EVALUATION OF NEURONAL NUMERICAL DENSITY BY DIRICHLET TESSELLATION

The technique that we describe aims at evaluating the numerical densit y of cells in highly heterogeneous regions, e.g., nuclei, layers or co lumns of neurones. Rather than counting the number of neuronal section s ('profiles') in a reference frame, we evaluated the 'free area' whic h lies around each profile. The X and Y coordinates of the neuronal pr ofiles within a microscopical section were measured by 2 linear transd ucers fastened to the moving stage of the microscope. These coordinate s were used by a computer programme that we developed to calculate the 'free area' around each neuronal profile. These areas are polygons th at 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 sect ion closest to that profile than to any other. When that area is large , the density is low. An individual value of cellular density = 1/(are a of Dirichlet polygon) could thus be assigned to each neuronal profil e. 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 c onfidence interval (CI) of mean polygon areas (standard deviation (SD) of polygon areas/square-root n , n being the number of cells) could b e calculated and used to determine the CI of the density of neuronal p rofiles. This value helped to predict the number of profiles which had to be counted in a particular area to obtain a given precision. The c oefficient of variation (CV) of the polygon areas is a dimensionless v alue, 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. W ith 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 3 3% suggested a regular distribution, and a value higher than 64% a clu stered distribution. Automatic isolation of cell clusters was made pos sible with Dirichlet tessellation; a cluster was defined as a group of contiguous cells, exhibiting similar numerical density, i.e., whose p olygons had similar surface area. The recognition of clusters was made in 3 steps: (a) the smallest polygon was isolated in the population a nd its area used as a reference; (b) the contiguous polygons were exam ined: they were admitted into the cluster if the ratio of their area w ith the reference area was below a given threshold; and (c) the cluste r was closed when the polygons at the border of the cluster were all l arger than the threshold. The cluster which had been isolated was remo ved 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 t hose which had neighbours both inside and outside the cluster. The den sity at the boundary was measured by a value of linear density (number of profiles/length of the boundary). To analyse layered or columnar s tructures, we counted the number of polygons intersected by 2 orthogon al straight lines, which had been manually drawn along the layers or t he columns.