Betti number ratios as quantitative indices for bone morphometry in three dimensions

Computational homology is an emerging mathematical tool for characterizing shapes of data. In this work, we present a methodology using computational homology for obtaining quantitative measurements of the connectivity in bone morphometry. We introduce the Betti number ratios as novel morphological descriptor for the classification of bone fine structures in three dimensions.

A total of 51 Japanese white rabbits were used to investigate the connectivity of bone trabeculae after the administration of alendronate in a tendon graft model in rabbits. They were divided into a control group C and an experimental group A. Knee joints specimens were harvested for examination of their bone trabecular structure by micro-CT. Applying the computational homology software to the reconstructed 3D image data, we extract the morphological feature of a steric bone structure as the Betti numbers set (β₀, β₁, β₂). The zeroth Betti number β₀ indicates the number of the connected components corresponding to isolated bone fragments. The first and second Betti numbers, β₁ and β₂, indicate the numbers of open pores and closed pores of bone trabeculae, corresponding to a 2D empty space enclosed by a 1D curve and a 3D empty space enclosed by a 2D surface, respectively.

We define the Betti number ratios β₁/β₀ and β₂/β₀ to better distinguish the two groups A and C in the scatter plots. Testing the discriminant function line for 29 data points of A (22 data points of C), the 17 points (resp. 18 points) are correctly classified into group A (resp. C). The accuracy rate is 35/51. The classification results in terms of the Betti number ratios are consistent with the histomorphometric measurements observed by medical doctors.

This study is the first application of computational homology to bone morphometry in three dimensions. We show the mathematical basis of the Betti numbers index which are useful in a statistical description of the topological features of sponge-like structures. The potential benefits associated with our method include both improved quantification and reproducibility for the stereology.