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

Alison K. Cheeseman and Edward R. Vrscay
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Alison K. Cheeseman: Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Edward R. Vrscay: Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

Mathematics, 2022, vol. 10, issue 5, 1-21

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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