Chemistry Reference
In-Depth Information
log-normal frequency distributions. Thus, the probit or
log probit model is identical to the sigmoid model but
has been transformed to a straight line (Finney, 1971).
In the earlier example of making the best fi t of data
to a model, it was assumed that variations at different
doses had equal importance or weight. In the probit
model, however, the observations are precise at the 50%
response point and lose precision as response either
increases or decreases. The least squares method has,
therefore, been modifi ed to provide a variable weight
to the deviation as a function of the level of response
and is called the “maximum likelihood” method. This
method allows similar statistical statements of varia-
tion for the sigmoid or probit models, as was described
for the linear model.
The logistic model is very similar to the probit
model except that the frequency distribution follows
the logistic curve rather than the probability curve.
Further reference to the logistic model is made in the
next section. The equation of the logistic function reads
P = 1/[1 + exp(−(a + bx))].
With some of the simple curvilinear relationships,
it was noted that simple adaptations could amend
the model to fi t a given experimental requirement.
For example, the simple logarithmic model may be
amended by adding the value of one to the dose so
that the curve will include the zero dose-zero effect
point. In a similar manner, the power-function model
could be amended to account for a baseline effect by
simply adding the value for the baseline effect as a
constant to the experimentally expected effect. Mod-
els may also be amended to account for a threshold
dose if the simple model required the zero dose-zero
effect point.
It is also possible that a dose-effect relationship
will not fi t the same model over the entire dose range.
There could be many biological reasons for the need to
adapt the model at one dose range to a different model
at another dose range, or at least to amend the con-
stants of the model as a function of dose. For exam-
ple, the homeostatic mechanism that alters effect may
be exhausted and inoperable at higher dose levels.
Enzymes may be inducible over certain dose ranges,
leading to altered metabolism and altered effects. The
chemical may be stored in depots with differing affi ni-
ties for the chemical and different elimination rates,
causing a multiphase change with time and/or dose.
Heterogeneity in the population being studied may
cause an abnormal frequency distribution. For exam-
ple, two populations differing in susceptibility may
overlap in such a way that a composite of the two dose-
response relationships would be required. Clearly,
the dose-effect or dose-response relationship may be
based on complex mechanisms that are not adaptable
to modifi cations in a practical biological model. The
experimenter must be alert to conditions that should
not be forced into a simple model.
The problem of adapting a model to provide
either for a threshold or for a zero dose-zero effect
point requires particular care. In many cases, the
experimental data itself will not be adequate to pro-
vide support for either possibility. In such cases, it is
necessary to rely on the theory underlying the model
or to refuse to extrapolate to conditions that cannot be
experimentally validated.
The problem of the baseline effect is in large part
the problem of the “normal” range. An individual may
exhibit some range of effect values when tested repeat-
edly in the absence of a dose. The normal range may be
considerably expanded when the experiment includes
many different individuals. For most clinical labora-
tory measurements, the normal range is considered to
be the 95% confi dence limits around the mean value
for a population sample that is considered to be either
“healthy” or representative of the total population. The
gray area between “normal” and “abnormal” may be
subjective at times, and the limits of normality require
careful defi nition.
The variation and magnitude of effect or response
at zero dose can greatly confound interpretations of
experiments. For example, an incremental increase of
5% response on top of a 0% response at zero dose may
be relatively easily validated on statistical grounds.
However, the same 5% incremental increase on top
of an existing baseline response of 50% may be very
diffi cult to differentiate from the “normal” variation
around the 50% baseline value.
Figure 11 illustrates the relationships between the
frequency distribution and the sigmoid model of a
dose-response relationship. The biological basis for this
model is the observed variability among individuals
within a given population. The frequency distribution
shows that a given effect will be demonstrated by a few
individuals in the population at low doses and that a
few individuals will be more resistant and will require
very high doses before demonstrating the same effect.
Most individuals will, however, demonstrate the effect
in the middle range of doses as illustrated in Figure
13. The relative susceptibility or resistance is consid-
ered to be related to genetic differences, health status,
differences in detoxifi cation or elimination rates, dif-
ferences in absorption rates, different volumes of dis-
tribution, or other biological variables that make one
individual different from another. These differences
are not suffi ciently great to consider any individual as
not belonging to the population, and the variation in a
large population will approximate that predicted by a
probability curve.
