Models for analyzing data in initiation-promotion studies.

The objective of this paper is to construct a class of models for analyzing data in initiation-promotion (IP) studies. After the application of an initiator in animals IP studies, histochemical and/or histopathologic criteria are used to define the foci that are postulated to be the origin of tumors. Thus, the dynamics of foci growth are of inherent interest in the study of the mechanism of carcinogenesis. In this paper, models to explain these dynamics are developed and can be used to differentiate among proposed mechanisms of tumor formation and promotion. Examples are given to illustrate useful concepts for analyzing data from IP studies.


Introduction
The initiation-promotion (IP) study has been used to evaluate the mechanism of tumor promotion in various systems including liver, skin, and bladder. Tumor promotion is important because it constitutes a crucial stage in tumor development. Cellular proliferation has been viewed as a major factor of influence of all stages of malignant hepatic transformation. In their studies of vinyl chloride (VC) on Wistar rat liver, Laib et al. (1) found that continuous exposure of adult rats to VC did not result in either an increase in the area of foci or incidence oftumors over that of the controls; however, more enzyme-altered single cells, which could be assumed to be initiated cells, were observed when compared with control animals. This contrasts with the observation by Laib et al. (2) that the induction ofpreneoplstic hepatocellular lesions and hepatocellular carcinomas in rats by VC is mainly restricted to an exposure in the early lifetime when an animal undergoes a rapid liver growth. This observation suggests that modeling only the number of initiated cells (I-cells) without taking into account the frequency and size of foci could be misleading. Therefore, both frequency and size offoci are important factors for modeling tumor formation.
In IP studies, a promoter is administered over a period oftime following the application of an initiator at a dose level too low to induce tumors by itself but high enough to initiate a normal cell. During the period of promotion, initiated cells undergo rapid multiplication, eventually leading to neoplastic growth. An attractive feature ofthe IP protocol is that the ability for a suspect carcinogen to initiate and promote can be determined. Use ofthe IP or IPI protocol has been suggested by Krewski (3) as an approach to obtain parameters in the two-stage model developed by Moolgavkar and Venzon (4).
Putative preneoplastic foci (islands) induced by the initiator are identified by various histochemical or morphologic markers. These foci have proliferative advantage over the normal hepatocytes. This advantage is manifested when the promoter is administered. Aproblem inacarcinogenic mechanism study using the IP protocol is that the foci identified by these markers may not be mechanistically related to tumor formation. Therefore, it is desirable to have a model toprovide a framework for evaluating the relationship between foci and tumors. Ifthe model fails to predict the observed tumor incidence given available dynamics data on foci, we may conclude thateither the model is not adequate or the foci as identified are not mechanistically related to tumors. On the other hand, if the model predicts the observed tumor incidence given the dynamic data on foci, one would be more confident in assuming that the identified foci are tumor precursors even though we still cannot be definite about their exact mechanistic relationship to the tumors. In constructing a model ofthe relationship between a preneoplastic entity (e.g., foci, nodules) and tumors, we will first investigate the number and distribution ofsize ofdetectable foci at any given timeafterapplicationofan initiator. We also discuss the usefulness of using the maximum sized focus in estimating IP potential for a suspect carcinogen. The probability ofa tumor is calculated using a model that highlights the progression of the foci/nodules to malignant tumors. Finally, the models are applied to data on hepatectomized and nonhepatectomized rats, which were initiated by diethylnitrosamine (DEN).
Basic Assumptions a)Attime, t = 0, anormalcellhasaprobability, A1, ofbeing initiatedwhenaninitiatorisapplied. Thebackgroundinitiating rate is assumed negligible compared with the rate induced by the initiator. b) Each I-cell has a random lifetime (i.e, time to mitosis) with a probability density function, f(t) and the lifetime distribution function, F(t). c) At mitosis, an I-cell is subjected to a homogenous birth and death process with probabilities ofbirth and death given respectively by b and d with b + d = 1. d) All cells go through this process independently of each other. Dewanji et al. (5) have studied the birth and death process of focal growth without considering the random time to mitosis. Chover and King (6) also studied the growth ofa focus by assuming that a focus grows as a pure birth process. Our model offers improvement over the previous papers by considering random time to mitosis. There are many competing mechanisms of cell proliferation: from a modeler's viewpoint, cell proliferation is characterized by an increase in mitotic rate or a decrease in cell loss, or both. Thus, it is useful to incorporate mitosis information in a model. Furthermore, if time to mitosis is assumed to be exponentially distributed and the mitotic rate is known, then our model requires a fewer number ofparameters to be estimated than models that do not explicitly incorporate the mitotic rate.
Under assumptions a and d, the number of cells initiated at time, t = 0, can be assumed to be a Poisson variable with mean equal to tINo, where No is the number of normal cells at the time when the initiator is applied. All the foci to be observed later are assumed to have originated from these I-cells. Therefore, the number of the (detectable) foci must be less than the number of I-cells and can be assumed to be a Poisson variable.

Model Development
Assume that a tumor is developed in the sequence of normal cells, I-cells, foci, nodules, and tumors. Some ofthese events are observable under the IP protocol under which animals are serially sacrificed. We now proceed to take a closer look at each ofthe three preneoplastic entities (I-cells, foci, and nodules) and their relationship to tumor formation. Eqs. (2) and (3) can be used to calculate expected focus size at any time, t. In general, the analytic solutions ofthese equations are difficult to obtain. However, a numerical method can always be used to obtain the solutions for any given f(t). To introduce useful concepts for analyzing data from IP studies, the case where the time to mitosis is exponentially distributed is considered.
When f(t) = Xexp (-Xt), where the 1 / X is the mean time to mitosis, Eq. (1) becomes: Differentiating and simplifying, we have: This is a Riccati equation with constant coefficients. By noting that X = X (b + d), the solution is readily found to be the well known birth-death process: A (t) = rB (t) (7) (8)

Size of Foci
Let X(t) be the size (number ofI-cells) ofa focus at time, t, that is originated from an I-cell at time, t = 0. Let G(s,t) be the probability generating function of X(t). Following a similar approach to that ofKarlin and Taylor (7) in which a pure birth process with a random cell life was considered, it can be shown that G(s,t) satisfies the integral equation: This integral equation is fundamental for calculating the probability distribution and moments of X(t) for any given density function oftime to mitosis. For instance, the expected value function m(t) and probability of extinction PO(t) respectively satisfy the integral equations: The probability, P,(t) = Pr [X(t) = k], that a focus has size, k, is given by: and Since Po(t) is the probability that a focus is extinct, the probability that a nonextinct focus has size, k, is given by: We note that Qk (t) is a geometric distribution with the parameter 1 -A(t). The mean and variance for the size ofa detectable focus can be easily calculated.
Ifwe assume that a focus becomes detectable when it contains at least s cells, then the probability for a focus to be detectable is:  Foran arbitary x 2s, D.(t) is the proportion of foci thathavea size exceeding xcells. From Figures and2, which compare the growth offoci with and withoutpartialhepatectomyandareconstructed using the parameters derived in the application below, given one sees that the partial hepatectomy group (X = 0.12) has about90% ofthe foci expectedtoexceed 10,OOOcells by time, t = l50days, whilethecontrolgroup(X = 0.02)haslesstdan 1% ofthe foci expected to exceed 10,000 cells even at 450 days. The probability for a detectable focus to have size k (k 2 s) is: Pk.s(t) = [l -A(0)l[A(t)lk-S-l,ks >0 It is ofpractical interest to know the statistical properties about the maximum focus because it is relatively easy to measure, and it can be measured sequentially over time. A comparison ofthe sizes ofthe maximum focus among groups over time can reveal their carcinogenic potential because, as is to be shown, the distribution of the maximum-sized focus involves both the number of I-cells (initiation potential) and size ofthe foci (promotional potential). Thus, the maximum-sized focus can serve as an index of initiation/promotion as opposed to the indices of initiation and promotion defined by Pitot et al. (8). It can also be used to assess the promotion potential of a promoter in an IP study where both the promoter-treated and control animals are subjected to the same dose of initiator.
For a given number of nonextinct foci. I(t) = n with n > 0, the size of the maximum focus, X,, has a conditional distribution: For the nonconditional case, X, has an approximate distribution, assuming that P[I(t) = 0) is negligible, Fm (i, t) = exp {-E lI (t)l LA (t)li I (18) where E[I(t)] is given by Eq. (7).
The expected value of X,(t) can be approximated by a finite number of terms in a convergent series (9):

Probability of Tumor
It is possible to construct a stochastic process oftumor growth ifour third basic assumption that an I-cell is subjected to a birthdeath process at mitosis is extended to include the possibility that, at mitosis, an I-cell also has a probability of producing an I-cell and a malignant cell. A stochastic model that incorporates the above assumption, as well as the birth-death of malignant cells, is given by Chen and Farland (10).
Since the objective of this paper is to study the kinetics/dynamics ofpreneoplastic lesions, it is ofinterest to investigate how the incidence rate ofthese preneoplastic lesions affects the occurrence of malignant tumors. Data useful for the modeling include information on the number of preneoplastic and neoplastic lesions per aninal and their size distribution over time. A procedure using the sequential data on preneoplastic and neoplastic lesions to construct a dose-response model is given by Chen and Moini (II). However, because of the lack of data and our desire to focus only on the IP-related issues in this presentation, we use a simple and heuristic model to illustrate the relationship between nodules and tumor incidence rates. An approach to model tumor incidence is to assume that only nodules can become tumors because nodules contain more advanced cell types, some ofwhich can progress to tumors (12,13). As an operational definition, nodules are defined here as foci that have a size 0.5 mm (about 6000 cells) or larger in diameter. It is implicitly assumed that once a focus advances to a nodule, it cannot be reversed. This definition is motivated by Rotstein et al. (14) and Farber and Sarma (13), in which they report that the size of nodules is 0.5 mm in diameter or larger and that a small percentage of hepatocyte nodules (termed ""persistent" nodules) may commit to the pathway ofevolution toward cancer. An important implication oftheir finding is that the sheer size (number ofcells) of a nodule is not the only determining factor of its potential to progress into a tumor; thus, it is not reasonable to assume that the rate ofconversion to tumor from a nodule is linearly proportional to the number of cells in a nodule.
In this paper, we operationally define a nodule by the size of a focus because data on nodules and tumors are not available. If the number of nodules and tumors and the rate of formation of nodules can be biologically determined, then one need not artificially define a nodule by the size of a focus. A nodule is defined as a focus that contains m (e.g., 6000) or more I-cells with an assumption that once a nodule is formed, it cannot be reversed. Under these assumptions, the rate for a nodule to occur is given by: In this section, we present some calculations that demonstrate the usefulness of the model and the importance of considering time to mitosis. We estimate below that X =0.12 (a mean cell life of about 8 days) if the liver is partially hepatectomized. It is assumed that X = 0.02 (a mean cell life of 50 days) ifthe liver is not partially hepatectomized. It should be noted that these values are only estimated from data available to us. The objective here is to demonstrate our models, not to provide accurate estimation of these parameters. Other values of X are also used in the calculations. Before the application of the models, some knowledge about the parameters is needed.
The ideal data for estimatng parameters are frequency and size of foci over time. Although there are many IP studies in the published literature, these ideal data are generally not available. Table 1 gives some data that are reconstructed from graphs in Scherer and Emmelot (15). These data are obtained after a single intraperitoneal injection of 10 mg/kg DEN to partially hepatectomized rats.
Under the experimental conditions described for the data in Table 1 Eq. (6), along with the parameters given above, is used to fit data in Table 1 by the least squares method. The parameters, X, b, and d, are respectively estimated to be 0.12, 0.89, and 0.11. The predicted numbers of foci per liver along with the observed values are given in Table 1.
In order to calculate the tumor incidence, the assumption is made that the rate oftransition from an I-cell to tumor isp = 1.7!) x o0-8 per day. The value pis selected such that the probability ofa tumor predicted by the model is lessthan 0.05 att = 250 days to reflect the observation than no tumors were detected by that day in animals exposed to a single dose (10mg/kg) ofDEN by intraperitoneal injection (15). Using the parameters given above, we proceed to make some application of the models.
Eq. (9) is used to calculate the expected size ofthe maximum focus at time, t = 10, 20, 50, and 75 days, after application of DEN (Table 2). We have calculated size ofmaximum focus only up to 75 days because size of a focus is known to increase exponentially only at early stages and then to level offas t increases (17). Since maximum focus is relatively easy to measure, it can be used to study the tumor promotion. As demonstrated in Table  2, the relative sizes ofthe maximum focus between the partially hepatectomized and nonhepatectomized groups are highly Assume that the probability for a nodule to become a tumor during the time interval (t, t + h) is ph + 0 (h), where p is the transition rate ofnodule to tumor. Thus, the hazard rate oftumor is given by: The probability of tumor by time, t, is given by: nonlinear over time. These calculations suggest that the ratio of maximum focus size between the treated and control animals may be used as a promotion index in an IP study where both the treated and control animals are subjected to the same initiation treatment. When comparing the promotion effect between two groups oftreated animals, the condition that both groups are subjected to the same initiation treatment is required because the probability distribution ofthe maximum focus size involves both the number of I-cells and their growth rate. Figure 3 shows the relationship between nodules and tumor incidences, when X = 0.12 and p = 3.9 x 10-7. A larger value of p is used to increase the visual effect ofthe graph. The incidence of nodules increases rapidly to peak at about 90 days after the DEN treatment and then decreases, reflecting the promotional effect of partial hepatectomy. On the other hand, the tumor incidence increases and then levels off, reflecting the response pattern of nodule incidence. If X is small, it is expected that both nodule and tumor incidence will increase over time. The implication of Figure 3 is that if a population is exposed to a promoter, the relative risk will increase and then level off, consistent with the general belief of what a promoter would do. Figure 4 compares the probability of a tumor for different values ofmean time to mitosis, 1/X, using Eq. (11). The effect of tumor promotion (i.e., an increasing value ofX) on tumor induction is clearly seen from these curves.  Discussion Pitot et al. (8) have introduced a method to quantitate relative initiatng and promoting potencies of hepatocarcinogenic agents. The same concept could be used to quantitate the relative carcinogenic potencies of environmental pollutants that generally occur as a complex mixture in waste sites. Determining the priority of the site cleanup often requires knowledge about the carcinogenic risk resulting from the potential exposure from such sites. Since the composition ofa complex mixture varies among sites, it is not practical to conduct a long-term bioassay for each site-specific mixture. A possible solution to this problem would be to perform IP studies on complex mixtures taken from these sites, calculate their initiating and promoting potencies, and compare these potencies to a reference mixture of which the carcinogenic, as well as its initiating and promoting potencies, are known. Our models could be used to construct indices of initiation and promotion for a compound or a mixture ofcompounds. Further research would be needed to investigate the feasibility of this approach.
A model that takes into account the random time to mitosis is proposed for analyzing data in the IP studies. The consideration of random time to mitosis is biologically realistic. It has been shown that the time at which cell division occurs is not fixed (18). An advantage ofour model is that the tumor promotion effect can be interpreted by parameters relating to time to mitosis. This advantage is seen in a special case when the time to mitosis is assumed to follow the exponential distribution for which only one parameter (i.e., X) need be specified. An increasing value of X coincides with the increase of promotional capability of the treatment. There is a research need to investigate whether or not the assumption of exponential time to mitosis is reasonable.
We have also modeled the tumor incidence on the basis of nodules that are operationally defined as islands that exceed 0.5 mm in diameter (about 6000 cells), the lower range ofthe size of nodules that are operationally defined as islands that exceed 0.5 mm in diameter (about 6000 cells), the lower range ofthe size of nodules reported in Farber and Sarna (13). A desirable property ofthis model is that, at most, only one tumor will be developed from an island. The model is used to demonstrate the relationship between nodule and tumor incidence rates and the need to obtain data on both nodules and tumors.
An interesting application of our model is that the expected size of the maximum-sized focus can be calculated and used to study the promotion potential of promoters that are given to animals after administration ofan initiator. Since the maximum sized focus is relatively easy to measure, there is a significant practical implication ofthis aspect ofthe model. By studying the statistical property of this particular focus, one may find that it provides a great amount ofinformation about the carcinogenicity of a compound.
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