Quantification of connectivity in cancellous bone, with special emphasis on 3-D reconstructions

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.