## 4 Discussion
In addition, a power arm attached to the bracket is another effective and convenient method to execute bodily movement [29; 30]. The power arm changes the acting point and the force is applied directly through the CR. The CR position in Table 3 could help orthodontists design personalized moment arms for patients with low maxillary sinuses. To the best of our knowledge, the average hydrostatic stress was first used to evaluate the orthodontic force. The stress distribution in the PDL offers a connection between the external forces applied to a tooth and the movement. The classic \"pressure-tension\" theory indicates that OTM occurs due to compression and tension within the periodontal tissues generated by orthodontic appliances [31]. A general theory of tooth movement states that in any given plane, the distance from the applied force to the center of resistance multiplied by the distance from the center of resistance to the center of rotation is a constant, which represents the distribution of the restraining stresses in the PDL [26]. Therefore, stress determines the speed and type of tooth movement. Hydrostatic stress has been used as an indicator of ideal orthodontic force in several studies [17]. They usually decide that the force is suitable to promote tooth movement based on the maximum stress not exceeding 5 kPa. In general, using the average value to evaluate the stress distribution is inaccurate. However, it could be used when the stress distributions are similar between different models as shown in Figure 5b. Therefore, the average stress could only be used to evaluate the stress level in bodily movement, in which the stress distribution is uniform. Orthodontic treatment is a highly individualized process. The orthodontic force has a significant relationship with the material properties of the periodontal tissue, which differs between individuals. Therefore, it is difficult to provide a standard for the force value. We promoted the compensation coefficient to evaluate the changes in forces under the influence of varying root penetration depths into the MSF. The force magnitude for teeth with different penetration depths (\(x>0\) mm) is equal to the compensation factors multiplied by the force needed for the tooth with an ordinary structure (\(x=0\) mm). The compensation coefficient is reduced based on the variation in the stress level to compensate for the loss of the supporting capacity caused by the sinking of the MSF. This is consistent with various case reports in which light constant force was used to achieve the tooth movement through the MSF [32]. It should be noted that the results of the FEA are directly related to the input data, for example, the anatomical structure of the sinus floor and the tooth root. The model in this study was constructed from patient tomography images. Although the results from the special patient model herein cannot be applied directly to all patients, it is reasonable to deduce that the linear relationship between the CR position and penetration depth and the quadratic relationship between the force magnitude and penetration depth generally exist. Another limitation is that the MSF was considered horizontal in this study. There are various complex anatomic relationships between maxillary posterior teeth and the MSF in clinical practice [6]. A statistical study revealed that with a more vertical extension of the MSF in front of the tooth being moved, one can expect a higher degree of tipping as compared to teeth moved through a more horizontal maxillary sinus [33]. More complex conditions, including the anatomy and tooth movement types, can be analyzed in the future using a similar method.