Chemistry Reference
In-Depth Information
appropriate, so that we can have some assurance of
its applicability both within and outside the range
of measured dose-response values. The models cur-
rently used for low-dose extrapolation of data on car-
cinogenicity are of two kinds: dichotomous models
and “time-to-occurrence” models. The dichotomous
models relate dose to the probability of the occurrence
(expected response) of a particular effect. The second
group of models relates “time-to-occurrence” of an
effect (i.e., duration of latency period) to dose.
The simplest dichotomous model that has found
practical application and that has a biological basis for
some effects is the linear or one-hit model. This model
is based on the concept “that the effect can be induced
after a single susceptible target has been hit by a single
biologically effective unit of dose” (Hoel
et al.
, 1975).
This concept leads to the following equation linking
the probability (P) of an effect to the dose (D):
best known being the log-probit model of Mantel and
Bryan (1961), the logistic model, and the model based
on the Weibull distribution (see, e.g., Chand and Hoel,
1974). These models have, however, little biological
basis, which is a serious limitation.
The use of such models involves a high degree of
uncertainty, as is well illustrated by a comparison
made by the FDA Advisory Committee on Protocols
for Safety Evaluation (1971) (see Table 1). Three mod-
els, the probit, logit, and one-hit model, were com-
pared under conditions of essentially equally good
fi t to experimental data. All three models gave iden-
tical responses of 50 and 16% at doses of 1 and 0.25,
respectively; however, when extrapolation was made
to responses as low as 1 in 10,000 or 1 in 1 million, the
predicted doses differed by factors of approximately
10
3
and 3 × 10
4
. Of all the models of this type, the one-
hit model is the most conservative and will give the
lowest doses corresponding to predetermined low
responses. This may give rise to great practical prob-
lems when using the one-hit model, particularly if the
substance considered has not been well established as
a carcinogen.
The “time-to-occurrence of effect” models are based
on the observation that the median latency period
for some effects, for example, cancer, induced by
some chemicals depends on the dose and increases
with decreasing dose, whereas the individual latency
P(D) =1−exp(−
y
D)
where
y
is an unknown parameter that has to be
determined from experimental data, if possible. For
small values of
y
D (i.e., for low doses) the equation
reduces to a simple expression;
P(D) ≈
y
D
because exp(−x) can be expanded into a series
exp(−x) =
1
−
x
+ x
2
/2!/−
x
3
/3!
+…, and when
x
is small,
all terms involving
x
can be neglected except the fi rst
(i.e.,
x
). A procedure for low-dose extrapolation based
on this model has been described, for example, by Hoel
et al.
(1975).
A closely related model is based on the multievent
theory of carcinogenesis proposed by Armitage and
Doll (1961) and leads to the following equation relat-
ing the probability of effect (P
D
) to the dose (D):
TABLE 1
Comparison of Three Models for Expected
Response as a Function of Dose
Dose (in multiples
Probit
Logistic
One-particle
of the Dose)
model (%)
model (%)
model (%)
16
98
96
100
8
93
92
>99
4
84
84
94
P(D) =1− e
−(λ
0
+λ
1
D+λ
2
D
2
+... λ
k
Dk
2
69
70
75
I
50
50
50
where
k
is the number of transitional events in the
carcinogenic process and
0.5
31
30
29
λ
1
,…are unknown posi-
tive parameters. At low doses, the higher order terms
in D
2
, D
3
, etc. may be neglected, and P(D) reduces to
λ
0
,
0.25
16
16
16
0.125
7
8
8
0.0625
2
4
4
0.040
I
P(D) ≅ 1 −e
−
(λ
0
+λ
1
D)
≅ λ
0
+ λ
1
D
0.022
0.0155
0.1
λ
o
now represents the background response.
A procedure for low-dose extrapolation based on this
model has been suggested in an International Sym-
posium on General Air Pollution and Human Health
with Special Reference to Long-Term Effects, held in
Stockholm, 8-11 March 1977 (Task Group on Air Pol-
lution and Cancer, 1978) and was also used by the Safe
Drinking Water Committee (NAS, 1977).
There are several other dichotomous models that
have been proposed for low-dose extrapolation, the
where
0.0144
0.00315
0.1
0.00144
0.1
0.00136
0.0001
0.000412
0.000001
0.0000098
0.0001
0.00000144
0.0001
0.00000016
0.000001
0.0000000144
0.000001
Adapted from FDA (1971).
