Percolation and Burgers' dynamics in a model of capillary formation

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.

Graph analysis of cell clusters forming vascular networks

This manuscript describes the experimental observation of vasculogenesis in chick embryos by means of network analysis. The formation of the vascular network was observed in the area opaca of embryos from 40 to 55 h of development. In the area opaca endothelial cell clusters self-organize as a primitive and approximately regular network of capillaries. The process was observed by bright-field microscopy in control embryos and in embryos treated with Bevacizumab (Avastin^®), an antibody that inhibits the signalling of the vascular endothelial growth factor (VEGF). The sequence of images of the vascular growth were thresholded, and used to quantify the forming network in control and Avastin-treated embryos. This characterization is made by measuring vessels density, number of cell clusters and the largest cluster density. From the original images, the topology of the vascular network was extracted and characterized by means of the usual network metrics such as: the degree distribution, average clustering coefficient, average short path length and assortativity, among others. This analysis allows to monitor how the largest connected cluster of the vascular network evolves in time and provides with quantitative evidence of the disruptive effects that Avastin has on the tree structure of vascular networks.

The major transitions of life from a network perspective

Many attempts have been made to understand the origin of life and biological complexity both at the experimental and theoretical levels but neither is fully explained. In an influential work, Maynard Smith and Szathmáry (1995) argued that the majority of the increase in complexity is not gradual, but it is associated with a few so-called major transitions along the way of the evolution of life. For each major transition, they identified specific mechanisms that could account for the change in complexity related to information transmission across generations. In this work, I propose that the sudden and unexpected improvement in the functionality of an organism that followed a major transition was enabled by a phase transition in the network structure associated with that function. The increase in complexity following a major transition is therefore directly linked to the emergence of a novel structure-function relation which altered the course of evolution. As a consequence, emergent phenomena arising from these network phase transitions can serve as a common organizing principle for understanding the major transitions. As specific examples, I analyze the emergence of life, the emergence of the genetic apparatus, the rise of the eukaryotic cells, the evolution of movement and mechanosensitivity, and the emergence of consciousness. Finally, I discuss the implications of network associated phase transitions to issues that bear relevance to the history, the immediate present and perhaps the future, of life.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

In silico modeling of interstitial lung mechanics: implications for disease development and repair

In this perspective, we first review some of the published literature on structural modeling of the mechanical properties of the lung parenchyma. Based on a recent study, we demonstrate why mechanical dysfunction accompanying parenchymal diseases such as pulmonary fibrosis and emphysema can follow a very different course from the progression of the underlying microscopic pathophysiology itself, particularly in the early stages. The key idea is related to the concept of percolation on elastic networks where the bulk modulus of the network suddenly changes when the fibrotic stiff regions or the emphysematous holes become suddenly connected across the network. We also introduce the concept of depercolation as a basis for the rational optimization of tissue repair. Specifically, we use these network models to predict the functional improvements that a hypothetical biological or tissue engineering repair could achieve. We find that rational targeted repair can have significant benefits over generic random repair. This concept may find application in the treatment of lung fibrosis, surgical, bronchoscopic, or biological lung volume reduction, or any future alveolar regeneration or tissue engineering solution to the repair of connective tissue damage of the lung.

Linking parenchymal disease progression to changes in lung mechanical function by percolation

RATIONALE

The mechanical dysfunction accompanying parenchymal diseases such as pulmonary fibrosis and emphysema may follow a different course from the progression of the underlying microscopic pathophysiology itself, particularly in the early stages. It is tempting to speculate that this may reflect the geographical nature of lung pathology. However, merely ascribing mechanical dysfunction of the parenchyma to the vagaries of lesional organization is unhelpful without some understanding of how the two are linked.

OBJECTIVES

We attempt to forge such a link through a concept known as percolation, which has been invoked to account for numerous natural processes involving transmission of events across complex networks.

METHODS

We numerically determined the bulk stiffness (corresponding to the inverse of lung compliance) of a network of springs representing the lung parenchyma. We simulated the development of fibrosis by randomly stiffening individual springs in the network, and the development of emphysema by preferentially cutting springs under the greatest tension.

MEASUREMENTS AND MAIN RESULTS

When the number of stiff springs was increased to the point that they suddenly became connected across the network, the model developed a sharp increase in its bulk modulus. Conversely, when the cut springs became sufficiently numerous, the elasticity of the network fell to zero. These two conditions represent percolation thresholds that we show are mirrored structurally in both tissue pathology and macroscopic computed tomography images of human idiopathic fibrosis and emphysema.

CONCLUSIONS

The concept of percolation may explain why the development of symptoms related to lung function and the development of parenchymal pathology often do not progress together.

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.

A Review of Vasculogenesis Models

Mechanical and chemical models of vasculogenesis are critically reviewed with an emphasis on their ability to predict experimentally measured quantities. Final remarks suggest a possibility to merge the capabilities of different models into a unified approach.