Irreversible inhibition occurs with organophosphorus insecticides and chemical warfare agents (see Chap. 10), which combine covalently with the active site of acetylcholinesterase; recovery of cholinesterase activity depends on the formation of new enzyme. Covalent binding of aspirin to cyclo-oxygenase (COX) inhibits the enzyme in platelets for their entire lifespan because platelets have no system for synthesising new protein; this is why low doses of aspirin are sufficient for antiplatelet action.

Selectivity

The pharmacologist who produces a new drug and the doctor who gives it to a patient share the desire that it should possess a selective action so that additional and unwanted (adverse) effects do not complicate the management of the patient. Approaches to obtaining selectivity of drug action include the following.

Modification of drug structure

Many drugs have in their design a structural similarity to some natural constituent of the body, e.g. a neurotransmitter, a hormone, a substrate for an enzyme; replacing or competing with that natural constituent achieves selectivity of action. Enormous scientific effort and expertise go into the synthesis and testing of analogues of natural substances in order to create drugs capable of obtaining a specified effect, and that alone (see Therapeutic index, below). The approach is the basis of modern drug design and it has led to the production of adrenoceptor antagonists, histamine receptor antagonists and many other important medicines.

But there are biological constraints to selectivity. Anticancer drugs that act against rapidly dividing cells lack selectivity because they also damage other tissues with a high cell replication rate, such as bone marrow and gut epithelium.

Selective delivery (drug targeting)

Simple topical application, e.g. skin and eye, and special drug delivery systems, e.g. intrabronchial administration of β₂-adrenoceptor agonist or corticosteroid (inhaled, pressurised, metered aerosol for asthma) can achieve the objective of target tissue selectivity. Selective targeting of drugs to less accessible sites of disease offers considerable scope for therapy as technology develops, e.g. attaching drugs to antibodies selective for cancer cells.

Stereoselectivity

Drug molecules are three-dimensional and many drugs contain one or more asymmetrical or chiral¹ centres in their structures, i.e. a single drug can be, in effect, a mixture of two non-identical mirror images (like a mixture of left- and right-handed gloves). The two forms, which are known as enantiomorphs, can exhibit very different pharmacodynamic, pharmacokinetic and toxicological properties.

For example, (1) the S form of warfarin is four times more active than the R form,² (2) the peak plasma concentration of S fenoprofen is four times that of R fenoprofen after oral administration of RS fenoprofen, and (3) the S, but not the R, enantiomorph of thalidomide is metabolised to primary toxins.

Many other drugs are available as mixtures of enantiomorphs (racemates). Pharmaceutical development of drugs as single enantiomers rather than as racemic mixtures offers the prospect of greater selectivity of action and lessens risk of toxicity.

Quantitative aspects

That a drug has a desired qualitative action is obviously all important, but is not by itself enough. There are also quantitative aspects, i.e. the right amount of action is required, and with some drugs the dose has to be adjusted very precisely to deliver this, neither too little nor too much, to escape both inefficacy and toxicity, e.g. digoxin, lithium, gentamicin. While the general correlation between dose and response may evoke no surprise, certain characteristics of the relation are fundamental to the way drugs are used, as described below.

Dose–response relationships

Conventionally, the horizontal axis shows the dose and the response appears on the vertical axis. The slope of the dose–response curve defines the extent to which a desired response alters as the dose is changed. A steeply rising and prolonged curve indicates that a small change in dose produces a large change in drug effect over a wide dose range, e.g. with the loop diuretic furosemide (used in doses from 20 mg to over 250 mg/day). By contrast, the dose–response curve for thiazide diuretics soon reaches a plateau, and the clinically useful dose range for bendroflumethiazide, for example, extends from 5 to 10 mg; increasing the dose beyond this produces no added diuretic effect, although it adds to toxicity.

Dose–response curves for wanted and unwanted effects can illustrate and quantify selective and non-selective drug action (see Fig. 8.1).

Fig. 8.1 Dose–response curves for two hypothetical drugs. For drug A, the dose that brings about the maximum wanted effect is less than the lowest dose that produces the unwanted effect. The ratio ED₅₀ (unwanted effect)/ED₅₀ (wanted effect) indicates that drug A has a large therapeutic index; it is thus highly selective in its wanted action. Drug B causes unwanted effects at doses well below producing its maximum benefit. The ratio ED₅₀ (unwanted effect)/ED₅₀ (wanted effect) indicates that the drug has a small therapeutic index: it is thus non-selective.

Potency and efficacy

A clear distinction between potency and efficacy is pertinent, particularly in relation to claims made for usefulness in therapeutics.

Potency

is the amount (weight) of drug in relation to its effect, e.g. if weight-for-weight drug A has a greater effect than drug B, then drug A is more potent than drug B, although the maximum therapeutic effect obtainable may be similar with both drugs.

The diuretic effect of bumetanide 1 mg is equivalent to that of furosemide 50 mg; thus bumetanide is more potent than furosemide but both drugs achieve about the same maximum effect. The difference in weight of drug administered is of no clinical significance unless it is great.

Pharmacological efficacy

refers to the strength of response induced by occupancy of a receptor by an agonist (intrinsic activity); it is a specialised pharmacological concept. But clinicians are concerned with therapeutic efficacy, as follows.

Therapeutic efficacy

or effectiveness, is the capacity of a drug to produce an effect and refers to the maximum such effect. For example, if drug A can produce a therapeutic effect that cannot be obtained with drug B, however much of drug B is given, then drug A has the higher therapeutic efficacy. Differences in therapeutic efficacy are of great clinical importance, usually more than potency.

Amiloride (low efficacy) can at best effect excretion of no more than 5% of the sodium load filtered by the glomeruli; there is no point in increasing the dose beyond that which achieves this, as this is its maximum diuretic effect. Bendroflumethiazide (moderate efficacy) can effect excretion of no more than 10% of the filtered sodium load no matter how large the dose. Furosemide can effect excretion of 25% and more of filtered sodium; it is a high-efficacy diuretic.

Therapeutic index

With progressive increases in dose, the desired response in the patient usually rises to a maximum beyond which further increases elicit no greater benefit but induce unwanted effects. This is because most drugs do not have a single dose–response curve, but a different curve for each action, wanted as well as unwanted. Increases in dose beyond that which gives the maximum wanted response recruit only new and unwanted actions.

A sympathomimetic bronchodilator might exhibit one dose–response relation for decreasing airway resistance (wanted) and another for increase in heart rate (unwanted). Clearly, the usefulness of any drug relates closely to the extent to which such dose–response relations overlap.

Ehrlich (see p. 162) introduced the concept of the therapeutic index or ratio as the maximum tolerated dose divided by the minimum curative dose, but the index is never calculated thus as such single doses cannot be determined accurately in humans. More realistically, a dose that has some unwanted effect in 50% of humans, e.g. in the case of an adrenoceptor agonist bronchodilator a specified increase in heart rate, is compared with that which is therapeutic in 50% (ED₅₀), e.g. a specified decrease in airways resistance.

In practice, such information is not available for many drugs but the therapeutic index does embody a concept that is fundamental in comparing the usefulness of one drug with another, namely, safety in relation to efficacy. Figure 8.1 expresses the concept diagrammatically.

Tolerance

Continuous or repeated administration of a drug is often accompanied by a gradual diminution of the effect it produces. A state of tolerance exists when it becomes necessary to increase the dose of a drug to get an effect previously obtained with a smaller dose, i.e. reduced sensitivity. By contrast, the term tachyphylaxis describes the phenomenon of progressive lessening of effect (refractoriness) in response to frequently administered doses (see Receptors, p. 75); it tends to develop more rapidly than tolerance.

The use of opioids readily illustrates tolerance, as witnessed by the huge doses of morphine that may be necessary to maintain pain relief in terminal care; the effect is due to reduced pharmacological efficacy (see above) at receptor sites or to down-regulation of receptors. Tolerance is acquired rapidly with nitrates used to prevent angina, possibly mediated by the generation of oxygen free radicals from nitric oxide; it can be avoided by removing transdermal nitrate patches for 4–8 h, e.g. at night, to allow the plasma concentration to fall.

Accelerated metabolism by enzyme induction (see p. 93) also leads to tolerance, as experience shows with alcohol, taken regularly as opposed to sporadically. There is commonly cross-tolerance between drugs of similar structure.

Failure of certain individuals to respond to normal doses of a drug, e.g. resistance to warfarin, vitamin D, constitutes a form of natural tolerance (see Pharmacogenetics, p. 101).

Bioassay and standardisation

Biological assay (bioassay) is the process by which the activity of a substance (identified or unidentified) is measured on living material: e.g. contraction of bronchial, uterine or vascular muscle. It is used only when chemical or physical methods are not practicable as in the case of a mixture of active substances, or of an incompletely purified preparation, or where no chemical method has been developed. The activity of a preparation is expressed relative to that of a standard preparation of the same substance.

Biological standardisation is a specialised form of bioassay. It involves matching of material of unknown potency with an international or national standard with the objective of providing a preparation for use in therapeutics and research. The results are expressed as units of a substance rather than its weight, e.g. insulin, vaccines.

Pharmacokinetics

To initiate a desired drug action is a qualitative choice but, when the qualitative choice is made, considerations of quantity immediately arise; it is possible to have too much or too little of a good thing. To obtain the right effect at the right intensity, at the right time, for the right duration, with minimal risk of unpleasantness or harm, is what pharmacokinetics is about.

Dosage regimens of long-established drugs grew from trial and error. Doctors learned by experience the dose, the frequency of dosing and the route of administration that was most likely to benefit and least likely to harm. But this empirical (‘suck it and see’) approach is no longer tenable. We now have an understanding of how drugs cross membranes to enter the body, how they are distributed round it in the blood and other body fluids, how they are bound to plasma proteins and tissues (which act as stores), and how they are eliminated from the body. Quantification of these processes paves the way for efficient development of dosing regimens.

Pharmacokinetics³ is concerned with the rate at which drug molecules cross cell membranes to enter the body, to distribute within it and to leave the body, as well as with the structural changes (metabolism) to which they are subject within it.

The discussion covers the following topics:

• Drug passage across cell membranes.

• Order of reaction or process (first and zero order).

• Time course of drug concentration and effect:

plasma half-life and steady-state concentration

therapeutic monitoring.

• The individual processes: absorption, distribution, metabolism (biotransformation), elimination.

Drug passage across cell membranes

Certain concepts are fundamental to understanding how drug molecules make their way around the body to achieve their effect. The first concerns the modes by which drugs cross cell membranes and cells.

Our bodies are labyrinths of fluid-filled spaces. Some, such as the lumina of the kidney tubules or intestine, connect to the outside world; the blood, lymph and cerebrospinal fluid are enclosed. Sheets of cells line these spaces, and the extent to which a drug can cross epithelia or endothelia is fundamental to its clinical use, determining whether a drug can be taken orally for systemic effect, and whether within the glomerular filtrate it will be reabsorbed or excreted in the urine.

Cell membranes are essentially bilayers of lipid molecules with ‘islands’ of protein, and they preserve and regulate the internal environment. Lipid-soluble substances diffuse readily into cells and therefore throughout body tissues. Adjacent epithelial or endothelial cells are linked by tight junctions, some of which are traversed by water-filled channels that allow the passage of water-soluble substances of small molecular size.

The jejunum and proximal renal tubule contain many such channels and are leaky epithelia, whereas the tight junctions in the stomach and urinary bladder do not have these channels and water cannot pass; they are termed tight epithelia. Special protein molecules within the lipid bilayer allow specific substances to enter or leave the cell preferentially, i.e. energy-utilising transporter processes, described later. The natural processes of passive diffusion, filtration and carrier-mediated transport determine the passage of drugs across membranes and cells, and their distribution round the body.

Passive diffusion

This is the most important means by which a drug enters the tissues and distributes through them. It refers simply to the natural tendency of any substance to move passively from an area of high concentration to one of low concentration. In the context of an individual cell, the drug moves at a rate proportional to the concentration difference across the cell membrane, i.e. it shows first-order kinetics (see p. 81); cellular energy is not required, which means that the process does not become saturated and is not inhibited by other substances.

The extent to which drugs are soluble in water or lipid is central to their capacity to cross cell membranes and depends on environmental pH and the structural properties of the molecule.

Lipid solubility is promoted by the presence of a benzene ring, a hydrocarbon chain, a steroid nucleus or halogen (-Br, -Cl, -F) groups. Water solubility is promoted by the presence of alcoholic (-OH), amide (-CO·NH₂) or carboxylic (-COOH) groups, or the formation of glucuronide and sulphate conjugates.

It is useful to classify drugs in a physicochemical sense into:

• Those that are variably ionised according to environmental pH (electrolytes) (lipid soluble or water soluble).

• Those that are incapable of becoming ionised whatever the environmental pH (un-ionised, non-polar substances) (lipid soluble).

• Those that are permanently ionised whatever the environmental pH (ionised, polar substances) (water soluble).

Drugs that ionise according to environmental pH

Many drugs are weak electrolytes, i.e. their structural groups ionise to a greater or lesser extent, according to environmental pH. Most such molecules are present partly in the ionised and partly in the un-ionised state. The degree of ionisation influences lipid solubility (and hence diffusibility) and so affects absorption, distribution and elimination.

Ionisable groups in a drug molecule tend either to lose a hydrogen ion (acidic groups) or to add a hydrogen ion (basic groups). The extent to which a molecule has this tendency to ionise is given by the dissociation (or ionisation) constant (K_a), expressed as the pK_a, i.e. the negative logarithm of the K_a (just as pH is the negative logarithm of the hydrogen ion concentration). In an acidic environment, i.e. one already containing many free hydrogen ions, an acidic group tends to retain a hydrogen ion and remains un-ionised; a relative deficit of free hydrogen ions, i.e. a basic environment, favours loss of the hydrogen ion from an acidic group, which thus becomes ionised. The opposite is the case for a base. The issue may be summarised:

• Acidic groups become less ionised in an acidic environment.

• Basic groups become less ionised in a basic (alkaline) environment and vice versa.

This in turn influences diffusibility because:

• Un-ionised drug is lipid soluble and diffusible.

• Ionised drug is lipid insoluble and non-diffusible.

Quantifying the degree of ionisation helps to express the profound effect of environmental pH. Recall that when the pH of the environment is the same as the pK_a of a drug within it, then the ratio of un-ionised to ionised molecules is 1:1. But for every unit by which pH is changed, the ratio of un-ionised to ionised molecules changes 10-fold. Thus, when the pH is 2 units less than the pK_a, molecules of an acid become 100 times more un-ionised and when the pH is 2 units more than the pK_a, molecules of an acid become 100 more ionised. Such pH change profoundly affects drug kinetics.

pH variation and drug kinetics

The pH partition hypothesis expresses the separation of a drug across a lipid membrane according to differences in environmental pH. There is a wide range of pH in the gut (pH 1.5 in the stomach, 6.8 in the upper and 7.6 in the lower intestine). But the pH inside the body is maintained within a limited range (pH 7.46 ± 0.04), so that only drugs that are substantially un-ionised at this pH will be lipid soluble, diffuse across tissue boundaries and so be widely distributed, e.g. into the central nervous system (CNS). Urine pH varies between the extremes of 4.6 and 8.2, and the prevailing pH affects the amount of drug reabsorbed from the renal tubular lumen by passive diffusion.

In the stomach, aspirin (acetylsalicylic acid, pK_a 3.5) is un-ionised and thus lipid soluble and diffusible. When aspirin enters the gastric epithelial cells (pH 7.4) it will ionise, become less diffusible and so will localise there. This ion trapping is one mechanism whereby aspirin is concentrated in, and so harms, the gastric mucosa. In the body aspirin is metabolised to salicylic acid (pK_a 3.0), which at pH 7.4 is highly ionised and thus remains in the extracellular fluid. Eventually the molecules of salicylic acid in the plasma are filtered by the glomeruli and pass into the tubular fluid, which is generally more acidic than plasma and causes a proportion of salicylic acid to become un-ionised and lipid soluble so that it diffuses back into the tubular cells. Alkalinising the urine with an intravenous infusion of sodium bicarbonate causes more salicylic acid to become ionised and lipid insoluble so that it remains in the tubular fluid, and passes into the urine. Treatment for salicylate (aspirin) overdose utilises this effect.

Conversely, acidifying the urine increases the elimination of the base amfetamine (pK_a 9.9) (see Acidification of urine, p. 126).

Drugs that are incapable of becoming ionised

These include digoxin and steroid hormones such as prednisolone. Effectively lacking any ionisable groups, they are unaffected by environmental pH, are lipid soluble and so diffuse readily across tissue boundaries. These drugs are also referred to as non-polar.

Permanently ionised drugs

Drugs that are permanently ionised contain groups that dissociate so strongly that they remain ionised over the range of the body pH. Such compounds are termed polar, for their groups are either negatively charged (acidic, e.g. heparin) or positively charged (basic, e.g. ipratropium, tubocurarine, suxamethonium) and all have a very limited capacity to cross cell membranes. This is a disadvantage with heparin, which the gut does not absorb, so that it is given parenterally. Conversely, heparin is a useful anticoagulant in pregnancy because it does not cross the placenta (which the orally effective warfarin does and is liable to cause fetal haemorrhage as well as being teratogenic).

The following are particular examples of the relevance of drug passage across membranes.

Brain and cerebrospinal fluid (CSF)

The capillaries of the cerebral circulation differ from those in most other parts of the body in that they lack the filtration channels between endothelial cells through which substances in the blood normally gain access to the extracellular fluid. Tight junctions between adjacent capillary endothelial cells, together with their basement membrane and a thin covering from the processes of astrocytes, separate the blood from the brain tissue, forming the blood–brain barrier. Compounds that are lipid insoluble do not cross it readily, e.g. atenolol, compared with propranolol (lipid soluble), and unwanted CNS effects are more prominent with the latter. Therapy with methotrexate (lipid insoluble) may fail to eliminate leukaemic deposits in the CNS.

Conversely lipid-soluble substances enter brain tissue with ease; thus diazepam (lipid soluble) given intravenously is effective within 1 min for status epilepticus, and effects of alcohol (ethanol) by mouth are noted within minutes; the level of general anaesthesia can be controlled closely by altering the concentration of inhaled anaesthetic gas (lipid soluble).

Placenta

Maternal blood bathes the chorionic villi, which consist of a layer of trophoblastic cells that enclose fetal capillaries. Their large surface area and the high placental blood flow (500 mL/min) are essential for gas exchange, uptake of nutrients and elimination of waste products. Thus a lipid barrier separates the fetal and maternal bloodstreams, allowing the passage of lipid-soluble substances but excluding water-soluble compounds, especially those with a molecular weight exceeding 600.⁴

This exclusion is of particular importance with short-term use, e.g. tubocurarine (mol. wt. 772) (lipid insoluble) or gallamine (mol. wt. 891) used as a muscle relaxant during caesarean section do not affect the infant; with prolonged use, however, all compounds will eventually enter the fetus to some extent (see Index).

Filtration

Aqueous channels in the tight junctions between adjacent epithelial cells allow the passage of some water-soluble substances. Neutral or uncharged, i.e. non-polar, molecules pass most readily because the pores are electrically charged. Within the alimentary tract, channels are largest and most numerous in jejunal epithelium, and filtration allows for rapid equilibration of concentrations and consequently of osmotic pressures across the mucosa. Ions such as sodium enter the body through the aqueous channels, the size of which probably limits passage to substances of low molecular weight, e.g. ethanol (mol. wt. 46). Filtration seems to play at most a minor role in drug transfer within the body except for glomerular filtration, which is an important mechanism of drug excretion.

Carrier-mediated transport

The membranes of many cells incorporate carrier-mediated transporter processes that control the entry and exit of endogenous molecules, and show a high degree of specificity for particular compounds because they have evolved from biological needs for the uptake of essential nutrients or elimination of metabolic products. Drugs that bear some structural resemblance to natural constituents of the body are likely to utilise these mechanisms.

Some carrier-mediated transport processes operate passively, i.e. do not require cellular energy, and this is facilitated diffusion, e.g. vitamin B₁₂ absorption. Other, energy-requiring processes move substrates into or out of cells against a concentration gradient very effectively, i.e. by active transport; they are subject to saturation, inhibition and induction (see p. 91).

The order of reaction or process

In the body, drug molecules reach their sites of action after crossing cell membranes and cells, and many are metabolised in the process. The rate at which these movements or changes take place is subject to important influences called the order of reaction or process. In biology generally, two orders of such reactions are recognised, and are summarised as follows:

• First-order processes by which a constant fraction of drug is transported/metabolised in unit time.

• Zero-order processes by which a constant amount of drug is transported/metabolised in unit time.

First-order (exponential) processes

In the majority of instances, the rates at which absorption, distribution, metabolism and excretion of a drug occur are directly proportional to its concentration in the body. In other words, transfer of drug across a cell membrane or formation of a metabolite is high at high concentrations and falls in direct proportion to be low at low concentrations (an exponential relationship).

This is because the processes follow the Law of Mass Action, which states that the rate of reaction is directly proportional to the active filtration masses of reacting substances. In other words, at high concentrations there are more opportunities for crowded molecules to interact with one another or to cross cell membranes than at low, uncrowded concentrations. Processes for which the rate of reaction is proportional to the concentration of participating molecules are first-order processes.

In doses used clinically, most drugs are subject to first-order processes of absorption, distribution, metabolism and elimination, and this knowledge is useful. The current chapter later describes how the rate of elimination of a drug from the plasma falls as the concentration in plasma falls, and the time for any plasma concentration to fall by 50% (t_½, the plasma half-life) is always the same. Thus, it becomes possible to quote a constant value for the t_½ of the drug. This occurs because rate and concentration are in proportion, i.e. the process obeys first-order kinetics.

Knowing that first-order conditions apply to a drug allows accurate calculations that depend on its t_½, i.e. time to achieve steady-state plasma concentration, time to elimination, and the construction of dosing schedules.

Zero-order processes (saturation kinetics)

As the amount of drug in the body rises, metabolic reactions or processes that have limited capacity become saturated. In other words, the rate of the process reaches a maximum amount at which it stays constant, e.g. due to limited activity of an enzyme, and any further increase in rate is impossible despite an increase in the dose of drug. In these circumstances, the rate of reaction is no longer proportional to dose, and exhibits rate-limited or dose-dependent⁵ or zero-order or saturation kinetics. In practice, enzyme-mediated metabolic reactions are the most likely to show rate limitation because the amount of enzyme present is finite and can become saturated. Passive diffusion does not become saturated. There are some important consequences of zero-order kinetics.

Alcohol

Alcohol is subject to first-order kinetics with a t_½ of about 1 h at plasma concentrations below 10 mg/dL (attained after drinking about two-thirds of a unit (glass) of wine or beer). Above this concentration the main enzyme (alcohol dehydrogenase) that converts the alcohol into acetaldehyde approaches and then reaches saturation, at which point alcohol metabolism cannot proceed any faster than about 10 mL or 8 g/h for a 70-kg man. If the subject continues to drink, the blood alcohol concentration rises disproportionately, for the rate of metabolism remains the same, as alcohol shows zero-order kinetics.

An illustration. Consider a man of average size who drinks about half (375 mL) a standard bottle of whisky (40% alcohol), i.e. 150 mL alcohol, over a short period, absorbs it and goes drunk to bed at midnight with a blood alcohol concentration of about 250 mg/dL. If alcohol metabolism were subject to first-order kinetics, with a t_½ of 1 h throughout the whole range of social consumption, the subject would halve his blood alcohol concentration each hour (see Fig. 8.2). It is easy to calculate that, when he drives his car to work at 08.00 hours the next morning, he has a negligible blood alcohol concentration (less than 1 mg/dL) though, no doubt, a hangover might reduce his driving skill.

Fig. 8.2 Changes in plasma concentration following an intravenous bolus injection of a drug in the elimination phase (the distribution phase, see text, is not shown). As elimination is a first-order process, the time for any concentration point to fall by 50% (t_½) is always the same.

But at these high concentrations, alcohol is in fact subject to zero-order kinetics and so, metabolising about 10 mL alcohol per hour, after 8 h the subject has eliminated only 80 mL, leaving 70 mL in his body and giving a blood concentration of about 120 mg/dL. At this level, his driving skill is seriously impaired. The subject has an accident on his way to work and is breathalysed despite his indignant protests that he last touched a drop before midnight. Banned from the road, on his train journey to work he will have leisure to reflect on the difference between first-order and zero-order kinetics (though this is unlikely!).

In practice. The example above describes an imagined event but similar cases occur in everyday therapeutics. Phenytoin, at low dose, exhibits a first-order elimination process and there is a directly proportional increase in the steady-state plasma concentration with increase in dose. But gradually the enzymatic elimination process approaches and reaches saturation, the process becoming constant and zero order. While the dosing rate can be increased, the metabolism rate cannot, and the plasma concentration rises steeply and disproportionately, with danger of toxicity. Salicylate metabolism also exhibits saturation kinetics but at high therapeutic doses. Clearly saturation kinetics is a significant factor in delay of recovery from drug overdose, e.g. with aspirin or phenytoin.

Order of reaction and t_½. When a drug is subject to first-order kinetics, the t_½ is a constant characteristic, i.e. a constant value can be quoted throughout the plasma concentration range (accepting that there will be variation in t_½ between individuals), and this is convenient. But if the rate of a process is not directly proportional to plasma concentration, then the t_½ cannot be constant. Consequently, no single value for t_½ describes overall elimination when a drug exhibits zero-order kinetics. In fact, t_½ decreases as plasma concentration falls and the calculations on elimination and dosing that are so easy with first-order elimination (see below) become more complicated.

Zero-order absorption processes apply to iron, to depot intramuscular formulations and to drug implants, e.g. antipsychotics and sex hormones.

Time course of drug concentration and effect

Plasma half-life and steady-state concentration

The manner in which plasma drug concentration rises or falls when dosing begins, alters or ceases follows certain simple rules, which provide a means for rational control of drug effect. Central to understanding these is the concept of half-life (t_½) or half-time.

Decrease in plasma concentration after an intravenous bolus injection

Following an intravenous bolus injection (a single dose injected in a period of seconds as distinct from a continuous infusion), plasma concentration rises quickly as drug enters the blood to reach a peak. There is then a sharp drop as the drug distributes round the body (distribution phase), followed by a steady decline as drug is removed from the blood by the liver or kidneys (elimination phase). If the elimination processes are first order, the time taken for any concentration point in the elimination phase to fall to half its value (the t_½) is always the same; see Figure 8.2. Note that the drug is virtually eliminated from the plasma in five t_½ periods.

The t_½ is the one pharmacokinetic value of a drug that it is most useful to know.

Increase in plasma concentration with constant dosing

With a constant rate infusion, the amount of drug in the body and with it the plasma concentration rise until a state is reached at which the rate of administration to the body is exactly equal to the rate of elimination from it: this is called the steady state. The plasma concentration is then on a plateau, and the drug effect is stable. Figure 8.3 depicts the smooth changes in plasma concentration that result from a constant intravenous infusion. Clearly, giving a drug by regularly spaced oral or intravenous doses will result in plasma concentrations that fluctuate between peaks and troughs, but in time all of the peaks will be of equal height and all of the troughs will be of equal depth; this is also called a steady-state concentration, as the mean concentration is constant.⁶

Fig. 8.3 Changes in plasma concentration during the course of a constant-rate intravenous infusion. (a) The infusion commences and plasma concentration rises to reach a steady state (plateau) in about 5 × t_½ periods. (b) The infusion rate is increased by 50% and the plasma concentration rises further to reach a new steady state that is 50% higher than the original steady state; the process takes another 5 × t_½ periods. (c) The infusion is decreased to the original rate and the plasma concentration returns to the original steady state in 5 × t_½ periods. (d) The infusion is discontinued and the plasma concentration falls to virtually zero in 5 × t_½ periods.

Time to reach steady state

It is important to know when a drug administered at a constant rate achieves a steady-state plasma concentration, for maintaining the same dosing schedule then ensures a constant amount of drug in the body and the patient will experience neither acute toxicity nor decline of effect. The t_½ provides the answer. Taking ultimate steady state attained as 100%:

in 1 × t_½ the concentration will be (100/2) 50%

in 2 × t_½ (50 + 50/2) 75%

in 3 × t_½ (75 + 25/2) 87.5%

in 4 × t_½ (87.5 + 2.5/2) 93.75%

in 5 × t_½ (93.75 + 6.25/2) 96.875%

of the ultimate steady state.

When a drug is given at a constant rate (continuous or repeated administration), the time to reach steady state depends only on the t_½ and, for all practical purposes, after 5 × t_½ periods the amount of drug in the body is constant and the plasma concentration is at a plateau (a–b in Fig. 8.3).

Change in plasma concentration with change or cessation of dosing

The same principle holds for change from any steady-state plasma concentration to a new steady state brought about by increase or decrease in the rate of drug administration. Provided the kinetics remain first order, increasing or decreasing the rate of drug administration (b and c in Fig. 8.3) gives rise to a new steady-state concentration in a time equal to 5 × t_½ periods.

Similarly, starting at any steady-state plasma concentration (100%), discontinuing the dose (d in Fig. 8.3) will cause the plasma concentration to fall to virtually zero in 5 × t_½ periods, as described in Figure 8.2.

Note that the difference between the rate of drug administration (input) and the rate of elimination (output) determines the actual level of any steady-state plasma concentration (as opposed to the time taken to reach it). If drug elimination remains constant and administration increases by 50%, in time the plasma concentration will reach a new steady-state concentration, which will be 50% greater than the original.

The relation between t_½ and time to reach steady-state plasma concentration applies to all drugs that obey first-order kinetics. This holds as much to dobutamine (t_½ 2 min), when it is useful to know that an alteration of infusion rate will reach a plateau within 10 min, as to digoxin (t_½ 36 h), when a constant daily oral dose will give a steady-state plasma concentration only after 7.5 days. This book quotes plasma t_½ values where they are relevant. Inevitably, natural variation within the population produces a range in t_½ values for any drug and the text quotes only single average t_½ values while recognising that the population range may be as much as 50% from the stated figure in either direction.

Some t_½ values appear in Table 8.1 to illustrate their range and implications for dosing in clinical practice.

Table 8.1 Plasma t_½ of some drugs

Drug	t_½
Adenosine	< 2 s
Dobutamine	2 min
Benzylpenicillin	30 min
Amoxicillin	1 h
Paracetamol	2 h
Midazolam	3 h
Tolbutamide	6 h
Atenolol	7 h
Dosulepin	25 h
Diazepam	40 h
Piroxicam	45 h
Ethosuximide	54 h