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Eur J Appl Physiol (2014) 114:1691–1702DOI 10.1007/s00421-014-2899-5

OrIgInAl ArtIclE

Interdependence of torque, joint angle, angular velocity and muscle action during human multi‑joint leg extension

Daniel Hahn · Walter Herzog · Ansgar Schwirtz

received: 22 november 2013 / Accepted: 20 April 2014 / Published online: 14 May 2014 © Springer-Verlag Berlin Heidelberg 2014

Results For contractions of increasing velocity, opti-mum knee angle shifted from 52 ± 7 to 64 ± 4° knee flex-ion. Furthermore, the curvature of the concentric force/torque–angular velocity relations varied with joint angles and maximum angular velocities increased from 866 ± 79 to 1,238 ± 132° s−1 for 90–50° knee flexion. normal-ised eccentric forces/torques ranged from 0.85 ± 0.12 to 1.32 ± 0.16 of their isometric reference, only show-ing significant increases above isometric and an effect of angular velocity for joint angles greater than optimum knee angle.Conclusions the findings reveal that force/torque produc-tion during multi-joint leg extension depends on the com-bined effects of angle and angular velocity. this finding should be accounted for in modelling and optimisation of human movement.

Keywords Knee joint torques · Maximum unresisted velocity · Multi-joint leg extension · torque–angle relationship · torque–velocity relationship

AbbreviationsAnOVA Analysis of varianceθ0 Optimum velocity-specific knee joint angleF0 Angle-specific isometric external forceFext External reaction forceF/T-θ-r Force/torque–angle relationF/T-θ-ω-r Force/torque–angle–velocity relationF/T-ω-r Force/torque–velocity relationl0 Optimum muscle lengthM0 Angle-specific isometric knee joint torqueMK Knee joint torqueMtc Muscle tendon complexrOM range of motionSD Standard deviation

Abstract Purpose Force and torque production of human muscles depends upon their lengths and contraction velocity. How-ever, these factors are widely assumed to be independent of each other and the few studies that dealt with interactions of torque, angle and angular velocity are based on isolated single-joint movements. thus, the purpose of this study was to determine force/torque–angle and force/torque–angular velocity properties for multi-joint leg extensions.Methods Human leg extension was investigated (n = 18) on a motor-driven leg press dynamometer while measuring external reaction forces at the feet. Extensor torque in the knee joint was calculated using inverse dynamics. Isometric contractions were performed at eight joint angle configura-tions of the lower limb corresponding to increments of 10° at the knee from 30 to 100° of knee flexion. concentric and eccentric contractions were performed over the same range of motion at mean angular velocities of the knee from 30 to 240° s−1.

communicated by Olivier Seynnes.

D. Hahn (*) Human Movement Science, Faculty of Sports Science, ruhr-Universität Bochum, gesundheitscampus, Haus nord nr. 10, 44801 Bochum, germanye-mail: daniel.hahn@rub.de

D. Hahn · A. Schwirtz Department of Biomechanics in Sports, Faculty of Sport and Health Science, technische Universität München, Munich, germany

W. Herzog Human Performance laboratory, Faculty of Kinesiology, University of calgary, calgary, canada

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vmax Maximum velocity of unloaded shortening of an (isolated) muscle

ωmax Maximum angular velocity during unrestricted leg extension

Introduction

Muscular force and power production depends on the instantaneous contractile conditions determined by muscle length and the rate and direction of length change. For iso-lated muscles this is expressed by the force–length (gor-don et al. 1966) and the force–velocity relations (Hill 1938; Katz 1939) and explained by cross-bridge cycling (Hux-ley 1957). Instead, for in vivo human muscles, maximum voluntary torque is expressed as a function of joint angle (T-θ-r) and angular velocity (T-ω-r). In addition to con-tractile properties of muscle fibres, the torque-output of in vivo human muscle action reflects the interaction of mus-cle architecture (e.g. Finni 2006), joint geometry (Krevo-lin et al. 2004), elasticity of the musculo-tendinous tissues (e.g. Kawakami et al. 2002) and neural activation (Pasquet et al. 2005, 2006). Although the contribution of each factor to the resulting torque is difficult to discriminate, the T-θ and T-ω relationships are important properties to character-ise in vivo human muscle function. they further represent subject-specific strength capability, which can be used as input for torque-driven models of human movement (King and Yeadon 2002) or serve as validation criteria for muscle-driven simulations (Delp et al. 2007; Pandy et al. 1990).

there is lot of research on the T-θ and T-ω relations of human muscles (Dudley et al. 1990; Kulig et al. 1984; Maganaris 2004; Pincivero et al. 2004; Seger and thor-stensson 2000; Webber and Kriellaars 1997; Westing et al. 1988; Wilkie 1950) and it is still widely assumed that joint angle and angular velocity can be considered as independ-ent regulators of torque-output that simply need scaling to the appropriate level of activation. Accordingly, the majority of these studies have focused only on joint angle or angular velocity effects separately. this approach has been criticised (Epstein and Herzog 2003; Forrester and Pain 2010; Huijing 1998) and there is numerous experi-mental evidence suggesting a complex interdependence

between length, velocity and the degree of activation for in vitro muscles (Abbott and Wilkie 1953; Bahler et al. 1968; Brown et al. 1996, 1999; granzier et al. 1989; Haan et al. 2003; Joyce et al. 1969; Krylow and Sandercock 1997; rack and Westbury 1969; Scott et al. 1996).

Similar data on in vivo human muscle function are rela-tively rare and almost limited to the knee extensor muscles. For concentric contractions with increasing velocity James et al. (1994) found that the convex shape of the length–ten-sion curve was lost. thorstensson et al. (1976) observed a systematic shift of optimal joint angle θ0 to more extended knee joint, i.e. the optimal muscle–tendon complex (Mtc) length became shorter when angular velocity increased. this change of the T-θ-r was confirmed by others (Fug-levand 1987; Marshall et al. 1990) and is associated with effects of Mtc series elasticity (Kawakami et al. 2002). Fuglevand (1987) was the first to describe an experimen-tally based torque–angle–angular velocity relationship (T-θ-ω-r) for concentric muscle action of the human knee extensor muscles. For extended knee joint positions, he found a plateau in the T-ω-r, indicating that Hill’s curve must be adapted to joint angle when used for modelling the T-ω-r. After a conversion of joint torque into tendon force Marshall et al. (1990) confirmed these findings and showed that maximum shortening velocity depended on muscle length. In a combined experimental–theoretical study on isolated concentric knee extensions (chow et al. 1999a, b, c) predictions of the measured knee torques were more accurate when Hill’s constants a and b were varied with muscle length.

Data on the interdependence of torque, joint angle and angular velocity during eccentric muscle action are even sparser. Westing et al. (1988) showed that eccen-tric torques varied within 0.9–1.18 of the isometric ref-erences depending on muscle length and velocity (see their table 1). However, these interactions were neither systematic nor analysed statistically. In a more recent study (Forrester and Pain 2010) no enhanced torques and no systematic interdependence of calculated fibre forces, muscle lengths and lengthening velocities were found but maximum voluntary eccentric joint torque decreased with increasing stretching velocity (Forrester and Pain 2010; Pain et al. 2013). this is in contrast to widely accepted

Table 1 Optimum knee joint angle θ0 for isometric muscle action (iso) and concentric muscle action (con) at given mean angular velocities

Values are mean ± SD. Superior lowercase letters indicate significant differences between the columns data and the datasets indicated by the let-ters. level of significance p ≤ .05

Muscle action (angular velocity [° s−1])

iso (0)a con (30)b con (60)c con (120)d con(180)e

θ0 [°] 52.2 ± 6.6 55.5 ± 7.3 58.4 ± 7.2 62.7 ± 5.0 63.8 ± 3.8

Significant to c, d, e d, e a, d, e a, b, c a, b, c

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knowledge that voluntary eccentric muscle action is largely unaffected by the speed of lengthening (Enoka 1996). Although several further studies of the same group (Forrester et al. 2011; King and Yeadon 2002; King et al. 2012; lewis et al. 2012; Pain and Forrester 2009; Yeadon et al. 2006) considered the influence of joint angle and angular velocity as well as biarticular effects on voluntary torque production of the ankle plantar flexors and knee extensors, they rather focused on obtaining fitted T-θ-v relationships for torque-driven modelling than analysing distinctive features in their measured data. nevertheless, modelled T-θ-v plots show that maximum angular veloc-ity ωmax varies with joint angle.

For joints other than the knee, data on the interdepend-ence of joint angle and angular velocity including eccentric muscle action are rare. Yeadon and King (2002) developed an exponential function to fit and extrapolate measured data of the ankle, knee, hip and shoulder to express joint torque as a function of joint angle and joint angular veloc-ity. Similarly, Anderson et al. (2007) presented a model predicting maximum voluntary joint torques as a function of joint angle and angular velocity for the ankle, knee and hip. However, they only studied three velocities of contrac-tion, thus their predicted T-θ-v relationships were smooth and they neither detected a shift in θ0 with increasing angu-lar velocity, nor a change in ωmax with joint angle. For the eccentric part, one velocity was considered only; therefore, it is impossible to derive a T-θ-ω-r.

Data of all experimental studies on voluntary human muscle function presented above were obtained from iso-lated joint activities, whereas everyday human movements involve multi-joint actions, such as leg extensions. thus, it is to question if observations of single-joint actions are valid for multi-joint activities. this is supported by find-ings of others (rahmani et al. 2001; Yamauchi et al. 2007) who found a linear instead of a hyperbolic force–velocity relation for multi-joint leg extension and it has further been demonstrated that torque–angle relations of ankle and knee joint differ between single- and multi-joint tasks (Hahn et al. 2011).

to our knowledge there are no data available on the combined effects of joint angle and angular velocity on voluntary force/torque production. therefore, the aim of this study was to investigate the interdependence of force/torque, joint angles, and angular velocity during multi-joint leg extension, which mimics the muscle action con-ditions of real-life movements. According to the literature we hypothesise that during concentric muscle action there is a shift of optimal joint angles towards leg extension with linear force–angle and torque–angle relations. For eccen-tric muscle action we hypothesise force and torque not to exceed the isometric references and to decrease with increasing angular velocity at least beyond a certain point.

Materials and methods

Subjects

Male subjects (n = 18; 30 ± 6.3 year, 1.81 ± 0.08 m, 77.9 ± 5.2 kg) without neuromuscular disorders or injuries participated in this study. Free, written informed consent was obtained and the study was conducted according to the Declaration of Helsinki and approved by the institutional review board for human research.

Experimental settings, determination of joint angles and angular velocity

Bilateral leg extensions were performed on a motor-driven leg press dynamometer (IsoMed2000, D&r Ferstl gmbH, germany). Subjects were placed on the dynamometer with the horizontal seat always reclined to 5° and the backrest reclined to 50°. the pelvis was secured by a safety belt and upper body by two safety belts and two shoulder pads. the footrest with the force plate was rotated by 15° from vertical towards plantar flexion and fixed. Foot placement always resulted in a vertical distance of 0.1 m between the heel and the height of the seat (Fig. 1). the T-θ-r was deter-mined over a rOM from 30 to 100° knee flexion (0° refers to the straight leg), and measurements were made in incre-ments of 10° knee angle (see Hahn et al. 2011). Individual dynamometer positions resulting in the desired joint angles were determined as described in Hahn et al. (2011) and in accordance with a two-segment model of the lower limb (Hahn et al. 2005). In addition, T-ω relations for shorten-ing and lengthening conditions were determined over the same rOM (30–100° knee flexion) at mean angular veloci-ties of the knee of 30, 60, 120, 180, and 240° s−1, respec-tively (the last velocity only for concentric muscle action). the two-segment model of the lower limb also accounted

Fig. 1 Experimental setting in the leg press dynamometer. the fig-ure shows a subject placed on the dynamometers seat with reflec-tive markers attached to the lower extremity and the force plates. Although EMg was recorded during the experiments, data are not presented here

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for the acceleration of the system; therefore, the kinematic parameters of the dynamometer that resulted in the desired mean angular velocities refer to the isovelocity phase of the dynamometer only. Due to a non-linear function between dynamometer translation and angular rotation of the knee, the mean angular velocity accounts for the fact that the instantaneous angular velocity increases through-out leg extension despite the constant linear speed of the dynamometer. to verify actual angular velocity condi-tions, the kinematics of all movements were measured using a motion analysis system (see “Data processing”). More detailed information on the model calculations can be found in our previous work (Hahn et al. 2005).

Experimental protocol

Subjects attended four sessions on different days, with at least one rest day between sessions. In two preparation sessions, subjects were familiarised and trained to per-form maximal voluntary isometric, concentric, and eccen-tric contractions. Subjects were instructed to develop their maximum force and then to maintain maximal effort for the duration of the test. Start and end of each contraction were clearly announced and verbal encouragement was given during contractions. In the third and fourth session subjects performed the test protocol for the determination of force/torque–angle and force/torque–velocity relations. the iso-metric tests were split into three sets consisting of eight iso-metric contractions each at a different angle configuration of hip, knee and ankle joints (Hahn et al. 2011). For each angle configuration, subjects had to make 3 repetitions, resulting in a total of 24 contractions. to avoid learning or sequence effects, isometric contractions at different knee angles were presented in a random order in any set. In the fourth session, subjects performed five sets of three concentric contractions and four sets of three eccentric contractions (total of 27 con-tractions), each set at a given mean angular velocity of the knee. Sets were presented in a random order and dynamic contractions were released only when subjects reached a 95 % preload of their angle-specific external reaction force. During both sessions subjects were given as much rest as requested, but a minimum rest of 3 and 5 min was strictly enforced between contractions and sets, respectively.

Data processing

In general procedures were identical to Hahn et al. (2011). In brief, external reaction forces (Fext) were measured for each leg by force plates with three-component force sen-sors (KIStlEr, Switzerland). Based on a slightly modified Plug-In-gait-markerset (charlton et al. 2004) a VIcOn MX-3 Motion-System (Vicon Motion Systems, UK) served

for measuring lower extremity kinematics. capturing fre-quency was 240 Hz and kinematic measurements were synchronised with force measurements by software. Joint torques for the knee (MK) were calculated by methods of inverse dynamics but are reported for sagittal plane only. to account for inertial effects the anthropometric model was scaled to each subject’s weight and body height by linear regression (Zatsiorsky et al. 1984). In addition, Zatiorsky’s segment inertial parameters were adjusted according to de leva (1996). Forces and joint torques were smoothed using a recursive fourth order Butterworth low-pass filter with cut-off frequencies of 7 and 6 Hz, respectively and results from the right leg were used for analysis. For the determi-nation of the force/torque–angle relations, peak resultant Fext was determined from smoothed force–time histories for each knee flexion angle. Subsequently, corresponding joint torques in the knee joint was taken at the same instant of time. Angle-specific Fext and MK of the dynamic condi-tions were taken from force/torque–time histories when the knee joint angles corresponded to the isometric trials, except for 30 and 100° knee flexion since theses angles cor-respond to the start and finish positions of dynamic mus-cle actions. Both, concentric and eccentric forces/torques were normalised through division by their angle-specific isometric references. this allows for analysis of the effects of shortening and stretching velocities on force/torque pro-duction independently of the absolute forces/torques, and allows for comparisons of these effects across joint angles and subjects.

Follow-up

For the determination of maximum angular velocity of the knee (ωmax) during multi-joint leg extension, eleven male subjects (31.1 ± 8.2 year, 1.79 ± 0.05 m, 75.9 ± 4.7 kg) were instructed to extend their unresisted legs as fast as possible. Experiments were done in the same seated leg press used for the previous experiments but contractions started without a preload from the resting state at maxi-mum knee flexion. the latter guaranteed for approximately 130–150 ms time for force development to reach near maximum angle-specific velocity between 90 and 50° knee flexion. Subjects performed at least six repetitions and angular velocity of the knee was measured using motion analysis and the markerset and camera system described above. the best (i.e. fastest) trial was used for analysis. In addition to the follow-up, ωmax was extrapolated from the originally measured force data by linear regression and Hills hyperbola (Hill 1938). For Hill’s approximation, the thermodynamic constant a was replaced by the function a = 0.16F0 + 0.18F (Hill 1964). this allowed for a com-parison between calculated and measured ωmax.

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Statistics

After determining the best trials of each subject (i.e. the isometric trials with the greatest leg extension force and the dynamic trials with the greatest area under the force–knee angle curve), mean values (±1SD) were calculated. normality of the data was tested using the Kolmogorov–Smirnov test, and a two-way repeated AnOVA with Bon-ferroni–Holm post hoc comparisons, was used to identify significant differences between torques at different joint angles and angular velocities. Further, repeated measures AnOVA with Bonferroni–Holm post hoc comparisons served for identifying differences between angle-specific ωmax. Statistical analysis was performed using SPSS 14.0 for Windows (SPSS, chicago, Il, USA) and the level of significance was set a priori to p ≤ .05.

Results

Force/torque–angle relations

With increasing knee joint flexion, single leg Fext of bilat-eral contractions decreased from 3,369 ± 575 n at 30° knee flexion to 1,015 ± 152 n at 100° knee flexion in a non-linear manner (Fig. 2). For isometric MK, we found an ascending–descending T-θ-r with a mean maximum torque of 281 ± 48 nm. Maximum knee torques occurred mostly between 50 and 60° of knee flexion, resulting in a mean optimum angle of 53 ± 7° of knee flexion. For concentric muscle action, we observed reduced torques and a shift of θ0 towards increased knee flexion with increasing speeds

of contraction (table 1). Moreover, concentric contractions appeared to flatten the T-θ-r for both, Fext and Mk (Fig. 3). Eccentric muscle action led to significant enhanced forces for joint flexion angles greater than θ0 with peak normal-ised Fext and MK of 1.15 ± 0.07 F0 and 1.32 ± 0.16 M0 both occurring at 90° knee flexion.

Force/torque–angular velocity relation

Fext and Mk decreased with increasing concentric angular velocity, and there was a clear angle dependence of the shape of the force/torque-ω-r (Fig. 4). In addition, ωmax obtained by extrapolation of the experimental data by lin-ear regression and according to Hills hyperbola increased with more flexed knee joints. As a result of this, maximum power output was shifted towards more flexed knee joint angles. the force/torque-ω-r calculated by linear regression and Hills hyperbolic function showed good agreement with experimental data. correlation coefficients ranged between r = .86 ± .12 and r = .94 ± .04 for linear regression and r = .87 ± .12 and r = .94 ± .03 for Hills hyperbolic func-tion. In addition, AnOVA revealed that r was not influ-enced by joint angle but by the type of regression with Hills hyperbola showing slightly better results than linear regres-sion. this was true for Fext as well as MK and for each knee flexion angle, expect for joint torques at 90° knee flexion.

the follow-up experiments showed that maximum angu-lar velocity of the knee decreased significantly (p < .001) with increasing knee joint flexion from 1,238 ± 132° s−1 at 50° knee flexion to 866 ± 79° s−1 at 90° knee flexion. In addition, ωmax was independent of maximum leg extension

Fig. 2 Exemplar force time records for the right leg of a single sub-ject as measured during bilateral isometric contractions at the knee joint angles tested. According to the legends of each trace, the exter-nal reaction force decreases from 30 to 100° knee flexion in a non-linear manner from top to bottom

Fig. 3 torque-angle relationship of the knee joint for isometric (iso) and concentric data (con) at given mean angular velocities of 30° s−1 up to 240° s−1 (sidewise triangles). According to the symbols, veloc-ity increases from top to bottom. T-θ properties for different veloci-ties do not show the same shape and reveal a shift of θ0 to the right. As indicated by cartoons, 0° knee flexion refers to the straight leg

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force F0. For six subjects who participated in the original and the follow-up study, the experimentally determined ωmax allowed for normalisation of the angular velocity from the original study. the maximum mean angular velocity of 240° s−1 from the original study therefore corresponds to approx. 0.25 ωmax. For F-ω relations normalised to F0 and ωmax, regressions using Hill’s hyperbolic function showed slightly higher correlation coefficients than linear regres-sions. However, differences between linear and hyperbolic regressions only became significant for knee joint flexion angles ≤70°. Except for 90° knee flexion, F0 obtained by extrapolation was always too low for linear and hyperbolic approximation, but was significantly better for Hill’s hyper-bolic approximation than linear regression.

For the eccentric contractions, there was a strong angle dependence of the velocity relation for both, Fext and MK. While angular velocity had no effect on force/torque for knee joint angles less than or equal to velocity-specific θ0, eccentric force/torque varied systematically with angu-lar velocity for knee joint angles greater than velocity-specific θ0. For joint angles less than or equal to θ0, Fext and MK ranged between 0.94 ± 0.08 and 1.17 ± 0.12 of the corresponding isometric values, with only MK at θ0 and ω = −34.3 ± 6.1° s−1 being significantly greater than its isometric reference. When joint angles exceeded θ0, there were significant and more consistent force/torque increases with stretching (Fig. 4). For example, at a 60° knee flexion angle, MK peaked at 1.24 ± 0.16M0 for ω = −32.0 ± 2.9° s−1, dropped to 1.12 ± 0.16M0 for ω = −59.0 ± 7.0° s−1 and did not show torques beyond

isometric for the remaining eccentric angular velocities (ω = −157.7 ± 35.0 and −247.5 ± 34.5° s−1). In con-trast, for the greatest knee joint angle tested (90° knee flexion) there was a significant increase in torque for all angular velocities. However, the torques changed signifi-cantly with angular velocity between 1.14 ± 0.15M0 and 1.32 ± 0.16M0 (table 2).

Force/torque–angle–velocity relation

combining the results described above allows for the con-struction of three-dimensional force/torque–angle–angular velocity plots. For the specific motion in the specific device used, these plots provide a more complete description of in vivo human muscle function by showing the interdepend-ence of force/torque, angle and angular velocity during multi-joint leg extension (Fig. 5).

Discussion

Our approach allowed for analysis of the combined effects of joint angle position, angular velocity and contraction type on force/torque production during multi-joint leg extensions. For concentric muscle action, the main findings indicate a flattened torque–angle relationship, a shift of the optimal joint angle, and an angle-dependent change in the shape of the force/torque–angular velocity relationship. In contrast to the literature, eccentric forces/torques sig-nificantly exceeded their angle-specific isometric reference

a b

Fig. 4 Force/torque–velocity relations for different joint angles whereas θ0 stands for optimum joint angle and numbers indicate knee joint flexion angle. note that each data point represents the group mean of angle-specific measurement at a given mean angular veloc-ity. Further note that due to the mechanics of multi-joint leg exten-sion (see “Methods” section and Hahn et al. 2005), instantaneous angle-specific velocities are partly higher and lower than predeter-mined mean angular velocities. For both concentric (positive x-axis

values) and eccentric muscle action (negative x-axis values) left side (a) represents normalised external reaction forces (Fext), whereas on the right side (b) knee joint torques (MK) are shown. Grey curves represent F-v properties for joint angles >θ0 (black) and light grey lines for joint angles <θ0. Since Fext θ0 during concentric contractions always occurred at the end of the leg extension during deceleration of the dynamometer, data was excluded from analysis so that there is no black line for concentric contractions in (a)

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values, and further depended on the angular velocity, at least at muscle lengths longer than optimal. these results are discussed in view of previous studies, and with the aim to identify possible mechanisms that can be used to explain the findings. We then attempt to identify the relevance and application of our results.

Force/torque–angle relations

the effect of joint positions on Fext for multi-joint leg extensions is well documented (Hugh-Jones 1947) and was discussed elsewhere (Hahn 2011). For isometric contrac-tions, the T-θ-r for MK agrees well with findings described in the literature using single-joint experiments (Pincivero et al. 2004). therefore, it is assumed that the quadri-ceps femoris muscles operate at maximum effort during multi-joint leg extensions. Our results from concentric

contractions presented in Fig. 3 further correspond to James et al. (1994) who found that the shape of single-joint quadriceps femoris length–tension curves is flattened with angular velocity. However, the shift of θ0 towards greater knee flexion for concentric contractions is in contrast to previous results obtained from single-joint or -muscle experiments (Brown et al. 1999; Fuglevand 1987; Marshall et al. 1990; de Haan et al. 2003). this result is also in con-trast to what might be expected from a mechanical point of view. Since MK decreased with increasing angular velocity, for a given joint angle one would expect series elastic ele-ment lengths to be shorter and fascicle lengths to be longer the faster the contraction (Forrester and Pain 2010). If opti-mal length is assumed to occur at a given fascicle length, then this should lead to smaller θ0 with increasing speeds of shortening. However, for decreasing levels of activation (Austin et al. 2010; Brown et al. 1999; guimaraes et al.

Table 2 Influence of joint angle and angular velocity on eccentric forces and torques normalised to their corresponding isometric values

Values are mean ± SD. reading from left to right shows the influence of joint angle at a given angular velocity, reading from top to bottom the influence of angular velocity at a given joint angle

* Indicates a significant increment of force or torque in comparison to corresponding isometric levels (p ≤ .05)

normalised force or torque at (velocity)

Knee joint flexion angle [°]

40 50 60 70 80 90

Fext (−30° s−1) 0.94 ± 0.08 1.03 ± 0.14 1.09 ± 0.12* 1.13 ± 0.12* 1.13 ± 0.09* 1.10 ± 0.10*

Fext (−60° s−1) 0.97 ± 0.09 0.95 ± 0.14 1.01 ± 0.10 1.10 ± 0.09* 1.14 ± 0.08* 1.15 ± 0.07*

Fext (−120° s−1) 1.03 ± 0.09 1.02 ± 0.14 0.89 ± 0.14 0.88 ± 0.12 1.05 ± 0.10 1.13 ± 0.11*

Fext (−180° s−1) 1.04 ± 0.09 1.05 ± 0.12 0.96 ± 0.13 0.85 ± 0.12 0.90 ± 0.07 1.02 ± 0.09

MK (−30° s−1) 1.01 ± 0.13 1.09 ± 0.16 1.24 ± 0.16* 1.26 ± 0.15* 1.25 ± 0.16* 1.23 ± 0.15*

MK (−60° s−1) 1.06 ± 0.18 0.98 ± 0.19 1.12 ± 0.16* 1.22 ± 0.13* 1.26 ± 0.14* 1.27 ± 0.14*

MK (−120° s−1) 1.09 ± 0.17 1.03 ± 0.19 0.97 ± 0.21 0.99 ± 0.16 1.20 ± 0.18* 1.32 ± 0.16*

MK (−180° s−1) 1.09 ± 0.13 1.04 ± 0.14 1.01 ± 0.19 0.93 ± 0.16 0.98 ± 0.15 1.14 ± 0.15*

Fig. 5 three-dimensional torque–angle–angular veloc-ity relation from group data to indicate the interdependence of normalised knee joint torque (y-axis), joint angle (z-axis), and angular velocity (x-axis). Positive x values represent concentric and negative values eccentric muscle action. All data points have been normalised to their corresponding angle-specific isometric maximum and the colour scale indicates normalised knee joint torque. Knee flexion of 0° refers to the straight leg

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1994; Huijing 1996), there is a shift of l0 towards longer muscle lengths. this shift defies force expectations based on filament overlap (Brown et al. 1999) and is thought to be caused by an increased ca2+ sensitivity at long muscle length (roszek et al. 1994; Stephenson and Wendt 1984).

However, there are other factors than ca2+ sensitivity that may have contributed to the results observed in Fig. 6. First, for a given constant speed of the leg press dynamom-eter, the instantaneous angular velocity increases through-out leg extension. therefore, the decrease in normalised Fext and MK with decreasing knee flexion may partly be explained with an increasing speed of shortening. never-theless, when eliminating the effects of velocity by com-paring two data points (see grey circles in Fig. 6b) with matching angular velocities (116 ± 9 vs. 118 ± 14° s−1), normalised MK still differs (0.55 ± 0.1M0 vs. 0.82 ± 0.1M0 at 60 vs. 90° knee angle). thus, other reasons than veloc-ity must affect the normalised Fext and MK and might be related to the fact that leg extensions mainly take place on the descending limb of the T-θ-r of the knee. therefore, additional cross-bridges can interact with actin when the leg is extended. Assuming that the difference in MK in the example above is caused by the time needed for additional cross-bridge bindings and their force production as well as Mtc series elasticity, one could argue that this effect is caused by changes in joint angle position. Second, the phenomenon of dynamic force depression (lee and Herzog 2003; McDaniel et al. 2010) has to be taken into account. Dynamic force depression is positively related to the amplitude of muscle shortening; therefore, force depres-sion increases with leg extension, being smallest at 90° and strongest at 50° knee flexion.

Force/torque–velocity relation

In accordance with the literature, Fext and MK decreased with increasing velocity (Fenn and Marsh 1935; Wilkie 1950) and the curvature of the F/T-ω relations varied as a function of joint angle, especially since the slope between the isometric data point and the first data point for concen-tric contractions was observed to change with joint angle (Fig. 4). In agreement with findings by others, this required Hill’s constants a and b to be adjusted with joint angles (chow and Darling 1999; de Haan et al. 2003; Fuglevand 1987; granzier et al. 1989; Krylow and Sandercock 1997). the mechanisms that contribute to this result have already been discussed in the section on the T-θ-r. Further, Hill’s hyperbolic force–velocity relationship gave a better fit to our results than linear regression, in contrast to Yamauchi et al. (2007) who reported better fits of their force–veloc-ity relation with linear compared to exponential functions. Others (Bosco et al. 1995; Pearson et al. 2004; rahmani et al. 2001) also found that linear regression fit lower limb multi-joint force–velocity data well, but did not compare their findings to exponential or hyperbolic functions. the linear force–velocity relation was recently explained by purely mechanical factors of segmental dynamics; how-ever, the modelling approach used was strongly simpli-fied with only one knee extensor muscle–tendon complex incorporated to the model (Bobbert 2012). However, since both hyperbolic and linear regressions correlated well with force/torque–velocity data (correlation coefficients from 0.86 ± 0.12 to 0.99 (no SD provided by Yamauchi et al. 2007), depending on joint angle they appear equally suited for representing force/torque–angular velocity properties of

a b

Fig. 6 complete sets of F/T-θ data for external forces (a) and knee joint torques (b) for isometric (iso) and concentric muscle action (con) at different mean angular velocities. the effects of different force/torque level due to joint angle were removed in these data sets by normalising the forces/torques to the corresponding angle-specific isometric data. Furthermore grey circles in (b) represent two data points with identical angular velocity eliminating any influence of

angular velocity that emerges from the mechanics of multi-joint leg extension (see “Methods” and Hahn et al. 2005). the different slopes therefore indicate a potential influence of factors like ca2+-sensitiv-ity, time for cross-bridge binding, Mtc elasticity and dynamic force depression on F/T-θ properties during multi-joint leg extension at dif-ferent velocities. As indicated by cartoons, 0° knee flexion refers to the straight leg

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multi-joint leg extensions, at least for velocities up to 0.25 ωmax (Fig. 7).

Velocity of unresisted shortening

According to previous reports vmax decreases with muscle length (Edman 1979; Huxley and Julian 1964; Krylow and Sandercock 1997) and using linear and hyperbolic regres-sions, our calculations showed that ωmax also decreased when the leg was straightened (i.e. with decreasing knee extensor length). However, this is in contrast to the results of the follow-up experiments where ωmax increased when the leg was straightened, and since ωmax was independent of the maximum isometric force F0, it is thought to rep-resent the actual ωmax of unresisted shortening (Yamauchi et al. 2007). Moreover, when estimating ωmax by regression

analysis, it varied strongly between different types of regression analysis and was overestimated when using the hyperbolic function but underestimated by linear regression (Fig. 7; table 3). this is in contrast to the study of Yamauchi et al. (2007) where maximal speed of the leg press and F0 as measured could be extrapolated accurately by linear regres-sion. However, compared to our results their unresisted shortening velocity was relatively slow (2.2 ms−1, which approximately corresponds to an ω of 600–700° s−1 as cal-culated by Hahn et al. 2005) and is doubted to represent the actual possible maximum speed of unresisted leg extension (Forrester et al. 2011; rahmani et al. 2004).

nevertheless, our results also suggest that neither linear extrapolation nor Hill’s hyperbolic relationship was able to predict ωmax properly from the small range of velocities tested. Hill’s hyperbolic force–velocity relationship was obtained for isolated muscle preparations tested at opti-mum muscle length, and activated by electrical stimulation. In contrast, Fext obtained during multi-joint leg extensions is not a direct measure of an isolated muscle property but rather represents results from a double transformation of muscle forces and torques from multiple muscles into a single leg extensor force Fext (Bobbert 2012; Zatsiorsky 2003). Based on our results, it is assumed that F-ω rela-tions in multi-joint leg extension have a hyperbolic shape until an unknown critical velocity is reached. Beyond that velocity, F-ω relations are supposed to have an inflection point so that Fext and MK quickly drop to zero and ωmax is reached, resulting in concave–convex F-ω relations similar to Marshall et al. (1990). Because of the angle-dependent variations in ωmax, it further appears that this critical veloc-ity increases with decreasing knee joint flexion. However, to confirm these interpretations on in vivo multi-joint mus-cle function, further research and experimental evidence by force/torque measurements over the full range of velocities is required (Forrester et al. 2011).

Eccentric force/torque–velocity relation

Finally, regarding eccentric contractions, there were three main findings. First, normalised eccentric Fext and

Fig. 7 Exemplar Fext-ω-r for a single subject normalised to maxi-mum isometric force F0 and maximum unresisted angular velocity ωmax. that means the values of 1.0 on the two axes represents the angle-specific isometric force and ωmax as obtained during unresisted shortening, respectively. As indicated by correlations and also by a residual analysis, measured data (black squares) fit slightly better to Hills hyperbolic function (grey line) than to linear regression (light grey line). Extrapolations show a systematic underestimation and overestimation of ωmax by linear and hyperbolic functions, respec-tively

Table 3 Maximum external reaction force F0 and velocity ωmax for unresisted shortening as extrapolated by linear and hyperbolic functions (n = 6)

Both, F0 and ωmax, are normalised to their corresponding values as measured during the follow-up experiment. Values are mean ± SD

Parameter extrapolated by regression type

Knee joint flexion angle [°]

50 60 70 80 90

F0 linear 0.84 ± 0.07 0.86 ± 0.07 0.89 ± 0.05 0.95 ± 0.07 0.99 ± 0.05

F0 Hill 0.91 ± 0.06 0.92 ± 0.05 0.93 ± 0.04 0.97 ± 0.06 1.00 ± 0.05

ωmax linear 0.32 ± 0.04 0.38 ± 0.06 0.42 ± 0.06 0.59 ± 0.13 0.78 ± 0.16

ωmax Hill 1.25 ± 0.30 1.66 ± 0.45 2.05 ± 0.49 3.40 ± 1.16 5.09 ± 1.30

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MK increased with knee joint flexion. Second, a non-sys-tematic dependence of eccentric force/torque on angu-lar velocity was observed for knee joint flexion angles greater than θ0 (Fig. 8). In agreement with the literature eccentric forces and torques for joint angles less or equal to θ0 were largely unaffected by changes in velocity and showed almost no significant increase in force (Enoka 1996; Webber and Kriellaars 1997; Westing et al. 1988). this finding is usually explained by neural inhibition (Duclay et al. 2011; gruber et al. 2009) and/or inappro-priate cross-bridge attachment at muscle lengths below l0 (Scott et al. 1996). When joint angles exceeded θ0 (≥60° knee flexion), a significant increase in force/torque of up to 1.32 ± 0.16M0 was observed. In some studies similar results to those found here were reported for cat soleus (Brown et al. 1996; Scott et al. 1996) and single fibres from frogs (granzier et al. 1989), and the origin of increas-ing eccentric forces for longer muscle lengths might be caused by decreases in the myofilament lattice spacing (Edman 1999). However, the higher eccentric forces/tor-ques at angles ≥θ0 may merely be a result of increasing stretch amplitude (lee and Herzog 2002). Furthermore, we cannot provide a good explanation for the decrease in eccentric forces with increasing stretching speeds at knee joint angles ≥θ0, a result that is in contrast to the gener-ally accepted findings in the literature (Seger and thor-stensson 1994; Westing et al. 1988). Possibly this result is caused by muscle inhibition to protect the quadriceps group against extreme tension, but further research on this find-ing is required. third, normalised eccentric MK always exceeded normalised Fext (table 2; Fig 8). this probably results from the fact that during stretch there was only inconsistent and small enhancement of ankle joint torque

(0.80 ± 0.15–1.17 ± 0.17M0_ankle) and no enhancement of hip joint torques (0.76 ± 0.15–1.03 ± 0.17M0_hip).

In conclusion, we found that force and torque produc-tion during multi-joint leg extension strongly depends on the instantaneous combination of joint angle, angular velocity and the type of muscle action. For concentric muscle action, increasing angular velocities flattened the force/torque–angle relationship and caused a shift of the optimum joint angle towards longer muscle lengths. Fur-thermore, the shape of force/torque–angular velocity rela-tionships varied with joint angle, whereby Hill’s hyperbola and linear functions showed equivalent approximations. However, linear and hyperbolic approximations failed in predicting force/torque production adequately for our multi-joint leg extension movements across the entire range of velocities. For eccentric muscle actions, force/torque did not exceed the corresponding isometric refer-ence value for muscle lengths shorter or equal to optimal, whereas eccentric force/torque for lengths beyond the opti-mal muscle length exceeded the corresponding angle-spe-cific isometric reference values by as much as 32 ± 16 %. Eccentric forces/torques further showed some non-system-atic changes with stretching speed that we cannot explain satisfactorily.

Summarising, future research should include experi-ments covering the entire range of muscle shortening velocities and should be aimed at elucidating the mecha-nisms underlying the findings presented here. notwith-standing missing explanations, real-life movements and rehabilitation exercises typically involve multi-joint move-ments, thus our results should be taken into consideration when modelling human movement or designing training concepts.

a b

Fig. 8 complete sets of normalised force/torque–angle data for external forces (a) and knee joint torques (b) for isometric (iso) and eccentric muscle action (ecc) at different mean angular velocities. From left to right the graphs indicate the influence of increasing

joint angles on eccentric force or torque enhancement. the distance between data points at a given joint angle indicates the influence of stretching speed on eccentric force or torque production. As indicated by cartoons, 0° knee flexion refers to the straight leg

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Conflict of interest the authors do not have a conflict of interest.

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