Vol. 86, Issue 3, 1025-1031, March 1999
The maximum shortening velocity of muscle should be scaled
with activation
John W.
Chow1 and
Warren G.
Darling2
1 Department of Kinesiology,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801; and
2 Department of Exercise Science,
University of Iowa, Iowa City, Iowa 52242
 |
ABSTRACT |
The purpose of this study was to determine
whether the maximum shortening velocity
(Vmax) in
Hill's mechanical model (A. V. Hill. Proc. R. Soc.
London Ser. B. 126: 136-195, 1938) should be
scaled with activation, measured as a fraction of the maximum isometric
force (Fmax). By using the
quick-release method, force-velocity (F-V) relationships of the wrist
flexors were gathered at five different activation levels
(20-100% of maximum at intervals of 20%) from four subjects. The
F-V data at different activation levels can be fitted remarkably well
with Hill's characteristic equation. In general, the shortening
velocity decreases with activation. With the assumption of nonlinear
relationships between Hill constants and activation level, a scaled
Vmax model was
developed. When the F-V curves for submaximal activation were forced to
converge at the
Vmax obtained
with maximum activation (constant
Vmax model), there were drastic changes in the shape of the curves. The differences in Vmax values
generated by the scaled and constant
Vmax models were
statistically significant. These results suggest that, when a Hill-type
model is used in musculoskeletal modeling, the
Vmax should be
scaled with activation.
force-velocity relationship; Hill model; wrist flexors
| |
INTRODUCTION |
A NUMBER OF MUSCLE MODELS have been proposed since the
1930s. The prominent models include Hill's mechanical model (10), the
cross-bridge model (11), the molecular model (9), and the
distribution-moment model (28). Among these models, Hill's mechanical
model
|
(1)
|
where
F is muscle force, Fmax is maximum
isometric force, v is muscle
shortening velocity, and a and
b are Hill constants, has stood the
test of time and still serves as the most convenient mathematical
description of the force-velocity (F-V) characteristic of muscle
contraction. One of the problems that has to be addressed when the
Hill-type muscle model (10) is used in muscle modeling is how to modify
the F-V relationship at submaximal activation (26, 30). The major issue
is the maximum velocity of shortening (Vmax) at
different activation levels (ALs). Some investigators assume that the
Vmax is constant
(2, 15, 22), whereas others assume that it decreases with activation
(4, 27). In other words, the former group assumes that the constants in
Hill's characteristic equation (Hill constants
a and
b) are constants regardless of the
activation, and the latter group considers the Hill constants functions
of the activation.
In a study of the F-V relationship in cat soleus muscle at different
stimulus rates, Joyce and Rack (14) found that the Vmax decreased
with the stimulus rate. Similar results were also reported by Mashima
et al. (18), who controlled the activation [measured by the
contractile force before the quick release (QR)] of the frog
semitendinosus muscle by varying the potassium concentration of Ringer
solution and the frequency of the stimulation. A review of the
literature indicates that only one attempt has been made to study the
F-V-activation (F-V-A) relationship by using human subjects. Zahalak et
al. (29) attempted to relate the surface electromyographic (EMG)
measurements to the F-V relationship in the forearm flexors. Subjects
in this study were trained to move their forearms at different constant
angular velocities before the experimental trials were conducted. A
decrease in Vmax
at submaximal activation was reported. However, these results represent values obtained by control over velocity rather than responses of the
muscle to different loads at different ALs. In view of the limited data
on the F-V-A relationship in human muscle under voluntary contraction,
the purpose of this study was to determine whether the
Vmax in Hill's
mechanical model should be scaled with activation by using a technique
different from the one used by Zahalak et al.
| |
METHODS |
Three male and two female paid volunteers with no history of persistent
joint disorders or musculoskeletal trauma served as subjects (age
18-25 yr) in this study. Informed consent was obtained from
subjects before their participation. (Only the data of four subjects
were considered adequate and used in subsequent analyses. See
explanation in Data reduction.)
Experimental setup.
During the trials, the subject sat behind a table in an upright
posture, and his or her left forearm was constrained by a plastic
polymer cast (Fig. 1). The left upper arm
was inclined at an ~70° angle. Two openings in the cast permitted
placement of EMG electrodes over the medial and lateral surfaces of the forearm. The center bore of a motor pulley was fitted to the shaft of a
potentiometer (model 534, Spectrol Electronics, City of Industry, CA,
nominal resistance 5 k
± 5%, nonlinearity <0.25%) located underneath the tabletop. Potentiometer measurements were highly reliable in repeated tests, and the resolution of angle measurement was
0.1°. The distance between the longitudinal axis of the handle (length = 10 cm, diameter = 2.5 cm) and the axis of rotation of the
motor pulley (AOR) was adjustable, ranging from 4.5 to 7.0 cm. The
shortest distance between the steel cable and the AOR was 9.5 cm. The
back support and the hold-down pulley were installed to prevent lateral
displacement of the wrist and upward tilt of the lateral side of the
motor pulley during the course of wrist flexion, respectively. The part
of the back support that made contact with the subject's forearm was 3 cm behind the AOR. The steel cable (diameter = 2 mm) had one end fixed
to the medial side of the motor pulley and the other end connected to a
load cell (model AWU250, Genesco Technology, Simi Valley, CA,
nonlinearity <0.05% full scale, hysteresis <0.03% full scale,
maximum capacity = 1,112 N). The load cell was in series with a weight
plate holder, located directly above an electromagnet fixed to the
floor. When the holder was held down by the active electromagnet and
the cable was taut (i.e., before the QR), both the center of the handle and the AOR were aligned with the longitudinal axis of the cast in an
overhead view. In other words, the wrist was at a neutral position
during the isometric contractions before the QR.
Although it was not the purpose of this study to relate the EMG to the
F-V relationship, muscular activities of the wrist flexors and
extensors were monitored so that the electrical activities during the
course of each trial could be examined. Two pairs of bipolar
silver-silver chloride surface electrodes (diameter = 8 mm,
center-to-center distance = 20 mm) were placed over the bellies of the
flexor (one-third of the distance from the medial humeral epicondyle on
a line joining the medial humeral epicondyle and the styloid process of
the radius) and extensor (one-third of the distance from the lateral
humeral epicondyle on a line joining the lateral humeral epicondyle and
the styloid process of the ulna) carpi radialis muscles approximately
parallel to the direction of the muscle actions. A reference electrode
was placed over the medial humeral epicondyle. EMG signals were
preamplified (model 301, Biocommunication Electronics, Madison, WI,
input impedance 1010 , common
mode rejection ratio = 140 db) before further amplification.
The tension of the steel cable and the angular position of the handle
were measured by the load cell and potentiometer, respectively. The
analog signals were filtered (20- to 1,000-Hz band pass for EMG and
1,000-Hz low pass for force and angle) and amplified by using a
general-purpose amplifier (model 205, Biocommunication Electronics)
before being digitized (12-bit resolution) at a sampling rate of 500 Hz
by using the Watscope data-collection system (Northern Digital,
Waterloo, ON).
The force measured by the load cell was displayed to the subject by
using a storage oscilloscope (model 5111, Tektronix, Beaverton, OR)
stationed on the tabletop next to the apparatus. The signal from the
load cell was branched off to the oscilloscope and displayed as a
horizontal line on the cathode ray tube screen. Considering a free body
diagram of the hand and summing the moments about the transverse axis
of the wrist joint, the total moments due to forces of the wrist
flexors must be equal to the moment due to the contact force acting
through the handle. Assuming the wrist flexors act as a single
equivalent muscle (3, 24), we obtain
|
(2)
|
where
FR is the force measured by the
load cell, dR is
the distance between the line of action of the steel cable and the AOR,
Fmus is the muscle force, and
dm is the moment
arm of the wrist flexors. Because
dR and
dm at a given
joint angle were constants, the position of the force line (load cell
force FR) was linearly related
to the muscle force [i.e.,
FR = Fmus(dm/dR)].
When there was no load attached to the load cell, the "0% force
line" was positioned to the top of the screen. It should be pointed
out that it is impossible to mimic a "zero-load" condition
because, even if the load cell and weight plate holder were removed
from the setup, the resistance due to the masses of the pulley, handle, and hand cannot be avoided. Because the inertial resistance at the
instant of release was the same in all trials and the absolute muscle
shortening velocity values were not the main focus of this study, the
inertial resistance should not affect the main results obtained in this study.
Data collection.
Before the experimental trials, the subject was asked to perform 3 sets
of 10 repetitions of wrist flexion against a load of 55.7 N as warm-up.
To align the wrist joint with the AOR, the positions of the forearm,
hand, and handle were adjusted such that the tip of the styloid process
of the ulna (1) was directly above the AOR. The data-collection session
started with three trials of maximum isometric contractions against a
very large resistance provided by the active electromagnet. In each
trial, the operator triggered the data collection once the maximum
deflection of the force line was attained. The average maximum
deflection (100% force line) was used to construct a template. The
template was a small card (2 × 8 cm) fixed on the left-hand side
of the cathode ray tube screen. Along the right edge of the card, the space between the 0 and 100% force lines, as it appeared on the screen, was divided into five equal spaces, and the divisions became
force lines at 20% intervals.
Fmax was computed as the average
of the maximum forces recorded in the first three trials.
The next three trials were maximum isometric extension contractions.
The operator applied a force to the back the subject's hand, and the
subject was encouraged to extend the hand with maximum effort. The
purpose of these three trials was to obtain the maximum EMG level of
the wrist extensors.
QR trials followed isometric trials. In each trial, the load (weight
plates and holder) was initially held down by the active electromagnet,
and the subject was asked to apply force on the handle and maintain the
effort at a particular force level. In other words, the subject
contracted the wrist flexors such that the force line remained at a
specific position indicated on the template. Once the force line stayed
at a target position, the operator triggered the data collection and
followed with the release of the load by switching off the electrical
supply to the electromagnet. In each trial, the digital data were
collected for 1.5 s and stored on the hard disk.
For the purpose of this study, an AL was defined as a fraction of the
Fmax (21). Five ALs, ranging from
20 to 100% at intervals of 20%, were employed in this study. Except
for 20% AL, six different loads were used for each AL. For all ALs,
the lightest load used was the weight of the weight plate holder plus
load cell, which was 11.15 N. In this study, the smallest increment in
the load was 5.6 N, which was not small enough to provide six different loads for 20% AL trials. Trials of the same AL were conducted as a
group from heavy to light loads. Trials progressed in the order of
100-20-80-40-60%. The order was designed to
minimize the effect of fatigue. The 20 and 40% AL trials served as
recovery periods after periods of intense muscle contractions (i.e.,
100 and 80% AL trials). Rest intervals between consecutive trials ranged from 1 to 5 min, depended on the AL. The data-collection session
ended with a maximum isometric flexion trial to test for the presence
of fatigue.
Data reduction.
Before the F-V data were computed, the forces attained immediately
before QR were examined first, and it was found that these deviated
from the expected forces by 7.17 ± 8.74 (SD) % of
Fmax on average in one subject.
The average deviations were 1.97 ± 2.42, 2.06 ± 2.86, 2.29 ± 2.70, and 3.13 ± 3.63% for the remaining four subjects,
respectively. Only data of these four subjects were used in subsequent analyses.
To compute the angular velocity, the time instant when the handle was
5° away from the release position (5° instant) was identified. The angular positions of the handle at the instants 4 ms before and
after the 5° instant were then identified. The angular velocity was
obtained by dividing the change in angular position by 8 ms. Angular
velocities of the handle at the instants of 2.5, 5, and 10° away
from the release location were also computed in one subject, and it was
found that the normalized F-V curves for these three positions were
qualitatively similar. The normalized F-V data obtained by using the
these three handle locations could be fitted well with the Hill
equation, and each set of F-V curves exhibited similar trends (e.g.,
decrease in Vmax
with decreasing activation).
It was assumed that the shortening velocity of the wrist flexors was
linearly related to the angular velocity of the hand. This assumption
is considered acceptable because an arc segment of a circle can be
approximated as a straight line if the angle defining the segment is
small. Because the highest angular velocity recorded was 491.2°/s,
the largest angular distance used for the calculation of angular
velocity in this study was 3.9° (i.e., 491.2°/s × 0.008 s). The tendon compliance was not considered because wrist flexors are
relatively small muscles and cannot generate sufficient force to
stretch the tendons appreciably. Lehman and Calhoun (16) estimated the
tendon stiffness of the flexor carpi radialis to be ~8 × 104 
105 N/m.
Hill's characteristic equation (Eq. 1) was fitted to each set of F-V data by using the
least squares technique (minimizing the sum of the squared force
residuals; see APPENDIX A). By
setting F in Eq. 1 equal to 0, the
Vmax was obtained
by the following equation
|
(3)
|
The
force and velocity data were then normalized as fractions of the
Fmax and
Vmax for the
100% AL, respectively. The normalized data were pooled, and Hill's
hyperbolic curves were fitted to the data of the same AL. The
coefficient of determination
(r2) for each
fitted curve was also computed (see APPENDIX
A).
For the QR trials, the average time elapsed from triggering the data
collection to the moment of releasing the load (average delay time) was
found to be 330 ± 155 ms. The EMG level was computed as the average
rectified integrated value during the delay, i.e., by dividing the area
under the rectified EMG curve by the respective delay time. For each
subject, the EMG values for the wrist flexors and extensors obtained
for each trial were expressed as percentages of the corresponding
average EMG values in maximum activation trials.
| |
RESULTS AND DISCUSSION |
In the last trials of the data-collection sessions, the four subjects
could produce, on average, 96.4% of their own
Fmax (SD = ±1.95%). Thus
fatigue was minimal. It was observed in most low load-high activation
QR trials that, very soon after the load was released (e.g., 46 ms in
Fig. 2), the activity of the wrist flexors
diminished for a short period of time (e.g., 54 ms in Fig. 2) and at
the same time there was a burst of activity in the extensors. The
diminution in muscle activity in the flexors and the burst of activity
in the extensors are typical examples of unloading and stretch
reflexes, respectively (17). The reflex responses produced no effects
on the results of this study because they occurred after the instant
when the angular velocity was measured. Because the angular velocity of
the hand was measured before voluntary or reflex intervention to the
unloading, we assumed that there was a high degree of repeatability of
responses even if there was only one trial for each load-activation
condition. Such repeatability was observed in preliminary tests of the
apparatus.
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Fig. 2.
Plot of raw data of a quick-release trial, maximum activation and
11.15-N load. Top to
bottom: load cell, potentiometer, and
EMG activity of flexors and extensors, respectively.
|
|
In QR experiments using isolated muscle preparation (8, 12), a typical
displacement (length change)-time curve rose sharply immediately after
the release of the load and was followed by a steady increase. The
initial slope, indicating rapid shortening, was attributable to the
series elastic component of the muscle. However, this was not observed
in the present study.
The EMG of the wrist extensors at different ALs indicated that the
degree of cocontraction increased with the activation of the agonist.
On average, the EMG of the wrist extensors at 20 and 100% ALs of the
flexors were ~1 and 7%, respectively. It was assumed that the
effects of the wrist extensors were negligible because of the low EMG
and because the maximal extension torque is usually much smaller than
the maximal flexion torque for the wrist joint.
The F-V curves at different ALs for subject
KR (Table 1) are considered
atypical compared with the results of the other subjects because the
maximal velocities were similar for the 40, 80, and 100% ALs. The
cause of such atypical results is not known. However, a more
homogeneous muscle fiber population in wrist flexors of subject KR is a possible explanation.
It may also be due to the single-trial protocol used in this study, and
the data collected from this subject were not reliable. The very large
vmax value (2,912°/s) predicted from regression analysis for the 20% AL was due to an outlier, indicated by an arrow in Fig.
3. (Note: To avoid confusion,
fmax and
vmax, expressed
as a fraction of Fmax and
Vmax,
respectively, denote the isometric force and estimated maximum
shortening velocity for submaximal ALs.)
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Fig. 3.
Normalized force-velocity data from all subjects
(n = 4). Curves were obtained by
fitting Hill's hyperbolic curves to data of same activation level by
using least squares technique (best-fit curves in Table 2). Each symbol
on abscissa represents 4 data points, 1 from each subject.
|
|
As indicated by the coefficient of determination
(r2) values,
the F-V data at different ALs can be fitted remarkably well with Hill's characteristic equation (Table 1). In general, the velocity of
shortening decreases with activation for a given load. For example,
when a horizontal line is drawn for a given normalized force in Fig. 3,
the shortening velocity decreased with the activation. These findings
are in agreement with results obtained in isolated muscle (21) but
contrary to results obtained in individual muscle fibers in which
maximum unloaded velocity is independent of AL (5, 23). The property of
constant Vmax is
not found in the whole muscle because of the different fiber and motor
unit types in a whole muscle and motor unit recruitment order. Because
type S motor units are slow contracting because they contain type I fibers and are recruited at low force output according to the size
principle (7), the wrist flexors, which have an even distribution of
fast- and slow-twitch fibers (13), are expected to contract more slowly
as the activation decreases.
Although individual differences are apparent in most of the
characteristic parameters (Table 1), it is noteworthy that, for all
subjects, almost all of the
a/Fmax
and
a/fmax
values at 100 and 80% ALs are in the neighborhood of 0.5 and 0.6. These values are greater than those obtained from isolated muscles (25)
and muscle fibers (6). (Note: the
a/Fmax
value, a ratio that has no units, indicates the curvature of the
hyperbolic curve. The smaller the
a/Fmax
value, the greater the curvature of the curve.) The only
exception is that Phillips and Petrofsky (21) reported a/Fmax
value values of 0.5 or above in three different cat muscles, lateral
and medial gastrocnemius and soleus. Interestingly, in another study
(19) the same authors reported values of ~0.25 in cat medial gastrocnemius.
When a curve is fitted to the normalized F-V data of each AL (Fig. 3),
the data point that was responsible for the huge 20% AL
vmax for
subject KR (indicated by an arrow in
Fig. 3) accounted for more than one-half of the sum of the squared
force residuals (SSR; see APPENDIX
A) of the 20% AL curve. It was decided to exclude
this data point from the determination of best-fit curves and
mathematical models. Despite the fluctuations in the 60 and 80% ALs,
both Hill constants of the best-fit curves (Table 2) show a trend of decrease with decreasing
activation. Both linear and nonlinear relationships between Hill
constants and activation have been reported in the literature (20, 21).
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Table 2.
Characteristics parameters of the force-velocity curves at different
activation levels obtained by best-fit, scaled maximum-velocity model,
and when constant maximum velocity is assumed
|
|
With the assumption of nonlinear relationships (2nd-order polynomials)
between Hill constants and AL, the following equations were obtained if
the shape of the normalized curve at 100% AL was retained (i.e., both
curves relating the Hill constants to AL will pass through the data
point: a = b = 59%, AL = 100%; curves in Fig.
4 and APPENDIX
B)
![<IT>a</IT> = (59 − <IT>y</IT><SUB>1</SUB>)[(<IT>A</IT> − <IT>x</IT><SUB>1</SUB>)/(100 − <IT>x</IT><SUB>1</SUB>)]<SUP>2</SUP> + <IT>y</IT><SUB>1</SUB>](/content/vol86/issue3/fulltext/1025/img004.gif)
|
(4)
|
![<IT>b</IT> = (59 − <IT>y</IT><SUB>2</SUB>)[(<IT>A</IT> − <IT>x</IT><SUB>2</SUB>)/(100 − <IT>x</IT><SUB>2</SUB>)]<SUP>2</SUP> + <IT>y</IT><SUB>2</SUB>](/content/vol86/issue3/fulltext/1025/img005.gif)
|
(5)
|
where
x1,
y1,
x2, and
y2 are constants
and A is the AL expressed as a
percentage. The scaled
Vmax model is
obtained by substituting Eqs. 4 and 5 into Eq. 1. By using all normalized data, the constants were
estimated by employing "carpet search" and least squares criteria. The values
x1,
y1,
x2, and
y2 were found to
be 94.2, 59.3, 70.3, and 72.0, respectively. The F-V curves at
different ALs generated by the scaled
Vmax model are
presented as solid lines in Fig. 5.
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Fig. 4.
Hill constants of best-fit curves at different activation levels
(symbols). Curves are Hill constant-activation level relations
generated by scaled maximum-velocity model.
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Fig. 5.
Comparison of force-velocity curves obtained by scaled (solid lines)
and constant (dashed lines) maximum-velocity models.
|
|
By setting the Hill constant a = Fmaxb or
fmaxb (see
Eq. 3) in the parameter-searching
process described in APPENDIX A, the
F-V curves were forced to converge at the
Vmax (constant
Vmax model). The
F-V curves at different ALs generated by the constant
Vmax model are
presented as dashed lines in Fig. 5. The
r2 values for the
constant Vmax
curves are about the same as the corresponding values for the curves
obtained by using the scaled Vmax model and
slightly smaller than the corresponding values for the best-fit curves
(Table 2). The r2
values of these curves indicate that the SSR is not very sensitive to
the changes in a and
b. However, by using
vmax as the
dependent variable, a 5 (ALs) × 3 (models) ANOVA revealed
significant differences in
vmax among the
three models, best fit, scaled
Vmax, and
constant Vmax
[F(degrees of freedom = 2, 8) = 5.05, P = 0.04]. A Duncan multiple-range test (P < 0.05)
showed that vmax
values from the constant
Vmax model were
significantly different from the corresponding values from the other
two models, and there was no significant difference in
vmax between the
best-fit and scaled
Vmax models.
The vmax values
of individual subjects and the comparison among models clearly
suggested that the
vmax was not
constant at different ALs. Although the limited samples available in
this study do not provide much insight into the fine details of the F-V-A relationship, these results extend the findings of Zahalak et al.
(29) concerning voluntary contractions of different velocities to
contractions in which the subject voluntarily controlled only the AL.
With reference to musculoskeletal modeling by using Hill's characteristic equation, these results suggest that, unless the movement being modeled requires low velocity of shortening of the
muscles, the vmax
should be scaled with activation. More data, especially from different
muscles, are needed to further address the issues raised in this study.
| |
APPENDIX A |
Curve Fitting by Using Hill's Characteristic Equation
With reference to Eq. 1, the SSR is
given by
|
(6)
|
where
n is the number of data points in a
set of F-V data. Curve fitting can be considered as searching for the
combination of Fmax,
a, and
b that will give a minimal SSR.
To start a search, the input for
Fmax,
a, and
b were 100-105% of
Fmax at intervals of 1%,
50-200% at intervals of 4%, and 100-400% at intervals of
4%, respectively. These initial inputs for
a and
b were based on the results of the
searches performed on several sets of data. In the subsequent rounds of
search, the inputs for the Fmax
remained the same. The inputs for a
and b were plus and minus 50 of the
respective values obtained in the previous round at intervals of 1%. A
search would be terminated when the combinations obtained in two
consecutive rounds of search were identical. In general, three to five
rounds of search were required before the optimal combination was found.
Let 
be the sample mean of the forces
|
(7)
|
and
SSY be the sum of the squares of deviation associated with
|
(8)
|
the
coefficient of determination
(r2) was
obtained by
|
(9)
|
| |
APPENDIX B |
Determination of the Relationship Between a Hill Constant and the
Activation Level
Mathematically, a parabola (2nd-order polynomial) in a two-dimensional
rectangular coordinate system
(x-y
plane) can be expressed as
|
(10)
|
where
(p,
q) is the coordinate of the
parabola's vertex and f is a constant
related to the focus of the parabola. With reference to Fig. 4, a
parabola passing through the data point for 100% AL (100, 59) is
|
(11)
|
Rearranging
the terms, we get
|
(12)
|
Substituting
Eq. 12 into Eq. 10, we have
![<IT>y</IT> = (59 − <IT>q</IT>)[(<IT>x</IT> − <IT>p</IT>)/(100 − <IT>p</IT>)]<SUP>2</SUP> + <IT>q</IT>](/content/vol86/issue3/fulltext/1025/img014.gif)
|
(13)
|
Therefore, a second-order polynomial passing through (100, 59)
will have the form of Eq. 13.
| |
ACKNOWLEDGEMENTS |
The authors thank Tony Wolff for technical assistance.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: J. W. Chow,
Dept. of Kinesiology, Univ. of Illinois at Urbana-Champaign, Urbana, IL
61801 (E-mail: j-chow1{at}uiuc.edu).
Received 28 April 1997; accepted in final form 12 November 1998.
| |
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