Research Article: The 3D Spatial Autocorrelation of the Branching Fractal Vasculature

Date Published: April 25, 2019


Author(s): Kevin J. Parker, Jonathan J. Carroll-Nellenback, Ronald W. Wood.


The fractal branching vasculature within soft tissues and the mathematical properties of the branching system influence a wide range of important phenomena from blood velocity to ultrasound backscatter. Among the mathematical descriptors of branching networks, the spatial autocorrelation function plays an important role in statistical measures of the tissue and of wave propagation through the tissue. However, there are open questions about analytic models of the 3D autocorrelation function for the branching vasculature and few experimental validations for soft vascularized tissue. To address this, high resolution computed tomography scans of a highly vascularized placenta perfused with radiopaque contrast through the umbilical artery were examined. The spatial autocorrelation function was found to be consistent with a power law, which then, in theory, predicts the specific power law behavior of other related functions, including the backscatter of ultrasound.

Partial Text

In optics, electromagnetics, and acoustics, the behavior of waves within a weakly inhomogeneous medium can be related to properties of the spatial correlation function, a statistical measure of the spatial patterns or fluctuations within the material [1,2]. Both the forward propagating wave and the waves scattered from the inhomogeneities have distributions that have been linked to the spatial patterns of the inhomogeneities. For example, under the Born approximation and assumptions of stationarity, a key integral formula relates the ensemble average measures of backscattered intensity to the material’s spatial correlation function as a 3D Fourier transform operation. A specific case emerges if we consider the parenchyma within organs such as the liver, prostate, or placenta to be the reference media, while the long, cylindrical fluid-filled, fractal branching vessels serve as the weak scattering sites.

The fetal side placental perfusion was adapted from dual lobular placental perfusion methods previously published [19–21]. Briefly, human placentae from normal term deliveries were obtained within 5–10 min of delivery and examined for tears and gross lesions. Within 25 min, the umbilical artery and vein were cannulated near the insertion of the cord on the surface of the chorionic plate with five French umbilical catheters for perfusion at 10–15 mL min−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% O2/75% N2/5% CO2 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.

The raw data set (shown in the maximum intensity projections in the left panel of Figure 4) was thresholded using a constant threshold limit of 180. The threshold was set at the minimum level necessary to zero out the poorly vascularized edges of the placenta, while maintaining well connected branches. Projections of the thresholded binary data can be viewed in the right panel of Figure 4. Also shown in the right panel of Figure 4 is the convex hull of the placenta (solid black line) as well as the boundary of the sampling region (dashed black line). The sampling region contains locations whereby small autocorrelation sample cubes (6 mm on a side, or 27 × 27 × 21 voxels) can be translated along any axis or diagonal by 6 mm without being outside the whole placenta’s boundary. For reference, the size of the autocorrelation cubes is shown in the bottom right corner of the right panel of Figure 4, and the surrounding region that contributes to the autocorrelation spectrum is shown by the red circle of radius 9 mm.

High resolution contrast CT studies of a normal placenta’s branching vasculature in 3D demonstrated a power law behavior for the 3D spatial autocorrelation function with an exponent close to −1.3. Correspondingly, a theoretical hypothesis about scattering from tissue was generated from a primitive cylindrical shape representing the plausible distribution of extracellular fluids and blood in long fluid-filled channels throughout normal soft tissue. Assuming a wide range of diameters of the cylindrical fluid spaces and a macroscopically isotropic distribution over some region of interest within the organ, the predicted backscatter is of the form of a power law with dimension greater than one. This matches some observations about the nature of vessels and measurements of backscatter from soft tissues and is consistent with the power law autocorrelation function measured experimentally from the placenta.




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