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, 18 (2), 136-51

Multifractal and Lacunarity Analysis of Microvascular Morphology and Remodeling

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Multifractal and Lacunarity Analysis of Microvascular Morphology and Remodeling

Daniel J Gould et al. Microcirculation.

Abstract

Objective: Classical measures of vessel morphology, including diameter and density, are employed to study microvasculature in endothelial membrane labeled mice. These measurements prove sufficient for some studies; however, they are less well suited for quantifying changes in microcirculatory networks lacking hierarchical structure. We demonstrate that automated multifractal analysis and lacunarity may be used with classical methods to quantify microvascular morphology.

Methods: Using multifractal analysis and lacunarity, we present an automated extraction tool with a processing pipeline to characterize 2D representations of 3D microvasculature. We apply our analysis on four tissues and the hyaloid vasculature during remodeling.

Results: We found that the vessel networks analyzed have multifractal geometries and that kidney microvasculature has the largest fractal dimension and the lowest lacunarity compared to microvasculature networks in the cortex, skin, and thigh muscle. Also, we found that, during hyaloid remodeling, there were differences in multifractal spectra reflecting the functional transition from a space filling vasculature which nurtures the lens to a less dense vasculature as it regresses, permitting unobstructed vision.

Conclusion: Multifractal analysis and lacunarity are valuable additions to classical measures of vascular morphology and will have utility in future studies of normal, developing, and pathological tissues.

Figures

Figure 1
Figure 1
A schematic of the box counting algorithm is shown here. Note, the grid has box width ε, and as ε decreases, the total number of boxes N(ε) increases (A). In the case of a fractal N(ε) is related to ε by the relation N(ε) ~ε−D0. D0 is defined as the box counting fractal dimension. B shows the plot of N(ε) vs. ε. The log-log plot is shown in C along with the linear fits to the plot. Two equations are shown in the upper right hand corner of figure 1C. The first one corresponds to the linear fit to all points and the second one corresponds to the linear fit to the data excluding the lowest and highest ε values. The lowest ε value corresponds to the width of the pixel in the image and the largest ε value corresponds to the bounding size of the image. Therefore, it is reasonable to define the linear region of the log-log plot as between the limiting values mentioned above. The R2 values in both cases are close to 1. The standard errors of the slopes of the lines based on linear regression are 0.10 and 0.09 respectively. The slopes of the lines are not significantly different. Hence, for this project, all points will be used to calculate the slope of the linear region (fractal dimension).
Figure 2
Figure 2
A flow chart of image contrast enhancement used in this paper is shown here. A is the maximum intensity projection image of a 17.5 um thick confocal z-stack of cortex vasculature at post natal day 6 (P6). The vessels are tagged by the fluorescent marker mCherry. B is the grayscale image obtained by maximum intensity projection of the z-stack used in A. C was obtained from B by applying an unsharp mask filter (radius 12). D is the binary image obtained from C by applying an intensity threshold of 10 using a MATLAB program (2). The original image A is then overlapped with the image D to check whether the thresholded image is a good representation of the original image. Regions in yellow show the overlap between the two images (A and D) in E. Importantly the vessel diameters from the original and the processed images do not differ significantly, thus the processing steps preserve critical information about the original vessel beds for analysis and comparison. This is reflected in the similarities in the histograms of the vessels (F) and the similarities of the average vessel diameters within each bin (G). Note the gaps on the vessel surface due to thresholding. Vessel density, box counting dimension, and lacunarity obtained by applying the set of operations mentioned above are listed in table 1. Scale bar = 50 micrometers.
Figure 3
Figure 3
Multifractal analysis of a straight line (A), checkerboard (B), Sierpinski carpet (C) and retinal vasculature (D). The generalized dimensions of the line in 3A were all equal to 1 (E). The generalized dimensions of the checkerboard (B) were all equal to 2 (F). C is a monofractal called a Sierpinski carpet with a fractal dimension of 1.79. The generalized dimensions of the carpet were equal to 1.79 (G). D is the image of retinal vessels taken from the database for retinal images called STARE (13). The vasculature is different from the three patterns discussed above in that the space filling property of the vasculature cannot be described by a single dimension as in the previous cases. A set of generalized dimensions is required to characterize the retinal vascular pattern (H). The retinal vasculature is an example of a multifractal found in nature. In fact, in nature most objects are quasi fractal and must be measured with generalized dimensions to fully capture their behavior. All monofractals represent a special case of multifractal behavior in which all generalized dimensions are equal.
Figure 4
Figure 4
Thresholded confocal images of microvessels in four different tissues taken from postnatal day 6 (P6) of Flk1-myr::mCherry mice. 4A is the thresholded image of the vasculature in the cortex, B is kidney, C is skin and D is muscle (vastus medialus). Scale bar = 50 micrometers.
Figure 5
Figure 5
A graph of lacunarity values L(ε) versus the box widths (ε) for various tissues is presented here. The lacunarity plots overlap in the case of skin and cortex, while the plots in the case of kidney and muscle tissues are well separated from each other and also from the plots of cortex and skin.
Figure 6
Figure 6
The multifractal spectra of the images of vascular patterns presented in figure 4 are shown here. The spectra corresponding to patterns with larger lacunarity are shifted to the lower α range and have lower maxima in comparison to patterns with smaller lacunarity. Dashed lines indicate values of α beyond which the R2 values associated with the linear fits in the evaluation of α fall below 0.95. f(α) values with α values whose R2 above 0.95 lie to the left of the dashed lines.
Figure 7
Figure 7
The homogeneity of a structure can be determined by the proximity of its data point to the 1:1 line on the α (0) – D0 plot and the symmetry of a multifractal spectrum can be assessed by the proximity of its widths to the 1:1 line on the α (0) − α(qi,+) versus α(qi,−) -α(0) plot. A shows that the patterns in figure 4 are heterogeneous and thus cannot be described by box counting dimension alone. B shows the asymmetry of the multifractal spectra of images from different tissue types. The spectra of cortex, skin, and muscle are more symmetric than the spectrum of kidney.
Figure 8
Figure 8
A–D are grayscale confocal microscope images of hyaloid vasculature in the retina of a Flk1-myr::mCherry mouse taken from P0 to P10. P refers to the postnatal stage of the mouse imaged, as in postnatal day 0 through day 10. E–H are images of the regions in figures A–C first cropped and dilated to occupy 512 × 512 pixels, then modified with an unsharp mask in Adobe CS2 with radius set to 12. I–L are binary images obtained from images D–F after applying an intensity threshold.
Figure 9
Figure 9
A shows the average diameters of the vessels in the images 8 I–L. The only significant differences between vessel diameters are between the P0 and P10 stage (P=0.006) and between the P7 and P10 stage (P=0.02). There is no significant difference in vessel diameter between the other images of vessels in the hyaloid (P>0.05). Here, there is significant correlation between vessel densities and box counting dimensions (B, R2=0.99). C shows the multifractal spectra of binary images 8 I–L. The spectrum at the top right corresponds to the hyaloid vasculature at P0. The pattern has the largest box counting dimension D0 and the smallest lacunarity. The pattern on day P10 has the smallest box counting dimension D0 and the largest lacunarity and is at bottom left. D shows the heterogeneity analysis and compares P0–P10. E is the symmetry comparison of the spectra of P0–P10 hyaloid vessels. 9D is the heterogeneity analysis of the spectra from the hyaloid vessels. The values for the hyaloid vessels are closer to the 1:1 line than in figure 7, and thus are less heterogeneous than the images of vessels from the other tissues. 9E demonstrates the symmetry of the curves. Here, the symmetry data points for P3 and P7 vessels lie closer to the 1:1 line than P0 and P10 vessels. This means P3 and P7 are more symmetric than P0 and P10. The higher asymmetry in the spectrum of P0 is partly due to the presence of more vessels and thus higher complexity in the image at P0. In P10 there is more empty space in the image of the vessels. Due to the relatively low contribution of the vessels to the image, the curve is asymmetric and the corresponding data point lies above the 1:1 line (more empty space than vessels causes this shift).

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