Ideal chain on a two-dimensional critical percolation cluster

Dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool is used to study tumor tissue perfusion. The results are then analyzed using percolation models. Percolation cluster geometry is depicted using the wash-in component of MRI contrast signal intensity. Fractal characteristics are determined for each two-dimensional cluster. The invasion percolation model is used to describe the evolution of the tumor perfusion front. Although tumor perfusion can be depicted rigorously only in three dimensions, two-dimensional cases are used to validate the methodology. It is concluded that the blood perfusion in a two-dimensional tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. For all the cases studied, the front starts to grow from the periphery of the tumor (where the feeding vessel was assumed to lie) and continues to grow toward the center of the tumor, accounting for the well-documented perfused periphery and necrotic core of the tumor tissue.

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Estimates of critical percolation probabilities for a set of two-dimensional lattices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100050325 ◽

1972 ◽

Vol 71 (1) ◽

pp. 97-106 ◽

Cited By ~ 16

Author(s):

D. G. Neal

Keyword(s):

Monte Carlo ◽

Two Dimensions ◽

Two Dimensional ◽

Critical Percolation ◽

Site Percolation

AbstractThis paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

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Exact factorization of correlation functions in two-dimensional critical percolation

Physical Review E ◽

10.1103/physreve.76.041106 ◽

2007 ◽

Vol 76 (4) ◽

Cited By ~ 10

Author(s):

Jacob J. H. Simmons ◽

Peter Kleban ◽

Robert M. Ziff

Keyword(s):

Correlation Functions ◽

Two Dimensional ◽

Critical Percolation ◽

Exact Factorization

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The percolation process on a tree where infinite clusters are frozen

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004199004326 ◽

2000 ◽

Vol 128 (3) ◽

pp. 465-477 ◽

Cited By ~ 9

Author(s):

DAVID J. ALDOUS

Keyword(s):

Binary Tree ◽

Percolation Cluster ◽

Critical Percolation ◽

Percolation Process ◽

Infinite Clusters ◽

The Law

Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t > ½ does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t > ½ has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer [12].

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