Métodos para la Determinación de los Precios de Transferencia

Vertebral geometry for continuum-level FE models can be built directly from individual specimen images (specimen-specific models) or from averaging measurements taken from specimens using statistical methods or parameterised models to give a generic model (Higgins et al., 2006).

In order to build the finite element mesh, there is a need to segment the volumes of interest relating to different materials. Segmentation of vertebrae may require identification of the trabecular bone, cortical shell, soft tissues and any supporting bone cement, which is often included as part of validation experiments. The identification of these regions can be challenging considering the variation in densities captured by microCT seen between specimens (Pahr and Zysset, 2009b). Segmentation can be performed in a number of ways such as manually ‘painting’ voxels, signal intensity based thresholding or region-based or edge-based numerical analysis of geometries (Leventon et al., 2000).

The accuracy of the resultant model is dependent on the accuracy of the segmentation. It is known that FE solutions are sensitive to the threshold values of bone used to segment the

bone mask, as this determines the amount of bone material captured, and therefore both geometry and material properties. The accuracy is also dependent on the original image quality. It is therefore important to consider the image resolution, contrast and noise as well as user interpretation. Image reconstruction software (ScanIP, Simpleware, UK) has been used successfully to segment vertebral bodies through thresholding according to signal intensity and morphological adjustment of masks to segment bone and bone cement (Tarsuslugil et al., 2014). In this method, density-dependent material properties were assigned to the full vertebrae, avoiding the need to separate the trabecular bone centrum from the cortex; this approach has been shown to provide models with material properties in good agreement with corresponding experiments (Tarsuslugil et al., 2014).

Volume meshing is conducted after segmentation, and can be done by matching the mesh element size to the size of the voxels in the segmented three dimensional models. Modern microCT scanners can provide very high resolution scans with voxel sizes less than 5µm, however voxel based finite element models based on such scans have very high number of elements, creating FE models which take a lot of computational time to solve. To overcome this issue scans can be taken at lower resolutions or high resolution scans can be down- sampled prior to mesh generation. Whilst this improves the practicality of the modelling, information about the trabecular architecture is lost, or not captured by the scan if the voxel resolution becomes larger than the trabeculae size. Yeni et al. showed by investigating the effect of scan resolution and reconstruction resolution that creating a coarse mesh from high resolution scans more accurately represented material properties and the mechanical response of the structure than scanning at a lower resolution to match the mesh resolution (Yeni et al., 2005). It is suggested that this may be due to the lower signal to noise ratio when a lower scan resolution is used, as well as some effect of the scan resolution decrease causing effective trabeculae thickening. When down-sampling or choosing a voxel resolution for the FE model it is important to understand the sensitivity of the model to the voxel resolution. A range of voxel resolutions can be seen in the literature for use in

specimen-specific vertebrae models and a range smoothing algorithms are applied to the models. For voxel based methods, element sizes typically range from 1 to 5 mm. Jones and Wilcox (Jones and Wilcox, 2007) conducted a mesh sensitivity analysis based on FE models and concluded that a 2x2x2mm3 was sufficient as errors created by other factors outweigh those created as a result of voxel size. Crawford et al. investigated the difference in predictive capability between a 3x3x3mm element sized mesh and a 1x1x1.5mm mesh, concluding that when developing models for a clinical application the variation seen in vertebral mechanical properties seen in the population are far greater than the differences in predicted values resulting from differences in mesh resolution (Crawford et al., 2003b). It is important to evaluate all such comparisons with the overall application in mind to apply context to the investigation. The applications in these studies, to predict vertebral stiffness and strength, match best to the applications intended in this thesis, and so therefore provide evidence on the mesh density required.

FE models are sensitive to geometry and material properties, so in order to most accurately represent the geometry of the vertebrae the smoothing algorithms can be used on the surface of the model using tetrahedral elements instead of hexahedral elements to create a smooth cortical shell and endplates (Jones and Wilcox, 2008).

As mentioned above, vertebral model geometries can also be derived from averaged data, taken either from the literature or experimental measurements, creating a non-specimen specific model. This effectively reduces the effect of inaccuracies due to the large patient variation seen in direct anatomical data, however also makes models less meaningful as they are more susceptible to errors in results appearing correct. It is currently more common to create specimen-specific models as these can be directly validated by experimental results, giving much more confidence in model outcomes (Jones and Wilcox, 2008).