Appendix
Determination of measurement parameters
The significance of several variables was defined as shown in Table 5. (Fig. 7).
Table 5 Some variables used in this study and their significance The above indicators for all cases in group 1 and group 2 were measured to find the median.
median_d_MD = 8.01 mm, median_d_LU = 8.17 mm, median_th_MD = 17.5°, median_cupsize = 52 mm. (Table 5).
Acetabular cup templating and the calculation of templating error
The program was executed to get the templating results and templating errors according to the following steps.
(1) For all the anteroposterior X-rays of the pelvis obtained from our hospital, a magnification of 8.09 pixels/mm was uniformly used;
(2) Let cup_size = 50 mm;
(3) Let d_MD = cup_size × median_d_MD/median_cupsize, d_LU = cup_size × median_d_LU/median_cupsize;
(4) In the dots_LU, a point closest to the d_LU from the boneedge_LU was found, named A;
(5) In the dots_MD, a point closest to the d_MD from the teardrop_BT was found, named B;
(6) The acetabular cup circle passed through A and B, so the acetabular cup center O was located on the mid-perpendicular line of AB; for the hemispherical acetabular cup, the ∠AOB was 180 − (18 + median_th_MD), for the sub-hemisphere/super hemisphere acetabular cup, if the center angle of the acetabular cup was angle_cup, then ∠AOB was angle_cup − (18 + median_th_MD); the diameter of the acetabular cup was calculated as d_cup_templating = L/sin(∠AOB/2), where L was the distance between AB. In addition, the coordinates of the center O of the acetabular cup was calculated and named as center_cup_templating; (Fig. 8).
(7) For the d_cup_templating measured in step (6), if the difference from the cup_size in step (2) was <0.3 mm, the process was continued to the next step; otherwise, let cup_size = d_cup_templating, and return to step (3), and continue the measurement;
(8) d_cup_templating was decimal, and the nearest double integer from the d_cup_templating was used as the final measurement result of acetabular cup size;
(9) By comparing the templating results of the acetabular cup and the results of the surgical record, the templating error of each case and the measurement accuracy of the entire sample were obtained.
Explanation of our templating method (Table 5)
To help understanding our templating method, we first simplify the templating conditions: all cups are hemispherical (the center angle of the cup is 180°), and the size of all the cups is about the same (close to the median 52 mm).
If it is known that a circle passes through two points A and B, it can be known that the center O of this circle is located on the perpendicular line of the line segment AB. If we also know ∠AOB, we can determine the position of the center O and the radius of this circle. In fact, there are two solutions for the circle center, which are located on both sides of the line segment, but considering the application scenario of the pelvic X-ray, the circle center O must be positioned below the line segment AB, so there is only one reasonable solution. To calculate the radius of the cup circle, we need to first determine A, B, and ∠AOB, and then determine O, and then the length of the AO is the radius of the cup.
On the AP X-ray of the pelvis, A is the lateral superior intersection of the cup and the inner wall of the acetabulum. The distance between it and the lateral edge of the acetabulum is d_LU. By measuring the d_LU on postsurgery X-ray of all 406 cases, the median of d_LU is obtained. When templating a new preoperative AP X-ray of the pelvis, as long as we know the coordinates of the lateral edge of the acetabulum and the contour of the nearby acetabular edge, combining with the value of median_d_LU, the ideal position of A can be estimated (A is located on this contour and its distance from the lateral edge of the acetabulum is median_d_LU). In a similar way, the ideal position of B, the medial inferior intersection of the cup and the inner wall of the acetabulum, can also be obtained.
By measuring the th_MD on postsurgery X-ray of all 406 cases, the median of this angle can be obtained (median_d_MD); the lateral uncoverage should not exceed 20%. It can be known that the average coverage ratio is 10%, then the ideal angle on the lateral side should be 180° × 10% = 18°, so the ideal ∠AOB should be equal to 180° − (18 + median_d_MD).
In this way, A, B and ∠AOB are obtained, and then the position and radius of the cup can be calculated.
The above is a simplified templating process. In clinical application, some cups are sub-hemisphere or super-hemisphere, we need to adjust the center angle of the cup, which will affect the value of ∠AOB (See templating step 6 in "Appendix"). In addition, median_d_LU and median_d_MD is suitable in the cases when the cup size is equal to median_cupsize. If the measured cup size is not equal to median_cupsize, we need to scale median_d_LU and median_d_MD (See templating step 3). Because we do not know the templating results in advance, the ideal value of cup_size/median_cupsize is also unknown, therefore, we let the computer try different cup_sizes repeatedly until the templating result becomes satisfactory (see templating step 7).