Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution

Version 1 : Received: 4 February 2022 / Approved: 8 February 2022 / Online: 8 February 2022 (13:59:05 CET)

A peer-reviewed article of this Preprint also exists.

Cheeseman, A.K.; Vrscay, E.R. Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution. Mathematics 2022, 10, 839. Cheeseman, A.K.; Vrscay, E.R. Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution. Mathematics 2022, 10, 839.

Journal reference: Mathematics 2022, 10, 839
DOI: 10.3390/math10050839

Abstract

Branching patterns are ubiquitous in nature, consequently over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties, however their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log-log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.

Keywords

Fractal dimension; self-similarity; box-counting method; sandbox method; fractal trees; canopies; vascular networks; branching structures

Subject

MATHEMATICS & COMPUTER SCIENCE, General Mathematics

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