Multistage Models of Carcinogenesis: An Approximation for the Size and Number Distribution of Late-Stage Clones

Multistage models have become the basic paradigm for modeling carcinogenesis. One model, the two-stage model of carcinogenesis, is now routinely used in the analysis of cancer risks from exposure to environmental chemicals. In its most general form, this model has two states, an initiated state and a neoplastic state, which allow for growth of cells via a simple linear birth-death process. In all analyses done with this model, researchers have assumed that tumor incidence is equivalent to the formation of a single neoplastic cell and the growth kinetics in the neoplastic state have been ignored. Some researchers have discussed the impact of this assumption on their analyses, but no formal methods were available for a more rigorous application of the birth-death process. In this paper, an approximation is introduced which allows for the application of growth kinetics in the neoplastic state. The adequacy of the approximation against simulated data is evaluated and methods are developed for implementing the approximation using data on the number and size of neoplastic clones.     

On the Exact Hazard and Survival Functions of the MVK Stochastic Carcinogenesis Model

The MVK two-stage carcinogenesis model is one of the most widely accepted mechanistic models in carcinogenesis modeling. However, due to a perceived difficulty in obtaining analytic solutions for the hazard and survival functions, approximations and numerical methods have been used to calculate these two fundamental quantities. This paper focuses on a special case of the homogeneous MVK model where the number of normal cells is constant. The probability generating function (pgf) for the number of tumor cells is derived, and the exact analytic solutions to the hazard and survival functions are obtained from the pgf.     

Quasiextinction Probabilities as a Measure of Impact on Population Growth

A probabilistic language based on stochastic models of population growth is proposed for a standard language to be used in environmental assessment. Environmental impact on a population is measured by the probability of quasiextinction. Density-dependent and independent models are discussed. A review of one-dimensional stochastic population growth models, the implications of environmental autocorrelation, finite versus "infinite" time results, age-structured models, and Monte Carlo simulations are included. The finite time probability of quasiextinction is presented for the logistic model. The sensitivity of the result with respect to the mean growth rate and the amplitude of environmental fluctuations are examined. Stochastic models of population growth form a basis for formulating reasonable criteria for environmental impact estimates.     

Human Salmonellosis: Estimation of Dose-Illness from Outbreak Data

The quantification of the relationship between the amount of microbial organisms ingested and a specific outcome such as infection, illness, or mortality is a key aspect of quantitative risk assessment. A main problem in determining such dose-response models is the availability of appropriate data. Human feeding trials have been criticized because only young healthy volunteers are selected to participate and low doses, as often occurring in real life, are typically not considered. Epidemiological outbreak data are considered to be more valuable, but are more subject to data uncertainty. In this article, we model the dose-illness relationship based on data of 20 Salmonella outbreaks, as discussed by the World Health Organization. In particular, we model the dose-illness relationship using generalized linear mixed models and fractional polynomials of dose. The fractional polynomial models are modified to satisfy the properties of different types of dose-illness models as proposed by Teunis et al . Within these models, differences in host susceptibility (susceptible versus normal population) are modeled as fixed effects whereas differences in serovar type and food matrix are modeled as random effects. In addition, two bootstrap procedures are presented. A first procedure accounts for stochastic variability whereas a second procedure accounts for both stochastic variability and data uncertainty. The analyses indicate that the susceptible population has a higher probability of illness at low dose levels when the combination pathogen-food matrix is extremely virulent and at high dose levels when the combination is less virulent. Furthermore, the analyses suggest that immunity exists in the normal population but not in the susceptible population.     

Stochastic models of industrial buying behaviour

This paper presents one of the first attempts to develop stochastic models of industrial buying behaviour. The Beta-binomial model, used to model consumer purchase incidence, also works well for industrial consumers. Purchase size is shown to be lognormally distributed suggesting that something like a log random walk process may be operating. However it proved impossible to combine these two stochastic models analytically and a less rich model, the Markov reservoir, was shown to fit the data providing useful insights into the dynamics of industrial markets as well as suggesting areas for further research.     

Apoptosis and Chemical Carcinogenesis

Long recognized as a normal component of organogenesis during development, apoptosis (programmed cell death) has recently been implicated in alterations of cell growth and differentiation. Tissue homeostasis is normally maintained by a balance between cell division and cell death, with apoptosis often functioning in complement to cell growth. Thus, antithetical parallels in chemical carcinogenesis can be drawn between apoptosis and the proliferative events more commonly addressed. While enhanced cell replication may contribute to an increased frequency of mutation, apoptosis within a tissue may counteract chemical carcinogenesis through loss of mutated cells. Many strong carcinogens act as tumor promoters, selectively expanding an initiated cell population advantageously over surrounding cells. Similarly, chemicals with a selective inhibition of apoptosis within an initiated population would offer a growth advantage. In contrast, chemicals causing selective apoptosis of initiated cells would be expected to have an anticarcinogenic effect. Selective apoptosis, in concert with cell-specific replication, may explain the unique promoting effects of different carcinogens such as the peroxisome-proliferating chemicals, phenobarbital, and 2,3,7,8-tetrachloro-dibenzo- p -dioxin (TCDD). Cell turnover, both cell growth and cell death, is central to the process of chemically induced carcinogenesis in animals and understanding its impact is a critical determinant of the relevance of chemically induced effects to man.     

Decision Making with Uncertain Judgments: A Stochastic Formulation of the Analytic Hierarchy Process*

In the analytic hierarchy process (AHP), priorities are derived via a deterministic method, the eigenvalue decomposition. However, judgments may be subject to error. A stochastic characterization of the pairwise comparison judgment task is provided and statistical models are introduced for deriving the underlying priorities. Specifically, a weighted hierarchical multinomial logit model is used to obtain the priorities. Inference is then conducted from the Bayesian viewpoint using Markov chain Monte Carlo methods. The stochastic methods are found to give results that are congruent with those of the eigenvector method in matrices of different sizes and different levels of inconsistency. Moreover, inferential statements can be made about the priorities when the stochastic approach is adopted, and these statements may be of considerable value to a decision maker. The methods described are fully compatible with judgments from the standard version of AHP and can be used to construct a stochastic formulation of it.     

Stochastic efficiency analysis with a reliability consideration

Stochastic Data Envelopment Analysis (DEA) models have been introduced in the literature to assess the performance of operating entities with random input and output data. A stochastic DEA model with a reliability constraint is proposed in this study that maximizes the lower bound of an entity׳s efficiency score with some pre-selected probability. We define the concept of stochastic efficiency and develop a solution procedure. The economic interpretations of the stochastic efficiency index are presented when the inputs and outputs of each entity follow a multivariate joint normal distribution.     

Accounting for Both Random Errors and Systematic Errors in Uncertainty Propagation Analysis of Computer Models Involving Experimental Measurements with Monte Carlo Methods

A Monte Carlo method is presented to study the effect of systematic and random errors on computer models mainly dealing with experimental data. It is a common assumption in this type of models (linear and nonlinear regression, and nonregression computer models) involving experimental measurements that the error sources are mainly random and independent with no constant background errors (systematic errors). However, from comparisons of different experimental data sources evidence is often found of significant bias or calibration errors. The uncertainty analysis approach presented in this work is based on the analysis of cumulative probability distributions for output variables of the models involved taking into account the effect of both types of errors. The probability distributions are obtained by performing Monte Carlo simulation coupled with appropriate definitions for the random and systematic errors. The main objectives are to detect the error source with stochastic dominance on the uncertainty propagation and the combined effect on output variables of the models. The results from the case studies analyzed show that the approach is able to distinguish which error type has a more significant effect on the performance of the model. Also, it was found that systematic or calibration errors, if present, cannot be neglected in uncertainty analysis of models dependent on experimental measurements such as chemical and physical properties. The approach can be used to facilitate decision making in fields related to safety factors selection, modeling, experimental data measurement, and experimental design.     

Estimation When a Parameter is on a Boundary

This paper establishes the asymptotic distribution of an extremum estimator when the true parameter lies on the boundary of the parameter space. The boundary may be linear, curved, and/or kinked. Typically the asymptotic distribution is a function of a multivariate normal distribution in models without stochastic trends and a function of a multivariate Brownian motion in models with stochastic trends. The results apply to a wide variety of estimators and models. Examples treated in the paper are: (i) quasi-ML estimation of a random coefficients regression model with some coefficient variances equal to zero and (ii) LS estimation of an augmented Dickey-Fuller regression with unit root and time trend parameters on the boundary of the parameter space.     

基于扩展广义随机时间Petri网的流程银行建模及等价性能分析

对业务流程进行建模和性能分析是流程银行研究的重要课题.通过对广义随机时间Petri网进行扩展,将其用于对流程银行的业务流程建模.同时,基于活动执行时间为正态分布的假设,给出了4种流程元模式的等价性能分析方法.在此基础上,进一步把活动执行时间扩展到任意分布的情况,给出了通用的等价性能分析方法.     

Modeling the Risks of Nonindigenous Species Introductions Using a Patch-Dynamics Approach Incorporating Contaminant Effects as a Disturbance

The establishment and spread of invasive or nonindigenous species has caused concern from stakeholders in affected areas, and has prompted many field and modeling studies. We used stochastic two species, circular three patch dynamic models to investigate the patterns of invasion and impacts upon the affected species. Both persistent and degradable toxicants were incorporated as parts of the model system to act as disturbance regimens. There is a clear series of patterns that result from these simulations. Competition increases population variability, but decreases the number of distinct outcomes possible from the same initial conditions. Isolation of the patch of the introduction was the main determinant of successful establishment through a process we call the beachhead effect. Coexistence of species was often possible in local patches, contrary to the analytical solutions of Lotka-Volterra equations and numerous modeling studies. Contaminants and their resultant disturbances are important as contributors to the stochastic nature of models. The stochasticity leads to a variety of outcomes from some sets of initial conditions. Different outcomes have different probabilities of occurrence and are dependent upon the specific initial conditions of the simulation. A clear pattern that is apparent is the "beachhead effect," where the invasive establishes a population within a relatively remote patch before migrating to the remainder of the landscape. We make predictions and provide specific research hypotheses as to the causes and effects of invasive species establishment, spread, and impacts.     

When are two multivariate random processes indistinguishable

Prediction error identification methods have been recently the objects of much study, and have wide applicability. The maximum likelihood (ML) identification methods for Gaussian models and the least squares prediction error method (LSPE) are special cases of the general approach. In this paper, we investigate conditions for distinguishability or identifiability of multivariate random processes, for both continuous and discrete observation time T. We consider stationary stochastic processes, for the ML and LSPE methods, and for large observation interval T, we resolve the identifiability question. Our analysis begins by considering stationary autoregressive moving average models, but the conclusions apply for general stationary, stable vector models. The limiting value for T → ∞ of the criterion function is evaluated, and it is viewed as a distance measure in the parameter space of the model. The main new result of this paper is to specify the equivalence classes of stationary models that achieve the global minimization of the above distance measure, and hence to determine precisely the classes of models that are not identifiable from each other. The new conclusions are useful for parameterizing multivariate stationary models in system identification problems. Relationships to previously discovered identifiability conditions are discussed.     

A Cellular Dynamics Model of Experimental Bladder Cancer: Analysis of the Effect of Sodium Saccharin in the Rat

To make the methodology of risk assessment more consistent with the realities of biological processes, a computer-based model of the carcinogenic process may be used. A previously developed probabilistic model, which is based on a two-stage theory of carcinogenesis, represents urinary bladder carcinogenesis at the cellular level with emphasis on quantification of cell dynamics: cell mitotic rates, cell loss and birth rates, and irreversible cellular transitions from normal to initiated to transformed states are explicitly accounted for. Analyses demonstrate the sensitivity of tumor incidence to the timing and magnitude of changes to these cellular variables. It is demonstrated that response in rats following administration of nongenotoxic compounds, such as sodium saccharin, can be explained entirely on the basis of cytotoxicity and consequent hyperplasia alone.     

An Improved Approximation to the Exact Solution of the Two-Stage Clonal Growth Model of Cancer

Multistage clonal growth models are of interest for cancer risk assessment because they can explicitly incorporate data on cell replication. Both approximate and exact formulations of the two stage growth model have been described. The exact solution considers the conditional probability of tumors arising in previously tumor-free animals; the approximate solution estimates total probability of tumor formation. The exact solution is much more computationally intensive when time-dependent cell growth parameters are included. The approximate solution deviates from the exact solution at high incidences and probabilities of tumor. This report describes a computationally tractable,'improved approximation'to the exact solution. Our improved approximation includes a correction term to adjust the unconditional expectation of intermediate cells based on the time history of formation of intermediate cells by mutation of normal cells (recruitment) or by cell division in the intermediate cell population (expansion). The improved approximation provided a much better match to the exact solution than the approximate solution for a wide range of parameter values. The correction term also appears to provide insight into the biological factors that contribute to the variance of the expectation for the number of intermediate cells over time.     

Approximation algorithms for the bus evacuation problem

We consider the bus evacuation problem. Given a positive integer B, a bipartite graph G with parts S and \(T \cup \{r\}\) in a metric space and functions \(l_i :S \rightarrow {\mathbb {Z}}_+\) and \({u_j :T \rightarrow \mathbb {Z}_+ \cup \{\infty \}}\), one wishes to find a set of B walks in G. Every walk in B should start at r and finish in T and r must be visited only once. Also, among all walks, each vertex i of S must be visited at least \(l_i\) times and each vertex j of T must be visited at most \(u_j\) times. The objective is to find a solution that minimizes the length of the longest walk. This problem arises in emergency planning situations where the walks correspond to the routes of B buses that must transport each group of people in S to a shelter in T, and the objective is to evacuate the entire population in the minimum amount of time. In this paper, we prove that approximating this problem by less than a constant is \(\text{ NP }\)-hard and present a 10.2-approximation algorithm. Further, for the uncapacitated BEP, in which \(u_j\) is infinity for each j, we give a 4.2-approximation algorithm.     

产品市场纵向差异的两阶段再制造供应链契约协调

针对常规产品与再制品销售市场的纵向差异和常规材料价格随机波动的情况,建立了"制造商-回收商"Stackelberg博弈模型,设计了考虑常规材料价格信息更新的两阶段回收费用共担再制造供应链契约协调机制,证明了两阶段可变契约的效益高于单阶段固定契约。算例分析表明两阶段可变契约的有效性:常规材料价格随机波动幅度越大,两阶段契约的改善程度越高;且协调机制能夠鼓励废旧产品的再制造,实现经济效益与社会效益的双赢。     

Stochasticity in Physiologically Based Kinetics Models: Implications for Cancer Risk Assessment

In case of low-dose exposure to a substance, its concentration in cells is likely to be stochastic. Assessing the consequences of this stochasticity in toxicological risk assessment requires the coupling of macroscopic dynamics models describing whole-body kinetics with microscopic tools designed to simulate stochasticity. In this article, we propose an approach to approximate stochastic cell concentration of butadiene in the cells of diverse organs. We adapted the dynamics equations of a physiologically based pharmacokinetic (PBPK) model and used a stochastic simulator for the system of equations that we derived. We then coupled kinetics simulations with a deterministic hockey stick model of carcinogenicity. Stochasticity induced substantial modifications relative to dose-response curve, compared with the deterministic situation. In particular, there was nonlinearity in the response and the stochastic apparent threshold was lower than the deterministic one. The approach that we developed could easily be extended to other biological studies to assess the influence of stochasticity at macroscopic scale for compound dynamics at the cell level.     

Bounding the scaling window of random constraint satisfaction problems

The model \(k\)-CSP is a random CSP model with moderately growing arity \(k\) of constraints. By incorporating certain linear structure, \(k\)-CSP is revised to a random linear CSP, named \(k\)-hyper-\({\mathbb F}\)-linear CSP. It had been shown theoretically that the two models exhibit exact satisfiability phase transitions when the constraint density \(r\) is varied accordingly. In this paper, we use finite-size scaling analysis to characterize the threshold behaviors of the two models with finite problem size \(n\). A series of experimental studies are carried out to illustrate the scaling window of the model \(k\)-CSP.