Correction of stereological parameters from biased samples on nucleated particle phases

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Abstract
Stereologists are aware that the experimental evaluation of component volume fractions and surface-to-volume ratios are subject to systematic errors whenever the requirements for cell identification impose the necessity for component-biased sectioning. Mathematical corrections of biased volume proportion data have recently been published; these corrections assume that the components under analysis are spherical, and that the nucleated particle phase is monodispersed. In this report, general methods for obtaining corrections of biased nuclear volume fraction data are set out for polydispersed phases of nucleated particles, in terms of the relevant shapes and joint size distribution of nucleus and cell; the scope and limitations of these methods are thereby discussed. Explicit corrections of an immediate applicability are obtained, together with their standard errors, for monodispersed phases where nucleus and cell are two dissimilar biaxial ellipsoids (spheroids). When nucleus and cell are two concentric and similar convex bodies of a certain class—to which triaxial ellipsoids belong—the corrections are shown to be very simple. The corrections for the spheroid–spheroid systems are easily accessible with the aid of a small programmable calculator, whereas those for the sphere–spheroid models are directly obtainable from two nomograms.