Fractal nature of regional myocardial blood flow heterogeneity

Spatial variation in regional flows within the heart, skeletal muscle, and in other organs, and temporal variations in local arteriolar velocities and flows is measurable even with low resolution techniques. A problem in the assessment of the importance of such variations has been that the observed variance increases with increasing spatial or temporal resolution in the measurements. This resolution-dependent variance is now shown to be described by the fractal dimension, D. For example, the relative dispersion (RD = SD/mean) of the spatial distribution of flows for a given spatial resolution, is given by: RD(m) = RD(mref).[m/mref]1-Ds where m is the mass of the pieces of tissue in grams, and the reference level of dispersion, RD(mref), is taken arbitrarily to be the RD found using pieces of mass mref, which is chosen to be 1 g. Thus, the variation in regional flow within an organ can be described with two parameters, RD(mref) and the slope of the logarithmic relationship defined by the spatial fractal dimension Ds. In the heart, this relation has been found to hold over a wide range of piece sizes, the fractal Ds being about 1.2 and the correlation coefficient 0.99. A Ds of 1.2 suggests moderately strong correlation between local flows; a Ds = 1.0 indicates uniform flow and a Ds = 1.5 indicates complete randomness.