Mattfeldt T
Institute of Pathology, University of Heidelberg, Federal Republic of Germany.
J Math Biol. 1987;25(6):685-95. doi: 10.1007/BF00275503.
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.
生物物体的绝对体积通常通过对一组详尽的系统切片进行体视学估计。常用的体积估计量(V)是切片内容的总和乘以切片之间的距离。对于随机起始的系统切片，最近已经表明，当投影物体长度与切片距离的比值(m)为整数时，(V)是无偏的（克鲁兹 - 奥里夫，1985年）。由于在现实世界中这个量不是整数，我们研究了在一般且现实的情况(m\in R)下(V)的性质。通过严格证明，在适当的采样条件下，三维空间中任意紧致集的(V)的无偏性得到了证明。对一般三轴椭球体的(V)的进一步性质的探索导致了一类具有共同形式结构的新的非初等实函数，我们将其称为(np)函数。椭球体中(V)的相对均方误差（(CE^2)）是一个振荡可微的(np)函数，对于整数(m)，它简化为已知结果(CE^2 = 1/(5m^4))。作为一个生物学实例，从系统切片估计了10个左心室及其内腔的绝对体积。结果表明，使用(m\in R)而不是(m\in N)进行重复系统切片的蒙特卡罗模拟得到了改进。与具有一些附加形状不规则性的椭球体几何模型一致，心室及其腔中的平均经验(CE)分别与(m^{-1.36})和(m^{-1.73})成比例。通过系统切片实现的显著方差减少被证明是对偶变量原理的几何体现。
