## 2 Materials and Methods  
A finite element model of the L3-L4 lumbar spine segment was created using the software ANSYS 14 (ANSYS Inc., Canonsburg, PA, USA). The model (Figure 1a) was developed using geometry from a physiologically accurate spinal model that included intervertebral discs and vertebrae (Zygote Media Group, Inc., American Fork, UT, USA). The material properties of the L3-L4 model shown in Table 1 were sourced from the literature [22; 23; 24; 25]. The model included six ligaments which were represented as 2-node tension-only spring elements (Figure 1b) and were approximated as nonlinear with insertion points approximated to typical anatomy [22]. The model was validated by comparing the range of motion against experimental data from a published in vitro study and finite element analysis results from the literature [25]. An unconstrained pure moment of 7.5 Nm was applied in four directions (flexion, lateral bending, extension, and axial rotation) to the superior endplate of T7, and the inferior endplate of L5 was fixed. The range of motion of the model was recorded and demonstrated to be within the published ranges in the literature [25]. Five finite element models of the L3-L4 segment were developed in this study: (a) healthy condition (Figure 2a); (b) after conservative percutaneous transforminal endoscopic discectomy (PTED) to correct lumbar disc herniation (LDH) (Figure 2b), with a volume of 581 mm\({}^{3}\) removed from the intervertebral disc; (c) after aggressive PTED to correct LDH (Figure 2c), with 2172 mm\({}^{3}\) removed from the disc; (d) after aggressive PTED to correct LDH with annulus fibrosus repair (Figure 2d), with 2172 mm\({}^{3}\) removed from the disc; and (e) after aggressive PTED with nucleus pulposous augmentation (2172 mm\({}^{3}\) removed) to correct LDH (Figure 2e), where the vacated space was filled with a polyurethane elastomer or hydrogel [28]. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Property** & **Modulus (MPa)** & **v** & **References** \\ \hline Cortical bone & 12,000 & 0.2 & [PERSON] et al., 1995 [23] \\ Cancellous bone & 300 & 0.2 & [PERSON] et al., 2003 [24] \\ Annulus fibrosus & Mooney-Rivlin & NA & \\  & cl = 0.18, \(c\) = 0.045 & [PERSON] et al., 2007 [25] \\ Nucleus pulposus & Mooney-Rivlin & NA & \\  & cl = 0.12, \(c\) = 0.03 & \\ Ligaments & Hyperelastic & NA & \\ Polyurethane elastomer & 4.7 MPa & 0.499 & [PERSON] et al. [26] \\ Hydrogel & 0.45 MPa & 0.2 & [PERSON] et al. [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of lumbar spine model. Figure 1: L3–L4 spine segment model: (a) finite element mesh; (b) model containing supraspinous ligament (SSL), interspinous ligament (ISL), capsular ligaments (CL), transverse process ligaments (TPL), posterior longitudinal ligament (PLL), anterior longitudinal ligament (ALL), and ligamentum flavum (LF). The material properties of the implants are shown in Table 1. The cortical bone, cancellous bone, and disc were modeled using 8-node solid elements. For the disc, 12 double cross-linked fibrous layers were embedded in the ground substance, and the fiber stiffness increased proportionally from the outermost layer to the innermost layer. The nucleus pulposus was modeled as an incompressible fluid using 8-node fluid elements. Mesh convergence was achieved when an unconstrained pure moment of 7.5 Nm applied to the superior endplate of L3 resulted in a change in displacement of the disc of less than 2%. The resulting model consisted of approximately 290,400 elements and 616,500 nodes. The interfaces between facet articular surfaces were treated as standard contact pairs at all levels. An unconstrained pure moment was applied to the superior endplate of L3 to simulate the physiological motions of a lumbar spine, which were reported as the maximum voluntary motions in an in vivo study by [PERSON] et al. (2015) (Table 2). The distal vertebra was restricted from all motion by rigidly anchoring the inferior endplate of L4.