Lung Cancer Mortality (1950-1999) among Eldorado Uranium Workers: A Comparison of Models of Carcinogenesis and Empirical Excess Risk Models

Lung cancer mortality after exposure to radon decay products (RDP) among 16,236 male Eldorado uranium workers was analyzed. Male workers from the Beaverlodge and Port Radium uranium mines and the Port Hope radium and uranium refinery and processing facility who were first employed between 1932 and 1980 were followed up from 1950 to 1999. A total of 618 lung cancer deaths were observed. The analysis compared the results of the biologically-based two-stage clonal expansion (TSCE) model to the empirical excess risk model. The spontaneous clonal expansion rate of pre-malignant cells was reduced at older ages under the assumptions of the TSCE model. Exposure to RDP was associated with increase in the clonal expansion rate during exposure but not afterwards. The increase was stronger for lower exposure rates. A radiation-induced bystander effect could be a possible explanation for such an exposure response. Results on excess risks were compared to a linear dose-response parametric excess risk model with attained age, time since exposure and dose rate as effect modifiers. In all models the excess relative risk decreased with increasing attained age, increasing time since exposure and increasing exposure rate. Large model uncertainties were found in particular for small exposure rates.     

Risk coefficient for gamma-rays with regard to solid cancer

A previous investigation has uncoupled the solid cancer risk coefficient for neutrons from the low dose estimates of the relative biological effectiveness (RBE) of neutrons and the photon risk coefficient, and has related it to two more tangible quantities, the excess relative risk (ERR1) due to an intermediate reference dose D1 = 1 Gy of gamma-rays and the RBE of neutrons, R1, against this reference dose. With tentatively assumed RBE values between 20 and 50 and in terms of organ-averaged doses--rather than the usually invoked colon doses--the neutron risk factor was seen to be in general agreement with the current risk estimate of the International Commission on Radiation Protection (ICRP). The present assessment of the risk coefficient for gamma-rays incorporates--in terms of the unchanged A-bomb dosimetry system, DS86--this treatment of the neutrons, but is otherwise largely analogous to the evaluation of the A-bomb data for the ICRP report and for the recent report of the United Nations Scientific Committee on the effects of ionizing radiation, UNSCEAR. The resulting central estimate of the lifetime attributable risk (LAR) for solid cancer mortality is 0.043/Gy for a working population (ages 25-65), and is nearly the same whether the age at exposure or the attained age model is used for risk projection. For a population of all ages 0.042/Gy is obtained with the attained age model and 0.068/Gy with the age at exposure model. The values do not include a dose and dose rate effectiveness factor (DDREF), and they are only half as large as the new UNSCEAR estimates of 0.082/Gy (attained age model and all ages) and 0.13/Gy (age at exposure model and all ages). The difference is only partly due to the more explicit treatment of the neutrons. It reflects also the fact that UNSCEAR has converted ERR into LAR in a way that differs from the ICRP procedure, and that it has summed the overall risk coefficient for solid tumor mortality and incidence from separate estimates for eight solid tumor categories, whereas the present study employs a combined computation for all solid tumors and uses the ICRP procedure for the conversion of ERR into LAR. The appendix gives results for the solid cancer incidence data.     

Gestational mutations and carcinogenesis

We present a mathematical formulation to evaluate the effects of gestational mutations on cancer risk. The hazard or incidence function of cancer is expressed in terms of the Probability Generating Function (PGF) of the number of normal and mutated cells at birth. Using Filtered Poisson Process Theory, we obtain the PGF for several models for the accumulation of gestational mutations. In particular, we develop expressions for the hazard function when one or two successive mutations could occur during gestation. We also calculate the hazard when the background gestational mutation rates are increased due to exposure to mutagens, such as prenatal radiation. To illustrate the use of our models, we apply them to colorectal cancer in the SEER database. We find that the proportion of cancer risk attributable to developmental mutations depends on age and that it could be quite significant when gestational mutation rates are high. The analysis of the SEER data also shows that gestational mutations could contribute to inter-individual variations in colorectal cancer risk.     

Promoting action of radiation in the atomic bomb survivor carcinogenesis data?

The age-time patterns of risk in the atomic bomb survivor data on incidence of solid cancers suggest an action of low-LET radiation not only on the initiating event but also on promotion in a biologically motivated model that allows for both actions. The favored model indicates a decrease of radiation risks with age at exposure due to the initiating effect and with time since exposure due to the promoting effect. These result in a relative risk that depends mostly on attained age for ages at exposure above 20 years. According to the model, a dose of 100 mGy is inducing about the same number of initiating events that occur spontaneously in 1 year. Assuming that several mutations are needed to obtain intermediate cells with growth advantage does not improve the quality of fit. The estimated promoting effect could be explained if the number of intermediate cells increases by 80% at 1 Gy, e.g. due to stimulated cell repopulation.     

Ionizing radiation and genetic risks. XI. The doubling dose estimates from the mid-1950s to the present and the conceptual change to the use of human data on spontaneous mutation rates and mouse data on induced mutation rates for doubling dose calculations

This paper provides an overview of the concept of doubling dose, changes in the database employed for calculating it over the past 30 years and recent advances in this area. The doubling dose is estimated as a ratio of the average rates of spontaneous and induced mutations in a defined set of genes. The reciprocal of the doubling dose is the relative mutation risk per unit dose and is one of the quantities used in estimating genetic risks of radiation exposures. Most of the doubling dose estimates used thus far have been based on mouse data on spontaneous and induced rates of mutations. Initially restricted to mutations in defined genes (with particular focus on the seven genes at which induced recessive mutations were studied in different laboratories), the doubling dose concept was subsequently expanded to include other endpoints of genetic damage. At least during the past 20 years, the magnitude of the doubling dose has remained unchanged at approximately 1 Gy for chronic low LET radiation exposures.One of the assumptions underlying the use of the doubling dose based on mouse data for predicting genetic risks in humans, namely, that the spontaneous rates of mutations in mouse and human genes are similar, is incorrect; this is because of the fact that, unlike in the mouse, the mutation rate in humans differs between the two sexes (being higher in males than in females) and increases with paternal age. Further, an additional source of uncertainty in spontaneous mutation rate estimates in mice has been uncovered. This is related to the non-inclusion of mutations which arise as germinal mosaics and which result in clusters of identical mutations in the following generation. In view of these reasons, it is suggested that a prudent way forward is to revert to the use of human data on spontaneous mutation rates and mouse data on induced mutation rates for doubling dose calculations as was first done in the 1972 BEIR report of the US National Academy of Sciences. The advantages of this procedure are the following: (i) estimates of spontaneous mutation rates in humans, which are usually presented as sex-averaged rates, automatically include sex differences and paternal age-effects; (ii) since human geneticists count all mutations that arise anew irrespective of whether they are part of a cluster or not, had clusters occurred, they would have been included in mutation rate calculations and (iii) one stays close to the aim of risk estimation, namely, estimation of the risk of genetic diseases in humans.On the basis of detailed analyses of the pertinent data, it is now estimated that the average spontaneous mutation rate of human genes (n=135 genes) is: (2.95+/-0.64)x10(-6) per gene and the average induced mutation rate of mouse genes (n=34) is: (0.36+/-0.10)x10(-5) per gene per Gy for chronic low LET radiation. The resultant doubling dose is (0.82+/-0.29) Gy. The standard error of the doubling dose estimate incorporates sampling variability across loci for estimates of spontaneous and induced mutation rates as well as variability in induced mutation rates in individual mouse experiments on radiation-induced mutations. We suggest the use of a rounded doubling dose value of 1 Gy for estimating genetic risks of radiation. Although this value is the same as that used previously, its conceptual basis is different and the present estimate is based on more extensive data than has so far been the case.     

Incidence of female breast cancer among atomic bomb survivors, Hiroshima and Nagasaki, 1950-1990

An incidence survey of the Life Span Study (LSS) population found 1093 breast cancers among 1059 breast cancer cases diagnosed during 1950-1990. As in earlier breast cancer surveys of this population, a linear and statistically highly significant radiation dose response was found. In the analysis, particular attention was paid to modification of radiation dose response by age at exposure (e) and attained age (a). Dose-specific excess relative risk (ERR(1Sv)) decreased with increasing values of e and a. A linear dose-response model analysis, with e and a as exponential age modifiers, did not conclusively discriminate between the two variables as modifiers of dose response. A modified isotonic regression approach, requiring only that ERR(1Sv) be monotonic in age, provides a fresh perspective indicating that both e and a are important modifiers of dose response. Exposure before age 20 was associated with higher ERR(1Sv) compared to exposure at older ages, with no evidence of consistent variation by exposure age for ages under 20. ERR(1Sv) was observed to decline with increasing attained age, with by far the largest drop around age 35. Possible explanations for these observations are discussed, along with research approaches that might provide more information.     

Inferring somatic mutation rates using the stop-enhanced green fluorescent protein mouse

A new method is developed for estimating rates of somatic mutation in vivo. The stop-enhanced green fluorescent protein (EGFP) transgenic mouse carries multiple copies of an EGFP gene with a premature stop codon. The gene can revert to a functional form via point mutations. Mice treated with a potent mutagen, N-ethyl-N-nitrosourea (ENU), and mice treated with a vehicle alone are assayed for mutations in liver cells. A stochastic model is developed to model the mutation and gene expression processes and maximum-likelihood estimators of the model parameters are derived. A likelihood-ratio test (LRT) is developed for detecting mutagenicity. Parametric bootstrap simulations are used to obtain confidence intervals of the parameter estimates and to estimate the significance of the LRT. The LRT is highly significant (alpha < 0.01) and the 95% confidence interval for the relative effect of the mutagen (the ratio of the rate of mutation during the interval of mutagen exposure to the rate of background mutation) ranges from a minimum 200-fold effect of the mutagen to a maximum 2000-fold effect.     

Estimation of genetic risks of exposure to ionizing radiation: status in the year 2000

This paper provides an overview of the advances in the estimation of genetic risks of exposure of human populations to ionizing radiation with particular emphasis on the advances during the last decade. Among the latter are: (a) an upward revision of the estimates of the baseline frequencies of Mendelian diseases (from 1.25 to 2.4%); (b) the conceptual change to the use of a doubling dose based on human data on spontaneous mutation rates and mouse data on induced mutation rates (from the one based entirely on mouse data on spontaneous and induced mutation rates, which was the case thus far); (c) the fuller development of the concept of mutation component (MC) and its application to predict the responsiveness of Mendelian and chronic multifactorial diseases to induced mutations; (d) the concept that the major adverse effects of radiation exposure of human germ cells are likely to be manifest as multi-system developmental abnormalities and (e) the concept of potential recoverability correction factor (PRCF) to bridge the gap between induced mutations studied in mice and the risk of genetic disease in humans. For a population exposed to low LET, chronic/low dose-rate irradiation, the current estimates of risk for the first generation progeny are the following (all estimates per million live born progeny per Gy of parental irradiation): autosomal dominant and X-linked diseases, approximately 750 to 1,500 cases; autosomal recessive, nearly zero; chronic multifactorial diseases, approximately 250 to 1,200 cases and congenital abnormalities, approximately 2,000 cases. The total risk per Gy is of the order of approximately 3,000 to 4,700 cases which represent approximately 0.4 to 0.6% of the baseline frequency of these diseases. The main message is that at low doses of radiation of interest in risk estimation, the risk of adverse hereditary effects is small.     

Dose-rate effects of ethyl methanesulfonate on survival and mutation induction in cultured Chinese hamster cells

The dose-rate effects of ethyl methanesulfonate (EMS) on the survival and induction of mutations in Chinese hamster Don cells were investigated. The most effective time of exposure to EMS for reducing the surviving fraction of cells was 4 h, shorter and longer exposure times being less effective. The threshold or minimal concentration of EMS giving a surviving fraction of 0.5 was 0.05 mg/ml. The minimal effective time of exposure to EMS for cell death was 1 h. Corrected survival curves showed that longer exposure times at lower dose rates of EMS had less cytotoxic effect than shorter exposure times at higher dose rates.After exposure of Don cells to various doses of EMS for various times, the frequencies of mutations resistant to 6-thioguanine (6TG) were measured. An exposure time of 4 h produced a lower mutation frequency than shorter or longer exposure times that resulted in the same surviving fraction of cells. An exposure time of 20 h produced the highest induced mutation frequency.This system using cultured Chinese hamster cells should be useful as a sensitive procedure for detecting the mutagenic actions of chemicals.     

Lung cancer risk of Mayak workers: modelling of carcinogenesis and bystander effect

Lung cancer mortality in the period of 1948-2002 has been analysed for 6,293 male workers of the Mayak Production Association, for whose information on smoking, annual external doses and annual lung doses due to plutonium exposures was available. Individual likelihoods were maximized for the two-stage clonal expansion (TSCE) model of carcinogenesis and for an empirical risk model. Possible detrimental and protective bystander effects on mutation and malignant transformation rates were taken into account in the TSCE model. Criteria for non-nested models were used to evaluate the quality of fit. Data were found to be incompatible with the model including a detrimental bystander effect. The model with a protective bystander effect did not improve the quality of fit over models without a bystander effect. The preferred TSCE model was sub-multiplicative in the risks due to smoking and internal radiation, and more than additive. Smoking contributed 57% to the lung cancer deaths, the interaction of smoking and radiation 27%, radiation 10%, and others cause 6%. An assessment of the relative biological effectiveness of plutonium was consistent with the ICRP recommended value of 20. At age 60 years, the excess relative risk (ERR) per lung dose was 0.20 (95% CI: 0.13; 0.40) Sv(-1), while the excess absolute risk (EAR) per lung dose was 3.2 (2.0; 6.2) per 10(4) PY Sv. With increasing age attained the ERR decreased and the EAR increased. In contrast to the atomic bomb survivors, a significant elevated lung cancer risk was also found for age attained younger than 55 years. For cumulative lung doses below 5 Sv, the excess risk depended linearly on dose. The excess relative risk was significantly lower in the TSCE model for ages attained younger than 55 than that in the empirical model. This reflects a model uncertainty in the results, which is not expressed by the standard statistical uncertainty bands.     

Combining Magnetic Sorting of Mother Cells and Fluctuation Tests to Analyze Genome Instability During Mitotic Cell Aging in Saccharomyces cerevisiae

Saccharomyces cerevisiae has been an excellent model system for examining mechanisms and consequences of genome instability. Information gained from this yeast model is relevant to many organisms, including humans, since DNA repair and DNA damage response factors are well conserved across diverse species. However, S. cerevisiae has not yet been used to fully address whether the rate of accumulating mutations changes with increasing replicative (mitotic) age due to technical constraints. For instance, measurements of yeast replicative lifespan through micromanipulation involve very small populations of cells, which prohibit detection of rare mutations. Genetic methods to enrich for mother cells in populations by inducing death of daughter cells have been developed, but population sizes are still limited by the frequency with which random mutations that compromise the selection systems occur. The current protocol takes advantage of magnetic sorting of surface-labeled yeast mother cells to obtain large enough populations of aging mother cells to quantify rare mutations through phenotypic selections. Mutation rates, measured through fluctuation tests, and mutation frequencies are first established for young cells and used to predict the frequency of mutations in mother cells of various replicative ages. Mutation frequencies are then determined for sorted mother cells, and the age of the mother cells is determined using flow cytometry by staining with a fluorescent reagent that detects bud scars formed on their cell surfaces during cell division. Comparison of predicted mutation frequencies based on the number of cell divisions to the frequencies experimentally observed for mother cells of a given replicative age can then identify whether there are age-related changes in the rate of accumulating mutations. Variations of this basic protocol provide the means to investigate the influence of alterations in specific gene functions or specific environmental conditions on mutation accumulation to address mechanisms underlying genome instability during replicative aging.     

Polytomous disease mapping to detect uncommon risk factors for related diseases

A statistical model for jointly analysing the spatial variation of incidences of three (or more) diseases, with common and uncommon risk factors, is introduced. Deaths for different diseases are described by a logit model for multinomial responses (multinomial logit or polytomous logit model). For each area and confounding strata population (i.e. age-class, sex, race) the probabilities of death for each cause (the response probabilities) are estimated. A specic disease, the one having a common risk factor only, acts as the baseline category. The log odds are decomposed additively into shared (common to diseases different by the reference disease) and specic structured spatial variability terms, unstructured unshared spatial terms and confounders terms (such as age, race and sex) to adjust the crude observed data for their effects. Disease specic spatially structured effects are estimated; these are considered as latent variables denoting disease-specic risk factors. The model is presented with reference to a specic application. We considered the mortality data (from 1990 to 1994) relative to oral cavity, larynx and lung cancers in 13 age groups of males, in the 287 municipalities of Region of Tuscany (Italy). All these pathologies share smoking as a common risk factor; furthermore, two of them (oral cavity and larynx cancer) share alcohol consumption as a risk factor. All studies suggest that smoking and alcohol consumption are the major known risk factors for oral cavity and larynx cancers; nevertheless, in this paper, we investigate the possibility of other different risk factors for these diseases, or even the presence of an interaction effect (between smoking and alcohol risk factors) but with different spatial patterns for oral and larynx cancer. For each municipality and age-class the probabilities of death for each cause (the response probabilities) are estimated. Lung cancer acts as the baseline category. The log odds are decomposed additively into shared (common to oral cavity and larynx diseases) and specic structured spatial variability terms, unstructured unshared spatial terms and an age-group term. It turns out that oral cavity and larynx cancer have different spatial patterns for residual risk factors which are not the typical ones such as smoking habits and alcohol consumption. But, possibly, these patterns are due to different spatial interactions between smoking habits and alcohol consumption for the first and the second disease.     

Using semiparametric‐mixed model and functional linear model to detect vulnerable prenatal window to carcinogenic polycyclic aromatic hydrocarbons on fetal growth

Prenatal exposure to carcinogenic polycyclic aromatic hydrocarbons (c‐PAHs) through maternal inhalation induces higher risk for a wide range of fetotoxic effects. However, the most health‐relevant dose function from chronic gestational exposure remains unclear. Whether there is a gestational window during which the human embryo/fetus is particularly vulnerable to PAHs has not been examined thoroughly. We consider a longitudinal semiparametric‐mixed effect model to characterize the individual prenatal PAH exposure trajectory, where a nonparametric cyclic smooth function plus a linear function are used to model the time effect and random effects are used to account for the within‐subject correlation. We propose a penalized least squares approach to estimate the parametric regression coefficients and the nonparametric function of time. The smoothing parameter and variance components are selected using the generalized cross‐validation (GCV) criteria. The estimated subject‐specific trajectory of prenatal exposure is linked to the birth outcomes through a set of functional linear models, where the coefficient of log PAH exposure is a fully nonparametric function of gestational age. This allows the effect of PAH exposure on each birth outcome to vary at different gestational ages, and the window associated with significant adverse effect is identified as a vulnerable prenatal window to PAHs on fetal growth. We minimize the penalized sum of squared errors using a spline‐based expansion of the nonparametric coefficient function to draw statistical inferences, and the smoothing parameter is chosen through GCV.     

Radiation effects on breast cancer risk: a pooled analysis of eight cohorts

Breast cancer incidence rates after radiation exposure in eight large cohorts are described and compared. The nature of the exposures varies appreciably, ranging from a single or a small number of high-dose-rate exposures (Japanese atomic bomb survivors, U.S. acute post-partum mastitis patients, Swedish benign breast disease patients, and U.S. infants with thymic enlargement) to highly fractionated high-dose-rate exposures (two U.S. tuberculosis cohorts) and protracted low-dose-rate exposure (two Swedish skin hemangioma cohorts). There were 1,502 breast cancers among 77,527 women (about 35,000 of whom were exposed) with 1.8 million woman-years of follow-up. The excess risk depends linearly on dose with a downturn at high doses. No simple unified summary model adequately describes the excess risks in all groups. Excess risks for the thymus, tuberculosis, and atomic bomb survivor cohorts have similar temporal patterns, depending on attained age for relative risk models and on both attained age and age at exposure for excess rate models. Excess rates were similar in these cohorts, whereas, related in part to the low breast cancer background rates for Japanese women, the excess relative risk per unit dose in the bomb survivors was four times that in the tuberculosis or thymus cohorts. Excess rates were higher for the mastitis and benign breast disease cohorts. The hemangioma cohorts showed lower excess risks suggesting ameliorating dose-rate effects for protracted low-dose-rate exposures. For comparable ages at exposure (approximately 0.5 years), the excess risk in the hemangioma cohorts was about one-seventh that in the thymus cohort, whose members received acute high-dose-rate exposures. The results support the linearity of the radiation dose response for breast cancer, highlight the importance of age and age at exposure on the risks, and suggest a similarity in risks for acute and fractionated high-dose-rate exposures with much smaller effects from low-dose-rate protracted exposures. There is also a suggestion that women with some benign breast conditions may be at elevated risk of radiation-associated breast cancer.     

Epithelial tissue architecture protects against cancer

We consider the design of colon crypts from the point of view of minimizing the likelihood of generation of cancerous mutations. A stochastic mathematical model (a finite branching process) is developed and fully analyzed. It is found that depending on the mutation rates, different designs are evolutionarily advantageous. If the mutation rates associated with stem cells are a lot higher than the mutation rates of daughter cells, then few stem cells per crypt is the evolutionarily optimal strategy. If the mutation rates of stem cells are of the same order of magnitude or lower than those for daughter cells, then having as many stem cells per crypt as possible is the desirable design. We also found that the optimal evolutionary strategy may work very well to protect the organism from cancer in the young age, but the same strategy becomes detrimental as the organism ages. It pushes the onset of cancer back in time, but it results in an elevated cancer initiation rates as the organism gets older. Our model quantifies the idea that cancer and aging are the two sides of one coin.     

Trends in childhood leukaemia in the Nordic countries in relation to fallout from atmospheric nuclear weapons testing.

OBJECTIVE--To obtain further information about the risks of childhood leukaemia after exposure to ionising radiation at low doses and low dose rates before or after birth or to the father''s testes shortly before conception. DESIGN--Observational study of trends in incidence of childhood leukaemia in relation to estimated radiation exposures due to fallout from atmospheric nuclear weapons testing during the 1950s and 1960s. SETTING--Nordic countries. SUBJECTS--Children aged under 15 years. MAIN OUTCOME MEASURES--Incidence rates of leukaemia by age at diagnosis, sex, country, and calendar year of diagnosis or year of birth; exposure category; relation between leukaemia and exposure for children aged 0-14 and 0-4 separately. RESULTS--During the high fallout period the average estimated dose equivalent to the fetal red bone marrow was around 140 mu Sv and the average annual testicular dose 140 mu Sv. There was little evidence of increased incidence of leukaemia among children born in these years. Doses to the red bone marrow of a child after birth were higher, and during the high exposure period children would have been subjected to an additional dose equivalent of around 1500 mu Sv, similar to doses received by children in several parts of central and eastern Europe owing to the Chernobyl accident and about 50% greater than the annual dose equivalent to the red bone marrow of a child from natural radiation. leukaemia incidence and red marrow dose was not related overall, but rates of leukaemia in the high exposure period were slightly higher than in the surrounding medium exposure period (relative risk for ages 0-14: 1.07, 95% confidence interval 1.00 to 1.14; for ages 0-4: 1.11, 1.00 to 1.24). CONCLUSIONS--Current predicted risks of childhood leukaemia after exposure to radiation are not greatly underestimated for low dose rate exposures.     

Analysis of thyroid cancer data from the Ukraine after ’Chernobyl’ using a two-mutation carcinogenesis model

The thyroid cancer data of children in the northern regions of the Ukraine after the reactor accident at Chernobyl were combined with thyroid dose measurements in the same regions and analysed using a two- mutation carcinogenesis model. The best fit was obtained for radiation acting as an initiating agent, i.e. on the first mutation of the model. The observed relatively high increase of thyroid cancer incidence after 1990 in children exposed to radiation released after the reactor accident could be ascribed to the high thyroid doses and the relatively low background thyroid cancer incidence in children. The maximum annual incidence is predicted to occur fairly soon after the reactor accident, i.e. about 10 years. For adults, the predicted relative increase of annual thyroid cancers is much lower than for children younger than 20 years. The modelling results are used to derive risk estimates for radiation-induced thyroid cancer. These risk estimates are dependent on age at exposure, follow-up time and the background thyroid cancer incidence. The calculated excess absolute risk for a population of all ages is about one-third of that currently used by ICRP, but for children the calculated absolute risks are about a factor of 3 higher than derived in other epidemiological studies. The model results indicate that the excess absolute radiation risk per unit dose for children is about the same as or a little lower than that for adults. Received: 11 May 1999 / Accepted: 30 December 1999     

Genetic progression and the waiting time to cancer

Cancer results from genetic alterations that disturb the normal cooperative behavior of cells. Recent high-throughput genomic studies of cancer cells have shown that the mutational landscape of cancer is complex and that individual cancers may evolve through mutations in as many as 20 different cancer-associated genes. We use data published by Sjöblom et al. (2006) to develop a new mathematical model for the somatic evolution of colorectal cancers. We employ the Wright-Fisher process for exploring the basic parameters of this evolutionary process and derive an analytical approximation for the expected waiting time to the cancer phenotype. Our results highlight the relative importance of selection over both the size of the cell population at risk and the mutation rate. The model predicts that the observed genetic diversity of cancer genomes can arise under a normal mutation rate if the average selective advantage per mutation is on the order of 1%. Increased mutation rates due to genetic instability would allow even smaller selective advantages during tumorigenesis. The complexity of cancer progression can be understood as the result of multiple sequential mutations, each of which has a relatively small but positive effect on net cell growth.