Interdependence of Torque, Joint Angle, Angular Velocity and Muscle Action During Human Multi-Joint Leg Extension

Home , Action (physics), Torque, Velocity

Eur J Appl Physiol (2014) 114:1691–1702 DOI 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

Abstract Results For contractions of increasing velocity, opti- Purpose Force and torque production of human muscles mum knee angle shifted from 52 7 to 64 4° knee flex- ± ± depends upon their lengths and contraction velocity. How- ion. Furthermore, the curvature of the concentric force/ ever, these factors are widely assumed to be independent of torque–angular velocity relations varied with joint angles each other and the few studies that dealt with interactions and maximum angular velocities increased from 866 79 1 ± of torque, angle and angular velocity are based on isolated to 1,238 132° s− for 90–50° knee flexion. Normal- ± single-joint movements. Thus, the purpose of this study ised eccentric forces/torques ranged from 0.85 0.12 ± was to determine force/torque–angle and force/torque– to 1.32 0.16 of their isometric reference, only show- ± angular velocity properties for multi-joint leg extensions. ing significant increases above isometric and an effect Methods Human leg extension was investigated (n 18) of angular velocity for joint angles greater than optimum = on a motor-driven leg press dynamometer while measuring knee angle. external reaction forces at the feet. Extensor torque in the Conclusions the findings reveal that force/torque produc- knee joint was calculated using inverse dynamics. Isometric tion during multi-joint leg extension depends on the com- contractions were performed at eight joint angle configura- bined effects of angle and angular velocity. This finding tions of the lower limb corresponding to increments of 10° should be accounted for in modelling and optimisation of at the knee from 30 to 100° of knee flexion. Concentric and human movement. eccentric contractions were performed over the same range of motion at mean angular velocities of the knee from 30 to Keywords Knee joint torques · Maximum unresisted 1 240° s− . velocity · Multi-joint leg extension · Torque–angle relationship · Torque–velocity relationship

Abbreviations Communicated by Olivier Seynnes. ANOVA Analysis of variance θ Optimum velocity-specific knee joint angle * 0 D. Hahn ( ) F Angle-specific isometric external force Human Movement Science, Faculty of Sports Science, Ruhr- 0 Universität Bochum, Gesundheitscampus, Haus Nord Nr. 10, Fext External reaction force 44801 Bochum, Germany F/T-θ-r Force/torque–angle relation e-mail: [email protected] F/T-θ-ω-r Force/torque–angle–velocity relation F/T-ω-r Force/torque–velocity relation D. Hahn · A. Schwirtz Department of Biomechanics in Sports, Faculty of Sport l0 Optimum muscle length and Health Science, Technische Universität München, Munich, M0 Angle-specific isometric knee joint torque Germany MK Knee joint torque Mtc Muscle tendon complex W. Herzog Human Performance Laboratory, Faculty of Kinesiology, ROM range of motion University of Calgary, Calgary, Canada SD Standard deviation

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vmax Maximum velocity of unloaded shortening of between length, velocity and the degree of activation for in an (isolated) muscle vitro muscles (Abbott and Wilkie 1953; Bahler et al. 1968;

ωmax Maximum angular velocity during unrestricted Brown et al. 1996, 1999; Granzier et al. 1989; Haan et al. leg extension 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- Introduction tively rare and almost limited to the knee extensor muscles. For concentric contractions with increasing velocity James Muscular force and power production depends on the et al. (1994) found that the convex shape of the length–ten- instantaneous contractile conditions determined by muscle sion curve was lost. Thorstensson et al. (1976) observed a length and the rate and direction of length change. For iso- systematic shift of optimal joint angle θ0 to more extended lated muscles this is expressed by the force–length (Gor- knee joint, i.e. the optimal muscle–tendon complex (MTC) don et al. 1966) and the force–velocity relations (Hill 1938; length became shorter when angular velocity increased. Katz 1939) and explained by cross-bridge cycling (Hux- This change of the T-θ-r was confirmed by others (Fug- ley 1957). Instead, for in vivo human muscles, maximum levand 1987; Marshall et al. 1990) and is associated with voluntary torque is expressed as a function of joint angle effects of MTC series elasticity (Kawakami et al. 2002). (T-θ-r) and angular velocity (T-ω-r). In addition to con- Fuglevand (1987) was the first to describe an experimen- tractile properties of muscle fibres, the torque-output of in tally based torque–angle–angular velocity relationship vivo human muscle action reflects the interaction of mus- (T-θ-ω-r) for concentric muscle action of the human knee cle architecture (e.g. Finni 2006), joint geometry (Krevo- extensor muscles. For extended knee joint positions, he lin et al. 2004), elasticity of the musculo-tendinous tissues found a plateau in the T-ω-r, indicating that Hill’s curve (e.g. Kawakami et al. 2002) and neural activation (Pasquet must be adapted to joint angle when used for modelling the et al. 2005, 2006). Although the contribution of each factor T-ω-r. After a conversion of joint torque into tendon force to the resulting torque is difficult to discriminate, the T-θ Marshall et al. (1990) confirmed these findings and showed and T-ω relationships are important properties to character- that maximum shortening velocity depended on muscle ise in vivo human muscle function. They further represent length. In a combined experimental–theoretical study on subject-specific strength capability, which can be used as isolated concentric knee extensions (Chow et al. 1999a, input for torque-driven models of human movement (King b, c) predictions of the measured knee torques were more and Yeadon 2002) or serve as validation criteria for muscle- accurate when Hill’s constants a and b were varied with driven simulations (Delp et al. 2007; Pandy et al. 1990). muscle length. There is lot of research on the T-θ and T-ω relations Data on the interdependence of torque, joint angle of human muscles (Dudley et al. 1990; Kulig et al. 1984; and angular velocity during eccentric muscle action are Maganaris 2004; Pincivero et al. 2004; Seger and Thor- even sparser. Westing et al. (1988) showed that eccen- stensson 2000; Webber and Kriellaars 1997; Westing et al. tric torques varied within 0.9–1.18 of the isometric ref- 1988; Wilkie 1950) and it is still widely assumed that joint erences depending on muscle length and velocity (see angle and angular velocity can be considered as independ- their Table 1). However, these interactions were neither ent regulators of torque-output that simply need scaling systematic nor analysed statistically. In a more recent to the appropriate level of activation. Accordingly, the study (Forrester and Pain 2010) no enhanced torques and majority of these studies have focused only on joint angle no systematic interdependence of calculated fibre forces, or angular velocity effects separately. This approach has muscle lengths and lengthening velocities were found but been criticised (Epstein and Herzog 2003; Forrester and maximum voluntary eccentric joint torque decreased with Pain 2010; Huijing 1998) and there is numerous experi- increasing stretching velocity (Forrester and Pain 2010; mental evidence suggesting a complex interdependence 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 1 Muscle action (angular velocity [° s− ]) iso (0)a con (30)b con (60)c con (120)d con(180)e

θ [°] 52.2 6.6 55.5 7.3 58.4 7.2 62.7 5.0 63.8 3.8 0 ± ± ± ± ± Significant to c, d, e d, e a, d, e a, b, c a, b, c

Values are mean SD. Superior lowercase letters indicate significant differences between the columns data and the datasets indicated by the letters. Level of significance± p .05 ≤ 1 3 Eur J Appl Physiol (2014) 114:1691–1702 1693 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- Fig. 1 Experimental setting in the leg press dynamometer. The fig- ure shows a subject placed on the dynamometers seat with reflec- ity ωmax varies with joint angle. tive markers attached to the lower extremity and the force plates. For joints other than the knee, data on the interdepend- Although EMG was recorded during the experiments, data are not ence of joint angle and angular velocity including eccentric presented here 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 Materials and methods torque as a function of joint angle and joint angular velocity. Similarly, Anderson et al. (2007) presented a model Subjects predicting maximum voluntary joint torques as a function of joint angle and angular velocity for the ankle, knee and Male subjects (n 18; 30 6.3 year, 1.81 0.08 m, = ± ± hip. However, they only studied three velocities of contrac- 77.9 5.2 kg) without neuromuscular disorders or injuries ± tion, thus their predicted T-θ-v relationships were smooth participated in this study. Free, written informed consent and they neither detected a shift in θ0 with increasing angu- was obtained and the study was conducted according to the lar velocity, nor a change in ωmax with joint angle. For the Declaration of Helsinki and approved by the institutional eccentric part, one velocity was considered only; therefore, review board for human research. it is impossible to derive a T-θ-ω-r. Data of all experimental studies on voluntary human Experimental settings, determination of joint angles muscle function presented above were obtained from iso- and angular velocity lated joint activities, whereas everyday human movements involve multi-joint actions, such as leg extensions. Thus, Bilateral leg extensions were performed on a motor-driven it is to question if observations of single-joint actions are leg press dynamometer (IsoMed2000, D&R Ferstl GmbH, valid for multi-joint activities. This is supported by find- Germany). Subjects were placed on the dynamometer with ings of others (Rahmani et al. 2001; Yamauchi et al. 2007) the horizontal seat always reclined to 5° and the backrest who found a linear instead of a hyperbolic force–velocity reclined to 50°. The pelvis was secured by a safety belt relation for multi-joint leg extension and it has further been and upper body by two safety belts and two shoulder pads. demonstrated that torque–angle relations of ankle and knee The footrest with the force plate was rotated by 15° from joint differ between single- and multi-joint tasks (Hahn vertical towards plantar flexion and fixed. Foot placement et al. 2011). always resulted in a vertical distance of 0.1 m between the To our knowledge there are no data available on the heel and the height of the seat (Fig. 1). The T-θ-r was deter- combined effects of joint angle and angular velocity on mined over a ROM from 30 to 100° knee flexion (0° refers voluntary force/torque production. Therefore, the aim of to the straight leg), and measurements were made in incre- this study was to investigate the interdependence of force/ ments of 10° knee angle (see Hahn et al. 2011). Individual torque, joint angles, and angular velocity during multi- dynamometer positions resulting in the desired joint angles joint leg extension, which mimics the muscle action con- were determined as described in Hahn et al. (2011) and in ditions of real-life movements. According to the literature accordance with a two-segment model of the lower limb we hypothesise that during concentric muscle action there (Hahn et al. 2005). In addition, T-ω relations for shorten- is a shift of optimal joint angles towards leg extension with ing and lengthening conditions were determined over the linear force–angle and torque–angle relations. For eccen- same ROM (30–100° knee flexion) at mean angular veloci- 1 tric muscle action we hypothesise force and torque not ties of the knee of 30, 60, 120, 180, and 240° s− , respec- to exceed the isometric references and to decrease with tively (the last velocity only for concentric muscle action). increasing angular velocity at least beyond a certain point. The two-segment model of the lower limb also accounted

1 3 1694 Eur J Appl Physiol (2014) 114:1691–1702 for the acceleration of the system; therefore, the kinematic for measuring lower extremity kinematics. Capturing fre- parameters of the dynamometer that resulted in the desired quency was 240 Hz and kinematic measurements were mean angular velocities refer to the isovelocity phase synchronised with force measurements by software. Joint of the dynamometer only. Due to a non-linear function torques for the knee (MK) were calculated by methods of between dynamometer translation and angular rotation of inverse dynamics but are reported for sagittal plane only. To the knee, the mean angular velocity accounts for the fact account for inertial effects the anthropometric model was that the instantaneous angular velocity increases through- scaled to each subject’s weight and body height by linear out leg extension despite the constant linear speed of the regression (Zatsiorsky et al. 1984). In addition, Zatiorsky’s dynamometer. To verify actual angular velocity condi- segment inertial parameters were adjusted according to de tions, the kinematics of all movements were measured Leva (1996). Forces and joint torques were smoothed using using a motion analysis system (see “Data processing”). a recursive fourth order Butterworth low-pass filter with More detailed information on the model calculations can be cut-off frequencies of 7 and 6 Hz, respectively and results found in our previous work (Hahn et al. 2005). from the right leg were used for analysis. For the determination of the force/torque–angle relations, peak resultant

Experimental protocol Fext was determined from smoothed force–time histories for each knee flexion angle. Subsequently, corresponding Subjects attended four sessions on different days, with joint torques in the knee joint was taken at the same instant at least one rest day between sessions. In two preparation of time. Angle-specific Fext and MK of the dynamic condi- sessions, subjects were familiarised and trained to per- tions were taken from force/torque–time histories when form maximal voluntary isometric, concentric, and eccen- the knee joint angles corresponded to the isometric trials, tric contractions. Subjects were instructed to develop their except for 30 and 100° knee flexion since theses angles cor- maximum force and then to maintain maximal effort for the respond to the start and finish positions of dynamic mus- duration of the test. Start and end of each contraction were cle actions. Both, concentric and eccentric forces/torques clearly announced and verbal encouragement was given were normalised through division by their angle-specific during contractions. In the third and fourth session subjects isometric references. This allows for analysis of the effects performed the test protocol for the determination of force/ of shortening and stretching velocities on force/torque pro- torque–angle and force/torque–velocity relations. The iso- duction independently of the absolute forces/torques, and metric tests were split into three sets consisting of eight iso- allows for comparisons of these effects across joint angles metric contractions each at a different angle configuration of and subjects. hip, knee and ankle joints (Hahn et al. 2011). For each angle configuration, subjects had to make 3 repetitions, resulting Follow‑up in a total of 24 contractions. To avoid learning or sequence effects, isometric contractions at different knee angles were For the determination of maximum angular velocity of the presented in a random order in any set. In the fourth session, knee (ωmax) during multi-joint leg extension, eleven male subjects performed five sets of three concentric contractions subjects (31.1 8.2 year, 1.79 0.05 m, 75.9 4.7 kg) ± ± ± and four sets of three eccentric contractions (total of 27 con- were instructed to extend their unresisted legs as fast as tractions), each set at a given mean angular velocity of the possible. Experiments were done in the same seated leg knee. Sets were presented in a random order and dynamic press used for the previous experiments but contractions contractions were released only when subjects reached a started without a preload from the resting state at maxi- 95 % preload of their angle-specific external reaction force. mum knee flexion.T he latter guaranteed for approximately During both sessions subjects were given as much rest as 130–150 ms time for force development to reach near requested, but a minimum rest of 3 and 5 min was strictly maximum angle-specific velocity between 90 and 50° knee enforced between contractions and sets, respectively. flexion. Subjects performed at least six repetitions and angular velocity of the knee was measured using motion Data processing analysis and the markerset and camera system described above. The best (i.e. fastest) trial was used for analysis. In

In general procedures were identical to Hahn et al. (2011). addition to the follow-up, ωmax was extrapolated from the In brief, external reaction forces (Fext) were measured for originally measured force data by linear regression and each leg by force plates with three-component force sen- Hills hyperbola (Hill 1938). For Hill’s approximation, the sors (KISTLER, Switzerland). Based on a slightly modified thermodynamic constant a was replaced by the function Plug-In-Gait-markerset (Charlton et al. 2004) a VICON a 0.16F 0.18F (Hill 1964). This allowed for a com- = 0 + MX-3 Motion-System (Vicon Motion Systems, UK) served parison between calculated and measured ωmax.

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Fig. 2 Exemplar force time records for the right leg of a single subject as measured during bilateral isometric contractions at the knee Fig. 3 Torque-angle relationship of the knee joint for isometric (iso) 1 joint angles tested. According to the legends of each trace, the exter- and concentric data (con) at given mean angular velocities of 30° s− 1 nal reaction force decreases from 30 to 100° knee flexion in a non- up to 240° s− (sidewise triangles). According to the symbols, veloc- linear manner from top to bottom ity increases from top to bottom. T-θ properties for different velocities 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 Statistics of contraction (Table 1). Moreover, concentric contractions

After determining the best trials of each subject (i.e. the appeared to flatten the T-θ-r for both, Fext and Mk (Fig. 3). isometric trials with the greatest leg extension force and Eccentric muscle action led to significant enhanced forces the dynamic trials with the greatest area under the force– for joint flexion angles greater than θ0 with peak normal- knee angle curve), mean values ( 1SD) were calculated. ised F and M of 1.15 0.07 F and 1.32 0.16 M ± ext K ± 0 ± 0 Normality of the data was tested using the Kolmogorov– both occurring at 90° knee flexion. Smirnov test, and a two-way repeated ANOVA with Bon- ferroni–Holm post hoc comparisons, was used to identify Force/torque–angular velocity relation significant differences between torques at different joint angles and angular velocities. Further, repeated measures Fext and Mk decreased with increasing concentric angular ANOVA with Bonferroni–Holm post hoc comparisons velocity, and there was a clear angle dependence of the served for identifying differences between angle-specific shape of the force/torque-ω-r (Fig. 4). In addition, ωmax ωmax. Statistical analysis was performed using SPSS 14.0 obtained by extrapolation of the experimental data by lin- for Windows (SPSS, Chicago, IL, USA) and the level of ear regression and according to Hills hyperbola increased significance was set a priori to p .05. 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 Results and Hills hyperbolic function showed good agreement with experimental data. Correlation coefficients ranged between Force/torque–angle relations r .86 .12 and r .94 .04 for linear regression and = ± = ± r .87 .12 and r .94 .03 for Hills hyperbolic func- = ± = ± With increasing knee joint flexion, single leg Fext of bilat- tion. In addition, ANOVA revealed that r was not influ- eral contractions decreased from 3,369 575 N at 30° enced by joint angle but by the type of regression with Hills ± knee flexion to 1,015 152 N at 100° knee flexion in a hyperbola showing slightly better results than linear regres- ± non-linear manner (Fig. 2). For isometric MK, we found an sion. This was true for Fext as well as MK and for each knee ascending–descending T-θ-r with a mean maximum torque flexion angle, expect for joint torques at 90° knee flexion. of 281 48 Nm. Maximum knee torques occurred mostly The follow-up experiments showed that maximum angu- ± between 50 and 60° of knee flexion, resulting in a mean lar velocity of the knee decreased significantly (p < .001) 1 optimum angle of 53 7° of knee flexion. For concentric with increasing knee joint flexion from 1,238 132° s− at ± 1 ± muscle action, we observed reduced torques and a shift of 50° knee flexion to 866 79° s− at 90° knee flexion. In ± θ0 towards increased knee flexion with increasing speeds addition, ωmax was independent of maximum leg extension

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Fig. 4 Force/torque–velocity relations for different joint angles values) and eccentric muscle action (negative x-axis values) left side whereas θ0 stands for optimum joint angle and numbers indicate knee (a) represents normalised external reaction forces (Fext), whereas on joint flexion angle. Note that each data point represents the group the right side (b) knee joint torques (MK) are shown. Grey curves mean of angle-specific measurement at a given mean angular veloc- represent F-v properties for joint angles >θ0 (black) and light grey ity. Further note that due to the mechanics of multi-joint leg exten- lines for joint angles <θ0. Since Fext θ0 during concentric contractions sion (see “Methods” section and Hahn et al. 2005), instantaneous always occurred at the end of the leg extension during deceleration of angle-specific velocities are partly higher and lower than predeter- the dynamometer, data was excluded from analysis so that there is no mined mean angular velocities. For both concentric (positive x-axis black line for concentric contractions in (a)

force F0. For six subjects who participated in the original isometric for the remaining eccentric angular velocities 1 and the follow-up study, the experimentally determined (ω 157.7 35.0 and 247.5 34.5° s− ). In con- = − ± − ± ωmax allowed for normalisation of the angular velocity from trast, for the greatest knee joint angle tested (90° knee the original study. The maximum mean angular velocity of flexion) there was a significant increase in torque for all 1 240° s− from the original study therefore corresponds to angular velocities. However, the torques changed signifi- approx. 0.25 ω . For F-ω relations normalised to F and cantly with angular velocity between 1.14 0.15M and max 0 ± 0 ω , regressions using Hill’s hyperbolic function showed 1.32 0.16M (Table 2). max ± 0 slightly higher correlation coefficients than linear regressions. However, differences between linear and hyperbolic Force/torque–angle–velocity relation regressions only became significant for knee joint flexion angles 70°. Except for 90° knee flexion,F obtained by Combining the results described above allows for the con- ≤ 0 extrapolation was always too low for linear and hyperbolic struction of three-dimensional force/torque–angle–angular approximation, but was significantly better for Hill’s hyper- velocity plots. For the specific motion in the specific device bolic approximation than linear regression. used, these plots provide a more complete description of in For the eccentric contractions, there was a strong angle vivo human muscle function by showing the interdepend- dependence of the velocity relation for both, Fext and MK. ence of force/torque, angle and angular velocity during While angular velocity had no effect on force/torque for multi-joint leg extension (Fig. 5). knee joint angles less than or equal to velocity-specific

θ0, eccentric force/torque varied systematically with angular velocity for knee joint angles greater than velocity- Discussion specific θ0. For joint angles less than or equal to θ0, Fext and M ranged between 0.94 0.08 and 1.17 0.12 Our approach allowed for analysis of the combined effects K ± ± of the corresponding isometric values, with only MK at of joint angle position, angular velocity and contraction 1 θ and ω 34.3 6.1° s− being significantly greater type on force/torque production during multi-joint leg 0 = − ± than its isometric reference. When joint angles exceeded extensions. For concentric muscle action, the main findings

θ0, there were significant and more consistent force/ indicate a flattened torque–angle relationship, a shift of the torque increases with stretching (Fig. 4). For example, at optimal joint angle, and an angle-dependent change in the a 60° knee flexion angle, MK peaked at 1.24 0.16M0 shape of the force/torque–angular velocity relationship. 1 ± for ω 32.0 2.9° s− , dropped to 1.12 0.16M0 for In contrast to the literature, eccentric forces/torques sig- = − ± 1 ± ω 59.0 7.0° s− and did not show torques beyond nificantly exceeded their angle-specific isometric reference = − ± 1 3 Eur J Appl Physiol (2014) 114:1691–1702 1697

Table 2 Influence of joint angle and angular velocity on eccentric forces and torques normalised to their corresponding isometric values Normalised force or torque Knee joint flexion angle [°] at (velocity) 40 50 60 70 80 90

1 Fext ( 30° s− ) 0.94 0.08 1.03 0.14 1.09 0.12* 1.13 0.12* 1.13 0.09* 1.10 0.10* − 1 ± ± ± ± ± ± Fext ( 60° s− ) 0.97 0.09 0.95 0.14 1.01 0.10 1.10 0.09* 1.14 0.08* 1.15 0.07* − 1 ± ± ± ± ± ± Fext ( 120° s− ) 1.03 0.09 1.02 0.14 0.89 0.14 0.88 0.12 1.05 0.10 1.13 0.11* − 1 ± ± ± ± ± ± Fext ( 180° s− ) 1.04 0.09 1.05 0.12 0.96 0.13 0.85 0.12 0.90 0.07 1.02 0.09 − 1 ± ± ± ± ± ± MK ( 30° s− ) 1.01 0.13 1.09 0.16 1.24 0.16* 1.26 0.15* 1.25 0.16* 1.23 0.15* − 1 ± ± ± ± ± ± MK ( 60° s− ) 1.06 0.18 0.98 0.19 1.12 0.16* 1.22 0.13* 1.26 0.14* 1.27 0.14* − 1 ± ± ± ± ± ± MK ( 120° s− ) 1.09 0.17 1.03 0.19 0.97 0.21 0.99 0.16 1.20 0.18* 1.32 0.16* − 1 ± ± ± ± ± ± M ( 180° s− ) 1.09 0.13 1.04 0.14 1.01 0.19 0.93 0.16 0.98 0.15 1.14 0.15* K − ± ± ± ± ± ± 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) ≤

Fig. 5 Three-dimensional torque–angle–angular velocity 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

values, and further depended on the angular velocity, at contractions presented in Fig. 3 further correspond to least at muscle lengths longer than optimal. These results James et al. (1994) who found that the shape of single-joint are discussed in view of previous studies, and with the aim quadriceps femoris length–tension curves is flattened with to identify possible mechanisms that can be used to explain angular velocity. However, the shift of θ0 towards greater the findings. We then attempt to identify the relevance and knee flexion for concentric contractions is in contrast to application of our results. previous results obtained from single-joint or -muscle experiments (Brown et al. 1999; Fuglevand 1987; Marshall Force/torque–angle relations et al. 1990; de Haan et al. 2003). This result is also in contrast to what might be expected from a mechanical point of

The effect of joint positions on Fext for multi-joint leg view. Since MK decreased with increasing angular velocity, extensions is well documented (Hugh-Jones 1947) and was for a given joint angle one would expect series elastic ele- discussed elsewhere (Hahn 2011). For isometric contrac- ment lengths to be shorter and fascicle lengths to be longer tions, the T-θ-r for MK agrees well with findings described the faster the contraction (Forrester and Pain 2010). If opti- in the literature using single-joint experiments (Pincivero mal length is assumed to occur at a given fascicle length, et al. 2004). Therefore, it is assumed that the quadri- then this should lead to smaller θ0 with increasing speeds ceps femoris muscles operate at maximum effort during of shortening. However, for decreasing levels of activation multi-joint leg extensions. Our results from concentric (Austin et al. 2010; Brown et al. 1999; Guimaraes et al.

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Fig. 6 Complete sets of F/T-θ data for external forces (a) and knee angular velocity that emerges from the mechanics of multi-joint leg joint torques (b) for isometric (iso) and concentric muscle action extension (see “Methods” and Hahn et al. 2005). The different slopes 2 (con) at different mean angular velocities. The effects of different therefore indicate a potential influence of factors like Ca +-sensitiv- force/torque level due to joint angle were removed in these data sets ity, time for cross-bridge binding, MTC elasticity and dynamic force by normalising the forces/torques to the corresponding angle-specific depression on F/T-θ properties during multi-joint leg extension at dif- isometric data. Furthermore grey circles in (b) represent two data ferent velocities. As indicated by cartoons, 0° knee flexion refers to points with identical angular velocity eliminating any influence of the straight leg

1994; Huijing 1996), there is a shift of l0 towards longer Force/torque–velocity relation muscle lengths. This shift defies force expectations based on filament overlap (Brown et al. 1999) and is thought to In accordance with the literature, Fext and MK decreased 2 be caused by an increased Ca + sensitivity at long muscle with increasing velocity (Fenn and Marsh 1935; Wilkie length (Roszek et al. 1994; Stephenson and Wendt 1984). 1950) and the curvature of the F/T-ω relations varied as a 2 However, there are other factors than Ca + sensitivity function of joint angle, especially since the slope between that may have contributed to the results observed in Fig. 6. the isometric data point and the first data point for concen- First, for a given constant speed of the leg press dynamom- tric contractions was observed to change with joint angle eter, the instantaneous angular velocity increases through- (Fig. 4). In agreement with findings by others, this required out leg extension. Therefore, the decrease in normalised Hill’s constants a and b to be adjusted with joint angles

Fext and MK with decreasing knee flexion may partly be (Chow and Darling 1999; de Haan et al. 2003; Fuglevand explained with an increasing speed of shortening. Never- 1987; Granzier et al. 1989; Krylow and Sandercock 1997). theless, when eliminating the effects of velocity by com- The mechanisms that contribute to this result have already paring two data points (see grey circles in Fig. 6b) with been discussed in the section on the T-θ-r. Further, Hill’s 1 matching angular velocities (116 9 vs. 118 14° s− ), hyperbolic force–velocity relationship gave a better fit to ± ± normalised M still differs (0.55 0.1M vs. 0.82 0.1M our results than linear regression, in contrast to Yamauchi K ± 0 ± 0 at 60 vs. 90° knee angle). Thus, other reasons than veloc- et al. (2007) who reported better fits of their force–velocity must affect the normalised Fext and MK and might be ity relation with linear compared to exponential functions. related to the fact that leg extensions mainly take place on Others (Bosco et al. 1995; Pearson et al. 2004; Rahmani the descending limb of the T-θ-r of the knee. Therefore, et al. 2001) also found that linear regression fit lower limb additional cross-bridges can interact with actin when the multi-joint force–velocity data well, but did not compare leg is extended. Assuming that the difference in MK in the their findings to exponential or hyperbolic functions.T he example above is caused by the time needed for additional linear force–velocity relation was recently explained by cross-bridge bindings and their force production as well purely mechanical factors of segmental dynamics; how- as MTC series elasticity, one could argue that this effect ever, the modelling approach used was strongly simpli- is caused by changes in joint angle position. Second, the fied with only one knee extensor muscle–tendon complex phenomenon of dynamic force depression (Lee and Herzog incorporated to the model (Bobbert 2012). However, since 2003; McDaniel et al. 2010) has to be taken into account. both hyperbolic and linear regressions correlated well with Dynamic force depression is positively related to the force/torque–velocity data (correlation coefficients from amplitude of muscle shortening; therefore, force depres- 0.86 0.12 to 0.99 (no SD provided by Yamauchi et al. ± sion increases with leg extension, being smallest at 90° and 2007), depending on joint angle they appear equally suited strongest at 50° knee flexion. for representing force/torque–angular velocity properties of

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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 regression. However, compared to our results their unresisted 1 shortening velocity was relatively slow (2.2 ms− , which 1 approximately corresponds to an ω of 600–700° s− as calculated 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 Fig. 7 Exemplar F -ω-r for a single subject normalised to maxi- ext obtained for isolated muscle preparations tested at optimum isometric force F0 and maximum unresisted angular velocity ωmax. That means the values of 1.0 on the two axes represents the mum muscle length, and activated by electrical stimulation. angle-specific isometric force and ωmax as obtained during unresisted In contrast, Fext obtained during multi-joint leg extensions shortening, respectively. As indicated by correlations and also by a is not a direct measure of an isolated muscle property but residual analysis, measured data (black squares) fit slightly better to Hills hyperbolic function (grey line) than to linear regression (light rather represents results from a double transformation of grey line). Extrapolations show a systematic underestimation and muscle forces and torques from multiple muscles into a overestimation of ωmax by linear and hyperbolic functions, respec- single leg extensor force Fext (Bobbert 2012; Zatsiorsky tively 2003). Based on our results, it is assumed that F-ω relations in multi-joint leg extension have a hyperbolic shape multi-joint leg extensions, at least for velocities up to 0.25 until an unknown critical velocity is reached. Beyond that

ωmax (Fig. 7). velocity, F-ω relations are supposed to have an inflection point so that Fext and MK quickly drop to zero and ωmax is Velocity of unresisted shortening reached, resulting in concave–convex F-ω relations similar to Marshall et al. (1990). Because of the angle-dependent

According to previous reports vmax decreases with muscle variations in ωmax, it further appears that this critical veloc- length (Edman 1979; Huxley and Julian 1964; Krylow and ity increases with decreasing knee joint flexion. However, Sandercock 1997) and using linear and hyperbolic regres- to confirm these interpretations on in vivo multi-joint mus- sions, our calculations showed that ωmax also decreased cle function, further research and experimental evidence by when the leg was straightened (i.e. with decreasing knee force/torque measurements over the full range of velocities extensor length). However, this is in contrast to the results is required (Forrester et al. 2011). of the follow-up experiments where ωmax increased when the leg was straightened, and since ωmax was independent Eccentric force/torque–velocity relation of the maximum isometric force F0, it is thought to represent the actual ωmax of unresisted shortening (Yamauchi Finally, regarding eccentric contractions, there were et al. 2007). Moreover, when estimating ωmax by regression three main findings. First, normalised eccentric Fext and

Table 3 Maximum external reaction force F0 and velocity ωmax for unresisted shortening as extrapolated by linear and hyperbolic functions (n 6) = Parameter extrapolated Knee joint flexion angle [°] by regression type 50 60 70 80 90

F linear 0.84 0.07 0.86 0.07 0.89 0.05 0.95 0.07 0.99 0.05 0 ± ± ± ± ± F Hill 0.91 0.06 0.92 0.05 0.93 0.04 0.97 0.06 1.00 0.05 0 ± ± ± ± ± ω linear 0.32 0.04 0.38 0.06 0.42 0.06 0.59 0.13 0.78 0.16 max ± ± ± ± ± ω Hill 1.25 0.30 1.66 0.45 2.05 0.49 3.40 1.16 5.09 1.30 max ± ± ± ± ± Both, F and ω , are normalised to their corresponding values as measured during the follow-up experiment. Values are mean SD 0 max ± 1 3 1700 Eur J Appl Physiol (2014) 114:1691–1702

Fig. 8 Complete sets of normalised force/torque–angle data for joint angles on eccentric force or torque enhancement. The distance external forces (a) and knee joint torques (b) for isometric (iso) and between data points at a given joint angle indicates the influence of eccentric muscle action (ecc) at different mean angular velocities. stretching speed on eccentric force or torque production. As indicated From left to right the graphs indicate the influence of increasing by cartoons, 0° knee flexion refers to the straight leg

M increased with knee joint flexion. Second, a non-sys- (0.80 0.15–1.17 0.17M ) and no enhancement of K ± ± 0_ankle tematic dependence of eccentric force/torque on angu- hip joint torques (0.76 0.15–1.03 0.17M ). ± ± 0_hip lar velocity was observed for knee joint flexion angles In conclusion, we found that force and torque produc- greater than θ0 (Fig. 8). In agreement with the literature tion during multi-joint leg extension strongly depends eccentric forces and torques for joint angles less or equal on the instantaneous combination of joint angle, angular to θ0 were largely unaffected by changes in velocity and velocity and the type of muscle action. For concentric showed almost no significant increase in force (Enoka muscle action, increasing angular velocities flattened the 1996; Webber and Kriellaars 1997; Westing et al. 1988). force/torque–angle relationship and caused a shift of the This finding is usually explained by neural inhibition optimum joint angle towards longer muscle lengths. Fur- (Duclay et al. 2011; Gruber et al. 2009) and/or inappro- thermore, the shape of force/torque–angular velocity rela- priate cross-bridge attachment at muscle lengths below l0 tionships varied with joint angle, whereby Hill’s hyperbola (Scott et al. 1996). When joint angles exceeded θ ( 60° and linear functions showed equivalent approximations. 0 ≥ knee flexion), a significant increase in force/torque of up However, linear and hyperbolic approximations failed to 1.32 0.16M was observed. In some studies similar in predicting force/torque production adequately for our ± 0 results to those found here were reported for cat soleus multi-joint leg extension movements across the entire (Brown et al. 1996; Scott et al. 1996) and single fibres range of velocities. For eccentric muscle actions, force/ from frogs (Granzier et al. 1989), and the origin of increas- torque did not exceed the corresponding isometric refer- ing eccentric forces for longer muscle lengths might be ence value for muscle lengths shorter or equal to optimal, caused by decreases in the myofilament lattice spacing whereas eccentric force/torque for lengths beyond the opti- (Edman 1999). However, the higher eccentric forces/tor- mal muscle length exceeded the corresponding angle-spe- ques at angles θ may merely be a result of increasing cific isometric reference values by as much as 32 16 %. ≥ 0 ± stretch amplitude (Lee and Herzog 2002). Furthermore, Eccentric forces/torques further showed some non-system- we cannot provide a good explanation for the decrease in atic changes with stretching speed that we cannot explain eccentric forces with increasing stretching speeds at knee satisfactorily. joint angles θ a result that is in contrast to the gener- Summarising, future research should include experi- ≥ 0, ally accepted findings in the literature (Seger andT hor- ments covering the entire range of muscle shortening stensson 1994; Westing et al. 1988). Possibly this result is velocities and should be aimed at elucidating the mecha- caused by muscle inhibition to protect the quadriceps group nisms underlying the findings presented here. Notwith- against extreme tension, but further research on this find- standing missing explanations, real-life movements and ing is required. Third, normalised eccentric MK always rehabilitation exercises typically involve multi-joint move- exceeded normalised Fext (Table 2; Fig 8). This probably ments, thus our results should be taken into consideration results from the fact that during stretch there was only when modelling human movement or designing training inconsistent and small enhancement of ankle joint torque concepts.

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