Vaccari D.A., Strom P.F., Alleman J.E. Environmental Biology for Engineers and Scientists

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applicable to models that require numerical solution as well, is to set all derivatives (accu-

mulation terms) to zero. The differential equations then become algebraic and are solved

as such. Do not confuse steady state with equilibrium. As long as a net reaction is occur-

ring in a compartment, it may still be balanced by ﬂux terms. For example, an organism

could be ingesting benzene on a daily basis and biotransforming it to phenol for excretion.

As long as the rate of ingestion equals the rate of reaction, there will be no accumulation

of benzene, and the system will be at steady state.

Equilibrium formally refers to a situation when the chemical potential of reactants

and products are equal for all reactions. In practical terms, equilibrium means that instead

of using the rate laws and mass transfer ﬂux equations to describe the reactions, one sub-

stitutes equi librium relationships such as equation (5.9) for reaction equilibrium or equa-

tion (18.1) for mass transfer equilibrium. Just as steady state does not mean ‘‘no reaction,’’

equilibrium does not mean ‘‘no reaction.’’ For example, chloroform in respired air may be

assumed to be in mass transfer equilibrium with its concentration in blood plasma, yet a

continuous transfer of the solute will continue as long as the air is changed continually.

In the next several sections some simple compartment models are developed, both to

illustrate the modeling process and because they have several important features that are

used to describe the fate and transport of toxins in biological systems.

18.7.1 Dynamic Model and the Half-Life

The most basic model, of course, is the one-compartment model, in which the compart-

ment represents a whole organism. As a hypothetical case, consider how to model a ﬁsh

that ingests zooplankton contaminated with a hydrocarbon. Having decided on the one-

compartment model, we have ﬁnished the ﬁrst step of the model development. We pos-

tulate only two processes: absorption by ingestion and elimination by kidney excretion.

Let us suppose that the hydrocarbon is biotransformed completely. We treat the hydro-

carbon and its metabolite as a single compound. Thus, it is not eliminated until the meta-

bolite is excreted. Finally, let us assume that the metabolite is removed by the kidney by

glomerular ﬁltration only and is not reabsorbed. Thus, the rate of excretion, r

, will be

negatively proportional to the concentration in the blood plasma:

¼k

C ð18:19Þ

where C is the concentration of hydrocarbon plus metabolite and k

is a coefﬁcient related

to the renal clearance rate. Note that this mass transfer process can be formulated as a rate

instead of a ﬂux. The same will be true for ingestion. The rate of absorption, r

, is the pro-

duct of the assimilation efﬁciency, a (the fraction of ingested toxicant that is absorbed),

the mas s of food ingested per unit time, W, and the average concentration of toxican t in

the food, C

(in units of mass of solute per unit mass of food):

¼ aWC

ð18:20Þ

This situation is shown schematically in Figure 18.8. Equations (18.19) and (18.20),

having units of [Mt

1

], must be multiplied by the volume before substitution into equa-

tion (18.13) along with equation (18.14). Canceling the volume yields

¼ aWC

 k

C ð18:21Þ

758 FATE AND T RANSPORT OF TOXINS

For the initial condition that C

¼ 0:0att ¼ 0:0, equation (18.21) can be solved

analytically:

C ¼

ð1  e

k

Þð18:22Þ

This represents a period of increasing body burden (the mass of contaminant contained

by an organism), as shown in Figure 18.8. If the contamination in the food supply were

suddenly eliminated, uptake would cease, and the only proce ss would be elimination. In

aquatic biology, this situation is called depuration. Depuration is used commercially

to eliminate contaminants from shellﬁsh grown in polluted waters by transferring them to

clean waters for a period prior to marketing. The model of equation (18.21) applies, with

set to zero. This leaves the following:

¼k

C ð18:23Þ

This is a ﬁrst-order decay model. The solution, starting from an initial concentration of

,is

C ¼ C

k

ð18:24Þ

Thus, ﬁrst-order decay results in an exponential decrease in concentration. This has

the important property that the concentration decreases by equal fractions in equal time

intervals. For example, the time it takes for the concentration to drop from 100 mg/L to

50 mg/L is the same as the time to go from 2.0 mg/L to 1.0 mg/L. The time it takes for

the concentration to drop by half is called the half-life (t

1=2

). It is used as a convenient

index of how rapidly a toxin is eliminated. The rate coefﬁcient gives the same information

but does not have such a tangible interpretation. In fact, the half-life and the rate coefﬁ-

cient can be computed from one another. From equation (18.24), using C=C

¼ 0:5 yields

1=2

lnð0 :5Þ

0:693

ð18:25Þ

0246

Time

Body Concentration

Uptake

Depuration

Exposure to

ambient pollution

Exposure

terminated

Steady-state exposure conc.

Figure 18.8 Schematic of a one-compartment model, and a plot of uptake and depuration of a

contaminant in the case of a ﬁxed environmental concentration C

PHARMAKOKINETIC MODELS 759

Other characteristics times, such as t

, the time required for a 90% reduction, could be

computed in a similar way. Keep in mind that the half-life and similar parameters can be

used only if the elimination can be approximated as ﬁrst order. (See Problem 18.3 for an

example where this is not the case.) Conveniently, this is true quite often, and half-lives

have been measured for a wide variety of compounds.

Example 18.6 Predatory animals living in a contaminated ecosystem are found to have

a tissue concentration of a toxin equal to 80 mg toxin per kilogram of tissue. Samples of

their usual prey were found to have a body burden of 10 mg/kg. Several animals are taken

into captivity and fed food without the toxin. After 100 days of this treatment, the body

burden is found to have decreased by 40%. Assuming that the animals were at a steady-

state body burden when collected and that the assimilation efﬁciency equals 1.0, est imate

the elimination rate coefﬁcient, the half-life, and the mass loading rate in the wild.

Answer Rearranging equation (18.24) gives us k

¼lnðC=C

Þ=t ¼ 0:511/(40 days) ¼

0.0128 day

1

. From equation (18.25), t

1=2

¼ 0:693=k

¼ 0:693=ð0:013 day

1

Þ¼54:3days.

Assuming steady state and that a ¼ 1:0, equation (18.21) becomes 0 ¼ WC

 k

Therefore,

W ¼ k

¼ð0:0128 day

1

80 mg toxin

kg body weight



kg food

10 mg toxin



¼ 0:102

kg food=day

kg body weight

The animal must eat one-tenth its body weight each day to maintain the stated

body burden.

18.7.2 Steady-State Model and Bioaccumulation

The uptake and retention of contaminants in tissues in excess of concentrations in

the source of the contaminant (such as food or water) is called bioaccumulation.In

Figure 18.8, uptake is being held constant, but elimination increases as concentration

increases. A situation is approached asymptotically where elimination balances uptake,

and the system will be at steady state. The concentration at which this will occur can

be determined in two equivalent ways, as mentioned above. One would be to take the

limit of equation (18.22) as t !1. The other is to set equation (18.21) to zero and solve

for C. Either way gives the same result for the steady-state concentration, C

. The ratio of

to C

is the bioaccumulation factor, K

(also designated BAF):

ð18:26Þ

Bioaccumulation can occur with other routes of exposure. One would develop differ-

ent models for K

if this were the case. In this case, since C

is expressed per unit body

weight and C

is per weight of food, K

will depend on the mass of food ingested per day

per body weight. The caloric requirement of organisms is related to their surface area;

therefore, smaller animals tend to have higher bioaccumulation by ingestion (although

other routes of exposure may be more important). It can also depend on other

things that inﬂuence metabolic rate, such as stress or temperature. If elimination occurs

760 FATE AND T RANSPORT OF TOXINS

predominantly by interphase mass transfer, as is often the case with xenobiotic com-

pounds, k

will be inversely related to the partition coefﬁcient, making K

directly related.

This accounts for the observation that bioaccumulation is strongly correlated with

, especially if the concentration in fatty tissues is used in place of whole body

concentration.

Example 18.7 In Example 18.6, what is the bioaccumulation factor?

Answer K

¼ 80 mg/kg/(10 mg/kg)¼ 8; also, K

¼ aW

¼ð1:0Þ (0.102 day

1

0.0128 day

1

¼ 8.

18.7.3 Equilibrium Model and Bioconcentration

Other mechanisms can give proportional bioaccumulation relationships. For example,

consider a one-compartment model in which both uptake and elimination are by mass

transfer between the environmental concentration, C

env

, and the concentration within the

organism, C

org

. For example, this might apply to the uptake of hydrocarbons from

the air by terrestrial animals or from the water for aquatic animals. In such cases, uptake

and elimination could be described by equation (18.12), with mass transfer coefﬁcients k

and k

for the respective rates r

and r

¼ k

ðK

env

 C

org

¼k

ðK

env

 C

org

ð18:27Þ

Assuming steady state, the mass balance leads to the relationship that the rate of uptake

equals the rate of elimination:

¼ r

ð18:28Þ

The process in which bioaccumulation occurs by direct uptake from the environment

is called bioconcentration. It is distinguished from bioaccumulation in that bioaccu-

mulation includes all routes of exposure, whereas bioconcentration only considers uptake

directly from the environmental medium in which the organism lives. The term biocon-

centration is usually reserved for aquatic systems.

Equations (18.27) and (18.28) can be combined and solved for the ratio of organism

concentration to environmental concentration, called the bioconcentration factor (K

also designated BCF). In this case,

org

env

þ k

¼ K

ð18:29Þ

Thus, for this model the bioc oncentration factor is equal to the partition coefﬁcient.

Again, it must be emphasized that both the bioaccumulation factor and the bioconcentra-

tion factor can be derived from different models using different assumptions. For example,

if we assume simple ﬁrst-order rate processes for uptake and elimination:

org

¼ k

 k

org

ð18:30Þ

PHARMAKOKINETIC MODELS 761

where in this case k

and k

are the uptake and elimination rate coefﬁcients, respectively,

then the bioconcentration factor would be written as

ð18:31Þ

Thus, the bioconcentration and bioaccumulation factors are not deﬁned uniquely in

terms of a particular model. More generally, they are deﬁned as the ratio of organism

concentration to the concentration in the environment and/or the food supply.

Whatever the theoretical mechanisms are to explain the bioconcentra tion factor,

experimental measurements have shown that it correlate s with K

. For example, mea-

surements with 64 organics in ﬁsh have been used to develop the following empirical

relationship:

log

¼ 0:76 log

 0:23 r

¼ 0:82 ð 18 :32Þ

Relationships such as this should be used cautiously, however. Although they are useful

over a wide range of K

, at a particular values the upper and lower 95% conﬁdence

levels can differ by a factor of 63. Furthermore, numerous correlations of this form

have been developed for different groups of chemicals and different species. Some studies

have correlated K

with aqueous solubility instead of K

. Several of these physicochem-

ical parameters are given in Table 18.5.

A more complete model would separately take into account the processes of environ-

mental uptake, ingestion, biotransform ation, and excretion:

org

¼ aWC

þ k

env

 k

org

ð18:33Þ

Assuming steady state, rear ranging, and substituting for K

and K

from equations

(18.26) and (18.31) shows the contribution to body burden by environmental uptake

and ingestion:

org

¼ K

env

þ K

ð18:34Þ

TABLE 18.5 Bioconcentration Factors and Physicochemical

Parameters for Some Compounds That Bioaccumulate

Compound Log K

Log

(mg/L)

Chlordane 5.51 4.05 0.056

DDT 5.98 4.78 0.0017

Dieldrin 5.48 3.76 0.022

Lindane 4.82 2.51 0.150

Chlorpyrifos 4.99 2.65 0.3

Cycexatin 5.38 2.79 <1.0

2,4-D 1.57 1.30 900

2,4,5-T 0.6 1.63 2.38

TCDD 6.15 4.73 0.0002

Source: LaGrega et al. (2001); original source D. W. Connell (1990),

Bioaccumulation of Xenobiotic Compounds, CRC Press, Boca Raton, FL.

762 FATE AND T RANSPORT OF TOXINS

Note that K

in this case should be interpreted as an accumulation factor for ingestion

alone. More generally, bioaccumulation includes all sources of contaminants.

18.7.4 Food Chain Transfer and Biomagniﬁcation

A bioaccumulation mode l such as equation (18.34) can be applied separately to each

trophic level in an ecosystem, with the food at one level being the organism concentration

at the lower level:

¼ K

C;i

env

þ K

B;i

i1

ð18:35Þ

Here C

is the concentration of toxin in the organisms of trophic level i, C

i1

is the con-

centration of the next-lower trophic level, K

is the bioconcentration factor for trophic

level i, and K

is the bioaccumulation factor for trophic level i.

For example, suppose that the environmental concentration of a toxic substance is

1 mg/kg and K

¼ K

¼ 10 for all trophic levels. The primary producers will have a con-

centration of 10 mg/kg due to bioconcentration alone. The herbivores will have C ¼

10  1 þ10  10 ¼ 110 mg/kg. The concentration in carnivores will be C ¼ 10  1 þ

10  110 ¼ 1110 mg/kg. Secondary carnivores will have a concentration 11,110 mg/kg.

An increase of tissue concentration of a toxic substance with increasing trophic level is

called biomagniﬁcation. Figure 18.9 illustrates the relationship for two food chains. Sub-

stances that bioaccumulate also tend to biomagn ify. Thus, the tendency to biomagnify is

usually also related to the K

. However, a low rate of elimination, especially by bio-

transformation, is also necessary. These conditions are particularly true for chlorinated

organics. Biomagniﬁcation can produce extremely high body burdens in organisms at

Water

0.02 ppm

Plankton

5.3 ppm; x265

Small fish

10.0 ppm; x500

Predator fish

1,700 ppm; x85,000

Pesticide: DDD

Pesticide: Toxaphene

Water

0.2 ppm

Planktonic crustaceans

73 ppm; x365

Goldfish

200 ppm; x1,000

Pelican

1,700 ppm; x8,500

Figure 18.9 Biomagniﬁcation of two organochlorine pesticides in aquatic food chains.

PHARMAKOKINETIC MODELS 763

the top of the food chain, the secondary carnivores or predators. The bioaccumulation

factor (and therefore, biomagniﬁcation) can be related to the bioconcentration factor by

a food chain multiplier (FM):

BAF ¼ FM  BCF ð18:36Þ

The U.S. EPA recommends using food chain multipliers based on K

and trophic

level as shown in Table 18.6. At log K

values higher than those in the table, the FM

becomes uncertain and may even decrease due to slow transport kinetics and bioavail-

ability. Thus, if log K

is 5.0, the bioaccumulation factor for carnivores (trophic

level 3) will be about twice the bioconcentration factor.

Because organochlorine pesticides are not only lipophilic and persistent but also toxic

by their nature, they are the substances most capable of causing harm by biomagniﬁ-

cation. Furthermore, biomagniﬁcation limits the effectiveness of pesticides. Since in

agriculture pesticides are often used to control plant pests, they are applied at levels

toxic to the base of the food pyramid. If the pesticide biomagniﬁes, this will neces sarily

result in more harm to predatory insects, which are the ones needed to control the pests.

Thus, ironic ally, it was sometimes found that applications of these substances resulted in

increased pest problems. For these reasons, organochlorine pesticides have been replaced

in many applications by other pesticides that do not biomagnify.

18.7.5 Multicompartment Models

If sufﬁcient information is available, or if contaminant behaviors that are more complex

are to be studied, multiple-compartment models can be used. For example, Figure 18.10

shows a model that could be used to predict narcosis by inhalation of a hydrocarbon. A

separate mass-balance equation must be written for each compartment. Each arr ow

represents a ﬂux, and a term must be added to the liver compartment for elimination

by biotransformation. Although it may not be possible to deﬁne a half-life or a bioaccu-

mulation factor for some of the more complex multicomponent systems, their behavior

often may be approximated by one of those parameters.

TABLE 18.6 Thoman’s Food Chain Multipliers

Trophic Level

Log K

23 4

3.5 1.0 1.0 1.0

4.0 1.1 1.0 1.0

4.5 1.2 1.2 1.2

5.0 1.6 2.1 2.6

5.5 2.8 5.9 11.0

6.0 6.8 21 67.0

6.5 19 45 100.0

Source: U.S. EPA (1994).

764 FATE AND T RANSPORT OF TOXINS

18.8 EFFECT OF EXPOSURE TIME AND MODE OF EXPOSURE

Toxic effects can be affected by the time distribution of exposure. Consider three cases:

(1) a single, instantaneous exposure; (2) a continuous, chronic exposure; and (3) periodic

or intermittant exposures. The difference between these modes will depend on the type of

effect, such as whether it is a local or a systemic toxin, whether or not there is a threshold,

and whether the damage can be repaired or accumulates. If there is a threshold, or if the

effect depends on the concentration of the toxicant or a metabolite in a compartment,

the effect would be correlated with the concentration in the compartment containing the

receptor. If the damage accumulates, an effect could be seen even at a low dose if sufﬁ-

cient prior doses were applied. Here we might expect the effect to correlate with the time

integral of the concentration in the compartment.

For the ﬁrst case, Figure 18.11 shows how concentr ation will vary for an instanta-

neous dose in a two-compartment model. The ﬁrst, or systemic, compartment, peaks

immediately. This is what would be found in the bloodstream if a substance were to be

injected intravenously by hypodermic syringe, for example. The systemic compartment

then clears by some process, such as the ﬁrst-order process illustrated. The second com-

partment represents one containing a receptor, such as neurons in the brain. It exchanges

toxicant only with the systemic compartment. Its concentration increases until it exceeds

that in the ﬁrst compartment. Then it decreases slowly by exchange with the ﬁrst compart-

ment. Here there would not be a great difference with type of effect, except that integrated

Absorption

Excretion

Storage

Bile

Liver

LungGI tract

Dermal

or eye

InhalationIngestion

ExhalationFeces

Bones and

tissues

Organs

Fat

Kidneys

BladderLung

SecretionsUrine

Blood and

Lymph

Glands

Distribution

Absorption

Excretion

Storage

Bile

Liver

LungGI tract

Dermal

or eye

InhalationIngestion

ExhalationFeces

Bones and

tissues

Organs

Fat

Kidneys

BladderLung

SecretionsUrine

Blood and

lymph

Glands

Distribution

Figure 18.10 Multicompartment model of an animal.

EFFECT OF EXPOSURE TIME AND MODE OF EXPOSURE 765

effects would take longer to appear. Biotransformation reactions are neglected in these

examples. If the system behaved more like a one-compartment model, the toxicant

would be cleared in a depuration process similar to that shown in the second part of

Figure 18.8.

Figure 18.12 shows a similar plot for the second two cases. Intermittent exposure

results in higher peak concentrations, which may cause a greater effect either if the effect

is concentration associated or if there is a threshold. On the other hand, the periodic

decrease in concentration may enable repair mechanisms to eliminate previous damage.

This has been observed in exposing rats to ozone. A concentration that caused pulmonary

edema when applied continuously had no effect when applied intermittently. However, if

damage is cumulative, the integral of both curves will be similar, as will be their effect.

An example is the application of organophosphate pesticides, which cumulatively deplete

acetocholinesterase. There is a repair mechanism, in the form of hydrolysis of the

enzyme-OP compound and replacement of the enzyme, but it can be overwhelmed at

moderate doses.

0.0

0.2

0.4

0.6

0.8

1.0

01234

Systemic

component

Peripheral

component

Time

Concentration

/dt = {K

*(C

-C

) - k

} / V

/dt = {-K

*(C

-C

)} / V

Figure 18.11 Typical concentration–time plot for two-compartment model. The systemic

compartment typically represents blood concentration. The peripheral compartment can represent

another tissue or organ, such as the brain or muscle. C and V are the concentration and volume,

respectively, of the compartments. In this simulation k

; K

, and V

¼ 1:0, and V

¼ 3:0.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 2.0 4.0 6.0

Time

Concentration

Figure 18.12 Concentration–time plot for continuous and intermittent exposure.

766

FATE AND TRANSP ORT OF TOXINS

Finally, the same dosing regimen could result in very different concentration–time

relationships, depending on the pharmacokinetics involved. Slow adsorption and elimina-

tion or the presence of a large storage compartment could smooth out changes in con-

centration. In Figure 18.13 the concentration in the peripheral compartment does not

vary greatly when the two compartments have the same volume (V

¼ V

¼ 1:0). But

when the peripheral compartment is much smaller (V

¼ 0:1), large concentration swings

occur.

PROBLEMS

18.1. The OSHA eight-hour inhalati on standard for phenol is approximately 19 mg/m

What is this in ppmv?

18.2. The OSHA eight-hour inhalation standard for chloroform is 50 ppmv, and its

Henry’s constant is 0.148 (conc./conc.). If wastewater ﬂowing through a sewer was

contaminated with 0.50 mg of chloroform per liter, the air space in the sewer and

in the manholes would eventually achieve equilibrium with the chloroform in the

water. Would the air concentration of the TCE in the manhole exceed the OSHA

limit?

18.3. Use equations (18.7) and (18.32) to estimate the octanol–water partition coefﬁ-

cient and the bioconcentration factor of chlordane. Compare to the values given in

Table 18.5.

18.4. Ethyl alcohol is eliminated from the body by a zero-order process. That is, it

is eliminated by a constant mass per unit time. Assume that the removal rate is

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

01234

Time

Concentration

= 0.1 V

= 1.0

Figure 18.13 Effect of differing pharmacokinetics on the time course of concentration in an

organism. k

, K

, and V

¼ 1:0, and V

¼ 1:0 or 0.1 as indicated.

PROBLEMS 767