Self-similarity and fractal irregularity in pathologic tissues.

The irregularity and self-similarity under scale changes are the main attributes of the morphologic complexity of cells and tissues, either normal or pathologic. In other words, the shape of a self-similar object does not change when scales of measure change because any part of it might be similar to the original object. Size and geometric parameters of an irregular object, however, differ when inspected at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how to analyze irregular shapes and structures in the physical and biologic sciences. Dominant influences have been the discovery by B.B. Mandelbrot of a new, practical geometry of nature, now called fractal geometry, and the continuous improvements in computational capabilities. The application of the principles of fractal geometry, unlike the conventional Euclidean geometry developed for describing regular and ideal geometric shapes practically unknown in nature, enables one to measure the fractal dimension, contour length, surface area, and other dimensional parameters of almost all irregular and complex biologic tissues. During the past decade, a large amount of experimental evidence has accumulated showing that even in biomedical sciences fractal patterns could be observed. Through several examples borrowed from the recent literature, we focus on the application of the fractal approach to measuring irregular and complex features of pathologic cells and tissues and also on its potential role in the understanding of tumor biology.