Introduction

Drug therapy is a dynamic process. When a drug product is administered, absorption usually proceeds over a finite time interval; and distribution, metabolism, and excretion (ADME) of the drug and its metabolites proceed continuously at various rates. The relative rates of these "ADME processes" determine the time course of the drug in the body, most importantly at the receptor sites that are responsible for the pharmacologic action of the drug.

The usual aim of drug therapy is to achieve and maintain effective concentrations of drug at the receptor site. However, the body is constantly trying to eliminate the drug and, therefore, it is necessary to balance absorption against elimination so as to maintain the desired concentration. Often the receptor sites are tucked away in a specific organ or tissue of the body, such as the central nervous system, and it is necessary to depend on the blood supply to distribute the drug from the site of administration, such as the gastrointestinal tract, to the site of action.

Since the body may be viewed as a very complex system of compartments, at first, it might appear to be hopeless to try to describe the time course of the drug at the receptor sites in any mathematically rigorous way. The picture is further complicated by the fact that, for many drugs, the locations of the receptor sites are unknown. Fortunately, body compartments are connected by the blood system, and distribution of drugs among the compartments usually occurs much more rapidly than absorption or elimination of the drug. The net result is that the body behaves as a single homogeneous compartment with respect to many drugs, and the concentration of the drug in the blood directly reflects or is proportional to the concentration of the drug in all organs and tissues. Thus, it may never be possible to isolate a receptor site and determine the concentration of drug around it, but the concentration at the receptor site usually can be controlled if the blood concentration can be controlled.

The objective of pharmacokinetics is to describe the time course of drug concentrations in blood in mathematical terms so that (i) the performance of pharmaceutical dosage forms can be evaluated in terms of the rate and amount of drug they deliver to the blood, and (ii) the dosage regimen of a drug can be adjusted to produce and maintain therapeutically effective blood concentrations with little or no toxicity. The primary objective of this chapter will be to describe the graph paper and calculator level, mathematical tools needed to accomplish these aims when the body behaves as a single homogeneous compartment and when all pharmacokinetic processes obey first-order kinetics.

On some occasions, the body does not behave as a single homogeneous compartment, and multicompartment pharmacokinetics is required to describe the time course of drug concentrations. In other instances certain pharmacokinetic processes may not obey firstorder kinetics, and saturable or nonlinear models may be required. More information about these complexities may be found on the Web site http://www.boomer.org/c7p4/. Additionally advanced pharmacokinetic analyses require the use of various computer programs, such as those listed on the Web site http://www.boomer.org/pkin/soft.html. Readers interested in such advanced topics are referred to a number of texts that describe these more complex pharmacokinetic models in detail (1-5) and to the Web site http://www.boomer.org/pkin/.

PRINCIPLES OF FIRST-ORDER KINETICS Definition and Characteristics of First-Order Processes

The science of kinetics deals with the mathematical description of the rate of the appearance or disappearance of a substance. One of the most common types of rate processes observed in nature is the first-order process in which the rate is dependent on the concentration or amount of only one component. An example of such a process is radioactive decay in which the rate of decay (i.e., the number of radioactive decompositions per minute) is directly proportional to the amount of undecayed substance remaining. This may be written mathematically as follows:

Rate of radioactivity decay / [undecayed substance] (1)

Rate of radioactive decay = ¿(undecayed substance) (2)

where k is a proportionality constant called the first-order rate constant.

Chemical reactions usually occur through collision of at least two molecules, very often in a solution, and the rate of the chemical reaction is proportional to the concentrations of all reacting molecules. For example, the rate of hydrolysis of an ester in an alkaline-buffered solution depends on the concentration of both the ester and hydroxide ion:

The rate of hydrolysis may be expressed as follows:

where k is the proportionality constant called the second-order rate constant.

But, in a buffered system, [OH-] is constant. Therefore, at a given pH, the rate of hydrolysis is dependent only on the concentration of the ester and may be written:

where k is the pseudo-first-order rate constant at the pH in question. (The pseudo-first-order rate constant, k , is the product of the second-order rate constant and the hydroxide ion concentration:

Fortunately, most ADME processes behave as pseudo-first-order processes—not because they are so simple, but because everything except the drug concentration is constant. For example, the elimination of a drug from the body may be written as follows:

Drug in body

Enzymes; membranes; pH; protein binding; etc.

Metabolized or excreted drug

If everything except the concentration of drug in the body is constant, the elimination of the drug will be a pseudo-first-order process. This may seem to be a drastic oversimplification, but most in vivo drug processes, in fact, behave as pseudo-first-order processes.

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