Resolution effect on the stereological estimation of surface and volume and its interpretation in terms of fractal dimensions

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Abstract
SUMMARY: Estimating surface and volume density of subcellular membrane systems at different magnifications yield different results. As the magnification is increased from × 18,000 to × 130,000 the estimates of surface density of endoplasmic reticulum and inner mitochondrial membranes increase by a factor of 3, whereas that for outer mitochondrial membranes increases only by 20%. The estimate of volume density of endoplasmic reticulum also increases by a factor of 3. No further increase is observed at magnifications above × 130,000 which is therefore called critical magnification. The findings are interpreted on the basis of the concept of fractals proposed by Mandelbrot, and the fractal dimensions of the membrane systems considered are estimated. This can lead to the derivation of resolution correction factors which permit measurements obtained at any magnification to be converted to estimates at critical magnification. These findings may explain, at least in part, the large discrepancy in the estimates of the surface of cytomembranes found in the literature.