Skeletal Class III malocclusions appear in various conditions and patterns. It is caused by abnormal growth of the jaws or growth disharmony, as overdevelopment of the mandible, underdevelopment of the maxilla, or a combination of both. There are many controversies concerning the treatment modalities and treatment timing of skeletal Class III malocclusions with respect to skeletal and dental discrepancy, age, and residual growth. A reduction in the growth of the maxilla is caused not only by the antero-posterior divergence but also by a transverse variation, resulting, in many cases, in posterior crossbites .
Haas reported on the orthopedic effect of RME, which produced a forward and downward tipping of the maxilla with concomitant downward and backward mandibular rotation . These orthopedic changes facilitated the correction of a mild Class III malocclusion. RME is effective for correction of transverse discrepancies and also for protraction of the maxilla by remodeling the nine circumaxillary sutures  .
When Yu et al. compared the effect of maxillary protraction with and without RME, they concluded that opening the mid-palatal suture using a RME appliance and directing the protraction force inferiorly from the occlusal plane, passing through the maxillary center of resistance and also through the apical portion of the first premolar, maxillary protraction that is similar to normal downward and forward growth of the maxilla can be effectively achieved .
In yet another study by Park et al. to analyze displacement and stress distribution in the maxilla during maxillary expansion followed by protraction using bone-borne and conventional tooth-borne palatal expanders and a facemask via three-dimensional finite element analysis, they concluded that the group with only facemask resulted in more anterior displacement of the maxilla than the combination of facemask and bone-borne expanders. Further, tooth-borne expanders reported more anterior movement of the maxilla when protraction was combined with expansion and compared with protraction without expansion. This is due to the synergistic effect when combined with the facemask as reported by Yu et al. .
The three-dimensional FEM used in the present study provides the freedom to simulate orthodontic force systems applied clinically and allows analysis of the response of the craniofacial skeleton to the orthodontic loads in three-dimensional space. The FEM has several advantages, and the study can be repeated as many times as the operator wishes .
Sequential CT scan images were taken at 1-mm intervals to reproduce finer and detailed aspects of the geometry. Previous studies reported sections at 10-mm [11, 12], 5-mm [13, 14], and 2.5-mm  intervals. This study differs from previous studies in the type of element used for meshing and solid tetrahedral elements used, each having 10 nodes, giving better stress transmissibility and bending deformations. Therefore, this model is a better representation of a real human skull than the previous models [13, 14].
Displacement of various craniofacial structures with rapid maxillary expansion by using finite element analysis
Maximum lateral displacement was at the incisal edge of the upper central incisor. The pyramidal displacement of maxilla was evident. The apex of the pyramid faced the nasal bone, and the base was located on the oral side. The posterosuperior part of the nasal cavity had moved minimally in the lateral direction, and the width of the nasal cavity at the floor of the nose increased markedly, whereas no significant lateral displacement was observed at the temporal, parietal, frontal, sphenoid, and occipital bones. The studies done by Haas et al., Jafari et al. , Cleall et al. , Davis and Kronman , and Iseri et al.  showed similar findings.
The maximum negative Y-displacement (backward displacement) was observed at the anterior zygomatic arch, indicating that this portion of the craniofacial complex had moved posteriorly. In a similar study done by Jafari et al. , the maximum negative Y-displacement was less compared to our study. The difference may be due to (i) CT scan images were taken at 5-mm interval, (ii) analysis of the complete skull was not considered as it was thought that the analysis of one half of the cranium will mirror similar results on the opposite side, and (iii) a known force was not applied.
The present study suggests that the resistance of the mid-palatal opening is probably not in the suture itself but in the surrounding structures of the sphenoid and zygomatic bones, thus confirming the studies done in the past by Isaacson and Ingram  and Wertz .
When the mid-palatal suture opens with RME therapy, the maxilla always moves downward and forward. This is probably due to the disposition of the maxilla-cranial sutures . This was demonstrated by anterior and downward displacement of point A, ANS, and prosthion. This is in accordance with previous studies [13, 14, 19, 20].
The pterygoid plates can bend only to a limited extent with pressure, and this confining effect of the pterygoid plates of the sphenoid minimizes the ability of the palatine bones to separate at the midsagittal plane .
An increase in nasal width has been demonstrated as a response to RME [1, 2, 19, 21]. The numerical results demonstrate that the width of the nasal cavity at the floor of the nose increased markedly compared to the superior part. This result is similar to Pavlin and Vukicevik’s study . Therefore, a combination of increase in nasal width, lowering of palatal plane, and probably straightening of the nasal septum after RME can help the patients with nasal stenosis, by increasing air flow.
This study also provided additional explanation about the bony tissue mechanical reactions, which are the first steps in the compound process of tissue response to jaw expansion.
Evaluation of stress distribution along craniofacial sutures
The von Mises stresses were used for this analysis because of the appropriateness and the validity of the von Mises theory of failure .
The present study showed that the mid-palatal suture had experienced high levels of stress (59.532 MPa). An interesting finding of this study was the presence of high stress all along the maxillary bone (58.7 MPa), radiating upward to deeper anatomic structures such as the body of the sphenoid bone.
Mechanical stresses elicited by exogenous forces are experienced and transmitted by sutures . Sutural strain patterns are similar between dry skull models and the same structures in vivo . Sutures have a range of mobility and certainly are not immovable joints when they are patent. Mechanical stresses experienced in sutures, with the right characteristics, can modulate sutural growth . This had been proved time and again. Our present study also confirmed this.
The present study concludes that mid-palatal suture shows a maximum von Mises stress followed by pterygomaxillary, nasomaxillary, and frontomaxillary sutures in the descending order of frequency. This is contrary to studies done by Jafari et al. . The difference observed may be attributed to the sample used for the generation of FE model and selection of nodes and elements. In the present study, a known force is applied instead of known displacement.
The study done by Shetty et al.  showed that in the anterior region, the forces spread superomedially along the frontal process of the maxilla and the medial orbital wall up to the junction of the nasal and lacrimal bones. The primary stresses in their study radiated laterally to the zygomaticomaxillary and the zygomaticofrontal sutures, which is contrary to our findings. This difference may be attributed to discretization process during FE model generation and selection of nodes and elements.
In this study, the lateral stresses mainly radiated to the zygomaticotemporal and the sphenozygomatic sutures and it has been demonstrated that the pattern of stress distribution was different along the various craniofacial sutures in response to RME. Both tensile and compressive stresses of variable magnitude were demonstrated along the same suture. The zygomaticomaxillary, zygomaticotemporal, and zygomaticofrontal sutures were associated with both tensile and compressive stresses. Sutural growth is accelerated by both tension and compression with appropriate parameters such as strain amplitude, rate, and dose .
The presence of differential strain patterns suggests the possibility of differential bone remodeling along the same suture. Differential strain patterns and magnitudes along the same suture were documented by Oberheim and Mao  who showed contrasting bone-strain patterns in zygomatic arch across zygomaticotemporal suture—i.e., tensile on its lateral surface and compressive on its medial surface—suggesting potentially differential growth responses of zygomatic arch with headgear therapy.
The absolute level of induced stresses greatly depends on bone elasticity and patient’s age. With the same orthopedic load, equivalent sutures of juvenile skulls experience significantly higher bone strain than adult skulls, suggesting that same mechanical force might have different biologic effects on immature and mature facial skeletons.
It may be noted that any variation in values between this study and other studies may be attributed to the sample used for the generation of FEM or the model generated on the computer (or both) or selection of the nodes and elements on the FEM (or both).