Monitoring skeletal muscle dynamics and modelling the nonlinear response
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A custom ultrasonic calliper was employed to monitor voluntary and externally excited muscle dynamics with synchronous 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 electromyography 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.
KeywordsUltrasonic monitoring of muscle dynamics Modelling of muscle dynamics Motor-spring-damper-ratchet lumped circuit Muscle response to external stimuli Post-tetanus autonomous contraction
In recent decades, it has become apparent that in vitro study of skeletal muscle does not adequately represent the in vivo situation. However, non-invasive monitoring of living skeletal muscle has been further refined, and applications are progressing. This approach also provides information on the dynamics of muscle contraction. Non-invasive monitoring includes ultrasonic imaging, near infrared spectroscopy, and electromyography (EMG). We have recently developed a high-resolution ultrasonic calliper (HRUC) for the non-invasive study of in vivo muscle dynamics [9, 10, 11, 12, 13, 14]. This system is employed here for monitoring of the gastrocnemius muscle (GM) of human subjects in vivo.
This HRUC has revealed a property of muscles that current muscle models cannot emulate. It has been observed that during a sustained isometric contraction the active muscle shortens while stretching the tendon. Then upon the following relaxation the muscle extends to a length longer than the initial length (overshoot) and then afterwards slowly recovers to the original length. Similar asymmetric property of gastrocnemius and biceps muscles is presented in our previous research works [11, 12]. The purpose of this paper is to explore possible mechanisms for this novel observed behaviour. A new model capable of accounting for the slow recovery to the initial length is presented.
The observations presented here focus on muscle dynamics under isometric (fixed-end) conditions and modelling adapted to emulate the experimental results obtained; muscle elongation and subsequent restoration to the original length follow isometric contraction. In some respects, these after-effects of contraction are similar to those first described by Kohnstamm  and Salmon  thereafter named “Katatonusversuch” (catatonus experiment), respectively, “catatonus”. A further refined discussion follows the presentation of the experiments and respective modelling.
Monitoring of the lateral expansion of muscle during contraction is performed by the observation of the time-of-flight (TOF) of ultrasonic waves travelling across the contracting and relaxing muscle. There is a small increase in the velocity of ultrasound under activation with respect to the relaxed muscle which is only 0.6% as demonstrated by Hossain et al. [9, 14]. This variation of the velocity is sufficiently small that it can be ignored, and the time-of-flight can be used to estimate the length variations, assuming a fixed averaged velocity for the results presented here.
The mid-longitudinal area of whole GM muscle is confirmed constant by experimental observations . The mid-longitudinal area remains constant over the length range (, chapter 2). Viscous properties are neglected as the monitoring is performed under the isometric conditions. The muscle is considered to be hyper-elastic, i.e. deformations are extremely large and strains are recoverable. Therefore no energy dissipation in a closed cycle of application and removal of stress . This ensures the strain energy density function to be a function of strain. The stress can thus be calculated by differentiation of the strain energy density function with respect to the strain tensor .
The major goal of this study is the comparison of the observed elementary muscle dynamics with those predicted by a suitably developed mechanical model. A. V. Hill’s classical mechanical model (Fig. 1a) is employed as a starting point for modelling. The technical elements lately utilized in mechanical lumped circuits for biomechanical modelling are usually idealized springs, viscous dampers (dashpots; , and force generators. Since neither the historical approach nor recently developed models were able to represent the experimental findings obtained here by HRUC monitoring, the modelling had to be adapted.
2 Materials and methods
2.1 Experimental procedures
The athletes were advised to initiate their effort when a ball was launched from a device in front of them. They were further instructed to pull their lower leg towards the back with all-out effort for as long as they could and end with a sudden release when they felt they could not maintain maximal effort. Contractions were isometric (fixed-end). Under the experimental conditions enforced by fixation of the lower leg (Fig. 3a), no joint angle change could be accomplished. Monitoring the GM belly diameter with the HRUC  was initiated with the release of the ball and continued for 30–50 s after the relaxation.
Additional procedures were followed to probe the nature of the droop, overshoot, and recovery of muscle belly diameter and corresponding estimated muscle length. In one instance, subjects were asked to hold their maximal effort for a specific duration (2–10 s) controlled with a visible signal (as shown in Fig. 3a), to determine if the overshoot and recovery were dependent on the duration of contraction. Maximal effort contractions lasting 2–7 s were also monitored with the ultrasonic transducers positioned orthogonal to the usual arrangement to see if orientation of the sensors affected the outcome (see Fig. 9 presented with the experimental results).
External electric stimulation 30 mA (symmetric bi-phasic supra-maximal stimulus of 762 µs duration) was used to avoid the neural control aspect of the response. This allowed quantification of the consequences of maximal activation of all motor units simultaneously. In addition, the neural processing time could be estimated. The time from initiation of the visual signal to the time of first muscle length change minus the time from initiation of the electrical signal to initiation of muscle length change would reveal this neural processing time. A healthy male subject with an age of 36 and BMI of 23.2 volunteered for this experiment. Muscle stimulating electrodes, adhesive 50 mm round surface electrodes (REF 282000, Schwa medico®), driven by an electronic stimulator (EMP 4 Pocket, Pierenkemper GmbH Wetzlar, Germany), were attached to the ends of the GM belly. Stimulation with 4–5 consecutive pulses was conducted on the right GM. The 4–5 consecutive pulses were used to compare the responses of the gastrocnemius muscle with the identical external electric stimulation. The average shortening, lengthening, and undershoot of 5.98, 6.94, and 2.65 mm are observed respectively, and a typical response for a single pulse is shown in Fig. 12a.
2.2 Data analysis
3 Results and evaluations
An example of data obtained from monitoring of the GM of an athlete is displayed in Fig. 3b. It shows the rapid shortening of the muscle on initiation of the contraction, a droop representing slight elongation over time then rapid elongation on relaxation when the athlete could no longer hold the maximal contraction. For each contraction, we measured the duration of the following phases: the delay between visual command and the onset of the muscle response (total reaction time T), the build-up of maximum voluntary isometric contraction (contraction phase C), maintained maximum voluntary isometric contraction (holding phase H), sudden withdrawal of effort (relaxation phase E with overshoot), and finally the recovery phase R following the overshoot with exponential return to baseline. In addition, the peak rate of rise of force was determined by differentiation and the return to baseline was quantified by a time constant for the exponential shortening associated with this phenomenon.
3.1 Mechanical model
In the approach presented here (Fig. 4), the model is based on a linear drive, which is used instead of a force generator to achieve a closer approximation to the linear micro-motor devices, action of actomyosin complex in sarcomeres, present in the actual muscle. In addition, springs and dampers are employed, as they are commonly utilized in lumped circuit modelling for biomechanical applications and mechanical modelling of viscoelastic materials. For the demonstrated monitoring, only isometric contractions are involved. Since limbs are not accelerated under these conditions, effects of inertia can only be caused by the deforming muscle. As finally demonstrated by the results of modelling, these effects are negligible and point masses are not needed for modelling.
The elements of the model are identified in Fig. 4a and some output variables are highlighted with measurement scales: force, lengths, and diameter. Input is also indicated as an output variable in order to relate the input to the output. Since the overall length of the muscle plus tendon cannot vary under fixed-end conditions, the respective ends attached to bones are therefore considered immovable. The output variables are introduced to allow identification of model properties, which can emulate the results obtained with human subjects.
Consistent with previous work [10, 13, 14], the tendon is modelled by a spring. The elastic behaviour can be modelled with an idealized spring and a hook transmitting extensional stress only (Fig. 4b). During an isometric contraction, the hook is not required for modelling. The hook is therefore and by principle of simplification not employed in the modelling.
To model the observed asymmetric nonlinear response, an idealized mechanical ratchet had to be introduced. This represents a refinement of the established modelling required to describe the experimental observations of the overshoot following the termination of muscle contraction and the subsequent recovery. The ratchet is mounted in series with a damper and both in parallel with a spring. Although ratchets have previously been employed in modelling, this has only been done to represent the motor mechanism [3, 22]. The current application here is in addition to the motor.
It should be mentioned that the special features implemented by A. V. Hill concerning the rubber like spring and contractile linear motor drive already indicate the need for non-symmetry and in today’s terms nonlinear elements. Additionally, the implementation of a novel mechanical element, a ratchet, is required to explain the overshoot and subsequent recovery by modelling with a lumped mechanical circuit.
3.2 Equations relating to modelling
Variations from each individual LN for the relaxed state where FN = 0 for all N representing the different indices C, D, M, S, T are denoted by ΔLN.
Each element of the mechanical lumped circuit can, respectively, transfer a push or pull relating to pairs of forces + FN and − FN aligned with the element. As suitable for the treatment of lumped circuits, where components with cross sections are modelled by lines, the stress is represented by FN only. Positive FN relates to pull and therefore tensile stresses for the respective element in the modelled muscle or tendon. An activated muscle will supply and be subject to a positive stress.
Springs S and B of the modelled muscle (Fig. 4a) would provide a so-called passive force under external elongation of the muscle. For internally provided forces caused by the linear motor drive in modelling, one of these springs, namely B, is essential for the observed and, respectively, modelled recovery phase (denoted by R in Fig. 3b) following activation.
The model presented here solely serves the purpose of describing isometric activities observed in this report. The model therefore contains no masses since kinetic forces were negligible.
In addition to the motor other idealized elements serving the purpose of modelling the monitored response with a minimum of components are introduced in the lumped mechanical circuit. The spring S introduced in the muscle changes length in proportion to the force generated by the motor drive. This simplifies any comparison of the here introduced lumped mechanical circuit driven by a linear motor element, as present in the real muscle, with already established lumped circuits employing idealized force generators. The spring S introduced here is also needed to describe the mechanical properties inherent to the muscle. In established modelling, the spring S is usually combined with the spring modelling the tendon, here denoted with T. Since S and T are connected in series this mainly leads to differences concerning the actual length of the muscle belly with respect to the tendon.
The spring constant kN is a parameter of the model that has to be adjusted to the monitored muscle according to the observed response.
The model presented in Fig. 7 achieves the overshoot following deactivation at normalized time 1 due to the ratchet. This is followed by an exponential recovery in close approximation to the experimental observations which is dependent on the properties of the damper. The combination of a ratchet and a damper in the model (Fig. 7) can also be interpreted as an idealized shock absorber with no resistance for dilatation and viscous damping for contraction. This equivalent alternative is of importance for the interpretation, since it avoids the features of a potentially infinite lengthening of the ratchet compensated by a, respectively, infinite shortening of the damper that could otherwise occur under cyclic motions. Such effects would of course only occur with idealized modelling components without length restrictions as commonly employed in lumped circuits.
3.3 Comparison of experimental results and modelling
Individual variability in the response can be programmed in the input function, including delay due to reaction time, hold time, and droop. One result is displayed in Fig. 8b. In Fig. 9a, the result observed for a better trained athlete exhibiting a more stable hold phase (less droop) is shown. The persistence of the overshoot in this circumstance would suggest that it is not a consequence of the droop. Also illustrated in Fig. 9b are the results of ultrasonic monitoring across the tibia bone, observed in orthogonal direction with respect to all other results of monitoring presented here. This indicates that the observation of the overshoot and recovery cannot be attributed to anisotropic deformation in lateral directions.
Since much faster activation and deactivation phases were observed when contraction was elicited with external electric excitation of the muscle (see results presented in 3.5 below), the observed slower voluntary response was attributed to the nervous system. Therefore, electrical lumped circuits were introduced for modelling of the fast responses related to activation and deactivation.
3.4 Electrical lumped circuit
The reaction delay and especially the observed finite slope of the rise and fall of force, as well as soft curved onsets, and similarly observable rounded off ends of rapid movements are all attributed to brain processing and subsequent signal transmission in the nerves. For the purpose of illustrating the respective features and effects happening on relatively short time scales, the data points derived from experimental monitoring are displayed on magnified scales in Fig. 10. It shows the initial rise related to activation and in Fig. 11 the rapid drop associated with deactivation. A delay of (657.37 ± 0.01) ms for the onset of the muscle response from the initiated visual command given at time t = 0 was observed for one of our subjects.
Variations of the muscle length near the initial and final inflection points of this contraction are observed to be smooth and approximately symmetric. As the nearly symmetric response is rounded of at the top and bottom, a high-order filter is required to derive the observed response from a rectangular input function. From respective simulations, it has been derived that a filter with an order of at least 30 is needed to reach a suitable output.
It appears rather unlikely though that filters of the order of at least 30 are physically present in the real nervous system. The delay from initiation of the visual signal to initiation of muscle response relies on the following: sensory perception of the signal, neural transit time to the brain, central processing, neural transit time to the muscles along the motor pathway, neuromuscular transmission, and electromechanical delay. The response caused by such filters can also be produced by parallel arranged delay elements with a suitable distribution of delay times, also called delay lines. The resulting alternative model with an equivalent kernel filter  is represented in Fig. 10c.
Parameters derived from the monitored muscle dynamics (MVIC: maximum voluntary isometric contraction)
657 ± 1
958 ± 1
Contraction speed (mm/s)
19 ± 0.5
22.6 ± 0.1
Holding slope (mm/s)
− 0.07 ± 0.01
0.82 ± 0.01
1162 ± 2
Relaxation speed (mm/s)
− 14 ± 0.5
19 ± 0.5
Recovery time constant τ (s)
4.40 ± 0.8
Max. shortening (mm)
− 6.36 ± 0.12
Max. lengthening (mm)
7.62 ± 0.18
3.5 Position of the high-order filter
Nevertheless, it has to be stated that besides of the differences in involved units concerning the response, mechanical and electrical lumped circuits are interchangeable . The two models are presented independently here to relate as closely as reasonably possible to actual active biological elements in the live muscle, the tendon, and the connected nervous system.
Figure 14b demonstrates that the holding phase is associated with a rising EMG signal for voluntary isometric contraction. The athlete does not reach a steady hold phase in the contractile response. The EMG signal reached an early peak value of 0.80 ± 0.01 then fell in the middle of the MVIC holding phase to 0.68 ± 0.01 before rising to a peak value of 0.99 ± 0.01. The corresponding values for muscle length were 0.99 ± 0.01, 0.84 ± 0.01, and 0.79 ± 0.01, respectively. This indicates that effort is maintained to achieve the demanded maximum voluntary isometric contraction and the decline in length deviation must be due to changes within the muscle.
To further quantify details related to the recovery from overshoot, the dependence of the recovery time constant on the duration of maximum voluntary isometric contraction has been determined. The results of this experiment are displayed in Fig. 14b. The athlete relating to the graph with the grey dots could not hold the longer contractions.
For the second athlete’s monitored activity, a rapid increase in the recovery time constant as contraction duration increased was observed for longer duration contractions until finally his maximum activity level is reached. The horizontal line indicates an averaged recovery time constant suitable for modelling of the range of holding times where strong deviations due to overload or fatigue are not present. For short holding times, up to typically 5 or 6 s, only a slight increase in τ is observed. A value for the recovery time constant of about 3 s is suitable. For the athlete able to hold a longer maximum time, a pronounced increase in the recovery time constant is observed, reaching 30 s. This study demonstrates that the overshoot and recovery show a pronounced increase, when the limits of performance are approached.
This reasoning is supported by monitoring with external electric stimulation where even relatively short contractions exhibit the overshoot and recovery, as shown in Fig. 12a. Since in this case the overshoot and recovery are observed without the activation from the brain, the observed changes must be caused by biomechanical or physical effects within the muscle as modelled here.
4 Discussion and conclusion
The results of monitoring muscle dynamics with an acoustic calliper are presented together with adapted modelling using a lumped mechanical circuit consisting of springs, dampers, ratchets, and an activating linear motor drive as elements. The lumped mechanical model combined with a high-order electronic filter attributed to the nervous system is capable of modelling the observed results of voluntary isometric contractions of the GM in close approximation. Additional corrections relating to the voluntary control of the hold phase under maximum voluntary isometric contraction are introduced empirically whenever observed. The reaction time of the monitored muscle following either an external command or an electrical stimulus is introduced as additional parameter to consider. Observations based on synchronous monitoring of EMG signals as well as on direct external electric stimulation show that the overshoot and subsequent recovery are a property of the muscle. Analysis of these observations supports the modelling and additionally relates the overshoot and recovery in a quantitative manner to preceding activities of the muscle, not yet included in the presented model. The observed dependence of the overshoot and recovery on the duration of the contraction phase is left to future more refined modelling. The overshoot and recovery were observed for all athletes and, although not reported here, has additionally also been observed in the biceps brachii muscle with the same high-resolution ultrasonic calliper.
The observed overshoot and recovery described here have not previously been reported. Representing this effect required a Poynting–Thomson element with a ratchet in series with a damper for modelling. With respect to A. V. Hill’s original modelling, only slight modifications regarding the bidirectional linear drive and the absence of a spring counteracting the original unidirectional drive were implemented together with an additional high-order electronic filter representing the nervous system capable of mimicking the results of in vivo measures for isometric contraction in close approximation for the contraction and hold phase. The elements additionally implemented here with respect to the original model of A. V. Hill, a ratchet in serial with a damper and both arranged in parallel with a spring, behave similar to a shock absorber. The energy transferred to such a system is effectively thermalized as well as demonstrated by any suspension of vehicles involving shock absorbers. The asymmetry caused by the ratchet acting as mechanical diode has so far neither been monitored nor been included in modelling of muscle dynamics based on lumped mechanical circuits. The influence of the novel components in modelling required by the performed monitoring is most pronounced for temporally extended muscle activities up to ultimate performance employed here for demonstration: an all-out muscle activity kept for the maximum time span possible for the performing athlete.
The earlier discussed historic catatonus experiment  also involves an involuntary isometric contraction but different to the conditions for the results presented here; this condition is terminated following the end of the all-out contraction. Subsequently, observed is a movement of the free limb caused by the involved relaxation of the activated muscle. This movement relates to shortening of the muscle and exhibits a time constant of a similar order of magnitude as observed here under extended isometric conditions.
Monitoring of the skeletal muscle dynamics presented here was solely performed under isometric conditions to demonstrate previously not modelled and so far not studied effects. No movement of limbs was involved; therefore, point masses were not required for modelling with lumped mechanical elements. If moving limbs and body movements are to be considered, suitable point masses could easily be added to the developed model (Fig. 4) in a similar manner as in the model of Makssoud et al.  shown in Fig. 1b. Idealized elements such as springs, dampers, and linear motors are already widely used in mechanical modelling of skeletal muscles. The only additional new element required in our developed model is a mechanical ratchet. As shown with this work, it can easily be added to already established skeletal muscle models.
Some additionally observed novel effects relate to possible fatigue. They have been quantitatively demonstrated for extended tetanus (Fig. 15a) but are not included in the modelling presented here. The extension of the model to include these responses requires more refined nonlinear effects than already introduced by the implemented mechanical ratchet and is currently under study and development.
The findings reported here clearly indicate that a not previously identified nonlinear behaviour is present in muscle dynamics. Among other effects this adds a new channel for the thermalization of energy in muscles' dynamic processes. It has to be left to future studies to relate the experimentally observed and modelled effects to the three-dimensionality of the real muscle. Additional research is also needed to refine the physiological role of the here presented nonlinear effect in muscle contraction and relaxation. In that respect it has been identified that the nonlinear element relates to biomechanical components of the muscle and cannot be attributed to the nervous system.
Some of the signal processing and data evaluation software were developed with support of the European Commission under the European Union 7th Framework Program within AISHAII (Aircraft Integrated Structural Health Assessment II, EU-FP7-CP 212912). Furthermore, helpful discussions with Brian MacIntosh, Human Performance Laboratory of the University of Calgary, are gratefully appreciated and acknowledged.
Compliance with ethical standards
Conflict of interest
Hereby, we state that we have submitted our work as an original article and have made substantial contributions to the following: The conception and design of the study, data acquisition, analysis and interpretation of data, drafting the article or revising it critically for important intellectual content. We have read and concur with the content in the manuscript, and the final version is submitted with the approval of all authors. We do not have any financial or personal relationships with other people or organizations that could inappropriately influence or bias our work. The material within this manuscript has not been and will not be submitted for publication elsewhere except as an abstract.
In all cases presented, human studies have been approved by the appropriate ethics committee according to requirements and have therefore been performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki. All persons involved in monitoring gave their informed consent prior to their inclusion in the study.
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