Making the most of what we have: application of extrapolation approaches in radioecological wildlife transfer models

We will never have data to populate all of the potential radioecological modelling parameters required for wildlife assessments. Therefore, we need robust extrapolation approaches which allow us to make best use of our available knowledge. This paper reviews and, in some cases, develops, tests and validates some of the suggested extrapolation approaches. The concentration ratio (CRproduct-diet or CRwo-diet) is shown to be a generic (trans-species) parameter which should enable the more abundant data for farm animals to be applied to wild species. An allometric model for predicting the biological half-life of radionuclides in vertebrates is further tested and generally shown to perform acceptably. However, to fully exploit allometry we need to understand why some elements do not scale to expected values. For aquatic ecosystems, the relationship between log10(a) (a parameter from the allometric relationship for the organism-water concentration ratio) and log(Kd) presents a potential opportunity to estimate concentration ratios using Kd values. An alternative approach to the CRwo-media model proposed for estimating the transfer of radionuclides to freshwater fish is used to satisfactorily predict activity concentrations in fish of different species from three lakes. We recommend that this approach (REML modelling) be further investigated and developed for other radionuclides and across a wider range of organisms and ecosystems. Ecological stoichiometry shows potential as an extrapolation method in radioecology, either from one element to another or from one species to another. Although some of the approaches considered require further development and testing, we demonstrate the potential to significantly improve predictions of radionuclide transfer to wildlife by making better use of available data. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
The assessment of the exposure of wildlife to ionising radiation for planned, existing and emergency scenarios requires predictions to be made of the transfer of a wide range of radionuclides to a diversity of species. Most models assessing the exposure of wildlife for regulatory purposes use a simple concentration ratio (CR womedia ) relating the whole organism activity concentration to that in an environmental medium (i.e. soil, air or water) (Beresford et al., 2008a). Other models use radionuclide biological half-lives and transfer from the diet (e.g. USDOE, 2002).
For many of the radionuclide-species combinations that require assessment, there are no empirical data. When empirical data are lacking, predictions are often made using transfer parameter values derived using extrapolation approaches, though clarity on the use of such approaches varies between publications/models. For instance, the CR wo-media value for an organism of 'similar taxonomy' may be assumed (e.g. a mammal value may be used to model birds if data for the latter are lacking) (e.g. Beresford et al., 2008b). Brown et al. (2013) recently evaluated how the most commonly used approaches performed against additional data now available . They concluded that the extrapolation approaches commonly used to date have under-predicted the empirical 95th percentile CR wo-media value as often as they have over-predicted. This highlights the need to treat results from these previous extrapolations with caution, but there have been various recent advances that may bring greater confidence in the application of extrapolation methods. In this paper, we assess our ability to extrapolate radioecological data of relevance to wildlife radiological assessments considering these recent advances and future potential.

Transfer from the diet to terrestrial vertebrates is a constant across species
The transfer coefficient was first proposed by  to describe the transfer of radiocaesium from the diet to the milk of dairy cattle. The authors defined the transfer coefficient as the ratio between the radiocaesium activity concentration in milk and the daily dietary radionuclide intake.  reported that this parameter exhibited less variability between individual animals within their experimental herd than when transfer was expressed as the total amount of Cs excreted in milk (represented as a percentage of intake).  subsequently defined the meat transfer coefficient as the ratio of the 137 Cs activity concentration in boneless meat to the dietary daily 137 Cs intake.
Following the publications of Ward and co-workers in the 1960s, the transfer coefficient was adopted as the basis for quantifying transfer to milk (F m , d l À1 or d kg À1 ), and meat and eggs (F f , d kg À1 ) for all radionuclides. By the late 1970s to early 1980s, transfer coefficient values were being recommended for most radionuclideanimal product combinations (e.g. Ng, 1982;Ng et al., 1977Ng et al., , 1979Ng et al., , 1982 and the International Atomic Energy Agency (IAEA) recommended their use (IAEA, 1994). These recommended values have been incorporated into many predictive food chain models (e.g. Brown and Simmonds, 1995;Müller and Pr€ ohl, 1993;USNRC, 1977;Yu et al., 2001).
On the basis of the many studies conducted over the approximately 50 years since the transfer coefficient concept was introduced, it has generally been accepted that transfer coefficients for smaller animals are higher than those for larger animals, and that those for adults are lower than those for young (and hence smaller) livestock. For instance, F f values for sheep meat recommended by IAEA (2010) for many radionuclides are circa one order of magnitude higher than those recommended for beef. Similarly, F m values for goat milk tend to be one order of magnitude higher than those recommended for cow milk. The use of transfer coefficients has also been suggested for wildlife (e.g. Thomas et al., 1994;MacDonald, 1996;Moss and Horrill, 1996) and some models use simple food chains to estimate radionuclide concentrations in wildlife (e.g. USDOE, 2002).
However, it has been suggested that much of the observed difference in F f,m values is a consequence of differences in dry matter intake, between animals of differing age or species, rather than any difference in radionuclide transfer (e.g. Smith and Beresford, 2005;Galeriu et al., 2007;Beresford et al., 2007). The rational for this is that the transfer coefficient is defined as: Consequently, the concentration ratio (CR meat-diet ) is equal to: Â Daily dry matter intake kg d À1 The above equations are for meat but a similar derivation can be performed for milk or eggs.
A between species similarity in CR values for animal derived food products should not be surprising, given that the concentrations of many elements in meat, or milk, are similar across species (Mertz, 1986(Mertz, , 1987. A particular advantage of being able to assume that the milk or meat CR for many radionuclides varies little between species is that generic values can be derived for animals for which no data are currently available. Recognising this, the IAEA (2010) summarised milk and meat CR values, as well as transfer coefficients.
We would also expect that CR values for wildlife would vary little between species  and would be similar to those of farm animals. To test this hypothesis, Table 1 presents Cs CR meat-diet values for seven herbivorous species of wild mammals and birds. The CR meat-diet values for these species are similar to those for the meat of farm animals in IAEA (2010), which presents a generic value of 0.39 based on data for four farm animal species. Since Cs is relatively homogenously distributed throughout the body tissues (Yankovich et al., 2010a), it can then be assumed that, for Cs CR meat-diet z CR org-diet (where CR org-diet is the ratio of the radionuclide activity concentration in the whole organism to that in its diet). Currently there are few data with which to test our hypothesis for the transfer of other radionuclides to wild animals. For many elements other than Cs, distribution is not homogenous throughout the body tissues. However, for such elements an assumption that the distribution within the body was similar across animal species would be reasonable (e.g. Sr accumulates in the bone of all vertebrates).

Allometry
Size affects rates of biological processes from cellular metabolism to population dynamics (Peters, 1983;Hoppeler and Weibel, 2005). The dependence of a biological variable (Y) on body mass (M) is typically characterised by an allometric scaling law. There are several allometric equations that can be proposed, the simplest being to assume that: where a and b (the allometric exponent) are constants, b is dimensionless and a has the units of the variable, Y, per mass to the power of Àb. Kleiber (1932) found that basal metabolic rate (measured as heat production) across 13 groups of mature animals, ranging from   (Kleiber, 1947); this has since become known as Kleiber's law.
There is considerable debate with regard to the numerical values for the allometric exponent, in particular whether it should be 0.75 or perhaps 0.67 for basal metabolic rate (e.g. West et al., 1997;Hoppeler and Weibel, 2005;Isaac and Carbone, 2010;Agutter and Tuszynski, 2011). The debate hinges on theoretical considerations, such as whether allometry of the metabolic rate relates to the fractal nature of energy distribution networks (West et al., 1999;Brown et al., 2002) with and an exponent of 0.75, or whether it is based on a consideration of surface area-to-volume ratios with an exponent of 0.67 (Farrell-Gray and Gotelli, 2005).
In discussing this issue with respect to radioecological models, Higley and Bytwerk (2007) suggested that, given other uncertainties in radioecological modelling, the exact value of the allometric scaling exponent 'may not be of critical importance' for practical (rather than theoretical) purposes. We support this suggestion and Fig. 1 demonstrates the relatively small effect of assuming M 0.67 rather than M 0.75 over a likely range of vertebrate live masses.

Homeothermic vertebrates
Many of the reported allometric relationships, such as dry matter food ingestion rates, water ingestion rates and inhalation rates, have been used in models of the radionuclide transfer to wildlife (e.g. Beresford et al., 2008a;Johansen et al., 2012), including the US Department of Energy's 'Graded Approach' as implemented in the RESRAD-Biota model (USDOE, 2002(USDOE, , 2004. However, in the 1970s, radionuclide biological half-life (T 1/2B ) was also shown to scale allometrically (e.g. Stara et al., 1971;Kitchings et al., 1976). The application of allometry to radioecological parameters has received revived attention during the development of models to predict the exposure of wildlife to radionuclides in terrestrial (Higley et al., 2003;Higley, 2010;Beresford et al., 2004;Sheppard, 2001) and aquatic ecosystems (Vives i Batlle et al., 2007Brown et al., 2004). USDOE (2002) presents T 1/2B allometric relationships for 16 elements in terrestrial/riparian vertebrates. When used in model inter-comparison exercises, allometric relationships from USDOE (2002), and similar models, produced results comparable to models using CR wo-media values (Beresford et al., 2009;Wood et al., 2009a). Of the allometric expressions describing radionuclide T 1/2B from USDOE (2002), a number have an exponent of approximately 0.25 (Cs, Co, Ra, Sb, Sr, U, Zn and Zr). Iodine has an exponent of 0.13 and H of 0.55, but other literature suggests these too should approximate to 0.25 (Galeriu et al., 2003;MacDonald, 1996). Therefore, it appears that for most radionuclides, T 1/2B is driven by metabolic rate and Beresford and Vives i Batlle (2013) demonstrated that this was logical by considering a simple radionuclide retention model (Sazykina, 2000) and Kleibers Law. Notable exceptions are Pu, Am, Ce and Eu, which USDOE (2002) suggest scale to circa 0.8. Sheppard (2001) proposed that, if it is accepted that there is an approximation of the exponent applicable for all elements (i.e. in the case of T 1/2B , circa 0.25), then only an estimation of the multiplicand is needed for any given element. Through algebraic derivation, Beresford and Vives i Batlle (2013) proposed a method of determining the multiplicand for the T 1/2B relationship: where a B is the multiplicand for the allometric relationship describing T 1/2B , a I is the multiplicand for the allometric relationship describing dry matter intake and f 1 is the fractional gastrointestinal absorption coefficient. By substituting this expression for a B in the equation describing the T 1/2B allometric relationship, Beresford and Vives i Batlle (2013) suggested that an estimate of T 1/ 2B could be derived as: Values of a I are relatively well documented for terrestrial vertebrates (e.g. Nagy, 2001), f 1 values are collated in publications (e.g. IAEA, 2010), and, if we accept that CR org-diet is a constant across Table 1 Caesium CR meat-diet values for wildlife; for both source references 'diet' concentration is determined from dried stomach contents.

Species Latin
Species common Mean ± SD a n Reference species (see above), then this parameter can also be sourced from IAEA (2010) for many radionuclides.

Cervus elaphus
Beresford & Vives i Batlle tested their hypothesis using data primarily summarised by Whicker and Schultz (1982), which tabulates T 1/2B estimates for a number of radionuclides and terrestrial organisms. Data were for mammals ranging from 0.01 kg (harvest mouse) to 80 kg (reindeer) and radionuclides of Co, Cs, I and Sr. A comparison of predicted T 1/2B values across all radionuclides made using a I values appropriate to the feeding type of each species with measured data yielded a linear regression (p < 0.001) with an R 2 value of 0.58, a slope of 1.4 and an intercept which was not significantly different from zero (Beresford and Vives i Batlle, 2013). Subsequent to Beresford and Vives i Batlle (2013), a review of radionuclide T 1/2B values has been conducted (this is contributing to IAEA MOdelling and DAta for Radiological Impact Assessments (MODARIA) programme activities, see: http://bit.ly/1e9Nxxq). Data identified in this review enable us to further test the Beresford and Vives i Batlle (2013) model and to consider a wider range of elements (Ag, Co, Cs, I, Na, Nb, Ru, Se, Sr and Zn) and species. The data used had to meet various criteria: animal mass had to be presented in the source reference; only data for adults were used; T 1/2B values had to be for the whole body except in the cases of I and Cs, for which thyroid and muscle data were also used, respectively, assuming these tissues reflected whole body loss rates; with the exception of I, Sr and Cs, only data for studies where the radionuclide had been ingested or administered orally were considered; for I, Sr and Cs, data from studies using intravenous administration were also used, as the weight of evidence suggested that these elements behave in a similar manner in the circulatory system after either oral or intravenous administration (Mayes et al., 1996).
A total of 123 T 1/2B values have been considered in this expanded model testing (source references are listed in Supplementary Materials); these included data originally considered by Beresford and Vives i Batlle (2013), though source references were consulted rather than relying on the Whicker and Schultz (1982) compilation. Data were largely for mammals (ranging from 8 g to 70 kg body mass), although limited data for ducks were also identified. When multiple components of loss were cited in the source reference, the longest T 1/2B value was used for comparison with predictions.
Radionuclide-specific parameter values used to make predictions are presented in Table 2. Estimates of f 1 were taken from IAEA (2010), which cites values from ICRP (2006) (2010) were, therefore, used together with typical dry matter intake rates from IAEA (1994) to estimate an average Sr CR org-diet across all five species for which F f data were available (cattle, goat, sheep, poultry and pig) in IAEA (2010). To convert CR meat-diet values to CR org-diet values, conversion factors from Yankovich et al. (2010a) were used; if Yankovich et al. contained no data for a given element, then conversion factors were estimated from  or the data of Barnett et al. (2014).
Values of a I were sourced from Nagy (2001), and both the 'all mammals' value (a I ¼ 0.057 d À1 kg 0.25 ) and values for the most appropriate taxonomic or feeding strategy group were used: carnivorous mammals (a I ¼ 0.027 d À1 kg 0.25 ); herbivorous mammals (a I ¼ 0.15 d À1 kg 0.25 ); omnivorous mammals (a I ¼ 0.077 d À1 kg 0.25 ); omnivorous birds (a I ¼ 0.119 d À1 kg 0.25 ). Values presented by Nagy, in grammes, were converted to kilogrammes for application here, using the multiplicative factor: (a I Â 1000 0.75 )/1000.
The majority of predictions were within an order of magnitude of the observed data: 97 of 110 predictions using the a I for all mammals and 107 of 123 predictions made using the most appropriate feeding group a I (Table 3). Allometry is a broad indicator only (Higley and Bytwerk, 2007) and hence predictions within an order of magnitude are considered acceptable here.
Using the a I for 'all mammals', nine predictions were more than an order of magnitude lower than the observed data; with the exception of one prediction (Cs in rabbit), these large underpredictions were for Nb (n ¼ 4) and Ru (n ¼ 4). These eight comparisons comprised all of the available data for Ru and Nb. Only four predictions were more than an order of magnitude higher than the observed data. All of these over predictions were for Sr; these data had not previously been considered in the initial evaluation of the model by Beresford and Vives i Batlle (2013). These observations are discussed further below.
When predictions were made using a I values for specific feeding strategies (see Table 3), the following were observed: Carnivorous mammals e Using the a I value for carnivorous mammals increased the predicted T 1/2B value compared to that predicted using the 'all mammals' a I value (Fig. 2). The predictions using the carnivorous mammal a I value were generally in better agreement with the observed data.
Herbivorous mammals e Using the a I value for herbivorous mammals decreased the predicted T 1/2B value compared to that predicted using the 'all mammals' a I value (Fig. 3). There was not a consistent trend across all of the elements to improve, or not, the level of agreement between predictions and observed data (e.g. for I predictions using the a I value for herbivorous mammals resulted in better agreement with the observed data than those using the 'all mammals' a I value, whereas the opposite was the case for Zn).
Omnivorous mammals e Using the a I value for omnivorous mammals decreased the predicted T 1/2B value compared to that predicted using the 'all mammals' a I value, but only by circa 25%.
Predictions for I were in good agreement for herbivorous and carnivorous mammals. For omnivores, predicted values tended to be <20% of the observed data values. Caesium data were the most numerous (n ¼ 55). If we omit an outlying value for rabbit, which was under-predicted by more than an order of magnitude, a linear regression of measured Cs T 1/2B values to predictions made using Table 2 Parameter values used to predict the biological half-life of radionuclides in mammals and birds (data sources are cited in text). There was a tendency to over-predict Sr T 1/2B values, some by more than an order of magnitude (Table 3). Those four values which were over-predicted by more than an order of magnitude were all from studies using intravenous administration or radiostronitum. The reported half-lives (see DiGregorio et al., 1978) were comparatively short compared to the two studies reporting Sr T 1/2B values following oral administration and also data from farm animal studies (Fesenko et al., 2015).
Predictions for Ag, Co, Se and Zn all had an acceptable level of agreement. Those for Na were within an order of magnitude of the observed T 1/2B values, but were all comparatively low.
Predictions for both Ru and Nb were poor, both elements being considerably under-predicted. In the cases of Ru and Nb, it is worth noting that the CR meat-diet values in IAEA (2010) are each based  (2015) 1e14 5 upon one observation only, although this is also the case for Ag and Se. To improve the predictions such that they are within an order of magnitude of the observed values, the CR meat-diet for Nb would need to increase by approximately 1e2 orders of magnitude and that for Ru by about two-orders of magnitude. This is plausible as IAEA (2010) presents CR meat-diet values for other elements which vary by 1e2 orders of magnitude between studies. The f 1 values used also impact on the predicted T 1/2B values for both Nb and Ru.
There are reported f 1 values for these elements, which are lower than those we have used in this assessment Beresford et al., 1998a); application of a lower f 1 would improve the predicted values. However, the Nb and Ru T 1/2B used here are from  and  respectively; the f 1 values estimated by these authors from experimental data were similar to those used to make our predictions. There was reasonable agreement between the predicted and measured Zn T 1/2B values for ducks; the ratio of predicted to measured values ranged from 0.2 to 1.5. However, T 1/2B values varied considerably more for the observed data (28e250 d; n ¼ 13) than the predicted values (41e42 d). The low variation in predictions is not surprising given that the mass of the ducks varied little (1.1e1.3 kg).
Overall, our ability to obtain reasonable predictions is, in part, dependent upon the quality of data available for the required input parameters. For many elements in IAEA (2010), CR meat-diet values are based upon few observations. Similarly, the correction factors to convert from CR meat-diet to CR org-diet are based on relatively few data, as exemplified by the need to derive them for Co, I and Sr. Investigation of the data used by Nagy (2001) to derive the allometric dry matter intake relationship for herbivores shows that they are dominated by relatively small species, with many of the larger species being marsupials.
It should also be acknowledged that the dry matter intake relationships presented by Nagy (2001) are for animals under field and not laboratory conditions, and that field metabolic rates are generally higher than basal metabolic rates determined for housed animals (Nagy, 2005). This may result in a tendency to under-predict T 1/2B for housed (i.e. experimental) animals. Overall, in the above assessment, there was a tendency to under-rather than over-predict, 103 of the 123 predictions using feeding group a I values were less than the observed value.
An assumption of the approach described here is that T 1/2B scales to the power of 0.25. For five elements in USDOE (2002) (Am, Ce, Eu, Pu and Th), T 1/2B scales to the power of 0.8. For all of these elements, International Commission on Radiological Protection publications (ICRP, 1979(ICRP, , 1981(ICRP, , 1988 are quoted as the data source, but unfortunately, we have not been able to find the data within these publications to independently verify the allometric equations presented by USDOE (2002). None of the elements scaling to circa 0.8 play an important biological role, and hence, it could, perhaps, be suggested that there is no reason for them to follow a metabolically driven uptake process (i.e. as signified by a mass scaling function of 0.25). However, we acknowledge that some of the elements which do scale as mass to the power of 0.25 are neither essential elements nor their analogues (e.g. U). Therefore, it is recommended that the reasons for the deviation of some elements from the mass scaling function of 0.25 be explored and/or the relationships presented by USDOE (2002) be independently verified.
For comparison with our predictions, we have used the longest T 1/2B value reported where multiple components of loss were observed. In doing so, we made no differentiation based on the magnitude of this loss component, even though in many instances, the long component of loss contributes relatively little to the total loss (<5%). It is questionable whether the longest T 1/2B always best represents metabolic turnover when three and four component loss equations are reported.
The assumption of a single long-component of loss, as currently used in most wildlife assessment models (e.g. USDOE, 2002;Avila et al., 2004), should, if the models are correctly parameterised, yield estimates of the equilibrium activity concentration in organisms which are conservative (i.e. they should be over-estimated compared to a model that assumes more than one loss component). However, if used in dynamic models, this assumption will predict slower changes in organism activity concentrations than would be observed in reality as a result of changes in activity concentrations in environmental media.

Application to reptiles
Although only proposed for homeothermic vertebrates, allometric models for T 1/2B , such as USDOE (2002), have been used to make predictions of radionuclide activity concentrations in (poikilothermic) reptiles and amphibians (e.g. Wood et al., 2009a;Beresford et al., 2010;Yankovich et al., 2010b;Johansen et al., 2012). Beresford and Wood (2014) evaluated to what extent the Beresford and Vives i Batlle (2013) model could be applied to reptiles. Similar criteria as used above for mammal and bird data were applied to identify T 1/2B values against which to compare predictions. In total 28 T 1/2B values for Cs, three for Sr and two for Ra were identified (see Beresford and Wood (2014)). Data for Cs were predominantly for different species of snake (Staton et al., 1974), with one value for a turtle species (Trachemys scripta scripta) (Scott et al., 1986). Data identified for Sr and Ra were all for T. scripta scripta (Scott et al., 1986;Hinton et al., 1992). Data covered reptiles with masses ranging from 0.02 to 1.5 kg. Most of the estimated T 1/2B values available were longer than the length of the studies from which they were derived. For instance, the maximum radiocaesium half-life determined by Staton et al. (1974) for snakes from their 63 d long study was 430 d. This will undoubtedly add some uncertainty to the reported T 1/2B values.
Using parameter values for mammals, all but one prediction was under-estimated; in the case of Cs, the under-estimates were by 1e2 orders of magnitude.
The exponent in the allometric biological half-life expressions is, in effect, defined by the exponent for the allometric model of metabolic rate (B r ) of circa 0.75 for mammals; the exponent for T 1/2B is '(1 e the exponent for B r )'. However, the exponent for the allometric model describing B r for reptiles has been shown to be in the range circa 0.80e0.92 (Nagy, 2005;Isaac and Carbone, 2010). Therefore, the exponent describing T 1/2B for reptiles should be in the range 0.08e0.20. From Nagy (2001), Beresford and Wood (2014) estimated exponents of 0.037 for snakes, derived from information for carnivorous reptiles, and 0.08 for turtles, derived from information for 'all reptiles'. These exponents, together with the relevant values of a I for reptiles (Nagy, 2001) and reptilespecific values of CR org-diet and f 1 (see Beresford and Wood (2014) for details of derivation), were used to estimate T 1/2B values to compare with the available data (Fig. 4). Only one of the 33 predictions deviates by more than a factor of six from the measured value; the one 'outlier', a prediction of the Sr T 1/2B in T. scripta scripta, was predicted to be 14 times higher than the measured value. However, T 1/2B allometric exponents applicable to reptiles result in a relatively low dependence of T 1/2B with mass (see Figs. 1  and 4). Over a mass range 0.1 ge1000 kg, the predicted biological half-lives for reptiles vary by less than a factor of four using an exponent of 0.08, and by less than a factor of two using an exponent of 0.037. For homeotherms, M 0.25 predicts T 1/2B values varying over circa 60-fold (Fig. 1). Given the small influence of mass on T 1/2B predictions for reptiles, Beresford and Wood (2014) suggested that if sufficient reported T 1/2B values are available for a given element, then it is likely that these would be applicable to any reptile. For instance all of the 28 reported values of Cs T 1/2B for reptiles considered by Beresford and Wood (2014), which covered a 50-fold mass range, were within a factor of 5 of the mean.
However, the relatively good agreement between predicted and measured T 1/2B in Fig. 4 demonstrates that if no reptile data are available for a given radionuclide, then the Beresford and Vives i Batlle model (2013), populated with reptile-specific parameter values, will give reasonable estimates.
There is some evidence that the dry matter intake rate of herbivorous reptiles (namely tortoises), scales to mass with an exponent of circa 0.75 (Franz et al., 2011). Therefore, we may expect the allometric relationship for T 1/2B for tortoise species to scale to circa 0.25, but we currently have no data to test this.

Aquatic organisms
For marine mammals, the allometric models developed by USDOE (2002) for terrestrial and riparian mammals and birds have been used within radioecological transfer models (e.g. Brown et al., 2004). As discussed above for reptiles, for most other aquatic organisms, this would not be a valid approach. However, the application of allometry to derive both biological half-life and CR values for marine organisms has been the subject of some studies (see review by Vives i Batlle et al., 2009). Table 4 presents allometric parameters describing T 1/2B for Tc, Cs, Pu and Am in marine organisms. To parameterises these models, data were available for fish, crustaceans, molluscs, macroalgae/ seaweed and plankton (Vives i Batlle et al., 2007Batlle et al., , 2008. The mean value of b for the T 1/2B relationships was 0.16 ± 0.03. Vives i Batlle et al. (2009) presented allometric T 1/2B expressions for Sr and I from previous studies. However, these expressions were taken from USDOE (2002) and were for terrestrial animals; they are not valid for most marine organisms (although they may be applicable to marine birds and mammals).
For CR, the mean (±SD) value of b for marine organisms, calculated across all those elements (Ac, Am, Ce, Cm, Mn, Pa, Pm/Eu, Po Pu, Ra, Ru, Th and Zr), was À0.26 ± 0.09 and the fitted relationship had an R 2 value >0.7 (Vives i Batlle et al., 2009) (see Table 5). For the individual radionuclides with an R 2 value in Table 5 greater than 0.7 (i.e. Am, Ce, Cm, Pm/Eu, Pu, Ra, Ru, and Th), Vives i Batlle et al. (2009) also found a significant linear relationship between log 10 (a) and log 10 (K d ) (i.e. the sediment-water distribution coefficient) (Fig. 5), showing particle-reactive (or sediment seeking) radionuclides to have the highest log10(a) values. The relationship between log10(a) and log(K d ) improved when Ru and Pm/Eu were removed though the authors had no scientific justification for doing this. With the addition of Ac, Mn, Pa, Po and Zr (which have R 2 values <0.7 in Table 5), the trend became weaker (log 10 (a) ¼ 0.38 Â log 10 (K d ) À 1.6; R 2 ¼ 0.50; n ¼ 13), but the statistical significance remained strong (p ¼ 0.007) (Vives i . For CR, if we accept that the exponent will approximate to À0.25 for cations, then the relationship between log 10 (a) and log(K d ) observed by Vives i Batlle et al. (2009) presents a potential opportunity to estimate CR values when data are lacking, assuming K d is known.
An anomaly when considering the allometric relationships available for marine organisms with those derived for terrestrial organisms is that the actinide and lanthanide elements scale with an exponent similar to other elements for marine organisms, whereas this is not the case for terrestrial organisms. A potential reason for this with respect to CR for marine organisms is the relationship between log 10 (K d ) and log 10 (a) which implies that the more particle-reactive the nuclide is, the more strongly it attaches to organic matter (e.g. food). As the ingestion of food is related to metabolism, and hence, to body mass via Kleiber's law, the resultant CR will scale allometrically. Obviously, this argument applies to animals, but not to plants. Pan and Wang (2008) have previously made similar suggestions with respect to metabolically driven allometry of the ingestion rate and the uptake of Cd and Zn by marine invertebrates. However, Vives i  suggest that for some elements, the process could also be surface-area driven (e.g. passive sorption of radionuclides on body surfaces, such as mollusc shells). Supporting this suggestion, it has been shown that dead cells of phytoplankton had the same uptake of transuranic elements as live cells, indicating that the uptake process is passive (Fisher et al., 1983). Similarly, Ginn and Fein (2008) observed that metal adsorption on bacteria was the same for many different bacteria species, suggesting that the surface area rather than the biology of the bacteria is important in metal transfer. For a perfectly isometrically scaled organism, all surface area-based properties change with mass to the power of 2 / 3 (Galilei, 1638), this would result in the CR scaling to M À0.33 instead of M À0.25 .

An alternative to the CR wo-media model for wildlife?
There are many cases where empirical data to derive CR wo-media are lacking. For instance, in the revised version of the ERICA Tool (released 2014 http://www.erica-tool.com/news/news-articles/ 2014/erica-assessment-tool-version-1.2/) (Brown et al., 2008), data were available for only 622 of 1521 required values. Furthermore, this parameter is highly variable as it incorporates many processes and will be largely determined by site-specific characteristics (e.g. Beresford et al., 2008a;Johansen et al., 2012;Wood et al., 2009bWood et al., , 2013Yankovich et al., 2010b).
Soil-to-plant transfer of elements of radiological interest has been related to plant evolutionary history, or phylogeny (Willey, 2010), including for Cs (Broadley et al., 1999;Willey et al., 2005),  Sr (Willey and Fawcett, 2005a), Ru (Willey and Fawcett, 2006), Cl (Willey and Fawcett, 2005b) and Co (Willey and Wilkins, 2008) (Fig. 6). Such phylogenetic relationships present a potential approach to enable predictions of transfer, with some scientific justification, for taxonomic groups for which there are no data either at the generic or site-specific level (Willey, 2010). The potential to derive phylogenetic relationships for radionuclide transfer to other organism types was demonstrated by Jeffree et al. (2010Jeffree et al. ( , 2013. Jeffree et al. suggested that the transfer of a number of radionuclides to marine teleost and chondrichthyan fishes, and to the amphioxus (fish like chordate) species, Branchiostoma lanceolatumis, was influenced by phylogeny. However, the work of Jeffree et al. was based upon the results of laboratory studies that only considered the exposure pathway from radionuclide contaminated seawater. Although this usefully removes the influences of many confounding factors, it is not directly applicable to environmental conditions as food chain transfer was excluded.  used the approach of Willey (2010) to analyse data of Cs transfer to freshwater fish species. Starting with the database on radionuclide transfer to freshwater organisms as described by Yankovich et al. (2013) (see also Copplestone et al., 2013), they compiled a data set of 597 entries covering 53 species of freshwater fish from 67 sites. The 53 fish species all belonged to the class Actinopterygii (ray-finned fishes), with 10 taxonomic orders, 14 families and 33 genera being represented in the dataset.
The Residual Maximum Likelihood (REML) fitting of a mixedmodel regression (as described by Willey, 2010) was used to analyse the dataset. The output consists of a mean value for each species on a common scale after REML adjustment (the fixed factor), taking account of the effect of the random factor (i.e. inter-site variation). This provides a method for statistically accounting for as much of the effect of site as possible within the collated data. The mean value output for each species provides a relative scaling value. The REML-adjusted means for different taxonomic groups of freshwater fish are presented in Table 6.
The results demonstrated differences in Cs transfer to freshwater fish based upon phylogenetically-derived taxonomic Table 5 Allometric parameters for CR (L kg À1 ) in marine organisms (adapted from Vives i Batlle et al., 2009); a and b are the constant and exponent for the allometric model. The authors describe those relationships with R 2 values of <0.7 as 'potentially approaching statistical significance'. For all elements n ¼ 9.    6. Extrapolated concentration ratios for radionuclides of chlorine, caesium, ruthenium, cobalt and strontium. Published REML-adjusted data (Broadley et al., 1999;Fawcett, 2005a,b, 2006;Willey et al., 2005;Willey and Wilkins, 2008), have been transformed to have a geometric mean of IAEA recommended transfer value for 'grass and herbs' (IAEA, 2014) then divided into phylogenetic groups of the Angiosperm Phylogeny Group III phylogeny for flowering plants (1 ¼ Monocotyledonous plants, 2 ¼ Rosids, 3 ¼ Caryophyllids, 4 ¼ Asterids. Geomeans with 95% confidence Intervals. Cl n ¼ 106 species, Ru n ¼ 114, Sr n ¼ 155, Cs n ¼ 256, Co n ¼ 241).
groupings, though they did not indicate a phylogenetic trend as previously seen for plants (e.g. Fig. 6). However, although a relatively large dataset was considered, data were only available for 53 of the total 11,952 freshwater species (Nelson, 2006), representing only 10 orders and one class. Earlier analyses, which have suggested phylogenetic relationships for the transfer of radionuclides to plants (Willey, 2010) and marine fish (Jeffree et al., 2010, have included species encompassing much wider evolutionary time scales (e.g. >500 million years in the case of marine fish).  suggested that the REML-adjusted means potentially provide a more refined approach than the CR wo-media model. By taking into account inter-site variation, the REMLadjusted means in effect provide a mechanism of accounting for site-specific variables (e.g. K concentrations in water in the case of Cs transfer to fish). To independently test this hypothesis, data from 27 Finnish lakes (STUK, 2012; http://bit.ly/1xDJQu4), which had not been used in the model calibration, were analysed. Data were available for four or more fish species at each lake with a total of 11 fish species being represented. As Perca fluviatilis was present at all 27 sites and was also well represented within the dataset used for the REML analysis, it was used as the 'known species' from which to calculate activity concentrations for the other species (treated as 'unknowns'). This allowed 100 predictions to be made.
There was relatively good agreement between predicted and measured values, with a linear regression fit yielding an R 2 of 0.83 (p < 0.001) and a slope (±standard error) of 0.98 ± 0.04 (p < 0.001) . The intercept was not significantly different to zero.
Predictions were considerably better than if the appropriate feeding group (i.e. benthic feeding, piscivorous or forage feeding), Table 6 REML-adjusted means for different taxonomic groups; these are relative values and not absolute values of CR wo-water (reproduced from .

Order
Family Genus Species geometric mean CR wo-water values from Yankovich et al. (2013) are used to predict the 137 Cs activity concentrations in the fish. The CR wo-water values generally resulted in under-predictions, with a linear regression of predicted-to-measured activity concentrations yielding a slope of only 0.31. In Fig. 7, we further test the model of  against recent data for three lakes in northern England . Across the three lakes, data were available for vendace (Coregonus albula), Northern pike (Esox lucius), ruffe (Gymnocephalus cernua), perch (Perca fluviatilis), roach (Rutilus rutilus) and brown tout (Salmo trutta). Replication for each species ranged from 6 to 20, although not all species were sampled in each lake. Perch was present at all sites and was hence used as our 'known' species. Activity concentrations in the other species were then predicted as: Cs concentration in 0 unknown 0 species ¼ REML À adjusted mean for unknown species REML À adjusted mean for perch

Â concentration in perch
The results of this comparison were again encouraging, yielding a significant linear regression (p < 0.03) with a slope of 0.82, an R 2 value of 0.58, and an intercept not significantly different from zero. Water data were not available for these lakes so a comparison of predictions with those using CR values was not possible. However, this demonstrates a potential advantage of the alternative ('REML') approach, in that water concentrations, which are prone to comparatively large temporal variations, are not required.

Stoichiometry and ionomics
The terms, 'ecological stoichiometry' and 'ionomics', are closely related. Ecological stoichiometry (or ecostoichiometry) is 'the study of the balance of chemical elements in components, interactions, and processes in ecosystems' (Sterner and Elser, 2002) and has its origins in aquatic ecology. Ionomics is the study of the total elemental composition of an organism, the ionome (Salt, 2004), and the term has been used mostly within terrestrial botany.

Previous and current uses
Early studies in aquatic ecosystems mainly used ecological stoichiometry to investigate food web dynamics (Lindeman, 1942;Lotka, 1925), and carbon cycling and energy flows (Odum, 1960). Redfield (1958) recognised that the elemental composition of seawater and plankton was fairly constant across wide oceanographic areas and that biological processes could be controlled by elements other than C, such as N and P and trace elements. Since then, most studies have continued to focus on C, N and P and primary producers. More recently, ionomics studies focussing on macro-and micronutrients have been used to explore plant phylogenetics and physiology White et al., 2012). Elements other than C, N and P have also been used in the natural sciences to trace the origins of organisms or materials in the environment. For example, elemental signatures have been used to identify the biological origin of raw materials of biodiesels (Paredes et al., 2014) and the geographic origin of a range of food products (Gonzalvez et al., 2009). Multivariate methods (e.g. principal component analysis (PCA) and discriminant analysis), based on elemental composition, have been used to distinguish between ecosystem components and trophic groups in a coastal ecosystem (Kumblad and Bradshaw, 2008), different angiosperm families in grassland habitats (White et al., 2012) and freshwater invertebrate taxa (Karimi and Folt, 2006). However, such approaches have not been widely used in radioecology.
In radioecology, it is known from field and experimental data that concentrations of certain elements/radionuclides are closely correlated to each other, to such a degree that they may be used as analogues for each other. The most common examples are radio-Sr/ Ca and radio-Cs/K. The concentration of Sr or Cs in fish can be estimated from the water concentrations of Ca or K, respectively (e.g. Kryshev and Ryabov, 2000;Smith et al., 2009) and the Sr concentration in milk can be predicted from Ca concentrations in the diet and milk (Beresford et al., 1998b). Similar relationships between Ca and P (e.g. Hendrixson et al., 2007) or Ra and Ba (e.g. Gonneea et al., 2013) offer potential for further exploration.
Another approach that has been used in radioecology is based on the earlier ecological theory of organic matter (carbon) and Fig. 7. A comparison of measured mean Cs concentrations in fish of different species from three different lakes in northern England with values predicted using measured data for perch and the REML-adjusted means (see Table 6). Error bars are minimum and maximum values, and the solid line represent the 1:1 relationship between predicted and measured values. energy flows in ecosystems. Radionuclide or element concentrations in ecosystem components are normalised to their C content, i.e. element:C ratios are calculated, based on the assumption that many elements are stoichiometrically related to the carbon content because of their role in metabolism and structural components of the organism (Elser et al., 2000;Bradshaw et al., 2012). Ecosystem models based on carbon flows are thus constructed as the basis for radionuclide/element transfer models, using CR values based on Cnormalised element concentrations (Kumblad et al., 2006;Bradshaw et al., 2012;Konovalenko et al., 2014).

Natural variation in element composition
A key issue in ecostoichiometry is whether an element is essential or non-essential. The ability of organisms to maintain constant body concentrations, despite changing concentrations in the environment and/or their resource supply, (homeostasis) is generally assumed to be weak for autotrophs and strong for heterotrophs (Sterner and Elser, 2002). Therefore, plant and algae stoichiometry is thought to more closely reflect that of the environment than animals, though this is much debated (e.g. Persson et al., 2010). The degree of homeostasis has been shown to vary depending on whether elements are macronutrients, essential micronutrients, or non-essential elements (Karimi and Folt, 2006;Bradshaw et al., 2012). Organisms often take up the necessary amounts of trace elements from their food within the 'window of essentiality' (Hopkin, 1989) to ensure essential levels but avoid toxic concentrations in the body.
These principles could be made use of when using ecostoichiometry in extrapolation. For example, for autotrophs, extrapolation from environmental concentrations or ratios is probably more appropriate, whereas for heterotrophs, extrapolation within taxonomic groups may be more relevant (Karimi and Folt, 2006;White et al., 2012). Higher trophic levels may have a more similar elemental composition to their food than lower trophic levels, and stoichiometric ratios may shift between abiotic-biotic components and from primary producers to primary consumers (Bradshaw et al., 2012). In cases of homeostasis, element ratios will be constrained by the biology and ecology of the organism/ecosystem. A better understanding of these natural ranges would greatly help in our ability to predict element concentrations.

Discussion
We will never have data to populate all of the potential radioecological modelling parameters for wildlife (or indeed human) assessments. Therefore, we need robust extrapolation approaches which allow us to make best use of our available knowledge. In this study, we have reviewed, developed, tested and validated some of these extrapolation approaches.
Application of the transfer coefficient concept has been widely adopted in radioecology to estimate activity concentrations in farm animal products; applicability to wildlife assessments has also been proposed. However, this parameter is, in part, dependent upon the dry matter intake of animals and this gives a false impression of differences in transfer between different species. The concentration ratio (CR product-diet or CR wo-diet ) is a more generic parameter, and should enable the more abundant data for farm animal species to be applied to wild species.
Allometric expressions describing radionuclide biological halflife have previously been proposed for some elements. Beresford and Vives i Batlle (2013) present an approach, based on allometry, to predict acceptable T 1/2B for a number of radionuclides and over orders of magnitude range in animal live-weight. However, we acknowledge that for some radionuclides (i.e. Am, Ce, Eu, Pu and Th), T 1/2B has been reported to scale to circa M 0.8 and not the circa 0.25, as assumed in Beresford and Vives i Batlle (2013). To fully exploit the potential of allometric models, the reasons for this need to be explored. Beresford and Wood (2014) demonstrated that the relationship derived by Beresford and Vives i Batlle (2013) for homoeothermic vertebrates is not directly applicable to poikilothermic reptiles. However, it is possible to parameterise the Beresford & Vives i Batlle model using reptile-specific values to obtain reasonable agreement between measured and predicted T 1/2B values. Similarly it is possible to adapt the model parameters for other animal groupings using published information (e.g. as we have demonstrated above for different categories of mammals).
For aquatic ecosystems, the relationship between log 10 (a) (on the allometric relationship for CR) and log(K d ) observed by Vives i Batlle et al. (2009) presents a potential opportunity to estimate CR values when data are lacking, assuming K d is known.
The commonly used CR wo-media approach to estimating the radionuclide activity concentrations in wildlife is open to criticism, as CR wo-media values can be highly variable, largely due to sitespecific factors. The analyses of available data using the REML analysis, as demonstrated by , should compensate for inter-site variation, assuming sufficient data are available for the analysis. For freshwater fish the outputs of the REML analysis predicted 137 Cs activity concentrations in a new independent dataset well. We recommend that this approach of producing relative values be further investigated and developed for other radionuclides and across a wider range of organisms and ecosystems. A disadvantage of the approach is that it requires relatively large datasets which must meet specific criteria. However, recent data compilations Yankovich et al., 2013;Copplestone et al., 2013) should enable similar analysis to be conducted for a number of elements for terrestrial, marine and freshwater species.
The ICRP (2009) suggested identifying a series of terrestrial, freshwater and marine sites from which samples of their Reference Animals and Plants (RAPs) could be sampled and analysed to serve as 'points of reference'. Such studies have been initiated and results are starting to be published . However, such data are highly site-specific, potentially limiting their wider applicability. The application of the REML approach to data, such as those presented by Barnett et al. (2014), to derive relative values for different organisms should provide a more generic set of 'reference data'. A collaboration of the TREE (http://www.ceh.ac.uk/tree) and COMET (www.comet-radioecology.org) projects is taking this area of research forward, including the focused sampling of species falling within the RAP definitions at a number of terrestrial sites in different countries (Japan, Norway, Spain, UK and Ukraine).
Ecological stoichiometry shows potential as an extrapolation method in radioecology, either from one element to another or from one species to another. This will be most successful when using data from elements, organisms (taxa, functional groups) and/ or ecosystems that are as similar as possible to each other. At the very least, stoichiometry could be used to set limits on the range of possible element concentrations; there are biological and ecological limits to many element concentrations/ratios, even if there is variability.
Multi-element datasets are becoming more common as analytical techniques (e.g. ICP-MS) become easier, faster and cheaper. In the near future we will have access to a much larger amount of data (e.g. Baxter et al., 2007; http://www.ionomicshub. org/home/PiiMS), which can be used to test stoichiometric assumptions and theories, and to develop extrapolation methods. Such multi-element datasets could be analysed to identify similarities in element profiles between or within taxa, trophic level, functional group, habitat, geographical region, season or year, depending on the temporal and spatial resolution of the data. This would identify the most promising groupings for which extrapolation methods could subsequently be tested, using simple element-specific ratios or predictive multivariate methods, such as those developed by the food industry. There, multi-element profiles are usually compared against known profiles using a range of pattern recognition methods (mostly multivariate techniques). Neural networks and regression trees have also been used to separate samples by fingerprinting techniques (see Gonzalvez et al., 2009) and many of these can also be used in a predictive rather than a descriptive or analytical way. The TREE project (http://www. ceh.ac.uk/tree) is taking forward this area of research, primarily for crop plants, but with some consideration of wildlife (and potentially farm animals).