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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 (yes-no or quantitative) is of interest. c. The dose-response 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 C-1 refers to the proportion (or probability) of animals dead in a yes-no 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 single-agent 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 single-agent dose-response curves. 177

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1 78 OR ~ N K! NO WATER AN D H EALTH Dose of B \\ so% :~ Lo%\ - - DoseofA FIGURE C-1 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 C-2, C-3, and C-41. If the line goes through rfa,b), as in Figure C-3, 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 C-2 b (a,b) b a FIGURE C-3 _'`~< r(a,b) FIGURE C-4 Examples of departure from simple dose additivity upon exposure to two materials. Figure C-2 shows synergism. Figure C-3 shows mixed synergism/antagonism. Figure C-4 shows antagonism.

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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) !~~) ra,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 dose-additive 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 response-additive 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 dose-response 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 dose-additive model. By way of contrast, consider two materials with dose-response 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.

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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 C-2, 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 230-03-87-027 ASA/EPA Conferences on the Interpretation of Environmental Data: Cur rent Assessment of Combined Toxicant Effects, May 5-6, 1986.