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

Abstract

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.

Blood flow rate and wall shear stress in seven major cephalic arteries of humans

“…The major arteries, including those assessed in the present meta‐analysis, buffer pulsatile pressures from the heart and minimise energy loss to reflected waves (Savage et al, ; Newberry et al, ), whereas the peripheral arteries distribute the blood according to the demands of the tissues and reduce the velocity to allow time for gas exchange (Zamir, ). It is clear that Murray’s Law applies well in the peripheral arteries that are the major source of resistance to flow (Womersley, ; Sherman, ; Zamir et al, ; Savage et al, ; Reneman et al, ; McGah & Capobianchi, ). Low resistance in the major arteries is indicated by a small drop in mean blood pressure of approximately 5 mmHg between the aortic root and the MCA (Reymond et al, ).…”

“…This equation ignores actual viscosity at the wall, which may be slightly lower than the blood in general, as well as velocity changes during the cardiac cycle, blunted velocity profiles that occur in major arteries, and effects due to curves and bifurcations (Reneman et al, ). However, flow in the measured arteries is essentially laminar (Winkel et al, ) and effectively Newtonian (McGah & Capobianchi, ). If a full parabolic profile is assumed, according to Poiseuille flow conditions, the WSS may be underestimated (Dammers et al, ; Reneman et al, ).…”