The first order statistics of backscatter from the fractal branching vasculature

TL;DR

This paper develops a fractal-based model for ultrasound backscatter from tissue vasculature, deriving a Burr distribution for echo envelope statistics that aligns with liver scan data.

Contribution

It introduces a novel fractal scattering framework that links vessel distribution to ultrasound speckle statistics, providing a closed-form theoretical formula.

Findings

The envelope histogram follows a Burr distribution.

The model accurately fits liver scan data.

Power law parameters relate to vessel density distribution.

Abstract

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.