An improved trabecular bone model based on Voronoi tessellation

BACKGROUND AND OBJECTIVE

Accurate numerical and physical models of trabecular bone, correctly representing its complexity and variability, could be highly advantageous in the development of e.g. new bone-anchored implants due to the limited availability of real bone. Several Voronoi tessellation-based porous models have been reported in the literature, attempting to mimic the trabecular bone. However, these models have been limited to lattice rod-like structures, which are only structurally representative of very high-porosity trabecular bone. The objective of this study was to provide an improved model, more representative of trabecular bone of different porosity.

METHODS

Boolean operations were utilized to merge scaled Voronoi cells, thereby introducing different structural patterns, controlling porosity and to some extent anisotropy. The mechanical properties of the structures were evaluated using analytical estimations, numerical simulations, and experimental compression tests of 3D-printed versions of the structures. The capacity of the developed models to represent trabecular bone was assessed by comparing some key geometric features with trabecular bone characterized in previous studies.

RESULTS

The models gave the possibility to provide pore interconnectivity at relatively low porosities as well as both plate- and rod-like structures. The mechanical properties of the generated models were predictable with numerical simulations as well as an analytical approach. The permeability was found to be better than Sawbones at the same porosity. The models also showed the capability of matching e.g. some vertebral structures for key geometric features.

CONCLUSIONS

An improved numerical model for mimicking trabecular bone structures was successfully developed using Voronoi tessellation and Boolean operations. This is expected to benefit both computational and experimental studies by providing a more diverse and representative structure of trabecular bone.

Characterization of microstructures of SAN foam core using micro-computed tomography

In this study, the microstructure of a SAN foam was imaged using a micro-CT scanner. Through image processing and analysis, variations in density, cell wall thickness and cell size in the foam were quantitatively explored. It is found that cells in the foam are not elongated in the thickness (or rise) direction of foam sheets, but rather equiaxed. Cell walls in the foam are significantly straight. Density, cell size and cell wall thickness all vary along the thickness direction of foam sheets. The low density in the vicinity of one face of foam sheets leads to low compressive stiffness and strength, resulting in the strain localization observed in our previous compressive tests. For M80, large open cells on the top face of foam sheets are likely to buckle in compressive tests, therefore being another potential contributor to the strain localization as well. The average cell wall thickness measured from 2D slice images is around 1.4 times that measured from 3D images, and the average cell size measured from 2D slice images is about 13.8% smaller than that measured from 3D images. The dispersions of cell wall thickness measured from 2D slice images are 1.16–1.20 times those measured from 3D images. The dispersions of cell size measured from 2D slice images are 1.12–1.36 times those measured from 3D images.

Influence from polydispersity on the morphology of Voronoi and equilibrium foams

Stochastic foam models are generated from Voronoi spatial partitioning, using the centers of equi-sized hard spheres in random periodic distributions as seed points. Models with different levels of polydispersity are generated by varying the packing of the spheres. Subsequent relaxation is then performed with the Surface Evolver software which minimizes the surface area for better resemblance with real foam structures. The polydispersity of the Voronoi precursors is conserved when the models are converted into equilibrium models. The relation between the sphere packing fraction and the resulting degree of volumetric polydispersity is examined and the relations between the polydispersity and a number of associated morphology parameters are then investigated for both the Voronoi and the equilibrium models. Comparisons with data from real foams in the literature indicate that the used method is somewhat limited in terms of spread in cell volume but it provides a very controlled way of varying the foam morphology while keeping it periodic and truly stochastic. The study shows several strikingly consistent relations between the spread in cell volume and other geometric parameters, considering the stochastic nature of the models.