Robust estimation of fractal measures for characterizing the structural complexity of the human brain: Optimization and reproducibility
Introduction
Fractal properties of the structure of an object capture its self-similarity in shape over a range of spatial scales or resolutions (Mandelbrot, 1967, Mandelbrot, 1983). Fractal measures (Hentschel and Procaccia, 1983, Henry et al., 2004, Lopes and Betrouni, 2009) have been used extensively as estimators of topological complexity in nature, e.g. in river basins (Cieplak et al., 1998), plant development (Corbit and Garbary, 1995), dendritic arborization of spinal cord neurons (Milosević and Ristanović, 2006) or the morphology and classification of α ganglion cells in the rat retina (Jelinek et al., 2011). A number of studies have used fractal measures to characterize the complexity of gray matter (GM) or white matter (WM) structures of the human brain (Bullmore et al., 1994, Zhang et al., 2006, Zhang et al., 2007). Several neuroimaging studies have explored fractal measures, for instance, in relation to cognitive changes and age (Mustafa et al., 2012), in relation to diffuse WM damage (Esteban et al., 2007) and GM damage (Esteban et al., 2009) in multiple sclerosis, or in relation to GM neurodegeneration in mild Alzheimer's disease (King et al., 2010). Fractal measures complement more standard quantitative analyses of brain structure based on gray matter volume, cortical thickness (Fischl and Dale, 2000) or voxel-based morphometry (Ashburner and Friston, 2000). Fractal measures describe topological characteristics based on scaling properties that can provide an estimate of the structural complexity of the object under study, and changes in structural complexity may occur independent of changes in volume. Phenomena related to morphological changes in the brain such as neural reorganization, plasticity or neuronal death may be amenable to fractal quantifications. However, little is known about the reproducibility of fractal measurements and their sensitivity to variations in key analysis parameters.
This study analyzes three fractal-based measures (Hentschel and Procaccia, 1983), the Kolmogorov capacity or fractal dimension, the information dimension and the correlation dimension. All rely on the box-counting algorithm, and key parameters of this algorithm that impact the reproducibility of these measures are explored. Fractal measures were computed for the pial surface, the cortical ribbon volume, the white matter volume, and the gray matter/white matter boundary. Two independently collected datasets were analyzed comprising 50 subjects with three separate scanning sessions and 24 subjects with four scanning sessions per subject. The reproducibility of fractal measures was assessed by computing intra-class correlations (Shrout and Fleiss, 1979), derived for volumes and surfaces covering whole hemispheres, as well as for parcellated gray matter regions.
Section snippets
Dataset A
Dataset A was collected at the Indiana University School of Medicine (Indianapolis, IN, USA) and consisted of 150 T1-weighted scans (NIfTI format) from 50 healthy controls (all males, 24 ± 3.2 years). For each subject, images were acquired during 3 successive visits scheduled 3 weeks apart on a 3.0 Tesla (T) MR scanner (Siemens TRIO, Germany) using a 12-channel head coil. Anatomical images were acquired using a T1-weighted MPRAGE sequence: TR/TE = 2300/2.91 ms, inversion time (TI) = 900 ms, FOV = 256 × 240 × 192
Results
For each scanner session of the two datasets, computation of Kolmogorov capacity dimension (D0), the information dimension (D1) and the correlation dimension (D2) was performed for seven objects, namely left and right pial surfaces, left and right ribbon, left and right white surfaces and white matter. The parameters evaluated were GridModes (min, avg and max), GridOffsets (from 1 to 20) and GridScales (from 3 to 9). The ICC was computed for all objects and all parameter configurations on each
Discussion
Fractal measures have been used in a number of studies to characterize the complexity of human brain structures. The approach may have some promises in identifying differences across individual subjects that index differences in behavioral or cognitive capacities, developmental stages, neurodegeneration or other disease-related processes. In order to be operationally useful, fractal estimates need to exhibit robustness and reproducibility, with high consistency between multiple acquisitions
Acknowledgment
JG acknowledges a Spanish Government grant, contract number E-28-2012-0504681. JG and OS acknowledge support from the James S. McDonnell Foundation. AS acknowledges support by the National Institutes of Health (P30 AG10133, R01 AG19771 and R01 LM011360) and the National Science Foundation (IIS-1117335).
Conflict of interest
The authors have no conflict of interest to disclose.
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