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Critical Review

# Dietary uptake models used for modeling the bioaccumulation of organic contaminants in fish^{†}^{‡}

Article first published online: 9 DEC 2009

DOI: 10.1897/07-462.1

Copyright © 2008 SETAC

Additional Information

#### How to Cite

Barber, M. C. (2008), Dietary uptake models used for modeling the bioaccumulation of organic contaminants in fish. Environmental Toxicology and Chemistry, 27: 755–777. doi: 10.1897/07-462.1

^{†}^{‡}Published on the Web 12/3/2007.

#### Publication History

- Issue published online: 9 DEC 2009
- Article first published online: 9 DEC 2009
- Manuscript Accepted: 13 SEP 2007
- Manuscript Received: 1 AUG 2007

### Keywords:

- Bioaccumulation;
- Model;
- Dietary uptake;
- Assimilation efficiencies;
- Fish

### Abstract

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

Numerous models have been developed to predict the bioaccumulation of organic chemicals in fish. Although chemical dietary uptake can be modeled using assimilation efficiencies, bioaccumulation models fall into two distinct groups. The first group implicitly assumes that assimilation efficiencies describe the net chemical exchanges between fish and their food. These models describe chemical elimination as a lumped process that is independent of the fish's egestion rate or as a process that does not require an explicit fecal excretion term. The second group, however, explicitly assumes that assimilation efficiencies describe only actual chemical uptake and formulates chemical fecal and gill excretion as distinct, thermodynamically driven processes. After reviewing the derivations and assumptions of the algorithms that have been used to describe chemical dietary uptake of fish, their application, as implemented in 16 published bioaccumulation models, is analyzed for largemouth bass (*Micropterus salmoides*), walleye (*Sander vitreus = Stizostedion vitreum*), and rainbow trout (*Oncorhynchus mykiss*) that bioaccumulate an unspecified, poorly metabolized, hydrophobic chemical possessing a log *K*_{OW} of 6.5 (i.e., a chemical similar to a pentachlorobiphenyl).

### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

Over the past 30 years, two basic modeling approaches have evolved to predict concentrations of persistent organic chemicals in fish. The first uses equilibrium-based distribution coefficients that are simply the ratios of steady-state chemical concentrations in fish to those in selected exposure media. Commonly used examples of these distribution coefficients include bioaccumulation factors (BAFs; i.e., the ratios of fish concentrations to water concentrations), biomagnification factors (BMFs; i.e., the ratios of fish concentrations to prey concentrations), and biota-sediment accumulation factors (BSAFs; i.e., the ratios of fish concentrations to sediment concentrations). The second approach explicitly describes chemical bioaccumulation using process-based differential equation models. Although concentrations of moderately hydrophobic chemicals (i.e., log *K*_{OW} < 5) can often be predicted accurately using equilibrium-based methods, such methods frequently fail to predict accurately observed concentrations of extremely hydrophobic chemicals (i.e., log *K*_{OW} > 6), which generally are the contaminants of greatest concern. For example, BAFs for chemicals with log *K*_{OW} > 6 can range over two or three orders of magnitude (e.g., see Fig. 12 in Arnot and Gobas [1]). Moreover, lipid-normalized BAFs for such chemicals can range over four orders of magnitude (e.g., see Fig. 13 in Arnot and Gobas [1]). Consequently, concentrations predicted using any particular BAF can be considerably above or below observed concentrations. Several factors can be identified to explain these discrepancies.

Lower-than-expected concentrations can result when the duration of exposure is insufficient to allow chemicals to equilibrate. Because bioaccumulation generally is treated as a linear, first-order process, one can estimate the time needed for chemicals to equilibrate between fish and their exposure media. For example, assuming a constant exposure, the time required for a chemical to achieve 95% of its equilibrium concentration within a fish is approximately 4.3-fold its elimination half-life [2]. Because elimination half-lives of poorly metabolized chemicals generally increase as their hydrophobicity increases, equilibration times also increase as functions of chemical hydrophobicity. Consequently, for persistent organic chemicals, such as polychlorinated biphenyls (PCBs) and dioxins that have elimination half-lives ranging from months to a year or more [3–6], equilibration times can range from one to five years. Even when exposure times are sufficient for equilibration, whole-body concentrations can be lower than those predicted by thermodynamic partitioning because of biodilution of the chemical that accompanies body growth, biotransformation of the compound, and the complexities of dynamic, rather than constant, aqueous and dietary exposures.

Equilibrium-based distribution coefficients also implicitly assume either that only one route of exposure dominates a fish's total accumulation process or that the relative contributions of multiple exposure pathways are essentially constant (i.e., one pathway can serve as a surrogate for the remaining pathways). For BCFs, the implicit assumption is that fish accumulate virtually all their body burdens directly from water via gill or, possibly, dermal exchange. Although direct aqueous uptake is the dominant route of accumulation for moderately hydrophobic chemicals, dietary uptake is the dominant pathway for extremely hydrophobic chemicals. This shift in the relative contributions of the direct aqueous and dietary pathways is largely a result of changes in the relative rates of exposure via these pathways as chemical hydrophobicities increase. Consider, for example, the relative magnitude of aqueous and dietary exposures for lake trout (*Salvelinus namaycush*) feeding on alewife (*Alosa pseudoharengus*) as a function of *K*_{OW}. Although the trout's direct gill exposure (GE; mol/d) is the product of its ventilation volume (*Q*_{v}; ml/d) and the chemical's ambient water concentration (*C*_{w}; mol/ml), its dietary exposure (DE; mol/d) is the product of its feeding rate (*F*_{w}; g wet wt/d) and the chemical's concentration in alewife (*C*_{p}; mol/g wet wt). Assuming that alewives have essentially equilibrated with the ambient water because of their small size and trophic habits, one can estimate the condition for which a trout's dietary exposure exceeds its aqueous exposure using the inequalities:

- (1)

Using data from Stewart et al. [7] and from Erickson and McKim [8], the ventilation to feeding ratio for a 1-kg trout would be approximately 10^{4.3} ml/g wet weight. If one then assumes that the alewife's BAF (BAF_{p}) can be estimated by the quantitative structure-activity relationship (QSAR) proposed by Mackay [9] (i.e., BAF_{p} = 0.048 *K*_{OW}), one would conclude that food is the trout's predominant route of exposure for any chemical with *K*_{OW} > 10^{5.6}.

Once the dietary pathway becomes a fish's dominant route of exposure, its accumulated whole-body chemical concentrations can actually exceed those predicted by thermodynamic partitioning. At least two factors are responsible for this phenomenon. Whereas the first of these factors is the fish's decreasing ability to excrete extremely hydrophobic chemicals across its gills, the second is the fish's ability to maintain high dietary diffusion gradients. These sustained diffusion gradients are the direct consequence of food digestion and the assimilation of digestion by-products. In particular, because the uptake of digestion by-products generally is more rapid than the uptake of hydrophobic contaminants, concentrations of hydrophobic chemicals in the fish's unassimilated gut contents and intestinal fluids can actually increase during food digestion and assimilation [10–15]. Not only do these sustained diffusion gradients facilitate continued dietary uptake, they also retard fecal excretion of previously accumulated chemical.

To overcome the limitations of equilibrium-based methods, numerous process-based differential equation models have been developed to predict chemical concentrations in fish as functions of aqueous and dietary exposures [12,16–33]. Conceptually, all these models explicitly or implicitly describe the bioaccumulation of chemicals within individual fish using a system of equations that is equivalent to

- (2)

- (3)

In Equation 2, *W*_{d} denotes the fish's dry body weight (g dry wt/fish), and *F*_{d}, *E*_{d}, *R*, EX, and SDA denote the fish's feeding, egestion, routine respiration, excretion, and specific dynamic action (i.e., the additional respiratory expenditure in excess of *R* required to assimilate food), respectively, in units of g dry wt/d. Although many physiologically based models for fish growth are formulated in terms of energy content (e.g., kcal/fish and kcal/d), Equation 2 generally applies to these bioenergetic models, because the energy densities of fish are estimated by their dry weights [34–36]. In Equation 3, *B*_{f} denotes the fish's chemical body burden (mol/fish); *J*_{g} and *J*_{i} denote the net chemical exchange (mol/d) across the fish's gills from the ambient water and across its intestine from food, respectively; and *J*_{m} denotes the gain or loss of chemical (mol/d) because of biotransformation. A fish's whole-body concentration (*C*_{f}; mol/g wet wt), which typically is used to formulate *J*_{g}, *J*_{i}, and *J*_{m}, is calculated algebraically as the ratio of the fish's body burden to its wet body weight, *W*_{w} = *W*_{dρdf}^{−1}, where _{ρdf} is the fraction of the fish's live weight that is dry organic matter and ash. Despite this underlying similarity, however, most published fish bioaccumulation models differ significantly from one another regarding how chemical exchanges from food and water are conceptually and mathematically represented.

Although Barber [37] recently reviewed the mathematical formulation and performance for 10 of the most widely used gill exchange models, a comparable review of the dietary exchange models used for fish bioaccumulation modeling does not exist except for excellent conceptual overviews by Landrum et al. [38], Wania and Mackay [39], Gobas and Morrison [40], and Mackay and Fraser [41]. Therefore, the goal of the present study is to review the derivation and important assumptions of dietary exchange models and to analyze how their predictions compare both to one another and to published laboratory and field results. Important issues considered herein include alternative interpretations of chemical assimilation efficiencies, analysis of factors influencing observed chemical assimilation efficiencies, use of dynamic versus steady-state kinetics in describing dietary chemical exchange, analysis of patterns of non-steady state BAFs and BMFs predicted by dynamic simulation models, analysis of the relative significance of dietary and gill uptake as predicted by different bioaccumulation models, analysis of the relative significance of growth and excretion as elimination processes, and ramifications of different parameterization schemes on model predictions. For comparable reviews of models used for predicting chemical bioaccumulation in oligochaetes and aquatic invertebrates, see Ram and Gillett [42] and Wang and Fisher [43], respectively.

The explicit presentation of the mathematical details of the models reviewed herein is deliberate and necessary to delineate basic model assumptions, to understand differences and similarities in model predications, and to evaluate potential model strengths and weaknesses. The *Appendix* summarizes the notation used to describe the models compared herein. Readers should remember, however, that this notation is adopted here for consistency of presentation and comparison and may differ from that used by the developers of the models being discussed. In addition to this notation, the acronyms AE and RC are used throughout the subsequent text for the terms assimilation efficiency and rate coefficient, respectively.

Last, numerous supporting data tables, figures, and references have been placed into a SETAC Supplemental Data Archive (http://dx.doi.org/10.1897/07-462.S1). These archived tables and figures are referenced herein with an uppercase S prefix (e.g., Table S1 and Fig. S1).

### REVIEW OF DIETARY EXCHANGE MODELS

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

#### AE models

The dietary uptake of chemicals by fish often has been modeled assuming that fish assimilate a constant fraction of the chemical that they ingest:

- (4)

where α_{c} is the fish's AE (dimensionless) for the chemical of interest, *C*_{p} is the chemical concentration (mol/g wet wt) in the fish's ingested prey, and *F*_{w} is the fish's wet weight consumption (g wet wt/d) (see, e.g., Penry [44]). Bioaccumulation models that have employed this formulation, however, can be separated into two distinct groups. The first group implicitly assumes that chemical AEs describe the net chemical exchanges between fish and their food [16,19,22–25,28–30,32,45–47]. These models describe a fish's chemical elimination either as a single, lumped parameter process that is independent of the fish's egestion rate or as a process that does not require an explicit fecal egestion term. The second group of models, on the other hand, assumes that chemical AEs describe gross chemical uptake [12,17,18,20,21,26,31,33,48–50]. These models explicitly formulate a fish's chemical elimination via fecal egestion, gill excretion, and metabolism as distinct processes. Most of the models in this second group have been developed by describing the exchange of dietary contaminants as a thermodynamically driven diffusion process. The remainder of this section will focus only on the first group of models. Discussion of the second group is presented in the following sections concerning thermodynamically based models. Readers also should consult Borgmann and Whittle [51] for an alternative discussion of the application and analysis of these two types of bioaccumulation models.

Assimilation efficiencies for Equation 4 often have been estimated using the following constant coefficient (i.e., α_{c}, ϕ_{ww}, *C*_{p}, and *k*_{2}) model proposed by Roberts et al. [52] and Bruggeman et al. [53]:

- (5)

- (6)

where ϕ_{ww} = *F*_{W}/*W*_{W} is the fish's specific feeding rate (g wet wt/g wet wt/d), *k*_{2} is the fish's lumped-process chemical elimination RC (1/d), and *t* is time (d). Equation 6, however, is the solution to Equation 5 only when the dietary exposure starts at time zero (*t*_{0}) and the fish's initial body concentration is zero (i.e., *C*_{f}(*t*_{0}) = 0). The general solution to Equation 5 is actually

- (7)

Although Equations 5 and 6 frequently have been used to describe the results of laboratory-exposure studies [53–58], model users should evaluate carefully the conditions associated with such studies before using their results to parameterize Equation 4. To illustrate this assertion, Equation 7 can be redifferentiated to obtain the following exponential equivalent of Equation 5:

- (8)

Using this equation, one can analyze the expected behavior of chemical AEs determined for consecutive time segments of a dietary exposure for which Equations 5 and 6 are assumed to apply. For example, let *t*_{1}, *t*_{2}, and *t*_{3} denote the times that a dietary exposure starts, is halfway over, and ends, respectively. During the first half of the exposure (i.e., *t*_{1} < *t* < *t*_{2}), the fish's bioaccumulation dynamics will be described by

- (9)

During the second half of the exposure (i.e., *t*_{2} < *t* < *t*_{3}), however, these dynamics are given by

- (10)

where and k̂_{2} denote the fish's chemical AE and elimination RC, respectively, that may require re-estimation. In other words, the fish's chemical AE and elimination RC might be only piecewise constants during the exposure interval of concern. If, indeed, Equation 6 describes the fish's concentration dynamics over the entire exposure (i.e., *t*_{1} < *t* < *t*_{3}), then the derivatives specified by Equations 9 and 10 must be equal when they are evaluated for *t* = *t*_{2}. This continuity condition requires that

- (11)

when the fish's initial whole-body concentration is zero, the preceding equation can be shown to reduce to

- (12)

Because the elimination RCs k̂_{2} and *k*_{2} typically are decreasing functions of the fish's body weight [10,21,28,37,59–61], if no growth occurs during the experiment, then k̂_{2} = *k*_{2}, which, in turn, implies that = α_{c}. If significant growth does occur, however, then k̂_{2} < *k*_{2} and < α_{c}. This prediction is completely consistent with the notion that AEs determined by fitting experimental data to Equation 6 must, in general, characterize net rather than gross AEs that necessarily must decrease during the course of a dietary exposure. In particular, consider the experimental situation in which one determines AEs for fish at 1 and 100 d after a dietary exposure begins by direct measurement of a chemical in the fish's ingested food and egested feces. After 1 d, virtually all the chemical in the fish's feces is truly unassimilated chemical. After 100 d, however, the fish's feces will contain not only unassimilated chemical but also previously assimilated chemical that is being excreted. This increased fecal chemical burden obviously will decrease the fish's calculated AE, because excreted chemical cannot be distinguished from unassimilated chemical.

To illustrate how different the chemical AEs α_{c} and can be when fish are growing, assume for simplicity that the chemical of interest is not metabolized. In this case, the fish's apparent elimination RC would be given by

- (13)

where ϵ_{g} denotes the fish's gill excretion RC (1/d), and γ = *W*_{w}^{−1}*dW*_{w}/*dt* is the fish's specific growth rate (g wet wt/g wet wt/d). Remember that because α_{c} is a net uptake parameter, no fecal elimination term is required for this expression of *k*_{2}. Although several excellent process-based models are available to estimate ϵ_{g}, for the sake of discussion this parameter is estimated simply by

- (14)

where ρ_{1f} is the fish's whole-body lipid fraction [37]. Using Equations 13 and 14, Equation 11 can be plotted as a function of *K*_{OW} and the fish's specific growth rate for two scenarios. The first scenario (Fig. 1) assumes that fish (10 g wet wt) are fed a constant ration of contaminated food for 60 d. The second scenario (Fig. 2) displays the longer-term behavior of Equation 11. For this scenario, fish (1 g wet wt) are fed a constant ration of contaminated food for one year. Both figures demonstrate that for Equation 6 to describe the concentration dynamics of growing fish over the entire exposure interval of concern, the fish's chemical AEs estimated for sequential exposure subintervals must be decreasing functions of time. This conclusion is consistent with the findings of Connolly [19], who discovered that to simulate accurately the PCB bioaccumulation in age-structured populations of winter flounder (*Pleuronectes americanus*) using constant AEs, calibrated efficiencies much lower than those suggested by short-term uptake studies had to be used. Similarly, Jones et al. [62] noted that chemical AEs estimated for long-term exposures of dioxins to rainbow trout (*Oncorhynchus mykiss*) were markedly lower than those estimated for shorter term exposures.

Graphic and statistical analyses of chemical AEs reported for Equations 4 to 6 have shown that they generally are nonlinearly correlated with a chemical's *K*_{OW}. Analyses by Thomann [29] and by Thomann et al. [31] have suggested that whereas AEs for low to moderately hydrophobic chemicals are either constant or, possibly, positively correlated with *K*_{OW}, AEs for chemicals with *K*_{OW} > 10^{6} are negatively correlated with *K*_{OW}. Because chemical AEs for gill uptake also display this pattern of correlation, Thomann et al. concluded that a fish's dietary and gill AEs could be estimated with the same empirical function of *K*_{OW}. Using published data from a variety of sources, Thomann et al. developed a generalized, piecewise function to estimate dietary and gill uptake efficiencies for steady-state bioaccumulation models that is equivalent to

- (15)

Fisk et al. [63] observed that growth-adjusted AEs also increased with increasing *K*_{OW} for low to moderately hydrophobic chemicals and decreased with increasing *K*_{OW} for extremely hydrophobic chemicals. Rather than using a piecewise function to describe this pattern, however, Fisk et al. characterized this relationship for juvenile rainbow trout using the quadratic regression

- (16)

Finally, following the example of Gobas et al. [11] and Clark et al. [48], who used hyperbolic regressions to estimate initial AEs for their thermodynamically based dietary exchange models, Muir et al. [64] characterized constant AEs of rainbow trout exposed to chlorinated dioxins and furans using the hyperbolic regression

- (17)

#### Thermodynamically based exchange models

Because chemical exchanges across the intestine are driven by diffusive gradients [13,48,65,66], dietary uptake models that employ net AEs cannot logically be consistent with thermodynamics unless those parameters are decreasing functions of the fish's total body concentration. To address this issue, several thermodynamically based models for describing a fish's chemical dietary exchange have been proposed. These models explicitly or implicitly assume that chemical dietary exchanges are driven either by the concentration difference between a common phase (e.g., water/plasma) within the fish and its food/feces or by the fugacity difference between mixed phases within the fish and its food/feces. Although the non-steady state models discussed here assume that these exchanges occur by simple molecular diffusion, other modes of diffusion also have been proposed as contributing or controlling mechanisms. An excellent review of these alternative modes of diffusion, which include both micelle-mediated and fat-flush diffusion, is given by Kelly et al. [67]. Whereas these alternative diffusion mechanisms may be important in explaining dietary exchanges of birds and mammals, their role in controlling dietary exchanges of fish is not well established. For example, Gobas et al. [13] concluded that intestinal absorption of organochlorines by goldfish (*Carassius auratus*) appeared to conform to a simple molecular diffusion model rather than to a lipid micelle-mediated diffusion model. This conclusion is consistent with findings showing that micelle diffusivities generally exhibit the Stokes-Einstein relationship (see, e.g., [68,69]) and that observed diffusivities of freely dissolved drugs are 10- to 30-fold faster than those of drug-micelles (see, e.g., [70]).

#### Dietary exchange based on fecal partitioning

To formulate their dietary exchange model, Barber et al. [12,17,18] focused on the simple mass-balance relationship

- (18)

where *F*_{w} and *E*_{w} denote the fish's daily wet-weight feeding and egestion, respectively, and *C*_{g} is the chemical's concentration (mol/g wet wt) in the fish's gut contents/feces. Because the transit time through the gastrointestinal tract is relatively slow, Barber et al. assumed that the chemical concentrations in the fish's aqueous body, intestinal fluids, and dry fecal matter essentially equilibrate with one another. Connell [71] made similar assumptions to analyze the ratio of a predator's chemical concentration to that of its prey. Using this assumption, a fish's chemical fecal elimination was formulated as

- (19)

where *C*_{af} and *C*_{ag} are the chemical's concentrations (mol/ml) in the aqueous fraction of the whole fish and gut contents/feces, respectively; *C*_{og} is the chemical's concentration (mol/g dry wt) in the dry organic fraction of the gut contents/feces; ρ_{ag} is the fraction of the gut contents/feces that is aqueous; *K*_{OC} is an organic carbon-water partition coefficient [72–75]; and *K*_{g} is the distribution coefficient from the gut contents' aqueous phase to the whole, wet-weight gut contents (ml/g wet wt).

To parameterize Equation 19, Barber et al. also assumed that chemical equilibration between a fish's internal aqueous, lipid, and nonlipid organic phases was rapid in comparison to chemical exchanges across its gill or intestine. Consequently, chemical concentrations in a fish's aqueous phase were formulated in terms of its whole-body chemical concentrations using the equation

- (20)

where ρ_{af}, ρ_{1f}, and ρ_{of} are the fractions of the whole fish that are aqueous, lipid, and nonlipid organic matter, respectively, and *K*_{f} is the distribution coefficient from the fish's aqueous phase to its whole body (ml/g wet wt). Although the observed percentage moisture of fish intestinal contents typically varies between 50 and 80% [76–80], Barber et al. assumed that ρ_{ag} = ρ_{af} because of rapid osmotic equilibration between the fish's whole-body and intestinal contents. With these additional assumptions, Equation 19 was then rewritten as

- (21)

When this equation is substituted into Equation 18, the fish's predicted net dietary uptake is given by

- (22)

It therefore follows that

- (23)

corresponds to a thermodynamic chemical AE that equals unity when an exposure begins (i.e., *C*_{f} = 0) and that decreases as the fish whole-body concentration increases. Although analyses by Gobas et al. [11] suggest that initial chemical AEs for fish typically vary between 0.10 and 0.43, short-term studies by Nichols et al. [15,81] reported observed initial AEs of greater than 0.95 for rainbow trout exposed to 2,2′,5,5′-tetrachlorobiphenyl. Similarly, Gobas et al. [14] reported the chemical AE for rainbow trout feeding on dry trout chow contaminated with 2,2′,4,4′,6,6′-hexachlorobiphenyl to be approximately 0.71. Tanabe et al. [3] reported that chemical AEs varied between 0.67 and 0.93 for carp (*Cyprinus carpio*) exposed to a variety of PCB congeners for 5 d. During 9-h, single-meal exposures, Burreau et al. [82] reported that chemical AEs varied between 0.35 and 0.90 for northern pike (*Esox lucius*) exposed to a variety of halogenated diaromatic compounds. Importantly, Burreau et al. specifically noted that most of their results exceeded the maximum uptake efficiency of 0.43 predicted by the Gobas et al. regression equation (see Eqn. 52).

Equation 23 can be simplified with respect to the fish's feeding and egestion by letting α_{f} = (*F*_{d} — *E*_{d})/*F*_{d} denote the fish's food AE and by letting ρ_{df}, ρ_{dg}, and ρ_{dp} denote the dry organic matter fractions of the fish, its gut contents/feces, and its prey, respectively. Adopting this notation and again assuming osmotic equilibration, it then follows that

- (24)

Consequently, a fish's thermodynamic AE can also be written as

- (25)

To discuss the qualitative or quantitative behavior of Equations 23 and 25, one must first acknowledge that a fish's whole-body concentration (*C*_{f}) is itself a function of both α_{t} and *C*_{p}. Therefore, consider the following general differential equations for a fish's whole-body chemical burden and concentration:

- (26)

- (27)

where β_{g}, δ_{g}, and δ_{m} are the fish's RCs (cm^{3}/d or g wet wt/d or ml/d) for gill uptake, gill excretion, and chemical metabolism, respectively; *C*_{w} is the concentration (mol/ml) of the truly dissolved chemical in the ambient water; and γ is, again, the fish's specific growth rate (g wet wt/g wet wt/d). Note that because the fish's thermodynamic AE α_{t} already encompasses fecal excretion, Equation 26 does not require an explicit fecal excretion term. When Equation 25 is substituted into Equation 26 and the result substituted into Equation 27, a fish's bioaccumulation dynamics are described by

- (28)

where *k*_{1} and *k*_{2} denote the fish's RCs of gill uptake (ml/g wet wt/d) and of total chemical elimination (1/d), respectively, and ϕ_{dd} = (ρ_{dp}*F*_{w})/(ρ_{df}*W*_{w}) is the fish's specific feeding rate expressed as g dry wt/g dry wt/d. If fish are exposed only to a constant concentration of dietary contaminants, it should be apparent that Equations 5 and 28 are mathematically equivalent except for the definition of their parameters. Consequently, the qualitative features of their solutions are identical, and from the perspective of statistically fitting experimental data, virtually no difference exists between the two models. From the process perspective, however, these models are fundamentally different, because Equation 5 is based on the assumption that the fish's net chemical AE is a fixed time constant whereas Equation 28 assumes that the fish's net chemical AE is an explicit function of its whole-body concentration.

Using Equations 25 and 28, the behavior of a fish's thermodynamic AE can be analyzed for a typical dietary-only laboratory exposure for which *t*_{0} = 0, *C*_{t}(0) = 0, and the fish's dietary exposure concentration (*C*_{p}) and specific feeding rate (ϕ_{ww}) are constant. If the fish's growth is small enough to allow *k*_{2} to be treated as a constant during the exposure period, then the solution to Equation 28 is

- (29)

This equation differs from Equation 6 (i.e., the solution of Eqn. 5 for the stated exposure conditions) by having an implied initial AE equal to one rather than the constant net AE (α_{c}) and by the way the elimination RC (*k*_{2}) is defined (i.e., the explicit inclusion of a fecal elimination term). When this solution is substituted into Equation 25, one can show that

- (30)

(For additional discussion, see Barber et al. [12].) The most obvious feature of Equation 30 is the prediction that a fish's thermodynamic chemical AE decreases exponentially as exposure time increases. Importantly, such trends have been observed by several authors [62,83,84] studying PCB and polychlorinated dioxin bioaccumulation in channel catfish (*Ictalurus punctatus*), coho salmon (*Oncorhynchus kisutch*), and rainbow trout. Another obvious feature of Equation 30 is the prediction that a fish's thermodynamic chemical AE should be inversely proportional to its specific feeding rate once sufficient time has elapsed.

Figures S1 through S3 display the behavior of Equation 25 when Equation 28 is numerically integrated for dietary exposures in which fish growth is significant (i.e., the fish's total elimination RC, *k*_{2}, is a function of time). Figure S1 displays the behavior of Equation 25 as a function of time and specific feeding rate for guppies (*Poecilia reticulata*) during dietary exposures that are analogous to those reported by Clark and Mackay [85]. Figure S2 displays the behavior of Equation 25 as a function of time and food concentration for guppies during a dietary exposure as described by Opperhuizen and Schrap [58]. Although Equation 25 appears to indicate that a fish's thermodynamic chemical AE is a hyperbolic function of food concentrations, the realized AEs displayed in Figure S2 clearly are independent of food concentrations, as is Equation 30. Such predictions are consistent with findings reported by Fisk et al. [63], Jones et al. [62], and Opperhuizen and Schrap [58] for exposures that do not result in lethal or sublethal effects. Figure S3 displays the behavior of Equation 25 as a function of time and *K*_{OW} for rainbow trout during dietary exposures that are analogous to those reported by Fisk et al. [63]. The predicted behavior of α_{t} as a function of *K*_{OW} displayed in Figure S3 is qualitatively the same as that predicted by the Muir et al. [64] hyperbolic regression for α_{c} (i.e., Eqn. 17). Although this figure's predicted α_{t} × *K*_{OW} relationships obviously are different from those predicted by the Thomann et al. [31] regression (i.e., Eqn. 15), obvious similarities exist in their qualitative behavior. In particular, AEs at day 60 for moderately hydrophobic chemicals seem to be constant, whereas AEs for extremely hydrophobic chemicals decrease with increasing *K*_{OW}.

The fecal partitioning model represented by Equations 18 to 22 is used by both the Food and Gill Exchange of Toxic Substances (FGETS) and Bioaccumulation and Aquatic System Simulator (BASS) bioaccumulation models that have been used to simulate both PCB and methylmercury bioaccumulation in a variety of fish communities and laboratory studies [12,17,86–91]. The FGETS model also has been used by Apeti et al. [92] to simulate metal bioaccumulation in American oysters (*Crassostrea virginica*).

#### Dietary exchange based on diffusion kinetics

Barber et al. [18] also formulated a diffusion-based model for dietary uptake and fecal excretion. According to this model,

- (31)

where *S*_{i} and *p*_{i} denote the fish's intestinal surface area (cm^{2}) and chemical permeability (cm/d), respectively. This equation again assumes that chemical concentrations in the aqueous and dry gut contents are equilibrated with one another. The total chemical concentration in the fish's gut contents was calculated algebraically (i.e., *C*_{g} = *B*_{g}/*G*_{w}) from the solutions of the following differential equations:

- (32)

- (33)

where *B*_{g} and *G*_{w} denote the total chemical burden (mol) and wet mass (g wet wt), respectively, of the gut contents and EV_{d} denotes the rate of dry-weight gastric evacuation (g dry wt/d) that is the sum of assimilation and egestion. Equation 33 follows from its dry weight counterpart:

- (34)

where *G*_{d} denotes dry mass (g dry wt) of the gut contents assuming osmotic equilibration between the fish and its gut contents (i.e., *G*_{d} = α_{df}*G*_{w}). The exponent ζ_{3} used to parameterize Equation 33 generally depends both on the characteristics of the food item being consumed and on the mechanisms that control gastrointestinal motility and digestion [93-95]. For example, when gut clearance is controlled by intestinal peristalsis, ζ_{3} theoretically equals one-half, because peristalsis is stimulated by circumferential pressure exerted by the intestinal contents that, in turn, is proportional to the square root of its mass. On the other hand, when surface area controls the rate of digestion, ζ_{3} should approximately equal either two-thirds or unity. If the fish consumes a small number of large-sized prey (e.g., a piscivore), ζ_{3} = 2/3 may be the appropriate surface area model. If the fish consumes a large number of smaller, relatively uniform-sized prey (e.g., a planktivore or drift feeder), however, ζ_{3} = 1 is more appropriate, because total surface area and total volume of prey become almost directly proportional to one another.

Parameterization of this model, however, was difficult. Barber et al. [18] found few studies that could be used to estimate accurately intestinal surface areas in a general way [96–102]. Additionally, objectively defining the diffusion layers needed to calculate the chemical permeabilities of the intestine also was problematic [103]. These facts, combined with the reasonable performance of the fecal partitioning model for describing both field and laboratory data [12], suggested that their fecal partitioning model was adequate for many, if not most, applications of the FGETS bioaccumulation model.

Although Nichols et al. [27] tried implementing a dietary exchange model analogous to the FGETS fecal partitioning model for a real time, physiologically based, toxicokinetic model of brook trout (*Salvelinus fontinalis*), they concluded that a kinetically based model was required. To develop such a model, Nichols et al. formulated a two-component, mass-balance model for chemical exchanges between the fish's gut contents and its intestinal tissues. For the gut contents, Nichols et al. assumed that

- (35)

where *P*_{i} is the transfer coefficient (cm^{3}/d) across the intestine and *K*_{gi} is a partition coefficient describing the equilibrium between the fish's gut contents and intestine. This equation is completely analogous to Equation 32 except for the way that the concentration gradients driving diffusion are formulated. Although the transfer coefficient *P*_{i} corresponds to *S*_{i}*P*_{i}, Nichols et al., rather than decomposing *P*_{i} into its components, simply calibrated this parameter to be directly proportional to the allometric power function describing a fish's postgastric intestinal surface area. For the mass balance of the intestinal tissue, Nichols et al. assumed that

- (36)

where *B*_{i} and *C*_{i} denote the intestine's chemical burden (mol/organ) and concentration (mol/g wet wt), respectively; *Q*_{i} is the perfusion rate (ml/d) of the intestinal tissue; and *C*_{ab} and *C*_{vb} denote the chemical's concentrations (mol/ml) in the arterial and venous blood, respectively. Nichols et al. successfully calibrated Equations 35 and 36 and accurately simulated observed concentrations of 2,3,7,8-tetrachlorodibenzo-*p* -dioxin in brook trout.

Nichols et al. [81,104] recently refined their original, physiologically based, toxicokinetic model for dietary uptake by treating the fish's gastrointestinal tract as four distinct compartments—the stomach, the pyloric caeca, the upper intestine, and the lower intestine—rather than as a single, undifferentiated compartment. Nichols et al. also elected to treat the transfer coefficient of each gastrointestinal component as the product of the component's surface area and permeability. To overcome the paucity of surface area data for these components, Nichols et al. assumed that these areas all scaled to body weight according to *S*_{ij} = σ_{ij}*W*_{w}^{0.75} and used data from Buddington and Diamond [105] to estimate the required coefficients σ_{ij}. The permeabilities of the four components, which were estimated by calibration, were approximately 320-fold smaller than the permeabilities that have been estimated for gill uptake and excretion. This large difference in gastrointestinal and gill permeabilities most likely is attributable to the relative effective thickness of the gastrointestinal tract and gill diffusion boundary layers. In particular, whereas the thickness of the gastrointestinal diffusion layer probably is on the order of millimeters (the thickness of the intestinal mucosa and a significant fraction of the food's bulk diameter) [67], the thickness of the gill's diffusion layer probably is only on the order of tens of microns (the thickness of the gill epithelium and a fraction of the gill's interlamellar distance).

#### Dietary exchange using fugacity-based diffusion

Gobas et al. [11] developed a dietary exchange model using a steady-state equivalent of Equations 32 and 35 expressed in terms of fugacities rather than conventional concentrations. In particular,

- (37)

where *f*_{f}, *f*_{g}, and *f*_{p} denote the fugacities (Pa) of the fish's whole body, gut contents, and food, respectively, and Z_{f}, Z_{g}, and Z_{p} denote the fugacity capacities (mol/m^{3}/Pa) of the fish's whole body, gut contents, and food, respectively. For this model, the fish's egestion (*E*_{w}) and feeding (*F*_{w}) must be expressed volumetrically rather than gravimetrically (i.e., m^{3}/d rather than g/d). Similarly, the intestinal transfer coefficient (*P*_{i}) must be expressed in units of m^{3}/d rather than as cm^{3}/d, as used in Equations 35 and 36. Consequently, the transport coefficients for fecal egestion, intestinal uptake, and food ingestion (*D*_{e}, *D*_{i}, and *D*_{p}, respectively) have units of mol/Pa/d. Solving Equation 37 for the fugacity of the fish's gut contents then yields

- (38)

When this expression is substituted into the fugacity equivalent of Equation 31, that is,

- (39)

Gobas et al. obtained the following expression for the fish's net dietary exchange:

- (40)

To compare Equation 40 with the models presented previously, the right-hand side of this equation can be rewritten in terms of conventional concentrations using the identity

- (41)

where *f*_{x} is the fugacity of *x* (Pa), *C*_{x} is the chemical's concentration (mol/m^{3}) in phase *x*, and *Z*_{x} is the fugacity capacity of *x* (mol/m^{3}/Pa). (Readers should note that in Gobas et al. [11], this relationship was misprinted as *C*_{x} = *f*_{x}*Z*_{x}^{−1}; likewise the units of *D*_{i} ≡ *D*_{G} were misprinted as having units of fugacity capacity.) Applying this conversion and factoring the result to resemble Equation 4, one observes that

- (42)

This expression can be further transformed by noting that the ratio of any two fugacity capacities defines a distribution coefficient for chemical partitioning between the phases of interest. In particular,

- (43)

Consequently, Equation 40 is also equivalent to

- (44)

- (45)

is a thermodynamic AE for a fish's dietary uptake. Comparing these latter two equations to Equations 22 and 23, it immediately follows that the thermodynamic AEs predicted by the Gobas et al. steady-state diffusion model differ from those predicted by the Barber et al. [12,17,18] fecal partitioning model only by the factor

- (46)

which corresponds to the fish's initial or maximum AE when *C*_{f} = 0. For other derivations of Equations 44 and 45, see Clark et al. [48], Borgmann and Whittle [51], and Kelly et al. [67].

Although Gobas et al. [11] could have formulated their intestinal transfer coefficients (*P*_{i} and *D*_{i} = *P*_{i}Z_{g}) as the product of the intestine's surface area and permeability [18,27], these transfer coefficients were represented as the resultant of an aqueous-phase and a lipid-phase transfer coefficient acting in series. In particular,

- (47)

- (48)

When the latter is substituted into the reciprocal form of Equation 46, that is,

- (49)

one can verify that

- (50)

where α_{0} is the initial gross chemical AE. To parameterize this expression, Gobas et al. [11] assumed that the distribution coefficients for the fish's gut contents and storage lipids could be estimated as the product of the component's organic carbon fraction and the partition coefficient of an appropriately selected organic reference material. Not surprisingly, octanol was chosen as this organic reference material. Using this assumption, Gobas et al. rewrote Equation 50 as

- (51)

where ρ_{og} denotes the organic fraction of the fish's gut contents. Using observed AEs for Atlantic salmon (*Salmo salar*), fathead minnow (*Pimephales promelas*), goldfish, guppy, rainbow trout, and redhorse sucker (*Moxostoma macrolepidotum*), Gobas et al. then derived the regression

- (52)

Using the same modeling framework, Clark et al. [48] re-estimated this relationship as follows:

- (53)

As originally published, the Gobas and Barber models also differed in the way that their developers parameterized the fish's gut content to whole-body partition coefficient (i.e., Eqn. 43). In particular, whereas Barber et al. [12,17,18] used Equations 19 and 20 to define this partition coefficient, Gobas et al. [11] simply assumed that

- (54)

where ρ_{1p} is the lipid fraction of food/prey contents. Although Gobas et al. originally assumed that the lipid fractions of a fish's food and feces could be treated as being equal, later model applications [20,21,26] modified this assumption. Morrison et al. [26], for example, estimated this partition coefficient using the expression

- (55)

where α_{1} = 0.83 is the fish's lipid AE. Finally, Arnot and Gobas [106] adopted a formulation that is similar to that of Barber et al. [12,17] but with the additional ability of treating a fish's gross food AE as the resultant of distinct AEs for lipid and nonlipid organic matter.

Versions of the Gobas et al. [11] model parameterized with Equation 52 have been used successfully to model the bioaccumulation dynamics of PCBs in Lake Ontario fish [20,21,107] and PCBs in winter flounder off the coast of New York and New Jersey (USA) [33]. Parameterized with other estimates of initial AEs (i.e., Eqn. 46), this model also has been used successfully to model bioaccumulation of PCBs in Lake Erie fish [26] and of PCB, dioxin, and dibenzofuran congeners in other assessments of Lake Ontario fish [49].

Even though the Gobas and Barber models can be conceptualized as using thermodynamic AEs (e.g., see Eqn. 28), it is more useful to treat these models in terms of their initial or gross uptake rates (*J*_{initial uptake}) and fecal excretion rates (*J*_{fecal excretion}). That is,

- (56)

Unlike Equations 22, 23, 44, and 45, when these models are expressed in this form, it is clear that feeding—or, more precisely, fecal excretion—continues to influence a fish's body burden even when the contamination of its prey becomes zero. Moreover, depending on the exposure scenario of concern, Equations 23, 25, and 45 can predict negative AEs; that is, dietary exposure pathways actually become net excretory pathways.

### MATERIALS AND METHODS

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

#### Statistical analysis of published AEs

Using data from the sources summarized in Table S1, a database was compiled to analyze the relationships between the chemical AEs and the variables that Equations 12, 15-17, 23, 25, 45, 46, and 51 predict to determine such efficiencies. All studies selected for analysis had experimental designs that assessed continuous daily exposures of freely feeding fish to natural or aquacultural diets. Studies that assessed the absorption efficiencies of force-fed fish to single or multiple artificial meals (contaminated gelatin capsules) were not included for analysis. Variables of interest included not only those that appear explicitly in these equations but also those that occur implicitly as determinants of a fish's current whole-body concentration. The variables considered for this analysis therefore were the chemical's *K*_{OW} and the fish's body weight, specific feeding rate, specific growth rate, and duration of exposure. Because of the expected complex interactions among these variables, however, this analysis simply focused on multiple linear regressions between the variables of interest and observed AEs that were transformed using the standard arcsinesquare root transformation for proportions. Linear regressions of all possible combinations of these variables were performed using a personal library of validated Fortran 95 subroutines (copies of this software [executable and source code] and database can be obtained from the author by request). To perform these regressions, feeding rates were back-calculated from reported growth rates as outlined in Barber [37] rather than using reported feeding or ration rates that were not expressed in common units across all studies. Back-calculated feeding rates used herein have units of g dry wt/g wet wt/d and are denoted as ϕ_{dw}. For the present study, all required *K*_{OW} values were estimated using the SPARC chemical properties estimation software (http://ibmlc2.chem.uga.edu/sparc/) [108]. Because chemical AEs often have been considered only as functions of *K*_{OW}, observed AEs also were fitted to the hyperbolic and quadratic regression models:

- (57)

- (58)

#### Analysis of bioaccumulation model predictions

To compare and contrast the aforementioned dietary uptake/exchange models, a simple Fortran 95 simulation program was developed to solve the following system of differential equations describing the bioaccumulation of an organic chemical during an arbitrary constant water exposure:

- (59)

- (60)

In these equations, *T* denotes the ambient water temperature (°C) that is assumed to be sinusoidal with a frequency of one year and BAF_{f} is the fish's BAF (ml/g wet wt). Whereas Equation 59 is simply a temperature-dependent version of the Parker-Larkin growth model [109], Equation 60 is a reformatted generalization of Equation 28. For thermodynamic dietary uptake models, α_{*} denotes the fish's initial chemical AE (α_{0}), and *k*_{ex} denotes the sum of the fish's RCs of gill and fecal excretion and chemical metabolism. For nonthermodynamic dietary uptake models, α_{*} corresponds to the fish's constant chemical AE (α_{c}), and *k*_{ex} corresponds to the sum of the fish's gill or lumped excretion RC and its rate of chemical metabolism. Equation 60 was then parameterized for a nonmetabolized contaminant using the formulations for α_{*} and *k*_{ex} summarized in Table 1.

Models 9, 11, and 15, which use the generalized gill exchange model proposed by Thomann [29], also were reparameterized to use ventilation volumes that were consistent with those used by other models of interest. These reparameterized models are designated as models 9R, 11R, and 15R, respectively. For the analyses herein, Equations 59 and 60 were solved and reinitialized in annual increments corresponding to the fish's expected spawning times to simulate biomass losses associated with spawning. Specific feeding rates ϕ_{ww}(*W*_{w}, *T*) were back-calculated from assumed specific growth rates γ(*W*_{w}, *T*) as outlined by Barber [37] using species-specific oxygen consumption rates, food AEs, specific dynamic action relationships, and moisture to lipid relationships. For this analysis, the fish species of concern were assumed to be piscivores that prey on nonspecific forage fish having a total body length 20% of the total body length of the fish species of concern [110–120]. Forage fish, in turn, are assumed to have bioaccumulated a steady-state, whole-body concentration of the chemical of interest. The BAFs of such forage fish were calculated using the steady-state solution of Equation 28; that is,

- (61)

where BCF_{i} = 0.25*K*_{OW} is the bioconcentration factor of the fish's assumed invertebrate prey, ϵ_{g} = *k*_{1}*K*_{p}^{−1} is the fish's gill excretion RC, and *K*_{p} = 0.05*K*_{OW} is the forage fish's bioconcentration factor assuming a 5% lipid content (see Mackay [9]). Forage-fish feeding rates were back-calculated from their estimated growth rates, as previously mentioned. Forage-fish gill uptake rates (*k*_{1}) were estimated using the allometric model

- (62)

(For additional discussion, see Barber [37] or Sijm et al. [61,121,122]; copies of this software [executable and source code] can be obtained from the author by request.)

Using Equations 59 to 61, realized BAFs, BMFs, AEs, gill uptake RCs, gill to diet uptake ratios, elimination half-lives, growth to excretion ratios, and gill to fecal excretion ratios were simulated for largemouth bass (*Micropterus salmoides*), walleye (*Sander vitreus = Stizostedion vitreum*), and rainbow trout that bioaccumulate a nonmetabolized, neutral, hydrophobic chemical with log *K*_{OW} = 6.5 (i.e., a chemical similar to a pentachloro-PCB). The physiological parameters and temperature regimes used for these simulations are summarized in Table S2. Because the principal biological controls on the rates of dietary and gill exchange (feeding, respiration, and growth) are dependent on body weight, the variability of the aforementioned bioaccumulation characteristics was investigated by allowing the specific growth exponents (*g*_{2}) to vary by ± 0.2 from the mean values reported in Table S2. Associated growth coefficients (*g*_{1}) were recalibrated, however, to enable each species to achieve its maximum body weight at its maximum age as reported in Table S2. This procedure addresses not only the variability of a species' bioaccumulation characteristics with respect to its growth rate but also the species' variability with respect to its back-calculated feeding rates. The resulting coefficients of variation for the specific growth rates of largemouth bass, walleye, and rainbow trout were 0.766, 0.662, and 0.485, respectively. Back-calculated specific feeding rates for largemouth bass, walleye, and rainbow trout had coefficients of variation equal to 0.313, 0.363, and 0.310, respectively. These values indicate that for growth and feeding, at least, the present study's analyses investigated a much larger range of parameter variability than typically is performed by sensitivity analyses that vary selected parameters by ±10% of their assumed mean values (see, e.g., [26,49,123]). It also is important to note that this approach to assessing model variability matches the same fish body weights with different water temperatures. This, in turn, varies the predictions of the coupled dietary and gill exchange models that are, in this analysis, temperature-dependent, allometric functions of the fish's body weight.

Model ID-Source | Initial or constant assimilation efficiency | Gill uptake rate k_{1} (ml·g[wet wt]^{−1}·d^{−1}) | Excretion rate k_{ex} (d^{−1}) |
---|---|---|---|

^{}^{a}AQUATOX estimates gill uptake efficiencies and the fish's chemical excretion rate coefficient using the piecewise functions^{}^{}^{}^{d}*K*_{gf}calibrated to be (1 — α_{f})*K*_{gf}= 0.0079_{ρ1f}^{−1}.^{}^{e}This equation is a modification of the Thomann and Connolly model [30] where chemical and oxygen uptake efficiencies are substituted for chemical and oxygen diffusivities, respectively, and where the chemical to oxygen uptake efficiency ratio equals 1.^{}^{f}This expression is a generalized parameterization proposed by Thomann [29] for the gill uptake model α_{g}*Q*_{v}*W*_{w}^{−1}. This parameterization assumes that the fish's routine specific oxygen consumption in units of g O_{2}/g wet wt/d is*R*= 8.65*W*_{w}^{−0.2}, that the fish's oxygen assimilation efficiency equals one, and that the dissolved oxygen concentration of the ambient water is 8.5 mg O_{2}/L. See Equations 16–21 in Thomann [29].^{}^{g}*K*_{gf}=*K*_{g}*K*_{f}^{−1}given by Equation 54.^{}^{h}Although this expression is the parameterization reported by Madenjian et al. [25], it is an incorrect implementation of the Thomann model that the authors identify as being used. See SETAC Supplemental Data Archive, (http://dx.doi.org/10.1897/07-462.S2).^{}^{i}*K*_{gf}=*K*_{g}*K*_{f}^{−1}given by Equation 55.^{}^{j}This model is a modification of Connolly [19] that accounts for kinetic limitations associated with slowly perfused tissues. In this model, τ = 0.15 denotes a correction factor for the slow transfer of chemical from lipids to blood, and ρ_{ab}= 0.05 denotes the fish's aqueous blood fraction.^{}^{k}*D*_{c}and*D*denote the aqueous diffusivities of the chemical of concern and oxygen, respectively, and*C*is the dissolved oxygen concentration.
| |||

1: AQUATOX [155] | α_{c} = 0.62 for YOY fish 0.92 for adult fish | α_{g}Q_{v}W^{−1} (See footnote a for α_{g}) | See footnote a |

2: Arnot and Gobas [106] | α_{0} = 1/(3.0 + 10^{−7}K_{OW}) | α_{g}Q_{v}W^{−1} Where α_{g} = (1.85 + 155 K^{−1}_{OW})^{−1} | k_{1}K^{−1}_{f} + α_{0}(1 - α_{f})ϕ_{dd}K_{g}K^{−1}_{f}^{b} |

3: Barber [17] | α_{0} = 1.0 | Q_{v}(1 - Θ_{B})W^{−1}_{w} | k_{1}K^{−1}_{f} + α_{0}(1 - α_{f})ϕ_{dd}K_{g}K^{−1}_{f}^{c} |

4: Barber et al. [12] | α_{0} = 1.0 | Q_{v}(1 - Θ_{B})W^{−1}_{w} | k_{1}K^{−1}_{f} + α_{0}(1 - α_{f})ϕ_{dd}K_{g}K^{−1}_{f}^{c} |

5: Borgmann and Whittle [51] | α_{0} = 0.75 | Assumed zero | α_{0}(1 - α_{f})ϕ_{ww}K_{gf}^{d} |

6: Borgmann and Whittle [51] | α_{c} = 0.75 | Assumed zero | 0.06W_{w}^{−0.58} |

7: Connolly [19] | α_{c} = 0.40 | R (CW_{w})^{−1 e} | k_{1}(_{ρ1f}K_{OW})^{−1} |

8: Connolly and Pedersen [10] | α_{c} = 0.60 | R (CW_{w})^{−1 e} | k_{1}(_{ρ1f}K_{OW})^{−1} |

9: Eby et al. [146] | α_{c} = 0.95 | Assumed zero | 1000α_{g}W_{w}^{−μ} where α_{g} = 0.5 ^{f} |

10: Gobas et al. [11, 20] | Equation 52 | [W_{w}P^{−1}_{w} + W_{w}(_{ρ1f}P_{1})^{−1}]^{−1} | k_{1}K^{−1}_{f} + α_{0}(1 - α_{f})ϕ_{dd}K_{g}K^{−1}_{f}^{g} |

11: Madenjian et al. [25] | α_{c} = 0.80 | 1000α_{g}W_{w}^{−μ} where α_{g} = 0.5 ^{f} | 1.0α_{g}W^{−μ}_{w}(_{ρ1f}K_{OW})^{−1 h} where α_{g} = 0.5 |

12: Morrison et al. [26] | α_{0} = 0.75 | α_{g}Q_{v}W^{−1}_{1} where α_{g} = 0.75 | k_{1}K^{−1}_{f} + α_{0}(1 - α_{f})ϕ_{dd}K_{g}K^{−1}_{f}^{i} |

13: Norstrom et al. [28] | α_{c} = 0.80 | α_{g}Q_{v}W^{−1}_{1} where α_{g} = 0.75 | 0.404W^{−0.58}_{w} |

14: QEAFDCN | α_{c} = 0.60 | R (CW_{w})^{−1 e} | k_{1}τ(ρ1f + _{ρ1f}K_{OW})^{−1 j} |

15: Thomann [29] | Equation 15 | 1000α_{g}W_{w}^{−μ} where α_{g} = α_{c}^{f} | k_{1}(_{ρ1f}K_{OW})^{−1} |

16: Thomann and Connolly [30] | α_{c} = 0.75 | D_{c}R (CDW_{w})^{−1 k} | k_{1}(_{ρ1f}K_{OW})^{−1} |

ID | Regression Model | Parameter | 95% CL | r^{2} |
---|---|---|---|---|

1 | β = a_{0} + a_{1} ln K_{ow} + a_{2}t + a_{3}W_{w} + a_{4}γ + a_{5}ϕ_{dw} | α_{0} = 0.553 | ±0.193 | 0.447 |

α_{1} = −0.0208 | ±0.00840 | |||

α_{2} = −0.00142 | ±0.000831 | |||

α_{3} = 0.00184 | ±0.000959 | |||

α_{4} = −17.2 | ±10.1 | |||

α_{5} = 94.6 | ±23.0 | |||

2 | β = a_{0} + a_{1} ln K_{ow} | α_{0} = 1.25 | ±0.175 | 0.118 |

α_{1} = −0.0282 | ±0.0101 | |||

3 | α_{c} = a_{0} + a_{1}logK_{ow} + a_{2}(log K_{ow})^{2} | α_{0} = 0.956 | ±0.516 | 0.122 |

α_{1} = −0.0688 | ±0.124* | |||

α_{2} = 0.000661 | ±0.00715* | |||

4 | 1/α_{c} = a_{0} + a_{1}K_{ow} | α_{0} = 4.22 | ±0.926 | <0.001 |

α_{1} = 1.11 × 10^{−14} | ±1.61 × 10^{−13}* | |||

5 | α_{c} = 1/(a_{0} + a_{1}K_{ow}) | α_{0} = 1.84 | NA | 0.080 |

α_{1} = 1.63 × 10^{−9} | NA |

### RESULTS AND DISCUSSION

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

#### Dietary AEs

Table 2 summarizes the results of the regression analyses conducted herein for studies summarized in Table S1. As already mentioned, many applications of bioaccumulation models have used net chemical AEs estimated using only the *K*_{OW} of the chemicals of concern (see, e.g., [29,31,64,124]). Besides *K*_{OW}, however, regression model 1 (see Table 2) clearly suggests that chemical AEs also should be treated as functions of the fish's body weight, total exposure time, specific growth rate, and specific feeding rate. Not only do these variables have regression coefficients that are significantly different from zero for *p* < 0.001, but also the *r*^{2} of the regression model 1 is 3.6- to 5.5-fold that of regression models 2, 3, and 5 and 5,400-fold that of regression model 4. The poorer performance of regression models 2 through 5 is not surprising given the scatter pattern of observed AEs displayed in Figure 3.

When regression model 1 is evaluated for its underlying database assuming log *K*_{OW} = 6.5 for all chemicals, the mean estimated AE is 0.524 (standard deviation [SD] = 0.170, *n* = 229). This value agrees well with the mean AEs predicted by the six models that use thermodynamically based dietary uptake algorithms (models 2-5, 10, and 12 in Table 1). In particular, the mean thermodynamic AEs predicted by these models for largemouth bass, walleye, and rainbow trout are 0.447 (SD = 0.163), 0.513 (SD = 0.146), and 0.483 (SD = 0.136), respectively (Tables S3-S5). The lowest and highest mean thermodynamic AEs are predicted by model 10 for largemouth bass (i.e., 0.240) and by model 12 for walleye (i.e., 0.703). These results raise questions concerning the appropriateness of using “high” constant AEs (models 1, 6, 9, 9R, 11, 11R, 13, and 16 in Table 1, for which 0.75 ≤ α_{c} ≤ 0.95) when simulating fish bioaccumulation. As mentioned previously, Connolly [19] concluded that to simulate accurately the PCB bioaccumulation in age-structured populations of winter flounder, calibrated AEs much lower than those suggested by shortterm uptake studies had to be used. To model accurately PCB concentrations in Lake Michigan (USA) rainbow trout, Madenjian et al. [125] had to employ a chemical AE of 0.5 rather than the value of 0.8 that had been assumed previously for lake trout [25]. The QSAR equations proposed by Thomann et al. [29,31] and by Muir et al. [64] to estimate chemical AEs also are at odds with the use of these high constant AEs.

An obvious shortcoming of regression model 1, however, is that it does not address the potential effects that chemical metabolism may have on observed AEs. This observation is important, because some of the chemicals included in this analysis are known to be metabolized by certain fish species (e.g., non-2,3,7,8-substituted dioxins and dibenzofurans). Inspection of Equations 26 through 30 suggests that chemical metabolism would be expected to affect chemical AEs in the same way that a fish's specific growth rate does. In particular, chemical metabolism, like growth, should speed up the fish's approach to steady state. Observed chemical AEs therefore would be expected to decrease, as reported by Muir and Yarechewski [126] and as predicted by the Gobas, BASS, and FGETS models.

Although using the Gobas and Barber models (i.e., Equations 23, 25, 30, 45, and 56) makes it relatively easy to understand how *K*_{OW} and a fish's body weight, growth and feeding rates, and total exposure times interact to determine its chemical AEs, it is less obvious how QSAR functions, such as Equations 15 and 16, arise. To demonstrate how such QSARs could arise, synthetic data sets were generated using the Gobas model (i.e., model 10 in Table 1) for constant dietary exposures to juvenile rainbow trout with initial body weight of 3 g wet wt. For each exposure, the trout's specific feeding rate, specific growth rate, and lipid fraction were assumed to be constant and equal to ϕ_{ww} = 0.015, γ = 0.012, and ρ_{1f} = 0.04, respectively. Fish were assumed to be fed a dry trout chow that possessed lipid and moisture fractions of 0.15 and 0.10, respectively, and that was contaminated with unspecified, nonmetabolized chemicals for which the log *K*_{OW} values ranged from four to nine. For each chemical, the trout's initial wholebody and food concentrations (μg/g wet wt) were assumed to be *C*_{f} = 0 and *C*_{p} = 1, respectively. Finally, the trout were assumed to be maintained in water held at 10°C with a dissolved oxygen content equal to 80% of saturation. Time series of the trout's expected whole-body concentrations then were generated by numerically by solving the following equation:

- (63)

assuming total exposure times ranging from 1 to 21 weeks. For each chemical and total exposure time, the trout's wholebody concentrations were then fitted to Equation 6 using the NL2SOL nonlinear optimization software [127]. The mean *r*^{2} of the resulting optimizations was 0.92 (SD = 0.15, *n* = 441). When the resulting α_{c} values are plotted as a function of *K*_{OW} and total exposure time (Fig. 4a), a distinct parabolic trend with respect to *K*_{OW} is observed even though the model that generated their underlying data predicts that thermodynamic AEs decrease monotonically with increasing *K*_{OW} (Fig. 4b). This parabolic QSAR probably results from describing a four-component process (i.e., gross dietary uptake, fecal excretion, gill excretion, and growth) as a two-component/parameter process. In particular, this simplification enables optimization algorithms, such as NL2SOL, to allocate the action of fecal excretion either to α_{c} as a net exchange parameter or to *k*_{2} as a pure elimination parameter. When Figure 4a is compared with Equation 52, it is clear that for chemicals with log *K*_{OW} < 6, α_{c} represents a net AE. On the other hand, for chemicals with log *K*_{OW} > 6, α_{c} clearly approximates the fish's initial gross AEs. When the preceding procedure is repeated using synthetic data sets generated by the Arnot and Gobas [106] model (i.e., model 2 in Table 1), a plateau-shaped QSAR relationship resembling that of Thomann et al. [31] (Eqn. 15) is obtained (Fig. S4).

#### Quantitative behavior of bioaccumulation models

*Estimated BAFs.* Predicted log BAFs range from 5.63 (i.e., model 2 for largemouth bass) to 6.34 (i.e., model 11 for largemouth bass). Although BAFs generally are positively correlated with fish lipid fractions, 6 of the 11 highest log BAFs (i.e., 6.13-6.34) were predicted for largemouth bass (see models 1, 9, 11, 13, 14, and 16) that had the lowest assumed lipid fraction (i.e., 0.0406 vs 0.0788 and 0.11). The most probable explanation for this result lies in the feed efficiencies (growth/consumption) predicted for these species. In particular, the mean lifetime dry-weight feed efficiencies (K = γ/ϕ_{dw}) predicted for largemouth bass, walleye, and rainbow trout are 0.408 (SD = 0.173), 0.666 (SD = 0.222), and 0.566 (SD = 0.104), respectively. The low feed efficiencies of largemouth bass imply that this species has the lowest capacity to biodilute the chemical assimilated from their diet. Because dietary uptake is predicted to be the dominant route of exposure for all models, this reduced capacity, in turn, results in largemouth bass having the highest predicted BAFs.

Readers who are familiar with applications of the Gobas-type bioaccumulation models may notice that BAFs predicted for models 2, 10, and 12 in Tables S3 to S5 appear to be inconsistent with those published by Gobas [20], Morrison et al. [26,49], and Arnot and Gobas [106]. For example, Morrison et al. [49] predicted log BAFs for Lake Ontario rainbow trout bioaccumulating penta-PCBs (i.e., PCBs 90/101, 105, 118, and 126) that ranged from 6.8 to 7.9. The average of these values was 7.4 (SD = 0.407), compared to the present study's value of 6.02 (SD = 0.0448). The mean log BAF predicted by Arnot and Gobas [106] for Lake Ontario salmonids bioaccumulating penta-PCBs was 6.97 (SD = 0.258), compared to the present study's value of 5.76 (SD = 0.0552). Similarly, the mean log BAFs predicted by Arnot and Gobas for Lake Erie largemouth bass and walleye bioaccumulating penta-PCBs were 6.75 (SD = 0.178) and 6.01 (SD = 0.149), respectively, compared to the present study's mean values of 5.63 (SD = 0.0627) and 5.67 (SD = 0.0832), respectively. The mean ratio of the BAFs from these four studies to those predicted herein is approximately 14.

Some of these discrepancies obviously can be attributed to differences in assumed *K*_{OW} values and ambient water temperatures. Some of the differences also can be attributed to differences in the way the predators' diets and prey concentrations are characterized. For example, Gobas [20], Morrison et al. [26,49], and Arnot and Gobas [106] have used steady-state, food-web models to describe predator-prey interactions, whereas the implementation of these Gobas-type models herein used a size-dependent, single prey-type model to characterize these interactions. Despite these obvious differences, the primary explanation for these model discrepancies may be traced to the studies' implicit feed efficiencies. Gobas [20], Morrison et al. [26,49], and Arnot and Gobas [106], like the authors of other studies [29,31,123,128], have used the following generalized growth equations:

- (64)

- (65)

to estimate specific growth rates of fish for steady-state bioaccumulation models. These equations originally were developed by Thomann [129] using data from Sheldon et al. [130]. In conjunction with these equations, coworkers of Gobas [20,26,49,106] estimated the feeding rate of fish using the generalized feeding model

- (66)

where *F*_{w} is, again, the fish's consumption (g wet wt/d). Using Equations 64 and 66 and assuming a nominal dry- to wet-weight conversion ratio of 0.25 for a fish's food, it follows that the feed efficiency (g wet wt growth/g dry wt consumption) for fish at 10°C would be given by

- (67)

This equation predicts that feed efficiencies decrease from K = 0.0760 for a young-of-year YOY fish of 0.25 g wet weight to K = 0.0463 for am adult of 5 kg wet weight. These feed efficiencies, however, are unexpectedly low not only in terms of general aquacultural standards but also the method used herein to back-calculate fish consumption using observed growth rates and standard metabolic relationships. Dry-weight feed efficiencies (K) typically reported by aquacultural studies generally range from 0.2 to 1.5, with a mean, unadjusted for species characteristics or experimental conditions, of approximately 0.83 (SD = 0.44; *n* = 24) (see Table S6). Unlike the predictions from Equation 67, the mean lifetime feed efficiencies of 0.408, 0.666, and 0.566 estimated herein for largemouth bass, walleye, and rainbow trout, respectively, agree reasonably well with these aquacultural observations.

Another, less obvious problem with Equations 64 and 65 that has been overlooked because of their applications with steady-state bioaccumulation models is the growth trajectories or age-at-weight relationships that they predict when used to parameterize their underlying allometric growth model; that is,

- (68)

If fish are recruited into a population of interest at *t*_{0} = 0 weighing 0.25 g wet wt, Equations 64 and 68 predict that 5-, 10-, and 15-year-old fish weigh 7, 54, and 223 g wet wt, respectively. Equation 64, however, has been used by steady-state, food-web models to describe the growth rate of 1.33 kg wet weight Lake Erie walleye [26], 2.41 kg wet weight Lake Ontario lake trout [20], and 5.43 kg wet weight Lake Ontario lake trout [49]. The predicted ages of these fish would have been 23.8, 27.3, and 33.2 years, respectively. These age-at-weight predictions agree poorly with the known life histories of these species. In fact, Equations 64 and 65 probably are poor descriptors of growth trajectories of most North American fish that have been studied. Using growth data compiled by Carlander [131–133], Barber [37] estimated the geometric mean specific growth rate of 68 species of North American freshwater fish species to be

- (69)

Growth exponents for the individual species, however, ranged from −1.61 to −1.14 × 10^{−5}, with associated growth coefficients varying from 1.66 × 10^{−3} to 4.03. A similar analysis of 317 primary growth studies for the BASS bioaccumulation and fish community model [17] database yielded the geometric mean

- (70)

for 89 freshwater fish species. Although Equations 69 and 70 yielded credible weights-at-age for many of the fish species summarized by Carlander [131–133] and were used herein to estimate forage-fish growth rates, these equations generally should be calibrated to the application at hand. For example, the calibration procedure adopted herein from the BASS bioaccumulation model [17,134] involves assigning each species of interest a maximum longevity (*a*_{max}), body weights for both YOY fish and fish at maximum age (*W*_{YOY} and *W*_{max}, respectively), and an empirically determined allometric growth exponent (γ_{2}). The species' allometric growth coefficient (γ_{1}) is then back-calculated, taking into account both assumed reproductive/spawning losses and any relevant temperature-growth relationships (see Table S2).

It should be noted that Gobas et al. [21] avoided the aforementioned issues with Equations 64 and 65 when investigating the time responses of Lake Ontario fish to PCB reductions by directly modeling chemical masses rather than concentrations and by using a predetermined age-class structure for the fish species of concern. This approach also was adopted by Gobas and Wilcockson [135]. More recently, deBruyn and Gobas [136] discuss the importance of having a closed, logically consistent, bioenergetics balance for analysis and prediction of steady-state BMFs.

*Estimated BMFs.* For this analysis, realized BMFs are defined as the ratio of a predator's whole-body concentration to that of its prey; that is,

- (71)

Predicted BMFs reported in Tables S3 to S5 ranged from 1.29 (i.e., model 2 for largemouth bass) to 7.05 (i.e., model 11 for largemouth bass). The mean interspecies BMF predicted by all models was 3.17 (SD = 1.25). Using data reported by Madenjian et al. [137], the mean observed BMF for Lake Michigan coho salmon bioaccumulating six different penta-PCB congeners was 1.96 (SD = 0.205, *n* = 12). Campfens and Mackay [107] used a steady-state, fugacity-based model to describe PCB bioaccumulation in Lake Ontario fish. Using data presented in Figure 5 of Campfens and Mackay [107], the BMF of Lake Ontario salmonids bioaccumulating penta-PCBs is estimated to be approximately 3.64. The BMF predicted by Gobas [20] for Lake Ontario salmonids bioaccumulating total PCBs was approximately 2.88. Hoekstra et al. [138] reported data that indicated the mean penta-PCB BMF for Arctic cod (*Boreogadus saida*) was approximately 2.78 (SD = 1.05). Dabrowska et al. [56] reported laboratory-derived BMFs for rainbow trout and yellow perch (*Perca flavescens*), feeding on both high- and low-lipid diets, that ranged from 2.4 to 8.1, with a mean of 4.9.

Although Morrison et al. [26,49] predicted BAFs for a variety of persistent organic chemicals in Lake Erie and Lake Ontario fish, BMFs cannot be calculated for all species investigated, because only a subset of the required BAFs were reported. Lake Ontario chinook salmon (*Oncorhynchus tshawytscha*), however, feeding predominately on YOY fish had predicted BMFs ranging from 30 to 50 for the five penta-PCB congeners (i.e., PCBs 90/101, 105, 118, and 126) studied. These elevated BMFs, however, may be attributable to the growth rate equation (i.e., Eqn. 64) used by Morrison et al. that yields unexpectedly low feed efficiencies, as discussed previously (see discussion concerning Eqns. 64–67).

#### Diet versus gill uptake

For this analysis, gill to diet uptake ratios were calculated as

- (72)

Predicted gill to diet uptake ratios ranged from zero (i.e., models 5, 6, 9, and 9R for all species) to 0.306 (i.e., model 3 for largemouth bass). Twenty-eight of 57 model-species combinations predicted ratios greater than 0.10 (i.e., models 2-4, 7, 8, 12-14, and 15R for all species and 11R for largemouth bass). Clearly, as BAF_{p} increases, predicted gill to diet uptake ratios decrease, and vice versa. For this analysis, the mean forage-fish BAF_{p} values for largemouth bass, walleye, and rainbow trout were 5.52 (SD = 0.0288), 5.52 (SD = 0.0308), and 5.46 (SD = 0.0114), respectively. The mean forage-fish BMFs associated with these values were 1.26, 1.26, and 1.10, respectively.

Several studies have estimated the relative contributions of gill and dietary uptake to a fish's total chemical bioaccumulation using either steady-state or dynamic simulation models [12,26,107,128,139-142]. These studies, however, have not yielded unambiguous answers to this question. A definitive answer has remained elusive not only because these studies have used different models to describe gill and dietary uptake but also because the models of interest have been parameterized very differently from one another. Table S7 summarizes gill to diet uptake ratios that have been estimated from actual laboratory experiments or by other model-based studies.

Because the predicted feeding rates and prey BAFs are identical across all models analyzed herein for any of the three species, differences in the predicted gill to diet uptake ratios summarized in Tables S3 to S5 simply reflect the ratios of the models' gill uptake rates to their assumed or predicted AEs. Model 3 predicted the highest mean gill uptake rates for all species (i.e., 1,310/d [SD = 373] for largemouth bass, 432/d [SD = 161] for walleye, and 545/d [SD = 198] for rainbow trout). Model 15 predicted the lowest mean gill uptake RC for largemouth bass (i.e., 102/d [SD = 19.5]), model 10 the lowest mean gill uptake RC for walleye (i.e., 66.0/d [SD = 28.4]), and model 16 the lowest mean gill uptake RC for rainbow trout (i.e., 85.8/d [SD = 31.1]). The interplay between gill uptake rates and dietary AEs are not necessarily obvious from a model's predicted gill to diet uptake ratios. For example, consider models 2 and 3 for largemouth bass. Although both models describe gill and dietary exchange as diffusion/thermodynamic processes and predict essentially the same mean gill to diet uptake ratio (i.e., 0.303 vs 0.306), this agreement is attained because model 3 predicts mean gill uptake rates and thermodynamic AEs that are approximately 1.8-fold those predicted by model 2 (i.e., 1,310/d vs 710/d and 0.582 vs 0.318, respectively).

Although only 4 of the 19 models (i.e., models 5, 6, 9, and 9R) analyzed herein assumed that gill uptake is zero, other models [22,45,47,143–145] also have made this assumption. Even though food uptake undoubtedly dominates gill uptake of most fish species for chemicals possessing log *K*_{OW} > 6, given the availability of numerous gill uptake models that can be readily parameterized (see, e.g., [37]), we know of no obvious reason to assume that gill uptake is zero or that it should be ignored. An analogy to support this claim can be found in fish bioenergetic growth models. In particular, although the fish's specific dynamic action is only 10 to 20% of the fish's ingestion, few researchers would find this fraction to be small enough to ignore it as an important process in understanding fish growth trajectories. Furthermore, as discussed elsewhere [37], many fish species routinely fast during certain periods of their life, such as when water temperatures reach some threshold or during spawning. Although dietary uptake and excretion are zero at these times, gill uptake and excretion obviously continue in parallel with the fish's oxygen consumption. The temperature-dependency models used by the Wisconsin Bioenergetics Model [22] to estimate fish feeding and respiration can be used to illustrate this situation. For example, when this model's suggested default parameters are used to simulate the feeding and oxygen consumption of blue-gill sunfish (*Lepomis macrochirus*) and yellow perch, one observes that feeding ceases not only well below the fish's maximum temperature tolerances but also well below their temperatures of maximum oxygen consumption.

#### Elimination half-lives

Predicted elimination half-lives range from 71.4 d (i.e., model 12 for largemouth bass) to 507 d (i.e., model 11 for walleye). Elimination half-lives of penta-PCBs and pentachlorodiphenyl ethers in rainbow trout of 300 to 900 g wet weight under laboratory conditions have been observed to range from 81 to 820 d [4,6]. Thirteen models predicted half-lives of less than six months (i.e., models 2-5, 7, 8, 9R, 10, 11R, 12, and 15R for largemouth bass and models 1 and 3 for rainbow trout). Eight models predicted half-lives of greater than one year (i.e., model 11 for largemouth bass and rainbow trout and models 9, 11, and 13-16 for walleye).

Using the exchange algorithms in model 9, Eby et al. [146] concluded that bloater (*Coregonus hoyi*) could excrete only 2 to 5% of their PCB concentration during any given year. In the present study, however, the mean elimination half-lives predicted by model 9 for walleye and rainbow trout were 384 d (SD = 109) and 305 d (SD = 113), respectively. These elimination half-lives imply that walleye and rainbow trout would be expected to excrete or biodilute approximately 50% of their PCB whole-body concentrations within a year.

#### Relative importance of growth and excretion

Predicted growth to excretion rate ratios ranged from 0.283 (i.e., model 12 for largemouth bass) to 5,950 (i.e., model 11 for rainbow trout). Fourteen models predicted ratios less than 1.0 (i.e., models 2-5, 7, 8, 9R, 10, 11R, 12, and 15R for largemouth bass; models 3 and 4 for walleye; and model 3 for rainbow trout). These 14 models, therefore, predict that actual excretion is more important than growth dilution in regulating a fish's whole-body chemical concentration. Fourteen models predicted ratios greater than 3.0 (i.e., model 11 for largemouth bass; models 9, 11, 13-16, and 15R for walleye; and models 9, 11, 14-16, and 15R for rainbow trout). Unlike the preceding group of models, these models suggest that whole-body chemical concentrations of contaminated fisheries could be remediated by managing for maximal fish growth rates [125,144,147]. Although model 11 predicts virtually no chemical excretion for any of the three species studied, these predictions are the results of Madenjian et al.'s [25] incorrect implementation of the Thomann gill exchange model [29] (SE-TAC Supplemental Data Archive (http://dx.doi.org/10.1897/07-462.S2)). Last, it should be noted that some bioaccumulation models have ignored chemical excretion entirely [45,143,144,147].

#### Relative importance of gill and fecal excretion

Unlike most of the models studied, models 2, 3, 4, 10, and 12 explicitly describe both gill and fecal chemical excretion. For these models, gill to fecal excretion rate ratios can be defined by

- (73)

which follows from Equation 28, allowing for initial AEs different from unity and where ϵ_{i} is the intestinal/fecal excretion rate coefficient (1/d). For models 3, 4, and 10, mean estimated gill to fecal excretion ratios range from a low of 0.282 (SD = 0.0358) for rainbow trout by model 4 to a high of 1.03 (SD = 0.144) for largemouth bass by model 3. For models 2 and 12, however, mean estimated gill to fecal excretion ratios range from a low of 7.35 (SD = 0.877) for rainbow trout by model 12 to a high of 12.2 (SD = 1.70) for largemouth bass by model 2. Thus, the relative predicted importance of gill and fecal excretion obviously is sensitive to how the fish's gill uptake RC (*k*_{1}), initial AE (α_{0}), and gut/fecal partitioning coefficient (*K*_{g}) are modeled or assigned. For example, because models 3 and 12 predict virtually the same mean gill uptake rate to initial AE ratio (i.e., *k*_{1}/α_{0} ≈ 430) for walleye, the large discrepancy in their predicted gill to fecal excretion rate ratios (i.e., 0.735 vs 7.94) is entirely attributed to the way that their gut/fecal partitioning coefficients are assigned. Similarly, although models 3 and 4 use the same parameterization scheme for α_{0} and *K*_{g}, they estimate gill uptake rates differently. In this case, assuming that gill perfusion is an important process in controlling gill uptake [8,37,148–154] results in these models predicting gill to fecal excretion rate ratios that differ by factors of 1.9 to 2.5.

The large gill to fecal excretion rate ratios predicted by model 2 are partially attributable to the way that Arnot and Gobas [106] parameterized their fecal partition coefficient:

- (74)

where ρ_{lg} is the lipid fraction of gut contents and β is a proportionality constant between *K*_{OW} and an appropriately chosen organic carbon partition coefficient (see the numerator of Eqn. 18 in Arnot and Gobas [106]). Although Equation 74 is conceptually sound, the parameterization scheme used by Arnot and Gobas to quantify this equation has one conceptual and mathematical shortcoming. In particular, Arnot and Gobas assumed that the fish's fecal composition could be calculated using the following system of equations:

- (75)

- (76)

- (77)

In these equations, ρ_{ig} denotes the fraction of component j in the fish's gut/feces (where j equals a, l, or o denotes the component as aqueous, lipid, or nonlipid organic matter, respectively), ρ_{jp} denotes the fraction of component j in the fish's prey/food, and α_{j} denotes the fish's AE of component j. If *M*_{jg} and *M*_{jp} denote the masses of component j in the fish's gut/feces and prey/food, respectively, then the fish's AE for this material (α_{j}) is obviously given by

- (78)

If one now arbitrarily selects Equation 77, it then follows that

- (79)

- (80)

- (81)

- (82)

One can easily verify, however, that Equations 75 and 76 also yield Equation 82. This situation therefore is equivalent to having only two equations and three unknowns. Consequently, Equations 75 through 77 only guarantee that the estimated fecal fractions conform to a necessary mass-balance constraint and sum to unity. These equations, however, guarantee nothing about the uniqueness of the estimated fractions themselves.

To understand the consequences of this result, consider the gut moisture content predicted by Equations 75 through 77 for rainbow trout using the food AEs proposed by Arnot and Gobas (i.e., α_{1} = 0.92, α_{0} = 0.60, and α_{a} = 0.25). Whereas the proximate composition of rainbow trout assumed herein is 11% lipid, 68.3% moisture, and 20.7% nonlipid organic matter, the composition of their prey is assumed to be 5% lipid, 75.7% moisture, and 19.3% nonlipid organic matter. The fecal composition predicted by Equations 75 through 77 for rainbow trout therefore would be 0.6% lipid, 87.5% moisture, and 11.9% nonlipid. This predicted moisture content, however, is unexpectedly high based on a number of published studies [76–80] and would imply a reduced capacity for fecal excretion. An additional constraint that could be imposed on Equations 75 through 77 is the assumption that fish feces are hydrated or dehydrated to a constant proportion ω of a fish's percentage moisture, as assumed by the BASS/FGETS fecal excretion models. Under this assumption, one can verify that the fish's water AE should be assigned using the following equations:

- (83)

- (84)

- (85)

Assuming the aforementioned proximate fish and prey compositions, lipid and nonlipid AEs, and complete osmotic equilibration (i.e., ω = 1), Equations 75 through 77 and 85 predict the fecal composition of rainbow trout to be 1.6% lipid, 68.2% moisture, and 30.2% nonlipid. This re-estimated proximate composition increases the trout's *K*_{g} from 3.26 × 10^{4} to 8.30 × 10^{4} and decreases the trout's predicted mean gill-to-fecal excretion ratio from 9.65 to 3.79.

The elevated gill to fecal excretion ratios predicted by model 12 compared with those predicted by models 3, 4, and 10 also are attributable primarily to the way that a fish's fecal partition coefficient (*K*_{g}) is estimated. In particular, Morrison et al. [26,49] assumed that

- (86)

where α_{1} denotes the fish's lipid AE (see Eqn. 10 in Morrison et al. [26]). Assuming Morrison et al.'s reported lipid AE of α_{1} = 0.83 and a prey lipid content of 5%, this equation predicts that *K*_{g} = 2.69 × 10 ^{4} for a chemical with log *K*_{OW} = 6.5. This value is even lower than the fecal partition coefficient predicted by the Arnot and Gobas model discussed immediately above. Because the latter *K*_{g} appears to be underestimated, it follows that the *K*_{g} of Morrison et al. also might be underestimated. This underestimation could be attributed to what some researchers would consider a conceptual problem with Equation 86. In particular, using Equation 78, this equation can also be rewritten as

- (87)

When expressed in this form, one can see that the gut's lipid mass is being normalized to the total mass of the fish's ingested prey. An alternative formulation, which is essentially a simplification of the Arnot and Gobas model [106], would have been

- (88)

### CONCLUSION

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

All the bioaccumulation models considered herein assume that net chemical AEs are either constants (e.g., models 1, 6–9, 9R, 11, 11R, 13, 14, and 16), QSAR functions (e.g., models 15 and 15R), or thermodynamic functions of a fish's consumption and egestion, chemical concentrations within it and its prey, and physicochemical properties of the contaminant of concern (e.g., models 2–5, 10, and 12). The present study's statistical analyses of published laboratory chemical AEs suggest that this basic bioaccumulation parameter should not be treated simply as a constant or QSAR function of *K*_{OW}. Instead, chemical AEs should be considered as functions of *K*_{OW} and the fish's body weight, growth and feeding rates, and duration of exposure. Based on the simulation analyses herein, thermodynamically based dietary exchange models (i.e., models 2–5, 10, and 12) can reasonably emulate the statistical patterns of AEs observed under diverse laboratory conditions. Although these models can be conceptualized as using thermodynamic AEs, treating these models in terms of their initial gross uptake efficiencies and fecal excretion rates is more useful. In this latter form, it is clear that feeding—or, more precisely, fecal excretion—continues to influence a fish's body burden even when dietary exposures become zero. Moreover, depending on the exposure scenario of concern, thermodynamic AEs (e.g., Eqns. 23, 25, and 45) actually can be negative when dietary exposure pathways become net excretory pathways.

The quantitative behavior of the combined dietary and gill exchange algorithms used by 16 widely cited bioaccumulation models/frameworks were analyzed for largemouth bass, walleye, and rainbow trout that bioaccumulate an unspecified, poorly metabolized, hydrophobic chemical possessing a log *K*_{OW} of 6.5 (i.e., a chemical similar to a pentachloro-PCB). Predictions of the present study's implementations of selected bioaccumulation models sometimes did not agree with results that the developers of those models have published previously. These differences were particularly apparent for certain steady-state bioaccumulation models. Detailed analysis revealed that these differences generally could be attributed to differences in model parameterization. Differences in fish growth rates, feeding rates, and feed efficiencies were critical in explaining apparent model discrepancies. For thermodynamically based models, differences in distribution coefficients describing the chemical partitioning between the aqueous and nonlipid organic matter of fish feces also were critical in understanding apparent model discrepancies.

Because no single bioaccumulation model can be expected to be the best model for all conditions and applications, model selection obviously should depend on the application being considered. For example, dynamic simulation models generally are not needed to predict bioaccumulation of chemicals having log *K*_{OW} < 5, because QSAR-based BAFs or BSAFs or steady-state solutions of differential equation-based bioaccumulation models are more than adequate. On the other hand, if the goal of analysis is to estimate expected recovery times of a fishery contaminated with extremely hydrophobic chemicals (e.g., PCBs, dioxins, etc.) as a function of alternative remediation scenarios, one obviously would want to select models that accurately predict bioaccumulation RCs that determine the fishery's realized and theoretical BAFs. Model selection also should be guided by how well prevailing scientific understanding is represented by the models' algorithms. Although it is widely accepted that chemical bioaccumulation in fish results from thermodynamic chemical exchange processes from food and water, many bioaccumulation modeling studies have employed formulations that ignore this commonly held understanding. In more narrowly focused studies, Burkhard [123] as well as Borgmann and Whittle [51] compared the relative abilities of thermodynamically based and nonthermodynamically based bioaccumulation models to predict observed fish chemical concentrations and BAFs under field conditions. Although both types of models could be parameterized to perform well, both studies concluded that fully thermodynamically based models performed better than their nonthermodynamically based counterparts. A third issue that should be considered when selecting or applying a bioaccumulation model is the model's internal consistency regarding the parameterization of interdependent processes. Four examples of this issue were identified and assessed in the present study. These include independent parameterization of fish growth, feeding, and metabolic rates that yield unrealistic feed efficiencies and size-at-age relationships; application of the generalized Thomann and Connolly [29] gill exchange model in conjunction with independently estimated oxygen consumption rates; using the Thomann and Connolly [29] gill exchange model to estimate chemical excretion while simultaneously assuming no gill uptake; and using lumped parameter allometric models that are independent of a fish's ingestion/egestion and oxygen consumption rates to estimate chemical excretion. Users also should be aware that using bioaccumulation models calibrated from observed data can be particularly vulnerable to parameter inconsistencies. Model calibration is seldom, if ever, a unique process, and multiple parameter combinations can produce comparable fits to an observed data set. For example, unexpectedly high calibrated AEs may be required to offset ignoring gill uptake or assuming higher-than-expected specific growth rates or lower-than-expected exposure concentrations.

### SUPPORTING INFORMATION

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

**Table S1.** Summary of studies whose assimilation efficiencies (AEs) were analyzed. Abbreviations for chemicals are: BB (brominated benzenes), CB (chlorinated benzenes), PAH (polyaromatic hydrocarbons), PAHH (polyaromatic heterocyclic hydrocarbons), PBB (polybrominated biphenyls), PCA (polychlorinated alkanes), PCB (polychlorinated biphenyls), PCDD (polychlorinated dibenzo-p-dioxins), PBDE (polybrominated diphenyl ethers), PCDF (polychlorinated dibenzofurans), and PCN (polychlorinated naphthalenes).

**Table S2.** Summary of physiological, morphological, and environmental parameters used to simulate BAFs for largemouth bass, walleye, and rainbow trout. Temperature functions assume that April 1 corresponds to *t* = 0 and that the coldest water temperatures occur on January 15.

**Table S3.** Mean and standard deviations for simulated BAFs, BMFs, assimilation efficiencies, gill uptake rates, gill-to-diet uptake ratios, apparent “elimination” half-lives (ln 2/*k*_{2}), growth-to-excretion rate ratios, and gill-to-fecal excretion rate ratios, for largemouth bass bioaccumulating a log *K*_{OW} = 6.5 chemical. See Table 2 for model identification.

**Table S4.** Mean and standard deviations for simulated BAFs, BMFs, assimilation efficiencies, gill uptake rates, gill-to-diet uptake ratios, apparent “elimination” half-lives (ln 2/*k*_{2}), growth-to-excretion rate ratios, and gill-to-fecal excretion rate ratios, for walleye bioaccumulating a log *K*_{OW} = 6.5 chemical. See Table 2 for model identification.

**Table S5.** Mean and standard deviations for simulated BAFs, BMFs, assimilation efficiencies, gill uptake rates, gill-to-diet uptake ratios, apparent “elimination” half-lives (ln 2/*k*_{2}), growth-to-excretion rate ratios, and gill-to-fecal excretion rate ratios, for rainbow trout bioaccumulating a log *K*_{OW} = 6.5 chemical. See Table 2 for model identification.

**Table S6.** Mean feed efficiencies (K g wet wt growth/g dry wt feed) for selected fish species under aquacultural conditions. The estimated interspecies mean, uncorrected for experimental conditions, is 0.832 (SD = 0.442; *n* = 25).

**Table S7.** Summary of the relative contributions of dietary and gill uptake. Abbreviations for chemicals are: DDT (dichloro-diphenyl-trichloroethane), PCB (polychlorinated biphenyls), PCDD (polychlorinated dibenzo-*p* -dioxins) and PCDF (polychlorinated dibenzofurans).

**Figure S1.** Assimilation efficiencies predicted by Equations 25 and 28 for guppies as a function of time and feeding rate during a dietary exposure as described by Clark and Mackay [5]. The fish's initial body weight is assigned to be 0.1 g wet wt and no growth is assumed to occur. The fish's food assimilation efficiency is assumed to be 0.80. The fish's moisture, lipid, and nonlipid organic fractions are assumed to be 0.76, 0.05, and 0.19, respectively. Food for the exposure is assumed to be TetraMin® fish food (approximately 6% moisture) that contains 0.2 μmol/g wet wt of chemical similar to a tetrachlorobiphenyl (i.e., molar weight equal to 300 g/mol and log *K*_{OW} = 6).

**Figure S2.** Assimilation efficiencies predicted by Equations 25 and 28 for guppies as a function of time and food concentrations during a dietary exposure as described by Opperhuizen and Schrap [24]. The fish's initial body weight is assigned to be 0.1 g wet wt and no growth is assumed to occur. The fish's specific feeding rate and food assimilation efficiency are assumed to be 0.02 g wet wt/g wet wt/d and 0.80, respectively. The fish's moisture, lipid, and nonlipid organic fractions are assumed to be 0.76, 0.05, and 0.19, respectively. Food for the exposure is assumed to be TetraMin® fish food (approximately 6% moisture) that contains a chemical similar to a tetrachlorobiphenyl (i.e., molar weight equal to 300 g/mol and log *K*_{OW} = 6).

**Figure S3.** Assimilation efficiencies predicted by Equations 25 and 28 for rainbow trout as a function of time and *K*_{OW} during a dietary exposure as described by Fisk et al. [14]. The fish's initial body weight and specific growth rate are assumed to be 3 g wet wt and 0.012 g wet wt/g wet wt/d, respectively. The fish's specific feeding rate and food assimilation efficiency are assumed to be 0.0175 g wet wt/g wet wt/d and 0.80, respectively. The fish's moisture, lipid, and nonlipid organic fractions are assumed to be 0.76, 0.05, and 0.19, respectively. Food for the exposure is assumed to be Martin's Feed Mills fish food (approximately 10% moisture) that contains PCBs.

**Figure S4.** Plot of assimilation efficiencies (AEs) predicted for juvenile rainbow trout bioaccumulating persistent organic chemicals under laboratory conditions similar to that of Fisk et al. [13]. For this figure, whole-body concentrations predicted by the Arnot and Gobas model (see Table 2 Model 2) for constant, dietary exposures of 1 μg/g wet wt lasting for total exposure times ranging from 1 week to 21 weeks were fitted to Equation 6 using the NL2SOL nonlinear optimization software [29]. The resulting average *r*^{2} equaled 0.96 (SD = 0.097; n = 441). Whereas Figure S4 a displays the fitted AEs, Figure S4 b displays the model's underlying thermodynamic AEs predicted by Equation 45. See text for details.

Tables, figures, and references found at DOI: 10.1897/07-462.S1 (4MB PDF).

### Acknowledgements

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

Sincere appreciation is extended to David Glaser, Frank Gobas, John Nichols, Dick Sijm, and Robert Swank for reviewing this work. Special thanks are given to John Nichols and Robert Swank for their insightful comments and analyses. Michael Cyterski provided the statistical support for this work, and Luis Suárez provided invaluable mathematical quality-assurance analysis and comments. This work also was greatly improved by comments and suggestions made by its four anonymous peer reviewers.

### REFERENCES

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

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### APPENDIX

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

*B*_{f} = chemical burden of whole fish (mol)

*B*_{g} = chemical burden of gut contents/feces (mol)

*B*_{i} = chemical burden of intestinal tissue (mol)

BAF_{f} = *C*_{f}/*C*_{w} = bioaccumulation factor of fish (ml/g wet wt)

BAF_{p} = *C*_{p}/*C*_{w} = bioaccumulation factor of fish's prey (ml/g wet wt)

BMF = *C*_{f}/*C*_{p} = biomagnification factor of fish (g wet wt/g wet wt)

*C*_{ab} = chemical concentration in arterial blood (mol/g wet wt)

*C*_{af} = chemical concentration in the aqueous phase of the fish (mol/ml)

*C*_{ag} = chemical concentration in the aqueous phase of the gut contents/feces (mol/ml)

*C*_{f} = chemical concentration in whole fish (mol/g wet wt)

*C*_{g} = chemical concentration in gut contents/feces (mol/g wet wt)

*C*_{i} = chemical concentration in intestinal tissue (mol/g wet wt)

*C*_{og} = chemical concentration in the organic phase of the gut contents/feces (mol/g wet wt)

*C*_{p} = chemical concentration in food/prey (mol/g wet wt)

*C*_{vb} = chemical concentration in venous blood (mol/g wet wt)

*C*_{w} = dissolved chemical concentration in water (mol/ml)

*D*_{e} = transport coefficient for fecal egestion (mol/Pa/d)

*D*_{p} = transport coefficient for food ingestion (mol/Pa/d)

*D*_{i} = transport coefficient for intestinal uptake (mol/Pa/d)

*E*_{d} = egestion/defecation rate (g dry wt/d or m^{3}/d)

*E*_{w} = egestion/defecation rate (g wet wt/d or m^{3}/d)

E^{V}_{d} = gastric evacuation rate (g dry wt/d)

EX = nitrogenous excretion rate (g dry wt/d)

*G*_{d} = weight of gut contents (g dry wt)

*G*_{w} = weight of gut contents (g wet wt)

*f*_{f} = fugacity of whole fish (Pa)

*f*_{g} = fugacity of gut contents/feces (Pa)

*f*_{p} = fugacity of food/prey (Pa)

*F*_{d} = feeding rate (g dry wt/d or m^{3}/d)

*F*_{w} = feeding rate (g wet wt/d or m^{3}/d)

*k*_{1} = gill uptake rate coefficient (1/d)

*k*_{2} = lumped-process elimination rate coefficient (1/d)

*k*_{ex} = joint excretion and metabolism rate coefficient (1/d)

*K*_{g} = distribution coefficient from the aqueous phase of the gut contents to the whole, wet-weight gut contents (ml/g wet wt)

*K*_{f} = distribution coefficient from the aqueous phase of the fish to its whole body (ml/g wet wt); also equivalent to the BCF of the fish

*K*_{xy} = partition coefficient for *C*_{x}/*C*_{y} at equilibrium

*J*_{g} = net chemical exchange across the gill (mol/d)

*J*_{i} = net chemical exchange across the intestine (mol/d)

*J*_{m} = chemical lost because of biotrans formation (mol/d)

*M*_{ag} = the aqueous mass of feces/gut contents (g)

*M*_{lg} = the lipid mass of feces/gut contents (g)

*M*_{og} = the nonlipid organic mass of feces/gut contents(g)

*M*_{ap} = the aqueous mass of food/prey (g)

*M*_{lp} = the lipid mass of food/prey (g)

*M*_{op} = the nonlipid organic mass of food/prey (g)

*P*_{i} = intestinal permeability (cm/d)

*P*_{i} = lumped transfer coefficient for intestinal uptake (cm^{3}/d or m^{3}/d)

*P*_{l} = lipid transfer coefficient for intestinal uptake acting in series (m^{3}/d)

*P*_{w} = aqueous transfer coefficient for intestinal uptake acting in series (m^{3}/d)

*Q*_{i} = perfusion volume of intestinal tissue (ml/d)

*Q*_{v} = ventilation volume (ml/d)

*R* = respiration rate (g dry wt/d)

*S*_{i} = total absorptive gut surface area (cm^{2})

SDA = specific dynamic action rate (g dry wt/d)

*t* = time (d)

*T* = water temperature (°C)

*W*_{d} = weight of fish (g dry wt)

*W*_{w} = weight of fish (g wet wt)

*Z*_{f} = fugacity capacity of whole fish (mol/m^{3}/Pa)

*Z*_{g} = fugacity capacity of gut contents/feces (mol/m^{3}/Pa)

*Z*_{p} = fugacity capacity of food/prey (mol/m^{3}/Pa)

α_{0} = initial gross chemical assimilation efficiency (dimensionless)

α_{a} = water assimilation efficiency (dimensionless)

α_{c} = net chemical assimilation efficiency (dimensionless)

α_{f} = food assimilation efficiency (dimensionless)

α_{l} = lipid assimilation efficiency (dimensionless)

α_{o} = nonlipid organic matter assimilation efficiency (dimensionless)

α_{t} = thermodynamic chemical assimilation efficiency (dimensionless)

β_{g} = gill uptake rate (ml/d)

γ = specific growth rate (g wet wt/g wet wt/d)

δ_{g} = gill/branchial excretion rate coefficient (g wet wt/d)

δ_{m} = chemical biotransformation rate coefficient (g[wet wt]/d)

ϵ_{g} = gill/branchial excretion rate coefficient (1/d)

ϵ_{i} = intestinal/fecal excretion rate coefficient (1/d)

ϵ_{m} = chemical biotransformation rate coefficient (1/d)

κ = feed efficiency (g wet wt growth/g dry wt feed)

ρ_{af} = aqueous fraction of whole fish (unitless)

ρ_{ag} = aqueous fraction of gut contents (unitless)

ρ_{ap} = aqueous fraction of food/prey (unitless)

ρ_{df} = dry matter fraction of whole fish (unitless)

ρ_{dg} = dry matter fraction of gut contents (unitless)

ρ_{dp} = dry matter fraction of food/prey (unitless)

ρ_{lf} = lipid fraction of whole fish (unitless)

ρ_{lg} = lipid fraction of gut contents (unitless)

ρ_{lp} = lipid fraction of food/prey contents (unitless)

ρ_{of} = nonlipid organic matter fraction of whole fish (unitless)

ρ_{og} = nonlipid organic matter fraction of gut contents (unitless)

ρ_{op} = nonlipid organic matter fraction of food/prey (unitless)

ϕ_{dd} = specific feeding rate (g dry wt/g dry wt/d)

ϕ_{dw} = specific feeding rate (g dry wt/g wet wt/d)

ϕ_{ww} = specific feeding rate (g wet wt/g wet wt/d)

### Supporting Information

- Top of page
- Abstract
- INTRODUCTION
- REVIEW OF DIETARY EXCHANGE MODELS
- MATERIALS AND METHODS
- RESULTS AND DISCUSSION
- CONCLUSION
- SUPPORTING INFORMATION
- Acknowledgements
- REFERENCES
- APPENDIX
- Supporting Information

Filename | Format | Size | Description |
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10.1897_07-462.S1.pdf | 3764K | Supplementary Materials | |

10.1897_07-462.S2.pdf | 98K | Supplementary Materials |

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