Cell-Oriented Modeling of In Vitro Capillary Development

On the evolution and development of morphological complexity: A view from gene regulatory networks

How does morphological complexity evolve? This study suggests that the likelihood of mutations increasing phenotypic complexity becomes smaller when the phenotype itself is complex. In addition, the complexity of the genotype-phenotype map (GPM) also increases with the phenotypic complexity. We show that complex GPMs and the above mutational asymmetry are inevitable consequences of how genes need to be wired in order to build complex and robust phenotypes during development. We randomly wired genes and cell behaviors into networks in EmbryoMaker. EmbryoMaker is a mathematical model of development that can simulate any gene network, all animal cell behaviors (division, adhesion, apoptosis, etc.), cell signaling, cell and tissues biophysics, and the regulation of those behaviors by gene products. Through EmbryoMaker we simulated how each random network regulates development and the resulting morphology (i.e. a specific distribution of cells and gene expression in 3D). This way we obtained a zoo of possible 3D morphologies. Real gene networks are not random, but a random search allows a relatively unbiased exploration of what is needed to develop complex robust morphologies. Compared to the networks leading to simple morphologies, the networks leading to complex morphologies have the following in common: 1) They are rarer; 2) They need to be finely tuned; 3) Mutations in them tend to decrease morphological complexity; 4) They are less robust to noise; and 5) They have more complex GPMs. These results imply that, when complexity evolves, it does so at a progressively decreasing rate over generations. This is because as morphological complexity increases, the likelihood of mutations increasing complexity decreases, morphologies become less robust to noise, and the GPM becomes more complex. We find some properties in common, but also some important differences, with non-developmental GPM models (e.g. RNA, protein and gene networks in single cells).

A preliminary study of endothelial cell migration during angiogenesis using a meshless method approach

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the "influence-domain" approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two-dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

Colloidal Gels for Guiding Endothelial Cell Organization via Microstructural Morphology

One of the fundamental challenges in vascular morphogenesis is to understand how the microstructural morphology of a 3D matrix can provide the spatial cues to organize the endothelial cells (ECs) into specific vascular structures. Colloidal gels can provide well-controlled distinct morphological matrices because these gels are formed by the aggregation of particles. By altering the aggregation mode, the spatial organization of the particles can be controlled to yield different microstructural morphology. To demonstrate this, colloidal aggregates and gels were developed by electrostatic interaction-mediated aggregation of cationic polyurethane (PU) colloidal particles by using low molecular weight electrolyte and polyelectrolyte to develop microstructurally different colloidal gels without altering their bulk elasticity. Compact dense colloidal aggregates with constricted voids were developed via electrolyte-mediated aggregation, whereas stranded branched networks with interconnected voids were formed via polyelectrolyte-mediated bridging interactions. Results show that the microstructure of aggregated colloids and gels can regulate EC organizations. Within endothelial matrices, ECs track the microstructure of particulate phase to interconnect with stranded colloidal network but cluster around compact colloidal aggregate. Similarly, in colloidal gels, ECs formed capillary-like structures by interconnecting along the stranded networks with enhanced cell-matrix interactions and increased cell extension but aggregated within the constricted voids of compact dense gel with enhanced cell-cell interaction. Both morphometric analysis and expression of EC markers corroborated the cell organizations in these gels. Using these colloidal gels, we demonstrated the role of 3D microstructural morphology as an important regulator for spatial guidance of ECs and simultaneously established the significance of colloidal gels as 3D matrix to regulate cellular morphogenesis.

Computational Modeling to Quantify the Contributions of VEGFR1, VEGFR2, and Lateral Inhibition in Sprouting Angiogenesis

Sprouting angiogenesis is a necessary process in regeneration and development as well as in tumorigenesis. VEGF-A is the main pro-angiogenic chemoattractant and it can bind to the decoy receptor VEGFR1 or to VEGFR2 to induce sprouting. Active sprout cells express Dll4, which binds to Notch1 on neighboring cells, in turn inhibiting VEGFR2 expression. It is known that the balance between VEGFR2 and VEGFR1 determines tip selection and network architecture, however the quantitative interrelationship of the receptors and their interrelated balances, also with relation to Dll4-Notch1 signaling, remains yet largely unknown. Here, we present an agent-based computer model of sprouting angiogenesis, integrating VEGFR1 and VEGFR2 in a detailed model of cellular signaling. Our model reproduces experimental data on VEGFR1 knockout. We show that soluble VEGFR1 improves the efficiency of angiogenesis by directing sprouts away from existing cells over a wide range of parameters. Our analysis unravels the relevance of the stability of the active notch intracellular domain as a dominating hub in this regulatory network. Our analysis quantitatively dissects the regulatory interactions in sprouting angiogenesis. Because we use a detailed model of intracellular signaling, the results of our analysis are directly linked to biological entities. We provide our computational model and simulation engine for integration in complementary modeling approaches.

The Molecular Basis of Radial Intercalation during Tissue Spreading in Early Development

Radial intercalation is a fundamental process responsible for the thinning of multilayered tissues during large-scale morphogenesis; however, its molecular mechanism has remained elusive. Using amphibian epiboly, the thinning and spreading of the animal hemisphere during gastrulation, here we provide evidence that radial intercalation is driven by chemotaxis of cells toward the external layer of the tissue. This role of chemotaxis in tissue spreading and thinning is unlike its typical role associated with large-distance directional movement of cells. We identify the chemoattractant as the complement component C3a, a factor normally linked with the immune system. The mechanism is explored by computational modeling and tested in vivo, ex vivo, and in vitro. This mechanism is robust against fluctuations of chemoattractant levels and expression patterns and explains expansion during epiboly. This study provides insight into the fundamental process of radial intercalation and could be applied to a wide range of morphogenetic events.

Review on experiment-based two- and three-dimensional models for wound healing

Traumatic and chronic wounds are a considerable medical challenge that affects many populations and their treatment is a monetary and time-consuming burden in an ageing society to the medical systems. Because wounds are very common and their treatment is so costly, approaches to reveal the responses of a specific wound type to different medical procedures and treatments could accelerate healing and reduce patient suffering. The effects of treatments can be forecast using mathematical modelling that has the predictive power to quantify the effects of induced changes to the wound-healing process. Wound healing involves a diverse and complex combination of biophysical and biomechanical processes. We review a wide variety of contemporary approaches of mathematical modelling of gap closure and wound-healing-related processes, such as angiogenesis. We provide examples of the understanding and insights that may be garnered using those models, and how those relate to experimental evidence. Mathematical modelling-based simulations can provide an important visualization tool that can be used for illustrational purposes for physicians, patients and researchers.

Mathematical modelling of angiogenesis using continuous cell-based models

In this work, we develop a mathematical formalism based on a 3D in vitro model that is used to simulate the early stages of angiogenesis. The model treats cells as individual entities that are migrating as a result of chemotaxis and durotaxis. The phenotypes used here are endothelial cells that can be distinguished into stalk and tip (leading) cells. The model takes into account the dynamic interaction and interchange between both phenotypes. Next to the cells, the model takes into account several proteins such as vascular endothelial growth factor, delta-like ligand 4, urokinase plasminogen activator and matrix metalloproteinase, which are computed through the solution of a system of reaction–diffusion equations. The method used in the present study is classified into the hybrid approaches. The present study, implemented in three spatial dimensions, demonstrates the feasibility of the approach that is qualitatively confirmed by experimental results.

Investigating in vitro angiogenesis by computer-assisted image analysis and computational simulation

In vitro assays that stimulate the formation of capillary-like structures by EC have become increasingly popular, because they allow the study of the EC's intrinsic ability to self-organize to form vascular-like patterns. Here we describe a widely applied protocol involving the use of basement membrane matrix (Matrigel) as a suitable environment to induce an angiogenic phenotype in cultured EC. EC differentiation on basement membrane matrix is a highly specific process, which recapitulates many steps in blood vessel formation and for this reason it is presently considered as a reliable in vitro tool to identify factors with potential antiangiogenic or pro-angiogenic properties. The morphological features of the obtained cell patterns can also be accurately quantified by computer-assisted image analysis and the main steps of such a procedure will be here outlined and discussed. The dynamics of in vitro EC self-organization is a complex biological process, involving a network of interactions between a high number of cells. For this reason, the combined use of in vitro experiments and computational modeling can represent a key approach to unravel how mechanical and chemical signaling by EC coordinates their organization into capillary-like tubes. Thus, a particularly helpful approach to modeling is also briefly described together with examples of its application.

Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro

In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to ECM mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the ECM, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.

Role of differential adhesion in cell cluster evolution: from vasculogenesis to cancer metastasis

Cell-cell and cell-matrix adhesions are fundamental to numerous physiological processes, including angiogenesis, tumourigenesis, metastatic spreading and wound healing. We use cellular potts model to computationally predict the organisation of cells within a 3D matrix. The energy potentials regulating cell-cell (JCC) and cell-matrix (JMC) adhesive interactions are systematically varied to represent different, biologically relevant adhesive conditions. Chemotactically induced cell migration is also addressed. Starting from a cluster of cells, variations in relative cell adhesion alone lead to different cellular patterns such as spreading of metastatic tumours and angiogenesis. The combination of low cell-cell adhesion (high JCC) and high heterotypic adhesion (low JMC) favours the fragmentation of the original cluster into multiple, smaller cell clusters (metastasis). Conversely, cellular systems exhibiting high-homotypic affinity (low JCC) preserve their original configuration, avoiding fragmentation (organogenesis). For intermediate values of JCC and JMC (i.e. JCC/JMC ∼ 1), tubular and corrugated structures form. Fully developed vascular trees are assembled only in systems in which contact-inhibited chemotaxis is activated upon cell contact. Also, the rate of secretion, diffusion and sequestration of chemotactic factors, cell deformability and motility do not significantly affect these trends. Further developments of this computational model will predict the efficacy of therapeutic interventions to modulate the diseased microenvironment by directly altering cell cohesion.

A cell-based model of extracellular-matrix-guided endothelial cell migration during angiogenesis

Angiogenesis, the formation of new blood vessels sprouting from existing ones, occurs in several situations like wound healing, tissue remodeling, and near growing tumors. Under hypoxic conditions, tumor cells secrete growth factors, including VEGF. VEGF activates endothelial cells (ECs) in nearby vessels, leading to the migration of ECs out of the vessel and the formation of growing sprouts. A key process in angiogenesis is cellular self-organization, and previous modeling studies have identified mechanisms for producing networks and sprouts. Most theoretical studies of cellular self-organization during angiogenesis have ignored the interactions of ECs with the extra-cellular matrix (ECM), the jelly or hard materials that cells live in. Apart from providing structural support to cells, the ECM may play a key role in the coordination of cellular motility during angiogenesis. For example, by modifying the ECM, ECs can affect the motility of other ECs, long after they have left. Here, we present an explorative study of the cellular self-organization resulting from such ECM-coordinated cell migration. We show that a set of biologically-motivated, cell behavioral rules, including chemotaxis, haptotaxis, haptokinesis, and ECM-guided proliferation suffice for forming sprouts and branching vascular trees.

Vascular networks due to dynamically arrested crystalline ordering of elongated cells

Recent experimental and theoretical studies suggest that crystallization and glass-like solidification are useful analogies for understanding cell ordering in confluent biological tissues. It remains unexplored how cellular ordering contributes to pattern formation during morphogenesis. With a computational model we show that a system of elongated, cohering biological cells can get dynamically arrested in a network pattern. Our model provides an explanation for the formation of cellular networks in culture systems that exclude intercellular interaction via chemotaxis or mechanical traction.

Multiobjective optimization based-approach for discovering novel cancer therapies

Solid tumors must recruit new blood vessels for growth and maintenance. Discovering drugs that block tumor-induced development of new blood vessels (angiogenesis) is an important approach in cancer treatment. The complexity of angiogenesis presents both challenges and opportunities for cancer therapies. Intuitive approaches, such as blocking VegF activity, have yielded important therapies. But there maybe opportunities to alter nonintuitive targets either alone or in combination. This paper describes the development of a high-fidelity simulation of angiogenesis and uses this as the basis for a parallel search-based approach for the discovery of novel potential cancer treatments that inhibit blood vessel growth. Discovering new therapies is viewed as a multiobjective combinatorial optimization over two competing objectives: minimizing the estimated cost of practically developing the intervention while minimizing the simulated oxygen provided to the tumor by angiogenesis. Results show the effectiveness of the search process by finding interventions that are currently in use, and more interestingly, discovering potential new approaches that are nonintuitive yet effective.

Computational discrete models of tissue growth and regeneration

Tissue growth and regeneration are fundamental processes underpinning crucial physiological and pathological conditions: ranging from normal blood vessel network development, response to stem cells therapy and cancers. Modelling of such biological phenomena has been addressed through mathematical and algorithmic approaches. The former implements continuous representations based on differential equations. The latter exploit operational descriptions in the form of computing programs to represent and execute the models. Within this area, models that define the cell as the fundamental unit of model development, as well as discrete representations of different model entities, are important to plan in vitro experiments and to generate new testable hypotheses. This article reviews the application of algorithmic discrete models, with a focus on tissue growth and regeneration phenomena in the context of health and disease. The review begins with an overview of basic concepts, problems and approaches of computational discrete models. This will include a discussion of basic assumptions and design principles. An overview of key cell-driven approaches and examples of applications in tissue growth and regeneration is provided. The specification, implementation and analysis of a model are illustrated with a hypothetical example, which mimics the branching and sprouting patterns observed in blood vessel network development. The article concludes with a discussion of current challenges and recommendations.

Mathematical modeling of the capillary-like pattern generated by adrenomedullin-treated human vascular endothelial cells in vitro

A recently proposed approach was used to model the self-organization into capillary-like structures of human vascular endothelial cells cultured on Matrigel. The model combines a Cellular Potts Model, considering cell adhesion, cytoskeletal rearrangement and chemotaxis, and a Partial Differential Equation model describing the release and the diffusion of a chemoattractant. The results were compared with the data from real in vitro experiments to establish the capability of the model to accurately reproduce both the spontaneous self-assembly of unstimulated cells and their self-organization in the presence of the pro-angiogenic factor adrenomedullin. The results showed that the model can accurately reproduce the self-assembly of unstimulated cells, but it failed in reproducing the adrenomedullin-induced self-organization of the cells. The extension of the model to include cell proliferation led to a good match between simulated and experimental patterns in both cases with predicted proliferation rates in agreement with the data of cell proliferation experiments.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Role of repulsive factors in vascularization dynamics

Capillary networks are essential in vertebrates to supply tissues with nutrients. Experiments of in vitro capillary formation show that endothelial cells randomly spread on a gel matrix autonomously organize to form vascular networks with a characteristic length independent of the initial cell density. A mathematical model based on free cell migration and on cell cross-talk mediated by soluble chemical factors has been recently proposed and explains the main dynamical and geometrical properties of the networks. We extend this model introducing the action of repulsive factors and we show that their activity results in a larger degree of reorganization of cellular matter and in more robust control over the size of the growing vascular network.

Dynamic mechanisms of blood vessel growth

The formation of a polygonal configuration of proto-blood-vessels from initially dispersed cells is the first step in the development of the circulatory system in vertebrates. This initial vascular network later expands to form new blood vessels, primarily via a sprouting mechanism. We review a range of recent results obtained with a Monte Carlo model of chemotactically migrating cells which can explain both de novo blood vessel growth and aspects of blood vessel sprouting. We propose that the initial network forms via a percolation-like instability depending on cell shape, or through an alternative contact-inhibition of motility mechanism which also reproduces aspects of sprouting blood vessel growth.

Cell elongation is key to in silico replication of in vitro vasculogenesis and subsequent remodeling

Vasculogenesis, the de novo growth of the primary vascular network from initially dispersed endothelial cells, is the first step in the development of the circulatory system in vertebrates. In the first stages of vasculogenesis, endothelial cells elongate and form a network-like structure, called the primary capillary plexus, which subsequently remodels, with the size of the vacancies between ribbons of endothelial cells coarsening over time. To isolate such intrinsic morphogenetic ability of endothelial cells from its regulation by long-range guidance cues and additional cell types, we use an in vitro model of human umbilical vein endothelial cells (HUVEC) in Matrigel. This quasi-two-dimensional endothelial cell culture model would most closely correspond to vasculogenesis in flat areas of the embryo like the yolk sac. Several studies have used continuum mathematical models to explore in vitro vasculogenesis: such models describe cell ensembles but ignore the endothelial cells' shapes and active surface fluctuations. While these models initially reproduce vascular-like morphologies, they eventually stabilize into a disconnected pattern of vascular "islands." Also, they fail to reproduce temporally correct network coarsening. Using a cell-centered computational model, we show that the endothelial cells' elongated shape is key to correct spatiotemporal in silico replication of stable vascular network growth. We validate our simulation results against HUVEC cultures using time-resolved image analysis and find that our simulations quantitatively reproduce in vitro vasculogenesis and subsequent in vitro remodeling.

Solving the advection-diffusion equations in biological contexts using the cellular Potts model

The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approximate the Laplacian. Directed spin flips in the CPM handle the advective movement of the fluid particles. A constraint on relative velocities in the fluid explicitly accounts for fluid viscosity. We use the CPM to solve various diffusion examples including multiple instantaneous sources, continuous sources, moving sources, and different boundary geometries and conditions to validate our approximation against analytical and established numerical solutions. We also verify the CPM results for Poiseuille flow and Taylor-Aris dispersion.

Endothelial microparticles affect angiogenesis in vitro: role of oxidative stress

Endothelium-derived microparticles have recently been described as a new marker of endothelial cell dysfunction. Increased levels of circulating microparticles have been documented in inflammatory disorders, diabetes mellitus, and many cardiovascular diseases. Perturbations of angiogenesis play an important role in the pathogenesis of these disorders. We demonstrated previously that isolated endothelial microparticles (EMPs) impair endothelial function in vitro, diminishing acetylcholine-induced vasorelaxation and nitric oxide production by rat aortic rings and simultaneously increasing superoxide production. Herein, using the Matrigel assay of angiogenesis in vitro and a topological analysis of the capillary-like network by human umbilical vein endothelial cells (HUVECs), we investigated the effects of EMPs on formation of the vascular network. All parameters of angiogenesis were affected by treatment for 48 h with isolated EMPs in a concentration of 105 but not 103 or 104 EMPs/ml. The effects included decreases in total capillary length (24%), number of meshes (45%), and branching points (36%) and an increase in mesh area (38%). The positional and topological order indicated that EMPs affect angiogenic parameters uniformly over the capillary network. Treatment with the cell-permeable SOD mimetic Mn(III)tetrakis(4-benzoic acid) porphyrin chloride (Mn-TBAP) partially or completely restored all parameters of angiogenesis affected by EMPs. EMPs reduced cell proliferation rate and increased apoptosis rate in time- and dose-dependent manners, and this phenomenon was also prevented by Mn-TBAP treatment. Our data demonstrate that EMPs have considerable impact on angiogenesis in vitro and may be an important contributor to the pathogenesis of diseases that are accompanied by impaired angiogenesis.