Syn Schmitt

High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models

High-frequency vibrations e.g., induced by legs impacting with the ground during terrestrial locomotion can provoke damage within tendons even leading to ruptures. So far, macroscopic Hill-type muscle models do not account for the observed high-frequency damping at low-amplitudes. Therefore, former studies proposed that protective damping might be explained by modelling the contractile machinery of the muscles in more detail, i.e., taking the microscopic processes of the actin–myosin coupling into account. In contrast, this study formulates an alternative hypothesis: low but significant damping of the passive material in series to the contractile machinery—e.g., tendons, aponeuroses, titin—may well suffice to damp these hazardous vibrations. Thereto, we measured the contraction dynamics of a piglet muscle–tendon complex (MTC) in three contraction modes at varying loads and muscle–tendon lengths. We simulated all three respective load situations on a computer: a Hill-type muscle model including a contractile element (CE) and each an elastic element in parallel (PEE) and in series (SEE) to the CE pulled on a loading mass. By comparing the model to the measured output of the MTC, we extracted a consistent set of muscle parameters. We varied the model by introducing either linear damping in parallel or in series to the CE leading to accordant re-formulations of the contraction dynamics of the CE. The comparison of the three cases (no additional damping, parallel damping, serial damping) revealed that serial damping at a physiological magnitude suffices to explain damping of high-frequency vibrations of low amplitudes. The simulation demonstrates that any undamped serial structure within the MTC enforces SEE-load eigenoscillations. Consequently, damping must be spread all over the MTC, i.e., rather has to be de-localised than localised within just the active muscle material. Additionally, due to suppressed eigenoscillations Hill-type muscle models taking into account serial damping are numerically more efficient when used in macroscopic biomechanical neuro-musculo-skeletal models.

Hill-type muscle model with serial damping and eccentric force–velocity relation

Hill-type muscle models are commonly used in biomechanical simulations to predict passive and active muscle forces. Here, a model is presented which consists of four elements: a contractile element with force-length and force-velocity relations for concentric and eccentric contractions, a parallel elastic element, a series elastic element, and a serial damping element. With this, it combines previously published effects relevant for muscular contraction, i.e. serial damping and eccentric force-velocity relation. The model is exemplarily applied to arm movements. The more realistic representation of the eccentric force-velocity relation results in human-like elbow-joint flexion. The model is provided as ready to use Matlab and Simulink code.

A forward dynamics simulation of human lumbar spine flexion predicting the load sharing of intervertebral discs, ligaments, and muscles

Determining the internal dynamics of the human spine’s biological structure is one essential step that allows enhanced understanding of spinal degeneration processes. The unavailability of internal load figures in other methods highlights the importance of the forward dynamics approach as the most powerful approach to examine the internal degeneration of spinal structures. Consequently, a forward dynamics full-body model of the human body with a detailed lumbar spine is introduced. The aim was to determine the internal dynamics and the contribution of different spinal structures to loading. The multi-body model consists of the lower extremities, two feet, shanks and thighs, the pelvis, five lumbar vertebrae, and a lumped upper body including the head and both arms. All segments are modelled as rigid bodies. 202 muscles (legs, back, abdomen) are included as Hill-type elements. 58 nonlinear force elements are included to represent all spinal ligaments. The lumbar intervertebral discs were modelled nonlinearly. As results, internal kinematics, muscle forces, and internal loads for each biological structure are presented. A comparison between the nonlinear (new, enhanced modelling approach) and linear (standard modelling approach, bushing) modelling approaches of the intervertebral disc is presented. The model is available to all researchers as ready-to-use C/C++ code within our in-house multi-body simulation code demoa with all relevant binaries included.

Electro-mechanical delay in Hill-type muscle models

In this study, we investigated to which extent Hill-type muscle models can explain the electro-mechanical delay (EMD). The EMD is a phenomenon that has been well examined in muscle experiments. The EMD is the time lag between a change in muscle stimulation and the subsequent measurable change in muscle force. A variety of processes as, e.g., signal conduction and interaction of contractile and elastic muscle structures contribute to the EMD. The relative contributions of the particular processes have not been fully unveiled so far. Thereto, we simulated isometric muscle contractions using two Hill-type muscle models. Their parameters were extracted from experiments on the cat soleus muscle. In agreement with literature data, predicted EMD values depend on muscle-tendon complex (MTC) length and increase when reducing MTC lengths. The highest EMD values (28 and 27 ms) occur at the lowest MTC length examined (78% of optimal length). Above optimal MTC length, we find EMD saturation (2 ms) in one model. In the other model, the EMD slightly re-increases up to 9 ms at the highest length examined (113% of optimal length). The EMD values predicted by the two models were then compared to EMD values found in the same experiments from which the muscle parameters were extracted. At optimal MTC length, the EMD values, mapping ion release and visco-elastic interactions, predicted by both models (3.5 and 5.5 ms) just partly account for the measured value (15.8 ms). The biggest share (about 9 ms) of the remaining 11 ms can be attributed to signal conduction along the nerve and on the muscle surface. Further potential sources of delayed force generation are discussed.

Novel approach for a precise determination of short-time intervals in ankle sprain experiments

The etiology of ankle sprain injury is still under debate. Therefore, diagnoses of ankle inversion experiments play an important role. Recent studies stress the importance of exact time measurements due to the short inversion period of around 70ms. This paper presents a novel approach using the vertical ground reaction force (vGRF) to determine the short-time intervals in ankle sprain experiments, which are present in the form of short periods from the beginning of the movement to its end and short latencies to following signals, e.g. EMG onset of peroneal muscles. We compare our method to electrogoniometry at the ankle which is considered as the gold standard. During the inversion movement the kinematic action at the ankle can be measured with electrogoniometry, whereas the vGRF quantifies the vertical dynamic reaction of the tested subject entirely. We observe a difference of DeltaT(f,0-->g,0)=10+/-0.5ms between the first observable vGRF response and the first observable electrogoniometer response following platform release. The end of the ankle inversion measured with electrogoniometry is DeltaT(f,1-->g,1)=3+/-0.5ms later than the maximal vGRF peak. The potential supplementary (mechanical) information of this novel approach compared to electrogoniometry and its ease of use, may be not only interesting for researchers when studying ankle sprain simulations but also for clinicians when testing functional ankle stability.

Comparative Sensitivity Analysis of Muscle Activation Dynamics

We mathematically compared two models of mammalian striated muscle activation dynamics proposed by Hatze and Zajac. Both models are representative for a broad variety of biomechanical models formulated as ordinary differential equations (ODEs). These models incorporate parameters that directly represent known physiological properties. Other parameters have been introduced to reproduce empirical observations. We used sensitivity analysis to investigate the influence of model parameters on the ODE solutions. In addition, we expanded an existing approach to treating initial conditions as parameters and to calculating second-order sensitivities. Furthermore, we used a global sensitivity analysis approach to include finite ranges of parameter values. Hence, a theoretician striving for model reduction could use the method for identifying particularly low sensitivities to detect superfluous parameters. An experimenter could use it for identifying particularly high sensitivities to improve parameter estimation. Hatzes nonlinear model incorporates some parameters to which activation dynamics is clearly more sensitive than to any parameter in Zajacs linear model. Other than Zajacs model, Hatzes model can, however, reproduce measured shifts in optimal muscle length with varied muscle activity. Accordingly we extracted a specific parameter set for Hatzes model that combines best with a particular muscle force-length relation.

Linking continuous and discrete intervertebral disc models through homogenisation

At present, there are two main numerical approaches that are frequently used to simulate the mechanical behaviour of the human spine. Researchers with a continuum-mechanical background often utilise the finite-element method (FEM), where the involved biological soft and hard tissues are modelled on a macroscopic (continuum) level. In contrast, groups associated with the science of human movement usually apply discrete multi-body systems (MBS). Herein, the bones are modelled as rigid bodies, which are connected by Hill-type muscles and non-linear rheological spring-dashpot models to represent tendons and cartilaginous connective tissue like intervertebral discs (IVD). A possibility to benefit from both numerical methods is to couple them and use each approach, where it is most appropriate. Herein, the basic idea is to utilise MBS in simulations of the overall body and apply the FEM only to selected regions of interest. In turn, the FEM is used as homogenisation tool, which delivers more accurate non-linear relationships describing the behaviour of the IVD in the multi-body dynamics model. The goal of this contribution is to present an approach to couple both numerical methods without the necessity to apply a gluing algorithm in the context of a co-simulation. Instead, several pre-computations of the intervertebral disc are performed offline to generate an approximation of the homogenised finite-element (FE) result. In particular, the discrete degrees of freedom (DOF) of the MBS, that is, three displacements and three rotations, are applied to the FE model of the IVD, and the resulting homogenised forces and moments are recorded. Moreover, a polynomial function is presented with the discrete DOF of the MBS as variables and the discrete forces an moments as function values. For the sake of a simple verification, the coupling method is applied to a simplified motion segment of the spine. Herein, two stiff cylindrical vertebrae with an interjacent homogeneous cylindrical IVD are examined under the restriction of purely elastic deformations in the sagittal plane.

Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction. A recently published muscle model, deriving the hyperbolic force-velocity relation from first-order mechanical principles, predicts different force-velocity operating points for different load situations. With a new approach, this model could be simplified and thus, transferred into a numerical simulation and a hardware experiment. Two types of quick release experiments were performed in simulation and with the hardware setup, which represent two extreme cases of the contraction dynamics: against a constant force (isotonic) and against an inertial mass. Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations. Interestingly, the analytical model not only predicts these extreme cases, but also additionally all contraction states in between. It was possible to validate these predictions with the numerical model and the hardware experiment. These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements. The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.

Nature as an engineer: one simple concept of a bio-inspired functional artificial muscle

The biological muscle is a powerful, flexible and versatile actuator. Its intrinsic characteristics determine the way how movements are generated and controlled. Robotic and prosthetic applications expect to profit from relying on bio-inspired actuators which exhibit natural (muscle-like) characteristics. As of today, when constructing a technical actuator, it is not possible to copy the exact molecular structure of a biological muscle. Alternatively, the question may be put how its characteristics can be realized with known mechanical components. Recently, a mechanical construct for an artificial muscle was proposed, which exhibits hyperbolic force-velocity characteristics. In this paper, we promote the constructing concept which is made by substantiating the mechanical design of biological muscle by a simple model, proving the feasibility of its real-world implementation, and checking their output both for mutual consistency and agreement with biological measurements. In particular, the relations of force, enthalpy rate and mechanical efficiency versus contraction velocity of both the constructs technical implementation and its numerical model were determined in quick-release experiments. All model predictions for these relations and the hardware results are now in good agreement with the biological literature. We conclude that the construct represents a mechanical concept of natural actuation, which is suitable for laying down some useful suggestions when designing bio-inspired actuators.

A two-muscle, continuum-mechanical forward simulation of the upper limb.

By following the common definition of forward-dynamics simulations, i.e. predicting movement based on (neural) muscle activity, this work describes, for the first time, a forward-dynamics simulation framework of a musculoskeletal system, in which all components are represented as continuous, three-dimensional, volumetric objects. Within this framework, the mechanical behaviour of the entire muscle–tendon complex is modelled as a nonlinear hyperelastic material undergoing finite deformations. The feasibility and the full potential of the proposed forward-dynamics simulation framework is demonstrated on a two-muscle, three-dimensional, continuum-mechanical model of the upper limb. The musculoskeletal model consists of three bones, i.e. humerus, ulna, and radius, an one-degree-of-freedom elbow joint, and an antagonistic muscle pair, i.e. the biceps and triceps brachii, and takes into consideration the contact between the skeletal muscles and the humerus. Numerical studies have shown that the proposed upper limb model is capable of predicting realistic moment arms and muscle forces for the entire range of activation and motion. Within the limitations of the model, the presented simulations provide, for the first time, insights into existing contact forces and their influence on the muscle fibre stretch. Based on the presented simulations, the overall change in fibre stretch is typically less than 3%, despite the fact that the contact forces reach up to 71% of the exerted muscle force. Movement-predicting simulations are achieved by minimising a nonlinear moment equilibrium equation. Based on the forward-dynamics simulation approach, an iterative solution procedures for position-driven (inverse dynamics) and force-driven scenarios have been proposed accordingly. Applying these methodologies to time-dependent scenarios demonstrates that the proposed methods can be linked to state-of-the-art control algorithms predicting time-dependent muscle activation levels based on principles of forward dynamics.