Mathematical analysis on effect of non-Newtonian behavior of blood on optimal geometry of microvascular bifurcation system

Abstract

The principle of minimum work is utilized to interpret the optimal geometry of the vascular bifurcation. The present paper deals with the energy expenditure due to the rheology of blood and that for maintaining the metabolic states of the blood cells and of the vessel wall. It is of interest to mention that the optimal radii of the main and branch vessels and the optimal branching angle have been found to be related to the rhelogical parameters (the shear thinning property, the shear dependent non-linear consistency index and the yield stress) of blood and the two parameters which represent the morphologic and metabolic states of the blood and the vessel wall. In the special case of symmetrical bifurcation, Newton–Raphson method is adopted to compute numerical values of optimal relative radii and branching angle and it is observed that the optimal values for relative radius and branching angle decrease with an increase in the yield stress or the Power law index (n). An increase in the non-linear consistency index (k) leads to an increase in the relative radius and branching angle. Other important result is that for blood as Bingham fluid, the value of relative radius (or branching angle) decreases (or increases) rapidly as the value of yield stress increases whereas the former one (or the latter one) slowly decreases (or increases) when blood behaves like a Herschel–Bulkley fluid. It is pertinent to pin-point out here that the yield stress value of blood plays a pivotal role on the relative radius and branching angle, which is the new information added to the literature for the first time. For a given value of morphologic and morpho-metabolic parameters, the relative radius and branching angle are less for Herschel-Bulkley fluid as compared to that for Bingham, Power law and Newtonian fluids respectively.