An optimization principle for vascular radius including the effects of smooth muscle tone

An optimization principle is proposed for the regulation of vascular morphology. This principle, which extends Murray's law, is based on the hypothesis that blood vessel diameter is controlled by a mechanism that minimizes the total energy required to drive the blood flow, to maintain the blood supply, and to support smooth muscle tone. A theoretical analysis reveals that the proposed principle predicts that the optimum shear stress on the vessel wall due to blood flow increases with blood pressure. This result agrees qualitatively with published findings that the fluid shear stress in veins is significantly smaller than it is in arteries.