Fractal dimension of trabecular bone: comparison of three histomorphometric computed techniques for measuring the architectural two-dimensional complexity

Trabecular bone has been reported as having two-dimensional (2-D) fractal c haracteristics at the histological level, a finding correlated with biomech anical properties. However, several fractal dimensions (D) are known and co mputational ways to obtain them vary considerably. This study compared thre e algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patie nts. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), interconnectiv ity index, and Euler-Poincare number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D-k), the dilatation method f or the Minkowski-Bouligand dimension (D-MB), and the sandbox for the mass-r adius dimension (D-MR) and lacunarity (L). Logarithmic relationships were o bserved between BV/TV and the fractal dimensions. The best correlation was obtained with D-MR and the lowest with D-MB. Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relat ionships were observed among the three fractal techniques which appeared hi ghly correlated. A cluster analysis of all histomorphometric parameters pro vided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut ); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the com plexity of the network (Tb.N and the three D's). A sole fractal dimension c annot be used instead of the classic 2-D descriptors of architecture; D rat her reflects the complexity of branching trabeculae. Computation time is al so, an important determinant when choosing one of these methods. Copyright (C). 2001 John Wiley & Sons, Ltd.