Pharmakokinetic Models

A model is a kind of analogy. It is a surrogate for a real process that is easier to analyze and experiment with. A mathematical model is thus a set of equations that behave similar to a real system. Often, the process of developing a mathematical model helps to develop our mental model, leading to improved understanding. Furthermore, the model may be useful in making predictions, and for designing engineered systems to obtain desired results.

Ultimately, we would like to predict actual toxic effects. The models described here, however, will only predict the exposure of an organism to substances in the environment and the concentration of various tissues that results. Our interest will first be in pharmacokinetic models, models of the behavior of toxic substances within organisms: how they are taken up, distributed and stored, biotransformed, and excreted. The same principles can then be used for environmental models, models of the fate and transport of toxic substances between and among compartments of the environment (air, water, soil, etc., and different organisms in the environment), which control exposure.

A common type of model is the compartment model, which represents organisms by one or more volumes characterized by a single concentration for each substance being modeled and which is physicochemically and biologically uniform. For example, an organism could be represented by three compartments: the gut, systemic fluids (blood), and target organs. The compartments are connected to each other and to the environment by transport mechanisms. Biochemical reactions create or consume the modeled substance within the compartments. Mathematical relations describe the rate at which each of these processes adds or removes material from each compartment. The process of model development can be broken down into the following steps:

1. Define the compartments that will make up the model.

2. Identify the transport mechanisms that move substances between compartments.

3. Identify the biochemical reactions producing or destroying the modeled substance in each compartment.

4. Write the mathematical description of the individual transport fluxes and reaction processes.

5. Link the processes with material balance considerations to produce the final mathematical form of the model.

6. Mathematically solve the model using the appropriate initial and boundary conditions.

7. Validate the model by comparing with experimental results.

Material balance expresses the law of conservation of matter. The balance of the matter that enters or leaves a compartment, or is created or destroyed within it, must result in an accumulation. This can be expressed in this way:

accumulation = X inputs + X outputs + X reactions (18.13)

The accumulation term is the change in mass, M, within the compartment, which in turn is the product of volume, V, and concentration, C. All the terms in equation (18.13) should have the units [M • t_1]. It is common to assume that the volume of the compartment is constant (no growth). In this case, the accumulation can be expressed in terms of the change in concentration in the compartment:

dt dt dt

The fluxes are the mass transport flows moving matter from one compartment to another, or to or from the environment. These include the various forms of absorption, such as passive diffusion [equation (18.11) or film theory equation (18.12)]. An important type of flux is advection, the result of a substance being carried into a compartment by fluid flow. Examples include transport of a toxin to the liver by blood flow, or water flowing through the gills of a fish. Advective flux, F, can be represented by the product of flow, Q, and concentration:

Advective flux can also describe uptake by ingestion, wherein the flow is the mass of food ingested per unit time, and the concentration is the average toxin concentration in the food. Fluxes must have positive signs for transport into a compartment, and negative for movement out.

The biochemical reactions are described using the methods from chemical kinetics. The general form of the reaction term is

where r is the reaction rate, that is, the mass produced or consumed per unit volume per unit time. The rate is described, in turn, by a rate law, which gives the dependence of the rate on other factors, especially concentrations of reactants. Often, simple elementary rate laws can be assumed. In some cases more complex kinetics may be assumed, such as a rate based on Michaelis-Menten kinetics [equation (5.36)], or one of the other rate equations from Section 5.3.1. Here we describe two of the simpler kinetic rate laws, which are often sufficiently accurate to describe many reactions. The first is the zero-order reaction, in which the rate is constant independent of the concentration of reactants:

The sign will be positive for production reactions and negative for destruction. An example of a zero-order reaction is the metabolism of ethanol in humans (Problem 18.1).

The other simple elementary rate law is the first-order reaction, in which the rate of reaction is proportional to the amount of reactant present:

First-order reaction is probably the most common rate law. The negative form is called first-order decay. Its use will lead us to several important parameters for describing toxicant behavior which are developed in the next two sections.

Models developed as just described are differential equations which, when solved with appropriate initial conditions, predict the concentration of toxin in a compartment vs. time. Simpler models can be solved analytically giving the result as a function. More complex models must often be solved numerically using a computer, giving the concentration vs. time as a table of values. Models that predict how results change with time are called dynamic models.

Often, if enough time is allowed to elapse, dynamic models show that concentrations approach a constant value. In the limit as time goes to infinity, all changes cease, and a steady-state condition exists. Mathematically, the steady-state condition could be found by taking the limit of an analytical solution as time goes to infinity. Even simpler, and applicable to models that require numerical solution as well, is to set all derivatives (accumulation 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 occurring 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 substitutes equilibrium relationships such as equation (5.9) for reaction equilibrium or equation (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 compartment 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 postulate only two processes: absorption by ingestion and elimination by kidney excretion. Let us suppose that the hydrocarbon is biotransformed completely. We treat the hydrocarbon and its metabolite as a single compound. Thus, it is not eliminated until the metabolite 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, re, will be negatively proportional to the concentration in the blood plasma:

where C is the concentration of hydrocarbon plus metabolite and ke 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, ra, is the product of the assimilation efficiency, a (the fraction of ingested toxicant that is absorbed), the mass of food ingested per unit time, W, and the average concentration of toxicant in the food, Cf (in units of mass of solute per unit mass of food):

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

At ie

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