Application of a color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment

Several hypotheses are established to describe the formation and progression of foci of altered hepatocytes (FAH). A common model of hepatocarcinogenesis is the mutation model (MM), which is based on the assumption that cells have to undergo multiple successive changes on their way from the normal to the malignant stage. This model describes growth and phenotype change of foci on the cellular level and is based on the assumption that single cells change their phenotype through mutation into the next stage and proliferate according to a linear stochastic birth-death process. In contrast, the color-shift model (CSM) was introduced by Kopp-Schneider et al. to describe that whole colonies of altered hepatocytes simultaneously alter their phenotype. In this paper two modifications of the color-shift model are considered which allow the growth rate to vary from focus to focus. All four models are compared with respect to their ability to predict number and radii of foci in a rat hepatocarcinogenesis experiment, in which rats were treated with the carcinogens N-nitrosomorpholine, 2-acetylaminofluoren and Phenobarbital. Maximum likelihood parameter estimates are given, and predicted and empirical FAH size distributions are visualized. The Cramer-von-Mises distance is used as a measure for the discrepancy between empirical and theoretical size distributions.     

Application of a two-phenotype color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment

The color-shift model (CSM) was introduced by Kopp-Schneider et al. [1] to describe formation and progression of foci of altered hepatocytes (FAH). It incorporates the field-effect hypothesis which postulates that entire colonies of altered hepatocytes simultaneously alter their phenotype. In the original CSM, FAH grow with deterministic growth rate and change their phenotype after an exponentially distributed waiting time. A modification of the original color-shift model (CSMbeta) is presented here in which the growth rate varies from focus to focus according to a beta distribution. The concept of an exponentially distributed waiting time to phenotype change is modified to the concept of a random radius at which phenotype changes and this radius is modelled as beta distributed. The original and the modified CSM are applied to data from an initiation-promotion rat hepatocarcinogenesis experiment with diethylnitrosomorpholine (DEN) and N-nitrosomorpholine (NNM), in which two phenotypes of FAH were observed in hematoxilin/eosin (H&E) stained liver sections. The Cramer-von-Mises Distance is used as a measure for the discrepancy between empirical and theoretical size distributions. Comparisons of model fit show that considerable improvement is obtained for CSMbeta compared to the original CSM.     

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].     

Calculation of the cumulative distribution function of the time to a small observable tumor

Multistage mathematical models of carcinogenesis (when applied to tumor incidence data) have historically assumed that the growth kinetics of cells in the malignant state are disregarded and the formation of a single malignant cell is equated with the emergence of a detectable tumor. The justification of this simplification is, from a mathematical point of view, to make the estimation of tumor incidence rates tractable. However, analytical forms are not mandatory in the estimation of tumor incidence rates. Portier et al. (1996b, Math. Biosci. 135, 129–146) have demonstrated the utility of the Kolmogorov backward equations in numerically calculating tumor incidence. By extending their results, the cumulative distribution function of the time to a small observable tumor may be numerically obtained.     

A stochastic carcinogenesis model incorporating multiple types of genomic instability fitted to colon cancer data

A generalization of the two-mutation stochastic carcinogenesis model of Moolgavkar, Venzon and Knudson and certain models constructed by Little [Little, M.P. (1995). Are two mutations sufficient to cause cancer? Some generalizations of the two-mutation model of carcinogenesis of Moolgavkar, Venzon, and Knudson, and of the multistage model of Armitage and Doll. Biometrics 51, 1278-1291] and Little and Wright [Little, M.P., Wright, E.G. (2003). A stochastic carcinogenesis model incorporating genomic instability fitted to colon cancer data. Math. Biosci. 183, 111-134] is developed; the model incorporates multiple types of progressive genomic instability and an arbitrary number of mutational stages. The model is fitted to US Caucasian colon cancer incidence data. On the basis of the comparison of fits to the population-based data, there is little evidence to support the hypothesis that the model with more than one type of genomic instability fits better than models with a single type of genomic instability. Given the good fit of the model to this large dataset, it is unlikely that further information on presence of genomic instability or of types of genomic instability can be extracted from age-incidence data by extensions of this model.     

Investigating the formation and growth of alpha-particle radiation-induced foci of altered hepatocytes: a model-based approach

The effect of alpha-particle radiation on the formation and increase in volume of preneoplastic liver lesions was investigated in an animal experiment. Mice were divided into four groups; two groups received different doses of the alpha-particle-labeled antibody (213)Bi-anti CD19 ((213)Bi-CD19), Thorotrast was administered to one group, and one group was left untreated. Hematoxylin and eosin-stained liver sections were evaluated for preneoplastic foci of altered hepatocytes 6, 12 and 17 months after treatment. The density and size distribution of focal transections were described by a mechanistic model for the formation and growth of foci of altered hepatocytes. The negative control and the (213)Bi-CD19 groups were combined to investigate the dose-response relationship for model parameters describing the formation and growth of foci of altered hepatocytes. Although (213)Bi-CD19 was given by single injection, the effect on formation of foci of altered hepatocytes lasted for the entire experiment. Likelihood-ratio tests comparing nested models showed that (213)Bi-CD19 increases the rates of both the formation and growth of foci of altered hepatocytes. Comparing the effects of Thorotrast with those of (213)Bi-CD19 revealed that Thorotrast had an effect similar to that of a low dose of (213)Bi-CD19, but the effect on focus formation was slightly smaller whereas the effect on focus growth was slightly higher for Thorotrast, in contrast to a low dose of (213)Bi-CD19.     

Early effects in chemical-induced rat liver carcinogenesis: an immunohistochemical study following exposure to 0.04% AAF

To investigate effects that distinguish AAF from incomplete carcinogens, the rate of cell death (apoptosis) and cell proliferation was studied at early stages of AAF induced rat liver carcinogenesis. Male Wistar rats were fed 0.04% AAF in the diet for 2, 6 and 16 weeks and immunohistochemical markers were measured in the liver. The formation of initiated cells and preneoplastic foci was followed by staining for GST-P (glutathione-S-transferase). GST-P-positive foci were present from 6 weeks on. Apoptosis was increased in the periportal area and in preneoplastic foci at all time points. Cell proliferation was enhanced in the periportal area in oval cells and in bile duct-like cells particularly at 2 and 6 weeks and mainly in GST-P positive foci at 16 weeks. Notably, more cells always proliferated than were eliminated. Other apoptosis-related markers like p53 and FAS/Apo-1 could not be demonstrated in either normal hepatocytes, preneoplastic foci or in hepatocytes from treated animals. Scattered bcl-2 positive cells were present in livers at 16 weeks of treatment. The two cell growth and differentiation related proto-oncogenes c-FOS and c-JUN were increased in all treated animals at early stages. If feeding was stopped after 6 weeks, livers did not recover significantly within the following 10 weeks. The results support the complex effects of AAF in rat liver carcinogenesis. Chronic toxicity locally impairs the balance between cell proliferation and cell death and induces morphological alterations that promote the growth of initiated cells.     

Spreading speeds as slowest wave speeds for cooperative systems

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.     

Asymptotic behavior of joint distributions of characteristics of a pair of randomly chosen individuals in discrete-time Fisher-Wright models with mutations and drift

This is a continuation of the series of articles (C.R. Rao, D.N. Shanbhag (Eds.), Handbook of Statistics 19: Stochastic Processes: Theory and Methods, Elsevier Science, Amsterdam, 2001 (Chapter 8); Math. Biosci. 175 (2002) 83; Math. Meth. Appl. Sci. 26 (2003) 1587; Adv. Appl. Probab. 36 (2004) 57) devoted to a study of the interplay between two of the main forces of population genetics, mutations and drift, in the Fisher-Wright model. We provide discrete-time versions of theorems describing asymptotic behavior of joint distributions of characteristics of a pair of individuals in this model; their continuous-time counterparts were presented in the previous papers. Furthermore, we show that imbalance index, introduced in Kimmel et al. (Genetics 148 (1998) 1921) and King et al. (Mol. Biol. Evol. 17(12) (2000) 1895) in the context of continuous-time models, may also be used in discrete-time models to detect past population growth.     

The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics, Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig’s result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

Early indicators of exocrine pancreas carcinogenesis produced by non-genotoxic agents

In the past 40 years the incidence of pancreatic cancer in many Western countries had increased. Since no single factor responsible for the development of pancreatic cancer has been identified, it is believed that non-genotoxic factors may play an important role in the pathogenesis of this highly fatal form of cancer. Focal abnormalities of acinar cells, referred to as atypical acinar cell foci or nodules, occur spontaneously in rats and some other species. Their incidence increases with age from zero at birth to about 75% in 2-year-old rats. These spontaneous lesions have a phenotype that cannot be distinguished from the putative, atypical preneoplastic, acinar cell foci induced in rat pancreas by the carcinogen azaserine. Unsaturated fat (corn oil) has been found to increase the incidence of atypical acinar cell nodules and adenomas in the pancreas of non-carcinogen-treated rats without influencing the weight of the pancreas. Furthermore, unsaturated fat has a specific promoting effect on the growth potential of atypical acinar cell foci and nodules induced in rat pancreas by azaserine, resulting in an increase in the number and size of these lesions. Rats fed raw soya flour or trypsin inhibitors develop an enlarged pancreas as a result of hypertrophy and hyperplasia. They also develop acidophilic atypical acinar cell foci and nodules, adenomas and adenocarcinomas after being fed full-fat raw soya flour for 2 years. It may be concluded from the observations in rat pancreas that non-genotoxic compounds or conditions that enhance pancreatic growth may be classified as non-genotoxic pancreatic tumour promoters. The observations with corn oil, however, indicate that there may be non-genotoxic compounds that specifically enhance growth of spontaneous initiated atypical acinar cell foci without causing hyperplasia of the pancreas. The possible mechanisms whereby unsaturated fat and trypsin inhibitors exert their effects on exocrine pancreatic carcinogenesis are discussed.     

Multiple limit cycles for predator-prey models

We construct a Gause-type predator-prey model with concave prey isocline and (at least) two limit cycles. This serves as a counter-example to the global stability criterion of Hsu [Math. Biosci. 39:1-10 (1978)].     

Decreased density of beta1-adrenergic receptors in preneoplastic and neoplastic liver lesions of F344 rats

There is some evidence that rodent hepatocarcinogenesis is accompanied by changes in the adrenergic responsiveness of liver cells to catecholamines. In this study, immunohistochemical expression of beta1-adrenergic receptors (beta1-ARs) has been examined in spontaneous and chemically induced preneoplastic and neoplastic liver lesions of female and male Fischer 344 rats. An antibody specific for beta1-AR subtype was used. The study was carried out on archival formalin-fixed and paraffin-embedded livers from rats used in a previous study of hepatocarcinogenesis. One control group given distilled water by gavage, and two experimental groups, one initiated with a single dose of diethylnitrosamine (DEN) and one initiated with DEN and continuously treated with phenobarbital (PB) were examined. Rats were sacrificed after 2, 4, 8 and 21 months of experimentation. All types of liver putative preneoplastic lesions examined (basophilic, glycogen-retaining, or mixed cell foci) show a lower density of beta1-ARs than the surrounding normal liver parenchyma, either in control and in DEN-treated or DEN+PB-treated rats. No immunostaining is detectable in several altered cell foci. Hepatocellular adenomas and hepatocellular carcinomas also show a very low density of beta1-ARs, extensive areas completely devoid of beta1-ARs being mingled with areas showing a weak immunostaining.     

Dynamics of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

Stilianakis and Seydel (Bull. Math. Biol., 1999) proposed an ODE model that describes the T-cell dynamics of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). Their model consists of four components: uninfected healthy CD4+ T-cells, latently infected CD4+ T-cells, actively infected CD4+ T-cells, and ATL cells. Mathematical analysis that completely determines the global dynamics of this model has been done by Wang et al. (Math. Biosci., 2002). In this note, we first modify the parameters of the model to distinguish between contact and infectivity rates. Then we introduce a discrete time delay to the model to describe the time between emission of contagious particles by active CD4+ T-cells and infection of pure cells. Using the results in Culshaw and Ruan (Math. Biosci., 2000) in the analysis of time delay with respect to cell-free viral spread of HIV, we study the effect of time delay on the stability of the endemically infected equilibrium. Numerical simulations are presented to illustrate the results.     

On simulating strongly interacting, stochastic population models. II. Multiple compartments

In Wick and Stelf [Math. Biosci. 187 (2004) 1], we showed how to simulate a pair of strongly interacting biological populations evolving stochastically over many orders-of-magnitude. Here we generalize the method to any (finite) number of compartments; transitions including births, deaths, progression through life-stages, and mitoses; and arbitrary rate functions. We illustrate the technique for a seven-compartment model of the cellular immune response to a viral infection.     

Proliferation of anchorage-dependent contact-inhibited cells: I. Development of theoretical models based on cellular automata

We report the development of new class of discrete models that can accurately describe the contact-inhibited proliferation of anchorage-dependent cells. The models are based on cellular automata, and they quantitatively account for contact inhibition phenomena occurring during all stages of the proliferation process: (a) the initial stage of "exponential" growth of cells without contact inhibition; (b) the second stage where cell colonies form and grow with few colony mergings; and (c) the final stage where proliferation rates are dominated by colony merging events. Model prediction are presented and analyzed to study the complicated dynamics of large cell populations and determine how the initial spatial cell distribution, the seeding density, and the geometry of the growth surface affect the observed proliferation rates. Finally, we present a model variant that can simulate contact-inhibited proliferation of asynchronous cell populations with arbitrary cell cycle-time distribution. The latter model can also compute the percentage of cells that are in a specific phase of their division cycle at a given time.     

A note on heroin epidemics

We show that the steady states of the White and Comiskey [E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (2007) 312-324.] model of heroin epidemics are stable.