This post is part of a tutorial series on morphometric indices. You may also be interested in Bone Volume Density (BV/TV) – Light Version and Choose The Right Volume Of Interest – Light Version posts.
Measuring the volume and surface of an object is an intuitive concept and these measures are considered base measures in characterizing three-dimensional objects such as trabecular bone. Where the surface delineates the boundary between object (i.e. bone) and background (i.e. marrow space), the volume enclosed by the surface describes the volume of the object (i.e. bone volume).
Although the concept is very basic, it is not that simple to compute accurate numbers from the microCT images, because the real life object is converted to a discrete voxel object during the scan process. Thus, the scanned object looks very edgy and the surface is not smooth at all (Figure 1A). If the surface of this volume is determined by summing up the faces that build the interface of the object to the background, the surface will always be overestimated. For this reason, it is important to smooth the surface to reach an object that represents the real life object as good as possible.
A common approach is to create a triangle mesh that represents the surface of the object. One of the most used algorithms is the so called Marching Cubes algorithm presented by Lorensen and Cline . This algorithm places triangles on the surface of the object and the triangles cut the sharp corners of the voxels (Figure 1B). This mesh is already a much smoother representation of the surface and summing the total area of all triangles should yield in a better value of the surface of the object. However, also this mesh does not really yield in a good representation of the surface of the real life object. To get an even better representation the triangle mesh can be smoothed. A simple way to do this is to iteratively translate the vertices (the corners of the triangles) perpendicular to the surface to reduce the angle of adjacent triangles. In Figure 1C, the surface mesh of Figure 1B was smoothed by 2 iteration steps. This new mesh has exactly the same number of triangles but looks much smoother than the mesh in Figure 1B. Summing the area of these triangles will certainly result in an accurate value for the total surface of the object.
All three meshes (Figure 1A – 1C) are models of the real life object and it is difficult to say which model best represents the real life object. It could well be that the real life object actually looks like the model in Figure 1A (i.e. if the sample is a 3D printed structure), however for natural materials (i.e. trabecular bone) this is very unlikely and a representation as shown in Figure 1D is probably the best approximation.
Meshes cannot only be created for the surface but can also be used to represent the whole sample. There is an extension of the Marching Cubes algorithm called Volumetric Marching Cubes . This algorithm basically creates tetrahedrons (or hexahedrons) to mesh the whole object. While in the inside of the object the elements basically all look the same, the elements on the surface can also be smoothed in the same way as for normal surface meshes. The volumetric meshes do have the advantage that they do not only allow for an accurate surface determination but also allow to compute an accurate volume. Furthermore, these meshes can be used as input files for finite element simulations.
The surface and volume computation can be done for any object in the same way. In the bone field, these measures are typically referred to as bone volume (BV) and bone surface (BS). From these measures, it is straight forward to derive the specific bone surface (BS/BV), a measure for the bone surface per given bone volume. In bone biology, this measure is used quite often, since it gives a measure on how many bone lining cells cover a given volume of bone.
Surface measures are not only used for the characterization of single phase objects but often to determine the interface of two objects, i.e. the contact surface of an implant in bone. In this case, the surface creation is done in the same way as for single phase objects but to determine the interface area only the triangles are summed that are part of both phases.
 Lorensen WE, Cline HE, Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Computer Graphics, 1987, 21(4):163-169.
 Müller R, Rüegsegger P, Three-dimensional fnite element modelling of non-invasively assessed trabecular bone structures. Medical Engineering & Physics, 1995, 17(2):126-33.