## 4 Discussion
The causes of pathologic bone formation are controversial, and various factors including sex, age, and surgical method have been presumed as causes of pathologic bone formation [12; 27; 28]. The authors believe that the major advantage of the sequential bone formation algorithm, compared to the simultaneous one, is showing intermediate bone formation progress, and this provides critical information for understanding the mechanism of bone formation. The extragraft bone (Figure 6) was generated along the directions of principal stress shown in Figure 10. Thus, when the sequential algorithm was used, the extragraft bone was initially generated on the surfaces of the vertebrae, it grew up to the wedge shape, and then, finally, extragraft bone bridging was formed. This progress showed good agreement with the clinical observation shown in Figure 2. These results support the hypothesis in this study. With the stepwise decreases in the maximum von Mises stresses on both the interbody cage and trabecular bone in extension with the sequential algorithm, slight changes in stresses were predicted in flexion (Figures 8 and 9). However, the simultaneous algorithm did not show changes in the mechanical behaviours of the spine during the progress of extragraft bone formation. At the fifth iteration in the extension loading condition with the sequential algorithm, extragraft bone bridging in the trabecular bone region was predicted, and the bridge extended to the cortical region at the 10 th iteration. Although the stress on the interbody cage increased with the initial generation of extragraft bone formation, a sudden decrease in the maximum stress on the interbody cage was predicted with extragraft bone bridge formation (at the fifth iteration) (Figure 8b). The maximum stress on the trabecular bone in the verteb decreased even with the initial extragraft bone formation (Figure 9b). The second step of the decrease in the maximum stress in the trabecular bone was shown with extragraft bone bridge formation (at the fifth step), and the third step of the decrease was shown with the expansion of bone bridging to the cortical bone region (at the 10 th step). Even though the inserted interbody cage affects bone adaptation in the trabecular bone of the vertebrae [22], extragraft bone formation should also be considered for the accurate prediction of bone adaptation in the vertebrae. There were several limitations to this study. For the bone modelling and remodelling simulation to predict the trabecular bone architecture, a large-scale FE model with millions of micrometre-level elements was used (50 \(\upmu\)m [29], 87.5 \(\upmu\)m [5], and 175 \(\upmu\)m [4]). However, in this study, the maximum number of elements used for simulating extragraft bone formation was 2576, and the mean volume of the elements used for simulating extragraft bone formation was approximately 0.0755 mm\({}^{3}\). Given that not only extragraft bone formation but also the motion segmental unit should be simulated together, the authors \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Loading** & \multicolumn{2}{c}{**Number of Iterations**} & **Computing** \\  & **Condition** & **Inner** & **Outer** & **Time (Days)** \\ \hline Simultaneous & Flexion & 124 & 1 & 1.6 \\ algorithm & Extension & 240 & 1 & 3.0 \\ \hline Sequential & Flexion & 24 & 6 & 1.8 \\ algorithm & Extension & 64 & 20 & 15.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computing time for simulating bone formation after ACDF using the simultaneous and sequential algorithms. used a very limited number of elements for simulating extragraft bone formation. A recent computer simulation study predicted bone adaptation with consideration of resorption in the presence of an excessive mechanical stimulus [30]. When a cage is inserted in the intradiscal space, bone subsidence at the interface between the vertebrae and the inserted cage might occur because of excessive mechanical stimuli, including stress, strain, or the SED.