Christian Rode

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.

Stable and robust walking with compliant legs

Bipedal walking could be implemented into a robot by mimicking spring-like leg behaviour. The fundamental model, describing human-like leg function in walking is the bipedal spring-mass model which was investigated in this study. We identified several types of walking patterns, e.g. symmetric and asymmetric walking, that accounts for high variability of gait. The aim of this study is the identification of a stiffness range which allows for stable walking with comparatively large robustness against perturbations. Walking with compliant legs was tested with the bipedal PogoWalker, demonstrating advantages of the underlying leg mechanics, i.e. walking without the necessity of precise leg adjustments, and acceptable impact forces.

Characterization of isovelocity extension of activated muscle: a Hill-type model for eccentric contractions and a method for parameter determination.

Tension in eccentric contractions of skeletal muscles is expressed with a Hill-type model with a term containing a tension-velocity relation and a second term which is supposed to account for effects of contraction history. Based on experimental data (rat gastrocnemius medialis) a method is derived to uniquely characterize the tension-velocity relation in the first term. Also, a description of the second term is derived and its single additional parameter is determined. The model is tested in simulations. The experimental data used to characterize the model indicates that the slope of tension in the second phase of eccentric contractions is independent of muscle length.

The effects of parallel and series elastic components on the active cat soleus force-length relationship

Typically, active muscle force is calculated by subtracting measured passive force from measured total force for corresponding whole muscle lengths (standard method). From a mechanical point of view, this requires a parallel elastic component (PEC) that is arranged in parallel to both the series elastic component (SEC) and the contractile component (CC). From a morphological point of view, however, the PEC should be rather in parallel to the CC, and both in series to the SEC (model [CC]). In this study, we investigated the differences in active muscle force estimated with these two different approaches and their impact on the interpretation of experiments. We measured passive forces without stimulation and total forces during supramaximal stimulation of six cat soleus muscles in end-held isometric contractions from lengths near active insufficiency to lengths close to inducing stretch damage. The active forces estimated with model [CC] reach about 10% higher maximum isometric forces and reveal about 10% longer optimal lengths of the CC compared to the standard method. Model choice affects the interpretation of the physiological working range and residual force enhancement. The active force-length relationships of the contractile component determined with model [CC] agree better with the theoretical sarcomere force-length relationship.

Three-Dimensional Muscle Architecture and Comprehensive Dynamic Properties of Rabbit Gastrocnemius, Plantaris and Soleus: Input for Simulation Studies

The vastly increasing number of neuro-muscular simulation studies (with increasing numbers of muscles used per simulation) is in sharp contrast to a narrow database of necessary muscle parameters. Simulation results depend heavily on rough parameter estimates often obtained by scaling of one muscle parameter set. However, in vivo muscles differ in their individual properties and architecture. Here we provide a comprehensive dataset of dynamic (n = 6 per muscle) and geometric (three-dimensional architecture, n = 3 per muscle) muscle properties of the rabbit calf muscles gastrocnemius, plantaris, and soleus. For completeness we provide the dynamic muscle properties for further important shank muscles (flexor digitorum longus, extensor digitorum longus, and tibialis anterior; n = 1 per muscle). Maximum shortening velocity (normalized to optimal fiber length) of the gastrocnemius is about twice that of soleus, while plantaris showed an intermediate value. The force-velocity relation is similar for gastrocnemius and plantaris but is much more bent for the soleus. Although the muscles vary greatly in their three-dimensional architecture their mean pennation angle and normalized force-length relationships are almost similar. Forces of the muscles were enhanced in the isometric phase following stretching and were depressed following shortening compared to the corresponding isometric forces. While the enhancement was independent of the ramp velocity, the depression was inversely related to the ramp velocity. The lowest effect strength for soleus supports the idea that these effects adapt to muscle function. The careful acquisition of typical dynamical parameters (e.g. force-length and force-velocity relations, force elongation relations of passive components), enhancement and depression effects, and 3D muscle architecture of calf muscles provides valuable comprehensive datasets for e.g. simulations with neuro-muscular models, development of more realistic muscle models, or simulation of muscle packages.

Trunk orientation causes asymmetries in leg function in small bird terrestrial locomotion.

In contrast to the upright trunk in humans, trunk orientation in most birds is almost horizontal (pronograde). It is conceivable that the orientation of the heavy trunk strongly influences the dynamics of bipedal terrestrial locomotion. Here, we analyse for the first time the effects of a pronograde trunk orientation on leg function and stability during bipedal locomotion. For this, we first inferred the leg function and trunk control strategy applied by a generalized small bird during terrestrial locomotion by analysing synchronously recorded kinematic (three-dimensional X-ray videography) and kinetic (three-dimensional force measurement) quail locomotion data. Then, by simulating quail gaits using a simplistic bioinspired numerical model which made use of parameters obtained in in vivo experiments with real quail, we show that the observed asymmetric leg function (left-skewed ground reaction force and longer leg at touchdown than at lift-off) is necessary for pronograde steady-state locomotion. In addition, steady-state locomotion becomes stable for specific morphological parameters. For quail-like parameters, the most common stable solution is grounded running, a gait preferred by quail and most of the other small birds. We hypothesize that stability of bipedal locomotion is a functional demand that, depending on trunk orientation and centre of mass location, constrains basic hind limb morphology and function, such as leg length, leg stiffness and leg damping.

Grounded running in quails: Simulations indicate benefits of observed fixed aperture angle between legs before touch-down

Many birds use grounded running (running without aerial phases) in a wide range of speeds. Contrary to walking and running, numerical investigations of this gait based on the BSLIP (bipedal spring loaded inverted pendulum) template are rare. To obtain template related parameters of quails (e.g. leg stiffness) we used x-ray cinematography combined with ground reaction force measurements of quail grounded running. Interestingly, with speed the quails did not adjust the swing legs angle of attack with respect to the ground but adapted the angle between legs (which we termed aperture angle), and fixed it about 30ms before touchdown. In simulations with the BSLIP we compared this swing leg alignment policy with the fixed angle of attack with respect to the ground typically used in the literature. We found symmetric periodic grounded running in a simply connected subset comprising one third of the investigated parameter space. The fixed aperture angle strategy revealed improved local stability and surprising tolerance with respect to large perturbations. Starting with the periodic solutions, after step-down step-up or step-up step-down perturbations of 10% leg rest length, in the vast majority of cases the bipedal SLIP could accomplish at least 50 steps to fall. The fixed angle of attack strategy was not feasible. We propose that, in small animals in particular, grounded running may be a common gait that allows highly compliant systems to exploit energy storage without the necessity of quick changes in the locomotor program when facing perturbations.

A new biarticular actuator design facilitates control of leg function in BioBiped3

Bioinspired legged locomotion comprises different aspects, such as (i) benefiting from reduced complexity control approaches as observed in humans/animals, (ii) combining embodiment with the controllers and (iii) reflecting neural control mechanisms. One of the most important lessons learned from nature is the significant role of compliance in simplifying control, enhancing energy efficiency and robustness against perturbations for legged locomotion. In this research, we investigate how body morphology in combination with actuator design may facilitate motor control of leg function. Inspired by the human leg muscular system, we show that biarticular muscles have a key role in balancing the upper body, joint coordination and swing leg control. Appropriate adjustment of biarticular spring rest length and stiffness can simplify the control and also reduce energy consumption. In order to test these findings, the BioBiped3 robot was developed as a new version of BioBiped series of biologically inspired, compliant musculoskeletal robots. In this robot, three-segmented legs actuated by mono- and biarticular series elastic actuators mimic the nine major human leg muscle groups. With the new biarticular actuators in BioBiped3, novel simplified control concepts for postural balance and for joint coordination in rebounding movements (drop jumps) were demonstrated and approved.

Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.

Striated muscle contraction requires intricate interactions of microstructures. The classic textbook assumption that myosin filaments are compressed at the meshed Z-disc during striated muscle fibre contraction conflicts with experimental evidence. For example, myosin filaments are too stiff to be compressed sufficiently by the muscular force, and, unlike compressed springs, the muscle fibres do not restore their resting length after contractions to short lengths. Further, the dependence of a fibres maximum contraction velocity on sarcomere length is unexplained to date. In this paper, we present a structurally consistent model of sarcomere contraction that reconciles these findings with the well-accepted sliding filament and crossbridge theories. The few required model parameters are taken from the literature or obtained from reasoning based on structural arguments. In our model, the transition from hexagonal to tetragonal actin filament arrangement near the Z-disc together with a thoughtful titin arrangement enables myosin filament sliding through the Z-disc. This sliding leads to swivelled crossbridges in the adjacent half-sarcomere that dampen contraction. With no fitting of parameters required, the model predicts straightforwardly the fibres entire force–length behaviour and the dependence of the maximum contraction velocity on sarcomere length. Our model enables a structurally and functionally consistent view of the contractile machinery of the striated fibre with possible implications for muscle diseases and evolution.

Force reduction induced by unidirectional transversal muscle loading is independent of local pressure.

Transversal unidirectional compression applied to muscles via external loading affects muscle contraction dynamics in the longitudinal direction. A recent study reported decreasing longitudinal muscle forces with increasing transversal load applied with a constant contact area (i.e., leading to a simultaneous increase in local pressure). To shed light on these results, we examine whether the decrease in longitudinal force depends on the load, the local pressure, or both. To this end, we perform isometric experiments on rat M. gastrocnemius medialis without and with transversal loading (i) changing the local pressure from 1.1-3.2Ncm(-2) (n=9) at a constant transversal load (1.62N) and (ii) increasing the transversal load (1.15-3.45N) at a constant local pressure of 2.3Ncm(-2) (n=7). While we did not note changes in the decrease in longitudinal muscle force in the first experiment, the second experiment resulted in an almost-linear reduction of longitudinal force between 7.5±0.6% and 14.1±1.7%. We conclude that the observed longitudinal force reduction is not induced by local effects such as malfunction of single muscle compartments, but that similar internal stress conditions and myofilament configurations occur when the local pressure changes given a constant load. The decreased longitudinal force may be explained by increased internal pressure and a deformed myofilament lattice that is likely associated with the decomposition of cross-bridge forces on the one hand and the inhibition of cross-bridges on the other hand.