  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, multi
scale collection of cylindrical scatterers governed by a power law distribution
relating 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.
