The multistage model of cancer development: some implications

Modeling of senescent cell dynamics predicts a late‐life decrease in cancer incidence

Current oncogenic theories state that tumors arise from cell lineages that sequentially accumulate (epi)mutations, progressively turning healthy cells into carcinogenic ones. While those models found some empirical support, they are little predictive of intraspecies age-specific cancer incidence and of interspecies cancer prevalence. Notably, in humans and lab rodents, a deceleration (and sometimes decline) of cancer incidence rate has been found at old ages. Additionally, dominant theoretical models of oncogenesis predict that cancer risk should increase in large and/or long-lived species, which is not supported by empirical data. Here, we explore the hypothesis that cellular senescence could explain those incongruent empirical patterns. More precisely, we hypothesize that there is a trade-off between dying of cancer and of (other) ageing-related causes. This trade-off between organismal mortality components would be mediated, at the cellular scale, by the accumulation of senescent cells. In this framework, damaged cells can either undergo apoptosis or enter senescence. Apoptotic cells lead to compensatory proliferation, associated with an excess risk of cancer, whereas senescent cell accumulation leads to ageing-related mortality. To test our framework, we build a deterministic model that first describes how cells get damaged, undergo apoptosis, or enter senescence. We then translate those cellular dynamics into a compound organismal survival metric also integrating life-history traits. We address four different questions linked to our framework: can cellular senescence be adaptive, do the predictions of our model reflect epidemiological patterns observed among mammal species, what is the effect of species sizes on those answers, and what happens when senescent cells are removed? Importantly, we find that cellular senescence can optimize lifetime reproductive success. Moreover, we find that life-history traits play an important role in shaping the cellular trade-offs. Overall, we demonstrate that integrating cellular biology knowledge with eco-evolutionary principles is crucial to solve parts of the cancer puzzle.

COMBINED ANALYSIS OF CANCER INCIDENCE AND LIFESPAN IN MICE EXPOSED TO CHRONIC LOW DOSE RATE RADIATION

The authors performed a combined analysis using the data obtained from continuous low dose rate irradiation experiments on mice conducted at the Institute for Environmental Sciences, namely, cancer incidence data and lifespan data. They estimated the length of cancer progression period, which is difficult to assess experimentally. The combined analysis showed that the mean cancer progression period is 173 d in the control group and 103 d in the irradiated group.

Profound synchrony of age-specific incidence rates and tumor suppression for different cancer types as revealed by the multistage-senescence model of carcinogenesis

The age-specific trend of cancer incidence rates, but not its magnitude, is well described employing the multistage theory of carcinogenesis by Armitage and Doll in combination with the senescence model of Pompei and Wilson. We derived empirical parameters of the multistage-senescence model from U.S. Surveillance, Epidemiology, and End Results (SEER) incidence data from 2000–2003 and 2010–2013 for The Cancer Genome Atlas (TCGA) cancer types. Under the assumption of a constant tumor-specific transition rate between stages, there is an extremely strong linear relationship (P < 0.0001) between the number of stages and the stage transition rate. The senescence tumor suppression factor for 20 non-reproductive cancers is remarkably consistent (0.0099±0.0005); however, five female reproductive cancers have significantly higher tumor suppression. The peak incidence rate for non-reproductive cancers occurs at a younger age for cancers with fewer stages and their carcinogenic stages are of longer duration. Driver gene mutations are shown to contribute on average only about a third of the carcinogenic stages of different tumor types. A tumor’s accumulated incidence, calculated using a two-variable (age, stage) model, is strongly associated with intrinsic cancer risk. During both early adulthood and senescence, the pace of tumor suppression appears to be synchronized across most cancer types, suggesting the presence of overlapping evolutionary processes.

A Multi-Step Model of Parkinson's Disease Pathogenesis

BACKGROUND

Parkinson's disease (PD) may result from the combined effect of multiple etiological factors. The relationship between disease incidence and age, as demonstrated in the cancer literature, can be used to model a multistep pathogenic process, potentially affording unique insights into disease development.

OBJECTIVES

We tested whether the observed incidence of PD is consistent with a multistep process, estimated the number of steps required and whether this varies with age, and examined drivers of sex differences in PD incidence.

METHODS

Our validated probabilistic modeling process, based on medication prescribing, generated nationwide age- and sex-adjusted PD incidence data spanning 2006-2017. Models of log(incidence) versus log(age) were compared using Bayes factors, to estimate (1) if a linear relationship was present (indicative of a multistep process); (2) the relationship's slope (one less than number of steps); (3) whether slope was lower at younger ages; and (4) whether slope or y-intercept varied with sex.

RESULTS

Across >15,000 incident cases of PD, there was a clear linear relationship between log(age) and log(incidence). Evidence was strongest for a model with an initial slope of 5.2 [3.8, 6.4], an inflexion point at age 45, and beyond this a slope of 6.8 [6.4, 7.2]. There was evidence for the intercept varying by sex, but no evidence for slope being sex-dependent.

CONCLUSIONS

The age-specific incidence of PD is consistent with a process that develops in multiple, discrete steps - on average six before age 45 and eight after. The model supports theories emphasizing the primacy of environmental factors in driving sex differences in PD incidence. © 2021 The Authors. Movement Disorders published by Wiley Periodicals LLC on behalf of International Parkinson and Movement Disorder Society.

Characteristic patterns of cancer incidence: Epidemiological data, biological theories, and multistage models

We investigate and classify several patterns in cancer incidence and relative risk data which persist across different countries and multiple published studies. We then explore biological hypotheses as well as many mathematical models in the literature that attempt to explain these patterns. A general modeling framework is presented which is general enough to model most of observed behaviors. It is our belief that this model has sufficient flexibility to be adapted to new information as it is discovered. As one application of this framework, we give a model for the effect of aging on the process of carcinogenesis.

Simulating Space Radiation-Induced Breast Tumor Incidence Using Automata

Estimating cancer risk from space radiation has been an ongoing challenge for decades primarily because most of the reported epidemiological data on radiation-induced risks are derived from studies of atomic bomb survivors who were exposed to an acute dose of gamma rays instead of chronic high-LET cosmic radiation. In this study, we introduce a formalism using cellular automata to model the long-term effects of ionizing radiation in human breast for different radiation qualities. We first validated and tuned parameters for an automata-based two-stage clonal expansion model simulating the age dependence of spontaneous breast cancer incidence in an unexposed U.S.

POPULATION

We then tested the impact of radiation perturbation in the model by modifying parameters to reflect both targeted and nontargeted radiation effects. Targeted effects (TE) reflect the immediate impact of radiation on a cell's DNA with classic end points being gene mutations and cell death. They are well known and are directly derived from experimental data. In contrast, nontargeted effects (NTE) are persistent and affect both damaged and undamaged cells, are nonlinear with dose and are not well characterized in the literature. In this study, we introduced TE in our model and compared predictions against epidemiologic data of the atomic bomb survivor cohort. TE alone are not sufficient for inducing enough cancer. NTE independent of dose and lasting ∼100 days postirradiation need to be added to accurately predict dose dependence of breast cancer induced by gamma rays. Finally, by integrating experimental relative biological effectiveness (RBE) for TE and keeping NTE (i.e., radiation-induced genomic instability) constant with dose and LET, the model predicts that RBE for breast cancer induced by cosmic radiation would be maximum at 220 keV/μm. This approach lays the groundwork for further investigation into the impact of chronic low-dose exposure, inter-individual variation and more complex space radiation scenarios.

Minimizing second cancer risk following radiotherapy: current perspectives

Secondary cancer risk following radiotherapy is an increasingly important topic in clinical oncology with impact on treatment decision making and on patient management. Much of the evidence that underlies our understanding of secondary cancer risks and our risk estimates are derived from large epidemiologic studies and predictive models of earlier decades with large uncertainties. The modern era is characterized by more conformal radiotherapy technologies, molecular and genetic marker approaches, genome-wide studies and risk stratifications, and sophisticated biologically based predictive models of the carcinogenesis process. Four key areas that have strong evidence toward affecting secondary cancer risks are 1) the patient age at time of radiation treatment, 2) genetic risk factors, 3) the organ and tissue site receiving radiation, and 4) the dose and volume of tissue being irradiated by a particular radiation technology. This review attempts to summarize our current understanding on the impact on secondary cancer risks for each of these known risk factors. We review the recent advances in genetic studies and carcinogenesis models that are providing insight into the biologic processes that occur from tissue irradiation to the development of a secondary malignancy. Finally, we discuss current approaches toward minimizing the risk of radiation-associated secondary malignancies, an important goal of clinical radiation oncology.

A computational model incorporating neural stem cell dynamics reproduces glioma incidence across the lifespan in the human population

Glioma is the most common form of primary brain tumor. Demographically, the risk of occurrence increases until old age. Here we present a novel computational model to reproduce the probability of glioma incidence across the lifespan. Previous mathematical models explaining glioma incidence are framed in a rather abstract way, and do not directly relate to empirical findings. To decrease this gap between theory and experimental observations, we incorporate recent data on cellular and molecular factors underlying gliomagenesis. Since evidence implicates the adult neural stem cell as the likely cell-of-origin of glioma, we have incorporated empirically-determined estimates of neural stem cell number, cell division rate, mutation rate and oncogenic potential into our model. We demonstrate that our model yields results which match actual demographic data in the human population. In particular, this model accounts for the observed peak incidence of glioma at approximately 80 years of age, without the need to assert differential susceptibility throughout the population. Overall, our model supports the hypothesis that glioma is caused by randomly-occurring oncogenic mutations within the neural stem cell population. Based on this model, we assess the influence of the (experimentally indicated) decrease in the number of neural stem cells and increase of cell division rate during aging. Our model provides multiple testable predictions, and suggests that different temporal sequences of oncogenic mutations can lead to tumorigenesis. Finally, we conclude that four or five oncogenic mutations are sufficient for the formation of glioma.

A new model of time scheme for progression of colorectal cancer

BACKGROUND

tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process. These models assumed that the mutation rate of genes is constant during different stages. However, it has been pointed that the subsequent driver mutations appear faster than the previous ones and the cumulative time to have more driver mutations grows with the growing number of gene mutations. Thus it is still a challenge to calculate the time when the first mutation occurs and to determine the influence of tumour size on the mutation rate.

RESULTS

In this work we present a general framework to remedy the shortcoming of existing models. Rather than considering the information of gene mutations based on a population of patients, we for the first time determine the values of the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient determined by our model are consistent with the predictions made by the published models. Our calculation showed that the values of biological parameters, such as the selective advantage coefficient, initial mutation rate and cell doubling time diversely depend on individuals. Our model has successfully predicted the values of several important parameters in cancer progression, such as the selective advantage coefficient, initial mutation rate and cell doubling time. In addition, experimental data validated our predicted initial mutation rate and cell doubling time.

CONCLUSIONS

The introduced new parameter makes our proposed model more flexible to fix various types of information based on different patients in cancer progression.

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.

Comparison of multi-stage dose-response mixture models, with applications

This article concerns the analysis of a stochastic model that we propose for the population that generates a response (response measure) to the dose with the multi-stage model. The parameter uncertainty is dealt with via random dose and random size of the population at risk. The response measure is modeled by a random sum of mixed Bernoulli random variables with arbitrary distribution for the mixing parameters. Some extensions of the model are defined by functionals of the infection probability, fulfilling some convexity properties. We analyze the response by stochastic comparisons under different stochastic relations on the random dosages and the random sizes of the population at risk; or on the random infection rates. We provide stochastic exact bounds of the mixture model for the response, using inequalities and the positive quadrant dependence. Numerical bounds of the response by a dose having a scalar value or having an exponential or uniform distributions are obtained. Some conclusions are derived: the lower estimation of the response measure in the increasing convex order sense by replacing the dosages by their means; effects of the variation of the dose on the magnitude of the probability distribution of the response; effects of parameter correlation on the degree of variability of the response to any random dose; the low-dose region assessment; and also, the classical multi-stage model is compared versus the mixture model featuring independence and versus that with positive quadrant dependence.

Similarities in the Age-Specific Incidence of Colon and Testicular Cancers

Colon cancers are thought to be an inevitable result of aging, while testicular cancers are thought to develop in only a small fraction of men, beginning in utero. These models of carcinogenesis are, in part, based upon age-specific incidence data. The specific incidence for colon cancer appears to monotonically increase with age, while that of testicular cancer increases to a maximum value at about 35 years of age, then declines to nearly zero by the age of 80. We hypothesized that the age-specific incidence for these two cancers is similar; the apparent difference is caused by a longer development time for colon cancer and the lack of age-specific incidence data for people over 84 years of age. Here we show that a single distribution can describe the age-specific incidence of both colon carcinoma and testicular cancer. Furthermore, this distribution predicts that the specific incidence of colon cancer should reach a maximum at about age 90 and then decrease. Data on the incidence of colon carcinoma for women aged 85–99, acquired from SEER and the US Census, is consistent with this prediction. We conclude that the age specific data for testicular cancers and colon cancers is similar, suggesting that the underlying process leading to the development of these two forms of cancer may be similar.

Age-specific incidence data indicate four mutations are required for human testicular cancers

Normal human cells require a series of genetic alterations to undergo malignant transformation. Direct sequencing of human tumors has identified hundreds of mutations in tumors, but many of these are thought to be unnecessary and a result of, rather than a cause of, the tumor. The exact number of mutations to transform a normal human cell into a tumor cell is unknown. Here I show that male gonadal germ cell tumors, the most common form of testicular cancers, occur after four mutations. I infer this by constructing a mathematical model based upon the multi-hit hypothesis and comparing it to the age-specific incidence data. This result is consistent with the multi-hit hypothesis, and implies that these cancers are genetically or epigenetically predetermined at birth or an early age.

Corrections to: ‘‘Age distribution of cancer in mice’’

We found a crucial error in an earlier paper on cancer in elderly mice, Age distribution of cancer in mice: the incidence turnover at old age (Pompei et al., 2001). That paper’s principal data set, the ED01 records, was scrambled when read and analyzed with a statistical software package. Having done our best to correct the error, and having subjected the data to a more exact extension of originally published methods, we arrive at conclusions significantly different from those proposed in the original article. What appeared to be a dramatic fall off of the cancer mortality rate in mice over 2 years of age is now found to be a continuation or flattening of approximately exponential growth. This new finding is entirely at odds with the old, and does not support our later work on humans. Two of this paper’s authors, F Pompei and R Wilson, contributed to the original article. We are informing authors who have cited our paper in the past and apologize deeply for any wasted time or lost work. We should have subjected the ED01 records to more error checks. We thank Jennifer Blank for helping us discover and correct this error. The ED01 records and our earlier research are available http://physics.harvard.edu/∼wilson/cancer&chemicals/ED01.

A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: approach

Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) long-term formalisms that track pre-malignant cell numbers throughout an entire lifetime but treat initial radiation dose-response simplistically and (2) short-term formalisms that provide a detailed initial dose-response even for complicated radiation protocols, but address its modulation during the subsequent cancer latency period only indirectly. We argue that integrating short- and long-term models is needed. As an example of this novel approach, we integrate a stochastic short-term initiation/inactivation/repopulation model with a deterministic two-stage long-term model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills pre-malignant cells; a pre-malignant cell generates a clone, which, if it survives, quickly reaches a size limitation; the clone subsequently grows more slowly and can eventually generate a malignant cell; the carcinogenic potential of pre-malignant cells decreases with age.

Cancer suppression at old age

Increased age is regularly linked with heightened cancer risk, but recent research suggests a flattening around age 80. We report that, independent of cancer site or time period, most incidence rates decrease in the more elderly and drop to or toward zero near the ceiling of human life span. For all major organ sites, male and female, we use 1979 to 2003 Surveillance, Epidemiology, and End Results registry records (8-26% of the U.S. population) to construct three sequential cross-sections at 10-year intervals, totaling 129 sets of age-specific cancer data. To compute incidence rates, we estimate older populations at risk with census counts and NIH life tables. This article provides both a minimal and a more comprehensive extension of Surveillance, Epidemiology, and End Results cancer rates to those above 85. Almost all cancers peak at age approximately 80. Generally, it seems that centenarians are asymptomatic or untargeted by cancers. We suggest that the best available justification for this pattern of incidence is a link between increased senescence and decreased proliferative potential among cancers. Then, thus far, as senescence may be a carcinogen, it might also be considered an anticarcinogen in the elderly. We model rising and falling incidence rates with a beta curve obtained by appending a linearly decreasing factor to the well-known Armitage-Doll multistage model of cancer. Taken at face value, the beta model implies that medical, diet, or lifestyle interventions restricting carcinogenesis ought to be examined for possible effects on longevity.

Symmetry, identifiability, and prediction uncertainties in multistage clonal expansion (MSCE) models of carcinogenesis

Many models of exposure-related carcinogenesis, including traditional linearized multistage models and more recent two-stage clonal expansion (TSCE) models, belong to a family of models in which cells progress between successive stages-possibly undergoing proliferation at some stages-at rates that may depend (usually linearly) on biologically effective doses. Biologically effective doses, in turn, may depend nonlinearly on administered doses, due to PBPK nonlinearities. This article provides an exact mathematical analysis of the expected number of cells in the last ("malignant") stage of such a "multistage clonal expansion" (MSCE) model as a function of dose rate and age. The solution displays symmetries such that several distinct sets of parameter values provide identical fits to all epidemiological data, make identical predictions about the effects on risk of changes in exposure levels or timing, and yet make significantly different predictions about the effects on risk of changes in the composition of exposure that affect the pharmacodynamic dose-response relation. Several different predictions for the effects of such an intervention (such as reducing carcinogenic constituents of an exposure) that acts on only one or a few stages of the carcinogenic process may be equally consistent with all preintervention epidemiological data. This is an example of nonunique identifiability of model parameters and predictions from data. The new results on nonunique model identifiability presented here show that the effects of an intervention on changing age-specific cancer risks in an MSCE model can be either large or small, but that which is the case cannot be predicted from preintervention epidemiological data and knowledge of biological effects of the intervention alone. Rather, biological data that identify which rate parameters hold for which specific stages are required to obtain unambiguous predictions. From epidemiological data alone, only a set of equally likely alternative predictions can be made for the effects on risk of such interventions.

On use of the multistage dose-response model for assessing laboratory animal carcinogenicity

We explore how well a statistical multistage model describes dose-response patterns in laboratory animal carcinogenicity experiments from a large database of quantal response data. The data are collected from the US EPA's publicly available IRIS data warehouse and examined statistically to determine how often higher-order values in the multistage predictor yield significant improvements in explanatory power over lower-order values. Our results suggest that the addition of a second-order parameter to the model only improves the fit about 20% of the time, while adding even higher-order terms apparently does not contribute to the fit at all, at least with the study designs we captured in the IRIS database. Also included is an examination of statistical tests for assessing significance of higher-order terms in a multistage dose-response model. It is noted that bootstrap testing methodology appears to offer greater stability for performing the hypothesis tests than a more-common, but possibly unstable, "Wald" test.