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OCR for page 177
APPENDIX C
Models of Response:
Dose Adclit~vity and Response
Aciclitivity
Two definitions of synergy have attained considerable currency, although
they are based on distinct and largely incompatible statistical concepts and
relate to different biologic views of interaction. This brief discussion is
intended only to clarify ideas. It is not a rigorous treatment.
For simplicity, assume that:
a. Only two agents need be considered.
b. Only a single response (yesno or quantitative) is of interest.
c. The doseresponse relationship is monotonic in both agents; that is, an
increase in dose of one or the other (or both) is never associated with
a decrease in response.
d. Study samples are big enough to ignore random variation.
The relations among the two agents and the single response can be charted
like a topographic map. Figure C1 refers to the proportion (or probability)
of animals dead in a yesno response, but could just as well show (for
example) average creatinine clearance or average weight loss.
Note that the response at the origin, where both doses are zero, is what
we usually consider "background" and that responses along the two axes
(where one or the other dose is zero) yield the usual singleagent dose
response curves: 10% dead at this dose, 20% dead at that dose, etc.
To simplify notation, let rfa,b) designate the response when agent A is
given at dose a, and agent B at dose b. Thus, r(O,O) indicates zero dose of
both agents and hence designates the background level of response, and
rta,O) and r(O,b) indicate zero doses of one or the other agent and hence
the singleagent doseresponse curves.
177
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1 78 OR ~ N K! NO WATER AN D H EALTH
Dose of B
\\ so%
:~
Lo%\


DoseofA
FIGURE C1 Probability of response (i.e., death) at joint exposure to two materials.
DOSE ADDITIVITY
Pick some point (a,b) in the figure where we are interested in determining
whether there is synergy. The response there is rka,b), and rta,b) falls on
some "topographic contour." Find the ends of the contour and connect them
by a straight line. The line can go through rfa,b), or it can be higher or
lower (Figures C2, C3, and C41.
If the line goes through rfa,b), as in Figure C3, the agents are said to
exhibit dose additivity at that point. For concreteness, if half the polo of A
plus half the Limo of B causes 10% (another LO of mortality, A and B are
said to exhibit dose additivity in that specific combination.
Note that A and B exhibit dose additivity for all combinations that produce
polo (or whatever) if and only if that dose contour is a straight line. Fur
thermore, A and B show dose additivity over the whole range of responses
(e.g., all the IDA) if and only if every contour is a straight line. The straight
lines need not be parallel, nor need they be equally spaced in any sense, but
this is still a very tight restriction.
b
:(a,b)
1 ,iLN
a
FIGURE C2
b
(a,b) b
a
FIGURE C3
_'`~<
r(a,b)
FIGURE C4
Examples of departure from simple dose additivity upon exposure to two materials. Figure C2
shows synergism. Figure C3 shows mixed synergism/antagonism. Figure C4 shows antagonism.
OCR for page 177
Appendix C 179
RESPONSE ADDITIVITY
Another definition of additivity is much closer to that used in other sectors
of statistics and mathematics: response at dose a,b is additive if it equals the
sum of the separate responses at a and b; rfa,b) = rfa,O) + r (O,b). This
definition is often modified for use with dichotomous responses, such as
cancer or no cancer and birth defects or no birth defects, in a way that reflects
concepts of statistical independence. For these responses, in the notation here
and with no allowance for a nonzero background, dose additivity is defined
as:
rfa,b) = rfa,O) + r(O,b)~1 rfa,O)]
or
rfa,b) = rfa,O) + r(O,b)rfa,O)r(O,b).
Allowing for background, r(O,O), by redefining rta,O), etc., as r'(a,O) =
rta,O)r(O,O)/1 r(O,O) gives:
r'(a,b) = r'(a,O) + r'(O,b) r'(a,O)r'(O,b).
Of course, if the background rate is nil, r(O,O) drops out, and 1  r(O,O)
= 1.0, and r'(a,b) = rta,b) for all a and b.
In the contour graph, draw horizontal and vertical lines from rta,b) to the
two axes, examine the four indicated points, and determine whether the
equation above is satisfied:
r(O,b)
!~°~°)
r¢a,b)
rta,OJ
If rta,b) is too big, A and B are said to be synergistic at that point; if rta,b)
is too small, A and B are antagonistic. It is clear from this graph that the
definition of synergy has been profoundly altered. Additivity is now defined
in terms of the corners of a rectangle, rather than in terms of an isocontour
plus the straight line connecting its endpoints. Response additivity seems to
be more tractable in the laboratory (as well as more tractable mathematically),
and it is somewhat less restrictive than dose additivity, but these apparent
benefits need to be specified more precisely and examined analytically.
Can we integrate the definitions? A and B can be additive in both senses
under some limited circumstances, which might be so restrictive as to be of
no practical value. In practice, we must choose one or the other.
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8O DRINKING WATER AND HEALTH
The model, or definition, that one uses for additivity and for departures
from additivity affects the interpretation of experimental data. For example,
exposure to two materials from a doseadditive point of view, each at a dose
of ~D50/2, would yield a 50% mortality response. The slope of the dose
response curves is of no consequence in this definition.
Now consider the same mixture from a responseadditive point of view.
If the actions of the two materials are independent, writing P(d~) for rta,O),
P(d2) for r(O,b), and P(d~,d2) for rfa,b), the expected result of the com
bination would be
P(d~,d2) = P(d~) + P(d21~1 P(d~]
or
P(dl,d2) = P(dl) + P(d2) P(dl)P(d2),
where P(di) is the probability of response at dose di (i = 1,2). Say further
that the two materials have a doseresponse curve that is convex upward,
with di = ~D50/2 = 0.4. Then response additivity would produce
P(dl,d2) = 0.4 + 0.4  (0.4 x 0.4) = 0.64,
rather than the 0.5 expected from a doseadditive model.
By way of contrast, consider two materials with doseresponse curves that
are concave upward, so that P(LD50/2) = 0. 1. Response additivity would
require that:
P(d~,d2) = 0.1 + 0.1  (0.1)~0.1) = 0.19,
a result that would be considered much lower than the 0.5 anticipated from
a doseadditive point of view.
For example, chemicals A and B are tested in various combinations with
results as shown below:
Cancer Incidence Percent at
Dose of A, ~g/kg
O IO 20
Dose of0 5% 19% 22%
B,~g/kg100 17% 27% 39%
200 25% 42% 61%
Are the effects of A and B additive, synergistic, or antagonistic with A at
10 g/kg and B at 100 ~g/kg?
Response additivity is easily tested inasmuch as 0.22/0.95 = 0.232 is less
than 0. 14/0.95 + 0. 12/0.95  (0. 14/0.951~0. 12/0.95) = 0.254. A and B
are antagonistic at these doses.
OCR for page 177
Appendix C 181
To check dose additivity, note that a 27% incidence does not occur on
either axis, but (by our assumptions) occurs at some dose higher than 20 g/
kg for A and higher than 200 g/kg for B. Thus, the straight line connecting
the ends of the 27% contour would be outside (on the far side of the origin
from) the point of interest, as in Figure C2, and A and B are synergistic at
these doses.
Which definition should be used? Each has substantial and valid uses, and
one should not want to proscribe either. What is needed, however, is to make
clear which definition is being used in each particular context. As Kodell
(1986) points out, "to the pharmacologist and toxicologist, the concept of
addition or 'additivity' can imply something about either the doses (concen
trations) or the responses (effects) of toxicants acting together. To the bio
statistician, addition of doses is in line with the concept of 'similar action,'
whereas addition of responses is related to the 'independence' of action. The
epidemiologist includes the concept of 'multiplication' of responses . . . that
. . . can be interpreted as a type of independence of action."
REFERENCE
Kodell, R., 1986. Modeling the joint action of toxicants: Basic concepts and approaches. EPA
2300387027 ASA/EPA Conferences on the Interpretation of Environmental Data: Cur
rent Assessment of Combined Toxicant Effects, May 56, 1986.