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Frank SA. Dynamics of Cancer: Incidence, Inheritance, and Evolution. Princeton (NJ): Princeton University Press; 2007.

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Dynamics of Cancer: Incidence, Inheritance, and Evolution.

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Chapter 9Carcinogens

Carcinogens shift age-incidence curves. Such shifts provide clues about the nature of cancer progression. For example, a carcinogen that influenced only a late stage in progression would have little effect if applied early in life, whereas a carcinogen that influenced only an early stage would have little effect if applied late in life. By various combinations of treatments, one can test hypotheses about the causes of different stages in progression.

The first section begins with the observation that incidence rises more rapidly with the duration of exposure to a carcinogen than with the dosage. Cigarette smoking provides the classic example, in which incidence rises with about the fifth power of duration and the second power of dosage.

The standard explanation for the relatively weaker effect of dosage compared with duration assumes that a carcinogen affects only a subset of stages. I contrast that standard theory with a variety of alternative explanations. For example, a model in which a carcinogen affects equally all stages also fits the data well. Overall, fitting different models to the data provides little insight.

The second section begins with the observation that lung cancer incidence changes little after the cessation of smoking but increases in continuing smokers. The standard explanation assumes that smoking does not affect the final transition in the sequence of stages of cancer progression. Among those who quit, nearly all subsequent cases arise from individuals who progressed to the penultimate stage while smoking, and await only the final transition. With one stage to go, incidence remains nearly constant over time.

I show once again that a model in which a carcinogen affects equally all stages also fits the data well. Although the data do not distinguish between theories, the various theories do set a basis for connecting how carcinogens influence mechanisms of cellular and tissue change, how those changes affect rates of transition in the stages of tumorigenesis, and how those rates of progression affect incidence curves.

The third section links different mechanistic hypotheses about carcinogen action to predicted shifts in age-incidence patterns. Those links between mechanism and incidence provide a way to test hypotheses about carcinogenic effects on the rate of transition between stages, on the number of stages affected, and on the particular order of affected transitions.

By altering both carcinogen treatment and animal genotype, one may test explicit hypotheses about carcinogenic action. For example, if a carcinogen is believed to cause a particular genetic change, then a knockout of that genotype should be less affected by the carcinogen when measured by age-incidence curves. Such tests can manipulate different components of progression and compare the outcomes to quantitative theories of incidence.

9.1 Carcinogen Dose-Response

Lung cancer incidence increases with roughly the fourth or fifth power of the number of years (duration) of cigarette smoking but with only the first or second power of the number of cigarettes smoked per day (dosage). The stronger response to duration than dosage occurs in nearly all studies of carcinogens. Peto (1977) concluded: "The fact that the exponent of dose rate is so much lower than the exponent of time is one of the most important observations about the induction of carcinomas, and everyone should be familiar with it—and slightly puzzled by it!"

In this section, I first summarize the background concepts and two studies of duration and dosage. I then consider five different explanations. The most widely accepted explanation is that cancer progresses through several stages, causing incidence to rise with a high power of duration, but that a carcinogen usually affects only one or two of those stages in progression, causing incidence to rise with only the first or second power of dosage. However, several alternative explanations also fit the data, so fitting provides little insight. In a later section, I discuss ways to formulate comparative tests. Such comparative tests may help to distinguish between alternative hypotheses and to reveal the processes by which carcinogens influence progression.


In the standard theory, the usual approximation of incidence is I(t) ≈ kuntn−1, where k is a constant, n is the number of rate-limiting transitions between stages that must be passed before cancer, u is the rate of transition between stages, and t is age. Suppose a carcinogen increases the rate of transition between some of the stages to u(1 + bd), where d is dosage and b scales dose level into an increment in transition rate.

If the carcinogen affects r of the transitions, then I(t) ≈ kun(1 + bd)rtn−1. Two further changes to this equation provide a more useful formula for studies of dosage and duration.

First, in examples such as cigarette smoking, the onset of carcinogen exposure does not begin at birth but at some age t0 at which smoking starts, so the duration of exposure is tt0 = τ.

Second, in empirical studies, one cannot directly estimate u, the baseline transition rate between stages, so the term kun = c enters in analysis only as a single constant, c. In different formulations, there will be different combinations of factors that together would be estimated as a single constant from data. I will use c to denote such constants, although the particular aggregate of factors subsumed by c may change from case to case.

With these assumptions, one may begin an analysis of dosage, d, and duration, τ, with an expression such as

Image ch9e1.jpg
or a suitably modified equation to match the particular problem.

If, as often assumed, moderate to large doses significantly increase transitions, then bd is much larger than one, and the transition rate becomes u(1 + bd) ≈ ubd. Incidence is then

Image ch9e2.jpg
with incidence rising as the rth power of dose, dr and the n−1st power of duration, τn−1. Here, c = kunbr, representing a single constant that may be varied or estimated from data.

Sometimes it is useful to study cumulative incidence, the summing up (integration) of incidence rates over the duration of exposure. This leads to the simple expression for cumulative incidence

Image ch9e3.jpg
Here, c differs from above but remains an arbitrary constant to vary or estimate from data.

In most empirical studies, incidence rises with a much lower power of dose than duration, r < n. This fact has led most authors to suggest that carcinogens typically affect only a subset of the transitions. For example, if one estimates r = 2 and n = 6, then one could interpret those results by concluding that the carcinogen affects two of the six transitions.

Later, I will suggest that this classic formulation of the theory may be misleading. In particular, the observation that the exponent on dosage is usually less than the exponent on duration does not necessarily imply that the carcinogen affects only a small number of transitions. However, the classic puzzle for the different responses to dosage and duration arises from the theory outlined here, so I use that theory as my starting point.

Cigarette Smoking

The classic study of cigarette smoking among British doctors estimated annual lung cancer incidence in the age range 40–79 as I(t) ≈ c(1+d/6)2τ4.5, where c is a constant, d is dosage measured as cigarettes per day, and τ = tt0 is duration of smoking with t as age and t0 = 22.5 as estimated age at which smoking starts (Doll and Peto 1978). If we use the expression for incidence in Eq. (9.1), then the estimate by Doll and Peto (1978) corresponds to r = 2 and n = 5.5.

Figure 9.1 shows the dose-response relationship for cigarette smoking, in which Doll and Peto (1978) fit a quadratic response curve. Subsequent authors have reiterated that lung cancer incidence increases with the first or second power of the number of cigarettes smoked per day (Zeise et al. 1987; Whittemore 1988; Freedman and Navidi 1989; Moolgavkar et al. 1989).

Figure 9.1. Dose-response for cigarette smoking, standardized for age.

Figure 9.1

Dose-response for cigarette smoking, standardized for age. The filled circle and error bars mark the mean and 90% confidence interval at various dosages. The solid line shows the quadratic fit given by Doll and Peto (1978) with incidence per 105 equal (more...)

Carcinogen Applied to Laboratory Rats

Peto et al. (1991) presented a large dose-response experiment in which they applied the carcinogen N-nitrosodiethylamine (NDEA) to laboratory rats. I summarized the details of this experiment and other laboratory studies in Section 2.5. Here, I repeat the main conclusions.

Peto et al. (1991) measured, for each dosage level, the median duration of carcinogen exposure required to cause a tumor. Suppose we fit an empirical relation for the cumulative incidence rate, CI, which is the total incidence over the duration of exposure (see Background above and Section 7.5). In empirical studies of dose-response, one typically observes that CI increases approximately with the rth power of dose and the nth power of duration, CI(τ) ≈ cdrtn. Then for the fixed level of cumulative incidence that occurs at the median duration to tumor development, t = m, we have CI(m)/cdrmn. Taking the logarithm of both sides and solving for log(m) yields

Image ch9e4.jpg
where k = CI(m)/c is a constant estimated from data. This equation is the expression of the classical Druckrey formula that I presented in Eq. (2.4).

Figure 9.2 shows that the results of Peto et al. (1991) fit closely to the Druckrey relation with n = 7 and the slope −r/n approximately −3/7, leading to an estimate of r = 3. This analysis again shows that incidence increases with a high power of duration and a relatively low power of dose.

Figure 9.2. Esophageal tumor dose-response line.

Figure 9.2

Esophageal tumor dose-response line. The circles show the observed durations of exposure required to cause one-half of the treatment group to develop a tumor. Each median duration is matched to the dosage level for the treatment group of rats. The line (more...)

Zeise et al. (1987) reviewed many other examples of dose-response relationships. In some cases, increasing dose causes a roughly linear rise in incidence; in other cases, incidence rises with dr, where d is dose rate and r > 1, usually near 2; in yet other cases, incidence rises at a rate lower than linear, with r < 1.

Perhaps only one pattern in dose-response studies recurs: the rise in incidence with dose is usually lower than the rise in incidence with duration of exposure, that is, r < n, as emphasized by Peto (1977).

Alternative Explanations

The observation that incidence rises more slowly with dosage than with duration plays a key role in the history of carcinogenesis studies and multistage theories. To give a sense of this history, I briefly list some alternative explanations. I also comment on how well different theories fit the observations: although fitting provides a weak mode of discrimination, it does provide a good point of departure for figuring out how to construct informative comparative tests. I delay discussion of tests until later in this chapter.


Suppose, as discussed above, that a carcinogen increases the rate of transition between stages to u(1 + bd), where d is dosage and b translates dose level into an increment in transition rate. If, for certain stages in progression, moderate to large doses significantly increase the transition rate, then bd is much larger than one, and the transition rate becomes u(1 + bd) ≈ ubd. For other stages not much affected by the carcinogen, bd is small, and u(1 + bd) ≈ u.

If a large increase in transition rate occurs for r of the stages, and the carcinogen has little effect on the other nr transitions, then as I showed in Eq. (9.3) above,

Image ch9e5.jpg
with the cumulative incidence rate rising as the rth power of dose, dr, and the nth power of duration, τn.

This explanation easily fits any case in which incidence increases exponentially with dosage and duration. However, the mathematics of curves provides no reason to believe that the number of steps affected by carcinogens can be inferred by measuring the empirical fit to the exponent on dosage.


Consider the most famous dose-response study: smoking among British doctors. Figure 9.1 shows the fit given by Doll and Peto (1978), in which the highest exponent of dose is two. From that fit, many authors have stated that lung cancer depends on the second power of dose, and thus the carcinogens in cigarette smoke affect only two stages in lung cancer progression. Against that explanation, the dashed curve in Figure 9.1 illustrates my calculation that a nearly equivalent fit for incidence can be obtained with a higher power of dose, in this case proportional to (1 + d/46)5.

The fact that one can fit a higher power of dose to those lung cancer data certainly does not mean that the carcinogens in cigarette smoke affect five stages of carcinogenesis rather than two. It does mean that the original fit to the second power of dose provides little evidence with regard to the number of stages affected.

In general, an expression in a lower power of dosage, d, will often fit the data about as well as an expression in a higher power of d over a moderate range of dosage (Zeise et al. 1987; Pierce and Vaeth 2003). In fitting data, one usually prefers the fit from the lower exponent because it is regarded as more parsimonious. However, when trying to infer biological mechanism, moderate distinctions between the goodness-of-fit of expressions that have various exponents on d do not provide strong evidence about the number of stages affected by a carcinogen.

In the remainder of this section, I present some examples and technical issues about dose-response curves for those readers who like to see the details (see also Pierce and Vaeth 2003). Suppose a carcinogen affects all n transitions equally. Then dosage raises incidence by k(1 + bd)n, where k is an arbitrary constant, and bd is the incremental increase in transition rate caused by dose d and scaling factor b. The expression for dosage can be expanded into a series of terms with increasing powers of d as

Image ch9e6.jpg
As bd declines, those terms with smaller exponents on d increasingly dominate the contribution of dosage, and so it would appear in the data as if the exponent on dose was small.

The smoking data in Figure 9.1 provide a good example. In those data, the exponent on duration suggests that n ≈ 6, that dosage varies over a range of about 0–40 cigarettes per day, and that incidence increases by a factor of about 50 over the range of dosage studied. Using those data to provide reasonable ranges for dosage and for the consequences on incidence, suppose that a carcinogen affects incidence by the expression k(1 + bd)r, with k = 1, b = 1/43.5, and r = 6. The solid curve in Figure 9.3 shows the dose-response effect.

Figure 9.3. Lower power dose-response curves match higher power curves when dose and response vary over intermediate scales.

Figure 9.3

Lower power dose-response curves match higher power curves when dose and response vary over intermediate scales. Here, dosage varies over 1–40 and relative incidence in response to exposure varies over 1–50, matching the ranges in the (more...)

In an empirical study, we would attempt to estimate the solid dose-response curve in Figure 9.3 from the data. The difficulty arises from the fact that we can get a good fit for r = 2, and that the fit improves relatively little for higher values of r. The figure shows example curves for r = 2 and r = 3 that fit very closely to the true curve. By the common statistical methods, one would usually choose the fit with the lower power of r = 2, noting that there is no statistical evidence that higher exponents fit the data significantly better.


Multistage analyses typically assume that, for each particular transition rate between stages, the carcinogen either has no effect or causes a linear rise in transition rate with increasing dose. Authors rarely discuss reasons for assuming a linear increase in transition rates with dose. A supporting argument might proceed as follows. Mutation rates often rise linearly with dose of a mutagen. If carcinogens act directly as mutagens, then carcinogens increase the rates of transition between stages in a linear way with dose.

Carcinogens may often act by processes other than direct mutagenesis. In particular, Cairns (1998) argued that carcinogens act mainly as mitogens, increasing the rate of cell division. Increased cell division does of course increase the accumulation of mutations, but does so differently from the mechanisms by which classical mutagens act. For example, the potentially mutagenic chemicals in cigarette smoke diffuse widely throughout the body, yet the carcinogenic effects concentrate disproportionately in the lungs. To explain this discrepancy in smokers between the distribution of chemical mutagens and the distribution of tumors, Cairns argued that the carcinogenic effects of smoke arise mostly from the irritation to the lung epithelia and the associated increase in cell division.

If carcinogens sometimes act primarily by increasing cell division, then we would need to know how mitogenic effects rise with dose. For example, doubling the number of cigarettes smoked might not double the rate at which epithelial stem cells divide to repair tissue damage. I do not know of data that measure the actual relation between mitogenesis and dose, but, plausibly, mitogenesis might rise with something like the square root of dose instead of increasing linearly with dose.

A diminishing increase in transition rates with dose would explain the observation that the exponent on dose is usually less than the exponent on duration. That observation is often expressed with the Druckrey equation that fits data from many studies of chemical carcinogenesis (Figures 2.11, 9.2). The Druckrey equation can be expressed as k = drmn, where k is a constant, d is the dose level, and m is the median duration of carcinogen exposure to onset of a particular type of tumor. Usually, r < n, that is, the exponent on dose is less than the exponent on duration. Peto (1977) mentioned that, for carcinomas, r/n is often about 1/2.

Now consider a simple multistage model with n stages and equal transition rates, u, between stages. Assume a carcinogen has the same effect on all stages, in which the transition rate is uf(d), where f(d) is a function of carcinogen dose, d. Then k = [f(d)]nmn, because the carcinogen has the same multiplicative effect on all n stages.

Suppose that the rise in transition rates diminishes with dose, for example, f(d) = da, with a < 1. Then the basic multistage model with all n transitions affected by a carcinogen leads to k = danmn. If a = r/n, then we have the standard Druckrey relation, k = drmn, which closely fits observations from many different experiments with a = r/n ≈ 1/2.

Alternatively, we could use the more plausible expression uf(d) = u(1 + bda), which leads to the multistage prediction k = (1 + bda)nmn. This expression is, on a log-log scale, log(m) = k3−log(1+bda), and may often fit the data well. For example, in the large carcinogen study shown in Figure 9.2, if we use Peto's (1977) suggested value of a = r/n = 1/2, with fitted values for two parameters of k3 = 1.01 and b = 16, we obtain a line that is almost exactly equivalent to the fit of the Druckrey formula shown in the figure.

The match of this diminishing effect theory to the observed relation in Figure 9.2 shows that the data fit equally well to a model in which the carcinogen affects only r < n of the stages in progression or a model in which the effects of carcinogen dose rise at a diminishing rate with increasing dose.

Diminishing effects of carcinogens with dose readily explain the observation that r < n. At present, little information exists about how widespread such diminishing effects may be. Carcinogenic acceleration of mitogenesis provides a plausible mechanism by which diminishing effects may arise, but additional mechanisms probably occur.


Individuals vary in their susceptibility to carcinogens. Heterogeneity in susceptibility arises from both genetic and environmental factors. Lutz (1999) suggested that heterogeneity may tend to linearize the dose-response curve, that is, to reduce the exponent on dosage in such curves. Lutz based his argument on a graph that illustrated how the aggregate dose-response curve may form when summed over individuals with different susceptibilities. To evaluate this idea, I describe a few specific quantitative models. These models suggest that heterogeneity can influence the dose-response curves, but heterogeneity does not provide a convincing explanation for the widely observed low exponent on dose.

Consider the following rough calculation to illustrate the effect of heterogeneity on the dose-response curve. Suppose a carcinogen affects the relative risk of cancer, S. Let S depend on bd, where d is the dose, and b is a factor that scales the effect of dose on relative risk.

Heterogeneity in individual susceptibility enters the analysis through individual variability in b, the scaling factor that translates dose into an increment in transition rate between stages of progression. We need the value of S averaged over the different individual susceptibilities in the population. Let the probability distribution for the values of b among individuals be f(b). The value of S for each level of susceptibility, b, must be weighted by the various probabilities of different values of b. The average value of S over the different values of b is

Image ch9e7.jpg
in which the distribution f(b) describes the level of heterogeneity, and S is a function of b.

The slope of the dose-response curve on a log-log scale provides the empirical estimate for r, the exponent on dosage. The observed dose-response curve is S*, so the log-log slope is

Image ch9e8.jpg

How does heterogeneity in individual susceptibility affect the shape of the dose-response curve? To study particular examples, we first need assumptions about the form of heterogeneity described by the distribution f(b). Figure 9.4 shows three probability curves for heterogeneity, ranging from wide variation (solid line) to essentially no heterogeneity (tall, short-dashed curve).

Figure 9.4. Distribution of individual susceptibility to carcinogens.

Figure 9.4

Distribution of individual susceptibility to carcinogens. For each individual, the consequence of carcinogen dose d scales with bd, where b is the individual's susceptibility to the carcinogen. This example uses the beta distribution to describe variation (more...)

Next, we need to assume particular shapes for the dose-response curve for a fixed level of susceptibility, that is, a fixed value of b. Figure 9.5 shows various examples. In the left panel, all the curves have a saturating response to high dose, above which relative risk no longer increases. In the right panel, risk continues to accelerate with increasing dose.

Figure 9.5. Relative risk, S, in response to dose, d.

Figure 9.5

Relative risk, S, in response to dose, d. The plots show dose varying from 0 to 40, to illustrate roughly the range of dosage in number of cigarettes per day. However, the consequences of dose always depend only on bd, where b scales the dose into the (more...)

Figure 9.6 illustrates how heterogeneity affects the aggregate dose-response pattern in the population. In panel (a), the short-dash curve shows the dose-response pattern when there is essentially no heterogeneity. Increasing heterogeneity alters the shape of the dose-response curve, illustrated by the long-dash and solid curves of panel (a).

Figure 9.6. Consequences of heterogeneity in individual susceptibility on carcinogen dose-response curves.

Figure 9.6

Consequences of heterogeneity in individual susceptibility on carcinogen dose-response curves. All curves derive from the response function shown in Figure 9.5a: in panels (a) and (b), the average value of susceptibility is b̄ = 0.05; (more...)

Figure 9.6b shows the log-log slopes of the aggregate dose-response curves, obtained by calculating the slopes of the curves in the panel above. These slopes provide the standard estimates for r, the exponent on dose in the dose-response relationship.

Figure 9.7 shows the same calculations, but for a base response curve that does not saturate at higher doses. In this case, heterogeneity always increases the slope of the dose-response curve.

Figure 9.7. Consequences of heterogeneity in individual susceptibility on carcinogen dose-response curves.

Figure 9.7

Consequences of heterogeneity in individual susceptibility on carcinogen dose-response curves. All curves derive from the response function shown in Figure 9.5b. Other assumptions match those described in Figure 9.6.

The consequences of heterogeneity follow general rules. When the base curve rises at an increasing rate, then heterogeneity causes an increase in value because, at each point, the average of higher and lower doses is greater than the value at that point. By contrast, when the base curve rises at a decreasing rate, then heterogeneity causes a decrease in value because, at each point, the average of higher and lower doses is less than the value at that point.

In summary, large increases in heterogeneity usually cause minor changes in the dose-response patterns. Those changes alter the details of the dose-response relationship in interesting ways, but probably do not explain the different effects of dosage and duration on incidence.


Precancerous stages in progression may proliferate by clonal expansion. The expanding clone of cells carries somatic mutations or other heritable changes. I described the theory of clonal expansion in Section 6.5.

Clonal expansion could explain the different observed exponents on dosage and duration. Suppose, for example, that cancer requires only two rate-limiting transitions. The first transition causes the affected cell to expand clonally. As the number of cells in the clone increases, the rate of transition to the second stage rises because of the greater number of target cells available. In a carcinogen exposure study, incidence would rise with an increasing exponent on duration because the target population of cells for the final transforming step would increase with time.

A two-stage model could fit a variety of exponents for duration of smoking (Gaffney and Altshuler 1988; Moolgavkar et al. 1989), including the exponent of n − 1 ≈ 4.5 reported by Doll and Peto (1978). The two-stage model could also fit the observed exponent on dosage of about two, because in a two-stage model the carcinogenic effects of smoking may influence two independent transformations.

Although the two-stage model cannot be ruled out, we do not know the exact nature of cancer progression and the rate-limiting steps that determine progression dynamics. I tend to favor other models for four reasons.

First, the ability of two-stage models to fit the data provides relatively little insight: with enough parameters and a mathematically flexible formulation, a model can be molded to a wide variety of data. Second, qualitative genetic evidence points to several rate-limiting steps in most adult-onset cancers (Chapter 3), although those data are not conclusive. Third, to explain the high observed exponents on age or duration, one must typically assume that clonal expansion is slow and steady over many years; bursts of clonal expansion over shorter periods do not match the observations so easily. Fourth, clonal expansion is more difficult to test experimentally than models that emphasize simple genetic or epigenetic changes to cells, because genomic changes can be manipulated and compared between treatments more easily than properties of clonal expansion.

The two-stage model may be limited and difficult to test. However, aspects of clonal expansion in multistage progression may play an important role in the patterns of incidence (Luebeck and Moolgavkar 2002). To move ahead, this idea requires useful comparative hypotheses that predict different outcomes based on measurable differences in the dynamics of clonal expansion.


Several theories fit the observed relatively low exponent on dosage and high exponent on duration. But a close fit by itself provides little evidence to distinguish one theory from another. Rather, one should use the alternative theories and fits to the data as a first step toward developing biologically plausible hypotheses and their quantitative consequences. Once those theories are understood, one can then try to formulate comparative tests that discriminate between the alternatives. I turn to potential comparative tests after I discuss a related topic in chemical carcinogenesis.

9.2 Cessation of Carcinogen Exposure

Lung cancer incidence of continuing smokers increases with approximately the fourth or fifth power of the duration of smoking (Doll and Peto 1978). By contrast, incidence among those who quit remains relatively flat after the age of cessation (Doll 1971; Peto 1977; Halpern et al. 1993).

In 1977, Richard Peto (1977) stated that the approximately constant incidence rate after smoking ceases "is one of the strongest, and hence most useful, observational restrictions on the formulation of multistage models for lung cancer." Peto argued that, in any model, the observed constancy in incidence after smoking has stopped "suggests that smoking cannot possibly be acting on the final stage" of cancer progression. There could, for example, be a particular gene or pathway that acts as a final barrier in progression and resists the carcinogenic effects of cigarette smoke.

In 2001, Julian Peto (2001) reiterated Richard's argument: "The rapid increase in the lung cancer incidence rate among continuing smokers ceases when they stop smoking, the rate remaining roughly constant for many years in ex-smokers (Halpern et al. 1993). The fact that the rate does not fall abruptly when smoking stops indicates that the mysterious final event that triggers the clonal expansion of a fully malignant bronchial cell is unaffected by smoking, suggesting a mechanism involving signaling rather than mutagenesis."

In this section, I discuss which stages of progression may be affected by the carcinogens in cigarette smoke. I begin by summarizing observations on how cancer incidence changes after the cessation of carcinogen exposure. I then consider two alternative explanations. First, the carcinogen may affect only a subset of stages in cancer progression; the particular stages affected determine how patterns of incidence change after cessation. Second, the carcinogen may affect all stages of progression; the different precancerous stages at which individuals cease exposure determine how patterns of incidence change after cessation. Both models fit the data reasonably well.

As we have seen often, fitting by itself does not strongly distinguish between competing hypotheses. I therefore introduce some comparative approaches that may provide a better way to test alternatives.


Figure 9.8a shows the flattening of the incidence curve upon cessation of smoking from data collected in the Cancer Prevention Study II of the American Cancer Society (Stellman et al. 1988). This figure summarizes data for 117,455 men who never smoked, 91,994 current smokers, and 136,072 former smokers. The top curve represents lifetime smokers who never quit. The four curves below it represent individuals who quit at different ages; the age at which smoking ceased coincides with the intersection of each curve with the top curve for lifetime smoking. The bottom curve shows incidence among those who never smoked.

Figure 9.8. Reduction in relative risk of lung cancer between men who continued to smoke and those who quit at different ages.

Figure 9.8

Reduction in relative risk of lung cancer between men who continued to smoke and those who quit at different ages. (a) Summary of data from Figure 1 of Halpern et al. (1993). The top curve shows those who continued to smoke. The lower curves show those (more...)

Figure 9.9a presents data from a cessation of smoking study in the UK (Peto et al. 2000). That study analyzed cumulative risk rather than incidence rate. Cumulative risk measures the lifetime probability of death from lung cancer at each age if no other causes of death were to occur. A flat incidence rate translates into a linear increase in cumulative risk with age. The plot shows that cessation of smoking reduces the upslope in cumulative risk, somewhere between linear (flat incidence) and the accelerating curve for those who continue to smoke. Thus, the pattern in Figure 9.9a matches the pattern in Figure 9.8a: an initial flattening of the incidence rate after cessation of smoking followed by a relatively slow rise later in life.

Figure 9.9. Reduction in relative risk of lung cancer between men who continued to smoke and those who quit at different ages.

Figure 9.9

Reduction in relative risk of lung cancer between men who continued to smoke and those who quit at different ages. (a) Redrawn from Figure 3 of Peto et al. (2000). Samples for this case-control design include 1465 case-control pairs in a 1950 study combined (more...)

Other studies report data on cessation of carcinogen exposure (reviewed by Day and Brown 1980; Freedman and Navidi 1989; Pierce and Vaeth 2003). I focus only on the smoking data, because those studies have the largest samples and have been discussed most extensively. I emphasize how to develop and test hypotheses rather than argue for a comprehensive explanation to cover all of the available data. In my opinion, the existing studies do not provide enough evidence to decide between competing hypotheses. Instead, the smoking data define the challenge for future studies.

Alternative Explanations

All theories must account for two observations. First, the relative risk of lung cancer decreases in those who quit compared with those who continue to smoke (Figures 9.8 and 9.9). Second, the rise in incidence with smoking fits an increase in incidence with roughly the second power of number of cigarettes smoked per day (dose).

I discuss two alternative formulations. First, most prior explanations fit the observations by positing that carcinogens in smoke affect only one or two stages in progression, leaving the other stages mostly unaffected.

Second, I show that the standard multistage model of progression also fits the observations very well. Previous authors rejected that standard model because they used the common approximation for incidence given by Armitage and Doll (1954), which in fact does not apply well to the problem of carcinogen exposure followed by cessation.


This idea was stated most clearly and perhaps originally by Armitage in the published discussion following Doll (1971). I quote from Armitage at length, because his words set the line of thinking that has dominated the subject. Note that, at the time, the dose-response curve was thought to be linear. Later work suggested that the response may in fact fit a curve that rises with the square of dose (Doll and Peto 1978). Here is what Armitage said:

The dose-response relationship seems to be linear, which suggests that the carcinogen affects the rate of occurrence of critical events at one stage, and one only, in the induction period. (If it affected two stages, one might have expected a quadratic relationship, and so on.) Does this crucial event happen early or late in the induction period? For example, in a six-stage process, are we thinking of an early stage, the first or second, or a late stage, the fifth or sixth?

The evidence here seems to conflict. One argument would suggest that a very early stage is involved. I am thinking of the delay of a generation or so between the increase in smoking in men around the First World War, and the rise in lung cancer mortality rates which was so marked 20 or 30 years later; and similarly the increase in cigarette smoking among women about the time of the Second World War, and the rise in lung cancer rates for females which has become so noticeable in the last few years. This long delay is what one would expect if a very early part of the process were involved rather than a very recent one.

On the other hand, the halt in the rise in risk quite soon after smoking stops suggests that a late stage is involved. Professor Doll's very ingenious treatment of the data on ex-smokers, in Tables 13 and 14, confirms the latter view. In a multi-stage process, if the first stage were involved, the rate after stopping smoking would continue to rise in the same way as for continuing smokers. If, on the other hand, the last stage were affected, one would expect the rate to drop immediately to the rate for nonsmokers. What seems to happen is a stabilization at the current rate until it is caught up by the rate for nonsmokers. That is precisely what one would expect if the next to last stage in a multi-stage process were affected.

I should be interested to know whether Professor Doll has considered this anomaly and can resolve it. Is it, for example, conceivable that two stages in a multi-stage process are affected ...?

Exactly how does incidence change when a carcinogen affects only one of n stages? Whittemore (1977) and Day and Brown (1980) presented approximate theoretical analyses. However, those approximations can be rather far off from the actual theoretical values. I prefer exact calculations as shown in the example of Figure 9.10. I describe in detail the results in Figure 9.10, because this particular model played an important role in the history of carcinogen studies. The model also provides general insight into multistage progression.

Figure 9.10. Theoretical incidence curves in response to carcinogen application followed by cessation.

Figure 9.10

Theoretical incidence curves in response to carcinogen application followed by cessation. The carcinogen affects only a single transition in a model with n = 6 steps. The legend shows the curve type for each of the i = 0,...,5 transitions, in which the (more...)

In Figure 9.10a,b, I used a basic n stage model in which a carcinogen increases the rate of the ith transition between stage i and stage i + 1. For example, if i = 0, then the carcinogen affects only the first transition between the baseline stage 0 and the first precancerous stage 1; if i = n−2, then the carcinogen affects only the penultimate transition between stage n − 2 and stage n − 1. The model in Figure 9.10 has n = 6 stages. The legend shows the line types that describe the outcome when the carcinogen affects the ith transition.

In Figure 9.10a,b, the carcinogen is applied only between age 0 and age 60, after which carcinogen application ceases. If the carcinogen affects one of the first three transitions, shown in Figure 9.10a, then incidence follows closely the curve that would result if the carcinogen was applied throughout life, from age 0 to age 80. With acceleration of an early stage, cessation has little effect on incidence because anyone who ultimately progresses to cancer has already passed the early stages by age 60.

Figure 9.10b shows the strong effect that cessation has on incidence when a carcinogen is applied from age 0 to age 60 and influences a later stage in progression. If the carcinogen affects the last transition, i = 5, then during carcinogen application, anyone who progresses to the fifth stage is almost immediately transformed into the final cancerous stage. Thus, the curve for i = 5 up to age 60 shows the incidence pattern for a five-stage model: the six stages of progression minus one stage that is not rate limiting in the presence of the carcinogen. After cessation, progression follows the full six rate-limiting stages, and so incidence instantly drops to the rate for a six-stage model.

If the carcinogen affects only the penultimate transition, i = 4, then during carcinogen application, individuals move very rapidly from stage 4 to stage 5, where they await the final transforming event at the normal, background rate. By essentially skipping a stage during carcinogen application, the incidence follows a five-stage model. After cessation, almost all new cancers arise from the pool of individuals in stage 5 who await the final transition. When transformation occurs by a single random event, the incidence rate remains flat over time. The final event is as likely to happen this year as next year or a later year. If the carcinogen affected only the third transition, i = 3, then after cessation most cancers would arise in the pool of individuals that require two further steps, causing incidence to increase only slowly with time as in a model with only two stages.

In Figure 9.10c,d, the carcinogen is applied only between age 25 and age 80. The carcinogen has relatively little effect when it increases the earliest transition, i = 0, because that transition has already occurred by age 25 in many of the individuals who ultimately progress to cancer. For the next transition, i = 1, fewer would have passed that step by age 25, and so more will be affected by the carcinogen. For the later steps, almost no one would have passed those steps by age 25, and so the carcinogen increases incidence equally for all of the later transitions.

In Figure 9.10e,f, the carcinogen is applied only between age 25 and age 60, after which carcinogen application ceases. This case matches the problem of cessation smoking, with onset of smoking in the first third of life and cessation in the last third of life. The patterns can be understood from the previous cases. If smoking affects only an early stage, then the earlier the stage, the less the effect, because the earliest stages are more likely to have been passed already before the onset of smoking and the acceleration of that stage. If smoking affects only a later transition, i, then after cessation, the pool of individuals most susceptible has ni steps remaining; if smoking affects the final transition, no excess pool of susceptibles exists, and incidence reverts to the background rate.

The first theoretical studies of smoking cessation considered models in which smoking affected only one stage (Whittemore 1977; Day and Brown 1980). The analyses I just presented improve the accuracy of such models over previous studies, but the main points hold from earlier work. After that early work, two observations affected subsequent analyses of smoking cessation. First, none of the curves in Figure 9.10 fit closely to data such as in Figure 9.8. Second, later studies of dose-response favored a quadratic fit to the data, leading many to suppose that smoking affects two stages in progression.

One can see from Figure 9.10e,f that a combination of the earliest transition, i = 0, and the penultimate transition, i = n − 2 = 4, provides the shapes needed to fit the data in Figure 9.8, and with two transitions affected, the overall incidence would be higher. Various authors fit the data in this way, sometimes weighting the role of those two stages differently (Day and Brown 1980; Brown and Chu 1987; Whittemore 1988).

Those fitted models based on two affected stages match the data reasonably well for both dose-response and incidence. In particular, one can easily explain the flattening of the incidence curves upon cessation by the penultimate transition and the later rise in incidence several years after cessation by the earliest transition.

The data and matching models tell a pleasing empirical and logical story. However, other plausible models also fit nicely to the data. The next section provides an example.


Armitage's quote shows that the linear or perhaps quadratic dose-response curve motivated the initial models in which smoke carcinogens affect only one or two stages of progression. Those assumptions about number of stages affected may over-interpret the data: one cannot draw firm biological conclusions about a molecular mechanism from a fitted exponent of a dose-response curve. In addition, the mathematical analyses of progression have in the past typically used approximations; those approximations do not capture key aspects of incidence curves and dose-response curves.

I decided to analyze how well the standard model of multistage progression fits the data, in which the carcinogens affect equally all n stages. I first fit the data in Figure 9.8a, giving the fitted curves shown in Figure 9.8b. To obtain those fitted curves, I began with the basic multistage model described earlier in the theory chapters. I took the following parameters as given based on previous studies or on common assumptions: the number of stages, n = 6; the number of independent cell lineages at risk, L = 108; the age at which smoking starts, 25 years; and the maximum age of the analysis, 80 years. Those parameters were not fit to the data but instead derived from extrinsic considerations.

I then used the following crude procedure to fit the model to the data. I set the cumulative lifetime risk of lung cancer for nonsmokers to 0.005 to match the lowest curve in Figure 9.8, which shows data for nonsmokers. I then fit the transition rate between stages per year, u, needed to match that nonsmoker incidence curve, resulting in the estimate u = 7.24 × 10−4. Given this value for the baseline transition rate, I next assumed that during exposure to smoke carcinogens, all transitions between stages rise to u(1 + bd), where d is dose, and bd is the increase in the transition rate caused by carcinogens. The value of b sets a proportionality constant for the effect of a given dose; without loss of generality, I used b = 1, because all calculations depend only on the product bd and not on the separate values of the two parameters.

I estimated the value of d = 1.187 to match the top curve, in order to obtain a lifetime cumulative risk for continuing smokers of 0.158. Finally, I assumed that, upon cessation of smoking, carcinogenic effects decay with a half-life of 5 years; this assumption prevents an unrealistic instantaneous decline in incidence immediately upon cessation.

This fitting procedure required estimation of only two parameters, u and d. The other values came from prior studies or common assumptions. The fit shown in Figure 9.8b provides a reasonable qualitative match to the observed patterns in Figure 9.8a; some deviation occurs at age 80—a few observations at this point cause some of the incidence curves to rise late in life. Better fit could be obtained by optimizing the fit procedure and by using additional parameters. But my point is simply that the basic multistage model gives a nice match to the data without the need for any special adjustment or refined fitting.

Originally, Armitage, Peto, and others rejected a model in which carcinogens affect all stages because the estimated exponent of the dose-response curve is between one and two. Does the model I used, with all stages affected, also match that observed dose-response relation?

To test the model fit to the observed dose-response curve, I focused on the estimated value of d, which in the standard models is proportional to dose. At the maximum age measured, in this case 80 years, I varied the cumulative lifetime risk for continuing smokers between the value for nonsmokers, 0.005, and the approximate observed value for lifetime smokers of 0.158. For each cumulative risk value (the response), I fit the d value (the dose) needed to match the cumulative risk. I then calculated the log-log slope of the dose-response curve, which turned out to be 1.84. Thus, the model provides a good match to the observed exponent on the dose-response relation. The earlier section, The Mathematics of Curves, and Figure 9.3 explain why a model with n = 6 steps can give an approximately quadratic dose-response curve.

I repeated the same fitting procedure for the data in Figure 9.9a. In those data, the maximum observed age is 75; otherwise, I used the same background assumptions as in the previous case. The shift in maximum observed age altered the two fitted parameters: u = 7.72 × 10−4 and d = 1.225. The model provides a close fit to the data (Figure 9.9b). The log-log slope of the dose-response curve is 1.84, as in the previous case.

In summary, a model with all stages affected fits the data reasonably well. The data do not provide any easy way to distinguish between this model, with all stages affected, and the earlier models in which the carcinogens affect only one or two stages. Perhaps the most striking difference arises in the carcinogenic increase in transition rate that one must assume: when the carcinogen affects all stages, the increase, d, is about 1.2, or 120 percent. This small increase in transition would be consistent with a moderate and continuous increase in cell division: the mitogenic effect perhaps caused by irritation. By contrast, when the carcinogen affects only one stage, the required increase in transition rate, d, may be around 70, and for two stages, d is probably around 8–10. Those large increases in transition seem too high for a purely mitogenic effect, and would therefore point to a significant role of direct mutagenesis in increasing progression.

Fitting models cannot decide between mitogenic and mutagenic hypotheses. In the next section, I discuss how to use the quantitative models as tools to formulate testable hypotheses.

9.3 Mechanistic Hypotheses and Comparative Tests

Two observations set the puzzle. First, cancer incidence rises more rapidly with duration of exposure than with dosage. In terms of lung cancer, incidence rises more rapidly with number of years of smoking than with number of cigarettes smoked per year. Second, lung cancer incidence remains approximately constant after cessation of smoking but rises in continuing smokers.

Traditional explanations suggest that carcinogens affect only a subset of stages in progression. Such specificity in carcinogenic effects would often lead to incidence patterns that fit the observations.

I discussed in the previous section how an alternative model in which carcinogens affect all stages also fits the observations. The fact that the observations can be fit by a model in which all stages are affected does not argue against the traditional explanation in which only a few stages are affected. Rather, the proper inference is that we need to be cautious about drawing firm conclusions about mechanism solely from models fit to age-incidence curves.

Further progress requires testing alternative hypotheses about the link between, on the one hand, how carcinogens affect the mechanisms of progression dynamics and, on the other hand, how perturbations of progression dynamics cause shifts in the age-onset curves. I focus on shifts in age-onset curves because carcinogenic perturbations are important only to the extent that they cause changes in incidence patterns.

In this section, I present alternative mechanistic hypotheses about how carcinogenic perturbation affects progression dynamics. I also consider the sorts of comparative tests that could distinguish between alternative mechanistic hypotheses.


Tumors arise when cell lineages evolve ways around the normal limits on tissue growth. Because tumors develop through evolutionary processes, we can classify the mechanisms of carcinogen action by the particular evolutionary processes that they affect.

Variation and selection comprise the most important evolutionary processes. For variation, I consider carcinogenic effects that act directly by mutagenesis, defined broadly to include karyotypic and epigenetic change. The different types of heritable change cause different spectra of variation and act at different rates. For selection, I divide mechanisms into three classes: mitogens directly increase cellular birth rate, anti-apoptotic agents directly reduce cellular death rate, and selective environment agents favor cell lineages predisposed to develop tumors. Those selective mechanisms may indirectly increase variation. For example, mitogens often increase mutation by raising the rates of DNA replication.

I do not use the common classification that divides the effects of carcinogens into initiation, promotion, and progression. That classification primarily arises from the tendency of certain agents, at certain doses, to have stronger effects when applied before or after other agents. Such patterns certainly exist and must, to some extent, be correlated with mechanism of action. Indeed, initiators do sometimes act as direct mutagens that cause particular mutations early in tumor formation, and promoters do often act as mitogens. But there are many exceptions with regard to the consistency of the patterns, and the connections to mechanism often remain vague and somewhat speculative (Iversen 1995).

My focus on variation and selection does not set a mutually exclusive alternative against the classical initiation-promotion-progression scheme. Instead, my emphasis on variation and selection simply puts the processes of tumor evolution ahead of the sometimes debatable patterns for the ordering of consequences under certain experimental conditions.

I place carcinogenic mechanism in the context of multistage progression, measured by shifts in age-onset curves. I therefore emphasize how certain mechanisms affect rate processes and the time course of tumor formation. For example: How does a carcinogenic agent affect the rate of transition between particular stages? How many stages does an agent affect? Does a particular agent have an effect only on tissues that have already progressed to a certain stage? Put concisely, the issues concern changes in rate, number of stages affected, and order of effects.

Mutagens: Increase Heritable Variation

I begin with background observations from the mouse skin model of chemical carcinogenesis (Slaga et al. 1996). I then interpret those observations in terms of hypotheses about rate, number, and order.


The first step in skin tumor development often appears to be a mutation to H-ras that causes an amino acid substitution at codon positions 12, 13, or 61 in the phosphate binding domain of the protein (Brown et al. 1990). Those substitutions can abrogate negative regulation of the Ras signal that stimulates cell division (Barbacid 1987).

Different carcinogens induce different spectra of mutation to H-ras isolated from papillomas or carcinomas of mouse skin. Table 9.1 shows the most frequent DNA base substitutions in response to four different carcinogens, measured in papillomas that did not progress to carcinomas. In this case, the carcinogens were applied in one dose at the start of treatment (an initiator), and most likely acted as direct mutagens. The initial treatment with one of the mutagens listed in Table 9.1 was followed by repeated application of a mitogen, TPA.

Table 9.1. Carcinogen-induced H-ras substitutions in mouse skin papillomas.

Table 9.1

Carcinogen-induced H-ras substitutions in mouse skin papillomas.

The observed substitution spectrum in response to an initial carcinogen probably results from two processes. First, the initial carcinogen treatment causes a particular spectrum of genetic changes. That primary spectrum depends on the biochemical action of the carcinogen with respect to DNA damage and repair. Second, among the variation caused by those initial changes, only certain mutations become amplified to form papillomas. In this case, selection amplified those cells that carry changes to the Ras protein and abrogation of negative regulation of mitogenic signals.

I summarized results on H-ras mutation (Table 9.1) to emphasize that different carcinogens often cause different spectra of heritable variation. Several other studies report carcinogen-specific spectra of heritable change to DNA sequence, epigenetic marks, or karyotypic alterations (reviewed by Lawley 1994; Turker 2003).

Mutation of H-ras appears to be a common early step of skin carcinogenesis in both mice and humans (Brown et al. 1995). Two alternative hypotheses could explain why H-ras mutations arise early in experimental studies of chemical carcinogenesis in mice. First, the particular carcinogens may produce a mutational spectrum that favors H-ras variation and selection. Second, amplification of H-ras mutation may be a favored early step in skin carcinogenesis, so that early change in H-ras is not strongly dependent on the particular spectrum of heritable change caused by a direct mutagen.

How do chemical carcinogens affect different stages of progression? The stage at which p53 mutations occur in skin carcinogenesis and the spectrum of mutations to that gene provide some clues (Brown et al. 1995; Frame et al. 1998). Burns et al. (1991) observed no p53 mutations in benign papillomas, an early stage in carcinogenesis, whereas they found that 25% of later stage carcinomas had p53 mutations. It may be that early p53 mutations are actually selected against in skin carcinogenesis. In three different studies that applied an initial mutagen to mouse skin, heterozygote p53+/− mice had fewer papillomas than did wild-type p53+/+ mice (Kemp et al. 1993; Greenhalgh et al. 1996; Jiang et al. 1999). Another study showed that p53+/− mice had a three-fold increase in progression of papillomas to carcinomas, demonstrating a causal role of p53 mutation in later stages of carcinogenesis (Brown et al. 1995).

In three different chemical carcinogen treatments of mouse skin, the particular spectrum of p53 mutations depended on the treatment. When an initial mutagen, DMBA, was followed by the mitogen, TPA, most p53 changes were loss of function mutations, including frameshifts, deletions, and the introduction of stop codons. Repeated application of DMBA led to five carcinomas with one deletion and four transversion mutations in p53. Repeated application of the mutagen MNNG led to four carcinomas with G A transitions in p53 (Brown et al. 1995).


I describe a series of hypotheses and tests to show how one might in principle connect particular mechanisms of carcinogen action to consequences for multistage carcinogenesis. Some of the tests may not be experimentally well posed or practical to do, but they should help to stimulate thought about how to develop new, more practical tests that provide information about mechanism.

Measuring a rate of transition directly is difficult, so I focus on the number of transitions and the order of effects.

Hypothesis for number of steps affected by a carcinogen.—

A treatment affects only a subset of rate-limiting steps.


Apply a mutagen continuously. If all steps are affected equally, then untreated and treated animals should have approximately the same slope of the incidence curve (log-log acceleration, LLA), because they have the same number of rate-limiting steps. The treated animals should, however, have a higher intercept for their age-incidence curve, because their transitions happen at a faster rate. If some transitions are more sensitive than others, then the LLA of the incidence curve should decrease with increasing dose because, as dose rises, an increasing number of steps should change from rate limiting to not rate limiting. The fewer the number of rate-limiting steps, the lower the LLA.

Hypothesis for mechanism of initial carcinogen treatment.—

The primary effect is mutation of the first rate-limiting step in multistage progression.


Compare age-onset curves in mice with wild-type H-ras and H-ras mutated in one of the carcinogenic codons, each mouse genotype either treated or not treated with a single dose of an initial carcinogen. To get enough tumors for comparison, the mice could have a cancer-predisposing genotypic background with changes distinct from the functional consequences of H-ras mutation. If the initial carcinogen treatment only has a tumorigenic effect through mutation of H-ras as the first rate-limiting step, then the untreated, wild-type mice would have to pass one more step than either of the other three treatments: mutated H-ras with or without initial carcinogen treatment and wild-type H-ras treated with an initial carcinogen. An additional rate-limiting step to pass should cause the slope of the incidence curve (LLA) to be one unit higher than in treatments that rapidly pass that step.

Hypothesis for order of stages affected.—

Certain carcinogens affect only a particular transition in an ordered series of stages of progression.


Suppose carcinogen A is thought to affect primarily an early stage, such as H-ras mutation in skin tumors, and carcinogen B is thought to affect primarily a late stage, such as p53 mutation in skin tumors. The following comparisons support the hypothesis. If A acts early and B acts late, then the difference in incidence between A early and A late is greater than the difference between B early and B late. If A acts early and B acts late, then the combination of A applied early and B applied late has a stronger effect than B applied early and A applied late.


These tests emphasize treatments that apply chemical carcinogens to altered animal genotypes, with age-incidence curves measured as the outcome and interpreted in the light of quantitative predictions of multistage theory.

Mitogens: Increase Cellular Birth Rate

Increased cell division raises the rate of tumor formation (reviewed by Peto 1977; Cairns 1998). Higher rates of tumorigenesis occur in response to irritation, wound healing, and chemical mitogens.

I first describe three hypotheses to explain the association between mitogenesis and carcinogenesis. Ideally, I would follow with tests that clearly distinguish between alternative hypotheses. However, given the current level of technology, it is not easy to define practical experiments that connect biochemical changes caused by mitogens to consequences for rates of tumorigenesis. With that difficulty in mind, I finish by laying a foundation for how to formulate tests as understanding and technology continue to improve.


Faster cell division balanced by increased cell death.—

In this case, the number of cells does not increase because tissue regulation balances cell birth and death, but the mitogen increases cell division and turnover. The faster rate of DNA replication increases the rate at which mutations occur (Cunningham and Matthews 1995).

Normally asymmetrically dividing cell lineages divide symmetrically.—

Epithelial stem cells sometimes divide asymmetrically. One daughter remains as a stem cell to provide for future renewal; the other daughter often initiates a rapidly dividing and short-lived lineage. Cairns (1975) suggested that in each asymmetric stem cell division, the stem lineage may retain the older DNA templates, with the younger copies segregating to the other daughter cell (supporting evidence in Merok et al. 2002; Potten et al. 2002; Armakolas and Klar 2006). If most mutations occur in the production of new DNA strands, then most mutations would segregate to the nonstem daughter lineage, and the stem lineage would accumulate fewer mutations per cell division. In addition, stem cells may be particularly prone to apoptosis in response to DNA damage, killing themselves rather than risking repair of damage (supporting evidence in Bach et al. 2000; Potten 1998).

If these processes reduce stem cell mutation rates, then carcinogens or other accidents that kill stem cells may have a large effect on the accumulation of mutations (Cairns 2002). In particular, lost stem cells must be replaced by normal, symmetric cell division with typical mutation rates that may be much higher than stem cell mutation rates. Thus, regeneration of stem cells following carcinogen exposure or during wound healing may cause increased mutation.

Clonal expansion of predisposed cell lineages.—

Once a mutation occurs, a mitogen may stimulate clonal expansion. An expanding clone increases the number of target cells for the next transition (Muller 1951). This increase in transition rate between stages does not require a rise in mutation rate per cell division, only an increase in the number of cells available for progressing to the next stage.


The mechanistic details of mitogenesis may be studied directly at the biochemical and cellular levels. However, I am particularly interested in the different ways in which mitogenesis shifts age-incidence curves. To study shifts in age incidence, one must analyze how mechanistic consequences of mitogenesis affect rates at which carcinogenic changes accumulate in cells.

The first two mechanistic hypotheses in the previous section focus on an increase in the mutation rate per cell; the third hypothesis focuses on an increase in the number of target cells susceptible for transition to the next stage. The two processes have different consequences for age-incidence curves.

Increase in mutation rate per cell.—

In this case, the mitogen acts like a mutagen. The particular hypotheses and tests from the section on direct mutagenesis apply.

Increase in number of target cells for next transition.—

More target cells cause a higher transition rate per unit time. The main difference from mutagenic agents arises from the time course over which the mutation rate increases. When a chemical agent causes an increased rate of mutation per cell, the rise in the mutation rate most likely occurs over a short period of time. By contrast, an increase in the number of target cells may happen slowly as a predisposed clone expands, causing a slow rise in the transition rate to the following stage.

In the theory chapters, I demonstrated a clear difference in how age-incidence curves shift in response to a change in transition rate. A quick rise in a particular transition abrogates a rate-limiting step and reduces the slope of the age-incidence curve. In an idealized model, each abrogation of a rate-limiting step reduces the slope by one unit. By contrast, a slow rise in a transition rate causes a slow rise in the slope of the age-incidence curve. Multiple rounds of slow clonal expansion can lead to high age-incidence slopes. (See Section 6.5, which describes the theory of clonal expansion.)

Increasing the dosage of a mitogen may cause more rapid clonal expansion. The theory predicts that the increase in the rate of clonal expansion causes a steeper rise in the slope of the incidence curve over a shorter period of time. If the rate of clonal expansion is not too fast, then longer duration of exposure to a mitogen may cause a sequence of clonal expansions as one transition follows another, leading to a steep rise in the slope of the incidence curve. At high doses and rapid rates of clonal expansion, transitions may occur so rapidly that the rate-limiting effects of a stage may be abrogated, causing a drop in the slope of the incidence curve.

Anti-Apoptotic Agents: Decrease Cellular Death Rate

Anti-apoptosis may act in at least two different ways. First, blocking cell death may allow mutations to accumulate at a faster rate, because apoptosis is an important mechanism for purging damaged cells. Second, absence of cell death may cause clonal expansion, with an increase in the number of target cells for the next transition.

I discussed in the previous sections some of the ways in which to study increased mutation rate per cell versus increased target size in an expanding clone of cells. It may be possible to complement those approaches by study of genotypes with loss of apoptotic function.

Selective Environment: Favors Predisposed Cell Lineages

The previous sections discussed carcinogens that directly cause mutations or directly affect cellular birth or death. This section focuses on carcinogens that change the competitive hierarchy between genetically or epigenetically variable cell lineages.

Consider, for example, an agent that kills cells by inducing apoptosis. That agent favors variant cell lineages that resist the induction of apoptosis. Clonal expansion of the anti-apoptotic lineages follows. Anti-apoptosis may often be an early step in carcinogenesis.

Variant cell lineages arise continuously. However, in the absence of a selective agent to expand clones of predisposed cells, variant cell lineages may have relatively little chance of completing progression. In this regard, selective agents may play a key role in raising cancer incidence. As always, variation and selection must complement each other in the evolutionary process of transformation.


A recent theory proposes that carcinogens may act as both mutagens and selective agents (Breivik and Gaudernack 1999b; Fishel 2001). In the presence of a mutagen that causes a certain type of DNA damage, selection may favor cells that lose the associated repair pathway. Cells that lack the appropriate repair response may not stop the cell cycle to wait for repair or may not commit apoptosis, whereas repair-competent cells often slow or stop their cycle during repair. Thus, repair-deficient cells could outcompete repair-competent cells, as long as the gain in survival or in the speed of the cell cycle offsets any loss in division efficacy caused by the increased accumulation of mutations.

In support of their theory, Breivik and Gaudernack (1999a) noted the association between the physical location of colorectal tumors and the loss of particular types of DNA repair. Proximal colorectal tumors tend to have microsatellite instability caused by loss of mismatch repair (MMR) genes. The MMR pathway repairs damage caused by methylating carcinogens. Breivik and Gaudernack (1999a) argue that methylating carcinogens often arise from bile acid conjugates that occur mainly in the proximal colorectum.

The argument for proximal tumors can be summarized as follows. Methylating carcinogens concentrate in the proximal colorectum. The MMR pathway repairs the damage caused by methylating agents. Those cells that lose the MMR repair pathway gain an advantage in the selective environment created by methylating agents, because MMR-deficient cells slow down less for repair or commit apoptosis less often than do MMR-competent cells.

By contrast, distal colorectal tumors tend to have chromosomal instability caused by loss of the mechanisms that maintain genomic integrity, such as the nucleotide excision repair (NER) pathway. Breivik and Gaudernack (1999a) argue that the bulky-adduct-forming (BAF) carcinogens may arise primarily from dietary and environmental factors and concentrate primarily in the distal colorectum.

The argument for distal tumors can be summarized as follows. BAF carcinogens concentrate in the proximal colorectum. The NER pathway primarily repairs the damage caused by BAF agents. Those cells that lose the NER repair pathway gain an advantage in the selective environment created by BAF agents, because NER-deficient cells slow down less for repair or commit apoptosis less often than do NER-competent cells.

By this theory, a carcinogen may act in three stages. First, direct mutagenesis creates variant cell lineages. Second, selection favors clonal expansion of variant cells that lose repair function for the type of mutagenic damage caused by the carcinogen. Third, direct mutagenesis of cells that lack associated repair processes may speed the rate at which subsequent transitions occur through the steps of multistage progression.


By test, I mean the ways in which to study the predicted consequences of a carcinogen for age-specific incidence. This section focuses on carcinogens that may act both as direct mutagens and as selective agents. No clear theory has been defined to formulate hypotheses for the relation between the dosage of such carcinogens and the patterns of age-specific incidence.

I can speculate a bit. As mentioned above, a directly mutagenic agent may have three separate effects: initial mutagenesis, secondary selective expansion of mutator clones, and tertiary mutagenesis.

Consider a particular mutagen and an associated DNA repair system that fixes the kind of damage caused by the mutagen. A knockout genotype with reduced or absent repair function should respond differently to the carcinogen when compared to the wild type. In particular, the incidence rate of the knockout should be insensitive to the initial mutagenesis directed at the repair system under study, because that repair system has already been mutated in the germline. The knockout should also be insensitive to clonal expansion, because in the knockout all cells share the loss of repair function and so there should be no selective advantage for relative loss of repair function. The knockout should be affected mainly by the tertiary mutagenesis.

Quantitative predictions could be developed for the relative incidence patterns in wild-type and knockout genotypes, using the methods of the earlier theory chapters. Those predictions could be tested in laboratory animals. Although such tests may not be easily accomplished, it is worthwhile to consider how to connect carcinogenic effects to mechanism, and mechanism to incidence. Ultimate understanding of cancer can only be achieved by understanding how factors influence the rates of progression, and how rates of progression affect incidence.

9.4 Summary

This chapter analyzed classical explanations for chemical carcinogenesis. Those explanations focused on how dosage and duration of chemical exposure may alter incidence. The classical explanations are not as compelling as they originally appeared. The problem arises from the ease with which alternative models can be fit to the data. To avoid the problems of fitting models to the data, I showed how one may frame quantitative hypotheses about chemical carcinogenesis as comparative predictions—the most powerful method for testing causal interpretations of cancer progression.

The next chapter turns to mortality patterns for the leading causes of death. I show that the quantitative tools I have developed to study cancer may help to understand the dynamics of progression for other age-specific diseases and the processes of aging.

Copyright © 2007, Steven A Frank.

This book, except where otherwise noted, is licensed under a Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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