Clements: “3357_c008” — 2007/11/9 — 18:10 — page 115 — #1 8 Models of Bioaccumulation and Bioavailability 8.1 OVERVIEW The object of this work will consist in the derivation of general mathematical relations from which it is possible, at least for practical purposes, to describe the kinetics of distribution of substances in the body. (Teorell 1937a) There are well-established ways to quantify bioaccumulation. Some mathematical models give parsi- monious description or restricted prediction for a particular exposure scenario. As reflected in the above quote from Teorell’s classic paper in which these methods were first introduced, such models are focused on practicality. More complicated models describe or predict bioaccumulation in a more general way based on physiological, biochemical, and anatomical features. Most simple models employ mathematical compartments while complicated models describe exchange among several interconnected physical or biochemical compartments. Many combine features of both modeling extremes.Although initially appearing as ajumble of competing approaches, this blend of approaches makes sense in ecotoxicology: some scenarios require a simple, pragmatic model but more involved models might be needed to capture the essential features of the situation under study. This is con- sistent with the general tenet that a model should be no more complicated than needed to answer the question being asked. For cases in which accurate description or prediction is paramount for many diverse species, contaminants, or conditions, a more complicated model incorporating physiolo- gical, biochemical, and anatomical features likely will be the best alternative. Otherwise, many simple models each of which only describes one relevant species/toxicant/condition combination would have to be employed. Similarly, estimations of contaminant bioavailability from relevant sources can be produced for a particular situation using simple empirical models or more generally by applying predictive models based on in-depth mechanistic knowledge. Like bioaccumulation models, which bioavailability approach is the most appropriate depends on the study goals and the desired generality of the results. 8.2 BIOACCUMULATION Most bioaccumulation models translate an external concentration to an internal concentration that is then related to an effect. The form of the model depends on the media containing the contaminant, qualities of the environment in which the exposuretakes place, and the qualities ofthe organism itself. Models for nonionic organic compound uptake via gills might incorporate general equations based on lipid solubility relationships. Models for weakly acidic or basic compounds ingested in food might incorporate pH Partition Theory using the Henderson–Hasselbalch equations. Models for dissolved metal uptake across gill surfaces might be formulated using free ion activity model (FIAM) or biotic ligand model (BLM) based relationships. For example, a FIAM relationship might be developed for a dissolved metal being taken up under different pH, temperature, or water quality conditions that 115 © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 116 — #2 116 Ecotoxicology: A Comprehensive Treatment influence chemical speciation. Allometric 1 power equations might be incorporated if the organism grows substantially during the period of bioaccumulation. Still other models need to accommodate the interactions between physical and biological qualities. For example, temperature’s influence on chlorpyrifos uptake is different for insect groups differing in respiratory strategy (Buchwalter et al. 2003). There are basic similarities among most models although a model might take slightly different forms depending on the exposure scenario and objectives of the modeling effort. These fundamental similarities will be highlighted in the next few pages. 8.2.1 UNDERLYING MECHANISMS The exact formulation of a bioaccumulation model depends on the underlying mechanisms of uptake, internal redistribution, and elimination. As detailed in Chapter 7, some compounds are taken up or eliminated by mechanisms that can be saturated or modified by competition with other compounds. Bioaccumulation of others is influenced by factors such as urine or gut pH, and inclusion of these factors in the associated model might berequired. Still other processes can be modifiedbyacclimation or damage. Some compounds are subject to internal breakdown but others are not. 8.2.2 ASSUMPTIONS OF MODELS AND METHODS OF FITTING DATA Models are developed based on three different formulations: rate-constant-based, clearance-volume- based, and fugacity-based formulations. All are equivalent in their basic forms, but each formulation has its own advantages and disadvantages. (Newman and Unger 2003) Models are based on mathematical expediency (descriptive models), processes and structures described in earlier chapters (mechanistic models), or a blending of both. The most common assump- tions revolve around reaction order for the relevant processes so it is worth taking a moment to review the fundamentals of zero, first, and mixed order reactions. Recollect that order for a reaction or pro- cess involving one reactant refers to the power to which concentration is raised in the differential equation describing the associated kinetics, for example, Zero order: dC dt =−kC 0 =−k, First order: dC dt =−kC 1 =−kC. The equations above are expressed for elimination so the change in internal concentration is denoted by −k or −kC: concentrations are decreasing with time. For uptake, the sign would be positive. The generic k denoted here is a simple proportionality or rate constant. Continuing the example with elimination, a plot of concentration (C t ) versus time (t) will produce a straight line for zero order processes: the absolute value of the slope of that line is an estimate of k. The zero order k has units of C/t. For first order processes, ln C t is plotted against t to produce a straight line and the absolute value of the slope is k. Alternatively, one could plot the ln(C t /C t=0 ) against time for processes following first order kinetics. The slope would then be k (Piszkiewicz 1977). The units for the first order k are 1/t; however, we will see that some slightly more involved bioaccumulation models apply a “first order k” for some processes that have different units. Saturation kinetics used to describe carrier-mediated membrane transport or enzyme-mediated breakdown of a compound are slightly more complicated. Let S be the compound being acted on, by 1 Allometry is the study of organism size and its consequences. © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 117 — #3 Models of Bioaccumulation and Bioavailability 117 either enzymatic conversion or transport by a carrier molecule, E be the enzyme or carrier molecule, and P be the product of enzyme conversion or the compound successfully transported across the membrane via the carrier: S + E k −1  k +1 ES k −2  k +2 E +P. The differential equation describing the change in substrate through time would be dC S dt =−k +1 C E C S +k −1 C ES , (8.1) where C S , C E , and C ES are the concentrations of S, E, and ES, respectively. If one were to plot the curve of C S versus time, the apparent order would depend on the initial C S and might change as C S changes through time. Above a certain C S , the enzyme or membrane transport system would be saturated and incapable of converting/transporting S any faster than a characteristic maximum velocity (V max ): the C S versus time curve would appear to conform to zero order kinetics above the saturation concentration. If one started with very low C S relative to the saturation concentration, the curve would appear to describe first order kinetics. The Michaelis–Menten equation predicting the rate (v) at which conversion occurs for such a process is the following: v = V max C S k m +C S , (8.2) where k m is the C S at which v is half of V max . Like concentration–time curves for zero and first order processes, there are several ways to estimate parameters for saturation kinetics, including the most commonly applied double-reciprocal (Lineweaver–Burk) plot and three less common plots (Eadie–Hofstee, Scatchard, and Woolf plots). Traditionally, a series of C S are established in mixture with the same concentration of enzyme and the rate of disappearance of S (or appearance of P) is measured for each concentration. The C S and conversion rates (v) are then plotted or fit by linear regression to estimate V max and k s . Raaijmaker (1987) discusses the statistical concerns associated with these transformations, concluding that the Woolf plot functions best. Lineweaver–Burk: 1 v = k m +C S V max C S = 1 V max + k m V max C S , (8.3) Eadie–Hofstee: v = V max − k m v C S , (8.4) Scatchard: v C S = V max k m − v k m , (8.5) Woolf : C S v = k m V max + C S V max . (8.6) Piszkiewicz (1977) provides details for deriving these transformations and relating one to the other. Of course, a nonlinear model can also be fit directly to the v versus C S data to parameterize the Michaelis–Menten model. If such fitting involves an iterative maximum likelihood approach, the estimates from one of the above linearizing plots might be used as initial values. It might be more convenient during some modeling efforts to assume zero order kinetics above a certain concentration and then first order when concentrations drop below saturation during a time course. Because the time it will take for the concentration to fall below saturation depends on the initial concentration, it would © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 118 — #4 118 Ecotoxicology: A Comprehensive Treatment be convenient to be able to estimate when one should shift from zero to first order computations. Wagner (1979) provides a convenient relationship for estimating the time to transition from zero to first order (t ∗ ) as a function of the initial concentration (C S0 ): t ∗ =  1 − (1/e) V max  C S0 + k m V max . (8.7) Box 8.1 Silver Transport across Membranes Exhibits Saturation Kinetics Because the silver ion, Ag + , is highly toxic to freshwater fishes, its transport across and effects on gills is a very active area of research. As one example, Bury et al. (1999) characterized silver transport by a Na + /K + -ATPase located on the basolateral membranes of gill cells using a conventional saturation kinetics model. This study is ideal for illustrating here the relevance of saturation kinetics for one feature of contaminant bioaccumulation. Bury et al. (1999) describe studies published preceding theirs in which the Ag + ion was shown to be transported into gill cells by a Na + channel subject to saturation kinetics. Because P-type ATPases 2 had been documented for other metals, Bury et al. hypothesized that Ag + might also be transported by ATPases in rainbow trout (Oncorhynchus mykiss) gill membrane vesicles. They removed gills from trout and produced membrane vesicles by a sequence of homogenization, agitation, and centrifugation steps. Radioactive silver ( 110m Ag) was then used to measure movement of Ag + into isolated gill cell membrane vesicles. Supporting the hypothesis of Bury and coworkers, the Ag + transport into the vesicles was found to be ATP dependent. Competitive inhibition was also apparent from experiments showing that Ag + transport slowed if Na + or K + concentrations were increased in the media surrounding the vesicles. Michaelis–Menten parameters were estimated by fitting the nonlinear Michaelis–Menten model (Equation 8.2) to vesicle uptake (nmolofAg + /mg protein/min) versus Ag concentration (µmol). The V max was 14.3 nmol/mg membrane protein/min and the K m was 62.6 µmol. An Eadie–Hofstee plot produced straight lines and also was used to fit these data by linear regression methods. Bury et al. concluded that there was a P-type ATPase transport mechanism for silver in trout gills and defined the characteristics of the saturation kinetics in vesicle preparations. 8.2.3 RATE CONSTANT-BASED MODELS [The assumption of a single compartment] may not be applied to all drugs. For most drugs, concentrations in plasma measured shortly after iv injection reveal a distinct distributive phase. This means that a measurable fraction of the dose is eliminated before attainment of distribution equilibrium. These drugs impart the characteristics of a multicompartment system upon the body. No more than two compartments are usually needed to describe the time course of drug in the plasma. These are often called the rapidly equilibrating or central compartment and the slowly equilibrating or peripheral compartment. (Gibaldi 1991) Without a doubt, the most commonly applied bioaccumulation model in ecotoxicology is a one-compartment model with onefirst order uptake and one first order elimination term.After gaining entry, the toxicant is assumed to instantly distribute itself uniformly within that compartment. There 2 P-typeATPases are one of three categories ofATPases. They are ubiquitous in living systems, facilitating cation transport for a variety of functions. The formation of a phosphorylated intermediate during transport leads to their designation as P-type. © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 119 — #5 Models of Bioaccumulation and Bioavailability 119 is no hysteresis, that is, the likelihood of a molecule of the toxicant leaving the compartment is independent of how long it has been in the compartment. The relevant model can be constructed easily with the information covered already. The elimination from the compartment would simply be the following: dC i dt =−k e C i , (8.8) where C i is the internal concentration (or amount) of the compartment and k e is the first order rate constant (1/t). As already discussed, the k e for such elimination can be estimated by fitting a linear regression line to ln C i at different times (ln C i,t ) versus time (t). The antilog of the y-intercept of the regression line can also be used to estimate the initial concentration in the compartment, C i,0 . Because this estimate can be biased, Newman (1995) provides a method of removing any bias from an estimated C i,0 . This differential equation (8.8) can be integrated to predict the concentration remaining in the compartment through time, perhapsafterasource has been removed 3 or the organism has been dosed once and then allowed to eliminate the toxicant: C i,t = C i,0 e −k e t . (8.9) Useful metrics associated with this simple elimination model include the biological half-life of the toxicant in the compartment (t 1/2 ) (Equation 8.10) and the mean residence (or turnover) time of a toxicant molecule in the compartment (τ ) (Equation 8.11): t 1/2 = ln 2 k e (8.10) τ = 1 k e . (8.11) Equation 8.12 can be used to model elimination if there are two components of elimination that remove the toxicant from one compartment: C i,t = C i,0 e −(k e,1 +k e,2 )t . (8.12) If a compartment were composed of two subcompartments with no exchange between them, the change in the total concentration or amount in the two combined subcompartments (1 and 2) could be estimated by C i,t = C 0,1 e −k e,1 t +C 0,2 e −k e,2 t . (8.13) Figure 8.1 depicts the change in total concentration (or amount) in such a situation. In that figure, the two subcompartments are designated “fast” and “slow.” A plot of ln C for the compartment that appears to be composed of the two subcompartments versus time of elimination will result in a curve composed of two linear segments. The linear segment for the laterportion of the total curve will reflect the change in concentration (or amount) in the slow subcompartment because the compound in the fast compartment will have been eliminated by that time in the course of depuration. The linear segment at the beginning of the depuration period will reflect the combined concentration in both the slow and fast subcompartments. The two elimination components can be modeled by nonlinear regression or a 3 An experimental design or action in which an organism containing a toxicant is removed to a clean environment where it can eliminate the toxicant is called a depuration design. The elimination of toxicant after movement to a clean environment is called depuration. © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 120 — #6 120 Ecotoxicology: A Comprehensive Treatment Time ln concentration Predominantly slow eliminatio n component Mixture of fast and slow elimination components Backstripped fast component ln C 0 , Slow ln C 0 , Fast FIGURE 8.1 The elimination of compound from a compartment composed of two subcompartments with no exchange of compound between compartments. Both subcompartments have one, first order elimination component. conventional backstripping method. To begin the backstripping approach, a line is fit to the portion of the curve that is predominantly associated with “slow” elimination. The y-intercept of the regression line is the ln C 0,Slow and the absolute value of the slope is the k e,Slow . The C 0,Fast and k e,Fast are then estimated using the data for the line segment associated with the combined concentrations in the two compartments through time and the regression model. The regression model for the slow component is used to predict how much of the concentration of compound measured during the initial linear phase of depuration was associated with the slow component. These predictions are subtracted from the observed concentrations to estimate the amount associated only with the fast elimination component. In essence, the concentration associated with the slow component is stripped away, leaving only that associated with the fast component. These “fast elimination” predictions for each data point would be distributed about the “backstripped” line depicted in Figure 8.1. A linear regression model is fit to these predictions, and the y-intercept and slope used as just described for the slow elimination component to predict C 0,Fast and k e,Fast . This same general procedure can also be used if more than two components are present. The uptake from an external source with a constant concentration (C x ) can be defined as the following: dC i dt = k u C x , (8.14) where k u is the first order rate constant for uptake (1/t). As we will see, the units of k u will change when Equations 8.8 and 8.14 are combined and applied to model bioaccumulation in many cases. Combining these two equations, bioaccumulation for a single compartment model can be described with the following: dC i dt = k u C x −k e C i . (8.15) The above equation can be integrated to yield the conventional single compartment model with one uptake and one elimination term, both of which conform to first order kinetics: C i,t = C x k u k e [1 − e −k e t ]. (8.16) © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 121 — #7 Models of Bioaccumulation and Bioavailability 121 A simple adjustment to Equation 8.16 provides some insight about conditions as concentrations inside the organism approach steady state conditions with the external source. As t gets very large, the bracketed term in Equation 8.16 approaches 1; the bracketed term falls out of the equation as concentrations approach steady state. If both sides of the equation are then divided by C S ,it becomes apparent that the quotient of the concentration in the organism at steady state and external concentration (C ∞ /C x ) is equal to k u /k e . If this model described uptake from water, the k u /k e would then be an estimate of the bioconcentration factor (BCF). It is important to note that the constants are conditional if this model described the kinetics for compartments of differentsizes. The size ofthebiologicalcompartmentandthatoftheexternalsource compartment influence the k u value. If two individuals of identical volume were placed into sources of different volumes but identical concentrations, the estimated values of k u would be different for each. Similarly, the k u values would be different for two different sized individuals exposed to the same concentration of contaminant contained in the same volume of media. In reality, the k u is the flux or clearance rate of the source by an individual of a specific size; therefore, the units of k u are flow/mass, for example, (mL/h)/g, for the equation to balance properly. (SeeAppendix 5 in Newman and Unger (2003) for a detailed dimensional analysis.) We will return to this important point of considering compartment volumes after a brief elaboration on single-compartment bioaccumulation models. The rudimentary model defined by Equation 8.15 or 8.16 can be changed to meet the needs of the modeler. Equation 8.16 can accommodate the confounding factor of the organism rep- resented by the compartment having an initial concentration (C 0 ) before the exposure being modeled C i,t = C x k u k e [1 − e −k e t ]+C i,0 e −k e t . (8.17) Two elimination (Equation 8.18) or uptake (Equation 8.19 for uptake from water and food) terms can be included in the model also: C i,t = C x k u k e1 +k e2 [1 − e −(k e1 +k e2 )t ], (8.18) where k e1 and k e2 are the elimination rate constants for processes 1 and 2: C i,t = C w k uw +αRC f k e [1 − e −k e t ], (8.19) where C w and C f are concentrations in water and food, respectively, α is amount of compound absorbed per amount of compound ingested, and R = the weight-specific ration. Obviously, the exact form of any model will change to the most convenient one as sources change but the general framework remains the same. Returning to our discussion of volumes, compartment volumes also become relevant if the organism was modeled as having two or more compartments that exchange compound. But how are these volumes measured? As a simple illustration, a dose of the compound (D) is applied in a one-compartment model and allowed enough time to evenly distribute in the compartment. The compartment is sampled to determine the concentration (C), and then the compartment volume (V) is estimated as D/C = V. The estimation of volumes becomes complicated if more than one compartment is involved. How volumes are handled in such a situation can be illustrated with the conventional, two-compartment models used often in pharmacology and ecotoxicology (Figure 8.2). With such multiple compartments, volumes are expressed as apparent or effective volumes of dis- tribution (V d ) and treated as mathematically defined compartments that might or might not be easily related to a physical compartment. The most common situation in which such volumes are employed © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 122 — #8 122 Ecotoxicology: A Comprehensive Treatment Time ln concentration C t =C A e −k A t + C B e −k B t D A B k BA k AB k A0 FIGURE 8.2 The estimation of compartment volumes for an organism modeled as two compartments with exchange between compartments and a single dose, D. First order microconstants are also derived from the macroconstants for concentration–time curve of the reference compartment (C 0,A , C 0,B , k A , k B ). In many cases, the reference compartment (A) is the blood (or plasma) and the peripheral compartment includes the tissues with which the compound exchanges with the blood. More complex models are often warranted and require more detailed computations. would be one in which a compound is introduced into the blood and the concentrations are then fol- lowed through time in the blood. The compound in the blood is envisioned as exchanging with some other compartment. The V d for the nonblood (peripheral) compartment is expressedinunitsofvolume of the blood (reference) compartment. For the two-compartment model depicted in Figure 8.2, the microconstants and volumes of distribution for the compartments can be estimated with the following relationships: k AB = C A k B +C B k A C A +C B (8.20) k A0 = k A k B k AB (8.21) k AB = k A +k B −k AB −k A0 (8.22) V dA = D C A +C B (8.23) V dB = V dA  k AB k BA  . (8.24) The steady-state V d for the entire organism consisting of the two compartments is the sum of V dA and V dB . It is important to note that the steady state V d reflects the amount of compound in a unit volume of the organism expressed in terms of the equivalent volume of the reference (source) compartment that would contain that same amount of compound. So, the volume of the peripheral compartment is expressed in units of the reference (blood) compartment. This holds true also if the source (reference) compartment was the water surrounding the organism; the steady-state V d for the organism compartment would be a measure of the BCF because it expresses the volume of organism compartment that holds an equivalent amount of the chemical as a unit volume of the water source compartment. 8.2.4 CLEARANCE VOLUME-BASED MODELS Often, especially in the fields of pharmaco-or toxicokinetics, bioaccumulation models are formulated in the context of clearance. Clearance is the volume of a compartment that is cleared of a substance per unit time. In these kinds of models, one compartment, such as the blood or plasma, is selected as © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 123 — #9 Models of Bioaccumulation and Bioavailability 123 the reference compartment and clearances are calculated relative to that compartment. Clearances can be expressed as volume/time or as mass-normalized clearances [(volume/time)/mass]. A quick glance back to our previous discussions of k u in bioaccumulation models will show that the k u was actually a clearance. Clearances (Cl) can be calculated for simple bioaccumulation models from terms already discussed above (i.e., Cl = k e V d ). By substitution, the rate constant-based model shown in Equation 8.16 can be converted to a clearance-based model: C i,t = C x V d [1 − e −(Cl/V d )t ]. (8.25) Following the reasoning used already for V d at steady state, it is easy to see again from Equation 8.25 that V d at steady state is equal to the quotient of the steady-state concentration in the organism (compartment) to the source. 8.2.5 FUGACITY-BASED MODELS Bioaccumulation models sometimes are formulated in terms of fugacity, the escaping tendency for a substance from a medium or phase. Formulations using fugacity have the distinct advantage that states and rates for all components in complex models can be expressed in the same units. This makes mass balance equations much more tractable than with other approaches (Mackay 1979). Fugacity (f ) is expressed as a pressure (e.g., units of Pascal or Pa) and is derived from concentrations (C in units of mol/m 3 ), i.e., f = Z/C where Z is the fugacity capacity of the phase [mol/(m 3 Pa)]. The fugacity capacity is “a kind of solubility or capacity of a phase to absorb the chemical” (Mackay 1979) so a modeled compound tends to accumulate in phases or compartments with high fugacity capacities. Because of this direct relationship between f and C, the steady state quotient of the concentration in one phase or compartment to that in another (e.g., the BCF) is also equal to the quotient of the fugacity capacities of the two compartments. Only one definition and two identities are needed to convert the simple bioaccumulation models shown in Equation 8.16 to a fugacity-based model. In fugacity-based models, the movement of a substance between compartments is expressed as a transport rate (N, mol/h) and is calculated with a transport constant (D, mol/h×Pa) and the difference in fugacities for the two compartments (f 1 −f 2 ): N = D[f 1 −f 2 ]. (8.26) The rate constant, D, can be used for a diverse range of relevant processes including chemical reactions, diffusion, or advection (Mackay 2004), making it a very convenient term in complex envir- onmental models. The conversion of k u and k e to terms used in fugacity modeling is straightforward (Gobas and Mackay 1987): k e = D 0 V 0 Z 0 , (8.27) k u = D 0 V 0 Z S , (8.28) where D 0 = transport constant for the organism being modeled, V 0 = the volume of the organism being modeled, Z 0 = a proportionality constant called the fugacity capacity for the organism, and Z S = the fugacity capacity for the source of the substance being accumulated. With these definitions, a simple fugacity-based model can be produced. C 0,t = C x Z 0 Z S [1 − e −[D 0 /(V 0 Z 0 )]t ] (8.29) © 2008 by Taylor & Francis Group, LLC Clements: “3357_c008” — 2007/11/9 — 18:10 — page 124 — #10 124 Ecotoxicology: A Comprehensive Treatment This model can be expressed in terms of fugacities instead of concentrations also (Gobas and Mackay 1987): f 0 = f S [1 − e −[D 0 /(V 0 Z 0 )]t ]. (8.30) Box 8.2 Unit and Real World Renderings with Fugacity Models Fugacity simplifies and clarifies the relationship between equilibrium concentrations in various fluids and solids. Rather than relate two concentrations using a partition coefficient , each concentration is independently related to fugacity, and the two fugacities equated. (Mackay 2004) It seemed to me that by reformulating equations in terms of fugacity, environmental mass balances could be done more easily, especially for systems involving disparate phases (Mackay 2004) With a toolbox containing f , Z, and D, the modeler has the key tools to quantify chemical fate in a vast variety of situations from transport across a cell membrane to estimating global distribution of chemical of commerce. (Mackay 2004) A brief look at the remarkable work of Don Mackay and his colleagues seems an appropriate way to illustrate the value and universal applicability of fugacity-based contaminant modeling. Fugacity-based models have been so influential that an entire issue of the journal Environmental Toxicology and Chemistry (vol. 23, no. 10) was recently dedicated to them. Beyond MacKay’s first paper introducing the fugacity concept for contaminant modeling (Mackay 1979), particu- larly useful or exemplary publications applying this approach include Cahill et al. (2003), Czub and McLachlan (2004), Gobas and Mackay (1987), Hickie et al. (1999), Mackay (1979, 2001), Mackay and Wania (1995), and Wania and Mackay (1995). As described above in Mackay’s own words, the great advantage of the fugacity approach is the ability to simplify models by simplifying units. Equations for different processes such as diffusion, advection, or chemical reaction could be made more consistent in this manner. Mackay also facilitated environmental modeling by developing a series of fugacity-based models of increasing complexity. These conceptual microcosms or “unit worlds” provided the starting point for addressing numerous real-world questions. The simpler unit worlds included a few compartments such as air, sediment, soil, and water. Among the more complicated unit worlds is a Level III fugacity model including air, soil, water, settled and suspended sediments, and fish (e.g., Mackay et al. 1985). Elaboration upon unit-world fugacity models proved an effective way to expedite application to diverse, real-world situations. The reader might also have begun to realize that the approach developed by Mackay fosters consilience and translation among levels of biological organization, a central theme of this book. Extensions of models based on the classic approach begun by Teorell (1937a,b) to include many abiotic phases and processes are possible but much more difficult than elab- oration of Mackay’s fugacity models. Some representative studies can be used to illustrate this point. As one example, the general fugacity-based bioaccumulation model of hydrophobic organic compounds by fish incorporates relationships between key rates and qualities, and the K ow (Gobas and Mackay 1987). The model assumed uptake from water via the gills of compounds that do not degrade. It estimated BCFs and displayed good agreement with experimental data. © 2008 by Taylor & Francis Group, LLC [...]... chemical-specific (Wania et al 1999) fugacity-based models have been applied successfully to model the global movement of organic contaminants Clearly, the fugacity-based approach pioneered by Mackay and co-investigators allows contaminant issues to be addressed at remarkably diverse temporal and spatial scales 8. 2.6 PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELS Another approach to modeling bioaccumulation... means of estimating F in such a situation is to use SI and feeding rates, as done by Kukkonen and Landrum (1995) in their assessment of assimilation of sediment-associated PAH and PCB congeners 8. 4 SUMMARY In this chapter, the basic quantitative approaches applied to bioaccumulation were described as well as the basic approaches to estimating bioavailability Rate constant-, clearance-, and fugacity-based... including a five environmental phase unit world was used to predict bioaccumulation in edible fish (Mackay et al 1 985 ) A fugacity model called ACC-HUMAN was applied in 2004 (Czub and McLachlan 2004) to predictions of PCB congener trophic transfer to human milk At a subcontinental scale, Mackay and Wania modeled the movement of organochlorine contaminants in the Arctic General (Wania and Mackay 1995) and chemical-specific... Wania, F and Mackay, D., A global distribution model for persistent organic chemicals, Sci Total Environ., 160/161, 211–232, 1995 Wania, F., Mackay, D., Li, Y.-F., Bidleman, T.F., and Strand, A. , Global chemical fate of α-hexachlorocyclohexane I Evaluation of a global distribution model, Environ Toxicol Chem., 18, 1390–1399, 1999 Yamaoka, Y., Nakagawa, T., and Uno, T., Statistical moments in pharmacokineties,... sediment-bound PAH and PCB congeners by benthic organisms, Aquat Toxicol., 32, 75–92, 1995 Ling, M.-P., Liao, C.-M., Tsai, J.-W., and Chen, B.-C., A PBTK/TD modeling-based approach can assess arsenic bioaccumulation in farmed Tilapia (Oreochromis massambicus) and human health risks, Int Environ Assess Man., 1, 40–54, 2005 Lydy, M.J and Landrum, P.F., Assimilation efficiency for sediment-sorbed benzo [a] pyrene... the rate constants for absorption (Equation 8. 36) from AUC analysis also suggest relative availabilities for the ingested materials Box 8. 4 Taking a Few Moments to Estimate Methylmercury Bioavailability The accumulation of methylmercury in fish has become a major issue relative to human exposure Consequently, methylmercury bioaccumulation dynamics and bioavailability have become important research themes... Comprehensive Treatment 130 The dose-adjusted AUC approach (Equation 8. 41) was applied to estimate oral bioavailability (F) for each catfish, using both noncompartment and compartment model–based methods The noncompartment-based method simply used the linear trapezoidal method to measure AUC The compartment model involved fitting of a triexponential model for the intra-arterial injected dose Cblood,t = A e−π... called either physiologically based pharmacokinetic (PBPK) or physiologically based toxicokinetic (PBTK) modeling Such models can be formulated in rate constant-, clearance-, or fugacity-based terms, but all PBPK have in common the creation of compartments and expressions of transfers among compartments based on real physical, biochemical, physiological, or anatomical compartments and processes As a. .. can describe mathematical compartments/processes or physical compartments/processes as in the case of PBPK (or PBTK) models • Noncompartment statistical moments methods can also generate extremely useful information about bioaccumulation processes and bioavailability • Procedures used to estimate bioavailability vary Each method has advantages and slight differences, making it important to understand... methods As an example, clearance after intravenous (iv) injection can be estimated to be simply Div /AUC (Medinsky and Klaassen 1996) Also, if a dose was administered and the MRT calculated, the first order rate constant ke can be estimated ke = 1 MRT © 20 08 by Taylor & Francis Group, LLC Clements: “3357_c0 08 — 2007/11/9 — 18: 10 — page 126 — #12 (8. 34) Models of Bioaccumulation and Bioavailability . rate-constant-based, clearance-volume- based, and fugacity-based formulations. All are equivalent in their basic forms, but each formulation has its own advantages and disadvantages. (Newman and. this approach include Cahill et al. (2003), Czub and McLachlan (2004), Gobas and Mackay (1 987 ), Hickie et al. (1999), Mackay (1979, 2001), Mackay and Wania (1995), and Wania and Mackay (1995). As. organic contaminants. Clearly, the fugacity-based approach pioneered by Mackay and co-investigators allows contaminant issues to be addressed at remarkably diverse temporal and spatial scales. 8. 2.6