# The 3D Spatial Autocorrelation of the Branching Fractal Vasculature

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## Abstract

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## 1. Introduction

#### Fractal Structures and Scattering

## 2. Theory of Scattering Applied to Fractal Branching Networks

#### 2.1. General Theory

#### 2.2. The Long Cylindrical Shape

_{z}but shown with finite thickness to make the graphic easier to draw and visualize). Its transform is symmetric and given by the Hankel transform of order zero $F\left(\rho \right)=\mathscr{H}\left\{f\left(r\right),\rho \right\}.$ A particular radius of value of ${q}_{0}$ is shown for reference.

## 3. Methods

^{−1}with a finger pump. Flow rate was adjusted to maintain fetal vessel pressure at ~60 mmHg. The fetal perfusate consisted of M199 media without phenol red (Gibco) modified by the addition of dextran (35–45 kDa; 30 mg/mL fetal), D-Glucose (2 mg/mL), sulfamethoxazole (80 µg/mL), trimethoprim (16 µg/mL) gentamicin (52 µg/mL), and heparin (20 USP IU/mL). The fetal perfusate was gassed with 20% O

_{2}/75% N

_{2}/5% CO

_{2}in a 250 mL vessel and bubbles trapped before delivery to the placenta. The cannulated placenta was placed in a plastic bag and immersed in a 37 °C water bath for an hour followed by Doppler and ultrasound elastography experiments as described in McAleavey, et al. [22]. At the conclusion of these experiments, the placenta was perfused with a 37 °C suspension of 30% barium sulfate in 1% agarose prepared from a 60% emulsion oral contrast suspension (Barium-Liquid E-Z-Pague; Bracco) diluted with a 2% agarose in water solution with a gelling temperature of 35 °C. Perfusion continued until no further change was apparent. The placenta was then immersed in 10% neutral buffered formalin for fixation before imaging with a Philips Brilliance 64 computerized axial tomography system. The slice dimension was 768 × 768 pixels, each 0.25 × 0.25 mm; slice thickness: 0.67 mm, spacing between slices: 0.33 mm.

## 4. Results

## 5. Discussion

#### 5.1. Consequence of the Model and Power Law

#### 5.2. Refinement of Theory vs. Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Particular Form of Fourier Transform Pairs

- (a)
- Fourier transform in one dimension:$$\begin{array}{c}F\left(s\right)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}f\left(x\right){\mathbf{e}}^{-i2\pi xs}dx}=\Im \left\{f\left(x\right)\right\}\\ f\left(x\right)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}F\left(s\right){\mathbf{e}}^{+i2\pi xs}ds}\end{array}$$
- (b)
- Fourier transform/Hankel transform for cylindrical symmetry:$$\begin{array}{c}F\left(\rho \right)=2\pi {\displaystyle \underset{0}{\overset{\infty}{\int}}f\left(r\right){J}_{0}\left(2\pi \rho r\right)rdr}=\mathscr{H}\left\{f\left(r\right)\right\}\\ f\left(r\right)=2\pi {\displaystyle \underset{0}{\overset{\infty}{\int}}F\left(\rho \right){J}_{0}\left(2\pi \rho r\right)\rho d\rho}\end{array}$$
- (c)
- Three-dimensional Fourier transform with spherical symmetry:$$\begin{array}{c}F\left(q\right)=\frac{2\pi}{q}{\displaystyle \underset{0}{\overset{\infty}{\int}}f\left(r\right)\mathrm{sin}\left(2qr\right)rdr}={\text{}}^{3DS}\Im \left\{f\left(r\right)\right\}\\ f\left(r\right)=\frac{2\pi}{r}{\displaystyle \underset{0}{\overset{\infty}{\int}}F\left(q\right)\mathrm{sin}\left(2qr\right)qdq}\end{array}$$

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**Figure 1.**A cylindrical function (

**a**) and its Hankel transform represented in 3D Fourier transform space (

**b**). Rotations around spherical coordinates similarly rotates the corresponding transform, leading to a spherically symmetric ensemble average (

**c**).

**Figure 2.**Differential volume element around point ${q}_{0}$ in 3D curvilinear coordinates; cylindrical coordinates in (

**a**), and spherical coordinates in (

**b**).

**Figure 4.**(

**a**) Maximum intensity projections of the raw data along the three primary axes. (

**b**) Projections of the binary thresholded data along the three primary axes. Shown also is the convex hull (solid black line) and sampling region (dashed black line). The blue square in the bottom right shows the 6 mm × 6 mm autocorrelation window and the red circle 9 mm in radius shows the region used to calculate the spectrum for each sample point.

**Figure 5.**The bottom row shows 4 of the 1000 regions and the surrounding neighborhood. The top row shows, for each region, the normalized autocorrelation for each displacement (dots) as well as the average for each radial bin for a range of $N=B\left(0\right)$ (red line).

**Figure 6.**Average normalized autocorrelation function for regions with 100 < N < 1000. The grey area is bounded by ±1 standard deviation across all sample locations included in this set.

**Table 1.**Average normalized autocorrelation power law fits as a function of upper and lower bounds on occupied voxels, excluding extremely empty or filled samples.

$\mathbf{Lower}\text{}\mathbf{Bound}\text{}\mathit{N}$ | $\mathbf{Upper}\text{}\mathbf{Bound}\text{}\mathit{N}$ | $\mathbf{Average}\text{}\mathbf{Power}\text{}\mathbf{Law}\text{}\mathit{\gamma}$ |
---|---|---|

10 | 1000 | −1.6 |

10 | 10,000 | −1.5 |

100 | 1000 | −1.3 |

100 | 10,000 | −1.3 |

100 | 100,000 | −1.3 |

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**MDPI and ACS Style**

Parker, K.J.; Carroll-Nellenback, J.J.; Wood, R.W.
The 3D Spatial Autocorrelation of the Branching Fractal Vasculature. *Acoustics* **2019**, *1*, 369-381.
https://doi.org/10.3390/acoustics1020020

**AMA Style**

Parker KJ, Carroll-Nellenback JJ, Wood RW.
The 3D Spatial Autocorrelation of the Branching Fractal Vasculature. *Acoustics*. 2019; 1(2):369-381.
https://doi.org/10.3390/acoustics1020020

**Chicago/Turabian Style**

Parker, Kevin J., Jonathan J. Carroll-Nellenback, and Ronald W. Wood.
2019. "The 3D Spatial Autocorrelation of the Branching Fractal Vasculature" *Acoustics* 1, no. 2: 369-381.
https://doi.org/10.3390/acoustics1020020