Asymmetrically Branching Microvascular Networks

Abstract

To examine the influences of asymmetrical bifurcations on such hemodynamics as blood flow, velocity, pressure, wall shear stress, and circumferential wall stress, a numerical simulation of the hemodynamics was made with the theoretical models for asymmetrical branching networks of the human retina. The dichotomous branching vessels at asymmetrical bifurcations are defined by a set of two indices: a decrement index DI S/M, the ratio (\( {{{{r_{\mathrm{ S}}}}} \left/ {{{r_{\mathrm{ M}}}}} \right.} \)) of the radii of the smaller daughter branch to the mother vessel, and a symmetry index \( S{I_{{{{\mathrm{ S}} \left/ {\mathrm{ L}} \right.}}}} \), the ratio (\( {{{{r_{\mathrm{ S}}}}} \left/ {{{r_{\mathrm{ L}}}}} \right.} \)) of the radii of the smaller daughter to the larger daughter branch. MCT values for the asymmetrically branching networks lie between 2.87 and 3.16 s. The values are in good agreement with the experimental data on human retinal circulation. The distribution of mean blood flow within the asymmetrical networks can be expressed as \( {f_{{\mathrm{ S}\uppsi, \mathrm{ L}\upomega}}}={1 \left/ {2} \right.}DI_{{{{\mathrm{ S}} \left/ {\mathrm{ M}} \right.}}}^{{2.85(\uppsi \rm{-}2)}}{{\left( {{{{D{I_{{{{\mathrm{ S}} \left/ {\mathrm{ M}} \right.}}}}}} \left/ {{S{I_{{{{\mathrm{ S}} \left/ {\mathrm{ L}} \right.}}}}}} \right.}} \right)}^{{2.85(\upomega -2)}}}{f_1} \) (ψ, ω ≥ 2, ω = g ‒ ψ), where \( {f_1} \) is the mean flow in the trunk vessel of the first generation and ψ and ω are the generation numbers of smaller and larger daughter branches, respectively. According to conservation of flow, the arteriovenous distributions of other hemodynamic variables within the asymmetrically branching networks are also given as a function of vessel radius in the following: blood flow, \( f={{\left( {{r \left/ {{{r_1}}} \right.}} \right)}^m}{f_1} \); flow velocity, \( \bar{v}={{\left( {{r \left/ {{{r_1}}} \right.}} \right)}^{m-2 }}{{\bar{v}}_1} \); wall shear stress, \( {\tau_{\mathrm{ w}}}={{{4\mu {{{({r \left/ {{{r_1}}} \right.})}}^{m-3 }}{{\bar{v}}_1}}} \left/ {{{r_1}}} \right.} \); and pressure drop, \( \Delta P=8\upmu \mathrm{ l}{{{{{{\left( {{r \left/ {{{r_1}}} \right.}} \right)}}^{m-4 }}{f_1}}} \left/ {{\left( {\pi r_1^4} \right)}} \right.} \), where m = 2.85 for the retinal vasculature is the bifurcation exponent and the apparent viscosity of blood is \( \mu ={0.043 \left/ {{{{{(1+{4.29 \left/ {r} \right.})}}^2}}} \right.} \) for arterioles and \( \mu ={0.046 \left/ {{{{{(1+{4.29 \left/ {r} \right.})}}^2}}} \right.} \) for venules. However, it is impossible to express intravascular pressure in a given vessel as a function of the vessel radius within a symmetrical or asymmetrical network without the sequential determinations of pressure reduction in the individual vessels from upstream to downstream.