A comparative study of muscle force estimates using Huxley's and Hill's muscle model

Determination of muscle forces in individual muscles is often essential to assess optimal performance of human motion. Inverse dynamic methods based on the kinematics of the given motion and on the use of optimisation approach are the most widely used for muscle force estimation. The aim of this study was to estimate how the choice of muscle model influences predicted muscle forces. Huxley's (1957, Prog Biophys Biop Chem. 7: 255–318) and Hill's (1938, Proc R Soc B. 126: 136–195) muscle models were used for determination of muscle forces of two antagonistic muscles of the lower extremity during cycling. Huxley's model is a complex model that couples biochemical and physical processes with the microstructure of the muscle whereas the Hill's model is a phenomenological model. Muscle forces predicted by both models are within the same range. Huxley's model predicts more realistic patterns of muscle activation but it is computationally more demanding. Therefore, if the overall muscle forces are to be assessed, it is reasonable to use a simpler implementation based on Hill's model.     

A new parametric approach for modeling hip forces during gait

An analytical parametric model was developed to estimate the natural biological variations in muscle forces and their effect on the hip forces subject only to physiological constraints and not predefined optimization criterion. Force predictions are based on the joint kinematics and kinetics of each subject, a previously published muscle model, and physiological constraints on the muscle force distributions. The model was used to determine the hip contact forces throughout the stance phase of gait of a subject with a total hip replacement (THR). The parametrically modeled peak hip force without antagonistic muscle activity varied from 2.7 to 3.2 Body Weights (mean 2.9 Body Weights), which agreed well with published in vivo measurements from instrumented THRs in other subjects. For every 10% increase in antagonistic activity, the mean peak hip force increased by 0.2 Body Weights. The parametric model allows one to examine the effect of specific muscle weaknesses or increased antagonistic muscle activity on the hip forces. The model also provides a tool for studying the effect of gait adaptations on hip forces, as predictions are based on each individual's gait data. Differences in peak forces between subjects can then be evaluated relative to the uncertainty in not knowing the precise muscle force distributions.     

Simultaneous prediction of muscle and contact forces in the knee during gait

Musculoskeletal models are currently the primary means for estimating in vivo muscle and contact forces in the knee during gait. These models typically couple a dynamic skeletal model with individual muscle models but rarely include articular contact models due to their high computational cost. This study evaluates a novel method for predicting muscle and contact forces simultaneously in the knee during gait. The method utilizes a 12 degree-of-freedom knee model (femur, tibia, and patella) combining muscle, articular contact, and dynamic skeletal models. Eight static optimization problems were formulated using two cost functions (one based on muscle activations and one based on contact forces) and four constraints sets (each composed of different combinations of inverse dynamic loads). The estimated muscle and contact forces were evaluated using in vivo tibial contact force data collected from a patient with a force-measuring knee implant. When the eight optimization problems were solved with added constraints to match the in vivo contact force measurements, root-mean-square errors in predicted contact forces were less than 10 N. Furthermore, muscle and patellar contact forces predicted by the two cost functions became more similar as more inverse dynamic loads were used as constraints. When the contact force constraints were removed, estimated medial contact forces were similar and lateral contact forces lower in magnitude compared to measured contact forces, with estimated muscle forces being sensitive and estimated patellar contact forces relatively insensitive to the choice of cost function and constraint set. These results suggest that optimization problem formulation coupled with knee model complexity can significantly affect predicted muscle and contact forces in the knee during gait. Further research using a complete lower limb model is needed to assess the importance of this finding to the muscle and contact force estimation process.     

A dynamic model of the forearm including fatigue

A biomechanical model of the forearm, consisting of 61 muscle-tendon systems or tendons and 8 sections, is presented. The model can be used to calculate the muscle forces when resultant of the external forces and the motion is known. Calculations are based on constraints of muscle forces, joint forces, contact forces, and tendon junctions, and a load sharing principle telling which of the feasible solutions are likely and which are not. Fatigue is accounted for by updating the upper limits of the muscle forces according to the loading history. As an example, the model is used to predict the load sharing between the fingers when they are pressed against a table with a given total force.     

A mass-length scaling law for modeling muscle strength in the lower limb

Musculoskeletal computer models are often used to study muscle function in children with and without impaired mobility. Calculations of muscle forces depend in part on the assumed strength of each muscle, represented by the peak isometric force parameter, which is usually based on measurements obtained from cadavers of adult donors. The aim of the present study was twofold: first, to develop a method for scaling lower-limb peak isometric muscle forces in typically-developing children; and second, to determine the effect of this scaling method on model calculations of muscle forces obtained for normal gait. Muscle volumes were determined from magnetic resonance (MR) images obtained from ten children aged from 7 to 13yr. A new mass-length scaling law was developed based on the assumption that muscle volume and body mass are linearly related, which was confirmed by the obtained volume and body mass data. Two musculoskeletal models were developed for each subject: one in which peak isometric muscle forces were estimated using the mass-length scaling law; and another in which these parameters were determined directly from the MR-derived muscle volumes. Musculoskeletal modeling and quantitative gait analysis were then used to calculate lower-limb muscle forces in normal walking. The patterns of muscle forces predicted by the model with scaled peak isometric force values were similar to those predicted by the MR-based model, implying that assessments of muscle function obtained from these two methods are practically equivalent. These results support the use of mass-length scaling in the development of subject-specific musculoskeletal models of children.     

Computational biodynamics of human knee joint in gait: from muscle forces to cartilage stresses

Using a validated finite element model of the intact knee joint we aim to compute muscle forces and joint response in the stance phase of gait. The model is driven by reported in vivo kinematics-kinetics data and ground reaction forces in asymptomatic subjects. Cartilage layers and menisci are simulated as depth-dependent tissues with collagen fibril networks. A simplified model with less refined mesh and isotropic depth-independent cartilage is also considered to investigate the effect of model accuracy on results. Muscle forces and joint detailed response are computed following an iterative procedure yielding results that satisfy kinematics/kinetics constraints while accounting at deformed configurations for muscle forces and passive properties. Predictions confirm that muscle forces and joint response alter substantially during the stance phase and that a simplified joint model may accurately be used to estimate muscle forces but not necessarily contact forces/areas, tissue stresses/strains, and ligament forces. Predictions are in general agreement with results of earlier studies. Performing the analyses at 6 periods from beginning to the end (0%, 5%, 25%, 50%, 75% and 100%), hamstrings forces peaked at 5%, quadriceps forces at 25% whereas gastrocnemius forces at 75%. ACL Force reached its maximum of 343 N at 25% and decreased thereafter. Contact forces reached maximum at 5%, 25% and 75% periods with the medial compartment carrying a major portion of load and experiencing larger relative movements and cartilage strains. Much smaller contact stresses were computed at the patellofemoral joint. This novel iterative kinematics-driven model is promising for the joint analysis in altered conditions.     

Optimization of skeletal configuration: studies of scoliosis correction biomechanics

A scheme for optimizing configurations in models of skeletal structures is presented. Use of the scheme is illustrated through determination of biomechanically optimal correction of a right-thoracic scoliosis by passive brace and active muscle forces. The locations and magnitudes of the passive brace forces, and the trunk muscle groups and their corresponding contraction intensity magnitudes that would optimally correct the geometric deformities of the spine were determined. The results suggest that, from a biomechanical viewpoint, both brace and muscle forces are capable of substantial correction of a model thoracic scoliosis. However, comparison of model results with long-term clinical results suggests that even under optimal conditions it is unlikely that scoliosis can be fully corrected by passive brace forces or active muscle contractions.     

Dynamic joint and muscle forces during knee isokinetic exercise

Isokinetic exercise has been commonly used in knee rehabilitation, conditioning and research in the past two decades. Although many investigators have used various experimental and theoretical approaches to study the muscle and joint force involved in isokinetic knee extension and flexion exercises, only a few of these studies have actually distinguished between the tibiofemoral joint forces and muscle forces. Therefore, the objective of this study was to specify, via an eletromyography(EMG)-driven muscle force model of the knee, the magnitude of the tibiofemoral joint and muscle forces acting during isokinetic knee extension and flexion exercises. Fifteen subjects ranging from 21 to 36 years of age volunteered to participate in this study. A Kin Com exercise machine (Chattecx Corporation, Chattanooga, TN, U.S.A.) was used as the loading device. An EMG-driven muscle force model was used to predict muscle forces, and a biomechanical model was used to analyze two knee joint constraint forces; compression and shear force. The methods used in this study were shown to be valid and reliable (r > 0.84 andp < 0.05). The effects on the tibiofemoral joint force during knee isokinetic exercises were compared with several functional activities that were investigated by earlier researchers. The muscle forces generated during knee isokinetic exercise were also obtained. Based on the findings obtained in this study, several therapeutic justifications for knee rehabilitation are proposed.     

Elastic Energy Storage and Radial Forces in the Myofilament Lattice Depend on Sarcomere Length

We most often consider muscle as a motor generating force in the direction of shortening, but less often consider its roles as a spring or a brake. Here we develop a fully three-dimensional spatially explicit model of muscle to isolate the locations of forces and energies that are difficult to separate experimentally. We show the strain energy in the thick and thin filaments is less than one third the strain energy in attached cross-bridges. This result suggests the cross-bridges act as springs, storing energy within muscle in addition to generating the force which powers muscle. Comparing model estimates of energy consumed to elastic energy stored, we show that the ratio of these two properties changes with sarcomere length. The model predicts storage of a greater fraction of energy at short sarcomere lengths, suggesting a mechanism by which muscle function shifts as force production declines, from motor to spring. Additionally, we investigate the force that muscle produces in the radial or transverse direction, orthogonal to the direction of shortening. We confirm prior experimental estimates that place radial forces on the same order of magnitude as axial forces, although we find that radial forces and axial forces vary differently with changes in sarcomere length.     

Validation of a biodynamic model of pushing and pulling.

Pushing and pulling during manual material handling can increase the compressive forces on the lumbar disc region while creating high shear forces at the shoe-floor interface. A sagittal plane dynamic model derived from previous biomechanical models was developed to predict L5/S1 compressive force and required coefficients of friction during dynamic cart pushing and pulling. Before these predictions could be interpreted, however, it was necessary to validate model predictions against independently measured values of comparable quantities. This experiment used subjects of disparate stature and body mass, while task factors such as cart resistance and walking speed were varied. Predicted ground reaction forces were compared with those measured by a force platform, with correlations up to 0.67. Predicted erector spinae and rectus abdominus muscle forces were compared with muscle forces derived from RMS-EMGs of the respective muscle groups, using a static force build-up regression relationship to transform the dynamic RMS-EMGs to trunk muscle forces. Although correlations were low, this was attributed in part to the use of surface EMG on subjects of widely varied body mass. The biodynamic model holds promise as a tool for analysis of actual industrial pushing and pulling tasks, when carefully applied.     

Load-sharing in the lumbosacral spine in neutral standing & flexed postures – A combined finite element and inverse static study

Understanding load-sharing in the spine during in-vivo conditions is critical for better spinal implant design and testing. Previous studies of load-sharing that considered actual spinal geometry applied compressive follower load, with or without moment, to simulate muscle forces. Other studies used musculoskeletal models, which include muscle forces, but model the discs by simple beams or spherical joints and ignore the articular facet joints.This study investigated load-sharing in neutral standing and flexed postures using a detailed Finite Element (FE) model of the ligamentous lumbosacral spine, where muscle forces, gravity loads and intra-abdominal pressure, as predicted by a musculoskeletal model of the upper body, are input into the FE model. Flexion was simulated by applying vertebral rotations following spine rhythm measured in a previous in-vivo study, to the musculoskeletal model. The FE model predicted intradiscal pressure (IDP), strains in the annular fibers, contact forces in the facet joints, and forces in the ligaments. The disc forces and moments were determined using equilibrium equations, which considered the applied loads, including muscle forces and IDP, as well as forces in the ligaments and facet joints predicted by the FE model. Load-sharing was calculated as the portion of the total spinal load carried along the spine by each individual spinal structure. The results revealed that spinal loads which increased substantially from the upright to the flexed posture were mainly supported by the discs in the upright posture, whereas the ligaments’ contribution in resisting shear, compression, and moment was more significant in the flexed posture.     

Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses

There are different opinions in the literature on whether the cost functions: the sum of muscle stresses squared and the sum of muscle stresses cubed, can reasonably predict muscle forces in humans. One potential reason for the discrepancy in the results could be that different authors use different sets of model parameters which could substantially affect forces predicted by optimization-based models. In this study, the sensitivity of the optimal solution obtained by minimizing the above cost functions for a planar three degrees-of-freedom (DOF) model of the leg with nine muscles was investigated analytically for the quadratic function and numerically for the cubic function. Analytical results revealed that, generally, the non-zero optimal force of each muscle depends in a very complex non-linear way on moments at all three joints and moment arms and physiological cross-sectional areas (PCSAs) of all muscles. Deviations of the model parameters (moment arms and PCSAs) from their nominal values within a physiologically feasible range affected not only the magnitude of the forces predicted by both criteria, but also the number of non-zero forces in the optimal solution and the combination of muscles with non-zero predicted forces. Muscle force magnitudes calculated by both criteria were similar. They could change several times as model parameters changed, whereas patterns of muscle forces were typically not as sensitive. It is concluded that different opinions in the literature about the behavior of optimization-based models can be potentially explained by differences in employed model parameters.     

A biomechanical model for the analysis of the cervical spine in static postures.

To gain a better understanding of the forces working on the cervical spine, a spatial biomechanical computer model was developed. The first part of our research was concerned with the development of a kinematic model to establish the axes of rotation and the mutual position of the head and vertebrae with regard to flexion, extension, lateroflexion and torsion. The next step was the introduction of lines of action of muscle forces and an external load, created by gravity and accelerations in different directions, working on the centre of gravity of the head and possibly a helmet. Although the results of our calculations should be interpreted cautiously in the present stage of our research, some conclusions can be drawn with respect to different head positions. During flexion muscle forces and joint reaction forces increase, except the force between the odontoid and the ligamentum transversum atlantis. This force shows a minimum during moderate flexion. The joint reaction forces on the levels C0-C1, C1-C2, and C7-T1 reach minimum values during extension, each in different stages of extension. Axial rotation less than 35 degrees does not need great muscle forces, axial rotation further than 35 degrees causes muscle forces and joint reaction forces to increase fast. While performing, lateral flexion muscle forces and joint reaction forces must increase rapidly to balance the head. We obtained some indications that the order of magnitude of the calculated forces is correct.     

Estimating the muscle forces generated in the human lower extremity when walking: a physiological solution

A model capable of estimating individual muscle activity during human lower extremity activity has been developed. This model was implemented and applied to normal gait. Both 3D cinematography and force platform records were collected to define the kinematics of the lower limb segments and the ground reaction forces. From these data the lengths, velocities, and moment arms of the muscles and the net muscle moments were calculated. These data were then used to estimate the output from a set of pattern generators which specified the neural input to and hence the force generated in each muscle. Results were obtained which demonstrate that it is feasible to construct a hierarchical physiological model which will estimate individual muscle forces. This statement must be tempered somewhat in that better anatomical and neurophysiological data must be made available and that this model, and all others attempting to estimate individual muscle forces, must be rigorously tested before being applied in a research, sports, or clinical setting.     

Subject-specific musculoskeletal modelling in patients before and after total hip arthroplasty*

The goal of this study was to define the effect on hip contact forces of including subject-specific moment generating capacity in the musculoskeletal model by scaling isometric muscle strength and by including geometrical information in control subjects, hip osteoarthritis and total hip arthroplasty patients. Scaling based on dynamometer measurements decreased the strength of all flexor and abductor muscles. This resulted in a model that lacked the capacity to generate joint moments required during functional activities. Scaling muscle forces based on functional activities and inclusion of MRI-based geometrical detail did not compromise the model strength and resulted in hip contact forces comparable to previously reported measured contact forces.     

The effects of electromyography-assisted modelling in estimating musculotendon forces during gait in children with cerebral palsy

Neuro-musculoskeletal modelling can provide insight into the aberrant muscle function during walking in those suffering cerebral palsy (CP). However, such modelling employs optimization to estimate muscle activation that may not account for disturbed motor control and muscle weakness in CP. This study evaluated different forms of neuro-musculoskeletal model personalization and optimization to estimate musculotendon forces during gait of nine children with CP (GMFCS I-II) and nine typically developing (TD) children. Data collection included 3D-kinematics, ground reaction forces, and electromyography (EMG) of eight lower limb muscles. Four different optimization methods estimated muscle activation and musculotendon forces of a scaled-generic musculoskeletal model for each child walking, i.e. (i) static optimization that minimized summed-excitation squared; (ii) static optimization with maximum isometric muscle forces scaled to body mass; (iii) an EMG-assisted approach using optimization to minimize summed-excitation squared while reducing tracking errors of experimental EMG-linear envelopes and joint moments; and (iv) EMG-assisted with musculotendon model parameters first personalized by calibration. Both static optimization approaches showed a relatively low model performance compared to EMG envelopes. EMG-assisted approaches performed much better, especially in CP, with only a minor mismatch in joint moments. Calibration did not affect model performance significantly, however it did affect musculotendon forces, especially in CP. A model more consistent with experimental measures is more likely to yield more physiologically representative results. Therefore, this study highlights the importance of calibrated EMG-assisted modelling when estimating musculotendon forces in TD children and even more so in children with CP.     

A three-dimensional mathematical model of the human masticatory system predicting maximum possible bite forces

A three-dimensional mathematical model of the human masticatory system, containing 16 muscle forces and two joint reaction forces, is described. The model allows simulation of static bite forces and concomitant joint reaction forces for various bite point locations and mandibular positions. The system parameters for the model were obtained from a cadaver head. Maximum possible bite forces were computed using optimization techniques; the optimization criterion we used was the minimizing of the relative activity of the most active muscle. The model predicts that at each specific bite point, bite forces can be generated in a wide range of directions, and that the magnitude of the maximum bite force depends on its direction. The relationship between bite force direction and its maximum magnitude depends on bite point location and mandibular position. In general, the direction of the largest possible bite force does not coincide with the direction perpendicular to the occlusal plane.     

Estimation of trunk muscle forces using the finite element method and in vivo loads measured by telemeterized internal spinal fixation devices.

Direct quantitative measurement of muscle forces is not possible. Forces in the trunk muscles were estimated for standing and flexion of the upper body using three-dimensional, nonlinear finite element models of the lumbar spine with and without an internal spinal fixation device. Muscle forces assumed were two pairs dorsally and one ventrally, each representing several muscles. Muscle forces in the model with internal fixators were varied in discrete steps until the implant loads calculated closely corresponded to those measured in a patient with an instrumented implant. The calculated angles between adjacent lumbar vertebrae were compared with corresponding values measured on X-ray films of a patient as well as with literature values and served as a second criterion for predicting muscle forces. For the model without an implant, the muscle forces of the first model were slightly varied until the lumbar spine shape and the intradiscal pressure were physiological. The abdomen was shown to have a considerable supporting function for flexion.     

Extension of a state-of-the-art optimization criterion to predict co-contraction

Most studies concerned with the prediction of muscle forces have tried to predict a physiologically reasonable, synergistic muscle behavior. In addition to the load sharing of synergistic muscles, co-contraction of antagonistic muscles also occurs. An extension to a standard quadratic criterion for the calculation of muscle forces is presented in this study. This extension however is not limited to quadratic optimization. The extension is applied to a planar, one degree of freedom model of the human knee. For this model an analytical solution is presented. With the extended criterion it was possible to predict and control the amount of co-contraction for the knee model. The enforced antagonistic muscle activity led to higher agonistic muscle activity. In the absence of an external load flexor and extensor muscles were activated. As a consequence the knee joint was preloaded. This might indicate that antagonistic muscle activity is generated to maintain or improve joint stability. In conclusion, this study presents a novel approach to predict co-contraction when using optimization techniques to determine muscle forces by introducing a shift parameter for the optimization criterion.     

Sensitivity of muscle and intervertebral disc force computations to variations in muscle attachment sites

Abstract

The current paper aims at assessing the sensitivity of muscle and intervertebral disc force computations against potential errors in modeling muscle attachment sites. We perturbed each attachment location in a complete and coherent musculoskeletal model of the human spine and quantified the changes in muscle and disc forces during standing upright, flexion, lateral bending, and axial rotation of the trunk. Although the majority of the muscles caused minor changes (less than 5%) in the disc forces, certain muscle groups, for example, quadratus lumborum, altered the shear and compressive forces as high as 353% and 17%, respectively. Furthermore, percent changes were higher in the shear forces than in the compressive forces. Our analyses identified certain muscles in the rib cage (intercostales interni and intercostales externi) and lumbar spine (quadratus lumborum and longissimus thoracis) as being more influential for computing muscle and disc forces. Furthermore, the disc forces at the L4/L5 joint were the most sensitive against muscle attachment sites, followed by T6/T7 and T12/L1 joints. Presented findings suggest that modeling muscle attachment sites based on solely anatomical illustrations might lead to erroneous evaluation of internal forces and promote using anatomical datasets where these locations were accurately measured. When developing a personalized model of the spine, certain care should also be paid especially for the muscles indicated in this work.