A phenomenological model and validation of shortening-induced force depression during muscle contractions

A review of the efforts to develop muscle and musculoskeletal models for biomechanics in the last 50 years

Both the Hill and the Huxley muscle models had already been described by the time the International Society of Biomechanics was founded 50 years ago, but had seen little use before the 1970s due to the lack of computing. As computers and computational methods became available in the 1970s, the field of musculoskeletal modeling developed and Hill type muscle models were adopted by biomechanists due to their relative computational simplicity as compared to Huxley type muscle models. Muscle forces computed by Hill type muscle models provide good agreement in conditions similar to the initial studies, i.e. for small muscles contracting under steady and controlled conditions. However, more recent validation studies have identified that Hill type muscle models are least accurate for natural in vivo locomotor behaviours at submaximal activations, fast speeds and for larger muscles, and thus need to be improved for their use in understanding human movements. Developments in muscle modelling have tackled these shortcomings. However, over the last 50 years musculoskeletal simulations have been largely based on traditional Hill type muscle models or even simplifications of this model that neglected the interaction of the muscle with a compliant tendon. The introduction of direct collocation in musculoskeletal simulations about 15 years ago along with further improvements in computational power and numerical methods enabled the use of more complex muscle models in simulations of whole-body movement. Whereas Hill type models are still the norm, we may finally be ready to adopt more complex muscle models into musculoskeletal simulations of human movement.

Numerical instability of Hill-type muscle models

Hill-type muscle models are highly preferred as phenomenological models for musculoskeletal simulation studies despite their introduction almost a century ago. The use of simple Hill-type models in simulations, instead of more recent cross-bridge models, is well justified since computationally 'light-weight'-although less accurate-Hill-type models have great value for large-scale simulations. However, this article aims to invite discussion on numerical instability issues of Hill-type muscle models in simulation studies, which can lead to computational failures and, therefore, cannot be simply dismissed as an inevitable but acceptable consequence of simplification. We will first revisit the basic premises and assumptions on the force-length and force-velocity relationships that Hill-type models are based upon, and their often overlooked but major theoretical limitations. We will then use several simple conceptual simulation studies to discuss how these numerical instability issues can manifest as practical computational problems. Lastly, we will review how such numerical instability issues are dealt with, mostly in an ad hoc fashion, in two main areas of application: musculoskeletal biomechanics and computer animation.

Innovative technique of minimally invasive closed reduction for impacted femoral neck fractures (MICRIF)

The moment arm of gluteus medius proportionated to distance from femoral head tends to be decreased postoperatively in valgus-impacted femoral neck fractures treated by in situ internal fixation. The aim of this article is to introduce a new gentle technique to correct the deformity. The innovative technique of Minimally Invasive Closed Reduction for Impacted Femoral neck fractures (MICRIF) mainly focused to disimpact valgus neck fractures into anatomical position. Patients were positioned on the fracture table to fix the hip joint in abduction and internal rotation. A 2.4-mm diameter Kirschner wire was inserted a few centimetres outside the iliac crest piercing the acetabular beak to enter the femoral head, followed by repositioning of the lower extremity from abduction into neutral. This method provides satisfactory anatomical reduction. Thereafter, a surgical implant was applied to osteosynthesize the reduced fracture. This simple technique effectively provides anatomical reduction in valgus impacted femoral neck fracture.

Muscle Function from Organisms to Molecules

Gaps in our understanding of muscle contraction at the molecular level limit the ability to predict in vivo muscle forces in humans and animals during natural movements. Because muscles function as motors, springs, brakes, or struts, it is not surprising that uncertainties remain as to how sarcomeres produce these different behaviors. Current theories fail to explain why a single extra stimulus, added shortly after the onset of a train of stimuli, doubles the rate of force development. When stretch and doublet stimulation are combined in a work loop, muscle force doubles and work increases by 50% per cycle, yet no theory explains why this occurs. Current theories also fail to predict persistent increases in force after stretch and decreases in force after shortening. Early studies suggested that all of the instantaneous elasticity of muscle resides in the cross-bridges. Subsequent cross-bridge models explained the increase in force during active stretch, but required ad hoc assumptions that are now thought to be unreasonable. Recent estimates suggest that cross-bridges account for only ∼12% of the energy stored by muscles during active stretch. The inability of cross-bridges to account for the increase in force that persists after active stretching led to development of the sarcomere inhomogeneity theory. Nearly all predictions of this theory fail, yet the theory persists. In stretch-shortening cycles, muscles with similar activation and contractile properties function as motors or brakes. A change in the phase of activation relative to the phase of length changes can convert a muscle from a motor into a spring or brake. Based on these considerations, it is apparent that the current paradigm of muscle mechanics is incomplete. Recent advances in our understanding of giant muscle proteins, including twitchin and titin, allow us to expand our vision beyond cross-bridges to understand how muscles contribute to the biomechanics and control of movement.

Size, History-Dependent, Activation and Three-Dimensional Effects on the Work and Power Produced During Cyclic Muscle Contractions

Muscles undergo cycles of length change and force development during locomotion, and these contribute to their work and power production to drive body motion. Muscle fibers are typically considered to be linear actuators whose stress depends on their length, velocity, and activation state, and whose properties can be scaled up to explain the function of whole muscles. However, experimental and modeling studies have shown that a muscle's stress additionally depends on inactive and passive tissues within the muscle, the muscle's size, and its previous contraction history. These effects have not been tested under common sets of contraction conditions, especially the cyclic contractions that are typical of locomotion. Here we evaluate the relative effects of size, history-dependent, activation and three-dimensional effects on the work and power produced during cyclic contractions of muscle models. Simulations of muscle contraction were optimized to generate high power outputs: this resulted in the muscle models being largely active during shortening, and inactive during lengthening. As such, the history-dependent effects were dominated by force depression during simulated active shortening rather than force enhancement during active stretch. Internal work must be done to deform the muscle tissue, and to accelerate the internal muscle mass, resulting in reduced power and work that can be done on an external load. The effect of the muscle mass affects the scaling of muscle properties, with the inertial costs of contraction being relatively greater at larger sizes and lower activation levels.

A modelling approach for exploring muscle dynamics during cyclic contractions

Hill-type muscle models are widely used within the field of biomechanics to predict and understand muscle behaviour, and are often essential where muscle forces cannot be directly measured. However, these models have limited accuracy, particularly during cyclic contractions at the submaximal levels of activation that typically occur during locomotion. To address this issue, recent studies have incorporated effects into Hill-type models that are oftentimes neglected, such as size-dependent, history-dependent, and activation-dependent effects. However, the contribution of these effects on muscle performance has yet to be evaluated under common contractile conditions that reflect the range of activations, strains, and strain rates that occur in vivo. The purpose of this study was to develop a modelling framework to evaluate modifications to Hill-type muscle models when they contract in cyclic loops that are typical of locomotor muscle function. Here we present a modelling framework composed of a damped harmonic oscillator in series with a Hill-type muscle actuator that consists of a contractile element and parallel elastic element. The intrinsic force-length and force-velocity properties are described using Bézier curves where we present a system to relate physiological parameters to the control points for these curves. The muscle-oscillator system can be geometrically scaled while preserving dynamic and kinematic similarity to investigate the muscle size effects while controlling for the dynamics of the harmonic oscillator. The model is driven by time-varying muscle activations that cause the muscle to cyclically contract and drive the dynamics of the harmonic oscillator. Thus, this framework provides a platform to test current and future Hill-type model formulations and explore factors affecting muscle performance in muscles of different sizes under a range of cyclic contractile conditions.

One of the primary functions of skeletal muscle is to generate work and power to move the body during locomotor tasks such as walking and running. Because it is difficult to measure muscle behaviour in living animals, most of what we know about how muscles perform this function is from experiments where the muscle is removed from the animal and studied under controlled laboratory conditions, or from computer simulations of such muscle contractions. Recent work has shown how internal mass within the muscle causes scale-dependent changes to contractile properties. This study demonstrates a forward-dynamic modelling framework that links a Hill-type muscle model to an oscillating external load. Scaling relations are developed to preserve the kinematic and dynamic similarity of the system to allow the model to be implemented from single fibre to whole muscle sizes. The model replicates contraction cycles that are typically seen in real muscles. The framework will allow the relative effects of history-dependent, internal mass and activation properties to be quantitatively evaluated for cyclic contractions.

Diminished abductor muscular strength in patients with valgus-impacted femoral neck fractures treated by internal fixation: Clinical study and biomechanical considerations

BACKGROUND

Valgus-impacted femoral neck fractures treated with internal fixation occasionally result in unsatisfactory postoperative locomotive function, partially due to muscle shortening and a decrease in the moment arm. This study quantifies the degree of diminished abduction strength both clinically and biomechanically.

METHODS

Fifteen patients were enrolled in this study. Twelve patients with fracture healed in valgus-impacted position were further evaluated. Muscular strength around hip was examined, and values between the nonoperated and operated side were compared and analyzed. For the biomechanical study, two three-dimensional models were prepared: model I (control model without displacement) and model II (simulated malunion of a 15° valgus-impacted fracture). Two sets of hip flexion angles in each of the models were simulated with flexion angles of 0° and 23°.

RESULTS

Mean and standard deviation values for muscle strength from the nonoperative/operative side among the valgus group are as follows: flexion strength was 9.2 ± 4.0/9.2 ± 3.2, extension strength was 5.8 ± 2.8/6.1 ± 3.2, abduction strength at 0° was 9.1 ± 3.7/7.4 ± 3.6, abduction strength at 10° was 6.7 ± 3.0/5.5 ± 2.2, and knee extension strength was 15.3 ± 6.2/15.1 ± 6.0 (kgf). When comparing values between the nonoperative and operative sides, statistical significance was only observed in abduction strength ( p < 0.01). The biomechanical models prove that valgus impaction decreases the moment arm by approximately 10% at both flexion angle.

CONCLUSIONS

A significant decrease in abductor strength at 0° and 10° was observed in the valgus-healed group. This may be related to a decrease in the moment arm. Further research should be done to define the acceptable limit of deformity for the satisfactory postoperative functioning.

Torque depression following active shortening is associated with a modulation of cortical and spinal excitation: a history-dependent study

The reduction in steady-state isometric torque following a shortening muscle action when compared to a purely isometric contraction at the same muscle length and level of activation is termed torque depression (TD). The purpose of this study was to investigate spinal and supraspinal neural responses during the TD state of a maximal voluntary activation of the ankle dorsiflexors. Thirteen subjects (10 male) were recruited for the study. To explore alterations in corticospinal excitability during voluntary muscle activation in the TD state, motor evoked potentials (MEPs), cervicomedullary motor evoked potentials (CMEPs), and maximal compound muscle action potentials (Mmax) were elicited during the isometric steady-state following active shortening (i.e., TD) and the purely isometric condition. A 15% reduction in steady-state isometric torque (P < 0.05) was observed following isokinetic shortening at 40°/sec. Although mean evoked responses (MEP and CMEP) were not different in the TD state as compared with purely isometric state, the changes in evoked responses were inversely related to one another depending on the level of TD These findings indicate that supraspinal and spinal responses are interrelated in the TD state. Furthermore, antagonist muscle coactivation during the isometric reference contraction was positively related to TD These findings suggest the possibility of a relationship between the central nervous system and TD in humans. Further work should be performed to definitively link TD to specific spinal interneurons.

Comparison of Post-Operative Muscular Strength Between Gamma Nailing and Hemiarthroplasty System in Femoral Intertrochanteric Fractures

Background:

The current study focuses on the comparison of postoperative muscular strength around the hip joint of patients with femoral intertrochanteric fractures treated either by cephalo-medullary (CM) nailing or a new bipolar hip prosthesis (BHP), an especially attached device to secure displaced greater trochanteric fragment.

Methods:

Twenty patients treated with CM nailing were age- and sex- matched with a control group of 20 patients treated with BHP. Maximum isometric forces at the bilateral hip joint were measured during the follow up period. Means of 3 measurements were represented.

Results:

The mean and standard deviation values (kg) of muscle strength at the non-operative/ operative side in the CM nailing group were as follows: flexion strength 9.5±4.7/8.5±4.9 (P=0.06), extension strength 6.2±3.5/5.5±3.7 (P=0.08), abduction strength at 0 degrees 7.7±3.5/6.2±2.8 (p=0.002), abduction strength at 10 degrees 5.5±2.0/4.2±2.0 (p=0.001). In the BHP group, mean and standard deviation values of muscle strength at the non-operative/ operative side were as follows: flexion strength 6.5±2.8/6.0±3.4 (P=0.08), extension strength 4.4±0.9/4.4±0.6 (P=0.83), abduction strength at 0 degrees 5.1±1.9/5.0±1.6 (p=0.12), and that at 10 degrees 4.7±1.4/4.6±1.3 (p=0.10).

Conclusion:

Our results demonstrate that CM nailing may cause a 25-30% decrease in postoperative muscle strength around the hip joint, particularly during hip abduction. With the new BHP, greater trochanter reduction is achieved allowing early weight bearing and maintaining strength in abduction. Surgeons should consider postoperative muscular strength as one of the necessary factors for selection of the appropriate surgical procedure.

Level of Evidence:

Therapeutic Level III.

An action potential-driven model of soleus muscle activation dynamics for locomotor-like movements

OBJECTIVE

The goal of this study was to develop a physiologically plausible, computationally robust model for muscle activation dynamics (A(t)) under physiologically relevant excitation and movement.

APPROACH

The interaction of excitation and movement on A(t) was investigated comparing the force production between a cat soleus muscle and its Hill-type model. For capturing A(t) under excitation and movement variation, a modular modeling framework was proposed comprising of three compartments: (1) spikes-to-[Ca(2+)]; (2) [Ca(2+)]-to-A; and (3) A-to-force transformation. The individual signal transformations were modeled based on physiological factors so that the parameter values could be separately determined for individual modules directly based on experimental data.

MAIN RESULTS

The strong dependency of A(t) on excitation frequency and muscle length was found during both isometric and dynamically-moving contractions. The identified dependencies of A(t) under the static and dynamic conditions could be incorporated in the modular modeling framework by modulating the model parameters as a function of movement input. The new modeling approach was also applicable to cat soleus muscles producing waveforms independent of those used to set the model parameters.

SIGNIFICANCE

This study provides a modeling framework for spike-driven muscle responses during movement, that is suitable not only for insights into molecular mechanisms underlying muscle behaviors but also for large scale simulations.

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.

Muscle force depends on the amount of transversal muscle loading

Skeletal muscles are embedded in an environment of other muscles, connective tissue, and bones, which may transfer transversal forces to the muscle tissue, thereby compressing it. In a recent study we demonstrated that transversal loading of a muscle with 1.3Ncm(-2) reduces maximum isometric force (Fim) and rate of force development by approximately 5% and 25%, respectively. The aim of the present study was to examine the influence of increasing transversal muscle loading on contraction dynamics. Therefore, we performed isometric experiments on rat M. gastrocnemius medialis (n=9) without and with five different transversal loads corresponding to increasing pressures of 1.3Ncm(-2) to 5.3Ncm(-2) at the contact area between muscle and load. Muscle loading was induced by a custom-made plunger which was able to move in transversal direction. Increasing transversal muscle loading resulted in an almost linear decrease in muscle force from 4.8±1.8% to 12.8±2% Fim. Compared to an unloaded isometric contraction, rate of force development decreased from 20.2±4.0% at 1.3Ncm(-2) muscle loading to 34.6±5.7% at 5.3Ncm(-2). Experimental observation of the impact of transversal muscle loading on contraction dynamics may help to better understand muscle tissue properties. Moreover, applying transversal loads to muscles opens a window to analyze three-dimensional muscle force generation. Data presented in this study may be important to develop and validate muscle models which enable simulation of muscle contractions under compression and enlighten the mechanisms behind.

Variation in isometric force after active shortening and lengthening and their mechanisms: a review

Introduction The isometric force history dependence of skeletal muscle has been studied along the last one hundred years. Several theories have been formulated to explain and establish the causes of the phenomenon, but not successfully, as they have not been fully accepted and demonstrated, and much controversy on such a subject still remains. Objective To present a systematic literature review on the dynamics of the mechanisms of force depression and force enhancement after active shortening and lengthening, respectively, identifying the key variables involved in the phenomenon, and to date to present the main theories and hypothesis developed trying to explaining it. Method The procedure of literature searching complied the major databases, including articles either, those which directly investigated the phenomena of force depression and force enhancement or those which presented possible causes and mechanisms associated with their respective events, from the earliest studies published until the year of 2010. Results 97 references were found according to the criteria used. Conclusion Based on this review, it is suggested that the theory of stress inhibition of actin-myosin cross-bridges is that better explain the phenomenon of force depression. Whereas regarding the force enhancement phenomenon, one theory have been well accepted, the increased number of actin-myosin cross-bridges in strong binding state influenced by the recruitment of passive elastic components, which hole is attributed to the titin filament.

Force enhancement and force depression in a modified muscle model used for muscle activation prediction

This article introduces history-dependent effects in a skeletal muscle model applied to dynamic simulations of musculoskeletal system motion. Force depression and force enhancement induced by active muscle shortening and lengthening, respectively, represent muscle history effects. A muscle model depending on the preceding contractile events together with the current parameters was developed for OpenSim software, and applied in simulations of standing heel-raise and squat movements. Muscle activations were computed using joint kinematics and ground reaction forces recorded from the motion capture of seven individuals. In the muscle-actuated simulations, a modification was applied to the computed activation, and was compared to the measured electromyography data. For the studied movements, the history gives a small but visible effect to the muscular force trace, but some parameter values must be identified before the exact magnitude can be analysed. The muscle model modification improves the existing muscle models and gives a more accurate description of underlying forces and activations in musculoskeletal system movement simulations.

A phenomenological muscle model to assess history dependent effects in human movement

Most musculoskeletal models used to analyze human movement utilize Hill-type muscle models that account for state dependent intrinsic muscle properties (e.g., force-length-velocity relationships), but rarely do these models include history dependent effects (e.g., force depression or enhancement). While the relationship between muscle shortening and force depression can be well characterized by muscle mechanical work, the relationship between muscle stretch and force enhancement is more complex. Further, it is not well known how these properties influence dynamic movements. Therefore, the goal of this study was to develop a modified Hill-type muscle model that incorporated stretch-induced force enhancement into a previously described model that included shortening-induced force depression. The modified muscle model was based on experimental data from isolated cat soleus muscles. Simulations of in situ muscle experiments were used to validate the model and simulations of a simple human movement task (counter-movement jumping) were used to examine the interactions of the history dependent effects. The phenomenological model of stretch-induced force enhancement was dependent on both the magnitude of stretch and relative length of the muscle fiber. Simulations of the in situ muscle experiments showed that the model could accurately reproduce force enhancement and force depression, as well as the complex additive relationship between these effects. Simulations of counter-movement jumping showed that a similar jump pattern could be achieved with and without history dependent effects and that a relatively minor change in muscle activation could mitigate the impact of these effects.

History effect and timing of force production introduced in a skeletal muscle model

Skeletal muscle modelling requires a detailed description of muscular force production. We have performed a series of experiments on mouse skeletal muscles to give a basis for an improved description of the muscular force production. Our previous work introduced a force modification in isometric phases, which was based on the work performed by or on the muscle during transient-length-varying contractions. Here, state-space diagrams were used to investigate the timing aspects of the force production. These show a dominant exponential nature of the force development in isometric phases of the contractions, reached after a non-exponential phase, assumed as an activation or deactivation stage and not further analysed here. The time constants of the exponential functions describing isometric force redevelopment after length variations appear to be related to the one for an initial isometric contraction, but depending on the previous history. The timing of force production calculated from the state-space diagrams was in agreement with the generally accepted muscle properties, thereby demonstrating the reliability of the method. A macroscopic muscular model consisting of a contractile element, parallel and series elastic elements was developed. The parameters from the experiment analysis, particularly the force modification after non-isometric contractions and the time constants, were reproduced by the simulations. The relationship between time constants introduced in a mechanistic model and the measured macroscale timings is discussed.

Common muscle synergies for control of center of mass and force in nonstepping and stepping postural behaviors

We investigated muscle activity, ground reaction forces, and center of mass (CoM) acceleration in two different postural behaviors for standing balance control in humans to determine whether common neural mechanisms are used in different postural tasks. We compared nonstepping responses, where the base of support is stationary and balance is recovered by returning CoM back to its initial position, with stepping responses, where the base of support is enlarged and balance is recovered by pushing the CoM away from the initial position. In response to perturbations of the same direction, these two postural behaviors resulted in different muscle activity and ground reaction forces. We hypothesized that a common pool of muscle synergies producing consistent task-level biomechanical functions is used to generate different postural behaviors. Two sets of support-surface translations in 12 horizontal-plane directions were presented, first to evoke stepping responses and then to evoke nonstepping responses. Electromyographs in 16 lower back and leg muscles of the stance leg were measured. Initially (∼100-ms latency), electromyographs, CoM acceleration, and forces were similar in nonstepping and stepping responses, but these diverged in later time periods (∼200 ms), when stepping occurred. We identified muscle synergies using non-negative matrix factorization and functional muscle synergies that quantified correlations between muscle synergy recruitment levels and biomechanical outputs. Functional muscle synergies that produce forces to restore CoM position in nonstepping responses were also used to displace the CoM during stepping responses. These results suggest that muscle synergies represent common neural mechanisms for CoM movement control under different dynamic conditions: stepping and nonstepping postural responses.

Computational Models for Neuromuscular Function

Computational models of the neuromuscular system hold the potential to allow us to reach a deeper understanding of neuromuscular function and clinical rehabilitation by complementing experimentation. By serving as a means to distill and explore specific hypotheses, computational models emerge from prior experimental data and motivate future experimental work. Here we review computational tools used to understand neuromuscular function including musculoskeletal modeling, machine learning, control theory, and statistical model analysis. We conclude that these tools, when used in combination, have the potential to further our understanding of neuromuscular function by serving as a rigorous means to test scientific hypotheses in ways that complement and leverage experimental data.