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Section IBasic principlesThe exponential function
An exponential function takes the form:
y=Anax.
Inthis relationship n is thebaseand x theexponent;Aand a are constants. Although
it is possible to use any base for our exponential function, the natural number e
is chosen for its mathematical properties. The exponential function, y=ex,isthe
only function that integrates and differentiates to itself, making manipulation of
relationships involving exponentials much easier than if another base were chosen.
The number e is irrational, it cannot be expressed as a fraction, and takes the value
2.716... where there is an infinite number of digits following the decimal point.
Exponentials are positive if therateof change of y increases or negative if therate
of change decreases as x increases; in the example above ‘a’ is positive for a posi-
tive exponential and negative for a negative exponential. Bacterial cell growth is an
example of a positive exponential relationship between the number of bacteria and
time; compound interest is a further example relating to the growth of an investment
with time. For a negative exponential we write:
y=Be−bx.
For the two examples above, A and B are constants that relate to the intercept on the y-
axis and a and b are rate constants that determine how steep the exponential curve is.
Inpharmacokinetics, we consider time, t, to be our independent variable (equivalent
to x) and plasma concentration, C, to be the dependent variable (equivalent to y).
The simple wash-out curve (where plasma concentration declines and approaches
zero) describes a negative exponential. This equation was given above:
C=C 0 e−kt.
The ‘steepness’ of the wash-out curve depends on the rate constant, k. If k is halved
and everything else remained constant, then the plasma concentration would take
twice as long to reach any given level; if k is doubled then the time for the plasma
concentration to reach a given level would halve. The wash-in curve also describes a
negative exponential; although plasma concentration increases with time, therateat
which it approaches its maximum value is decreasing with time, making it a negative
exponential (Figure6.2).
Asymptote.Theoretically, a negative exponential process approaches its steady-
state value evermore closely without actually reaching it. This steady-state value is
termed the asymptote. For the wash-out curve this asymptote is zero; for the wash-in
curve during a constant infusion it is the steady-state concentration reached, which
is determined by the infusion rate and the clearance of the drug (Figure6.2). Ifwe
consider the wash-out curve, after one half-life plasma concentration has fallen by
50%, after two half-lives it has fallen another (50/2)%, that is, a further 25% and
after five half-lives the process will be 50+ 25 +12.5+6.25+3.125=96.875%
complete, thus in practice five half-lives represents the approximate time needed to