Optimum Geometric Bifurcation under Pulsating Flow Assuming Minimum Energy Consumption in Cardiovascular System, an Extension on Murray’s Law

In a bifurcation including a mother artery and two daughter arteries, the energy drop is minimum, if, the cube of the radius of the mother artery equals the sum of the cube of the radii of daughter arteries. This is the expression of Murray’s law (or cubic law) assuming the flow is steady. In this paper, an extension of Murray’s law is investigated using the minimum energy hypothesis, totally analytical for pulsating flow. In addition to the two terms that Murray considered in his calculations, there is additional energy to move fluid toward and back in the pulsating flow. This additional energy is calculated and added to two other parts of energy in Murray’s analysis, and then optimized. The relationships for diameters and the angle between daughter arteries are extended. The effect of frequency and Womersley number have appeared as coefficients in the relations. According to the results, the most difference between Murray’s law for both diameters and the angle between daughter arteries, and the relationship derived in the present paper, occurs in Womersley number between 2 and 5. For a special case which in the daughter arteries have the same diameter, the power of diameters varies up from 3 to 3.2. Also, for this special case, there is maximum 6 degrees difference with Murray’s law for the angle between daughter arteries. In short, the obtained relations, assuming pulsating flow, do not yield very different results from Murray's law assuming steady flow.