Discrete element framework for modelling extracellular matrix, deformable cells and subcellular components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an 'agent', meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory

On the foundations of cancer modelling: selected topics, speculations, & perspectives

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution

Modeling and simulation of multi-cellular systems using hybrid FEM/Agent-based approaches

Muchas de las propiedades biomecánicas de los organismos multicelulares surgen directamente de las interacciones entre células. Las células de los órganos y tejidos interactúan entre sí y con su entorno de diferentes formas. Debido a este hecho, es fundamental analizar cómo estas interacciones se traducen como propiedades mecánicas a nivel del tejido. Por ejemplo, las adhesiones entre células determinan la rigidez aparente de una capa epitelial. Las interacciones célula-matriz pueden además determinar la formación de muchas estructuras biológicas y su morfología. Estos sistemas multicelulares no se pueden considerar como estructuras estáticas ya que sufren constantes cambios causados por la proliferación, la reorganización o la migración celular. Por lo tanto, es necesario estudiar la dinámica de la célula y las interacciones individuales para comprender plenamente cómo funcionan los fenómenos a escalas superiores, desde el desarrollo de tejidos hasta el crecimiento de tumores.Recientemente, el uso de enfoques basados en agentes se ha vuelto muy popular para modelar sistemas multicelulares. Los modelos basados en agentes representan células como entidades individuales. Estos modelos son especialmente adecuados para estudiar fenómenos biofísicos que ocurren a nivel celular. Aquí las interacciones célula-célula se pueden simular directamente de forma mecanicista. Además, estos modelos capturan realmente bien las heterogeneidades presentes en las estructuras biológicas. Por otra parte, los modelos continuos se utilizan comúnmente en problemas de escalas mayores. A diferencia de los modelos basados en agentes, en estos no representan células como entidades individuales, sino que se definen leyes constitutivas para modelar procesos biológicos, físicos y químicos. Por lo tanto, las propiedades celulares se promedian usando parámetros macroscópicos, y estos modelos a menudo trabajan con la densidad celular en lugar de entidades celulares separadas. En cualquier caso, los modelos continuos presentan una buena escalabilidad y una excelente representación de fenómenos físicos particulares como el transporte masivo y las transmisiones de fuerza en medios continuos.En esta tesis, se exploran las posibilidades que los enfoques híbridos pueden ofrecer para desarrollar nuevos modelos de sistemas multicelulares. Se presentan dos modelos híbridos diferentes que combinan un modelo basado en agentes y un modelo continuo. Ambos enfoques tienen en común que el modelo continuo se resuelve utilizando el método de los elementos finitos. También se muestra, siguiendo este patrón de diseño, cómo resolver varias de las limitaciones intrínsecas de cada tipo individual de modelo.En primer lugar, se presenta un modelo híbrido para simular la mecánica epitelial monocapa. Este modelo se centra en el modelado de las interacciones mecánicas célula-célula y célula-sustrato, pero también en la topología y morfología de los tejidos. Con este enfoque se reproducen tejidos epiteliales proliferativos, movimientos celular colectivo y procesos de migración. El segundo modelo presentado en esta tesis se ha diseñado para simular agregados celulares en entornos tridimensionales. Se estudian las interacciones mecánicas entre células, pero este modelo se centra especialmente en analizar cómo afecta el transporte de oxígeno a las células en un proceso de agrupamiento en 3D.Finalmente, se comparan los resultados de ambos modelos con datos experimentales de otros autores y se discuten los beneficios de combinar diferentes tipos de modelos. Se demuestra que los enfoques híbridos que se proponen en este trabajo son capaces de simular una amplia variedad de sistemas multicelulares. De hecho, son particularmente útiles para estudiar cómo algunos fenómenos emergen de las interacciones celulares individuales a escalas biológicas más grandes.<br /

Simulating tissue mechanics with Agent Based Models: concepts and perspectives

International audienceIn this paper we present an overview of agent based models that are used to simulate mechanical and physiological phenomena in cells and tissues, and we discuss underlying concepts, limitations and future perspectives of these models. As the interest in cell and tissue mechanics increase, agent based models are becoming more common the modeling community. We overview the physical aspects, complexity, shortcomings and capabilities of the major agent based model categories: lattice-based models (cellular automata, lattice gas cellular automata, cellular Potts models), off-lattice models (center based models, deformable cell models, vertex models), and hybrid discrete-continuum models. In this way, we hope to assist future researchers in choosing a model for the phenomenon they want to model and understand. The article also contains some novel results

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

Bridging from single to collective cell migration: A review of models and links to experiments

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.Comment: 39 pages, 5 figure

Multiscale modeling of layer formation in epidermis

The mammalian skin epidermis is a stratified epithelium composed of multiple layers of epithelial cells that exist in appropriate sizes and proportions, and with distinct boundaries separating each other. How the epidermis develops from a single layer of committed precursor cells to form a complex multilayered structure of multiple cell types remains elusive. Here, we construct stochastic, three-dimensional, and multiscale models consisting of a lineage of multiple cell types to study the control of epidermal development. Symmetric and asymmetric cell divisions, stochastic cell fate transitions within the lineage, extracellular morphogens, cell-to-cell adhesion forces, and cell signaling are included in model. A GPU algorithm was developed and implemented to accelerate the simulations. These simulations show that a balance between cell proliferation and differentiation during lineage progression is crucial for the development and maintenance of the epidermal tissue. We also find that selective intercellular adhesion is critical to sharpening the boundary between layers and to the formation of a highly ordered structure. The long-range action of a morphogen provides additional feedback regulations, enhancing the robustness of overall layer formation. Our model is built upon previous experimental findings revealing the role of Ovol transcription factors in regulating epidermal development. Direct comparisons of experimental and simulation perturbations show remarkable consistency. Taken together, our results highlight the major determinants of a well-stratified epidermis: balanced proliferation and differentiation, and a combination of both short- (symmetric/asymmetric division and selective cell adhesion) and long-range (morphogen) regulations. These underlying principles have broad implications for other developmental or regenerative processes leading to the formation of multilayered tissue structures, as well as for pathological processes such as epidermal wound healing

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

Multi-Scale Mathematical Modeling of Prion Aggregate Dynamics and Phenotypes in Yeast Colonies

Prion diseases are a multi-scale biological phenomenon that requires understanding intracellular processes as well as how cells interact with each other and their environment. In mammals, prion diseases are progressive, untreatable, and fatal. Yeast prion phenotypes are harmless and reversible, which suggests a deep understanding of the reversal of prion phenotypes in yeast may be informative to mammalian diseases. In yeast, the loss of some prion phenotypes appears to be stochastic and spatially dependent, suggesting a cell-based model of yeast prion dynamics would be a powerful tool for comparisons with experimental results and hypothesis generation. In this work, we consider the components necessary to develop such a model that depicts both the biochemical-, intracellular-, and colony-level scales in yeast prion phenotypes. We first review the literature of mathematical models of the intracellular processes of prion disease. We then review common approaches to cell-based modeling of multicellular systems and how they have led to biological insights in other systems. This chapter ends with a discussion of future studies aimed at motivating how these two types of models can be coupled to produce multi-scale models of prion phenotypes