Abstract A mathematical method based on computations of residual absolute value sums (RAVS) was developed for the quantitative analysis of tremor-like perturbations of knee angle during the gait cycle. The method was tested on simulation data created by adding sinusoidal tremor of varying frequency and amplitude to the knee-angle graph of a healthy test subject. The method was then applied to compare knee tremor reduction, with and without auditory rhythm, in a group of &quot;ve traumatically brain-injured patients with gait hemiparesis. Deviations from normal gait performance due to tremor were assessed by using self-comparison to a 17th-degree regression polynomial of each subject&apos;s own motion-, time-, and point-normalized knee-angle curve. With rhythmic cueing, the &quot;ve subjects had a statistically signi&quot;cant RAVS-measured mean tremor reduction of 39.5$22.6% (t&quot;!3.91; p&quot;0.017)

Gary P Kenyon

M H Thaut

Michael H Thaut

CiteSeerX

*Corresponding author. Tel.: #1-970-491-7384; fax: #1-970-491-
7541.
E-mail address: mthaut@lamar.colostate.edu (M.H. Thaut).
Journal of Biomechanics 33 (2000) 1319}1323
Technical note
A measure of kinematic limb instability modulation
by rhythmic auditory stimulation
Gary P. Kenyon, Michael H. Thaut*
Department of Electrical Engineering, Colorado State University, Center for Research in NeuroRehabilitation, Fort Collins, CO 80523, USA
Accepted 21 March 2000
Abstract
A mathematical method based on computations of residual absolute value sums (RAVS) was developed for the quantitative
analysis of tremor-like perturbations of knee angle during the gait cycle. The method was tested on simulation data created by adding
sinusoidal tremor of varying frequency and amplitude to the knee-angle graph of a healthy test subject. The method was then applied
to compare knee tremor reduction, with and without auditory rhythm, in a group of "ve traumatically brain-injured patients with gait
hemiparesis. Deviations from normal gait performance due to tremor were assessed by using self-comparison to a 17th-degree
regression polynomial of each subject's own motion-, time-, and point-normalized knee- angle curve. With rhythmic cueing, the "ve
subjects had a statistically signi"cant RAVS-measured mean tremor reduction of 39.5$22.6% (t"!3.91; p"0.017). ( 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Tremor; Gait; Rhythm; Knee position analysis; Polynomial regression
1. Introduction
This study proposes a method of analysis for lower
extremity tremor based on the measure of residual abso-
lute value sums (RAVS) of the knee angle versus time
curve. The method was used to quantify tremor and the
tremor reduction resulting from rhythmic cueing of walk-
ing, based on self-comparison to the 17th-degree regres-
sion polynomial of the knee joint angle versus time curve.
Auditory rhythm has been previously shown to improve
gait kinematic parameters in subjects with hemiparetic
gait disorders (Prassas et al., 1997; Thaut et al., 1997,
1999).
Applications of traditional waveform analyses to
tremor data, e.g. accelerometry with Fourier analysis
(Frost, 1978), have shown limited usefulness for a variety
of reasons, including the di$culty of standardizing move-
ment-related tremor for quanti"cation (Deuschl et al.,
1991), misinterpretation of multiple irregular peaks in
a tremor spectrum (Gresty and Buckwell, 1990), low
correlation with other tremor measures, the tendency to
overestimate tremor improvement (Bain et al., 1993), the
necessity of matching the data-processing algorithm to
the data characteristics (Timmer et al., 1996), and the
absence of information relating to the amplitude and
duration of tremor or the coincidence of tremor with
well-de"ned stages of a movement sequence. Further-
more, while Fourier analysis can yield useful information
about the frequency content of the tremor, in gait the
positional perturbation of the knee angle has functional
importance for the stability of the movement. Therefore,
a waveform analysis showing tremor reduction based on
positional data would yield information clinically and
functionally more useful for gait assessment and therapy.
2. Methods
2.1. Normalization of data
Trial variations in walking speed were normalized by
expressing time duration as a fraction of one complete
gait cycle. To normalize for variations in the knee-angle
range of motion of a given subject, the value of the
minimum joint angle over the gait cycle was subtracted
from each data angle,
h
.*/
"minMh
j
NN
j/1
, (1)
hI
i
"h
i
!h
.*/
, i"1, 2,2, N (2)
0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 0 7 7 - 4
Fig. 1. This graph shows the error (as de"ned by Eq. (5)) in approximat-
ing the normal subject's knee angle over the gait cycle by a regression
polynomial, with the degree of the regression polynomial as the inde-
pendent variable. The approximation error is less than 1% when the
degree of the regression polynomial is 15 or greater, and is monotoni-
cally decreasing.
and each shifted angle was then divided by the maximum
shifted angle
hI
.!9
"maxMhI
j
NN
j/1
, (3)
hK
i
"
hI
i
hI
.!9
, i"1, 2,2, N, (4)
thus creating a dimensionless angle scale ranging in value
from 0 to 1.
Each gait cycle was normalized to 1001 data points
using an interpolation routine, since the RAVS proced-
ure is dependent upon the number of points in the data
sets. On the average, this increased the size of the data
sets by a factor of four, and this level of interpolation has
been shown to model human movement with "ner res-
olution (Kenyon and Thaut, 2000).
2.2. Regression comparison reference
A tremor comparison reference was developed from
a regressed version of each subject's personal knee-joint
angle versus time curve, consistent with the assumption
that tremor and other positional perturbations are super-
imposed on each subject's unique gait pattern. The
choice of the degree for the regression polynomial was
made on the basis of three procedures, each of which left
the degree of the regression polynomial as a variable.
First, the regression polynomial was tested for `goodness
of "ta to a normal subject's knee-angle curve. The accu-
racy of the approximating regression polynomial was
de"ned by Eq. (5), where e represents the percent error
measure, hK is the normalized knee angle, and o is the
corresponding knee angle after polynomial regression.
e"
&N
i/1
DhK
i
!o
i
D
&N
i/1
DhK
i
D
]100. (5)
Second, it was required that the regression polynomial
be a good approximation to the underlying non-pertur-
bed gait patterns upon which tremor and other perturba-
tions were superimposed. Third, the degree of the
regression polynomial was `tuneda to give comparison
results most consistent with the idealized changes within
the tremor-simulated data sets, using tremor amplitude
as the primary indicator of tremor severity.
2.3. Dexnition of the RAVS tremor measure M
The RAVS tremor measure M was de"ned as follows.
First, a `residuala at a given data point was de"ned as the
di!erence in value of the normalized knee angle and the
regressed curve. The absolute values of the residuals were
summed over the entire data set, then this sum was
divided by the number of data points. In equation form
M"
1
N
N
+
i/1
DhK
i
!o
i
D (6)
where N"total number of normalized data points in the
gait cycle. The resulting number may be thought of as the
average tremor over the entire gait cycle. The percent
change in tremor severity *¹ over two measurements
(indicated by subscripts 1 and 2) was given by
*¹"
M
2
!M
1
M
1
]100. (7)
2.4. Simulation
The validity of the RAVS measure M was tested by
comparing changes in tremor severity, as calculated by
Eq. (7), to tremor-amplitude changes in simulated data
sets, which were constructed by mathematically adding
sinusoidal tremor to the weight-bearing phase of one gait
cycle of a 54-year-old male with normal gait. Ninety-
seven simulations were created (including the zero
tremor case) using tremor frequencies in the 4}15 Hz
range (increments of 1Hz), and tremor amplitudes in the
0.5}4.03 range (increments of 0.53). The frequency range
of 4}15Hz includes the most widely documented results
for tremor frequency, for a variety of tremor types, re-
ported in the literature (Brooks et al., 1981; Deuschl et al.,
1991; Elble et al., 1990; Frost, 1978; Ghika et al., 1993;
Gresty and Buckwell, 1990; Jankovic and Frost, 1981;
Marshall and Walsh, 1956; Timmer et al., 1996). The
amplitude range, intended to represent barely noticeable
to severe knee-joint tremors, was equal at its maximum
to 6% of the normal subject's total knee-angle range, and
included the tremor ranges of the "ve experimental sub-
jects.
1320 G.P. Kenyon, M.H. Thaut / Journal of Biomechanics 33 (2000) 1319}1323
Table 1
Summary of the results of applying the RAVS measure of tremor change to the 97 stimulated data sets (12 frequencies and 8 nonzero amplitudes, plus
the zero amplitude case); (a) Shows tremor change calculated directly from amplitude change (Eq. (9)); (b) Shows mean tremor change over the entire
frequency range as calculated by the RAVS method; (c) Shows the percent deviation of the Part (b) values from Part (a); (d) gives coe$cients of
variation for the percent deviations in Part (c).
A1 A2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
(a) Ideals (percent amplitude change)
0.5 !100.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
1.0 !100.0 !50.0 0.0 50.0 100.0 150.0 200.0 250.0 300.0
1.5 !100.0 !66.7 !33.3 0.0 33.3 66.0 100.0 133.3 166.7
2.0 !100.0 !75.0 !50.0 !25.0 0.0 25.0 50.0 75.0 100.0
2.5 !100.0 !80.0 !60.0 !40.0 !20.0 0.0 20.0 40.0 60.0
3.0 !100.0 !83.3 !66.7 !50.0 !33.3 !17.0 0.0 16.7 33.3
3.5 !100.0 !85.7 !71.4 !57.1 !42.9 !28.6 !14.3 0.0 14.3
4.0 !100.0 !87.5 !75.0 !62.5 !50.0 !37.5 !25.0 !12.5 0.0
(b) Mean tremor change by RAVS method
0.5 !67.6 0.0 95.7 192.3 288.8 385.0 480.9 576.3 671.3
1.0 !83.4 !48.9 0.0 49.3 98.6 147.7 196.6 245.4 293.9
1.5 !88.9 !65.8 !33.0 0.0 33.0 65.9 98.7 131.3 163.8
2.0 !91.6 !74.2 !49.6 !24.8 0.0 24.7 49.4 73.9 98.3
2.5 !93.3 !79.4 !59.6 !39.7 !19.8 0.0 19.8 39.4 59.0
3.0 !94.4 !82.8 !66.3 !49.7 !33.1 !16.5 0.0 16.4 32.8
3.5 !95.2 !85.2 !71.0 !56.8 !42.5 !28.3 !14.1 0.0 14.0
4.0 !95.8 !87.0 !74.6 !62.1 !49.6 !37.1 !24.7 !12.3 0.0
(c) Percent deviation of means from ideals
0.5 32.4 0.0 !4.3 !3.9 !3.7 !3.8 !3.8 !3.9 !4.1
1.0 16.6 2.2 0.0 !1.4 !1.4 !1.5 !1.7 !1.9 !2.0
1.5 11.1 1.4 0.9 0.0 !0.9 !1.6 !1.3 !1.5 !1.7
2.0 8.4 1.0 0.7 0.8 0.0 !1.1 !1.2 !1.5 !1.7
2.5 6.7 0.8 0.6 0.7 0.8 0.0 !1.2 !1.4 !1.7
3.0 5.6 0.7 0.6 0.7 0.7 3.0 0.0 !1.8 !1.6
3.5 4.8 0.6 0.5 0.6 0.9 1.1 1.4 0.0 !1.8
4.0 4.2 0.6 0.5 0.7 0.9 1.1 1.3 1.5 0.0
(d) Coe$cients of variation
0.5 2.9 0.0 4.9 4.8 4.8 4.9 4.9 5.1 5.2
1.0 1.4 2.5 0.0 2.4 2.4 2.5 2.6 2.8 2.9
1.5 0.9 1.7 1.6 0.0 1.7 1.8 1.9 2.1 2.3
2.0 0.7 1.3 1.2 1.3 0.0 1.5 1.7 1.9 2.0
2.5 0.6 1.0 1.0 1.1 1.2 0.0 1.6 1.7 1.9
3.0 0.5 0.9 0.9 1.0 1.1 1.3 0.0 1.7 1.9
3.5 0.4 0.7 0.8 0.9 1.0 1.3 1.4 0.0 1.8
4.0 0.4 0.7 0.8 0.9 1.0 1.2 1.4 1.6 0.0
It was assumed that, under conditions of constant
tremor duration, tremor amplitude is a valid indicator of
the functional severity of tremor, and that a change in
tremor severity is best re#ected by a change in tremor
amplitude. Thus, for the purposes of the simulated data
sets, percent change in tremor severity corresponded to
the percent change in tremor amplitude, and was cal-
culated as
*A"
A
2
!A
1
A
1
]100. (8)
The results for the change in tremor *¹ calculated by the
RAVS method were then compared to the idealized am-
plitude changes *A.
2.5. Pilot study with xve subjects
The RAVS tremor measure was applied to data col-
lected from "ve subjects to assess tremor activity and
changes in tremor activity under the cueing conditions of
with and without RAS. Subjects were 3 right-hemiparetic
and 2 left-hemiparetic patients with traumatic brain in-
jury (TBI) (3 female, 2 male; mean age 32$7 yr). Two
subjects had mild, 1 had moderate, and 2 subjects had
severe lower limb spasticity (Brunnstrom, 1970). All sub-
jects had visually observable tremors only during the
weight-bearing phase of the stride cycle. RAS was pre-
sented free-"eld as a metronome click (1000Hz, 20ms
plateau), which was frequency-matched to the step
G.P. Kenyon, M.H. Thaut / Journal of Biomechanics 33 (2000) 1319}1323 1321
Table 2
Individual and group results for the RAVS tremor measure computed
for gait data collected without (M
1
) and with (M
2
) a Rhythmic Audi-
tory Stimulus (RAS) for subjects with traumatic brain injury (TBI).
Tremors ranging from very moderate (M
1
"0.76) to severe
(M
1
"5.99) showed signi"cant improvement with RAS
Subject M
1
M
2
Tremor change (%)
1 0.76 0.26 !66.1
2 5.12 4.80 !6.3
3 0.80 0.56 !30.1
4 2.55 1.41 !44.7
5 5.99 2.98 !50.3
Mean 3.04 2.00 !39.5
Standard deviation 2.42 1.88 22.6
Fig. 2. For the two conditions of with and without rhythmic auditory
stimulus (RAS), the dashed lines indicate the right knee angle over one
complete gait cycle and the solid lines are the corresponding degree 17
regression polynomials. The RAVS method in essence measures the
change, from one case to another, of the magnitude of the areas between
the solid and dashed lines.
frequency recorded and computed for the trial without
RAS. One full gait cycle was recorded (using 60-Hz
videocameras) for three-dimensional digitized video
motion analysis.
3. Results
3.1. Regression polynomial
The degree of the regression polynomial was set at 17
because at this value, the regression polynomial con-
verged to the normal subject's knee-angle curve with less
than 1% error (Fig. 1), as de"ned by Eq. (5). It also
converged to the normal knee angle upon which tremor
had been superimposed in the 97 simulation data sets,
with a mean error of 0.70$0.65%. Degree 17 also result-
ed in the overall lowest values for the percent deviations
from the simulation results (Table 1).
3.2. Subject group
For the subject group, the RAVS-measured tremor
was reduced by 39.5$22.6% (Table 2) when RAS was
applied. This change was statistically signi"cant using
dependent sample t-tests (t"!3.914; p"0.017) as well
as a non-parametric analysis using the Wilcoxon Signed
Ranks Test (Z"!2.023; p"0.043).
3.3. Illustrative example
Graphs for one subject (Fig. 2) show the regression
polynomial and the actual knee}angle curve for the cases
of with and without RAS. Without RAS, the RAVS
tremor measure has a value of 5.99 (by Eq. (6)), and with
RAS a value of 2.98. Thus the resulting change in tremor,
as calculated by Eq. (7), is !50.3%, indicating a 50.3%
decrease in tremor.
4. Discussion
The RAVS method applied to our study sample quan-
ti"ed and con"rmed that tremor was reduced during
rhythmic cueing. Considering motor control as an optim-
ization problem of spatiotemporal precision (Harris and
Wolpert, 1999), trajectory smoothing during RAS may be
explained as a consequence of adding temporal stability
and precision to the movement through the entrainment
of the motor response to the rhythmic timekeeper.
The RAVS method, derived as a mathematical tool to
quantify tremor severity, showed good results in comput-
ing tremor strength and changes in tremor as well as
being sensitive enough to analyze degrees of tremor se-
verity from very mild to very pronounced. Quanti"cation
of tremor change is an essential and new feature of this
method, since no measure of tremor reduction has been
found in the literature. Furthermore, unlike acceler-
ation-based analyses, RAVS analysis uses position data
itself rather than data derived from position, which
allows analysis of duration and amplitude of tremor, and
of correlations of tremor activity with movement land-
marks of functional signi"cance.
From a purely theoretical point of view, it is note-
worthy that a direct mathematical correspondence exists
between the amplitude of pure sinusoidal tremor and its
RAVS-measure. For a tremor of frequency f Hz and
amplitude A, with *t as the time interval between RAVS
data points, it can be shown, by relating both tremor
amplitude and the RAVS measure of tremor to the area
1322 G.P. Kenyon, M.H. Thaut / Journal of Biomechanics 33 (2000) 1319}1323
contained by the tremor curve, that the RAVS measure is
proportional to the tremor amplitude:
M"
2A
Npf*t
. (9)
This result suggests that changes in tremor *¹ computed
by the RAVS method should correspond closely with
changes in tremor amplitude, assuming constant dura-
tion and location of tremor (i.e., weight-bearing phase)
over trials.
The RAVS method may be particularly useful when
analyzing pathological gait, because comparisons are
made to each subject as their own `normalizeda reference
instead of comparing pathological gait patterns to aver-
aged gait norms established with other study groups.
Although the patient group of this study had tremor
con"ned to the stance phase of the gait cycle, the RAVS
method was also tested against simulation data sets with
variable tremor duration over the entire gait cycle, with
similar results as for the variable-amplitude simulations.
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A measure of kinematic limb instability modulation by rhythmic auditory stimulation

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A measure of kinematic limb instability modulation by rhythmic auditory stimulation

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