Radiation Protection Dosimetry Advance Access originally published online on August 8, 2008
Radiation Protection Dosimetry 2008 131(3):316-330; doi:10.1093/rpd/ncn181
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Uncertainties in internal doses calculated for mayak workers—a study of 63 cases
1 Los Alamos National Laboratory, Los Alamos, NM, USA
2 Southern Urals Biophysics Institute, Ozersk, Russia
* Corresponding author: guthrie{at}lanl.gov
Received March 4, 2008, amended June 2, 2008, accepted June 10, 2008
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ABSTRACT |
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TOP
ABSTRACT
1. INTRODUCTIONThis study makes use of 63 cases of Mayak workers exposed to Pu-239 with autopsy data and some late-time urine bioassay data. In addition, air-concentration data—used to construct monthly average values—are available for each case, which provide the time dependence and potential magnitudes of normal inhalation intakes for each case. The purpose of the study is to develop and test Bayesian methods of dose calculation for the Mayak workers. The first part of the study was to quantitatively characterise the uncertainties of the bioassay data. Then, starting with three different published biokinetic models, the data are fit by varying intake and model perturbation parameters, e.g., parameters influencing the lung, thoracic lymph nodes, liver and bone retention. Statistical self-consistency arguments are used to check the measurement uncertainty parameters within the Poisson–lognormal model. The second part of the study is to set up and test Bayesian dose calculations, which use the point determinations of biokinetic parameters from the study cases within a discrete, empirical Bayes approximation. The main conclusion of the study is that these methods are now ready to be applied to the entire Mayak worker population.
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1. INTRODUCTION |
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In the effort to increase the accuracy and reliability of epidemiological risk estimates, it is important to calculate the uncertainties for the doses used in an epidemiological analysis. Internal dose is inferred from bioassay measurements using biokinetic models. This process is indirect and it is well recognised that the calculated internal doses have considerable uncertainty. The dominant uncertainty is sometimes the uncertainty (or variability) in the values of biokinetic model parameters, which can be studied by detailed comparisons of model predictions with data for a collection of representative cases with extensive data. This paper discusses such a comparative study, which in turn determines Bayesian prior probability distributions for biokinetic model parameters. Examples of dose calculations using such biokinetic prior distributions will be presented.
This study makes use of 63 cases of Mayak workers exposed to Pu-239 with complete autopsy data and some late-time urine bioassay data. In addition, air-concentration data—used to construct monthly average values—are available, which provide the time dependence and potential magnitudes of normal inhalation intakes for each case. Starting with a basic biokinetic model, the data are fit by varying intake and model perturbation parameters, e.g., parameters influencing the lung, thoracic lymph nodes, liver and bone retention. Point determinations of each of the fitting parameters are made for each case by minimising the appropriate 
2 function. As a result, biokinetic parameter values for each case are determined under various assumptions (data uncertainties, number of parameters varied, etc.) for three published biokinetic models. The results of these calculations are captured in collections of biokinetic model tables in a standard format that are used as the biokinetic prior input files for Bayesian dose calculations. The Bayesian dose calculations use the same codes that have been used for large-scale calculations of the Los Alamos plutonium workforce for over a decade.
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Using these biokinetic priors, a prior distribution on intake amount obtained from the air-concentration measurements, and the likelihood functions associated with the measured quantities, the posterior probability distribution of tissue doses up to time of death may be calculated. The use of different biokinetic modelling schemes allows an estimation of dose uncertainty caused by the lack of knowledge of the biokinetic prior. These numerical results can be used to probabilistically generate tissue doses from their posterior distributions for various biokinetic priors for use in epidemiological studies.
The Bayesian methodology answers the question ‘Given the data and using agreed-upon forward models, what is the dose, including uncertainty?’ It also allows straightforward hypothesis testing, e.g., given the data, what is the posterior probability of model A versus model B?
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2. MAYAK TEST DATA SET |
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The data set used in this study consisted of urine bioassay data and post mortem tissue data of 63 former Mayak workers(1–3). No association between radiation and primary cause of death was observed in any subject in the first group of 42 cases. The cases of death of the remaining 21 cases, however, were lung cancer (8), liver cancer (8), bone-related cancers (4) and connective tissue cancer (1). The subjects in the first group of 42 were male (32) and female (10), smoker (21) and non-smoker (21). The subjects had worked in workplaces dealing with plutonium reprocessing, conversion and metal finishing. Complete data were available for 41 and 18 cases in the two groups, respectively.
The autopsy data included the lung, pulmonary lymph nodes, liver, skeleton and other body (including soft tissues). No uncertainties were given, although urine bioassay data consisted of a series of replicate measurements on successive days that allowed calculation of mean and standard deviation of the mean for each observation. An example of bioassay data as provided by the Southern Urals Biophysics Institute (SUBI) is shown in Table 1.
The workers' age at start of exposure varied from 18 to 43, with a median of 25. The number of years of exposure varied from 1 to 43, with a median of 18. The total number of individual urine measurements was 1126. The number of composite urine bioassay measurements was 211, with a median number of observations of 3 to 4.
Other very important data consist of a work history for each worker divided into time periods where work was done at specified locations, together with the time fraction at each concurrent location. The measured or estimated air concentration at each location during each time period(4) is used to estimate the monthly intake, taking into account respiratory protection and respiratory zones (position of air monitor versus worker's position). This information used to calculate nominal quantitative inhalation intakes as a function of time for each worker.
The basic biokinetic model calculation is for a single acute intake. The time pattern of actual intakes is represented numerically as a succession of acute monthly intakes. A lognormal model of uncertainty is assumed, with the measured air concentrations and calculated monthly intakes assumed to represent mean values. The lognormal median is the mean (measured value) divided by the lognormal mean/median ratio A(S) 
exp (S2/2). In this paper, S is assigned a rather arbitrary large value of 4 (GSD = exp(S) = 55). Figure 1 shows an example of intakes determined from air-concentration data for Case 1, which involved work from only 1949 to 1951.
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For simplicity, the monthly intake data are assumed to be independent with the assumed lognormal S, even though independence is unlikely for intakes at this time resolution. These data (1) define the time profile of intake and (2) when summed, define the prior probability distribution of total intake amount (referred to in this paper using the symbol I), which is again assumed to be lognormal. The time profile of the intake is assumed to be constant, and the intake uncertainty is assumed to result only from the uncertainty in the total intake amount. In summing monthly intakes, the mean and variance of the sum of all intakes is calculated and used to characterise another lognormal, which is the prior probability distribution of total intake amount. As can be understood from the central limit theorem of statistics, the sum-of-intakes distribution becomes normal (S small) in the limit as more and more intake components are summed.
For the analysis presented here, the time profile of the intake and the uncertainty of the total intake amount are needed. The fact that the continuous intake is approximated as a succession of monthly acute intakes makes no difference when the times of interest in the biokinetic modelling are well separated from the times of intake. Regarding the uncertainty parameterisation, perhaps the simplest parameterisation would have been to assign the same lognormal uncertainty S to the total intake amount for all cases. This would miss the fact that for cases with many air-concentration measurements, the uncertainty should be less, because of the effect of averaging independent quantities. What has been instead done, assuming independent intakes for convenience on the same month-by-month time scale used to characterise the intake amounts, is perhaps somewhat better, but certainly not accurate either. Because of the potential importance of the air-concentration measurements, this work clearly should be iterated with a more accurate uncertainty parameterisation. This will involve identifying the actual independent measurements that contribute to the total intake and characterising their relative uncertainties. Information as shown in Figure 1 is available for all 63 cases. Use of this air-concentration data may be very important in the future because it might be used to estimate doses in lieu of any other data(5).
For Case 1, the mean total intake from the data shown in Figure 1 is about 4 MBq. As discussed in Section 9, the lognormal S describing uncertainty and variability of the air-concentration intakes is assumed to be 4. For Case 1, the total intake then has S = 3.64 (S of the sum-of-intakes lognormal). This distribution of total intake may be used as another data point in the minimum-2 determinations of biokinetic model parameters, although this paper does not investigate this possibility further. The distribution of total intake determined in this way is also used as a prior probability distribution of intake amount for Bayesian dose calculations.
In addition to the intake amount as a function of time, Figure 1 shows intake material characterised by the measured Khokhryakov transportability parameter SKhok (6). This provides information about the type of material involved in terms of solubility in the lung fluids. At present, the cases were divided into three categories (A, B and C), as is discussed in Appendix A, although this information is currently not being used to refine dose estimates.
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3. MINIMUM 2 FITTING
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The determination of the biokinetic model parameters used the first group of 41 complete cases that did not die of radiation-related diseases. The approach of this study is to start with an accepted plutonium biokinetic model that defines the basic model structure and serves as the unperturbed model. As the number of bioassay data are relatively small (compared with the total number of biokinetic parameters), only a small number of parameters can be determined from the data. Multiplicative perturbation parameters, e.g., flung, fLN, fliver and fbone, denoted collectively by the symbol f, are defined as that influence the retention in the lung, lymph nodes, liver and bone (value of 1 implies no change). The data are used to determine the values of these parameters and the intake amount I by the condition that
2 be a minimum with respect to variations of these parameters. The Poisson–lognormal uncertainty model is used(7). The values assumed for the lognormal geometric standard deviation are shown in Table 2. The justification for these choices will be discussed in Section 6.
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For the autopsy data, the measurement uncertainty is assigned a nominal value of
minmeas as given in Table 2. For the urine data, the measurement uncertainty is taken as
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| (1) |
irep is the standard deviation of the replicate measurements (shown in Table 1 for Case 1) increased by a factor Frep = 2 because of the small number of replicates used (median value of 3–4 replicates for the 42 cases)(7) and minmeas = 10 mBq as given in Table 2.
If the measurement uncertainty is small compared with the measurement value,
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2 is assumed to be
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| (3) |
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| (4) |
If the measurement uncertainty is large compared with the measured value, and inequality (2) is not satisfied(6),
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| (5) |
itot is given by
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| (6) |
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| (7) |
This uncertainty model can be used with the normal–lognormal version of the exact Poisson–lognormal likelihood function(8). A more unified approach might be to do straightforward maximum-likelihood fitting, making use of analytical approximations to the exact Poisson (or normal)–lognormal likelihood function in order to calculate the likelihood function. However, the quantitative interpretation of 2/NDF, where NDF is the number of degrees of freedom (number data minus number parameters) that follows, would be missing using the pure maximum-likelihood approach.
Intra-individual uncertainty is calculated for each case using Monte Carlo generation of a large number (usually 30) of data sets about the initial fit values using the assumed normal–lognormal distributions:
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imeas the measurement uncertainty standard deviation. The nonlinear fits were repeated for all these alternate realisations of the data in order to obtain an estimate of fit parameter variations caused by measurement and normalisation uncertainties. |
4. BASIC BIOKINETIC MODELS AND PERTURBATION PARAMETERS |
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The zeroth-order biokinetic model of the lung for the Leggett and Luciani models was the Human Respiratory Tract Model (HRTM)(9) with its Type S default absorption parameters. All mechanical removal rates and all absorption rates from the HRTM were kept constant at the default values, except those shown in Table 3.
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A more complex model structure (e.g. introducing a bound compartment in the lung(10)) can certainly be used, although it was thought to be unnecessary for this initial study. An additional type-M intake component has also been investigated(11), which adds one additional parameter—the intake amount for the type-M component. The choice of biokinetic model perturbation scheme involves expert judgement and another iteration of this work involving other experts might well use a somewhat different parameterisation. The different dose results that would be obtained represent the effect of uncertainty of the biokinetic prior probability distribution.
The systemic models considered are the Leggett (2005) systemic model(12), the Luciani and Polig (2000) systemic model(13) and the Khokhryakov model(6,14). The Luciani and Polig model is similar to Leggett's model, but has a single blood compartment rather than two. Khokhryakov's model (see Appendix A) is a simpler, empirically based model. The basic models are modified by varying rate constants, either singly or groupings of rate constants are varied in concert.
The nonlinear fitting method used required unconstrained parameters. The constraint of positivity is easily incorporated using a exponential transformation (x unconstrained, exp(x) positive). For the Leggett and Luciani models, two positive blood uptake parameters fliver and fbone, are utilised in such a way that the total transfer from blood to all tissues is constrained to be constant.1 The total transfer rate from blood to tissues, given by
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| (9) |
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| (10) |
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To modify the metabolism of liver and bone, fliver and fbone were varied and at the same time, but inversely, rates involving long-term liver and bone retention, as shown in Table 5. Other possibilities, such as having an inverse multiplier with some other power (e.g., fliver–2 rather than fliver–1), were also investigated, but the scheme shown in Table 5 seemed to provide the best results for the fits with zero degree of freedom that will be discussed in Section 5. Again, expert judgement is involved. It would seem that the scientifically justified approach would be to have independent experts suggest model perturbation schemes, eliminate those that do not provide good enough fits with zero degrees of freedom and assume the resulting biokinetic priors are equally likely realisations to be used in averaging dose results.
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5. SUMMARY OF FIT RESULTS—LEGGETT, LUCIANI AND KHOKHRYAKOV MODELS |
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The fitting process involved varying parameters to minimise
2. Software to accomplish this was especially developed for this project and is described elsewhere(11). The fitting without a type-M intake component involved varying up to five parameters: intake amount, flung, fLN, fliver and fbone, and with these five parameters it should theoretically be possible (barring a singularity of the matrix JTJ, where J is the Jacobean matrix of partial derivatives of calculated data with respect to parameters) to fit five data points exactly. The data points chosen for fits with zero degree of freedom (NDF = the number of data points minus the number of parameters = 0) were the last (positive) urine and the lung, lymph nodes, liver and bone autopsy data. The remaining data, unused for zero-degree-of-freedom fits, were the ‘other tissues’ autopsy data and additional urine data.
For the Khokhryakov model, which is discussed in Appendix A, there were three fit parameters: intake, Khokhryakov transportability and transfer to lymph nodes. In the Khokhryakov case, the fitted data were urine data and lung, lymph node and systemic autopsy data.
In the fitting procedure, two independent fits were made with different parameter starting values for the nonlinear minimisation to check independence of starting values. Mostly, the final 2/NDF obtained was nearly the same. In any case, the fit with the smaller value of 2/NDF was selected for use.
Nine different modelling approaches are summarised in Table 6 for 41 cases with complete data where the cause of death was not radiation-related.
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Fits 0 and 8 used the Leggett systemic model without perturbation. Fit 8 had an additional ICRP-66 type-M intake component. Figures 2 and 3 show the distribution over 41 cases of values of long-term lung absorption and mechanical transport to the lymph nodes for fits 0 and 8.
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Fits 2 and 5, using the Leggett and Luciani models, involved zero degrees of freedom and functioned as a verification of the parameterisation scheme for perturbing the basic biokinetic model and the nonlinear fitting process itself. For NDF = 0,
2/NDF in Table 6 means just 2. In these cases, the data were fit perfectly, as expected. Fits 1, 3 and 4, using Leggett, Khokhryakov and Luciani models used the same parameters, but by including all data had at least one degree of freedom (average of 2.6 degrees of freedom).
Fits 6 and 7 are similar to fit 0 except in regard to the assumed intake profile. Fit 6 used the Khokhryakov exponential fits to intake profile(6). Fit 7 assumed that the entire intake occurred as a single acute intake at the beginning of work.
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6. DISCUSSION OF UNCERTAINTY PARAMETERS |
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This paper focuses on the lognormal parameterisation of the normalisation uncertainty for tissue measurements and the effect of having a small number of replicate samples for urine measurements. A discussion of measurement uncertainties for this data is given by Krahenbuhl et al.(15).
The fact that the values of 2/NDF shown in Table 6 are consistent with 1 demonstrates the self-consistency of the assumed uncertainty parameters. Also that fit 0 (no inter-individual variability of systemic model) is equally as good as fit 1 suggests that variability of systemic model parameters between individuals in the population is small compared with normalisation and measurement uncertainty.
To further investigate the hypothesis that, for the autopsy data, the variability between individuals in the population is small compared with normalisation and measurement uncertainty, distributions of the ratios of liver/bone, other/bone and other/liver are made for the 41 cases without radiation-related disease,2 which should then reflect normalisation and measurement uncertainty, e.g.,
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| (11) |
The distributions for the three ratios are shown in Figure 4 (the straight line implies lognormal distribution).
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The solutions of the three simultaneous equations similar to Equation (11) gives values comparable to the S value assumptions shown in Table 2.
The other/bone curve is the poorest fit to a lognormal; however, the definition of ‘other’ tissues was variable, and in some cases more tissues were measured and included than in others.
Other evidence that implies that inter-individual variability of liver and bone can be neglected is shown in Figures 5 and 6. Figure 5 shows the inter-individual variability for 41 cases of the blood-to-liver and blood-to-bone transfer fractions xl and xb, defined by Equation (10).
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Figure 5 shows that the unperturbed Luciani and Polig model is not centred relative to this population, which accounts for it not producing statistically satisfactory fits. In Figure 6, the uncertainty of liver and bone fractions for Case 1, obtained by repeating the fits for 30 alternate realisations of the data, is shown.
From Figures 5 and 6, it is apparent that uncertainty is approximately the same as the variability plus uncertainty, which implies that variability of these parameters is relatively small in this population. This is one of the important conclusions of this study that the unmodified Leggett 2005 systemic model is adequate to describe Mayak workers unaffected by radiation-related disease.
In the remainder of this section, the uncertainty of urine measurement data is discussed. The uncertainty of urine measurement data is obtained from the standard deviation of the average of the replicate samples. Using this standard deviation from the replicate data directly, large values of 2/NDF were observed. A theoretical/Monte Carlo study(6) has shown that this is to be expected because the number of replicate data is small (the median of number of replicate measurements is 3–4). The standard deviations actually used are calculated from Equation (3).
As recommended in the above-mentioned reference(6), two distributions need to be examined. Figure 7 shows the distribution of the replicate data standard deviation for the urine data (all 63 cases) (before multiplying by factor Frep = 2) together with the parameter minmeas from Table 2 appearing in Equation (3).
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Similarly, Figure 8 shows the distribution of the standard deviation of the natural logarithm of the replicate urine data (all 63 cases) together with the parameter S from Table 2.
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These plots show that only a small fraction of cases would be strongly affected by the replacements of the parameters S = 0.1 and
minmeas = 10 mBq/day in Equation (7) with 0. The effect of non-zero values of these parameters is to improve the statistical self-consistency of the urine data as discussed in Ref. (6). |
7. BAYESIAN METHOD |
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In the Bayesian method, rather than making point estimates of the inferred quantities (e.g., tissue doses) by minimising
2, probability distributions are calculated using Bayes theorem. The choice of biokinetic model, which determines the biokinetic model function, Fi (f), is very important. Sometimes the uncertainty surrounding biokinetic model assumptions dominates all other sources of uncertainty.
Without loss of generality, the possible choices of biokinetic model parameters f may be enumerated by an integer biokinetic type index l. In Bayesian statistics, a prior probability distribution that defines the ‘universe of possibilities’ is necessary for all parameters. Optimally such a prior probability distribution is itself based on data. The 41 minimum 2 determinations of biokinetic model parameters f summarised in Table 6 form the basis for such prior probability distributions of biokinetic type. The posterior distribution of intake and biokinetic type (I,l) given the data is then determined from the bioassay data using Bayes theorem.
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| (12) |
As the bioassay data are independent, the combined likelihood function is given by the product of the likelihood functions for the N individual measurements:
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| (13) |
By definition, the likelihood function for the ith measurement is
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The discrete prior probability distribution of different values of l in Equation (12) is assumed to be constant. The prior probability of intake amount I is the lognormal distribution discussed in Section 2.
The posterior probability of any function of the parameters may be calculated by one-dimensional integration or summation. For example, the posterior probability of biokinetic type l is given by
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| (15) |
The posterior probability of dose D is given by
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| (16) |
The expectation value of the dose is given by
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| (17) |
Dose calculation software quality assurance test cases are given on the Los Alamos National Laboratory (LANL) Bayesian web site(18).
In Table 6, several different hypotheses about biokinetic models are considered. A natural question is, ‘Which one of these modeling hypothesis is best supported by the data?’. A data set that applies uniformly to each of these cases is urine data and lung, lymph node and systemic autopsy data for the 41 cases without evidence of radiation-related disease. Applying Bayes theorem with a uniform prior on hypotheses and this data set, the posterior probability of modelling hypothesis Hk is determined by the relative size of the denominator in Equation (17)(19,20):
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| (18) |
The above equation is applied using the data for each of the 41 cases individually, and the averages and standard deviations of the 41 results are as shown in Figure 9.
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The data for all hypotheses shown in Figure 9 consist of all urine, undifferentiated systemic body burden, lung and lymph nodes. The ICRP-60 approach uses type M and S and particle sizes of 1, 5 and 10 µm AMAD (six biokinetic models). The ICRP approach is characterised by large excursions in goodness of fit, some of the 41 cases fit well and others very poorly. The conclusion from Figure 9 is that the ICRP approach is not as good as other approaches, and of these, approach 8 (additional type-M intake) seems best.
The comparison of the Bayesian posterior expectation value and the urine measurements is as shown in Figure 10 for Case 1 using all data (biokinetic prior from modelling approach 8 leg).
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The discreteness of the assumed monthly intakes from Figure 1 is unrealistic, but makes no difference in calculations of bioassay quantities well removed in time.
The posterior probability of biokinetic type is given by Equation (13). The calculated posterior distribution of biokinetic type for Case 1 is shown in Figure 11.
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Biokinetic type index is the original case number used in the point determinations of biokinetic parameters. The case under consideration is Case 1, which quite naturally has the highest posterior probability. With weak bioassay data, the posterior probability of biokinetic type is the same as the prior probability and is completely uniform.
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8. DOSE CALCULATIONS—EXAMPLES |
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An example of a dose calculation is shown in Figure 12. The dose quantity is the cumulative absorbed dose to lung alveolar-interstitial (AI) tissues up to time of death. Two data sets are considered: all data and using only the last urine data point. The prior probability distribution of intake amount is obtained from the air-concentration data shown in Figure 1 (mean total intake = 4 MBq, S = 3.64, time profile as shown). Cumulative posterior probability distributions are shown in Figure 12 calculated using the Los Alamos ID code(21).
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It is clear that the dose uncertainty depends on the amount and quality of bioassay data and that it can be quite large. However, for basic epidemiology, what is needed is the collective posterior mean dose(22) for a study cohort. With a given prior, the posterior mean dose is a single number without uncertainty. In order to illustrate uncertainties relevant for epidemiology, Table 7 shows the collective posterior mean tissue doses for the 41 cases using different biokinetic priors and four tissues. These calculations were done using the Los Alamos UF code(18).
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Table 8 is similar to Table 7, but it shows collective cohort tissue doses when only the last urine data point is used. Often, only one or perhaps a few urine bioassay data are available for the epidemiology study.
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As these examples show, the cohort dose biases may be quite significant. The way to reduce bias and uncertainty is to obtain new high-precision data or to narrow the prior probability distributions by making use of additional information, such as subject health status(23) and refined workplace information.
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9. EFFECTS OF RADIATION-RELATED DISEASE |
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Table 9 shows the fit results for the additional cases not considered so far where the cause of death was radiation-related disease. The same modelling approaches, parameters and data as given in Table 6 were used.
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The striking feature of this data is that none of the modelling approaches are satisfactory as evidenced by the large values of
2/NDF. Table 10 shows a comparison of liver/bone ratios directly from autopsy results between groups of cases with no radiation-related disease and radiation-related disease. The liver metabolism is much different for the second group(23).
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The conclusion is that dose assessments for cases with radiation-related disease will require specialised biokinetic models.
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10. DISCUSSION |
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This work is concerned with developing and testing dose and dose uncertainty calculation methodologies for use in the Mayak epidemiology studies. To determine a Bayesian prior on biokinetic parameters, the approach is to select a basic biokinetic model and a perturbation scheme. Using a data set with complete data for each case, possibly subdivided into externally defined categories (e.g. radiation-related disease or not, smoker/non-smoker, male/female), minimum
2 fits for each case are made. The collection of these biokinetic model fits constitute a discrete, empirical Bayes prior for biokinetic type that can be used in Bayesian dose calculations for other situations that are judged to be similar. A number of these biokinetic priors have been constructed using different assumptions, such as basic biokinetic model and model perturbation scheme, in order to allow quantitative estimation of the effect of uncertainty in regards to the biokinetic prior. This uncertainty of the prior is discussed further in Ref. (22). When no bioassay data or only late-time urine data exist, the posterior dose distributions using these priors will be quite broad. These uncertainties and related posterior mean biases need to be taken into account in epidemiology studies. The only way to reduce uncertainty and bias is to have more or better bioassay data or a more accurate and informative prior. A more informative prior might be achieved by subdividing the populations so that smaller variations exist (e.g. by segregating by workplace and categorising material solubility in lungs) or by other means.
In this paper, the Bayesian biokinetic prior has been simplified by allowing variations only of certain key biokinetic parameters and doing point determinations of these parameter values. This is not the ultimate Bayesian method, which would allow all parameters to vary and, using the data from these test cases, determine their joint posterior distribution. This would be a daunting project at this time. The empirical Bayes method employed here allows an initial assessment of the effects of variable and uncertain biokinetics, as well as measurement uncertainty, on the epidemiology.
Because true values of internal dose that are needed for epidemiology are not directly measurable, the estimation of these true values is complex and indirect, and subjectivity is usually involved in estimation of the prior probability distributions, it would seem to be important to have estimates of internal doses done independently by different groups.
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FUNDING |
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This work was part of the United States–Russian Joint Coordinating Committee for Radiation Effects Research (JCCRER) Project 2.5 and was funded under a Cooperative Agreement with the United States Department of Energy Office of International Health Programs (HS-14), Health Safety and Security Division (HSS). Funding to pay the Open Access publication charges for this article was provided by the United States Department of Energy contract for the management and operation of Los Alamos National Laboratory.
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APPENDIX 1 |
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KHOKHRYAKOV MODEL
The Khokhryakov model is attractive relative to the other models discussed here in that it has a smaller number of parameters, all of which are rather directly related to actual measurement data. This model is primarily based on the plutonium injection studies(5). The starting point is an expression for the urine and faecal excretion rates as a function of time following an injection of a unit amount of plutonium. A five-term exponential fit is used as follows:
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| (A.1) |
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Conservation of material implies that the total time-integrated excretion is equal to the injected amount
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| (A.2) |
A compartmental model that reproduces these excretion rates has five compartments with input transfer fractions ai/xi and transfer rates to accumulated urine of xi and another five compartments with input transfer fractions bi/yi and transfer rates to accumulated faeces of yi. This 10-compartment model serves as an empirical systemic model, representing retention and excretion of material introduced into the blood.
The complete Khokhryakov model includes three more compartments representing (1) slowly clearing material in lungs, (2) pulmonary lymph nodes and (3) material that is fixed in the lungs. The 
pathway (see Figure A1) represents fast lung clearance to blood. Early time faecal excretion, involving transfer of material from airways to gastrointestinal tract, is not modelled. The complete model is shown in Figure A1(14).
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In this paper, the three parameters of the lung clearance model have been empirically fit in terms of the single ‘Khokhryakov transportability parameter’ SK (in %), using the results given in Table 4 of Ref. (5):
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| (A.3) |
Transportability was experimentally determined from the dynamics of alpha-activity dialysis through a semi-permeable membrane in Ringer's physiological solution. The numeric value of measured transportability, SKmeas, is the percentage of plutonium relative to the total filter activity measured after 2 days of dialysis.
Using the formulas given in Equation (A.3), the model has three parameters, the initial deposition amount, SK, and the transfer to lymph nodes Kn. The fitted values of SK are correlated, as expected, with SKmeas. The correlation is shown in Figure A2 (40 out of the 42 cases are shown).
The cases may be subdivided into categories A, B and C defined by SK 
0.5, 0.5 < SK 1.5 and 1.5 < SK. For the 32 out of the 42 cases, the assignments of transportability based on dissolution measurements and the fit values of SK result in the same categorisation. Results of fitting the 42 cases using the Khokhryakov model are shown in Table A2.
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In the Khokhryakov model, there is a single systemic burden, consisting of the sum of contents of the 10 empirical systemic compartments. This systemic burden is fit to the sum of measured liver, bone and ‘other’ autopsy quantities. Doses to individual organs are obtained by scaling the calculated systemic burden by the factors (from the autopsy data) shown in Table A3 for Case 1 (liver 29.7%, bone 64.7%, other 5.6%). Red Marrow is always assumed to have 0.77% of the systemic burden.
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The reason for the factors of 0.25 is the ICRP-30 bone-dosimetry assumption(24) that 25% of the bone surface alpha decays deposit energy in bone surface and 25% deposit energy in Red Marrow (half of alpha activity in trabecular rather than cortical bone and that fraction deposits half of its energy in Red Marrow).
Other parameters important for dosimetry calculations with the Khokhryakov model are shown in Table A4. For the Leggett and Luciani models, the SEE matrices as calculated by the SEECAL program were used(25).
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The energy absorbed per decay is assumed to be 5 MeV using a quality factor of 20 for equivalent dose.
Table A5 shows a comparison with DOSES2000(26) of cumulative absorbed dose until death for systemic organs calculated using the Khokhryakov model. The average values of the ratio over 41 cases without radiation-related disease (for which DOSES2000 doses were available) are close to unity.
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ACKNOWLEDGEMENTS |
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G.M. and R.G. acknowledge helpful discussions with the members of the National Council on Radiation Protection Committee considering uncertainties in internal dosimetry, in particular, Iulian Apostoai, Alan Birchall, Andre Bouville, George Sgouros, Owen Hoffman, Tony James, Kim Kearford, Rich Leggett, David Pawel, Gus Potter, Dick Tooey and Wes Bolch. The work has been continuously reviewed and approved by Institutional Review Board (IRB) of the Southern Urals Biophysics Institute and subsequently the Human Subjects Research Review Board (HSRBB) of Los Alamos National Laboratory to ensure the welfare and privacy of the human subjects.
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FOOTNOTES |
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1 Suggested by R. Leggett.
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[Abstract/Free Full Text] - Khokhryakov V. F., Suslova K. G., Vadim V. V., Romanov S. A., Zoya S. M., Kudryavtseva T. I., Filipy R. E., Miller S. C., Krahenbuhl M. P. The development of the plutonium lung clearance model for exposure estimation of the Mayak Production Association, Nuclear Plant Workers. Health Phys (2002) 82:425–431.[CrossRef][Web of Science][Medline]
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