Analytical and quasi-Bayesian methods as development of the iterative approach for mixed radiation biodosimetry

Analytical and quasi-Bayesian methods as development of the iterative approach for mixed... The present paper proposes two methods of calculating components of the dose absorbed by the human body after exposure to a mixed neutron and gamma radiation field. The article presents a novel approach to replace the common iterative method in its analytical form, thus reducing the calculation time. It also shows a possibility of estimating the neutron and gamma doses when their ratio in a mixed beam is not precisely known. Keywords Biological dosimetry · Bayesian · Nuclear accident · Cytogenetics · Radiation · Biodosimetry · Dose assessment Introduction The process of dose assessment through a dicentric assay requires the presence of dose–response calibration curves. Radiation biodosimetry refers to the analysis of biologi- Such curves are produced by exposing human blood in vitro cal changes in a particular tissue after exposure to ionizing to several doses of radiation under carefully controlled con- radiation, and allows for an estimation of absorbed doses in ditions. It is implicitly assumed that in vivo and in vitro irra- an exposed person based on this analysis. The quantifying diations of peripheral blood lymphocytes produce similar measurement of chromosome alterations, mainly dicentrics alterations in the cells. Accordingly, the dicentric frequency in lymphocytes, is the most widely used method for biologi- observed in vivo can be converted to absorbed dose by com- cal dose assessment (IAEA 2011). The aim of radiation bio- paring it with the dose–response calibration curve obtained dosimetry by cytogenetic methods is to calculate doses and in vitro. The shape of a dose–response curve is influenced the associated confidence intervals to exposed (or suspected by the linear energy transfer, LET, for any particular radia- to have been exposed) individuals after a radiation accident tion quality. For low-LET radiation, the relation between or incident. Assessing the absorbed dose based on the dicen- absorbed dose and dicentric chromosome frequency can be tric assay in peripheral blood lymphocytes is a very sensitive expressed as a combination of linear and quadratic terms method that can be used when measured doses are not avail- (Eq. 1), while for high-LET radiation the dose dependence is able. The analysis is performed in the lymphocytes during linear (Eq. 2). The linear and quadratic terms are consistent their first mitosis after radiation exposure. The advantage with the single- and two-track model of dicentric formation of the method is the low spontaneous frequency of dicentric by both radiation qualities (IAEA 2011; Kellerer and Rossi chromosome aberrations in healthy individuals, indicating 1978): that this indicator is caused mainly by ionizing radiation. Y D = Y +  D +  D , (1) g g 0 g g * Iwona Słonecka Y D = Y +  D , (2) n n 0 n slonecka@clor.waw.pl where Y (x = g is for gamma and x = n for neutron radiation) Central Laboratory for Radiological Protection, Konwaliowa is the frequency of dicentrics, D is the absorbed dose, Y is x 0 7, 03-194 Warszawa, Poland the background frequency of dicentrics (dose independent), Faculty of Physics, Warsaw University of Technology, α and β are the linear coefficients, and γ is the quadratic Warszawa, Poland coefficient. Ex-Polon Laboratory, Łazy, Poland PGE EJ 1 Sp. z o. o., Warszawa, Poland Vol.:(0123456789) 1 3 196 Radiation and Environmental Biophysics (2018) 57:195–203 In the case of exposure to mixed neutron and gamma radi- irradiation by a mixed neutron and gamma radiation field ation, n + γ (e.g., after a nuclear reactor accident), biological with doses D and D . Parameters Y , α, β and γ are usually n g 0 dosimetry is more complex than in the case of exposure to a found as a result of a regression analysis (defining the so- single type of radiation. Mixed radiation fields are composed called calibration curve). The parameter y can be written of particles with varying biological effectiveness, which as a ratio of u/w, where u represents the number of chromo- have different biological effects on the body. Therefore, some aberrations and w is the number of cells in the ana- there is a strong need to calculate not only the total absorbed lyzed sample. dose, but also its components, D and D , separately. Unfor- Having the fit parameters of the calibration curves, (Y , n g 0 tunately, there is no visible difference between dicentrics α, β, and γ), the Y(D ) functions can be used to estimate the induced by the two types of radiation, and therefore, it is doses (D ) and/or the frequency of chromosome aberrations not possible to directly discriminate between dicentrics pro- (Y ) after exposure to gamma radiation, neutron radiation, or duced by gamma radiation and those produced by neutrons. to a mixed neutron and gamma radiation field. In cases when Discrimination is only possible by quantifying the separate the ratio of neutron to gamma absorbed dose is estimated by dose components and using the additivity assumption in the physical methods (Eq. 4): production of chromosomal damages (IAEA 2011; Kellerer and Rossi 1978).  = , (4) Dose estimation using biodosimetry methods for a mixed neutron and gamma radiation field can be found in the lit- it is possible to calculate the separate neutron and gamma erature (IAEA 2011; Brame and Groer 2003; Szłuińska radiation doses using the iterative method (IAEA 2011) et al. 2005; Fornalski 2014). The most common approaches mentioned earlier. In this method, the doses and the chro- are the classical iterative method, promoted widely by the mosome aberration frequencies are estimated by a manual International Atomic Energy Agency (IAEA 2011), and iterative approach, meaning that the values are precisely the increasingly used Bayesian methods (Brame and Groer determined. 2003; Ainsbury et al. 2013a, b; Pacyniak et al. 2014). The iterative method is adopted in many accredited laboratories Iterative method worldwide, including the Central Laboratory for Radiologi- cal Protection (CLOR), Poland. This method is generally The iterative method involves performing several series well-known due to its simplicity and relative accuracy. of calculations using the same input data. Each consecu- Bayesian methods are also becoming more and more popu- tive series gives more accurate results, until the results of lar. However, they are not yet commonly used probably due the next steps do not change significantly any more. In this to the much more advanced mathematical approach required. method, it is initially assumed that all aberrations found in Thus, some intermediate methods, which will improve the the analyzed sample, y , are caused by neutron radiation, i.e., iterative approach without being as complex as the Bayes- f using the recurrence relation, Y = y . Thus, the neutron ian approach, are developed and tested in the present paper. n f i=0 dose can be defined directly from Eq.  2, as: Y − Y n 0 D = . (5) i+1 Materials and methods th The dose from gamma radiation in the (i + 1) step, D , i+1 Assessment of neutron and gamma doses is calculated with the use of the actual D value and the i+1 neutron/gamma dose ratio, ρ, according to Eq. 6: In practice, it is assumed that the number of dicentrics i+1 in an analyzed sample is characterized by a Poisson dis- D = , (6) i+1 tribution, and that the observed alterations are the sum of those induced by neutron and gamma radiation. Thus, the also formulated as the recurrence equation. With informa- dose–response relationship for a mixed neutron and gamma tion about the gamma dose, the dicentric frequency due to radiation field may be described as a combination of Eqs.  1 gamma radiation, Y , can be obtained from Eq. 1: i+1 and 2: Y = Y + D + D . (7) 2 g 0 g i+1 i+1 g i+1 Y D , D = Y +  D +  D +  D ≡ y = , n+g n g 0 n g g f (3) The dicentric frequency caused by neutrons (Y ) is then i+1 which is usually called a combined linear-quadratic equa- obtained (Eq. 8) by subtracting the gamma dicentric fre- tion for the frequency of chromosome aberrations y after quency from the measured aberrations, y . f f 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 197 Y = y − Y . n f g (8) i+1 i+1 The neutron dose is estimated using Eq. 5 and the above steps are repeated using recurrence equations from Eqs. 5–8. With this algorithm, all parameters are calculated iteratively until their values are stable ( D ≅ D ). x x i i−1 Obviously, the more repetitions done, the more accurate the achieved values will be. All the fitting parameters of the calibration curves (Y , β, γ, and α) as well as the ratio of doses (ρ) should be calculated in advance. However, a significant disadvantage of the iterative method is the large effect of the propagation of uncertainty. The presented itera- tive algorithm is usually time-consuming and requires many Fig. 1 Distributions of gamma and neutron doses for a sample repetitions to obtain final results. Hence, it was proposed to with 53 dicentrics in 19 analyzed cells obtained with the analytical transform the iterative method into its analytical description, method. θ = 0.653 is marked with a dotted line which was originally introduced by Fornalski (2014). Analytical method where x = {g, n}, and  =  ,  ,  ,  ,  ,  are y Y f 0 uncertainties of parameters ε = {α, β, γ, y , Y , θ}. Uncertain- The analytical method (Fornalski 2014; Pacyniak et  al. f 0 2015) was proposed to automate the iterative algorithm used ties of D and D can be calculated from Eq.  11 finding g n partial derivatives of D . for calculating the absorbed doses in mixed radiation fields and exclude the propagation of uncertainty effect. In this Quasi‑Bayesian (Q‑B) method approach, the neutron/gamma dose ratio, ρ (Eq.  4), must be known. However, because ρ varies from zero to infinity, When the ratio of doses is not precisely known, the iterative it can conveniently be replaced by θ (Eq.  9), which cor- responds to the contribution of the gamma dose to the total and analytical methods cannot be used. In this case, the prob- ability density function (PDF) can be utilized to estimate the dose and is more practical in use than ρ. It is normalized to the range of [0, 1] and is defined as (Brame and Groer 2003): most probable value of θ (or ρ). Such a PDF, which is widely used in Bayesian statistics, is called a prior probability p(θ) or g p(ρ). In the most common cases it can be approximated by a = = . (9) Gaussian distribution for θ (or ρ) with a standard deviation of D + D  + 1 g n σ (or σ ), for example: θ ρ With Eqs. 3 and 9, D and D can be simply calculated as g n � � ⎡ ⎤ a function of θ (Fornalski 2014): 𝜃 − 𝜃 ⎢ ⎥ p(𝜃 )= exp − , √ (12) � � � � ⎢ ⎥ 2 2𝜎 2𝜋𝜎 1− 1− 𝜃 𝜃 ⎣ ⎦ + +4 y − Y −  + ( ) f 0 D () = (10) ⎨ 2 1− D () = D () where 𝜃 represents the expected value of θ with the uncer- n g tainty σ . The Gaussian distribution is usually selected because the estimation of θ (or ρ) is done using the classi- It is assumed that all constants (α, β, γ, y , Y and θ) are f 0 cal Gaussian regression method. However, one can also use precisely known from experimental data with proper uncer- other priors reflecting the knowledge on θ (or ρ); this issue tainties. The solution of Eq. 10 can be presented graphically will be discussed later. in terms of dose and θ (Fig. 1). Having sufficient information on the dose ratio, one can Finally, the uncertainties associated with the estimated try to estimate the neutron and gamma doses, as well as the doses,  , in the presented method can be calculated using dicentric frequencies. To enhance the classical method for the the delta method (as it was used in the current work): 2 2 2 2 2 2 D D D D D D x x x x x x 2 2 2 2 2 2 (11) =  +  +  +  +  +  , x    y Y y Y f 0 1 3 198 Radiation and Environmental Biophysics (2018) 57:195–203 possibility that θ is not given by the value, but by the prior Finally, dose D can be estimated from the maximum of probability density function (PDF), which is the first step to the distributions given in Eq. 15or calculated from the first Bayesian reasoning, one has to transform classical relation- derivative equation: ships into probability distributions (put this way the method dP(D ) can also be called the Bayesian–frequentist hybrid method). = 0. (16) dD However, contrary to the full-Bayesian method, the multipli- cation of prior and likelihood functions is avoided. Thus, the The dose uncertainties  , obtained in the presented proposed quasi-Bayesian method is somewhere between the method can be assessed using the approach described for the analytical (classical) method and the full-Bayesian method. analytical method (Eq. 11). In this case, one needs to deter- The proposed quasi-Bayesian method uses fixed values for mine the equation for D directly from Eq. 15 (or generally the dose response parameters α, β, γ, and Y , which can be from Eq. 14) using numerical methods, because an analytical assessed beforehand by means of maximum likelihood estima- solution is usually impossible. After that, the calculated D tion, the least squares method, or even by the robust Bayesian is included into Eq. 11 to get the uncertainty value, finding regression method (Fornalski 2014). The Q-B method does not partial derivatives of D . This approach was used in the cur- use the uncertainties of those parameters directly in the dose rent work to calculate all uncertainties in the quasi-Bayesian calculations, but only in the assessment of dose uncertainty. method (see Table 1). This approach is much easier than using probability densities of all parameters [which are used in the full-Bayesian method Prior distribution functions (Brame and Groer 2003)], and can be efficiently utilized in the case when the parameter values are known from earlier The PDF used in the Q-B method (see example given in estimations. Eq. 12) should reflect the actual knowledge about the dose In practice, Eqs. 3 and 9 must be solved to find distributions ratios. To select the proper PDF, information about θ (or of θ in a mixed neutron and gamma radiation field (Eq.  13) ρ), such as, for example, the expected value, needs to be (Fornalski 2014): considered. Based on all the available information, the prior � � g function (in some cases with its scale and shape parameters) D = g g 1 2 D + y − Y −D −D ( ) g f 0 g g ⎪  should be used to maximize the PDF for the considered θ 2 (13) ⎨ � � −4( Y +D −y )− 0 n f (ρ) parameter. As it was mentioned earlier, the most often D = √ n n ⎩  −4( Y +D −y )−+2D used Gaussian distribution (Eq. 12) may be substituted by 0 n f n other PDFs. For the present work, different functions were tested, both symmetrical and unsymmetrical ones. In special The two different designations of θ result from the fact that cases, even a non-informative prior can be used, which does θ is not precisely described by its value, but by the prior prob- not specify any exact information, but instead only defines ability function. Thus, using changing variables, the probabil- a very general way of parameter search. This approach can ity distribution of the dose can be generally written as: be used if detailed information about the dose ratio is miss- ing. Non-informative priors can also be used in situations P D = p  (D ) ⋅ ≅ p() ⋅ const. (14) x x x when it is assumed that one type of radiation contributes most significantly to the investigated biological effect, but Changing variables (Eq. 14) jointly with the expressions in the knowledge about the percentage value of ρ does not exist Eq. 12 (which exemplifies a potential PDF for θ; here it is a (Pacyniak et al. 2015). normal distribution, but other distributions can be used as well, All results presented below were obtained with the as seen below) and Eq. 13 gives a system of two probability Gaussian PDF (Eq. 12). distributions for D and D (Eq. 15). g n � � D Statistical test E ⎡ g ⎤ ̂ n − 𝜃 ⎢ D + y −Y −𝛽 D −𝛾 D ⎥ ⎪ g ( f 0 g g) 1 𝛼 P(D )= √ exp − ⎢ ⎥ ⎪ g 2 2 𝜎 2 𝜋𝜎 𝜃 𝜃 To verify the accuracy of the proposed methods, results were ⎢ ⎥ ⎣ ⎦ tested using E test (Pacyniak et al. 2015). √ 2 ⎛ ⎞ . 2 (15) ⎨ 𝛽 −4𝛾 Y +𝛼 D −y −𝛽 ⎡ ( n ) ⎤ 0 f ⎜ ̂⎟ √ − 𝜃 ⎪ D − D ⎢ ⎜ ⎟ ⎥ ref M 𝛽 −4𝛾 Y +𝛼 D −y −𝛽 +2𝛾 D ⎝ ( 0 n f) n ⎠ 1 E = , ⎢ ⎥ n P(D )= √ exp − (17) n 2 2 2 ⎪ 2 𝜎 2 𝜋𝜎 ⎢ ⎥ 𝜃 𝜃  + ref M ⎢ ⎥ ⎣ ⎦ where D is the dose from the reference source (here bio- ref logical doses), D is the dose assessed by the proposed 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 199 1 3 Table 1 Comparison of data from the literature (biological, based on the classical iterative approach, and reference values determined experimentally by the use of sophisticated physical instru- ments) with results obtained with the analytical and quasi-Bayesian methods described in the present work Source of data Y * [dic/cell] Y * [dic/cell] ρ Cells Dicentrics Biological doses Reference doses Analytical Quasi-Bayesian: Gaussian n g prior D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] n g n g n g n g IAEA Y = 0.0005 + 0.832D Y = 0.0005 + 0.0164D + 0.0492D 0.67 100 120 1.21 1.82 Unknown 1.21 ± 0.15 1.81 ± 0.25 1.21 ± 0.15 1.81 ± 0.25 HPA Y = 0.0005 + 0.83D Y = 0.0005 + 0.014D + 0.076D 0.67 100 100 1.00 1.50 Unknown 0.98 ± 0.13 1.47 ± 0.20 0.98 ± 0.13 1.47 ± 0.20 NRPB Y = 0.835D Y = 0.0142D + 0.0759D 5.78 34 100 3.50 ± 0.30 0.60 ± 0.06 3.42 0.59 3.48 ± 0.70 0.60 ± 0.15 3.48 ± 0.70 0.60 ± 0.15 1.87 40 108 3.00 ± 0.30 1.59 ± 0.15 3.42 1.83 2.98 ± 0.52 1.59 ± 0.30 2.98 ± 0.53 1.59 ± 0.31 0.64 28 85 2.30 ± 0.30 3.70 ± 0.40 2.60 4.04 2.35 ± 0.40 3.67 ± 0.62 2.35 ± 0.40 3.67 ± 0.62 0.53 10 37 2.40 ± 0.40 4.60 ± 0.80 2.60 4.89 2.44 ± 0.62 4.59 ± 1.16 2.43 ± 0.62 4.59 ± 1.16 IRSN Y = 0.9D Y = 0.023D + 0.054D 5.78 50 125 2.70 ± 0.20 0.46 ± 0.04 3.42 0.59 2.75 ± 0.48 0.48 ± 0.11 2.75 ± 0.48 0.48 ± 0.11 1.87 55 202 3.80 ± 0.30 2.04 ± 0.14 3.42 1.83 3.78 ± 0.56 2.02 ± 0.34 3.78 ± 0.56 2.02 ± 0.34 0.64 14 36 2.10 ± 0.40 3.30 ± 0.60 2.60 4.04 2.12 ± 0.55 3.31 ± 0.87 2.12 ± 0.55 3.31 ± 0.87 0.53 19 53 2.10 ± 0.30 3.90 ± 0.50 2.60 4.89 2.08 ± 0.44 3.92 ± 0.83 2.08 ± 0.44 3.92 ± 0.83 A Y = 0.001 + (0.603 ± 0.079) Y = 0.001 + (0.0187 ± 0.0056) 0.60 61 129 2.20 3.70 1.83 3.03 2.21 ± 0.31 3.68 ± 0.51 2.21 ± 0.31 3.68 ± 0.51 D D + (0.0527 ± 0.0046)D 2.08 278 185 1.10 0.50 0.85 0.41 1.06 ± 0.17 0.51 ± 0.09 1.06 ± 0.17 0.51 ± 0.09 2.08 50 81 2.50 1.20 1.80 0.87 2.52 ± 0.53 1.22 ± 0.27 2.52 ± 0.53 1.21 ± 0.27 B Y = 0.001 + (0.638 ± 0.018) Y = 0.001 + (0.0371 ± 0.0085) 0.60 46 114 2.40 3.90 1.83 3.03 2.35 ± 0.33 3.91 ± 0.54 2.35 ± 0.33 3.91 ± 0.54 D D + (0.0547 ± 0.0039)D 2.08 151 112 1.10 0.50 0.85 0.41 1.11 ± 0.14 0.53 ± 0.08 1.11 ± 0.14 0.53 ± 0.08 2.08 67 128 2.50 1.20 1.80 0.87 2.76 ± 0.40 1.33 ± 0.22 2.76 ± 0.40 1.33 ± 0.22 C Y = 0.001 + (0.832 ± 0.031) Y = 0.001 + (0.0128 ± 0.0031) 0.60 100 186 1.60 2.70 1.83 3.03 1.63 ± 0.18 2.71 ± 0.31 1.63 ± 0.18 2.71 ± 0.31 D D + (0.0640 ± 0.0022)D 2.08 100 81 1.00 0.50 0.85 0.41 0.95 ± 0.14 0.46 ± 0.08 0.95 ± 0.15 0.46 ± 0.08 2.08 100 144 1.70 0.80 1.80 0.87 1.67 ± 0.22 0.80 ± 0.13 1.67 ± 0.22 0.80 ± 0.13 Uncertainties represent 95% confidence intervals. * Presented uncertainties were taken from the original literature. If they were not available, it was assumed that the following standard values could be used to calculate the dose uncertainties in the analytical and QB methods: σα = 0.03, σβ = 0.003, σY = 0.0001, σγ = 0.0016. Data were taken from IAEA (2011) (in table marked as IAEA—International Atomic Energy Agency), Szłuińska et al. (2005) (HPA—Health Protection Agency), Voisin et al. (1997) (NRPB—National Radiation Protection Broad and IRSN—Institut de Radioprotection et de Sûreté Nucléaire, formerly IPSN—Institut de Protection et de Sûreté Nucléaire), and Voisin et al. (2004) (A, B, C—unknown laboratories) 200 Radiation and Environmental Biophysics (2018) 57:195–203 method, σ and σ are their uncertainties, respectively ref M (see Table 1). In the case when σ is not available, it was ref assumed here that σ = 10% D , which is typical in this ref ref type of biodosimetric assessments. If E ≤ 1, the result is satisfactory. Any result is classified as an outlier if the E value is greater than 1. Computational program As part of the project, a mobile phone application includ- ing the described algorithm was created. The application is designed for devices with the Android operating system and was written using the Android Studio Integrated Devel- opment Environment (IDE). The graphs were implemented using the GraphView open-source library. The structure of the program is relatively simple, as it relies on the implementation of a basic Android graphics component (TabLayout), which allows the user to choose between one of the methods discussed above (iterative, ana- lytical or quasi-Bayesian). The program allows the user to change the dose–response curve parameters and select one of the methods to assess absorbed dose. This can be done by selecting the dose–response curves tab from the drop-down menu located at the top right-hand corner of the menu. An important feature of the app is the possibility of drawing the prior function (for the Bayesian and quasi-Bayesian meth- ods), allowing the user to quickly and efficiently choose the Fig. 2 Screenshot of the computational program which uses the pre- sented methods distribution and its parameters. The program includes a user-friendly interface, which allows the automation of all calculations. The calculation measured in the sample were used to make the calculations. algorithms for determining absorbed doses were imple- Table 1 shows the corresponding data and biological doses for mented in accordance with the theory. There were no gamma and neutron radiation from the literature mentioned approximations done in any of the methods, only the final above, as well as the results of the dose assessments obtained results displayed on the screen are rounded (with a 1/1000 by applying the described analytical and quasi-Bayesian meth- precision). The program is constantly being improved, and ods. Note that the biological doses taken from the literature work is underway to enhance its responsiveness. It is avail- were obtained using the iterative procedure. Reference values able for testing on the website: http://www.clor.waw.pl/publi of doses and the neutron/gamma dose ratios were determined kacje .html. A sample screenshot of the program is shown experimentally by the use of sophisticated physical instru- in the Fig. 2. ments. These values are typically not available during a real accident and were made available by the exercise organizers for comparison only. In some cases, the uncertainty is not Results given in the literature and, accordingly, could not be added to Table 1. For the calculations performed in the present paper, In this work, a number of published dose estimates from bio- the uncertainty of θ is assumed to be equal to 0.02. dosimetry studies have been used for comparison with the dose estimates obtained by means of the proposed analytical and quasi-Bayesian methods. More specifically, to verify the Discussion validity and credibility of these methods, mixed gamma and neutron radiation doses were calculated using data from the In the present paper two methods are proposed to assess peer-reviewed literature: from IAEA (2011), Szłuińska et al. biodosimetric doses, which offer reasonable alternatives to (2005), Voisin et al. (1997), and Voisin et al. (2004). The fitted the widely used iterative and Bayesian methods. coefficients of the dose–response curves, the values for θ or ρ, the number of analyzed cells and the number of dicentrics 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 201 The first analytical method is a straightforward math- values. Thus, both methods provide results which are fully ematical development of the iterative method. It substan- comparable with those the classical method. The biggest dif- tially reduces the propagation of uncertainties effect and ference between the biological doses, and the doses obtained is generally faster. Moreover, it reduces the probability of from the proposed methods using the Gaussian distribution errors because it only requires to solve a set of equations as the PDF is visible for both the neutron and gamma doses, instead to perform a series of calculations. The second Q-B for samples A, B and C, especially B3 and C2. For sample method is substantially different because of the implemen- B3, the E value equals 0.55 for the neutron dose, and 0.52 tation of prior functions. This may provide more realistic for the gamma dose. For sample C2 in the analytical method, results, because the knowledge about θ (or ρ) is usually not E is 0.29 for the neutron dose, while for the gamma dose, very precise. Therefore, the Gaussian prior is usually used, equals 0.42. For quasi-Bayesian method it is 0.28 and 0.42, but other priors can be also appropriate. Choice of the prior respectively. Interestingly, the above values do not seem to will however influence the uncertainty which can be non- be caused by PDF selection and are rather method-inde- symmetrical. This choice is sometimes necessary, due to pendent. This is so especially for cases where the original the limited knowledge about θ (or ρ) where some additional dose uncertainties are unavailable and thus, an uncertainty information might increase the reliability of the deduced of 10% D had to be assumed in the calculations. Because ref parameter value, in some range of values. Table 1 demonstrates that for samples A, B, and C there is In the Q-B method, the influence of other prior distribu- also a quite significant difference between the biological and tions such as gamma, cauchy, beta and geometric distribu- reference doses, it is concluded that the analytical and Q-B tions was also evaluated here. It turned out that for the same methods work also well for these cases, and large E values θ value (the maximum of the distribution) all those PDFs must have some other reasons. give comparable (practically the same) results. The only It is generally not possible to identify one method as the difference is the uncertainty that is caused by the shape of best one, since the problem depends on the exact situation. distribution—generally the uncertainty is larger when the While the analytical method provides almost identical point PDF is wider. estimates as the Q-B method, the uncertainties obtained with Table 1 shows doses obtained using both of the proposed the analytical method are slightly lower than those obtained methods, the analytical and the Q-B method with a Gauss- with the Q-B method, which is probably due to the com- ian distribution, together with the original values from the pletely different methodology of both approaches, mostly literature. Additionally, the accuracy factor, E , given by the representation of θ as the PDF in the Q-B method. This Eq. 17 provides a measure for the relative goodness of all might suggest that the uncertainties calculated by the Q-B assessments (Figs. 3, 4). For all results E is much below method are more realistic than those calculated by the ana- 1; this indicates only small deviations from the reference lytical method. Fig. 3 E values for neutron radiation 1 3 202 Radiation and Environmental Biophysics (2018) 57:195–203 Fig. 4 E values for gamma radiation for all comments and Ms. Aleksandra Powojska for checking the English Conclusion in this manuscript. In the case of mixed radiation exposure, the iterative method Open Access This article is distributed under the terms of the Crea- is widely used for biodosimetry. In the present study, two tive Commons Attribution 4.0 International License (http://creat iveco alternative statistical methods of estimating neutron and mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- tion, and reproduction in any medium, provided you give appropriate gamma absorbed doses in a mixed radiation field from meas- credit to the original author(s) and the source, provide a link to the ured chromosomal aberration frequencies were investigated. Creative Commons license, and indicate if changes were made. Both the analytical and the Q-B method proposed here have their own advantages and disadvantages. The first-proposed References method is based on the classical iterative approach and is nothing more than its analytical description; for this reason Ainsbury EA, Vinnikov VA, Puig P, Higueras M, Maznyk NA, Lloyd the results obtained are the same, as shown in Table 1. Thus, DC, Rothkamm K (2013a) Review of Bayesian statistical analysis methods for cytogenetic radiation biodosimetry, with a practical the iterative method can be replaced by the analytical one example. Radiat Prot Dosim 162:1–12 which is more convenient in use because it does not need Ainsbury EA, Vinnikov VA, Puig P, Maznyk NA, Rothkamm K, Lloyd to perform a series of iterations; instead it only requires to DC (2013b) CytoBayesJ: Software tools for Bayesian analysis of insert the appropriate data into the given formulas. This clas- cytogenetic radiation dosimetry data. Mutation Res 756:184–191 Brame RS, Groer PG (2003) Bayesian methods for chromosome dosim- sical method is a very simple and fast calculation method, etry following a criticality accident. Radiat Prot Dosim 104:61–63 but requires knowledge of the neutron/gamma dose ratio. In Fornalski KW (2014) Alternative statistical methods for cytogenetic the Q-B method, this parameter is described by the probabil- radiation biological dosimetry. Cornell University Library, arXiv. ity distribution (prior function), similarly to Bayesian meth- org/abs/1412.2048 International Atomic Energy Agency (2011) Cytogenetic dosimetry: ods. Therefore, it can be used when the exact dose ratio is applications in preparedness for and response to radiation emergen- unknown without requiring complicated Bayesian statistics. cies. IAEA in Austria, September 2011. https://www -pub.iaea.org/ The statistical methods proposed in the present work were MTCD/publi catio ns/PDF/EPR-Biodo simet ry%25202 011_web.pdf programmed as computational algorithms which can easily Kellerer AM, Rossi HH (1978) A generalized formulation of dual radia- tion action. Radiat Res 75:471–488 be used in cytogenetic analyses. Additionally, the methods Pacyniak I, Fornalski KW, Kowalska M (2014) Employment of Bayesian are presented here in easy-to-use forms, which can be coded and Monte Carlo methods for biological dose assessment following even as Excel spreadsheet formulas. The required computer accidental overexposures of people to nuclear reactor radiation. In codes are provided as an electronic supplement material. proceeding of: the second international conference on radiation and dosimetry in various fields of research, University of Nis (Serbia), 27–30 May 2014, pp 49–52 Acknowledgements This paper is part of a master thesis written by the Pacyniak I, Fornalski KW, Kowalska M (2015) Alternatywne metody first author at the Physics Faculty at Warsaw University of Technology, obliczania dawek pochłoniętych w biologicznej dozymetrii promien- Poland, under scientific supervision of Dr. Maria Kowalska, Central iowania mieszanego n + γ. Postępy Fizyki Tom 66(zeszyt 1–4):13– Laboratory for Radiological Protection, as a part of scientific program 22 (in Polish) SP/J/16/143339/8, “Technologies Supporting Development of Safe Nuclear Power Engineering”. Authors wish to thank Dr. Maria Kowalska 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 203 Szłuińska M, Edwards AA, Lloyd DC (2005) Statistical methods for bio- Voisin P, Roy L, Hone PA, Edwards AA, Lloyd DC, Stephan G, Romm logical dosimetry. In: G Obe, Vijayalaxmi S (eds) Chromosomal H, Groer PG, Brame R (2004) Criticality accident dosimetry by alterations. Methods, results and importance in human health. chromosomal analysis. Radia Prot Dosim 110:443–447 Springer, Berlin Heidelberg Voisin P, Lloyd D, Edwards A (1997) Chromosome aberrations scoring for biological dosimetry in a criticality accident. Radiat Prot Dosim 70:467–470 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Radiation and Environmental Biophysics Springer Journals

Analytical and quasi-Bayesian methods as development of the iterative approach for mixed radiation biodosimetry

Free
9 pages

Loading next page...
 
/lp/springer_journal/analytical-and-quasi-bayesian-methods-as-development-of-the-iterative-Vrg8mEE0kx
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by The Author(s)
Subject
Physics; Biological and Medical Physics, Biophysics; Effects of Radiation/Radiation Protection; Environmental Physics; Monitoring/Environmental Analysis; Ecosystems
ISSN
0301-634X
eISSN
1432-2099
D.O.I.
10.1007/s00411-018-0745-6
Publisher site
See Article on Publisher Site

Abstract

The present paper proposes two methods of calculating components of the dose absorbed by the human body after exposure to a mixed neutron and gamma radiation field. The article presents a novel approach to replace the common iterative method in its analytical form, thus reducing the calculation time. It also shows a possibility of estimating the neutron and gamma doses when their ratio in a mixed beam is not precisely known. Keywords Biological dosimetry · Bayesian · Nuclear accident · Cytogenetics · Radiation · Biodosimetry · Dose assessment Introduction The process of dose assessment through a dicentric assay requires the presence of dose–response calibration curves. Radiation biodosimetry refers to the analysis of biologi- Such curves are produced by exposing human blood in vitro cal changes in a particular tissue after exposure to ionizing to several doses of radiation under carefully controlled con- radiation, and allows for an estimation of absorbed doses in ditions. It is implicitly assumed that in vivo and in vitro irra- an exposed person based on this analysis. The quantifying diations of peripheral blood lymphocytes produce similar measurement of chromosome alterations, mainly dicentrics alterations in the cells. Accordingly, the dicentric frequency in lymphocytes, is the most widely used method for biologi- observed in vivo can be converted to absorbed dose by com- cal dose assessment (IAEA 2011). The aim of radiation bio- paring it with the dose–response calibration curve obtained dosimetry by cytogenetic methods is to calculate doses and in vitro. The shape of a dose–response curve is influenced the associated confidence intervals to exposed (or suspected by the linear energy transfer, LET, for any particular radia- to have been exposed) individuals after a radiation accident tion quality. For low-LET radiation, the relation between or incident. Assessing the absorbed dose based on the dicen- absorbed dose and dicentric chromosome frequency can be tric assay in peripheral blood lymphocytes is a very sensitive expressed as a combination of linear and quadratic terms method that can be used when measured doses are not avail- (Eq. 1), while for high-LET radiation the dose dependence is able. The analysis is performed in the lymphocytes during linear (Eq. 2). The linear and quadratic terms are consistent their first mitosis after radiation exposure. The advantage with the single- and two-track model of dicentric formation of the method is the low spontaneous frequency of dicentric by both radiation qualities (IAEA 2011; Kellerer and Rossi chromosome aberrations in healthy individuals, indicating 1978): that this indicator is caused mainly by ionizing radiation. Y D = Y +  D +  D , (1) g g 0 g g * Iwona Słonecka Y D = Y +  D , (2) n n 0 n slonecka@clor.waw.pl where Y (x = g is for gamma and x = n for neutron radiation) Central Laboratory for Radiological Protection, Konwaliowa is the frequency of dicentrics, D is the absorbed dose, Y is x 0 7, 03-194 Warszawa, Poland the background frequency of dicentrics (dose independent), Faculty of Physics, Warsaw University of Technology, α and β are the linear coefficients, and γ is the quadratic Warszawa, Poland coefficient. Ex-Polon Laboratory, Łazy, Poland PGE EJ 1 Sp. z o. o., Warszawa, Poland Vol.:(0123456789) 1 3 196 Radiation and Environmental Biophysics (2018) 57:195–203 In the case of exposure to mixed neutron and gamma radi- irradiation by a mixed neutron and gamma radiation field ation, n + γ (e.g., after a nuclear reactor accident), biological with doses D and D . Parameters Y , α, β and γ are usually n g 0 dosimetry is more complex than in the case of exposure to a found as a result of a regression analysis (defining the so- single type of radiation. Mixed radiation fields are composed called calibration curve). The parameter y can be written of particles with varying biological effectiveness, which as a ratio of u/w, where u represents the number of chromo- have different biological effects on the body. Therefore, some aberrations and w is the number of cells in the ana- there is a strong need to calculate not only the total absorbed lyzed sample. dose, but also its components, D and D , separately. Unfor- Having the fit parameters of the calibration curves, (Y , n g 0 tunately, there is no visible difference between dicentrics α, β, and γ), the Y(D ) functions can be used to estimate the induced by the two types of radiation, and therefore, it is doses (D ) and/or the frequency of chromosome aberrations not possible to directly discriminate between dicentrics pro- (Y ) after exposure to gamma radiation, neutron radiation, or duced by gamma radiation and those produced by neutrons. to a mixed neutron and gamma radiation field. In cases when Discrimination is only possible by quantifying the separate the ratio of neutron to gamma absorbed dose is estimated by dose components and using the additivity assumption in the physical methods (Eq. 4): production of chromosomal damages (IAEA 2011; Kellerer and Rossi 1978).  = , (4) Dose estimation using biodosimetry methods for a mixed neutron and gamma radiation field can be found in the lit- it is possible to calculate the separate neutron and gamma erature (IAEA 2011; Brame and Groer 2003; Szłuińska radiation doses using the iterative method (IAEA 2011) et al. 2005; Fornalski 2014). The most common approaches mentioned earlier. In this method, the doses and the chro- are the classical iterative method, promoted widely by the mosome aberration frequencies are estimated by a manual International Atomic Energy Agency (IAEA 2011), and iterative approach, meaning that the values are precisely the increasingly used Bayesian methods (Brame and Groer determined. 2003; Ainsbury et al. 2013a, b; Pacyniak et al. 2014). The iterative method is adopted in many accredited laboratories Iterative method worldwide, including the Central Laboratory for Radiologi- cal Protection (CLOR), Poland. This method is generally The iterative method involves performing several series well-known due to its simplicity and relative accuracy. of calculations using the same input data. Each consecu- Bayesian methods are also becoming more and more popu- tive series gives more accurate results, until the results of lar. However, they are not yet commonly used probably due the next steps do not change significantly any more. In this to the much more advanced mathematical approach required. method, it is initially assumed that all aberrations found in Thus, some intermediate methods, which will improve the the analyzed sample, y , are caused by neutron radiation, i.e., iterative approach without being as complex as the Bayes- f using the recurrence relation, Y = y . Thus, the neutron ian approach, are developed and tested in the present paper. n f i=0 dose can be defined directly from Eq.  2, as: Y − Y n 0 D = . (5) i+1 Materials and methods th The dose from gamma radiation in the (i + 1) step, D , i+1 Assessment of neutron and gamma doses is calculated with the use of the actual D value and the i+1 neutron/gamma dose ratio, ρ, according to Eq. 6: In practice, it is assumed that the number of dicentrics i+1 in an analyzed sample is characterized by a Poisson dis- D = , (6) i+1 tribution, and that the observed alterations are the sum of those induced by neutron and gamma radiation. Thus, the also formulated as the recurrence equation. With informa- dose–response relationship for a mixed neutron and gamma tion about the gamma dose, the dicentric frequency due to radiation field may be described as a combination of Eqs.  1 gamma radiation, Y , can be obtained from Eq. 1: i+1 and 2: Y = Y + D + D . (7) 2 g 0 g i+1 i+1 g i+1 Y D , D = Y +  D +  D +  D ≡ y = , n+g n g 0 n g g f (3) The dicentric frequency caused by neutrons (Y ) is then i+1 which is usually called a combined linear-quadratic equa- obtained (Eq. 8) by subtracting the gamma dicentric fre- tion for the frequency of chromosome aberrations y after quency from the measured aberrations, y . f f 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 197 Y = y − Y . n f g (8) i+1 i+1 The neutron dose is estimated using Eq. 5 and the above steps are repeated using recurrence equations from Eqs. 5–8. With this algorithm, all parameters are calculated iteratively until their values are stable ( D ≅ D ). x x i i−1 Obviously, the more repetitions done, the more accurate the achieved values will be. All the fitting parameters of the calibration curves (Y , β, γ, and α) as well as the ratio of doses (ρ) should be calculated in advance. However, a significant disadvantage of the iterative method is the large effect of the propagation of uncertainty. The presented itera- tive algorithm is usually time-consuming and requires many Fig. 1 Distributions of gamma and neutron doses for a sample repetitions to obtain final results. Hence, it was proposed to with 53 dicentrics in 19 analyzed cells obtained with the analytical transform the iterative method into its analytical description, method. θ = 0.653 is marked with a dotted line which was originally introduced by Fornalski (2014). Analytical method where x = {g, n}, and  =  ,  ,  ,  ,  ,  are y Y f 0 uncertainties of parameters ε = {α, β, γ, y , Y , θ}. Uncertain- The analytical method (Fornalski 2014; Pacyniak et  al. f 0 2015) was proposed to automate the iterative algorithm used ties of D and D can be calculated from Eq.  11 finding g n partial derivatives of D . for calculating the absorbed doses in mixed radiation fields and exclude the propagation of uncertainty effect. In this Quasi‑Bayesian (Q‑B) method approach, the neutron/gamma dose ratio, ρ (Eq.  4), must be known. However, because ρ varies from zero to infinity, When the ratio of doses is not precisely known, the iterative it can conveniently be replaced by θ (Eq.  9), which cor- responds to the contribution of the gamma dose to the total and analytical methods cannot be used. In this case, the prob- ability density function (PDF) can be utilized to estimate the dose and is more practical in use than ρ. It is normalized to the range of [0, 1] and is defined as (Brame and Groer 2003): most probable value of θ (or ρ). Such a PDF, which is widely used in Bayesian statistics, is called a prior probability p(θ) or g p(ρ). In the most common cases it can be approximated by a = = . (9) Gaussian distribution for θ (or ρ) with a standard deviation of D + D  + 1 g n σ (or σ ), for example: θ ρ With Eqs. 3 and 9, D and D can be simply calculated as g n � � ⎡ ⎤ a function of θ (Fornalski 2014): 𝜃 − 𝜃 ⎢ ⎥ p(𝜃 )= exp − , √ (12) � � � � ⎢ ⎥ 2 2𝜎 2𝜋𝜎 1− 1− 𝜃 𝜃 ⎣ ⎦ + +4 y − Y −  + ( ) f 0 D () = (10) ⎨ 2 1− D () = D () where 𝜃 represents the expected value of θ with the uncer- n g tainty σ . The Gaussian distribution is usually selected because the estimation of θ (or ρ) is done using the classi- It is assumed that all constants (α, β, γ, y , Y and θ) are f 0 cal Gaussian regression method. However, one can also use precisely known from experimental data with proper uncer- other priors reflecting the knowledge on θ (or ρ); this issue tainties. The solution of Eq. 10 can be presented graphically will be discussed later. in terms of dose and θ (Fig. 1). Having sufficient information on the dose ratio, one can Finally, the uncertainties associated with the estimated try to estimate the neutron and gamma doses, as well as the doses,  , in the presented method can be calculated using dicentric frequencies. To enhance the classical method for the the delta method (as it was used in the current work): 2 2 2 2 2 2 D D D D D D x x x x x x 2 2 2 2 2 2 (11) =  +  +  +  +  +  , x    y Y y Y f 0 1 3 198 Radiation and Environmental Biophysics (2018) 57:195–203 possibility that θ is not given by the value, but by the prior Finally, dose D can be estimated from the maximum of probability density function (PDF), which is the first step to the distributions given in Eq. 15or calculated from the first Bayesian reasoning, one has to transform classical relation- derivative equation: ships into probability distributions (put this way the method dP(D ) can also be called the Bayesian–frequentist hybrid method). = 0. (16) dD However, contrary to the full-Bayesian method, the multipli- cation of prior and likelihood functions is avoided. Thus, the The dose uncertainties  , obtained in the presented proposed quasi-Bayesian method is somewhere between the method can be assessed using the approach described for the analytical (classical) method and the full-Bayesian method. analytical method (Eq. 11). In this case, one needs to deter- The proposed quasi-Bayesian method uses fixed values for mine the equation for D directly from Eq. 15 (or generally the dose response parameters α, β, γ, and Y , which can be from Eq. 14) using numerical methods, because an analytical assessed beforehand by means of maximum likelihood estima- solution is usually impossible. After that, the calculated D tion, the least squares method, or even by the robust Bayesian is included into Eq. 11 to get the uncertainty value, finding regression method (Fornalski 2014). The Q-B method does not partial derivatives of D . This approach was used in the cur- use the uncertainties of those parameters directly in the dose rent work to calculate all uncertainties in the quasi-Bayesian calculations, but only in the assessment of dose uncertainty. method (see Table 1). This approach is much easier than using probability densities of all parameters [which are used in the full-Bayesian method Prior distribution functions (Brame and Groer 2003)], and can be efficiently utilized in the case when the parameter values are known from earlier The PDF used in the Q-B method (see example given in estimations. Eq. 12) should reflect the actual knowledge about the dose In practice, Eqs. 3 and 9 must be solved to find distributions ratios. To select the proper PDF, information about θ (or of θ in a mixed neutron and gamma radiation field (Eq.  13) ρ), such as, for example, the expected value, needs to be (Fornalski 2014): considered. Based on all the available information, the prior � � g function (in some cases with its scale and shape parameters) D = g g 1 2 D + y − Y −D −D ( ) g f 0 g g ⎪  should be used to maximize the PDF for the considered θ 2 (13) ⎨ � � −4( Y +D −y )− 0 n f (ρ) parameter. As it was mentioned earlier, the most often D = √ n n ⎩  −4( Y +D −y )−+2D used Gaussian distribution (Eq. 12) may be substituted by 0 n f n other PDFs. For the present work, different functions were tested, both symmetrical and unsymmetrical ones. In special The two different designations of θ result from the fact that cases, even a non-informative prior can be used, which does θ is not precisely described by its value, but by the prior prob- not specify any exact information, but instead only defines ability function. Thus, using changing variables, the probabil- a very general way of parameter search. This approach can ity distribution of the dose can be generally written as: be used if detailed information about the dose ratio is miss- ing. Non-informative priors can also be used in situations P D = p  (D ) ⋅ ≅ p() ⋅ const. (14) x x x when it is assumed that one type of radiation contributes most significantly to the investigated biological effect, but Changing variables (Eq. 14) jointly with the expressions in the knowledge about the percentage value of ρ does not exist Eq. 12 (which exemplifies a potential PDF for θ; here it is a (Pacyniak et al. 2015). normal distribution, but other distributions can be used as well, All results presented below were obtained with the as seen below) and Eq. 13 gives a system of two probability Gaussian PDF (Eq. 12). distributions for D and D (Eq. 15). g n � � D Statistical test E ⎡ g ⎤ ̂ n − 𝜃 ⎢ D + y −Y −𝛽 D −𝛾 D ⎥ ⎪ g ( f 0 g g) 1 𝛼 P(D )= √ exp − ⎢ ⎥ ⎪ g 2 2 𝜎 2 𝜋𝜎 𝜃 𝜃 To verify the accuracy of the proposed methods, results were ⎢ ⎥ ⎣ ⎦ tested using E test (Pacyniak et al. 2015). √ 2 ⎛ ⎞ . 2 (15) ⎨ 𝛽 −4𝛾 Y +𝛼 D −y −𝛽 ⎡ ( n ) ⎤ 0 f ⎜ ̂⎟ √ − 𝜃 ⎪ D − D ⎢ ⎜ ⎟ ⎥ ref M 𝛽 −4𝛾 Y +𝛼 D −y −𝛽 +2𝛾 D ⎝ ( 0 n f) n ⎠ 1 E = , ⎢ ⎥ n P(D )= √ exp − (17) n 2 2 2 ⎪ 2 𝜎 2 𝜋𝜎 ⎢ ⎥ 𝜃 𝜃  + ref M ⎢ ⎥ ⎣ ⎦ where D is the dose from the reference source (here bio- ref logical doses), D is the dose assessed by the proposed 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 199 1 3 Table 1 Comparison of data from the literature (biological, based on the classical iterative approach, and reference values determined experimentally by the use of sophisticated physical instru- ments) with results obtained with the analytical and quasi-Bayesian methods described in the present work Source of data Y * [dic/cell] Y * [dic/cell] ρ Cells Dicentrics Biological doses Reference doses Analytical Quasi-Bayesian: Gaussian n g prior D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] D [Gy] n g n g n g n g IAEA Y = 0.0005 + 0.832D Y = 0.0005 + 0.0164D + 0.0492D 0.67 100 120 1.21 1.82 Unknown 1.21 ± 0.15 1.81 ± 0.25 1.21 ± 0.15 1.81 ± 0.25 HPA Y = 0.0005 + 0.83D Y = 0.0005 + 0.014D + 0.076D 0.67 100 100 1.00 1.50 Unknown 0.98 ± 0.13 1.47 ± 0.20 0.98 ± 0.13 1.47 ± 0.20 NRPB Y = 0.835D Y = 0.0142D + 0.0759D 5.78 34 100 3.50 ± 0.30 0.60 ± 0.06 3.42 0.59 3.48 ± 0.70 0.60 ± 0.15 3.48 ± 0.70 0.60 ± 0.15 1.87 40 108 3.00 ± 0.30 1.59 ± 0.15 3.42 1.83 2.98 ± 0.52 1.59 ± 0.30 2.98 ± 0.53 1.59 ± 0.31 0.64 28 85 2.30 ± 0.30 3.70 ± 0.40 2.60 4.04 2.35 ± 0.40 3.67 ± 0.62 2.35 ± 0.40 3.67 ± 0.62 0.53 10 37 2.40 ± 0.40 4.60 ± 0.80 2.60 4.89 2.44 ± 0.62 4.59 ± 1.16 2.43 ± 0.62 4.59 ± 1.16 IRSN Y = 0.9D Y = 0.023D + 0.054D 5.78 50 125 2.70 ± 0.20 0.46 ± 0.04 3.42 0.59 2.75 ± 0.48 0.48 ± 0.11 2.75 ± 0.48 0.48 ± 0.11 1.87 55 202 3.80 ± 0.30 2.04 ± 0.14 3.42 1.83 3.78 ± 0.56 2.02 ± 0.34 3.78 ± 0.56 2.02 ± 0.34 0.64 14 36 2.10 ± 0.40 3.30 ± 0.60 2.60 4.04 2.12 ± 0.55 3.31 ± 0.87 2.12 ± 0.55 3.31 ± 0.87 0.53 19 53 2.10 ± 0.30 3.90 ± 0.50 2.60 4.89 2.08 ± 0.44 3.92 ± 0.83 2.08 ± 0.44 3.92 ± 0.83 A Y = 0.001 + (0.603 ± 0.079) Y = 0.001 + (0.0187 ± 0.0056) 0.60 61 129 2.20 3.70 1.83 3.03 2.21 ± 0.31 3.68 ± 0.51 2.21 ± 0.31 3.68 ± 0.51 D D + (0.0527 ± 0.0046)D 2.08 278 185 1.10 0.50 0.85 0.41 1.06 ± 0.17 0.51 ± 0.09 1.06 ± 0.17 0.51 ± 0.09 2.08 50 81 2.50 1.20 1.80 0.87 2.52 ± 0.53 1.22 ± 0.27 2.52 ± 0.53 1.21 ± 0.27 B Y = 0.001 + (0.638 ± 0.018) Y = 0.001 + (0.0371 ± 0.0085) 0.60 46 114 2.40 3.90 1.83 3.03 2.35 ± 0.33 3.91 ± 0.54 2.35 ± 0.33 3.91 ± 0.54 D D + (0.0547 ± 0.0039)D 2.08 151 112 1.10 0.50 0.85 0.41 1.11 ± 0.14 0.53 ± 0.08 1.11 ± 0.14 0.53 ± 0.08 2.08 67 128 2.50 1.20 1.80 0.87 2.76 ± 0.40 1.33 ± 0.22 2.76 ± 0.40 1.33 ± 0.22 C Y = 0.001 + (0.832 ± 0.031) Y = 0.001 + (0.0128 ± 0.0031) 0.60 100 186 1.60 2.70 1.83 3.03 1.63 ± 0.18 2.71 ± 0.31 1.63 ± 0.18 2.71 ± 0.31 D D + (0.0640 ± 0.0022)D 2.08 100 81 1.00 0.50 0.85 0.41 0.95 ± 0.14 0.46 ± 0.08 0.95 ± 0.15 0.46 ± 0.08 2.08 100 144 1.70 0.80 1.80 0.87 1.67 ± 0.22 0.80 ± 0.13 1.67 ± 0.22 0.80 ± 0.13 Uncertainties represent 95% confidence intervals. * Presented uncertainties were taken from the original literature. If they were not available, it was assumed that the following standard values could be used to calculate the dose uncertainties in the analytical and QB methods: σα = 0.03, σβ = 0.003, σY = 0.0001, σγ = 0.0016. Data were taken from IAEA (2011) (in table marked as IAEA—International Atomic Energy Agency), Szłuińska et al. (2005) (HPA—Health Protection Agency), Voisin et al. (1997) (NRPB—National Radiation Protection Broad and IRSN—Institut de Radioprotection et de Sûreté Nucléaire, formerly IPSN—Institut de Protection et de Sûreté Nucléaire), and Voisin et al. (2004) (A, B, C—unknown laboratories) 200 Radiation and Environmental Biophysics (2018) 57:195–203 method, σ and σ are their uncertainties, respectively ref M (see Table 1). In the case when σ is not available, it was ref assumed here that σ = 10% D , which is typical in this ref ref type of biodosimetric assessments. If E ≤ 1, the result is satisfactory. Any result is classified as an outlier if the E value is greater than 1. Computational program As part of the project, a mobile phone application includ- ing the described algorithm was created. The application is designed for devices with the Android operating system and was written using the Android Studio Integrated Devel- opment Environment (IDE). The graphs were implemented using the GraphView open-source library. The structure of the program is relatively simple, as it relies on the implementation of a basic Android graphics component (TabLayout), which allows the user to choose between one of the methods discussed above (iterative, ana- lytical or quasi-Bayesian). The program allows the user to change the dose–response curve parameters and select one of the methods to assess absorbed dose. This can be done by selecting the dose–response curves tab from the drop-down menu located at the top right-hand corner of the menu. An important feature of the app is the possibility of drawing the prior function (for the Bayesian and quasi-Bayesian meth- ods), allowing the user to quickly and efficiently choose the Fig. 2 Screenshot of the computational program which uses the pre- sented methods distribution and its parameters. The program includes a user-friendly interface, which allows the automation of all calculations. The calculation measured in the sample were used to make the calculations. algorithms for determining absorbed doses were imple- Table 1 shows the corresponding data and biological doses for mented in accordance with the theory. There were no gamma and neutron radiation from the literature mentioned approximations done in any of the methods, only the final above, as well as the results of the dose assessments obtained results displayed on the screen are rounded (with a 1/1000 by applying the described analytical and quasi-Bayesian meth- precision). The program is constantly being improved, and ods. Note that the biological doses taken from the literature work is underway to enhance its responsiveness. It is avail- were obtained using the iterative procedure. Reference values able for testing on the website: http://www.clor.waw.pl/publi of doses and the neutron/gamma dose ratios were determined kacje .html. A sample screenshot of the program is shown experimentally by the use of sophisticated physical instru- in the Fig. 2. ments. These values are typically not available during a real accident and were made available by the exercise organizers for comparison only. In some cases, the uncertainty is not Results given in the literature and, accordingly, could not be added to Table 1. For the calculations performed in the present paper, In this work, a number of published dose estimates from bio- the uncertainty of θ is assumed to be equal to 0.02. dosimetry studies have been used for comparison with the dose estimates obtained by means of the proposed analytical and quasi-Bayesian methods. More specifically, to verify the Discussion validity and credibility of these methods, mixed gamma and neutron radiation doses were calculated using data from the In the present paper two methods are proposed to assess peer-reviewed literature: from IAEA (2011), Szłuińska et al. biodosimetric doses, which offer reasonable alternatives to (2005), Voisin et al. (1997), and Voisin et al. (2004). The fitted the widely used iterative and Bayesian methods. coefficients of the dose–response curves, the values for θ or ρ, the number of analyzed cells and the number of dicentrics 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 201 The first analytical method is a straightforward math- values. Thus, both methods provide results which are fully ematical development of the iterative method. It substan- comparable with those the classical method. The biggest dif- tially reduces the propagation of uncertainties effect and ference between the biological doses, and the doses obtained is generally faster. Moreover, it reduces the probability of from the proposed methods using the Gaussian distribution errors because it only requires to solve a set of equations as the PDF is visible for both the neutron and gamma doses, instead to perform a series of calculations. The second Q-B for samples A, B and C, especially B3 and C2. For sample method is substantially different because of the implemen- B3, the E value equals 0.55 for the neutron dose, and 0.52 tation of prior functions. This may provide more realistic for the gamma dose. For sample C2 in the analytical method, results, because the knowledge about θ (or ρ) is usually not E is 0.29 for the neutron dose, while for the gamma dose, very precise. Therefore, the Gaussian prior is usually used, equals 0.42. For quasi-Bayesian method it is 0.28 and 0.42, but other priors can be also appropriate. Choice of the prior respectively. Interestingly, the above values do not seem to will however influence the uncertainty which can be non- be caused by PDF selection and are rather method-inde- symmetrical. This choice is sometimes necessary, due to pendent. This is so especially for cases where the original the limited knowledge about θ (or ρ) where some additional dose uncertainties are unavailable and thus, an uncertainty information might increase the reliability of the deduced of 10% D had to be assumed in the calculations. Because ref parameter value, in some range of values. Table 1 demonstrates that for samples A, B, and C there is In the Q-B method, the influence of other prior distribu- also a quite significant difference between the biological and tions such as gamma, cauchy, beta and geometric distribu- reference doses, it is concluded that the analytical and Q-B tions was also evaluated here. It turned out that for the same methods work also well for these cases, and large E values θ value (the maximum of the distribution) all those PDFs must have some other reasons. give comparable (practically the same) results. The only It is generally not possible to identify one method as the difference is the uncertainty that is caused by the shape of best one, since the problem depends on the exact situation. distribution—generally the uncertainty is larger when the While the analytical method provides almost identical point PDF is wider. estimates as the Q-B method, the uncertainties obtained with Table 1 shows doses obtained using both of the proposed the analytical method are slightly lower than those obtained methods, the analytical and the Q-B method with a Gauss- with the Q-B method, which is probably due to the com- ian distribution, together with the original values from the pletely different methodology of both approaches, mostly literature. Additionally, the accuracy factor, E , given by the representation of θ as the PDF in the Q-B method. This Eq. 17 provides a measure for the relative goodness of all might suggest that the uncertainties calculated by the Q-B assessments (Figs. 3, 4). For all results E is much below method are more realistic than those calculated by the ana- 1; this indicates only small deviations from the reference lytical method. Fig. 3 E values for neutron radiation 1 3 202 Radiation and Environmental Biophysics (2018) 57:195–203 Fig. 4 E values for gamma radiation for all comments and Ms. Aleksandra Powojska for checking the English Conclusion in this manuscript. In the case of mixed radiation exposure, the iterative method Open Access This article is distributed under the terms of the Crea- is widely used for biodosimetry. In the present study, two tive Commons Attribution 4.0 International License (http://creat iveco alternative statistical methods of estimating neutron and mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- tion, and reproduction in any medium, provided you give appropriate gamma absorbed doses in a mixed radiation field from meas- credit to the original author(s) and the source, provide a link to the ured chromosomal aberration frequencies were investigated. Creative Commons license, and indicate if changes were made. Both the analytical and the Q-B method proposed here have their own advantages and disadvantages. The first-proposed References method is based on the classical iterative approach and is nothing more than its analytical description; for this reason Ainsbury EA, Vinnikov VA, Puig P, Higueras M, Maznyk NA, Lloyd the results obtained are the same, as shown in Table 1. Thus, DC, Rothkamm K (2013a) Review of Bayesian statistical analysis methods for cytogenetic radiation biodosimetry, with a practical the iterative method can be replaced by the analytical one example. Radiat Prot Dosim 162:1–12 which is more convenient in use because it does not need Ainsbury EA, Vinnikov VA, Puig P, Maznyk NA, Rothkamm K, Lloyd to perform a series of iterations; instead it only requires to DC (2013b) CytoBayesJ: Software tools for Bayesian analysis of insert the appropriate data into the given formulas. This clas- cytogenetic radiation dosimetry data. Mutation Res 756:184–191 Brame RS, Groer PG (2003) Bayesian methods for chromosome dosim- sical method is a very simple and fast calculation method, etry following a criticality accident. Radiat Prot Dosim 104:61–63 but requires knowledge of the neutron/gamma dose ratio. In Fornalski KW (2014) Alternative statistical methods for cytogenetic the Q-B method, this parameter is described by the probabil- radiation biological dosimetry. Cornell University Library, arXiv. ity distribution (prior function), similarly to Bayesian meth- org/abs/1412.2048 International Atomic Energy Agency (2011) Cytogenetic dosimetry: ods. Therefore, it can be used when the exact dose ratio is applications in preparedness for and response to radiation emergen- unknown without requiring complicated Bayesian statistics. cies. IAEA in Austria, September 2011. https://www -pub.iaea.org/ The statistical methods proposed in the present work were MTCD/publi catio ns/PDF/EPR-Biodo simet ry%25202 011_web.pdf programmed as computational algorithms which can easily Kellerer AM, Rossi HH (1978) A generalized formulation of dual radia- tion action. Radiat Res 75:471–488 be used in cytogenetic analyses. Additionally, the methods Pacyniak I, Fornalski KW, Kowalska M (2014) Employment of Bayesian are presented here in easy-to-use forms, which can be coded and Monte Carlo methods for biological dose assessment following even as Excel spreadsheet formulas. The required computer accidental overexposures of people to nuclear reactor radiation. In codes are provided as an electronic supplement material. proceeding of: the second international conference on radiation and dosimetry in various fields of research, University of Nis (Serbia), 27–30 May 2014, pp 49–52 Acknowledgements This paper is part of a master thesis written by the Pacyniak I, Fornalski KW, Kowalska M (2015) Alternatywne metody first author at the Physics Faculty at Warsaw University of Technology, obliczania dawek pochłoniętych w biologicznej dozymetrii promien- Poland, under scientific supervision of Dr. Maria Kowalska, Central iowania mieszanego n + γ. Postępy Fizyki Tom 66(zeszyt 1–4):13– Laboratory for Radiological Protection, as a part of scientific program 22 (in Polish) SP/J/16/143339/8, “Technologies Supporting Development of Safe Nuclear Power Engineering”. Authors wish to thank Dr. Maria Kowalska 1 3 Radiation and Environmental Biophysics (2018) 57:195–203 203 Szłuińska M, Edwards AA, Lloyd DC (2005) Statistical methods for bio- Voisin P, Roy L, Hone PA, Edwards AA, Lloyd DC, Stephan G, Romm logical dosimetry. In: G Obe, Vijayalaxmi S (eds) Chromosomal H, Groer PG, Brame R (2004) Criticality accident dosimetry by alterations. Methods, results and importance in human health. chromosomal analysis. Radia Prot Dosim 110:443–447 Springer, Berlin Heidelberg Voisin P, Lloyd D, Edwards A (1997) Chromosome aberrations scoring for biological dosimetry in a criticality accident. Radiat Prot Dosim 70:467–470 1 3

Journal

Radiation and Environmental BiophysicsSpringer Journals

Published: Jun 4, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off