Branching Systems of Fractal Vascular Trees

Abstract

The branching systems in our body (vascular and bronchial trees) and those in the natural world (plants, trees, and rivers) are characterized by a fractal nature: self-similar branching patterns and recursive bifurcations. These branching networks have the increasing density of branches toward the terminals with decreases in branch radius to the –Dth power: D is termed the fractal dimension. We have devised the primary expression ${N_{\mathrm{ b}}}(r)={{\left( {{r \left/ {{{r_{\mathrm{ o}}}}} \right.}} \right)}^{{-D-\alpha }}}$ that provides the number of branches in a group with a radius r in a tree, where r _o is the radius of the stem and α is the exponent in the branch length–radius relation. In the branching network, the mean blood flow rate and velocity in a given vessel with radius r can be expressed as ${F_{\mathrm{ b}}}(r)={F_{\mathrm{ b}\mathrm{ o}}}{{\left( {{r \left/ {{{r_{\mathrm{ o}}}}} \right.}} \right)}^{{D+\alpha }}}$ and ${U_{\mathrm{ b}}}(r)={U_{\mathrm{ b}\mathrm{ o}}}{{\left( {{r \left/ {{{r_{\mathrm{ o}}}}} \right.}} \right)}^{{D+\alpha -2}}}$, where F _bo is the total flow through the stem vessel of the network. Analogously, various hydrodynamic parameters, such as wall shear rate, shear stress, and intravascular pressure, are written as a function of vessel radius in a given position within the branching network. The validity of these expressions was verified by the comparison between the outcomes from the simulation and in vivo measurements from various vascular beds. For the power law, the so-called Murray’s law, it is clarified that the bifurcation exponent is equal to the sum of the fractal dimension and the branch length exponent. For allometric studies of the vascular system in mammalians, the distribution of the arteriolar ends of the capillaries in any organ is uniform independently of animal size, and then the difference in body size of mammals is attributable to the number of the basic units of the capillary and the surrounding tissue. Finally, an infarction index, the ratio of the number of the terminal branches downstream from an obstructed artery to that of the total terminal branches of a vascular tree, is also used to quantify the degree to which an organ has ischemic damage.