Robust estimation of fractal measures for characterizing the structural complexity of the human brain: Optimization and reproducibility

Abstract

High-resolution isotropic three-dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape, volume and topological complexity. In particular, methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions. The usefulness of such measures for characterizing individual differences in brain structure critically depends on their within-subject reproducibility in order to allow the robust detection of between-subject differences. This study analyzes key analytic parameters of three fractal-based methods that rely on the box-counting algorithm with the aim to maximize within-subject reproducibility of the fractal characterizations of different brain objects, including the pial surface, the cortical ribbon volume, the white matter volume and the grey matter/white matter boundary. Two separate datasets originating from different imaging centers were analyzed, comprising, 50 subjects with three and 24 subjects with four successive scanning sessions per subject, respectively. The reproducibility of fractal measures was statistically assessed by computing their intra-class correlations. Results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility. Highest reproducibility with intra-class correlations in the range of 0.9–0.95 is achieved with the correlation dimension. Further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region-specific patterns of individual variability. These results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure, for instance in studies of neurological diseases that do not allow repeated measurements or for disease-course longitudinal studies.

Cortical complexity as a measure of age-related brain atrophy

“…Specifically, whereas the box-counting algorithm usually uses a fixed grid scan to count if the boxes are filled or not, using the dilation algorithm with a cube is functionally identical to computing the box-counting algorithm using a sliding grid scan (i.e., if the grid was shifted in alignment with the structure, and the average of all shifted counts was taken, see Figure 2A). While a sliding grid space has been used previously (e.g., Goñi et al, 2013), the 3D-convolution operation but can be calculated much faster as it is less computationally demanding.…”

Evaluation of the 3D fractal dimension as a marker of structural brain complexity in multiple‐acquisition MRI

“…Moreover, we here compute FD estimates on global tissue segmentations. Given the increasingly sophisticated brain parcellation methods, however, region‐ and substructure‐specific fractal analysis is also being developed and is likely to yield interesting additional information, especially in the clinical context (see Eickhoff, Yeo, & Genon, ; Glasser et al, ; Goñi et al, ; Madan, ; Madan & Kensinger, ; Ruiz de Miras et al, ). As such, future work is warranted to expand upon the utility of fractal analysis for empirical neuroimaging, specifically with respect to clinical applications.…”

Evaluation of the 3D fractal dimension as a marker of structural brain complexity in multiple-acquisition MRI

“…This is in accordance with a recent reliability study of brain morphology estimates in two openaccess data sets by Madan and Kensinger (Madan and Kensinger, 2017) who found that regional fractal dimensionality as computed by both dilation and box-counting methods was generally very high and comparable to the reliability of gyrication indices, while it was in fact superior to volumetric measures such as cortical thickness. Similarly, Goñi and colleagues analyzed the fractal properties of the pial surface, the gray matter / white matter boundary and the cortical ribbon and white matter volumes in MRI data from dierent imaging centers and found a high within-subject reproducibility with regionspecic patterns of individual variability (Goñi et al, 2013). While there is thus converging evidence for the robustness of fractal analysis in neuroimaging, these studies used parcellation-and surfacebased methods, and T2WI were not analyzed.…”

“…Given the increasingly sophisticated brain parcellation methods, however, region-and substructure-specic fractal analysis is also being developed and is likely to yield interesting additional information, especially in the clinical context (see e.g. Goñi et al 2013;Glasser et al 2016;Madan and Kensinger 2017;Ruiz de Miras et al 2017;Madan 2018).…”