## 4 Discussion  
The following characteristics must be considered to model the biomechanics of the spine: its linear geometry, the joints at each intervertebral level, and its multiaxial movements [32]. Therefore, the mathematical model developed within this study uses an articulated model. As previously mentioned, articulated models represent the spine as rigid geometric transformations from one vertebra to the other. Typically, in these types of models, the first vertebra is stationary, while the previous ones guide each successive vertebra [20]. However, the methodology to find the minimum energy for the spring-mass system required that the position of the two extremity vertebrae was pre-determined and fixed. As a result, the mathematical model uses an articulated spine model that can include 20 masses (from S1 to C6) and requires a reference point by each vertebra: the apex of its spinous process previously identified by the VM. A preliminary study evaluated the capability to quantify the Cobb and intervertebral angles. The Cobb angles are the gold standard parameter to evaluate spinal biomechanics. On the other hand, the intervertebral angles are not a standard, but they can be used to evaluate the model's performance compared to the radiographs. Height patients of a Portuguese hospital with ages ranging from 22 to 55 years old participated in the study. Although the biomechanical changes in the spine are more prone to happen with age, they are a consequence of modifications in the vertebrae alignment. The biomechanical spinal changes affect the vertebrae positioning and, consequently, the spinous processes' positioning. By identifying the 3D position of their apex, we were able to model the vertebrae configuration independently of their age. The integrated system (VM and the new mathematical model) identified and modelled the 20 vertebrae. The results suggest that the position of C6 in the sagittal plane slightly influences the model performance. The 95% confidence interval limits decreased by customising the C6 rotation angle for each patient according to the radiographic data. This is because the literature did not include the effects of Ankylosing Spondylis or the inter-individual variability. Note that the errors associated with the measuring of the 3D position of the fluorescent marks were already quantified in previous studies performed by the authors [PERSON] et al. in 2015; [PERSON] et al. in 2018. The mentioned studies focused on the development and validation of the measuring performance of the VM device. It was statistically proven to provide accurate results (errors in the order of 1 mm). The resolution in the order of 1 mm is satisfactory since this value is much smaller than the diameter of the fluorescent marks (around 3 mm), as well as smaller than the anatomical dimension of the spinous processes. A previous study measured the height of the spinous processes (vertical distance between their upper and lower edges) of L1, third lumbar vertebra (L3) and L5 in 15 patients. The average height found for each of the spinous processes was 20.5 mm, 20.2 mm and 14.1 mm, respectively [19]. Although this parameter is conditioned by several factors (such as the type of vertebra, genetics and associated pathologies), it always exceeds 1 mm, meaning that the VM measurement error does not compromise the measure of the 3D position of the spinous processes. Regarding the mathematical model presented in the current paper, we focused on the analysis of the differences between the angles provided by the model and the angles measured in the radiographs. The Bland-Altman plot analysis \"only defines the intervals of agreements, it does not say whether those limits are acceptable or not. Acceptable limits must be defined a priori, based on clinical necessity, biological considerations or other goals\" [49]. This project was carried out in collaboration with the rheumatologic department of a Portuguese hospital, which provided insightful feedback of the results.