Prediction of Individual Muscle Forces Using Lagrange Multipliers Method — A Model of the Upper Human Limb in the Sagittal Plane: II. Numerical Experiments and Sensitivity Analysis

Abstract

Using the method of Lagrange multipliers an analytical solution of the optimization problem formulated for a two-dimensional, 3DOF model of the human upper limb has been described in Part I of this investigation. The objective criterion used is the following: Σ c_iF_i ², where F_i-s are the muscle forces modelled and c_i-s are unknown weight factors. This study is devoted to the numerical experiments performed in order to investigate which sets of the weight factors may predict physiologically reasonable muscle forces and joint reactions. A sensitivity analysis is also presented. The influence of: the gravity forces, different external loads applied to the hand, changes of the weight factors and of joint angle on the optimal solution is studied. A general conclusion may be drawn: using the above mentioned objective criterion, practically all motor tasks performed by the human upper limb may be described if the c_i-s are properly chosen. These weight factors generally depend on the joint moments and must be different (their magnitudes as well as their signs) for agonistic muscles and for their antagonists.     

Modulation of shoulder muscle and joint function using a powered upper-limb exoskeleton

Robotic-assistive exoskeletons can enable frequent repetitive movements without the presence of a full-time therapist; however, human-machine interaction and the capacity of powered exoskeletons to attenuate shoulder muscle and joint loading is poorly understood. This study aimed to quantify shoulder muscle and joint force during assisted activities of daily living using a powered robotic upper limb exoskeleton (ArmeoPower, Hocoma). Six healthy male subjects performed abduction, flexion, horizontal flexion, reaching and nose touching activities. These tasks were repeated under two conditions: (i) the exoskeleton compensating only for its own weight, and (ii) the exoskeleton providing full upper limb gravity compensation (i.e., weightlessness). Muscle EMG, joint kinematics and joint torques were simultaneously recorded, and shoulder muscle and joint forces calculated using personalized musculoskeletal models of each subject’s upper limb. The exoskeleton reduced peak joint torques, muscle forces and joint loading by up to 74.8% (0.113 Nm/kg), 88.8% (5.8%BW) and 68.4% (75.6%BW), respectively, with the degree of load attenuation strongly task dependent. The peak compressive, anterior and superior glenohumeral joint force during assisted nose touching was 36.4% (24.6%BW), 72.4% (13.1%BW) and 85.0% (17.2%BW) lower than that during unassisted nose touching, respectively. The present study showed that upper limb weight compensation using an assistive exoskeleton may increase glenohumeral joint stability, since deltoid muscle force, which is the primary contributor to superior glenohumeral joint shear, is attenuated; however, prominent exoskeleton interaction moments are required to position and control the upper limb in space, even under full gravity compensation conditions. The modeling framework and results may be useful in planning targeted upper limb robotic rehabilitation tasks.     

About weight factors in the non-linear objective functions used for solving indeterminate problems in biomechanics.

A lot of non-linear objective criteria are applied for solving the indeterminate problems formulated for different biomechanical models--most of them can be covered by the expression [formula in text]. It might be noted, however, that most of the suggested criteria are not applicable if considerable antagonistic co-contractions exist. This could be an effect of treating the agonistic muscles and their respective antagonists in one and the same manner in the objective function. Using a completely inverse approach (the muscle forces are supposed to be known quantities) and a simple 1DOF model (actuated by three agonistic muscles and one corresponding antagonist) it has been shown which values of the weight factors c(i) may predict different levels of muscle forces from the two antagonistic groups. Three hypothetical border variants for magnitudes of the muscle forces are considered (flexor muscles are only active, extensor muscles are only active, considerable co-contraction of flexors and extensors exists). The main conclusions are: the signs of c(i) at agonistic muscles have to be opposite to the c(i) signs at their antagonists; the signs of the weight factors depend on the direction of the net external joint moment; the closer c(i) to zero, the bigger force will be predicted in the ith muscle.     

Investigation of the Peculiarities of Two-joint Muscles Using a 3 DOF Model of the Human Upper Limb in the Sagittal Plane: an Optimization Approach

Abstract

The purpose of this paper is an investigation of the peculiarities of biarticular muscles by means of modelling and analytical solution of the indeterminate problem. The basic model includes 10 muscle elements performing flexio/extensio in the shoulder, elbow and wrist. Four of them are biarticular muscles. Two modifications of the model with only monoarticular muscles are developed. The indeterminate problem is solved analytically using the objective criterion σc_iF_i ² where F( is the module of the i-th muscle force and Cj is a weight coefficient. The predicted muscle forces, joint reactions and moments are compared in-between the basic model and its two modifications for different joint angles, external loading and weight coefficients. The main conclusions are: it is impossible to formulate strict advantages of the biarticular muscles under quasistatical conditions, their peculiarities depend on limb position, external loading and neural control; in general, monoarticular muscles are more powerful than biarticular ones; the biarticular muscles fine tune muscle coordination, their control is more precise and graceful; the presence of biarticular muscles leads to an increase of the joint reactions and moments, thus stabilizing the limb.     

Critical analysis of musculoskeletal modelling complexity in multibody biomechanical models of the upper limb

The inverse dynamics technique applied to musculoskeletal models, and supported by optimisation techniques, is used extensively to estimate muscle and joint reaction forces. However, the solutions of the redundant muscle force sharing problem are sensitive to the detail and modelling assumptions of the models used. This study presents four alternative biomechanical models of the upper limb with different levels of discretisation of muscles by bundles and muscle paths, and their consequences on the estimation of the muscle and joint reaction forces. The muscle force sharing problem is solved for the motions of abduction and anterior flexion, acquired using video imaging, through the minimisation of an objective function describing muscle metabolic energy consumption. While looking for the optimal solution, not only the equations of motion are satisfied but also the stability of the glenohumeral and scapulothoracic joints is preserved. The results show that a lower level of muscle discretisation provides worse estimations regarding the muscle forces. Moreover, the poor discretisation of muscles relevant to the joint in analysis limits the applicability of the biomechanical model. In this study, the biomechanical model of the upper limb describing the infraspinatus by a single bundle could not solve the complete motion of anterior flexion. Despite the small differences in the magnitude of the forces predicted by the biomechanical models with more complex muscular systems, in general, there are no significant variations in the muscular activity of equivalent muscles.     

The sensitivity of muscle force predictions to changes in physiologic cross-sectional area

The mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses. For more than a century, researchers have expressed muscle size by its 'physiological cross-sectional area' (PCSA). Researchers mathematically calculating muscle and joint forces typically use some expression of a muscle's PCSA to constrain the solution to one which is reasonable (i.e. a solution in which small muscles may not have large forces, and large muscles have large forces when expected or when there is significant electromyographic activity). It is obvious that muscle mass (and therefore any expression of PCSA) varies significantly from person to person, even in individuals of similar weight and height. Since it is not practical to predict the PCSA of each muscle in a living subject's limb or trunk, it is important to generally understand the sensitivity of muscle force solutions to possible variations in PCSA. We used nonlinear optimization techniques to predict 47 muscle forces and hip contact forces in a living subject. The PCSA (volume/muscle fiber length) of each of 47 lower limb muscle elements from two cadaver specimens and the 47 PCSA's reported by pierrynowski were input into an optimization algorithm to create three solution sets. The three solutions were qualitatively similar but at times a predicted muscle force could vary as much as two to eight times. In contrast, the joint force solutions were within 11% of each other and, therefore, much less variable.(ABSTRACT TRUNCATED AT 250 WORDS)     

A two-muscle,continuum-mechanical forward simulation of the upper limb

By following the common definition of forward-dynamics simulations, i.e. predicting movement based on (neural) muscle activity, this work describes, for the first time, a forward-dynamics simulation framework of a musculoskeletal system, in which all components are represented as continuous, three-dimensional, volumetric objects. Within this framework, the mechanical behaviour of the entire muscle–tendon complex is modelled as a nonlinear hyperelastic material undergoing finite deformations. The feasibility and the full potential of the proposed forward-dynamics simulation framework is demonstrated on a two-muscle, three-dimensional, continuum-mechanical model of the upper limb. The musculoskeletal model consists of three bones, i.e. humerus, ulna, and radius, an one-degree-of-freedom elbow joint, and an antagonistic muscle pair, i.e. the biceps and triceps brachii, and takes into consideration the contact between the skeletal muscles and the humerus. Numerical studies have shown that the proposed upper limb model is capable of predicting realistic moment arms and muscle forces for the entire range of activation and motion. Within the limitations of the model, the presented simulations provide, for the first time, insights into existing contact forces and their influence on the muscle fibre stretch. Based on the presented simulations, the overall change in fibre stretch is typically less than 3%, despite the fact that the contact forces reach up to 71% of the exerted muscle force. Movement-predicting simulations are achieved by minimising a nonlinear moment equilibrium equation. Based on the forward-dynamics simulation approach, an iterative solution procedures for position-driven (inverse dynamics) and force-driven scenarios have been proposed accordingly. Applying these methodologies to time-dependent scenarios demonstrates that the proposed methods can be linked to state-of-the-art control algorithms predicting time-dependent muscle activation levels based on principles of forward dynamics.     

An open source lower limb model: Hip joint validation

Musculoskeletal lower limb models have been shown to be able to predict hip contact forces (HCFs) that are comparable to in vivo measurements obtained from instrumented prostheses. However, the muscle recruitment predicted by these models does not necessarily compare well to measured electromyographic (EMG) signals. In order to verify if it is possible to accurately estimate HCFs from muscle force patterns consistent with EMG measurements, a lower limb model based on a published anatomical dataset (Klein Horsman et al., 2007. Clinical Biomechanics. 22, 239-247) has been implemented in the open source software OpenSim. A cycle-to-cycle hip joint validation was conducted against HCFs recorded during gait and stair climbing trials of four arthroplasty patients (Bergmann et al., 2001. Journal of Biomechanics. 34, 859-871). Hip joint muscle tensions were estimated by minimizing a polynomial function of the muscle forces. The resulting muscle activation patterns obtained by assessing multiple powers of the objective function were compared against EMG profiles from the literature. Calculated HCFs denoted a tendency to monotonically increase their magnitude when raising the power of the objective function; the best estimation obtained from muscle forces consistent with experimental EMG profiles was found when a quadratic objective function was minimized (average overestimation at experimental peak frame: 10.1% for walking, 7.8% for stair climbing). The lower limb model can produce appropriate balanced sets of muscle forces and joint contact forces that can be used in a range of applications requiring accurate quantification of both. The developed model is available at the website https://simtk.org/home/low_limb_london.     

Muscle compensation strategies to maintain glenohumeral joint stability with increased rotator cuff tear severity: A simulation study

Rotator cuff tear (RCT) in older adults may cause decreased muscle forces and disrupt the force balance at the glenohumeral joint, compromising joint stability. Our objective was to identify how increased RCT severity affects glenohumeral joint loading and muscle activation patterns using a computational model. Muscle volume measurements were used to scale a nominal upper limb model’s peak isometric muscle forces to represent force-generating characteristics of an average older adult male. Increased RCT severity was represented by systematically decreasing peak isometric muscle forces of supraspinatus, infraspinatus, and subscapularis. Five static postures in both scapular and frontal planes were evaluated. Results revealed that in both scapular and frontal planes, the peak glenohumeral joint contact force magnitude remained relatively consistent across increased RCT severity (average 1.5% and −4.2% change, respectively), and a relative balance of the transverse force couple is maintained even in massive RCT models. Predicted muscle activations of intact muscles, like teres minor, increased (average 5–30% and 4–17% in scapular and frontal planes, respectively) with greater RCT severity. This suggests that the system is prioritizing glenohumeral joint stability, even with severe RCT, and that unaffected muscles play a compensatory role to help stabilize the joint.     

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.     

Fibre operating lengths of human lower limb muscles during walking

Muscles actuate movement by generating forces. The forces generated by muscles are highly dependent on their fibre lengths, yet it is difficult to measure the lengths over which muscle fibres operate during movement. We combined experimental measurements of joint angles and muscle activation patterns during walking with a musculoskeletal model that captures the relationships between muscle fibre lengths, joint angles and muscle activations for muscles of the lower limb. We used this musculoskeletal model to produce a simulation of muscle-tendon dynamics during walking and calculated fibre operating lengths (i.e. the length of muscle fibres relative to their optimal fibre length) for 17 lower limb muscles. Our results indicate that when musculotendon compliance is low, the muscle fibre operating length is determined predominantly by the joint angles and muscle moment arms. If musculotendon compliance is high, muscle fibre operating length is more dependent on activation level and force-length-velocity effects. We found that muscles operate on multiple limbs of the force-length curve (i.e. ascending, plateau and descending limbs) during the gait cycle, but are active within a smaller portion of their total operating range.     

Upper limb loadings of gait with crutches

Long-term crutch users and patients with arthritis are particularly susceptible to upper limb joint degeneration during aided gait. The function of the walking aid for stability, support, and restraint/propulsion must be optimized with the upper limb loadings caused by the aids. Post-operative total hip replacement (THR) patients, tibial fracture, and paraplegic subjects using sticks and elbow crutches were analyzed in this study. Elbow and shoulder joint centers and aid orientations were monitored simultaneously in three dimensions and combined with aid forces to determine upper limb moment loadings. Three loading effects were observed: tendency for the aids to cause 1) the elbow to flex and shoulder to extend, 2) the elbow and shoulder to extend, and 3) the shoulder to abduct. Moment values of up to 0.10 Nm per body weight (BW) causing the shoulder to extend were measured, i.e., of similar magnitude to the moments at the hip in unaided gait. A modification of the elbow crutch, designed to improve medial-lateral stability, was unsuccessful in use due to wrist instability. This reinforced the requirement that crutch designs integrate the aid's function in gait with the ability of the upper limb joints to balance the applied loads.     

Theoretical requirements for the interpretation of signals of intramuscular tension

A model of a simple hinge joint is developed which allows the expression of intramuscular tension in terms of the angle at the joint and an applied force. A particular intramuscular tension does not specify uniquely the size of the force applied to a limb. For the nervous system to use signals of intramuscular tension from the agonist muscle to specify the applied force it requires information about (i) the angle at the joint, (ii) the angle between the line of applied force and the line of muscular contraction, and (iii) the relation between the axis of joint rotation and gravity.Sensory information about intramuscular tension can be perceived. This theoretical anlysis shows that central processing of signals of intramuscular tension may be required to provide a unique indicator of the external forces on a limb.     

Hierarchical genetic algorithm versus static optimization-investigation of elbow flexion and extension movements

The applicability of static optimization (and, respectively, frequently used objective functions) for prediction of individual muscle forces for dynamic conditions has often been discussed. Some of the problems are whether time-independent objective functions are suitable, and how to incorporate muscle physiology in models. The present paper deals with a twofold task: (1) implementation of hierarchical genetic algorithm (HGA) based on the properties of the motor units (MUs) twitches, and using multi-objective, time-dependent optimization functions; and (2) comparison of the results of the HGA application with those obtained through static optimization with a criterion "minimum of a weighted sum of the muscle forces raised to the power of n". HGA and its software implementation are presented. The moments of neural stimulation of all MUs are design variables coding the problem in the terms of HGA. The main idea is in using genetic operations to find these moments, so that the sum of MUs twitches satisfies the imposed goals (required joint moments, minimal sum of muscle forces, etc.). Elbow flexion and extension movements with different velocities are considered as proper illustration. It is supposed that they are performed by two extensor muscles and three flexor muscles. The results show that HGA is a suitable means for precise investigation of motor control. Many experimentally observed phenomena (such as antagonistic co-contraction, three-phasic behavior of the muscles during fast movements) can find their explanation by the properties of the MUs twitches. Static optimization is also able to predict three-phasic behavior and could be used as practicable and computationally inexpensive method for total estimation of the muscle forces.     

A型肉毒毒素联合肌电生物反馈治疗脑卒中后上肢肌痉挛的临床疗效分析

目的:分析A型肉毒毒素与肌电生物反馈联合用于治疗脑卒中后上肢肌痉挛的临床疗效。方法:选择2014年10月至2016年12月辽宁本溪市中心医院和北京博爱医院收治的84例脑卒中后上肢肌痉挛患者,并将其随机分为观察组和对照组,每组42例。两组患者均首先接受常规康复训练,随后对照组加用肌电生物反馈治疗,观察组患者在对照组的基础之上注射A型肉毒毒素。采用改良Ashworth痉挛量表(MAS)比较两组治疗前后的上肢肌痉挛改善情况,采用Fugl-Meyer评定量表评价患者上肢运动功能,采用量角器测量治疗前后患者的腕关节主动活动范围,采用改良的Barthel指数评价患者治疗前后的日常生活能力。结果:治疗后,观察组的上肢痉挛改善总有效率显著高于对照组(P0.05);治疗2周和4周时,两组的Fugl-Meyer评分、腕关节主动活动范围、改良的Barthel指数(MBI)均较治疗前显著升高(P0.05),且观察组治疗后2周和4周的Fugl-Meyer评分、腕关节主动活动范围、改良的Barthel指数(MBI)均显著高于对照组(P0.05)。结论:A型肉毒毒素与肌电生物反馈联合用于治疗脑卒中后上肢肌痉挛的临床疗效显著,可有效降低患者上肢痉挛状态,改善上肢和腕部运动能力,提高患者的日常生活能力。     

The effects of the antagonist muscle force on intersegmental loading during isokinetic efforts of the knee extensors

Hamstrings activation when acting as antagonists is considered very important for knee joint stability. However, the effect of hamstring antagonist activity on knee joint loading in vivo is not clear. Therefore, the purpose of this study was to examine the differences in antagonistic muscle force and their effect on agonist muscle and intersegmental forces during isokinetic eccentric and concentric efforts of the knee extensors. Ten males performed maximum isokinetic eccentric and concentric efforts of the knee extensors at 30 degrees s(-1). The muscular and tibiofemoral joint forces were then estimated using a two-dimensional model with and without including the antagonist muscle forces. The antagonist moment was predicted using an IEMG-moment model. The predicted antagonist force reached a maximum of 2.55 times body weight (BW) and 1.16 BW under concentric and eccentric conditions respectively. Paired t-tests indicated that these were significantly different (p<0.05). A one-way analysis of variance indicated that when antagonist forces are included in the calculations the patella tendon, compressive and posterior shear joint forces are significantly higher compared to those calculated without including the antagonist forces. The anterior shear force was not affected by antagonist activity. The antagonists produce considerable force throughout the range of motion and affect the joint forces exerted at the knee joint. Further, it appears that the antagonist effect depends on the type of muscle action examined as it is higher during concentric compared to eccentric efforts of the knee extensors.