The role of cell death in the growth of preneoplastic lesions: a Monte Carlo simulation model

Abstract. A variety of experimental and clinical examples of preneoplasia demonstrate that regression of early lesions is common. This paper examines the hypothesis that early lesions operate under the identical growth kinetics of 'late' lesions (neoplasms), but that kinetic features favouring continuous growth in established lesions tend to favour extinction of lesions composed of small numbers of cells. Growth simulations of early lesions were produced using the Monte Carlo method, a technique demanding intensive computations. With the advent of powerful personal computers, this technique is now widely available to biologists. Simulating growth under conditions of cell loss similar to those observed in established tumours, the model predicts that the great majority of initiated cell clusters are expected to reach extinction within a few cell doubling times, and most early (promoted) lesions would not likely progress to the size of a clinically detectable lesion within the life span of the host organism. These Monte Carlo simulations provide a model of initiated cell growth consistent with the recently demonstrated role of early lesion cell death in the development of human lymphomas and in transgenic mice expressing the bcl-2 oncogene. The model demonstrates that small increments in the intrinsic cell loss probability in even the earliest progenitors of malignancy can strongly influence the subsequent development of neoplasia from initiated foci.     

Information dynamics in carcinogenesis and tumor growth

The storage and transmission of information is vital to the function of normal and transformed cells. We use methods from information theory and Monte Carlo theory to analyze the role of information in carcinogenesis. Our analysis demonstrates that, during somatic evolution of the malignant phenotype, the accumulation of genomic mutations degrades intracellular information. However, the degradation is constrained by the Darwinian somatic ecology in which mutant clones proliferate only when the mutation confers a selective growth advantage. In that environment, genes that normally decrease cellular proliferation, such as tumor suppressor or differentiation genes, suffer maximum information degradation. Conversely, those that increase proliferation, such as oncogenes, are conserved or exhibit only gain of function mutations. These constraints shield most cellular populations from catastrophic mutator-induced loss of the transmembrane entropy gradient and, therefore, cell death. The dynamics of constrained information degradation during carcinogenesis cause the tumor genome to asymptotically approach a minimum information state that is manifested clinically as dedifferentiation and unconstrained proliferation. Extreme physical information (EPI) theory demonstrates that altered information flow from cancer cells to their environment will manifest in-vivo as power law tumor growth with an exponent of size 1.62. This prediction is based only on the assumption that tumor cells are at an absolute information minimum and are capable of "free field" growth that is, they are unconstrained by external biological parameters. The prediction agrees remarkably well with several studies demonstrating power law growth in small human breast cancers with an exponent of 1.72+/-0.24. This successful derivation of an analytic expression for cancer growth from EPI alone supports the conceptual model that carcinogenesis is a process of constrained information degradation and that malignant cells are minimum information systems. EPI theory also predicts that the estimated age of a clinically observed tumor is subject to a root-mean square error of about 30%. This is due to information loss and tissue disorganization and probably manifests as a randomly variable lag phase in the growth pattern that has been observed experimentally. This difference between tumor size and age may impose a fundamental limit on the efficacy of screening based on early detection of small tumors. Independent of the EPI analysis, Monte Carlo methods are applied to predict statistical tumor growth due to perturbed information flow from the environment into transformed cells. A "simplest" Monte Carlo model is suggested by the findings in the EPI approach that tumor growth arises out of a minimally complex mechanism. The outputs of large numbers of simulations show that (a) about 40% of the populations do not survive the first two-generations due to mutations in critical gene segments; but (b) those that do survive will experience power law growth identical to the predicted rate obtained from the independent EPI approach. The agreement between these two very different approaches to the problem strongly supports the idea that tumor cells regress to a state of minimum information during carcinogenesis, and that information dynamics are integrally related to tumor development and growth.     

The Role of Chromosome Missegregation in Cancer Development: A Theoretical Approach Using Agent-Based Modelling

Many cancers are aneuploid. However, the precise role that chromosomal instability plays in the development of cancer and in the response of tumours to treatment is still hotly debated. Here, to explore this question from a theoretical standpoint we have developed an agent-based model of tissue homeostasis in which to test the likely effects of whole chromosome mis-segregation during cancer development. In stochastic simulations, chromosome mis-segregation events at cell division lead to the generation of a diverse population of aneuploid clones that over time exhibit hyperplastic growth. Significantly, the course of cancer evolution depends on genetic linkage, as the structure of chromosomes lost or gained through mis-segregation events and the level of genetic instability function in tandem to determine the trajectory of cancer evolution. As a result, simulated cancers differ in their level of genetic stability and in their growth rates. We used this system to investigate the consequences of these differences in tumour heterogeneity for anti-cancer therapies based on surgery and anti-mitotic drugs that selectively target proliferating cells. As expected, simulated treatments induce a transient delay in tumour growth, and reveal a significant difference in the efficacy of different therapy regimes in treating genetically stable and unstable tumours. These data support clinical observations in which a poor prognosis is correlated with a high level of chromosome mis-segregation. However, stochastic simulations run in parallel also exhibit a wide range of behaviours, and the response of individual simulations (equivalent to single tumours) to anti-cancer therapy prove extremely variable. The model therefore highlights the difficulties of predicting the outcome of a given anti-cancer treatment, even in cases in which it is possible to determine the genotype of the entire set of cells within the developing tumour.     

Proliferation kinetics of endothelial and tumour cells in three mouse mammary carcinomas

Abstract. An autoradiographic study of three corded mouse tumours is reported. The proliferation characteristics of both tumour cells and endothelial cells were studied. The doubling time of these three tumours differed by a factor of 2.6 but there was only a small difference in the intermitotic time. All three tumours showed a very high cell loss factor (˜0.80) and the differences in growth rate resulted mainly from differences in the growth fraction .
The endothelial cell proliferation rates differed markedly in the three tumours, with labelling indices ranging from 18% in the faster tumours to 4.5% in the slowest. The potential doubling times for endothelium, calculated from these values, were much slower than the tumour cell cycle time or the tumour potential doubling time, but were two to four times faster than the volume doubling time of the tumour.
It appears likely that the endothelial proliferation rate influences the growth fraction, but similar high cell loss factors can occur in tumours with a four-fold difference in endothelial cell production rates. Inadequate branching of blood vessels seems likely to be at least as important as inadequate production of endothelial cells. It is not possible to determine whether slow tumour cell production evokes a slower endothelial growth or vice versa.     

A quantitative study of growth variability of tumour cell clones in vitro

Abstract.   Objectives : In this study, we quantify growth variability of tumour cell clones from a human leukaemia cell line. Materials and methods : We have used microplate spectrophotometry to measure growth kinetics of hundreds of individual cell clones from the Molt3 cell line. Growth rate of each clonal population has been estimated by fitting experimental data with the logistic equation. Results : Growth rates were observed to vary between different clones. Up to six clones with growth rates above or below mean growth rate of the parent population were further cloned and growth rates of their offspring were measured. Distribution of growth rates of the subclones did not significantly differ from that of the parent population, thus suggesting that growth variability has an epigenetic origin. To explain observed distributions of clonal growth rates, we have developed a probabilistic model, assuming that fluctuation in the number of mitochondria through successive cell cycles is the leading cause of growth variability. For fitting purposes, we have estimated experimentally by flow cytometry the average maximum number of mitochondria in Molt3 cells. The model fits nicely observed distributions in growth rates; however, cells in which mitochondria were rendered non-functional (ρ⁰ cells) showed only 30% reduction in clonal growth variability with respect to normal cells. Conclusions : A tumour cell population is a dynamic ensemble of clones with highly variable growth rates. At least part of this variability is due to fluctuations in the initial number of mitochondria in daughter cells.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

Using a stem cell and progeny model to illustrate the relationship between cell cycle times of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them

There is increasing evidence that the growth of human tumours is driven by a small proportion of tumour stem cells with self-renewal properties. Multiplication of these cells leads to loss of self-renewal and after division for a finite number of times the cells undergo programmed cell death. Cell cycle times of human cancers have been measured in vivo and shown to vary in the range from two days to several weeks, depending on the individual. Cells cultured directly from tumours removed at surgery initially grow at a rate comparable to the in vivo rate but continued culture leads to the generation of cell lines that have shorter cycle times (1–3 days). It has been postulated that the more rapidly growing sub-population exhibits some of the properties of tumour stem cells and are the precursors of a slower growing sub-population that comprise the bulk of the tumour. We have previously developed a mathematical model to describe the behaviour of cell lines and we extend this model here to describe the behaviour of a system with two cell populations with different kinetic characteristics and a precursor–product relationship. The aim is to provide a framework for understanding the behaviour of cancer tissue that is sustained by a minor population of proliferating stem cells.     

Phenotypic variations of TRAIL sensitivity in cloned populations of prostate cancer cells

Factors that regulate the induction of apoptosis of tumour cells are potential candidates for therapeutic intervention for the majority of cancers. Studying modifiers of apoptotic responses, such as members of the tumour necrosis factor receptor superfamily, may give clues as to how induction of apoptosis in tumours could be maximized to enhance the benefit of treatment regimes. Tumour necrosis factor‐related apoptosis‐inducing ligand (TRAIL) is a promising anti‐tumour molecule since its activity is specific for tumour cell populations. TRAIL binds to death receptors, inducing apoptosis in susceptible cells. The mechanisms which determine whether tumour cells are susceptible to TRAIL are unclear, and several mechanisms have been proposed, including expression of osteoprotegerin (OPG), decoy receptors, and factors that affect intracellular signalling of pro‐apoptotic molecules, such as c‐FLIP. Here we show that experiments to modulate the activity of one of these factors, OPG, by over‐expression and also by stable knockdown of OPG expression, alters the TRAIL sensitivity of PC3 prostate cancer cells. However we show that some observed effects, which appear to support the hypothesis that OPG prevents TRAIL‐induced apoptosis of tumour cells, may be due to variation of the TRAIL response of sub‐clones of tumour cells, even within a cloned population. These results highlight potential limitations of experiments designed to test contribution of factors affecting intrinsic apoptosis susceptibility using cloned tumour cell populations. J. Cell. Biochem. 104: 1452–1464, 2008. © 2008 Wiley‐Liss, Inc.     

Endothelial cell differentiation of human breast tumour stem/progenitor cells

Breast tumour stem cells have been reported to differentiate in the epithelial lineage but a cross-lineage potential has not been investigated. We aimed to evaluate whether breast tumour stem cells were able to differentiate also into the endothelial lineage. We isolated and cloned a population of breast tumour stem cells, cultured as mammospheres that expressed the stem markers nestin and Oct-4 and not epithelial and endothelial differentiation markers, and formed serially transplantable tumours in SCID mice. When cultured in the presence of serum, mammosphere-derived clones differentiated in the epithelial lineage. When cultured in the presence of VEGF, the same clones were also able to differentiate in the endothelial lineage acquiring endothelial markers and properties, such as the ability to organize in Matrigel into capillary-like structures. In the transplanted tumours, originated from mammospheres, we demonstrate that some of the intratumour vessels were of human origin, suggesting an in vivo endothelial differentiation of mammosphere-derived cells. Finally, endothelial cell clones originated from mammospheres were able, when implanted in Matrigel in SCID mice, to form after 7 days a human vessel network and, after 3–4 weeks, an epithelial tumour suggesting that in the endothelial-differentiated cells a tumourigenic stem cell population is maintained. In conclusion, the results of the present study demonstrate that stem cells of breast cancer have the ability to differentiate not only in epithelial but also in endothelial lineage, further supporting the hypothesis that the tumour-initiating population possesses stem cell characteristics relevant for tumour growth and vascularization.     

CELL LOSS FROM EXPERIMENTAL TUMOURS

The rate of loss of cells from a tumour may be estimated by measuring the cell production rate and comparing this with the rate at which cells are observed to be added to the tumour volume. An attractive method of measuring cell production rate is by the simultaneous measurement of a thymidine-labelling index and the duration of the DNA synthetic period. A theoretical treatment of this method in exponential populations is given and the necessary assumptions are indicated. Estimates of cell loss have been made for a number of experimental tumours using available published data. It is shown that in some cases cell loss is an important, perhaps even dominant, factor determining tumour growth rate and the shape of tumour growth curves.     

Hybrid mathematical model of glioma progression

Objectives:  Gliomas are an important form of brain cancer, with high mortality rate. Mathematical models are often used to understand and predict their behaviour. However, using current modeling techniques one must choose between simulating individual cell behaviour and modeling tumours of clinically significant size.
Materials and Methods:  We propose a hybrid compartment-continuum-discrete model to simulate glioma growth and malignant cell invasion. The discrete portion of the model is capable of capturing intercellular interactions, including cell migration, intercellular communication, spatial cell population heterogeneity, phenotype differentiation, epigenetic events, proliferation, and apoptosis. Combining this with a compartment and continuum model allows clinically significant tumour sizes to be evaluated.
Results and Conclusions:  This model is used to perform multiple simulations to determine sensitivity to changes in important model parameters, specifically, the fundamental length parameter, necrotic cell degradation rate, rate of cell migration, and rate of phenotype transformation. Using these values, the model is able to simulate tumour growth and invasion behaviour, observed clinically. This mathematical model provides a means to simulate various tumour development scenarios, which may lead to a better understanding of how altering fundamental parameters can influence neoplastic progression.     

Concentration dependence of variability in growth rates of microtubules

Growth and shortening of microtubules in the course of their polymerization and depolymerization have previously been observed to occur at variable rates. To gain insight into the meaning of this prominent variability, we studied the way in which its magnitude depends on the growth rate of experimentally observed and computer-simulated microtubules. The dynamic properties of plus-ended microtubules nucleated by pieces of Chlamydomonas flagellar axonemes were observed in real time by video-enhanced differential interference contrast light microscopy at differing tubulin concentrations. By means of a Monte Carlo algorithm, populations of microtubules were simulated that had similar growth and dynamic properties to the experimentally observed microtubules. By comparison of the experimentally observed and computer-simulated populations of microtubules, we found that 1) individual microtubules displayed an intrinsic variability that did not change as the rate of growth for a population increased, and 2) the variability was approximately fivefold greater than predicted by a simple model of subunit addition and loss. The model used to simulate microtubule growth has no provision for incorporation of lattice defects of any type, nor sophisticated geometry of the growing end. Thus, these as well as uncontrolled experimental variables were eliminated as causes for the prominent variability.     

Tumour size can have an impact on the outcomes of epidemiological studies on second cancers after radiotherapy

Obtaining a correct dose–response relationship for radiation-induced cancer after radiotherapy presents a major challenge for epidemiological studies. The purpose of this paper is to gain a better understanding of the associated uncertainties. To accomplish this goal, some aspects of an epidemiological study on breast cancer following radiotherapy of Hodgkin’s disease were simulated with Monte Carlo methods. It is demonstrated that although the doses to the breast volume are calculated by one treatment plan, the locations and sizes of the induced secondary breast tumours can be simulated and, based on these simulated locations and sizes, the absorbed doses at the site of tumour incidence can also be simulated. For the simulations of point dose at tumour site, linear and non-linear mechanistic models which predict risk of cancer induction as a function of dose were applied randomly to the treatment plan. These simulations provided for each second tumour and each simulated tumour size the predicted dose. The predicted-dose–response-characteristic from the analysis of the simulated epidemiological study was analysed. If a linear dose–response relationship for cancer induction was applied to calculate the theoretical doses at the simulated tumour sites, all Monte-Carlo realizations of the epidemiological study yielded strong evidence for a resulting linear risk to predicted-dose–response. However, if a non-linear dose–response of cancer induction was applied to calculate the theoretical doses, the Monte Carlo simulated epidemiological study resulted in a non-linear risk to predicted-dose–response relationship only if the tumour size was small (<?1.5 cm). If the diagnosed breast tumours exceeded an average diameter of 1.5 cm, an applied non-linear theoretical-dose–response relationship for second cancer falsely resulted in strong evidence for a linear predicted-dose relationship from the epidemiological study realizations. For a typical distribution of breast cancer sizes, the model selection probability for a resulting predicted-dose linear model was 61% although a non-linear theoretical-dose–response relationship for cancer induction had been applied. The results of this study, therefore, provide evidence that the shapes of epidemiologically obtained dose–response relationships for cancer induction can be biased by the finite size of the diagnosed second tumour, even though the epidemiological study was done correctly.     

Oral Tolerance to Cancer Can Be Abrogated by T Regulatory Cell Inhibition

Oral administration of tumour cells induces an immune hypo-responsiveness known as oral tolerance. We have previously shown that oral tolerance to a cancer is tumour antigen specific, non-cross-reactive and confers a tumour growth advantage. We investigated the utilisation of regulatory T cell (Treg) depletion on oral tolerance to a cancer and its ability to control tumour growth. Balb/C mice were gavage fed homogenised tumour tissue – JBS fibrosarcoma (to induce oral tolerance to a cancer), or PBS as control. Growth of subcutaneous JBS tumours were measured; splenic tissue excised and flow cytometry used to quantify and compare systemic Tregs and T effector (Teff) cell populations. Prior to and/or following tumour feeding, mice were intraperitoneally administered anti-CD25, to inactivate systemic Tregs, or given isotype antibody as a control. Mice which were orally tolerised prior to subcutaneous tumour induction, displayed significantly higher systemic Treg levels (14% vs 6%) and faster tumour growth rates than controls (p<0.05). Complete regression of tumours were only seen after Treg inactivation and occurred in all groups - this was not inhibited by tumour feeding. The cure rates for Treg inactivation were 60% during tolerisation, 75% during tumour growth and 100% during inactivation for both tolerisation and tumour growth. Depletion of Tregs gave rise to an increased number of Teff cells. Treg depletion post-tolerisation and post-tumour induction led to the complete regression of all tumours on tumour bearing mice. Oral administration of tumour tissue, confers a tumour growth advantage and is accompanied by an increase in systemic Treg levels. The administration of anti-CD25 Ab decreased Treg numbers and caused an increase in Teffs. Most notably Treg cell inhibition overcame established oral tolerance with consequent tumor regression, especially relevant to foregut cancers where oral tolerance is likely to be induced by the shedding of tumour tissue into the gut.     

Promotion of initiated cells by radiation-induced cell inactivation

Cells on the way to carcinogenesis can have a growth advantage relative to normal cells. It has been hypothesized that a radiation-induced growth advantage of these initiated cells might be induced by an increased cell replacement probability of initiated cells after inactivation of neighboring cells by radiation. Here Monte Carlo simulations extend this hypothesis for larger clones: The effective clonal expansion rate decreases with clone size. This effect is stronger for the two-dimensional than for the three-dimensional situation. The clones are irregular, far from a circular shape. An exposure-rate dependence of the effective clonal expansion rate could come in part from a minimal recovery time of the initiated cells for symmetric cell division.     

Mathematical Determination of Cell Population Doubling Times for Multiple Cell Lines

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.     

Mathematical model for chemotherapeutic drug efficacy in arresting tumour growth based on the cancer stem cell hypothesis

OBJECTIVE: Cancer stem cells have been identified as the growth root for various malignant tumours and are thought to be responsible for cancer recurrence following treatment. MATERIALS AND METHODS: Here, a predictive mathematical model for the cancer stem cell hypothesis is used to understand tumour responses to chemotherapeutic drugs and judge the efficacy of treatments in arresting tumour growth. The impact of varying drug efficacies on different abnormal cell populations is investigated through the kinetics associated with their decline in response to therapy. RESULTS AND CONCLUSIONS: The model predicts the clinically established 'dandelion phenomenon' and suggests that the best response to chemotherapy occurs when a drug targets abnormal stem cells. We compare continuous and periodic drug infusion. For the latter, we examine the relative importance of the drug cell-kill rate and the mean time between successive therapies, to identify the key attributes for successful treatment.     

Chromatin barcodes as biomarkers for melanoma

The major barrier to effective cancer therapy is the presence of genetic and phenotypic heterogeneity within cancer cell populations that provides a reservoir of therapeutically resistant cells. As the degree of heterogeneity present within tumours will be proportional to tumour burden, the development of rapid, robust, accurate and sensitive biomarkers for cancer progression that could detect clinically occult disease before substantial heterogeneity develops would provide a major therapeutic benefit. Here, we explore the application of chromatin conformation capture technology to generate a diagnostic epigenetic barcode for melanoma. The results indicate that binary states from chromatin conformations at 15 loci within five genes can be used to provide rapid, non‐invasive multivariate test for the presence of melanoma using as little as 200 μl of patient blood.     

A mathematical and computational approach for integrating the major sources of cell population heterogeneity

Several approaches have been used in the past to model heterogeneity in bacterial cell populations, with each approach focusing on different source(s) of heterogeneity. However, a holistic approach that integrates all the major sources into a comprehensive framework applicable to cell populations is still lacking.In this work we present the mathematical formulation of a cell population master equation (CPME) that describes cell population dynamics and takes into account the major sources of heterogeneity, namely stochasticity in reaction, DNA-duplication, and division, as well as the random partitioning of species contents into the two daughter cells. The formulation also takes into account cell growth and respects the discrete nature of the molecular contents and cell numbers. We further develop a Monte Carlo algorithm for the simulation of the stochastic processes considered here. To benchmark our new framework, we first use it to quantify the effect of each source of heterogeneity on the intrinsic and the extrinsic phenotypic variability for the well-known two-promoter system used experimentally by Elowitz et al. (2002). We finally apply our framework to a more complicated system and demonstrate how the interplay between noisy gene expression and growth inhibition due to protein accumulation at the single cell level can result in complex behavior at the cell population level.The generality of our framework makes it suitable for studying a vast array of artificial and natural genetic networks. Using our Monte Carlo algorithm, cell population distributions can be predicted for the genetic architecture of interest, thereby quantifying the effect of stochasticity in intracellular reactions or the variability in the rate of physiological processes such as growth and division. Such in silico experiments can give insight into the behavior of cell populations and reveal the major sources contributing to cell population heterogeneity.     

Brain tumour stem cells: the undercurrents of human brain cancer and their relationship to neural stem cells

Conceptual and technical advances in neural stem cell biology are being applied to the study of human brain tumours. These studies suggest that human brain tumours are organized as a hierarchy and are maintained by a small number of tumour cells that have stem cell properties. Most of the bulk population of human brain tumours comprise cells that have lost the ability to initiate and maintain tumour growth. Although the cell of origin for human brain tumours is uncertain, recent evidence points towards the brain's known proliferative zones. The identification of brain tumour stem cells has important implications for understanding brain tumour biology and these cells may be critical cellular targets for curative therapy.