Fractal analysis of dendrites morphology using modified Richardson's and box counting method

Fractal analysis has proven to be a useful tool in analysis of various phenomena in numerous naturel sciences including biology and medicine. It has been widely used in quantitative morphologic studies mainly in calculating the fractal dimension of objects. The fractal dimension describes an object's complexity: it is higher if the object is more complex, that is, its border more rugged, its linear structure more winding, or its space more filled. We use a manual version of Richardson's (ruler-based) method and a most popular computer-based box-counting method applying to the problem of measuring the fractal dimension of dendritic arborization in neurons. We also compare how these methods work with skeletonized vs. unskeletonized binary images. We show that for dendrite arborization, the mean box dimension of unskeletonized images is significantly larger than that of skeletonized images. We also show that the box-counting method is sensitive to an object's orientation, whereas the ruler-based dimension is unaffected by skeletonizing and orientation. We show that the mean fractal dimension measured using the ruler-based method is significantly smaller than that measured using the box-counting method. Whereas the box-counting method requires defined usage that limits its utility for analyzing dendritic arborization, the ruler-based method based on Richardson's model presented here can be used more liberally. Although this method is rather tedious to use manually, an accessible computer-based implementation for the neuroscientist has not yet been made available.