## 3 Results  
About 18 min was required for simulating each flexion and extension. When the simultaneous algorithm was used, an external iteration for importing the elements surrounding the formed elements was not imposed, but an internal iteration for calculating the new material properties in the predefined region was imposed. A total of 124 calculations were conducted for flexion, and 240 calculations were conducted for extension loading conditions. When the sequential algorithm was used, 6 and 20 external iterations were imposed for the flexion and extension loading conditions, respectively. Moreover, about 24 and 64 calculations were conducted for flexion and extension, respectively, in the internal iteration. Thus, the calculation times were 1.6 and 3 days for flexion and extension, respectively, when the simultaneous algorithm was used. The times were 1.8 and 15.9 days for flexion and extension, respectively, when the sequential algorithm was used.  
When the simultaneous algorithm was used, extragraft bone bridging and extragraft bone formation connecting the superior and inferior vertebral bodies were generated in both the flexion and extension loading conditions. However, extragraft bone bridging was generated in only the extension loading condition when the sequential algorithm was used (Figures 5 and 6). The volume of bone formation was 15.82 mm\({}^{3}\) in flexion and 28.36 mm\({}^{3}\) in extension, when the simultaneous algorithm was used. The volumes of the initial bone formation were 2.55 and 13.9 mm\({}^{3}\) in flexion and extension, respectively, when the sequential algorithm was used. They increased to 3.57 and 60.4 mm\({}^{3}\) in flexion and extension, respectively (Figure 7).  
Figure 5: Bone formation during flexion using the two algorithms.  
When extragraft bone formation was not generated, the maximum von Mises stresses on the interbody cage were 24.2 and 31.4 MPa under flexion and extension, respectively (Figure 8). The generation of extragraft bone bridging using the simultaneous algorithm led to decreases in the maximum stresses to 17.0 and 12.6 MPa under flexion and extension, respectively. When the sequential algorithm was used, initial extragraft bone formation increased the maximum stress to 31.1 and 37.7 MPa under flexion and extension, respectively. The stress decreased with the generation of extragraft bone bridging in extension, while the stress decreased slightly with the progress of the iterations in flexion. The maximum stresses at the final steps were 30.5 and 13.5 MPa under flexion and extension, respectively.  
The predicted maximum von Mises stresses on the trabecular bone were 3.0 and 4.7 MPa in flexion and extension, respectively, when extragraft bone formation was not generated (Figure 9). The stress in flexion decreased to 0.6 MPa after extragraft bone formation using the simultaneous algorithm, while it decreased to 2.3 MPa after bone formation using the sequential algorithm. The stress in extension decreased to 0.7 MPa after extragraft bone formation using the simultaneous algorithm. During extragraft bone formation using the sequential algorithm, the stepwise decrease in the stress was predicted. The predicted stress was 3.0, 2.0, 1.2, 0.8, 0.7, and 0.8 MPa at the 1 st, 2 nd, 5 th, 10 th, 11 th, and 20 th iterations, respectively.  
Figure 6: Bone formation during extension using two algorithms: (**a**) simultaneous and (**b**) sequential algorithms.  
Figure 7: Bone formation using simultaneous and sequential algorithms during (**a**) flexion and (**b**) extension.