#### 2.1. General Theory

In the theories of weak scattering from random media, it has been shown [

13,

14,

15] that the differential scattering cross section per unit volume

${\sigma}_{d}\left(k\right)$ and the spatial correlation

$b\left(\widehat{r}\right)$ function of the inhomogeneities are related by

where

$k$ is the wavenumber,

$\widehat{r}$ is the vector separation between two points within the ensemble average, and

$A$ is a constant. Assuming the correlation function is isotropic and simply dependent on separation distance

$r$, the volume integral reduces to

similar to the integral found in Equations (4), (9), and (11) from Parker [

16] covering both acoustic and electromagnetic scattering under the Born approximation.

Significantly, Equations (1) and (2) conform the 3D Fourier transform in spherical coordinates with spherical symmetry, denoted as

${}^{3DS}\mathfrak{I}\left\{\right\}$ [

17,

18], and shown as follows using Bracewell’s convention

Thus, for isotropic 3D distributions, Equation (1) can be written as

Equation (4) demonstrates the important interpretation of scattering as a 3D spatial Fourier transform of the correlation function, assumed to be isotropic in this case. Thus, it is necessary to determine the spatial correlation function of the branching vasculature, beginning with a canonical cylindrical element, and this is determined in the next section.

#### 2.2. The Long Vessel

We represent a long vascular channel as a cylinder of radius

a,

where κ

_{0} is the fractional variation in density plus incompressibility, assumed to be

$\ll 1$ consistent with the Born formulation,

F(ρ) represents the Hankel Transform, which is the 2 dimensional Fourier transform of a radially symmetric function, and

ρ is the spatial frequency [

17,

19].

Assuming the fluid-filled cylinder is long in the

$z$-axis, then the shape is one-dimensional and its autocorrelation can be obtained from the inverse Hankel transform of the square of the shape’s Hankel transform

where

ρ is the spatial frequency. Furthermore, we assume that cylinders of this kind exist within a fractal geometry in an isotropic pattern. Therefore, copies of this are interrogated by the forward wave over all possible angles. Thus, we find that [

3] the spherically symmetric, isotropic transform

B_{s}(q) is given by

Equation (7) represents the ensemble average 3D transform of an isotropic set of cylinders of some radius. However, we must also consider the cross-correlation of an ensemble of these elements and all other (including larger and smaller) elements within a fractal structure needs to be derived. The simplest assumption is that each generation of cylinders has an autocorrelation function with itself that has been determined (above), and that within the ensemble average the cross terms with all other branches (larger and smaller within the fractal structure) is simply a small constant that is nearly invariant with position and therefore can be neglected except for spatial frequencies nearing zero. Under that very simplistic assumption, the overall autocorrelation function is given by the sum (or integral in the continuous limit) of the different sizes’ correlation functions over all generations of branches, weighted by their relative numbers (number density in the continuous limit). Fractal structures in 3D and volume filling in 3D may be characterized by number density functions

N(a) that are represented by

N_{0}/a

^{b}, where

a is the characteristic radius of the canonical element,

N_{0} is a global constant, and

b is the power law coefficient [

20].

Averaging over all sizes, in the 3D transform domain

and for our specific model, this becomes

where

${f}_{1}\left(b\right)$ is a function of

$b,$ and with reference to Equation (4), we find that the predicted backscatter is

Also, the autocorrelation function $B\left(r\right)$ is found from the inverse Fourier transform of Equation (9) to be $B\left(r\right)=C\cdot f\left(b\right)/{r}^{\gamma}$ and where $\gamma =b-3$ for the convergence of the inverse transform integral, $5>b>3$, and where $C$ is a constant. Thus, for example, if $b=3/2,$ an eigenfunction of the transform occurs: ${}^{3DS}\mathfrak{I}\left\{1/{r}^{3/2}\right\}=c/{q}^{3/2}.$

To combine these key relationships and restate them in terms of the fractal dimension D,

Table 1 provides a summary of the theories.