A cellular automaton model for the migration of glioma cells

We present a study of in vitro cell migration in two dimensions as a first step towards understanding the mechanisms governing the motility of glioma cells. Our study is based on a cellular automaton model which aims at reproducing the kinetics of a lump of glioma cells deposited on a substrate of collagen. The dynamical effects of cell attraction and motion inertia are introduced through adequate automaton rules. We compare the density profiles given by the model to those obtained experimentally. The result of the best fit indicates a substantial cell-cell attraction due to cell-cell communication through gap junctions (or chemotaxis) and negligible inertia effects during migration. Tracking of individual migrating cells indicates highly convoluted cell trajectories.     

A cellular automaton model for microcarrier cultures

In order to achieve high cell densities anchoragedependent cells are commonly cultured on microcarriers, where spatial restrictions to cell growth complicates the determination of the growth kinetics. To design and operate large-scale bioreactors for microcarrier cultures, the effect of this spatial restriction to growth, referred to as contact inhibition, must be decoupled from the growth kinetics. In this article, a cellular automaton approach is recommended to model the growth of anchorage-dependent cells on microcarriers. The proposed model is simple to apply yet provides an accurate representation of contact-inhibited cell growth on microcarriers. The distribution of the number of neighboring cells per cell, microcarrier surface areas, and inoculation densities are taken into account with this model. When compared with experimental data for Vero and MRC-5 microcarrier cultures, the cellular automaton predictions were very good. Furthermore, the model can be used to generate contact-inhibition growth curves to decouple the effect of contact-inhibition from growth kinetics. With this information, the accurate determination of kinetic parameters, such as nutrient uptake rates, and the effects of other environmental factors, such as toxin levels, may be determined. (c) 1994 John Wiley & Sons, Inc.     

A simple cellular automaton model for influenza A viral infections

Viral kinetics have been extensively studied in the past through the use of spatially homogeneous ordinary differential equations describing the time evolution of the diseased state. However, spatial characteristics such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading. In order to investigate the influence of spatial heterogeneities on viral spread, a simple 2-D cellular automaton (CA) model of a viral infection has been developed. In this initial phase of the investigation, the CA model is validated against clinical immunological data for uncomplicated influenza A infections. Our results will be shown and discussed.     

A probability cellular automaton model for hepatitis B viral infections

The existing models of hepatitis B virus (HBV) infection dynamics are based on the assumption that the populations of viruses and cells are uniformly mixed. However, the real virus infection system is actually not homogeneous and some spatial factors might play a nontrivial role in governing the development of HBV infection and its outcome. For instance, the localized populations of dead cells might adversely affect the spread of infection. To consider this kind of inhomogeneous feature, a simple 2D (dimensional) probability Cellular Automaton model was introduced to study the dynamic process of HBV infection. The model took into account the existence of different types of HBV infectious and non-infectious particles. The simulation results thus obtained showed that the Cellular Automaton model could successfully account for some important features of the disease, such as its wide variety in manifestation and its age dependency. Meanwhile, the effects of the model's parameters on the dynamical process of the infection were also investigated. It is anticipated that the Cellular Automaton model may be extended to serve as a useful vehicle for studying, among many other complicated dynamic biological systems, various persistent infections with replicating parasites.     

A cellular automaton model for tumour growth in inhomogeneous environment

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.     

Novel cellular tracks of migrating Dictyostelium cells

After Dictyostelium cells were settled on a coverslip and allowed to migrate freely on the surface, they were stained with fluorescently labeled Concanavalin A. Tracks with distinct patterns that consist of dots and short fibers were observed behind the cells. In this study, we refer to these tracks as "cellular tracks", CTs for short. We characterized the biological effect of CTs on cell behavior and development. CTs decreased the strength of cell-substratum adhesion, increased the velocity of cell migration, but did not affect growth of cells. CTs also promoted cell aggregation. When pre-aggregation cells touched the CTs of other cells, they avoided or orthogonally crossed them, but did not migrate along them. These observations suggest that the CTs of pre-aggregation cells prompts cells to disperse uniformly on substratum and may enable cells to sense cell density. On the other hand, when aggregation-competent cells touched the CTs of other aggregation-competent cells, a half of them migrated along the CTs. Pre-aggregation cells did not migrate along the CTs of aggregation-competent cells. The CTs of aggregation-competent cells may help the cells to aggregate toward the aggregation center.     

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.     

An evolutionary hybrid cellular automaton model of solid tumour growth

We propose a cellular automaton model of solid tumour growth, in which each cell is equipped with a micro-environment response network. This network is modelled using a feed-forward artificial neural network, that takes environmental variables as an input and from these determines the cellular behaviour as the output. The response of the network is determined by connection weights and thresholds in the network, which are subject to mutations when the cells divide. As both available space and nutrients are limited resources for the tumour, this gives rise to clonal evolution where only the fittest cells survive. Using this approach we have investigated the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. The results show that the oxygen concentration affects the selection pressure, cell population diversity and morphology of the tumour. A low oxygen concentration in the tissue gives rise to a tumour with a fingered morphology that contains aggressive phenotypes with a small apoptotic potential, while a high oxygen concentration in the tissue gives rise to a tumour with a round morphology containing less evolved phenotypes. The tissue oxygen concentration thus affects the tumour at both the morphological level and on the phenotype level.     

A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth

We report numerical results from a 2D cellular automaton (CA) model describing the dynamics of the in vitro cultivated multicellular spheroid obtained from EMT6/Ro (mammary carcinoma) cell line. Significantly, the CA model relaxes the often assumed one-to-one correspondence between cells and CA sites so as to correctly model the peripheral mitotic boundary region, and to enable the study of necrosis in large avascular tumours. By full calibration and scaling to available experimental data, the model produces with good accuracy experimentally comparable data on a range of bulk tumour kinetics and necrosis measures. Our main finding is that the metabolic production of H⁺ ions is not sufficient to cause central necrosis prior to the sub-viable nutrient-deficient stage of tumour development being reached. Thus, the model suggests that an additional process is required to explain the experimentally observable onset of necrosis prior to the non-viable nutrient-deficient point being reached.     

Coexistence of sexual and asexual conspecifics: a cellular automaton model

Most previous models of populations mixed for reproductive mode have omitted important local interactions between sexual and asexual individuals. We propose a cellular automaton model where local rules focus on fertilization and colonization. This model produces rich sets of data which are then studied by means of spatial statistics. Results point to the fixation of one of the two reproductive modes in the landscape. However, some examples of coexistence of sexual and asexual conspecifics over long periods of time are also found. This model is an example of a CA that diverges from its mean field approximation. The formation of sexual and asexual clusters reduces effective colonization rate in the CA and may account for this behaviour.     

A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype

We present a cellular automaton model of clonal evolution in cancer aimed at investigating the emergence of the glycolytic phenotype. In the model each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. This implies that cells might react differently to the environment and when space and nutrients are limited only the fittest cells will survive. With this model we have investigated the impact of the environment on the growth dynamics of the tumour. In particular, we have analysed the influence of the tissue oxygen concentration and extra-cellular matrix density on the dynamics of the model. We found that the environment influences both the growth and the evolutionary dynamics of the tumour. For low oxygen concentration we observe tumours with a fingered morphology, while increasing the matrix density gives rise to more compact tumours with wider fingers. The distribution of phenotypes in the tumour is also affected, and we observe that the glycolytic phenotype is most likely to emerge in a poorly oxygenated tissue with a high matrix density. Our results suggest that it is the combined effect of the oxygen concentration and matrix density that creates an environment where the glycolytic phenotype has a growth advantage and consequently is most likely to appear.     

Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. In vitro experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least-resistance paths of cells and intrabranch homotype attraction, we also predict nontrivial coupling between the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that sophisticated in silico tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.     

Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning

We present a stochastic cellular automaton model for the behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic patterning. This "agent-oriented" model represents cells by points on a lattice that obey rules motivated by experimental findings. The "cells" follow these rules as autonomous agents, interacting with other cells and with the microenvironments cell activities produce. The rules include random cell motion, production and lateral deposition of a substrate adhesion molecule (SAM, corresponding to fibronectin), production and release of a diffusible growth factor ("activator," corresponding to TGF-beta) that stimulates production of the SAM, and another diffusible factor ("inhibitor") that suppresses the activity of the activator. We implemented the cellular automaton on a two-dimensional (2D) square lattice to emulate the quasi-2D micromass culture extensively used to study patterning in avian limb bud precartilage cells. We identified parameters that produce nodular patterns that resemble, in size and distribution, cell condensations in leg-cell cultures, thus establishing a correspondence between in vitro and in silico results. We then studied the in vitro and in silico micromass cultures experimentally. We altered the standard in vitro micromass culture by diluting the initial cell density, transiently exposing it to exogenous activator, suppressing the inhibitor, and constitutively activating fibronectin production. We altered the standard in silico micromass culture in each case by changing the corresponding parameter. In vitro and in silico experiments agreed well. We also used the model to test hypotheses for differences in the in vitro patterns of cells derived from chick embryo forelimb and hindlimb. We discuss the applicability of this model to limb development in vivo and to other organ development.     

A cellular automaton model examining the effects of oxygen,hydrogen ions and lactate on early tumour growth

Some tumours are known to exhibit an extracellular pH that is more acidic than the intracellular, creating a ‘reversed pH gradient’ across the cell membrane and this has been shown to affect their invasive and metastatic potential. Tumour hypoxia also plays an important role in tumour development and has been directly linked to both tumour morphology and aggressiveness. In this paper, we present a hybrid mathematical model of intracellular pH regulation that examines the effect of oxygen and pH on tumour growth and morphology. In particular, we investigate the impact of pH regulatory mechanisms on the cellular pH gradient and tumour morphology. Analysis of the model shows that: low activity of the Na \(^+\) /H \(^+\) exchanger or a high rate of anaerobic glycolysis can give rise to a “fingering” tumour morphology; and a high activity of the lactate/H \(^+\) symporter can result in a reversed transmembrane pH gradient across a large portion of the tumour mass. Also, the reversed pH gradient is spatially heterogenous within the tumour, with a normal pH gradient observed within an intermediate growth layer within the spheroid. We also include a fractal dimension analysis of the simulated tumour contours, in which we compare the fractal dimensions of the simulated tumour surfaces with those found experimentally via photomicrographs.     

Banded vegetation: development under uniform rainfall from a simple cellular automaton model

Banded vegetation communities are known from semi-arid and arid landscapes in many parts of the world, in grasslands, shrublands, and woodlands. The origin of the distinctive patterning has been the subject of speculation, a common view being that banding evolves through the decline of more complete vegetation cover because of climatic deterioration or through grazing disturbance. A simple model based on cellular automata is employed to test the hypothesis that plausible mechanisms of water partitioning in spatially unstructured plant communities can bring about the development of banding. It is shown that without any climatic change or external disturbance, strongly developed banding can emerge from an initially random distribution of plants. Physical processes underlying the water partitioning, some of which remain unresearched, are discussed, and management implications noted.