Correction of bias in the estimation of cell volume fraction from histology sections

Accurate determination of the fraction of a tissue's volume occupied by cells is critical for studying tissue development, pathology, and biomechanics. For example, homogenization methods that predict the function and responses of tissues based upon the properties of the tissue's constituents require estimates of cell volume fractions. A common way to estimate cellular volume fraction is to image cells in thin, planar histologic sections, and then invoke either the Delesse or the Glagolev principle to estimate the volume fraction from the measured area fraction. The Delesse principle relies upon the observation that for randomly aligned, identical features, the expected value of the observed area fraction of a phase equals the volume fraction of that phase, and the Glagolev principle relies on a similar observation for random rather than planar sampling. These methods are rigorous for analysis of a polished, opaque rock sections and for histologic sections that are thin compared to the characteristic length scale of the cells. However, when histologic slices cannot be cut sufficiently thin, a bias will be introduced. Although this bias - known as the Holmes effect in petrography - has been resolved for opaque spheres in a transparent matrix, it has not been addressed for histologic sections presenting the opposite problem, namely transparent cells in an opaque matrix. In this note, we present a scheme for correcting the bias in volume fraction estimates for transparent components in a relatively opaque matrix.