1
Setting the Scene: Concepts, Nature of Drugs, and Quality of Results

1.1 Introduction

This book is about the time course of drug action and how that relates to concentrations of the chemicals concerned in body fluids, principally plasma. Obviously relevant are drug properties such as solubility and pKa, properties induced by the pharmaceutical formulation, the physiological and biochemical influences such as metabolism and excretion, and the interaction of drugs and disease, because drugs both treat disease in animals and humans, and are affected differently by diseased and healthy organs. Drugs in this context include xenobiotics old and new, molecules small and large, and both natural and synthetic examples and importantly refers to the so-called 'biologicals' - products of the modern pharmaceutical industry with its focus on peptides, synthetic proteins, and oligonucleotides. It also includes certain molecules of a similar nature that are of importance because of their toxicity, while having little or no therapeutic significance.

1.2 Concepts and terminology: disposition and pharmacokinetics

Drug disposition is a collective term used to describe the processes of drug absorption, distribution, metabolism, and excretion, often given the acronym ADME, which is extended in some literature to LADME, recognizing L for liberation of a drug from its dosage form, a process that can be crucial to the successful development and use of many drugs. The importance of toxicological studies in the development of new drugs is reflected by the extension of ADME to ADMET. Promising drug candidates have been withdrawn because of unfavourable toxicological data - a process referred to as 'attrition'. A further acronym that will be encountered frequently is DMPK, drug metabolism and pharmacokinetics (PK), an area of study that clearly overlaps with ADME.

PK is somewhat more conceptual in nature, being the study of the rates of the drug disposition processes. By subjecting the observed changes, for example, in plasma concentrations as a function of time to mathematical equations (models), pharmacokinetic parameters such as elimination half-life (t½), volume of distribution (V?), and plasma clearance (CL) can be derived. Pharmacodynamics (PD) is the study of how the drug interacts with the body to cause an effect, and increasingly models are being developed that integrate quantitatively pharmacokinetics and dynamics (PK-PD) as discussed briefly later and in subsequent chapters, particularly Chapter 5.

Drug disposition can be considered to be a 'natural' science, whilst the mathematical modelling employed in PK is a 'formal' science (Patel, 2013). When natural science requires measurement of a phenomenon, there will be inevitable errors of observation leading to uncertainty about the result. Furthermore, when biological systems are studied, there will be variability as a result of differences between the samples, be they in vitro or in vivo - often referred to as 'biological variation'. The variation is likely to be less in studies with laboratory animals, such as an inbred colony of rats, than in human beings with their wide diversity of genetic differences and exposure to various agents that might influence the result. Being a formal science, mathematics is expected to provide a definitive answer; however, pharmacokinetic models not only use inputs from the natural sciences but also often make assumptions, approximations, and may be only applicable under clearly defined conditions. Thus, the results of mathematical modelling of pharmacokinetic data will also have attendant degrees of uncertainty. This chapter examines some of the issues that measurement and modelling raise as these need to be controlled and any errors minimized to ensure the most reliable outcomes.

1.3 Pharmacokinetic parameters

Pharmacokinetic parameters are the variables that are used for modelling, although this may be little more than a description of the time course of a substance in the body. At a first level of approximation, there are only a few key ones that need consideration: half-life and its attendant rate constant to which it is inversely proportional (Equation 3.9), systemic clearance, apparent volume of distribution, and the fraction of an extravascular dose that reaches the systemic circulation (F, often referred to as bioavailability). These parameters are the tools of the trade and like any other tools can be misused or used inappropriately. It is of interest that these tools have moved between disciplines over time, with half-life having been used with radioactivity before it found its place in DMPK, and more recently, bioavailability has become commonly used in relation to plant feeding and pesticides, such that these terms are now commonplace in conversational language.

1.3.1 Half-life

Half-life can be defined as the time that it takes for a defined concentration of drug to fall to half that value, analogous to the use of the term in describing the time course of decay of a radioactive substance. Radioactive decay is random, and the rate of decay is directly proportional to the number of unstable nuclei. This is exponential decay, a first-order rate process, and the rate of decay can be calculated from knowledge of the decay (rate) constant, ? (Equation 3.6). The half-life of a first-order reaction is constant and can be derived from any arbitrarily chosen concentration on the decay curve (Section 3.3). Because the elimination of most drugs is exponential when they are used at therapeutic doses, it makes perfect sense to describe their elimination kinetics in terms of half-lives.

A really useful feature of half-life is that, for many people, it is much easier to visualize what is happening in terms of the changes in drug concentrations, or amount of drug in the body, as a function of time. The simplest of models is that which looks solely at the elimination of drug after administration. However, declining concentrations from an instant high concentration cannot realistically occur, because the drug has to be put there in the first place, and even after intravenous bolus doses a finite time is required for the concentration to become homogeneous throughout the plasma. The practical way to reduce this potential error is to measure plasma concentrations as a function of time and to extrapolate to zero time to obtain the intercept on the concentration axis - this gives another very important variable, C0, which is referred to as the initial concentration even though it is impossible to measure it directly. The half-life can then be obtained from the time it takes for the concentration to fall to C0/2. This does not address potential problems of inhomogeneity of plasma, which can be an issue, particularly in behavioural studies (Section 5.4.4).

By definition, 50% of an administered intravenous bolus dose will be eliminated in the duration of one half-life, a further 25% in the next half-life, and so on, until <1% remains after seven half-lives have elapsed. As discussed later, half-life is a useful predictor of when the plasma concentrations of a drug reach their maximum during therapy with multiple doses.

A pedant may say that drugs do not 'have' half-lives, as this is not an inherent property of a drug, but arises as a result of several factors, including the chemical nature of the drug and the environment that it is in. However, under similar conditions, some drugs have longer half-lives than others, so that a drug may be chosen for clinical use or further development as drug, on the basis of its elimination half-life relative to those of other potential candidates.

1.3.1.1 Non-linear kinetics

There is nothing mystical about the fact that drugs exhibit first-order kinetics at therapeutic doses. It is reasonable to assume that the processes of elimination are not saturated by large amounts of drug, and that any enzymes and/or transporters that are involved are in such excess, relative to their substrates, that that their concentrations are not rate-limiting. Under these circumstances, the rate of elimination is determined purely by the amount/concentration of drug or xenobiotic. Even with newly developed drugs, those exhibiting first-order kinetics will have been chosen over any candidates that do not. However, it must be stressed that exponential decay is unique to first-order reactions (often referred to as linear kinetics, every other kind being non-linear) and so the concept of half-lives is of limited value with drugs such as phenytoin and ethanol that are not eliminated according to first-order kinetics at the doses that are usually administered. Furthermore, drugs that show linear kinetics at therapeutic doses often do not at higher doses when, for example, drug-metabolizing enzymes may become saturated with drug, such as in the case of drug overdose or when the drug metabolizing capacity is reduced because of disease or age.

It is important to ensure that the data show linear kinetics if first-order equations are to be used for modelling. It may not be sufficient to claim that a plot of ln(concentration) versus time can be fitted to a straight...