Research Paper J Vasc Res 2000;37:282–296 The Fractal Nature of Myocardial Blood Flow Emerges from a Whole-Organ Model of Arterial Network Daniel A. Beard James B. Bassingthwaighte University of Washington, Center for Bioengineering, Seattle, Wash., USA Received: October 10, 1998 Accepted after revision: March 23, 2000 Dr. James B. Bassingthwaighte University of Washington Bioengineering Department, Box 357962 Seattle, WA 98195-7962 (USA) Tel. +1 206 685 2005, Fax +1 206 685 2651, E-Mail jbb@bioeng.washington.edu ABC Fax + 41 61 306 12 34 E-Mail karger@karger.ch www.karger.com © 2000 S. Karger AG, Basel Accessible online at: www.karger.com/journals/jvr Key Words Regional blood flows in heart W Spatial distribution W Autocorrelation W Power-law washout W Fractal dimension W Arterial-arteriolar network W Microvasculature W Vasoregulation W Coronary blood vessels W Capillary-tissue exchange Abstract Mammalian hearts exhibit a heterogeneous spatial dis- tribution of blood flows, but flows in near-neighbor regions correlate strongly. Also, tracer 15 O-water wash- out 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 con- struct a mathematical model of the coronary arterial net- work 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 ob- served in nature, with maximal resistances in small ves- sels. In addition, the washout of intravascular tracer showed tails with power law slopes that fitted h(t) = at –·–1 with the exponents · = 2 for the reconstructed net- works compared with those from experimental outflow concentration-time curves with · = 2.1 B 0.3. Thus, we concluded that the fractal nature of spatial flow distribu- tion in the heart, and of temporal intravascular washout, are explicable in terms of the morphometry of the coro- nary network. Copyright © 2000 S. Karger AG, Basel Introduction Substances carried in the blood are dispersed in their traversal of the vascular network by a number of pro- cesses: (1) different parallel paths have different transit times, a result of heterogeneities in path lengths, vessel volumes, and flows [1]; (2) intravascular velocity profiles are dispersive because substances or particles located near the center of the tube travel faster than those near the wall [2–4]; (3) axial diffusion contributes to spreading along a tube while radial diffusion reduces the degree of axial spreading due to velocity gradients [5]; (4) erythrocyte rotation in blood augments molecular diffusion [6]; (5) pulsatile flow, eddies, and vortices in curved and branching vessels spread transit time distribution [7, 8]. Downloaded by: University of Washington 128.208.17.4 - 6/20/2013 2:54:21 AM