Chappard D, Legrand E, Haettich B, Chalès G, Auvinet B, Eschard J P, Hamelin J P, Baslé M F, Audran M
GEROM--LHEA: Laboratoire d'Histologie-Embryologie, CHU & Faculté de Médecine, 49045 Angers Cédex, France.
J Pathol. 2001 Nov;195(4):515-21. doi: 10.1002/path.970.
Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three 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 patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed 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 relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided 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 complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.
据报道，小梁骨在组织学水平上具有二维（2-D）分形特征，这一发现与生物力学特性相关。然而，已知有几种分形维数（D），且获取它们的计算方法差异很大。本研究在同一系列骨活检样本上比较了三种算法，以获得柯尔莫哥洛夫、闵可夫斯基-布利冈和质量-半径分形维数。研究了与二维小梁结构的组织形态计量学描述符之间的关系。从148名男性骨质疏松症患者身上获取骨活检样本。计算骨体积（BV/TV）、小梁特征（Tb.N、Tb.Sp、Tb.Th）、支柱分析、星体积（骨髓腔和小梁）、连通性指数和欧拉-庞加莱数。采用盒计数法获得柯尔莫哥洛夫维数（D(k)），采用膨胀法获得闵可夫斯基-布利冈维数（D(MB)），采用沙盒法获得质量-半径维数（D(MR)）和孔隙率（L）。观察到BV/TV与分形维数之间存在对数关系。与D(MR)的相关性最佳，与D(MB)的相关性最低。孔隙率与骨髓腔描述符（ICI、星体积、Tb.Sp）相关。在三种分形技术之间观察到线性关系，它们似乎高度相关。对所有组织形态计量学参数进行聚类分析得到一棵树，其中有三组描述符：小梁（Tb.Th、支柱）；骨髓腔（欧拉数、ICI、Tb.Sp、星体积、L）；以及网络复杂性（Tb.N和三个D）。不能用单一的分形维数替代经典的二维结构描述符；D更能反映分支小梁的复杂性。在选择这些方法之一时，计算时间也是一个重要的决定因素。
