Brain volume estimation from serial section measurements: a comparison of methodologies

Estimation of brain volume from serial sections typically involves using a rectangular. Cavalieri's, parabolic (Simpson's), or a trapezoidal rule to integrate numerically a curve of cross-sectional area measurements plotted against section number. We practically compare the efficacy of each of these methods using mathematical simulations of regularly- and irregularly-shaped "brain volumes" as well as actual morphometric measures from brain regions. There are no meaningful differences between the various estimates when many sections are used--with fewer sections. Cavalieri's estimator is most accurate. This confirms previous theoretical reports demonstrating the efficiency and accuracy of the Cavalieri estimator of volume, particularly when few sections are analyzed. While the Cavalieri approach provides a better approximation of volume under some circumstances, it requires equally spaced sections. We therefore describe methods for the estimation of brain volume from unequally spaced sections, including an estimator based on the fitting of piece-wise parabolic curves to the data. We outline a series of guidelines for the use of these mathematical rules in the estimation of brain volume from serial sections.