Nonsymmetrical Bifurcations in Arterial Branching

I N T R O D U C T I O N It has long been suspected that the branching angles of blood arteries in the cardiovascular system are based on well-defined physiological principles. The cardiovascular system being basically a fluid-conducting system, these physiological principles are generally sought in the context of fluid dynamics. Several principles have so far been examined (Murray, 1926 b; Kamiya and Togawa, 1972; Zamir, 1976 a), and the results were recently put together for comparison (Zamir, 1976 b). Two of these principles propose that the branching angles are such that the lumen surface or lumen volume of the vessels involved in an arterial junction is minimum, and by simple analysis it then follows that the optimum branching angles are determined by the radii of the vessels involved. Another two principles propose that the branching angles are such that the pumping power required to drive the flow or the drag force on the endothelial surface is minimum, and then find, as a consequence, that the optimum angles depend on the flow in as well as the radii of the vessels involved. In the case of a symmetrical bifurcation the flow in each of the branches is known to be half that in the parent vessel, and therefore, comparison of the four optimality principles is possible. In the case of nonsymmetrical bifurcations however, such comparison is not possible unless one knows the relation between the flow in and the radius of a blood artery in general. This question has been studied in some detail in the past (Murray, 1926 a; Rodbard, 1975; Zamir, 1977), and a possible conclusion seems to be that the radius of a blood artery is proportional to the cube root of the flow which that artery is designed to convey, but the constant of proportionality may be different in different parts of the system. The purpose of the present study is to use this result in combination with those on branching angles in order to extend and apply these optimality studies to nonsymmetrical bifurcations. This is an important step J. GEN. PHYSIOL. @ The Rockefeller University Press 9 0022-1295/78/12/0837-0951.00. 837 Volume 72 December 1978 837-845 on Jne 4, 2017 D ow nladed fom Published December 1, 1978