Flow-correlated dilution of a regular network leads to a percolating network during tumor-induced angiogenesis

Percolation in a distorted square lattice

This paper presents a Monte Carlo study of percolation in a distorted square lattice, in which the adjacent sites are not equidistant. Starting with an undistorted lattice, the position of the lattice sites are shifted through a tunable parameter α to create a distorted empty lattice. In this model, two occupied neighboring sites are considered to be connected to each other in order to belong to the same cluster, if the distance between them is less than or equal to a certain value, called connection threshold d. While spanning becomes difficult in distorted lattices as is manifested by the increment of the percolation threshold p_{c} with α, an increased connection threshold d makes it easier for the system to percolate. The scaling behavior of the order parameter studied through relevant critical exponents, and the fractal dimension d_{f} of the percolating cluster at p_{c} suggest that this new type of percolation may belong to the same universality class as ordinary percolation. This model can be very useful in various realistic applications since it is almost impossible to find a natural system that is perfectly ordered.

Physical determinants of vascular network remodeling during tumor growth

The process in which a growing tumor transforms a hierarchically organized arterio-venous blood vessel network into a tumor specific vasculature is analyzed with a theoretical model. The physical determinants of this remodeling involve the morphological and hydrodynamic properties of the initial network, generation of new vessels (sprouting angiogenesis), vessel dilation (circumferential growth), vessel regression, tumor cell proliferation and death, and the interdependence of these processes via spatio-temporal changes of blood flow parameters, oxygen/nutrient supply and growth factor concentration fields. The emerging tumor vasculature is non-hierarchical, compartmentalized into well-characterized zones, displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size, and transports drug injections efficiently. Implications for current theoretical views on tumor-induced angiogenesis are discussed.