Design-based estimation of surface area in thick tissue sections of arbitrary orientation using virtual cycloids

Surface area is a first-order stereological parameter with important biological applications, particularly at the intersection of biological phases. To deal with the inherent anisotropy of biological surfaces, state-of-the-art design-based methods require tissue rotation around at least one axis prior to sectioning. This paper describes the use of virtual cycloids for surface area estimation of objects and regions in thick, transparent tissue sections cut at any arbitrary (convenient) orientation. Based on the vertical section approach of Baddeley et al., the present approach specifies the vertical axis as the direction of sectioning (i.e. the direction perpendicular to the tissue section), and applies computer-generated cycloids (virtual cycloids) with their minor axis parallel to the vertical axis. The number of surface-cycloid intersections counted on focal planes scanned through the z-axis is proportional to the surface area of interest in the tissue, with no further assumptions about size, shape or orientation. Optimal efficiency at each x-y location can be achieved by three virtual cycloids orientated with their major axes (which are parallel to the observation planes) mutually at an angle of 120 degrees. The major practical advantage of the present approach is that estimates of total surface area (S) and surface density (SV) can be obtained in tissue sections cut at any convenient orientation through the reference space.