Murray’s law and the bifurcation angle in the arterial micro-circulation system and their application to the design of microfluidics

Abstract

Murray’s law which is related to the bifurcations of vascular blood vessels states that the cube of a parent vessel’s diameter equals the sum of the cubes of the daughter vessels’ diameters $D_{0}^{3} = D_{1}^{3} + D_{2}^{3} ,\,\alpha = D_{0}^{3} /\left( {D_{1}^{3} + D_{2}^{3} } \right) = 1,$ where D ₀, D ₁, and D ₂ are the diameters of the parent and two daughter vessels, respectively and α is the ratio). The structural characteristics of the vessels are crucial in the development of the cardiovascular system as well as for the proper functioning of an organism. In order to understand the vascular circulation system, it is essential to understand the design rules or scaling laws of the system under a homeostatic condition. In this study, Murray’s law in the extraembryonic arterial bifurcations and its relationship with the bifurcation angle (θ) using 3-day-old chicken embryos in vivo has been investigated. Bifurcation is an important geometric factor in biological systems, having a significant influence on the circulation in the vascular system. Parameters such as diameter and bifurcation angle of all the 140 vessels tested were measured using image analysis softwares. The experimental results for α (= 1.053 ± 0.188) showed a good agreement with the ratio of 1 for Murray’s law. Furthermore, the diameter relation α approached the theoretical value of 1 as the diameter of parent vessel D ₀ decreased below 100 μm. The bifurcation angle θ decreased as D ₀ increased and vice versa. For the arterial bifurcations of chicken embryos tested in this study, the bifurcation pattern appears to be symmetric (D ₁ = D ₂). The bifurcation angle exhibited a nearly constant value of 77°, close to the theoretical value of 75° for a symmetric bifurcation.