Chemistry Reference
In-Depth Information
When a dose-effect relationship is observed to fi t the
sigmoid model, it might be postulated that the effect
results from the cumulative reaction of many individ-
ual receptors having a frequency distribution of sus-
ceptibility. In other words, individual receptors, cells,
or fi bers may each demonstrate an all-or-none reaction
at a given dose. If the cumulative reactions produce
a type of effect that can be graded in a continuum, a
dose-effect relationship of the sigmoid type may be
constructed. If, however, the cumulative responses
produce an effect in the individual that is also all-or-
none, a population of individuals will be required to
produce a dose-response relationship for the effect.
A feature of the probability curve, and consequently
the corresponding frequency distribution, is that the
response is asymptotic to the dose scale (x-axis). That
is, by defi nition, a small fraction of the population will
respond to even the smallest of doses, and, conversely,
a small fraction that will never respond, even to the
highest of doses. In practice, such observations may
not exist, or they may not be seen because of limita-
tions in the size of the population. This feature is illus-
trated in curves 10A and 10B. In curve 10A, there is a
fi nite dose, often called the threshold dose, that must
be exceeded before any response is observed, whereas
curve l0B begins at zero dose-zero response.
Many
Resistant
individuals
Sensitive
individuals
Majority of
individuals
Minimal
effect
Maximal
effect
Average
effect
Few
Mild
Response to same dose
Severe
S.D. = Standard deviation
1SD (68%).
2SD (95%).
Low
response
Mean
response
High
response
FIGURE 13
The top graph illustrates that within a population,
most responses to a toxicant are similar; however, a wide variance
of responses may be encountered, some individuals are susceptible
and others are resistant. As demonstrated above, a graph of indi-
vidual responses can be depicted as a bell-shaped standard distribu-
tion curve. As illustrated in the bottom curve, dose responses are
commonly presented as mean + 1 SD (standard deviation), which
incorporates 68% of the individuals. The variance may also be pre-
sented as two standard deviations, which incorporates 95% of the
responses. A large standard deviation indicates great variability of
response.
3.1 Biological Basis for Modeling
The biological basis for the threshold is usually con-
sidered to be one or more of the following mechanisms:
The frequency distribution may fi t a symmetrical
curve given by the equation, y = exp(−x
2
), and is then
known as a normal distribution. However, many fre-
quency distributions in nature are observed to have a
distribution skewed in the direction of higher doses.
This skewness is particularly evident for effects caused
by a mean dose that is close to zero. The restriction that
zero dose is limiting for zero effect creates an inherent
skewness to the frequency distribution. When dose is
plotted on a log scale, however, the restriction of the
zero dose is removed from the dose scale, and it is
observed that many distribution frequencies in nature
become normalized (i.e., more symmetrical) when log
dose is used. Such frequency distributions are often
called log-normal distributions.
Thus, the sigmoid model of the dose-response rela-
tionship has a biological basis depending on a normal
or log-normal distribution of susceptibility of individ-
uals to a given effect at different doses. It is important
to remember that the effect in this case is an all-or-
none event (i.e., the individual is either showing the
effect or not: there is no gradation of the effect under
consideration).
1. The specifi c biological mechanism is not capable
of producing the effect at or below the threshold
dose.
2. Homeostatic, defense, and/or repair mechanisms
are capable of countering the effect at the thresh-
old dose in even the most susceptible subject.
3. The biological factors of absorption, transport,
metabolism, and elimination do not allow a
suffi cient dose to reach the site of effect for the
most susceptible subject of the population.
4. Any quantitative effect smaller than the effect
of the threshold dose will not subsequently
progress to that effect.
The biological basis for the no-threshold model is that
the smallest possible dose (i.e., one molecule or one par-
ticle) may interact with a cell or cellular component, initi-
ating a chain of events that ultimately produces the effect
of concern. Pathogenetic mechanisms involving such a
chain of events, usually involving self-replication, have
been described for carcinogenesis and mutagenesis.
The basic problem of low-dose extrapolation is the
selection of a mathematical model that is biologically
