The technique that we describe aims at evaluating the numerical density of cells in highly heterogeneous regions, e.g., nuclei, layers or columns of neurones. Rather than counting the number of neuronal sections ('profiles') in a reference frame, we evaluated the 'free area' which lies around each profile. The X and Y coordinates of the neuronal profiles within a microscopical section were measured by 2 linear transducers fastened to the moving stage of the microscope. These coordinates were used by a computer programme that we developed to calculate the 'free area' around each neuronal profile. These areas are polygons that cover the plane of the section without interstice or overlap, i.e., realize a tessellation of the section plane ('Dirichlet tessellation'). Each polygon contains one neuronal profile and the area of the section closest to that profile than to any other. When that area is large, the density is low. An individual value of cellular density = 1/(area of Dirichlet polygon) could thus be assigned to each neuronal profile. Coloured density maps were obtained by attributing a colour to each polygon according to its area. Those maps were useful to demonstrate the presence of neuronal clusters (columns, layers, nuclei, etc.). A confidence interval (CI) of mean polygon areas (standard deviation (SD) of polygon areas/square-root n , n being the number of cells) could be calculated and used to determine the CI of the density of neuronal profiles. This value helped to predict the number of profiles which had to be counted in a particular area to obtain a given precision. The coefficient of variation (CV) of the polygon areas is a dimensionless value, which is not affected by atrophy, shrinkage or stretching of the section, but is sensitive to restricted cell loss. When profiles are regularly spaced, the CV is low; it is high when they are clustered. With computer simulation (Monte-Carlo testing) we established that the CVs ranged from 33% to 64% (P < 0.05) when the profiles were randomly distributed according to a Poisson point process. A value lower than 33% suggested a regular distribution, and a value higher than 64% a clustered distribution. Automatic isolation of cell clusters was made possible with Dirichlet tessellation; a cluster was defined as a group of contiguous cells, exhibiting similar numerical density, i.e., whose polygons had similar surface area. The recognition of clusters was made in 3 steps: (a) the smallest polygon was isolated in the population and its area used as a reference; (b) the contiguous polygons were examined: they were admitted into the cluster if the ratio of their area with the reference area was below a given threshold; and (c) the cluster was closed when the polygons at the border of the cluster were all larger than the threshold. The cluster which had been isolated was removed from the map and the process was iterated until all the cells had been tested. Profiles at the boundary of the cluster were defined as those which had neighbours both inside and outside the cluster. The density at the boundary was measured by a value of linear density (number of profiles/length of the boundary). To analyse layered or columnar structures, we counted the number of polygons intersected by 2 orthogonal straight lines, which had been manually drawn along the layers or the columns.
