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Distinguishing between Models of Carcinogenesis: The Role of Clonal Expansion

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FUNDAMENTAL AND APPLIED TOXICOLOGY 17,60 l-6 13 ( 199 1)
Distinguishing between Models of Carcinogenesis: The Role of Clonal Expansion
ANNETTE KOPP-SCHNEIDER
* AND CHRISTOPHERJ.PORTIER~
*Abteilung Biostatistik, Institut
fir
Epidemiologie und Biometrie, Deutsches Krebsforschungszentrum, Im Neuenheimer Feld 280, D-6900 Heidelberg, Federal Republic
of
Germany; and tstatistics and Biomathematics Branch. National Institute
of
Environmental Health Sciences, P.O. Box 12233, Research Triangle Park, North Carolina 27709 Received December 27, 1990; accepted May 6, 1991 Distinguishing between Models of Carcinogenesis: The Role of Clonal Expansion.
KOPP-
SCHNEIDER, A., AND PORTIER, C. J. ( 199 1) . Fundam. Appl. Toxicol. 17,60 l-6 13. Several mul- tistage models have been proposed to describe the process of carcinogenesis. The model attributed to Armitage and Doll ( 1954, Brit. J. Cancer 8, 1-12) assumes that a normal cell is transformed into a malignant cell by k > 1 fundamental biological events. The model usually attributed to Moolgavkar and colleagues ( 1979. Math. Biosci. 47,54-i?‘; 198 1, J. Natl. Cancer Inst. 66, 1037- 1052) assumes that carcinogenesis is a two-stage process, taking into account the ability of pre- malignant cells to proliferate. In this paper, we investigate our ability to distinguish between these models using animal carcinogenicity data. Three different approaches are considered: one based on actual tumor incidence data from National Toxicology Program experiments and two based on simulated data. The results show that both models will adequately fit most tumor incidence data. This implies that we must be cautious in accepting the biological basis of either of these models simply because it “fits the data.” Suggestions for experimental designs and toxicological endpoints (other than tumor incidence) which might provide for better discrimination between models are given. 0
1991 Society of Toxicology.
One well-documented aspect of carcinogenesis Considerable attention has been focused on is that part of the process of going from phe- the role of cell division and/or increased re- notypically normal cells to frank neoplasia re- sistance to terminal differentiation as impor- quires that cells pass through several different tant factors in modeling the carcinogenic phenotypes or mutational “stages” (Boyd and mechanism (Neyman and Scott, 1967; Mool- Barrett, 1990; Slaga et al., 1980; Farber, 1984; gavkar, 1983; Greenfield
et
al., 1984; Thors- Weinberg, 1989). As our understanding of the lund et al., 1987; Thorslund, 1988; Bogen, carcinogenic process has improved in recent 1989; Kopp and Portier, 1989; Moolgavkar et years, various “multistage” theories of carci- al., 1990; Portier and Kopp-Schneider, 1990; nogenesis have evolved. Virtually all of these Ames and Gold, 1990; Cohen and Ellwein, models assume that cells go through a series 1990). For risk assessment purposes, the of irreversible alterations in genetic expression. model which has received the most attention These produce a heritable change in the phe- is a two-stage model of carcinogenesis which notype and behavior of cells. The most prom- allows for clonal expansion of cells in the first inent model of this type is the Armitage-Doll altered state (usually referred to as initiated (AD) multistage model (Armitage and Doll, cells). The support for this model is mostly 1954) which is illustrated by Fig. 1. due to work by Moolgavkar and colleagues
601 0272-0590/91 3.00
Copyright 0 1991 by the Society of Toxicology. All rights of reproduction in any form reserved.
602
KOPP-SCHNEIDER AND PORTIER
(--
A
tage 1 Cells p1
‘I
tage 2 Cells P 2
t
Progression to Tumor
FIG.
1. The Armitage-Doll model. In this model, mu- tation rates are denoted p, and are expressed as number of mutations per cell per unit of time and are possibly related to dose and age.
(Moolgavkar and Venzon, 1979; Moolgavkar and Knudson, 1981) and is referred to as the MVK model. This model is illustrated by Fig. 2. A statistical question which arises whenever different models are suggested is whether these models can be distinguished on the basis of obtainable data. The major difference between the AD model and the MVK model is the number of stages and the ability of subpopu- lations of cells to clonally expand. These two issues are closely related since, by increasing the number of stages in the process, we can account for alterations in the shape and mag- nitude of the tumor incidence curve over time. In biological assays related to elucidating sub- populations of cells in the carcinogenic process (Saga
et
al., 1980; Farber, 1984), it is observed that certain chemical compounds when given in certain sequence have higher carcinogenic- ity than the two compounds given either singly or in reverse order. These results support a two-stage model but clearly do not rule out a larger number of stages (Boyd and Barrett, 1990). The second compound in the sequence, usually referred to as the promoter, enhances the growth characteristics of preneoplastic le- sions in a way which is consistent with the increased carcinogenicity. Results of this type support the concept that increased carcino- genicity is due directly to the increased number of initiated cells resulting from promotion. This causal relationship has not been estab- lished beyond doubt (Hoe1
et al.,
1988). This paper explores the ability of statistical methods to differentiate between these two models using the type of tumor incidence data which are commonly available from animal carcinogenicity experiments. Our focus is on the ability to correctly reject the assumption that carcinogenesis is due to a large number
Normal Cell 5 c--:l- Initiated Cell 6 p2 Malignant Cell I- Progression to Tumor
FIG.
2. The Moolgavkar-Venzon-Knudson model. In this model, mutation rates are denoted p,, the birth rate of initiated cells is denoted /3, and the death/differentiation rate of initiated cells is denoted 6. AI1 rates are expressed as number of events per cell per unit of time and are pos- sibly related to dose and age.
CLONAL EXPANSION IN MULTISTAGE MODELS
603
of mutational changes, i.e., to reject the AD model, when chemical effects are largely due to the selective expansion of initiated cells, and to correctly rule out effects on the expansion of initiated cells, i.e., to reject the MVK model, when chemical effects are due to multiple mu- tational changes. Multistage Models of Cancer Historically, the first mathematical multi- stage model of carcinogenesis was the model proposed by Nordling ( 1953). His model as- sumed that a normal cell is transformed into a neoplastic cell by
k
2 1 fundamental bio- logical events. Later Armitage and Doll ( 1954, 196 1) modified this model by assuming that an ordered sequence of occurrence of these fundamental events is required. Generally these biological events are held to invoke damage to the genetic material such as mu- tations at specific gene loci. It was shown later (Cook et al., 1969) that this model failed to fit incidence curves of certain human tumors (e.g., prostate cancer). Generally, the AD model is viewed as not being biologically plausible because it does not account for the replication and differentiation of cells. It also seems implausible that a single “hit” could transform a normal cell into a malignant one. For this reason Neyman ( 1958) proposed a two-stage mutation mech- anism of carcinogenesis where it was assumed that a hit on a normal cell turns this cell im- mediately into a first-order mutant subject to three time-independent risks: division, death, or second-order mutation. The second-order mutant cells were considered cancer cells. In a revision of this model to explain data found by Shimkin and Polissar ( 1955), Neyman and Scott ( 1967) assumed that a normal cell hit by a carcinogen changes into a primary first- order mutant which is subject to only two risks: the risk of division and the risk of death. Thus, the primary first-order mutant is not subject to the risk of secondary mutation. However, at the division of this first-order mutant, two secondary first-order mutants are produced which are subject to three risks: di- vision, death, or second-order mutation. Moolgavkar and Venzon ( 1979) modified a two-stage model of carcinogenesis proposed by Neyman ( 1958 ) to be applicable to the case of adult tumor onset. They treated this new model in the framework of filtered Poisson processes (Parzen, 1962 ) . Moolgavkar and Knudson ( 198 1) illustrated the use of this two- stage model by assessing the fit of this model to incidence data on human cancers assuming different growth patterns for normal stem cells of different tissues. Recently, Moolgavkar and Luebeck ( 1990) investigated the fit of the model to incidence data from animal experi- ments. In the literature, this model is usually referred to as the Moolgavkar-Venzon- Knudson (MVK) model. METHODS
Tumor Onset Times in Multistage Models
In the AD model, a cell in the ith stage is transformed into a cell in the (i + I )st stage with rate p, (in almost all applications this rate is assumed to be constant over time or piecewise-constant over time) and the kth stage is the stage of malignancy. In these circumstances, the probability of at least one cell out of a population of X, cells having undergone k transformations and thus being malignant by time
t,
i.e., the cumulative distribution function of the time
T
of the first occurrence of a malignant cell. is well approximated by P(
T d t) = I -
exp(-oltk), where cy = &w ’ - . fir-,/k (Whittemore and Keller, 1978) and k is an integer. The AD model is a special case of the Weibull model for which
P( T s t ) = I -
exp( -
cd)
where, unlike
k,
fl can bc any positive real number. In the MVK model. let
X(t), Y(t),
and
Z(t) be
the number of normal, initiated. and malignant cells, respec- tively. In a small time interval
[t
,
t
+ h) a normal cell is subject to the risk of dividing into one normal and one initiated cell with rate pi(t). An initiated cell in the same small interval is subject to three risks: it may divide into two initiated cells with rate p(t). it may differentiate or die with rate 6(t). or it may divide into one initiated and one malignant cell with rate fi2(
t).
The system starts with X(0) = X0 and Y(0) = Z(0) = 0. Although an explicit solution for the hazard function of the time,
T,
of first occurrence of a malignant cell can be given in this case,
604
KOPP-SCHNEIDER AND PORTIER this solution still needs numerical integration (Moolgavkar et al., 1988; Portier and Kopp-Schneider, 1990). Thus, in general, the approximation given by Whittemore and Keller ( 1978) is used, I’( T <
t)
= 1 - exp( -E[Z(
t)]),
where for constant rates p, , p2, 0, and 6 and no stochastic growth in the normal cell population,
E[Z(t)l = xow2 ewaN (6 - (j)f - 1 (P - a)* -. P e7* yt - I Y2
for r~ := p,wz and y := /3 - 6. Whittemore and Keller justified their approximation by stating that the number of detectable tumors follows a Poisson distribution. This approximation has been shown to be close to the exact cumulative distribution function of T as long as neither normal nor initiated cells are proliferating excessively (Kopp and Portier, 1989). It is recognized that this ap- proximation could lead to slight bias in our results (Mool- gavkar and Dewanji, 1988; Kopp and Portier, 1989), but the practical limitations of our analysis preclude the use of the exact (computer-intensive) computation derived by Moolgavkar et al. ( 1988) or that derived by Portier and Kopp-Schneider ( 1990) and we use the above ap- proximation for the MVK model. In summary, we used log(1 - P(T<
t))
=
-dk
and
[ADI
log( 1 - P( 7-G
t))
= -X,&L ey’ - -ft - 1 Y2 IMVKl as the hazard rate for the time to tumor onset in the AD resp. the MVK model. Historical control data. To assess he fit of these models to bioassay data, we use data from the National Toxicology Program (NTP). The exact description of the data was given by Portier, et al. (1986) and Portier and Bailer ( 1989). In brief, the data consist of tumor counts from 15 tissues and organs for 2670 male Fischer 344 rats, 27 19 female Fischer 344 rats, 2692 male B6C3F1 mice, and 2836 female B6C3FI mice from the untreated control groups of 57 carcinogenicity experiments. Most control groups consisted of 50 animals and all are from 2-year studies. There are up to 16 distinct necropsy times for each sex-species combination, of which 8-10 are near study termination. The data come from NTP carcinoge- nicity experiments with a technical report number greater than 193 and with the laboratory’s pathology diagnosis approved as of March 1983. Simulation data. Monte Carlo methods were used to generate time-to-tumor data for the AD model and ap proximated time-to-tumor data for the MVK model. X0 was chosen to be 10’ and 01 and k (in the AD model) as well as r.~ nd y (in the MVK model) were chosen such that the lifetime tumor probability varied from 0.3 to 1. This large lifetime risk is a relatively high level of tumor production, common for a high dose of a carcinogen. We used high risk levels in this analysis, because our ability to distinguish between models will increase with increasing lifetime risk. Reduced ability of distinction could be ex- pected for smaller lifetime tumor probabilities. Statistical Methods Statistical methods for the evaluation of the historical control data. The historical control data include multiple times of sacrifice; therefore we use the likelihood methods derived by Portier ( 1986) to obtain parameter estimates. We assess he goodness-of-fit of the distribution function of the tumor onset times to the historical control data. When assessing goodness-of-fit for the historical control data, we use the standard
x2
goodness-of-fit test (Cramer, 1971, Chap. 30). The application of the standard
X*
goodness-of-fit test to the historical control data was de- scribed in Portier et nl. ( 1986). The degrees of freedom for the
x2
test are s - r - 1, where s is the number of sacrifice times and r is the number of parameters in the model (r = 1 for the AD model, because the number of stages, k, is treated as being fixed, and r = 2 for the MVK and the Weibull model). Statistical methods for the evaluation of the simulation data. Animal carcinogenicity experiments were simulated under the assumption that no animals died during the course of the experiment, that tumors were observable on a daily basis, and that the experiment was terminated after 2 years (730 days). Assuming daily observations of the animals, the log likelihood is given by
730
logL= Cx,log[P(T<i)-P(T<i- I)] t=, where i = 0, ,730 denotes the days, N the total number of animals in the study, xi the number of animals getting the tumor on the ith day, and x. = x, + x2 + * * * + xN. In simulations looking at smaller experiments, boot- strapping procedures are used. The first procedure is as follows: Given a data set of, say, 100 tumor onset times, we use maximum likelihood methods to estimate model parameters. The value of the likelihood at the maximum, Lo, is retained. Given the estimated model parameters and the number of animals, we simulate bioassay results 100 times. For each simulated experiment, we fit the model to the data using maximum likelihood methods, again
CLONAL EXPANSION IN MULTISTAGE MODELS
605
saving the maximum likelihood values, L,, , Liw. The rank of the srcinal likelihood relative to the 100 likeli- hoods derived above is determined and the model is re- jected if the rank is below 11 (i.e., only 10 of the 100 likelihood values are smaller). That is, if L(,,. , L,,m, are the 100 likelihoods, but sorted in ascending order, we reject the fit of the model if & < L, ,,,. The rank of the srcinal likelihood, &. within the simulated likelihoods, divided by the number of simulations ( 100 in our case) gives us an estimate of the probability that the observed data come from the estimated model. A similar procedure has been proposed elsewhere ( Efron, 1982 ) The second procedure concerns the fit of the MVK model relative to that of the AD model and the Weibull model. This procedure is related to the procedure proposed by Wahrendorf et al. ( 1987): data from one ofthe models are generated and the parameters of the three models (the MVK. the AD. and the Weibull model) are estimated by maximum likelihood methods from those data. This pro- cedure is repeated 1000 times and the number of times in which the likelihood of the MVK model, L,,,, is greater than the likelihood of the AD model,
L,,,
is,recorded. We also record the number of times in which LMvK is greater than the likelihood of the Weibull model, Lwr,. The proportion gives us an estimate ofthe probability that the MVK model will fit the data better than the AD or the Weibull model. The use of the likelihood function for nonnested models is not a standard statistical procedure. With the exception of the work by Wahrendorf et a[. ( 1987 ). there is little statistical theory available to support its use. However. their work encouraged us to use this procedure in this context, since no standard statistical procedure is available for the comparison of nonnested models discussed in this manuscript and no other proce- dure has been suggested. It should be noted that there are several drawbacks to the use of these simulation methods in this context. For the procedure to test the goodness-of-fit, we find that the Type I error is below the nominal value: that is. in most cases, rather than the fit of the “true” model being rejected 10% of the time, it was rejected less often, generally never. In practical terms, this means we are possibly accepting the fit of the wrong model more often than would occur if our test had the correct nominal Type 1 error. Another problem arises when we attempt to estimate model parameters from simulated data. It is possible to generate data sets which are inappropriate for estimating these model parameters (only early-life tumors or no tu- mors at all). For simulated cases where the cumulative incidence was small, this problem occurred quite often. One other method could have been employed to distin- guish between the multistage models discussed here. We could have divided the data into two groups. One set of data could be used for estimating the model parameters and the other could be used to validate the estimated pa- rameters. However, this validation step requires the same goodness-of-fit procedure applied in our analysis. Consid- ering that the validation data set will be only half as large as the whole data set and noting our lack of ability to distinguish between models with the full data set, it is un- likely that this approach would yield different results.
RESULTS An approach to testing the relationship be- tween the MVK model and the AD model is to study differences in the goodness-of-fit of the two models to tumor incidence data. If the proliferation of premalignant cells is critical to the probability of tumor formation (i.e., the data come from the MVK model), then we should be able to reject the fit of the AD model more often than we can reject the fit of the MVK model. Although this would not estab- lish that clonal expansion of initiated cells was necessary for carcinogenesis, it would at least support such a concept. Correspondingly, if the number of stages to malignancy is more important than clonal expansion of premalig- nant cells (i.e., the data come from an AD model), we should be able to reject the MVK model more often than the AD model. In order to test which model is more closely associated with the tumor incidence data, we considered the large data set of untreated con- trol animals from NTP carcinogenicity studies as described under Methods. Model parame- ters for the MVK model and the AD model were estimated for 69 different sex-species- tumor combinations using NTP data and the methods outlined by Portier
et al. (
1986). Goodness-of-fit of the estimated model to the data was tested using a standard goodness-of- fit test as described under Methods, testing at the 10% level. The results of the goodness-of- fit tests are given in Table 1 for both models and for all sites. The column labeled “MVK” provides the results of the goodness-of-fit test for the MVK model. Similarly, the column labeled “AD’ gives the results of the goodness- of-fit test for the AD model. Finally, the col- umn labeled “stages” gives the range of stages in the AD model which adequately fit the data.

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