Hill-type Muscle Model Research Articles

A three-dimensional musculoskeletal model of the chimpanzee (Pan troglodytes) pelvis and hind limb

Musculoskeletal models have become important tools for studying a range of muscle-driven movements. However, most work has been in modern humans, with few applications in other species. Chimpanzees are facultative bipeds and our closest living relatives, and have provided numerous important insights into our own evolution. A chimpanzee musculoskeletal model would allow integration across a wide range of laboratory-based experimental data, providing new insights into the determinants of their locomotor performance capabilities, as well as the origins and evolution of human bipedalism. Here, we described a detailed three-dimensional (3D) musculoskeletal model of the chimpanzee pelvis and hind limb. The model includes geometric representations of bones and joints, as well as 35 muscle-tendon units that were represented using 44 Hill-type muscle models. Muscle architecture data, such as muscle masses, fascicle lengths and pennation angles, were drawn from literature sources. The model permits calculation of 3D muscle moment arms, muscle-tendon lengths and isometric muscle forces over a wide range of joint positions. Muscle-tendon moment arms predicted by the model were generally in good agreement with tendon-excursion estimates from cadaveric specimens. Sensitivity analyses provided information on the parameters that model predictions are most and least sensitive to, which offers important context for interpreting future results obtained with the model. Comparisons with a similar human musculoskeletal model indicate that chimpanzees are better suited for force production over a larger range of joint positions than humans. This study represents an important step in understanding the integrated function of the neuromusculoskeletal systems in chimpanzee locomotion.

Phenomenological models of the dynamics of muscle during isotonic shortening

We investigated the effectiveness of simple, Hill-type, phenomenological models of the force-length-velocity relationship for simulating measured length trajectories during muscle shortening, and, if so, what forms of the model are most useful. Using isotonic shortening data from mouse soleus and toad depressor mandibulae muscles, we showed that Hill-type models can indeed simulate the shortening trajectories with sufficiently good accuracy. However, we found that the standard form of the Hill-type muscle model, called the force-scaling model, is not a satisfactory choice. Instead, the results support the use of less frequently used models, the f-max scaling model and force-scaling with parallel spring, to simulate the shortening dynamics of muscle.

The activation time-course of contractile elements estimated from in vivo fascicle behaviours during twitch contractions

To better understand the cascade from neural activation up to force production within in vivo contracting muscle-tendon units, we estimated activation of contractile elements from experimentally measured human fascicle length change and force using a Hill-type muscle model. The experiment was conducted with respect to twitch contractions of the tibialis anterior muscle at three joint angles. As muscle contractile element force is a function of its length and velocity, the activation of contractile elements was calculated using a Hill-type muscle model and measured data. The results were able to reproduce the continuous rising activation of contractile elements after termination of electromyographic activity, the earlier shift of peak activation in time compared to twitch force, and the differences in time-course activation at three different joint angles. These findings are consistent with the predicted change in the activation of contractile elements from previous reports. Also, the results suggest that the time-course of the activation of contractile elements was greatly influenced by the change in force generating capacities related to both length and velocity, even in fixed end contractions, which could result from muscle-tendon interaction.

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.

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.

Determining all parameters necessary to build Hill-type muscle models from experiments on single muscles.

Characterizing muscle requires measuring such properties as force-length, force-activation, and force-velocity curves. These characterizations require large numbers of data points because both what type of function (e.g., linear, exponential, hyperbolic) best represents each property, and the values of the parameters in the relevant equations, need to be determined. Only a few properties are therefore generally measured in experiments on any one muscle, and complete characterizations are obtained by averaging data across a large number of muscles. Such averaging approaches can work well for muscles that are similar across individuals. However, considerable evidence indicates that large inter-individual variation exists, at least for some muscles. This variation poses difficulties for across-animal averaging approaches. Methods to fully describe all muscle's characteristics in experiments on individual muscles would therefore be useful. Prior work in stick insect extensor muscle has identified what functions describe each of this muscle's properties and shown that these equations apply across animals. Characterizing these muscles on an individual-by-individual basis therefore requires determining only the values of the parameters in these equations, not equation form. We present here techniques that allow determining all these parameter values in experiments on single muscles. This technique will allow us to compare parameter variation across individuals and to model muscles individually. Similar experiments can likely be performed on single muscles in other systems. This approach may thus provide a widely applicable method for characterizing and modeling muscles from single experiments.

Hill-type muscle model parameters determined from experiments on single muscles show large animal-to-animal variation.

Models built using mean data can represent only a very small percentage, or none, of the population being modeled, and produce different activity than any member of it. Overcoming this "averaging" pitfall requires measuring, in single individuals in single experiments, all of the system's defining characteristics. We have developed protocols that allow all the parameters in the curves used in typical Hill-type models (passive and active force-length, series elasticity, force-activation, force-velocity) to be determined from experiments on individual stick insect muscles (Blümel et al. 2012a). A requirement for means to not well represent the population is that the population shows large variation in its defining characteristics. We therefore used these protocols to measure extensor muscle defining parameters in multiple animals. Across-animal variability in these parameters can be very large, ranging from 1.3- to 17-fold. This large variation is consistent with earlier data in which extensor muscle responses to identical motor neuron driving showed large animal-to-animal variability (Hooper etal. 2006), and suggests accurate modeling of extensor muscles requires modeling individual-by-individual. These complete characterizations of individual muscles also allowed us to test for parameter correlations. Two parameter pairs significantly co-varied, suggesting that a simpler model could as well reproduce muscle response.

인체 하지부 근골격계 모델 및 의자에서 일어서는 동작 시 근력 예측

An analytical model for a human body is important to predict muscle and joint forces. Because it is difficult to estimate muscle or joint forces from a human body, the objective of this study is the development of a reliable analytical model for a human body to evaluate the lower extremity muscle and joint forces. The musculoskeletal system of the human lower extremity is modeled as a multibody system employing the Hill-type muscle model. Muscle forces are determined to minimize energy consumption, and we assume that motion is constrained in the sagittal plane. Muscle forces are calculated through an equilibrium analysis while rising from a seated position. The musculoskeletal model consists of four segments. Each segment is a rigid body and connected by frictionless revolute joints. Muscles of the lower extremity are simplified to seven muscles with those that are not related to the sagittal plane motion are ignored. Muscles that play a similar role are combined together. The results of the present study are compared with experimental results to validate the lower extremity model and the assumptions of the present study.

A comparison of two Hill-type skeletal muscle models on the construction of medial gastrocnemius length-tension curves in humans in vivo

Human length-tension curves are traditionally constructed using a model that assumes passive tension does not change during contraction (model A) even though the animal literature suggests that passive tension can decrease (model B). The study's aims were threefold: 1) measure differences in human medial gastrocnemius length-tension curves using model A vs. model B, 2) test the reliability of ultrasound constructed length-tension curves, and 3) test the robustness of fascicle length-generated length-tension curves to variations between the angle and fascicle length relationship. An isokinetic dynamometer manipulated and measured ankle angle while ultrasound was used to measure medial gastrocnemius fascicle length. Supramaximal tibial nerve stimulation was used to evoke resting muscle twitches. Length-tension curves were constructed using model A {angle-torque [A-T((A))], length-torque [L-T((A))]} or model B {length-torque [L-T((B))]} in three conditions: baseline, heel-lift (where the muscle was shortened at each angle), and baseline repeated 2 h later (+2 h). Length-tension curves constructed from model B differed from those produced via model A, indicated by a significant increase in maximum torque (≈23%) when using L-T((B)) vs. L-T((A)). No parameter measured was different between baseline and +2 h for any method, indicating good reliability when using ultrasound. Length-tension curves were unaffected by the heel-lift condition when using L-T((A)) or L-T((B)) but were affected when using A-T((A)). Since the muscle model used significantly alters human length-tension curves, and given animal data indicate model B to be more accurate when passive tension is present, we recommend that model B should be used when constructing medial gastrocnemius length-tension curves in humans in vivo.

Work partitioning of transversally loaded muscle: experimentation and simulation

Skeletal muscles are surrounded by other muscles, connective tissue and bones, which may transfer transversal forces to the muscle belly. Simple Hill-type muscle models do not consider transversal forces. Thus, the aim of this study was to examine and model the influence of transversal muscle loading on contraction dynamics, e.g. on the rate of force development and on the maximum isometric muscle force (Fim). Isometric experiments with and without transversal muscle loading were conducted on rat muscles. The muscles were loaded (1.3 N cm− 2) by a custom-made plunger which was able to move in transversal direction. Then the muscle was fully stimulated, the isometric force was measured at the distal tendon and the movement of the plunger was captured with a high-speed camera. The interaction between the muscle and the transversal load was modelled based on energy balance between the (1) work done by the contractile component (CC) and (2) the work done to lift the load, to stretch the series elastic structures and to deform the muscle. Compared with the unloaded contraction, the force rate was reduced by about 25% and Fim was reduced by 5% both in the experiment and in the simulation. The reduction in Fim resulted from using part of the work done by the CC to lift the load and deform the muscle. The response of the muscle to transversal loading opens a window into the interdependence of contractile and deformation work, which can be used to specify and validate 3D muscle models.

Sensitivity of model predictions of muscle function to changes in moment arms and muscle–tendon properties: A Monte-Carlo analysis

Hill-type muscle models are commonly used in musculoskeletal models to estimate muscle forces during human movement. However, the sensitivity of model predictions of muscle function to changes in muscle moment arms and muscle–tendon properties is not well understood. In the present study, a three-dimensional muscle-actuated model of the body was used to evaluate the sensitivity of the function of the major lower limb muscles in accelerating the whole-body center of mass during gait. Monte-Carlo analyses were used to quantify the effects of entire distributions of perturbations in the moment arms and architectural properties of muscles. In most cases, varying the moment arm and architectural properties of a muscle affected the torque generated by that muscle about the joint(s) it spanned as well as the torques generated by adjacent muscles. Muscle function was most sensitive to changes in tendon slack length and least sensitive to changes in muscle moment arm. However, the sensitivity of muscle function to changes in moment arms and architectural properties was highly muscle-specific; muscle function was most sensitive in the cases of gastrocnemius and rectus femoris and insensitive in the cases of hamstrings and the medial sub-region of gluteus maximus. The sensitivity of a muscle's function was influenced by the magnitude of the muscle's force as well as the operating region of the muscle on its force-length curve. These findings have implications for the development of subject-specific models of the human musculoskeletal system.

A multibody biomechanical model of the upper limb including the shoulder girdle

The aim of this work is to propose a robust musculoskeletal model of the upper limb to serve as the basis for the study of different types of shoulder pathologies, including the use of anatomical or reverse prostheses. The multibody biomechanical model is defined by seven rigid bodies constrained by the sternoclavicular, acromioclavicular, and glenohumeral joints, each modeled as a three d.o.f. spherical joint; the humeroulnar and radioulnar joints, each modeled as one d.o.f. hinge joint; and the scapulothoracic articulation, modeled by two holonomic constraints that allow the scapula to glide over the thorax. The muscle system includes 21 muscles described by 37 individual segments using the obstacle-set method. The muscle contraction dynamics is represented by the Hill-type muscle model, being the activation of each muscle unknown. The muscle force sharing is a redundant problem in which an optimization technique is applied to find the muscle activations, and the corresponding muscle forces, by minimizing an objective function that represents muscle energy consumption. The fulfillment of the equations of motion of the biomechanical model are enforced and the stability of the glenohumeral joint and the scapulothoracic articulation is also imposed, thus providing two sets of constraints for the optimal problem. The validation of the model is carried out by comparing the results from an acquired motion, the abduction of the arm, with available data in the literature and with EMG data.

Method for determining musculotendon parameters in subject-specific musculoskeletal models of children developed from MRI data

Accurate knowledge of muscle-tendon parameters in biomechanical models is critical for accurate simulation and analyses of human movement. An excellent example of this is the creation of subject-specific models from magnetic resonance imaging (MRI). When Hill-type muscle models are used to calculate muscle forces, the determination of muscle attachment points, optimal fiber length, tendon slack length and maximum isometric force all have a significant influence on the joint moment-angle behavior of the model.

Identification of Individual Muscle Length Parameters from Measurements of Passive Joint Moment Around the Ankle Joint

In this study, we proposed a method for estimating muscle length parameters on an individual basis from measured data based on a musculo-skeletal structure with a Hill-type muscle model. Passive joint moments of the ankle were measured for 4 healthy subjects in two different knee positions, i.e., in knee flexion and extension, using manual measurement apparatus. Estimation of muscle length parameters based on measurement data from each subject was performed using a two-dimensional musculo-skeletal model of the lower limb with a Hill-type muscle model. Predicted passive joint moment properties using the estimated parameters were consistent with measured data. The estimated length parameters of muscles differed between subjects depending on their passive joint moment properties. These observations suggest that the proposed estimation method is available to identify muscle length parameters required for a quantification of ankle joint function and the musclo-skeletal model analyses on an individual basis.

A physiologically based, multi-scale model of skeletal muscle structure and function.

Models of skeletal muscle can be classified as phenomenological or biophysical. Phenomenological models predict the muscle’s response to a specified input based on experimental measurements. Prominent phenomenological models are the Hill-type muscle models, which have been incorporated into rigid-body modeling frameworks, and three-dimensional continuum-mechanical models. Biophysically based models attempt to predict the muscle’s response as emerging from the underlying physiology of the system. In this contribution, the conventional biophysically based modeling methodology is extended to include several structural and functional characteristics of skeletal muscle. The result is a physiologically based, multi-scale skeletal muscle finite element model that is capable of representing detailed, geometrical descriptions of skeletal muscle fibers and their grouping. Together with a well-established model of motor-unit recruitment, the electro-physiological behavior of single muscle fibers within motor units is computed and linked to a continuum-mechanical constitutive law. The bridging between the cellular level and the organ level has been achieved via a multi-scale constitutive law and homogenization. The effect of homogenization has been investigated by varying the number of embedded skeletal muscle fibers and/or motor units and computing the resulting exerted muscle forces while applying the same excitatory input. All simulations were conducted using an anatomically realistic finite element model of the tibialis anterior muscle. Given the fact that the underlying electro-physiological cellular muscle model is capable of modeling metabolic fatigue effects such as potassium accumulation in the T-tubular space and inorganic phosphate build-up, the proposed framework provides a novel simulation-based way to investigate muscle behavior ranging from motor-unit recruitment to force generation and fatigue.

A real-time EMG-driven musculoskeletal model of the ankle

The real-time estimation of muscle forces could be a very valuable tool for rehabilitation. By seeing how much muscle force is being produced during rehabilitation, therapists know whether they are working within safe limits in their therapies and patients know if they are producing enough force to expect improvement. This is especially true for rehabilitation of Achilles tendon ruptures where, out of fear of overloading and causing a rerupture, minimal therapy is typically done for eight weeks post-surgery despite animal studies that show that low-level loading is beneficial. To address this need, we have developed a biomechanical model that allows for the real-time estimation of forces in the triceps surae muscle and Achilles tendon. Forces are estimated using a Hill-type muscle model. To account for differences in neuromuscular control of each subject, the model used EMGs as input. To make this clinically useful, joint angles were measured using electrogoniometers. A dynamometer was used to measure joint moments during the model calibration stage, but was not required during real-time studies. The model accounts for the force-length and force-velocity properties of muscles, and other parameters such as tendon slack length and optimal fiber length. Additional parameters, such as pennation angle and moment arm of each muscle in the model, vary as functions of joint angle. In this paper, the model is presented and it application is demonstrated in two subjects: one with a healthy Achilles tendon and a second 6 months post Achilles tendon rupture and repair.

The influence of modeling hypothesis and experimental methodologies in the accuracy of muscle force estimation using EMG-driven models

This paper discusses some methodological questions regarding the application of EMG-driven models to estimate muscle forces, for the triceps surae performing isometric contractions. Ankle torque is estimated from a Hill-type muscle model driven by EMG data, collected from the three components of triceps surae and tibialis anterior. Ankle joint torque is synchronously collected from a dynamometer, which is compared to the sum of each muscle force multiplied by the respective ankle moment arm. A protocol consisting of two steps of low and medium/high loads is used. Raw EMG signal is processed and used as the input signal for the muscle model. The difference between simulated and dynamometer measured torque is calculated as the RMS error between the two curves. A set of nominal muscle model parameters is initially chosen from literature (e.g., OpenSim), which allows observing the characteristics of the error distribution. One possibility to improve model accuracy is using individual muscle parameters. We investigated the effect of applying simple scale factors to the nominal muscle model parameters and using ultrasound for estimating muscle maximum force. Other questions regarding muscle model improvements are also addressed, such as using a nonlinear formulation of activation dynamics and variable pennation angle. Surface EMG signals acquisition and processing can also affect force estimation accuracy. Electrodes positioning can influence signal amplitude, and the one-channel EMG may not represent actual excitation for the whole muscle. We have shown that high density EMG reduces, in some cases, the torque estimation error.

EMG‐driven modeling approach to muscle force and joint load estimations: Case study in knee osteoarthritis

It is important to know the magnitude and patterns of joint loading in people with knee osteoarthritis (OA), since altered loads are implicated in onset and progression of the disease. We used an EMG-driven forward dynamics model to estimate joint loads during walking in a subject with knee OA and a healthy control subject. Kinematic, kinetic, and surface EMG data were used to predict muscle forces using a Hill-type muscle model. The muscle forces were used to balance the frontal plane moment to obtain medial and lateral condylar loads. Loads were normalized to body weight (BWs) and the mean of three trials taken. The OA subject had greater medial and lower lateral loads compared to the control subject. Seventy-five to 80% of the total load was borne on the medial compartment in the control subject, compared to 90-95% in the OA subject. In fact, complete lateral unloading occurred during midstance for the OA subject. Loading for the healthy subject was consistent with the data from instrumented knee studies. In the future, the model can be used to analyze the impact of various interventions to reduce the loads on the medial compartment in people with knee OA.

On the relevance of structure preservation to simulations of muscle actuated movements

In this work, we implement a typical nonlinear Hill-type muscle model in a structure-preserving simulation framework and investigate the differences to standard simulations of muscle-actuated movements with MATLAB/Simulink. The latter is a common tool to solve dynamical problems, in particular, in biomechanic investigations. Despite the simplicity of the examples used for comparison, it becomes obvious that the MATLAB/Simulink integrators artificially loose or gain energy and angular momentum during dynamic simulations. The relative energy error of the MATLAB/Simulink integrators related to a very low actual muscle work can naturally reach large values, even higher than 100%. But also during periods with large muscle work, the relative energy error reaches up to 2%. Even in simulations with very small time steps, energy and angular momentum errors are still present using MATLAB/Simulink and can (at least partially) be responsible for phase errors in long-term simulations. This typical behaviour of commercial integrators is known to increase for more complex models or for computations with larger time steps, whose use is crucial for efficiency, especially in the context of optimal control simulations. In contrast to that, time-stepping schemes being derived from a discrete variational principle yield discrete analogues of the Euler-Lagrange equations and Noethers theorem. This ensures that the structure of the system is preserved, i.e. the simulation results are symplectic and momentum consistent and exhibit a good energy behaviour (no drift).

다물체 모델링과 근의 특성을 이용한 무릎뼈에 가해지는 압력의 불확실성 추정 연구

The goal of this study is to estimate the force acting on the knee joint in the human body by using the Hilltype muscle model based on a musculoskeletal model of the human lower extremity in the sagittal plane. For estimating the force applied, the human leg is modeled using multi-body modeling. This leg model comprises biarticular muscles acting on two joints of the upper and lower limbs, and the muscles include some of the major muscles such as the hamstring. In order to analyze the uncertainty of the applied forces acting on the knee joint, statistical distributions of human body, leg part, parameters are required and to obtain the parameter's statistical characteristic of the part sample survey method is employed. Finally, by using the sensitivity information of the parameters, the force acting on the knee joint can be estimated.