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The synergetic effect of pelvic rotation and X-ray offset on radiographic angles of the acetabular cup

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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.

The traditional methods for calculating the radiographic angles of the acetabular cup are based on the ellipse projection of the opening circle of the cup on radiographs. However, with varying locations of the X-ray source and pelvis rotations about different axes, the outline of this ellipse projection will change, and accordingly, the traditional method and calculating results will be inaccurate. In this study, a cone model for three-dimensional circle-to-two-dimensional ellipse projection is utilized to incorporate the effect of X-ray offset and quantify the relationships of the radiographic angles of the cup with the true orientation of the cup and pelvic rotations in either pelvic or hip anteroposterior radiographic situation.

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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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Acknowledgments

Anonymous reviewers are appreciated very much for their help to improve this manuscript.

Funding

This study was partially funded by the Fundamental Research Funds for the Central Universities (FRF-TP-16-045A1).

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Author notes

  1. Jing-Xin Zhao and Xiu-Yun Su contributed equally to this work.

Authors and Affiliations

  1. Department of Orthopaedics, Chinese PLA General Hospital, Beijing, 100853, China

    Jing-Xin Zhao, Xiu-Yun Su, Guo-Fei Sun, Li-Cheng Zhang & Pei-Fu Tang

  2. Department of Orthopaedics, Beijing Tsinghua Changgung Hospital, School of Clinical Medicine, Tsinghua University, Beijing, 102218, China

    Zhe Zhao

  3. School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Ruo-Xiu Xiao

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  1. Jing-Xin Zhao
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Correspondence to Li-Cheng Zhang or Pei-Fu Tang.

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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

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