3D finite element models of shoulder muscles for computing lines of actions and moment arms

Accurate representation of musculoskeletal geometry is needed to characterise the function of shoulder muscles. Previous models of shoulder muscles have represented muscle geometry as a collection of line segments, making it difficult to account for the large attachment areas, muscle–muscle interactions and complex muscle fibre trajectories typical of shoulder muscles. To better represent shoulder muscle geometry, we developed 3D finite element models of the deltoid and rotator cuff muscles and used the models to examine muscle function. Muscle fibre paths within the muscles were approximated, and moment arms were calculated for two motions: thoracohumeral abduction and internal/external rotation. We found that muscle fibre moment arms varied substantially across each muscle. For example, supraspinatus is considered a weak external rotator, but the 3D model of supraspinatus showed that the anterior fibres provide substantial internal rotation while the posterior fibres act as external rotators. Including the effects of large attachment regions and 3D mechanical interactions of muscle fibres constrains muscle motion, generates more realistic muscle paths and allows deeper analysis of shoulder muscle function.     

Modeling the effects of musculoskeletal geometry on scapulohumeral muscle moment arms and lines of action

Abstract

Biomechanical investigations examining shoulder function commonly observe a high degree of inter-individual variability in muscle activity and kinematic patterns during static and dynamic upper extremity exertions. Substantial differences in musculoskeletal geometry between individuals can alter muscle moment arms and lines of action that, theoretically, alter muscle activity and shoulder kinematics. The purposes of this research were to: (i) quantify model-predicted functional roles (moment arms, lines of action) of the scapulohumeral muscles, (ii) compare model predictions to experimental data in the literature, and (iii) evaluate sensitivity of muscle functional roles due to changes in muscle attachment locations using probabilistic modeling. Monte Carlo simulations were performed to iteratively adjust muscle attachment locations at the clavicle, scapula, and humerus of the Delft Shoulder and Elbow Model in OpenSim. Muscle moment arms and lines of action were quantified throughout arm elevation in the scapular plane. In general, model-predicted moment arms agreed well with the reviewed literature; however, notable inconsistencies were observed when comparing lines of action. Variability in moment arms and lines of action were muscle-specific, with 2 standard deviations in moment arm and line of actions as high as 25.8?mm and 18.8° for some muscles, respectively. Moment arms were particularly sensitive to changes in attachment site closest to the joint centre. Variations in muscle functional roles due to differences in musculoskeletal geometry are expected to require different muscle activity and movement patterns for upper extremity exertions.     

Effects of rotator cuff tears on muscle moment arms: a computational study

Rotator cuff tears cause morphologic changes to cuff tendons and muscles, which can alter muscle architecture and moment arm. The effects of these alterations on shoulder mechanical performance in terms of muscle force and joint strength are not well understood. The purpose of this study was to develop a three-dimensional explicit finite element model for investigating morphological changes to rotator cuff tendons following cuff tear. The subsequent objectives were to validate the model by comparing model-predicted moment arms to empirical data, and to use the model to investigate the hypothesis that rotator cuff muscle moment arms are reduced when tendons are divided along the force-bearing direction of the tendon. The model was constructed by extracting tendon, cartilage, and bone geometry from the male Visible Human data set. Infraspinatus and teres minor muscle and tendon paths were identified relative to the humerus and scapula. Kinetic and kinematic boundary conditions in the model replicated experimental protocols, which rotated the humerus from 45 degrees internal to 45 degrees external rotation with constant loads on the tendons. External rotation moment arms were calculated for two conditions of the cuff tendons: intact normal and divided tendon. Predicted moment arms were within the 1-99% confidence intervals of experimental data for nearly all joint angles and tendon sub-regions. In agreement with the experimental findings, when compared to the intact condition, predicted moment arms were reduced for the divided tendon condition. The results of this study provide evidence that one potential mechanism for the reduction in strength observed with cuff tear is reduction of muscle moment arms. The model provides a platform for future studies addressing mechanisms responsible for reduced muscle force and joint strength including changes to muscle length-tension operating range due to altered muscle and tendon excursions, and the effects of cuff tear size and location on moment arms and muscle forces.     

Three-dimensional model to predict muscle forces and their relation to motor variances in reaching arm movements

A three-dimensional (3-D) arm movement model is presented to simulate kinematic properties and muscle forces in reaching arm movements. Healthy subjects performed reaching movements repetitively either with or without a load in the hand. Joint coordinates were measured. Muscle moment arms, 3-D angular acceleration, and moment of inertias of arm segments were calculated to determine 3-D joint torques. Variances of hand position, arm configuration, and muscle activities were calculated. Ratios of movement variances observed in the two conditions (load versus without load) showed no differences for hand position and arm configuration variances. Virtual muscle force variances for all muscles except deltoid posterior and EMG variances for four muscles increased significantly by moving with the load. The greatly increased variances in muscle activity did not imply equally high increments in kinematic variances. We conclude that enhanced muscle cooperation through synergies helps to stabilize movement at the kinematic level when a load is added.     

Estimation of instantaneous moment arms of lower-leg muscles

Muscle moment arms at the human knee and ankle were estimated from muscle length changes measured as a function of joint flexion angle in cadaver specimens. Nearly all lower-leg muscles were studied: extensor digitorum longus, extensor hallucis longus, flexor digitorum longus, flexor hallucis longus, gastrocnemius lateralis, gastrocnemius medialis, peroneus brevis, peroneus longus, peroneus tertius, plantaris, soleus, tibialis anterior, and tibialis posterior. Noise in measured muscle length was filtered by means of quintic splines. Moment arms of the mm. gastrocnemii appear to be much more dependent on joint flexion angles than was generally assumed by other investigators. Some consequences for earlier analyses are mentioned.     

Comparison between line and surface mesh models to represent the rotator cuff muscle geometry in musculoskeletal models

Accurate muscle geometry (muscle length and moment arm) is required to estimate muscle function when using musculoskeletal modelling. In shoulder, muscles are often modelled as a collection of independent line segments, leading to non-physiological muscles trajectory, especially for the rotator cuff muscles. To prevent this, a surface mesh model was developed and validated against 7 MRI positions in one participant. Mean moment arm errors was 11.4% for the line vs. 8.8% for the mesh model. While the model with independent lines led to some non-physiological trajectories, the mesh model gave lower misestimations of muscle lengths and moment arms.     

Shoulder muscle function depends on elbow joint position: an illustration of dynamic coupling in the upper limb

Shoulder muscle function has been documented based on muscle moment arms, lines of action and muscle contributions to contact force at the glenohumeral joint. At present, however, the contributions of individual muscles to shoulder joint motion have not been investigated, and the effects of shoulder and elbow joint position on shoulder muscle function are not well understood. The aims of this study were to compute the contributions of individual muscles to motion of the glenohumeral joint during abduction, and to examine the effect of elbow flexion on shoulder muscle function. A three-dimensional musculoskeletal model of the upper limb was used to determine the contributions of 18 major muscles and muscle sub-regions of the shoulder to glenohumeral joint motion during abduction. Muscle function was found to depend strongly on both shoulder and elbow joint positions. When the elbow was extended, the middle and anterior deltoid and supraspinatus were the greatest contributors to angular acceleration of the shoulder in abduction. In contrast, when the elbow was flexed at 90°, the anterior deltoid and subscapularis were the greatest contributors to joint angular acceleration in abduction. This dependence of shoulder muscle function on elbow joint position is explained by the existence of dynamic coupling in multi-joint musculoskeletal systems. The extent to which dynamic coupling affects shoulder muscle function, and therefore movement control, is determined by the structure of the inverse mass matrix, which depends on the configuration of the joints. The data provided may assist in the diagnosis of abnormal shoulder function, for example, due to muscle paralysis or in the case of full-thickness rotator cuff tears.     

Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines

We present a robust and computationally inexpensive method to estimate the lengths and three-dimensional moment arms for a large number of musculotendon actuators of the human lower limb. Using a musculoskeletal model of the lower extremity, a set of values was established for the length of each musculotendon actuator for different lower limb generalized coordinates (joint angles). A multidimensional spline function was then used to fit these data. Muscle moment arms were obtained by differentiating the musculotendon length spline function with respect to the generalized coordinate of interest. This new method was then compared to a previously used polynomial regression method. Compared to the polynomial regression method, the multidimensional spline method produced lower errors for estimating musculotendon lengths and moment arms throughout the whole generalized coordinate workspace. The fitting accuracy was also less affected by the number of dependent degrees of freedom and by the amount of experimental data available. The spline method only required information on musculotendon lengths to estimate both musculotendon lengths and moment arms, thus relaxing data input requirements, whereas the polynomial regression requires different equations to be used for both musculotendon lengths and moment arms. Finally, we used the spline method in conjunction with an electromyography driven musculoskeletal model to estimate muscle forces under different contractile conditions, which showed that the method is suitable for the integration into large scale neuromusculoskeletal models.     

A model to predict canine pelvic limb musculoskeletal geometry

We developed a model to predict the three-dimensional canine pelvic limb muscular geometry (i.e., all muscle moment arms during any instant in gait). Forty-one muscle origins and insertions, as well as external landmarks (to obtain anthropometric dimensions) were marked on both pelvic limbs of five dogs and digitized on biplanar radiographs. Reference frames in the pelvis, femur, and tibia established the three-dimensional coordinates of each origin, insertion, and landmark. A set of dimensionless 'scaled coordinates' was created by dividing the actual origin and insertion coordinates by selected anthropometric dimensions of each animal. Combining scaled coordinates from all ten limbs produced an averaged 'template' of scaled coordinates. To provide limited validation of the scaling procedure, we measured the anthropometric dimensions between externally palpable landmarks of two additional pelvic limbs. The anthropometric dimensions were multiplied by the averaged template coordinates to calculate two new sets of hindlimb muscle coordinates within the three bony reference frames. The two limbs then were dissected, muscle endpoints were marked, and biplanar radiographs of each of the limb segments were digitized. The actual coordinates so obtained were similar to those predicted by the template and anthropometric measures.     

Modelling the mechanical effect of muscles with large attachment sites: Application to the shoulder mechanism

A general theory is described for deriving the mechanical effect of muscles with large attachment sites. In a cadaver experiment the complete attachment sites and bundle distribution of 16 muscles of the shoulder mechanism were recorded. These data were used to calculate the mechanical effect, i.e. the resulting force and moment vector, for a large number (200) and a reduced number (maximal 6) of muscle lines of action. The resulting error between both representations is small. The number of muscle lines of action in the reduced representation depends on the shape of the attachment site and muscle architecture. An important feature of this method is that the necessary number of muscle lines of action is determined afterwards. In the often used centroid line approach the number of muscle lines of action and partitioning of muscles is determined before recording the geometry, leading to unverifiable results.     

Measuring morphological parameters of the pelvic floor for finite element modelling purposes

The goal of this study was to obtain a complete data set needed for studying the complex biomechanical behaviour of the pelvic floor muscles using a computer model based on the finite element (FE) theory. The model should be able to predict the effect of surgical interventions and give insight into the function of pelvic floor muscles. Because there was a lack of any information concerning morphological parameters of the pelvic floor muscle structures, we performed an experimental measurement to uncover those morphological parameters. Geometric parameters as well as muscle parameters of the pelvic floor muscles were measured on an embalmed female cadaver. A three-dimensional (3D) geometric data set of the pelvic floor including muscle fibre directions was obtained using a palpator device. A 3D surface model based on the experimental data, needed for mathematical modelling of the pelvic floor, was created. For all parts of the diaphragma pelvis, the optimal muscle fibre length was determined by laser diffraction measurements of the sarcomere length. In addition, other muscle parameters such as physiological cross-sectional area and total muscle fibre length were determined. Apart from these measurements we obtained a data set of the pelvic floor structures based on nuclear magnetic resonance imaging (MRI) on the same cadaver specimen. The purpose of this experiment was to discover the relationship between the MRI morphology and geometrical parameters obtained from the previous measurements. The produced data set is not only important for biomechanical modelling of the pelvic floor muscles, but it also describes the geometry of muscle fibres and is useful for functional analysis of the pelvic floor in general. By the use of many reference landmarks all these morphologic data concerning fibre directions and optimal fibre length can be morphed to the geometrical data based on segmentation from MRI scans.These data can be directly used as an input for building a mathematical model based on FE theory.     

Chimpanzee thumb muscle cross sections, moment arms and potential torques, and comparisons with humans.

This study investigates the morphological basis of differences between humans and chimpanzees in the kinematical and dynamical parameters of the musculature of the thumb. It is partly intended to test an hypothesis that human thumb muscles can exert significantly greater torques, due to larger muscle cross-sectional areas or to longer tendon moment arms or to both. We focus on the estimation of the potentials of thumb muscles to exert torques about joint axes in a sample of eight chimpanzee cadaver hands. The potential torque of a muscle is estimated by taking the product of a muscle's physiological cross-sectional area (an estimator of force) with its dynamical moment arm (derived from the slope of tendon excursion versus joint angular displacement, obtained during passive movements of cadaver thumb joints). Comparison of our results with similar data obtained for humans at the same Mayo Clinic laboratory shows significant differences between humans and chimpanzees in potential torque of most thumb muscles, those of humans generally exhibiting larger values. The primary reason for the larger torques in humans is that their average moment arms are significantly longer, permitting greater torque for a given muscle size. An additional finding is that chimpanzees and humans differ in the direction of secondary thumb metacarpal movements elicited by contraction of some muscles, as shown by differences in moment arm signs for a given movement in the same muscle. The differences appear to be related to differences in the musculo-skeletal structures of the trapeziometacarpal joint.     

Approximating complex musculoskeletal biomechanics using multidimensional autogenerating polynomials

Computational models of the musculoskeletal system are scientific tools used to study human movement, quantify the effects of injury and disease, plan surgical interventions, or control realistic high-dimensional articulated prosthetic limbs. If the models are sufficiently accurate, they may embed complex relationships within the sensorimotor system. These potential benefits are limited by the challenge of implementing fast and accurate musculoskeletal computations. A typical hand muscle spans over 3 degrees of freedom (DOF), wrapping over complex geometrical constraints that change its moment arms and lead to complex posture-dependent variation in torque generation. Here, we report a method to accurately and efficiently calculate musculotendon length and moment arms across all physiological postures of the forearm muscles that actuate the hand and wrist. Then, we use this model to test the hypothesis that the functional similarities of muscle actions are embedded in muscle structure. The posture dependent muscle geometry, moment arms and lengths of modeled muscles were captured using autogenerating polynomials that expanded their optimal selection of terms using information measurements. The iterative process approximated 33 musculotendon actuators, each spanning up to 6 DOFs in an 18 DOF model of the human arm and hand, defined over the full physiological range of motion. Using these polynomials, the entire forearm anatomy could be computed in <10 μs, which is far better than what is required for real-time performance, and with low errors in moment arms (below 5%) and lengths (below 0.4%). Moreover, we demonstrate that the number of elements in these autogenerating polynomials does not increase exponentially with increasing muscle complexity; complexity increases linearly instead. Dimensionality reduction using the polynomial terms alone resulted in clusters comprised of muscles with similar functions, indicating the high accuracy of approximating models. We propose that this novel method of describing musculoskeletal biomechanics might further improve the applications of detailed and scalable models to describe human movement.     

Static optimization underestimates antagonist muscle activity at the glenohumeral joint: A musculoskeletal modeling study

Static optimization is commonly employed in musculoskeletal modeling to estimate muscle and joint loading; however, the ability of this approach to predict antagonist muscle activity at the shoulder is poorly understood. Antagonist muscles, which contribute negatively to a net joint moment, are known to be important for maintaining glenohumeral joint stability. This study aimed to compare muscle and joint force predictions from a subject-specific neuromusculoskeletal model of the shoulder driven entirely by measured muscle electromyography (EMG) data with those from a musculoskeletal model employing static optimization. Four healthy adults performed six sub-maximal upper-limb contractions including shoulder abduction, adduction, flexion, extension, internal rotation and external rotation. EMG data were simultaneously measured from 16 shoulder muscles using surface and intramuscular electrodes, and joint motion evaluated using video motion analysis. Muscle and joint forces were calculated using both a calibrated EMG-driven neuromusculoskeletal modeling framework, and musculoskeletal model simulations that employed static optimization. The EMG-driven model predicted antagonistic muscle function for pectoralis major, latissimus dorsi and teres major during abduction and flexion; supraspinatus during adduction; middle deltoid during extension; and subscapularis, pectoralis major and latissimus dorsi during external rotation. In contrast, static optimization neural solutions showed little or no recruitment of these muscles, and preferentially activated agonistic prime movers with large moment arms. As a consequence, glenohumeral joint force calculations varied substantially between models. The findings suggest that static optimization may under-estimate the activity of muscle antagonists, and therefore, their contribution to glenohumeral joint stability.     

Muscle moment arm and normalized moment contributions as reference data for musculoskeletal elbow and wrist joint models

A geometric musculoskeletal model of the elbow and wrist joints was developed to calculate muscle moment arms throughout elbow flexion/extension, forearm pronation/supination, wrist flexion/extension and radial/ulnar deviation. Model moment arms were verified with data from cadaver specimen studies and geometric models available in the literature. Coefficients of polynomial equations were calculated for all moment arms as functions of joint angle, with special consideration to coupled muscles as a function of two joint angles. Additionally, a “normalized potential moment (NPM)” contribution index for each muscle across the elbow and wrist joints in four degrees-of-freedom was determined using each muscle's normalized physiological cross-sectional area (PCSA) and peak moment arm (MA). We hypothesize that (a) a geometric model of the elbow and wrist joints can represent the major attributes of MA versus joint angle from many literature sources of cadaver and model data and (b) an index can represent each muscle's normalized moment contribution to each degree-of-freedom at the elbow and wrist. We believe these data serve as a simple, yet comprehensive, reference for how the primary 16 muscles across the elbow and wrist contribute to joint moment and overall joint performance.     

Moment-generating capacity of upper limb muscles in healthy adults

Muscle strength and volume vary greatly among individuals. Maximum isometric joint moment, a standard measurement of strength, has typically been assessed in young, healthy subjects, whereas muscle volumes have generally been measured in cadavers. This has made it difficult to characterize the relationship between isometric strength and muscle size in humans. We measured maximum isometric moments about the shoulder, elbow, and wrist in 10 young, healthy subjects, ranging in size from a 20th percentile female to a 97th percentile male. The volumes of 32 upper limb muscles were determined from magnetic resonance images of these same subjects, and grouped according to their primary function. The maximum moments produced using the shoulder adductors (67.9+/-28.4 Nm) were largest, and were approximately 6.5(+/-1.2) times greater than those produced using the wrist extensors (10.2+/-4.6 Nm), which were smallest. While there were substantial differences in moment-generating capacity among these 10 subjects, moment significantly covaried with muscle volume of the appropriate functional group, explaining between 95% (p<0.0001; shoulder adductors) and 68% (p=0.004; wrist flexors) of the variation in the maximum isometric joint moments among subjects. While other factors, such as muscle moment arms or neural activation and coordination, can contribute to variation in strength among subjects, they either were relatively constant across these subjects compared to large differences in muscle volumes or they covaried with muscle volume. We conclude that differences in strength among healthy young adults are primarily a consequence of variation in muscle volume, as opposed to other factors.     

A three-dimensional model of the rat hindlimb: musculoskeletal geometry and muscle moment arms

As a first step towards developing a dynamic model of the rat hindlimb, we measured muscle attachment and joint center coordinates relative to bony landmarks using stereophotogrammetry. Using these measurements, we analyzed muscle moment arms as functions of joint angle for most hindlimb muscles, and tested the hypothesis that postural change alone is sufficient to alter the function of selected muscles of the leg. We described muscle attachment sites as second-order curves. The length of the fit parabola and residual errors in the orthogonal directions give an estimate of muscle attachment sizes, which are consistent with observations made during dissection. We modeled each joint as a moving point dependent on joint angle; relative endpoint errors less than 7% indicate this method as accurate. Most muscles have moment arms with a large range across the physiological domain of joint angles, but their moment arms peak and vary little within the locomotion domain. The small variation in moment arms during locomotion potentially simplifies the neural control requirements during this phase. The moment arms of a number of muscles cross zero as angle varies within the quadrupedal locomotion domain, indicating they are intrinsically stabilizing. However, in the bipedal locomotion domain, the moment arms of these muscles do not cross zero and thus are no longer intrinsically stabilizing. We found that muscle function is largely determined by the change in moment arm with joint angle, particularly the transition from quadrupedal to bipedal posture, which may alter an intrinsically stabilizing arrangement or change the control burden.     

Inversion–eversion moment arms of gastrocnemius and tibialis anterior measured in vivo

In this study, the frontal plane moment arms of tibialis anterior (TA) and the lateral and medial heads of gastrocnemius (LG and MG) were determined using ultrasonography of ten healthy subjects. Analysis of variance was performed to investigate the effects of frontal plane angle, muscle activity, and plantarflexion angle on inversion–eversion moment arm for each muscle. The moment arms of each muscle were found to vary with frontal plane angle (all p<0.001). TA and LG exhibited eversion moment arms when the foot was everted, but MG was found to have a slight inversion moment arm in this position. As the ankle rotated from 0° to 20° inversion, the inversion moment arm of each increased, indicating that the three muscles became increasingly effective inverters. In neutral position, the inverter moment arm of MG was greater than that of LG (p=0.001). Muscle activity had a significant effect on both LG and MG moment arm at all frontal plane positions (all p0.005). These results demonstrate the manner in which frontal plane moment arms of gastrocnemius and TA differ across the frontal plane range of motion in healthy subjects. This method for assessing muscle action in vivo used in this study may prove useful for subject-specific planning of surgical treatments for frontal plane foot and ankle deformities.     

Musculotendon lengths and moment arms for a three-dimensional upper-extremity model

Generating muscle-driven forward dynamics simulations of human movement using detailed musculoskeletal models can be computationally expensive. This is due in part to the time required to calculate musculotendon geometry (e.g., musculotendon lengths and moment arms), which is necessary to determine and apply individual musculotendon forces during the simulation. Modeling upper-extremity musculotendon geometry can be especially challenging due to the large number of multi-articular muscles and complex muscle paths. To accurately represent this geometry, wrapping surface algorithms and/or other computationally expensive techniques (e.g., phantom segments) are used. This paper provides a set of computationally efficient polynomial regression equations that estimate musculotendon length and moment arms for thirty-two (32) upper-extremity musculotendon actuators representing the major muscles crossing the shoulder, elbow and wrist joints. Equations were developed using a least squares fitting technique based on geometry values obtained from a validated public-domain upper-extremity musculoskeletal model that used wrapping surface elements (Holzbaur et al., 2005). In general, the regression equations fit well the original model values, with an average root mean square difference for all musculotendon actuators over the represented joint space of 0.39 mm (1.1% of peak value). In addition, the equations reduced the computational time required to simulate a representative upper-extremity movement (i.e., wheelchair propulsion) by more than two orders of magnitude (315 versus 2.3 s). Thus, these equations can assist in generating computationally efficient forward dynamics simulations of a wide range of upper-extremity movements.