A cellular automaton model for the proliferation of migrating contact-inhibited cells

Equations describing semi-confluent cell growth (I) Analytical approximations

A set of differential equations with analytical solutions are presented that can quantitatively account for variable degrees of contact inhibition on cell growth in two- and three-dimensional cultures. The developed equations can be used for comparative purposes when assessing contribution of higher-order effects, such as culture geometry and nutrient depletion, on mean cell growth rate. These equations also offer experimentalists the opportunity to characterize cell culture experiments using a single reductive parameter.

MIL-CELL: a tool for multi-scale simulation of yeast replication and prion transmission

The single-celled baker's yeast, Saccharomyces cerevisiae, can sustain a number of amyloid-based prions, the three most prominent examples being [URE3], [PSI+], and [PIN+]. In the laboratory, haploid S. cerevisiae cells of a single mating type can acquire an amyloid prion in one of two ways (i) spontaneous nucleation of the prion within the yeast cell, and (ii) receipt via mother-to-daughter transmission during the cell division cycle. Similarly, prions can be lost due to (i) dissolution of the prion amyloid by its breakage into non-amyloid monomeric units, or (ii) preferential donation/retention of prions between the mother and daughter during cell division. Here we present a computational tool (Monitoring Induction and Loss of prions in Cells; MIL-CELL) for modelling these four general processes using a multiscale approach describing both spatial and kinetic aspects of the yeast life cycle and the amyloid-prion behavior. We describe the workings of the model, assumptions upon which it is based and some interesting simulation results pertaining to the wave-like spread of the epigenetic prion elements through the yeast population. MIL-CELL is provided as a stand-alone GUI executable program for free download with the paper. MIL-CELL is equipped with a relational database allowing all simulated properties to be searched, collated and graphed. Its ability to incorporate variation in heritable properties means MIL-CELL is also capable of simulating loss of the isogenic nature of a cell population over time. The capability to monitor both chronological and reproductive age also makes MIL-CELL potentially useful in studies of cell aging.

Rules of Engagement: A Guide to Developing Agent-Based Models

Agent-based models (ABM), also called individual-based models, first appeared several decades ago with the promise of nearly real-time simulations of active, autonomous individuals such as animals or objects. The goal of ABMs is to represent a population of individuals (agents) interacting with one another and their environment. Because of their flexible framework, ABMs have been widely applied to study systems in engineering, economics, ecology, and biology. This chapter is intended to guide the users in the development of an agent-based model by discussing conceptual issues, implementation, and pitfalls of ABMs from first principles. As a case study, we consider an ABM of the multi-scale dynamics of cellular interactions in a microbial community. We develop a lattice-free agent-based model of individual cells whose actions of growth, movement, and division are influenced by both their individual processes (cell cycle) and their contact with other cells (adhesion and contact inhibition).

A computational model of liver tissue damage and repair

Drug induced liver injury (DILI) and cell death can result from oxidative stress in hepatocytes. An initial pattern of centrilobular damage in the APAP model of DILI is amplified by communication from stressed cells and immune system activation. While hepatocyte proliferation counters cell loss, high doses are still lethal to the tissue. To understand the progression of disease from the initial damage to tissue recovery or death, we computationally model the competing biological processes of hepatocyte proliferation, necrosis and injury propagation. We parametrize timescales of proliferation (α), conversion of healthy to stressed cells (β) and further sensitization of stressed cells towards necrotic pathways (γ) and model them on a Cellular Automaton (CA) based grid of lattice sites. 1D simulations show that a small α/β (fast proliferation), combined with a large γ/β (slow death) have the lowest probabilities of tissue survival. At large α/β, tissue fate can be described by a critical γ/β* ratio alone; this value is dependent on the initial amount of damage and proportional to the tissue size N. Additionally, the 1D model predicts a minimum healthy population size below which damage is irreversible. Finally, we compare 1D and 2D phase spaces and discuss outcomes of bistability where either survival or death is possible, and of coexistence where simulated tissue never completely recovers or dies but persists as a mixture of healthy, stressed and necrotic cells. In conclusion, our model sheds light on the evolution of tissue damage or recovery and predicts potential for divergent fates given different rates of proliferation, necrosis, and injury propagation.

A particle-based model for endothelial cell migration under flow conditions

Endothelial cells (ECs) play a major role in the healing process following angioplasty to inhibit excessive neointima. This makes the process of EC healing after injury, in particular EC migration in a stented vessel, important for recovery of normal vessel function. In that context, we present a novel particle-based model of EC migration and validate it against in vitro experimental data. We have developed a particle-based model of EC migration under flow conditions in an in vitro vessel with obstacles. Cell movement in the model is a combination of random walks and directed movement along the local flow velocity vector. For model calibration, a set of experimental data for cell migration in a similarly shaped channel has been used. We have calibrated the model for a baseline case of a channel with no obstacles and then applied it to the case of a channel with ridges on the bottom surface, representative of stent strut geometry. We were able to closely reproduce the cell migration speed and angular distribution of their movement relative to the flow direction reported in vitro. The model also reproduces qualitative aspects of EC migration, such as entrapment of cells downstream from the flow-disturbing ridge. The model has the potential, after more extensive in vitro validation, to study the effect of variation in strut spacing and shape, through modification of the local flow, on EC migration. The results of this study support the hypothesis that EC migration is strongly affected by the direction and magnitude of local wall shear stress.

Modeling Living Cells Within Microfluidic Systems Using Cellular Automata Models

Several computational models, both continuum and discrete, allow for the simulation of collective cell behaviors in connection with challenges linked to disease modeling and understanding. Normally, discrete cell modelling employs quasi-infinite or boundary-less 2D lattices, hence modeling collective cell behaviors in Petri dish-like environments. The advent of lab- and organ-on-a-chip devices proves that the information obtained from 2D cell cultures, upon Petri dishes, differs importantly from the results obtained in more biomimetic micro-fluidic environments, made of interconnected chambers and channels. However, discrete cell modelling within lab- and organ-on-a-chip devices, to our knowledge, is not yet found in the literature, although it may prove useful for designing and optimizing these types of systems. Consequently, in this study we focus on the establishment of a direct connection between the computer-aided designs (CAD) of microfluidic systems, especially labs- and organs-on-chips (and their multi-chamber and multi-channel structures), and the lattices for discrete cell modeling approaches aimed at the simulation of collective cell interactions, whose boundaries are defined directly from the CAD models. We illustrate the proposal using a quite straightforward cellular automata model, apply it to simulating cells with different growth rates, within a selected set of microsystem designs, and validate it by tuning the growth rates with the support of cell culture experiments and by checking the results with a real microfluidic system.

Novel surface modification of three-dimensional bacterial nanocellulose with cell-derived adhesion proteins for soft tissue engineering

Bacterial nanocellulose (BNC) is a natural polymer composed of glucose units with an important application as a two and three-dimensional scaffold for tissue engineering. However, as a polysaccharide, BNC does not have the biological signals of protein biomaterials. Therefore, this paper aims to develop a novel methodology to biomimic soft extracellular matrix (ECM) chemistry on to 3D BNC using the bioengineering of fibroblasts (the cells responsible for producing and regenerating the ECM) to immobilise adhesion proteins such as collagen and fibronectin. Modified 3D BNC (Mod-BNC) biomaterials were morphologically, thermally, and chemically characterised, and furthermore, the cell response was analysed by adhesion studies using atomic force microscopy (AFM), XTT assay, and confocal microscopy. Cell-derived proteins were deposited on the BNC nanoribbon network to modify its surface. The contact angle was increased from 40° to 60°, reducing the wettability of the biomaterial, and during thermogravimetry, the proteins in Mod-BNC exhibited an enhanced thermal stability because of the interactions between themselves and BNC. Chemical and immunocytochemistry analyses confirmed the presence of collagen type I and fibronectin on 3D BNC. These proteins activate integrin adhesion pathways that generate stronger cell adhesions. AFM experiments showed higher forces and energies on modified biomaterials, and moreover, the cells that adhered on to Mod-BNC exhibited higher mitochondrial activity and higher cell populations per cubic millimetre than non-modified surfaces (NMod-BNC). Accordingly, it was established that this novel methodology is robust and able to biomimic the chemical surface of soft ECM and immobilise cell-derived adhesion proteins from fibroblast; moreover, the Mod-BNC exhibited better cell response than NMod-BNC because of the biological signals in 3D BNC.

A Model for Adult Organ Resizing Demonstrates Stem Cell Scaling through a Tunable Commitment Rate

Many adult organs grow or shrink to accommodate different physiological demands. Often, as total cell number changes, stem cell number changes proportionally in a phenomenon called "stem cell scaling". The cellular behaviors that give rise to scaling are unknown. Here we study two complementary theoretical models of the adult Drosophila midgut, a stem cell-based organ with known resizing dynamics. First, we derive a differential equations model of midgut resizing and show that the in vivo kinetics of growth can be recapitulated if the rate of fate commitment depends on the tissue's stem cell proportion. Second, we develop a 2D simulation of the midgut and find that proportion-dependent commitment rate and stem cell scaling can arise phenomenologically from the stem cells' exploration of physical tissue space during its lifetime. Together, these models provide a biophysical understanding of how stem cell scaling is maintained during organ growth and shrinkage.

In silico characterization of cell-cell interactions using a cellular automata model of cell culture

BACKGROUND

Cell proliferation is a key characteristic of eukaryotic cells. During cell proliferation, cells interact with each other. In this study, we developed a cellular automata model to estimate cell-cell interactions using experimentally obtained images of cultured cells.

RESULTS

We used four types of cells; HeLa cells, human osteosarcoma (HOS) cells, rat mesenchymal stem cells (MSCs), and rat smooth muscle A7r5 cells. These cells were cultured and stained daily. The obtained cell images were binarized and clipped into squares containing about 10⁴ cells. These cells showed characteristic cell proliferation patterns. The growth curves of these cells were generated from the cell proliferation images and we determined the doubling time of these cells from the growth curves. We developed a simple cellular automata system with an easily accessible graphical user interface. This system has five variable parameters, namely, initial cell number, doubling time, motility, cell-cell adhesion, and cell-cell contact inhibition (of proliferation). Within these parameters, we obtained initial cell numbers and doubling times experimentally. We set the motility at a constant value because the effect of the parameter for our simulation was restricted. Therefore, we simulated cell proliferation behavior with cell-cell adhesion and cell-cell contact inhibition as variables. By comparing growth curves and proliferation cell images, we succeeded in determining the cell-cell interaction properties of each cell. Simulated HeLa and HOS cells exhibited low cell-cell adhesion and weak cell-cell contact inhibition. Simulated MSCs exhibited high cell-cell adhesion and positive cell-cell contact inhibition. Simulated A7r5 cells exhibited low cell-cell adhesion and strong cell-cell contact inhibition. These simulated results correlated with the experimental growth curves and proliferation images.

CONCLUSIONS

Our simulation approach is an easy method for evaluating the cell-cell interaction properties of cells.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

An in silico prediction tool for the expansion culture of human skeletal muscle myoblasts

Regenerative therapy using autologous skeletal myoblasts requires a large number of cells to be prepared for high-level secretion of cytokines and chemokines to induce good regeneration of damaged regions. However, myoblast expansion culture is hindered by a reduction in growth rate owing to cellular quiescence and differentiation, therefore optimization is required. We have developed a kinetic computational model describing skeletal myoblast proliferation and differentiation, which can be used as a prediction tool for the expansion process. In the model, myoblasts migrate, divide, quiesce and differentiate as observed during in vitro culture. We assumed cell differentiation initiates following cell-cell attachment for a defined time period. The model parameter values were estimated by fitting to several predetermined experimental datasets. Using an additional experimental dataset, we confirmed validity of the developed model. We then executed simulations using the developed model under several culture conditions and quantitatively predicted that non-uniform cell seeding had adverse effects on the expansion culture, mainly by reducing the existing ratio of proliferative cells. The proposed model is expected to be useful for predicting myoblast behaviours and in designing efficient expansion culture conditions for these cells.

Modeling cell adhesion and proliferation: a cellular-automata based approach

BACKGROUND

Cell adhesion is a process that involves the interaction between the cell membrane and another surface, either a cell or a substrate. Unlike experimental tests, computer models can simulate processes and study the result of experiments in a shorter time and lower costs. One of the tools used to simulate biological processes is the cellular automata, which is a dynamic system that is discrete both in space and time.

METHOD

This work describes a computer model based on cellular automata for the adhesion process and cell proliferation to predict the behavior of a cell population in suspension and adhered to a substrate. The values of the simulated system were obtained through experimental tests on fibroblast monolayer cultures.

RESULTS

The results allow us to estimate the cells settling time in culture as well as the adhesion and proliferation time. The change in the cells morphology as the adhesion over the contact surface progress was also observed. The formation of the initial link between cell and the substrate of the adhesion was observed after 100 min where the cell on the substrate retains its spherical morphology during the simulation. The cellular automata model developed is, however, a simplified representation of the steps in the adhesion process and the subsequent proliferation.

CONCLUSION

A combined framework of experimental and computational simulation based on cellular automata was proposed to represent the fibroblast adhesion on substrates and changes in a macro-scale observed in the cell during the adhesion process. The approach showed to be simple and efficient.

Models, measurement and inference in epithelial tissue dynamics

The majority of solid tumours arise in epithelia and therefore much research effort has gone into investigating the growth, renewal and regulation of these tissues. Here we review different mathematical and computational approaches that have been used to model epithelia. We compare different models and describe future challenges that need to be overcome in order to fully exploit new data which present, for the first time, the real possibility for detailed model validation and comparison.

Analysis of impedance-based cellular growth assays

MOTIVATION

Impedance-based technologies are advancing methods for measuring proliferation of adherent cell cultures non-invasively and in real time. The analysis of the resulting data has so far been hampered by inappropriate computational methods and the lack of systematic data to evaluate the characteristics of the assay.

RESULTS

We used a commercially available system for impedance-based growth measurement (xCELLigence) and compared the reported cell index with data from microscopy. We found that the measured signal correlates linearly with the cell number throughout the time of an experiment with sufficient accuracy in subconfluent cell cultures. The resulting growth curves for various colon cancer cells could be well described with the empirical Richards growth model, which allows for extracting quantitative parameters (such as characteristic cycle times). We found that frequently used readouts like the cell index at a specific time or the area under the growth curve cannot be used to faithfully characterize growth inhibition. We propose to calculate the average growth rate of selected time intervals to accurately estimate time-dependent IC50 values of drugs from growth curves.

CONTACT

nils.bluethgen@charite.de

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

An agent-based model approach to multi-phase life-cycle for contact inhibited, anchorage dependent cells

Cellular agent-based models are a technique that can be easily adapted to describe nuances of a particular cell type. Within we have concentrated on the cellular particularities of the human Endothelial Cell, explicitly the effects both of anchorage dependency and of heightened scaffold binding on the total confluence time of a system. By expansion of a discrete, homogeneous, asynchronous cellular model to account for several states per cell (phases within a cell's life); we accommodate and track dependencies of confluence time and population dynamics on these factors. Increasing the total motility time, analogous to weakening the binding between lattice and cell, affects the system in unique ways from increasing the average cellular velocity; each degree of freedom allows for control over the time length the system achieves logistic growth and confluence. These additional factors may allow for greater control over behaviors of the system. Examinations of system's dependence on both seed state velocity and binding are also enclosed.

Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance

A tumour is a heterogeneous population of cells that competes for limited resources. In the clinic, we typically probe the tumour by biopsy, and then characterize it by the dominant genetic clone. But genotypes are only the first link in the chain of hierarchical events that leads to a specific cell phenotype. The relationship between genotype and phenotype is not simple, and the so-called genotype to phenotype map is poorly understood. Many genotypes can produce the same phenotype, so genetic heterogeneity may not translate directly to phenotypic heterogeneity. We therefore choose to focus on the functional endpoint, the phenotype as defined by a collection of cellular traits (e.g. proliferative and migratory ability). Here, we will examine how phenotypic heterogeneity evolves in space and time and how the way in which phenotypes are inherited will drive this evolution. A tumour can be thought of as an ecosystem, which critically means that we cannot just consider it as a collection of mutated cells but more as a complex system of many interacting cellular and microenvironmental elements. At its simplest, a growing tumour with increased proliferation capacity must compete for space as a limited resource. Hypercellularity leads to a contact-inhibited core with a competitive proliferating rim. Evolution and selection occurs, and an individual cell's capacity to survive and propagate is determined by its combination of traits and interaction with the environment. With heterogeneity in phenotypes, the clone that will dominate is not always obvious as there are both local interactions and global pressures. Several combinations of phenotypes can coexist, changing the fitness of the whole. To understand some aspects of heterogeneity in a growing tumour, we build an off-lattice agent-based model consisting of individual cells with assigned trait values for proliferation and migration rates. We represent heterogeneity in these traits with frequency distributions and combinations of traits with density maps. How the distributions change over time is dependent on how traits are passed on to progeny cells, which is our main enquiry. We bypass the translation of genetics to behaviour by focusing on the functional end result of inheritance of the phenotype combined with the environmental influence of limited space.

A multi-paradigm modeling framework to simulate dynamic reciprocity in a bioreactor

Despite numerous technology advances, bioreactors are still mostly utilized as functional black-boxes where trial and error eventually leads to the desirable cellular outcome. Investigators have applied various computational approaches to understand the impact the internal dynamics of such devices has on overall cell growth, but such models cannot provide a comprehensive perspective regarding the system dynamics, due to limitations inherent to the underlying approaches. In this study, a novel multi-paradigm modeling platform capable of simulating the dynamic bidirectional relationship between cells and their microenvironment is presented. Designing the modeling platform entailed combining and coupling fully an agent-based modeling platform with a transport phenomena computational modeling framework. To demonstrate capability, the platform was used to study the impact of bioreactor parameters on the overall cell population behavior and vice versa. In order to achieve this, virtual bioreactors were constructed and seeded. The virtual cells, guided by a set of rules involving the simulated mass transport inside the bioreactor, as well as cell-related probabilistic parameters, were capable of displaying an array of behaviors such as proliferation, migration, chemotaxis and apoptosis. In this way the platform was shown to capture not only the impact of bioreactor transport processes on cellular behavior but also the influence that cellular activity wields on that very same local mass transport, thereby influencing overall cell growth. The platform was validated by simulating cellular chemotaxis in a virtual direct visualization chamber and comparing the simulation with its experimental analogue. The results presented in this paper are in agreement with published models of similar flavor. The modeling platform can be used as a concept selection tool to optimize bioreactor design specifications.

Cellular automata approach to corrosion and passivity phenomena

Our research on employing the cellular automata methodology to corrosion and passivation phenomena is reviewed. Examples of a peculiar pit development are found and presented. The diffusion rate in the corroding medium is argued and shown in the simulation results to affect mainly the characteristic length scale for the corrosion process. New data for the pitting corrosion development on a planar interface are presented and discussed.

COMPUTATIONAL FLUID DYNAMICS FOR TISSUE ENGINEERING APPLICATIONS

Hydrodynamic cellular environment plays an important role in translating engineered tissue constructs into clinically useful grafts. However, the cellular fluid dynamic environment inside bioreactor systems is highly complex and it is normally impractical to experimentally characterize the local flow patterns at the cellular scale. Computational fluid dynamics (CFD) has been recognized as an invaluable and reliable alternative to investigate the complex relationship between hydrodynamic environments and the regeneration of engineered tissues at both the macroscopic and microscopic scales. This review describes the applications of CFD simulations to probe the hydrodynamic environment parameters (e.g., flow rate, shear stress, etc.) and the corresponding experimental validations. We highlight the use of CFD to optimize bioreactor design and scaffold architectures for improved ex-vivo hydrodynamic environments. It is envisioned that CFD could be used to customize specific hydrodynamic cellular environments to meet the unique requirements of different cell types in combination with advanced manufacturing techniques and finally facilitate the maturation of tissue-engineered constructs.

Quantification and regulation of cell migration

Cell migration is essential for many physiological and pathological processes that include embryonic development, the immune response, wound healing, angiogenesis, and cancer metastasis. It is also important for emerging tissue engineering applications such as tissue reconstitution and the colonization of biomedical implants. By summarizing results from recent experimental and theoretical studies, this review outlines the role played by growth factors or substrate-adhesion molecules in modulating cell motility and shows that cell motility can be an important factor in determining the rates of tissue formation. The application of cell motility assays and the use of theoretical models for analyzing cell migration and proliferation are also discussed.

Expansion of adipose mesenchymal stromal cells is affected by human platelet lysate and plating density

The composition of mesenchymal stromal cells (MSCs) changes in the course of in vitro culture expansion. Little is known how these cell preparations are influenced by culture media, plating density, or passaging. In this study, we have isolated MSCs from human adipose tissue in culture medium supplemented with either fetal calf serum (FCS) or human platelet lysate (HPL). In addition, culture expansion was simultaneously performed at plating densities of 10 or 10,000 cells/cm(2). The use of FCS resulted in larger cells, whereas HPL significantly enhanced proliferation. Notably, HPL also facilitated expansion for more population doublings than FCS (43 ± 3 vs. 22 ± 4 population doubling; p < 0.001), while plating density did not have a significant effect on long-term growth curves. To gain further insight into population dynamics, we conceived a cellular automaton model to simulate expansion of MSCS. It is based on the assumptions that the number of cell divisions is limited and that due to contact inhibition proliferation occurs only at the rim of colonies. The model predicts that low plating densities result in more heterogeneity with regard to cell division history, and favor subpopulations of higher migratory activity. In summary, HPL is a suitable serum supplement for isolation of MSC from adipose tissue and facilitates more population doublings than FCS. Cellular automaton computer simulations provided additional insights into how complex population dynamics during long-term expansion are affected by plating density and migration.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Computational modeling of epithelial tissues

There is an extensive literature on the computational modeling of epithelial tissues at all levels from subcellular to whole tissue. This review concentrates on behavior at the individual cell to whole tissue level, and particularly on organizational aspects, and provides an indication of where information from other areas, such as the modeling of angiogenesis, is relevant. The skin, and the lining of all of the body cavities (lung, gut, cervix, bladder etc) are epithelial tissues, which in a topological sense are the boundary between inside and outside the body. They are thin sheets of cells (usually of the order of 0.5 mm thick) without extracellular matrix, have a relatively simple structure, and contain few types of cells. They have important barrier, secretory and transport functions, which are essential for the maintenance of life, so homeostasis and wound healing are important aspects of the behavior of epithelial tissues. Carcinomas originate in epithelial tissues.There are essentially two approaches to modeling tissues--to start at the level of the tissue (i.e., a length scale of the order of 1 mm) and develop generalized equations for behavior (a continuum approach); or to start at the level of the cell (i.e., a length scale of the order of 10 µm) and develop tissue behavior as an emergent property of cellular behavior (an individual-based approach). As will be seen, these are not mutually exclusive approaches, and they come in a variety of flavors.

An automaton model for the cell cycle

WE CONSIDER AN AUTOMATON MODEL THAT PROGRESSES SPONTANEOUSLY THROUGH THE FOUR SUCCESSIVE PHASES OF THE CELL CYCLE: G1, S (DNA replication), G2 and M (mitosis). Each phase is characterized by a mean duration τ and a variability V. As soon as the prescribed duration of a given phase has passed, the transition to the next phase of the cell cycle occurs. The time at which the transition takes place varies in a random manner according to a distribution of durations of the cell cycle phases. Upon completion of the M phase, the cell divides into two cells, which immediately enter a new cycle in G1. The duration of each phase is reinitialized for the two newborn cells. At each time step in any phase of the cycle, the cell has a certain probability to be marked for exiting the cycle and dying at the nearest G1/S or G2/M transition. To allow for homeostasis, which corresponds to maintenance of the total cell number, we assume that cell death counterbalances cell replication at mitosis. In studying the dynamics of this automaton model, we examine the effect of factors such as the mean durations of the cell cycle phases and their variability, the type of distribution of the durations, the number of cells, the regulation of the cell population size and the independence of steady-state proportions of cells in each phase with respect to initial conditions. We apply the stochastic automaton model for the cell cycle to the progressive desynchronization of cell populations and to their entrainment by the circadian clock. A simple deterministic model leads to the same steady-state proportions of cells in the four phases of the cell cycle.

Correcting mean-field approximations for birth-death-movement processes

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair of an organism; on the macroscale, such effects are important in the sustainability of a population in its environment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may arise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative impact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological descriptions, such as the logistic model, are used to model population growth. In this work we outline some methods for analyzing exclusion processes which include agent proliferation, death and motility in two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description for these types of processes is of logistic form; we show that, under certain parameter conditions, such systems may display large deviations from the mean field, and suggest computationally tractable methods to correct the logistic-type description.

Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs

Mathematic models help interpret experimental results and accelerate tissue engineering developments. We develop in this paper a hybrid cellular automata model that combines the differential nutrient transport equation to investigate the nutrient limited cell construct development for cartilage tissue engineering. Individual cell behaviors of migration, contact inhibition and cell collision, coupled with the cell proliferation regulated by oxygen concentration were carefully studied. Simplified two-dimensional simulations were performed. Using this model, we investigated the influence of cell migration speed on the overall cell growth within in vitro cell scaffolds. It was found that intense cell motility can enhance initial cell growth rates. However, since cell growth is also significantly modulated by the nutrient contents, intense cell motility with conventional uniform cell seeding method may lead to declined cell growth in the final time because concentrated cell population has been growing around the scaffold periphery to block the nutrient transport from outside culture media. Therefore, homogeneous cell seeding may not be a good way of gaining large and uniform cell densities for the final results. We then compared cell growth in scaffolds with various seeding modes, and proposed a seeding mode with cells initially residing in the middle area of the scaffold that may efficiently reduce the nutrient blockage and result in a better cell amount and uniform cell distribution for tissue engineering construct developments.

A 3D hybrid model for tissue growth: the interplay between cell population and mass transport dynamics

To provide theoretical guidance for the design and in vitro cultivation of bioartificial tissues, we have developed a multiscale computational model that can describe the complex interplay between cell population and mass transport dynamics that governs the growth of tissues in three-dimensional scaffolds. The model has three components: a transient partial differential equation for the simultaneous diffusion and consumption of a limiting nutrient; a cellular automaton describing cell migration, proliferation, and collision; and equations that quantify how the varying nutrient concentration modulates cell division and migration. The hybrid discrete-continuous model was parallelized and solved on a distributed-memory multicomputer to study how transport limitations affect tissue regeneration rates under conditions encountered in typical bioreactors. Simulation results show that the severity of transport limitations can be estimated by the magnitude of two dimensionless groups: the Thiele modulus and the Biot number. Key parameters including the initial seeding mode, cell migration speed, and the hydrodynamic conditions in the bioreactor are shown to affect not only the overall rate, but also the pattern of tissue growth. This study lays the groundwork for more comprehensive models that can handle mixed cell cultures, multiple nutrients and growth factors, and other cellular processes, such as cell death.

Potential and bottlenecks of bioreactors in 3D cell culture and tissue manufacturing

Over the last decade, we have witnessed an increased recognition of the importance of 3D culture models to study various aspects of cell physiology and pathology, as well as to engineer implantable tissues. As compared to well-established 2D cell-culture systems, cell/tissue culture within 3D porous biomaterials has introduced new scientific and technical challenges associated with complex transport phenomena, physical forces, and cell-microenvironment interactions. While bioreactor-based 3D model systems have begun to play a crucial role in addressing fundamental scientific questions, numerous hurdles currently impede the most efficient utilization of these systems. We describe how computational modeling and innovative sensor technologies, in conjunction with well-defined and controlled bioreactor-based 3D culture systems, will be key to gain further insight into cell behavior and the complexity of tissue development. These model systems will lay a solid foundation to further develop, optimize, and effectively streamline the essential bioprocesses to safely and reproducibly produce appropriately scaled tissue grafts for clinical studies.

Computational fluid dynamics for improved bioreactor design and 3D culture

The complex relationship between the hydrodynamic environment and surrounding tissues directly impacts on the design and production of clinically useful grafts and implants. Tissue engineers have generally seen bioreactors as 'black boxes' within which tissue engineering constructs (TECs) are cultured. It is accepted that a more detailed description of fluid mechanics and nutrient transport within process equipment can be achieved by using computational fluid dynamics (CFD) technology. This review discusses applications of CFD for tissue engineering-related bioreactors -- fluid flow processes have direct implications on cellular responses such as attachment, migration and proliferation. We conclude that CFD should be seen as an invaluable tool for analyzing and visualizing the impact of fluidic forces and stresses on cells and TECs.

Optimal design of metabolic flux analysis experiments for anchorage-dependent mammalian cells using a cellular automaton model

Metabolic flux analysis (MFA) is widely used to quantify metabolic pathway activity. Typical applications involve isotopically labeled substrates, which require both metabolic and isotopic steady states for simplified data analysis. For bacterial systems, these steady states are readily achieved in chemostat cultures. However, mammalian cells are often anchorage dependent and experiments are typically conducted in batch or fed-batch systems, such as tissue culture dishes or microcarrier-containing bioreactors. Surface adherence may cause deviations from exponential growth, resulting in metabolically heterogeneous populations and a varying number of cellular "nearest neighbors" that may affect the observed metabolism. Here, we discuss different growth models suitable for deconvoluting these effects and their application to the design and optimization of MFA experiments employing surface-adherent mammalian cells. We describe a stochastic two-dimensional (2D) cellular automaton model, with empirical descriptions of cell number and non-growing cell fraction, suitable for easy application to most anchorage-dependent mammalian cell cultures. Model utility was verified by studying the impact of contact inhibition on the growth rate, specific extracellular flux rates, and isotopic labeling in lactate for MCF7 cells, a commonly studied breast cancer cell line. The model successfully defined the time over which exponential growth and a metabolically homogeneous growing cell population could be assumed. The cellular automaton model developed is shown to be a useful tool in designing optimal MFA experiments.

Computational modeling of combined cell population dynamics and oxygen transport in engineered tissue subject to interstitial perfusion

This work presents a computational model of tissue growth under interstitial perfusion inside a tissue engineering bioreactor. The model accounts both for the cell population dynamics, using a model based on cellular automata, and for the hydrodynamic microenvironment imposed by the bioreactor, using a model based on the Lattice-Boltzmann equation and the convection-diffusion equation. The conditions of static culture versus perfused culture were compared, by including the population dynamics along with oxygen diffusion, convective transport and consumption. The model is able to deal with arbitrary complex geometries of the spatial domain; in the present work, the domain modeled was the void space of a porous scaffold for tissue-engineered cartilage. The cell population dynamics algorithm provided results which qualitatively resembled population dynamics patterns observed in experimental studies, and these results were in good quantitative agreement with previous computational studies. Simulation of oxygen transport and consumption showed the fundamental contribution of convective transport in maintaining a high level of oxygen concentration in the whole spatial domain of the scaffold. The model was designed with the aim to be computationally efficient and easily expandable, i.e. to allow straightforward implementability of further models of complex biological phenomena of increasing scientific interest in tissue engineering, such as chemotaxis, extracellular matrix deposition and effect of mechanical stimulation.

Multi-scale modeling of a wound-healing cell migration assay

A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhibit contact inhibition of migration and contact-limited proliferation. The continuum model includes the two dominant mechanisms and characteristics of cell migration and proliferation, using a cell diffusivity function that decreases with cell density and a logistic proliferative growth term. The discrete model arises naturally from the continuum model. Individual cells are simulated as continuous-time random walkers with nearest-neighbor transitions, together with a birth/death process. The migration and proliferation parameters are determined by analysing individual mice 3T3 fibroblast cell trajectories obtained during the development of a confluent cell monolayer and in a wound healing assay. The population-scale model successfully predicts the shape and speed of the traveling wave, while the discrete model is also successful in capturing the contact inhibition of migration effects.

Cell population dynamics modulate the rates of tissue growth processes

The development and testing of a discrete model describing the dynamic process of tissue growth in three-dimensional scaffolds is presented. The model considers populations of cells that execute persistent random walks on the computational grid, collide, and proliferate until they reach confluence. To isolate the effect of population dynamics on tissue growth, the model assumes that nutrient and growth factor concentrations remain constant in space and time. Simulations start either by distributing the seed cells uniformly and randomly throughout the scaffold, or from an initial condition designed to simulate the migration and cell proliferation phase of wound healing. Simulations with uniform seeding show that cell migration enhances tissue growth by counterbalancing the adverse effects of contact inhibition. This beneficial effect, however, diminishes and disappears completely for large migration speeds. By contrast, simulations with the "wound" seeding mode show a continual enhancement of tissue regeneration rates with increasing cell migration speeds. We conclude that cell locomotory parameters and the spatial distribution of seed cells can have profound effects on the dynamics of the process and, consequently, on the pattern and rates of tissue growth. These results can guide the design of experiments for testing the effectiveness of biomimetic modifications for stimulating tissue growth.

Development and validation of computational models of cellular interaction

In this paper we take the view that computational models of biological systems should satisfy two conditions - they should be able to predict function at a systems biology level, and robust techniques of validation against biological models must be available. A modelling paradigm for developing a predictive computational model of cellular interaction is described, and methods of providing robust validation against biological models are explored, followed by a consideration of software issues.

Artificial life simulation of self-assembly in bacteriophage by movable finite automata

This paper presents a model which is based on biological research using the movable finite automata (MFA) on a self-assembly of T4 phage, and exhibits the results of artificial life simulation. In the previous work, Thompson and Goel [Artificial Life, Addison Weley, 1989, pp. 317-340; Biosystems 18 (1985) 23; J. Theor. Biol. 131 (1988) 351] presented the movable finite automata (MFA) which has a capability of moving on finite automata, and simulated on a computer. They were represented individual rectangular boxes, however, the results of simulation was different from real T4 phage. We propose the sphere model as a protein structure, and simulate the self-assembly of the entire structure of the T4 phage on a computer.

Attachment, proliferation, and migration of marrow stromal osteoblasts cultured on biomimetic hydrogels modified with an osteopontin-derived peptide

We prepared oligo(poly(ethylene glycol) fumarate) (OPF) hydrogels modified with a rat osteopontin-derived peptide (ODP), Asp-Val-Asp-Val-Pro-Asp-Gly-Arg-Gly-Asp-Ser-Leu-Ala-Try-Gly (DVDVPDGRGDSLAYG), as well as Gly-Arg-Gly-Asp-Ser (GRGDS) and investigated the modulation of marrow stromal osteoblast function on the peptide-modified hydrogels. Osteoblast attachment was competitively inhibited by a soluble peptide suggesting that the interaction of osteoblasts with the hydrogel was ligand specific. The proliferation index of osteoblasts relative to the initial seeding density was similar on the hydrogels modified with ODP (1.18+/-0.13) and GRGDS (1.27+/-0.12). However, fibroblasts proliferated faster on GRGDS-modified hydrogels than on ODP-modified hydrogels as evidenced by the proliferation indices of 4.89+/-0.03 and 2.42+/-0.16, respectively. A megacolony migration assay conducted for 3 days with a seeding density of 53,000 cells/cm(2) showed that osteoblasts migrated to a longer distance on ODP-modified hydrogels (0.23+/-0.06 mm/day) than on hydrogels modified with GRGDS (0.15+/-0.02 mm/day). In addition, osteoblasts migrated faster than fibroblasts seeded at the same density on ODP-modified hydrogels (0.15+/-0.11 mm/day). The migration of osteoblasts on the peptide-modified hydrogels was dependent on the peptide concentration of the hydrogels resulting in an increased migration distance with increasing the peptide concentration for the concentrations tested. These results show that OPF-based biomimetic hydrogels hold promise for modulating cell proliferation and migration for specific applications by altering the specific ligand and its concentration in the hydrogels.

Temporal variations in cell migration and traction during fibroblast-mediated gel compaction

Current models used in our laboratory to assess the migration and traction of a population of cells within biopolymer gels are extended to investigate temporal changes in these parameters during compaction of mechanically constrained gels. The random cell migration coefficient, micro (t) is calculated using a windowing technique by regressing the mean-squared displacement of cells tracked at high magnification in three dimensions with a generalized least squares algorithm for a subset of experimental time intervals, and then shifting the window interval-by-interval until all time points are analyzed. The cell traction parameter, tau(0)(t), is determined by optimizing the solution of our anisotropic biphasic theory to tissue equivalent compaction. The windowing technique captured simulated sinusoidal and step changes in cell migration superposed on a persistent random walk in simulated cell movement. The optimization software captured simulated time dependence of compaction on cell spreading. Employment of these techniques on experimental data using rat dermal fibroblasts (RDFs) and human foreskin fibroblasts (HFFs) demonstrated that these cells exhibit different migration-traction relationships. Rat dermal fibroblast migration was negatively correlated to traction, suggesting migration was not the driving force for compaction with these cells, whereas human foreskin fibroblast migration was positively correlated to traction.

Adhesion and migration of marrow-derived osteoblasts on injectable in situ crosslinkable poly(propylene fumarate-co-ethylene glycol)-based hydrogels with a covalently linked RGDS peptide

Marrow-derived osteoblasts were cultured on poly(propylene fumarate-co-ethylene glycol) (P(PF-co-EG)) based hydrogels modified in bulk with a covalently linked RGDS model peptide. A poly(ethylene glycol) spacer arm was utilized to covalently link the peptide to the hydrogel. Three P(PF-co-EG) block copolymers were synthesized with varying poly(ethylene glycol) block lengths relative to poly(ethylene glycol) spacer arm. A poly(ethylene glycol) block length of nominal molecular weight 2000 and spacer arm of nominal molecular weight 3400 were found to reduce nonspecific cell adhesion and show RGDS concentration dependent marrow-derived osteoblast adhesion. A concentration of 100 nmol/mL RGDS was sufficient to promote adhesion of 84 +/- 17% of the initial seeded marrow-derived osteoblasts compared with 9 +/- 1% for the unmodified hydrogel after 12 h. Cell spreading was quantified as a method for evaluating adhesivity of cells to the hydrogel. A megacolony migration assay was utilized to assess the migration characteristics of the marrow-derived osteoblasts on RGDS modified hydrogels. Marrow-stromal osteoblasts migration was greater on hydrogels modified with 100 nmol/mL linked RGDS when compared with hydrogels modified with 1000 nmol/mL linked RGDS, while proliferation was not affected. These P(PF-co-EG) hydrogels modified in the bulk with RGDS peptide are potential candidates as in situ forming scaffolds for bone tissue engineering applications.

Spreading, proliferation, and differentiation of the epidermis after wounding a cichlid fish, Hemichromis bimaculatus

A large superficial wound has been experimentally provoked in the cichlid fish Hemichromis bimaculatus to study the interactions between the epidermal cells and the substrate on which they spread, on the one hand, and the restoration of the subepidermal tissues and the epithelial-mesenchymal interactions preceding scale regeneration, on the other hand. The re-epithelialization process, e.g., migration, spreading, differentiation, and proliferation of the epidermal cells, has been followed step by step, using light, scanning and transmission electron microscopy, and tritiated thymidine incorporation, until complete reorganization of the healing epidermis. Wound healing is fast (500 microm/hr) and proceeds centripetally from the wound margins. The epidermal cells spread on a wound surface which is composed of two different matrices: the remains of basement membrane materials covering the scale-pockets, and collagen fibrils of cut dermal strips. Even though both matrices favour cell spreading and attachment, migrating cells show a different behaviour. The re-epithelialization of the wound follows an orderly sequence similar to amphibian and mammalian wound healing, i.e., a "leap frog" mechanism of cell locomotion involving three epidermal layers. The basal layer cells, which spread on the substrate, and the superficial layer cells which protect the epidermis, differentiate first. Whatever the type of substrate over which the epithelium spreads (basement membrane material or collagen fibrils), the epidermal basal layer cells differentiate as soon as they become attached. The incorporation of tritiated thymidine has revealed that there is no proliferation in the healing epidermis until after complete closure of the wound, but that the rapid re-epithelialization of the large surface requires the recruitment of epidermal cells at the wound margins. The present study offers new data on the dynamics of re-epithelialisation and on the resistance of cichlid skin to such wounds. It is also clearly shown that the epidermal basal layer cells differentiate rapidly, a step which is interpreted as the first stage of epithelial-mesenchymal interactions that will lead to scale regeneration.