## I

Fig. 7.3

The t'/2 may be used to predict the manner in which plasma concentration alters in response to starting, altering or ceasing drug administration. These events are illustrated in Figure 7.3 and the subsequent text.

### Increases in plasma concentration

When a drug is infused at a constant rate the amount in the body and with it the plasma concentration rise until a state is reached at which the rate of administration of drug to the body is exactly equal to the rate of elimination. This is called the steady state: when it is attained the amount of drug in the body remains constant, i.e. the plasma concentration is on a plateau, and a stable drug effect can be assumed. Figure 7.3 depicts the smooth changes in plasma concentration that result from a constant i.v. infusion. Clearly if a drug is given by intermittent oral or intravenous dose, the plasma concentration will fluctuate between peaks and troughs, but in time all the peaks will be of equal height and all the troughs will be of equal depth; this is also called a steady-state concentration, since the mean concentration is constant.7

7 The peaks and troughs can be of practical importance with drugs of low therapeutic index, e.g. aminoglycoside antibiotics, and it may be necessary to monitor both for safe and effective therapy.

### Time to reach steady state

If a drug is administered by constant-rate i.v. infusion it is important to know when steady state has been reached, for maintaining the same dosing schedule will then ensure a constant amount of drug in the body and the patient will experience neither acute toxicity nor decline of effect. The t[/2 provides the answer: with the passage of each tj/ period of time, the plasma concentration rises by half the difference between the current concentration and the ultimate steady-state (100%) concentration. Thus:

in 1 x t^, the concentration will reach (100/2) 50%, in 2 x t\ (50 + 50/2) 75%, in 3 x t\ (75 + 25/2) 87.5%, in 4 x t'/2 (87.5 + 12.5/2) 93.75% in 5 x ty2 (93.75 + 6.25/2) 96.875% of the ultimate steady state.

When a drug is given at a constant rate (continuous or intermittent) the time to reach steady state depends only on the t'/j and. for all practical purposes, after 5 x t'/^ the amount of drug in the body will be constant and the plasma concentration will be at a plateau.

### Changes in plasma concentration

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. Thus when the rate of administration is altered to cause either a rise or a fall in plasma concentration, a new steady-state concentration will eventually be reached and it will take a time equal to 5 x t!/2 to reach the new steady state.

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

### Decline in plasma concentration

Since tl/2 is the time taken for any plasma concentration to decline by one-half, starting at any steady-state (100%) plasma concentration, in 1 x t]/ the plasma concentration will fall to 50%, in 2 x t^ to 25%, in 3 x t% to 12.5%, in 4 x t'/2 to 6.25% and in 5 x t|/2 to 3.125% of the original steady-state concentration.

Hence the can predict the rate and extent of decline in plasma concentration after dosing is discontinued. The relation between t/2 and time to reach steady-state plasma concentration applies to all drugs that obey first-order kinetics, as much to dobutamine (tj/ 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\ 36 h) when a constant (repeated) dose will give a steady-state plasma concentration only after 7.5 days. Plasma t|/2 values are given in the text where they seem particularly relevant. Inevitably, natural variation within the population produces a range in t\ values for any drug. For clarity only, single average t/2 values are given while recognising that the population range may be as much as 50% from the stated figure in either direction.

A few t/, values are listed in Table 7.1 so that they can be pondered upon in relation to dosing in clinical practice.

Biological effect t\ is the time in which the biological effect of a drug declines by one half. With drugs that act competitively on receptors (a- and (3-adrenoceptor agonists and antagonists) the biological effect tl/2 can be provided with reasonable accuracy.

 TABLE 7.1 Plasma t'/2 of some drugs Drug t'4 adenosine < 2 sec dobutamine 2 min benzytpenicillin 30 min amoxycillin 1 h paracetamol 2 h midazolam 3 h tolbutamide 6 h atenolol 7 h dothiepin (dosulepin) 25 h diazepam 40 h Piroxicam 45 h ethosuximrde 54 h