Voronoi tessellation to study the numerical density and the spatial distribution of neurones
Voronoi tessellation to study the numerical density and the spatial distribution of neurones
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DOI：
10.1016/s0891-0618(00)00064-8
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发表时间：
2000-10-01
影响因子：
2.8
通讯作者：
Godefroy, G
中科院分区：
文献类型：
--
作者：
Duyckaerts, C;Godefroy, G
标准形态测定程序所需的规则性和各向同性条件通常在中枢神经系统 (CNS) 中无法满足，因为细胞以高度复杂的方式分布。对神经元或神经胶质细胞平均数值密度的评估没有考虑拓扑异质性，因此错过了它包含的信息。密度的局部测量可以通过评估“一个细胞的数字密度”来获得，即比率 1/（细胞占据的体积）。该体积是距离该单元比距离任何其他单元更近的空间区域。它具有多面体的形状，称为沃罗诺伊（或狄利克雷）多面体。在二维中，沃罗诺伊多面体是一个多边形，其边位于与相邻单元的中间距离。 Voronoi 多边形是连续的，并且它们的集合填充空间而没有间隙或重叠，即，它们执行“镶嵌”，当使用相同的颜色填充相似大小的多边形时，可能会产生密度图。使用 Voronoi 多边形可以计算平均数值密度的置信区间，从而使统计比较成为可能。镶嵌还提供有关空间分布的信息；当单元规则分布时，Voronoi 多边形的面积变化不大。相反，当存在细胞簇时，会发现小多边形和大多边形。多边形区域的变异系数是对其变异性的客观测量，有助于定义“规则”、“聚集”和“随机”分布。当细胞聚集时，小多边形是连续的，并且可以通过简单的算法客观地识别。 Voronoi 曲面细分很容易在二维中执行。平均而言，多边形的面积乘以截面的厚度等于相应多面体的体积。理论上可行且已发布算法的 3D 镶嵌仍有待适应组织学工作。 (C) 2000 Elsevier Science B.V. 保留所有权利。
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. (C) 2000 Elsevier Science B.V. All rights reserved.