The sum of all fiber tracts represented by lines in a three dimensional space and color-coded according to their orientation is usually referred to as tractogram.
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These methods allow for the visualization of neural fiber tracts based on a direction-sensitive MR measurement of the diffusion of free water. The most accurate macroscopic geometrical representation of the neural structure of the brain, by means of MRI, is provided by diffusion tensor imaging (DTI) based fiber tracking 26 and its further developments. However, due to limitations in spatial resolution, caused by MRI basic conditions such as signal strength and scan time, both methods are only rough approximations to investigate the underlying complex structure of nerve fiber bundles. Secondly, the fractal dimension is obtained from a skeletonized version of a T 1-weighted white matter mask 25. Firstly, fractal analysis of the entire binary WM mask 24, which is directly obtained from a T 1-weighted structural Magnetic Resonance Imaging (MRI) scan. When analyzing changes in FD of white matter structure, two strategies are mainly used. Hence, a quantification of structural changes is of highest interest. However, age-related changes of the brain structure are closely linked to mental fitness and cognitive performance 23. Diffusion tensor imaging revealed that the fractional anisotropy (FA), a measure for the anisotropy of the diffusion process, increases in the first three decades and decreases in the following decades 19, 20, 21, 22. Cerebral white matter is subject to structural changes during the entire life-span, which has been shown in several MRI studies 18, 19. Structural changes of the brain, that concern all of us, are due to a normal aging process and effects the brain on many levels including vascularization 14, atrophy 15, changes in cortical and subcortical regions 16, 17 and changes in myelination. Considering these studies, FD seems to be a sensitive biomarker for changes in the geometrical complexity of the brain. Fractal analysis of the brain’s geometrical structure was correlated with pathologies such as Multiple Scleroses 11, 12 or Alzheimer’s disease 13, suggesting that structural alterations, specifically in white matter structure, can be captured by changes in FD.
The complex structures of the brain have been extensively investigated using fractal methods focusing on the brain’s surface 6, the geometrical complexity of white matter 7, changes in the structure of the cerebrovascular system 8 due to pathologies and for analyzing the complex structure of neural networks 9, 10.
In the last two decades, a huge variety of applications utilizing fractal analysis have been published on different biomedical fields from the analysis of DNA base sequences 2 and the classification of biological structures 3 to the description of microvascularity in gliomas 4, 5. For a natural structure or an image, the fractal dimension cannot be calculated exactly, but is usually approximated as the ratio of change in detail with change in scale, plotted in a double logarithmic plot where the slope provides an estimate of FD. In his pioneering work “How long is the coast of Britain 1?” Mandelbrot paved the way for the field of fractal analysis, a mathematical framework describing geometrical structures that cannot be characterized sufficiently with Euclidian geometry. With this, FD can describe geometrical features such as self-similarity or space-filling properties of textures or structures that are obtained by stochastic processes. A fractal can be defined as a set, whose fractal dimension ( FD) is a non-integer value between the topological dimension, which is zero for a point, one for a curve and two for a plane, and its embedding dimension. In the late sixties Benoit Mandelbrot developed a mathematical concept to describe geometrical structures, denoted as fractals, whose measured metric properties (length, area or volume) depend on the scale of measurement.
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