ECOTOXICOLOGY: A Comprehensive Treatment - Chapter 8 potx

Continuing the example with elimination, a plot of concentration Ct versus time t will produce a straight line for zero order processes: the absolute value of the slope of that line is a

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 a jumble 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 exposure takes place, and the qualities of the 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

influence chemical speciation Allometric1power 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 UNDERLYINGMECHANISMS

The exact formulation of a bioaccumulation model depends on the underlying mechanisms of uptake, internal redistribution, and elimination As detailed inChapter 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 be required Still other processes can be modified by acclimation

or damage Some compounds are subject to internal breakdown but others are not

8.2.2 ASSUMPTIONS OFMODELS ANDMETHODS OFFITTINGDATA

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 = −kC0= −k,

First order: dC

dt = −kC1= −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 (Ct) 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 Ctis plotted against t to produce a straight line and the absolute value of the slope is k Alternatively, one could plot the ln (Ct/Ct =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.

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

ES

E+ P

The differential equation describing the change in substrate through time would be

dCS

where CS, CE, and CESare the concentrations of S, E, and ES, respectively If one were to plot the

curve of CS versus time, the apparent order would depend on the initial CS and might change as

CS changes through time Above a certain CS, the enzyme or membrane transport system would

be saturated and incapable of converting/transporting S any faster than a characteristic maximum

velocity (Vmax): the CSversus time curve would appear to conform to zero order kinetics above the

saturation concentration If one started with very low CS 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= VmaxCS

km+ CS

where km is the CS at which v is half of Vmax 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 CSare 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 CS and conversion rates (v) are then plotted or fit by linear regression to estimate Vmaxand ks Raaijmaker (1987) discusses the statistical concerns associated with these transformations, concluding that the Woolf plot functions best

Lineweaver–Burk: 1

v =km+ CS

VmaxCS = 1

Vmax + km

VmaxCS

Eadie–Hofstee: v = Vmax−kmv

CS

Scatchard: v

CS =Vmax

km − v

km

Woolf : CS

v = km

Vmax + CS

Vmax

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 CSdata 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

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)

Vmax



CS0+ km

Vmax

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 ATPases2 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 (110mAg) 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 (nmol of Ag+/mg protein/min) versus

Ag concentration (µmol) The Vmaxwas 14.3 nmol/mg membrane protein/min and the Kmwas 62.6µmol 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 RATECONSTANT-BASEDMODELS

[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 one first 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-type ATPases are one of three categories of ATPases 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.

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:

dCi

where Ciis the internal concentration (or amount) of the compartment and keis the first order rate constant (1/t) As already discussed, the kefor such elimination can be estimated by fitting a linear

regression line to ln Ci at different times (ln Ci,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, Ci,0 Because this estimate can be biased, Newman (1995) provides a method of removing any bias

from an estimated Ci,0 This differential equation (8.8) can be integrated to predict the concentration remaining in the compartment through time, perhaps after a source has been removed3or the organism has been dosed once and then allowed to eliminate the toxicant:

Ci,t = Ci,0e−ket

Useful metrics associated with this simple elimination model include the biological half-life of

the toxicant in the compartment (t1/2) (Equation 8.10) and the mean residence (or turnover) time of

a toxicant molecule in the compartment (τ) (Equation 8.11):

t1/2=ln 2

ke

(8.10)

τ = 1

ke

Equation 8.12 can be used to model elimination if there are two components of elimination that remove the toxicant from one compartment:

Ci,t= Ci,0e−(ke,1+ke,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

Ci,t= C0,1e−ke,1t + C0,2e−ke,2t (8.13) Figure 8.1depicts 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 later portion 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.

Predominantly slow

eliminatio

n component

Mixture of fast and slow elimination components

Backstripped fast component

ln C0,Slow

ln C0,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 C0,Slow and the absolute value of the slope is the ke,Slow The C0,Fast and ke,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 C0,Fastand ke,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 (Cx) can be defined as the following:

dCi

where kuis the first order rate constant for uptake (1/t) As we will see, the units of kuwill 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:

dCi

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:

Ci,t= Cx

ku

ke

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 CS, it becomes apparent that the quotient of the concentration in the organism at steady state and external

concentration (C∞/Cx) is equal to ku/ke If this model described uptake from water, the ku/kewould 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 different sizes The size of the biological compartment and that of the external source

compartment influence the kuvalue If two individuals of identical volume were placed into sources

of different volumes but identical concentrations, the estimated values of ku would be different

for each Similarly, the kuvalues 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 kuis

the flux or clearance rate of the source by an individual of a specific size; therefore, the units of kuare flow/mass, for example, (mL/h)/g, for the equation to balance properly (See Appendix 5 inNewman 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 (C0) before the exposure being modeled

Ci,t= Cx

ku

ke[1 − e−ket ] + Ci,0e−ket

Two elimination (Equation 8.18) or uptake (Equation 8.19 for uptake from water and food) terms can be included in the model also:

Ci,t = Cx

ku

ke1+ ke2[1 − e−(ke1+ke2)t], (8.18)

where ke1and ke2are the elimination rate constants for processes 1 and 2:

Ci,t = Cwkuw+ αRCf

where Cw and Cf 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 (Vd) 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

ation Ct=CAe −kAt+CBe −kBt

B

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 (C0,A, C0,B, kA, kB) 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 Vdfor the nonblood (peripheral) compartment is expressed in units of volume

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:

kAB= CAkB+ CBkA

CA+ CB

(8.20)

kA0= kAkB

kAB

(8.21)

VdA= D

CA+ CB

(8.23)

VdB= VdA

kAB

kBA



The steady-state Vd for the entire organism consisting of the two compartments is the sum of

VdAand VdB It is important to note that the steady state Vdreflects 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 Vdfor 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

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

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 kuin bioaccumulation models will show that the kuwas actually a clearance Clearances (Cl) can be calculated for simple bioaccumulation models from terms already discussed above (i.e., Cl = keVd) By substitution, the rate constant-based model

shown in Equation 8.16 can be converted to a clearance-based model:

Ci,t= CxVd[1 − e−(Cl/Vd)t] (8.25)

Following the reasoning used already for Vdat steady state, it is easy to see again from Equation 8.25

that Vd 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/m3), i.e., f = Z/C where Z is the fugacity capacity of the phase [mol/(m3Pa)] 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 (f1−f2):

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 kuand keto terms used in fugacity modeling is straightforward (Gobas and Mackay 1987):

ke= D0

V0Z0

where D0= transport constant for the organism being modeled, V0= the volume of the organism

being modeled, Z0 = a proportionality constant called the fugacity capacity for the organism, and

ZS= the fugacity capacity for the source of the substance being accumulated With these definitions,

a simple fugacity-based model can be produced

C0,t= Cx

Z0

ZS

This model can be expressed in terms of fugacities instead of concentrations also (Gobas and Mackay 1987):

f0= fS[1 − e−[D0/(V0Z0)]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 Kow (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