Analysis of a Model for the Morphological Structure of Renal Arterial Tree: Fractal Structure

Abstract

One of the fields of applied mathematics is related to model analysis. Biomedical systems are suitable candidates for this field because of their importance in life sciences including therapeutics. Here we deal with the analysis of a model recently proposed by Espinoza-Valdez et al. (2010) for the kidney vasculature developed via angiogenesis. The graph theory allows one to model quantitatively a vascular arterial tree of the kidney in sense that (1) the vertex represents a vessels bifurcation, whereas (2) each edge stands for a vessel including physiological parameters. The analytical model is based on the two processes of sprouting and splitting angiogeneses, the concentration of the vascular endothelial growth factor (VEGF), and the experimental data measurements of the rat kidneys. The fractal dimension depends on the probability of sprouting angiogenesis in the development of the arterial vascular tree of the kidney, that is, of the distribution of blood vessels in the morphology generated by the analytical model. The fractal dimension might determine whether a suitable renal vascular structure is capable of performing physiological functions under appropriate conditions. The analysis can describe the complex structures of the development vasculature in kidney.