# Fractal Analysis: Angiogenesis

The fractal dimension D and the lacunarity are frequently used to characterize complex vascular networks, although they are not perfect fractals. Angiogenetic sprouting results in network with increasing complexity and irregularity which represents fractal-like features. Fractal analysis seems to be a sensitive tool to characterize increasing complexity and may provide additional information on the cellular networks.

For example, fractal analysis was applied to a skeleton extracted from an image of in vitro angiogenesis (A).

The fractal dimension was calculated by the common box-counting method as D = 1.15, while lacunarity was 0.54.

Analyzing a second image (B) with more complex structures resulted in the fractal dimension of D = 1.48 and a lacunarity of 0.46. Analysis of a third image (C) resulted in the fractal dimension of D = 1.48 and a lacunarity of 0.53.

The results reflect the visual impression. Skeleton A is rather regular and fractal dimension approaches a value of 1 which is that of a straight line. The topological parameters as outlined on the angiogenesis sites are highly correlated to the results for fractal dimension. For instance, the total length of skeleton A is 2.8 mm, while B and C accounted for 10.4 and 11.5 mm, respectively.

As shown for the total skeleton length, the topological results for image B and C have only small differences, reflected by the nearly same fractal dimension, but image C seems to be more complex than image B.

This is best reflected by the different lacunarity and fractal analysis may therefore indeed account for additional results not seen in analyzes of topological parameters.