Bioconcentration, bioaccumulation, biomagni ﬁ cation and trophic magni ﬁ cation: a modelling perspective

We present a modelling perspective on quantifying metrics of bio-uptake of organic chemicals in ﬁ sh. The models can be in concentration, partition ratio, rate constant (CKk) format or fugacity, Z and D value (fZD) format that are shown to be exactly equivalent, each having it merits. For most purposes a simple, parameter-parsimonious one compartment steady-state model containing some 13 parameters is adequate for obtaining an appreciation of the uptake equilibria and kinetics for scienti ﬁ c and regulatory purposes. Such a model is ﬁ rst applied to the bioaccumulation of a series of hypothetical, non-biotransforming chemicals with log K OW (octanol – water partition ratio) values of 4 to 8 in 10 g ﬁ sh ranging in lipid contents to deduce wet-weight and lipid normalized concentrations, bioaccumulation and biomagni ﬁ cation factors. The sensitivity of biomagni ﬁ cation factors to relative lipid contents is discussed. Second, a hypothetical 5 species linear food chain is simulated to evaluate trophic magni ﬁ cation factors (TMFs) showing the critical roles of K OW and biotransformation rate. It is shown that lipid normalization of concentrations is most insightful for less hydrophobic chemicals (log K OW < 5) when bio-uptake is largely controlled by respiratory intake and equilibrium (equi-fugacity) is approached. For more hydrophobic chemicals when dietary uptake kinetics dominate, wet weight concentrations and BMFs are more insightful. Finally, a preferred strategy is proposed to advance the science of bioaccumulation using a combination of well-designed ecosystem monitoring, laboratory determinations and modelling to con ﬁ rm that the perceived state of the science contained in the models is consistent with observations.


Introduction
Bioaccumulation of organic substances is an important component of chemical risk assessment for both scientic and regulatory purposes. Chemical concentrations in biota that are orders of magnitude larger than those in water and air are important for several reasons. Such large concentrations may adversely affect organisms across food webs, especially if internal concentrations reach toxic levels. Rather than measure the usually low concentrations in water or air it may be preferable to measure the relatively higher concentrations in biota resulting from bioaccumulation, but this requires information on the magnitude and determinants of these relative concentrations. Studies of bioaccumulation fall generally into one of the following categories: ecosystem monitoring, laboratory tests under controlled conditions, mass balance modelling, and in vivo and in vitro ADME studies. In this contribution we address insights that can be gained from modelling focusing primarily on aquatic organisms that respire in water and sediments but recognizing that similar principles apply to airbreathing mammals, birds, and reptiles. In Table 1 we dene the bio-uptake factors and terminology employed herein and widely used in modelling and monitoring studies. [1][2][3][4][5] The bioconcentration factor (BCF) expresses the increase in concentration, but with little or no increase in fugacity as measured in tests such as OECD 305. 6 The bioaccumulation factor (BAF) includes a further concentration increase as well as a fugacity increase. It can be viewed as the product of the BCF and a 'multiplier' dependent on the BAF of the diet and the ratio of the rates of dietary uptake and respiratory uptake. 7 The bio-magnication factor BMF is essentially the ratio of the BAFs of the predator and the prey and may involve an increase in both concentration and fugacity. The TMF as the slope of the log concentration vs. trophic position is related to the mean BMF of the species comprising the food web. Considerable literature exists on these factors and especially BMFs and TMFs that may yield the highest concentrations and exposures. [8][9][10][11][12] In the terminology of MacDonald et al., 13 a BCF represents solvent switching from water to lipid at a constant fugacity, while BAF, BMF and TMF represent additional solvent depletion as the ingested lipid solvent is hydrolysed causing an increase in fugacity.
In Table 1 the water concentration may be of whole water or (as in this study) only truly dissolved chemical. Biotic concentrations may be expressed as wet weight or lipid normalized quantities or they may be specic to dened tissues. Trophic magnication factors are generally obtained from the slope of a plot of log lipid normalised concentrations vs. trophic position or level, the latter being deduced from 15 N measurements. 8,14,15 It is obviously critical to dene the concentration units derived from the mass balance equations for comparison with monitoring data. The most commonly used units are whole body (wet weight) C FW and lipid-normalized C FL concentrations where C FL equals C FW /L, and L is the lipid content. Here, subscript F refers to the organism (sh), W to wet weight and L to lipid normalised.
Homogenizing the whole sh, and using a standard method of lipid extraction may be used to determine these concentrations. It can be experimentally demanding to homogenize large sh, thus it may be convenient to analyse only part of the carcass; for example, a llet that is largely muscle and is preferred for human consumption. The concentration in a llet can be signicantly different from that of the whole sh and this difference also applies to lipid-normalized concentrations because the lipid content of the llet is usually much lower than that of the whole sh. For example, Niimi and Oliver 16 obtained such data for PCBs in rainbow trout and showed that for the more recalcitrant congeners (penta-to deca-chloro) the muscle concentrations C FM (ng g À1 ww) averaged a factor of 3.84 lower than that of the whole sh concentrations C FW (ng g À1 ww). This is largely attributable to differences in the reported lipid content of 9.8% in the whole sh and 1.7% in muscle, a factor of 5.76. For example, if C FW is 100 ng g À1 ww then C FM may be approximately 100/3.84 or 26 ng g À1 ww. The corresponding lipid-normalized wet-weight and muscle concentration will be respectively 26/0.098 or 1020 ng g À1 lw and 26/0.017 or 1532 ng g À1 lw, a factor of 5.76/3.84 or 1.5 difference. Lipid-normalized muscle or llet concentrations are thus likely to be signicantly larger than lipid-normalized wet-weight concentrations for substances such as PCBs.
Another complication is the simplifying assumption that hydrophobic chemicals partition only to lipids. It is generally accepted that there is appreciable partitioning into other nonlipid phases such as protein. For example, the ratio of protein to lipid partition coefficients with respect to water being of the order of 0.03 1 implying that less chemical partitions into protein tissue compared with lipid. It follows that if the llet protein content is much higher than that of the lipid, much of the solute will reside in the protein phase. The lipid-normalized concentration would then be over-estimated. Finally, there can be concerns about the accuracy of lipid content measurements, especially at low lipid levels as may occur in planktonic organisms of low lipid content for which the lipid-normalized concentrations are much larger than wet weight concentrations and are very sensitive to errors in lipid measurement. Endo, have demonstrated that partitioning to different biotic phases is more accurately evaluated using Abraham or poly-parameter LFER methods rather than simple octanol-water partitioning (K OW ). We assume, however, that for the present screening-level purposes, the lipid-water partition coefficient is approximately equal to the octanol-water partition coefficient K OW , recognizing that this is a signicant simplication and does not apply to all chemicals, nor to all lipids.
Lipid-normalized concentrations prove to be very convenient when comparing concentrations between species in food webs, in part because they are proportional to fugacities, thus the equilibrium status of a chemical between water, sediment and organisms comprising food webs can be revealed by comparing Table 1 Definitions for BCF, BAF, BMF and TMF

Bio-uptake factors Denition
Bioconcentration factor (BCF) Ratio of sh to water concentrations with no dietary intake Bioaccumulation factor (BAF) Ratio of sh to water concentrations with dietary intake Biomagnication factor (BMF) Ratio of sh to diet concentrations Trophic magnication factor (TMF) Averaged BMF over a food web of several trophic levels lipid normalized or organic carbon normalized concentrations. Obviously, C FW for one sh should not be compared with C FL for another. This issue can become critical when calculating BMFs and TMFs that necessarily involve comparison of concentrations between prey and predator sh. Common practice is to measure and report both wet-weight and lipid-normalized concentrations along with the identity of the tissues analysed and the measured lipid contents.

Evolution of bio-uptake models
Models of bioaccumulation have evolved considerably from the early bioconcentration work of Neely et al. 20 More comprehensive dietary and respiratory uptake models such as that of Thomann 21 address biomagnication in which the predators achieve higher concentrations than their prey. 1,2 An issue common to all environmental models is the optimal number of compartments. To achieve greater delity to reality the number of compartments may be increased, but this is at the expense of requiring additional parameters and their associated uncertainties, especially those controlling inter-compartment transport and partitioning. A balance is needed between complexity and parsimony as dictated by the nature of the application as discussed by several authors. 22,23 The currently preferred strategy is to use as parsimonious a model as possible consistent with satisfying the modelling objective. It is common to refer to one-compartment models in which there is no attempt to describe differences in internal distributions. Rates of input by respiration and dietary uptake are dened using a gross input rate and uptake efficiencies. This implies the existence of an external compartment in which there is a split between absorbed and non-absorbed chemical. It can be argued that the simple one-compartment model actually contains three compartments in which only the splitting performance of the two external compartments is dened, thus simplifying the model. If the chemical is subject to biotransformation in the gut as described by Lo and Gobas 10 or the chemical properties change in response to pH variation during respiration as described by Erikson et al., [24][25][26] then it is essential to dene the mass balances in these 'external' compartments.
For toxicokinetic and toxicodynamic or PBPK models in which uptake or effects in a target organ are evaluated it is essential to include that compartment explicitly. 17,27 An extreme example is the recent model by Larisch et al. 28 that treats 10 internal organs and 3 external organs.
We believe a consensus has emerged that for many practical purposes a simple and parameter-parsimonious one compartment steady-state model with chemical uptake efficiencies is adequate to obtain an inherent appreciation of the dynamic uptake and loss processes as shown in Fig. 1. Exceptions to these processes are likely to occur when sh forage in regions that have particularly high or low contaminant concentrations, when spawning, or when losing large amounts of body mass in the winter.
The primary compartment of concern is the body, the organs, and the tissues, but it is essential to address chemical transport 'splitting' in the gut and the gill cavity as separate absorption efficiencies. The processes corresponding to the arrows in Fig. 1 can be expressed in conventional concentrationpartition ratio, rate constant (CKk) or in fugacity-Z value, D value (fZD) format, but they are, or should be, algebraically equivalent.
The conventional differential and steady-state equations for uptake in CKk format are given in eqn (1)-(4), the various parameters being dened in Table 2.
Mass balance uptake differential equation in CKk format is: where C FW is the wet-weight sh concentration, t is time, C W is the water concentration, C D is the diet concentration, k R is the respiration uptake constant, k D is the diet uptake constant, k V is the respiration output constant, k E is the egestion output constant, k M is the biotransformation rate constant, and k G is the growth rate constant. Integration from an initial sh concentration of zero and constant inputs yields eqn (2). At steady-state this reduces to eqn (3) and the resulting BMF is given by eqn (4). where k T is the sum of the rate constants for all loss processes, namely (k V + k E + k M + k G ). At steady-state when k T t [ 1, C FW approaches (k R C W + k D C D )/k T and a bioaccumulation factor can be calculated as C FW /C W and a biomagnication factor as C FW /C D .
For hydrophobic chemicals when C D is large and k D [ k R C W /C D , BMF W approaches k D /k T .
Further, for a slowly metabolized, hydrophobic chemical in a slow-growing sh k E [ (k V + k M + k G ), BMF W approaches k D /k E dened here as Q C the ratio of dietary uptake and egestion rate constants.
The analogous uptake equations in fZD format are as follows, where D T is the sum of the loss D values.  Transport parameters Q values diet/egestion

Calculated quantities
Eqn (1)-(4) Eqn (5)-(8) Fish fugacity f F (Pa) 0.00018 When D T t/V F Z F [ 1 and steady state is approached, yielding eqn (7): For less hydrophobic chemicals when D R [ D D , f F approaches f W , equilibrium applies, BCF W and BAF W approach Z F /Z W and are approximately equal to L$K OW where L is the sh lipid content. BMF W becomes less relevant because dietary uptake is unimportant.
On the contrary, for a persistent, hydrophobic chemical in a slow-growing sh with dietary uptake, is dened as the ratio of dietary uptake and egestion D values. BAF W , C W and f W become less relevant because respiratory uptake is relatively unimportant. The lipid normalized uptake metric BMF L approaches Q f and is the ratio of the sh and diet fugacities. As before, the wet weight BMF W approaches Q C and Q f $(Z F /Z D ) or approximately Q f $(L F /L D ). The two Q values are thus not equal and depend on the relative lipid contents of the sh (L F ) and diet (L D ). Q f directly expresses the increase in fugacity corresponding to biomagnication, while Q C expresses the corresponding increase in concentration.
Substitution of the various rate constants and D values in Table 2 into eqn (3) and (7) demonstrates the exact equivalence of the two formats for a chemical of moderate hydrophobicity. The steady-state eqn (3) and (7) are most readily interpreted, and are of most interest for both scientic and regulatory purposes. It is relatively straightforward to apply the basic equations to multiple organisms in food webs with dened dietary preferences and to organisms that respire in sediments and the water column. The principal challenge is to obtain accurate values for the various equilibrium and rate parameters and dietary preferences.
For hydrophobic substances, the egestion loss rate constant and D value are particularly important, since as discussed later egestion along with biotransformation play a critical role in determining the extent of biomagnication. The most rigorous approach is to dene the input diet and output feces compositions and rates and as relative quantities of materials such as lipids and non-lipids including protein, carbohydrate, inert brous material, and water and assign partition ratios relative to water for each material. An example is the Arnot and Gobas 1 model that treats three materials, lipids, non-lipid organic matter (NLOM) and water in both diet and feces. Larisch et al. 28 treat ve materials. The capacity of the feces to absorb and transport the chemical is inevitably lower than that of the ingested diet by a factor typically ranging from 3 to 10. This factor is primarily determined by the quantities of lipid transported in food and feces, thus a simple and very approximate approach is to suggest, as in Table 2, a multiple Q in the range 3 to 10 by which the egestion rate constant k E or D value D E is less than that for the food. Inspection of the steady-state equation shows that for a persistent hydrophobic chemical in a sh that is not growing, the BMF will approach Q. It is noteworthy that Q C in the CKk format is generally not equal to Q f in the fZD format, thus the BMF expressed as a whole body or wet weight concentration ratio is generally unequal to the fugacity ratio. In principle, it is possible and potentially attractive to dene a Q for each material and calculate a lumped Q C or Q f to deduce the egestion rate.
If there is no dietary uptake and the principal loss is by ventilation, a bioconcentration factor (BCF) can be calculated as k R /k V and equilibrium (equi-fugacity) is approached. If the very simplistic assumption is made that octanol and lipids have similar solvent properties for the chemical and lipids are the only absorbing phase, this BCF can also be estimated as the product of the sh lipid content and the octanol-water partition coefficient, namely L F $K OW . If k R is known, k V can then be estimated as approximately k R /BCF or k R /(L F $K OW ). This BCF is effectively a thermodynamic partition ratio, however, it may be affected by weight gain (growth) or loss. For screening level purposes a lipid content L of 5% is oen used thus a BCF of 5000 corresponds to a K OW of 100 000 or log K OW of 5. We accept the simplistic nature of this approach in that it applies only to a subset of chemicals. Other chemicals partition to other phases by electrostatic interactions, protein binding, and covalent bonding.

Relationships between the CKk and fDZ mass balance equation formats
To illustrate the equivalence and relative merits of the two formats a simple spreadsheet was compiled to calculate the biouptake quantities from selected input parameters for a specimen chemical of log K OW of 6.0, Henry's constant 10 Pa m 3 mol À1 and molar mass 100 g mol À1 in a 10 g sh of lipid content 10% with a growth rate constant of 0.0025 days. The sh is exposed to water at a concentration of 0.001 g m À3 and a diet of lipid content 0.05 g g À1 with a fugacity 1.5 times that of the water. The chemical is subject to biotransformation with a rate constant of 0.01 days À1 . Respiratory and dietary uptake parameters were taken from allometric relationships used by Arnot and Gobas 1 but quantities were rounded off to facilitate interpretation.
Calculations were done in both CKk and fZD formats independently and yield identical results as shown in Table 2, namely a sh wet weight concentration C FW of 182 g m À3 or 1.82 mol m À3 . The corresponding lipid normalized concentrations are a factor of 10 greater. The equilibrium BCF of the sh is 100 000, the BAf W is 181 800, and the BAF L is 1 818 000. The calculated BMF W is 2.42 and BMF L is 1.21. The fugacity of the chemical in water is 0.1 mPa, the diet is 0.15 mPa and the sh is 0.18 mPa, thus the sh to diet fugacity ratio is 1.2, equal to the BMF L , showing modest biomagnication.
The uptake processes are: respiration 10% and diet 90% with a total rate of 0.05 mg per day i.e. 0.5 mmol per day. Loss processes are: respiration 17%, egestion 41%, biotransformation 35% and growth dilution 8%. At steady state the body burden is 1.81 mg or 18.1 mmol. The half-lives for uptake and loss are both 25.2 days. The total input and loss rates are 0.05 mg per day, thus the residence time of the chemical in the sh is 36 days.
It is notable that the assumed ratio of dietary uptake to egestion rate parameters Q C for the CKk format is 6 while Q f for the fZD format is 3. This difference is attributable to the different lipid contents of the diet and sh since Q f is Q C $(L D / L F ). Q C and Q f represent limiting maximum BMFs on a concentration and fugacity or lipid normalized basis respectively as is apparent from eqn (4) and (7). For example, increasing log K OW to 8 and setting biotransformation and growth rates to zero result in a BMF W of 5.98, approaching Q C of 6 and a BMF L of 2.99, approaching Q f of 3. These Q values are critical determinants of BMFs for very hydrophobic chemicals. As K OW increases, dietary uptake becomes the dominant input process and respiration becomes negligible. The sh is then unaffected by the concentration in water except that this water concentration controls concentrations at lower trophic levels.
Inspection of these results suggests that the CKk format is easier to understand and apply. Concentration ratios can, however, become very large and difficult to interpret and relative concentrations between sh and diet items can be misleading since both wet weight and lipid normalized concentrations can be used. This format proves to be most preferred for conditions under kinetic control as applies to hydrophobic chemicals.
The fZD format may be initially more difficult to apply, but it can provide additional insights into the bio-uptake process by revealing the relative equilibrium status between water, sediment and various aquatic species. Bio-uptake metrics expressed as fugacity ratios generally lie in the range 1 to 10 and are more easily interpreted. This format is most relevant when conditions are largely controlled by equilibrium processes as applies to less hydrophobic chemicals. Since BMF L factors and fugacity ratios are equivalent, either can be used to characterize trophic magnication in food webs, however this implies that all partitioning is into lipids and in this simple case that lipids are equivalent to octanol.
Connolly and Pederson 29 rst demonstrated this fugacity increase in monitoring data. This was followed by Gobas and colleagues 30,31 who demonstrated experimentally that lipid digestion causes a fugacity increase in the digestive system and this elevated fugacity is transmitted into the body, causing biomagnication. As noted earlier, this process of fugacity increase can be viewed as being caused by 'solvent depletion'. 13 The fugacity format thus captures the fundamental cause of bioconcentration as being driven by differences in Z values (i.e. the capacity to absorb the chemical, dependent on solubility) between water and the sh, the ratio of which is a partition ratio. It also addresses the fundamental cause of bio-magnication reecting a reduction in Z value in ingested food during lipid digestion and corresponding increase in fugacity.
Ultimately, since both formats yield identical results either or both can be used.

Mass-balance equations applied to bioaccumulation in a simple predator-prey system
We now discuss several implications of the models comprising these equations, especially the sensitivity of desired outcomes to the parameter values of the selected chemical. A signicant advantage of having available a simple and robust validated model is that the implications of changes in parameters can be explored mathematically rather than by actual, demanding, and expensive testing. To illustrate these implications we compile simple bioaccumulation models employing realistic ranges of properties of typical organisms and chemical parameters. The models are outlined in a recent study of TMFs to predict TMFs and BMFs. 9 We suggest typical values for rate constants as a function of sh species and the chemical. Uptake and loss parameters are given in Table 3 for ve species using allometric correlations suggested by Arnot and Gobas 1 as a function of sh mass and temperature. These rate constants (days À1 ) are dened in Table 2. Typical numerical values are included and applied initially to a 10 g sh i.e. smelt, namely k R (respiratory uptake) 470, k D (dietary uptake) 0.063, k G (growth dilution) 0.0025, and k E (egestion) 0.0105, conveniently estimated for screening level purposes as a factor Q of 6 less than k D . Dietary and respiratory assimilation efficiencies and the respiratory loss rate constant k V are K OW dependent and are estimated using parameters from the Arnot-Gobas model. 1 Values of k M are later selected arbitrarily.
We rst model the simple bioaccumulation of a series of hypothetical, non-biotransforming chemicals with log K OW values of 4, 5, 6, and 8 in three smelt of different lipid contents View Article Online exposed to contaminant in the same diet and the same respired water. These predator smelt occupy a functional trophic level (TL) of 2.0 and are designated either as lean (L) with a lipid content of 2%, mean (M) with 10%, and fat (F) with 20%. These rather extreme lipid contents are selected to facilitate interpretation of results. The water concentration is 1 mg m À3 , the diet has a lipid content of 10% and occupies a functional TL of 1.0 and is thus in equilibrium with the water as controlled by the lipid-water partition coefficient that is assumed to equal the octanol-water partition coefficient K OW . These dietary concentrations are designated C DW and C DL on a wet-weight and lipidweight basis respectively. An arbitrary fugacity Z-value Z W of 0.01 mol m À3 Pa À1 in water is assumed corresponding to a Henry's law constant of 100 Pa m 3 mol À1 to enable fugacities to be calculated and compared for the water and sh. In each case the wet-weight sh concentrations C FW and the lipidnormalized concentrations C FL i.e. C FW /L are deduced and compared. The bioaccumulation factors on a wet weight basis (BAF W ) and on a lipid-weight basis (BAF L ) are calculated as C FW / C W and C FL /C W respectively. The biomagnication factors on a wet-weight basis (BMF W ) and on a lipid-weight basis (BMF L ) are calculated as C FW /C DW and C FL /C DL respectively. Also calculated are the percentages of uptake by diet and respiration and the percentages that each loss rate constant contributes to the total loss rate constant, thus identifying the dominant rate constant(s) and the half-times for uptake and clearance.
A selection of the results is given in Table 4 that gives calculated values of C FW and C FL for values of log K OW of 4, 5, 6, and 8. In these simulations zero biotransformation is assumed i.e., k M is zero, but is varied later.
The columns on the le (log K OW of 4.0) show that the lipid normalized concentration C FL is fairly constant (10.8 to 11.3) as are values of BAF L because the lipid is close to equilibrium with the water, but the wet weight concentrations C FW vary considerably (0.23 to 2.15) depending on the lipid content of the sh. The sh fugacities are directly proportional to values of BAF L . This proportionality applies only if lipid is the only sorbing phase. The ratio of sh to diet fugacity equals BMF L and varies from 1.08 to 1.13 indicating near-equilibrium and negligible biomagnication. The percentage chemical input from the diet is minimal (11.9%) because of the low concentration in the diet. The primary loss process is by ventilation, k V , (95 to 99%) which depends inversely on lipid content, thus the rate constant for total loss is smallest for the fat sh, but the absolute rates of input and loss are equal in all three cases. Bioaccumulation factors on a wet-weight basis (BAF W ) range from 226 to 2150 reecting the high variability in C FW . Wet-weight bio-magnication factors (BMF W ) vary similarly from 0.23 to 2.15. The total rate constant for loss ranges from 2.4 to 0.25 days À1 corresponding to short uptake and loss times of 0.3 to 2.8 days. The fat sh is slower to approach steady state because of its greater capacity for chemical. In this case it is clearly preferable to interpret the bioaccumulation phenomena on an equilibrium basis using C FL or fugacity, as is normal practice recommended by Borga et al. 8 and Burkhart et al. 4 Use of wet-weight parameters can obscure the interpretation.
The columns in the centre-le with a higher value of log K OW of 5.0 show that C FL and C FW now both vary considerably. Input is mainly or primarily from diet (57%) and remaining input is from respiration, reecting the higher concentration in the diet. The primary loss processes are by ventilation (k V ) which depends inversely on lipid content, and egestion (k E ) which is independent of lipid content because the absolute rate is C FW -$k E . The rate constant for total loss is now less sensitive to sh lipid content and corresponds to uptake and loss half-times from 2.8 to 19 days. The ratio of sh to diet fugacities (BMF L ) now ranges from 2.21 to 1.51 indicating a higher fugacity in the Table 4 Results of bioaccumulation estimations of a series of chemicals in 3 predator smelt differing in lipid contents, namely 'lean', 2%, 'mean' 10% and 'fat' 20% with properties given in Table 3. Note that BAF L equals f F /f W and BMF L equals f F /f D . The percentage contribution of each process to the overall loss rate constant k T are included sh than that in the water in all cases and greater bio-magnication. In this case either C FW or C FL , or both, can be employed to interpret the bioaccumulation and trophic magnication. The columns on the centre right (log K OW of 6.0) show that C FW now becomes more constant (182 to 440) and C FL is now more variable (9127 to 2194). Diet is now responsible for 93% of the input because of its high concentration. There is appreciable biomagnication with BMF W ranging from 1.8 to 4.4 while BMF L ranges from 9.1 to 2.2 i.e. the effect of lipid content reverses the BMF trend. The primary loss processes are by ventilation and egestion. The uptake and loss half-times range from 19 to 46 days.
The columns on the right with log K OW of 8 represent an extreme condition of super-hydrophobicity 15,32 in which the diet is responsible for 99.8% of the inputs. The wet-weight concentrations are now nearly constant and the lipid-weight concentrations are highly variable. The primary loss processes are now by egestion (k E ) and growth (k G ) which are independent of lipid content, thus the rate constant for total loss ($0.006 days À1 ) is also independent of lipid content and corresponds to a long half-time of 117 days. In this case it is clearly preferable to interpret the bioaccumulation phenomena using C FW rather than C FL because lipid normalization introduces an unnecessary variability. BMF W ranges narrowly from 3.24 to 3.35 to 3.36 but the trend reverses for BMF L with values of 16.2 to 3.35 to 1.68. C FL is now highly variable while C FW is fairly constant. For these conditions, it can be argued that it is preferable to use wetweight concentrations and ratios.
These calculations illustrate several important features of BMFs and thus of TMFs. First, the wet-weight BMF W is C FW /C DW , while the lipid-weight BMF L is C FL /C DL , thus BMF W is BMF L (L F /L D ) where L D and L F are the lipid contents of the diet and sh respectively. When a sh consumes a lean diet and L F /L D is <1.0 then BMF L will exceed the BMF W . On the contrary, when the sh is fattier than its diet the opposite occurs. Only when L F and L D are equal are the two BMFs equal as is apparent for the 'mean lipid' content sh. A BMF L exceeding 1.0 in a predator may be attributable, not to biomagnication, but to a lean diet. At high values of K OW lipid normalisation distorts the predator prey relationships if lipid contents are variable. These results suggest that when interpreting biomagnication data, both BMF W and BMF L should be inspected with the expectation that BMF L will be more useful for less hydrophobic chemicals under equilibrium control and BMF W more useful for highly hydrophobic chemicals under kinetic control.
Second, it may seem counter-intuitive that for highly hydrophobic substances lipid content is inconsequential because most of the chemical probably resides in the lipid phases. This insensitivity to lipid content arises because the sh concentration is controlled by the rate constant for loss k T , and the principal contributing rate constants are independent of lipid content. Only k V depends directly on lipid content, but it is small and insignicant, contributing less than 5% to the losses when log K OW is 8.
Third, a common and correct justication for lipid normalization is that it reects the fugacity of the sh relative to the fugacity of the water and the diet. For the data in Table 2 BMF L equals the ratio of the sh fugacity to that of the water and food. This ratio is close to 1.0 for the le-hand columns, but there is an increase in fugacity for the middle columns reecting the expected biomagnication. For the columns on the right the fugacities in the sh are highly variable, specically 16.2 to 1.68 times that of the water and food. Fugacities thus vary greatly depending on lipid levels of the predator. The biomagnication is not a result of thermodynamic partitioning, it is caused by the kinetic effect of a low and fairly constant value of k T and the high and constant diet concentration that yield constant values of C FW but highly variable values of C FL . Fugacities then become less relevant indicators of bioaccumulation and biomagnication. In addition, varied lipid contents of the diet may impact various rate processes of the predator sh, including the diet ingestion rate, the growth rate, and ultimately the BMF. These types of impacts should be considered by modelers, however further discussion of this topic is beyond the scope of this study.
Simple bioaccumulation with biotransformation. To address biotransformation we again apply the simple bioaccumulation model to a series of chemicals with the same range of log K OW values in the same 3 sh and again with the same water and diets, but biotransformation is introduced as an arbitrarily selected rate constant k M of 0.01 days À1 corresponding to a halflife of approximately 70 days. The importance of this half-life has been reviewed by Arnot et al. 33,34 and Goss et al. 19 have suggested it as a direct metric of bioaccumulation potential.
The results in Table 5 are similar to those in Table 4 but biotransformation causes a reduction in all concentrations. The effect is greatest for the hydrophobic chemicals that have relatively slower non-biotransformation loss processes. Specically, for log K OW of 4 the sh concentrations are similar to those in Table 1. For log K OW of 5 and 6 the concentrations in the fat sh are lower by a factor of up to 2 because k M contributes up to 40% of the losses. For log K OW of 8, k M becomes the dominant (63%) loss rate constant and concentrations fall by nearly a factor of 3. These reductions in concentration are the direct result of k M causing an increase in k T .
For log K OW of 8 the BMF W values are all approximately 1.2 but the BMF L values vary from 6.0 to 1.22 to 0.61 and are clearly being distorted by the varying lipid contents. The introduction of biotransformation as a signicant loss process provides an additional incentive to avoid lipid normalization for very hydrophobic substances because the variation in C FW values is considerably less than those of C FL . Variation in lipid content for highly hydrophobic chemicals is inconsequential for steady state bioaccumulation and biomagnication, but it does affect the time required to reach steady state and this may be reected in concentrations in ecosystems. It is thus interesting to explore how these assertions are reected in food web simulations.

Linear food chain model to determine the TMF with and without biotransformation
A simple 5 species, 5 trophic level obligate food chain is dened in which the phytoplankton are assigned a TL of 1 while other species 2 to 5 have an exclusive diet of the species below. The sh properties listed in Table 3 are deduced from Arnot-Gobas 1 correlations and are regarded as typical values for the masses of the specic species with a Q C value i.e. k D /k E ratio of 6. The model was run as before for log K OW of 4, 5, 6, and 8 and the concentrations C FW and C FL were calculated. Logarithmic values of both concentrations were plotted and regressed against trophic level to obtain a slope (b). The TMF is then calculated using the conventional method, namely the TMF is the antilog of the regression slope, b, of log C on TL. 8,35,36 The results are presented in Fig. 2 as a series of TMF plots using both log C FW and log C FL as a function of TL (lower and upper plots respectively) from which TMF W and TMF L can be obtained, expecting that since TMF is essentially an average BMF, the same general conclusions will apply as discussed earlier for the lean, mean, and fat sh.
There is a signicant difference between wet-weight (C FW and TMF W ) and lipid-normalized (C FL and TMF L ) values. For less hydrophobic substances lipid normalization is clearly desirable because of the greater constancy of the C FL values. As K OW increases, the lipid-normalized lines have increased slopes   Table 4 but including biotransformation with a rate constant of 0.01 days À1 corresponding to a half-life of approximately 70 days and become more variable. In contrast, the wet-weight lines become less variable and approach straight lines indicating constancy in BMF W values. In contrast, for the more hydrophobic chemicals an improved regression is obtainable using TMF W . In general, the two TMF values are unequal. Part of the differences between the two TMFs arises because of systematic changes in lipid content (L) with TL. Generally, if L increases with TL the C FW and C FL lines tend to converge. Only if the lipid contents of all species are equal will the lines be parallel and TMF W and TMF L are equal. A simple numerical example illustrates this dependence of TMF on lipid variations.
For a 4-species linear food chain with a constant BMF W of 3.0, the C FW values for a hypothetical example could be 1, 3, 9 and 27. If the lipid contents are equal, both TMF W and TMF L will be 3.0. If the lipid contents increase from 0.05 at TL of 1 by multiples of 1.2 to 0.06, 0.072 and 0.0864, the TMF W is unchanged at 3.0 but the TMF L decreases to 2.5 which is 3.0/1.2. Similarly, if the lipid contents decrease by a factor of 1.2, TMF L increases to 3.6. These results suggest that lipid normalization is desirable for species that are approaching equilibrium with water with relatively insignicant dietary uptake, with fast respiratory exchange dominating. The opposite applies to hydrophobic chemicals that biomagnify appreciably. Lipid normalization can change the slope of the log C F on TL regression line, causing TMF W and TMF L to diverge. This effect is most important for substances that biomagnify and are likely subjects of regulatory TMF evaluations.
We again suggest that both TMF W and TMF L be calculated from monitoring data. Their ratio is an indication of a systematic variation of L with TL. This ratio can be regarded as a trophic dependence on lipid content (TDL). If the ratio TMF L / TMF W and TDL is 1.0 there is no systematic dependence of TMF on L. A TDL < 1.0 indicates an increase in L with TL and a TDL > 1.0 indicates a decrease in L with increasing TL. An important implication is that if TMFs are to be used in a regulatory context it must be appreciated that TMF L is a function of both bio-magnication and systematic changes in species lipid contents. In practice, large values of both TMFs indicate appreciable biomagnication and either or both can be used.
In Fig. 2 above, TMF W exceeds TMF L by a factor of approximately 1.6 indicating an increase in L with TL as is apparent from the lipid contents in Table 3.
Effect of biotransformation on TMFs. The simple food chain model was run for hydrophobic substances of varying biotransformation rate constants k M which was applied to all species. Fig. 3 shows plots of log C FW as a function of TL for k M values ranging from 0 to 0.05 days À1 . The corresponding TMFs are given. As expected, an increase in k M reduces concentrations, BMFs and TMFs, especially at higher trophic levels.
It is notable that the slopes of the log C FW vs. TL lines are most variable at high trophic levels. This is because the rate constants for other losses are slower for large sh thus the introduction of a specic k M has a greater effect on k T and thus on the BMF W . This raises an interesting possibility that if validated model is available then TMF W data could be used to deduce biotransformation rate constants.

The key role of dietary and respiratory uptake rates
This analysis suggests that a major factor inuencing bioaccumulation and biomagnication is the relative quantities of chemical taken up from water by respiration and from diet. Estimates of these rates require data on the volumetric or mass ows of water and food and the assimilation efficiencies in the gills and GI tract. Correlations and data 37 are available for these processes but improved estimates would be invaluable. A related issue is the location of biotransformation as being either somatic or intestinal as discussed by Gobas and Lo. 10,31 In a simple one compartment model this may be of little importance but for more detailed multi-compartment models this can be critical, especially if QSARs are used to estimate biotransformation rate constants that differ in location and thus reactive environments within the sh. This can cause variations in predicted BCF and BMF values.
We believe that the three superhydrophobic permethylcyclosiloxanes D4, D5 and D6 serve as good examples because they cover a range in hydrophobicity from log K OW of 6.98 for D4 to 8.09 for D5 and 8.87 for D6. These chemicals have been extensively studied in recent years. For D4, D5 and D6 respiratory uptake and loss are negligible thus it can be argued that the chemical observed in high trophic level species such as trout reached their destination entirely by dietary uptake from lower trophic level (planktonic) species, with water playing a negligible role as a source. Greater respiratory uptake is expected for D4, relative to D5 and D6, because of greater water solubility and lower to organic carbon partitioning coefficient (K OC ). This was tested using the model by setting the concentration in the plankton, then setting the water concentration as zero for all other species. As expected, this resulted in a negligible change in concentrations, BMFs and TMFs. A possible implication is that the source of these chemicals in lakes such as Lake Mjosa 15 and marine ecosystems such as Tokyo Bay 36 is not the dissolved chemical in water, it is the contaminated suspended biomass discharged from waste water treatment plants. Clearly more efficient contaminant removal is desirable because the discharged biomass is a direct source of diet to the resident biota. In the case of D5 this explains why contamination in sh is apparently not fully mitigated by the View Article Online expected fast volatilization from the surfaces of these water bodies. Improved parameters for uptake from planktonic and other small organisms would be desirable because they play a critical role as the source of contamination.

Discussion
We rst discuss the issue of whether or not to lipid normalize. In 1995 Hebert and Keenleyside 38 published a paper aptly titled "To normalize or not to normalize? Fat is the question." The authors acknowledged the value of lipid normalization when interpreting concentrations of hydrophobic contaminants in biota, but they pointed out that lipid normalization can lead to erroneous conclusions and direct interpretation of whole-body or wet-weight concentrations may be preferable. They cited three examples: concentrations of polychlorinated biphenyls (PCB) in herring gull (Larus argentatus) eggs in the Great Lakes, concentrations of hexachlorobenzene (HCB) in forage sh in the St. Clair and Detroit Rivers, and a hypothetical example of a hydrophobic contaminant in two species of sh. They concluded that both whole-body and lipid-normalized data should be examined and interpreted, as well as an alternative analysis of covariance (ANCOVA) approach. The paper has been cited over 100 times since 1995 testifying to its general acceptance. However, lipid normalization remains widely recommended and practiced, especially when assessing biomagnication and trophic magnication. 8,39 The results from the present simulations support their assertion that both wet weight and lipid normalised concentrations should be used. Second, the results presented here suggest that BMFs and TMFs of very hydrophobic chemicals subject to biotransformation are primarily dependent on K OW and biotransformation half-lives. This is entirely consistent with the ndings by Walters et al. 40 in their 'global synthesis' of over 1500 TMF measurements, most of which used lipid-normalized data. We suggest, however, that TMFs may be best determined using both lipid normalized concentrations and whole sh (wetweight) concentrations. A benet of the latter approach is that it avoids problems of lipid determination, especially when the organism has a very low lipid content or when llets or muscle are the sampled media. It also avoids the dependence of BMFs on relative lipid contents of the predator and prey. Regressions of log C FW vs. TL may be more robust than those using C FL . An uncontentious conclusion is that it is desirable when processing monitoring data to obtain and evaluate both wet-weight and lipid normalised concentrations, and since there is no added cost, costs for lipid determination could be reduced.
Inclusion of benthic organisms in food webs is oen essential because the subject chemicals may have partitioned into sediments and persisted there for a prolonged period of time. Estimation of pore water concentrations and hence sediment and pore water fugacities is fraught with uncertainties. It is likely that in many cases the prevailing sediment fugacity exceeds the water column fugacity because of organic carbon mineralization, thus fugacities at low trophic levels may be uncertain. 41 Models can be useful for exploring the effect of sediment/water fugacity ratios as discussed by Mackay et al. 9 and Kim et al. 35 The deductions presented above assume that the basic uptake eqn (1) is correct and no other factors substantially inuence the uptake. The assumption that K OW equals the lipid-water partition coefficient is questionable but this should not affect the general conclusions concerning trends in hydrophobicity. Of course, real food webs are more complex and variable than those discussed here but we believe that the same principles may apply.
It must be appreciated that TMFs will differ when using wetweight and lipid-normalised concentrations, especially if lipid contents vary throughout the food web. The differences in TMFs can be signicant and in extreme cases one TMF may indicate trophic magnication whereas another TMF derived from the same data may indicate trophic dilution. This is most likely for hydrophobic chemicals that are appreciably biotransformed. In most cases, the two methods should yield similar, but unequal TMFs characterizing the extent of trophic magnication. For highly hydrophobic substances the calculation and evaluation of fugacities can be fraught with difficulties because concentrations are not inuenced by lipid content. It is believed that these principles may apply to numerous hydrophobic substances including permethylcyclosiloxanes, phthalate esters, and halogenated hydrocarbons. Obviously when using BMFs and TMFs for regulatory purposes it is essential to appreciate the uncertainties introduced by these issues.
Finally, we comment on a semantic issue that it may be asserted that some chemicals biomagnify whereas others do not. Inspection of the uptake equation suggests that ALL chemicals biomagnify because k D inevitably exceeds k E and their ratio approaches Q. The reason that relatively hydrophilic chemicals do not apparently biomagnify is that the increase in fugacity caused by lipid digestion is mitigated by the relatively fast losses by ventilation or biotransformation. The experimental determinations of increases in fugacity in the GIT resulting from 'solvent depletion' by Gobas and colleagues 42 present a compelling case for the fugacity increase in the GIT that is inevitably transmitted into the body of the sh, however, that increase may be subsequently dissipated by loss processes from the body that restore the body fugacity to a lower value similar to that of the food or water.

Conclusions: the continued evolution of bioaccumulation models; needs and priorities
In conclusion, we suggest that improvements in the science and modelling of bioaccumulation would benet from advances in several subject areas.
Since bioconcentration is basically a partitioning phenomenon, there is a need for more data on partition ratios to all relevant tissues by experimental determinations, QSAR development and fundamental computational methods, including the inuences of temperature and ionization especially for cations that can be highly toxic. Such developments would enhance the use of tissue-specic parameters in PBPK models. Partitioning from sediment solids and pore water into benthic organisms can have a high degree of uncertainty, especially when water column concentrations are responding to changes in chemical emission rates. It would be useful to document passive sampling and other methods of determining sediment/ water fugacity ratios. 41 Chemical biotransformation or metabolism rates play a critical role in determining bioaccumulation, thus measurements and correlations need further improvement, including the differences between somatic (body) and gastrointestinal rates 10 that occur in very different milieu.
Determination of trophic magnication in ecosystems must consider the effects of spatial and temporal variability, sediment-water fugacity ratios, reproductive losses and changing diets with growth.
There should be continuing consideration of the number and nature of sh compartments required for scientic and regulatory purposes, including consideration of the required accuracy and the degree of parsimony justied. The full implications of using steady state models as distinct from dynamic models also requires evaluation.
The various uptake and loss parameters are fundamentally linked to the organism's bioenergetics, thus consistency between these variables should be sought. There is a need for improvements in estimation methods for processes such as dietary and respiratory assimilation efficiencies and extents of digestion of various food items. The use of different Q values for each digested material should be explored.
As described here the model gives only point estimates of output quantities. Although incorporation of methods to evaluate probability and uncertainty would be useful it would be impractical for this paper because of the complexity and food web specicity. The probability and uncertainty associated with the model is a function of numerous variables that are dependent upon the sampled or dened food web under consideration. There is thus an obligation for the modeler to quantify the perceived accuracy of the results of each specic food web, especially as being best qualied to identify important parameters and covariances. There is increasing emphasis on presenting probabilistic estimates of model output using Monte Carlo and Bayesian approaches. A discussion of these methods, while beyond our scope here, deserves increased attention, especially when models are used for regulatory purposes. Nonetheless, Bayesian methods may readily be applied to a sampled food web using methods provided elsewhere (Powell et al. 36,43 ).
Overall, it seems likely that a preferred strategy to advance the science of bioaccumulation is to continue ecosystem monitoring, laboratory determinations of the bio-uptake processes and model development, preferably in concert and supported by regulatory incentives. It is obvious that quantitative estimations of bioconcentration, bioaccumulation, bio-magnication and trophic magnication as used in regulatory programs are essential for exposure and risk assessment, especially for hydrophobic substances. Ultimately, regulatory decisions are best justied using data from well-designed and carefully interpreted monitoring programs, preferably using benchmark chemicals. Models can play an invaluable complementary role to monitoring by conrming that the perceived state of the science contained in the models is consistent with the ecosystem observations.

Conflicts of interest
There are no conicts to declare.