Abstract
The objective of this study is to investigate the synergetic effect of the pelvic rotation and X-ray offset on the radiographic anteversion/inclination (RA/RI) angles of the acetabular cup using a mathematical model. A cone model for establishing the spatial relationship between a three-dimensional (3D) circle and its two-dimensional (2D) elliptical projection is utilized to quantify the relationship between the 3D RA/RI angles of the cup and their 2D counterparts with different types of pelvic rotations in pelvic/hip anteroposterior radiographs. The results reveal that the effect of inlet/outlet views on the 2D RA angle is similar to that of iliac/obturator views. The permissible ranges of pelvic rotation for the 2D RA angle with an acceptable bias are the 3D space formed by the limits of triple axial rotations. For a specified acceptable bias of the 2D RA angle, these ranges are almost equal between pelvic and hip radiographs. The combined inlet/obturator or outlet/iliac views can maintain the 2D RA angle of a pelvic radiograph within the same range of acceptable bias as that of a hip radiograph. For a 2D RA angle with an acceptable bias, the permissible range of pelvic rotation needs to be evaluated with equal attention in both radiographs.
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Abbreviations
- 2D:
-
Two-dimensional
- 3D:
-
Three-dimensional
- AP:
-
Anteroposterior
- RA:
-
Radiographic anteversion
- RI:
-
Radiographic inclination
- THA:
-
Total hip arthroplasty
- r :
-
The radius of the cup
- (o, p, q):
-
The coordinate of the X-ray beam source
- (x 0, y 0, z 0):
-
The coordinate of the center of the cup
- [k 1, i 1, j 1]:
-
The normal vector of the cup before rotation
- [k, i, j]:
-
The normal vector of the axis of the cup after rotation
- [l, m, n]:
-
The normal vector of the space line
- α :
-
The x-axial rotation of the cup
- β :
-
The y-axial rotation of the cup
- γ :
-
The z-axial rotation of the cup
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Anonymous reviewers are appreciated very much for their help to improve this manuscript.
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This study was partially funded by the Fundamental Research Funds for the Central Universities (FRF-TP-16-045A1).
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Zhao, JX., Su, XY., Zhao, Z. et al. The synergetic effect of pelvic rotation and X-ray offset on radiographic angles of the acetabular cup. Med Biol Eng Comput 57, 2359–2371 (2019). https://doi.org/10.1007/s11517-019-02041-7
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DOI: https://doi.org/10.1007/s11517-019-02041-7