Symmetry

1. Symmetry James Richards, modified by W. Rose • 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 to form a 2x2 matrix S: S = MTM The eigenvalues and eigenvectors of S are computed.

7. Eigenvector Analysis To simplify the calculation process, we applied singular value decomposition (SVD) to the translated matrix M to determine the eigenvalues and eigenvectors of S=MTM, since SVD performs the operations of multiplying M by its transpose and extracting the eigenvectors. Note that the singular values of M are the non-negative square roots of the eigenvalues of S=MTM (as stated by Labview help).

8. Eigenvector Analysis Each row of M is then rotated by (minus) the angle formed between the eigenvector and the X-axis, so that the points lie around the X-axis (Eq. (2)): where and ex and ey are the x and y components of the (largest) eigenvector of S, and a 4-quadrant inverse tangent function is used.

9. Eigenvector Analysis The variability of the rotated points in the X and Y directions is then calculated. The Y-axis variability is the variability perpendicular to the eigenvector, and the X-axis variability is the variability along the eigenvector. Compute the ratio of the variability about the eigenvector to the variability along the eigenvector. This number will always be between 0 and 1. The ratio is subtracted from 1, giving the Trend Symmetry, which will be between 1 and 0.

10. Eigenvector Analysis Trend Symmetry = 1.0 indicates perfect symmetry Trend Symmetry = 0.0 indicates lack of symmetry. The Trend Symmetry will be 1 if the ratio of variabilities is 0. This will occur if and only if the rotated points all lie on the X axis (which means the variability along Y is zero). The Trend Symmetry will be 0 if the ratio of variabilities is 1. This will occur if the rotated points vary as much in the Y direction as they do in the X direction.

11. Additional measures of symmetry: • 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.

12. Eigenvector Analysis Trend Symmetry: 0.948 Range Amplitude Ratio: 0.79, Range Offset:0

13. Eigenvector Analysis Range Amplitude Ratio: 2.0 Trend Symmetry: 1.0, Range Offset: 19.45

14. Eigenvector Analysis Range Offset: 10.0 Trend Symmetry: 1.0, Range Amplitude Ratio: 1.0

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

16. Eigenvector Analysis

17. Final Adjustment #1 • Determining Phase Shift and the Maximum Trend Symmetry: • Shift one waveform in 1-percent increments (e.g. sample 100 becomes sample 1, sample 1 becomes sample 2…) and recalculate the trend symmetry for each shift. The phase offset is the shift which produces the largest value for trend symmetry. The associated maximum trend symmetry value is also noted.

18. Final Adjustment #2 • Trend Symmetry (TS), as defined so far, is unaffected if one of the waves is multiplied by -1. • Therefore Trend Symmetry, as computed, does not distinguish between symmetry and anti-symmetry. • We can modify Trend Symmetry to distinguish between symmetric and anti-symmetric waveforms:

19. Final Adjustment #2 • TSmod = 1.0 indicates perfect symmetry. • TSmod = -1.0 indicates perfect antisymmetry. • TSmod = 0.0 indicates complete lack of symmetry.

20. Symmetry Example

21. Symmetry Example…Hip Joint Unbraced Braced Amputee

22. Symmetry Example…Knee Joint Unbraced Braced Amputee

23. Symmetry Example…Ankle Joint Unbraced Braced Amputee

24. Normalcy Example

25. Braced Amputee Unbraced

26. Braced Amputee Unbraced

27. Unbraced Braced Amputee

28. Unbraced Braced Amputee

29. Unbraced Braced Amputee

30. Unbraced Braced Amputee

31. Simpler way to compute Trend Symmetry No need for SVD or Eigen-routines. Can be done in an Excel spreadsheet. Compute 2x2 covariance matrix: Note s12=s21. Eigenvalues of S are the values of which satisfy:

32. Simpler way to compute Trend Symmetry Eigenvalues of S are the values of which satisfy :

33. Ratio of smaller to larger eigenvalue: where Then