Medical & Biological Engineering & Computing

, Volume 57, Issue 11, pp 2359–2371 | Cite as

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

  • Jing-Xin Zhao
  • Xiu-Yun Su
  • Zhe Zhao
  • Ruo-Xiu Xiao
  • Guo-Fei Sun
  • Li-Cheng ZhangEmail author
  • Pei-Fu TangEmail author
Original Article


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.

Graphical abstract

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.


Acetabular cup 2D radiographic angle 3D radiographic angle Pelvic rotation X-ray offset 









Radiographic anteversion


Radiographic inclination


Total hip arthroplasty



The radius of the cup

(o, p, q)

The coordinate of the X-ray beam source

(x0, y0, z0)

The coordinate of the center of the cup

[k1, i1, j1]

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


The x-axial rotation of the cup


The y-axial rotation of the cup


The z-axial rotation of the cup



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

Funding information

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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Supplementary material

11517_2019_2041_MOESM1_ESM.docx (3.4 mb)
ESM 1 (DOCX 3502 kb)


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Copyright information

© International Federation for Medical and Biological Engineering 2019

Authors and Affiliations

  1. 1.Department of OrthopaedicsChinese PLA General HospitalBeijingChina
  2. 2.Department of Orthopaedics, Beijing Tsinghua Changgung Hospital, School of Clinical MedicineTsinghua UniversityBeijingChina
  3. 3.School of Computer and Communication EngineeringUniversity of Science and Technology BeijingBeijingChina

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