## 4 Discussion  
Bone formation after spinal fusion is critical for a successful surgical outcome [6; 26]. There are two pathways in bone formation, endochondral and intramembranous ossification. However, it is not clear which one is the major mechanism of extragraft bone formation. If endochondral ossification is a major pathway of bone formation during the fracture healing process, intragraft bone formation should be observed more frequently than extragraft bone formation or bridging because the region of intragraft bone formation is filled with graft bone. However, [PERSON] et al. reported that extragraft bone bridging had 100% sensitivity and a negative predictive value for diagnosing solid fusion; thus, extragraft bone bridging plays a greater role in securing stability with the formation of a solid union compared with intragraft bone bridging [6]. Moreover, [PERSON] et al. showed that the force applied to the superior verteb was mainly transmitted not via the graft bone inside the cage but via the implanted cage itself [22]. Therefore, it is not expected that graft bone inside the cage sustains mechanical loading. However, the bone surrounding the implanted cage could be stimulated by the external load, such as during the bending moment, and the load could lead to extragraft bone formation. Thus, the authors believe  
Figure 8: Changes in the maximum von Mises stress on the interbody cage when simulating bone formation using two algorithms during **(a)** flexion and **(b)** extension. Figure 9: Changes in the maximum von Mises stress on the trabecular bone when simulating bone formation using two algorithms during **(a)** flexion and **(b)** extension. that the hypothesis that the mechanism behind the generation of extragraft bone formation and bridging is intramembranous ossification, and that mechanical stimuli mainly affect extragraft bone formation after spinal implantation, is supported. That is why only extragraft bone formation was predicted in this study. Changes in the trabecular bone structure or density based on Wolff's law occur according to the magnitudes of the stress and strain and their directions [1,5]. When the simultaneous algorithm was used, stress bearings through the potential bone generation region were predicted in both flexion and extension (Figure 10,c), and they resulted in the generation of extragraft bone bridging (Figure 6). However, flexion loading only contributed to initial extragraft bone formation, not to growth, when the sequential algorithm was used. Thus, discontinuous stress bearing was shown in flexion (Figure 10b) with the sequential algorithm, and it predicted less volume of generated bone compared to the volume predicted using the simultaneous algorithm. Therefore, the authors believe that a simultaneous algorithm could overestimate extragraft bone formation because of an ill-defined design space. Even though the simultaneous algorithm has the advantage of reducing the calculation time (Table 1), the computing time can be reduced with a decrease in the number of elements of the FE model or an increase in the number of analysis processors. Figure 10: Directions of the maximum principal stress in the predefined potential region of the extragraft bone formation region during **(a,b)** flexion **(c,d)** and extension. To our knowledge, the first study to simulate bone formation after spinal surgery using an implanted device was conducted by [PERSON] et al. [15; 16]. That study conducted the simulation of heterotopic ossification formation after cervical disc arthroplasty using a simultaneous algorithm. Even though the simulation results in the previously published studies were consistent with clinical observations, there may have been overestimated bone formation caused by a load sharing pathway in the simultaneous algorithm, as shown in the results of this study. Therefore, the authors believe that the sequential algorithm introduced in this study could help the accurate prediction of bone formation, even for heterotopic ossification.