symmetry

Symmetry

Definition: Both limbs are behaving identically

Measures of Symmetry Symmetry Index

Symmetry Ratio

Statistical Methods

James Richards, modified by W. Rose

Symmetry Index

SI when it = 0, the gait is symmetricalDifferences 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.

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LR

LR

XX

XXSI

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.

%100*L

R

X

XSR

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: areview. 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.

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

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 = MTMThe eigenvalues and eigenvectors of S are computed.


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).


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.


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.

𝑇𝑟𝑒𝑛𝑑𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦=1−𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑎𝑏𝑜𝑢𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑎𝑙𝑜𝑛𝑔𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟


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.


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.


Trend Symmetry: 0.948 Range Amplitude Ratio: 0.79, Range Offset:0



Range Amplitude Ratio: 2.0 Trend Symmetry: 1.0, Range Offset: 19.45

𝑅 . 𝐴 .𝑅 .=𝐿𝑒𝑓𝑡 𝑙𝑖𝑚𝑏𝑀𝑎𝑥 .−𝑀𝑖𝑛 .h𝑅𝑖𝑔 𝑡 𝑙𝑖𝑚𝑏𝑀𝑎𝑥 .−𝑀𝑖𝑛 .


Range Offset: 10.0 Trend Symmetry: 1.0, Range Amplitude Ratio: 1.0

𝑅𝑎𝑛𝑔𝑒𝑂𝑓𝑓𝑠𝑒𝑡=𝑀𝑒𝑎𝑛 (𝐿𝑒𝑓𝑡 )−𝑀𝑒𝑎𝑛( h𝑅𝑖𝑔 𝑡)


Trend Symmetry: 0.979 Range Amplitude Ratio: 0.77 Range Offset: 2.9°

Raw flexion/extension waveforms from an ankle

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.

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:

Final Adjustment #2

TSmod = 1.0 indicates perfect symmetry.

TSmod = -1.0 indicates perfect antisymmetry.

TSmod = 0.0 indicates complete lack of symmetry.

Symmetry Example

Hip Joint Trend Symmetry

Phase Shift (% Cycle

Max Trend Symmetry

Range Amplitude

Range Offset

95% CI 0.98 – 1.00 -3.1 – 2.9 0.99 – 1.00 0.88 - 1.16 -5.99 – 5.66

Unbraced 1.00 1 1.00 0.95 4.21

Braced 1.00 0 1.00 1.02 4.73

Amputee 1.00 -1 1.00 0.88 -0.72

Symmetry Example…Hip Joint

Braced AmputeeUnbraced

Knee Joint

Trend Symmetry


Max Trend Symmetry

Range Amplitude

Range Offset

95% CI 0.97 – 1.00 -2.6 – 2.5 0.99 – 1.00 0.87 - 1.16 -8.95 - 10.51

Unbraced 1.00 0 1.00 1.03 5.28

Braced 1.00 -1 1.00 0.99 6.40

Amputee 0.98 -1 0.99 0.91 4.15

Symmetry Example…Knee Joint

Braced AmputeeUnbraced

Ankle Joint

Trend Symmetry


Max Trend Symmetry

Range Amplitude

Range Offset

95% CI 0.94 – 1.00 -2.62 – 2.34 0.96 – 1.00 0.75 - 1.32 -6.4 – 7.0

Unbraced 0.98 -1 0.98 1.03 -2.96

Braced 0.73 -4 0.79 0.53 5.84

Amputee 0.58 4 0.61 1.27 0.48

Symmetry Example…Ankle Joint

Unbraced Braced Amputee

Normalcy Example

Hip Joint Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.98 – 1.00 -3.1 – 2.9 0.99 – 1.00 0.88 - 1.16 -5.99 – 5.66

Right hip

Unbraced 1.00 2 1.00 0.85 -14.91

Braced 0.99 3 1.00 0.90 -14.20

Amputee 0.97 -4 1.00 0.92 -8.08

Left hip

Unbraced

Braced

Amputee


Hip Joint Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.98 – 1.00 -3.1 – 2.9 0.99 – 1.00 0.88 - 1.16 -5.99 – 5.66

Right hip

Unbraced

Braced

Amputee

Left hip

Unbraced 1.00 2 1.00 0.91 -19.28

Braced 0.99 4 1.00 0.91 -19.09

Amputee 0.99 -2 1.00 1.06 -7.52


Knee Joint

Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.97 – 1.00 -2.6 – 2.5 0.99 – 1.00 0.87 - 1.16 -8.95 – 10.51

Right knee

Unbraced 0.99 1 0.99 1.12 -11.89

Braced 0.98 3 0.99 1.07 -13.22

Amputee 0.96 -2 0.99 0.97 -7.45

Left knee

Unbraced

Braced

Amputee


Knee Joint

Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.97 – 1.00 -2.6 – 2.5 0.99 – 1.00 0.87 - 1.16 -8.95 – 10.51

Right knee

Unbraced

Braced

Amputee

Left knee

Unbraced 0.99 1 0.99 1.11 -16.35

Braced 0.97 4 0.99 1.10 -18.80

Amputee 0.98 -2 1.00 1.08 -10.78


Ankle Joint

Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.94 – 1.00 -2.62 – 2.34 0.96 – 1.00 0.75 - 1.32 -6.4 – 7.0

Right ankle

Unbraced 0.90 -2 0.94 1.48 1.33

Braced 0.65 -4 0.72 0.77 9.04

Amputee 0.80 -5 0.98 1.40 4.30

Left ankle

Unbraced

Braced

Amputee


Ankle Joint

Trend Normalcy


Max Trend Normalcy

Range Amplitude

Range Offset

95% CI 0.94 – 1.00 -2.62 – 2.34 0.96 – 1.00 0.75 - 1.32 -6.4 – 7.0

Right ankle

Unbraced

Braced

Amputee

Left ankle

Unbraced 0.93 -1 0.95 1.49 4.62

Braced 0.94 2 0.95 1.51 3.53

Amputee 0.11 -11 0.76 1.14 4.15


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:

Simpler way to compute Trend Symmetry

Eigenvalues of S are the values of which satisfy :

Ratio of smaller to larger eigenvalue:

where

Then

symmetry

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