Pharmacology for Anaesthesia and Intensive Care

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6 Mathematics and pharmacokinetics

Concentration

C 0

time

time

ln(Concentration)

ln(C 0 )

−k

(a)

(b)

Figure 6.6.Semilogarithmic transformation of the exponential wash-out curve.(a) An

exponential decrease in plasma concentration of drug against time after a bolus dose of a drug

displaying single-compartment kinetics. (b) A natural logarithmic scale on the y-axis produces

astraight line (C 0 is the plasma concentration at time, t=0).

concentration to fall from C to C/e.Half-life (t 1 / 2 )issimilar in that it is the time taken

for the plasma concentration to fall from C to C/2. If we think about the straight line

relationship between ln(C) and time, then (ln(C) – ln(C/2))/t 1 / 2 will give the gradient

of the line as does (ln(C)−ln(C/e))/τ.Since ln(C)−ln(C/2) is the same as ln(C÷

C/2), i.e. ln(2) and ln(C)−ln(C/e) is the same as ln (C÷C/e), i.e. ln(e) or 1, this gives

the relationship:

ln(2)/t 1 / 2 = 1 /τ

ln(2)·τ=t 1 / 2.

Wepreviously noted that ln(2) has a value of 0.693, so half-life is shorter than the

time constant by a factor of 0.693. In one time constant the plasma concentration

will have fallen to C/eor0.37C, that is, 37% of its original value.