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Journal of Peking University (Health Sciences) ›› 2026, Vol. 58 ›› Issue (1): 115-125. doi: 10.19723/j.issn.1671-167X.2026.01.015

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Calculation method of key articulator parameters based on mandibular movement trajectory

Shenyao SHAN1,2, Yongtao YANG2, Wenbo LI2, Aonan WEN1, Zixiang GAO1, Xiangyi SHANG2, Yong WANG1,2,*(), Yijiao ZHAO1,2,*()   

  1. 1. Center for Digital Dentistry, Peking University School and Hospital of Stomatology & National Center for Stomatology & National Clinical Research Center for Oral Diseases & National Engineering Research Center of Oral Biomaterials and Digital Medical Devices & Beijing Key Laboratory of Digital Stomatology & NHC Research Center of Engineering and Technology for Computerized Dentistry, Beijing 100081, China
    2. Institute of Medical Technology, Peking University Health Science Center, Beijing 100191, China
  • Received:2025-10-11 Online:2026-02-18 Published:2025-12-10
  • Contact: Yong WANG, Yijiao ZHAO
  • Supported by:
    the National Natural Science Foundation of China(82271039); Beijing Natural Science Foundation(L232100); Beijing Natural Science Foundation(L242132); National Key Research and Development Plan of China(2022YFC2405401)

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Abstract:

Objective: To explore a mathematical method for calculating key articulator parameters based on mandibular movement trajectory data, and to compare the results of this method with reference values provided by existing foreign mandibular movement recording system, thereby establishing an algorithmic basis for developing a domestic mandibular movement recording system. Methods: Twenty healthy volunteers (7 males, 13 females) meeting inclusion criteria were recruited, with a mean age of (31±8) years. Mandibular movement data during protrusive and left/right lateral movements were recorded using the JMA Optic foreign mandibular movement recording system. A reference plane coordinate system was established using reverse engineering software, the multi-source maxillofacial data were integrated, and the coordinate systems were then unified. The condylar apex, medial condylar pole, lateral condylar pole, condylar center, empirical hinge axis point, and mandibular incisor point were selected as reference points for mandibular movement trajectories. Three-dimensional movement trajectories were generated for each reference point to calculate the sagittal condylar inclination (SCI), transverse condylar inclination (TCI), immediate side shift (ISS), incisal guidance inclination and canine guidance inclination. The calculation results from different reference points served as distinct experimental groups. Reference values provided by the JMA Optic system were used as the control group for comparative analysis. Results: The SCI values of all the experimental groups were significantly higher than that of the control group (P < 0.001), with a systematic positive bias of approximately 3.1°, though the limits of agreement were relatively narrow. The TCI results varied depending on the reference point: Only the condylar apex group (5.7°±6.1°) was significantly lower than the control group (9.2°±6.6°) (t=5.023, P < 0.001). Differences between the remaining groups and the control group were not statistically significant. The empirical hinge axis point group showed the smallest mean bias and the narrowest limits of agreement, indicating optimal consistency with the control group's TCI. The ISS values were 0.0 (0.0) mm in all the groups. The incisal guidance inclination of the mandibular incisor point group (43.1°±8.6°) was significantly lower than that of the control group (50.6°±13.7°) (t=3.749, P=0.001) with poor consistency. However, the canine guidance inclination of the mandibular incisor point group showed no statistically significant difference compared with the control group (t=-1.873, P=0.069), with acceptable consistency. Conclusion: This study proposed a mathematical method for calculating key articulator parameters based on mandibular movement trajectory data, with a clear and traceable computational pathway. The proposed method showed acceptable consistency with the JMA Optic system algorithm in calculating TCI, ISS, and canine guidance inclination, but poor consistency in calculating SCI and incisal guidance inclination. The selection of reference points directly influenced the results of parameter calculation. This mathematical method provided a reliable theoretical foundation for achieving precise, personalized articulator parameter settings.

Key words: Mandibular movement trajectory, Articulator parameters, Sagittal condylar inclination, Transversal condylar inclination, Incisal guidance inclination

CLC Number: 

  • R783.3

Figure 1

Standard maxillary fork three-dimensional model A, the red dots indicate the relationship between three mark points and the standard maxillary fork model; B, the coordinate system for the standard maxillary fork model."

Figure 2

3D multi-source data integration process for the maxillofacial region A, select the labial region of the anterior teeth as the registration area, unify the coordinate systems of Model_fork and Model_upper; B, select the maxillary dental arch region as the registration area, unify the coordinate systems of Model_upper and Model_max, load ".tfm" file to align the coordinate systems of Model_lower and Model_fork; C, select the upper surface region of the maxillary fork as the registration area, unify the coordinate systems of Model_fork and Model_ref, load ".tfm" file to align all 3D models to the coordinate system where Model_ref resides."

Figure 3

Establishment of a reference plane coordinate system and selection of reference points A, procedure for establishing a reference plane coordinate system; B, select reference points based on the reference plane coordinate system, select mandibular incisor point on Model_lower, select bilateral condylar apex, medial condylar pole, lateral condylar pole and condylar center on Model_man, select bilateral empirical hinge axis points on Model_face."

Figure 4

Mandibular movement trajectory based on reference points"

Figure 5

Calculate key articulator parameters based on reference point trajectories Using trajectories of left condylar center and mandibular incisor point as examples to demonstrate the mathematical calculation method of SCI, TCI, ISS, IGI and CGI. SCI, sagittal condylar inclination; TCI, transversal condylar inclination; ISS, immediate side shift; IGI, incisal guidance inclination; CGI, canine guidance inclination."

Table 1

Comparison of left and right SCI, TCI, ISS among experimental groups and the control group"

Group n SCI/(°)，${\bar x}$±s TCI/(°)，${\bar x}$±s ISS/mm, M (IQR)
Left Right t P Left Right t P Left Right Z P
EG_condylar apex 20 39.7±7.3 40.0±8.5 -0.270 0.790 6.6±5.1 4.8±7.1 1.129 0.273 0.0(0.0) 0.0(0.0) 0.000 >0.999
EG_medial condylar pole 20 39.8±7.1 39.9±8.0 -0.191 0.851 9.0±5.6 6.6±7.5 1.453 0.162 0.0(0.0) 0.0(0.0) 0.000 >0.999
EG_lateral condylar pole 20 40.0±7.2 40.3±8.4 -0.214 0.833 11.0±5.3 9.1±5.8 1.374 0.185 0.0(0.0) 0.0(0.0) 0.000 >0.999
EG_condylar center 20 39.9±7.1 40.1±8.2 -0.240 0.813 10.7±5.5 8.6±6.5 1.367 0.188 0.0(0.0) 0.0(0.0) 0.000 >0.999
EG_empirical hinge axis point 20 39.9±7.4 40.5±8.9 -0.346 0.733 10.2±5.6 8.9±6.3 1.068 0.299 0.0(0.0) 0.0(0.0) 0.000 >0.999
CG 20 37.3±8.6 36.5±9.1 0.639 0.531 9.8±6.5 8.6±6.8 1.019 0.321 0.0(0.0) 0.0(0.0) -0.272 0.785

Table 2

Comparison of left and right canine guidance inclination in EG_mandibular incisor point and the control group"

Group n Canine guidance inclination/(°)，${\bar x}$±s
Left Right t P
EG_mandibular incisor point 20 33.9±9.3 32.7±6.5 0.900 0.379
CG 20 29.7±11.5 32.8±8.7 -1.611 0.124

Table 3

Pairwise comparisons of SCI, TCI, ISS between experimental groups and the control group"

Group SCI/(°)，${\bar x}$±s t P TCI/(°)，${\bar x}$±s t P ISS/mm, M (IQR) Z P
EG_condylar apex 39.8±7.8 -4.635 < 0.001 5.7±6.1 5.023 < 0.001 0.0(0.0) -1.633 0.102
EG_medial condylar pole 39.8±7.5 -4.586 < 0.001 7.8±6.7 1.971 0.056 0.0(0.0) -1.633 0.102
EG_lateral condylar pole 40.1±7.7 -5.206 < 0.001 10.1±5.6 -1.213 0.233 0.0(0.0) -1.633 0.102
EG_condylar center 40.0±7.6 -4.963 < 0.001 9.7±6.0 -0.655 0.517 0.0(0.0) -1.633 0.102
EG_empirical hinge axis point 40.2±8.1 -5.318 < 0.001 9.5±5.9 -0.539 0.593 0.0(0.0) -1.633 0.102
CG 36.9±8.8 9.2±6.6 0.0(0.0)

Figure 6

Bland-Altman plots for consistency of key articulator parameters between experimental and control groups SCI, sagittal condylar inclination; TCI, transversal condylar inclination; IGI, incisal guidance inclination; CGI, canine guidance inclination; EG, experimental group; CG, control group."

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