In reality, processes of elimination begin as soon as the
bolus dose (d) of drug is administered, the drug being cleared
at a rate Cls(total systemic clearance). In practice, blood is
sampled at intervals starting shortly after administration
of the dose. Clsis determined from a plot of plasma concentra-
tion vs. time by measuring the area under the plasma concen-
tration vs. time curve (AUC). (This is estimated mathematically
using a method called the trapezoidal rule – important in drug
development, but not in clinical practice.)
If the one-compartment, first-order elimination model holds,
there is an exponential decline in plasma drug concentration,
just as at the end of the constant rate infusion (Figure 3.2a). If
the data are plotted on semi-logarithmic graph paper, with
time on the abscissa, this yields a straight line with a negative
slope (Figure 3.2b). Extrapolation back to zero time gives the
concentration (c 0 ) that would have occurred at time zero, and
this is used to calculate Vd:
Half-life can be read off the graph as the time between any
point (c 1 ) and the point at which the concentration c 2 has
decreased by 50%, i.e. c 1 /c 2 2. The slope of the line is the
elimination rate constant, kel:
t1/2andkelare related as follows:
Vdis related partly to characteristics of the drug (e.g. lipid sol-
ubility) and partly to patient characteristics (e.g. body size,
t
l(^12) kk
2 0 693
/
n
el el
.
k
Cl
el V
s
d
V d
d c
0
Cl
d
sAUC
plasma protein concentration, body water and fat content). In
general, highly lipid-soluble compounds that are able to pen-
etrate cells and fatty tissues have a larger Vdthan more polar
water-soluble compounds.
Vddetermines the peak plasma concentration after a bolus
dose, so factors that influence Vd, such as body mass, need to
be taken into account when deciding on dose (e.g. by express-
ing dose per kg body weight). Body composition varies from
the usual adult values in infants or the elderly, and this also
needs to be taken into account in dosing such patients (see
Chapters 10 and 11).
Vdidentifies the peak plasma concentration expected
following a bolus dose. It is also useful to know Vdwhen
considering dialysis as a means of accelerating drug
elimination in poisoned patients (Chapter 54). Drugs with a
large Vd(e.g. many tricyclic antidepressants) are not removed
efficiently by haemodialysis because only a small fraction of
the total drug in the body is present in plasma, which is the
fluid compartment accessible to the artificial kidney.
If both Vdandt1/2are known, they can be used to estimate
the systemic clearance of the drug using the expression:
Note that clearance has units of volume/unit time (e.g.
mL/min),Vdhas units of volume (e.g. mL or L ), t1/2has units
of time (e.g. minutes) and 0.693 is a constant arising because
ln(0.5)ln 20.693. This expression relates clearance to Vd
andt1/2, but unlike the steady-state situation referred to above
during constant-rate infusion, or using the AUC method fol-
lowing a bolus, it applies only when a single-compartment
model with first-order elimination kinetics is applicable.
Cl
V
s t
d
/
0693
12
.
REPEATED(MULTIPLE) DOSING 13
Key points
- The ‘one-compartment’ model treats the body as a
single, well-stirred compartment. Immediately
following a bolus dose D, the plasma concentration
rises to a peak (C 0 ) theoretically equal to D/Vdand then
declines exponentially. - The rate constant of this process (kel) is given by Cl/Vd.
kelis inversely related to t1/2, which is given by 0.693/kel.
Thus,Cl0.693Vd/t1/2. - Repeated bolus dosing gives rise to accumulation
similar to that observed with constant-rate infusion,
but with oscillations in plasma concentration rather
than a smooth rise. The size of the oscillations is
determined by the dose interval and by t1/2. The steady
state concentration is approached (87.5%) after three
half-lives have elapsed.
REPEATED (MULTIPLE) DOSING
If repeated doses are administered at dosing intervals much
greater than the drug’s elimination half-life, little if any accu-
mulation occurs (Figure 3.3a). Drugs are occasionally used in[Drug] in plasma(a) TimeLog [Drug] in plasma(b) Time
Figure 3.2:One-compartment model. Plasma concentration–time
curve following a bolus dose of drug plotted (a) arithmetically
and (b) semi-logarithmically. This drug fits a one-compartment
model, i.e. its concentration falls exponentially with time.