Monte Carlo simulation of dose coefficients for a fish eye lens model exposed to monoenergetic electrons

Journal of Environmental Radioactivity 199–200 (2019) 7–15

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Monte Carlo simulation of dose coefficients for a fish eye lens model exposed to monoenergetic electrons

T

P. Nogueiraa,∗, M. Hillerb, M.-O. Austa a b

Thünen Institute of Fisheries Ecology, Herwigstrasse 31, 27572, Bremerhaven, Germany Stolberg, Germany

A B S T R A C T

Vision is an important sense for the majority of the wildlife species, affecting their ability to find food and escape predation. Currently, no study on radiation induced cataract frequency on the fish eyes lens has been done. However, any thorough future study of this subject will require more accurate dose estimates for the fish eye lens than those currently available. For this purpose, the eye lens absorbed dose per unit fluence conversion coefficients for electron irradiation were calculated using the MCNPX Monte Carlo radiation transport code package. All results were validated against three different fish voxel models. The discrepancies between model results mainly originate from the different fish eye dimensions used in the different studies and in two of the cases the lack of a defined eye lens region. The dose conversion coefficients calculated in this work can be used to estimate the dose to the fish eye lens based on the activity concentration of the surrounding water. The model developed in this work has also demonstrated that the mathematical models still have several advantages over the voxel models.

1. Introduction Vision is an important sense for the majority of wildlife species, affecting their ability to find food and escape predation. Thus, diseases that affect the vision such as cataracts, a clouding of the eye lens that reduces the light transmission to the retina (Riordan-Eva and Cunningham, 2011), may reduce the fitness of wild life species. For example, fishes affected by cataracts are unable to find food (Wall and Richards, 1992). However, cataracts are rare in wild animals in comparison with domestic animals, possibly due to the decrease in fitness (Keymer, 1977; Mousseau and Møller, 2013). This also applies to fish, with exception of the blind species living in the deep ocean, deep channels, and underground waters (Guthrie, 1986). The etiology of cataracts is not fully understood, however several studies demonstrated that they can be caused by ultraviolet radiation, dehydration, malnutrition, and oxidative stress sources such as ionizing radiation (Kumar et al., 2013; Minassian et al., 1989; Otake et al., 1991; Taylor et al., 1988). The radiation-induced posterior subcapsular cataract in humans have long been documented as a major ocular complication (Rohrschneider, 1932). The lens of the eye is considered as one of the most radiosensitive human tissues and several epidemiological studies indicate that the threshold for cataract induction may be much lower than the 1.5 Gy assumed before from the studies of Miller et al. (1967), Nefzger et al. (1969) and Otake and Schull (1982), or that there may be no threshold at all (Ainsbury et al., 2009; Nogueira et al., 2011). These

∗

findings motivated the recent developments of updated and newly developed eye lens models and dosimeters for the human eye, e.g. Behrens et al. (2009), Landauer (2016), Nogueira et al. (2011). In the past five years, various studies have been dedicated to analyse the relation between radiation-induced cataracts and the exposure ionizing radiation for wildlife. Mousseau and Møller (2013) studied the frequency of cataracts in birds from the Chernobyl exclusion zone and found not only an elevated frequency of cataracts but also a relationship between bird abundance, background radiation, and cataracts. Based on the work of Møller and Mousseau (2007a)and Møller and Mousseau (2007b) and, Mousseau and Møller (2013) suggest also that ionizingradiation, by increasing the frequency of cataracts and other radiation exposure effects, is decreasing the bird abundance in the regions with higher background radiation levels. Similarly, Lehmann et al. (2016) found an elevated frequency of cataracts in eyes of bank voles (Myodes glareolus) collected from the natural population in the Chernobyl Exclusion Zone. Note however, that the methods employed in the previous works of Mousseau and Møller have been questioned by other authors, e.g. Beresford et al. (2012), Garnier-Laplace et al. (2013) and Deryabina et al. (2015). To the best of the authors knowledge, there are no studies on the frequency of radiation induced cataracts in wild aquatic animals. There are however several studies on ionizing radiation induced cataracts in laboratory experiments with zebrafish, e.g. Geiger et al. (2006), and cataractogenesis in farm fish due to other effects such as Ultraviolet radiation exposure; nutritional levels of methionine, tryptophan,

Corresponding author. E-mail address: [email protected] (P. Nogueira).

https://doi.org/10.1016/j.jenvrad.2018.12.021 Received 10 July 2018; Received in revised form 20 December 2018; Accepted 20 December 2018 0265-931X/ © 2018 Published by Elsevier Ltd.

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images of wildlife, were developed and currently several of the ICRP RAP models have one or more voxel model equivalents. From these voxel models, three models were developed for fish: 1) Caffrey (2012) developed a voxel model of a flat fish and used it to perform absorbed fraction calculations of both internal and external irradiations; 2) Hess (2014) and Ruedig et al. (2014) developed a voxel model of a trout based on MRI images of a specimen caught at the Rainbow Trout Farm in Sandy, OR, USA, in the segmentation process of this model, the entire trout eye was defined as one; 3) in parallel to the previous work Martinez et al. (2014) developed a voxel model of a rainbow trout, in this model both the eye regions and eye lens were separately identified. As previously mentioned, small organs representation in a voxel model, are greatly dependent on the size of the voxels and may not be very well represented by them. Despite the great resolution of the fish voxel models developed, this could be to some extent the case for the fish eye lens. Following the ICRP Publication 110 (2010) approach for the human eye lens dose coefficients calculation, a mathematical model for the fish eye lens was developed to address this issue and improve the dosimetry. To simulate the transport of the beta particles through the matter and score the energy deposition, the Monte Carlo code MCNPX was used. Generally the transport of beta particles in Monte Carlo simulations uses several approaches to reduce the computational time, the MCNPX uses the “condensed history” method developed by Berger (1963) and implemented in the code through a class I scheme (Kawrakow, 2000; Reynaert et al., 2002). Using the mathematical model of the fish eye presented in this work, absorbed dose in fish eye lens in an external exposure scenario were calculated. The fish eye is simulated to be exposed to an electron radiation source homogeneously distributed in the surrounding water. To evaluate the water content on the self-absorption of the electron source, and the hypothesis that an increase of the water contents concentration will decrease the dose in the eye lens, three different water compositions, and two densities were used. In addition to the mathematical model developed, further calculations were performed using the voxel model for the rainbow trout developed by Martinez et al. (2014) for an external exposure scenario. Dose coefficients obtained with the mathematical model were compared to dose coefficients determined with the voxel model.

riboflavin, zinc, manganese and the amino acid histidine, e.g. Bjerkas (2010), Waagbo et al. (2010), Remø et al. (2014) and Remø et al. (2017). Beta particles are of main concern when addressing directly ionizing radiation to the eye lens, due to their high local energy deposition, that results in high, localized doses. In the terrestrial environment, beta particles are slightly attenuated by the air, with beta particles from typical sources having ranges of a few hundreds of centimetres in air. As an example, a beta particle resulting from the 90Sr decay chain (0.546 MeV) has a range of about 180 cm. In the aquatic environment however, due to attenuation in water, a beta particle with the same energy has a range of only 0.195 cm. A common approach for the calculation of dose coefficients is the use of anthropomorphic computational phantoms coupled with particle transport using Monte Carlo calculations. This allows for a computational model precisely representing absorption properties, anatomical dimensions and geometries. In the past, very simple shapes have been used to define the phantoms. For humans, e.g. the Medical Internal Radiation Dose (MIRD) phantom family was mainly defined by basic geometric shapes like spheres, cuboids, and cylinders (Snyder et al., 1969). Thanks to the increase in available computational power, the complexity of the phantoms has greatly increased over the past 40 years. Nowadays, anthropomorphic computational models are based on computed tomography (CT) and magnetic resonance imaging (MRI) scans, and thus allow for the definition of the anatomical geometry with great precision. There are however some exceptions, in particular, the representation of small organs, like the eye lens and thin tissues, like skin. Those organs are not very well represented in the voxel phantoms, as the voxel dimension exceeds the size of the organ (in case of the eye lens) or the thickness of the tissue (in case for the skin) (Kim et al., 2016; Yeom et al., 2013). Because of these limitations, the International Commission on Radiological Protection (ICRP) is using and recommending a mathematical eye lens model as reference model for dosimetric calculations to the lens of the eye (ICRP 116, 2010). "If the man is adequately protected then the other living things are also protected”, a statement that ruled ICRP recommendations since ICRP Publication 26 (1977), has been replaced by a dedicated radiation protection scheme for the environment with ICRP Publication 108 (2008a). The main objective of the radiological protection system for the environment is to prevent or reduce the frequency of deleterious radiation effects to insure the biological diversity, the health and status of natural habitats, communities and ecosystems (Pentreath, 2009). For this purpose, reference animals and plants (RAP) were defined. The list of RAP include deer, rat, duck, bee, worm, pine tree, and grass for the terrestrial environment; trout, flat fish, crab, frog, and seaweed for the aquatic environment. Computational models were shown for all RAP in ICRP Publication 108 and used to calculate the absorbed fraction and the corresponding dose coefficients for 75 radionuclides. Based on the same simplicity that ruled the early anthropomorphic human phantoms, all computational models for RAP were defined using simple, mostly ellipsoidal, bodies. This simplistic approach is very effective in terms of time-, work- and cost-saving when in use, however it undermines the anatomical accuracy of the dosimetric calculations. For radiation exposure, this inaccuracy can result: 1) In an underestimation of dose and the derived risk factors, that ultimately might result in an inadequate protection of the wild life. 2) In an overestimation of the dose to an organism, that might lead to a higher than necessary cost in a possible remediation of the environment (Ruedig et al., 2014). The concepts laid out for the protection of the environment made in ICRP Publication 108 (2008a) were extended and published in ICRP 136 (2017), which supersedes ICRP Publication 108 (2008a) by introducing two main changes. Firstly, the nuclear decay data was updated using the values published ICRP Publication 107 (2008b) and secondly, the radioactive progeny was considered in the calculation of dose coefficients. Over the last decade, complex voxel models, based on CT and MRI

2. Materials and methods 2.1. Geometry and elemental composition of the mathematical fish eye model The geometry of the mathematical fish eye model was based on a fish eye schema (Fish, 2012). The dimensions were based on the Devlin et al. (2012) allometric study of the relation between the lateral eye diameter and fish length for the coho salmon (Oncorhynchus kisutch) and the ICRP reference trout length. Devlin et al. (2012) was selected as it provided a smaller relative deviation of the relation between the lateral eye diameter and the fish length than the Howland et al. (2004) allometric study, when both allometric models were applied to estimate the lateral eye diameter in the voxel fish models of Caffrey (2012), Hess (2014) and Martinez et al. (2014). The fish model was simulated to be immersed in a water body with a dimension of 30.00 × 30.00 × 30.00 cm3. A total of three different water compositions were investigated: 1) pure water, 2) the mean composition of saltwater according to Castro and Huber (2010) and 3) the mean composition of river water in Europe according to Holland (1978). For the pure water and river water, the water density was assumed to be 1 kg m−3, for the saltwater, the water density was set after Beicher (2000) to be 1.030 kg m−3. The elemental composition and densities used for the definition of the tissues of the eye, as well as additional materials used in the Monte Carlo simulations are shown in Table 1. The values used for the eye lens 8

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Table 1 Elemental composition and density of the tissues used in the fish eye computational model, as well as additional materials used in the Monte Carlo simulations. The colours in the last column refer to the colours in Figs. 1 and 2. Region

Cornea Anterior chamber Eye Lens Vitreous humor Retina and Choroid Sclera Skin Water, Europea Water, Pure Water, Rivera Water, Salta a

Density

1.06 1.00 1.07 1.00 1.07 1.07 1.09 1.00 1.00 1.00 1.00

Elemental composition (weight fraction in %) H

C

N

O

Na

P

S

Cl

K

10.3 11.2 9.6 11.2 10.0 10.0 10.0 11.2 11.2 11.0 10.7

10.9 – 19.5 – 14.6 14.6 19.9 – – – –

3.5 – 5.7 – 4.5 4.5 4.2 – – – –

75.1 88.8 64.6 88.8 70.6 70.6 65.0 88.8 88.8 89.0 86.0

– – 0.1 – – – 0.2 – – – 1.1

– – 0.1 – – – 0.1 – – – –

0.2 – 0.3 – 0.3 0.3 0.2 – – – –

– – 0.1 – – – 0.3 – – – 1.9

– – – – – – 0.1 – – – 0.1

Mg

Colour

– – – – – – – – – – 0.1

Blue Light Blue Green Light Blue Orange Red Yellow – – – –

Elements with weight fractions < 0.1% not shown.

experience by the authors (Nogueira et al., 2011). The default MCNPX cross section libraries MCPLIB04 for photons and EL03 for electrons were used (Pelowitz, 2008).

were taken from ICRP Publication 89 (2002), the remaining tissues were based on ICRP Publication 23 (1975) and ICRU Report 46 (1992). The eye model was integrated into the ICRP Publication 108 (2008a) reference trout model at approximated positions of the eye in the trout, (Fig. 2). The trout model is represented by an ellipsoid with dimensions of 50 × 8 × 6 cm3. This allows for modelling the attenuation of the radiation by the body of the fish, compared to the eye model immersed in the water solely. All three voxel fish models discussed above include a representation for the fish eye. In terms of mass and volume, these models are significantly different from the model developed in this work, though. The main characteristics and dimensions of these eye models are compared to the eye model developed here (Table 2).

2.2.1. Voxel phantom calculations For validation purposes, the absorbed dose in the eye and eye lens were also calculated using the rainbow trout voxel model by Martinez et al. (2014) in the scenario with pure water (Fig. 3). The eye in this model is sectioned into the lenses of the eye and the rest of the eye. The full, uncompressed voxel model with a dimension of 0.0146484 × 0.0146484 × 0.1 cm3 per voxel was employed. The voxel lattice consists of 299,892,737 voxels in total with 9,773,093 (3.3%) being tissue voxels, with the remainder surrounding voxel's being filled with water. The eye lens consists of a total of 5274 voxels and the eye (without the lens) of 94,643 voxels. The voxel phantom was immersed in a water body with a dimension of 115.00 × 115.00 × 129.00 cm3, large enough to surround the fish model with a layer of water, at least three times the mean free path of the source particles, to ensure backscatter is included. Scoring was performed employing the same tally type as for the mathematical model (section 2.2).

2.2. Monte Carlo simulations The Monte Carlo calculations were performed using the general purpose Monte Carlo particle transport code MCNPX, version 2.6 (Pelowitz, 2008) in coupled photon:electron transport mode. The computation of the absorbed dose was performed using a surface crossing estimator that computes the energy of the particles leaving the region of interest and subtracts it from the energy of the particles entering the region of interest (MCNP *F8 tally). To limit the computational time, an energy cut-off of 10 keV for electrons and 1 keV for photons was used. The number of electron substeps per energy step (ESTEP) used was kept at the MCNPX default of 3. The number of particles simulated used variated between 1·108 and 2·109. No variance reduction techniques were used. The Integrated Tiger Series (ITS) style electron energy indexing algorithm was used in view of its superiority over the traditional MCNPX method (Chibani and Li, 2002; Jeraj et al., 1999) and previous

2.3. Relative deviation The relative deviation was used to compare the results calculated using different water compositions, densities as well as the differences between the mathematical model and the voxel model results. The relative deviation, rel, was calculated using equation (1):

rel = 100⋅

(a − b) b

(1)

With a and b the values to compare. Fig. 1. Model of the Fish eye developed in this work as represented in its implementation for MCNPX (left) based on a general schema of a fish eye (right), adapted from http:// www.cyphos.com/forums/showthread.php?t=29128 and the Devlin et al. (2012) allometric study of the relation between the length and lateral eye diameter.

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Fig. 2. Coronal section (left) and transverse section (right) MCNPX geometry plots of the fish eye model, developed in this work, integrated into ICRP Publication 108 (2008a) reference trout model.

of the volume of a sphere and the sphere diameter. Then, the Devlin et al. (2012) allometric function used in this work was also used to calculate the voxel eye models lateral diameters (Table 3). The relative deviation between the values varies between 7% and 26%, and shows that the Devlin et al. (2012) allometric model overestimates the lateral eye diameter. This however, can be due to an inappropriate estimation of the eye lateral diameter based on the geometrical function of a sphere. Furthermore, a possible overestimation of the eye lateral only explains part of volume difference. Taking into account that the length of the fish in the voxel models is significantly smaller than the ICRP reference length, it seems likely that the main reason for the discrepancies found between this works model and the other models is the use of small fishes for the model development, and consequently smaller eyes. Because of the beta particles low penetration ability, the absorbed dose per unit fluence varies significantly with depth and results in a non-uniformity in the spatial distribution of the radiation on the eye lens (Nogueira et al., 2011). As a consequence of the non-uniformity, for low energy electrons, the variation in the volume of the eye lens will have a significant impact on the absorbed dose in the eye lens. This was observed when comparing this work's model with the model by Martinez et al. (2014).

To estimate if the relative deviation was statistically different, an uncertainty of three standard deviations was considered, because it covers 99.7% of the distribution values. The relative uncertainty of the deviation σrel was estimated, applying the general equation (2) to the results of the simulation:

a − b ⎞ σa2 + σb2 σ 2 σrel = ⎛ ⋅ + ⎛ b⎞ 2 ⎝b⎠ ⎝ b ⎠ (a − b)

(2)

With σa and σb the uncertainties of a and b. 2.4. Dosimetric quantities All results of the Monte Carlo calculation are dose coefficients in units of dose per source particle. The results where then normalized to the volume by dividing the calculated absorbed energy in the eye lens per source particle by the water volume surrounding the models, resulting in a dose coefficient independent of the volume considered and the activity present in the water, given in units of μGy·L−1·particle−1. Absorbed fractions were calculated for certain values. The absorbed fraction is a fundamental quantity for the estimation of internal doses and it takes into account the fraction of the energy that is emitted by a source organ to the energy that is absorbed in a target organ. It is calculated by dividing the absorbed energy in the eye lens by the initial electron energy (Loevinger et al., 1991).

3.2. Comparison of water compositions and densities To evaluate the influence of the composition and the density of the water surrounding the model on the attenuation of the beta particles, the absorbed dose in the fish eye lens using different water composition and densities was compared in terms of their relative deviation (Table 4). Comparing pure and river water composition, relative deviations of 26% and 10%, where found for 0.1 MeV and 0.2 MeV electron energies, respectively. For 0.4 MeV and 0.5 MeV the relative deviation is below 10% and for all energies above 0.5 MeV it is smaller than 4%. In all cases the relative deviation was smaller than the three-sigma statistical uncertainty propagation (Fig. 4).

3. Results and discussion 3.1. Model dimensions The volume of the eye model developed in this work is a factor of 3.4 higher than the volume of the eye in the model developed by Martinez et al. (2014), a factor of 3.7 higher than in the model by Hess (2014), and a factor of 13.3 higher than the one in the model by Caffrey (2012) (Table 2). To understand this discrepancy, the diagonal dimensions of the voxel eye models were estimated, based on the relation

Table 2 Difference in the composition and dimensions of different eye models. Values of the model shown in this work were multiplied by two, to obtain the value for both eyes, which can be compared to the other. Reference

Species

Organ

Organ dimension 3

Hess (2014) Caffrey (2012) Martinez et al. (2014)

Trout Limanda Trout

This work

Salmon

Eyes Eyes Eyes Eyes Eyes Eyes

whole Choroid whole lens whole lens

Fish dimension

Note

Volume (cm )

Mass (g)

Size (cm)

Mass (g)

1.936 0.54 2.09 0.1 7.170 0.536

2.07152 0.574 2.2363 0.107 7.479 0.574

35.5 17.6 28.6

658 1024 222.7

50.0

1260

10

No lens No lens

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Fig. 3. A Coronal section and transverse section of the rainbow trout voxel model, left and right respectively. Organs are marked by numbers and are shown in different colours. The eyes are shown in orange, labelled 7, the eye lens in dark violet, labelled 16. For other organs, see Martinez et al. (2014). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Table 3 Estimated eye lateral diameter for each fish voxel model and the corresponding value calculated by the Devlin et al. (2012) allometric study.

Hess (2014) Caffrey (2012) Martinez et al. (2014)

Trout Limanda Trout

Fish length (cm)

Model eye lateral diameter (mm)

Devlin (2012) eye lateral diameter (mm)

Relative deviation (%)

35.5 17.6 28.6

12.273 8.019 12.590

15.564 9.645 13.504

−26.82 −20.28 −7.26

3.3. Dose coefficients for the fish eye lens

A relative deviation between pure water and sea water of 40%–7% was found for 0.1 MeV, 0.2 MeV, 0.4 MeV and 0.7 MeV. For all other energies the relative deviation was below 4%. In some cases, the relative deviations are below the three-sigma uncertainty (Fig. 5). Finally, the absorbed dose in river water and sea water were compared. For 0.1 MeV, 0.2 MeV and 0.4 MeV, the relative differences varied between 16% and 18% (Fig. 6). For 0.2 MeV and 0.5 MeV it was 7% and 8%, respectively, and for 0.7 MeV and energies above, the relative deviation was below 4%. In all cases the relative deviation was smaller than the three-sigma uncertainty. Tanking in account that an uncertainty of three standard deviations contains 99.7% of the results and that all relative deviations calculated are below this value, we conclude that differences in water content and density studied in this work will not significantly influence the absorbed dose in the fish eye values.

The mean absorbed dose in the eye lens was calculated for discrete energies between 0.1 MeV and 4 MeV using the mathematical model developed in this work (Fig. 7, Table 4). To validate the mathematical model calculations, the mean absorbed dose of the eye lens was also calculated using the model developed by Martinez et al. (2014) (Fig. 7, Table 4). Because of to the high uncertainty of the calculation, results for 0.1 MeV and 0.2 MeV source electrons are not presented for the voxel model after Martinez et al. (2014). Doses obtained with the mathematical model are smaller than those obtained using the voxel model across most electron energies. The relative difference of dose decreases with increasing source energies. The dose in the voxel model is about 219% higher for 0.5 MeV electrons and 93% higher for 1 MeV electrons. The doses are almost identical at 2 MeV with the dose in the voxel phantom being 8% higher than the dose in the mathematical phantom. For 4 MeV electrons the dose in the

Table 4 Absorbed dose for the eye lens, calculated for three different surrounding water contents, absorbed fraction of electron energy for pure water surrounding and respective statistical uncertainty (expressed as one estimated standard deviation, divided by the estimated mean value). Energy (MeV

Absorbed dose per litre per source particle (μGy·L−1·particle−1)

Absorbed fraction of electron energy

This work model

0.1 0.2 0.4 0.5 0.7 1.0 1.5 2.0 4.0

Martinez et al. (2014)

This work model

Pure water

Uncertainty (%)

River water

Uncertainty (%)

Sea water

Uncertainty (%)

Pure Water

Uncertainty (%)

Pure water

Uncertainty (%)

4.99E-09 1.44E-08 5.95E-08 9.38E-08 4.66E-07 3.02E-06 1.72E-05 4.93E-05 2.78E-04

15.73 9.15 4.94 4.71 4.25 2.41 1.29 0.88 0.63

3.65E-09 1.59E-08 6.27E-08 1.02E-07 4.46E-07 2.97E-06 1.72E-05 4.89E-05 2.74E-04

18.21 8.78 4.83 4.51 4.35 2.43 1.29 0.88 0.63

2.97E-09 1.34E-08 5.17E-08 9.46E-08 4.31E-07 2.99E-06 1.68E-05 4.77E-05 2.70E-04

20.33 9.51 5.32 4.60 4.44 2.43 1.31 0.89 0.64

– – 1.01E-07 3.00E-07 1.34E-06 5.83E-06 2.41E-05 5.37E-05 2.20E-04

– – 15.00 7.23 5.10 2.49 2.37 1.77 1.26

8.95E-08 1.29E-07 2.66E-07 3.36E-07 1.19E-06 5.41E-06 2.06E-05 4.42E-05 1.25E-04

15.73 9.15 4.94 4.71 4.25 2.41 1.29 0.88 0.63

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10 5

Relative deviation

0 -5 -10 -15 -20

River vs. Pure water

-25 -30 0

0.5

1

1.5

2 2.5 Energy (MeV)

3

3.5

4

4.5

Fig. 4. Relative deviation between absorbed doses to the fish eye lens calculated in river and pure water surrounding water contents. The error bars correspond to the uncertainty propagation of the calculation using formula 2 with a statistical three-sigma uncertainty.

10 5 0 Relative deviation

-5 -10 -15 -20 -25 -30

Sea vs. Pure water

-35 -40 -45 0

0.5

1

1.5

2 2.5 Energy (MeV)

3

3.5

4

4.5

Fig. 5. Relative deviation between absorbed doses to the fish eye lens calculated in sea water and pure water surrounding water contents. The error bars correspond to the uncertainty propagation of the calculation using formula 2 with a statistical three-sigma uncertainty.

results of the eye models by Caffrey (2012) and Hess (2014). Both Caffrey (2012) and Hess (2014) have performed simulations for electrons energies between 0.1 MeV and 4 MeV. However, they only present values for 0.1 MeV and 4 MeV energies, because all other energies show uncertainties above 10%, possibly due to the limited computational power available. For the 4 MeV electron exposure, the absorbed fraction for the Hess (2014) model is approximately 33% greater than the results presented here. Contrary, the Caffrey (2012) model results are 64% smaller compared to this work. The reason for these discrepancies is that neither model explicitly defines an eye lens region. Hess (2014) only considers the energy deposition in the whole eye volume. The results of Caffrey (2012) are relative to the energy deposition in the eye choroid, not the eye lens as in this work here. For 0.1 MeV electrons energies, both Caffrey (2012) and Hess (2014) simulations have scored zero absorbed fractions in the eye region, possibly due to the simulation of a relative low number of particles (2·106) to reduce the computational time. For comparison, calculations using the mathematical model developed in this work were

voxel phantom is 21% lower than the dose in mathematical model. The high relative deviations of doses found for low energies are probably due to the geometrical differences arousing from the different fish eye sizes, eye lens depth and use of CT imaging of a dead fish lying on their sides for the development of the voxel model, in opposition to the natural state of swimming in the water. A possible reason for the crossing of the absorbed dose curves calculated with different models is the positioning of the eye lens relative to the eye surface and the consequent average depth of the eye lens. Nogueira et al. (2011) observed a similar result when comparing two human eye-lens mathematical models and concluded, that due to the shallow location of the eye lens, there is higher dose for lower electron energies and that for higher energies, the production of secondary particles results in an increase of the energy deposition with the depth. 3.4. Absorbed fractions The dose coefficients calculated in this work were converted to absorbed fractions (Table 4), to allow direct comparison with the 12

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Fig. 6. Relative deviation between absorbed doses to the fish eye lens calculated in sea water and river water surrounding water contents. The error bars correspond to the uncertainty propagation of the calculation using formula 2 with a statistical three-sigma uncertainty.

Fig. 7. Mean absorbed dose of the eye lens normalized per litre per source particle, calculated using the mathematical eye lens model developed in this work and the trout model by Martinez et al. (2014). In both calculations the surrounding of the fish was pure water.

performed also with only 2·106 source particles, equally resulting in no score in the tally region, supporting the hypothesis of too few source particles in the works of Caffrey (2012) and Hess (2014).

Assuming

• the average activities of

3.5. The Baltic Sea dose

•

To demonstrate the use of the new model results, they were applied to calculate the 90Sr and 137Cs (the two major anthropogenic radionuclides in the Baltic Sea (Nielsen et al., 1999)) beta particle doses in the eye lens of a wild fish living in the Baltic Sea using equation (3):

Dose = Activity·Dose Coefficient⋅Life expectancy

•

90 Sr and 137Cs in the southern Baltic Sea of 0.008 Bq·L−1 and 0.036 Bq·L−1, respectively (Zalewska and Suplińska, 2013), dose coefficient's of 1.5·10−7 μGy L−1·particle−1 and 6.0·10−6 μGy L−1·particle−1, respectively, interpolating the results of this work (Table 4) and a life expectancy of 6 years for the trout (ICRP 108, 2008a),

doses in the eye lens of 0.227 μGy and 40.871 μGy are calculated for Sr and 137Cs, respectively. These values are quite low compared with the current accepted threshold of 0.5 Gy for radiation induced cataracts in the human eye lens (ICRP, 2007). However, they don't include the additional doses from the 137Cs gammas, and the natural radionuclides.

(3)

90

With

• the activity in the surrounding given in Bq·L , • the dose coefficient being calculated using the mathematical model developed in this work and • the life expectancy being the average life span of the fish species in −1

4. Conclusion

question.

The estimation of the absorbed dose per unit fluence to the fish eye 13

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lens was able with an improved precision compared to previous works. The main reason for the improvement is the usage of a novel mathematical model of a fish eye that was implemented in the ICRP reference trout model. Using the MCNPX Monte Carlo code, calculations were performed for discrete energies between 0.1 MeV and 4 MeV. The results were then presented in form of dose coefficients, allowing to quickly estimate the dose to the fish eye lens based on the energy of an electron emitted by a radionuclide and the respective activity concentration in the surrounding water. This was demonstrated for the dose calculation to a fish living in the southern Baltic Sea. To evaluate the influence of the surrounding water composition, the absorbed dose was calculated for three different water compositions: pure water, river water and sea water; and two different densities. The results showed that the water content and density plays only a small role in the external doses coefficient for electrons. Generally only small relative differences were found between the absorbed doses in the eye lens with different surrounding water contents, and in all the cases the relative deviation was within an uncertainty of three standard deviations. The trout voxel model developed by Martinez et al. (2014) was also used to calculate the absorbed doses in the eye lens. Discrepancies in the dose coefficients ranging from 8% to 220% were found between voxel and mathematical model, most likely originating from geometrical differences in the models and differences in the placement of the eye lens in relation to the eye surface. Furthermore, absorbed fractions were calculated using the dose coefficients estimated with the model presented here. The absorbed fractions were directly compared to the absorbed fractions presented by Caffrey (2012) and Hess (2014). The comparison showed considerable discrepancies between the calculated dose coefficients for 4 MeV. However, since both voxel models do not have a segmented eye lens region, it can be concluded, that the reason for the discrepancy is that different regions within the models were scored. The mathematical model developed in this work has not only been validated, but also demonstrated to have three major advantages over the voxel models presented so far: First, the model development is not dependent on finding a sample fish of equal dimensions to the reference trout. Second, it requires a significantly smaller amount of computational power, time, and resources for the model development as well as for computational applications using the model. And third, it provides a more accurate dose, taking into account the ICRP reference trout dimensions. The results presented here show, that the mathematical model provides the ideal balance between accuracy and resource requirements; this is in agreement with the findings by Martinez et al. (2014).

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