Surface Curvatures of Trabecular Bone Microarchitecture H. JINNAI, 1 H. WATASHIBA, 1 T. KAJIHARA, 1 Y. NISHIKAWA, 2 M. TAKAHASHI, 1 and M. ITO 3 1 Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Kyoto, Japan 2 Structural Biophysics Laboratory, The Institute of Physical and Chemical Research (RIKEN), Hyogo, Japan 3 Department of Radiology, Nagasaki University School of Medicine, Nagasaki, Japan Microstructure of trabecular bone has been examined with a particular emphasis on surface curvatures in two-phase (tra- becular and intertrabecular space— i.e., marrow space) structures. Three trabecular bone samples, quantified as “plate-like,” “rod-like,” and a mixture of these two structural elements according to the structure model index (SMI), were subjected to analysis based on (differential) geometry. A correspondence between the SMI and the mean curvature was found. A method to measure surface curvatures is pro- posed. The gaussian curvatures averaged over the surfaces for the three analyzed bone structures were all found to be negative, demonstrating their surfaces to be, on average, hyperbolic. In addition, the Euler–Poincare ´ characteristics and the genus, both characterizing topological features of bone connectivity, were estimated from integral gaussian curvature (Gauss–Bonnet theorem). The three bone micro- structures were found to be topologically analogous to spheres with one to three handles. (Bone 30:191–194; 2002) © 2002 by Elsevier Science Inc. All rights reserved. Key Words: Trabecular bone microstructure; Differential geom- etry; Surface curvatures; Parallel surface method; Hyperbolic surface; Three-dimensional (3D) image analysis. Introduction It has been assumed that bone microstructure is as important as bone mass in preventing fractures. But only recently have images of bone microstructures, especially trabecular bone structure, become available in three dimensions (3D), using various so- phisticated imaging techniques (e.g., micromagnetic resonance imaging [micro-MRI] and microcomputed tomography [micro- CT]). 1,2,7 Structural analysis on the basis of these 3D images opens new possibilities in the assessment of bone architecture and related mechanical strength of trabecular bone. Various parameters characterizing the structural features of bone microstructure have been proposed to determine the essen- tial features closely related to the biomechanical competence of trabecular bone. Among the morphometric parameters, the basic ones (e.g., tissue volume [TV], bone volume [BV], and bone surface [BS]), are obtained directly once the trabecular surface is found in the 3D image. In addition to these basic indices, trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), and trabecular number (Tb.N) are often used to characterize bone microstructure. 12,16,17 They are often estimated indirectly based on the assumption that the microstructure essentially consists of either a plate element (a “plate model”) or a rod element (a “rod model”). 16 Hildebrand et al. recently developed new methods to estimate these parameters without any assumptions directly from the 3D image. 4 A well-designed morphometric parameter, the “structure model index” (SMI), was proposed by Hildebrand and Ru ¨egseg- ger 5 to characterize the amount of plate and rod elements com- posing the bone microstructure. The SMI is based on the idea that the structure volume, V, depends on a linear measure, r. The surface area, S, is the derivative of the volume with respect to r. A simple representation of this is expressed by V = kr e (e  1). Parameter e describes the type of the structure. r is taken to be the half thickness (or the radius) of the structural elements, assumed to be constant over the entire structure. For example, e = 1 gives a plate with a surface area given by k. A rod has e = 2 and k is proportional to the length of the rod, respectively. A sphere has e = 3. Hildebrand and Ru ¨egsegger considered a mixed structure consisting of both plate and rod elements. 5 They defined the SMI as SMI  6  S'  V/S 2 (where S' is the derivative with respect to r), which is 0, 3, and 4 for an ideal plate, rod, and sphere, respectively. In the present study, we propose a methodology to fully characterize surface geometries on the basis of the differential geometry (i.e., the classical Hilbert theory 3 ). To the best of our knowledge, surface curvatures (i.e., the mean and gaussian cur- vatures of the surface of the trabecular bone microstructure) have never been measured. We note that these parameters have a solid mathematical foundation. In addition, the topology of the bone surface described by the Euler–Poincare ´ characteristic, , was estimated according to the Gauss–Bonnet theorem. 3,6 The Euler– Poincare ´ characteristic represents a way to describe the complex- ity of network-type structures and is often used to quantify the connectivity of trabecular bone. It is typically calculated from the voxel-based data set of a 3D reconstruction. 15 As shown, the mean curvature and gaussian curvatures are closely related, respectively, to the SMI and Euler number. Materials and Methods Three human iliac bone specimens were obtained from three women (a 60-year-old [sample A], a 64-year-old [sample B], and a 61-year-old [sample C]) undergoing total hip replacement. Their clinical diagnosis was degenerative osteoarthritis of the hip. Iliac bone samples were obtained from a region 4 –5 cm behind and 2–3 cm below the anterior superior illiac spine with a trephine of 10 mm diameter using a transiliac approach. 18 Address for correspondence and reprints:Dr. Hiroshi Jannai, Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto 606-8585, Japan. E-mail: hjinnai@ipc. kit.ac.jp Bone Vol. 30, No. 1 January 2002:191–194 191 © 2002 by Elsevier Science Inc. 8756-3282/02/$22.00 All rights reserved. PII S8756-3282(01)00672-X