Two-stage and Weibull models for carcinogenesis applied to the ED01 discontinued dosing data.

The two-stage clonal expansion model for a single, less-than-lifetime period of dosing is formulated and applied to the liver and bladder tumor data from the ED01 study. The model successfully predicts liver tumor incidence for time points beyond termination of dosing with 2-acetylaminofluorene, but it is unsuccessful for bladder tumor incidence. A discontinued dosing version of the Weibull model is proposed and is shown to predict successfully both liver and bladder tumor incidences for time points after termination of dosing.


Introduction
The ED,,1 study (1) was conducted at the National Center for Toxicological Research (NCTR) for the purpose of establishing the shape of the tumorigenic doseresponse curve for 2-acetylaminofluorene (2-AAF) at low doses (i.e., doses in the neighborhood of a 1% response rate). It was found that for the two primary carcinogenic responses, hepatocellular neoplasms of the liver and carcinomas of the bladder, distinctly different dose-response relationships occurred under a continuous dosing regimen. Whereas the tumorigenic dose response in the liver was nearly linear, that in the bladder was decidedly nonlinear. These contrasting shapes were observed at various points during the lifetimes of the BALB/c female mice that were used as test animals (1).
One aspect of the ED,1 study that, in addition to its size, sets it apart from most other carcinogenesis bioassays is its inclusion of groups of animals for which dosing with the test chemical was discontinued several months prior to sacrifice. These groups are useful for advancing theories about the carcinogenic process. Indeed, Littlefield et al. (2) postulated the precursor role prevalence data to be consistent with both a multistage model with four total stages, two stages being dose related, and a multistage model with six total stages, only one stage being dose related. Freedman and Navidi (6) included the ED,,1 data among various data sets that they examined in the context of the multistage model. Whereas their data selection and assumptions were slightly different from those of Brown and Hoel (5), Freedman and Navidi (6) found a multistage model with seven total stages, two being dose related, to give the best prediction of liver tumor response rates in the discontinued treatment groups. They were unable to find a version of the multistage model that would fit a selected portion of the continuous dosing bladder tumor data and successfully predict bladder tumor responses in the discontinued dosing groups. Chiang and Conforti (7) postulated a new model for time to onset of the tumorigenic response, including a version to describe discontinued dosing data. Working with only the EDO, bladder tumor data, they were unable to predict consistently the observed responses in the discontinued dosing groups. Although previous investigators have been able to describe successfully the EDO, discontinued dosing liver tumor responses within the framework of the multistage model, none has yet modeled successfully the EDo0 bladder tumor data for discontinued dosing. Indeed, it has been shown that acceptable model fitting for continuous dosing data cannot be taken to imply correctness of the model for extrapolating to discontinued dosing situations (6,7). The purpose of this paper is to investigate the currently popular two-stage clonal expansion model (8,9) with respect to its ability to describe the EDO, discontinued dosing data. The success of such modeling would perhaps enable the characterization of 2-AAF-induced liver and bladder tumorigenesis in terms of initiation, promotion, and completion (progression) (10). At any rate, successful prediction of bladder tumor responses for discontinued dosing groups by the two-stage clonal expansion model would add to its growing acceptance for carcinogenesis dose-response modeling.

Two-Stage Clonal Expansion Model
In recent years there has been movement away from the multistage model toward a model with only two distinct stages (e.g., two mutations or other genetic events) but with proliferation of cells that have undergone the first stage. One popular formulation of the model was proposed by Moolgavkar and his co-workers (8,11). Another widely recognized version of the twostage clonal expansion model has been developed and applied by Greenfield et al. (9). As pointed out by Bogan (12), the original two-stage model formulated by Armitage and Doll (13) provided for cell proliferation. However, the most widely studied and applied version of the multistage model does not (4). Whereas the growth of cells that have undergone the first stage in the two-stage clonal expansion model is exponential, Bogen (12) argued that geometric growth appeared to provide a better representation of certain data sets. In this paper; the two-stage model with exponential cell growth will be studied, with the terminology of Thorslund et al. (10) being used to characterize carcinogenic action. Specifically, carcinogenic action will be modeled in terms of initiation, promotion, and completion (progression), where initiation and completion refer to the occurrence of the firstand second-stage events, respectively, in the two-stage model, and promotion refers to the growth of initiated cells.
For continuous exposure to a carcinogenic agent at a dose rate d between time 0 and time t, the instantaneous hazard (or age-specific incidence) rate of the two-stage model with exponential growth of initiated cells may be written as t h(d,t) = I X<(d,u) 2(d,t) N(u) 0 t exp( I g(d,v)dv) du, O<t u where X, and k2represent, respectively, the rates of occurrence of the first-and second-stage genetic events, 6 represents the net birth minus death rate of firststage cells, and N is the number of normal cells at risk of a flrst-stage event. For a single period of dosing at a constant rate d from time 0 to time t1, followed by a period of discontinued dosing up to time t, the cumulative hazard for the twostage clonal expansion model under the above formulation is (see appendix) {expIcI3(t-tj)i l 1 -a3(t-tl)}, tI < t.
With this notation, the cumulative hazard under continuous dosing would be designated by H(d,t;0,t). Vari-ous special cases of the two-stage clonal expansion model may be obtained by setting selected coefficients of d(pi) equal to zero in the continuous and discontinuous versions of the cumulative hazard function. The pure initiator model arises when P2 = P = 0, the pure completer model when = l = 0, and the pure promoter model when P = 2= 0 (14). The initiator-completer model arises when P = 0, the initiator-promoter model when 12 = 0, and the promoter-completer model when 1P = 0.
Due to the fact that the rates of occurrence of the two genetic events appear as a product in the two-stage clonal expansion model, they are not, in general, separately identifiable. For continuous dosing data only, the models for pure initiator and pure completer are indistinguishable, although they can be distinguished with discontinued dosing data (14). The same is true for the initiator-promoter and promoter-completer models. Thus, if discontinued dosing data are available, then the pure initiator, pure completer, initiator-promoter, and promoter-completer models can all be resolved in terms of parameter estimation and model fit. However, if parameters are estimated and goodness-of-fit is ascertained using continuous data only, then extrapolation to situations of discontinued dosing, although it may be carried out, is very dependent on which of the indistinguishable continuous-dosing models is assumed. A different extrapolation problem arises for the initiatorcompleter model and for the full initiator-promotercompleter model. Although these models are identifiable for both continuous dosing and discontinued dosing data, the continuous dosing version of neither model can be extrapolated to a discontinued dosing situation, since individual parameters in the genetic event rates are nonidentifiable. Only the pure promoter model is uniquely identifiable irriespective of the dosing pattern.

EDO, Liver and Bladder Tumor Data
The ED,, study (1) was conducted in six animal rooms. Each was set up as a replicate of the experiment, to be loaded sequentially as animals became available from the NCTR breeding colony. For the various continuous dose groups (0, 30, 35, 45, 60, 75, 100, and 150 ppm 2-AAF), interim sacrifices were scheduled at 9, 12, 14, 15, 16, 17, and 18 months on study, with the earliest sacrifices being omitted in the lowest dose groups. Due to higher than expected survival and lower than expected bladder tumor rates by the time of the 18-month sacrifice in the first three rooms, some of the later scheduled sacrifices (including 18 months) were rescheduled for 24 months in the last three rooms. Thus, the ED0,, study's 18-month sacrifice data came from the first three rooms only, while its 24-month sacrifice data came from the last three rooms only.
For the four highest dose levels of 2-AAF, discontinued dosing was included in the study's design. For each of these dose levels, groups of animals received 2-AAF for either 9, 12, or 15 months after which dosing was discontinued until sacrifice. Sacrifices were all originally scheduled for 18 months, but as indicated above, those in the last three r ooms wer e rescheduled for 24 months.
Previous modelers of the EDO, tumor data have used various versions of the observed data for modeling purposes. Brown and Hoel (5) used only the liver tumor prevalence data observed in sacrificed animals, implicitly assuming that liver tumors were not related to the deaths of animals that did not survive to scheduled sacrifice. Chiang and Conforti (7) used bladderl tumor rates observed in combined sacrificed and dead/moribund animals, also implicitly assuming no relationship between bladder tumors and the deaths of the animals. Freedman and Navidi (6) used bladderl and liver tumor prevalence data from sacrificed animals as if the data were incidence data, i.e., they assumed that tumor s observed at a sacrifice arose during the month of sacrifice. Since the two-stage model with clonal expansion is, as are all other models of tumorigenesis, a model for the distribution of time to onset of tumors, it is important that the tumor data used for model fitting represent as closely as possible the observed distribution of time to tumor onset.
In this paper, the nonparametric estimator of Kodell et al. (15) is used to adjust the observed tumor data with respect to censoring by competing risks. Similar acljustments have been proposed by Dinse and Lagakos (16) andl Turnbull and Mitchell (17). With these methods, each animal's cause of death needs to have been determined by the examining pathologist. One of the strengths of the ED(,, pathology data is that cause of death was reported for each animal that was removed as deadl or moribund from the experiment. Hence, an adjustment forc death from competing risks could be made. With the method of Kodell et al. (15), the distribution of time to tumor onset is estimated nonparametrically for each dose group separately. Thus it is possible to obtain dose-response information at any point in time by using the value of the time to tumor distribution function from each dose group at that time. In the present study, the time points of primary interest were 18 and 24 months, the sacrifice times in the discontinued dosing groups. Both hepatocellular adenomas and carcinomas were included as liver tumors, while only carcinomas were included as bladder tumors.
The data used for model fitting are given in Tables 1  and 2 for liver and bladder tumors, respectively. P represents the nonparametric estimate of the probability of tumor onset by time T (18 or 24 months), X is the number of animals observed with tumors up to time T, and N = X/P is the "effective" sample size. That is, for given P, N is the number of animals that would have been put on test initially, if time to tumor onset of each animal could have been observed, and if exactly X animals of the N would have developed tumors by time T. Data on all dead, moribund, and sacrificed animals are reflected in the numbers in Tables 1 and 2, even though only times corresponding to sacrifices at 18 and 24 months have been selected for evaluation.
Due to the nesting of the 18-and 24-month sacrifices 'D represents close of 2-AAF in parts per million; T, represents time of termination of dosing expressed in months; T represents time of observation of time-to-tumor onset distribution expressed in months. 'P is the nonparametric estimate of the probability of tumor onset by time T; X is the actual number of tumors observed up to time T; N=X/P is the "effective" sample size.
within rooms (and, consequently, much of the data on dead/moribund animals), it was decided to estimate time to tumor distributions using combinations of three ani-mal riooms, grouped according to common sacrifice time. Although this approach confounds room differences with evaluation time differences, it does provide a starting "D represents dose of 2-AAF in parts per million; T1 r epresents time of termination of dosing expressed in months; T represents time of observation of time-tumor-onset distribution expressed in months. 'P is the nonparametric estimate of the probability of tumor onset by time T; X is the actual number of tumors observed up to time T; N=X/P is the "effective" sample size. point for investigating potential lack of fit of models. That is, if prediction of tumor rates for discontinued dosing groups based on models fit to continuous dosing data should break down due to room differences, it would still be possible to perform such predictions individually for 18 and 24 months (i.e., within groups of rooms). Whereas using tumor data from all six rooms taken together overcomes the confounding just mentioned, any model breakdown that might be due to room differences could not be so identified. Certainly for collapsed data, tumor probability estimates at 18 months are heavily influenced by the first three rooms and those at 24 months by the last three rooms.

Model Fitting
The nonlinear regression procedure (NLIN) in SAS (18) was used for fitting various versions of the twostage model with cell proliferation to the tumor data in Tables 1 and 2. Weighted least squares was employed, with reciprocals of estimated variances being used for weights. The weighted sum of squares that was minimized is where Pi and Ni are the values of the nonparametric estimate of the probability of tumor onset by time T, and the "effective" sample size, respectively, from Table  1 or Table 2 for the ith group; 1IN; is a correction for continuity; Pi is the predicted probability of tumor, 1-exp {-H(D,T;0,T1)}, from the two-stage clonal expansion model for the ith group; and m is the number of distinct groups (distinct combinations of D, T, and T in Table 1 or Table 2) used for fitting the model. Since the sum of squares minimized is asymptotically a chi-square random variable, the estimation method is a modified minimum chi-square method.
Computationally, the Gauss-Newton method, the Marquardt method, and the derivative-free multivariate secant method (designated as DUD in NLIN) were used to determine the direction and distance for each succeeding iteration. All parameters were constrained to be nonnegative using BOUNDS statements.
To assess goodness-of-fit, a chi-square statistic like the one minimized was used, except that predicted values of Pi obtained from the model, rather than observed values, Pi, were used to calculate variances in the denominator. If the fit of the model to a set-of data points used to estimate the model's parameters was being assessed, then the degrees of freedom for chi-square were calculated as the difference between the number of data points and the number of estimated parameters. If the fit of the model to a separate set of data points not used for fitting the model was being assessed, then the degrees of freedom for chi-square were taken was the number of data points.

Results for Two-Stage Clonal Expansion Model
For each of the two tumor types, the first step was to fit various versions of the model to all 40 data points in order to ascertain which versions should be eliminated and which warranted further investigation. Those submodels of the general model that fit acceptably to all the data points were further studied by fitting them to only the 16 continuous dosing data points and then assessing their predictive ability for the 24 discontinued dosing data points, where possible.
For the liver tumor data (Table 1), the full, sixparameter model fit acceptably to all 40 data points (p = 0.76), as did the four-parameter initiator-promoter model (p = 0.40) and the five-parameter initiatorcompleter model (p = 0.70). All other submodels demonstrated significant lack of fit (p < 0.0001). The estimated parameters of the six-parameter model are given in Table 3, and the predicted probabilities of tumor are presented in Table 4, where the observed probabilities from Table 1 are reported for comparison purposes. As stated previously in "Two-Stage Clonal Expansion Model," neither the full initiator-promoter-completer model nor the initiator-completer model can be extrapolated from a continuous to a discontinued dosing situation due to a lack of identifiability of parameters. Thus the only model left for such extrapolation was the initiatorpromoter model. That model gave an acceptable fit to the 16 continuous dosing data points (p = 0.71) and also fit well when extrapolated to the 24 discontinued dosing data points (p = 0.68). However, as previously noted, the continuous dosing version of the initiator-promoter model is indistinguishable from the promoter-completer model. And the promoter-completer model, as would be expected since it did not flt the 40 data points initially, demonstrated a signiflcant lack of flt when extrapolated from the continuous to the discontinued dosing data (p > 0.00001).
It is not possible to choose which, if any, of the three models that flt acceptably to all 40 liver tumor data points is the "correct" model. That is, the present bioassay data do not contain the refined biological information necessary to choose one model over the others. In a sense, it is gratifying that the full two-stage model with colonal expansion, the initiator-promoter-completer model, flts the data. However, it is not very satisfying that two submodels, the initiator-promoter and initiator-completer models, are approximately equally acceptable, since they represent two different biological mechanisms. From a statistical point of view, the initiator-promoter model might be preferred, since it has the fewest parameters and since it was possible to demonstrate for that model successful extrapolation from the continuous to the discontinued dosing situation. The importance of discontinued dosing experiments is high-   bD represents dose of 2-AAF in parts per million; T, represents time of termination of dosing expressed in months; T represents time of observation of time-to-tumor-onset distribution expressed in months. eP (obs) is the nonparametric estimate of the probability of tumor onset from Table 1. P (pred) is the predicted probability of tumor onset from the two-stage colonal expansion model. lighted by the rejection of the promoter-completer model based on the discontinued dosing data, whereas that model is indistinguishable with continuous dosing data from the very successful but mechanistically very different initiator-promoter model.
For the bladder tumor data ( Table 2), none of the seven versions of the two-stage clonal expansion model fit acceptably (p < 0.00001), whether all 40 data points or just the 16 continuous dosing data points were used. Table 3 gives the estimated parameters for the full, sixparameter model fitted to the 40 data points in Table 2. The predicted probabilities of tumor for the sixparameter model are given in Table 5, along with the observed probabilities from Table 2.
It is quite disappointing that the two-stage clonal expansion model would not fit even the 16 continuous dos-ing data points. Several possible remedies were investigated in an attempt to achieve an acceptable fit of the model to these data. The pharmacokinetic dose transformation of Whittemore et al. (19), d' = aldk (1 + a2dk), was tried for k = 1,2, which resulted in the estimation of one additional parameter (a2). This is the familiar Michaelis-Menten transformation when k = 1. A power transformation, do = dk, also was attempted for k = 2 to 4. This was used first throughout the full model and then only in the cell proliferation function. None of these remedies provided an appreciable improvement in the fit of the model.
It has been the experience of other investigators with other mechanistically derived models that the EDO, liver tumor data are consistent with a variety of such models (5,6), but that the EDO1 bladder tumor data are not ad- 'P (obs) is the nonparametric estimate of the probability of tumor onset from Table 2. P (pred) is the predicted probability of tumor onset from the two-stage clonal expansion model. equately described by any of them (6,7). It appears that the two-stage clonal expansion model as formulated in this paper may be characterized the same way. For this reason, a different approach is proposed in the next section.

Weibull-Type Model for Discontinued Dosing
The model to be proposed here is more empirically based than biologically based. Although the Weibull distribution can arise naturally in the theory of systems reliability, its application to biological problems has consisted mostly of empirical curve fitting. In the analysis of toxicity data, both dose and time have been modeled separately as Weibull random variables. When dose and time have appeared together in Weibull-type models, generally time has been considered the random variable (20). The commonly used continuous dosing version of the multistage model of Armitage and Doll (4) may be considered a Weibull model in time, with the polynomial function in dose representing a scale parameter.
Peto et al. (20) used two versions of a Weibull model to describe carcinogenesis data on nitrosamines in a study with a large number of rodents. In this discussion, these two models will be distinguished according to whether background is modeled as "additive" or "in- where a, f, w, and k are all nonnegative constants. For liver tumors that occurred under a continuous dosing regimen, Peto et al. (20) found that the additive background model described the data best, there being an appreciable background incidence of liver tumors. For esophageal tumors observed under continuous dosing, Peto et al. (20) found that the independent background model fit best; in fact, the observed background incidence of esophageal tumors was zero, so that a was zero in the model for esophageal tumors.
The parallels between the nitrosamine-induced liver tumors and the EDO1 study's 2-AAF induced liver tumors and between the nitrosamine-induced esophageal tumors and the EDO1 study's bladder tumors suggest that a Weibull model with additive background might fit the EDO1 liver tumor data and that a Weibull model with independent background might fit the EDO1 bladder tumor data. But how does one generalize the Weibull model to the discontinued dosing situation? Unlike mechanistically derived models, the Weibull model does not have a natural extension to discontinued dosing.
One generalization of the Weibull model to discontinued dosing was given by Carlborg (21). He modeled the EDO, tumor prevalence data using only what is described above as the independent background model. His generalization to discontinued dosing included a component for duration of dosing, but did not account for when the dosing occurred. Carlborg (21) successfully fit a five-parameter model to the EDO, liver tumor prevalence data, excluding a portion of the low-response continuous dosing data, but including all the discontinued dosing data. However, Carlborg (21) did not fit the bladder tumor data on dose and time simultaneously. Apparently, the model would not fit.
Freedman and Navidi (6) looked briefly at fitting the Weibull model to EDO, bladder tumor prevalence data.
They were unable to obtain an acceptable fit, but noted a tendency for the dose exponent, w, to exceed the time exponent, k, which they noted to be inconsistent with the multistage model. Freedman and Navidi (6) quoted the argument of Brown and Hoel (22) that a factorable hazard model (like the Weibull or multistage) will not fit the continuous dosing bladder tumor data over all dose and time points. Brown and Hoel (22) made a similar but weaker statement about the liver tumor data, which Carlborg (21) was able to fit acceptably, along with the discontinued dosing data, using a factorable Weibull model. The model that will now be proposed is a discontinued dosing extension of the Weibull model used by Peto et al. (20). First consider additive background. Assume initially that dosing occurs only between time to and t1 and at a constant rate, d. The model can easily be generalized to multiple periods of constant exposure, as will be discussed later. Let X represent a latency parameter, that is, a minimum amount of time that must elapse before a tumor can be observed. Suppose that to <. < ti. Then U(d,t) 0 9 t < ir 9 and D(d,t) (sd) (t-r)t, T < t < tit where l has been replaced by P1i for notational purposes.
If administered dose is abruptly discontinued, there is no reason to assume that the effective dose will immediately vanish. To generalize to time t > ti, suppose that the general form of H(d,t) is the same but that time to tumor is increased by a location shift of the distribution to the right and that the dose is effectively reduced by a scale factor. That is, let HI(d,t) -(,p2d)v (t-'r-c)w , t1 < t where c > 0 and P2 < P. In order for this generalization to be valid, at time t1, the condition (so-d)' (tl--s)' -(c*kd) (t'c must be satisfied. This implies that c must be equal to (t1-t)[1 -t(a+Pjd)/(a + 12d)1w'k], which in turn implies that v(d,t) = j(e*jSd) VA(t-tj) + (c.pd)V/k (t1-c)JI , t1 < t This generalized Weibull model (with to < t < t1) can be applied to the EDO1 liver and bladder tumor data. The independent background version is derived similarly and may be expressed as O ,P t < -, )k, X < t < ti l(pdv) 1/k kt-tl) Other versions of this generalized Weibull model depend on the relative magnitude of t with respect to to and tl. For 0 < t < to, the additive background model has B(d,t) v k(to-r) + (c*ud) (t1-t0) + a*2d)v/k(t-tk)kst, t I with an analogous expression for the independent background model. If t1 < t, then the additive background model's cumulative hazard is H(d,t) = I(.02d)/k(t- ¶)I -(* 02d)V(t-r)k 9 t, C t with an analogous expression for the independent background model. With regard to multiple periods of constant exposure, the cumulative hazard can easily be derived in the same way as for a single exposure period. It will have a similar appearance, but with additional terms inside the bracketed expression raised to the kth power.

Results for Weibull Discontinued Dosing Model
For the liver tumor data (Table 1), the additive background model fit acceptably to the 40 data points (p = 0.43), as did the independent background model (p = 0.63). The estimated value of t was essentially zero (7.2 x 10-17) for the additive model, whereas the estimated value was 0.81 for the independent model. Parameter estimates for the additive model, for which the prediction of background tumor rates was slightly better, are given in Table 6. The estimated power on dose (w = 1.7) and the estimated power on time (k = 7.7) are reasonably close to values (2 and 7, respectively) found by Freedman and Navidi (6) for the dose and time powers in the multistage model. Table 7 compares the predicted probabilities of the six-parameter additive background model to the observed probabilities from Table 1. cGoodness-of-fit p-value = 0.47.
For the bladder tumor data (Table 2), the indepen-background model to the observed probabilities from dent background model fit acceptably to the 40 data Table 2. points (p = 0.47), whereas the additive background The results in Tables 6, 7, and 8 indicate good fits and model did not (p < 10-7). Practically all the lack of fit of predictive ability of the given Weibull models within the additive background model appeared to be in the the range of observed doses (D), stopping times (T1), and background itself. Parameter estimates for the inde-observation times (T). In order to evaluate the predicpendent model are given in Table 6. The relative mag-tive ability of the models at points outside the range of nitudes of the estimated power on dose (w = 6.1) and data used for fitting, observation times T = 15 and 33 the estimated power on time (k = 2.2) are inconsistent for continuous dosing were selected for prediction. The with a multistage formulation in administered dose results are given in Table 9. The 15-month predictions (wherein the power on time must equal or exceed the appear consistent with the observed data, which is itpower on dose), a phenomenon noted by Freedman self characterized by considerable variability about the and Navidi (6) in reference to Carlborg (21). The same is lower dose rates. However, at 33 months the predictrue for the two-stage model with clonal expansion; tions appear to break down. The liver tumor model is however, in both cases, a judicious transformation of characterized by overprediction at the middle dose levdose might improve the fit. Table 8 compares the pre-els, while the bladder tumor model is characterized by dicted probabilities of the six-parameter independent substantial underprediction at 75 and 100 ppm.   cP (obs) is the nonparametric estimate of the probability of tumor onset from Table 2. P (pred) is the predicted probability of tumor onset from the Weibull discontinued dosing model. hPi(obs) is the nonparametric estimate of the probability of tumor onset by time T based on data from rooms 141,142, 143 combined; P2(obs) is the corresponding estimate from rooms 144,145,146 combined; P (pred) is the predicted probability of tumor onset from the Weibull models described in Table 6.

Discussion
It is disappointing that the two-stage model with proliferation of initiated cells would not fit the bladder tumor data from the EDO1 study. The experience here, like that of previous investigators (6,7,22), indicates that the relationship between dose of 2-AAF and time to blad-der tumor might be too complex to be explained by mechanistic models so far postulated. In the present case, even allowing for the Michaelis-Menten and other transformations of the administered dose, an acceptable fit was not obtained.
The fact that the two-stage clonal expansion model did fit the EDO1 study's liver tumor data is interesting 1 and helps to narrow down the possible characterizations of liver carcinogenesis within the context of the two-stage clonal expansion model. Although a complete characterization was not possible, at least the simplest notions of pure initiator, pure promoter, and pure completer could be eliminated. The importance of discontinued dosing experiments is highlighted by the fact that the promoter-completer model was rejected, while the initiator-promoter model was not, the two models being indistinguishable for continuous data only. It is probably not advisable to make too much of the twostage clonal expansion model's fit to the liver tumor data. For one thing, various other models have pirovided equally acceptable fits. For another, the model's failure to fit the bladdcer tumor data makes its biological interpretation less compelling.
With respect to the Weibull-type model that (lid fit acceptably to the 18-and 24-month data on both liver and bladder tumors, what conclusions can be drawn? Unfortunately, this model too broke down when extrapolated to tumor probabilities at 33 months. However, the good fits to the discontinued dosing data an(l the successful extrapolation to 15 months of continuous dosing give confidence that good predictions can be ma(le for both liver and bladdcer tumor probabilities up to 24 months, regardless of when dosing stops. The importance of discontinued dosing experiments is again emphasized. Whether a model is mechanistically based or simply empirically based, its ability or inability to explain discontinued dosing tumor data will priovide a good measure of its predictive capability beyond the experimental setting. In addition to the discontinued dosing version of the Weibull-type model presented here, there are other formulations that might be useful. One such model that was investigated in this study but not reported had only four parameters. Instead of the additional scale parameter 12 being introduced, the original parameter 13 was retained, but the dose was transformed to the time-weighted-average dose. This model gave an acceptable fit for both liver and bladdeer tumors. However, the bladder tumor fit was not as good as that of the model reported here. The advantage of such a model would be its extrapolation potential to discontinued dosing situations based on parameters estimated firom only continuous dosing data. We  Of primary interest in this paper is the cumulative hazard function for t1 < t. This is found by summing together the integrals of the above three instantaneous hazards, where the limits of integration on w are, respectively, 0 to t(, to to ti, and t1 to t. [expf(oc3 T NdU)(t1 t)) 11 a1 +d + .