PIECEWISE POLYNOMIAL APPROXIMATIONS TO THE ICRP 116 EFFECTIVE DOSE COEFFICIENTS: PHOTONS AND NEUTRONS

Abstract

We present fitting equations for estimating effective dose per unit fluence at any photon energy between 10 keV and 10 GeV and any neutron energy between 0.001 eV and 10 GeV. These new equations are based on the latest radiation protection quantities for external radiation exposure found in International Commission on Radiological Protection (ICRP) Publication 116 and incorporate the latest definition of effective dose as described in ICRP Publication 103. The ICRP 116 dose coefficients were fit to piecewise polynomial functions. A total of 8 irradiation geometries were considered: the six in ICRP 116 and two additional geometries presented elsewhere in the literature. The fitting functions generally reproduce the ICRP 116 data to within 3% or better. The functions were used to modify the Monte Carlo N-Particle radiation transport code version 6 (MCNP6) and were applied to a sample problem. The results are intended to be used as a basis for revising the American National Standards Institute/American Nuclear Society 6.1.1-1991 standard.

INTRODUCTION

The American Nuclear Society (ANS) standards committee working group ANS-6.1.1 published an American National Standards Institute (ANSI) standard in 1991 titled ‘Neutron and Gamma-Ray Fluence-to-Dose Factors’⁽¹⁾. This standard, which superseded an earlier version issued in 1977⁽²⁾, provides recommendations for computing effective dose equivalent for individuals exposed to external gamma-ray and neutron fields. The ANSI/ANS-6.1.1-1991 standard was meant to be used by shielding designers in cases where the particle fluence spectra are known and the irradiation conditions can be readily approximated by one of several standard exposure geometries. Emphasis was placed on gamma rays and neutrons because these are the types of radiation most commonly encountered by radiation workers. Specifically, ANSI/ANS-6.1.1-1991 provides equations for estimating the male–female averaged effective dose equivalent per unit fluence (or fluence-to-dose factors) for gamma rays with energy between 10 keV and 12 MeV and for neutrons with energy between 0.025 eV and 14 MeV. Effective dose equivalent can then be calculated by integrating the evaluated fluence-to-dose factors over the fluence spectrum.

The analytic equations in ANSI/ANS-6.1.1-1991 are based on fluence-to-dose factors taken from International Commission on Radiological Protection (ICRP) Publication 51⁽³⁾ which are tabulated at discrete particle energies. The rationale for using analytic equations was to provide a standard method for determining the fluence-to-dose factors at energies not listed in the tables. In such cases, one clear option is to interpolate the log–log data using a cubic spline. This approach works well as it offers a smooth function passing through all the data points. A drawback, however, is that cubic splines have a rather complicated explicit form (for $k$ data points there are $k-1$ intervals for a total of $4k-4$ spline coefficients). Fluence-to-dose factors are often tabulated at 50 or more particle energies so that more than 196 coefficients are needed to interpolate the data. For this reason, the authors of ANSI/ANS-6.1.1-1991 opted to fit the fluence-to-dose factors to simpler piecewise polynomial functions. The success of the analytic fits described in ANSI/ANS-6.1.1-1991 is such that they have since been incorporated into shielding codes such as the Monaco with Automated Variance Reduction using Importance Calculations (MAVRIC)⁽⁴⁾ module of the Standardized Computer Analyses for Licensing Evaluation (SCALE) 6.2⁽⁵⁾ package as well as the Monte Carlo N-Particle radiation transport code version 6 (MCNP6)⁽⁶⁾.

Despite its utility, the ANSI/ANS-6.1.1-1991 standard was formally withdrawn in 2001 and is now designated as a historical standard because the ANS working group failed to issue a revision within 10 years⁽⁷⁾. The field of radiation protection has since undergone several paradigm shifts as described below. As a result, the ANS aims to revive and revise the standard.

ANSI/ANS-6.1.1-1991 refers to effective dose equivalent as defined in ICRP Publication 26⁽⁸⁾. This quantity, however, was replaced in 1991 by effective dose upon the issuance of ICRP Publication 60⁽⁹⁾. ICRP 60 was superseded in 2007 by ICRP Publication 103⁽¹⁰⁾ which redefined effective dose as a gender-averaged quantity with new tissue-weighting factors. At the same time, the computational reference phantoms used for calculating the fluence-to-dose coefficients (previously called ‘factors’) were evolving⁽¹¹⁾ from the mathematical equation-based stylized phantoms to the more realistic, image-based voxel phantoms described in ICRP Publication 110⁽¹²⁾. It was not until 2010 that all of these changes were incorporated into new radiation protection quantities for external radiation exposure in ICRP Publication 116⁽¹³⁾. ICRP 116 superseded ICRP Publications 74⁽¹⁴⁾ and 51⁽³⁾. The fluence-to-dose coefficients tabulated in ICRP 116 represent the most current values recommended by the ICRP for radiation protection purposes and cover a wider range of particles, energies, and irradiation geometries than given in ICRP 74.

The purpose of the current paper is to provide updated equations similar to those in ANSI/ANS-6.1.1-1991 for estimating effective dose per unit fluence for photons with energy between 10 keV and 10 GeV and for neutrons with energy between 0.001 eV and 10 GeV. These equations were derived by fitting the ICRP 116 fluence-to-dose coefficients (henceforth referred to as dose coefficients) to piecewise polynomial functions.

MATERIALS AND METHODS

Data source

Dose coefficients for photons and neutrons were obtained from Annexes A.1 and A.5 of ICRP 116⁽¹³⁾. A corrigenda to Annex A.1 was published in 2015⁽¹⁵⁾ which resulted in small changes to the photon dose coefficients (maximum change was 1.7%). Annexes A.1 and A.5 list dose coefficients for several standard irradiation geometries consisting of idealized uniform parallel beams covering the whole body along the antero-posterior (AP), postero-anterior (PA), left-lateral (LLAT), right-lateral (RLAT) and 360° rotational (ROT) directions of incidence as shown in Figure 1. ICRP 116 also considers a fully isotropic (ISO) exposure geometry in which the particle fluence per unit solid angle is independent of direction and location. As such, the ISO exposure geometry represents the case where a worker is immersed in a uniform cloud of radioactive gas. The ICRP 116 dose coefficients are normalized by the incident particle fluence and are tabulated in units of pSv cm² for 55 photon energies between 10 keV and 10 GeV and 68 neutron energies between 0.001 eV and 10 GeV.

Figure 1.

The 8 standard irradiation geometries considered in this work: antero-posterior (AP), postero-anterior (PA), left-lateral (LLAT), right-lateral (RLAT), 360° rotational (ROT), fully isotropic (ISO), cranial (CRA) and caudal (CAU). The images are of the adult male hybrid phantom developed by the National Cancer Institute and University of Florida⁽¹⁶⁾. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

The ICRP 116 dose coefficients are based on Monte Carlo calculations performed using the ICRP voxel phantoms representing the adult reference male and reference female. These phantoms contain over 140 segmented organs and tissues as described in ICRP 110⁽¹²⁾. For validation purposes, the ICRP Dose CALculation (DOCAL) task group independently calculated the dose coefficients using different Monte Carlo radiation transport codes. The photon dose conversion coefficients were calculated using the codes EGSnrc, MCNPX and GEANT4. The neutron dose conversion coefficients were calculated using the codes PHITS, FLUKA, MCNPX and GEANT4. In general, good agreement was observed between the codes within the statistical uncertainties of the calculations (typically <5%). The dose coefficients tabulated in ICRP 116 were calculated by averaging the values computed using each Monte Carlo code and then smoothing the data using a cubic spline. The resulting spline coefficients were not presented as reference data. The ICRP 116 dose coefficients are defined as recommended reference values. This means that the coefficients should be taken as exact values and, by definition, do not have a specified uncertainty (despite the fact that the basis of these values include sources of uncertainty).

For completeness, we also considered cranial (CRA) and caudal (CAU) irradiation geometries for both photons and neutrons (Figure 1). The dose coefficients for these irradiation geometries, which were not included in ICRP 116, were taken from two recent publications by Veinot et al. (2015)⁽¹⁷⁾ and Veinot et al. (2016)⁽¹⁸⁾. The CRA and CAU dose coefficients were calculated using methods similar to ICRP 116 (but with only the MCNP6 code) and were tabulated at the same photon and neutron energies. One minor discrepancy is that ICRP 116 mistakenly lists one of the prominent gamma-ray lines of ⁶⁰Co as having energy 1.117 MeV, whereas Veinot et al.⁽¹⁷⁾ correctly uses 1.17 MeV. Personal communication with the ICRP revealed that this was only a typo. Unfortunately, this typo was not corrected in the 2015 corrigenda⁽¹⁵⁾.

Polynomial regression model

In this article, we derive piecewise polynomial fitting functions for the dose coefficients which approximate the interpolating cubic spline while having a simpler analytic form which will be more convenient for some applications. For our piecewise fit, we partition the energy domain into $n$ intervals. On each energy interval $i=1,2,\dots ,n$ our fitting function takes the following analytical form:

$$\begin{array}{cc}{h}_{p,g}(\overline{E})& =\exp (c_{i,m}{X}^m+c_{i,m-1}{X}^{m-1}\\ & +\dots +c_{i,1}X+c_{i,0}),\end{array}$$ (1)

where,

• $h_{p,g}\left(\overline{E}\right)$ = fluence-to-effective dose coefficient at energy $\overline{E}$ in units of pSv cm² for particles of type, p, and exposure geometry, g;

• $\overline{E}$ = kinetic energy of particles of type, p, in units of MeV;

• $X=\ln (\overline{E})$;

• $m$ = degree of the polynomial fit; and

• $c_{i,j}$ = jth polynomial coefficient determined by least-squares fitting of the data on energy interval, i.

The functional form described in equation (1) is the same as was used in ANSI/ANS-6.1.1-1991. ANSI/ANS-6.1.1-1991 considered piecewise fits with polynomials up to degree 4 and up to 3 energy intervals. In this work, we investigated fits with polynomials up to degree 5 and up to 5 energy intervals because the ICRP 116 dose coefficients cover a larger energy range. For thoroughness, we also explored another functional form for the dose coefficients that was first described by Harvey and Mill⁽¹⁹⁾ and which involved nine parameters determined by non-linear fitting over the whole energy range⁽²⁰⁾. However, the Harvey and Mill function was ultimately abandoned in favor of the piecewise polynomial functional form because we found that it offered similar or better performance in all cases. For clarity and brevity, this article focuses only on the results for the piecewise polynomial fits.

The piecewise polynomial fits to the photon and neutron dose coefficients were obtained using the following general approach:

1. A log–log transformation was applied to the dose coefficients tabulated in the literature at $k$ discrete energies.

2. To improve the data fitting, 10 new data points were uniformly added between each of the original log-transformed data using cubic-spline interpolation with the not-a-knot endpoint conditions⁽²¹⁾. These new points helped ensure that the analytical fit performed well throughout the entire energy domain, and not just at the original data points. The total number of data points, $l$, used for the polynomial regression was therefore $10\left(k-1\right)+k$.

3. The energy domain was partitioned into $n$ energy intervals, with the knot points selected from the set of $k$ discrete energies for which the dose coefficients were originally tabulated. The set of possible knot points has $k-2$ elements because the lower and upper endpoints are fixed for the first and last intervals, respectively. For the photon case, we also excluded the energies 0.511, 0.662, 1.17, 1.33 and 6.129 MeV as possible knots so that the interval endpoints would be at round values. Using block notation, the interval partitioning is represented as $\mathbf{X}={\left[{\mathbf{X}}_{\mathbf{1}}^{\mathit{T}}{\mathbf{X}}_{\mathbf{2}}^{\mathit{T}}{\mathbf{X}}_{\mathbf{3}}^{\mathit{T}}\cdots {\mathbf{X}}_{\mathbf{n}}^{\mathit{T}}\right]}^T,$ where the subvectors ${\mathbf{X}}_{\mathbf{i}}={\left[X_{i,1}{X}_{i,2}{X}_{i,3}\cdots X_{i,{\omega }_i}\right]}^T$ contain monotonically increasing entries which are the natural logarithm of the ${\omega }_i$ particle energies belonging to interval $i$. Similarly, we define a partitioning of the dose coefficients as $\mathbf{d}={\left[{\mathbf{d}}_{\mathbf{1}}^{\mathit{T}}{\mathbf{d}}_{\mathbf{2}}^{\mathit{T}}{\mathbf{d}}_{\mathbf{3}}^{\mathit{T}}\cdots {\mathbf{d}}_{\mathbf{n}}^{\mathit{T}}\right]}^T$, where each subvector ${\mathbf{d}}_{\mathbf{i}}={\left[d_{i,1}{d}_{i,2}{d}_{i,3}\cdots d_{i,{\omega }_i}\right]}^T$ has entries which are the natural logarithm of the corresponding dose coefficients at the energies in ${\mathbf{X}}_{\mathbf{i}}$.

4. Polynomials up to degree $m$ are fit to the cubic-spline data for each energy interval using least-squares optimization with a continuity constraint placed on the polynomials at the knot points. A continuity constraint was not placed on the derivatives at the knot points. Details on this method are provided in the next section. Using block notation, the polynomial coefficients for each interval are given by $\mathbf{c}={\left[{\mathbf{c}}_{\mathbf{1}}^{\mathit{T}}{\mathbf{c}}_{\mathbf{2}}^{\mathit{T}}{\mathbf{c}}_{\mathbf{3}}^{\mathit{T}}\cdots {\mathbf{c}}_{\mathit{n}}^{\mathit{T}}\right]}^T$, where each ${\mathbf{c}}_{\mathbf{i}}={\left[c_{i,0}{c}_{i,1}{c}_{i,2}\cdots c_{i,m}\right]}^T$ is a subvector containing the m + 1 polynomial coefficients for interval $i$.

5. The resulting analytical fit was evaluated at 10 000 points distributed logarithmically in the energy domain. Goodness of fit was determined by calculating the relative residuals at these points compared to cubic-spline interpolation of the log–log data. The spline was the same as that used to generate the intermediate points in step 2.

6. A brute force search was conducted for the optimal piecewise polynomial fit by iterating through all possible ways of partitioning the energy domain into $n$ intervals. The number of ways in which one can select $n-1$ knot points out of $k-2$ elements is given by the binomial coefficient $\left(k-2\right)!/[\left(n-1\right)!\left(k-n-1\right)!]$. For 4 intervals, the number of possible partitions was 17 296 for the photon case (excluding 5 energies) and 45 760 for the neutron case.

7. The optimal piecewise polynomial fit was chosen so as to minimize the fit criterion $\mathrm{\Psi }$ defined as follows:

   $$\mathrm{\Psi }=\frac{1}{2}(M^{*}+\hat{M}),$$ (2)

   where $M^{*}$ and $\hat{M}$ are the maximum and mean absolute relative residual in percent calculated across the entire energy domain, respectively.

Calculation formalism

The MATLAB (The Mathworks, Inc., Natick, MA) numerical computing software was used to perform the polynomial regression. (In this article, certain commercially available software are identified to foster understanding. Such identification does not imply recommendation or endorsement by the National Institutes of Health or National Institute of Standards and Technology, nor does it imply that they are the best available for the purpose.) For each possible energy partition, the function lsqlin was used to solve a constrained least-squares problem of the form:

$$\begin{array}{c}{\min }_{\mathbf{c}}\frac{1}{2}\left|\right|\mathbf{Ac}-\mathbf{d}|{|}_2^2\hfill \\ \mathrm{with}\mathrm{the}\mathrm{continuity}\mathrm{constraint}{\mathbf{A}}_{\mathbf{eq}}\mathbf{c}=0,\hfill \end{array}$$ (3)

where the vectors $\mathbf{c}$ and $\mathbf{d}$ are as described above. This optimization problem was solved using the interior-point algorithm with the default tolerance settings. The solution of equation (3) is the $n\left(m+1\right)\times 1$ vector $\mathbf{c}$ containing the coefficients for the optimal polynomials on each interval which fit the data in the least-squares sense while satisfying the continuity constraint. The optimization problem in equation (3) was setup through an appropriate choice of matrices $\mathbf{A}$ and ${\mathbf{A}}_{\mathbf{eq}}$. The matrix $\mathbf{A}$ is a $l\times n(m+1)$ block diagonal matrix given by the following equation:

$$\mathbf{A}=\left[\begin{array}{ccccc}{\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}({\mathbf{X}}_1)& 0& 0& \cdots & 0\\ 0& {\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}({\mathbf{X}}_2)& 0& \cdots & 0\\ 0& 0& {\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}({\mathbf{X}}_3)& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}({\mathbf{X}}_{\mathbf{n}})\end{array}\right],$$ (4)

where $l$ is the total number of data points, $m$ is the degree of the polynomial fit, $n$ is the number of energy intervals, the ${\mathbf{X}}_{\mathbf{i}}$ are the subvectors of the block vector $\mathbf{X}$ as previously described, and ${\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}(\mathbf{\alpha })$ refers to the Vandermonde matrix for any vector $\mathbf{\alpha }={\left[{\alpha }_1{\alpha }_2{\alpha }_3\cdots {\alpha }_j\right]}^T$ of length $j$ and is defined as follows:

$${\stackrel{-}{\mathbf{V}}}_{\mathbf{m}}(\mathbf{\alpha })=\left[\begin{array}{ccccc}1& {\alpha }_1& {\alpha }_1^2& \cdots & {\alpha }_1^m\\ 1& {\alpha }_2& {\alpha }_2^2& \cdots & {\alpha }_2^m\\ 1& {\alpha }_3& {\alpha }_3^2& \cdots & {\alpha }_3^m\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\alpha }_j& {\alpha }_j^2& \cdots & {\alpha }_j^m\end{array}\right]\equiv \left[\begin{array}{c}{\mathbf{V}}_{\mathbf{m}}({\alpha }_1)\\ {\mathbf{V}}_{\mathbf{m}}({\alpha }_2)\\ {\mathbf{V}}_{\mathbf{m}}({\alpha }_3)\\ ⋮\\ {\mathbf{V}}_{\mathbf{m}}({\alpha }_j)\end{array}\right].$$ (5)

Continuity at the interval endpoint points was enforced through a $(n-1)\times n(m+1)$ constraint matrix given by the following equation:

$${\mathbf{A}}_{\mathbf{eq}}=\left[\begin{array}{ccccc}{\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_1)& -{\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_1)& 0& 0& 0\\ 0& {\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_2)& -{\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_2)& 0& 0\\ 0& 0& {\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_3)& -{\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_3)& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & -{\mathbf{V}}_{\mathbf{m}}({\tilde{X}}_{n-1})\end{array}\right],$$ (6)

where, the ${\tilde{X}}_i$ denote the natural logarithm of the knot energies.

RESULTS AND DISCUSSION

Piecewise polynomial fits

We investigated piecewise polynomial fits using polynomials up to degree 5 and up to 5 energy intervals and recorded $M^{*},\hat{M},\mathrm{and}\mathrm{\Psi }$ for each case. In general, we observed that the residuals are improved by increasing the polynomial degree and by increasing the number of energy intervals. However, as seen in Figures 2 and 3, there was diminishing improvement when using polynomials with degrees higher than 4 and more than 4 energy intervals. We, therefore, selected these settings for our analytic fits in order to minimize the number of fit parameters. The optimal energy intervals varied with particle type and exposure geometry. Tables 1 and 2 provide the polynomial coefficients for our analytic fits of the dose coefficients for photons and neutrons, respectively. Illustrations of the analytic fits for each of the exposure geometries are shown in Figures 4 and 5. The relative residuals in the dose coefficients calculated using the analytic fits are shown in Figures 6–9. For the photon case, our fitting functions generally reproduce the data to within 3% or better. Similar performance was observed for the neutron case, except for the CRA and CAU exposure geometries for which the errors were as high as 6 and 7%, respectively, at some energies. The mean (standard deviation) of $\hat{M}$ across all geometries was 0.50% (0.14%) and 1.1% (0.26%) for the photon and neutron cases, respectively. These errors are acceptable in light of the fact that the tabulated coefficients were based on Monte Carlo calculations which in some cases had statistical errors as high as ~5%. The analytical fits in ANSI/ANS-6.1.1-1991 had a reported accuracy of 5 and 10% for photons and neutrons, respectively.

Figure 2.

Maximum relative residual in the photon dose coefficients as calculated using piecewise polynomial fits as a function of: (a) the polynomial degree using 4 energy intervals and (b) the number of energy intervals using polynomials of degree 4. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 3.

Maximum relative residual in the neutron dose coefficients as calculated using piecewise polynomial fits as a function of: (a) the polynomial degree using 4 energy intervals and (b) the number of energy intervals using polynomials of degree 4. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Table 1.

Polynomial fit coefficients for photons using polynomials with degree 4 and 4 energy intervals.

Photon energy interval, i	Polynomial coefficients
	hE($\overline{E}$) = exp(ci,4X4 + ci,3X3 + ci,2X2 + ci,1X + ci,0) pSv cm2, with $\mathrm{X}=\ln (\overline{E})$ and $\overline{E}$ in MeV
	ci,4	ci,3	ci,2	ci,1	ci,0
AP exposure
10 keV < $\overline{E}$ ≤ 60 keV	1.79206e−02	5.07505e−01	3.51950e+00	9.58716e+00	8.34462e+00
60 keV < $\overline{E}$ ≤ 1.5 MeV	2.72646e−02	7.21895e−02	−4.00876e−02	7.89978e−01	1.49025e+00
1.5 MeV < $\overline{E}$ ≤ 40 MeV	−8.73781e−03	6.67348e−02	−1.93658e−01	8.63207e−01	1.48715e+00
40 MeV < $\overline{E}$ ≤ 10 GeV	−2.52906e−06	5.21780e−03	−1.30408e−01	1.12864e+00	1.11776e+00
PA exposure
10 keV ≤ $\overline{E}$ ≤ 40 keV	−1.28988e+00	−2.25617e+01	−1.44219e+02	−3.98735e+02	−4.05015e+02
40 keV < $\overline{E}$ ≤ 0.2 MeV	−3.66734e−01	−3.36124e+00	−1.12227e+01	−1.52346e+01	−7.31742e+00
0.2 MeV < $\overline{E}$ ≤ 30 MeV	1.05692e−04	1.43217e−02	−7.73081e−02	8.64483e−01	1.33436e+00
30 MeV < $\overline{E}$ ≤ 10 GeV	−1.67936e−03	5.09057e−02	−5.90924e−01	3.17712e+00	−1.79033e+00
LLAT exposure
10 keV ≤ $\overline{E}$ ≤ 0.15 MeV	6.46764e−02	1.04513e+00	5.73972e+00	1.38278e+01	1.09613e+01
0.15 MeV < $\overline{E}$ ≤ 1.5 MeV	4.94681e−02	1.14866e−01	−3.90470e−02	9.37956e−01	1.15119e+00
1.5 MeV < $\overline{E}$ ≤ 100 MeV	−1.08020e−02	9.97585e−02	−3.10545e−01	1.13306e+00	1.11936e+00
100 MeV < $\overline{E}$ ≤ 10 GeV	−1.52971e−03	5.06977e−02	−6.38654e−01	3.71190e+00	−3.17705e+00
RLAT exposure
10 keV ≤ $\overline{E}$ ≤ 0.3 MeV	−3.43461e−02	−2.66898e−01	−5.36725e−01	1.01345e+00	1.40381e+00
0.3 MeV < $\overline{E}$ ≤ 1.5 MeV	1.89325e−01	2.91949e−01	−3.17803e−02	9.40528e−01	1.08940e+00
1.5 MeV < $\overline{E}$ ≤ 60 MeV	−8.82843e−03	8.34299e−02	−2.60780e−01	1.08804e+00	1.08649e+00
60 MeV < $\overline{E}$ ≤ 10 GeV	−1.57061e−03	5.20004e−02	−6.55144e−01	3.81602e+00	−3.35419e+00
ROT exposure
10 keV ≤ $\overline{E}$ ≤ 30 keV	−1.15146e−01	−1.79724e+00	−1.08301e+01	−2.85011e+01	−2.86971e+01
30 keV < $\overline{E}$ ≤ 0.15 MeV	−2.71345e−01	−2.75067e+00	−1.01161e+01	−1.52169e+01	−8.38780e+00
0.15 MeV < $\overline{E}$ ≤ 40 MeV	8.95307e−04	7.28264e−03	−7.42847e−02	8.73194e−01	1.30056e+00
40 MeV < $\overline{E}$ ≤ 10 GeV	−1.28974e−03	4.17916e−02	−5.18781e−01	2.99795e+00	−1.81646e+00
ISO exposure
10 keV ≤ $\overline{E}$ ≤ 30 keV	−1.24929e−01	−1.89274e+00	−1.10522e+01	−2.82818e+01	−2.80615e+01
30 keV < $\overline{E}$ ≤ 0.15 MeV	−2.55230e−01	−2.58412e+00	−9.47239e+00	−1.40998e+01	−7.86666e+00
0.15 MeV < $\overline{E}$ ≤ 40 MeV	1.25145e−03	6.07652e−03	−7.53498e−02	9.16475e−01	1.16375e+00
40 MeV < $\overline{E}$ ≤ 10 GeV	−5.73308e−04	2.35790e−02	−3.52459e−01	2.43656e+00	−1.21347e+00
CRA exposure
10 keV ≤ $\overline{E}$ ≤ 40 keV	8.11863e−01	1.33717e+01	8.06240e+01	2.12251e+02	2.03784e+02
40 keV < $\overline{E}$ ≤ 3 MeV	1.50061e−02	1.50040e−02	−9.17971e−02	1.08946e+00	4.73628e−01
3 MeV < $\overline{E}$ ≤ 100 MeV	−6.14364e−03	6.46867e−02	−2.31905e−01	1.21229e+00	4.72717e−01
100 MeV < $\overline{E}$ ≤ 10 GeV	−1.06893e−03	3.71901e−02	−5.03377e−01	3.48931e+00	−3.85302e+00
CAU exposure
10 keV ≤ $\overline{E}$ ≤ 40 keV	7.07653e−01	1.11885e+01	6.48724e+01	1.64253e+02	1.50258e+02
40 keV < $\overline{E}$ ≤ 1 MeV	3.04947e−02	8.84698e−02	3.23621e−02	1.19361e+00	−2.96157e−01
1 MeV < $\overline{E}$ ≤ 5 MeV	2.45940e−01	−7.75707e−01	7.06598e−01	9.95972e−01	−2.96157e−01
5 MeV < $\overline{E}$ ≤ 10 GeV	7.81416e−04	−2.06934e−02	1.46293e−01	6.00179e−01	2.89531e−01

Table 2.

Polynomial fit coefficients for neutrons using polynomials with degree 4 and 4 energy intervals.

Neutron energy interval, i	Polynomial coefficients
	hE($\overline{E}$) = exp(ci,4X4 + ci,3X3 + ci,2X2 + ci,1X + ci,0) pSv cm2, with $\mathrm{X}=\ln (\overline{E})$ and $\overline{E}$ in MeV
	ci,4	ci,3	ci,2	ci,1	ci,0
AP exposure
0.001 eV ≤ $\overline{E}$ ≤ 500 eV	2.51074e−04	1.32512e−02	2.42578e−01	1.85948e+00	7.13380e+00
500 eV < $\overline{E}$ ≤ 0.9 MeV	−3.89293e−03	−6.51919e−02	−2.61617e−01	5.48205e−01	5.68112e+00
0.9 MeV < $\overline{E}$ ≤ 150 MeV	−2.16227e−03	3.74430e−02	−2.49051e−01	6.29077e−01	5.68962e+00
150 MeV < $\overline{E}$ ≤ 10 GeV	2.79543e−02	−8.00455e−01	8.48513e+00	−3.92096e+01	7.24460e+01
PA exposure
0.001 eV ≤ $\overline{E}$ ≤ 5 keV	2.01039e−04	1.02475e−02	1.76788e−01	1.25946e+00	4.88644e+00
5 keV < $\overline{E}$ ≤ 1.5 MeV	1.94183e−03	3.72375e−03	−8.47487e−03	7.37584e−01	4.97996e+00
1.5 MeV < $\overline{E}$ ≤ 400 MeV	−1.87858e−03	4.15517e−02	−3.10627e−01	9.83065e−01	4.92769e+00
400 MeV < $\overline{E}$ ≤ 10 GeV	−8.91407e−03	3.04515e−01	−3.82532e+00	2.12186e+01	−3.76353e+01
LLAT exposure
0.001 eV ≤ $\overline{E}$ ≤ 500 eV	2.92852e−04	1.55245e−02	2.87145e−01	2.23677e+00	7.22935e+00
500 eV < $\overline{E}$ ≤ 0.7 MeV	−3.82951e−03	−6.39464e−02	−2.51555e−01	5.85509e−01	4.66236e+00
0.7 MeV < $\overline{E}$ ≤ 400 MeV	−1.15855e−03	2.64702e−02	−2.11988e−01	8.02127e−01	4.73865e+00
400 MeV < $\overline{E}$ ≤ 10 GeV	−1.55014e−02	4.99497e−01	−5.96623e+00	3.15853e+01	−5.63890e+01
RLAT exposure
0.001 eV ≤ $\overline{E}$ ≤ 500 eV	2.90196e−04	1.53620e−02	2.83832e−01	2.21011e+00	6.98660e+00
500 eV < $\overline{E}$ ≤ 0.9 MeV	−3.63442e−03	−6.05756e−02	−2.32188e−01	6.29815e−01	4.54024e+00
0.9 MeV < $\overline{E}$ ≤ 400 MeV	−9.16619e−04	2.34147e−02	−2.01781e−01	8.32350e−01	4.56134e+00
400 MeV < $\overline{E}$ ≤ 10 GeV	−1.57801e−02	5.10727e−01	−6.12429e+00	3.25231e+01	−5.83657e+01
ROT exposure
0.001 eV ≤ $\overline{E}$ ≤ 1 keV	2.21480e−04	1.15484e−02	2.06638e−01	1.53779e+00	5.60471e+00
1 keV < $\overline{E}$ ≤ 0.5 MeV	−4.31283e−03	−7.21095e−02	−3.04718e−01	4.14299e−01	4.99341e+00
0.5 MeV < $\overline{E}$ ≤ 130 MeV	1.44204e−03	5.77617e−03	−1.66846e−01	6.91776e−01	5.14411e+00
130 MeV < $\overline{E}$ ≤ 10 GeV	9.91467e−03	−2.84329e−01	3.02941e+00	−1.39575e+01	2.94223e+01
ISO exposure
0.001 eV ≤ $\overline{E}$ ≤ 2 keV	1.99009e−04	1.02971e−02	1.81545e−01	1.32220e+00	4.65049e+00
2 keV < $\overline{E}$ ≤ 0.3 MeV	−5.22289e−03	−8.92349e−02	−4.12414e−01	1.52315e−01	4.51769e+00
0.3 MeV < $\overline{E}$ ≤ 100 MeV	2.62399e−03	−3.88017e−03	−1.36617e−01	6.94491e−01	4.90314e+00
100 MeV < $\overline{E}$ ≤ 10 GeV	6.06836e−03	−1.72306e−01	1.82722e+00	−8.29768e+00	1.95654e+01
CRA exposure
0.001 eV ≤ $\overline{E}$ ≤ 10 keV	1.80697e−04	9.34506e−03	1.67173e−01	1.24213e+00	2.07707e+00
10 keV < $\overline{E}$ ≤ 50 MeV	5.11516e−03	1.93689e−03	−1.47486e−01	6.47558e−01	3.06930e+00
50 MeV < $\overline{E}$ ≤ 500 MeV	7.51009e−02	−1.46236e+00	1.03825e+01	−3.12447e+01	3.79569e+01
500 MeV < $\overline{E}$ ≤ 10 GeV	9.79103e−03	−2.37728e−01	1.96642e+00	−5.43012e+00	6.05623e+00
CAU exposure
0.001 eV ≤ $\overline{E}$ ≤ 1 keV	2.17037e−04	1.13342e−02	2.05298e−01	1.55912e+00	4.01875e+00
1 keV < $\overline{E}$ ≤ 1 MeV	−5.24528e−03	−8.66156e−02	−3.63719e−01	4.67649e−01	3.78218e+00
1 MeV < $\overline{E}$ ≤ 400 MeV	−9.43058e−03	1.14197e−01	−4.52364e−01	9.74242e−01	3.78218e+00
400 MeV < $\overline{E}$ ≤ 10 GeV	3.07730e−02	−8.62204e−01	8.86641e+00	−3.90656e+01	6.73531e+01

Figure 4.

Effective dose coefficients for photons for the 8 exposure geometries. The data points represent the tabulated dose coefficients taken from ICRP 116⁽¹⁵⁾ or Veinot et al.⁽¹⁷⁾. The lines represent the analytic functions described in Table 1. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 5.

Effective dose coefficients for neutrons for the 8 exposure geometries. The data points represent the tabulated dose coefficients taken from ICRP 116⁽¹³⁾ or Veinot et al.⁽¹⁸⁾. The lines represent the analytic functions described in Table 2. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 6.

Comparison between the analytic function for the photon dose coefficients described in Table 1 and a cubic spline interpolating the tabulated data in the literature for the AP, PA, LLAT and RLAT exposure geometries. The data points represent the photon energies taken from the dose coefficient tables in ICRP 116⁽¹⁵⁾. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 9.

Comparison between the analytic function for the neutron dose coefficients described in Table 2 and a cubic spline interpolating the tabulated in the literature for the ROT, ISO, CRA and CAU exposure geometries. The data points represent the neutron energies taken from the dose coefficient tables in ICRP 116⁽¹³⁾ or Veinot et al.⁽¹⁸⁾. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 7.

Comparison between the analytic function for the photon dose coefficients described in Table 1 and a cubic spline interpolating the tabulated in the literature for the ROT, ISO, CRA and CAU exposure geometries. The data points represent the photon energies taken from the dose coefficient tables in ICRP 116⁽¹⁵⁾ or Veinot et al.⁽¹⁷⁾. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Figure 8.

Comparison between the analytic function for the neutron dose coefficients described in Table 2 and a cubic spline interpolating the tabulated in the literature for the AP, PA, LLAT and RLAT exposure geometries. The data points represent the neutron energies taken from the dose coefficient tables in ICRP 116⁽¹³⁾. This figure can be viewed in color in the online version of Radiation Protection Dosimetry.

Computing effective dose

A radiation worker may be exposed to both photon and neutrons fields. If the incident particle fluence spectra are known, then the effective dose, $E$, in units of mSv can be calculated as follows:

$$\begin{array}{c}E={10}^{-9}{\sum }_{\mathrm{particle}\mathrm{type}p}{\sum }_{\mathrm{geometry}\mathrm{type}g}\hfill \\ \times {\int }_{{\overline{E}}_{low,p,g}}^{{\overline{E}}_{high,p,g}}{h}_{p,g}(\overline{E}){\phi }_{p,g}(\overline{E})\mathrm{d}\overline{E},\hfill \end{array}$$ (7)

where,

• $h_{p,g}\left(\overline{E}\right)$ = fluence-to-effective dose coefficient for particles of type, p, with energy, $\overline{E}$, and exposure geometry, g, in units of pSv cm²;

• ${\phi }_{p,g}\left(\overline{E}\right)$ = incident fluence energy spectrum in units of cm⁻² at energy, $\overline{E}$, for particles of type, p, and exposure geometry, g;

• ${\overline{E}}_{low,p,g}$ = lowest energy (MeV) in the incident spectrum of particle type, p, for exposure geometry, g; and

• ${\overline{E}}_{high,p,g}$ = highest energy (MeV) in the incident spectrum of particle type, p, for exposure geometry, g.

The values of the photon or neutron dose coefficients can be readily evaluated using the analytical fits described in this article for the appropriate choice of irradiation geometry. If the irradiation geometry is unknown, then a conservative approach is to choose the geometry which results in the largest effective dose. For photons and neutrons below 10 MeV this is the AP exposure geometry. The analytic functions provided in this article assume a uniform whole-body exposure and should not be used for the case of partial-body exposures.

EXAMPLE DOSE CALCULATION

As an example, we demonstrate how to calculate the dose rate received by a worker facing a 5.0 × 10³ MBq source of ¹³⁷Cs at a distance of 1 m. We assume that ¹³⁷Cs results in a 0.662 MeV gamma ray in 85% of the decays. We also assume that the source can be approximated as a point and that it is contained inside a spherical tungsten container (density 19.3 g cm⁻³) with a radius of 0.3 cm. We compute the worker dose rate using two methods: (1) Hand calculation using a look-up table of buildup factors obtained from the literature and (2) Monte Carlo radiation transport simulation using MCNP6⁽⁶⁾. For the Monte Carlo approach, the analytic functions from this article were incorporated directly into the MCNP6 source code. The source code was then compiled using GNU FORTRAN (GCC 4.7.0) to produce a customized executable with eight new standard dose functions (DFs) for each of the exposure geometries. The new dose functions could be selected using the DF card in MCNP6 with the IC parameter set to an integer between 61 and 68 (for AP, PA, LLAT, RLAT, ROT, ISO, CRA and CAU geometries, respectively). As in the standard version of MCNP6, the ANSI/ANS-6.1.1-1991 DFs could be called by setting the IC parameter to an integer between 31 and 34 (for AP, PA, LAT or average of LLAT and RLAT, and ROT geometries, respectively).

Hand calculation

The particle fluence as a function of radial distance $r$ is given by $\phi \left(r\right)=BS\exp (-\mu r)/4\pi r^2$, where B is the buildup factor, S is the source strength and $\mu$ is the linear attenuation coefficient for 0.662 MeV gamma rays in tungsten. From the NIST XCOM database⁽²²⁾ we see that the total mass attenuation coefficient for tungsten without coherent scattering is 9.22 × 10⁻² cm² g⁻¹. When we multiply this value by the density of tungsten we find that the linear attenuation coefficient is (9.22 × 10⁻² cm² g⁻¹)(19.3 g cm⁻³) = 1.78 cm⁻¹. The tungsten shielding, therefore, has a thickness of (0.3 cm)(1.78 cm⁻¹) = 0.534 mean free paths. Using tables in the literature, we can then estimate the buildup factor for 0.662 MeV gamma rays in 0.534 mean free paths of tungsten as 1.21⁽²³⁾. We find that the particle fluence at 1 m is as follows:

$$\begin{array}{c}\phi (100\mathrm{cm})=\frac{\left(1.21\right)\left(5.0\times {10}^9\mathrm{Bq}\right)\left(0.85\right)\left(3600\frac{\mathrm{s}}{\mathrm{h}}\right)}{4\pi {(100\mathrm{cm})}^2}\hfill \\ \times \exp (-(1.78{\mathrm{cm}}^{-1})(0.3\mathrm{cm}))\hfill \\ \approx 8.637\times {10}^7{\mathrm{h}}^{-1}{\mathrm{cm}}^{-2}\end{array}$$

In order to calculate dose rate using equation (7) we also need the fluence-to-effective dose coefficient for 0.662 MeV photons. Using the analytic fit in Table 1 for the AP exposure geometry we find that this is 3.169 pSv cm². Hence, using equation (7) we find that the estimated dose rate is (10⁻⁹ mSv/pSv)(3.169 pSv cm²)(8.637 × 10⁷ h⁻¹ cm⁻²) = 0.274 mSv h⁻¹.

MCNP6 calculation

An input deck was generated to run the example problem in MCNP6 with 15 million source particles. The average effective dose rate per particle emitted at 1 m was calculated using a F2 surface flux tally with a DF modifier for the AP exposure geometry. The result of the modified tally was 6.266 × 10⁻¹⁴ Sv h⁻¹ per particle emitted per second with a negligible statistical uncertainty. This result was then multiplied by the source activity in becquerels and the branching fraction to yield a dose rate of 0.266 mSv h⁻¹. For comparison, the result when using the ANSI/ANS-6.1.1-1991 analytic functions (IC parameter on the DF card set to 31) was 0.274 mSv h⁻¹. The small difference can be explained in part by the fact that the AP dose coefficient for 0.662 MeV gamma rays is 1.7% larger when using the ANSI/ANS-6.1.1-1991 analytic function compared to the function in Table 1 of this article. The analytic function in Table 1 accurately reproduces the dose coefficient tabulated at 0.662 MeV in ICRP 116 to within 0.04%.

CONCLUSIONS

Piecewise polynomial functions were derived for estimating the effective dose coefficients for photons with energy between 10 keV and 10 GeV and for neutrons with energy between 0.001 eV and 10 GeV. Piecewise polynomial functions were provided for 8 different exposure geometries. These analytic functions are analogous to the functions provided in ANSI/ANS-6.1.1-1991 standard except that they cover a wider energy range, more exposure geometries and incorporate the most recent ICRP 103 definition of effective dose. The fits described in this work are based on data taken from the literature which were calculated using the most recent ICRP 110 reference phantoms. These results may be used as a basis for revising the ANSI/ANS-6.1.1-1991 historical standard. The fitting functions were incorporated directly into the MCNP6 as selectable DFs for modifying a flux tally and applied in a sample problem involving a point source. Our implementation of the functions in MCNP6 was tested through a comparison with a direct calculation. In the future, the modified MCNP6 code can be used to estimate the effective dose rate around more complicated systems, such as a nuclear reactor, without the need to use a computational phantom.