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E Cancer Models One ultimate purpose of laboratory investigations of potential carcinogens is to predict the human health effects of mixtures to which the population is ex- posed environmentally. Mathematical models have been applied throughout the history of toxicity testing to describe dose-response relationships that might allow extrapolating from the high doses used in the laboratory to the low doses generally encountered in the environment. Understanding of the biologic mechanisms of carcinogenesis has advanced, and newer models describe dose- response relationships in a manner that reflects newer knowledge of the bio- logic bases of disease. When the carcinogenic effects of a mixture, instead of single chemicals, are being considered, the concern is usually over whether interactions might occur between components of the mixture and yield toxicity greater than what would be expected, given knowledge of the toxicity of the individual materials. Models of interaction observed at the high doses used in laboratory bioassays of carcinogenicity might predict additivity at lower, environmental doses. The purpose of this appendix is to detail and discuss that phenomenon. When mix- ture components are tested for carcinogenicity in a laboratory bioassay, inclu- sion of high-dose multiple-exposure groups might yield little useful informa- tion, if human environmental exposure to the mixture occurs only at low doses. In such a case, the results of this appendix suggest that cancer predicted from bioassays of the individual mixture components can yield a good estimate of the cancer risk associated with human exposure to the mixture. The basis for and qualification of that conclusion are elaborated in the next section. 185

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186 APPENDIX E IMPLICATIONS OF THE MULTISTAGE MODEL One approach to estimating the cancer risk associated with exposure to mul- tiple carcinogenic agents at low doses is to assume that the excess risks are additive, that is, that the excess risk of the mixture is equal to the sum of the excess risks of the components. The multistage model, originally developed by Berenblum-Shubik (1949) and Armitage and Doll (1954), provides a basic theoretical framework that can be extended to account for exposure to multiple agents. The following section describes how the multistage model can be generalized to take into account exposure to multiple agents. The generalized model is used to analyze the relationship between the extent of interaction observed in a bioassay and the departure from additivity at environmental exposures. On the basis of this examination, general principles concerning the most efficient approaches for estimating the joint effects of multiple agents at environmental doses are discussed. DERIVATION OF THE MULTISTAGE MODEL THAT ACCOUNTS FOR EXPOSURE TO MULTIPLE CARCINOGENIC AGENTS The multistage model assumes that a cell becomes a cancer cell by progress- ing through an ordered sequence of stages. The number of stages is assumed to be some small number k, and the cell is neoplastic once it completes the kth stage. Following the approach taken by Whittemore and Keller (1978), we define Pift) = probability that a cell is in the ith stage at time t, the age of the organism, and kift) = transition rate from i to the (ith + 1) stage at time t. The product of these two terms, [\ift)~[Pift)], is the instantaneous probabil- ity that a cell leaves the in stage for the (ith + 1) stage at time t. The rate of transition out of the ith stage at time t is defined as dPitt)/dt. This instantaneous rate of transition is equal to the instantaneous probability of transformation of a cell from the (ith _ 1) to the ith stage minus the instantaneous probability of a transformation from the 'ah to the dish + 1) stage. This set of conditions defines the set of simple differential equations: dPo(~)/~ = [ -Xo(~)][Po(~)] Pit)/ = [—Jo)] [Pi(~)] + [\i-, (I)] [Pi- ~ (I)] ~Pk(~)/~ = [yak- ~ (I)] [Pk—~ (I)] Pot)= ~ Pi(0) = 0, i = I, . . ., k-l, (E-1) Pk(0) = 0, which can be be solved to obtain Pift). To account for exposure to m carcinogenic agents, we assume the transition rate of a cell from the ith to the (i + 1) stage is

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APPENDIX E m Ni(t) = Hi + ~ [9ijxj(t)]. J=1 where Xj(t) = exposure to jut agent at time t, of = background transition rate for i th stage, and id = unit exposure transition rate forge agent on th stage. ~7 (E-2) Equation E-2 assumes that each of the molecules of the different agents in the mixture acts independently of the others with regard to the probability of causing a cell transformation. Suppose we also assume the following: · The time required for a cell in its kth or final, malignant transformed state to grow into a death-causing tumor is approximately constant and equal to w. · The probability that a given individual cell causes a tumor is very small. · An organ contains N cells of a specified type, each capable of being the origin of the tumor causing death. · N is vely large. · Each of the cells acts independently with regard to becoming transformed and ultimately leading to a tumor. Then the age-specific death rate due to a specified tumor type in a particular organ can be expressed, to a close approximation, as hate = N dot—W) = N[&k_lft—w)][Pk-~(t—W)], (E-3) which is the instantaneous conditional probability that deaths from tumor will occur at time t given that the animal is alive just before time t. The uncondi- tional probability of death due to that tumor by age t in the absence of compet- . · . ng rest as IS P(t) = 1—exp { - | h(V)dv } = 1—exp { - | N[dPk(V—W)Idv]dv}. (E~) We assume that exposures are constant and continuous over a lifetime, as is often the case in chronic carcinogenesis experiments. That means that Fit) = xj, j = 1, 2, . . ., m and that the transition rates are constants. In this restricted case, Equation E-2 has the simplified form, NiLt) = m Hi = [se +.Z dijXj]- J=1 (E-S)

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188 APPENDIX E Note that this assumes that the exposure level x; at time t determines \(t), as opposed to some other value as a part of the exposure history, such as cumula- tive exposure. Substituting this definition for the ith-stage transition rate in Equation E-1 and integrating both sides of the equation yields the iterative solution, t Pl(t) = TV= hot o P2(t) = 1 \1 ~vdv = \1 ~ 2{ I~ vk—1 Pk(t) = ~ k—1 Ok—2 · o (E-6) . A0 dv = II ~ t (k—1)! j=o J k! With these results, the age-specific death rate defined in Equation E-3 and the probability of death due to that tumor defined in Equation E-4 are expressible as k—1 k—1 in(t) = N t II ) OCR for page 185
APPENDIX E ~ 89 )\s = tats ski, j Pi = Hi + aim, (E-9) A = ~j + dj2X2- Substituting these values into Equation E-8 yields the expression Path = ~—exp ~—Nick ~ fIo`~`xi + ~Bj~x~(cej + ~j2x2~/cYjajk! l,fE-10) which may be written in the more concise form P(t) = 1—exp ~—A(1 + B~x~1 + B2X2~}, (E-11) where k—~ A = Ntk II oak!, and s=u Be = hi~/ai and B2 = dj2/~j are the relative transition rates for the first and second agents, respectively. If the risk of death from nontumor causes is negligible to time t, Equation E-1 1 represents the probability of a tumor death by time t. Using this model, the next section describes the dose dependence of the synergistic effect. ESTIMATION OF LARGEST SYNERGISTIC EFFECT DETECTABLE IN 2 x 2 BALANCED-DESIGN EXPERIMENT The experimental design most often used for estimating the joint effects of multiple agents is the balanced 2 x 2 form consisting of four experimental groups: control, a single exposure group for each of two agents, and a joint exposure group at the same doses of the single agents. With that type of design, it is shown how large an interaction effect can be measured in a practical bio- assay under the assumption that the dose-response relationship of two agents can be represented by the specific form of the generalized multistage model depicted in Equation E- 1 1. That degree of interaction will represent a practical upper bound on the interaction of two agents at experimental doses, because greater interaction cannot be measured within the constraints of the chosen experimental design. On the basis of this degree of interaction, we assess the rate of error that is introduced by assuming that the joint risk is additive at environmental exposures. The approach establishes an error rate for additivity that is as great as can be directly measured from a 2 x 2 bioassay design defined by the example under the assumption that the multistage model in its generalized form is a valid representation of the joint carcinogenic effect of two agents. In the example that follows, it is assumed that there are 50 animals per exposure group.

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190 APPENDIX E To obtain an extreme interaction effect, we impose several additional condi- tions on our hypothetical bioassay and use them to estimate values of the parameters in Equation E-1 1: · The background rate is so low that we expect to see no tumors in the control group in most experiments. · The single-exposure doses of each agent are chosen to yield identical responses in the bioassay. This condition may be expressed within the context of the model depicted in Equation E-11 as BE = B2x2, which, for ease of notation, we set equal to the constant V. · The doses ofthe single agents are chosen to be as low as possible such that an observed result that would be equal to the expected value of the response would indicate a statistically significant increase in tumor rates over the con- trols. That is done so that a response can be measured for each agent, but the greatest possible leeway is left for measuring the joint effect of two agents. Assuming zero tumors in the control group, because the background rate is low, five or more tumors are needed in a test group for the response to be considered statistically significant at the 0.05 level, according to a Fisher exact 2 x 2 test. To meet that condition, we select our exposure dose so that the probability that the tumor increase is statistically significant is 0.5. That im- plies that the probability of obtaining four or fewer tumors in a single-agent test group is also 0.50. The probability that a single animal will develop a tumor at the single-agent dose under the conditions of the experiment is obtained from Equation E-11 by setting BE = 0 and B2x2 = V, or vice versa. This yields a response expressed in the form P=1—expel—A(1+V)~. (E-12) The probability of four or fewer tumors is obtained from a binomial distribu- tion with n = 50 and P defined as in Equation E-12. The probability is set equal to 0.5 so that the value of A(1 + V) can be estimated. The relationship may be expressed as 4 ~ (50) [1—e-A(1 + V)s] e-A(l + V)(50 - s) = 0.5 s=0 which is solved numerically to obtain the solution, A(1 + V) = 0.097282. , (E-13) (E- 14) To estimate values of the individual terms A and V, we must impose an addi- tional condition on the bioassay: · Interaction cannot be measured quantitatively if the joint response is 100%. To obtain the maximal measurable interactive effect, we assume that the joint interactive effect will yield a 100% response 50% of the time, so a

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APPENDIX E 191 large interaction effect would be measurable 50% of the time. The probability that all 50 animals will respond at the joint exposure dose is obtained from the binomial distribution with P = 1—exp {—A(1 + V)2} and n = 50, where P is obtained from Equation E-l l when BE = B2x2 = V. The relationship may be expressed as [1—e-A(1 + V)23so = 0 S which has the solution (E-15) A(1 + v)2 =—in [1—(o.5)l/s°] = 4.28546. (E-16) The numerical values for A and V can be obtained by solving Equations E-14 and E-16 simultaneously. Dividing Equation E-16 by Equation E-14 gives the result A(1 + V)2 A(1 + V) = 1 + V= 0 og87524862 = 44.05187. V= 44.05187 - 1 = 43.05187, and A = 0.097282/44.05187 = 0.0022084. To obtain estimates of values of the response-model parameters B1 and B2, the forms with which Equation E-l l is expressed, one additional condition is needed: exposure doses x1 and x2, which would give the single-agent re- sponses, are required. They represent a scaling factor to redefine the equation for specific exposure potencies. To obtain a specific equation, we arbitrarily assume that the low-dose response to each of the single agents resulted from exposures at x1 = 0.1 and x2 = 0.2; agent 1 is twice as potent as agent 2. Because BE = B2x2 = V, it follows that B1 = 43.05187/0.1 = 430.5187, and B2 = 43.05187/0.2 = 215.2593. Substituting the numerical estimates of A, B1, and B2 into Equation E-11 yields the following dose-response model: P(x1,X2) or P(x1, X2) = = 1—exp { - 0.002208~1 + 430.5187xl)~1 + 215.2593x21}, (E-17) . 1—exp{ - [0.002208 + 0.9507x1 + 0.4754x2 + 204.6552xlx2]}.(E-18)

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192 APPENDIX E Given this model, it is possible to estimate the risk at any exposures x~ end x2. The model derived here predicts an interaction effect that is one of the largest that can be estimated from a 2 x 2 balanced-design bioassay with 50 or fewer animals per group. If designs as assumed here were used to estimate joint effects, interactions of the magnitude assumed here would be among the largest observed. * ESTIMATION OF JOINT EFFECTS OF ENVIRONMENTAL EXPOSURE WITH PREVIOUSLY DERIVED NUMERICAL MODEL The model depicted in Equation E- 18 will be considered to be the true under- lying dose-response relationship. That model has a large interaction-synergis- tic effect built into it; it is about the largest that could be measured with a classic 2 x 2 balanced design with 50 animals per exposure group and that would allow the statistical estimation of the multistage model. In many situations, the results of multiple-agent exposure experiments are not available, so estimates of cancer risk associated with environmental expo- sures must be based on the results of single-agent expenments. In this section, we evaluate the error that the assumption of low-dose additiv- ity could introduce. For the purpose of the evaluation, the true underlying dose-response relationship for a single agent is assumed to be known and to be the marginals from Equation E-18. The term "marginals" refers to the re- sponse obtained from exposure to one agent when there is no exposure to the second agent. Thus, the excess cancer risk associated with exposure to the first or second agent is P*(x~) = P(x~,O)—P(O,O) or (E- 19) P*(X2) = P(O,x2) - P(O,O), respectively, with P(xl,x2) as defined in Equation E-18. Assuming additivity, the estimated joint excess risk associated with both exposures is P*(x~,x2) = P*(X1) + P*(X21- (E-20) The error introduced by that assumption can thus be evaluated by comparing P*(x~,x2) with the true excess risk of P*(x~, X2) = P(X1, X2)—P(O,O). (E-2 1) *To our knowledge, an interaction of the magnitude of this example has never been observed; however, an infinite interaction is theoretically consistent with theory underlying the multistage model. A bioassay design similar to that defined here would not, however, allow the estimation of a multistage model with more extreme interaction than that given here.

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APPENDIX E 193 The true increased risk associated with exposure to one agent conditional on exposure to a second agent may be written as: P*(x1~x2) = P(x1,X2)—P(O,X2), and (E-22) P*(x2~xl) = P(xl~x2)—P(X1,°~. We can evaluate the error introduced by the approximations for various val- ues of x~ end x2. The results ofthis analysis are shown in Table E-1. From Table E-1, the following trends are noted: · The synergistic effect is extreme at high doses (i.e., x~ = 0.1, x2 = 0.21. Each agent by itself gives about a 10% response, but jointly they yield a 99% response. In this case, the additivity assumption clearly does not hold. · If one agent remains at a high dose (i.e., x~ = 0. 1) and the other is reduced by 2 or more orders of magnitude, the additivity approximation is reasonably good. The same is true for the conditional risk for the high-exposure agent. However, the conditional risk for the low-exposure agent is 40 times higher than predicted by the additivity assumption at smaller exposures. · If exposure to both agents is reduced by 2 orders of magnitude, the addi- tivity assumption is reasonably good. If exposure to both agents is reduced by 4 orders of magnitude, there is virtually no difference between the true risk and that predicted by the additivity assumption. PRACTICAL IMPLICATIONS DISCERNIBLE FROM NUMERICAL INVESTIGATION OF GENERALIZED MULTISTAGE MODEL If the previous examples were representative of the types of dose-response relationships encountered in practice, the following conclusions can be drawn: · Agents with high environmental exposure such as background radia- tion, cigarette smoke, and some workplace exposures must be investigated very carefully, because, if they act on the same cell type in the same organ, the potential for a strong synergistic effect is great. · The excess cancer risk at low doses from an agent that acts on the same cell type in the same organ as another agentts) to which exposure at high levels occurs (e.g., cigarette smoke) could be seriously underestimated in an animal bioassay, because the bioassay ignores the effects of the other agentts), such as cigarette smoke, on the estimated augmented risk. · When all environmental exposures are 3-4, or more, orders of magnitude below that associated with observable effects in bioassays or in epidemiology studies, additivity assumptions can provide a reasonable approximation of the . . . ~ joint risk.

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194 o Cal 8 - I ·> 4 - o A: i_ o o Cal m I Go Day ~ no <: us on - * Pa Cat ~ ~ . ~ ~4 a' ~ ~ ~ 4 - * Pa =~ o=: a _ ~;~ - O ~ ~ ·~ *d En =- 3 o Z .Cd ~ ~ o o s: ~: ca Ct C~ X s: ~ X - X o o o t— ~] ao X ~ o o~ ~ ~ _ ~ 1 ~o 1 1 1 1 1 1 1 o o o o o o o ~ _ _4 _4 _ _~ ~4 X X X X X X X ~ ~ oo o ~ ~ ~ ~ U~ . . . . . . . c~ ~ ~ c~ cr ~ ox _ _ ~ ~ 1 1 1 1 o o o o X X ~ C~ oo oo oo 1 1 ~r o X X X n . . . 1 1 o o ~N . ~ ~ ~ o o o ~ ~ ~ O ~ o o ~ o . . . . . ~D 1 1 1 1 1 1 1 o o o o o o o ~ ~ ~ _4 ~ ~ _, X X X X X X X n ~ _ o 0 0 0 0 . . ~ cr. ox ~ ~ ~ ~ _ _ _ ~ ~ (-~ \O 1 1 1 1 1 1 1 o o o o o o o _ _4 ~ _1 ~ ~ ~ X X X X X X X ~4 ~ ~ ~ ~ o ~ ~ C~ C~ ~ C~ ~ — ox ~o ~ o ~ ~ ~ o o ~ o ~ o o o C~ o U) _ o o o o o X X X X X _ o~ ~ ~ ~ oo ~ ~ o o . . . . . ~ ~ ox ~ ~ o o X X o o _ _ _ ~ ~ ~ ', 1 1 1 1 1 1 1 o o o o o o o ~ ~4 _' ~ ~ _. ~ X X X X X X X X ~ C~ ~ 0 00 . . . . . . . ~ ~ ~ cr, ~ C~ ~ _ C4 ~ ~ ~ ~ ~ 1 1 1 1 1 1 1 0 0 0 0 0 0 0 X X X X X X X _ _ _ _ _ ~ V} 1 1 1 1 1 1 1 0 0 0 0 0 0 0 ~4 ~ ~ ~ ~ ~ ~ X X X X X X X .

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196 APPENDIX E TABLE E-2 Effect of Background Rate on Accuracy of Additivity Approximation Under the Assumotion of the Multistage Model Defined in Equation E-1 1 a Deviation of Additivity True Background Exposure Level Approximation from True Risk Rate for Both Agents after Subtracting Background, % 0.1 0.01 -0.73 0.001 -0.10 0.01 0.01 -3.39 0.001 -0.34 0.001 0.01 - 1 1.23 0.001 - 1.18 such effects at low doses is not amenable to direct investigation, because of the low response rates generally associated with low doses. This section identifies conditions under which the additivity assumption can be expected to hold at sufficiently low doses. MULTISTAGE MODEL Consider first the multistage model in the case of joint exposure to two chemicals Cat and C2 given in Equation E-8. The probability that a tumor will occur by time t after exposure tom and C2at doses off end x2, respectively, is given by: k k—~ ~ P(x~,x2,t) = 1—exp —Nk! ,.IIo Nii ~ Nflk!t ~ Ail, for small values of xl and x2, where (E-23) 2 Ai = Hi + [3ijXj J=1 (E-24) denotes the transition intensity function for the ith stage (i = 0, 1, . . . , k—14. At any fixed time t, it follows from Equation E-23 that the excess risk at low doses may be written as H(xl,x2) = P(X1,X2)—P(O,O) ~ clx1 + C2X2, (E-25)

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APPENDIX E where /97 cj = (Ntk~lk! ~ (H Pi )3ij > 0 ' ° Hi (E-26) Because the excess risk for chemical Cj is approximately CjXj for sufficiently small xi, it follows from Equation E-25 that the excess risks for C ~ and C2 are additive at these doses with the multistage model in Equation E-23. Note that the same result also demonstrates that the excess risk associated with each chemical will be nearly linear in dose at these doses. ADDITIVE-BACKGROUND MODELS For exposure to single chemicals, Crump et al. (1976) have established other conditions under which low-dose linearity will obtain. The same argument may be invoked in the case of joint exposure to two substances C, and C2, to demonstrate low-dose linearity and hence additivity of the excess risks associ- ated with C, and C2 separately. Specifically, suppose that the joint response rate can be expressed as P(x,, x2) = H(x, + at, x2 + 62), (E-27) where H is an increasing function of both x~ and x2 with both first partial derivatives Offhand D2H strictly positive. Here, b~ > 0 and 62 > 0 denote the effective background doses of x~ end x2, respectively, leading to a spontaneous- response rate P(O,O) = H(~,621. A Taylor expansion of P(x~,x2) shows that, for sufficiently small x~ and x2, Taxi, X2) ~ c~x~ + C2 X2, (E-28) where Ci = DiH (~ ,62) > 0- The model in Equation E-27 has been termed an additive-background model, because the doses x~ and x2 of Cat and C2 are con- sidered to act in an additive manner in conjunction with the effective back- ground doses b~ and 62- MIXED-BACKGROUND MODELS For single chemicals, the result of Crump et al. (1976) was further general- ized by Hoel ( 1980) to include combinations of both additive and independent background. In the present context, combinations of independent and additive background may be represented by the model P(x~,x2) = ~ + (1—y)H(x~ + 6,,X2 + 62), (E-29)

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198 APPENDIX E where 0 < By ~ 1 denotes the probability of a spontaneous tumor's occurring independently of those induced by Cat or C2 or the effective background doses b~ and 62. The same argument as used for the additive-background model in Equation E-27 may be used to show, for sufficiently small x~ and x2, that Il~x~,x2) ~ C/X + c~x2 under the mixed-background model in Equation E-29, with cj* j= 1,2. (E-30) = (1 MULTIPLICATIVE-RISK MODELS In some cases, the interaction between two substances might be well de- scribed by a multiplicative model in which the relative risk associated with the mixture is the product of the relative risks associated with the components. Let pa = P(x~, 0) and pet = P(0, X2) denote the respective probabilities of tumor occurrence when chemicals Cat and C2 are administered separately, and let p00 = P(0,0) denote the spontaneous-response rate. The corresponding relative risks associated with chemicals C ~ and C2 are thus given by rut = p ~ 0/p00 and rot = p0~/p00, respectively. Letting pi ~ = P(x~ ,x2), the relative risk rat ~ = pi ~/p00 when both C, and C2 are administered simultaneously is given by rat = rare (E-31) with the multiplicative-risk model (Siemiatycki and Thomas, 19811. With this model, the excess risks associated with C, and C2 will be approxi- mately additive when both r,0 and rO, are sufficiently close to 1. To see this, note that, from Equation E-3 1, r,, - 1 ~ (rut - 1) + (rot - 1) because log r ~ r—1 for r near unity. Thus, (E-32) PI ~ Poo ~ ~ Pro—Poo) + ~ Pot—poo), (E-33) when the relative risks rut and rO, are close to 1. This implies that, for two substances whose interaction follows a multiplicative-risk model, as with can- cer of the oral cavity induced by smoking and alcohol consumption (Rothman and Keller, 1972; Tuyns et al., 1977), the excess risks will be effectively addi- tive at doses where the relative risks are sufficiently small (e.g., less than 1.051. PREDICTION OF RISK AT LOW DOSES To predict the potential carcinogenic effects of joint exposure to two or more substances at low doses, it is necessary to extrapolate downward from higher doses that induce measurable rates of response (Brown, 1984; Clayson and

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APPENDIX E 199 Krewski, 19861. When data on the effects of joint exposures are available, that may be done by modeling tumor-occurrence rates as a function of x~ and x2 of chemicals Cat and C2. The response surface described by the fitted model may then be used to extrapolate to lower doses. The preceding section identified models under which the excess risks associ- ated with simultaneous exposures to C ~ and C2 would be additive at sufficiently low doses. When those conditions are satisfied, it is sufficient to predict the excess risks associated with Cat and C2 separately, because the excess risk associated with joint exposure to the two substances is well approximated by the sum of the excess risks associated with exposures to C, and C2 alone. That may be done with separate dose-response curves for C, and C2, so that the prediction of low-dose risks does not require mapping the response surface associated with joint exposure to the two substances in this case. When joint- exposure data are available, they may be used not only to identify interactions that can exist at moderate to high doses, but also to assess the extent to which such interactions can be reduced at lower doses. To make full use of joint- exposure data in predicting risks at low doses, we require a model for the probability of tumor induction P(x~, x2) by time t after exposure to chemicals C ~ and C2 at constant closes x~ end x2. For example, the multistage model in Equa- tion E- 1 1 is of the form P(x~,x2) = 1—exp ~—A(1 + B~x~1 + B2x2~} = 1—exp ~—(GO + Next + e2x2 + e3x~x21), (E-34) where the His are subject to the nonlinear constraint e0e3 = Aim, as well as the linear constraints Eli ' O (i = 0,1,2,3~. To avoid nonlinear constraints in fitting models of the form of Equation E-34, the linear constraints Eli ' O may be invoked, as has been done in fitting the multistage model to data involving exposure to only a single substance (Krewski and Van Ryzin, 1981; Armitage, 19821. The linear constraints repre- sent a broader class of models than the nonlinear constraints arising in the derivation of the model, but greatly simplify the statistical problems involved in fitting the models to experimental data. (Note that, although the linear con- straints admit the possibility that Ed or e2 is zero, upper confidence limits on values of these parameters will necessarily be greater than zero, so low-dose linearity is preserved.) More generally, we consider the model P(x~, x2) = 1 —exp ~—~ eO + ~ (ei~xt~ + e2 x2) k k ~ + ~ ~ e3ixi~xi2i, (E-35)

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200 APPENDIX E where the values of parameters are constrained to be nonnegative. To fit this model to data on a response surface P(x~, x2) that is highly curvilinear, it might be necessary to use a value of k in excess of 1 or 2. The model involves (k + 1~2 unknown parameters, so experiments with at least this many exposure groups are necessary. With k = 3, for example, at least 16 treatment groups would be required. The process of describing the response surface would be simpler, if a suffi- ciently flexible model involving fewer parameters were used. For example, the four-parameter logistic model P(x~,x2) = [1 + exp (-(GO + Text + e2x2 + e3x~x2~-i (E-36) might be used to describe the response surface P(x~, x2) within the obse~vable- response range. In analogy with the simple linear extrapolation procedure pro- posed by Van Ryzin (1980), excess risks at low doses could then be predicted by extrapolating along straight lines joining the response surface at points P(xt,x1) and the origin P(O,O), where (xt, x~ is chosen so that the excess risk 7r~xt, x1) = 7r*, where 7r* is chosen to be as small as possible, yet still within the experimentally measurable response range (e.g., 0.01 c or* c 0.101. The points P(xt,x!) may be estimated by fitting a model, such as that in Equation E-36, to experimental data involving joint exposures to C, and C2. SUMMARY Empirical tests of joint action in laboratory settings, which typically are based on high doses, provide little insight into the corresponding effects at low environmental doses. The information available on the mechanisms of tumor induction by chemicals strongly suggests that their relationship with dose is not linear. Quantitative models of the carcinogenic process, such as the multistage model of Armitage and Doll, also reflect such nonlinearity in the response to mixtures. Although a multiplicative exposure effect sometimes dominates at high doses, further exploration of this model indicates that the joint effect will be additive (that is, close to the sum of the individual effects) at sufficiently low doses. A newer model, that of Moolgavkar and Knudson (1981), is more bio- logically specific and yields essentially the same conclusion. (The conclusion depends on the assumption that the augmented risks of the chemical are small, with respect to the natural background rate of the tumor.) Additivity at low doses was also demonstrated under a general class of additive background models and under the multiplicative risk model when the relative risk for each component in the mixture is small. In addition, it can be shown for a broad range of mathematical dose-response models that the joint risk associated with a complex mixture can be determined on the basis of the background risk and the risks associated with the individual components. The models for which that is true include many widely used dose-

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APPENDIX E 201 response models, such as the linear, multiplicative, logistic, and multistage models. The results suggest the utility and desirability of identifying appropn- ate models. Prediction of low-dose risks generally requires the extrapolation of results obtained at higher doses that induce measurable tumor-response rates. When the excess risks associated with exposures to mixture components can be rea- sonably considered to be additive, that can be done by downward extrapolation of the dose-response curves for the individual components. Otherwise, it will be necessary to model the tumor-response surface associated with the joint exposure to two or more carcinogens. In general, the use of joint-exposure data is to be preferred, in that it provides some empirical evidence on the magnitude of any interactions that might be present even at lower doses. REFERENCES Armitage, P. 1982. The assessment of low-dose carcinogenicity. Biometrics 28(Suppl.): 119-129. Armitage, P., and R. Doll. 1954. The age distribution of cancer and a multi-stage theory of carcino- genesis. Br. J. Cancer8:1-12. Berenblum, I., and PA. Shubik. 1949. An experimental study of the initiating stage of carcinogenesis, and a re-examination of the somatic cell mutation theory of cancer. Br. J. Cancer 3: 100- 118. Brown, C. C. 1984. High- to low-dose extrapolation in animals, pp.57-79. In J. V. Rodricks and R. G. Tardiff, Eds. Assessment and Management of Chemical Risks. ACS Symposium Series 239. Ameri- can Chemical Society, Washington, D.C. Clayson, D. B., and D. Krewski. 1986. The concept of negativity in experimental carcinogenesis. Mutat. Res. 167:233-240. Crump, K. S., D. G. Hoel, C. H. Langley, and R. Peto. 1976. Fundamental carcinogenic processes and their implications for low dose risk assessment. Cancer Res. 36:2973-2979. Hoel, D. G. 1980. Incorporation of background in dose-response models. Fed. Proc. 39:73-75. Krewski, D., and J. Van Ryzin. 1981. Dose response models for quantal response toxicity data, pp. 201-231. In M. Csorgo, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, Eds. Statistics and Related Topics. North-Holland, New York. Moolgavkar, S. H., and A. G. Knudson, Jr. 1981. Mutation and cancer: A model for human carcino- genesis. J. Natl. Cancer Inst. 66: 1037-1052. Rothman, K., and A. Keller. 1972. The effect of joint exposure to alcohol and tobacco on risk of cancer of the mouth and pharynx. J. Chron. Dis. 25:711-716. Siemiatycki, J., and D. C. Thomas. 1981. Biological models and statistical interactions: An example from multistage carcinogenesis. Intl. J. Epidemiol. 10:383-387. Tuyns, A. J., G. Pequignot, and O. M. Jensen. 1977. Oesophagal cancer in Ille-et-Vilaine in relation to alcohol and tobacco consumption: Multiplicative risks. Bull. Cancer 64:45-60. (In French; English summary.) Van Ryzin, J. 1980. Quantitative risk assessment. J. Occup Med. 22:321-326. Whittemore, A., and J. B. Keller. 1978. Quantitative theories of carcinogenesis. SIAM Rev. 20: 1-30.