# Liver Backscatter and the Hepatic Vasculature’s Autocorrelation Function

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

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

#### 2.1. General Theory

#### 2.2. The Long Vessel

_{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].

_{s}(q) is given by

_{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].

## 3. Methods

#### 3.1. Experimental Animals

#### 3.2. Micro-CT

## 4. Results

## 5. Discussion

#### 5.1. Comparison of Measures

#### 5.2. Implications for Scattering from Liver

#### 5.3. Resolving Uncertainties and Future Work

## 6. Conclusions

^{1.8}scattering relation will be observed within the scale of wavenumbers corresponding to the structure elements. This experimental result from spatial autocorrelation measurements is consistent with the general framework for fractal branching structures [21] and also with recent results determined in the human placenta [22]. It is an important first step to measure baseline parameter values in normal soft tissues. Once established, the important question of parameter sensitivity to pathologies will need careful examination, as subtle changes in tissue morphology then alter the backscatter and enable detection and staging of disease by ultrasound well before gross anatomical changes are imaged.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Micro-CT volume rendering of the high contrast liver vasculature created by thresholding; two views corresponding to rotation of the 3D structure are shown. Voxel dimensions are 16.5 microns.

**Figure 2.**The top row shows, for each region, the normalized autocorrelation for each displacement (dots) as well as the average overall vector displacements in 3D, for each radial bin for a range of N = B(0) (red line). The bottom row shows intensity projections for four sub-volumes of the 1000 autocorrelation windows, with their surrounding neighborhoods.

**Figure 3.**(

**a**) Plot showing the number of boxes N(l) that are required to enclose the fractal structure (threshold = 80) as a function of scale (L/l) on log-log plot. L is the largest length of the structure, 17 mm. (

**b**) Plot similar to (a) showing the surface area and volume (threshold = 80) of the boxes that conform to the vasculature. (

**c**) Plot showing the fit to the autocorrelation function for 10 < N < 1000 (threshold = 80). (

**d**) Autocorrelation function for 10 < N <1000 for threshold = 60. Results from liver 1 and 2 are shown in each.

Name | Symbol | Equation | Notes |
---|---|---|---|

Fractal dimension | $D$ | $N\left(l\right)~{l}^{-D}$ | $\mathrm{box}\text{}\mathrm{counting}\text{}\mathrm{with}\text{}\mathrm{scale}\text{}l;\text{}D3$ |

Autocorrelation | $C\left(r\right)$ | $C\left(r\right)~{C}_{0}/{r}^{\left(3-D\right)}$ | $r\text{}\mathrm{is}\text{}\mathrm{autocorrelation}\text{}\mathrm{lag}\text{}\mathrm{in}\text{}\mathrm{spherical}\text{}\mathrm{coordinates};\text{}r0;\text{}D3$ |

3D spherical Fourier transform | ${}^{3DS}\mathfrak{I}\left\{\right\}$ | ${}^{3DS}\mathfrak{I}\left\{1/{r}^{\left(3-D\right)}\right\}~1/{q}^{D}$ | $q\text{}\mathrm{is}\text{}\mathrm{spatial}\text{}\mathrm{frequency};\text{}1D3$ |

Scattering differential cross section | ${\sigma}_{d}\left(k\right)$ | ${\sigma}_{d}\left(k\right)~{k}^{\left(4-D\right)}$ | $k\text{}\mathrm{is}\text{}\mathrm{wavenumber},\text{}\mathrm{derived}\text{}\mathrm{from}\text{}\mathrm{Fourier}\text{}\mathrm{transform}\text{}\mathrm{of}\text{}C\left(r\right);\text{}D3$ |

**Table 2.**Power law fits to autocorrelation functions can vary with the threshold used to generate the binary vasculature structures, and the number of points N above threshold contained in the selected autocorrelation voxels.

Liver | Threshold | N, Lower Bound | N, Upper Bound | Power Law | Regions |
---|---|---|---|---|---|

1 | 60 | 10 | 100 | −0.91 | 49 |

60 | 100 | 1000 | −1.09 | 172 | |

60 | 1000 | 10000 | −0.79 | 279 | |

80 | 10 | 100 | −1.28 | 104 | |

80 | 100 | 1000 | −1.23 | 210 | |

80 | 1000 | 10000 | −0.84 | 126 | |

2 | 60 | 10 | 100 | −1.16 | 50 |

60 | 100 | 1000 | −1.22 | 187 | |

60 | 1000 | 10000 | −0.87 | 253 | |

80 | 10 | 100 | −1.28 | 80 | |

80 | 100 | 1000 | −1.32 | 219 | |

80 | 1000 | 10000 | −0.87 | 137 |

**Table 3.**Approximate values of D from different metrics. The estimated slope γ is from log-log plots of the metric vs. scale.

Method | Dependence | Estimates |
---|---|---|

Autocorrelation | D = 3 − γ | 2.2 > D > 1.8 |

Surface area | D = 2 + γ | 2.1 |

Volume | D = 3 + γ | 2.2 |

Box counting | D = γ | 2.2 |

Backscatter (predicted) | ∼k^{4−D} | k^{1.8} |

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**MDPI and ACS Style**

Carroll-Nellenback, J.J.; White, R.J.; Wood, R.W.; Parker, K.J.
Liver Backscatter and the Hepatic Vasculature’s Autocorrelation Function. *Acoustics* **2020**, *2*, 3-12.
https://doi.org/10.3390/acoustics2010002

**AMA Style**

Carroll-Nellenback JJ, White RJ, Wood RW, Parker KJ.
Liver Backscatter and the Hepatic Vasculature’s Autocorrelation Function. *Acoustics*. 2020; 2(1):3-12.
https://doi.org/10.3390/acoustics2010002

**Chicago/Turabian Style**

Carroll-Nellenback, Jonathan J., R. James White, Ronald W. Wood, and Kevin J. Parker.
2020. "Liver Backscatter and the Hepatic Vasculature’s Autocorrelation Function" *Acoustics* 2, no. 1: 3-12.
https://doi.org/10.3390/acoustics2010002