Assessing the effects of metabolism of environmental agents on cancer tumor development by a two-stage model of carcinogenesis.

By combining the Michaelis-Menten kinetics of metabolism with the two-stage model of Moolgavkar and Knudson (1981) and the extended two-stage model of carcinogenesis proposed by Tan and Gastardo (1985), this paper proceeds to investigate the effects of metabolism of carcinogens on cancer tumor development. It is shown that the nonlinear kinetics of metabolism of carcinogens affect the dose-response relationship mainly through the mutation rates. If the initiator is affected by metabolism, then the metabolism of promoters has very little or negligible effects of the expected incidences and the number of tumors.


Introduction
In assessing effects ofenvironmental agents on cancer development, it is important to note that the biological dose inside the cell is quite different from the exposure dose, and it is the biological dose that is directly responsible for cancer development. For example, Hoel, Kaplan, and Anderson (1) have shown that it is not the exposed dose but the DNA adduct of agents that gives a linear dose-response curve for small doses. By using Michaelis-Menten kinetics, Van Ryzin and Rai (2) and Van Ryzin (3) have shown that for the Weibull model, the one hit model, the multistage model, and the approximate multihit model, the nonlinear kinetics of metabolism of carcinogens have significant impact on doseresponse relationships in risk assessment. Further, as shown by Van Ryzin (4), in risk assessment, different models give very different results.
To provide a mathematical description of the carcinogenic process which can be used to interpret the results of experimental animal and human epidemiologic studies, Moolgavkar and Venzon (5) and Moolgavkar and Knudson (6) proposed a two-stage model of carcinogenesis. They modeled only two stages because no more than two distinct stages have been experimentally demonstrated. This model assumes that a malignant tumor develops from a normal stem cell after two cellular changes such as activation of cellular oncogenes; it dif-fers from the commonly used Armitage-Doll multistage model (7,8) in that the two-stage model includes stochastic birth and death processes to describe cell proliferation and differentiation of both normal stem cells and premalignant initiated cells (i.e., cells that have undergone only the first cellular change). By assuming different tissue growth patterns, Moolgavkar and Knudson (6) showed their model could fit incidence curves of all human cancers, while the Armitage-Doll model could only fit most tumors of adult onset. In addtion, Moolgavkar (9) and Tan and Gastardo (10) have shown that the Moolgavkar-Venzon-Knudson (MVK) two-stage model provides an explanation for the results of initiation-promotion animal carcinogenesis experiments, the initiator affecting the rate of occurrence of the first cellular change and the promoter affecting the proliferation rates of the initiated cells. The discovery of antioncogenes (11) provides biological support for the MVK model. As noted by Moolgavkar (12), pedigree analyses have shown that human cancers in some families are transmitted in an autosomal-dominant fashion.
Cytogenetic analyses of these hereditary cancers have revealed that particular genes are deleted. Thus, in contrast to oncogenes, it is the inactivation of these antioncogenes that leads to malignancy. Examples of antioncogenes include the retinoblastomas rb gene on chromosome 13 (13)(14)(15) and the Wilm's tumor wm gene on chromosome llp (16)(17)(18).
Since it is definitely desirable to use biologically supported models of careinogenesis to perform risk assessments of carcinogens, in this paper, we proceed to assess effects of metabolism of environmental agents by combining the Michaelis-Menten kinetics of metabolism of carcinogens with the two-stage model of Moolgavkar and Knudson (6) and the extended two-stage model of Tan and Gastardo (10).

Nonlinear Kinetics of Metabolism of Carcinogens and Carcinogenesis
As a well-documented example, it has been observed that mouse skin, when first treated by an initiator such as 7,12-dimethylbenz[a]anthracene (DMBA) and then followed by a promoter such as 12-O-tetradecanoylphorbol-13-acetate (TPA), gives rise to papillomas that may further progress with a very low rate of conversion to yield squamous cell carcinomas (malignant conversion) (19); however, Hennings et al. (20) reported that initiators such as N-methyl-N'-nitro-N-nitroso-guanidine (MNNG) or 4-nitroquinoline-N-oxide (r-QO), but not promoters, would induce carcinomas from papillomas. These results suggest different effects of metabolism of initiators and promoters. In terms of the twostage model of Moolgavkar and Knudson (6), initiators are associated with the mutation rates, while promoters are related to proliferation and differentiation rate of initiated cells.

Effects of Metabolism of Carcinogens That Are Initiators
To initiate carcinogenesis, carcinogens are first converted metabolically into chemically reactive forms that bind covalently to DNA adducts, leading to DNA lesions. The DNA lesions may be repaired (normal), or not repaired (die), or mismatched repaired, which leads to mutations (21). Recent experimental results of molecular biology have confirmed this theory for initiation of carcinogenesis. For example, Zarbl, Sukumar, and Barbacid (22) reported that, by injecting nitrosomethylurea (NMU) into the breast of female rats, NMU binds with DNA. Such a binding induces a G (guanine) to A (adenine) base transition at codon 12 of the ras gene, thus initiating the carcinogenesis process (initiation process).
To assess effects of metabolism of a carcinogen that is an initiator, we let C, M, and DM denote the carcinogen, the chemically activated metabolite of C and the DNA adduct, respectively. As illustrated in Gehring and Blau (23) and Hoel, Kaplan and Anderson (1), C may either be excreted or activated electrophilically to produce M; similarly, M is either detoxicated (deleted from the cell) or covalently bound to DNA to yield DNA adduct leading to DNA lesion. It is the mismatched repaired DNA lesion (error-prone repair) that is linearly related to the mutation rate al(I) of normal stem cells induced by mutagens and carcinogens. Let for DM DNA lesion. Let ao be the spontaneous mutation rate of normal stem cells. As illustrated in Trosko and Chang (21), spontaneous mutation is probably caused by error-prone replication of normal DNA, independently of induction of mutation by mutagens and carcinogens. It follows that one may express the mutation rate ox1 of normal stem cells by oX1 = aoo + P[DM], where ,B is a constant.
To relate [DM] to the exposed dose [C] of the initiator C, we assume Michaelis-Menten kinetics for both the activation process and the covalent binding process, but first-order kinetics for detoxication and other eliminating processes. Assuming steady-state condition for the metabolism, then, as shown in Van Ryzin where (VA, KA) are the Michaelis-Menten constants for the activation process; (VB, KB) are the Michaelis-Menten constants for the covalent binding process; and C1 and C2 are functions of detoxication rates and rates of other eliminating processes. This gives where y = ClC2VAVBI(KAKB) and 8 = KB + C1VA/(KAKB).

Effects of Metabolism of Carcinogens That Are Promoters
The exact mechanism of how promoters increase cell proliferation remains illusive. However, a rough picture painted by molecular biologists seems to suggest that promoters facilitate the release of active oxygen species (2, HO, 02*, and H202) or free radicals or organic peroxides and their degradation products, which may mediate the induction of poly (ADP)-ribosylation of nuclear proteins for cell proliferation and macromolecular synthesis (24)(25)(26)(27). For these electrophilic processes and/ or enzymatic processes, one may again assume Michaelis-Menten kinetics. Assuming first-order kinetics for detoxication processes and other elimination processes, the exposed dose [C] is then related to the biological dose [B] by [B] [C], where -y and 8 are constants that are functions of Michaelis-Menton constants, detoxication rates, and rates of other eliminating processes.
Let b1-dl be the difference between cell proliferation rate and cell differentiation rate of initiated cells. The above results then suggest that b1 and bd is the natural background difference of cell proliferation rate and cell differentiation rate ofinitiated cells.

Assessing Effects of Metabolism of Carcinogens by a Two-Stage Model of Carcinogenesis
In this section we illustrate how to use the two-stage model of Moolgavkar and Knudson (6) and the extended model of Tan and Gastardo (10) to assess effects of metabolism of carcinogens on cancer tumor development. Specifically, we shall illustrate how the metabolism of carcinogens affects the expected incidence rate and the expected number of tumors by using the two-stage models of Moolgavkar and Knudson (6) and the extended two-stage model of Tan and Gastardo (10). Note that the Tan-Gastardo extended model appears to provide a realistic model for many human cancers, including, for example, breast and ovary cancers (28). This is expected, since for breast and ovary cancers, hormone (estrogen) levels are different over different time intervals, so that menarche, menopause, and the time of first pregnancy provide natural partitions of the lifetime interval.

Assessing Effects of Metabolism by the Two-Stage Model of Moolgavkar and Knudson
Let the first and second mutation rates be ao1 and t2, respectively, and let the birth rate and the death rate for intermediate cells be b and d, respectively. Then for small a2, the expected incidence function X(t) is given approximately by To illustrate how the nonlinear kinetics of metabolism of carcinogens affect cancer tumor development, we assume that a carcinogen with concentration c is applied during [O,t] and that this carcinogen affects only the first mutation rate (initiation process) so that a1 is now replaced by al + ccxll, where e = yc/(l + bc) is the biological dose (i.e., concentration of DNA adduct) and c is the exposed dose. If 8 = 0 and/or the metabolism is not acting, then both A(t) and ,u(t) for fixed t are linear functions of c; on the other hand, if the Michaelis-Menten nonlinear kinetic is acting so that y 4 0 and 8 + 0, then X(t) and ,u(t) for fixed t are nonlinear functions of c. If the carcinogen affects also a2 and/or birth and death, then by replacing a2 and bd by at2 + ea22 and (bd) + ce, respectively, one may readily assess the effects of nonlinear kinetics of carcinogens on X(t) and R(t).

Assessing Effects by the Extended Two-Stage Model of Carcinogenesis
Let the time interval [0, t] be partitioned by Ij = [ti1, t0), j= 1, . . . k -1 and Ik = [tk-l, tk] with to = 0 and tk = t. For the jth interval, Ij, assume that the first and second mutation rates are aolj and a2j, respectively, and that the birth rate and the death rate for the intermediate cells are given, respectively, by bj and dj.
Then, for small a2j, the expected incidence X(tk) at t = tk is given approximately by: where X(s) is the expected number of normal cells time s given a large number of normal cells at s = [For proof, see (6)].
The expected number ,u(t) and the variance V(t) tumors at time t are given, respectively, by: Let c be the exposed dose of carcinogen over the Ij intervaf. Replacing ait by ao +cja0j, where Cj = -ycj! (1 + bcj), if au in the Ij interval is affected by the carcinogen and replacing bjdj by (bjdj) + ej, if the birth rate and death rate in Ij is affected by the carcinogen, one may evaluate the effects of nonlinear kinetics of metabolism of carcinogens on X(tk) and tOk) at tk.

Some Numerical Results
To illustrate the effects of metabolism of environmental agents on cancer tumor development, we generated some data by computer. Two cases are considered: In case 1, the time consists of one time interval of length 55 units; in case 2, the time is divided into two time intervals with length 15 and 40 units. Thus, case 1 is related to the Moolgavkar-Knudson two-stage model, while case 2 is related to the extended two-stage model of Tan and Gastardo (10). In generating data, wue follow Moolgavkar and Venzon (5) and Tan and Gastardo (10) to assume logistic growth for normal cells with    Table 1 parameter values for the above two cases. Note that for case 2, we considered only the situations of initation and initiation followed by promotion; we did not present situations of promotion only because effects of promotion are negligible if initiator is not applied before promotion (9,10,37). To determine effects of choice of different parameter values, we have done computations for many other sets of parameters than those given in Table 1 (38). Since the results are quite similar, we present only numerical results for parameters given in Table 1.
Using parameter values of Table 1, we computed the expected incidences and the expected numbers of tumors; some of the results are given in Tables 2 and 3 to illustrate some basic characteristics of the model and its consequences. From these results the following observations are made: For initiators, if metabolism is not acting, then both the incidences and the expected numbers of tumors are linearly related to exposed dose of initiators. This is predicted from formulas given previously. If metabolism is functioning and if the carcinogen is an initiator, then the dose-response curves are no longer linear; for cases where y/8 < 1, although the incidences and the expected number of tumors for c > 0 are considerably greater than those for c = 0 (no initiator), little changes in incidences and expected tumors are observed for different c > 0 values. On the other hand, if y18 > 1, then both the incidences and expected number of tumors increase monotonically as c increases.
For initiation and promotion experiments, if the initiator is not affected by metabolism, then both the incidences and the expected numbers of tumors are affected by promoters; furthermore, metabolism of promoters would reduce significantly the cancer incidence rates and the expected number of tumors. On the other hand, if initiator is affected by metabolism, then the metabolism of promoters have little effect. This is expected since metabolism of initiators would significantly reduce the number of initiated cells while the function of promoters is to facilitate cell proliferation of