## 2 Methods
### Establishing Input Conditions  
In the first step, input conditions were established. These included pre-defined anatomical and physical characteristics. Respect for the anatomical differences between vertebrae and intervertebral discs is relevant to the relationship between rigid and elastic elements. Regarding the anatomical properties, the statistical dimensions and orientation of the vertebrae were obtained from the literature and are presented in Table 1[37; 38; 39]. For a better understanding, Figure 2 provides the vertebrae components and measures considered in different perspectives. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c c c} \hline  & **S1** & **L5** & **L4** & **I3** & **L2** & **L1** & **T12** & **T11** & **T10** & **T9** & **T8** & **T7** & **T6** & **T5** & **T4** & **T3** & **T2** & **T1** & **C7** & **C6** \\ \hline EPW (mm) & 48 & 48 & 48 & 46 & 44 & 42 & 42 & 39 & 35 & 33 & 30 & 29 & 28 & 27 & 26 & 26 & 27 & 28 & 23 & 20 \\ \hline EPD (mm) & 34 & 34 & 35 & 35 & 35 & 35 & 33 & 32 & 31 & 29 & 28 & 27 & 26 & 24 & 23 & 22 & 20 & 18 & 17 \\ \hline EPA (mm\({}^{2}\)) & - & 1218 & 1273 & 1290 & 1197 & 1117 & 1024 & 945 & 834 & 754 & 664 & 603 & 552 & 495 & 444 & 412 & 400 & 376 & 280 & 290 \\ \hline VBH (mm) & 23 & 23 & 24 & 24 & 24 & 24 & 23 & 21 & 20 & 19 & 18 & 17 & 16 & 16 & 16 & 16 & 14 & 13 & 11 \\ \hline SCW (mm) & - & 27 & 25 & 24 & 24 & 24 & 22 & 19 & 18 & 18 & 18 & 17 & 17 & 17 & 17 & 18 & 19 & 22 & 25 & 26 \\ \hline SCD (mm) & - & 20 & 19 & 17 & 18 & 19 & 18 & 16 & 15 & 16 & 16 & 16 & 16 & 16 & 16 & 15 & 16 & 15 & 15 \\ \hline PDW (mm) & - & 16 & 14 & 10 & 8 & 9 & 9 & 10 & 9 & 8 & 7 & 6 & 6 & 6 & 7 & 8 & 8 & 7 & 6 \\ \hline PDH (mm) & - & 16 & 15 & 14 & 15 & 16 & 17 & 16 & 15 & 14 & 12 & 12 & 12 & 11 & 12 & 11 & 10 & 7 & 7 \\ \hline PDIt (\({}^{\circ}\)) & - & 5 & 4 & 3 & 3 & 3 & 5 & 9 & 7 & 8 & 12 & 11 & 8 & 8 & 8 & 9 & 8 & 8 & 11 & 6 \\ \hline SPW (mm) & 5 & 5 & 6 & 7 & 6 & 7 & 5 & 5 & 5 & 5 & 5 & 4 & 4 & 5 & 4 & 5 & 5 & 7 & 6 & 7 \\ \hline SPD (mm) & 27  
The characteristics considered for the intervertebral disks are presented in Table 2.  
The previous information is for an average height of 167.8 cm [37]. Another consideration taken in the present model is that the alignment of the vertebrae along the spine would provide the minimum potential energy to the whole system. The Levenberg-Marquardt (LM) algorithm solves this optimisation problem [40]. The energy of each intervertebral disc is the sum of the potential elastic energy of the three springs. To find the total potential energy (_TPE_) of the system, the potential elastic energy of all intervertebral discs is added, as shown below. \[TPE=\sum_{i=1}^{23}1/2k_{i}\Big{[}(\Delta I_{A})_{i}^{2}+(\Delta I_{B})_{i}^{2 }+(\Delta I_{C})_{i}^{2}\Big{]}\]  
As mentioned in Table 2, \(k_{i}\) is the stiffness of the intervertebral disc, which is the same for springs \(Ai\), \(Bi\), and \(Ci\). \(\Delta I\) is the change in the spring size. The height of the correspondent disc gives the initial size and is equal to springs \(Ai\), \(Bi\), and \(Ci\). In the coordinated system adopted, X is the transversal axis, Y represents the anteroposterior axis, and Z is the vertical axis [30]. Constraints were also included regarding the maximum rotation of the vertebrae in the sagittal plane (\(\pm 75^{\circ}\)) and in the transversal plane (\(\pm 45^{\circ}\)). Those values are extreme enough; however, they include all the rotation possibilities.