Objective: Vessel lumen calibres and flow rates are thought to be related by mathematical power laws, reflecting the optimization of cardiac versus metabolic work. While these laws have been confirmed indirectly via measurement of branch calibres, there is little data confirming power law relationships of flow distribution to branch calibres at individual bifurcations.
Approach: Flow rates and diameters of parent and daughter vessels of the internal carotid artery terminal bifurcation were determined, via robust and automated methods, from 4D phase-contrast magnetic resonance imaging and 3D rotational angiography of 31 patients.
Main results: Junction exponents were 2.06 ± 0.44 for relating parent to daughter branch diameters (geometrical exponent), and 2.45 ± 0.75 for relating daughter branch diameters to their flow division (flow split exponent). These exponents were not significantly different, but showed large inter- and intra-individual variations, and with confidence intervals excluding the theoretical optimum of 3. Power law fits of flow split versus diameter ratio and pooled flow rates versus diameters showed exponents of 2.17 and 1.96, respectively. A significant negative correlation was found between age and the geometrical exponent (r = -0.55, p = 0.003) but not the flow split exponent. We also found a dependence of our results on how lumen diameter is measured, possibly explaining some of the variability in the literature.
Significance: Our study confirms that, on average, division of flow to the middle and anterior cerebral arteries is related to these vessels' relative calibres via a power law, but it is closer to a square law than a cube law as commonly assumed.