Monte Carlo results for the 3-poly test for animal carcinogenicity experiments.

By using a two stage model of carcinogenesis, we generated Monte Carlo studies to assess the efficiency and robustness of the 3-poly test for animal carcinogenicity experiments. The Monte Carlo results indicate that the 3-poly test is quite powerful for detecting the carcinogenic effects of complete carcinogens, moderate promoters, and initiators with moderate or large effect, but, in some cases, it is less powerful for weak initiators or weak promoters. As expected, the 3-poly test is insensitive to the toxicity of many agents.

To assess risk of environmental agents by animal carcinogenicity experiments in the past, statisticians would carry out trend tests to determine if the agents were carcinogenic. One well-known carcinogenicity test is the Armitage-Cochran test (1,2). Portier and Bailer (3) pointed out that such tests are subjected to serious biases when agents are toxic or when there are causes of death other than cancer. To correct for such biases, Portier (4)], one may wonder if the method is robust and efficient under other realistic models of carcinogenesis. To answer these questions, we generated Monte Carlo studies of this test by using a two-stage model of carcinogenesis as described by Moolgavkar and Knudson (5). The basic reason we used this model is that it has strong biological support [see Tan (4)]. Because the two-stage model and its extensions are complex enough to involve stochastic proliferation and differentiation of initiated cells and yet simple enough to be applicable to many data sets, this model has been suggested as the basic model for assessing risk of environmental agents (6).
We will briefly describe the model and how Monte Carlo data can be generated; we will then apply the 3-poly methods to the data. To assess the efficiency and robustness of the test, we will compute the Monte Carlo size and the power of the test. Finally, we will discuss the results and some of the issues relevant to the method.

The Model and Generation of Monte Carlo Data
To generate Monte Carlo studies, animal carcinogenicity experiments were carried out in which healthy animals were exposed to different dose levels of the carcinogen at to-0 and are followed up until time tM = tk.
In the group of animals exposed to the carcinogen with dose level z, let a,(jlz) and a2(jIlz) denote the numbers of animals which died at time j without and with cancerous tumors, respectively; similarly, in the group of animals exposed to the carcinogen with dose level z, let b1(z) and b2(z) be the numbers of terminal sacrifices without and with cancerous tumors, respectively.
To generate Monte Carlo data from such an experiment, we assign D as the random variable for the time to death, Tas the random variable for the time to the onset of cancer tumors, and Z as the variable for the dose level. Given Z=z, let o0(tjz), O(tls,z), (t > s), and ko(tlz) be the incidence func- (3) For j < t < j+1, the survival functions associated with cx0(tl z), P,0(tl s,z), and ko(tl z) are given respectively by Bo(tli) = exp{-f P (xk,z) d} = exp -I 1z(uk)-f0(xk,z)d4 (5) for j >i, and where a (i) = J1 a. (xlz)dx, f3z(jls) = j0 t(xs x for j-12 s, and AZW = .f17lo(xIzkPc Assume that 300(tls,z) = 00(tli,z) for i-l < s < i and that during one time unit, the event of death and the event of developing cancerous tumors are independent of each other. Then Pz(tIs) = Pz(t Ii) for i-I < s < i < t, and the probabilities for generating a,(jIz), a2(jlz), b1(z), b2(z) are given respectively by PiCIz)=Pr{DE i-1, j), T 2 j ID 2 j-1, Z=z} =[AzUl)-Az(j)] Azy) (7) P2(jlz)= P {D E [j-1,j),T <jp 2 j-1, The Two-stage Model of Carcinogenesis Let v,(tlz) and v2(tlz) denote the mutation (8) rates at time t of the first event and the second event, respectively, for animals exposed to the carcinogen with dose level z. Let @(y,ws,t) = 4)(s,t) be the probability generating function of the number of initiated (9) cells (I cells) and cancerous tumor cells (T cells) at time t, given one initiated cel arising from a normal stem cell at time s. The twostage model ofcarcinogenesis specifies that (10) Ao ( (7) and Portier and Bailer (3) proposed a 3-poly test to determine if the agents are carcinogenic. The first step of this test is to compute the weight co. for each animal and adjust the estimates of quantal response by using this weight. A modified X2-test is then derived by substituting the adjusted estimates of quantal response into the Cochran-Armitage trend test. Specifically, the procedure is given in the following steps.
If co,. is the weight for the j animal in the group of animals exposed to the carcinogen with dose level di and t.-is the time of death including sacrifice of this animal, then based on the Armitage-Doll multistage model, Bailer and Portier (7) derived co as CO =(t1tM)k where tM is the termination time of the experiment. For practical purposes, k was taken as 3, thus giving the terminology of the 3-poly test.
If S..is the indicator function defined by 6i.= 1 if'the animal died (including sacrifice) with tumors at tijand 8,., =0 if the animal died without tumors at t4,, and ni is the number of animals in the group exposed to the carcinogen with dose z,, then the adjust- Thus, one rejects the null hypothesis at level a to conclude that the agent is car- the upper a% of the X2 distribution with one degree of freedom. The p-value of the test is the probability that %2 > X2 where x denotes a X2 distribution with one degree of freedom. Portier

Generation of Monte Carlo Studies
In this section we will assume some parameter values and use the model to generate Monte Carlo data to assess the efficiency and robustness of the 3-poly test. To illustrate the results, we will assume four dose levels z=0, 25, 50, and 100.

Selection of Parameter Values
To make the model more realistic, we will select the parameter values from estimates in published papers and discuss these parameters below.
Survivalparameters. Because Portier et al. (8) have shown that the survival functions of Fischer rats are best fitted by the modified Weibull model, we chose the incidence of death without tumors by Theparameters ofgrowth oftumor cells. Following Tan and Chen (9), we assume that for each tumor cell arising at time s, the probability that this tumor cell will divide for the first time during [t,t+ At) is YT(sIt) At + O(At). Given that a tumor cell divides during [t,t+ 1), we assume that at the end of cell turnover this tumor cell either gives rise to two tumor cells with probability aT(t) or dies with probability P T(t). For the Monte Carlo studies, we assume that YT (slt)=YT, aT(t)=aT, and P T (t)= T Then, as shown in Tan and Chen (9), the probability that a tumor cell at time t will eventually develop into a cancerous tumor is given by q=1-(BT/aT). For the generation of data, we take q = 0.80.
Cancer parameters. Because mutations take place only during cell division, we will follow Chen and Farland (10) to assume that for mutations, cell proliferation, and differentiation to take place, cells must first enter into the division stage from the resting To model the proliferation and mutation of I cells, we followed Chen and Farland (10) to assume that for animals exposed to the carcinogen with dose level z, the probability that each I cell will divide during [t,t+ At) is y1(z)At +o(At) = (yO+y1z)At+o(At). Given that an I cell divides during [t,t+ At), we assume that at the end of cell turnover, this I cell will either give rise to two I cells with probability a1, one I cell and one tumor cell with probability PI, or die with probability V, (ai + I + VI = 1). Then, as shown in Tan and Chen (9) For the generation of Monte Carlo data, we take yo=1-e015 _71=0, 0.5, 1, 2, ai=0.52, 0I3=0.479999, and V 1=10-7 with q=0.8. Note that if yr=O then y1(z)=y0 is independent of z so that the carcinogen has no effects on the proliferation of I cells. It follows that y1 provides a measure for the increased proliferation of I cells due to the action of the carcinogen. We will thus use 7y as a measure for the promoting effects of the carcinogen. Finally, for the mutation rates, we take v2(z)=10-7. Also, we follow Moolgavkar et al. (11) to assume Vj(z)=p__eO12log91+z). For generation of data, we take p10=10-7, 012=0, 0.1, and 0.92. Note that if 012=0, then v1(z)=p10 is independent of z. It follows that 012 provides a measure for the increased mutation rate over the spontaneous rate due to the action of the carcinogen. Thus we will use 012 as a measure for the initiating effects of the carcinogen. With the parameter value given above, we have used the Monte Carlo model to generate independent random samples. For each random sample, we assume four dose levels 0, 25, 50, and 100; the sample size is assumed to be 50 to comply with the sample size given by the example in Bailer and Portier (7). Five hundred independent random samples were generated under each condition. The Monte Carlo p-values for the case in which fzt is a gamma density are shown in Table 1 Table 2.
The following observations are made from the results of Tables 1 and 2: -If y1>0 and 012 >0 so that the carcinogen is a complete carcinogen (i.e., both an initiator and promoter), then the 3-poly test appears to be quite powerful in detecting the carcinogenic effects of the agent in question.
*lf yl>0 is moderate or large so that the carcinogen is a promoter with moderate or large effect, the 3-poly test is quite powerful in detecting the carcinogenic effects of the agent in question, even though the effect of an initiator is very small. On the other hand, if 012=0 and 'y>°is small so that the carcinogen is a weak promoter without the presence of an initiator, then the 3-poly test may not be a powerful test for detecting the carcinogenic effect of the agent unless the toxicity of the agent is not strong or the proliferation rate of the normal stem cells is zero (i.e., f (t)= 1). In any case, however, the 3-poly test is at least as good as the Armitage-Cochran test. This is expected since the Armitage-Cochran test may not detect the carcinogenic effects of the chemical, even in situations in which the carcinogen is a promoter with moderate effect, due to the toxicity of the agent. *If 012>0 is moderate or large so that the carcinogen is an initiator with moderate or large effect, the 3-poly test is quite powerful in detecting the carcinogenic effects of the agent in question, regardless of whether a promoter is present or not. On the other hand, if y1=0 and 012>0 is small so that the carcinogen is a weak initiator without the presence of promoting agents, then in some cases, the p-values can be very small, depending on the toxicity of the agent. In these cases, the 3-poly test may not be a powerful test for detecting the carcinogenic effects of the agent. In any case, however, the 3-poly test is at least as powerful as the Armitage-Cocharan test, which is expected because the Armitage-Cochran test may not detect the carcinogenic effects of the agent, even in situations in which the carcinogen is an initiator with moderate effect, due to the toxicity of the agent. *From Tables 1 and 2, we observe that when Ho is true (i.e., the agent is not carcinogenic), the probability (the p-values) of rejecting Ho ranges from 0.0001->0.09.
Furthermore, this probability is <0.05 only when there is high toxicity (in this case, there were probably few tumor responses). These results suggest that at the 0.05 level, Ho may be rejected even though Ho is true.
Note that Ho is true if, and only if, yI=°a nd 012=0. Thus, if Hois true, the proliferation rate of the I cells is the same over different dose levels, and the mutation rate from normal stem cells to I cells equals the spontaneous rate of this mutation, regardless ofwhether the carcinogen is present. *Results in Tables 1 and 2 indicate that the decision reached by the 3-poly test is quite robust with respect to the toxicity of the carcinogen. This suggests that the 3-poly test has achieved its intended purpose for adjusting for the toxicity effects of the agent.

Conclusions and Discussion
To adjust for the toxic effects or competing death from the carcinogen, Bailer and Portier (1O) and Portier and Bailer (3) have proposed a 3-poly test to determine the carcinogenic effects of the agent. Because this test is based on the dassical Armitage-Doll model of carcinogenesis and because the latter model does not take into account the cell proliferation of the normal stem cells and the intermediate cells (initiated cells) and is not supported by modern cancer biology [see Tan (4)], one may wonder how the 3-poly test would perform under some real situations. To answer these questions, we have generated Monte Carlo studies by using the two-stage model of carcinogenesis. The main reason that we chose the two stage model is that it has strong biological Environmental Health Perspectives * Volume 104, Number 8, August 1996 support and is simple enough to be applicable to many data sets (4,6).
Our results have indicated that, as it was intended, the 3-poly test is very insensitive to the toxic effects and competing death of the carcinogen. In all cases, when the agent is not carcinogenic, the p-values are, in general, quite small so that the type-I error of the test is quite small. Thus, the test would not reject the null hypothesis that the agent is not carcinogenic whenever the agent is truly not carcinogenic. If the agent is a complete carcinogen, a moderate promoter, or an initiator with moderate effect, the 3-poly test is usually very powerful for testing the carcinogenic effects of the agent. However -iiL .4. iiii agent is a weak initiator and there are no promoters present, then the power of the test could be very small in some cases, depending on the toxicity of the agent. Similarly, if the agent is a weak promoter and there are no initiators present, then depending on the toxicity of the agent, the 3-poly test may not be a powerful test for detecting the carcinogenic effects of the agent, in some cases. In any case, the 3-poly test is at least as powerful as the Armitage-Cochran test.
From this analysis, one may conclude that the 3-poly test, as proposed by Portier and Bailer (3) is quite robust and powerful for detecting carcinogenic effects of complete carcinogens, moderate promoters, and initiators with moderate or large effect. It is less powerful for detecting carcinogenic effects of the agent if the agent is a weak initiator in the absence ofpromoters or a weak promoter in the absence of initiators (in some cases). Because the mutation rate of the second mutational event is very small (e.g., 10-7/cell in the generated data), these results may be a consequence of the fact that a weak initiator can only give rise to a very small number of initiated cells, which can hardly be sustained in the population in the absence of promoters. Similarly, if the promotional effect of the promoter is very weak, the second mutational event can hardly take place among the small number of initiated cells that arise from spontaneous mutation of the first event in the absence of initiators. Transporting waste and recyclables by rail has been a reality for a decade, a concept with great potential but few success stories. Now that's changing. New projects are coming on line, and the pioneers have data to discuss and personal experiences to share. Join this unique gathering of railroad industry officials and solid waste professionals to address mutual opportunities, debate challenging issues, and explore the real world "how and why" of railhaul.
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