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

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. Conclusions: 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.