A Reinvestigation of the Extended Counting Method for Fractal Analysis of the Pial Vasculature

A Reinvestigation of the Extended Counting Method for Fractal Analysis of the Pial Vasculature To the Editor: A recent article published in the Journal of Cerebral Blood Flow & Metabolism (Hermán et al., 2001) documented the findings of fractal branching pattern on the pial vascular networks in the cat brain. Using the standard box-counting method (Cross, 1997) and an extended counting method (Sandau and Kurz, 1997), the authors reported similar fractal dimensions of approximately 1.3 for both the arterial and venous vasculature of brain cortex from six cats. These fractal dimensions were found to be invariant under geometric transformations, such as translation and rotation of the raw images. After a further assessment of accuracy for the two fractal methods using known fractal objects, the authors concluded that the fractal model seemed to be correct in describing the scale-invariant bifurcation patterns in the pial vascular networks of cats. Although we agree that the pial vascular networks appear random, such that a quantitative characterization of the arborization of complex vasculature seems difficult using traditional Euclidean geometry, we are specifically concerned about the suitability of the extended counting method used by the authors for fractal analysis. Before discussing the pitfalls of the extending counting method, we first take a brief look at the procedure used by Herman et al. After intensity correction of a digitized video image taken from suprasylvian gyrus, the vasculature was defined by manual tracing of the image’s laser printout. The manual tracing was further digitized at a spatial resolution of 600 dots per inch, after which the resulting digital image was skeletonized to yield a one pixel-wide vascular trace. For fractal analysis using the extended counting method, a large square box of width q0 with a square grid of small boxes of width q1 inside was overlaid onto the skeletonized image (Fig. 1). The number of small boxes covering any part of the vasculature was counted as N. Subsequently, the large box was moved to a new place and the counting process repeated until every part of the vessel structure had been visited. The extended counting fractal dimension DXCM, alternatively termed the capacity dimension, was then calculated as