## 1 Introduction  
Maxillary constriction is a frequent disorder in the structure of the facial skeleton [1, 2, 3, 4]. The occurrence of the disorder in the entire population is estimated at 9.4%. Almost 30% of orthodontically treated adult patients are affected by this abnormality in which half of the lower dental arch overlaps the upper arch [1, 5]. Complete crossbite occurs in 13-24% of the European children population, 7% in American and less than 1-2% in African children [6].  
A popular method for treating this disorder is rapid maxillary expansion using various appliances as sources of force [7, 8, 9, 10]. The treatment aims at obtaining optimal values of the width of the anterior and posterior upper dental arch, which leads to stable and functional occlusion [11, 12] as well as improves respiratory functions [13, 14]. [PERSON] et al. [15] suggest that rapid maxillary expansion in growing patients with maxillary hypoplasia may support proper condylar growth and facial symmetry. Orthodontic, orthodontic and surgical, as well as surgical methods are used in the treatment of crossbite.  
The type of therapy used for expanding the palatal suture should be strictly dependent on the patient's age [1; 8; 9; 10]. The elasticity of the facial skeleton and the possibility of orthopaedic operation to expand the bone base of the maxilla decreases with age. According to various authors [3; 5] the upper age limit for orthodontic treatment is 14-18 years. As regards the time of closing the craniofacial sutures and a possibility for performing orthodontic operations, sexual dimorphism is taken into account by some researchers. [PERSON] and [PERSON] report that the closure of maxillary bone sutures occurs at the age of 14-15 in women and 15-16 in men [16].  
The first reports on the subject of palatal expansion using an orthodontic method come from the first half of the 19 th century. [PERSON] and [PERSON] reported on palatal expansion using a C-shaped spring device [17]. In 1860, [PERSON] expanded the maxilla in a 14-year-old girl, and his report was described in the periodical \"Dental Cosmos\" [18]. After bone growth is complete, osteotomy is required at sites of increased bone resistance in the midface. In adult patients, an orthodontic expansion therapy only results in deflection of the alveolar process of the maxilla, with little or no orthopaedic effect [19; 20]. The SARME (surgically assisted rapid maxillary expansion) therapy is the method of choice. As a result of the complex structure and mechanics of the facial skeleton numerous patterns of osteotomy and corticotomy of the midface were developed. The precursor of surgically assisted rapid palatal expansion (SARPE) was probably [PERSON], who in 1938 described osteotomy of the palatal suture combined with fixing an appliance for maxillary expansion [17; 20]. The first osteotomies in SARME were performed only to the palatal suture. The types of osteotomies evolved in the following years. [PERSON] and [PERSON] believe that the palatal suture is not the only bone resistance site for proper maxillary expansion in the SARME therapy [21]. [PERSON] et al. are of the opinion that only the palatal suture requires incision, thus conditioning maxillary expansion in adult patients [20]. To date, no consensus has been reached on how to perform osteotomy of the midface using methods of surgically assisted rapid maxillary expansion [20].  
The purpose of the following analysis is to present stress distributions and displacement patterns of specific structures in the area of the facial skeleton in surgically assisted rapid maxillary expansion therapy using a tooth-borne appliance. By comparing five different osteotomy variants and one without surgical intervention, the best model for making an incision in the maxilla will be determined. Five types of osteotomy of the midface and expansion of the appliance at a level of 0.5 mm were modelled using the finite element analysis.