Application of Fractal Analysis to Evaluate the Rat Brain Arterial System
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LEX SYSTEMS BIOPHYSICS
Application of Fractal Analysis to Evaluate the Rat Brain Arterial System V. S. Kopylovaa, *, S. E. Boronovskiya, and Ya. R. Nartsissova aInstitute
of Cytochemistry and Molecular Pharmacology, Moscow, 115404 Russia *e-mail: [email protected]
Received November 29, 2019; revised November 29, 2019; accepted February 27, 2020
Abstract—Vascular networks possess properties of self-similarity, which allows one to consider them as stochastic fractals. The box-counting method based on calculations along the vessel centerlines is traditionally used to evaluate the parameters of the fractal structure. Such an algorithm does not allow one to consider structural differences between different bifurcation levels of the system, characterized by the natural property of changing the blood vessel caliber. In this case, the discrepancy between the values of the fractal dimension may exceed 20%. In this paper, an approach that allows one to avoid underestimating the complexity of the system for low bifurcation orders and large vessels is proposed. Based on the constructed arterial tree of the rat brain, it was shown that the fractal dimension increases with an increase in the values of both bifurcation exponent and length coefficient. The obtained values most fully reflect the properties of the arterial tree considering the real geometry of the vessels; they are proposed for use in estimating three-dimensional vascular networks. Keywords: arterial system, bifurcation of blood vessel, computer modeling, fractal dimension DOI: 10.1134/S0006350920030100
INTRODUCTION The main function of the vascular system is to provide all cells of the body with oxygen and other vital metabolites. For this to occur most effectively, the arterial tree should be a branching system, while bifurcation is the most common form of division at each step [1]. Due to the extreme complexity and multilevel topology of the vascular network, there is no unequivocal opinion about which parameters to use for describing the structure of blood vessels. In addition, there is a demand for a criterion of normal development that allows one to diagnose diseases. To solve these problems, fractal analysis has been used to evaluate various healthy and pathological circulatory systems [2, 3]. Vascular networks are not strictly fractal, since they do not exhibit large-scale invariance in the infinite range; however, they have self-similarity properties, since the branching process is the same at each stage, therefore, the vascular system is fractal in nature and can be considered as a pseudofractal [4]. It was shown that at least the arterial system of the brain is a combination of two components: a capillary network that fills the space uniformly and a branched fractal structure of larger vessels [5]. In fractal geometry, the properties of self-similar structures observed in a wide spectrum of successive bifurcations are quantified using fractal dimension (FD), which is a measure of the complexity of structures [6]. It can be considered
as a qu
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