The Fractal Nature of Myocardial Blood Flow Emerges from a Whole-Organ Model of Arterial Network

Abstract

Mammalian hearts exhibit a heterogeneous spatial distribution of blood flows, but flows in near-neighbor regions correlate strongly. Also, tracer ¹⁵O-water washout after injection into the inflow shows a straight log-log relationship between outflow concentration and time. To uncover the role of the arterial network in governing these phenomena, morphometric data were used to construct a mathematical model of the coronary arterial network of the pig heart. The model arterial network, built in a simplified three-dimensional representation of tissue geometry, satisfies the statistical morphometric data on segment lengths, diameters and connectivities reported for real arterial networks. The model uses an avoidance algorithm to position successive vascular segments in the network. Assuming flows through the network to be steady, the calculated regional flow distributions showed (1) the degree of heterogeneity observed in normal hearts; (2) spatial self-similarity in local flows; (3) fractal spatial correlations, all with the same fractal dimension found in animal studies; (4) pressure distributions along the model arterial network comparable to those observed in nature, with maximal resistances in small vessels. In addition, the washout of intravascular tracer showed tails with power law slopes that fitted h( t) = at^–α–1 with the exponents α = 2 for the reconstructed networks compared with those from experimental outflow concentration-time curves with α = 2.1 ± 0.3. Thus, we concluded that the fractal nature of spatial flow distribution in the heart, and of temporal intravascular washout, are explicable in terms of the morphometry of the coronary network.