INDIVIDUAL ASSESSMENT OF MUSCLE FATIGUE AND ITS RELATION TO

THE N-REPETITION MAXIMUM

Andreas Schrempf(a), Daniel Hametner(a) and Armin Blaha(b)

(a) Upper Austria University of Applied Sciences,School of Applied Health and Social Sciences, Garnisonstrasse 21, 4020 Linz, Austria

(b) Sportland OÖ Olympiazentrum, Auf der Gugl 30, 4020 Linz, Austria(a)andreas.schrempf@fh-linz.at, daniel.hametner@fh-linz.at

(b)armin.blaha@spantec.at

ABSTRACTThe main focus of this paper comprises the assessment ofmuscle fatigue by means of time-domain and frequency-domain parameters. Based on a simple biomechanicalmodel time-domain parameters can be computedusing measurement data recorded with an isometricmuscle contraction experiment of the knee extensormuscles. Frequency-domain parameters are computed fromsurface electromyographic recordings. Both time-domainparameters and frequency-domain parameters are comparedto the experimentally determined individual N-repetitionmaximum of healthy subjects. In contrast to the changeof the median frequency it turns out that fatigue time isable to quantify muscle fatigue and can be considered as agood predictor for the individual N-repetition maximum.

Keywords: Rehabilitation engineering, Biomedical systemmodeling, Simulation

1. INTRODUCTIONThe individual adjustment of the training intensity duringstrength training plays a crucial role in medical applicationslike strength rehabilitation of chronic low back pain ormetabolic bone disease patients as well as with preven-tive measures and sport applications. Training intensity ofmuscle strengthening is usually quantified as a portion ofthe 1-Repetition Maximum (1-RM) or by the N-RepetitionMaximum (N-RM), where N denotes a number. The N-RM is the weight on a training device an individual can liftexactly for N repetitions.The number N of repetitions depends strongly on the healthstatus of the person. Strength training with healthy subjectcommonly starts at N = 10, for chronic low back painpatients N = 12− 15 whereas for high-risk patients likepersons with metabolic bone disease N = 20 is proposed(Smeets et al., 2006).An 1-repetition maximum test may increase injury risk anda common N-repetition maximum test with several trialsis stressful, in many cases inaccurate and time-consuming.With specialized devices an isometric contraction measure-ment can be obtained more safely compared to an 1-RMtest (Figure 1). From the isometric measurements individualparameters representing muscle fatigue can be obtained. In

Figure 1. Isometric muscle contraction measurement.

this paper the aim is to assess muscle fatigue by meansof time-domain parameters obtained from a biomechanicalmodel together with measurement data from an isometricmuscle contraction experiment as well as by frequency-domain parameters obtained from surface electromyography(sEMG) - De Luca (1997). Furthermore the correlationbetween these muscle-fatigue parameters and the individualN−RM will be investigated, which in turn should allow topredict the individual training intensity.

2. BIOMECHANICAL MODEL

As depicted in Figure 2 a general biomechanical model ofmuscle strengthening consists of two parts, a mechanicalmodel which represents the training device itself and themechanics of the muscoskeletal system, and a physiologicalmodel which describes the generation of muscle forceand the associated muscle fatigue (Schrempf et al., 2008).Hereby ϕ(t) is the joint-angle (shoulder, elbow, knee, hip,lumbar joint), ω(t) the corresponding angular velocity, Θ

summarizes the inertia of the moving body parts and thetraining device, Mm the moment generated by the musclesand Mload the loading moment of the training device. Theactive muscle moment Mm depends on the muscle activationa(t) and is given by

Mm = rL(ϕ(t))Fm(a(t),ϕ(t),ω(t)) (1)

Proceedings of the European Modeling and Simulation Symposium, EMSS 2009Vol II - ISBN 978-84-692-5415-8 173

x

Physiological

Model

Mechanical Modelmoment arm

load moment

Muscle−

force

Figure 2. Biomechanical model.

with the moment arm rL and the muscle force Fm. Theloading moment depends on the weight m selected fortraining as

Mload = rloadmg, (2)

where rload is the effective moment arm of the load actingon the joint and g denotes the gravity constant. The muscleforce Fm is modeled by (Zajac, 1989)

Fm(a(t),ϕ(t),ω(t)) = Fmaxa(t) f it(t) f f l(ϕ(t)) f f v(ω(t))(3)

where Fmax denotes the maximum voluntary isometric mus-cle force, a(t) muscle activation, f it(t) muscle fitness, f f lthe force-length relation and f f v the force-velocity relationof the considered muscles which depend on the extensionangle ϕ(t) and the angular velocity ω(t) respectively.Hereby muscle activation a(t) and muscle fitness f it(t) aswell as functions f f at , f f l , and f f v are within the range of[0,1].

3. ISOMETRIC CONTRACTION MEASUREMENT

During isometric muscle contraction muscle length andconsequently angle ϕ remains constant at ϕ0 and henceω̇ = ω = 0 and f f v(ϕ0) = 1. By means of a measurementsystem joint position can be adjusted in neutral positionsuch that optimal muscle fiber length f f l(ϕ0) = 1 can beachieved. Considering an isometric muscle contraction inneutral position the joint muscle moment Mm is equal tothe load moment Mload (see Figure 2) which in turn can bemeasured by a sensor. Hence the measured isometric jointmoment Mmeas is given by

Mmeas(t) = Mmaxa(t) f it(t). (4)

Depending on the generated muscle force the isometricmeasurement can be either an isometric maximal voluntarycontraction (MVC) experiment or an isometric sub-maximalvoluntary contraction (S-MVC) experiment.Whereas the MVC experiment starts with 100% muscleactivation, the isometric S-MVC experiment consists oftwo phases, where in the first phase the aim is to providea constant sub-maximal joint moment Msub as long aspossible. If it is not possible to maintain Msub any longer(t = t1), the experiment enters the second phase. In thesecond phase the aim is to provide a maximum possiblejoint moment until it falls below a predefined threshold (seeFigure 3).

0 10 20 30 40 50 60−100

0

100

200

300

400

500

t [sec]

Mm

eas [N

m]

70% Mmax

40% Mmax

PHASE 1 PHASE 2

t1

Figure 3. Isometric S-MVC experiment at Msub = 260Nm(approximately 70% of Mmax).

4. MUSCLE FATIGUE MODELDuring the isometric S-MVC experiment (ω = 0, ϕ = ϕ0,f f l = 1, f f v = 1) the measured joint moment is given by eq.(4). According to Riener et al. (1996) muscle fitness can bemodeled by

d f it(t)dt

=( f itmin− f it(t))a(t)

Tf at+

(1− f it(t))(1−a(t))Trec

.

(5)This first-order relation describes muscle fatigue (first term)as well as recovery (second term). If the muscle is activatedby 100% (a(t) = 1) then the fitness function decreases andno recovery is possible. On the other hand if there is nomuscle activation (a(t) = 0) recovery of muscle fitness takesplace. The corresponding time constants are Tf at and Trecrespectively. The minimum fitness is given by f itmin.At end of phase 1 muscle activation reaches a(t) = 1 andduring the second phase t ≥ t1 muscle activity remains ata(t) = 1. According to the fact, that the recovery time Trecis much longer than the fatigue time Tf at i.e. Trec � Tf atand (1−a(t)) < 1 for a sub-maximal contraction, it can beconcluded that the second term in (5) is less important thanits first term and hence eq. (4) can be approximately solvedtogether with eq. (5) by

Mmeas(t) = Mmax

[f itmin +

(Msub

Mmax− f itmin

)×

exp(− (t− t1)

Tf aths(t− t1)

)]. (6)

Hereby hs(t− t1) denotes the Heaviside-function with

hs(t− t1) =

{0 , t < t11 , t ≥ t1

.

5. ASSESSMENT OF MUSCLE FATIGUEIsometric muscle contraction experiments for the knee-extensors (m. quadriceps) were performed with 18 youngathletes of age 17.6± 1.7 years, height 1.76± 0.06m andweight 67.3± 7.7kg (mean ± SD). The isometric jointmoment was measured by means of a force-sensor withsampling frequency of 2kHz.Fitting model eq. (6) to the acquired measurement dataMmeas(t) and estimating the unknown parameters Mmax,f itmin and Tf at by solving a nonlinear least-squares problem,muscle fatigue was assessed by model parameters f itmin and

Proceedings of the European Modeling and Simulation Symposium, EMSS 2009Vol II - ISBN 978-84-692-5415-8 174

Table 1. Results of the experimental study. The body height of subjects is defined by H.# H [m] Mmax[Nm] Tf at [s] f itmin N NRM [kg] ∆ fmed [Hz] ∆ fmed

∆t

[Hzs

]5 1.72 394.3 42.2 0.00 13 55.00 -26.7 -0.676 1.69 260.8 74.8 0.00 14 45.00 -22.7 -0.587 1.88 337.9 46.6 0.00 8 55.00 -19.8 -0.498 1.85 302.6 70.4 0.00 12 55.00 -25.0 -0.469 1.80 380.4 26.2 0.18 7 60.00 -28.1 -0.72

10 1.70 305.2 53.7 0.00 9 55.00 -19.5 -0.5711 1.75 286.7 35.0 0.00 6 50.00 -22.9 -0.5812 1.69 224.1 50.3 0.00 11 40.00 -29.2 -0.7113 1.72 355.2 29.0 0.00 8 50.00 -13.2 -0.3214 1.79 354.1 64.5 0.00 14 50.00 -18.6 -0.4615 1.68 204.2 61.9 0.21 9 45.00 -11.2 -0.2016 1.79 322.9 46.0 0.00 12 50.00 -30.8 -0.6917 1.84 243.4 61.5 0.00 10 40.00 -21.5 -0.5118 1.73 351.8 30.8 0.00 9 50.00 -22.6 -0.57

Tf at . Measurement data of 4 persons was removed since theS-MVC experiment was not performed correctly i.e. in thefirst constant phase of the experiment the variance of thejoint-moment was too high or the length of the second phasewas too short for the estimation of Tf at .For comparison additionally to the muscle-moment mea-surement Mmeas(t) the decrease of the median frequency∆ fmed(t) and the corresponding slope ∆ fmed/∆t of therecorded surface EMG was computed, which is establishedas a common measure of muscle fatigue in literature -Mathur et al. (2005). An array of 12 EMG electrodes wasused hereby for measurement of the sEMG. The EMG-datawas sampled with 1.2kHz and filtered by a 50Hz notchfilter and a bandpass between 2-500Hz.The results of the experimental study are summarized inTable 1. In most of the cases parameter f itmin is around 0,which agrees perfectly with values documented in Rieneret al. (1996). Significant information is represented by time-constant Tf at where the relation between N/Tf at and thequotient Mmax/(NRM ·g ·H) is depicted in the left subplotof Figure 4. Furthermore the relation between the slope of

Table 2. Linear regression analysis. A significant linearrelation is marked by an Asterisk.

Model P-value Err.-Var.Mmax

NRM ·g ·H= 0.2052+0.7032 · N

Tf at0.0001 ∗ 5.7E−4

Mmax

NRM ·g ·H= 0.3840−0.0013 · N∣∣∣ ∆ fmed

∆t

∣∣∣ 0.3579 2.0E−3

the change of the median frequency ∆ fmed/∆t of the surfaceEMG and the quotient Mmax/(NRM ·g ·H) is represented bythe right subplot in Figure 4. A linear regression analysisfor both relations was performed computing the coefficientestimates of the linear model, the 95% confidence intervalsfor the coefficient estimates, the residuals, the R2 statis-tic, the F statistic, its P−value and the error variance.Hereby a P−value of 0.05 was accepted as the level ofsignificance. The parameters of the linear model and theP−value of the regression analysis are summarized in Table2. Local muscle fatigue was assessed by the relative changeof the median frequency ∆ fmed [Hz] between subsequentelectrodes in the EMG array. For each subject the time-history of the amplitudes of the relative change of themedian frequency was coded by a color-map, plotted andinterpolated for all 12 array-electrodes (see Figure 5).

Figure 4. Measurement data (squares, circles) and linearmodel (line) between Mmax/(NRM · g ·H) and N/Tf at andN/|∆ fmed/∆t| respectively.

6. CONCLUSIONS

Assessing muscle-fatigue by the time-domain parametersMmax and Tf at allows for the quantification of mus-cle fatigue. The relation between N/Tf at and quotientMmax/(NRM · g ·H) shows a significant linear dependence(see Figure 4). This in turn allows directly to predict theindividual training intensity by means of the NRM fromMmax and estimated fatigue parameter Tf at . From linearmodel in Table 2 it can be concluded that

NRM =Mmax

g ·H1

b1 +b2 ·N

Tf at

Proceedings of the European Modeling and Simulation Symposium, EMSS 2009Vol II - ISBN 978-84-692-5415-8 175

Figure 5. Interpolated map of fatigue-parameter ∆ fmed [Hz] for all 12 electrodes of the EMG-array. Hot colors indicate regionsof high muscle fatigue, cold colors indicate regions of low muscle fatigue.

with b1 = 0.2052 and b2 = 0.7032. For any fixed numberN of repetitions the individual N−RM is higher for slowtime constants Tf at (high values) and higher maximumisometric joint moments Mmax (see Figure 6). The time-

Figure 6. Relation between the N-RM and time-domainparameters Mmax and Tf at .

domain parameters Mmax and Tf at can be simply obtainedfrom an isometric S-MVC experiment which allows for asafe and time-efficient measurement of the joint-moment,since the size of the given sub-maximal moment Msub canbe simply adopted to the health status of the person.In contrast to time-domain parameters Mmax and Tf at onlya qualitative assessment of muscle fatigue can be obtainedby the change of the median frequency fmed of the recordedsurface EMG, since a significant linear relation do notexist (see Table 2). As well known from Iguchi (2008) areduction of the median frequency with increasing musclefatigue can be observed. In principle it can be concluded,that the higher the slope |∆ fmed/∆t| of the reduction of themedian frequency, the faster fatigue occurs and the smallerthe N−RM for any fixed N would be. However, a quantita-tive assessment of muscle fatigue cannot be provided, sincethere is no significant relation between ∆ fmed/∆t and theN−RM (see Figure 4). But the assessment of muscle fatigue

based on the median frequency allows for a localizationof fatigue, since regions of high muscle fatigue can beidentified (see Figure 5).

ACKNOWLEDGMENTSWe gratefully acknowledge Thomas Minarik, Michael Samsand Gerold Schoßleitner for their helpful involvement insetting-up the experiments and performing measurements.Furthermore the support of the Sportland OÖ Olympiazen-trum is gratefully acknowledged.

REFERENCESDe Luca, C., 1997. The use of surface electromyography

in biomechanics. Journal of Applied Biomechanics 13,135–163.

Iguchi, M., 2008. Low frequency fatigue in human quadri-ceps is fatigue dependent and not task dependent. Journalof Electromyography and Kinesiology 18, 308–316.

Mathur, S., Eng, J., MacIntyre, D., 2005. Reliability ofsurface EMG during sustained contractions of the quadri-ceps. Journal of Electromyography and Kinesiology 15,102–110.

Riener, R., Quintern, J., Schmidt, G., 1996. Biomechanicalmodel of the human knee evaluated by neuromuscularstimulation. Journal of Biomechanics 29 (9), 1157–1167.

Schrempf, A., Habelsberger, W., Hutter, D., Brunner, S.,Goritschnig, M., Nagl, K., 2008. Model-based Predictionof the individual N-Repetition Maximum with Appli-cation to Physical Rehabilitation. Proceedings of the17th World Congress, The International Federation ofAutomatic Control, Seoul, Korea, July 6-11, 6658–6663.

Smeets, R., Vlaeyen, J., Kester, A., Knottnerus, J., 2006.Reduction of pain catastrophizing mediates the outcomeof both physical and cognitive-behavioural treatment inchronic low back pain. The Journal of Pain 7 (4), 261–271.

Zajac, F., 1989. Muscle and tendon properties: Models, scal-ing, and application to biomechanics and motor control.Critical Reviews in Biomechanical Engineering 17, 359–411.

Proceedings of the European Modeling and Simulation Symposium, EMSS 2009Vol II - ISBN 978-84-692-5415-8 176

AUTHORS BIOGRAPHYAndreas Schrempf was born in Schladming, Austria

and went to the Johannes KeplerUniversity of Linz, where he stud-ied Mechatronics and obtained hismaster degree and PhD in 1998 and2004 respectively. After working as asenior researcher at the competencecenter in mechatronics he movedto the Upper Austria University of

Applied Sciences, where currently he is professor forbiomechanics at the School of Applied Health and So-cial Sciences, department of medical technology in Linz.His research interests concern modeling and simulation inbiomechanics and biomedical engineering as well as systemidentification and control theory.

Daniel Hametner was born in Amstetten, Austriaand went to the Upper Austria Uni-versity of Applied Sciences, wherehe studied Medical Technology at theSchool of Applied Health and SocialSciences and obtained his diplomadegree in 2008. Currently he’s work-ing in the field of railway transporta-

tion, where he’s engaged with the development of doorentrance systems. His field of activity involves the spec-ification, development and testing of complex software andelectronic components.

Armin Blaha was born in Vienna, Austriaand went to the University of Ap-plied Sciences Technikum Wien,where he studied Sports-Equipment-Technology and obtained his diplomadegree in 2005. After working at theUpper Austrian Olympic Sports Cen-ter, were he was responsible for the

development of Feedback systems for athletes, he foundedthe Spantec GmbH in 2008. With his company he focuseson research and development of Feedback systems speciallydesigned for elderly people.

Proceedings of the European Modeling and Simulation Symposium, EMSS 2009Vol II - ISBN 978-84-692-5415-8 177