On the design of the coronary arterial tree: a generalization of Murray's law

Murray's law has been generalized to provide morphometric relationships amo ng various subtrees as well as between a feeding segment and the subtree it perfuses. The equivalent resistance of each subtree is empirically determi ned to be proportional to the cube of a subtree's cumulative arterial lengt h (L) and inversely proportional to a subtree's arterial volume (V)raised t o a power of approximately 2.6. This relationship, along with a minimizatio n of a cost function, and a linearity assumption between flow and cumulativ e arterial length, provides a power law relationship between V and L. These results, in conjunction with conservation of energy, yield relationships b etween the diameter of a segment and the length of its distal subtree. The relationships were tested based on a complete set of anatomical data of the coronary arterial trees using two models. The first model, called the trun cated tree model, is an actual reconstruction of the coronary arterial tree down to 500 mu m in diameter. The second model, called the symmetric tree model, satisfies all mean anatomical data down to the capillary vessels. Ou r results show very good agreement between the theoretical formulation and the measured anatomical data, which may provide insight into the design of the coronary arterial tree. Furthermore, the established relationships betw een the various morphometric parameters of the truncated tree model may pro vide a basis for assessing the extent of diffuse coronary artery disease.