underground mining and on different mathematical models that are used to fit the data. Diagrams of the excess relative rate per 100 working level months in its dependence on age at exposure and age attained are shown to be a useful tool to elucidate the influence that is due to the choice of the model, and to assess the differences between the data from the major western cohorts and those from the Czech uranium miners. It is seen that the influence of the choice of the model is minor compared to the difference between the data sets. The results are used to derive attributable lifetime risks and probabilities of causation for lung cancer following radon progeny exposures.

A survey of the Czechoslovak follow-up of lung cancer mortality in uranium miners

The major Czechoslovak cohort of uranium miners (S-cohort) is surveyed in terms of diagrams illustrating dependences on calendar year, age, and exposure to radon and radon progeny. An analysis of the dose dependence of lung cancer mortality is performed by nonparametric and, subsequently, by parametric methods. In the first step, two-dimensional isotonic regression is employed to derive the lung cancer mortality rate and the relative excess risk as functions of age attained and of lagged cumulated exposure. In a second step, analytical fits in terms of relative risk models are derived. The treatment is largely analogous to the methods applied by the BEIR IV Committee to other major cohorts of uranium miners. There is a marked dependence of the excess risk on age attained and on time since exposure. A specific characteristic of the Czechoslovak data is the nonlinearity of the dependence of the lung cancer excess risk on the cumulated exposure; exposures on the order of 100 working level months or less appear to be more effective per working level month than larger exposures but, in the absence of an internal control group, this cannot be excluded to be due to confounders such as smoking or environmental exposures. A further notable observation is the association of larger excess risks with longer protraction of the exposures.

A nonparametric method for the derivation of alpha/beta ratios from the effect of fractionated irradiations

Multifractionation isoeffect data are commonly analysed under the assumption that cell survival determines the observed tissue or tumour response, and that it follows a linear-quadratic dose dependence. The analysis is employed to derive the alpha/beta ratios of the linear-quadratic dose dependence, and different methods have been developed for this purpose. A common method uses the so-called Fe plot. A more complex but also more rigorous method has been introduced by Lam et al. (1979). Their method, which is based on numerical optimization procedures, is generalized and somewhat simplified in the present study. Tumour-regrowth data are used to explain the nonparametric procedure which provides alpha/beta ratios without the need to postulate analytical expressions for the relationship between cell survival and regrowth delay.

The reverse protraction factor in the induction of bone sarcomas in radium-224 patients

More than 50 bone sarcomas have occurred among a collective of about 800 patients who had been injected in Germany after World War II with large activities of radium-224 for the intended treatment of bone tuberculosis and ankylosing spondylitis. In an earlier analysis [H. Spiess and C. W. Mays, in Radiation Carcinogenesis. (C. L. Sanders et al., Eds.) pp. 437-450. USAEC Symposium Series 29, CONF-720505, 1973] it was concluded that, at equal mean absorbed doses in the skeleton, patients with longer exposure time had a higher incidence of bone sarcomas. The previous analysis was based on approximations; in particular, it did not account for the varying times at risk of the individual patients. In view of the implications of a reverse protraction factor for basic considerations in radiation protection, the need was therefore felt to reevaluate the data from the continued follow-up by more rigorous statistical methods. A first step of the analysis demonstrates the existence of the reverse dose-rate effect in terms of a suitably constructed rank-order test. In a second step of the analysis it is concluded that the data are consistent with a linear no-threshold dose dependence under the condition of constant exposure time, while there is a steeper than linear dependence on dose when the exposure times increase proportionally to dose. A maximum likelihood fit of the data is then performed in terms of a proportional hazards model that includes the individual parameters, dose, treatment duration, and age at treatment. The fit indicates proportionality of the tumor rates to mean skeletal dose with an added factor (1 + 0.18.tau), where tau is the treatment time in months. This indicates that a protraction of the injections over 15 months instead of 5 months doubles the risk of bone sarcoma.

Kinetics of recovery from sublethal radiation damage in four murine tumors

The kinetics of repair of sublethal radiation damage (SLD) was studied in four transplantable C3H mouse tumors, i.e. mammary carcinoma AT17, fibrosarcoma SSK2, and squamous cell carcinomas AT51 and AT478. Tumors were irradiated with 4 fractions of 300 kV X-rays given under local hypoxia at intervals ranging from 0 to 6 h. Radiation response was measured by growth delay, which was directly analyzed using a general curve description based on the extended linear-quadratic model (exponential repair kinetics). In contrast to existing methods all growth delay values were utilized to estimate the alpha/beta ratios and the half-times as well as their confidence limits in a non-linear least squares analysis. The half-times were 42, 44, 54 and 31 min, respectively. It is concluded that repair of SLD is virtually complete after 5 h in these tumors. This is also due to the relatively small proportion of repairable damage in these tumors reflected in their alpha/beta values, which were 38, 30, 54 and 42 Gy, respectively.

Lung carcinomas in Sprague-Dawley rats after exposure to low doses of radon daughters, fission neutrons, or gamma rays

The effectiveness of radon-daughter inhalation and irradiation with fission neutrons and gamma rays in the induction of lung carcinomas in Sprague-Dawley rats at low doses is compared. Earlier reports which compared radon-daughter inhalations and neutron irradiations over a wider range of doses were based on dosimetry for the radon-daughter inhalations which has recently been found to be faulty. In the present analysis, low-dose experiments were designed to derive revised equivalence ratios between radon-daughter exposures, and fission neutron or gamma irradiations. The equivalence is approximately 15 working level months (WLM) of radon daughters to 10 mGy of neutrons (the earlier value was 30 WLM to 10 mGy). The relative biological effectiveness (RBE) of neutrons is 50 or more at a gamma-ray dose of 1 Gy. In these experiments with low doses and exposures, the lifetime incidences can be estimated from the raw incidences, while the derivation of the time dependence of the prevalence is essential for the estimation of RBE values and equivalence ratios.

An epidemiological assessment of lens opacifications that impaired vision in patients injected with radium-224

The incidence of lens opacifications that impaired vision (cataract) was analyzed among 831 patients who were injected with known dosages of 224Ra in Germany shortly after World War II. The dependence of the incidence on dosage, i.e., injected activity per unit body weight, and on time after treatment was determined. The observations are equally consistent with proportionality of the incidence of cataract to the square of dosage or with a linear dependence beyond a threshold of 0.5 MBq/kg. The possibility of a linear dependence without threshold was strongly rejected (P less than 0.001). The analysis of temporal dependences yielded a component that was correlated with the injected amount of 224Ra and a component that was uncorrelated. The former was inferred by a maximum likelihood analysis to increase approximately as the square of the time after treatment. The component unrelated to the treatment was found to increase steeply with age and to become dominant within the collective of patients between age 50 and 60. The relative magnitudes of the two components were such that a fraction of 55 to 60% of the total of 58 cataracts had to be ascribed to the dose-related incidence. Impaired vision due to cataract was diagnosed before age 54 in 25 cases. In terms of injected activity per unit body weight no dependence of the sensitivity on age was found; specifically there was no indication of a faster occurrence of the treatment-related cataracts in patients treated at older ages.

Comparison of the induction of pulmonary neoplasms in Sprague-Dawley rats by fission neutrons and radon daughters

Pulmonary carcinomas were recorded in a life-span experiment of male Sprague-Dawley rats exposed to fission neutrons. Mortality-corrected prevalences are obtained by the method of isotonic regression. In a second part of the paper a comparison is made with data obtained earlier for radon-daughter inhalations in the same strain of rats. A simultaneous maximum likelihood analysis is applied jointly to all experimental groups from the radon inhalation and the fission neutron study. The dependence of the resulting coefficients for the different groups on absorbed dose or inhalation dose permits a derivation of equivalence ratios. At low doses the equivalence ratio is 3 WLM (working level months) of radon-daughter exposure to 1 mGy of fission neutrons. At higher doses the equivalence ratio decreases. The neutron data are also utilized to derive mortality-corrected lifetime incidences of pulmonary carcinomas in the exposed animals. At low doses the relation is consistent with linearity, but sublinearity (dose exponent less than 1) cannot be excluded.

Induction of mammary neoplasms in the ACI rat by 430-keV neutrons, X-rays, and diethylstilbestrol

Mammary tumorigenesis was studied in female ACi rats after treatment with X-irradiation or neutron-irradiation, with or without diethylstilbestrol (DES) treatment. The mortality-corrected cumulative tumor rate based on all mammary neoplasms and the mortality-corrected incidence based on the first neoplasms only have been derived. In non-DES-treated animals, at the relatively high radiation doses studied, all dose-effect relationships were consistent with relative biological effectiveness (RBE) values slightly in excess of 10. In DES-treated rats definite findings were observed at neutron doses as low as 0.01 Gy (1 rad). The dose-effect relationship in DES-treated rats showed a strong sublinearly (dose exponent less than 1) at low neutron doses. RBE values in DES-treated rats increased in inverse proportion to the square root of the neutron dose, and exceeded 100 at a neutron dose of 0.01 Gy (1 rad).

Induction of mammary neoplasms in the Sprague-Dawley rat by 430keV neutrons and X-rays

Female noninbred Sprague-Dawley rats were exposed to single doses of 0.28, 0.56, and 0.85 gray (Gy = 1 J/kg or 100 rads) of X-rays or 0.001, 0.004, 0.016, and 0.064 Gy of 430-keV neutrons at 62 +/- 1 days of age and were then observed over the rest of their lives for the appearance of mammary neoplasia. As mammary neoplasms were detected, they were removed and given a classification of adenocarcinoma(s) (AC) or fibroadenoma(s) (FA) after microscopic study. All irradiated groups exhibited an increased incidence of mammary neoplasia. The tumor rate increased steeply with age of the animals, and the effect of the irradiation could be adequately described as a forward shift in time of the spontaneous incidence. The cumulative prevalence was derived from first neoplasms only, and a formalism was presented that makes it possible to derive the integral tumor rate from all neoplasms in all animals. Mortality-corrected cumulative prevalences and integral tumor rates as a function of age were given for the different doses and separately for FA and AC. The mammary FA response and the total mammary neoplastic response (including both FA and AC) were approximately proportional to the absorbed dose of X-rays or the square root of the neutron dose. The relative biological effectiveness (RBE) of the neutrons increased with decreasing dose and reached values exceeding 100 at a neutron dose of 1 mGy; the single dose of 1 mGy of neutrons produced a significant increase of the tumor rate that corresponded to a foward shift or roughly 35 days of the spontaneous incidence. The AC, taken separately, were subject to considerable statistical uncertainties due to their small numbers. However, their RBE-dose dependence was consistent with that for the FA and, even at the highest dose studied, the RBE value exceeded 10. The nonrandom development of multiple FA within individual animals appeared to be the result of differences in susceptibility to radiation. However, mammary FA and AC within individual animals were not statistically correlated.