Morphology of the growing sprouts with endothelial cell proliferation dependent on VEGF concentration but triggered by local strain.

<p>In plate A we indicate the morphology of the growing sprout (by using different colours) for different values of maximum proliferation (<i>M</i>_<i>P</i>) and limit VEGF concentration (<i>L</i>_<i>V</i>). In these simulations the proliferation rate increases linearly with the VEGF concentration until it reaches its maximum value <i>M</i>_<i>P</i> at the concentration <i>L</i>_<i>V</i>, but is only different from zero where the strain is larger than a cut-off <i>S</i>_<i>m</i> = 0.05. The colours indicate the morphology of the observed sprouts for the corresponding parameters: blue dots correspond to situations where the sprout breaks, the green dots correspond to well formed sprouts without appreciable thickening of the parental vessel, and the red dots correspond to deformed vessels (i.e. triangular sprouts). We observe that the sprout breaks for low proliferation rates, but there is a very large region of the parameter space with well formed sprouts. In plates B1, B2, B3 and B4 we plot examples of the morphologies observed (the color of the border in these plates follows the same code as in plate A). The parameters used to obtain the morphologies depicted in B1, B2, B3 and B4 are indicated by arrows in plate A.</p

Morphology of the growing sprouts with endothelial cell proliferation dependent on local strain but triggered by VEGF.

<p>In plate A we indicate the morphology of the growing sprout (by using different colours) for different values of maximum proliferation (<i>M</i>_<i>P</i>) and limit strain (<i>L</i>_<i>S</i>). In these simulations the proliferation rate increases linearly with the strain until it reaches its maximum value <i>M</i>_<i>P</i> at the strain <i>L</i>_<i>S</i>, but is only different from zero where the concentration of angiogenic factor is larger than a cut-off, i.e. <i>V</i> > <i>V</i>_<i>m</i> = 0.05. The colours indicate the morphology of the observed sprouts for the corresponding parameters: blue dots correspond to situations where the sprout breaks, the red dots correspond to deformed vessels (vessels with variable thickness in this case), and the green dots correspond to well formed sprouts without appreciable thickening of the parental vessel. We observe that the sprout breaks for low proliferation rates, and it becomes deformed for large proliferation rates; the region of parameters that produce good sprouts is localised in a narrow vertical band. In plates B1, B2, B3 and B4 we plot examples of the morphologies observed (the color of the border in these plates follows the same code as in plate A). The parameters used to obtain the morphologies depicted in B1, B2, B3 and B4 are indicated by arrows in plate A.</p

The Force at the Tip - Modelling Tension and Proliferation in Sprouting Angiogenesis

<div><p>Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art.</p></div

Morphology of the growing sprouts with endothelial cell proliferation dependent VEGF concentration.

<p>In plate A we indicate the morphology of the growing sprout (by using different colours) for different values of maximum proliferation (<i>M</i>_<i>P</i>) and limit VEGF concentration (<i>L</i>_<i>V</i>). In these simulation the proliferation rate increases linearly with the VEGF concentration until it reaches the value <i>M</i>_<i>P</i> at the VEGF concentration <i>L</i>_<i>V</i>. The colours indicate the morphology of the observed sprouts for the corresponding parameters: blue dots correspond to situations where the sprout breaks, the green dots correspond to well formed sprouts without appreciable thickening of the parental vessel, the orange dots correspond to well formed sprouts with appreciable thickening of the parental vessel, and the red dots correspond to deformed vessels (triangular sprouts in this case). We observe that the sprout breaks for low proliferation rates, but there is a large region of the parameter space with well formed sprouts. There is however extensive parental vessel thickening. In plates B1, B2, B3 and B4 we plot examples of the morphologies observed (the color of the border in these plates follows the same code as in plate A). The parameters used to obtain the morphologies depicted in B1, B2, B3 and B4 are indicated by arrows in plate A.</p

Formation of a single sprout without endothelial cell proliferation.

<p>Plate A: The contractile force field that is created by the tip cell is oriented along the direction of the VEGF gradient. Here we consider this force distribution, akin to the traction force field observed by <i>in vitro</i> experiments [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004436#pcbi.1004436.ref020" target="_blank">20</a>]. The typical size of the force region is the same as the size of an endothelial cell. Plate B: The centre of the force field is located at the border of the original vessel. In this figure a sprout is already forming. The local intensity of the traction force exerted by the tip cell is indicated. The maximum traction force in this case is 3.0 KPa. Plate C: Length of the sprout after 14.5 hours plotted as a function of the maximum tip cell traction force and adhesion coefficient. After that period of time we observe that the sprout almost does not increase its size. The values for adhesion and traction tested are chosen randomly within the studied interval. The length of the sprouts increases with the traction force applied. For large values of adhesion and traction force the tip cell pinches off.</p

Morphology of the growing sprouts with low adhesion between endothelial cells.

<p>The two plates correspond to the cases of proliferation regulated by VEGF concentration (Plate A), and proliferation regulated by VEGF concentration but triggered by strain (Plate B). The colours indicate the morphology of the observed sprouts for each set of parameters. In both cases the black dots correspond to situations where vessel grows very slowly, not being able to reach a few cell widths in length after 14.5 hours. The red dots correspond to deformed vessels (triangular or with variable thickness), the orange dots correspond to sprouts that are straight but with a significantly thickened parental vessel, and the green dots correspond to well formed sprouts without appreciable thickening of the parental vessel. We observe that the proliferation is able to extend the short sprouts.</p

Representation of the terms included in Eq (1).

<p>The evolution of the order parameter depends on two processes: the endothelial cell proliferation and the movement of endothelial cells of the capillary on the ECM. The latter results from three main mechanisms that are implemented in our phase-field model. First, the constitution of a capillary wall (cells resist being isolated and leaving the vessel); we model this term as a surface tension. Second, the endothelial cells occupy regions of stiff tissue with high strains; this term is proportional to the difference between the tissue rigidities, but also proportional to the the local strain. Finally the cells can be pulled by the adhesion forces.</p