Vol. 88, Issue 3, 917-925, March 2000
Development of a mathematical model that predicts optimal
muscle activation patterns by using brief trains
Jun
Ding1,
Anthony S.
Wexler2, and
Stuart A.
Binder-Macleod3
1 Interdisciplinary Graduate Program in
Biomechanics and Movement Science,
2 Department of Mechanical Engineering, and
3 Department of Physical Therapy, University
of Delaware, Newark, Delaware 19716
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ABSTRACT |
Because muscles must be repetitively activated during
functional electrical stimulation, it is desirable to identify the
stimulation pattern that produces the most force. Previous experimental
work has shown that the optimal pattern contains an initial
high-frequency burst of pulses (i.e., an initial doublet or triplet)
followed by a low, constant-frequency portion. Pattern optimization is particularly challenging, because a muscle's contractile
characteristics and, therefore, the optimal pattern change under
different physiological conditions and are different for each person.
This work describes the continued development and testing of a
mathematical model that predicts isometric forces from fresh and
fatigued muscles in response to brief trains of electrical pulses. By
use of this model and an optimization algorithm, stimulation patterns
that produced maximum forces from each subject were identified.
functional electrical stimulation; fatigue; human quadriceps
femoris muscle; doublets; variable-frequency trains
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INTRODUCTION |
ELECTRICAL STIMULATION can be used to evoke skeletal
muscle contractions that enable patients with central nervous system damage to perform functional movements such as ambulation (2, 19, 21).
The practical use of functional electrical stimulation (FES) is limited
by muscle fatigue, which develops far more rapidly during this
exogenous activation of skeletal muscle than during activation by the
central nervous system (25). The frequency of the stimulation train
affects the forces produced (1) and the rate of muscle fatigue (3).
Traditionally, during FES, skeletal muscles are activated with
constant-frequency trains (CFTs), where the pulses within each train
are separated by regular interpulse intervals (IPIs; Fig.
1). More recently, however, it has also been shown that
changing the pattern of the pulses within a brief train can markedly
affect force production and fatigue (4, 7). Therefore, it is important
to identify the stimulation pattern that maximizes force.
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Fig. 1.
Schematic representation of 3 stimulation train types. Bottom
train (CFT30), constant-frequency train with all interpulse
intervals (IPIs) equal to 30 ms; middle train (VFT30),
variable-frequency train with an initial doublet of 5 ms and remaining
pulses equally spaced by 30 ms; top train (DFT30), DFT with
5-ms doublets separated by intervals of 30 ms. Each train's name is
based on duration of its longer IPI (e.g., VFT50 contains an initial
doublet of 5 ms followed by pulses equally spaced by 50 ms).
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One stimulation pattern that has been shown to produce more force from
a fatigued muscle than any CFT is a variable-frequency train (VFT) that
begins with a doublet (i.e., 2 pulses separated by a brief IPI; Fig. 1)
(6). However, the specific pattern of activation that maximizes the
force varies from person to person (20) and, even for the same
individual, depends on the physiological conditions of the muscle, such
as the level of fatigue (6) or length of the muscle (24). Numerous
tests, therefore, would be required to find the optimal patterns for
each patient experimentally. One way to assist the search for the
optimal pattern is to use mathematical models that can predict forces
under a range of physiological conditions. Such models could markedly
reduce the number of tests needed to identify the optimal pattern for
each individual.
Although many models have been developed to explore different aspects
of muscle contraction (9-11, 17, 28, 29, 33), most are complex (8,
10, 28) and few have the ability to predict force responses to a wide
range of stimulation patterns (13, 32). Our laboratory has recently
developed a two-step model that predicts force responses to a wide
range of stimulation patterns under various physiological conditions
(13). This model is governed by two differential equations with four
free parameters. It successfully predicted the forces in response to
CFTs and VFTs with various IPIs for human quadriceps femoris muscles.
However, a major limitation of this model is that separate sets of
parameter values were needed to predict the forces produced in response to CFTs and VFTs when the muscles were fatigued. This resulted in a
nonunique representation of each muscle by the model. The model,
therefore, cannot be used to identify the pattern that produces the
greatest forces.
The present study first outlines the modification of our two-step model
to characterize each muscle with one set of parameter values regardless
of stimulation pattern. Next, this new model is used to identify the
activation pattern that produces the greatest forces. This new model
suggested an activation pattern that we have not previously tested as
the optimal for fatigued muscles. Finally, experimental results are
presented that verify the model's predictions.
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METHODS |
Development of R Model
Modification of the two-step model.
Previously, we developed a two-step mathematical model for predicting
isometric muscle forces that contained two differential equations and
four free parameters (13). Briefly, the two differential equations are
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(1)
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and
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(2)
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representing
the chemical and physical events that result in force development.
Equation 1 characterizes the rate-limiting step to the
formation of the Ca2+-troponin complex
(CN, unitless); i identifies each
stimulation pulse within the train (e.g., 1st, 2nd, or 3rd pulse);
c (in ms) is the time constant controlling the rise and
decay of CN; t (in ms) is the time
since the first pulse of the train; ti
(in ms) is the time of the ith stimulation pulse; n
is the total number of stimulation pulses before t (its maximum
value is 6 in this study). F (in N) in Eq. 2 represents the
whole muscle force, the development of which is driven by
CN/(1 + CN), the force producing cross bridges, and is scaled by parameter A
(in N/ms). The second term in Eq. 2 accounts for the force
decay over two time constants, 1 (in ms) and
2 (in ms); 1 represents the time
constant in the absence of actin-myosin cross bridges, and
2 represents the extra friction between actin and myosin resulting from the presence of actin-myosin cross bridges. In Eqs.
1 and 2, CN and F are the two
dependent variables and c, A, 1,
and 2 are free parameters.
As one of the simplest mathematical representations of muscle force,
our previous two-step model overlooked many physiological details, such
as the nonlinear summation of Ca2+ transients in single
muscle fibers stimulated with two closely spaced pulses (15). Because
VFTs contain doublets (closely spaced pulses), this could explain why
the two-step model needed separate sets of parameter values to predict
CFTs and VFTs. Similarly, we noted that our previous model did not
accurately predict responses to high-frequency CFTs, which contain
closely spaced pulses.
Duchateau and Hainaut (16) investigated the force summation from human
adductor pollicis muscles triggered by paired stimuli at different IPIs
ranging from 5 to 200 ms. They showed that the forces produced by the
two-pulse trains were greater than the sum of two individual twitches
and that this force enhancement from the second pulse was highest when
the IPI was 5 ms and declined exponentially with increases of the IPI.
The enhanced force of the paired stimuli was suggested to be due to the
intensified release of Ca2+ by the second pulse (15). In
our new model, a factor Ri is introduced to
account for this nonlinear summation. We, therefore, modified the model
by adding a factor modifying Eq. 1
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(3)
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where
Ri is a scaling term that accounts for the
differences in the degree of activation by each pulse relative to the
first pulse of the train. The analytic solution of Eq. 3 is
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(4)
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where
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(5)
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for
i > 1 and R1 = 1. The magnitude of the
enhancement is characterized by R0, and its
duration is characterized by c. This new model, termed
the R model, has two differential equations (Eqs. 2 and 3) and is governed by five free, constant parameters R0, c, A, 1,
and 2.
Parameter identification.
Data from a previous study of the human quadriceps femoris muscles (13)
were used to develop and test our new R model. Pilot studies
showed that a fixed value of 20 ms for c was sufficient for human quadriceps muscles under different physiological conditions. The remaining four free parameters must be identified for each individual under each physiological condition. We believe that trains
or train combinations that exhibited tetanic forces, as well as a clear
rise and fall in force for each pulse, would give the model the most
information. Ten trains [CFTs with IPIs of 20 (CFT20), 30 (CFT30), 40 (CFT40), 70 (CFT70), and 100 ms (CFT100) and VFTs with IPIs
of 10 (VFT10), 70 (VFT70), 80 (VFT80), 90 (VFT90), and 100 ms
(VFT100)] were tested to determine which train or train combinations would be best to parameterize the model. We found that two
VFTs, one with an initial 5-ms IPI followed by constant IPIs of 10 ms
(VFT10) and the other with an initial 5-ms IPI followed by constant
IPIs of 100 ms (VFT100), were the best train combinations to
parameterize the model for nonfatigued muscles among the trains or
train combinations tested. The combination of CFT100 and VFT100 was
found to be the best for fatigued muscles. In our previous work (13,
32), parameter identification was performed using SAAM II, a
compartment-modeling software package developed by the University of
Washington. In this work we developed our own technique. First,
1 was identified as described previously (13). Briefly,
Eq. 2 became dF/dt = 
F/1
by setting CN = 0. By taking the
force decay at the end of the force-time response and performing a
linear regression fitting of lnF vs. t, 1 was
obtained from the slope of the fit. The remaining three parameters were
identified by employing the following objective function
![<IT>G</IT><SUB>1</SUB> (<IT>A</IT>, <IT>R</IT><SUB>0</SUB>, &tgr;<SUB>2</SUB>) = <LIM><OP>∑</OP><LL><IT>i</IT></LL></LIM> [F<SUP>pred</SUP> (<IT>t</IT><SUB><IT>i</IT></SUB>; <IT>A</IT>, <IT>R</IT><SUB>0</SUB>, &tgr;<SUB>2</SUB>) − F<SUP>obs</SUP> (<IT>t</IT><SUB><IT>i</IT></SUB>)]<SUP>2</SUP>](/content/vol88/issue3/fulltext/917/img006.gif)
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(6)
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where
F pred, the forces predicted by Eqs. 2 and 3 at time ti, are a function of the
four parameters A, R0, 1,
and 2. Fobs are the observed forces at
time ti. G1 was
minimized using CFSQP (23), which numerically identifies the optimum
values for the three parameters (A, R0,
2) by employing feasibility set techniques coupled to
sequential quadratic programming.
We next tested the model's ability to predict the six-pulse
stimulation pattern that produced the greatest force-time integral. CFSQP was also used to maximize the force-time integral by varying the
IPIs separating each pulse within the six-pulse train. The force-time
integral was approximated by
![<IT>G</IT><SUB>2</SUB> (<OVL>IPI</OVL>) = <LIM><OP>∑</OP><LL><IT>i</IT></LL></LIM> <FR><NU>[F<SUP>pred</SUP> (<IT>t</IT><SUB><IT>i</IT></SUB>; <OVL>IPI</OVL>) + F<SUP>pred</SUP> (<IT>t</IT><SUB><IT>i</IT>+1</SUB>; <OVL>IPI</OVL>)]</NU><DE>2</DE></FR> (<IT>t</IT><SUB><IT>i</IT>+1</SUB> − <IT>t</IT><SUB><IT>i</IT></SUB>)](/content/vol88/issue3/fulltext/917/img007.gif)
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(7)
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is the set of five IPIs in the model stimulation and
(ti + 1 ti) is the time between the ith and
the (i + 1)th pulse. The minimum duration allowed for each IPI
was 5 ms, because pilot testing showed that reducing the IPI to <5 ms
had no effect on force or produced a decline in force. The present
model, however, is not capable of demonstrating this decline in force
at very short IPIs. In addition, pilot testing of the model showed that
for fatigued muscles a stimulation pattern that contained doublets
throughout (i.e., not only in the beginning of the train) should
produce the greatest forces for some muscles. This pattern, denoted as
a DFT (Fig. 1), has rarely been tested previously (20, 26). The
following experiments, therefore, were conducted to verify the
predictions of the model by use of a wide range of six-pulse
stimulation patterns.
Testing of the R Model
Twelve subjects were recruited for this part of study. Each subject
signed an informed consent form approved by the University of Delaware
Human Subject Review Board and participated in one testing session.
The experimental setup was similar to that described previously (5).
Briefly, subjects were seated on a force dynamometer with their hips
flexed to ~75°, and their knees were flexed to 90°. Two
stimulating electrode pads were placed over the quadriceps muscle. The
force transducer was positioned against the anterior aspect of the leg
proximal to the lateral malleolus. The stimulus intensity was set to
elicit a force equal to 20% of the subject's maximum voluntary
isometric contraction when stimulated with a six-pulse, 100 pulse/s
train. Force data from the force dynamometer's force transducer were
entered into a personal computer via an analog-to-digital converter
board and digitized at a rate of 200 Hz.
Stimulation protocols.
Isometric muscle performance was tested using CFTs, VFTs, and DFTs when
the muscles were fresh and fatigued (Fig. 1). All trains had six pulses
(5 IPIs), because studies have shown that most functional movement
employed a brief activation pattern (18). To cover a wide range of
frequencies, the nine CFTs had all IPIs equal to 10, 30, 50, 70, 90, 100, 110, 130, or 150 ms. These trains are referred to as CFT10, CFT30,
CFT50, CFT70, CFT90, CFT100, CFT110, CFT130, and CFT150, respectively.
Comparable VFTs had an initial IPI of 5 ms, and the remaining pulses
were equally spaced by 10, 30, 50, 70, 90, 100, 110, 130, or 150 ms.
These trains are referred to as VFT10, VFT30, VFT50, VFT70, VFT90,
VFT100, VFT110, VFT130, and VFT150, respectively. Six DFTs had the
three 5-ms doublets separated by 30, 50, 70, 90, 110, and 130 ms (i.e., the 2nd and 4th IPIs were long). These trains are referred to as DFT30,
DFT50, DFT70, DFT90, DFT110, and DFT130, respectively.
The nonfatigue protocol consisted of the 24 testing trains presented in
random order, which were then repeated in reverse order, for a total of
48 trains. Trains were delivered once every 10 s to avoid muscle
fatigue. The forces produced in response to the second occurrence of
each train were compared with the forces produced in response to the
first occurrence to determine whether the muscles became fatigued
during the nonfatigue protocol. None of the subjects showed a
substantial (>5%) decline in the force of the second occurrence.
This nonfatigue testing lasted ~10 min. The subjects were then
allowed 5 min of rest before the fatigue test.
To produce fatigue, a 13-pulse CFT with IPIs of 25 ms (CFT25) was
delivered to the muscle once every 1 s for a total of 100 contractions.
Testing commenced the next second after the fatigue-producing stimulation. Trains continued to be delivered in 1-s intervals. The 24 testing trains were delivered with each train preceded by two
fatigue-producing stimulations (i.e., two 13-pulse CFT25). This
procedure was used to control for prior activation history of the
muscle and to maintain a consistent level of fatigue throughout the
fatigue testing (Fig. 2). Thus the fatigue
protocol consisted of the 100 fatigue-producing trains, 24 testing
trains, and 48 intervening fatigue-producing trains for a total of 172 trains and lasted ~3 min.
Data management.
For the nonfatigued data, the two occurrences of each of the 24 stimulation patterns were averaged to minimize the effects of previous
activation history on the muscle's responses to each train type.
TESTING THE MODEL'S ABILITY TO PREDICT FORCES.
The force responses to all CFTs, VFTs, and DFTs were used to test the
model. For each subject, the model was first parameterized, as
described above using G1 (Eq. 6) and CFSQP
employing the VFT10-VFT100 combination for the nonfatigued muscles and
CFT100-VFT100 combination for the fatigued muscles. The parameterized
model was then used to predict the force responses to all other
stimulation patterns. Two tests were used to examine the model's
ability in predicting the force responses to a wide range of
stimulation patterns. First, to compare the shape of the force-time
response to each stimulation pattern, a correlation coefficient was
calculated by comparing the measured and predicted forces at 5-ms
intervals. Any phase shift or disagreement in shape between the
predicted and measured data lowers the correlation coefficient. Second,
for each subject, the force-time integral was calculated for each train
tested. The measured force-time integrals were then plotted against the predicted force-time integrals for each condition (i.e., nonfatigued and fatigued). Linear regression trend lines, with intercepts of zero,
were used to compare force response across all the stimulation patterns
and determine how well the model predicts the force-time integrals. A
perfectly accurate model would have a slope and R2
of 1 (the identity line).
TESTING THE MODEL'S ABILITY TO PREDICT OPTIMAL PATTERNS.
Then, with the model parameterized for each subject under each
condition, the optimizer (Eq. 7) was used to predict the
activation pattern that produces the greatest force-time integral. The
upper limit of each IPI was set at 200 ms, and the lower limit was set at 5 ms. Because the program varies each IPI in a continuous manner and
can test every possible combination of IPIs while searching for the
optimal pattern, we first categorized the predicted optimal pattern to
allow comparisons of the predicted results to our experimental data.
Predicted optimal trains that had all five IPIs of similar value
(standard deviation <10% of the mean) were called CFTs. Trains that
had a short IPI (
10 ms) in the beginning of the train and the other
four IPIs of similar value were labeled VFTs. Trains that had three
short IPIs separated by two long IPIs (>10 ms) were labeled DFTs. All
short IPIs of the predicted optimal trains were equal to 5 ms except
one (7 ms), and all predicted optimal trains fell into one of these
three categories. Next, the long IPIs within each train (i.e., all 5 IPIs for CFT, the last 4 IPIs for VFT, and the 2 long IPIs for DFT)
were averaged, and this average was used to designate the optimal IPI.
The optimal IPI, along with the train type, was assigned as the optimal
train pattern. The model's ability in predicting the activation
pattern that produced the greatest force-time integral was examined by
comparing the train type and the IPI of the predicted optimal pattern
with those of the experimental optimals.
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RESULTS |
The model was tested with data collected from nonfatigued (n = 12) and fatigued (n = 12) human quadriceps femoris muscles. One
subject's fatigued data were not included into group analysis, because
her fatigued forces were too low, resulting in an unacceptably high
signal-to-noise ratio. Subjective evaluations of the force-time responses showed that the model was able to predict the force responses
of the muscle to a wide range of stimulation patterns in the
nonfatigued and fatigued states (Figs. 3
and 4). The correlation coefficients
between the measured and predicted force-time responses, averaged
across subjects, generally showed that the model more accurately
predicted the force profiles when the muscles were nonfatigued (median = 0.94, SD = 0.06) than when they were fatigued (median = 0.88, SD = 0.05; Table 1). In general, for
all three train types when the muscles were nonfatigued, the higher the average frequency (i.e., the shorter the IPI), the greater the correlation coefficient. When fatigued, CFTs and VFTs with 70- to
100-ms IPIs showed the highest correlation coefficients.
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Fig. 3.
Measured and predicted forces from a typical subject's nonfatigued
quadriceps muscle. A: model was first parameterized by fitting
to force responses to VFT10 and VFT100. exp, Experimental; mod,
modeled. B-J: experimental and predicted force responses
to other CFTs, VFTs, and DFTs.
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Fig. 4.
Measured and predicted forces from a subject's fatigued quadriceps
muscle (same subject presented in Fig. 3). A: model
was first parameterized by fitting to force responses to CFT100 and
VFT100. B-J: experimental and predicted force responses to
other CFTs, VFTs, and DFTs.
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For nonfatigued and fatigued conditions, the best-fit trend lines for
modeled force-time integrals vs. experimental data had slopes close to
1. The model accounted for ~87 and 75% of the variance of the
force-time integrals of the experimental data for nonfatigued and
fatigued muscles, respectively (Fig. 5).
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Fig. 5.
Comparisons of force-time integrals between measured and predicted
forces of 12 nonfatigued (A) and 11 fatigued (B) human
quadriceps muscles.
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The activation pattern that produced the greatest force-time integral
was predicted for each subject under each condition (i.e., nonfatigued
and fatigued) and compared with the experimental data (Table
2). For the experimental data, because of
the physiological variability in the force responses of a muscle tested
repeatedly with the same stimulation pattern, trains were considered to
be producing the maximum force-time integral from the muscle if they produced forces that were within 5% of the greatest force produced by
any of the trains. For 20 of 23 comparisons (i.e., data from 12 nonfatigued and 11 fatigued subjects), the model and the experimental measurements identified the same train type as producing the maximum force-time integral (Table 2). In addition, for 18 of the 20 matches
described above, the model predicted the optimal IPI within 10 ms of
the experimentally observed optimum (Fig.
6).
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Fig. 6.
Model's ability to predict optimal pattern for 1 typical subject under
nonfatigued (A) and fatigued (B) conditions.  ,  ,
and  , Experimental force-time integrals produced by CFTs, VFTs, and
DFTs, respectively, plotted against long-duration IPI within each
train. Arrows, optimal patterns predicted by model (i.e., CFT66 and
DFT130, respectively).
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Similar to our two-step model, fatigue produced a significant decrease
in the parameter value A (4.8 ± 1.5 and 1.1 ± 0.5 for nonfatigued and fatigued muscles, respectively) but significant increases in 1 (47.3 ± 7.6 and 89.2 ± 18.6 ms for
nonfatigued and fatigued muscles, respectively) and 2
(703.4 ± 383.6 and 1,564.5 ± 646.5 ms for nonfatigued and
fatigued muscles, respectively). The new parameter,
R0, showed a significant increase when muscles were
fatigued (0.9 ± 0.4 and 7.2 ± 3.7 for nonfatigued and
fatigued muscles, respectively).
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DISCUSSION |
This new R model predicted accurately the force response to a
wide range of stimulation patterns when muscles were fresh or fatigued.
Subjective and objective evaluations of the force response showed that
the predicted force responses closely matched the forces that were
measured (Figs. 3-5, Table 1). In addition, the R model
successfully identified the optimal pattern for each individual under
different physiological conditions (Table 2, Fig. 6).
The present model was refined from our previous two-step model (13).
Its structural simplicity and predictive ability remained intact, but
the effect of short-term activation history on the force production
(Ri in Eq. 3) was taken into
consideration. Previous study in our laboratory has shown that for
nonfatigued and highly potentiated human quadriceps femoris muscles the
force produced by the doublet in the VFT was slightly less than that produced by the first two pulses in the CFT, whereas when muscles were
fatigued the doublet produced more forces than the first two pulses in
the CFT (6). Thus values of R0 close to 1 for nonfatigued muscles and much greater than 1 for fatigued muscles were
expected. The differences in the values of R0
between the nonfatigued and fatigued conditions are also consistent
with the fact that, for most subjects, CFTs were optimal for the
nonfatigued condition and DFTs or VFTs were optimal for the fatigued
condition (Table 2). The major improvement over the two-step model is
that the R model could characterize each individual muscle with
only one set of parameter values regardless of stimulation pattern, allowing the model to predict the optimal patterns for each person under each condition. In addition, the present R model
predicted more accurately the responses to high-frequency CFTs than our previous model.
An interesting finding was the identification of the DFT as the optimal
pattern for the activation of the majority of subjects when muscles
were fatigued. As an example, for one subject's fatigued muscle, the
model and experimental data suggested that DFT130 produced the greatest
force-time integral among all trains tested (Fig. 6B). The
firing pattern of DFTs (i.e., repetitive doublets) has been observed in
motoneuron discharge patterns of human trapezius muscles during
voluntary contractions (22). In our study the minimum value allowed for
the doublets was 5 ms, because pilot testing showed that trains with
doublet values <5 ms produced equivalent or less force than trains
with doublet values of 5 ms. In addition, Karu and colleagues (20)
showed that a 5-ms interval produced the greatest force-time integral
for two-pulse trains. Maxwell and colleagues (26) examined the
isometric force outputs from human quadriceps muscles stimulated with
doublets varying from 2.5 to 20 ms. All DFTs tested produced greater
force-time integrals and peak forces than a CFT50, and the best DFTs
had a doublet of 5 ms.
In this study we also explored the optimal interdoublet interval for
DFTs that produced the greatest force-time integral. Our experimental
data showed that the DFT110, which had a train duration similar to the
CFT50 and VFT50 (235, 250, and 205 ms, respectively), was the most
common optimal activation pattern for the fatigued muscles (Table 2).
Interestingly, Kudina and Alexeeva (22) showed that repetitive doublets
fired most frequently at an interdoublet interval of 110 ms for human
trapezius muscles during voluntary contractions. In contrast, Karu and
colleagues (20) suggested 50-80 ms as the interdoublet interval to
produce the greatest force-time integral per pulse for human quadriceps muscles. The DFTs in their study had a fixed train duration (3 s) but a
different number of pulses. In contrast, our present study used only
six-pulse DFTs. Different optimal patterns may have emerged if we used
longer trains or controlled the train duration rather than the number
of pulses.
Two mechanisms have been proposed for the enhancement in force seen
with doublets: increased Ca2+ transient (15) and increased
stiffness (30) by the second pulse in the doublet. Impaired
excitation-contraction coupling (31) and decreased muscle stiffness
(12) with fatigue could make the positive effects of a doublet on force
production more effective. In addition, unlike VFTs that have a doublet
only in the beginning of the train, DFTs have doublets throughout the train. The positive effect of doublets would be used to advantage with
each doublet.
Conclusion
The present model predicts isometric forces for human quadriceps
femoris muscles in response to a wide range of stimulation patterns by
use of brief trains of electrical pulses and under a variety of
physiological conditions. Interestingly, this model predicted that DFTs
produced greater force-time integrals than traditionally used patterns
under fatigued condition. The predictions of the model were verified
with experimental data, demonstrating the predictive power and
usefulness of this mathematical muscle model.
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ACKNOWLEDGEMENTS |
This study was supported by National Institute of Child Health and
Human Development Grant HD-36797.
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FOOTNOTES |
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: S. A. Binder-Macleod, 315 McKinly Laboratory, University of Delaware, Newark,
DE 19716 (E-mail: sbinder{at}udel.edu).
Received 27 August 1999; accepted in final form 3 November 1999.
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