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Carcinogenesis Advance Access originally published online on September 14, 2006
Carcinogenesis 2007 28(2):479-487; doi:10.1093/carcin/bgl173
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© The Author 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Stochastic modelling of colon cancer: is there a role for genomic instability?

Mark P. Little* and Guangquan Li

Department of Epidemiology and Public Health, Imperial College Faculty of Medicine London W2 1PG, UK

*To whom all correspondence should be addressed. Tel: +44 0 20 7594 3312; Fax: +44 0 20 7402 2150; Email: mark.little{at}imperial.ac.uk


   Abstract
Top
 Abstract
Introduction
 Materials and methods
 Results
 Discussion
 References
 
Three stochastic models of genomic instability recently developed by Little and Wright (Math. Biosci., (2003) 183, 111–34), with two, three and five stages, and the two-stage genomic instability model of Nowak et al. (Proc. Natl Acad. Sci. USA, (2002) 99, 16226–16231) are compared with the four-stage model proposed by Luebeck and Moolgavkar (Proc. Natl Acad. Sci. USA, (2002) 99, 15095–15100) that does not assume such an instability mechanism. All models are fitted to US colon cancer incidence data. The best fitting models are the two-stage model of Nowak et al. and the two-stage model of Little and Wright, with the four-stage model of Luebeck and Moolgavkar not markedly inferior. The fits of the three-stage and five-stage models are somewhat worse (P < 0.05), the five-stage model fitting particularly poorly (P < 0.01). Both optimal genomic instability models predict cellular mutation rates that are at least 10 000 times higher after genomic destabilization, for both sexes. Therefore, the results of this paper are somewhat at variance with those of previous analyses of Little and Wright in suggesting that equivalently good fit may be obtained by models that do not assume a role for genomic destabilization in the induction of colon cancer as for those that do.

Abbreviations: AIC, Akaike information criterion; APC, adenomatous polyposis coli; CIN, chromosomal instability; GI, genomic instability; HNPCC, hereditary non-polyposis colon cancer; MIN, microsatellite instability; SEER, Surveillance, Epidemiology and End Results.


   Introduction
Top
 Abstract
 Introduction
Materials and methods
 Results
 Discussion
 References
 
There is much biological data suggesting that the initiating lesion in the multistage process leading to cancer might be one involving a destabilization of the genome resulting in elevation of mutation rates, reviewed by Morgan (1,2). Loeb (3,4) has presented evidence that an early step in carcinogenesis is mutation in a gene controlling genome stability. Stoler et al. (5) showed that there are 11 000 mutations per carcinoma cell for a number of different cancer types, again implying that genomic destabilization is an early event in carcinogenesis. In particular, there is strong evidence of such an early genomic destabilization event for colon cancer (3–5). However, the question of whether chromosomal instability is the initiating event in carcinogenesis, in particular colon cancer, is controversial. Tomlinson and Bodmer (6) argue that cancer is an evolutionary process, and that the observed accumulation of chromosomal and other damage in colon cancers may simply be the result of selection for cells with growth advantage.

Recently two papers have appeared proposing formulations of stochastic carcinogenesis models that incorporate genomic instability (GI) (7,8), both applied to colon cancer. In contrast, Luebeck and Moolgavkar (9) have recently proposed a four-stage stochastic model positing inactivation of the adenomatous polyposis coli (APC) gene followed by a high-frequency event, possibly positional in nature, an extension of the two-stage clonal expansion model of Moolgavkar and Venzon (10) and Knudson (11; the so-called MVK model); this model does not assume GI. The paper of Little and Wright (8) proposed a general class of carcinogenesis models that includes as special cases the models proposed by Luebeck and Moolgavkar (9) and Nowak et al. (7). The model of Little and Wright (8) also generalizes the class of so-called generalized MVK models developed by Little (12), and which in turn therefore generalizes the two-mutation model of Moolgavkar and Venzon (10) and Knudson (11). The model is close in spirit to the model of Mao et al. (13).

In this paper we shall compare the goodness of fit to US Surveillance, Epidemiology and End Results (SEER; 14) colon cancer data of three models developed by Little and Wright (8) with ones recently proposed by Nowak et al. (7) and Luebeck and Moolgavkar (9). By so doing we hope to assess the evidence for chromosomal instability in relation to the initiation of colon cancer.


   Materials and methods
Top
 Abstract
 Introduction
 Materials and methods
Results
 Discussion
 References
 
Description of the model
The general class of models fitted are precisely those developed by Little and Wright (8). This class includes as special cases the models proposed by Nowak et al. (7) and Luebeck and Moolgavkar (9). The category of models proposed by Little and Wright (8) assumes that cells can acquire two sorts of mutation, those associated with progression to a malignant phenotype (‘cancer-stage’ mutations), and those associated with successive destabilization of the genome (‘destabilizing’ mutations). With acquisition of successively more destabilizing mutations the cancer-stage mutation rates are generally higher, corresponding to the genome destabilization that is characteristic of GI.

Specifically, the model supposes that at age t there are X(t) susceptible stem cells, each subject to mutation to a type of cell carrying an irreversible cancer-stage mutation at a rate of M(0,0)(t). The cells in the stem cell compartment can also acquire a destabilizing mutation at a rate A(0,0)(t). Thereafter the cells in compartment I(i,j) with i cancer-stage mutations and j destabilizing mutations divide into two such cells at a rate G(i, j)(t); at a rate D(i, j)(t) they die or differentiate. Each such cell can also divide into an equivalent daughter cell and another cell with an additional cancer-stage mutation at a rate M(i, j)(t). In addition, each such cell can also divide into an equivalent daughter cell and another cell with an additional destabilizing mutation, at a rate A(i, j)(t). There are assumed to be a total of k cancer-stage mutations required for a cell to become malignant. Likewise, there are assumed to be m destabilizing mutations. Once a cell has acquired all m such destabilizing mutations it is assumed to remain at the m th destabilizing mutation level. This model is illustrated schematically in Figure 1. The acquisition of carcinogenic (cancer-stage) mutations amounts to moving horizontally (left to right) in Figure 1, whereas acquisition of destabilizing mutations amounts to moving vertically (top to bottom) in this figure. The asymmetric cell divisions associated with most of the cancer-stage and destabilizing mutations [all except (i, j) = (0,0)], in which each cell produces a daughter cell identical to the parent and another carrying an additional mutation, should be contrasted with the symmetric cell divisions associated with the cell proliferation processes [with rates G(i, j)], in which each cell produces two identical daughter cells. The two-mutation model of Moolgavkar and Venzon (10) and Knudson (11) corresponds to the case k = 2, m = 0, while the generalized MVK model with K stages developed by Little (12) amounts to the case k = K, m = 0.


Figure 1
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  		Fig. 1 Schematic diagram of generalized MVK model with k cancer-stage mutations and m destabilizing mutations.

 
Data and model fitting
Incidence data on colon cancer, excluding rectal cancer, were derived from the SEER registry data for the period 1973–2002, using the CD-ROM supplied by SEER (14). The cases were grouped into 5-year age intervals (0–4, 5–9, ..., 80–84, 85+) separately by sex, cancer registry and race, for each year of follow-up. For the purposes of this analysis only data for whites was used, and data for ages 85 and over was ignored, because of doubts as to the completeness of ascertainment in this group. In younger age groups it is possible that there are a disproportionate number of familial genetic colon cancers [e.g. hereditary non-polyposis colon cancer (HNPCC)], in which presumably one or more of the necessary mutations for development of colon cancer is carried by all stem cells, so that one would not expect any of these models to fit this age group well. In view of this for certain of the model fits we shall omit those aged <20. Mid-year population estimates provided by SEER were also employed. Table I displays the data used for the analysis in summary form.


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  		Table I Summary SEER (14) Caucasian colon cancer incidence data, for the years 1973–2002, used in model fits

 
Five models were considered. The first three were models fitted by Little and Wright (8), those with two cancer-stage mutations and one destabilizing mutation, three cancer-stage mutations and one destabilizing mutation, and five cancer-stage mutations and two destabilizing mutations (Table III). In contrast to Little and Wright (8), who made no assumptions regarding the stem cell number, for most model fits (all except those of Table II) we assume that there are 108 colon stem cells, as estimated by Potten et al. (15). The fourth model was close to that proposed by Nowak et al. (7). This model had two cancer-stage mutations and one destabilizing mutation (Table IV). The parameter M(1,0) was also set to 0, and all other parameters were free to vary, subject to certain constraints imposed to improve parameter stability. This model is illustrated schematically in Figure 2. It should be emphasized that Nowak et al. (7) never fitted their model to population-based cancer data, and indeed never provided an explicit mathematical form of the model; nevertheless, our fitted model is close to the scheme proposed by Nowak et al. (7). Because not all the biological parameters are identifiable (M.P. Little, G. Li and W.F. Heidenreich, manuscript in preparation) we set the intermediate cell proliferation rate G(1,1) = 8.92 year–1, a value for the pre-cancerous (adenoma) proliferation rate determined by Herrero-Jimenez et al. (16) in model fits to human colon cancer data. The fifth model was close to that fitted by Luebeck and Moolgavkar (9). This model had four cancer-stage mutations and no destabilizing mutations (Table V). In this model the intermediate cell proliferation parameters G(i,0) and D(i,0) were set to 0 for the first two compartments (i = 1,2). Luebeck and Moolgavkar imposed the constraint M(0,0)(t) = M(1,0)(t), as did we. As above, for this model not all the biological parameters are identifiable (17; see also M.P. Little, G. Li and W.F. Heidenreich, manuscript in preparation), so we set the intermediate cell proliferation rate G(3,0) = 8.92 year–1, a value determined by Herrero-Jimenez et al. (16) as above. The model is parameterized somewhat differently from that of Luebeck and Moolgavkar for model fitting purposes, but is otherwise equivalent. This model is illustrated schematically in Figure 3. As implied by Figure 3, it should be noted that apart from the first mutation all mutational steps are asymmetric, in other words each mutation produces a cell that is identical to the parent cell and a daughter cell that carries an extra mutation, as also assumed by Luebeck and Moolgavkar (9). This is in contrast to the mechanism assumed in the model of Armitage and Doll (18), in which only a single mutated daughter cell results. Fitting of the models was achieved by maximum likelihood, with adjustment for overdispersion (19). (There is modest overdispersion in the data: the variance is inflated over that to be expected from the Poisson distribution by ~20–30%.) Further details of model fitting are given in the paper of Little and Wright (8). In contrast to the model fitting of Little and Wright (8), multiplicative adjustments to the hazard function for calendar year and registry are used; these result in a somewhat better fit than the multiplicative adjustments for birth cohort and registry previously used (8). We centred the calendar year adjustment at 1998 and the registry at Connecticut, these being the categories of each variable with the largest number of cases. For purposes of comparison with these models, we also fitted a simple empirical model, in which the colon cancer rate in age group i was assumed to be given by exp[{theta}i]. These empirical models were fitted by quasi-likelihood techniques to males and females separately, with adjustment for overdispersion (19). All other modelling assumptions (multiplicative adjustment for registry, calendar year) are as above. The results are reported in Table II, and are also used as the basis for the plots of ‘observed’ data (and 95% CI) in Figures 4 and 5. Goodness of fit of all mechanistic models was determined by 1000 Monte Carlo simulations of numbers of cases predicted by this empirical model, simulations being taken from a negative binomial distribution with mean that predicted by the empirical model and variance given by the mean multiplied by the observed inflation derived from the quasi-likelihood fits (for males and females separately). The scaled deviance, a measure of aggregate goodness of fit (19), for each mechanistic model fit was compared against the distribution of deviances from these Monte Carlo simulations; P-values reported in Table II reflect the proportion of simulated deviances above that of the model in question.


Figure 2
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  		Fig. 2 Schematic diagram of model of Nowak et al. (7), similar to a generalized MVK model with two cancer-stage mutations and one destabilizing mutation.

 


Figure 3
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  		Fig. 3 Schematic diagram of model of Luebeck and Moolgavkar (9), similar to a generalized MVK model with four cancer-stage mutations and no destabilizing mutations.

 


Figure 4
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  		Fig. 4 Cancer hazards predicted by models of Nowak et al. (7) (with two cancer-stage mutations and one destabilizing mutation), of Luebeck and Moolgavkar (9) (with four cancer-stage mutations and no destabilizing mutations), and of Little and Wright (8; with two cancer-stage mutations and one destabilizing mutation, three cancer-stage mutations and one destabilizing mutation, five cancer-stage mutations and two destabilizing mutations), with stem cell population fixed to 108 cells, refitted to SEER (14) colon cancer data, and observed data [with 95% CI, adjusted for overdispersion (19)].

 


Figure 5
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  		Fig. 5 As for Figure 4, but showing cancer rates and model fits up to age 40.

 


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  		Table II Deviance (and degrees of freedom) and AIC in fits of various generalized MVK models to SEER (14) colon cancer incidence data

 


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  		Table III Coefficients of optimal 2-stage and 3-stage generalized MVK modelsa

 


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  		Table IV Coefficients of optimal generalized MVK model with two cancer-stage mutations and one destabilizing mutationa

 
Table II gives the deviance statistics (19) for fits of the five models to the male and female data separately. Tables III–V give the optimal parameters for the four near-optimal models (we do not give parameter estimates for the five cancer stage two destabilization model because, as we shall see, it fits less well); 95% confidence intervals (CI) are evaluated in all cases by parametric bootstrap techniques (20), using 199 samples, with adjustment for overdispersion. The Akaike information criterion (AIC) (21) (based on minimizing the quantity: scaled deviance + 2 x number of fitted parameters) is employed to select the optimal model.


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  		Table V Coefficients of optimal generalized MVK model with four cancer-stage mutations and no destabilizing mutationsa

 

   Results
Top
 Abstract
 Introduction
 Materials and methods
 Results
Discussion
 References
 
As can be seen from Table II and Figures 4 and 5, the fits of all five models to the data are reasonably good at ages 30 and above. For most of the models the fit is acceptable, as determined from the simulated deviance distribution (P > 0.05). However, for the three and five cancer-stage models there are indications of overall lack of fit (P < 0.05) to the male data; the lack of fit is most significant for the five cancer-stage model (P < 0.01). Looked at another way, for all but 2 of the 66 datapoints corresponding to ages ≥30 (11 age points x 2 sexes x 3 models), the predicted incidence for the three optimal models (both the two cancer stage one destabilization models and Luebeck and Moolgavkar model) lie within the observed 95% CI, which is of course the number of outliers one would expect by chance in this number of independent observations. However, the three- and five-stage models of Little and Wright (8) show some discrepancy at younger ages, particularly for females, where the hazard is generally consistently [and statistically significantly (P < 0.05)] underestimating the observed incidence (Figures 4 and 5). The AIC indicates that for each sex separately and for both sexes together the best model is that of Nowak et al. (7), followed by the two-stage model of Little and Wright (8); the fit of the four-stage model of Luebeck and Moolgavkar (9) is not much inferior. Table II also indicates that these conclusions are minimally affected by excluding those under the age of 20; the optimal model remains the two-stage model of Nowak et al. (7), followed by the four-stage model of Luebeck and Moolgavkar (9), followed by the two-stage model of Little and Wright (8), which all give pretty much equivalent fit. At least in this restricted dataset the three-stage model of Little and Wright (8) gives fit not much inferior to the other three models (Table II) (p = 0.03 for males, p > 0.1 for females) (results not shown).

Table II shows that fixing the stem cell number at 108 has quite a marked effect on some of the model fits. In particular, the three and five cancer-stage models of Little and Wright (8) fit somewhat worse when the stem cell number is fixed at this value (Table II).

Tables III and IV show that the cancer-stage mutation rates are in general very substantially elevated in those parts of the model associated with genomic destabilization, both for the two and three cancer-stage models of Little and Wright (8) and for the model of Nowak et al. (7). For example, for the two-stage model of Little and Wright (8) the cancer-stage mutation rates after the first destabilizing mutation, M(i,1), are higher by a factor of ~17 500 (males) and 86 000 (females) than the mutation rates with no destabilizing mutation, M(i,0). For the three-stage model of Little and Wright (8) the cancer-stage mutation rates after the first destabilizing mutation, M(i,1), are higher by a factor of ~82 000 (males) and 79 700 (females) than the mutation rates with no destabilizing mutation, M(i,0). Likewise, the model of Nowak et al. (7) predicts cancer-stage mutation rates after the first destabilizing mutation, M(0,1), to be higher by a factor of ~280 000 (males) to 390 000 (females) than the mutation rates with no destabilizing mutation, M(0,0).

As shown in Table III, the mutation and cell proliferation rates predicted by the two and three-stage models of Little and Wright (8) are similar both for males and females, although there is a slight difference in cancer-stage mutation rates at the first destabilization level, M(i,1), for the two-stage model, which are higher by a factor of 4 for females compared with males. For the model of Nowak et al. (7) there are few if any substantive differences between the sexes in mutation and cell proliferation rates (Table IV). However, the model of Luebeck and Moolgavkar (9) predicts very different cancer-stage mutation rates between the sexes (Table V); in particular M(2, 0) is higher by an order of magnitude for males compared with females, and M(3, 0) is lower for males than for females by approximately the same factor.

The multiplicative adjustments for registry and calendar are both highly statistically significant (P < 0.00001 for registry and for calendar year, for males and females). Additional model fits in which the multiplicative adjustments were either for (i) birth cohort and calendar year, (ii) cancer registry, birth cohort and calendar year or (c) cancer registry and birth cohort yielded very similar parameter estimates (generally within 5% of those given here) (results not shown).


   Discussion
Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
References
 
The results of this paper demonstrate that three models, the two-stage models of Little and Wright (8) and Nowak et al. (7) and the four-stage model of Luebeck and Moolgavkar (9) yield pretty much equivalent fit. These findings are somewhat at variance with those of previous analyses of Little and Wright (8), in suggesting that models that do not assume a role for genomic destabilization in the induction of colon cancer may fit as well as models that do. The conclusions are not much affected by excluding those aged <20 years, in which group a higher proportion of heritable genetic colon cancers might be expected.

It appears that analyses of data that contains information only in relation to the age distribution of cancer do not possess the power to discriminate between models and hence to confirm or to falsify the hypothesized involvement of genomic instability in colon cancer. It is possible that additional analysis, incorporating for example quantitative information on exposure to various mutagenic agents (e.g. ionizing radiation) would have the power to discriminate between these models.

It is still controversial whether transmissible GI is an initiating event for colon cancer. Tomlinson and Bodmer (6) argue that cancer is an evolutionary process, and that the observed accumulation of chromosomal and other damage in colon cancers may simply be the result of selection for cells with growth advantage, with mutations ‘piggy-backing’ on this process of selection. Breivik (22,23) presents evidence that GI arises as a result of selection of cells in relation to specific mutagens in the environment; in particular he argues that the tissue specificity of chromosomal instability (CIN) and microsatellite instability (MIN) within the colon may result from adaptive selection associated with exposure to different agents. The fact that the two cancer stage GI model developed by Little and Wright (8) and the GI model of Nowak et al. (7), fit as well as, but no better than, that of Luebeck and Moolgavkar (9) suggests that, based on the fit of these models to the SEER colon cancer data, there is little evidence for or against the involvement of GI in colon cancer.

In general models predict similar mutation and proliferation rates between the sexes. A notable exception is the model of Luebeck and Moolgavkar (9), which predicts much higher mutation rates for males than for females in the penultimate stage, offset by much lower mutation rates for males in the final stage. These predictions are precisely in accord with the predictions made by Luebeck and Moolgavkar in fits of their model to a slightly different dataset (as discussed below). It is well known that germline mutation rates are much greater for men than for women, for reasons associated with the number of cell divisions each relevant cell incurs (24). Evidence for gender differences in somatic cell mutation rates is more equivocal, although there is evidence in experimental mice of elevated mutation rates for females compared with males at the hprt locus (25). There is suggestive evidence of involvement of hormonal factors in the aetiology of colon cancer in humans (26,27); hormonal factors are known to modulate cell proliferation in a number of tissues (27).

The elevation in mutation frequencies implied by the two genetic instability models, which is of the order of 20 000–80 000 for the two cancer-stage model of Little and Wright (8), and of the order of 300 000–400 000 in the model of Nowak et al. (7) (Tables III and IV), is large but not necessarily inconsistent with the biological data. Loeb and colleagues (4,28) derive estimates of numbers of gross chromosomal abnormalities in human tumours, which are of the order of 10 000–100 000-fold elevated compared with normal tissue. Mutation rates 1000–100 000-fold elevated have been observed in bacteria with defects in the dnaQ (DNA polymerase coding) gene (29,30).

As indicated above, the model that we describe as that of Luebeck and Moolgavkar (9) is close to that originally fitted, using a slightly different (but statistically equivalent) parameterization from that of Luebeck and Moolgavkar (9). We fitted this model to a somewhat different dataset, of colon cancers for whites in the SEER data for the years 1973–2002, whereas Luebeck and Moolgavkar fitted their model to whites and blacks (separately) for colon cancer in the SEER data over the years 1973–1996. We excluded those aged ≥85, because of doubts as to the completeness of ascertainment in this group; Luebeck and Moolgavkar included this age group. In contrast to the analysis of Little and Wright (8; see Materials and methods), we multiplicatively adjusted the cancer rates for registry and calendar year, similar to the adjustments used by Luebeck and Moolgavkar (9). It is possible that the differences in the underlying data employed may have some bearing on the very slightly different parameter estimates that we derive from those of Luebeck and Moolgavkar (9). For example, we estimate M(2,0)/G(3,0) to be 8.2 for males and 0.80 for females (Table V), whereas Luebeck and Moolgavkar (9) estimate it to be 8.0 for males and 0.7 for females. Likewise we estimate M(0,0) = M(1,0) to be 1.36 x 10–6 for males and 1.33 x 10–6 for females (Table V), whereas Luebeck and Moolgavkar (9) estimate these values as 1.4 x 10–6 for males and 1.3 x 10–6, respectively. Likewise, we estimate G(3,0) – D(3,0) to be 0.14 for males and 0.13 for females (Table V), whereas Luebeck and Moolgavkar (9) estimate these values as 0.15 and 0.13 respectively.

Nowak et al. (7) describe their colon cancer model as one of tumour initiation or induction. Nowak et al. (7) did not fit the model to data, but relied on numerical simulations assuming plausible parameter values. We have treated it here as a complete model of colon cancer. It may be, as is often assumed (8,12,31), that the duration of the cancer stage or stages between the induction of carcinogenic mutation and the detection of clinically overt colon cancer is relatively short, in which case the difference between these two interpretations of the model may be slight.

In contrast to earlier work (8), for most model fits we fixed the colon stem cell number at 108 cells, a value recently estimated by Potten et al. (15); it is thought that the number of stem cells is not likely to be more than an order of magnitude either side of this estimate (Professor Chris Potten, personal communication). Fixing the stem cell number has quite a marked effect on some of the model fits. In particular, the three and five cancer-stage models of Little and Wright (8) fit somewhat worse when the stem cell population is fixed at this value. When the stem cell population is allowed to vary, the predicted numbers of stem cells for both these models is arguably implausibly small, no more than ~103 for the three-stage model and no more than ~102 for the five-stage model (8).

An assumption made by most carcinogenesis models (7–10,12) is that cells are statistically conditionally independent, so that the cell populations may be described by a branching process. This is assumed for analytic tractability, but it is difficult to test. To the extent that it is known that cells communicate with each other via cell surface markers and otherwise, it is unlikely to be precisely true. One tissue in which, because of its spatial structure, this assumption may break down is the colon. The colon and small intestine are structured into crypts, each crypt containing some thousands of cells, and organized so that the stem cells are at the bottom of the crypt (32,33). There is evidence that there may be more than one stem cell at the bottom of each crypt (34). The progeny of stem cells migrate up the crypt and continue to divide, becoming progressively more differentiated. The differentiated cells eventually reach the top of the crypt where they are shed into the intestinal lumen. Potten and Loeffler (32) and Nowak and colleagues (33,35) have postulated similar models for cancers of the small intestine and colon taking account of the linear structure of the crypts, and in which necessarily the assumption of conditional independence breaks down. Set against this conventional view, there is evidence that early dysplastic cells originate in the inter-cryptal region, and spread laterally and downward to form new crypts that first connect to pre-existing crypts and eventually replace them (36).

A perhaps surprising feature is the simplicity of the models that we fit. In particular all models assume time constant mutation rates, and take no account of dietary or genetic factors, or occupational exposures. There are well-established familial risk factors for colon cancer, in particular familial adenomatous polyposis (FAP; 37). There are also well-known dietary risk factors, high fat and low fibre diets and diets deficient in fruit and vegetables, as also certain other chronic diseases, in particular ulcerative colitis and Crohn's disease (37). Ionizing radiation is also known to be associated with colon cancer (38). The fact that these relatively simple models can adequately account for the shape of the colon cancer age-incidence curve is also reflected by the comparatively modest overdispersion [by ~20% (males), 31% (females)]. For any random variable X({theta}), with mean and variance given by µ({theta}) = E{theta}[X({theta})] and v({theta}) = var{theta}[X({theta})] (functions of some parameter vector {theta}, which we can assume to be a random variable also) it is the case that var[X] = E[var{theta}[X]] + var[E{theta}[X]] = E[v({theta})] + var[µ({theta})]. For a Poisson random variable v({theta}) = var{theta}[X({theta})] = E{theta}[X({theta})] = µ({theta}), so that var[X] = E[X] + var[µ({theta})], i.e. the variance exceeds the mean by an amount equal to the variance of the mean. The fact that the variance is no more than ~30% greater than the mean therefore implies that there is relatively modest variation in the mean not accounted for by the fitted models. These facts imply that genetic, dietary and environmental factors have a relatively modest role at the population level. This is not to say that these well known risk factors (37,38) for colon cancer do not exist, only that they affect relatively small proportions of the general US population.

There are known to be at least two distinct phenotypes underlying GI. CIN is the dominating phenotype of cancer cells, and is characterized by numerical and structural aberrations of the genome. Cell cycle checkpoint genes such as TP53 (39), oncogenes such as ras and myc (40,41) and mitotic-spindle checkpoint genes such as hBUB1 (42) have all been implicated in generating this phenotype. The MIN phenotype on the other hand, is characterized by DNA polymerase errors, and typically results in a profusion of short DNA sequences, called microsatellites, being scattered throughout the genome. It is thought to be caused by defects in DNA mismatch repair genes such as MSH2 or MLH1 (43,44). There is evidence that the GI pathway, CIN or MIN, a cell goes down may be determined by the selective pressures exerted by specific carcinogens (45). The GI model devised by Little and Wright (8) can to some extent incorporate a multiplicity of pathways, and indeed this model formally embraces many of the other multiple pathway models that have been proposed (31,46), for example by setting certain mutation rates to 0 and others to very large values. Its flexibility is exemplified by this paper, which demonstrates that the models of Luebeck and Moolgavkar (9) and Nowak et al. (7) are special cases of it. Nevertheless, the multiple pathways possible within this model do not easily embrace models that assume multiple types of GI, such as might arise from models of colon cancer that include MIN and CIN destabilization pathways. Development of mechanistic models incorporating such features is the subject of ongoing research with European collaborators.

Epigenetic events, that is to say heritable changes (at the level of somatic cells) in gene expression that do not involve alterations in DNA sequence, are believed to play an important role, at least as important as that played by conventional mutational events, in both initiation and progression of many cancers (47). Methylation abnormalities are a particular sort of epigenetic event particularly common in tumour cells. Hypo-methylation is observed in a number of tumour cell lines, and is linked to activation of nearby oncogenes, in particular the RAS family (47,48); it is now thought that such loss of DNA methylation is a driving force of chromosomal instability (49). On the other hand addition of methyl groups to gene promoter regions (hyper-methylation) results in inactivation of certain tumour suppressor genes and silencing of DNA mismatch repair genes such as hMLH1 (50). In contrast to mutational events, methylation changes are potentially reversible, depending on environmental conditions (51). Our models make no assumption on the nature of the heritable modification carried by the cell population, and in the absence of such reversal of methylation status (whether hypo- or hyper-methylation) such epigenetic events have the same status as genetic ones. However, there are serious technical difficulties in incorporating reversible epigenetic events into our modelling framework (8), and a fortiori into other models thereby embraced by it (7,9,10,12).

Epistatic interactions between genes are known to take place, although it is not clear what their role is in relation to colon cancer (52). It is known that patients with FAP, who go on to develop colon cancer in most cases, carry mutated alleles of the APC gene, and inherit susceptibility according to Mendelian (monogenic) principles (52). However, many other genes are known to be associated with some elevation of risk for colon cancer (e.g. 52,53), and the interaction of these genes is as yet unclear. The modelling framework that we use (8) allows for relatively complex sequences of interactions of genetic or epigenetic events (subject to the limitations discussed above), and is appreciably more complex in this respect than the simple linear sequence of gene mutations assumed in most other cancer models (7,9,10,12), and which it embraces; more complex interactions may require further elaboration of the model, along the lines discussed above.


   Acknowledgments
 
This work was funded partially by the European Commission under contracts FIGD-CT-2000-0079 (NDISC) and FI6R-CT-2003-508842 (RISC-RAD). The authors are grateful for the helpful comments of a referee and of Dr Georg Luebeck, Professor Eric Wright and Professor Chris Potten.

Conflict of Interest Statement: None declared.


   References
Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 

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Received April 12, 2006; revised August 4, 2006; accepted September 2, 2006.


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