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

Blood flow rate ( Q ˙ ) in relation to arterial lumen radius (ri ) is commonly modelled according to theoretical equations and paradigms, including Murray's Law ( Q ˙ ∝ r i 3 ) and da Vinci's Rule ( Q ˙ ∝ r i 2 ). Wall shear stress (τ) is independent of ri with Murray's Law (τ ∝  r i 0 ) and decreases with da Vinci's Rule (τ ∝  r i - 1 ). These paradigms are tested empirically with a meta-analysis of the relationships between Q ˙ and ri in seven major arteries of the human cephalic circulation from 19 imaging studies in which both variables were presented. The analysis shows that Q ˙ ∝ r i 2.16 and τ ∝  r i - 1.02 , more consistent with da Vinci's Rule than Murray's Law. This meta-analysis provides standard values for Q ˙ , ri and τ in the human cephalic arteries that may be a useful baseline in future investigations. On average, the paired internal carotid arteries supply 75%, and the vertebral arteries supply 25%, of total brain blood flow. The internal carotid arteries contribute blood entirely to the anterior and middle cerebral arteries and also partly to the posterior cerebral arteries via the posterior communicating arteries of the circle of Willis. On average, the internal carotid arteries provide 88% of the blood flow to the cerebrum and the vertebral arteries only 12%.