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 1 3 1692 Eur J Appl Physiol (2014) 114:1691–1702 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 let- ters. 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 veloc- ity.