The first order statistics of backscatter from the fractal branching vasculature

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