ABSTRACT
Four biopsies of human trabecular bone from the femur, radius and vertebral body with a bone volume fraction (BV/TV) between 5 and 35% were scanned with CT (μCT 40, SCANCO Medical AG)
1 INTRODUCTION
With increasing computational power, nonlinear micro finite element (μFE) models are becoming more and more suitable for the investigation of the mechanical competence of full bone specimens and trabecular bone structures. Currently, nonlinear μFE modeling of trabecular bone is performed in vivo on peripheral skeletal sites using scans from HRpQCT (MacNeil 2008) and μMRI (Zhang 2013) or in vitro using high resolution μCT scans. Especially μCT based μFE models of full bones are computationally very demanding and often require the usage of special solution algorithms and supercomputers (Adams 2004). In principle, μFE models of trabecular bone can be generated by means of direct conversion of segmented CT voxels into hexahedral finite elements or by means of marching cube based meshing algorithms, generating a tetrahedral mesh of the bone microstructure (Fig. 1). Direct voxel conversion is a very efficient way to generate μFE models since no sophisticated meshing algorithm has to be used. However, a vast amount of elements is needed in order to accurately describe the geometry leading to huge FE models with a high number of degrees of freedom (DOF). In order to reduce model size, the image voxels could be coarsened, however trabecular connectivity will be lost in regions with low bone volume fraction (Ulrich 1998). Volume
using a 12μm isotropic spatial resolution. After resampling to 24μm, cubic sub-regions with a side length of 5 mm were extracted from the middle of the μCT scans and further used for the analysis (Fig. 2). The bone cubes were segmented using the single-level threshold of IPL (SCANCO Medical AG). Hexahedral (hex) μFE models were generated by converting the segmented image voxels into linear isotropic eight-node hexahedral finite elements. Linear (tet) and quadratic (tet2) tetrahedral μFE models with different mesh sizes were created using the volume conserving “+FE Free” meshing algorithm (Simpleware Ltd.). The targeted element sizes were 20, 40, 60, 80 and 100μm, however the meshing algorithm was allowed to locally decrease the element size to capture small details (see Fig. 3). In all model types, all finite elements were assigned the same isotropic Young’s modulus of 10GPa, a Poisson’s ratio of 0.3 and an elasto-plastic material model with tension-compression asymmetry (Cast Iron Plasticity, Abaqus 6.12). The tissue level yield strains were chosen with 0.41% in tension, 0.83% in compression (Bayrakter 2004) and the post yield modulus was set to 0.1GPa. The displacements of the bottom nodes of the bone cubes were restrained in all three directions while the top nodes of the bone cubes were restrained perpendicular to the z direction and displaced in the z direction by 1% apparent strain, simulating a compression test until apparent level yielding. All analyses were performed using nonlinear geometry in Abaqus 6.12 (Dassault Systems). The nominal stress was then plotted against the nominal strain and the yield
point evaluated using a 0.2% offset method. The apparent level stiffness, yield stress and yield strain was compared across the different model types. Since all quadratic tet2 models, regardless of element size, predicted the same apparent mechanical behavior, the results of all other models were compared with the quadratic tet2 model type.
