## 2 Materials and Methods
### Finite Element Modeling and Validation  
A finite element (FE) model of the lumbar spine, which has the highest incidence of spinal disease, was developed and used in this study. The FE model used a three-dimensional eight-node hexahedral element for the anterior vertebral body and intervertebral disc, and a three-dimensional five-node tetrahedral element for the posterior vertebral body. A one-dimensional beam element (tension only) was applied to each ligament. Table 1 lists the material properties of the cortical and trabecular bone, ligaments, intervertebral disc, and the spinal implants of the lumbar spine, which were applied to the FE model [27; 32; 33; 34; 35; 36; 37]. The lordosis of the FE model was set to 45\({}^{\circ}\), which is the same as that of a normal lumbar spine. Furthermore, a part of the sacrum was modeled for evaluation of the intervertebral disc between lumbar 5 to sacrum 1. For each implant of the spinal fixation system, the pedicle screw diameter was 3.0 mm, the spinal cage dimensions were 10.0 \(\times\) 29.0 \(\times\) 10.0 mm\({}^{3}\), the spinal rod diameter was 5.5 mm, and the bone graft was  
The loading conditions were applied according to spinal motions, that is, flexion, extension, lateral bending, and rotation, and different follower loads and moments were set for each motion. Follower load was applied similarly to the cadaver experimental study of [PERSON] et al. Follower load is a compressive load that occurs along the axis of the lumbar. In the FE analysis, the middle node of the top and bottom end plates of each segment is coupled with the surface of the endplate, and the nodes are connected through a connector. By applying the load in the direction of gravity from the top of the sacrum, the follower load was applied along the axis of the lumbar. The boundary condition was applied as a fixed support for the bottom of the sacrum. In the contact conditions, most of the lumbar FE model, such as cortical-tribecular bone, cortical bone-IVD, and anterior bone-posterior bone, were adjusted bonded conditions; soft contact and a frictional coefficient of 0.1 were used only for the facet joint. Figure 2 shows the detailed loading and boundary conditions [32; 33]. Figure 1: Finite element (FE) model of lumbar spine and spinal implants: (**a**) isometric view of FE model of lumbar spine and spinal implants; (**b**) rear view of FE model of lumbar spine and spinal implants; (**c**) isometric view of spinal implants. Figure 2: Loading and boundary conditions for simulation of six spinal motions: (**a**) flexion; (**b**) extension; (**c**) lateral bending; (**d**) rotation. To verify the reliability of the current FE model, it was compared to the results of cadaver experiments and FE analysis in the papers of [PERSON] et al. and [PERSON] et al. The FE model of this study is difficult to compare to the lumbar of the human body. There are many restrictions on conducting experiments with cadavers. Therefore, the FE model was verified through comparison of the lumbar cadaver experiment results of conventional studies. The results of the cadaver experiment and FE analysis of conventional studies were compared for the FE model validation of the current study. In the verification of the intact FE model, the material properties shown in Table 1 were used, and a pure moment was applied to each lumbar motion (flexion 8 Nm, extension 6 Nm, lateral bending 6 Nm, and rotation 4 Nm). Boundary condition was set as a fixed support at the bottom of the sacrum, as shown in Figure 2 below. The FE model of the current study was satisfied within standard deviation in all motions in comparison to the cadaver experiment results of [PERSON] et al., and the maximum error was \(-2.18\) degree in lumbar 5-Sacrum 1 under flexion-extension. A maximum error of 3.80 degree was investigated in lumbar 4-5 under lateral bending with FE analysis results of [PERSON] et al., and the maximum error was \(-2.21\) degree in lumbar 5-Sacrum 1 under extension-flexion with [PERSON] et al.