Clinical pharmacokinetics is usually associated with therapeutic drug monitoring (TDM), and its subsequent utilisation. When TDM is used appropriately, it has been demonstrated that patients suffer fewer side effects than those who are not monitored (Reid et al., 1990). Although TDM is a proxy outcome measure, a study with aminoglycosides (Crist et al., 1987) demonstrated shorter hospital stays for patients where TDM was used. Furthermore, a study on the use of anticonvulsants (McFadyen et al., 1990) showed better epilepsy control in those patients where TDM was used. A literature review of the cost-effectiveness of TDM concluded that emphasis on just cost is inappropriate and clinical relevance should be sought (Touw et al., 2007). There are various levels of sophistication for the application of pharmacokinetics to TDM. Knowledge of the distribution time and an understanding of the concept of steady state can facilitate determination of appropriate sampling times.

For most drugs that undergo first-order elimination, a linear relationship exists between dose and concentration, which can be used for dose adjustment purposes. However, if the clearance of the drug changes as the concentration changes (e.g. phenytoin), then an understanding of the drug’s pharmacokinetics will assist in making correct dose adjustments.

More sophisticated application of pharmacokinetics involves the use of population pharmacokinetic data to produce initial dosage guidelines, for example nomograms for digoxin and gentamicin, and to predict drug levels. Pharmacokinetics can also assist in complex dosage individualisation using actual patient specific drug level data.

Given the wide range of clinical situations in which pharmacokinetics can be applied, pharmacists must have a good understanding of the subject and of how to apply it to maximise their contribution to patient care.

Basic concepts

Volume of distribution

The apparent volume of distribution (V_d) may be defined as the size of a compartment which will account for the total amount of drug in the body (A) if it were present in the same concentration as in plasma. This means that it is the apparent volume of fluid in the body which results in the measured concentration of drug in plasma (C) for a known amount of drug given, that is:

This relationship assumes that the drug is evenly distributed throughout the body in the same concentration as in the plasma. However, this is not the case in practice, since many drugs are present in different concentrations in various parts of the body. Thus, some drugs which concentrate in muscle tissue have a very large apparent volume of distribution, for example digoxin. This concept is better explained in Fig. 3.2.

Fig. 3.2 Distribution: more of drug A is distributed in the tissue compartment resulting in a higher apparent volume of distribution than drug B, where more remains in the plasma.

The apparent volume of distribution may be used to determine the plasma concentration after an intravenous loading dose:

(1)

Conversely, if the desired concentration is known, the loading dose may be determined:

(2)

In the previous discussion, it has been assumed that after a given dose a drug is instantaneously distributed between the various tissues and plasma. In practice this is seldom the case. For practical purposes it is reasonable to generalise by referring to plasma as one compartment and tissue as if it were another single separate compartment. However, in reality there will be many tissue subcompartments. Thus, in pharmacokinetic terms the body may be described as if it were divided into two compartments: the plasma and the tissues.

Figure 3.3 depicts the disposition of a drug immediately after administration and relates this to the plasma concentration–time graph.

Fig. 3.3 (A) Two-compartment model showing two phases in the plasma concentration–time profile. (B) Representation of a two-compartment model showing distribution of drug between plasma and tissue compartments.

Initially, the plasma concentration falls rapidly, due to distribution and elimination (α phase). However, when an equilibrium is reached between the plasma and tissue (i.e. the distribution is complete), the change in plasma concentration is only due to elimination from the plasma (β phase), and the plasma concentration falls at a slower rate. The drug is said to follow a two-compartment model. However, if distribution is completed quickly (within minutes), then the α phase is not seen, and the drug is said to follow a one-compartment model.

The practical implications of a two-compartment model are that any sampling for monitoring purposes should be carried out after distribution is complete. In addition, intravenous bolus doses are given slowly to avoid transient side effects caused by high peak concentrations.

Elimination

Drugs may be eliminated from the body by a number of routes. The primary routes are excretion of the unchanged drug in the kidneys, or metabolism (usually in the liver) into a more water soluble compound for subsequent excretion in the kidneys, or a combination of both.

The main pharmacokinetic parameter describing elimination is clearance (CL). This is defined as the volume of plasma completely emptied of drug per unit time. For example, if the concentration of a drug in a patient is 1 g/L and the clearance is 1 L/h, then the rate of elimination will be 1 g/h. Thus, a relationship exists:

(3)

Total body elimination is the sum of the metabolic rate of elimination and the renal rate of elimination. Therefore:

Thus, if the fraction eliminated by the renal route is known (f_e), then the effect of renal impairment on total body clearance can be estimated.

The clearance of most drugs remains constant for each individual. However, it may alter in cases of drug interactions, changing end-organ function or autoinduction. Therefore, it is clear from equation (Eq.) (3) that as the plasma concentration changes so will the rate of elimination. However, when the rate of administration is equal to the rate of elimination, the plasma concentration is constant (C^ss) and the drug is said to be at a steady state.

At steady state:

At the beginning of a dosage regimen the plasma concentration is low. Therefore, the rate of elimination from Eq. (3) is less than the rate of administration, and accumulation occurs until a steady state is reached (see Fig. 3.1).

(4)

It is clear from Eq. (3) that as the plasma concentration falls (e.g. on stopping treatment or after a single dose), the rate of elimination also falls. Therefore, the plasma concentration–time graph follows a non-linear curve characteristic of this type of first-order elimination (Fig. 3.4). This is profoundly different from a constant rate of elimination irrespective of plasma concentration, which is typical of zero-order elimination.

Fig. 3.4 First-order elimination.

For drugs undergoing first-order elimination, there are two other useful pharmacokinetic parameters in addition to the volume of distribution and clearance. These are the elimination rate constant and elimination half-life.

The elimination rate constant (k_e) is the fraction of the amount of drug in the body (A) eliminated per unit time. For example, if the body contains 100 mg of a drug and 10% is eliminated per unit time, then k_e = 0.1. In the first unit of time, 0.1 × 100 mg or 10 mg is eliminated, leaving 90 mg. In the second unit of time, 0.1 × 90 mg or 9 mg is eliminated, leaving 81 mg. Elimination continues in this manner. Therefore:

(5)

Combining Eqs. (3) and (5) gives

and since

then

Therefore,

(6)

Elimination half-life (t_1/2) is the time it takes for the plasma concentration to decay by half. In five half-lives the plasma concentration will fall to approximately zero (see Fig. 3.4).

The equation which is described in Fig. 3.4 is

(7)

where C₁ and C₂ are plasma concentrations and t is time.

If half-life is substituted for time in Eq. (7), C₂ must be half of C₁.

Therefore,

(8)

There are two ways of determining k_e, either by estimating the half-life and applying Eq. (8) or by substituting two plasma concentrations in Eq. (7) and applying natural logarithms:

In the same way as it takes approximately five half-lives for the plasma concentration to decay to zero after a single dose, it takes approximately five half-lives for a drug to accumulate to the steady state on repeated dosing or during constant infusion (see Fig. 3.1).

This graph may be described by the equation

(9)

where C is the plasma concentration at time t after the start of the infusion and C^ss is the steady state plasma concentration. Thus (if the appropriate pharmacokinetic parameters are known), it is possible to estimate the plasma concentration any time after a single dose or the start of a dosage regimen.

Absorption

In the preceding sections, the intravenous route has been discussed, and with this route all of the administered drug is absorbed. However, if a drug is administered by any other route it must be absorbed into the bloodstream. This process may or may not be 100% efficient.

The fraction of the administered dose which is absorbed into the bloodstream is the bioavailability (F). Therefore, when applying pharmacokinetics for oral administration, the dose or rate of administration must be multiplied by F. Bioavailability F is determined by calculating the area under the concentration time curve (AUC). The rationale for this is described below.

The rate of elimination of a drug after a single dose is given by Eq. (3). By definition the rate of elimination is amount of drug eliminated per unit time.

The amount eliminated in any one unit of time

As previously explained, CL is constant. Therefore,

from start until zero is actually the area under the plasma concentration–time curve (Fig. 3.5).

Fig. 3.5 The area under the concentration time curve (AUC) is the sum of C × dt.

After a single i.v. dose the total amount eliminated is equal to the amount administered D i.v.

Therefore, or

However, for an oral dose the amount administered is

As CL is constant in the same individual,

Rearranging gives

In this way, F can be calculated from plasma concentration–time curves.

Dosing regimens

From the preceding sections, it is possible to derive equations which can be applied in clinical practice.

From Eq. (1) we can determine the change in plasma concentration ΔC immediately after a single dose:

(10)

where F is bioavailability and S is the salt factor, which is the fraction of active drug when the dose is administered as a salt (e.g. aminophylline is 80% theophylline, therefore S = 0.8).

Conversely, to determine a loading dose:

(11)

At the steady state it is possible to determine maintenance dose or steady state plasma concentrations from a modified Eq. (4):

(12)

where T is the dosing interval.

Peak and trough levels

For oral dosing and constant intravenous infusions, it is usually adequate to use the term ‘average steady state plasma concentration’ (average C^ss). However, for some intravenous bolus injections it is sometimes necessary to determine peak and trough levels (e.g. gentamicin).

At the steady state, the change in concentration due to the administration of an intravenous dose will be equal to the change in concentration due to elimination over one dose interval:

Within one dosing interval the maximum plasma concentration (C^ss_max) will decay to the minimum plasma concentration (C^ss_min) as in any first-order process.

Substituting C^ss_max for C₁ and C^ss_min for C₂ in Eq. (7):

where t is the dosing interval.

If this is substituted into the preceding equation:

Therefore,

(13)

(14)

Interpretation of drug concentration data

The availability of the technology to measure the concentration of a drug in plasma should not be the reason for monitoring. There are a number of criteria that should be fulfilled before therapeutic drug monitoring is undertaken. These are:

• the drug should have a low therapeutic index;

• there should be a good concentration–response relationship;

• there are no easily measurable physiological parameters.

In the absence of these criteria being fulfilled, the only other justification for undertaking TDM is to monitor adherence or to confirm toxicity. When interpreting TDM data, a number of factors need to be considered.

Sampling times

In the preceding sections, the time to reach the steady state has been discussed. When TDM is carried out as an aid to dose adjustment, the concentration should be at steady state. Therefore, approximately five half-lives should elapse after initiation or after changing a maintenance regimen, before sampling. The only exception to this rule is when toxicity is suspected. When the steady state has been reached, it is important to sample at the correct time. It is clear from the discussion above that this should be done when distribution is complete (see Fig. 3.3).

Dosage adjustment

Under most circumstances, provided the preceding criteria are observed, adjusting the dose of a drug is relatively simple, since a linear relationship exists between the dose and concentration if a drug follows first-order elimination (Fig. 3.6A). This is the case for most drugs.

Fig. 3.6 Dose–concentration relationships: (A) first-order elimination, (B) capacity-limited clearance, (C) increasing clearance.

Capacity limited clearance

If a drug is eliminated by the liver, it is possible for the metabolic pathway to become saturated, since it is an enzymatic system. Initially the elimination is first-order, but once saturation of the system occurs, elimination becomes zero-order. This results in the characteristic dose–concentration graph seen in Fig. 3.6B. For the majority of drugs eliminated by the liver, this effect is not seen at normal therapeutic doses and only occurs at very high supratherapeutic levels, which is why the kinetics of some drugs in overdose is different from normal. However, one important exception is phenytoin, where saturation of the enzymatic pathway occurs at therapeutic doses. This will be dealt with in the section on phenytoin.

Increasing clearance

The only other situation where first-order elimination is not seen is where clearance increases as the plasma concentration increases (Fig. 3.6C). Under normal circumstances, the plasma protein binding sites available to a drug far outnumber the capacity of the drug to fill those binding sites, and the proportion of the total concentration of drug which is protein bound is constant. However, this situation is not seen in one or two instances (e.g. valproate and disopyramide). For these particular drugs, as the concentration increases the plasma protein binding sites become saturated and the ratio of unbound drug to bound drug increases. The elimination of these drugs increases disproportionate to the total concentration, since elimination is dependent on the unbound concentration.

Therapeutic range

Wherever TDM is carried out, a therapeutic range is usually used as a guide to the optimum concentration. The limits of these ranges should not be taken as absolute. Some patients may respond to levels above or below these ranges, whereas others may experience toxic effects within the so-called therapeutic range. These ranges are only adjuncts to dose determination, which should always be done in the light of clinical response.

Clinical applications

Estimation of creatinine clearance

Since many drugs are renally excreted, and the most practical marker of renal function is creatinine clearance, it is often necessary to estimate this in order to undertake dosage adjustment in renal impairment. The usual method is to undertake a 24-h urine collection coupled with a plasma creatinine measurement. The laboratory then estimates the patient’s creatinine clearance. The formula used to determine creatinine clearance is based upon the pharmacokinetic principles in Eq. (3).

The rate of elimination is calculated from the measurement of the total amount of creatinine contained in the 24-h urine sample divided by 24, that is,

Using this rate of excretion and substituting the measured plasma creatinine for C^ss in Eq. (4), the creatinine clearance can be calculated.

However, there are practical difficulties with this method. The whole process is cumbersome and there is an inevitable delay in obtaining a result. The biggest problem is the inaccuracy of the 24-h urine collection.

An alternative approach is to estimate the rate of production of creatinine (i.e. rate in) instead of the rate of elimination (rate out). Clearly this has advantages, since it does not involve 24-h urine collections and requires only a single measure of plasma creatinine. There are data in the literature relating creatinine production to age, weight and sex since the primary source of creatinine is the breakdown of muscle.

Therefore, equations have been produced which are rearrangements of Eq. (4), that is,

Rate of production is replaced by a formula which estimates this from physiological parameters of age, weight and sex.

It has been shown that the equation produced by Cockcroft and Gault (1976) appears to be the most satisfactory. A modified version using SI units is shown as

where F = 1.04 (females) or 1.23 (males).

There are limitations using only plasma creatinine to estimate renal function. The modification of diet in renal disease (MDRD) formula can be used to estimate glomerular filtration rate (eGFR). This formula uses plasma creatinine, age, sex and ethnicity (Department of Health, 2006).

eGFR = glomerular filtration rate (mL/min per 1.73 m²).

The MDRD should be used with care when calculating doses of drugs, as most of the published dosing information is based on Cockcroft and Gault formula. In patients with moderate to severe renal failure, it is best to use the Cockcroft and Gault formula to determine drug dosing.

Digoxin

A two-compartment model best describes digoxin disposition (see Fig. 3.3), with a distribution time of 6–8 h. Clinical effects are seen earlier after intravenous doses, since the myocardium has a high blood perfusion and affinity for digoxin. Sampling for TDM must be done no sooner than 6 h post-dose, otherwise an erroneous result will be obtained.

Elimination

Digoxin is eliminated primarily by renal excretion of unchanged drug (60–80%), but some hepatic metabolism occurs (20–40%). The population average value for digoxin clearance is:

However, patients with severe congestive heart failure have a reduced hepatic metabolism and a slight reduction in renal excretion of digoxin:

Ideal body weight should be used in these equations.

Absorption

Digoxin is poorly absorbed from the gastro-intestinal tract, and dissolution time affects the overall bioavailability. The two oral formulations of digoxin have different bioavailabilities:

Practical implications

Using population averages it is possible to predict plasma concentrations from specific dosages, particularly since the time to reach the steady state is long. Population values are only averages, and individual values may vary. In addition, a number of diseases and drugs affect digoxin disposition.

As can be seen from the preceding discussion, congestive heart failure, hepatic and renal diseases all decrease the elimination of digoxin. In addition, hypothyroidism increases the plasma concentration (decreased metabolism and renal excretion) and increases the sensitivity of the heart to digoxin. Hyperthyroidism has the opposite effect. Hypokalaemia, hypercalcaemia, hypomagnesaemia and hypoxia all increase the sensitivity of the heart to digoxin. There are numerous drug interactions reported of varying clinical significance. The usual cause is either altered absorption or clearance.

Theophylline

Theophylline is an alkaloid related to caffeine. It has a variety of clinical effects including mild diuresis, central nervous system stimulation, cerebrovascular vasodilatation, increased cardiac output and bronchodilatation. It is the last which is the major therapeutic effect of theophylline. Theophylline does have some serious toxic effects. However, there is a good plasma concentration–response relationship.

Plasma concentration–response relationship

• <5 mg/L: no bronchodilatation ¹

• 5–10 mg/L: some bronchodilatation and possible anti-inflammatory action

• 10–20 mg/L: optimum bronchodilatation, minimum side effects

• 20–30 mg/L: increased incidence of nausea, vomiting ² and cardiac arrhythmias

• >30 mg/L: cardiac arrhythmias, seizures.