Abstract
Regional blood flows in the heart muscle are remarkably heterogeneous. It is very likely that the most important factor for this heterogeneity is the metabolic need of the tissue rather than flow dispersion by the branching network of the coronary vasculature. To model the contribution of tissue needs to the observed flow heterogeneities we use arterial trees generated on the computer by constrained constructive optimization. This method allows to prescribe terminal flows as independent boundary conditions, rather than obtaining these flows by the dispersive effects of the tree structure. We study two specific cases: equal terminal flows (model 1) and terminal flows set proportional to the volumes of Voronoi polyhedra used as a model for blood supply regions of terminal segments (model 2). Model 1 predicts, depending on the number N term of end-points, fractal dimensions D of perfusion heterogeneities in the range 1.20 to 1.40 and positively correlated nearest-neighbor regional flows, in good agreement with experimental data of the normal heart. Although model 2 yields reasonable terminal flows well approximated by a lognormal distribution, it fails to predict D and nearest-neighbor correlation coefficients r 1 of regional flows under normal physiologic conditions: model 2 gives D = 1.69 ± 0.02 and r 1 = −0.18 ± 0.03 (n = 5), independent of N term and consistent with experimental data observed under coronary stenosis and under the reduction of coronary perfusion pressure. In conclusion, flow heterogeneity can be modeled by terminal positions compatible with an existing tree structure without resorting to the flow-dispersive effects of a specific branching tree model to assign terminal flows.
Highlights
It is widely accepted that regional blood flows in organs and tissues, including the heart, the lung and skeletal muscles, are remarkably heterogeneous (Yipintsoi et al, 1973; Glenny and Robertson, 1990; Iversen and Nicolaysen, 1995; Sonntag et al, 1996)
Regional flows per gram of uniformly sized tissue pieces have been documented to vary 6–10-fold with only small fluctuations over time (King and Bassingthwaighte, 1989), and recent experimental results suggest that perfusion heterogeneity is maintained even down to microscopic levels (Matsumoto et al, 1996; Glenny et al, 2000; Bauer et al, 2001; Kalliokoski et al, 2001)
We have studied two models for the distribution of terminal flows: Model 1 employs equal terminal flows and predicts, depending on the specific setting of the number Nterm of end-points, fractal dimensions of perfusion heterogeneities in the range from 1.20 to 1.40 and positively correlated nearest-neighbor regional flows in good agreement with experimental data observed under normal physiological conditions
Summary
It is widely accepted that regional blood flows in organs and tissues, including the heart, the lung and skeletal muscles, are remarkably heterogeneous (Yipintsoi et al, 1973; Glenny and Robertson, 1990; Iversen and Nicolaysen, 1995; Sonntag et al, 1996). On a double logarithmic scale Eq 1 yields a linear relationship between the logarithm of RD and the logarithm of piece size with slope (1 Ϫ D), i.e., RD exhibits self-similarity upon scaling with respect to the size of sample pieces. In this sense, Eq 1 represents a fractal relationship and the parameter D, which is a global measure of heterogeneity, is identified as a spatial fractal dimension (Mandelbrot, 1983; Bassingthwaighte et al, 1994). An average D of ف1.2 has been found (Bassingthwaighte et al, 1989), suggesting that heterogeneity is not random: Flows in adjacent tissue samples are positively correlated
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