Abstract The location of the hip joint centre (HJC) is of great importance in the field of gait analysis. Since it is not easily palpable, its location is often predicted using more easily attainable anatomical data, such as the pelvic width. In this study, relevant anatomical data was obtained from CT scans of 158 subjects of various ages and both genders. Regression equations were fitted using what were identified to be the best predictors of the HJC location and an analysis on the difference in regression equations between age and gender groups was performed. It was found that the x, y and z coordinates of the HJC could be predicted with a predictive root mean square deviation (RMSD) of 4.78, 3.59 and 4.68 mm respectively. It was also found that by generating separate regression equations for the y and z coordinates in separate age groups, the RMSD was reduced, indicating an improvement in predictability. The use of regression equations has been shown to be an effective and consistent method in predicting the location of the HJC. Introduction The hip joint centre (HJC) is an important feature in gait analysis, used for kinetic and kinematic calculations. The HJC is not easily palpable, unlike the prominent bony landmarks such as the anterior superior iliac spine (ASIS) and the posterior superior iliac spine (PSIS). For this reason, the location of the HJC must be calculated using the positions of other landmarks. Previous studies have employed two main methods for calculating the position of the HJC. The first is a functional method (Camomilla et al., 2006) and the second is a predictive method (Bell et al., 1989; Harrington et al., 2007) using regression equations. Despite work on the predictive method using data from both adults and children (Bell et al., 1989; Harrington et al., 2007), little has been done to produce separate regression equations for separate age groups or genders. The present study focuses on using the predictive method and assesses the differences in the REs calculated for different groups. New regression equations are proposed and compared to those presented by Harrington et al., (2007). Materials and Methods Data from a total of 158 subjects was taken for the study. This included 60 adults aged 25-­‐
40, 30 male and 30 female, 60 teenagers aged 16-­‐19, 30 male and 30 female, as well as 38 children aged 5-­‐11, 25 male and 13 female. CT scans were taken of each subject with a number of 1mm slices in the coronal, sagittal and transverse planes. A dedicated program was used to locate bony landmarks including the HJC from the CT scan slices in each subject. The locations of these landmarks were transformed into a pelvic-­‐embedded coordinate system in accordance with the one proposed in Davis et al., (1991). The y-­‐coordinates have been adjusted so that they represent the distance from the origin, resulting in a positive value for the y-­‐coordinate of both the left and right HJC. For each subject the data from the CT scans was used to calculate ASIS separation (SW), pelvic width (PW) pelvic depth (PD) defined as the distance between the midpoint of the two ASIS and the two PSIS and clinical leg length (LL). The x, y and z coordinates (HJCx, HJCy and HJCz respectively) of both the left and right HJC were recorded. All length measurements throughout the study are presented in millimetres (mm). A preliminary qualitative analysis of the relationship between each of the coordinates of the HJC and each of the independent variables (SW, PW, PD and LL) was performed to determine the most suitable form of the predictive equations. Scatter plots for each coordinate against each independent variable were obtained (Figure 1.) The scatter plots showed overall linear trends, suggesting regression equations of the form HJCx = a + bSW + cPW + dPD + eLL similar to those presented in Harrington et al., (2007). -­‐10 HJCx (mm) -­‐20 5-­‐11 -­‐30 16-­‐19 -­‐40 25-­‐40 Harrington Chidren -­‐50 Harrington Adults -­‐60 -­‐70 60 80 100 PD 120 (mm) 140 160 180 Figure 1. Scatter plot of x coordinates of HJCs against PD. Each age group is represented with separate markers and the data from Harrington et al., (2007) is shown for comparison. The best predictors for each coordinate of the HJC were determined by grouping the data by age group, gender and both age group and gender. An initial group of best predictors for each coordinate was selected based on the predictive R2 value of the regressions. Further analyses were performed on the resulting predictive equations to select the best regression. This took into account statistical significance, judged by p-­‐values, and root mean square deviation (RMSD). RMSD was calculated by dividing the predictive sum of squares of the residuals and taking the square root. An emphasis was placed on comparing the predictive capabilities of the potential models. The significance level was set at p < 0.05. After selecting the form of each regression, an analysis of covariance (ANCOVA) was used to determine if separate regressions should be used for separate age groups and/or genders. Results Each coordinate was given a list of possible best predictors, one of which was chosen for the regression equations (Table 1). The regression equation for the x coordinate was chosen to include SW as well as PD because of the decrease in RMSD when SW was added. The y coordinate regression was chosen to include PW and PD because it had a low RMSD. SW was not included because it was found to be statistically insignificant upon being brought into the model. Any of the four sets of predictors for the z coordinate would have been suitable but PW and LL were chosen for the sake of comparison with the regression equation presented by Harrington et al., (2007). 2
Table 1. Multiple regressions with predictive R values, RMSD values and p-­‐values. This list of potential best 2
predictors was chosen based on R values. All regressions were calculated with a non-­‐zero intercept. Coordinate X X Y Y Y Y Y Y Z Z Z Z . Predictors PD SW, PD SW PW PD SW, PW PW, PD SW, PW, PD LL PW, LL PD, LL PW, PD, LL p-­‐values R-­‐Sq Pred RMSD Constant Coefficient 1 Coefficient 2 0.73 5.20 0.001 0.000 0.77 4.78 0.027 0.000 0.000 0.66 7.05 0.003 0.000 0.84 4.82 0.174 0.000 0.88 4.13 0.000 0.000 0.91 3.68 0.000 0.000 0.000 0.91 3.59 0.000 0.000 0.000 0.91 3.60 0.000 0.620 0.000 0.80 4.92 0.000 0.000 0.81 4.68 0.002 0.000 0.000 0.82 4.67 0.000 0.000 0.000 0.82 4.60 0.000 0.001 0.000 Coefficient 3 0.000 0.000 The regression equations generated using the best predictors from Table 1 are listed below. HJCx = – 4.31 + 0.11 SW – 0.46 PD, R2pred = 0.78, RMSD = 4.78 HJCy = 8.37 + 0.13 PW + 0.30 PD, R2pred = 0.91, RMSD = 3.59 HJCz = – 5.88 – 0.09 PW – 0.06 LL, R2pred = 0.82, RMSD = 4.68 As an example interpretation of Table 2, the constant term in the regression equation for HJCx did not have a statistically significant difference after separating the data by age group because this p-­‐value (0.655) was above the significance level, p < 0.05. However, the coefficients of SW and PD did differ significantly when separating the data by age group because the p-­‐values (0.029 and 0.041 respectively) were below the significance level. Despite there being a statistically significant difference in the coefficients, the mean change RMSD was negligible, only 0.05mm. From this, it is not worth generating separate regression equations for separate age groups. Table 2. ANCOVA values for each coordinate’s regression equation. A positive ΔRMSD value indicates a decrease in mean RMSD from the regression equation generated from all subjects, suggesting an improvement in the model’s predictive capability. p-­‐values p-­‐values p-­‐values Coordinate Data Set Factor Factor SW*Factor PD*Factor ΔRMSD All Age Gender 0.655 0.392 0.029 0.074 0.041 0.003 0.05 0.13 5-­‐11 Gender 0.356 0.105 0.038 1.07 X 16-­‐19 Gender 0.000 0.002 0.011 -­‐0.07 25-­‐40 Gender 0.611 0.709 0.726 -­‐0.29 Male Female Age Age 0.053 0.003 0.004 0.894 0.033 0.016 -­‐0.02 0.49 25-­‐40 Gender 0.932 0.535 0.377 -­‐0.36 Male Female Age Age 0.319 0.003 0.974 0.970 0.322 0.007 0.37 -­‐0.07 25-­‐40 Gender 0.722 0.850 0.750 -­‐0.21 Male Female Age Age 0.120 0.107 0.816 0.005 0.502 0.107 0.94 -­‐0.17 Coordinate Data Set Factor Factor PW*Factor PD*Factor ΔRMSD All Age Gender 0.002 0.277 0.824 0.737 0.055 0.217 0.21 0.00 5-­‐11 Gender 0.986 0.537 0.597 0.68 Y 16-­‐19 Gender 0.016 0.554 0.041 0.12 Coordinate Data Set Factor Factor PW*Factor LL*Factor ΔRMSD All Age Gender 0.000 0.014 0.131 0.984 0.331 0.107 0.41 0.10 5-­‐11 Gender 0.914 0.005 0.005 1.50 Z 16-­‐19 Gender 0.463 0.031 0.362 -­‐0.14 It was found to be beneficial to generate separate regressions for separate age groups when predicting HJCy and HJCz only. This was because the mean RMSD decreased by 0.21 and 0.41 mm respectively. A further split of the 5-­‐11 age group by gender proved beneficial for both HJCy and HJCz leading to a total reduction in mean RMSD of 0.68 and 1.50 mm respectively. Regression equations for each coordinate were generated with separate regressions for appropriate subdivisions that were found to improve predictability (Table 3) based on the previous ANCOVA. Table 3. Regression equations for HJCx, HJCy and HJCz for appropriate subdivisions of the data that resulted in greater predictivity. X Y Z Age Gender Regression Equation All All 5-­‐11 5-­‐11 5-­‐11 16-­‐19 All All All Male Female All HJCx = – 4.31 + 0.11 SW – 0.46 PD HJCy = 8.37 + 0.13 PW + 0.30 PD HJCy = 7.95 + 0.11 PW + 0.34 PD HJCy = 7.22 + 0.11 PW + 0.35 PD HJCy = 7.31 + 0.08 PW + 0.40 PD HJCy = 32.90 + 0.09 PW + 0.19 PD 25-­‐40 All 5-­‐11 5-­‐11 5-­‐11 16-­‐19 25-­‐40 All All All Male Female All All HJCy = 17.00 + 0.11 PW + 0.28 PD HJCz = 5.88 – 0.09 PW – 0.06 LL HJCz = –11.00 – 0.08 PW – 0.05 LL HJCz = – 9.38 – 0.10 PW – 0.05 LL HJCz = – 12.90 – 0.05 PW – 0.05 LL HJCz = – 39.40 – 0.08 PW – 0.02 LL HJCz = – 7.87 – 0.07 PW – 0.06 LL Discussion The method of finding the best predictors resulted in regression equations of a very similar form to those presented by Harrington et al., (2007). HJCy was found to be best estimated using the same predictors. Four sets of best predictors for HJCz were found to be approximately as good as each other, one of which was used by Harrington et al., (2007). HJCx was best predicted using PD in both studies, with the addition of a second predictor in the present study not used by Harrington et al., (2007), namely SW. In addition to the similarities in the predictors used, there was a distinct similarity in the coefficients of the HJCy and the HJCz regression equations in particular. The regressions used by both Harrington et al., (2007) and the present study were compared (Table 4). Table 4. Comparison of the independently determined regression equations. Note that the y and z coordinates used by Harrington et al., (2007) have been swapped to match the convention used in the present study, allowing for easy comparison. X Y Z Proposed HJCx = 5.42 – 0.36 PD HJCy = 8.37 + 0.13 PW + 0.30 PD HJCz = – 5.88 – 0.09 PW – 0.06 LL Harrington et al., (2007) HJCx = – 9.9 – 0.24 PD HJCy = 7.9 + 0.16 PW + 0.28 PD HJCz = – 7.1 – 0.16 PW – 0.04 LL The only significant difference was the constant used in the regression equation for HJCx, which differed by over 15 mm. This could be explained by the use of markers placed on the subjects’ bodies (Harrington et al., 2007). This would have led to an extra 14 mm recorded for certain lengths resulting from the addition of the radii of the markers and the width of the plates they were fixed to. From this, it is suggested that both regression equations for HJCx are quite similar but a slightly different process of measurement was used. In addition to the noticeable similarities between the HJCy and HJCz regressions, an ANCOVA on the regressions was performed to examine the statistical significance of the difference between the regressions of each study (Table 5). It was found that the difference between both the coefficients and the constants in the HJCx regressions were statistically significant, with p-­‐values well below the significance level of p < 0.05. This was to be expected as there was a noticeable difference in the constants of the HJCx regressions presented in each study, and to a lesser extent, a difference in the PD coefficients. The differences in the HJCy and HJCz regressions between the studies were all statistically insignificant, which was also to be expected after noticing the high degree of similarity in the regressions. Table 5. ANCOVA values describing the statistical significance of the differences between the regressions presented in the present study and by Harrington et al., (2007). p-­‐values Coordinate Predictor 1 Predictor 2 Study Predictor 1*Study Predictor 2*Study X PD 0.001 0.004 Y PW PD 0.903 0.427 0.792 Z PW LL 0.809 0.270 0.459 The similarities between the independently determined regression equations suggest that this predictive method is in fact a suitable method for estimating the location of the HJC. Conclusion Linear regression equations to locate the HJC were found to be an effective prediction method. The best predictors were judged by the predictive R2, predictive root mean square deviation (RMSD) and statistical significance values of various regressions. The best predictors were found to include different combinations of ASIS separation (SW), pelvic width (PW), pelvic depth (PD) and clinical leg length (LL) for each coordinate. The following regression equations are proposed for locating the HJC: HJCx = – 4.31 + 0.11 SW – 0.46 PD HJCy = 8.37 + 0.13 PW + 0.30 PD HJCz = – 5.88 – 0.09 PW – 0.06 LL A comparison of regression equations generated independently and from independent data (Harrington et al., 2007) showed that the same predictors, measured in a similar fashion, resulted in similar regression equations. In addition, the differences in the regressions were found to be statistically insignificant. References Bell, A.L., Pedersen, D.R., Brand, R.A., 1990. A comparison of the accuracy of several hip center location prediction methods. Journal of Biomechanics 23, 617–621. Camomilla, V., Cereatti, A., Vannozzi, G., Cappozzo, A., 2006. An optimized protocol for hip joint centre determination using the functional method. Journal of Biomechanics 39, 1096–1106. Harrington, M.E., Zavatsky, A.B., Lawson, S.E.M., Yuan, Z., Theologis, T.N., (2007). Prediction of the hip joint centre in adults, children, and patients with cerebral palsy based on magnetic resonance imaging. Journal of Biomechanics 40, 595–602