# Three-Dimensional Anatomical Analysis of Muscle-Skeletal Districts
## 4 Results
### Single Exam Evaluation: Vertebral Spine Characterisation  
For the analysis of a _single exam_, as the main example, we considered the spine district. The CT data used for the spine characterisation are a sub-part of the _Large Scale Vertebrae Segmentation challenge_ (VerSe), which provides a common benchmark for current and future spine-processing algorithms [2]. The data were multi-site acquired using multiple CT scanners and present a variety of FoVs (including cervical, thoracolumbar, and cervical-thoracolumbar scans), a mix of sagittal and isotropic reformations, and cases with vertebral fractures, metallic implants, and foreign materials [2]. The annotations in the dataset are twofold and present the 3D coordinate locations of the vertebral centroids and voxel-level labels as segmentation masks. Twenty-six vertebrae (C1 to L5 and the transitional T13 and L6) are annotated with labels from 1 to 24, along with labels 25 and 28 for L6 and T13, respectively. This CT dataset contains 160 image series of 141 patients including segmentation masks of 1725 fully visualised vertebrae [49]. The metadata includes the annotation of vertebral fractures obtained through [PERSON]'s method, the indication of foreign material, and the measurement of lumbar Bone Mineral Density (BMD) per patient age (Figure 6a). For our purposes, we used only one type of CT scan, to be able to compare the results obtained in the tissue evaluation. Figure 6b shows the age distribution of the subjects considered in our work. The segmentation of the vertebral bones localises the three main functional components of each vertebra (i.e., vertebral body, vertebral arch, and transverse processes) from a patient-specific perspective. The first step is the computation of the distance distribution of each vertex composing the vertebral surface from the centroid of the vertebral body. The idea is to grow a sphere centred in the centroid and to assign to each vertex of the bone a value of the distance equal to the radius of the first sphere containing it. For efficiency, the sphere's growth is executed by computing the Euclidean distance \(d(\mathbf{p}_{i},\mathbf{c}):=\|\mathbf{p}_{i}-\mathbf{c}\|_{2}\) of each vertex \(\mathbf{p}_{i}\) from the vertebral body centroid \(\mathbf{c}\) (Figure 7). Figure 6: (**a**) Red dots represent the Bone Mineral Density (BMD) behaviour as a function of subjects’ age; the \(x\)–axis presents subjects’ age, the \(y\)–axis BMD values. (**b**) Age distribution of the considered subjects. Figure 7: Distance distribution of a vertebral bone. Each vertex is associated with the value of its distance from the vertebral body centroid. The unit measure of the colour map is millimetres. Then, we computed the _probability density_ from the distribution of the distances associated with the vertices of each bone. In this way, we transformed the 3D information, such as the distance of the vertex from the centroid, into 1D information. In different subjects, the same vertebra presents a similar behaviour of the probability density curve. Thus, we leveraged the inflexion points of the probability density curve as the distance thresholds. Indeed, the vertex that has a distance less than the first inflexion point (\(T_{1}\)) is considered a part of the vertebral body. The vertex whose distance belongs to the range between the first and the second inflexion point (\(T_{2}\)) is considered a part of the vertebral arch. All the vertices that have a distance greater than the second inflexion point are classified as part of the transverse or spinous processes, and the distance value considered as a threshold is the third inflexion point (\(T_{3}\)) (Figure 8). Once the vertebral bone has been segmented and labelled in its functional sub-parts, we considered, as a geometrical parameter, the distance values corresponding to the inflexion points (i.e., \(T_{1}\), \(T_{2}\), \(T_{3}\)). Indeed, the threshold used for vertebral segmentation is the geometrical property that permits the classification of the surface vertices.