The pattern of coronary arteriolar bifurcations and the uniform shear hypothesis

By minimizing the cost function, which is the sum of the friction power loss and the metabolic energy proportional to blood volume, Murray derived an optimal condition for a vascular bifurcation. Murray's law states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughters. We tested Murray's law against our data of pig's maximally vasodilated coronary arteriolar blood vessels at bifurcation points in control and hypertensive ventricles. Data were obtained from 7 farm pigs, 4 normal controls and 3 with right ventricular hypertrophy induced by stenosis of a pulmonary artery. Data on coronary arteriolar bifurcations were obtained from histological specimens by optical sectioning. The experimental results show excellent agreement with Murray's law in control and hypertensive hearts. Theoretically, we show that Murray's law can be derived alternatively as a consequence of the uniform vessel-wall shear strain rate hypothesis and a fluid mechanics equation based on conservation of mass and momentum. Conversely, the fluid mechanical equation, together with Murray's law, established as an empirical equation of actual measurements implies the uniformity of the shear strain rate of the blood at the vessel wall throughout the arterioles. The validity of these statements is discussed.