Y + E,
The cancer risk model described herein first appeared in 1982 (Downs, T. and Frankowski, R. Influence of repair processes on dose-response models. Drug Metabolism Reviews 13:839-852, 1982). The model provides a plausible mechanism with biological underpinnings that might reasonably explain observed phenomena in cancer dose-response studies that are not being addressed by the risk models currently in use. This article is a summary and explanation of the features of the model that relate to hormesis.
Data from several saccharin studies served in large part to motivate the original research, and were used to illustrate beneficial low-dose effects that could be explained by the model. The model itself is a function of linear ratios of the administered dose x. Linear ratios arise naturally from non-linear Michaelis-Menten kinetics, as described in the next section.
The one-hit cancer risk model is used to illustrate the hormetic concepts brought out by linear ratios. The one-hit model is modified to account for repair and for non-linear kinetics, by using a linear ratio of the administered dose instead of the actual administered dose, and by expressing the probability that a "hit" is repaired as a linear ratio of the administered dose.
The phenomenon of hormesisor low-dose beneficial effectshas been widely observed and accepted. Yet none of the cancer risk models currently in wide use have attempted to accommodate hormetic effects, The modified one-hit model developed herein does not attempt to account for elimination or detoxification, but nevertheless it provides plausible explanations for certain classes of observed hormetic phenomena associated with experimental testing of carcinogenic substances in animals.
Non-linear Michaelis-Menten kinetics provide the basis for the linear ratio form for the effective dose of the carcinogenic test substance in question. Suppose the test substance (call it X) must undergo metabolization through an enzymatic reaction with an enzyme E to create a reversible complex intermediate reactive metabolite, M. The reversible metabolite then irreversibly yields the final reactive metabolite Y, which is capable of carcinogenic activity, and the bound enzyme E is released. The reaction may be expressed symbolically as:
(1) X + E ÷ MY + E,
Michaelis-Menten kinetics dictate that, under steady-state conditions, the effective dose, a, which is the amount of Y created per unit time, has the algebraic form:
(2) 
= dy/dt = ax/(cx + d),
where t denotes time, x is the administered dose
in some suitable units such as mg/kg/day, y is the
concentration of Y, and a, c and d are positive constants
determined by the available amount of the enzyme E and
the rate constants of the kinetic reactions. Equation (2)
gives the steady-state effective dose corresponding to a
given constant and continuously administered dose x.
If, in addition to the enzymatic reaction (1) there
is a background substance S, different from X,
which undergoes the same type of enzymatic reaction , with
the same enzyme, to yield the same carcinogenic agent
Y, then the amount of Y created per unit time will
have the form
(3) = dy/dt = (ax+b)/(cx+d),
where a, b, c and d are positive constants,
determined now by the amount of enzyme available,
the concentration of the background substance, the
rate constants for the kinetic reaction with X, and the
rate constants for the kinetic reaction with S. The
expression (3) for is called a linear ratio, the ratio of two
linear functions of x.
In developing (3) it was assumed that the background substance S used the same enzyme and
produced the same intermediate metabolite as the test
substance X. Examination of the three remaining cases
(same enzyme and different metabolite, different enzyme
and same metabolite, and both different) shows that
the effective dose in each case has the form (3), so long
as the final carcinogenic agent Y is the same
throughout. The coefficients a, b, c, and d vary from case to case,
but they are always positive.
The linear function = ax+b is a special case of
the general linear ratio (3), and is obtained from (3)
by setting c=0, d=1.
The linear ratio form (3) for the effective dose ranges from b/d when x is zero to a/c as x becomes infinite:
= (ax+b)/(cx+d) = b/d when x = 0,
so b/d is the "spontaneous effective dose" when the administered dose x is 0.
As the concentration x becomes very large the
ax and cx terms dominate the numerator and
denominator, respectively, of 
, and the administered dose
approaches a/c:
= (ax+b)/(cx+d)a/c as x
.
The linear ratio a is thus bounded from b/d,
when x= 0, to a/c when x. The lower bound b/d can
be considered as accounting for"spontaneous tumors of
the sort due to Y when the administered dose x is
zero. Linear ratios of the form (3) thus automatically
account for spontaneous tumors not due to the actions of the
test substance X.
The ratio a/c is an upper bound to the effective dose, allowing a plausible explanation of the frequently observed situation wherein not all test animals get cancer, even when exposed to extremely high administered doses of the test substance.
As the administered dose x increases
monotonically from zero to infinity, the effective dose varies
monotonically from b/d to a/c along the branch of a
hyperbolic curve. It might be thought that the effective dose
would always have to be an increasing function of x,
but this is not necessarily so. Suppose for example that
the test substance X employs the same enzyme as a
background substance S, but the background substance
is metabolized to a different carcinogenic agent, Z, with
a different target organ, than the agent Y for X. Then
X and S compete for the enzyme, and an increase in
the dose x will result in an increased chance of tumors
due to Y, but in a decreased chance of tumors due to Z.
Thus linear ratio forms for effective doses afford a
possible explanation for another frequently observed
phenomena among carcinogens: increased tumor incidence at
one site, and decreased incidence at another site.
Symbolically, the effective dose of X for the Y tumors
increases along the hyperbolic curve from b/d to a/c as x
increases from zero to infinity, but the effective dose of
the background substance for the Z tumors
decreases with increasing x.
The linear ratio whose general form is given by
(3) will be a steadily decreasing, constant, or a
steadily increasing function of x according as the
so-called determinant of : ad-bc, is less than, equal to, or
greater than zero, respectively . Equivalently, if b/d > a/c
then the linear ratio of (3) decreases steadily from b/d
to a/c as x increases from zero to infinity; if b/d = a/c
then the linear ratio maintains this constant common value
as x increases from zero to infinity; and if b/d < a/c
then the linear ratio increases steadily from b/d to a/c as
x increases from zero to infinity.
On the basis of the effective dose , the test
substance is thus beneficial, has no effect, or is
detrimental at a particular site according as the effective dose for
the test substance at that site is a decreasing, constant,
or increasing function of x, according as, in turn,
the determinant ad-bc is negative, zero, or positive. And,
the determinant is negative, zero, or positive according as
b/d is less than, equal to, or greater than a/c.
It is widely accepted that the DNA repair process
is saturable and that Michaelis-Menten kinetics are
applicable. Accordingly, it will be assumed that the
probability of DNA repair, or repair rate,
, for a lesion caused
by the reactive metabolite Y is a linear ratio of the
effective dose a, lies between zero and one, and thus has
the form:
(4)= (e + f)/(g + h), for some e, f, g and
h, where 0
eg and 0fh.
The inequalities 0eg and 0fh ensure thatlies between zero and one. The mathematical and
kinetic properties developed above for the effective dose
a carry over, without exception, to similar properties for
the repair rate. Thus the repair ratewill decrease,
stay constant, or increase, as increases, according as
its determinant, eh-fg, is negative, zero, or positive. In
the first case the test substance has the effect of
inhibiting repair. In the second case, the repair rate is
independent of, and unaffected by changes in, the effective dose,
and in the third case repair is enhanced by an increase in
the effective dose. We refer to this latter case as
adaptive repair. The probability of repair varies from q/(q+s)
when x is zero, to p/(p+r) when x is infinite. The
determinant of the repair rateis ps-qr. Remarks analogous to
those for the effective dose of a substance apply also to
the repair rate: thus, a non-carcinogenic substance
may appear to be carcinogenic at some tissue site merely
by competing with an anti-carcinogen for an enzyme, and
a substance that is carcinogenic at one site may appear
to be anti-carcinogenic at another.
If we substitute the right-side of (3) for
in
equation (4), we infer thatis itself a linear ratio in x, as given
in (5) below:
(5)= [px + q]/[(p+r)x + (q+s)], where
1, and
p=(ae+cf), q=(be+df), p+r=(ag+ch), q+s=(bg+dh).
Thus it is readily verified from (5) that the
determinant foris ps-qr=(ad-bc)(eh-fg), the product of
the determinants of (3) and (4). From this fact one
can conclude, for example, that if the effective dose
is a decreasing function of the administered dose x, and
the repair rateis also a decreasing function of
, then ad-bc and eh-fg are both negative, so their product
is positive, and the repair ratemust be an
increasing function of x.
The probability that a DNA lesion will not be
repaired is 1-, which is also a linear ratio in x,
with determinant qr-ps equal to the negative of the
determinant of:
(6) 1 -= [rx + s]/[(p+r)x + (q+s)], where
1 -1, and
p=(ae+cf), q=(be+df), p+r=(ag+ch), q+s=(bg+dh).
Imagine that particles of the test substance or
its metabolites are impinging on or "hitting", a section of
a DNA molecule in a cell and causing some type of
pre-cancerous lesions. Let n be the number of hits on
the target tissue, and assume that n is Poisson
distributed with mean given by the linear ratio in (3), where
any multiplicative potency factor has been incorporated
into the linear ratio. The one-hit model without
adaptive repair postulates that the probability of cancer
induction is the probability of at least one hit. This probability
is the same as one minus the probability of no hits, so
that the probability of cancer for an administered dose x is
(7) Px = 1 - exp[-] ( linear ratio one-hit
model without repair).
The standard linear one-hit model, where
Px = 1 - exp[-ax-b], is a special case of (7) obtained by taking
c=0, d=1 in the linear ratio expression (3) for . The
linear ratio one-hit model (7) can be modified to account
for adaptive repair by requiring that there be at least
one non-repaired hit for cancer induction. If the number
of hits n is Poisson distributed with parameter , and
the number of repairs v is binomially distributed
with parameters n and , whereis given by (5), then it
can be shown that the number of unrepaired hits, n-v,
is Poisson distributed with parameter (1-).
Therefore, the probability Px of cancer in the presence of repair
is the probability that n-v is at least one, so that
(8) Px = 1 - exp[- (1-)] (linear ratio one-hit
model modified for repair).
The influence of the repair rate on the response
Px is illustrated hypothetically in Figure 1. Each of the
three dose-response curves in the figure was calculated
from the modified one-hit model (8) with given, for each
of the three curves, by
= (10x + 10)/(x + 10)
while is given for the top, middle, and
bottom curves, respectively by
= (85x + 9)/(100x + 10)
= (95x + 9)/(100x + 10)
= (99x + 9)/(100x + 10).
The condition for the response Px in (8) to be a decreasing function of the administered dose x can be found by differentiating (8) with respect to x, and finding those values of x which make the derivative negative. This condition may be expressed as:
(1) '/(1-) >
'/ ,
where and are given by (3) and (5), and
'
and ' are their derivatives with respect to x. For the
top curve of Figure 1 the repair rate function is such that
the condition (9) cannot be satisfied for any values of x.
For the middle curve the condition (9) is satisfied for
values of x close to 0, and for the bottom curve the condition
is satisfied for a wider range of values of x.
Figure 1.
Hypothetical one-hit dose-response curves when the repair rate is a function of the administered dose.