Small artery occlusion: a theoretical approach to the definition of coronary architecture and resistance by a branching tree model.

The aim of this study was to evaluate the theoretical possibility of assessing the architecture of the coronary small artery circulation in vivo by the analysis of the increments in the total resistance of the vascular system caused by the progressive occlusion of the terminal vessels as would be possible in animal experiments by microsphere embolization. Different distributions of the resistance values of all the vessels have been obtained in branching tree models by means of only two parameters: (a) the resistance ratio between daughter and parent vessels at each branching site (KO); and (b) the resistance ratio between the two daughter vessels at each branching site (KV). Simulation of branching tree occlusion has been performed under two main conditions of resistance distribution: (1) symmetric resistance distribution, characterized by equal KO values at each branching site of the same level and by KV = 1, that is, equal resistance values for the vessel of the same level; and (2) asymmetric resistance distribution in dichotomous branching trees, wherein all the vessels may have different values of resistance; the variability in these values has however been restricted on the basis of physiological considerations. The analysis of the function of the total resistance vs the number of occluded vessels, obtained by a simulated progressive occlusion of the terminal vessels in these two systems, gives the following results: (1) in a symmetric branching tree, discontinuities are present in the occlusion function which permit identification of both the architecture and the resistance value of each single vessel of any unknown vascular tree; and (2) in the asymmetric model, the function does not allow a direct definition of the branching architecture and the values of resistance of each vessel; however, also in this case, any branching tree can be analyzed by means of a nonlinear optimization procedure which produces an equivalent symmetric branching tree. We conclude that, theoretically, the analysis of the occlusion function represents a valuable indirect approach to the quantitative study of the coronary microcirculation as well as of other vascular districts under steady flow conditions.