This post is part of a tutorial series on morphometric indices. You may also be interested in Bone Volume Density (BV/TV) – Light Version, Choose The Right Volume Of Interest – Light Version and The surface of an object. Bone surface (BS) and specific bone surface (BS/BV) posts.
In basic bone research, trabecular thickness (Tb.Th) and trabecular spacing (Tb.Sp) are key measures characterizing the three-dimensional (3D) structure of cancellous bone. Several studies have been performed to identify the contribution of these indices to bone strength and on how they are altered with age and by pharmaceutical intervention (i.e. [1-12]).
In the past, classic stereological methods were used to estimate three-dimensional morphometric indices, such as trabecular thickness or trabecular spacing. With these methods, measures taken from two-dimensional (2D) images were used to compute 3D values based on assumptions of the structure shape. For rod-like bone samples (i.e. trabecular bone from vertebrae) a rod model and for plate-like bone samples (i.e. trabecular bone from femoral head) a plate model was used to estimate an average 3D thickness and separation . However, it has been shown that derived Tb.Th and Tb.Sp using the plate model are often smaller than when measured directly in three dimensions and may yield in biased results [13-15]. For this reason, it is important to assess 3D morphometric indices from 3D images using direct 3D algorithms.
In 1997, Hildebrand and Rüegsegger proposed a new method for the model-independent assessment of thickness and spacing in three-dimensional images . The base idea of this approach is to fit maximal spheres inside an object. The voxels are then set to the values of the largest sphere that contains this voxel.
Figure 1 shows an example of a small portion of a trabecular bone (A) and the corresponding thickness labeled image (B). The color in this thickness labeled image corresponds to the local thickness of the structure. In Figure 1 C, all voxels are set to semi-transparent and only voxels representing a thickness of 200 µm are shown in solid. With this thickness definition, thickness is defined locally in each voxel. The mean thickness of the structure (i.e. Tb.Th) is then calculated as the volume weighted average of the local thicknesses.
Figure 1: A) Trabecular bone sample (3x3x3 mm3). B) Same as A), where the colors codes the local thickness (0 – 320 µm). C) Color coded trabecular bone in semi-transparent and all voxels that are assigned to a value of 200 µm are shown in yellow. D) Histogram of thickness distribution.
The mean trabecular thickness (Tb.Th) can be a good measure for comparing the thickness of trabecular structures. However, as this is a scalar this measure may not be able to describe all structural changes. We can think of two samples with identical Tb.Th values, where one sample is composed of struts with uniform thickness and the other sample is composed of thick vertical struts that are interconnected by thin horizontal trabeculae. Thus, it could be useful to rather compare the histograms (Figure 1D), where the two cases could clearly be distinguished.
This local thickness method cannot only be used to compute the thickness of structures but when applied to the inverse of the structure, it can also be used to estimate the average separation of the struts. Figure 2 shows a trabecular bone sample where on the left trabecular thickness and on the right trabecular spacing is shown in fading colors.
Figure 2: Animation showing color coding of Tb.Th (left) and Tb.Sp (right)
Furthermore, this method can be used to calculate the average trabecular number (Tb.N). Trabecular number is taken as the inverse of the mean distance between the mid-axes of the structure to be examined. The mid-axes are assessed from the binary 3D image using the 3D distance transformation and extracting the center points of non-redundant spheres which fill the structure completely. The mean distance between the mid-axes is then determined in analogy to the Tb.Sp calculation, i.e. the average separation between the mid-axes is calculated.
The concept for measuring thickness and separation can be applied to any porous structre and is used in many different fields. Depending on the application, the one or the other measure is of higher importance. For instance, in food industry, it is important that fat foams do have a certain porosity, which results in a light intense taste. In engineered scaffolds (i.e. HA/TCP-scaffolds) that are seeded with cells it is important to know the cavity diameters as well as the hole diameter connecting the pores. If these diameters are too small, the cells may not be able to penetrate the structure and will cover only the outside of the scaffold. Similar problems are presented in the oil and gas industry, where the porosity and the pore interconnections in rocks are used for flow simulations.
There are many more fields where the concept of measuring thickness and separation can be applied. They are important measures characterizing three-dimensional porous structures, however, as all morphometric indices, they are just a small piece of a puzzle and only the combination of several indices will lead to a full picture. In the last two examples, it is important to also quantify connectivity density, a measure that will be covered in one of our next posts.
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