Studies on depth-of-field effects in microscopy supported by numerical simulations

Micrographs are two-dimensional (2D) representations of three-dimensional (3D) objects. When the depth-of-field of a micrograph is comparable with or larger than the characteristic dimension of objects within the micrograph, measured 2D parameters (e.g. particle number density, surface area of particles, fraction of open space) require stereological correction to determine the correct 3D values. Here, we develop a stereological theory using a differential approach to relate the 3D volume fraction and specific surface area to the 2D projected area and perimeter fractions, accounting for the influence of depth-of-field. The stereological theory is appropriate for random isotropic arrangements of non-interpenetrating particles and is valid for convex geometries (e.g. spheres, spheroids, cylinders). These geometrical assumptions allow the stereological formulae to be expressed as a set of algebraic equations incorporating a single parameter to describe particle shape that is tightly bounded between 1.5 pi and 2 pi. The stereological theory may also be applied to arrangements of interpenetrating convex particles, and for this case, the resulting stereological formulae become identical to the formulae previously presented by Miles. To test the accuracy of the stereological theory, random computational arrangements of non-interpenetrating and interpenetrating spheres or cylinders are analysed, and the projected area and perimeter fractions are numerically determined as a function of depth-of-field. The computational results show very good agreement with the theoretical predictions over a broad range of depth-of-field, volume fraction and particle geometry for both non-interpenetrating and interpenetrating particles, demonstrating the overall accuracy of the stereological theory. Applications of the stereological theory towards analysis of biological tissues and extracellular matrix are discussed.