Vol.:(0123456789) SN Applied Sciences (2019) 1:1538 | https://doi.org/10.1007/s42452-019-1560-7 Research Article Monitoring skeletal muscle dynamics and modelling the nonlinear response M. Zakir Hossain 1,2,3 · Julian Grill 2 · Wolfgang Grill 1,2 Received: 2 July 2019 / Accepted: 22 October 2019 © Springer Nature Switzerland AG 2019 Abstract A custom ultrasonic calliper was employed to monitor voluntary and externally excited muscle dynamics with synchro- nous electromyography. The activation, hold, and relaxation phases of the gastrocnemius muscle were monitored for maximum voluntary isometric contraction. Muscle belly shortening occurred during contraction and a post-contractile overshoot (lengthening) and subsequent exponential recovery of muscle dimension to the baseline were observed. Both the overshoot and recovery are attributed to the muscle as suggested by combined monitoring including elec- tromyography and modelling with a lumped mechanical circuit containing idealized elements, such as a bidirectional linear motor unit, a ratchet, dampers, and springs. The rapid contraction and relaxation phases require a high-order filter or alternatively a kernel filter, attributed to the nervous system as suggested by external electric stimulation, which resulted in faster rise and relaxation times. The respective response function is modelled with an electrical lumped circuit. Together with empirically adjusted reaction times and corrections for droop in the hold phase, the monitored response is represented in close approximation by the combined electrical and mechanical lumped circuits. The refined combinatory model includes a ratchet as a novel nonlinear mechanical element. In combination with determined model parameters, it provides a refined evaluation scheme capable to model monitored muscle dynamics in physical activity in close approximation. Keywords Ultrasonic monitoring of muscle dynamics · Modelling of muscle dynamics · Motor-spring-damper-ratchet lumped circuit · Muscle response to external stimuli · Post-tetanus autonomous contraction 1 Introduction Skeletal muscle dynamics have been an intriguing field of study for centuries, having already been reported in 1667 [5, 18, 25]. Lumped circuit modelling of muscle dynamics was introduced by Hill [8]. This model illustrated in Fig. 1a and its adaptations have frequently been used to study muscle properties, allowing advances in understanding of fundamental dynamic properties. Recently more complex models have been used in an attempt to model proper- ties not suitably represented by the basic Hill model, like the one proposed by Makssoud et al. [20] illustrated in Fig. 1b. Hill’s historical three-element muscle model [7, 8] has an active contractile element E, which is a linear motor drive only capable of providing pull (shortening stress). It has two springs S 1 and S 2 with S 1 in series with the con- tractile element and S 2 parallel to both. These springs are considered capable of transmitting pull only, resisting elongation. Figure 1b represents the model proposed by Makssoud et al. [20]. In addition to the classic contractile and elastic elements, it contains two mass-spring-damper arrangements. Both models connect to bones at each end (grey boxes). Further historical perspectives of modelling * Wolfgang Grill, wg@analogspeed.de | 1 Faculty of Physics and Earth Sciences, University of Leipzig, Leipzig, Germany. 2 ASI Analog Speed Instruments GmbH, Burgweg 8, 61460 Koenigstein im Taunus, Germany. 3 Scientific Soccer School, Calgary, Canada.