Symmetry

1. Symmetry • Definition: • Both limbs are behaving identically • Measures of Symmetry • Symmetry Index • Symmetry Ratio • Statistical Methods

2. Symmetry Index • SI when it = 0, the gait is symmetrical • Differences are reported against their average value. If a large asymmetry is present, the average value does not correctly reflect the performance of either limb Robinson RO, Herzog W, Nigg BM. Use of force platform variables to quantify the effects of chiropractic manipulation on gait symmetry. J Manipulative Physiol Ther 1987;10(4):172–6.

3. Symmetry Ratio • Limitations: relatively small asymmetry and a failure to provide info regarding location of asymmetry • Low sensitivity Seliktar R, Mizrahi J. Some gait characteristics of below-knee amputees and their reflection on the ground reaction forces. Eng Med 1986;15(1):27–34.

4. Statistical Measures of Symmetry • Correlation Coefficients • Principal Component Analysis • Analysis of Variance • Use single points or limited set of points • Do not analyze the entire waveform Sadeghi H, et al. Symmetry and limb dominance in able-bodied gait: a review. Gait Posture 2000;12(1):34–45. Sadeghi H, Allard P, Duhaime M. Functional gait asymmetry in ablebodied subjects. Hum Movement Sci 1997;16:243–58.

5. Eigenvector Analysis The measure of trend symmetry utilizes eigenvectors to compare time-normalized right leg and left leg gait cycles in the following manner. Each waveform is translated by subtracting its mean value from every value in the waveform. for every ith pair of n rows of waveform data

6. Eigenvector Analysis Translated data points from the right and left waveforms are entered into a matrix (M), where each pair of points is a row. The rectangular matrix M is premultiplied by its transpose (MTM) to form a square matrix S, and the eigenvectors are derived from the square matrix S. To simplify the calculation process, we applied a singular value decomposition (SVD) to the translated matrix M to determine the eigenvectors, since SVD performs the operations of multiplying M by its transpose and extracting the eigenvectors.

7. Eigenvector Analysis Each row of M is then rotated by the angle formed between the eigenvector and the X-axis (u) so that the points lie around the X-axis (Eq. (2)):

8. Eigenvector Analysis The variability of the points is then calculated along the X and Y-axes, where the Y-axis variability is the variability about the eigenvector, and the X-axis variability is the variability along the eigenvector. The trend symmetry value is calculated by taking the ratio of the variability about the eigenvector the variability along the eigenvector, and subtracting it from 1.0. A value of 1.0 indicates perfect symmetry, and a value of 0.0 indicates asymmetry.

9. Eigenvector Analysis • We also calculated two additional measures of symmetry between waveforms. • Range amplitude ratio quantifies the difference in range of motion of each limb, and is calculated by dividing the range of motion of the right limb from that of the left limb. • Range offset, a measure of the differences in operating range of each limb, is calculated by subtracting the average of the right side waveform from the average of the left side waveform.

10. Eigenvector Analysis Expressed as ratio of the variance about eigenvector to the variance along the eigenvector Trend Symmetry: 0.948 Range Amplitude Ratio: 0.79, Range Offset:0

11. Eigenvector Analysis Expressed as a ratio of the range of motion of the left limb to that of the right limb Range Amplitude Ratio: 2.0 Trend Symmetry: 1.0, Range Offset: 19.45

12. Eigenvector Analysis Calculated by subtracting the average of the right side waveform from the average of the left side waveform Range Offset: 10.0 Trend Symmetry: 1.0, Range Amplitude Ratio: 1.0

13. Eigenvector Analysis Raw flexion/extension waveforms from an ankle Trend Symmetry: 0.979 Range Amplitude Ratio: 0.77 Range Offset: 2.9°

14. Eigenvector Analysis

15. Final Adjustment #1 • Determining Phase Shift and the Maximum Trend Symmetry Index: • Shift one waveform in 1-percent increments (e.g. sample 100 becomes sample 1, sample 1 becomes sample 2…) and recalculate the trend similarity for each shift. The phase shift is determined by identifying the index at which the smallest value for trend similarity occurs. The maximum trend similarity value independent of original phase position is also identified in this process.

16. Final Adjustment #2 • Modifications to Trend Symmetry Index to accommodate mirrored waveforms: • Assign the sign of the eigenvector slope to the TSI value. A modified TSI value of 1.0 indicates perfect symmetry in like oriented waveforms, while a TSI value of -1 indicates perfect symmetry in reflected waveforms. A TSI value of 0.0 still indicates asymmetry.

17. Symmetry Example

18. Symmetry Example…Hip Joint Unbraced Braced Amputee

19. Symmetry Example…Knee Joint Unbraced Braced Amputee

20. Symmetry Example…Ankle Joint Unbraced Braced Amputee

21. Normalcy Example

22. Braced Amputee Unbraced

23. Braced Amputee Unbraced

24. Unbraced Braced Amputee

25. Unbraced Braced Amputee

26. Unbraced Braced Amputee

27. Unbraced Braced Amputee