Invariant Characteristics of Carcinogenesis.

Carcinogenic modeling is aimed at mathematical descriptions of cancer development in aging. In this work, we assumed that a small fraction of individuals in the population is susceptible to cancer, while the rest of the population is resistant to cancer. For individuals susceptible to cancer we adopted methods of conditional survival analyses. We performed computational experiments using data on pancreatic, stomach, gallbladder, colon and rectum, liver, and esophagus cancers from the gastrointestinal system collected for men and women in the SEER registries during 1975-2009. In these experiments, we estimated the time period effects, the birth cohort effects, the age effects and the population (unconditional) cancer hazard rates. We also estimated the individual cancer presentation rates and the individual cancer resistance rates, which are, correspondingly, the hazard and survival rates conditioned on the susceptibility to cancer. The performed experiments showed that for men and women, patterns of the age effects, the individual cancer presentation rates and the individual cancer resistance rates are: (i) intrinsic for each cancer subtype, (ii) invariant to the place of living of the individuals diagnosed with cancer, and (iii) well adjusted for the modifiable variables averaged at a given time period. Such specificity and invariability of the age effects, the individual cancer presentation rates and the individual cancer resistance rates suggest that these carcinogenic characteristics can be useful for predictive carcinogenic studies by methods of inferential statistics and for the development of novel strategies for cancer prevention.

Heuristic modeling of carcinogenesis for the population with dichotomous susceptibility to cancer: a pancreatic cancer example.

At present, carcinogenic models imply that all individuals in a population are susceptible to cancer. These models either ignore a fall of the cancer incidence rate at old ages, or use some poorly identifiable parameters for its accounting. In this work, a new heuristic model is proposed. The model assumes that, in a population, only a small fraction (pool) of individuals is susceptible to cancer and decomposes the problem of the carcinogenic modeling on two sequentially solvable problems: (i) determination of the age-specific hazard rate in individuals susceptible to cancer (individual hazard rate) from the observed hazard rate in the population (population hazard rate); and (ii) modelling of the individual hazard rate by a chosen "up" of the theoretical hazard function describing cancer occurrence in individuals in time (age). The model considers carcinogenesis as a failure of individuals susceptible to cancer to resist cancer occurrence in aging and uses, as the theoretical hazard function, the three-parameter Weibull hazard function, often utilized in a failure analysis. The parameters of this function, providing the best fit of the modeled and observed individual hazard rates (determined from the population hazard rates), are the outcomes of the modeling. The model was applied to the pancreatic cancer data. It was shown that, in the populations stratified by gender, race and the geographic area of living, the modeled and observed population hazard rates of pancreatic cancer occurrence have similar turnovers at old ages. The sizes of the pools of individuals susceptible to this cancer: (i) depend on gender, race and the geographic area of living; (ii) proportionally influence the corresponding population hazard rates; and (iii) do not influence the individual hazard rates. The model should be further tested using data on other types of cancer and for the populations stratified by different categorical variables.

Web tool for estimating the cancer hazard rates in aging.

A computational approach for estimating the overall, population, and individual cancer hazard rates was developed. The population rates characterize a risk of getting cancer of a specific site/type, occurring within an age-specific group of individuals from a specified population during a distinct time period. The individual rates characterize an analogous risk but only for the individuals susceptible to cancer. The approach uses a novel regularization and anchoring technique to solve an identifiability problem that occurs while determining the age, period, and cohort (APC) effects. These effects are used to estimate the overall rate, and to estimate the population and individual cancer hazard rates. To estimate the APC effects, as well as the population and individual rates, a new web-based computing tool, called the CancerHazard@Age, was developed. The tool uses data on the past and current history of cancer incidences collected during a long time period from the surveillance databases. The utility of the tool was demonstrated using data on the female lung cancers diagnosed during 1975-2009 in nine geographic areas within the USA. The developed tool can be applied equally well to process data on other cancer sites. The data obtained by this tool can be used to develop novel carcinogenic models and strategies for cancer prevention and treatment, as well as to project future cancer burden.

Prognostic estimator of survival for patients with localized and extended pancreatic ductal adenocarcinoma.

The 18,352 pancreatic ductal adenocarcinoma (PDAC) cases from the Surveillance Epidemiology and End Results (SEER) database were analyzed using the Kaplan-Meier method for the following variables: race, gender, marital status, year of diagnosis, age at diagnosis, pancreatic subsite, T-stage, N-stage, M-stage, tumor size, tumor grade, performed surgery, and radiation therapy. Because the T-stage variable did not satisfy the proportional hazards assumption, the cases were divided into cases with T1- and T2-stages (localized tumor) and cases with T3- and T4-stages (extended tumor). For estimating survival and conditional survival probabilities in each group, a multivariate Cox regression model adjusted for the remaining covariates was developed. Testing the reproducibility of model parameters and generalizability of these models showed that the models are well calibrated and have concordance indexes equal to 0.702 and 0.712, respectively. Based on these models, a prognostic estimator of survival for patients diagnosed with PDAC was developed and implemented as a computerized web-based tool.

Basic equations and computing procedures for frailty modeling of carcinogenesis: application to pancreatic cancer data.

Modeling of cancer hazards at age t deals with a dichotomous population, a small part of which (the fraction at risk) will get cancer, while the other part will not. Therefore, we conditioned the hazard function, h(t), the probability density function (pdf), f(t), and the survival function, S(t), on frailty α in individuals. Assuming α has the Bernoulli distribution, we obtained equations relating the unconditional (population level) hazard function, hU (t), cumulative hazard function, HU (t), and overall cumulative hazard, H0, with the h(t), f(t), and S(t) for individuals from the fraction at risk. Computing procedures for estimating h(t), f(t), and S(t) were developed and used to fit the pancreatic cancer data collected by SEER9 registries from 1975 through 2004 with the Weibull pdf suggested by the Armitage-Doll model. The parameters of the obtained excellent fit suggest that age of pancreatic cancer presentation has a time shift about 17 years and five mutations are needed for pancreatic cells to become malignant.

Breast cancer incidence in black and white women stratified by estrogen and progesterone receptor statuses.

BACKGROUND: There is increasing evidence that breast cancer is a heterogeneous disease presented by different phenotypes and that white women have a higher breast cancer incidence rate, whereas black women have a higher mortality rate. It is also well known that white women have lower incidence rates than black women until approximately age 40, when rate curves cross over and white women have higher rates. The goal of this study was to validate the risk of white and black women to breast cancer phenotypes, stratified by statuses of the estrogen (ER) and progesterone (PR) receptors. METHODOLOGY/PRINCIPAL FINDINGS: SEER17 data were fractioned by receptor status into [ER+, PR+], [ER-, PR-], [ER+, PR-], and [ER-, PR+] phenotypes. It was shown that in black women compared to white women, cumulative age-specific incidence rates are: (i) smaller for the [ER+, PR+] phenotype; (ii) larger for the [ER-, PR-] and [ER-, PR+] phenotypes; and (iii) almost equal for the [ER+, PR-] phenotype. Clemmesen's Hook, an undulation unique to women's breast cancer age-specific incidence rate curves, is shown here to exist in both races only for the [ER+, PR+] phenotype. It was also shown that for all phenotypes, rate curves have additional undulations and that age-specific incidence rates are nearly proportional in all age intervals. CONCLUSIONS/SIGNIFICANCE: For black and white women, risk for the [ER+, PR+], [ER-, PR-] and [ER-, PR+] phenotypes are race dependent, while risk for the [ER+, PR-] phenotype is almost independent of race. The processes of carcinogenesis in aging, leading to the development of each of the considered breast cancer phenotypes, are similar in these racial groups. Undulations exhibited on the curves of age-specific incidence rates of the considered breast cancer phenotypes point to the presence of several subtypes (to be determined) of each of these phenotypes.

A heuristic solution of the identifiability problem of the age-period-cohort analysis of cancer occurrence: lung cancer example.

BACKGROUND: The Age-Period-Cohort (APC) analysis is aimed at estimating the following effects on disease incidence: (i) the age of the subject at the time of disease diagnosis; (ii) the time period, when the disease occurred; and (iii) the date of birth of the subject. These effects can help in evaluating the biological events leading to the disease, in estimating the influence of distinct risk factors on disease occurrence, and in the development of new strategies for disease prevention and treatment. METHODOLOGY/PRINCIPAL FINDINGS: We developed a novel approach for estimating the APC effects on disease incidence rates in the frame of the Log-Linear Age-Period-Cohort (LLAPC) model. Since the APC effects are linearly interdependent and cannot be uniquely estimated, solving this identifiability problem requires setting four redundant parameters within a set of unknown parameters. By setting three parameters (one of the time-period and the birth-cohort effects and the corresponding age effect) to zero, we reduced this problem to the problem of determining one redundant parameter and, used as such, the effect of the time-period adjacent to the anchored time period. By varying this identification parameter, a family of estimates of the APC effects can be obtained. Using a heuristic assumption that the differences between the adjacent birth-cohort effects are small, we developed a numerical method for determining the optimal value of the identification parameter, by which a unique set of all APC effects is determined and the identifiability problem is solved. CONCLUSIONS/SIGNIFICANCE: We tested this approach while estimating the APC effects on lung cancer occurrence in white men and women using the SEER data, collected during 1975-2004. We showed that the LLAPC models with the corresponding unique sets of the APC effects estimated by the proposed approach fit very well with the observational data.

Extension of cox proportional hazard model for estimation of interrelated age-period-cohort effects on cancer survival.

In the frame of the Cox proportional hazard (PH) model, a novel two-step procedure for estimating age-period-cohort (APC) effects on the hazard function of death from cancer was developed. In the first step, the procedure estimates the influence of joint APC effects on the hazard function, using Cox PH regression procedures from a standard software package. In the second step, the coefficients for age at diagnosis, time period and birth cohort effects are estimated. To solve the identifiability problem that arises in estimating these coefficients, an assumption that neighboring birth cohorts almost equally affect the hazard function was utilized. Using an anchoring technique, simple procedures for obtaining estimates of interrelated age at diagnosis, time period and birth cohort effect coefficients were developed.As a proof-of-concept these procedures were used to analyze survival data, collected in the SEER database, on white men and women diagnosed with LC in 1975-1999 and the age at diagnosis, time period and birth cohort effect coefficients were estimated. The PH assumption was evaluated by a graphical approach using log-log plots. Analysis of trends of these coefficients suggests that the hazard of death from LC for a given time from cancer diagnosis: (i) decreases between 1975 and 1999; (ii) increases with increasing the age at diagnosis; and (iii) depends upon birth cohort effects.The proposed computing procedure can be used for estimating joint APC effects, as well as interrelated age at diagnosis, time period and birth cohort effects in survival analysis of different types of cancer.

Weibull-like Model of Cancer Development in Aging.

Mathematical modeling of cancer development is aimed at assessing the risk factors leading to cancer. Aging is a common risk factor for all adult cancers. The risk of getting cancer in aging is presented by a hazard function that can be estimated from the observed incidence rates collected in cancer registries. Recent analyses of the SEER database show that the cancer hazard function initially increases with the age, and then it turns over and falls at the end of the lifetime. Such behavior of the hazard function is poorly modeled by the exponential or compound exponential-linear functions mainly utilized for the modeling. In this work, for mathematical modeling of cancer hazards, we proposed to use the Weibull-like function, derived from the Armitage-Doll multistage concept of carcinogenesis and an assumption that number of clones at age t developed from mutated cells follows the Poisson distribution. This function is characterized by three parameters, two of which (r and λ) are the conventional parameters of the Weibull probability distribution function, and an additional parameter (C(0)) that adjusts the model to the observational data. Biological meanings of these parameters are: r-the number of stages in carcinogenesis, λ-an average number of clones developed from the mutated cells during the first year of carcinogenesis, and C(0)-a data adjustment parameter that characterizes a fraction of the age-specific population that will get this cancer in their lifetime. To test the validity of the proposed model, the nonlinear regression analysis was performed for the lung cancer (LC) data, collected in the SEER 9 database for white men and women during 1975-2004. Obtained results suggest that: (i) modeling can be improved by the use of another parameter A- the age at the beginning of carcinogenesis; and (ii) in white men and women, the processes of LC carcinogenesis vary by A and C(0), while the corresponding values of r and λ are nearly the same. Overall, the proposed Weibull-like model provides an excellent fit of the estimates of the LC hazard function in aging. It is expected that the Weibull-like model can be applicable to fit estimates of hazard functions of other adult cancers as well.

Estimation of hazard functions in the log-linear age-period-cohort model: application to lung cancer risk associated with geographical area.

An efficient computing procedure for estimating the age-specific hazard functions by the log-linear age-period-cohort (LLAPC) model is proposed. This procedure accounts for the influence of time period and birth cohort effects on the distribution of age-specific cancer incidence rates and estimates the hazard function for populations with different exposures to a given categorical risk factor. For these populations, the ratio of the corresponding age-specific hazard functions is proposed for use as a measure of relative hazard. This procedure was used for estimating the risks of lung cancer (LC) for populations living in different geographical areas. For this purpose, the LC incidence rates in white men and women, in three geographical areas (namely: San Francisco-Oakland, Connecticut and Detroit), collected from the SEER 9 database during 1975-2004, were utilized. It was found that in white men the averaged relative hazard (an average of the relative hazards over all ages) of LC in Connecticut vs. San Francisco-Oakland is 1.31 +/- 0.02, while in Detroit vs. San Francisco-Oakland this averaged relative hazard is 1.53 +/- 0.02. In white women, analogous hazards in Connecticut vs. San Francisco-Oakland and Detroit vs. San Francisco-Oakland are 1.22 +/- 0.02 and 1.32 +/- 0.02, correspondingly. The proposed computing procedure can be used for assessing hazard functions for other categorical risk factors, such as gender, race, lifestyle, diet, obesity, etc.

A Generalized Beta model for the age distribution of cancers: application to pancreatic and kidney cancer.

The relationships between cancer incidence rates and the age of patients at cancer diagnosis are a quantitative basis for modeling age distributions of cancer. The obtained model parameters are needed to build rigorous statistical and biological models of cancer development. In this work, a new mathematical model, called the Generalized Beta (GB) model is proposed. Confidence intervals for parameters of this model are derived from a regression analysis. The GB model was used to approximate the incidence rates of the first primary, microscopically confirmed cases of pancreatic cancer (PC) and kidney cancer (KC) that served as a test bed for the proposed approach. The use of the GB model allowed us to determine analytical functions that provide an excellent fit for the observed incidence rates for PC and KC in white males and females. We make the case that the cancer incidence rates can be characterized by a unique set of model parameters (such as an overall cancer rate, and the degree of increase and decrease of cancer incidence rates). Our results suggest that the proposed approach significantly expands possibilities and improves the performance of existing mathematical models and will be very useful for modeling carcinogenic processes characteristic of cancers. To better understand the biological plausibility behind the aforementioned model parameters, detailed molecular, cellular, and tissue-specific mechanisms underlying the development of each type of cancer require further investigation. The model parameters that can be assessed by the proposed approach will complement and challenge future biomedical and epidemiological studies.

A novel approach for analysis of the log-linear age-period-cohort model: application to lung cancer incidence.

A simple, computationally efficient procedure for analyses of the time period and birth cohort effects on the distribution of the age-specific incidence rates of cancers is proposed. Assuming that cohort effects for neighboring cohorts are almost equal and using the Log-Linear Age-Period-Cohort Model, this procedure allows one to evaluate temporal trends and birth cohort variations of any type of cancer without prior knowledge of the hazard function. This procedure was used to estimate the influence of time period and birth cohort effects on the distribution of the age-specific incidence rates of first primary, microscopically confirmed lung cancer (LC) cases from the SEER9 database. It was shown that since 1975, the time period effect coefficients for men increase up to 1980 and then decrease until 2004. For women, these coefficients increase from 1975 up to 1990 and then remain nearly constant. The LC birth cohort effect coefficients for men and women increase from the cohort of 1890-94 until the cohort of 1925-29, then decrease until the cohort of 1950-54 and then remain almost unchanged. Overall, LC incidence rates, adjusted by period and cohort effects, increase up to the age of about 72-75, turn over, and then fall after the age of 75-78. The peak of the adjusted rates in men is around the age of 77-78, while in women, it is around the age of 72-73. Therefore, these results suggest that the age distribution of the incidence rates in men and women fall at old ages.