A 3-D model of tumor progression based on complex automata driven by particle dynamics

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.     

Morphogenetic Processes: Application to Cambial Growth Dynamics

Both the physiological and the pathological morphogenetic processes that we can meet in embryogenesis, neogenesis and degenerative dysgenesis present common features: they are ruled by three different kinds of mechanisms, one related to cell migration, the second to cell differentiation and the third to cell proliferation. We deal here with an application to the cambial growth which essentially involves the third type of mechanism. Woody plants produce secondary tissue (secondary xylem and phloem) from a meristematic tissue called vascular cambium, responsible for the radial growth of a tree. This paper focuses on the formation of secondary xylem, considered in two dimensions in a cross-section framework. A new discrete modelling approach is used, based on the cellular scale, in order to attain a more accurate understanding of how the elementary microscopic behaviour of each cell takes part in the macroscopic morphogenesis. The mathematical model essentially uses an occurrence method simulating the main features of radial growth with simple geometric rules, such as Thom's division rule (Thom,1972)to account for the cell proliferation. The study applies to concrete instances in which the changes made in the geometrical cellular patterns of the vascular cambium clearly affect the shape of the tree, as in Pinus radiata (D. Don.).     

Implementing vertex dynamics models of cell populations in biology within a consistent computational framework

The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cell–cell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable.     

Cellular Automata Modeling of Stem‐Cell‐Driven Development of Tissue in the Nervous System

Mathematical and computational modeling enables biologists to integrate data from observations and experiments into a theoretical framework. In this review, we describe how developmental processes associated with stem‐cell‐driven growth of tissue in both the embryonic and adult nervous system can be modeled using cellular automata (CA). A cellular automaton is defined by its discrete nature in time, space, and state. The discrete space is represented by a uniform grid or lattice containing agents that interact with other agents within their local neighborhood. This possibility of local interactions of agents makes the cellular automata approach particularly well suited for studying through modeling how complex patterns at the tissue level emerge from fundamental developmental processes (such as proliferation, migration, differentiation, and death) at the single‐cell level. As part of this review, we provide a primer for how to define biologically inspired rules governing these processes so that they can be implemented into a CA model. We then demonstrate the power of the CA approach by presenting simulations (in the form of figures and movies) based on building models of three developmental systems: the formation of the enteric nervous system through invasion by neural crest cells; the growth of normal and tumorous neurospheres induced by proliferation of adult neural stem/progenitor cells; and the neural fate specification through lateral inhibition of embryonic stem cells in the neurogenic region of Drosophila.     

Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids

Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1–100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.     

Quantitative Modeling of Tumor Dynamics and Radiotherapy

Cancer is a complex disease, necessitating research on many different levels; at the subcellular level to identify genes, proteins and signaling pathways associated with the disease; at the cellular level to identify, for example, cell-cell adhesion and communication mechanisms; at the tissue level to investigate disruption of homeostasis and interaction with the tissue of origin or settlement of metastasis; and finally at the systems level to explore its global impact, e.g. through the mechanism of cachexia. Mathematical models have been proposed to identify key mechanisms that underlie dynamics and events at every scale of interest, and increasing effort is now being paid to multi-scale models that bridge the different scales. With more biological data becoming available and with increased interdisciplinary efforts, theoretical models are rendering suitable tools to predict the origin and course of the disease. The ultimate aims of cancer models, however, are to enlighten our concept of the carcinogenesis process and to assist in the designing of treatment protocols that can reduce mortality and improve patient quality of life. Conventional treatment of cancer is surgery combined with radiotherapy or chemotherapy for localized tumors or systemic treatment of advanced cancers, respectively. Although radiation is widely used as treatment, most scheduling is based on empirical knowledge and less on the predictions of sophisticated growth dynamical models of treatment response. Part of the failure to translate modeling research to the clinic may stem from language barriers, exacerbated by often esoteric model renderings with inaccessible parameterization. Here we discuss some ideas for combining tractable dynamical tumor growth models with radiation response models using biologically accessible parameters to provide a more intuitive and exploitable framework for understanding the complexity of radiotherapy treatment and failure.     

A Sub-Cellular Viscoelastic Model for Cell Population Mechanics

Understanding the biomechanical properties and the effect of biomechanical force on epithelial cells is key to understanding how epithelial cells form uniquely shaped structures in two or three-dimensional space. Nevertheless, with the limitations and challenges posed by biological experiments at this scale, it becomes advantageous to use mathematical and ‘in silico’ (computational) models as an alternate solution. This paper introduces a single-cell-based model representing the cross section of a typical tissue. Each cell in this model is an individual unit containing several sub-cellular elements, such as the elastic plasma membrane, enclosed viscoelastic elements that play the role of cytoskeleton, and the viscoelastic elements of the cell nucleus. The cell membrane is divided into segments where each segment (or point) incorporates the cell''s interaction and communication with other cells and its environment. The model is capable of simulating how cells cooperate and contribute to the overall structure and function of a particular tissue; it mimics many aspects of cellular behavior such as cell growth, division, apoptosis and polarization. The model allows for investigation of the biomechanical properties of cells, cell-cell interactions, effect of environment on cellular clusters, and how individual cells work together and contribute to the structure and function of a particular tissue. To evaluate the current approach in modeling different topologies of growing tissues in distinct biochemical conditions of the surrounding media, we model several key cellular phenomena, namely monolayer cell culture, effects of adhesion intensity, growth of epithelial cell through interaction with extra-cellular matrix (ECM), effects of a gap in the ECM, tensegrity and tissue morphogenesis and formation of hollow epithelial acini. The proposed computational model enables one to isolate the effects of biomechanical properties of individual cells and the communication between cells and their microenvironment while simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue.     

Modelling evolutionary cell behaviour using neural networks: Application to tumour growth

In this paper, we present a modelling framework for cellular evolution that is based on the notion that a cell’s behaviour is driven by interactions with other cells and its immediate environment. We equip each cell with a phenotype that determines its behaviour and implement a decision mechanism to allow evolution of this phenotype. This decision mechanism is modelled using feed-forward neural networks, which have been suggested as suitable models of cell signalling pathways. The environmental variables are presented as inputs to the network and result in a response that corresponds to the phenotype of the cell. The response of the network is determined by the network parameters, which are subject to mutations when the cells divide. This approach is versatile as there are no restrictions on what the input or output nodes represent, they can be chosen to represent any environmental variables and behaviours that are of importance to the cell population under consideration. This framework was implemented in an individual-based model of solid tumour growth in order to investigate the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. Our results show that the oxygen concentration affects the tumour at the morphological level, but more importantly has a direct impact on the evolutionary dynamics. When the supply of oxygen is limited we observe a faster divergence away from the initial genotype, a higher population diversity and faster evolution towards aggressive phenotypes. The implementation of this framework suggests that this approach is well suited for modelling systems where evolution plays an important role and where a changing environment exerts selection pressure on the evolving population.     

Murine tumor suppressor models

Tumor suppressor genes have been shown to be necessary for proper maintenance of cell growth control. Inactivation of these genes in the germline of humans is linked to inherited cancer predisposition. Moreover, sporadically arising human tumors often have somatic mutations in tumor suppressor genes. During the past few years, advances in molecular and cellular biology have led to the creation of animal models that have germline mutations of various tumor suppressor genes. Such mice potentially represent important animal models for familial cancer predisposition syndromes, and the study of the tumorigenesis process has been greatly assisted by their development. Such models have also demonstrated the importance of tumor suppressor function in embryonic development. In this review, we describe mice with inactivated germline tumor suppressor genes that are genetically analogous to 10 different inherited cancer syndromes in humans. We describe the variable usefulness of the mutant mice as models for human disease.     

A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic, and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles and follow the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell size and shape that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with a constant-thickness rim of growing cells at the cluster edge, as well as the emergence of short-range ordering and solid-like behavior. A detailed analysis of the model reveals a scale invariance of the growth and provides insight into the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: it is independent of dimension, it can be parameterized using classical elastic properties (Poisson’s ratio and Young’s modulus), and it can easily be extended to incorporate multiple cell types and general substrate geometries.     

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.     

A life-like virtual cell membrane using discrete automata

A framework is presented that captures the discrete and probabilistic nature of molecular transport and reaction kinetics found in a living cell as well as formally representing the spatial distribution of these phenomena. This particle or agent-based approach is computationally robust and complements established methods. Namely it provides a higher level of spatial resolution than formulations based on ordinary differential equations (ODE) while offering significant advantages in computational efficiency over molecular dynamics (MD). Using this framework, a model cell membrane has been constructed with discrete particle agents that respond to local component interactions that resemble flocking or herding behavioural cues in animals. Results from simulation experiments are presented where this model cell exhibits many of the characteristic behaviours associated with its biological counterpart such as lateral diffusion, response to osmotic pressure gradients, membrane growth and cell division. Lateral diffusion rates and estimates for the membrane modulus of elasticity derived from these simple experiments fall well within a biologically relevant range of values. More importantly, these estimates were obtained by applying a simple qualitative tuning of the model membrane. Membrane growth was simulated by injecting precursor molecules into the proto-cell at different rates and produced a variety of morphologies ranging from a single large cell to a cluster of cells. The computational scalability of this methodology has been tested and results from benchmarking experiments indicate that real-time simulation of a complete bacterial cell will be possible within 10 years.     

Cellular mechanobiology and cancer metastasis

The primary cause of cancer treatment failure is invasion and metastasis, and invading tumor cells utilize many of the motility patterns that have been documented for normal morphogenesis. Recently, the role of mechanical forces in guiding various tissue and cell movements in embryonic development has been systematically analyzed with new experimental and computational methods. The tissue and cellular mechanobiology approach also holds promise for increasing the understanding of tumor invasion. In fact, the mechanical stiffness of tumors has correlated with invasiveness, and manipulation of extracellular matrix (ECM) stiffness in vitro has suppressed the cancer phenotype. Several important signaling molecules reside on the cytoskeleton, which is affected by external stress imparted by the ECM, and deformation of the nucleus can trigger the activation of certain genes. All these observations suggest that a synthesis of the biology of cancer cell invasion and cellular mechanobiology may offer new targets for the treatment of malignant disease. Accordingly, sensitive and relevant in vivo models and methods to study cancer mechanobiology are needed.     

Epidermal Patterning Genes Impose Non-cell Autonomous Cell Size Determination and have Additional Roles in Root Meristem Size Control

The regulation of cellular growth is of vital importance for embryonic and postembryonic patterning. Growth regulation in the epidermis has importance for organ growth rates in roots and shoots, proposing epidermal cells as an interesting model for cellular growth regulation. Here we assessed whether the root epidermis is a suitable model system to address cell size determination. In Arabidopsis thaliana L., root epidermal cells are regularly spaced in neighbouring tricho- (root hair) and atrichoblast (non-hair) cells, showing already distinct cell size regulation in the root meristem. We determined cell sizes in the root meristem and at the onset of cellular elongation, revealing that not only division rates but also cellular shape is distinct in tricho- and atrichoblasts. Intriguingly, epidermal-patterning mutants, failing to define differential vacuolization in neighbouring epidermal cell files, also display non-differential growth. Using these epidermal-patterning mutants, we show that polarized growth behaviour of epidermal tricho- and atrichoblast is interdependent, suggesting non-cell autonomous signals to integrate tissue expansion. Besides the interweaved cell-type-dependent growth mechanism, we reveal an additional role for epidermal patterning genes in root meristem size and organ growth regulation. We conclude that epidermal cells represent a suitable model system to study cell size determination and interdependent tissue growth.     

Information-Theoretic Analysis of the Dynamics of an Executable Biological Model

To facilitate analysis and understanding of biological systems, large-scale data are often integrated into models using a variety of mathematical and computational approaches. Such models describe the dynamics of the biological system and can be used to study the changes in the state of the system over time. For many model classes, such as discrete or continuous dynamical systems, there exist appropriate frameworks and tools for analyzing system dynamics. However, the heterogeneous information that encodes and bridges molecular and cellular dynamics, inherent to fine-grained molecular simulation models, presents significant challenges to the study of system dynamics. In this paper, we present an algorithmic information theory based approach for the analysis and interpretation of the dynamics of such executable models of biological systems. We apply a normalized compression distance (NCD) analysis to the state representations of a model that simulates the immune decision making and immune cell behavior. We show that this analysis successfully captures the essential information in the dynamics of the system, which results from a variety of events including proliferation, differentiation, or perturbations such as gene knock-outs. We demonstrate that this approach can be used for the analysis of executable models, regardless of the modeling framework, and for making experimentally quantifiable predictions.     

The molecular biology of platelet-derived growth factor

PDGF is a connective tissue mitogen that has been associated with clotted blood serum for at least 300 million years. It regulates the expression of cell cycle "early genes" in normal fibroblasts. Induction of early genes is preceded by stimulation of a tyrosine-specific kinase. The putative structural gene for PDGF has been acquired by an acutely transforming retrovirus and is expressed in many connective tissue tumors. Further work is needed to determine whether (i) production of PDGF by tumor cells confers a proliferative advantage on these cells, (ii) tyrosine-specific phosphorylations mediate the induction of cell cycle early genes by PDGF, and (iii) products of cell cycle early genes play any functional role in the 10-12 hr chain of events that culminates in replicative DNA synthesis and cell division. In the meantime, these very issues represent candidate functions for other viral oncogenes and their cellular homologs. Some of these genes could act at the onset of the mitogenic cascade by causing the production of automitogenic growth factors. Others may function in the interior of the cascade by promoting tyrosine-specific phosphorylations. Still others may be mutated or rearranged homologs of cell cycle early genes whose expression is normally modulated by extracellular growth factors.     

Towards computational models of cells for environmental toxicology

This paper outlines an approach to the development of computational models of cells for marine environmental toxicology. Exposure of cells to pollutants can lead to lysosomal damage and dysfunction, augmented autophagy, cellular dysfunction and atrophy and ultimately tissue pathology and organ damage. The application of carbon and nitrogen based models of intra cellular vesicular traffic for simulating the autophagic and lysosomal response of the hepatopancreatic digestive cells of marine molluscs is described. Two numerical models of the vesicular transport of carbon and nitrogen in the cell are presented. These demonstrate the importance of endocytotic uptake as a driver of lysosomal dynamics and the need to recognize and model it as a discrete process. Conceptual and mathematical models of the toxic impact of polycyclic aromatic hydrocarbons on the digestive gland are presented. The role of experimental research and the need to integrate it with modelling is highlighted.