Because of the heterogeneous nature of the mass density distribution, we modeled tissue as a random medium with continuously fluctuating mass density. The most comprehensive way to describe such a medium is by its correlation function (i.e., mass density correlation function).
25–27 We used a three-parameter functional family, the Whittle-Matérn (WM) function, to quantify the correlation functional form.
28 This versatile functional family covers essentially all of the commonly used correlation functions, including the power law, exponential, stretched exponential, and the Gaussian function. The functional family of WM is formulated as
11:
where
K{.} is a modified Bessel function of the second type,
An is the amplitude of the RI fluctuation,
D is the “shape factor” determining the type of the function,
27 and
Ln is the length scale of the correlation function whose exact meaning depends on
D. When 0 <
D < 3
, the correlation function can be described by a power law and the tissue is organized as a fractal.
25 Given the linear relationship between RI and mass density,
D is the mass density fractal dimension
Dm. When 3 <
D < 4
, the functional form becomes a stretched exponential function. For
D = 4,
Bn (
rd) is an exponential function, where
Bn (
Ln) =
Bn (0)
−1. When
D approaches infinity, the function turns into a Gaussian form.