Odgaard A, Gundersen H J
Department of Orthopaedic Surgery, Aarhus University Hospital, Denmark.
Bone. 1993 Mar-Apr;14(2):173-82. doi: 10.1016/8756-3282(93)90245-6.
The connectivity of cancellous bone attracts increasing attention, as it has been hypothesized that a primary reason for decreasing strength and stiffness in osteoporosis is caused by a loss of trabecular elements and consequently a loss in connectivity. The Euler characteristic has in a few previous articles been used to express cancellous bone connectivity, but there are severe problems in using the Euler characteristic uncritically. The Euler characteristic of a three-dimensional structure is a topological invariant, which reports the number of particles of a structure plus the number of enclosed cavities minus the connectivity. As such, one must know the number of components and the number of enclosed cavities in order to use the Euler characteristic as an expression of connectivity. Another difficulty of using the Euler characteristic is that due to edge effects the Euler characteristic of an excised specimen provides a biased estimate of the Euler characteristic of the region from which the specimen was taken. In this article the intuitive concept of connectivity is given a precise mathematical definition, and a basic topological method for quantifying the connectivity of cancellous bone is presented. The method uses the Euler characteristic, but the above-mentioned problems are controlled. The development of the method and the practical implementation is based on a set of topological notes. It must be stressed that the method is free from assumptions concerning trabecular architecture, and that the method is unbiased. This is in contrast to previously presented methods. The unbiased and model-free method is used on a series of 3-D reconstructions of cancellous bone specimens, and it is demonstrated that the connectivity of cancellous bone is not simply related to volume fraction (density), and that biased and model-based 2-D methods aimed at determining connectivity do not have any general relationship to connectivity in cancellous bone.
松质骨的连通性引起了越来越多的关注，因为据推测，骨质疏松症中强度和刚度降低的一个主要原因是小梁结构的丧失，进而导致连通性丧失。在之前的几篇文章中，欧拉特征已被用于表示松质骨的连通性，但不加批判地使用欧拉特征存在严重问题。三维结构的欧拉特征是一个拓扑不变量，它表示结构的粒子数加上封闭腔的数量减去连通性。因此，为了使用欧拉特征来表示连通性，必须知道组件的数量和封闭腔的数量。使用欧拉特征的另一个困难在于，由于边缘效应，切除标本的欧拉特征会对所取标本区域的欧拉特征提供有偏差的估计。在本文中，连通性的直观概念被赋予了精确的数学定义，并提出了一种量化松质骨连通性的基本拓扑方法。该方法使用欧拉特征，但上述问题得到了控制。该方法的开发和实际实现基于一组拓扑笔记。必须强调的是，该方法不受关于小梁结构的假设的影响，并且该方法是无偏差的。这与之前提出的方法形成对比。这种无偏差且无模型的方法应用于一系列松质骨标本的三维重建，结果表明松质骨的连通性与体积分数（密度）并非简单相关，并且旨在确定连通性的有偏差且基于模型的二维方法与松质骨的连通性没有任何普遍关系。
