Summary Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell‐to‐cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (Lv) of linear objects based on random intersections with a two‐dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object–probe intersections, including isotropic‐uniform‐random (IUR) sections, vertical‐uniform‐random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate Lv and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.

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SUMMARY An efficient sampling procedure is presented for estimation of total line length per unit volume Lv. It involves the following steps: (1) choose a vertical axis in the specimen, and cut the specimen to obtain VUR vertical slices of constant thickness Δ such that parallel planes of the slices contain the vertical direction; (2) observe the projected image of a vertical slice using transmission microscopy such that beam direction is perpendicular to the slice; (3) count the number of intersections of the projected images of the lineal features of interest with cycloid‐shaped test lines whose minor axis is perpendicular to the vertical axis. The expected value of the number of intersections per unit length prj is related to Lv as follows: Thus, Lv can be estimated from the measurements performed on the projected images of VUR vertical slices.

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The fracture toughness of Al-Zn-Mg-Cu-based 7XXX aluminum alloys decreases with an increase in the extent of recrystallization. In this contribution, the fracture path of plane-strain fracture-toughness specimens of 7050 alloy (a typical alloy of the 7XXX series) is quantitatively characterized as a function of degree of recrystallization, specimen orientation, and aging condition. The fracture path is quantitatively correlated to fracture toughness, and the bulk microstructural attributes estimated via stereological analysis. In the companion article, these quantitative data are used to develop and verify a multiple-fracture micromechanism-based model that relates the fracture toughness to a number of microstructural parameters of the partially recrystallized alloy plate.

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