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Great advances have been made in identifying the biochemical factors and intracellular signaling pathways that mediate the control of angiogenesis.

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This review focuses on work that explores the biophysical aspect of angiogenesis regulation. Specifically, we discuss the role of cell-generated forces, counterforces from the extracellular matrix, and mechanical forces associated with blood flow and extravascular tissue activity in the regulation of angiogenesis. Because angiogenesis occurs in a mechanically dynamic environment, future investigations should aim at understanding how cells integrate chemical and mechanical signals so that a rational approach to controlling angiogenesis will become possible.

In this regard, computational models that incorporate multiple epigenetic factors to predict capillary patterning will be useful. However, this mechanism still functions at rather short ranges, on the order of the cell size, as shown by in vitro experiments in ref. In some tissues other type of mechanical cues also drive anastomoses events at larger ranges. In the retina, for example, vessels grow above a layer of astrocytes that provide guidance and support In this situation, filopodia mediate the contact between the endothelial cells and the astrocytes network underneath, guiding the neo-vessels to almost replicate its structure.

In this way, the astrocytes network plays an important role in driving vessel anastomoses in the retina. However, in tissues where there is no underlying cell network for vessels to replicate, and where they explore the space between cells, we question how the vessels can be guided toward other vessels at larger scales in order to be close enough for mechanical communication to be relevant. In this work, we will demonstrate that pro-angiogenic factors, such as VEGF, play the decisive role in the process of anastomosis and in shaping the final network.

There is strong experimental data that supports this hypothesis. In sprouting angiogenesis, these factors have a pivotal role in guiding the sprouts.

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Namely, VEGF in matrices is able to promote migration and anastomoses 24 , 45 , 46 , So much so that VEGF micro-patterns in 3D matrices are able to guide mouse endothelial cells in vivo to form anastomoses and functional vessels along those patterns Moreover, VEGF micro-beads also drive the formation of functional vessels in their vicinity in vivo 49 , In a non-pathological setting, hypoxic cells in vivo will be producing VEGF until they are irrigated, and they can only be irrigated after there are anastomoses in their surroundings.

Therefore VEGF will keep being produced until nearby vessels merge and form loops capable of carrying blood. We will demonstrate that this mechanism can also account for an increased resilience of the network morphology towards endothelial cell proliferation and chemotactic response. To address the capability of VEGF gradients to drive anastomoses in a neo-vessel network, we present an extensive and systematic 2D and 3D computational study of vessel formation in a tissue.

The 2D model can be applied to angiogenesis in quasi two-dimensional vascular networks, such as networks in the retina, chick chorioallantoic membrane CAM assays, or other in vivo membranes.

Mechanical and Chemical Signaling in Angiogenesis

On the other hand, the 3D model can be applied to angiogenesis in 3D scaffolds and in vivo tissues, as the adipose tissue, and in tumor angiogenesis. In the last two decades angiogenesis has been modelled using different approaches. Initially, researchers implemented continuous descriptions of angiogenesis that had the density of vessels as output 51 , 52 , 53 , 54 , 55 , These models were not aimed at obtaining the vascular network or the blood flow rate, but to provide an indication of the advancement of the irrigated region.

Angiogenesis was also modelled using more detailed discrete agent-based models that take into account the interaction between every single cell in the system 57 , Discrete models have been coupled to tumor growth models, used to describe neo-vascularisation in the retina, and to study in detail the interaction between endothelial cells and their micro-environment 59 , 60 , 61 , 62 , 63 , 64 , Nevertheless, the large number of parameters and rules often difficult the full exploration of the parameter space in these models.

Recently, hybrid models that combine a continuous description of the vessel sprout with a cell-based approach for tip cell creation and movement were developed 22 , 26 , These models allow the study of the shape of large vessel sprout networks with a smaller number of parameters. Nevertheless the role of chemical factors in regulating anastomoses formation in 2D and in 3D has not yet been studied.

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For reviews on the literature of angiogenesis modelling see for example 67 , 68 , 69 , 70 , In the next section we will introduce the computational model used. In the Results and Discussion section we will present and compare the results of the 2D and 3D models regarding the capability of VEGF to drive anastomoses formation.

It will be stressed the role of this mechanism in determining the morphology of the vascular networks, rendering it less dependent on endothelial stalk cell proliferation rates or on the migration velocity of tip cells. Finally we draw the conclusions of the work in the last section.

The angiogenesis mathematical model implemented is a natural extension of the model introduced in 22 , and further explored in 40 , 72 , 73 , Here, we will review, in an abbreviated way, this mathematical model underlining the two major improvements that are introduced: the description of the cells in hypoxia using a second order parameter, and the estimation of tissue irrigation to couple it with hypoxia regulation.

The value of the order parameters in different domains. The model dynamics prevents the superposition between the positive domains of the two order parameters. Due to the Heaviside function, the proliferation only occurs inside the capillary. Often hypoxic cells have a low motility, especially if they are part of an epithelial tissue. Therefore, we consider in the simulation that tissue cells are static and do not change their shape.

Nevertheless, the simulation methods used permit to introduce motility and deformation on the hypoxic cells, which will be explored in future work. The concentration of angiogenic factor is kept at a constant value T s at the center of all active hypoxic cells.

During the simulation, the sources will be deactivated as the vessel network grows and as they are irrigated. The endothelial tip cells are represented by agents with radius R c. A new agent is introduced whenever there is a minimum angiogenic factor concentration T c , a minimum angiogenic factor gradient G m , and if the tip cell candidate is located at a distance larger than 4 R c from all other tip cells.

The implementation of this more complex Delta-Jagged-Notch mechanism is out of the scope of the present work but it is expected to be introduced in future model developments. All the activated tip cells migrate by chemotaxis, and their velocity is aligned with the local angiogenic factor gradient. A tip cell is not able to enter inside a tissue cell independently if the cell is in hypoxia or not : when the distance between the tip cell and a tissue cell is lower than 2 R c , the radial component of the velocity is set to zero and the tip cell moves in the azimuthal direction around the tissue cell.

Blood is able to flow once there are anastomoses in the vascular network. In order to estimate the irrigated regions in the tissue, we first identify the vessels where the blood is flowing. We start by extracting the medial lines of the vascular structure, thus finding the network bifurcation sites and the length of every vessel. The extraction method used for this purpose is detailed in In the simulation there is a single input node at the top of the simulation box and a single exit node at the bottom of the simulation box.

With the aim of addressing the capability of VEGF to drive anastomosis formation, we compare the results of the model obtained with two alternative rules concerning the tissue angiogenic factor production. When the new vessels are capable of delivering oxygen to the tissue, the newly irrigated cells deactivate their hypoxia mechanism and cease the production of angiogenic factors. Therefore several models in the literature 10 , 14 , 60 , 81 use estimates of tissue irrigation in order to determine which cells are producing angiogenic factors. To mimic this mechanism we consider a simplified rule to identify the VEGF producing cells:.

Schematic representation of the two different rules for hypoxia regulation. In the simulation with Rule 1 a cell in hypoxia will deactivate its production of VEGF when the blood circulation reaches a site at a distance from the cell shorter than the oxygen diffusion length. In the model with Rule 2, the VEGF production will cease if any vessel reaches a point closer to the cell than the oxygen diffusion length, independently of whether there is blood circulating in that vessel or not.

The endothelial cells are represented in red, the tip cells are the yellow circles and the angiogenic factor is represented in light blue. The cells in hypoxia are represented as darker blue circles. The arrows indicate the vessels with blood flow. This is the simplest rule guaranteeing that anastomoses events are required for hypoxic cell deactivation. However, many other models focus on the ability of the angiogenic factor to drive vessel growth and not anastomoses 22 , 40 , 83 , These models implement a different rule for tissue cell deactivation:.

In short, with Rule 2 chemical guidance promotes sprouting but is not associated with tissue irrigation, while in Rule 1 it is.

Comparing the morphology of the networks formed under these two rules, we show how important it is for tissue irrigation to regulate VEGF production to create functional vascular networks in 2D and in 3D. However, in the present work a detailed calculation of tissue irrigation will not be required, since our aim is to demonstrate that the need for anastomoses formation to stop VEGF production in tissues as in Rule 1 will determine several characteristics of the observed vascular morphology. Rule 1 keeps the model general, with all anastomoses providing the same amount of irrigation to the tissue.

With a full calculation of tissue irrigation we would observe that the irrigation provided by each anastomosis would depend on its location within the vessel network. This systematic study of vessel density in growing vasculatures as a function of the geometry of the initial network and of the oxygen pressure is very relevant, but it is outside the scope of the present article. In this way, the final vasculatures obtained by running the simulation with Rule 1 will be directly compared to the simulations where the angiogenic factor drives growth but not anastomoses using Rule 2.

We will then draw conclusions regarding the capacity of VEGF gradients to drive vessel anastomoses.

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These hypoxic cells produce the angiogenic factor which diffuses slowly in the tissue. When the angiogenic factor concentration at the vessel initially located at one edge of the box is larger than T c , a new tip cell is introduced in the simulation. This tip cell moves in the direction of the local angiogenic factor gradient forming a new sprout. Progressively, new tip cells are activated and a ramified network is produced see Fig. As the tip cells migrate in the tissue they are able to contour the hypoxic cells, diverting their movement direction as they approach these cells.

Vessel growth in two dimensions. Evolution of a vascular system with an initial capillary at the bottom and a random distribution of hypoxic cells green circles , which drive the chemotactic response by ECs. Four time points are represented.

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The first row of figures correspond to the vessel network grown under the Rule 1 for hypoxic cell deactivation, while the second row of images correspond to the vessel network grown under Rule 2. On the final column the irrigated vessels are identified in red. The number of anastomoses and the number of vessels with circulating blood are much higher when Rule 1 is implemented.

Mechanical and Chemical Signaling in Angiogenesis | Cynthia Reinhart-King | Springer

The morphology of the obtained networks is strikingly dependent on the rule for hypoxic cell deactivation. With Rule 2, the hypoxic cells start deactivating once the vessels pass nearby, leading to an open network with fewer vessels and anastomoses. The vessels are straighter and directed towards the hypoxic tissue.

As a result, only few vessels are able to carry circulating blood. On the other hand, with Rule 1, the number of branches is higher. The vessels surround several cells in hypoxia, irrigating them. The resulting vessel network morphology is more lattice-like instead of tree-like. Many more vessels carry blood flow. The observed vasculature also suggests that with Rule 1 regions with more hypoxic cells have a higher density of vessels, while regions with less cells have a lower vessel density.