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6 Mathematics and pharmacokineticsRelationship between constants
The rate constant for elimination (kelor k 10 ), clearance (Cl) and volume of distribution
of the central compartment (Vd for one-compartment and V 1 for two-and three-
compartment models) are all constant values for any one model and are closely
related:
kel=Cl/V 1 or Cl=V 1 ·kel.
Inthe one compartment model the time constant for elimination,τ,isgiven by:
τ= 1 /kel.
The relationship between time constant and clearance is therefore:
τ=V 1 /Cl.
Clearance is determined both by V 1 andτ;drugs with the same clearance can have
very different volumes of distribution and time constants, it is the ratio of the two
that is important. In non-compartment models, the mean residence time is equiv-
alent to time constant and clearance is the ratio of dose given to the area under the
concentration time curve.Multiple doses and infusions
Loading dose, infusion rate and dose interval
Ifthe initial volume of distribution (V 1 ) and the required plasma concentration are
known then the dose needed to give a particular plasma concentration can be cal-
culated:
Dose=V 1 ×required concentration.
Ifwewant to maintain this particular plasma level, this first dose is known as the
loading dose; repeated doses must be then given or the drug can be given by a
constant infusion. The frequency of drug dosing depends on the rate at which drug
is removed from plasma either by distribution or elimination. After one half-life,
the concentration will have fallen to half of the initial value. If this concentration is
acceptable as the minimum therapeutic concentration, then the dose frequency is
equal to one elimination half-life (Figure6.13). If the rate of removal is high, then the
dosing interval must be short and doses given frequently; if the rate of removal is slow,
then the dosing interval can be longer and doses given less frequently. The frequency
and size of dose at the start of treatment influences the rate at which a steady state
can be reached; for certain drugs, such as amiodarone, frequent initial dosing is
replaced by less frequent and lower doses as the peripheral compartments become
saturated with drug. Although many drugs have a high therapeutic index and large
doses are well tolerated, some drugs have a narrow therapeutic index and maximum
dose is restricted by adverse side effects. Thus dosing schedules are determined both
bythe pharmacokinetics and the pharmacodynamics of a drug.