Statistical Methods in Medical Research 4ed

412 Further regression models
Dose
D
Central compartment
concentration X C
volume V C
k E
k CT
k TC
Tissue compartment
concentration X T
volume V T
Fig. 12.7 Schematic representation of a two-compartment model. The k parameters are the constants
governing the rate of transfer between the compartments or excreted from the body.
go back to the tissues. Each compartment is ascribed a notional volume, VC and
VT, and the drug concentration in each is XC and XT. The system can be
modelled by two linear differential equations. These can be solved to give, for
example,
XC ˆ Ae at ‡ Be bt , …12:28†
where t is the time since the injection of the bolus and A, B, a and b can be
expressed in terms of the underlying parameters given in Fig. 12.7. Once (12.28)
has been identified it can be used as f …x, b† in (12.23). This form of equation has
been developed greatly in the area of population pharmacokinetics and elsewhere;
see Davidian and Giltinan (1995). A minor point which should not be
overlooked when designing experiments that are to be analysed with this form of
equation is that if one of a or b is large, then the only information on the
corresponding term of (12.28) will reside in observations taken very soon after
the administration of the drug. If few or no observations are made or are possible
in this interval, then it may be impossible to obtain satisfactory estimates for the
parameters.
Analyses based on compartmental models are widely and successfully used
and their basis in a differential equation gives them a satisfying underpinning in
non-statistical theory. However, while they may be more appealing than simple
empirical models with no such basis, the model indicated in Fig. 12.7 is a gross
simplification and should not be interpreted too literally. It is not uncommon for
a compartmental volume to be estimated to exceed greatly the volume of the
whole patient.
Models which arise from differential equations are common in the modelling
of growth of many kinds, from the growth of cells to that of individuals or
populations. Appleton (1995) gives an interesting illustration of the application
of a model based on a system of differential equations to mandibular growth in
utero. This author also provides an interesting discussion of the nature of
statistical modelling.
Curves that can arise from differential equations include: