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

Murray's law has been generalized to provide morphometric relationships among various subtrees as well as between a feeding segment and the subtree it perfuses. The equivalent resistance of each subtree is empirically determined to be proportional to the cube of a subtree's cumulative arterial length (L) and inversely proportional to a subtree's arterial volume (V) raised to a power of approximately 2.6. This relationship, along with a minimization of a cost function, and a linearity assumption between flow and cumulative arterial length, provides a power law relationship between V and L. These results, in conjunction with conservation of energy, yield relationships between 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 truncated tree model, is an actual reconstruction of the coronary arterial tree down to 500 µm in diameter. The second model, called the symmetric tree model, satisfies all mean anatomical data down to the capillary vessels. Our 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 between the various morphometric parameters of the truncated tree model may provide a basis for assessing the extent of diffuse coronary artery disease.