Fractal analysis and imaging of the proximal nephron cell

Abstract

Cells of the S1 proximal renal tubule were examined to determine whether their peculiar shapes are a result of certain constructs of fractal mathematics. Morphometric measurements of the cell perimeter were made at several levels of cell height by measuring the intercellular boundaries that appear on electron micrographs of tubule cross sections. When the measurements were made over a range of scale lengths, the fractal dimension, D, of the cell perimeter was found to increase from 1.3 near the cell apex to 1.78 near the cell base. The length of scale was found to range between 8 and 0.4 micron and to represent the approximate dimensions of actual cell processes. Fractal patterns that conformed to the measured parameters were then constructed from a fractal generator composed of budlike formations that originated near the cell apex and that increased in number and decreased in size with cell depth according to a fractal scaling. It was found that the fractal rule of keeping a constant relative scale could be maintained between budding processes but, to obtain patterns that resemble biological structure, the processes must be positioned randomly on the cell periphery. It is shown that when the relative sizes of the buds decrease exponentially and their numbers increase geometrically, the perimeter can grow to the correct length without overlap. This suggests that patterns of the cell periphery corresponding to different levels of cell height obey a law of scale but occur randomly in a way that increases to high fractal dimension or near plane-filling values at the cell base. The fractal patterns that correspond to the measured fractal dimensions can be assembled into a three-dimensional model that closely resembles the known shape of the proximal tubule cell.