Estimation of surface area from vertical sections

Authors

  • A. J. Baddeley,

    1. School of Mathematics, University of Bath, U. K.
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    • School of Mathematics, University of Bath, U. K.

  • H. J. G. Gundersen,

    1. School of Mathematics, University of Bath, U. K.
    2. Stereologic and Electron Microscopic Diabetes Research Laboratory, University Institute of Pathology and Second University Clinic of Internal Medicine, Åhous Kommunehospital, DK-8000 Aarhus C, Denmark.
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  • L. M. Cruz-Orive

    Corresponding author
    1. School of Mathematics, University of Bath, U. K.
    2. Anatomisches Institut der Universität, Postfach 139, CH-3000 Bern 9, Switzerland.
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CSIRO Division of Mathematics and Statistics, P. O. Box 218, Lindfield, Sydney, NSW 2070, Australia.

SUMMARY

‘Vertical’ sections are plane sections longitudinal to a fixed (but arbitrary) axial direction. Examples are sections of a cylinder parallel to the central axis; and sections of a flat slab normal to the plane of the slab. Vertical sections of any object can be generated by placing the object on a table and taking sections perpendicular to the plane of the table.

The standard methods of stereology assume isotropic random sections, and are not applicable to this kind of biased sampling. However, by using specially designed test systems, one can obtain an unbiased estimate of surface area.

General principles of stereology for vertical sections are outlined. No assumptions are necessary about the shape or orientation distribution of the structure. Vertical section stereology is valid on the same terms as standard stereological methods for isotropic random sections. The range of structural quantities that can be estimated from vertical sections includes Vv, Nv, Sv and the volume-weighted mean particle volume v̄v, but not Lv.

There is complete freedom to choose the vertical axis direction, which makes the sampling procedure simple and ‘natural’. Practical sampling procedures for implementation of the ideas are described, and illustrated by examples.

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