## 4.1.2. Toxicokinetics

Author: Joop Hermens, Nico van Straalen

Reviewers: Kees van Gestel, Philipp Mayer

Learning objectives:

You should be able to

- mention the underlying assumptions of the kinetic models for bioaccumulation
- understand the basic equations of a one compartment kinetic bioaccumulation model
- explain the differences between one- and two-compartment models
- mention which factors affect the rate constants in a compartment model

Key words: Bioaccumulation, toxicokinetics, compartment models

In the section “Bioaccumulation”, the process of bioaccumulation is presented as a steady state process. Differences in the bioaccumulation between chemicals are expressed via, for example, the bioconcentration factor BCF. The BCF represents the ratio of the chemical concentration in, for instance, a fish versus the aqueous concentration at a situation where the concentrations in water and fish do not change in time.

(1)

where:

Caq concentration in water (aqueous phase) (mg/L)

Corg concentration in organism (mg/kg)

The unit of BCF is L/kg.

Kinetic models

Steady state can be established in a simple laboratory set-up where fish are exposed to a chemical at a constant concentration in the aqueous phase. From the start of the exposure (time 0, or t=0), it will take time for the chemical concentration in the fish to reach steady state and in some cases, this will not be established within the exposure period. In the environment, exposure concentrations may fluctuate and, in such scenarios, constant concentrations in the organism will often not be established. Steady state is reached when the uptake rate (for example from an aqueous phase) equals the elimination rate. Models that include the factor time in describing the uptake of chemicals in organisms are called kinetic models.

Toxicokinetic models for the uptake of chemicals into fish are based on a number of processes for uptake and elimination. An overview of these processes is presented in Figure 1. In the case of fish, the major process of uptake is by diffusion from the surrounding water compartment via the gill to the blood. Elimination can be via different processes: diffusion via the gill from blood to the surrounding water compartment, via transfer to offspring or eggs by reproduction, by growth (dilution) and by internal degradation of the chemical (biotransformation).

Figure 1. Uptake and elimination processes in fish and the rate constants (k) for each process. Reproduced from Van Leeuwen and Vermeire (2007) by Wilma IJzerman.

Kinetic models to describe uptake of chemicals into organisms are relatively simple with the following assumptions:

First order kinetics:

Rates of exchange are proportional to the concentration. The change in concentration with time (dC / dt ) is related to the concentration and a rate constant (k):

(2)

One compartment:

It is often assumed that an organism consists of only one single compartment and that the chemical is homogeneously distributed within the organism. For “simple” small organisms this assumption is intuitively valid, but for large fish this assumption looks unrealistic. But still, this simple model seems to work well also for fish. To describe the internal distribution of a chemical within fish, more sophisticated kinetic models are needed, similar to the ones applied in mammalian studies. These more complex models are the “physiologically based toxicokinetic” (PBTK) models (Clewell, 1995; Nichols et al., 2004)

Equations for the kinetics of accumulation process

The accumulation process can be described as the sum of rates for uptake and elimination.

(3)

Integration of this differential equation leads to equation 4.

(4)

Corg concentration in organisms (mg/kg)

Caq concentration in aqueous phase (mg/L)

kw uptake rate constant (L/kg·day)

ke elimination rate constant (1/day)

t time (day)

(dimensions used are: amount of chemical: mg; volume of water: L; weight of organism: kg; time: day); see box.

Box: The units of toxicokinetic rate constants The differential equation underlying toxicokinetic analysis is basically a mass balance equation, specifying conservation of mass. A mass balance implies that the amount of chemical is expressed in absolute units such as mg. If Q is the amount in the animal and F the amount in the environmental compartment the mass balance reads: where is the uptake rate constant and k2 the elimination rate constant, both with dimension time-1. However, it is often more practical to work with the concentration in the animal (e.g. expressed in mg/kg). This can be achieved by dividing the left and right sides of the equation by w, the body-weight of the animal and defining Cint = Q/w. In addition, we define the external concentration as Cenv = F/V, where V is the volume (L or kg) of the environmental compartment. This leads to the following formulation of the differential equation: Beware that Cenv is measured in other units (mg per kg of soil, or mg per litre of water) than Cint (mg per kg of animal tissue). To get rid of the awkward factor V/w it is convenient to define a new rate constant, k1: This is the uptake rate constant usually reported in scientific papers. Note that it has other units than : it is expressed as kg of soil per kg of animal tissue per time unit (kg kg-1 h-1), and in the case of water exposure as L kg-1 h-1. The dimension of k2 remains the same whether mass or concentrations are used (time-1). We also learn from this analysis that when dealing with concentrations, the body-weight of the animal must remain constant. References Moriarty, F. (1984) Persistent contaminants, compartmental models and concentration along food-chains. Ecological Bulletins 36: 35-45. Skip, B., A.J. Bednarska, & R. Laskowski (2014) Toxicokinetics of metals in terrestrial invertebrates: making things straight with the one-compartment principle. PLoS ONE 9(9): e108740.

Equation 4 describes the whole process with the corresponding graphical representation of the uptake graph (Figure 2).

Figure 2. The basic equation for uptake of a chemical from the aqueous phase to a fish: one compartment model with first order kinetics.

The concentration in the organism is the result of the net process of uptake and elimination. At the initial phase of the accumulation process, elimination is negligible and the ratio of the concentration in the organism is given by:

(5)

(6)

Steady state

After longer exposure time, elimination becomes more substantial and the uptake curve starts to level off. At some point, the uptake rate equals the elimination rate and the ratio Corg/Caq becomes constant. This is the steady state situation. The constant Corg/Caq at steady state is called the bioconcentration factor BCF. Mathematically, the BCF can also be calculated from kw/ke. This follows directly from equation 4: after long exposure time (t), becomes 0 leading to

(7)

Elimination

Elimination is often measured following an uptake experiment. After the organism has reached a certain concentration, fish are transferred to a clean environment and concentration in the organism will decrease in time. Because this is also a first order kinetic process, the elimination rate will depend on the concentration in the organism (Corg) and the elimination rate constant (ke) (see equation 8). Concentration will decrease exponentially in time (equation 9) as shown in Figure 3A. Concentrations are often transformed to the natural logarithmic values (ln Corg) because this results in a linear relationship with slope -ke.(equation 10 and figure 3B).

(8)

(9)

(10)

where Corg(t=0) is the concentration in the organism when the elimination phase starts.

The half-life (T1/2 or DT50) is the time needed to eliminate half the amount of chemical from the compartment. The relationship between ke and T1/2 is: T1/2 = (In 2) / ke. The half-life increases when ke decreases.

Figure 3. Elimination of a chemical from fish to water: one compartment model. Left: concentrations given on a normal scale. Right: concentration expressed as the natural logarithm to enable linear regression against time to yield the rate constant as the slope.

Multicompartment models

Very often, organisms cannot be considered as one compartment, but as two or even more compartments (Figure 4A). Deviations from the one-compartment system usually are seen when elimination does not follow an exponential pattern as expected: no linear relationship is obtained after logarithmic transformation. Figure 4B shows the typical trend of the elimination in a two-compartment system. The decrease in concentration (on a logarithmic scale) shows two phases: phase I with a relatively fast decrease and phase II with a relatively slow decrease. According to the linear compartment theory, elimination may be described as the sum of two (or more) exponential terms, like:

(11)

where

ke(I) and ke(II) represent elimination rate constants for compartment I and II,

F(I) and F(II) are the size of the compartments (as fractions)

Typical examples of two compartment systems are:

- Blood (I) and liver (II)
- Liver tissue (I) and fat tissue (II)

Elimination from fat tissue is often slower than from, for example, the liver. The liver is a well perfused organ while the exchange between lipid tissue and blood is much less. That explains the faster elimination from the liver.

Figure 4. Elimination of chemical from fish to water (top) in a two-compartment model (bottom).

Examples of uptake curves for different chemicals and organisms

Figure 5 gives uptake curves for two different chemicals and the corresponding kinetic parameters. Chemical 2 has a BCF of 1000, chemical 1 a BCF of 10,000. Uptake rates (kw) are the same, which is often the case for organic chemicals. Half-lives (time to reach 50 % of the steady state level) are 14 and 140 hours. This makes sense because it will take a longer time to reach steady state for a chemical with a higher BCF. Elimination rate constants also differ a factor of 10.

In figure 6, uptake curves are presented for one chemical, but in two organisms of different size/weight. Organism 1 is much smaller than organism 2 and reaches steady state much earlier. T1/2 values for the chemical in organisms 1 and 2 are 14 and 140 hours, respectively. The small size explains this fast equilibration. Rates of uptake depend on the surface-to-volume ratio (S/V) of an organism, which is much higher for a small organism. Therefore, kinetics in small organisms is faster resulting in shorter equilibration times. The effect of size on kinetics is discussed in more detail in Hendriks et al. (2001) and in the Section on Allometric Relationships.

Figure 5. Uptake curves for two chemicals, having different properties, in the same organism.

Figure 6. Uptake curves for the same chemical in two organisms having different sizes; organism 2 is much bigger than organism 1.

Bioaccumulation involving biotransformation and different routes of uptake

In equation 2, elimination only includes gill elimination. If other processes such as biotransformation and growth are considered, the equation can be extended to include these additional processes (see equation 12).

(12)

For organisms living in soil or sediment, different routes of uptake may be of importance: dermal (across the skin), or oral (by ingestion of food and/or soil or sediment particles). Mathematically, the uptake in an organism in sediment can be described as in equation 13.

(13)

Corg concentration in organisms (mg/kg)

Caq concentration in aqueous phase (mg/L)

Cs concentration soil or sediment (mg/kg)

kw uptake rate constant from water (L/kg/day)

ks uptake rate constant from soil or sediment (kgsoil/kgorganism/day)

ke elimination rate constant (1/day)

t time (day)

(dimensions used are: amount of chemical: mg; volume of water: L; weight of organism: kg; time: day)

In this equation, kw and ks are the uptake rate constants from water and sediment, ke is the elimination rate constant and Caq and Cs are the concentrations in water and sediment or soil. For soil organisms, such as earthworms, oral uptake appears to become more important with increasing hydrophobicity of the chemical (Jager et al., 2003). This is because the concentration in soil (Cs) will become higher than the porewater concentration Ca for the more hydrophobic chemicals (see section on Sorption).

References

Clewell, H.J., 3rd (1995). The application of physiologically based pharmacokinetic modeling in human health risk assessment of hazardous substances. Toxicology Letters 79, 207-217.

Hendriks, A.J., van der Linde, A., Cornelissen, G., Sijm, D. (2001). The power of size. 1. Rate constants and equilibrium ratios for accumulation of organic substances related to octanol-water partition ratio and species weight. Environmental Toxicology and Chemistry 20, 1399-1420.

Jager, T., Fleuren, R., Hogendoorn, E.A., De Korte, G. (2003). Elucidating the routes of exposure for organic chemicals in the earthworm, Eisenia andrei (Oligochaeta). Environmental Science and Technology 37, 3399-3404.

Nichols, J.W., Fitzsimmons, P.N., Whiteman, F.W. (2004). A physiologically based toxicokinetic model for dietary uptake of hydrophobic organic compounds by fish - II. Simulation of chronic exposure scenarios. Toxicological Sciences 77, 219-229.

Van Leeuwen, C.J., Vermeire, T.G. (Eds.) (2007). Risk Assessment of Chemicals: An Introduction. Springer, Dordrecht, The Netherlands.