Force enhancement and force depression in a modified muscle model used for muscle activation prediction

Journal of Electromyography and Kinesiology xxx (2013) xxx–xxx

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Journal of Electromyography and Kinesiology

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Force enhancement and force depression in a modified muscle modelused for muscle activation prediction

1050-6411/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jelekin.2013.02.008

⇑ Corresponding author at: KTH Mechanics, Osquars backe 18, 100 44 Stockholm,Sweden. Tel.: +46 87907719.

E-mail address: [email protected] (E.M. Gutierrez-Farewik).URL: http://www.mech.kth.se/mech/info_staff.jsp?ID=200 (E.M. Gutierrez-Fare-

wik).

Please cite this article in press as: Kosterina N et al. Force enhancement and force depression in a modified muscle model used for muscle activatidiction. J Electromyogr Kinesiol (2013), http://dx.doi.org/10.1016/j.jelekin.2013.02.008

Natalia Kosterina a, Ruoli Wang a, Anders Eriksson a, Elena M. Gutierrez-Farewik a,b,⇑a KTH Mechanics, Osquars backe 18, 100 44 Stockholm, Swedenb Karolinska Institutet, Department of Women’s and Children’s Health, Sweden

a r t i c l e i n f o

Article history:Received 27 June 2012Received in revised form 22 February 2013Accepted 25 February 2013Available online xxxx

Keywords:Muscle modelForce depressionForce enhancementActivation modificationMusculoskeletal systemHeel-raiseSquatElectromyography

a b s t r a c t

This article introduces history-dependent effects in a skeletal muscle model applied to dynamic simula-tions of musculoskeletal system motion. Force depression and force enhancement induced by active mus-cle shortening and lengthening, respectively, represent muscle history effects. A muscle model dependingon the preceding contractile events together with the current parameters was developed for OpenSimsoftware, and applied in simulations of standing heel-raise and squat movements. Muscle activationswere computed using joint kinematics and ground reaction forces recorded from the motion captureof seven individuals. In the muscle-actuated simulations, a modification was applied to the computedactivation, and was compared to the measured electromyography data. For the studied movements,the history gives a small but visible effect to the muscular force trace, but some parameter values mustbe identified before the exact magnitude can be analysed. The muscle model modification improves theexisting muscle models and gives a more accurate description of underlying forces and activations inmusculoskeletal system movement simulations.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Musculoskeletal modelling is widely used for motion analysisand movement prediction. An accurate muscle model is an essen-tial factor for reliable musculoskeletal system simulation of dy-namic tasks. Forward dynamics simulations allow examination ofmuscle properties that cannot be measured. Among the developedmusculo-skeletal models, very few studies consider muscle historyphenomena in dynamic simulations (Ettema, 2002; McGowanet al., 2010) despite the need to investigate the influence of his-tory-dependent effects (Winters, 1995). The existing muscle mod-els are limited due to complexity of the underlying processes andgaps in knowledge. Based on previous research (Kosterina et al.,2008, 2009, 2011) improvements to a common muscle model byadding a history dependence during active length variations havebeen suggested. Residual force enhancement (FE) induced by ac-tive stretching and force depression (FD) induced by active short-ening are complex phenomena, but can be relatively wellpredicted by a simple formula as these history effects can be

described by the mechanical work accumulated or performed dur-ing the movements (Kosterina et al., 2008, 2009).

Force history effect induced by muscle length variation hasbeen studied by many groups during the last six decades, (Abbottand Aubert, 1952; Herzog and Leonard, 1997; Lee and Herzog,2003; Lou et al., 1998; Marechal and Plaghki, 1979; Morganet al., 2000; Power et al., 2012; Sugi and Tsuchiya, 1988). A studyof voluntary contractions of large human muscles in vivo hasshown that the activation reduces along with FE after stretch con-tractions (Seiberl et al., 2012), which shows that the motor controlsystem can adjust according to the history effect. Activation mod-ification was observed along with the force modification in intactmuscle contraction, (Altenburg et al., 2008; Hahn et al., 2007;Oskouei and Herzog, 2006). Both manifestations of the history ef-fect, i.e. force and activation modifications, are essential althoughthe reported conclusions diverge (Pinniger and Cresswell, 2007;Seiberl et al., 2012; Tilp et al., 2009). The correct relation betweenforce and activation modifications during voluntary contractionswould further describe the muscle history effect. Regarding thequantitative significance of FD and FE, these phenomena have beensuggested to lead to alterations in the neural activation andchanges in muscle activation rather than producing smaller or lar-ger forces (Seiberl et al., 2012). Oskouei and Herzog (2006) foundthat less activation was required to obtain a certain submaximalforce level after muscle stretch and proposed that muscle historyphenomena are related to metabolic cost of the contractions.

on pre-

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Fig. 1. Schematic diagram of activation calculations. Kinematics of a movementsymbolically represented by L⁄ demand a muscular force F⁄ in each muscle.OpenSim calculates the corresponding activation A without consideration of historyeffects, through the CMC algorithm, representing the inverse of Eq. (1). Whenconsidering the history effect dFmod, a modified activation Amod creates the forceFmod which with dFmod sums up to F⁄. The present work aims to calculate thedifference dAmod, defined by Amod = A � dAmod.

2 N. Kosterina et al. / Journal of Electromyography and Kinesiology xxx (2013) xxx–xxx

Muscles’ characteristic force–length and force-shortening veloc-ity relationships play important roles in the mechanics of musclebehaviour and are measured and estimated using various tech-niques. Ultrasonography and magnetic resonance imaging allowfairly accurate estimations of muscle length (Maganaris, 2001),and the musculotendon complex length can be estimated from ana-tomical and joint kinematic data (Fukunaga et al., 2001). However,muscle forces in situ have only been measured in human adductorpollicis muscle (Lee and Herzog, 2002; Ruiter et al., 1998). Muscleforce measurements in vivo are invasive and restricted to superficialstructures such as Achilles’ tendon (Finni et al., 1998) and intact ten-dons (Schuind et al., 1992). To estimate muscle force, the torque cre-ated from a muscle group can be measured in vivo (Altenburg et al.,2008; Hahn et al., 2007; Lee et al., 1999) and the load sharing prob-lem solved, though the solution will depend on the optimization de-sign and criteria (Kaphle and Eriksson, 2008).

Due to the measurement limitations, a musculoskeletal modelcan be validated, at least in part, by several methods, e.g., by com-paring muscle forces calculated from inverse dynamics with dyna-mometric measurements for a muscle group (McGowan et al.,2010). Another method consists of comparing calculated muscleactivation with the recorded electromyography signal (EMG)(Lloyd and Besier, 2003). Computed activation is an approach todescribe muscle excitation, alluding to which muscles are involvedand to which extent. The EMG signal measures the stimulation di-rectly under the electrode placement, regardless of the resultingmuscle activation. Despite the differences between activation andstimulation, these quantities are commonly used interchangeablyin musculo-skeletal modelling.

Hatze (1977) proposed an equation describing how muscleforce and activation are related. A modified form of this relationby Thelen et al. (2003) is:

F ¼ ðA � F lvðl; _lÞ þ FpassiveðlÞÞ � cosðaÞ; ð1Þ

where F is a steady-state muscle force, A the activation, Flv the ac-tive force–length–velocity surface, Fpassive the passive force, a thepennation angle at the steady-state muscle length l for a given mus-cle, and _l denotes time derivative of length.

Muscle activation, A, can be calculated for an existing musclemodel using experimental data by solving A from a computed forceF⁄ under given l, _l and muscle architecture, using Eq. (1). A proce-dure of this type is used in OpenSim (Delp et al., 2007) to computea muscle force F⁄ from kinematics, and then ultimately an activa-tion A. This inverse calculation of activation from kinematics andexternal forces implicitly assumes an equation of the form in Eq.(1), regardless of history. The calculation is performed for eachconsidered muscle.

History effects from previous muscular work can be added tothe force calculated from the activation according to, Kosterinaet al. (2009):

Fþ ¼ F þ dFmod; ð2Þ

with F+ the total force consisting of the activated force F from Eq. (1)and a history-dependent modification dFmod. When this modifiedforce is used in an inverse dynamics setting, this demands a corre-sponding modification of the calculation of activation from thecomputed force. As the force modification dFmod at a certain time in-stance is completely defined by history, only the remainder of aneeded force F⁄ must be created by the activation. Using Eq. (1),one can find the modified activation as:

Amod ¼F� � dFmod

cosðaÞ � FpassiveðlÞ� �

� 1

F lvðl; _lÞ; ð3Þ

which creates the force F⁄ by using the expressions in Eqs. (1) and(2).

Please cite this article in press as: Kosterina N et al. Force enhancement and fodiction. J Electromyogr Kinesiol (2013), http://dx.doi.org/10.1016/j.jelekin.2013

The situation considered in the present work is shown in Fig. 1.The basis for the figure is that a force F⁄ (any of the forces in thestudied system) is needed to create the recorded kinematics, sym-bolically represented by L⁄. Without consideration of history, as inOpenSIM, an activation would be calculated by CMC, which is theinverse of Eq. (1) — or Eq. (3) with dFmod = 0. When consideringthe force modification, only the force Fmod = F⁄ � dFmod need be cre-ated by the activation, and the modified activation Amod is calcu-lated from Eq. (3). This activation Amod used in Eq. (1) creates theforce Fmod to which is added dFmod leading to the total force F⁄ con-sistent with kinematics. As will be further detailed below, themodified activation Amod can be seen as a subtraction of a quantitydAmod from the A calculated by the included CMC algorithm.

Our study hypotheses were that (i) introducing the history ef-fect in a musculoskeletal system improves the modelling of move-ment, that (ii) the motor control activation in human movementconsiders the history effect, and that and (iii) activation Amod con-forms to EMG data better than the OpenSim output A, which wouldtend to verify (i). The present work thereby introduces a method totest a musculoskeletal system modelling in the open-source soft-ware OpenSim. A series of experiments on human movementwas conducted to test the effects of this modified muscle model,using the model for all muscles in a studied subject, keeping trackof the force history of all muscles, and using an evaluation of acti-vation based on Eq. (3), rather than on the inverse of Eq. (1).

2. Methods

Seven healthy adults (six females and one male, age29 ± 3 years, weight 56.1 ± 6.9 kg, height 1.63 ± 0.04 m) partici-pated in this study. The subjects participated with informed con-sent; they were physically active, but none of them participatedin any competitive exercise training. The study was approved bythe local Ethics Committee.

2.1. Experimental setup

A maximum voluntary isometric contraction (MVIC) test wasperformed for EMG scaling. The subjects sat upright in an isometricdynamometer chair with the hip, knee and ankle fixed at 90�, 60�and 90� (neutral), respectively (Örtqvist et al., 2007). The calf andthe foot were tightly fastened to restrict joint motion. The isomet-ric condition was defined as a fixed-end contraction where themuscle–tendon complex was held at constant length. SurfaceEMG signals (Motion Laboratory System, Baton Rouge, LA) were re-corded for the subjects for the rectus femoris (RecFem), tibialis

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N. Kosterina et al. / Journal of Electromyography and Kinesiology xxx (2013) xxx–xxx 3

anterior (TibAnt), medial head of the gastrocnemius muscle (Med-Gas) and soleus (Sol) bilaterally according to the SENIAM protocol(http://www.seniam.org).

A series of tasks was chosen to place a high load on the fivementioned muscles. The participants performed a dynamic taskafter the MVIC test: 3 cycles of standing heel-raise (rise up anddown on toes) with a 3 s delay in the upper position, and 3 cyclesof squats. The subjects were examined using an 8–camera motioncapture system (Vicon MX40, Oxford, UK) and two force-plates(Kister, Winterthur, Switzerland). Sixty-four reflective markers(9 mm) were placed bilaterally on bony landmarks based on a con-ventional full-body marker set (Vicon Plug-in-Gait), plus a multi-segment foot model marker set (Stebbins et al., 2006).

One subject performed an additional series of squats bothslowly and quickly, and both with and without an additional loadof 40 kg, held at the shoulders.

2.2. Data processing

During experiments, EMG was sampled at 1000 Hz. Data werefiltered using a bidirectional second order Butterworth filter witha cutoff frequency of 3 Hz in Matlab (version R2011a, The Math-Works, Natick, USA). EMG for each muscle was normalized fromzero to one based on the minimum and maximum values for thatmuscle during the MVIC and the performed movements.

2.3. Musculoskeletal model

A generic musculoskeletal model with 14 segments, 23 degrees-of-freedom and 96 musculotendon actuators was used to generatethe simulation in OpenSim 2.4 (Delp et al., 2007). The head, arms,and torso were modelled as a single rigid body which articulatedwith the pelvis via a ball-and-socket back joint. Each hip was mod-elled as a ball-and-socket joint, each knee as a hinge joint, each an-kle, subtalar and metatarsophalangeal joint as a revolute joint(Anderson and Pandy, 1999; Delp et al., 1990).

A subject-specific simulation of heel-raise and squat was gener-ated. The model was scaled to each subject based on the experi-mental marker set placed on anatomical landmarks. The inversekinematics algorithms solved for joint kinematics that minimizedthe differences between experimental marker and virtual markerpositions. Dynamic inconsistency between the measuredgroundreaction forces and the kinematics was resolved by applying smallexternal forces and torques (i.e. residuals) to the torso and makingsmall adjustments to the model mass properties and kinematics(Delp et al., 2007). Constraints on muscle activation were pre-de-fined based on the EMG records for the relevant muscles, primarilyto maintain the load-sharing between synergetic muscles. Therange of the activations was outlined as filtered normalized EMGsignal ±10% of its maximum value. Computed muscle control(CMC) (Thelen et al., 2003) with the pre-defined constraints onthese muscles’ activations was used to find a set of actuator activa-tions, A, implicitly using the A � F relation of Eq. (1). The con-straints are required to track the kinematics and to be generallyconsistent with experimental EMG patterns. Static optimizationwas used in CMC to determine the muscle forces and to minimizea cost function at every timestep t for the set of N muscles:

JðtÞ ¼XN

i¼1

V i½AiðtÞ�2;

where Vi is the volume of muscle i and Ai(t) is the activation of mus-cle i (Happee, 1994; Thelen and Anderson, 2006).

Please cite this article in press as: Kosterina N et al. Force enhancement and fordiction. J Electromyogr Kinesiol (2013), http://dx.doi.org/10.1016/j.jelekin.2013

2.4. Muscle model modification

The muscle force modification in Eq. (2) during and after non-isometric contractions is expressed with a simple formula:

dFmod ¼ Khist �W � A; ð4Þ

where Khist is the history coefficient, and W is the mechanical workperformed by or on the muscle during the dynamic movement (de-scribed by the sign of W). Due to lack of information about the his-tory coefficient for the studied human muscles, Khist was varied inorder to test the benefit of implementing the muscle force modifi-cation. Considering that force enhancement and force depressionreaches up to 30% of maximum isometric force (Lee and Herzog,2003; Ruiter et al., 1998), the value Khist = �10 m�1 was assumedto be appropriate for all muscles.

The force modification described by Kosterina et al. (2009) wasstudied for fully active muscle. We scaled the force modification tomuscle activation A in order to eliminate the history effect whenthe muscle is destimulated (Julian and Morgan, 1979; Morganet al., 2000). The modified activation was calculated according toEq. (3) and appeared as:

Amod ¼ A� dAmod ¼ A� dFmod

cosðaÞ � F lv: ð5Þ

The mechanical work was evaluated at a current time by a numer-ical integration over the passed time range of the force multipliedby shortening velocity:

W ¼X

F � _l � Dt; ð6Þ

where F and _l are quantities in the time series of intervals Dt. Thesame modification was used for all muscles, using the individualwork histories. This means that Eqs. (4)–(6) were used for eachmuscle independently.

The new functionality was added to the generalOpenSim soft-ware using an application programming interface (API). A plug-inwas written in C++ using dynamically linked libraries to calculateFmod and Amod using built-in classes and objects. The plug-in wasthen used in the graphical user interface for simulated dynamicalmotions.

2.5. Data analysis

To test the accuracy of the muscle model modification, simu-lated quantities were compared to experimental data. EMG datawas compared to both muscle activation (A) and modified activa-tion (Amod). The differences between the datasets (EMG – A andEMG – Amod) were calculated. The root-mean squared errors wereextracted (RMSE). The RMSE quantity was calculated for every at-tempt, then mean and standard deviations for the groups of indi-vidual attempts were computed for both squat and heel-raisemovements.

3. Results

Heel-raise and squat data were analysed. Due to technical prob-lems, squat data for two of the tested subjects could not be ana-lysed, so the results are based on n = 7 subjects for heel-raise andn = 5 subjects for squat. Two pairs of muscles with the highest nor-malized forces during the specified movements were chosen. Thesemuscles during a heel-raise were MedGas and Sol, and duringsquat TibAnt and RecFem.

The muscle activations during a heel-raise and a squat are plot-ted along with the EMG signal in Figs. 2a and 4a. The RMSE be-tween EMG and A, and between EMG and Amod are presented inFigs. 2b and 4b. The modified activation Amod was closer to the

ce depression in a modified muscle model used for muscle activation pre-.02.008

http://www.seniam.org



EMG signal than the OpenSim output A for MedGas, Sol and TibAntmuscles but not for RecFem.

The muscle force F (calculated with A and Eq. (1)) was plottedalong with the modified muscle force F+ = F + dFmod, Fig. 3. Theforce modification appeared negative or positive depending onthe preceding muscle length variation. Quantitatively, the modifi-cation was up to 15% of the force value.

Different conditions of squat performance were compared in apilot-study. A subject was asked to squat slowly and quickly, thendo the same with two 20 kg loads held at the shoulders. Forcetraces for RecFem muscles are presented in Fig. 5. EMG signal, acti-vation and force modification appeared larger for quick and loadedsquats than for slow and unloaded, correspondingly. The resultsfrom the additional motion analysis illustrate that force modifica-tion is practically non-existent in slow motion with low loads, butpresent when motion is loaded or fast.

The plug-in used for additional analysis, i.e. for calculating ofmodified muscle activation and force, takes approximately 10% oftime used by CMC analysis.

4. Discussion

The main consequence of this study is the improvement of askeletal muscle model by adding the history effect to the muscleactivation using the possibility to add new functionality to theopen-source software OpenSim. The modified activation was closerto experimentally observed activation in three out of four studiedmuscles.

The major function of a muscle is to transform an electrical sig-nal into length variation through contractile force generation. Thecomputed muscle control, CMC, (Thelen and Anderson, 2006)makes it possible to compute activation that results in force F,which in turn can be modified by adding a history componentdFmod, Eq. (4). However, force F + dFmod would lead to a new move-ment, so activation modification was introduced instead, Fig. 1.Since muscle memory has mainly been investigated in electricallystimulated muscles and muscle fibres in vitro, modification of acti-vation has not been described in detail. Recent studies of voluntarycontractions of human muscles (Altenburg et al., 2008; Oskoueiand Herzog, 2006; Seiberl et al., 2012) have shown presence ofactivation modification along with the force modification. Consid-ering muscle history phenomena, the brain adjusts the signalaccording to a desired motion. Thus muscle activation decreasesafter lengthening and increases after shortening. One may

0 1 2 3 40

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0 1 2 3 40

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(a)Fig. 2. Simulation of muscle excitation compared to EMG signal during one heel-raise cymuscles. (a) Example of normalized EMG signal – thin grey line, muscle activation – dasheresiduals between muscle activation and EMG signal (triangles) and modified muscle ac


speculate that metabolic cost of contractions changes dependingon the preceding muscle history, with e.g. decrease of metaboliccost during eccentric contractions and compensation for forcedepression during concentric contractions.

In this study, CMC analysis was used to evaluate the set of mus-cular forces necessary for reproducing recorded movements, there-fore it was assumed F � F⁄ and only activation modification�dAmod

was considered, Eq. (5), Fig. 1. In fact, both modifications of activa-tion and force must be included and future studies should beaimed at defining a correlation between dA and dF. These couldbe represented in a new formula giving a more accurate descrip-tion for skeletal muscle model:

F þ ð1� cÞ � dFmod ¼ ððA� c � dAmodÞ � F lvðl; _lÞ þ FpassiveðlÞÞ � cosðaÞ;ð7Þ

where c 2 [0,1] defines a balance between dFmod, Eq. (4), and dAmod,Eq. (5), induced by active muscle length variations and motor con-trol. In the current study the value c = 1 was assumed, i.e. the brainhas full control of the force generation and adjusts the activation le-vel so that the force is not influenced by the length variations. Anexample of c = 0 is when the central nervous system does not con-sider the history effect, for instance when in vitro stimulation at aconstant activation is performed. Referring to studies wherein bothactivation and force were modified after active stretch (Oskouei andHerzog, 2006; Seiberl et al., 2012), one may speculate that coeffi-cient c might possess any value between 0 and 1 depending onthe extent to which the brain predicts and responds to the activemuscle length variation leading to the history effect. This specula-tion could thus indicate the degree to which the motor control sys-tem is familiar with the movement performed.

The amount of FD and FE in most published studies has beenassociated with the shortening magnitude, (Abbott and Aubert,1952; Bullimore et al., 2007; Herzog and Leonard, 1997; Louet al., 1998; Schachar et al., 2004) or speed of shortening (Herzogand Leonard, 1997; Lee and Herzog, 2003; Marechal and Plaghki,1979; Morgan et al., 2000; Sugi and Tsuchiya, 1988). However, astrong association between FD and the mechanical work per-formed by the muscle has been identified (Herzog et al., 2000;Josephson and Stokes, 1999; Kosterina et al., 2008, 2009). Thereis, on the other hand, no generally accepted predictor of FE. Bulli-more et al. (2007) and Hisey et al. (2009) have found that, startingfrom a certain stretch magnitude, FE does not depend on thelengthening parameters but on the current muscle length.Kosterina et al. (2009) observed a linear relation between FE and

MedGas Soleus0

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(b)cle with �3 s delay in the upper position. The results are given for MedGas and Sol

d black line, modified muscle activation – solid thick black line. (b) Root mean squaredtivation and EMG signal (circles), mean ± std, n = 7.



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Fig. 3. Simulation of muscle force (dashed black line) and modified muscle force (solid black line) are plotted above, traces of normalized muscle length plotted below. (a)Example of heel-raise cycle with �3 s delay in the upper position, the results are given for MedGas and Sol muscles. (b) Example of squat, the results are given for RecFem andTibAnt muscles. The force modification (grey line) is negative during shortening–stretch cycles (MedGas, Sol, TibAnt) and positive during stretch–shortening cycle (RecFem).


the mechanical work for a subrange of physiological stretch mag-nitudes. Even though the relationship between FE and mechanicalwork throughout a wide range of motion is not precisely known,the same function for FE and FD have been incorporated here, aswas also done in several other muscle models (Ettema and Meijer,2000; Forcinito et al., 1998).

For computing the force modification, there is one constant re-quired, Khist, Eq. (4), but there are no data for tested human mus-cles. In a previous study, Kosterina et al. (2009) derived a valueof the history coefficient for mouse muscles, �3 m�1 for extensordigitorum longus and �4 m�1 for soleus muscle. Herzog et al.(2000) obtained Khist = �10 m�1 for cat soleus muscle. Assumingthat force modification is linearly related with W � A, Khist = -�10 m�1 was chosen among tested values of the scaling coeffi-cient. The decision was based on the amount of forcemodification which lies in a range between �30% and 20% of max-imum isometric force, MVIC (Lee and Herzog, 2003; Ruiter et al.,1998), and the tested movements do not actually demand a

0 10

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(a)Fig. 4. Simulation of muscle excitation compared to EMG signal during one squat. The resline, muscle activation – dashed black line, modified muscle activation – solid thick black(triangles) and modified muscle activation and EMG signal (circles), mean ± std, n = 5.


substantial effort. A more correct value for Khist in human muscledemands further work and may differ for different muscles. Thisparameter can be obtained from in vitro experiments, and mostlikely will be a muscle-specific characteristic but not necessarilyperson-specific (Kosterina et al., 2009).

The modified activation Amod matches measured EMG signalbetter than A for Sol and MedGas during a heel-raise (Fig. 2) andTibAnt during a squat, but not for RecFem (Fig. 4). The inconstancyof the error tendencies (Figs. 2b and 4b) is probably due to the opti-mization procedure in OpenSim. A closer look at the muscle activa-tion A in Figs. 2a and 4a shows that its peak is always below theEMG peak. For this reason, Amod after stretch is inferior to A. It isnoted that the optimization of load-sharing in OpenSim presentlydoes not consider the history effect, which is not a negligiblesource of error. Stronger control constraints for muscle activationsbased on EMG signal might invalidate this irregularity but theintention was to give more freedom for the activation and identifythe trend of the modification.

RecFem TibAnt0

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(b)ults are given for TibAnt and RecFem muscles. (a) Example of EMG signal – thin greyline. (b) Root mean squared residuals between muscle activation and EMG signal



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Fig. 5. Simulation of RecFem muscle force during one squat cycle. Muscle force F –dashed black line, modified muscle force Fmod – solid black line, force modificationdFmod – grey line.


The diversity is also explained by muscle length variations dur-ing the movement. The muscle force was plotted with probablemodified forces, and the results show that dFmod increases forcefor RecFem but decreases for MedGas, Sol and TibAnt, Fig. 3. Inthe studied motions, Sol, MedGas and TibAnt muscles performshortening–stretch cycles, while RecFem muscle does the opposite:it first stretches and then shortens during a squat. Force modifica-tion reflects this variation: muscle shortening induces FD, andmuscle lengthening induces FE; dFmod P 0 during stretch–shorten-ing and dFmod 6 0 during shortening–stretch contraction. The mod-ification tends to change the force in the expected direction, butthe technique does not allow us to measure muscle forces and val-idate the improvement. Moreover, Eq. (7) should be used in inversedynamics and CMC analysis to keep an equilibrium between theforces and kinematics.

The question of additivity of history effects induced by stretch-ing and shortening was raised several times. Herzog and Leonard(2000) and Lee et al. (2001) have shown that FD and FE are additivein shortening–stretch cycles, but FE disappears in stretch–shorten-ing cycle. Contradicting results are shown for soleus and extensordigitorum longus (EDL) muscles, where FE and FD seem to be addi-tive for EDL but not for soleus muscle Kosterina et al. (2009),Fig. 3). Bullimore et al. (2008) have shown a remaining FE effectafter shortening when performing equal distances of stretch andshortening on frog muscle fibres. The last experimental setup israther similar to the present study, where muscles stretch andlengthen on the same amount during cyclic movements and thereis a time delay between the length ramps. Therefore, the hypothe-sis that shortening following stretch eliminates initial FE wasneglected.

The force modification introduced in musculoskeletal modellingmay be most relevant in sport and impact biomechanics since theeffect of muscle history increases with speed of motion and load.While it is difficult to specify the absolute value of the history cor-rection, the magnitude of force modification was somewhat low inthe tested motions. The additional analysis was included to illus-trate that force modification may be negligible in slow, unloadedmovements, but non-negligible in fast motions with high loads. Itis interesting that fast movements lead to larger force modification,but maximally stimulated muscles demonstrate smaller history


effect after quick ramps (Kosterina et al., 2008, 2009). This can beexplained by accompanied increase of muscle activation in fastmovements which scales the influence of the modification compo-nent, Eq. (4). The observation may have important implications in,for instance, sports or collision simulations.

5. Conclusions

Skeletal muscle model modification representing muscle forcedepression induced by active shortening and force enhancementinduced by active lengthening was introduced in simulations ofmusculoskeletal system motion by a simple formula. Though cur-rent techniques and limited experimental data do not enable usto validate the result fully, the added formulation improved thedescription of skeletal muscle force and showed the importanceof the modification in demanding tasks.

Conflict of interest statement

No authors had any proprietary, financial, professional or otherpersonal conflicts of interest that may have influenced this study.

Acknowledgement

The authors gratefully acknowledge financial support from theSwedish Research Council.

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Natalia Kosterina received her PhD degree in Engi-neering Mechanics in 2012 from the Royal Institute ofTechnology, Sweden, and Masters degree in AppliedMath and Mechanics from Saint-Petersburg State Uni-versity, Russia. Her doctoral research area was numeri-cal simulation of muscular force generation, particularlyinvestigation of muscle history effect and implementa-tion of this phenomenon in muscle and musculoskeletalmodels.

Ruoli Wang received her PhD degree in 2012 in Engi-neering Mechanics from the Royal Institute of Tech-nology, Stockholm, Sweden, and has a Master’s degreefrom Southeast University, China. Her research interestsinclude applying gait analysis, analytical biomechanicstheory and computational modelling and simulation tounderstand human movement, in particular, focusing inthe consequences of lower limb musculoskeletalimpairments and injuries.

Anders Eriksson is a professor of Structural Mechanicsat the Royal Institute of Technology in Stockholm,Sweden. His research background is primarily related tonumerical simulations of complex load-carrying sys-tems, with applications in both engineering and bio-logical systems. His research in biomechanical systemshas been concerned with muscle physiological models,with the biological–mechanical coupling in tissues, andwith optimization of human movements in numericalcontexts. Sports mechanics is frequently used asexample problems.

Elena Gutierrez-Farewik (‘Lanie’) is an associate pro-fessor of Biomechanics at KTH Mechanics, the RoyalInstitute of Technology in Stockholm, Sweden. She isalso affiliated with the Motion Analysis Laboratory atthe Karolinska Institutet and University Hospital. Shehas a PhD in orthopedics and a master’s degree in bio-medical engineering. Her research interests are inanalysis and simulation of human movement, particu-larly in persons with motion disorders.



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Force enhancement and force depression in a modified muscle model used for muscle activation prediction