Biological Cellular Morphometry-Fractal Dimensions, Lacunarity and Multifractals

This paper describes two fractal dimensions ( D ), one related to length (the capacity dimension) and another related to mass (the mass dimension) and illustrates how they may be determined. The concept of lacunarity and how it can be measured is also presented. These methods are applied to the borders of biological cells. Results illustrate that the magnitude of the capacity dimension corresponds to our intuitive concept of complexity and that this dimension is a useful tool for quantitatively studying morphological cellular differentiation in cultured glial cells. A major problem for quantitative fractal geometry is presented, namely, that two fractal objects may be morphologically very different, but have the same capacity dimension. Results are presented that suggest that the mass dimension and/or lacunarity may distinguish between such objects. An analysis of the data suggests that the borders of biological cells may be multifractal in the sense that the mass dimension varies locally along the border. Finally, the spectrum of fractal dimensions of defined multifractals and cell borders is presented and discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.