Orthogonal triplet probes: an efficient method for unbiased estimation of length and surface of objects with unknown orientation in space

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Abstract
If the axis of anisotropy of oriented, tubular or lamellar objects is unknown, the unbiased stereological estimation of length density and surface density (Lv and Sv) requires counts on sections with isotropic uniform random (IUR) orientation. It is shown theoretically that in homogeneous, anisotropic specimens the precision of Lv and Sv estimation is considerably augmented if IUR-oriented sets of three mutually perpendicular sections (orthogonal triplet probes=ortrips) are used instead of three directionally independent IUR sections. The mechanism of variance reduction results from a negative covariance between sections within ‘ortrips’ and corresponds to the antithetic variate principle of Monte Carlo work. Heterogeneity decreases the efficiency of the ortrip method, but this effect can often be counteracted by systematic sampling of ortrips within specimens. Practical estimation of length and surface area of the highly anisotropic, tubular myocardial capillaries per tissue volume in the left ventricles of eight normal, adult, male perfusion-fixed Wistar rats provided estimates of excellent precision with CEs of 3.3% (Lv) and 2.1% (Sv) of the group mean. The method will hopefully allow stereologists dealing with arbitrary anisotropic structures to apply the same simple, efficient, and unbiased sampling designs that have long been used in the study of liver, lung, and kidney tissue.

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