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Surface curvature and Minkowski tensors. a-b) The minimum (í µí¼ 1 ) and maximum (í µí¼ 2 ) principal curvatures and associated principal directions on a torus model. c-e) The definitions of the mean (í µí°»), Gaussian (í µí°¾), and net (í µí°·) curvatures as functions of the principal curvatures. The top row visualizes the curvatures of the torus, while the bottom row depicts some small sections of a trabecular bone interface. f) A visualization of the components used in the computation of the Minkowski tensors of a coarse torus model, showing the position vectors (í µí±̅ ) and normal vectors (í µí± ̅), as well as the expressions for the tensors considered in this study.
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A bstract
The organization and shape of the microstructural elements of trabecular bone govern its physical properties, are implicated in bone disease, and can serve as blueprints for biomaterial design. To devise fundamental structure-property relationships, it is essential to characterize trabecular bone from the perspective of geometry, the math...
Contexts in source publication
Context 1
... rank MT are briefly considered in section 2.6. For a 3D body, six relevant rank-two MT are defined ( Figure 1 and Supplementary Note 1). As an example, the tensor í µí± 0 2,0 (í µí°µ) is a measure of the spatial distribution of mass for a solid body í µí°µ, in some sense analogous to the moment of inertia tensor. ...
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... quantify the intra-specimen anisotropy changes, we decomposed 100 trabecular specimens into a set of smaller components. In order to maintain representative trabecular substructures, we used a 3×3×3 cubic grid for this spatial subdivision (Supplementary Figure 10b). We computed the two translation-invariant Minkowski tensors í µí± 1 0,2 and í µí± 2 0,2 on the resulting 2700 substructures, enabling a local characterization of the ellipticity with respect to those tensors. ...
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... a mesh as a doubly-connected edge list (DCEL), the boundary faces are those faces with at least one half-edge that appears only once in list (i.e., it is not shared with another face). A visual representation of the boundary face labeling is provided in Supplementary Figure 10a. ...
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... domain-wise Minkowski analysis was performed on spatially decomposed meshes. The mesh was subdivided into a 3×3×3 cubic grid, and all faces were assigned a label based on the location in the grid, resulting in a total of 28 different labels (27 labels for the grid domains and one label for the boundary, see Supplementary Figure 10b). The relative difference between the local and global DA was defined as: defines the ratio of the minimal to the maximal eigenvalue of the tensor í µí± í µí±£ í µí±,í µí± , while 〈⋅〉 refers to the average value. ...
