Voronoi tessellation to study the numerical density and the spatial distribution of neurones

The conditions of regularity and isotropy, required by standard morphometric procedures, are generally not fulfilled in the central nervous system (CNS) where cells are distributed in a highly complex manner. The evaluation of the mean numerical density of neuronal or glial cells does not take into account the topographical heterogeneity and thereby misses the information that it contains. A local measurement of the density can be obtained by evaluating the 'numerical density of one cell', i.e. the ratio 1/(the volume that the cell occupies). This volume is the region of space that is closer to that cell than to any other. It has the shape of a polyhedron, called Voronoi (or Dirichlet) polyhedron. In 2-D, the Voronoi polyhedron is a polygon, the sides of which are located at mid-distance from the neighbouring cells. The Voronoi polygons are contiguous and their set fills the space without interstice or overlap, i.e. they perform a 'tessellation' that may yield a density map when the same colours are used to fill polygons of similar sizes. The use of Voronoi polygons allows computing the confidence interval of a mean numerical density that makes statistical comparisons possible. The tessellation also provides information concerning spatial distribution; the areas of the Voronoi polygons do not vary much when the cells are regularly distributed. On the contrary, small and large polygons are found when cellular clusters are present. The coefficient of variation of the polygon areas is an objective measurement of their variability and helps to define 'regular', 'clustered' and 'random' distributions. When cells are clustered, small polygons are contiguous and may be objectively identified by simple algorithms. Voronoi tessellations are easily performed in 2-D. On an average the area of a polygon times the thickness of the section equals the volume of the corresponding polyhedron. 3-D tessellations that are theoretically possible and for which algorithms have been published remain to be adapted to histological works.