A Modification of Murray's Law for Shear-Thinning Rheology

This study reformulates Murray's well-known principle of minimum work as applied to the cardiovascular system to include the effects of the shear-thinning rheology of blood. The viscous behavior is described using the extended modified power law (EMPL), which is a time-independent, but shear-thinning rheological constitutive equation. The resulting minimization problem is solved numerically for typical parameter ranges. The non-Newtonian analysis still predicts the classical cubic diameter dependence of the volume flow rate and the cubic branching law. The current analysis also predicts a constant wall shear stress throughout the vascular tree, albeit with a numerical value about 15-25% higher than the Newtonian analysis. Thus, experimentally observed deviations from the cubic branching law or the predicted constant wall shear stress in the vasculature cannot likely be attributed to blood's shear-thinning behavior. Further differences between the predictions of the non-Newtonian and the Newtonian analyses are highlighted, and the limitations of the Newtonian analysis are discussed. Finally, the range and limits of applicability of the current results as applied to the human arterial tree are also discussed.