Many mathematical models have been developed for estimating the cancer risk resulting from an exposure to ionising radiation. One such set of models is developed by the BEIR (Committee on the Biological Effects of Ionizing Radiations, BEIR 2006). The BEIR VII committee has derived risk models both for cancer incidence and for cancer mortality. The models take into account the cancer site, sex, age at the exposure and attained age. Presently, low dose rates and small doses are believed to yield a relatively lower cancer risk compared to high dose rates and large doses. This reduction in risk is accounted for by the dose and dose rate reduction factor (DDREF).

Age-dependent mortality rates are used for subsequent assessment of lifetime cancer risk. Risk models are presented for leukaemia, solid cancers in some organs and for all solid cancers combined. For all these cancer types the BEIR VII committee derived absolute and relative risk models: in the absolute risk model excess cancer risk from radiation is independent of the background cancer risk (i.e., cancers from other causes than radiation) and in the relative risk model the radiation risk is proportional to the background cancer risk.

Many factors, e.g., limitations in epidemiologic data for radiation-induced cancer, contribute to the uncertainty of risk estimation. The BEIR VII committee suggests that the risk estimates should be regarded with a healthy scepticism, placing more emphasis on the magnitude of the risk. The committee estimates that the excess cancer mortality due to radiation can be estimated within a factor of two (at 95% confidence level). For leukaemia the corresponding factor is four. For individual solid cancer sites the risk estimation may have large uncertainties, up to an order of magnitude or more (BEIR 2006).

For solid cancers the models of the excess relative and absolute risk (ERR and EAR, respectively) at attained age t are of the form (BEIR 2006)


where e is age at exposure, e* = (e - 30 a)/10 a when e < 30 a, and e* is equal to zero for e ≥ 30 a. Attained age is t, and D is the organ or tissue equivalent dose. betaS, gamma and eta are fitting parameters of the model

For leukaemia the ERR and EAR models are of the form


where t - e is the time elapsed after the exposure and D is the equivalent dose in bone marrow. theta, delta and phi are fitting parameters.

The relative risks are non-zero only after a latency period. The BEIR VII committee assumes that solid cancers have a latency period of 5 years. For leukaemia the latency period is 2 years. These values are used in PCXMC as a default.

In the risk assessment model of BEIR the relative and absolute risk models are given the weights of 0.7 and 0.3, respectively (the weighting is done on a logarithmic scale as suggested by BEIR (2006)). For the lung cancer these weights are reversed. For the breast cancer only the absolute model is used and for the thyroid cancer only the relative risk model is used. The DDREF division is done after the weighting.

The excess risk values are the basis of the lifetime risk estimates. The lifetime risks can be assessed with various quantities. This software uses three different quantities:

  • Risk of exposure-induced death (REID)

  • Loss of life expectancy (LLE)

  • Loss of life expectancy per radiation induced fatal cancer (LLE/REID).

The definitions of these quantities are (Thomas et al. 1992)




Here, µc(t | e, D) is the mortality rate at age t due to death cause c, given that the subject was alive at the age of exposure e and the corresponding dose at that age was D. In the excess relative risk model


where µc(t) is the background mortality rate related to the death cause c. In the excess absolute risk model


S(t | e, D) is the conditional probability that the subject is alive at age t, given a dose D at the age e. For an unexposed subject the probability of surviving to age t is S(t | e). These conditional survival functions are calculated from mortality statistics, cancer mortality rates and, for S(t | e, D), the risk models. Specifically,


where µ(t | e, D) is the death rate for all causes combined. Thus, the software accounts for the reduction in S caused by the radiation exposure, and the site-specific REID estimates for different cancers can be added. The sum obtained reflects the total risk from the exposure in question. The lower limit in the integrals (8) and (9) is tau  = e + lambda, where lambda is the latency period in years. In PCXMC, the upper integration limit has been set to 120 years instead of infinity in these equations.

The concept of REID originates from cohort analysis techniques: death rates in the exposed and in the unexposed cohorts are compared. A null hypothesis corresponds to a statistically negligible difference between the two groups. A positive REID value indicates excess deaths in the exposed cohort.

The loss of life expectancy (LLE) is the difference between the expectation of life for a person exposed at age e and of an unexposed person who was alive at that age. LLE/REID describes the average length of life lost per excess cancer death. More information on the quantities can be found in Thomas et al (1992) and Vaeth and Pierce (1990).

The BEIR VII -committee simplified the definition of REID by replacing S(a | e, D) by S(e). The resulting risk estimate was called the lifetime attributable risk (LAR) (BEIR 2006). Also, the concept of excess lifetime risk (ELR) is sometimes used in the literature. For practical purposes, at typical dose levels encountered in x-ray diagnostics, REID, ELR and LAR can be interpreted to present the excess radiation-induced cancer risk. Their numerical values are close enough to be interpreted identical considering the uncertainties involved in the models.

The BEIR VII -committee recommends to use the dose and dose rate reduction factor value DDREF = 1.5 for solid cancers and DDREF = 1 for leukaemia. By default, these values are used in this software. If the equivalent doses are large (several tens or hundreds of mSv), the risk estimates should be regarded with care: it might be more appropriate to use DDREF = 1 also for solid cancers in such cases. This can be achieved by multiplying the solid cancer REID estimates by 1.5. Moreover, the risk models describe the stochastic effects of ionising radiation – if deterministic effects are possible, the risk estimates should be interpreted carefully.

The necessary input data for the calculation include the sex-specific mortality and cancer incidence rates. In the default data sets in PCXMC these data are mostly given in five year intervals and are assumed to be constant within the age intervals. The Finnish mortality data are from Statistics Finland databank (  and the cancer mortality and incidence data are from Finnish Cancer Registry (, retrieved on  March 20, 2007. The Euro-American and Asian mortality and cancer incidence data are from ICRP publication 103 (ICRP, 2007). Chronic lymphocytic leukaemia is believed to be unrelated to radiation exposure. Therefore it is excluded from the ICRP 103 leukaemia mortality data. The Finnish mortality and incidence data include all leukaemia types, leading to a slightly overestimated leukaemia risk. This overestimation is much less than the uncertainty present in the risk model for leukaemia.

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