## 2 Materials and Methods  
### Finite Element Model Construction of the Lumbar Spine  
A three-dimensional, nonlinear finite element model of the human lumbar spine was created using the commercial software ANSYS 14.5 (ANSYS Inc., Canonsburg, PA, USA). The material properties of the intact lumbar spine model (INT) were investigated in previous studies [14; 15; 16; 17]. The geometrical structure of our INT model was constructed from the computer tomographic images of the lumbar spine in a 19-year-old male. Each coronal plane slice was obtained by the 3 mm interval and enlarged to identify the different regions of the tissues. The edge of the spinal disc and the geometry of the nucleus were measured and referred to [PERSON] et al.'s study [18]. Of the total disc area, 30-50% was defined as the nucleus and the rest was assumed as the disc annulus. The range of motion (ROM) of the INT was compared with that of cadaveric specimens in in vitro testing. The reliability and similar stiffness were validated [19]. The present mesh density was selected based on the convergence test and the model validated in our previous studies [14; 16]. Overall, the discrepancy between the in vitro tests and our finite element simulation was within one standard deviation. The model also contained intervertebral discs, endplates, posterior bony elements, and seven ligaments. These intervertebral discs were composed of a ground substance, hyperelastic annulus fibrosus, and incompressible nucleus pulposus with 12 double-cross-linked fiber layers embedded in the ground substance (Table 1).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Material** & **Young’s Modulus (MPa)** & **Poisson’s Ratio** & **Area (mm\({}^{2}\))** & **Element Type** \\ \hline \multirow{8}{*}{Bone} & \multirow{4}{*}{Critical} & Ex = 11,300, Gx = 3800 & V\({}_{xy}\) = 0.484 & \multirow{8}{*}{Critical} \\  & & Ey = 11,300, Gy = 5400 & V\({}_{yx}\) = 0.203 & \\  & & Ez = 22,000, Gz = 5400 & V\({}_{yx}\) = 0.203 & \\ \cline{2-5}  & \multirow{4}{*}{Cancellous} & Ex = 140, Gx = 48.3 & \multirow{4}{*}{Solid 185} \\  & & Ey = 140, Gy = 48.3 & & \\  & & Ez = 200, Gz = 49.3 & & \\ \cline{2-5}  & \multirow{2}{*}{Posterior bone} & 3500 & 0.25 & \\ \cline{2-5}  & & Cartilaginous endplates & 24 & 0.4 & \\ \hline \multirow{8}{*}{Disc} & Nucleus pulposus & 1666.7 & \multirow{8}{*}{Solid 185} \\  & & C\({}_{10}\) = 0.42 & & \\ \cline{1-1} \cline{2-5}  & Ground substance & C\({}_{1}\) = 0.105 & & \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{Annulus fibers} & \multirow{2}{*}{550} & 0.76 & \\  & & Second & 495 & 0.5928 & Link 10 \\ \cline{1-1}  & & Third & 412.5 & 0.4712 & \\ \cline{1-1} \cline{2-5}  & Innermost & 357.5 & 0.3572 & \\ \hline \multirow{2}{*}{Facet joint} & \multirow{2}{*}{Facet joint} & \multirow{2}{*}{Contact 174} \\  & & & Target 170 \\ \hline \multirow{2}{*}{Interface (Implants/bone)} & \multirow{2}{*}{Contact 174} \\  & & & & Target 170 \\ \hline \multirow{8}{*}{Ligaments} & Anterior longitudinal & 7.8 & 24 & \\ \cline{2-5}  & Posterior longitudinal & 10 & 14.4 & \\ \cline{1-1} \cline{2-5}  & Transverse & 10 & 3.6 & \\ \cline{1-1} \cline{2-5}  & Flavum & 15 & 40 & Link 10 \\ \cline{1-1} \cline{2-5}  & Interspinous & 10 & 26 & \\ \cline{1-1} \cline{2-5}  & Supraspinous & 8 & 23 & \\ \cline{1-1} \cline{2-5}  & Capsular & 7.5 & 30 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties used in the finite element model.