Branching exponent heterogeneity and wall shear stress distribution in vascular trees
American Journal of Physiology
A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D 1)z = (D 2)z + (D 3)z;D 1 is the parent vessel diameter andD 2 and D 3 are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance of uniform shear stress. A mean z of 3 has been considered confirmation of such a control mechanism. The objective of the present study was to evaluate the consequences of a heterogeneous distribution of z values about the mean with regard to this uniform shear stress hypothesis. Simulations were carried out on model structures otherwise conforming to the criteria consistent with uniform shear stress whenz = 3 but with varying distributions of z. The result was that when there was significant heterogeneity inz approaching that found in a real arterial tree, the coefficient of variation in shear stress was comparable to the coefficient of variation in z and nearly independent of the mean value of z. A systematic increase in mean shear stress with decreasing vessel diameter was one component of the variation in shear stress even when the mean z = 3. The conclusion is that the influence of shear stress in determining vessel diameters is not, per se, manifested in a mean value of z. In a vascular tree having a heterogeneous distribution in zvalues, a particular mean value of z (e.g.,z = 3) apparently has little bearing on the uniform shear stress hypothesis.