Percolation

Looking at angiogenesis through a new point: Percolation theory

Alireza Mehdizadeh MD PhD student of medical physics

Shiraz University of Medical Sciences(SUMS)

INTRODUCTION 

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Angiogenesis is a complex process that includes the activation, proliferation, and migration of endothelial cells (EC), formation of vascular tubes and networks, and linkage to the pre-existing vascular networks (1). This process is implicated in some conditions, summarized in table 1.

Some pathologic conditions in which angiogenesis may be involved. 

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Conditions in which angiogenesis must get stimulated. Myocardial ischemia Peripheral ischemia Wound healing

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Conditions in which angiogenesis must get inhibited. Tumor growth Diabetic retinopathy Retinopathy of prematurity Rheumatoid arthritis Atherosclerotic plaques

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Among these extensive investigations, most of them have focused on stimulation of new sprout formation or inhibition of it but a little attention to other aspects of these processes has been paid. One of these aspects is the shape of new vessels, their branching, their anastomosis and their remodeling. Formation of connecting pathways between pre-existing vessels and a demand area such as a solid tumor must be through a media which is composed of extracellular matrix (ECM) components and different cells in ECM.

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ECM is composed of two basic components. First, proteins and its derivatives such as glycoproteins and proteoglycans. Second, water and soluble substances. EC at the tip of the new sprout must migrate among these cells and high molecular weight (HMW) components of ECM, like collagen fibers and glycoproteins and proteoglycans, to achieve their targets.

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Thus it must migrate through pathways that have lower resistance against their movement, figure 1. The shape of fixed structures of this media between a preexisting vessel a solid tumor, like HMW components of ECM and cells in ECM, determines the pathways through which EC can migrate easily and by this way, the final shape of newly formed vessels can be predicted. Migration of EC through this media which has some lower resistant pathways to migration is similar to flow of fluid through a porous media, figure 2 (9).

Figure 1. Histological organization of the retina. a-c In mice, retinal vessels arise from the optic nerve around birth then extend radially in the superficial retina over 7-10 days to reach the periphery. d-f, similar distribution of neurons, blood vessels and glia in a mouse retina. arrow indicates the optic nerve. d, Radially orientated ganglion cell axons (labelled blue for leptin receptor) exit the eye through the optic nerve (arrow). e, Fluorescent dextran (red) angiogram of adult retina; blood vessels also radiate from the optic nerve (white arrow) to the periphery. f, Retinal astrocyte meshwork labelled for glial fibrillary acidic protein (green) resembling that of blood vessels. g-h, Schematic representation of possible pathways (grey) through which sprout tip from parent capillary wall can migrate. Grey routes indicate low resistant pathways among fixed structures (black) of a media surrounding a parent capillary vessel. These pathways function as tracks for migration of sprout tip. Final morphology of vascularization is mimicry of possible pathways for tip migration.

Figure 2. Flow of fluid trough a porous media.

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From this point of view, theories which were used in other branch of sciences especially in Physics, mathematics and engineering to predict the path way of fluid flow through a porous media (10) could be applied for angiogenesis too. One of the most applicable methods is Percolation theory, as discussed below.

Percolation theory 

A representative question (and the source of the name) is as follows. Assume we have some porous material and we pour some liquid on top. Will the liquid be able to make its way from hole to hole and reach the bottom? We model the physical question mathematically as a three-dimensional network of n points (or vertices) the connections (or edges) between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1–p, and we assume they are independent. We ask for a given p, what is the probability that an open path exists from the top to the bottom? Mostly we are interested in the behavior for large n.

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As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is “does an infinite open cluster exist?” That is, is there a path of connected points of infinite length "through" the network? In this case we may use Kolmogorov's zero-one law to see that, for any given p, the probability that an infinite cluster exists is either zero or one. Since this probability is increasing, there must be a critical p (denoted by pc) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for n as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of p.

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In some cases pc may be calculated explicitly. For example, for the square lattice in two dimensions, pc = 1 / 2, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten (11) in the early '80s. More often than not, pc cannot be calculated. For example, pc is not known in three dimensions. However, it turns out that calculating pc is not necessarily the most interesting thing to do. The universality principle states that the value of pc is connected to the local structure of the graph, while the behavior of clusters below, at and above pc are invariants of the local structure, and therefore, in some sense are more natural quantities to consider.

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This model is Bernoulli percolation. In this model all bonds are independent. This model is also called bond percolation by physicists (12). In chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials. During the last three decades, Percolation theory, an extensive mathematical model of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science as well as geography. In mathematics, percolation theory describes the behavior of connected clusters in a random graph. (10).

HYPOTHESIS 

In our model, we consider surrounding environment of an assumed capillary vessel as a porous media. The fixed structures of this media act as scaffolding and the other parts of it act as pores within this scaffolding. To move any cell within this media, they must pass through this porous media to reach their targets.

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Thus there must be an open path from original location of a cell and its target toward which it migrates. The probability of existence of an open path, according to Percolation theory and the universality principle, depends on the structure of the porous media. If there are more connections within this media, the pc is low and the probability that an open path exists is high. In contrast, if the media gets more solid and has fewer connections, the pc gets higher and the probability of existence of an open path gets lower.

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Thus, a cell can not reach its target through this media easily. During angiogenesis, EC at the tip of the new sprout migrates through its surrounding media, acts as a porous media. Thus, if this porous media has a lot of connections, the probability that an open path for migration of EC exists is high. Otherwise, EC can not reach its target area easily through this solid media.

DISCUSSION 

Moving of cells within an environment needs scaffolding up on which they can migrate. The components of scaffolding depend on its environment. For example, in liver, ECM fibers act as scaffolding up on which hepatocytes migrate to renew themselves, after any injury caused necrosis of hepatocytes but has not altered lobular architecture. If this scaffolding alters, hepatocytes can not move regularly and they arise regenerative nodules, like cirrhosis. Arising of these nodules reflects the altered reconstruction of lobular architecture (13).

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During angiogenesis, tip EC also needs scaffolding that they can migrate through its pores. For example, in retina, neuron processes and glial cells are fixed in their places. Thus, it seems that they act as scaffolding in retinal layer for migration of sprout tip during angiogenesis. In normal condition, tip EC, passes through the pores within that scaffolding. Thus, retinal vasculature is mimicry of structure of glial cells and neuron processes. In diabetic retinopathy, neuronal, glial and pericyte damage might precede angiogenesis (6). Thus, that porous media surrounding the vessels changes and formation of new vessel sprouts is not similar to the primary vasculature.

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Also, a circular avascular zone at the fovea (6) indicates that this environment is solid enough that does not allow vessels to growth within it. Thus it can be concluded that this media has very high pc that the probability of the existence of an open path toward the center of the fovea is very low or even zero.In other conditions such as tumor growth, in McDougall et al. (2006), the probability of growth of the sprout tip in three possible directions in a twodimensional grid has been evaluated (14). That two-dimensional grid has been used for flow calculation from a parent vessel toward a tumor.

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Regardless the factors influence the probability of growth of the sprout tip through one connection of that grid, it seems that if that grid has more connections, the probability of arising an open path between parent vessel and tumor gets higher. Thus, by changing that environment and making it more solid which has fewer connections or pores, one can make the probability of existing an open path through which a new sprout migrates, lower. Thus, Percolation theory can be applied in this condition too.

CONCLUSION 

Local structure of surrounding media of a cell influences migration routes of it, like EC during angiogenesis. Thus, to supply blood to a target area, one can change the physical environment of a capillary network to a suitable physical one which allows new sprout tips to growth toward a target area more efficiently. Also, to inhibit angiogenesis, by changing the physical environment and making parent vessel’s environment more solid (e.g., reducing the pores within ECM) we might reach our goals.

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Acknowledgement 

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My colleagues in this project are Afsoon Fazelzadeh and Amir Norouzpour from Sadra–Sina Interdisciplinary Research Group, Mashad University of Medical Sciences (MUMS) We gratefully acknowledge Professor Chaplain for his valuable comments and discussion.