Central X-ray beam correction of radiographic acetabular cup measurement after THA: an experimental study

Abstract

Purpose

Accurate assessment of cup orientation on postoperative radiographs is essential for evaluating outcome after THA. However, accuracy is impeded by the deviation of the central X-ray beam in relation to the cup and the impossibility of measuring retroversion on standard pelvic radiographs.

Method

In an experimental trial, we built an artificial cup holder enabling the setting of different angles of anatomical anteversion and inclination. Twelve different cup orientations were investigated by three examiners. After comparing the two methods for radiographic measurement of the cup position developed by Lewinnek and Widmer, we showed how to differentiate between anteversion and retroversion in each cup position by using a second plane. To show the effect of the central beam offset on the cup, we X-rayed a defined cup position using a multidirectional central beam offset. According to Murray’s definition of anteversion and inclination, we created a novel corrective procedure to balance measurement errors caused by deviation of the central beam.

Results

Measurement of the 12 different cup positions with the Lewinnek’s method yielded a mean deviation of $1.8{^{\circ }}$ (95 % CI 1.3–2.3) from the original cup anteversion. The respective deviation with the Widmer/Liaw’s method was $3.2{^{\circ }}$ (95 % CI 2.4–4.0). In each case, retroversion could be differentiated from anteversion with a second radiograph. Because of the multidirectional central beam offset (${\pm }5$ cm) from the acetabular cup in the cup holder ($25{^{\circ }}$ anteversion and $36{^{\circ }}$ inclination), the mean absolute difference for anteversion was $3.9{^{\circ }}$ (range $-4.2{^{\circ }}$ to $+4.0{^{\circ }})$ and $0.7{^{\circ }}$ (range $-1.2{^{\circ }}$ to $+1.2{^{\circ }})$ for inclination. The application of our novel mathematical correction of the central beam offset reduced deviation to a mean difference of $0.4{^{\circ }}$ for anteversion and $0.3{^{\circ }}$ for inclination.

Conclusion

This novel calculation for central beam offset correction enables highly accurate measurement of the cup position.

Appendix

Vertical and horizontal correction for central beam deviation

To assess and correct cup position, we used a self-designed Microsoft Excel sheet (see supplementary material) and the calculations shown below. The following parameters are required and exemplarily measured in Fig. 5. For X-ray setting of pelvic radiographs, we added a 3D sketch (see supplementary material) that shows how to measure the vertical and horizontal X-ray offset angle.

$$\begin{aligned}&s = \text {small diameter of the cup}\\&l = \text {long diameter of the cup}\\&\hbox {RI} = \text {radiographic inclination}\\&X = \text {horizontal offset of X-ray}\\&Y = \text {vertical offset of X-ray} \end{aligned}$$

Radiographic anteversion (RA) is calculated using s and l. Radiographic inclination (RI) is measured as the angle between the horizontal line of the pelvis and the continued line of the long diameter of the cup. The horizontal and vertical offset angle $(X_{\mathrm{angle}},Y_{\mathrm{angle}})$ is calculated by means of the trigonometric relation between the X-ray offset (X, Y) and the focus object distance $(\hbox {FO}=85\,\hbox {cm})$.

$$\begin{aligned}&\hbox {RA} = \hbox {arcsin}\,(13/52\,\hbox {mm}) = 14.5{^{\circ }}\\&\hbox {RI} = 34.5{^{\circ }}\\&X_{\mathrm{angle}} = \hbox {arctan}\,(X/\hbox {FO}) = \hbox {arctan}\, (83/850\,\hbox {mm}) = 5.6{^{\circ }}\\&Y_{\mathrm{angle}} = \hbox {arctan}\,(Y/\hbox {FO})= \hbox {arctan}\,(59/850\,\hbox {mm}) = 4.0\,{^{\circ }} \end{aligned}$$

Horizontal correction-first step

Radiographic anteversion (RA) and inclination (RI) are transformed to the anatomical definition (AA and AI) according to Murray:

$$\begin{aligned} \hbox {AA}= & {} \hbox {arctan }(\hbox {tan RA} / \hbox {sin RI}) \\= & {} \hbox {arctan} (\hbox {tan} 14.5{^{\circ }} / \sin 34.5{^{\circ }}) = 24.5{^{\circ }}\\ \hbox {AI}= & {} \hbox {arccos }(\cos \hbox {RI} \cdot \cos \hbox {RA}) \\= & {} \hbox {arccos} (\cos 34.5{^{\circ }} \cdot \cos 14.5{^{\circ }}) = 37.1{^{\circ }} \end{aligned}$$

The error due to horizontal offset is balanced by adding the angle of horizontal X-ray offset to anatomical anteversion (AA):

$$\begin{aligned} \hbox {AA}_{\mathrm{Xcorr}} = \hbox {AA} + X_{\mathrm{angle}} = 24.5{^{\circ }} + 5.6{^{\circ }} =30.1{^{\circ }} \end{aligned}$$

Anatomical inclination (AI) is not affected by horizontal X-ray offset; thus, correction is not necessary.

Vertical correction-second step

Horizontally corrected anteversion $(\hbox {AA}_{\mathrm{Xcorr}})$ is transformed to the operative definition $(\hbox {OA}_{\mathrm{Xcorr}})$ according to Murray:

$$\begin{aligned} \hbox {OA}_{\mathrm{Xcorr}}= & {} \hbox {arctan} (\sin \hbox {AA}_{\mathrm{Xcorr}} \cdot \tan \hbox {AI}) \\= & {} \hbox {arctan} (\sin 30.1{^{\circ }} \cdot \tan 37.1) = 20.8{^{\circ }} \end{aligned}$$

By subtracting the angle of vertical X-ray offset ($Y_{\mathrm{angle}})$ from operative anteversion ($\hbox {OA}_{\mathrm{Xcorr}})$, the error due to horizontal offset is balanced, and horizontally and vertically corrected operative anteversion ($\hbox {OA}_{\mathrm{XYcorr}})$ is calculated:

$$\begin{aligned} \hbox {OA}_{\mathrm{XYcorr}} = \hbox {OA}_{\mathrm{Xcorr}}- Y_{\mathrm{angle}}= 20.8{^{\circ }} - 4.0{^{\circ }} =16.8{^{\circ }} \end{aligned}$$

Operative inclination is not influenced by vertical X-ray offset and calculated according to Murray from horizontally corrected anteversion $(\hbox {AA}_{\mathrm{Xcorr}})$ and anatomical inclination (AI).

$$\begin{aligned} \hbox {OI}_{\mathrm{XYcorr}}= & {} \hbox {arcsin} (\hbox {cosAA}_{\mathrm{Xcorr}} \cdot \sin \hbox {AI}) \\= & {} \hbox {arcsin} (\cos 30.1{^{\circ }} \cdot \sin 37.1{^{\circ }}) = 31.5{^{\circ }} \end{aligned}$$

Finally, horizontally and vertically corrected operative definition of anteversion $(\hbox {OA}_{\mathrm{XYcorr}})$ and inclination $(\hbox {OI}_{\mathrm{XYcorr}})$ is retransformed to the radiographic or the anatomical definition according to Murray. Because we set the anatomical definition with our cup holder, we calculated anatomical anteversion and inclination to validate the corrective procedure:

$$\begin{aligned} \hbox {AA}_{\mathrm{XYcorr}}= & {} \hbox {arctan} (\sin \hbox {OA}_{\mathrm{XYcorr}}/ \tan \hbox {OI}_{\mathrm{XYcorr}}) \\= & {} \hbox {arctan} (\sin 16.8{^{\circ }} / \tan 31, 5{^{\circ }}) =25,3{^{\circ }}\\ \hbox {AI}_{\mathrm{XYcorr}}= & {} \hbox {arctan} (\tan \hbox {OA}_{\mathrm{XYcorr}} / \sin \hbox {AA}_{\mathrm{XYcorr}})\\= & {} \hbox {arctan} (\tan 16.8{^{\circ }} / \sin 25,3{^{\circ }}) = 35,2{^{\circ }} \end{aligned}$$