Drug Targeting Organ-Specific Strategies

346 13 Pharmacokinetic/Pharmacodynamic Modelling in Drug Targeting
In Eq. 13.15, the squared standard deviations (variances) act as ‘weights’ of the squared
residuals.The standard deviations of the measurements are usually not known, and therefore
an arbitrary choice is necessary. It should be stressed that this choice may have a large influence
of the final ‘best’ set of parameters. The scheme for appropriate weighting and, if appropriate,
transformation of data (for example logarithmic transformation to fulfil the requirement
of homoscedastic variance) should be based on reasonable assumptions with respect
to the error distribution in the data, for example as obtained during validation of the
plasma concentration assay. The choice should be checked afterwards, according to the procedures
for the evaluation of goodness-of-fit (Section 13.2.8.5).
Usually, the standard deviation of the measurement is dependent on the magnitude of the
concentration. The most commonly applied assumption is that the standard deviation is proportional
to the concentration, which is either the measured concentration (also referred to
as ‘data-based weighting’), or the calculated concentration (model-based weighting).
Many software packages provide only a limited selection of weighting procedures. The
most commonly applied weighting procedure is based on the assumption that the standard
deviation of the concentration is proportional to the measured concentration. In that case,
the objective function may be simplified to:
OBJ = Σ (Cmeas, i – Ccalc, i) 2
n
i = 1
(C meas, i) 2
(13.16)
Alternatively, the following objective function may be used, assuming that the errors in the
measured concentrations are log-normally distributed:
OBJ = Σ [ ln (Cmeas, i) – ln (Ccalc, i)] 2
n
i = 1
(13.17)
Since both Eq. 13.16 and 13.17 assume a constant coefficient of variation of the measurement
error, these equations provide similar (but not identical) results.
13.2.8.3 Searching the Best-fitting Set of Parameters
The best-fitting set of parameters can be found by minimization of the objective function
(Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several
minimization algorithms can be applied, for example, Simplex, Gauss–Newton, and the
Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively.
For further reference, excellent papers and books are available [18].
The fitting procedure may be performed by any suitable minimization algorithm. In theory,
the final parameter set depends only on the objective criterion (Section 13.2.8.2) and is
not dependent on the minimization algorithm, nor on the initial set of parameter values (except
for rounding-off errors). However, in practice, the minimization algorithm may fail to
reach the minimum of the objective function, and may end in a local minimum. In this respect,
minimization algorithms may vary widely. An algorithm which is insensitive to the
choice of the initial estimates, is said to be robust, which is a highly desirable property.To lower
the risk of convergence to a local minimum of the objective function, the convergence criterion
(or stop criterion, for example the relative improvement of the objective function)