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The issue of speckle statistics from ultrasound images of soft tissues suchas the liver has a long and rich history. A number of theoreticaldistributions, some related to random scatterers or fades in optics and radar,have been formulated for pulse-echo interference patterns. This work proposesan alternative framework in which the dominant echoes are presumed to resultfrom Born scattering from fluid filled vessels that permeate the tissueparenchyma. These are modeled as a branching, fractal, self-similar, multiscale collection of cylindrical scatterers governed by a power law distributionrelating the number of branches at each radius. A deterministic accounting ofthe echo envelopes across the scales from small to large is undertaken, leadingto a closed form theoretical formula for the histogram of the envelope of theechoes. The normalized histogram is found to be related to the classical Burrdistribution, with the key power law parameter directly related to that of thenumber density of vessels vs. diameter, frequently reported in the range of 2to 4. Examples are given from liver scans to demonstrate the applicability ofthe theory.

The first order statistics of backscatter from the fractal branching vasculature