Vascular dimensions of the cerebral arteries follow the principle of minimum work

Abstract

Background and purpose: The principle of minimum work is a parametric optimization model for the growth and adaptation of arterial trees. It establishes a balance between energy dissipation due to frictional resistance of laminar flow (shear stress) and the minimum volume of the vascular system, implying that the radius of the vessel is adjusted to the cube root of the volumetric flow. The purpose of this study is to verify whether the internal carotid artery system obeys the principle of minimum work.

Methods: Measurements of the radius of parent and branch segments of the internal carotid, anterior, and middle cerebral arteries were performed on analog angiographs chosen at random from a set classified as normal. The branch angles were measured from lateral projections in bifurcations of the anterior cerebral artery. The relation of the calibers of parent and branch vessels was analyzed.

Results: The area ratio of the bifurcations (N = 174) was 1.2 +/- 0.4 (mean +/- SD). The equation (r0)n = (r1)n + (r2)n was solved for n, resulting in n = 2.9 +/- 0.7 (mean +/- SD, N = 157). Optimum proportions between the radii of parent (r0) and branch (r1 and r2) vessels in the internal carotid artery system were verified in normal carotid angiographs up to four branch generations, according to the theoretical equation r0(3) = r1(3) + r2(3) (r = 0.989, N = 174). No clear correlation was found between the measured branch angles, the relative branch cross-sectional area, and the theoretical optimum angles.

Conclusions: This study demonstrates that the process of branching of the internal carotid artery system obeys the principle of minimum work, as the diameter exponent approximates 3. The principle of minimum work establishes strict functional relations between volumetric flow, flow velocity, and vessel radius. This model was extended to parametric optimization of branch angles, which has proved irrelevant in terms of functional optimization. Our results corroborate this finding. Shear stress-induced endothelial mediation seems to be the regulating mechanism for the maintenance of this optimum vessel design. The magnitude of wall shear stress is the same at every point in a vascular network obeying the principle of minimum work, because the flow rate influences the shear stress proportionally to the third power of the vessel radius. This observation has implications for understanding the remodeling of the cerebral vascular network in the presence of arteriovenous malformations and for the pathogenesis of saccular aneurysms.