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

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. 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.