Properties of Arterial Trees

Although arterial trees in the various organs have markedly differing morphologies, they are all similar in several important respects: They are space-filling, asymmetrical, optimal in some sense, and self-similar, and their terminal elements, the capillaries, are all equivalent. Space-filling denotes the notion — dictated by the physics of diffusion, a delivery mechanism effective over only very short distances (tens of microns) — that the arterial tree must supply all viable tissues with the required amount of nutrients and oxygen via the blood. Thus, if one envisions a single inlet (trunk of the tree) to a 3D space, the tree must branch in such a way that the capillary density is relatively uniform throughout the organ. The general appearance of arterial trees, therefore, is characterized by several large distribution branches that carry high flows of blood to tissue volumes remote from the inlet, and a larger number of smaller delivering branches or subtrees that carry smaller volumes of blood from the high-flow conduits to the capillaries.

Teleological principles would suggest that natural processes are directed toward the end of physiological efficiency, and it is generally acknowledged that the structure of arterial trees has evolved toward some sort of optimality. In his classic book,

Thompson suggested that the cost of operating a given section of arteries is a combination of the cost of the power required to overcome its resistance and the cost of supplying the blood to fill it [13], In 1842 Sir Charles Bell had pointed out that the laws of hydraulics, which take into account only the first of these factors, are inadequate to explain the structure of the arterial tree and that the cost of blood must be taken into account, For example, if blood were free and the diameter of all arteries were doubled, quadrupling their volume, the work of the heart would be reduced to one-sixteenth. Conversely, if blood were more expensive than it is, the total arterial volume would have evolved to become smaller, increasing the work of the heart. A variety of cost functions could conceivably drive this evolution, but the four most often suggested have been the power required to pump blood through a bifurcation; the shear or drag force exerted by the blood on the endothelial vessel lining; the total blood volume; and the total lumen surface [4,14-17], Given three segment diameters at a bifurcation, the magnitude of each of these cost functions depends on the branching angles 01 and 02; or given two branching angles, the cost functions vary with the diameters. The diameters and angles that produce the minimum value of the assumed cost function are the optimal geometrical parameters. Although it is beyond the scope of this chapter to summarize all the research in this area, Zamir found, for the case of two human coronary arterial trees when considering a cost function combining all four of the above factors, that the branching angles and branch diameters were strikingly close to optimal across the entire range of bifurcation indices. In general, total (01 plus 02) branching angles, which would be expected to approach 90° to minimize shear stress, tend to be somewhat below that on average — around 57° for rat and 70° for human coronary arterial trees [4,14,16],

Symmetric bifurcations tend to have smaller branching angles, whereas highly asymmetric (small) side branches tend to come off closer to 90°.

0 0

Post a comment