Surface curvatures of trabecular bone microarchitecture.

Microstructure of trabecular bone has been examined with a particular emphasis on surface curvatures in two-phase (trabecular 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 proposed. 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-Poincaré characteristics and the genus, both characterizing topological features of bone connectivity, were estimated from integral gaussian curvature (Gauss-Bonnet theorem). The three bone microstructures were found to be topologically analogous to spheres with one to three handles.

Interfacial and topological measurements of bicontinuous polymer morphologies.

Bicontinuous morphologies are ubiquitous in nature and occur at various length scales. Topological features of two such morphologies arising in an ordered block copolymer at equilibrium and a polymer blend during spinodal decomposition are measured from three-dimensional images. Interfacial curvature, coordination number, and interjunction distance distributions exhibit remarkable similarity in these systems, despite vastly different length scales. A channel coordination of 3 is dominant in both morphologies, and topological measurements such as the Euler-Poincaré characteristic and genus are reported.