Multiple equilibrium states in a micro-vascular network

We use a simple model of micro-vascular blood flow to explore conditions that give rise to multiple equilibrium states in a three-node micro-vascular network. The model accounts for two primary rheological effects: the Fåhræus-Lindqvist effect, which describes the apparent viscosity of blood in a vessel, and the plasma skimming effect, which governs the separation of red blood cells at diverging nodes. We show that multiple equilibrium states are possible, and we use our analytical and computational tools to design an experiment for validation.     

Modeling of flow‐induced shear stress applied on 3D cellular scaffolds: Implications for vascular tissue engineering

Novel tissue‐culture bioreactors employ flow‐induced shear stress as a means of mechanical stimulation of cells. We developed a computational fluid dynamics model of the complex three‐dimensional (3D) microstructure of a porous scaffold incubated in a direct perfusion bioreactor. Our model was designed to predict high shear‐stress values within the physiological range of those naturally sensed by vascular cells (1–10 dyne/cm²), and will thereby provide suitable conditions for vascular tissue‐engineering experiments. The model also accounts for cellular growth, which was designed as an added cell layer grown on all scaffold walls. Five model variants were designed, with geometric differences corresponding to cell‐layer thicknesses of 0, 50, 75, 100, and 125 µm. Four inlet velocities (0.5, 1, 1.5, and 2 cm/s) were applied to each model. Wall shear‐stress distribution and overall pressure drop calculations were then used to characterize the relation between flow rate, shear stress, cell‐layer thickness, and pressure drop. The simulations showed that cellular growth within 3D scaffolds exposes cells to elevated shear stress, with considerably increasing average values in correlation to cell growth and inflow velocity. Our results provide in‐depth analysis of the microdynamic environment of cells cultured within 3D environments, and thus provide advanced control over tissue development in vitro. Biotechnol. Bioeng. 2010; 105: 645–654. © 2009 Wiley Periodicals, Inc.     

The epitheliome: agent-based modelling of the social behaviour of cells

We have developed a new computational modelling paradigm for predicting the emergent behaviour resulting from the interaction of cells in epithelial tissue. As proof-of-concept, an agent-based model, in which there is a one-to-one correspondence between biological cells and software agents, has been coupled to a simple physical model. Behaviour of the computational model is compared with the growth characteristics of epithelial cells in monolayer culture, using growth media with low and physiological calcium concentrations. Results show a qualitative fit between the growth characteristics produced by the simulation and the in vitro cell models.     

Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumour growth, linking the relatively harmless avascular growth phase and the potentially fatal vascular growth phase. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration, proliferation; degradation of tissue; new capillary vessel (sprout) formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is blood flow associated with the nascent network, the subsequent growth of the network evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix and perfusion-related haemodynamic forces in a manner that may be described as both adaptive and dynamic. In this paper we present a mathematical model which simultaneously couples vessel growth with blood flow through the vessels--dynamic adaptive tumour-induced angiogenesis (DATIA). This new mathematical model presents a theoretical and computational investigation of the process and highlights a number of important new targets for therapeutic intervention. In contrast to earlier flow models, where the effects of perfusion (blood flow) were essentially evaluated a posteriori, i.e. after generating a hollow network, blood flow in the model described in this paper has a direct impact during capillary growth, with radial adaptations and network remodelling occurring as immediate consequences of primary anastomoses. Capillary network architectures resulting from the dynamically adaptive model are found to differ radically from those obtained using earlier models. The DATIA model is used to examine the effects of changing various physical and biological model parameters on the developing vascular architecture and the delivery of chemotherapeutic drugs to the tumour. Subsequent simulations of chemotherapeutic treatments under different parameter regimes lead to the identification of a number of new therapeutic targets for tumour management.     

The Vital Role of Blood Flow-Induced Proliferation and Migration in Capillary Network Formation in a Multiscale Model of Angiogenesis

Sprouting angiogenesis and capillary network formation are tissue scale phenomena. There are also sub-scale phenomena involved in angiogenesis including at the cellular and intracellular (molecular) scales. In this work, a multiscale model of angiogenesis spanning intracellular, cellular, and tissue scales is developed in detail. The key events that are considered at the tissue scale are formation of closed flow path (that is called loop in this article) and blood flow initiation in the loop. At the cellular scale, growth, migration, and anastomosis of endothelial cells (ECs) are important. At the intracellular scale, cell phenotype determination as well as alteration due to blood flow is included, having pivotal roles in the model. The main feature of the model is to obtain the physical behavior of a closed loop at the tissue scale, relying on the events at the cellular and intracellular scales, and not by imposing physical behavior upon it. Results show that, when blood flow is considered in the loop, the anastomosed sprouts stabilize and elongate. By contrast, when the loop is modeled without consideration of blood flow, the loop collapses. The results obtained in this work show that proper determination of EC phenotype is the key for its survival.     

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.     

Numerical evaluation reveals the effect of branching morphology on vessel transport properties during angiogenesis

Blood flow governs transport of oxygen and nutrients into tissues. Hypoxic tissues secrete VEGFs to promote angiogenesis during development and in tissue homeostasis. In contrast, tumors enhance pathologic angiogenesis during growth and metastasis, suggesting suppression of tumor angiogenesis could limit tumor growth. In line with these observations, various factors have been identified to control vessel formation in the last decades. However, their impacts on the vascular transport properties of oxygen remain elusive. Here, we take a computational approach to examine the effects of vascular branching on blood flow in the growing vasculature. First of all, we reconstruct a 3D vascular model from the 2D confocal images of the growing vasculature at postnatal day 5 (P5) mouse retina, then simulate blood flow in the vasculatures, which are obtained from the gene targeting mouse models causing hypo- or hyper-branching vascular formation. Interestingly, hyper-branching morphology attenuates effective blood flow at the angiogenic front, likely promoting tissue hypoxia. In contrast, vascular hypo-branching enhances blood supply at the angiogenic front of the growing vasculature. Oxygen supply by newly formed blood vessels improves local hypoxia and decreases VEGF expression at the angiogenic front during angiogenesis. Consistent with the simulation results indicating improved blood flow in the hypo-branching vasculature, VEGF expression around the angiogenic front is reduced in those mouse retinas. Conversely, VEGF expression is enhanced in the angiogenic front of hyper-branching vasculature. Our results indicate the importance of detailed flow analysis in evaluating the vascular transport properties of branching morphology of the blood vessels.     

A cellular automaton model for tumour growth in inhomogeneous environment

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

Angiogenesis and vascular remodelling in normal and cancerous tissues

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.     

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.     

Evaluating the interaction of a tracheobronchial stent in an ovine in-vivo model

Tracheobronchial stents are used to restore patency to stenosed airways. However, these devices are associated with many complications such as stent migration, granulation tissue formation, mucous plugging and stent strut fracture. Of these, granulation tissue formation is the complication that most frequently requires costly secondary interventions. In this study a biomechanical lung modelling framework recently developed by the authors to capture the lung in-vivo stress state under physiological loading is employed in conjunction with ovine pre-clinical stenting results and device experimental data to evaluate the effect of stent interaction on granulation tissue formation. Stenting is simulated using a validated model of a prototype covered laser-cut tracheobronchial stent in a semi-specific biomechanical lung model, and physiological loading is performed. Two computational methods are then used to predict possible granulation tissue formation: the standard method which utilises the increase in maximum principal stress change, and a newly proposed method which compares the change in contact pressure over a respiratory cycle. These computational predictions of granulation tissue formation are then compared to pre-clinical stenting observations after a 6-week implantation period. Experimental results of the pre-clinical stent implantation showed signs of granulation tissue formation both proximally and distally, with a greater proximal reaction. The standard method failed to show a correlation with the experimental results. However, the contact change method showed an apparent correlation with granulation tissue formation. These results suggest that this new method could be used as a tool to improve future device designs.     

Microenvironmental Variables Must Influence Intrinsic Phenotypic Parameters of Cancer Stem Cells to Affect Tumourigenicity

Since the discovery of tumour initiating cells (TICs) in solid tumours, studies focussing on their role in cancer initiation and progression have abounded. The biological interrogation of these cells continues to yield volumes of information on their pro-tumourigenic behaviour, but actionable generalised conclusions have been scarce. Further, new information suggesting a dependence of tumour composition and growth on the microenvironment has yet to be studied theoretically. To address this point, we created a hybrid, discrete/continuous computational cellular automaton model of a generalised stem-cell driven tissue with a simple microenvironment. Using the model we explored the phenotypic traits inherent to the tumour initiating cells and the effect of the microenvironment on tissue growth. We identify the regions in phenotype parameter space where TICs are able to cause a disruption in homeostasis, leading to tissue overgrowth and tumour maintenance. As our parameters and model are non-specific, they could apply to any tissue TIC and do not assume specific genetic mutations. Targeting these phenotypic traits could represent a generalizable therapeutic strategy across cancer types. Further, we find that the microenvironmental variable does not strongly affect the outcomes, suggesting a need for direct feedback from the microenvironment onto stem-cell behaviour in future modelling endeavours.     

Coordinating cell behaviour during blood vessel formation

The correct development of blood vessels is crucial for all aspects of tissue growth and physiology in vertebrates. The formation of an elaborate hierarchically branched network of endothelial tubes, through either angiogenesis or vasculogenesis, relies on a series of coordinated morphogenic events, but how individual endothelial cells adopt specific phenotypes and how they coordinate their behaviour during vascular patterning is unclear. Recent progress in our understanding of blood vessel formation has been driven by advanced imaging techniques and detailed analyses that have used a combination of powerful in vitro, in vivo and in silico model systems. Here, we summarise these models and discuss their advantages and disadvantages. We then review the different stages of blood vessel development, highlighting the cellular mechanisms and molecular players involved at each step and focusing on cell specification and coordination within the network.     

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.     

Characterization of tissue scaffolds for time-dependent biotransport criteria – a novel computational procedure

This study aims to establish a new computational framework that allows modeling transient oxygen diffusion in tissue scaffolds more efficiently. It has been well known that the survival of cells strongly relies on continuous oxygen/nutrient supply and metabolite removal. With optimal design in scaffold architecture, its ability to sustain long distance oxygen supply could be improved considerably. In this study, finite element based homogenization procedure is first used to characterize the initial effective biotransport properties in silico. These initial properties are proper indicators to prediction of the on-going performance of tissue scaffolds over time. The transient model by adopting an edge-based smoothed finite element method with combination of mass-redistributed method is then established to more efficiently simulate the transient oxygen transfer process in tissue scaffolds. The proposed new method allows large time steps to model the oxygen diffusion process without losing numerical accuracy, thereby enhancing the computational efficiency significantly, in particular for the design optimization problems which typically require numerous analysis iterations. A number of different scaffold designs are examined either under net diffusion without cell seeding, or under cellular oxygen/nutrient uptake with or without considering cell viability. The association between the homogenized effective diffusivity of net scaffold microstructures and corresponding transient diffusion and time-dependent cellular activities is divulged. This study provides some insights into scaffold design.     

A computational model of intussusceptive microvascular growth and remodeling

Non-sprouting angiogenesis, also known as intussusceptive angiogenesis (IA), is an important modality of blood vessel morphogenesis in growing tissues. We present a novel computational framework for simulation of IA to answer some of the questions concerning the underlying mechanisms of the remodeling process. The model relies on mechanical interactions between blood and tissue, includes the structural maturation of the vessel wall, and is controlled by stimulating or inhibiting chemical agents. The model provides a simple explanation for the formation of microvessels and bifurcations from capillaries via IA, allowing for both maintenance and avoidance of shunt vessels. Detailed hemodynamic and transport properties for oxygen, metabolites or growth factors can be predicted. The model is an in silico framework for testing certain conceptual ideas about the mechanisms of intussusceptive growth and remodeling, in particular those related to mechanical and transport phenomena.     

Effect of cell seeding and mechanical loading on vascularization and tissue formation inside a scaffold: A mechano-biological model using a lattice approach to simulate cell activity

Achieving successful vascularization remains one of the main problems in bone tissue engineering. After scaffold implantation, the growth of capillaries into the porous construct may be too slow to provide adequate nutrients to the cells in the scaffold interior and this inhibits tissue formation in the scaffold core. Often, prior to implantation, a controlled cell culture environment is used to stimulate cell proliferation and, once in place, the mechanical environment acting on the tissue construct is determined by the loading conditions at the implantation site. To what extent do cell seeding conditions and the construct loading environment have an effect on scaffold vascularization and tissue growth? In this study, a mechano-biological model for tissue differentiation and blood vessel growth was used to determine the influence of cell seeding on vascular network development and tissue growth inside a regular-structured bone scaffold under different loading conditions. It is predicted that increasing the number of cells seeded homogeneously reduces the rate of vascularization and the maximum penetration of the vascular network, which in turn reduces bone tissue formation. The seeding of cells in the periphery of the scaffold was predicted to be beneficial for vascularization and therefore for bone growth; however, tissue formation occurred more slowly during the first weeks after implantation compared to homogeneous seeding. Low levels of mechanical loading stimulated bone formation while high levels of loading inhibited bone formation and capillary growth. This study demonstrates the feasibility of computational design approaches for bone tissue engineering.     

Computational Modeling of Fluid Flow and Intra-Ocular Pressure following Glaucoma Surgery

Background

Glaucoma surgery is the most effective means for lowering intraocular pressure by providing a new route for fluid to exit the eye. This new pathway is through the sclera of the eye into sub-conjunctival tissue, where a fluid filled bleb typically forms under the conjunctiva. The long-term success of the procedure relies on the capacity of the sub-conjunctival tissue to absorb the excess fluid presented to it, without generating excessive scar tissue during tissue remodeling that will shut-down fluid flow. The role of inflammatory factors that promote scarring are well researched yet little is known regarding the impact of physical forces on the healing response.

Methodology

To help elucidate the interplay of physical factors controlling the distribution and absorption of aqueous humor in sub-conjunctival tissue, and tissue remodeling, we have developed a computational model of fluid production in the eye and removal via the trabecular/uveoscleral pathways and the surgical pathway. This surgical pathway is then linked to a porous media computational model of a fluid bleb positioned within the sub-conjunctival tissue. The computational analysis is centered on typical functioning bleb geometry found in a human eye following glaucoma surgery. A parametric study is conducted of changes in fluid absorption by the sub-conjunctival blood vessels, changes in hydraulic conductivity due to scarring, and changes in bleb size and shape, and eye outflow facility.

Conclusions

This study is motivated by the fact that some blebs are known to have ‘successful’ characteristics that are generally described by clinicians as being low, diffuse and large without the formation of a distinct sub-conjunctival encapsulation. The model predictions are shown to accord with clinical observations in a number of key ways, specifically the variation of intra-ocular pressure with bleb size and shape and the correspondence between sites of predicted maximum interstitial fluid pressure and key features observed in blebs, which gives validity to the model described here. This model should contribute to a more complete explanation of the physical processes behind successful bleb characteristics and provide a new basis for clinically grading blebs.     

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.).