Monson’s sphere theory characterizes the arrangement of teeth in a three-dimensional space and has been used to reconstruct occlusal curves during oral rehabilitation [4, 14]. This theory has also been used to evaluate occlusal curves in digital models [5, 15]. However, the results of studies on Monson’s sphere are controversial because of differences in research methods and participant age and race. In addition, research on Monson’s spheres in Chinese young adult females with individual normal occlusion is insufficient. Therefore, further investigation of Monson’s spheres in China is needed to provide information for individualized occlusal adjustment. This study took advantage of digital models to analyze the characteristics of Monson’s spheres in Chinese young adult females.

Yuelin et al. [12]. attempted to substitute the Monson’s sphere with the curve of Spee on cephalometric films; however, the results may be inaccurate owing to the distortion of the teeth on the films. Wu et al. [13]. determined the radius of Monson’s sphere by measuring the distance between the midpoint of the condylar anterior surface and the glabella point using cone-beam computed tomography. This method was so simplified that detailed dentition information could be ignored. Ferrario et al. [9]. transformed three-dimensional coordinates into two-dimensional data to calculate the Monson’s sphere radius. Kagaya et al. [10]. calculated the Monson’s sphere radius in Japanese adults by restricting the sphere’s center to one of the coordinate planes. Both these methods sacrificed part of the three-dimensional information of the Monson’s sphere to some extent. Nam et al. [11]. developed a best-fit algorithm to produce a Monson’s sphere using selected points on digital models. This method made full use of the three-dimensional coordinates of the selected points. However, Nam et al. [11]. did not consider the incisors as part of Monson’s sphere; and abandoned the incisors when selecting points that differed from Monson’s theory. As part of the entire dentition, incisal guidance is relevant to mandibular movement and temporomandibular joint development [16, 17]. Thus, it is more reasonable to include incisors when researching Monson’s spheres. The present study adopted a method similar to that of Nam et al. but added points on the incisors. Furthermore, unlike previous studies, this study analyzed the deviation of each selected point from its relative sphere surface. This study included 51 Chinese young adult females with individual normal occlusion (mean age 19.08 ± 1.28 years). The mean Monson’s sphere radius was 79.60 ± 14.13 mm, which was smaller than the original 4-inch value measured by Monson [2]. The dentition of these subjects was less worn than that in Monson’s study, which was based on dry skulls with apparent dental attrition. The level of attrition might be one reason for the differences in Monson’s values in most subsequent studies. In addition, Ferrario et al. [9] reported a three-dimensional occlusal curvature in adolescents (12–14 years of age) of approximately 80 mm, which was smaller than the approximately 101 mm reported in adults (19–22 years of age). Ferrario et al. suggested that age significantly affects the radius of occlusal curvature. Some studies have shown that the intermolar arch width continues to decrease in adulthood [18, 19] and the buccolingual inclination of the mandibular molars varies among different age groups [20, 21]. These dentoalveolar changes indicated long-term alterations of natural dentition, which must be considered when oral rehabilitation or orthodontic treatment is performed for patients of different ages. Therefore, the results of this study may apply only to young adults.

The mean radius of 51 females included in the present study was 79.60 ± 14.13 mm. Actually, this study also collected 7 males and calculated their radii of Monson’s spheres. The individual Monson’s sphere radii in these seven males were 57.12 mm, 80.60 mm, 97.54 mm, 99.37 mm, 102.13 mm, 126.88 mm, and 148.16 mm, respectively. It seems that the male Monson’s sphere was larger than that of the female. However, limited by the sample size of males, this study did not perform a statistical comparison between the sexes. There are some controversies regarding whether occlusal curves vary between sexes. Some studies on the Curve of Spee, which represents the occlusal curve in the sagittal plane, showed that its depth was not affected by gender [22,23,24]. However, these studies ignored the transverse differences in dentition, which might affect the occlusal curve in three-dimensional space. The transverse dimension of the dental arch is smaller in females [25, 26]. Kagaya et al. [10]. and Nam et al. [11]. reported that the Monson’s sphere radius was significantly smaller in females than that in males, indicating the potential presence of sexual dimorphism of the occlusal curve in three-dimensional space. Thus, sexual dimorphism of the occlusal curve should be considered when planning oral rehabilitation or orthodontic therapy for the different sexes. The results of the present study may not be appropriate for males and a larger sample size of males is required for further investigation.

The radius of Monson’s sphere in Chinese young adult females was 79.60 ± 14.13 mm, smaller than the results in some other countries. Kagaya et al. [10]. calculated the radius of the sphere in healthy Japanese young adults. The result was not normally distributed, and the median sphere radius was 110.6 mm. The large variability in their results may have been due to the undefined overbite and overjet in their sample. Nam et al. reported a mean radius among Korean young adult females (n = 27) of 100.70 ± 19.73 mm based on digital models [11]. However, the authors did not define the relationships between the canines and molars or anterior overbite and overjet. Dentitions with different types of sagittal relationships differ in arch forms [27], which may affect Monson’s sphere. The present study included participants with individual normal occlusion and critically restricted the inclusion criteria to minimize their impact on the results. However, skeletal variations still exist even in normal occlusion, and dentoalveolar compensation appears according to the skeletal patterns [28, 29]. Monson’s theory suggests that the features of the occlusal sphere are related to the craniofacial structure. Monson thought that the sphere center was located at the glabella point and that the sphere passed bilaterally through the centers of the condyles. Recently, Casazza et al. [30] proposed a regression formula to allow individualized estimation of the radius of a Spee curve based on lateral cephalograms. Their study demonstrated the mathematical relationships between the occlusal curves and craniofacial structures. Considering the craniofacial differences between Chinese and Caucasians [7], the present study provides individualized information on the three-dimensional occlusal curves in Chinese young adult females. Apart from the Monson’s sphere radius in the Chinese population, further information regarding the relationship between craniofacial structures and occlusal curves remains to be explored based on digital models combined with CBCT, which also reflects the limitations of this study.

As shown in Table 3, the mean deviation of the selected points to their relative spheres was 0.38 ± 0.30 mm, with maximum deviations outside and inside the sphere of 0.93 ± 0.25 mm and 0.95 ± 0.30 mm, respectively. These deviations indicated that ideal dentition with perfect occlusion conforming to Monson’s sphere may not exist in nature. Although the deviation seemed too large to allow precise occlusal adjustment in the clinical setting, the information on the sphere obtained in this study can be used as a instruction for tooth alignment and cusp location.

Regarding the mean deviation of the selected points presented in Table 4; Fig. 2, most of the lingual cusps deviated more than the buccal cusps in the posterior teeth, which might have resulted from the lingual inclination of the crown [31]. While this inclination, described as torque in orthodontics, has been extensively studied, the deviation of the cusps has not yet been explored. Combined with digital software, the deviations found in the present study can be used as a new method to instruct tooth alignment in orthodontics. Figure 2 also shows that the most deviated point was the mesiolingual cusp of the first permanent molar, and that the distance between the buccal and lingual cusps of the first permanent molar was also the most apparent. The mesiolingual cusp of the first permanent molar is anatomically higher than the mesiobuccal cusp [32]. Besides, as the main functional cusp bearing occlusal force, the buccal cusps of the first permanent teeth may be much more worn than other teeth. These factors may induce an apparent distance between the buccal and lingual cusps of the first permanent molar, as shown in Fig. 2.

As shown in Table 5; Fig. 3, the most deviated points outside the spheres were located mainly at the edges of the first incisors, the buccal cusps of the first permanent molars and the lingual cusps of the second permanent molars. The most deviated points inside the sphere were mainly at the lingual cusps of the first permanent molars. Three reasons may explain these types of distribution. First, the lingual inclinations of the clinical crowns from anterior to posterior progressively increase in the mandible [31, 33], which results in the main distribution of the lingual cusps of the second permanent molars outside Monson’s spheres. Secondly, the morphological differences in the mesiobuccal and mesiolingual cusps of the first permanent molars, as mentioned above, may result in a higher distribution of most deviated points outside and inside the spheres at the buccal and lingual cusps, respectively. Besides, the point at incisal edge is one of the highest points and most anterior point of the curve of Spee, which might result in the deviation and distribution showing in Figs. 2 and 3. Therefore, the deviation of selected points at the incisal edges and buccal and lingual cusps of the molars required increased attention when it comes to tooth alignment during orthodontic therapy or occlusal reconstruction. When 3D software is used to align teeth virtually, the Monson’s sphere could be used as a kind of template and the deviations of all the points could help to adjust the locations of different incisal edges and cusps. Further exploration remains to be done for that application.

This study investigated the characteristics of Monson’s sphere in Chinese young adult females with individual normal occlusions. The results provided further information on the three-dimensional occlusal curve for orthodontic treatment and oral rehabilitation in Chinese young adult females. However, due to the limited sample size, further investigation is needed. In addition, it is necessary to collect craniofacial data from CBCT for integration with existing digital models for in-depth research.