Engineering, 2012, 5, 16-19
doi:10.4236/eng.2012.410B005 Published Online October 2012 (http://www.SciRP.org/journal/eng)
A mathematical Model to Predict Transition-to-Fatigue
During Isometric Exercise on Muscles of the Lower
Extremities
Jorge Garza-Ulloa1, Huiying Yu1, T. Sarkodie-Gyan1, Pablo Rangel1, Olatunde Adeoye1,
Noe Vargas Hernandez2
1Department of Electri cal an d Computer Engineering, UTEP, El Pas o, Texas, USA
2Mechanical Engineering Department, UTEP, El Paso, Texas, USA
Received 2012
ABSTRACT
Surface Electromyography (sEMG) activities of the four muscles were studied from twelve healthy subjects to analyze muscle fati-
gue. Data were recorded while subjects performed isometric exercises for a period of time until fatigue. The signal was segmented
with 5000 samples to enable the evolutionary process. Based on the mean power spectrum and Median Frequency (MDF) of each
segment, we developed a methodology that is able to detect the signal into a meaningful sequence of Non-Fatigue to Transi-
tion-to-Fatigue. By identifying this transitional fatigue stage, it is possible to predict when fatigue will occur, which provides the
foundation of the automated system that has the potential to aid in many applications of our lives, including sports, rehabilitation and
ergonomics.
Keywords: Surface Electromyograph y; Transition-to-Fatigue; Signal Processing; Median Frequency and Power Spectrum;
Polynomial Regr es s ion Models
1. Introduction
Localized muscle fatigue occurs after a prolonged, relatively
strong muscle activity. This is referred to as the process of a
declin e in force du ring a su stain ed muscle activity; the inability
to exert any more force or power defined as physiological
fatigue [1-3]. Localized muscle fatigue cause serious injury
when the level of fatigue is high [8] . Muscles that are fatigued
absorb less energy before they are stretched to such a degree it
causes injuries. Three parts of localized muscle fatigue are
descri bed on the elec tr omyogrphi c signa l: Non-Fatigue, Transition-
to-Fatigue, and Fatigue [4 -6,10,11,19-21] . Non-Fatigue is when
the fresh muscle is able to exert its maximum force. Fatigue
relates to the onset of fatigue during a muscle contraction.
Transition-To-Fatigue is the step between them and its
detection can prevent muscle injury where the measurement is
inherently lost. Once the onset of Transition-to-Fatigue is
detected, what follows is a progressive process until fatigue
onset is achieved. By identifying this transitional fatigue stage,
it is possible to predict when fatigue will occur, which provides
the foundation of the automated system that has the potential to
aid in many applications of our lives, including sports,
rehabilitation and ergonomics [12].
Due to the variability of inter-person muscle characteristics,
there is no simple function or method for muscle load and
timing that defines a precise muscle fatigue threshold [2,9].
Surface electromyography (sEMG) technique records electrical
activities of the muscle as a non-invasive technology where
signals can be analyzed to detect muscles on Transition-to -
Fatigue by examining the changes in measurements; as a
highlight to predict the ons et of the fatigue class. It was recently
found that the changes due to fatigue in the sEMG signal are
detected as increased in amplitude and decreased on frequency
[8].The electrical impulse is carried down the motor neuron to
the muscle. Muscl e fatigue c auses Mot or Unit (MU) recruitment
and the MU firing rate increases as a function of the elapsed
time suggesting that the recruitment of MU firing rates
correlates with sEMG amplitude of the motor unit action
potential (MUAP) detected [7,8].
It is common to study the sEMG in both the time and frequency
domains. The Fourier Transform allows representation as a
function of frequency rather than time, revealing its individual
frequency components. Using Fast Fourier Transform (FFT) in
equation 2, as a method for calculating the discrete Fourier
Transform, is suitable for use in stationary signals, stochastic
process whose joint probability distribution does not change
when shifted in time or space. Two of the most common
frequency-dependent features in sEMG analysis are the mean
frequency (MNF) and median frequency (MDF) [8,14-18].
These two features are mostly applicable in sustained contraction:
the mean frequency (MNF) is the average frequency of the
power spectrum and is defined as its first-order moment; and
the median frequency (MDF) is the frequency at which the
spectrum is divided into two parts of equal power as indicated
[13].
In this study, a mathematical model was used to predict the
stage of Transition-to-Fatigue during Isometric Exercise using
sEMG segmented assessment for the lower extremities. The
data were segmented with 5,000 samples. The MDF of each
segment were calculated and modeled with a regression poly-

J. GARZA-ULLOA ET AL.
17
nomial [5]. The slope of each segment was also calculated for
detection of Transition-to-Fatigue stage.
2. Modeling Equations
The following modeling equations applied in this study also
shown in a sequence order in Figure 1(a).
The sEMG raw data consist of the sum of Motor Unit Action
Potential (MUAP) [6] shown in equation (1):
1
0
()()() ()
N
r
xnhrenrwn
−
=
= −+
∑ (1)
where
()xn
is modeled EMG signal,
()en
is point processed
represents the firing impulse,
()hr
represents the MUAP,
()
wn zero mean addictive white Gaussian noise and N is the
number of motor unit firings.
By applying Fast Fourier Transform (FFT) to the sEMG raw
data, the Discrete Fourier Transform (DFT) is computed as
indicated using equation (2), where the result is used for de-
compose the signal into various frequency components as mag-
nitude and angle.
12/
0
() 0,,1
Nikn N
kn
xxne kN
π
−−
=
== −
∑
(2)
where
01
,,
N
xx
−

are complex nu mbers to calcu lat e th e P ower
Spectrum.
MDF is defined as that frequency that divides the power
density spectrum into two regions having the same amount of
power and was calculated by using a bisection search method
[7] .
00
1
() ()()
2
MDF
MDF
Pd pdpd ωω ωω ωω
∞∞
= =
∫∫ ∫
(3)
Polynomial Regression Models was used as an alternative
when transformations cannot linearize the relationship.
01
kj
j
j
yx β βε
=
=++
∑ (4)
Finally, calculat e the linear slop e (m) o f each s egment:
21 21
21
()
for
()
yy
m xx
xx
−
= ≠
−
(5)
3. Exp erimenta l Pro cedu r e
3.1. Participants
Twelve healthy subjects (age: 26.7 ± 6.67 years; height: 172.4
a
sEMG sensors placement
and iso m e tr ic exer c ise
b
Sequence of data processing and mathemati-
cal model to p redict Transition-to-Fatigue
Figure 1. D ata acqu is ition and d at a p r oc e s si ng.
± 8.46 cm; mass: 86.0 ± 17.29 kg; BMI: 28.8 ± 3.60) volun-
teered to participate in this study. This research was approved
by the UTEP’s Institutional Review Board (IRB) for human
subj ects’ studies. The experiment al procedure was explained to
the subjects and all participants were asked to sign a written
informed consent before testing.
3.2. Data Acqusitio n
EMG signals were reco rded with BTS® bipolar chloride surface
EMG (sEMG) electrodes. Following skin abrasion with an
alcohol soaked cotton pad, electrodes were placed on the
respective muscle bellies: Soleus (Sol), Tibialis Anterior (TA),
Gastrocnemius Later alis ( G L), and Vast us Lateralis (VL).
Each subject was asked to repeat the isometric exercise (see
Figure 1(a)) in the following sequence of four tests: one minute
followed by one minute break, another one minute followed by
one minute break, two minutes followed by two minutes break,
and final three minutes. Subjects can stop anytime as their ca-
pabilities. From the 12 healthy subjects, five could complete all
four tests and seven had to stop after the third test. The first two
tests were were-up task, and the third and/or fourth tests were
analyzed to pr edi ct the Tran sition-to-Fat igue (fatigue task).
3.3. Data Analysis
The EMG signals were stored on computer with a sampling
frequency of 2000 Hz. Fast Fourier Trans for m was th en app lied
into the raw data using equation (2), to calculate the Power
Spectrum for consecutive 2.5-s window (i.e. The signal was
segmented with 5000 samples) throughout the whole fatigue
task. From equation (3), the mean power spectrum was calcu-
lated in order to obtain the Median Frequency (MDF). The
Polynomial Regression Models in equation (4) were calculated
for each 2.5-s window or segment, and the value of this linear
slope in equation (5) and the number of the consecutive seg-
ments were used to predict the Transition-to-Fatigue stage.
Figure 2(b) illustrates the data processing and analysis proce-
dure.
4. Results
Figure 2 illustrates the prediction of Transition-to-Fatigue
stage of the eight muscles on one subject. It was not found the
Transition-to -Fatigue on right TA, left GL, and left Sol during
the fatigue test. The Transition-to-Fatigue was defined with the
decrease in the MDF of the EMG signals that typically occurs
towards the end of exhausting isometric contractions. The first
muscle Transition-to-Fatigue occurred on this subject was right
VL, followed by right GL, right Sol, left VL, and left TA. The
decreased MDF (negative slope) in the Transition-to-Fatigue
stage varies from -68.38 to -82.15. All twelve subjects’ data
EMG analyses were shown in Table 1.
5. Discussion
Generally, different EMG activities were observed between
subjects with regards to the slope and time length of the MDF
studies. However, this study also investigated that Vastus
Llateralis (VL) was the most predomin antly affected muscles

J. GARZA-ULLOA ET AL.
18
Figure 2. Transition-to-Fatigue stage of eight muscles on one subject.
Table 1. Transition-to-fatigue of twelve healthy subjects.
Time for Transition-to-Fatigue occurred with associated slope
Muscle Subject 1 Subject 2 Subject 4 Subjec t 6 Subjec t 7
Time (sec) Slope Time (sec) Slope Time (sec) Slope Time (sec) Slope Time (sec) Slope
Right TA - - - - - - - - - -
Right VL 22.5 -73.2 - - 57.5 -69.58 - - 155 -35.59
Righ t GL - - - - - - - - - -
Right Sol - - - - - - - - - -
Left TA - - 12.5 -46.18 - - - - - -
Left VL - - - - 40 -65.68 - - - -
Left GL - - - - - - - - - -
Left Sol - - 45 -58.99 - - 125 -80.35 - -
Muscle Subject 8 Subject 9 Subjec t 10 Subject 11 Subject 12
Time (sec) Slope Time (sec) Slope Time (sec) Slope Time (sec) Slope Time (sec) Slope
Right TA - - - - - - - - 7 2 .5 -82.30
Right VL - - 37.5 -79.34 20 -38.27 20 -68.37 70 -86.60
Righ t GL - - - - 27.5 -72.57 27.5 72.57 - -
Right Sol - - 45 -85.76 - - 32.5 -64.55 57.5 -74.80
Left TA 70 -46.01 42.5 -88.74 37.5 -82.15 37.5 -82.15 - -
Left VL - - - - 32.5 -80.65 32.5 -80.65 25 -73.94
Left GL - - - - - - - - - -
Left Sol - - - - - - - - - -
Note: T here was no Transition-to-Fatigue found in subject 3, and 5 on any muscles.

J. GARZA-ULLOA ET AL.
19
during Transition-to-Fatigue in this isometric exercise compared
with other three muscles. There were seven subjects which
showed Transition-to-Fatigue on VL, followed by Tibialis
Anterior with six subjects and the Soleus with five subjects.
Gastrocnemius Lateralis was the least affected muscle in this
isometric exercise.
The muscle act iviti es were dif feren t between fo ur muscl es of
both sides in a subject. The example subject in figure 1 shows
pronounced decrease in MDF on right-side VL, GL and Sol for
a long period of time (last for 10 segments on the right VL), but
not in the left-side muscles.
This methodology is based on mathematical models to
evaluate muscle Tran sitio n-to-Fatigu e du ri ng i sometric exer ci se
using a surface Electromyography segmented assessment for
the lower extremities. From this study we conclude that:
• Detection of the Transition-to-Fatigue stage is important
because it will soon progress the fatigue onset. By
identifying this transitional fatigue stage, it is possible to
predict when fatigue will occur;
• The segmented data assessment was useful methodology
for the detection of Transition-to-Fatigue;
• Muscle activities can vary across the subjects due to
anthropometric differences, but also vary from different
muscles in a subject’s left and right side of the legs.
• This methodology also provides insight into the
contributions that functional differences between muscles
have on lower extremity disorders as well as serve as an
index of underlying change in neuromuscular function
before injury and in conjunction with injury treatment and
rehabilitation;
• Future researchers should examine these muscles in a
clinical and sports population as well as in response to
specific interventions.
6. Acknowledgements
The authors would like to express thanks to all the subjects who
participates in this study. The authors would like to thank the
Stern Foundation for providing the funds for this research
study.
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