Volume estimation of biological objects by systematic sections

The absolute volume of biological objects is often estimated stereologically from an exhaustive set of systematic sections. The usual volume estimator V is the sum of the section contents times the distance between sections. For systematic sectioning with a random start, it has been recently shown that V is unbiased when m, the ratio between projected object length and section distance, is an integer number (Cruz-Orive 1985). As this quantity is no integer in the real world, we have explored the properties of V in the general and realistic situation m epsilon R. The unbiasedness of V under appropriate sampling conditions is demonstrated for the arbitrary compact set in 3 dimensions by a rigorous proof. Exploration of further properties of V for the general triaxial ellipsoid leads to a new class of non-elementary real functions with common formal structure which we denote as np-functions. The relative mean square error (CE2) of V in ellipsoids is an oscillating differentiable np-function, which reduces to the known result CE2 = 1/(5m4) for integer m. As a biological example the absolute volumes of 10 left cardiac ventricles and their internal cavities were estimated from systematic sections. Monte Carlo simulation of replicated systematic sectioning is shown to be improved by using m epsilon R instead of m epsilon N. In agreement with the geometric model of ellipsoids with some added shape irregularities, mean empirical CE was proportional to m-1.36 and m-1.73 in the cardiac ventricle and its cavity. The considerable variance reduction by systematic sectioning is shown to be a geometric realization of the principle of antithetic variates.