Mathematical models use varying parameter strategies to represent paralyzed muscle force properties: a sensitivity analysis

Applying a muscle fatigue model when optimizing load-sharing between muscles for short-duration high-intensity exercise: A preliminary study

Introduction: Multiple different mathematical models have been developed to represent muscle force, to represent multiple muscles in the musculoskeletal system, and to represent muscle fatigue. However, incorporating these different models together to describe the behavior of a high-intensity exercise has not been well described. Methods: In this work, we adapted the three-compartment controller (3CCr) muscle fatigue model to be implemented with an inverse-dynamics based optimization algorithm for the muscle recruitment problem for 7 elbow muscles to model a benchmark case: elbow flexion/extension moments. We highlight the difficulties in achieving an accurate subject-specific approach for this multi-level modeling problem, considering different muscular models, compared with experimental measurements. Both an isometric effort and a dynamic bicep curl were considered, where muscle activity and resting periods were simulated to obtain the fatigue behavior. Muscle parameter correction, scaling and calibration are addressed in this study. Moreover, fiber-type recruitment hierarchy in force generation was added to the optimization problem, thus offering an additional novel muscle modeling criterion. Results: It was observed that: i) the results were most accurate for the static case; ii) insufficient torque was predicted by the model at some time points for the dynamic case, which benefitted from a more precise calibration of muscle parameters; iii) modeling the effects of muscular potentiation may be important; and iv) for this multilevel model approach, the 3CCr model had to be modified to avoid reaching situations of unrealistic constant fatigue in high intensity exercise-resting cycles. Discussion: All the methods yield reasonable estimations, but the complexity of obtaining accurate subject-specific human models is highlighted in this study. The proposed novel muscle modeling and force recruitment criterion, which consider the muscular fiber-type distinction, show interesting preliminary results.

An approach to a muscle force model with force-pulse amplitude relationship of human quadriceps muscles

BACKGROUND

Recent advanced applications of the functional electrical stimulation (FES) mostly used closed-loop control strategies based on mathematical models to improve the performance of the FES systems. In most of them, the pulse amplitude was used as an input control. However, in controlling the muscle force, the most popular force model developed by Ding et al. does not take into account the pulse amplitude effect. The purpose of this study was to include the pulse amplitude in the existing Ding et al. model based on the recruitment curve function.

METHODS

Quadriceps femoris muscles of eight healthy subjects were tested. Forces responses to stimulation trains with different pulse amplitudes (30-100 mA) and frequencies (20-80 Hz) were recorded and analyzed. Then, specific model parameter values were identified by fitting the measured forces for one train (50 Hz, 100 mA). The obtained model parameters were then used to identify the recruitment curve parameter values by fitting the force responses for different pulse amplitudes at the same frequency train. Finally, the extended model was used to predict force responses for a range of stimulation pulse amplitudes and frequencies.

RESULTS

The experimental results indicated that our adapted model accurately predicts the force-pulse amplitude relationship with an excellent agreement between measured and predicted forces (R²=0.998, RMSE = 6.6 N).

CONCLUSIONS

This model could be used to predict the pulse amplitude effect and to design control strategies for controlling the muscle force in order to obtain precise movements during FES sessions using intensity modulation.

An Energetic Model of Low Frequency Isometric Neuromuscular Electrical Stimulation

The objective of this study was to evaluate whether an adapted Hill-type model of muscle energetics could account for the relatively high energy turnover observed during low frequency isometric Neuromuscular Electrical Stimulation (NMES). A previously validated Hill-based model was adapted to estimate the energy consumption due to muscle activation, force maintenance and internal shortening of the muscle during isometric NMES. Quadriceps muscle model parameters were identified for 10 healthy subjects based on the experimentally measured torque response to isometric stimulation at 8 Hz. Model predictions of torque and energy consumption rates across the stimulation range 1-12 Hz were compared with experimental data recorded from the same subjects. The model provided estimates in close agreement with the experimental values for the group mean energy consumption rate across the frequency range tested, (R adj (2) = 0.98), although prediction of individual data points for all frequencies and all subjects was more variable, (R adj (2) = 0.70). The model suggests that approximately one-third of the energy between 4 and 6 Hz is due to shortening heat. The model provides a means of identifying optimal therapeutic stimulation patterns for sustained incremental oxygen uptake at minimum torque output for a given muscle and provides insight into the energetic components involved.

Slow motor neuron stimulation of locust skeletal muscle: model and measurement

The isometric force response of the locust hind leg extensor tibia muscle to stimulation of a slow extensor tibia motor neuron is experimentally investigated, and a mathematical model describing the response presented. The measured force response was modelled by considering the ability of an existing model, developed to describe the response to the stimulation of a fast extensor tibia motor neuron and to also model the response to slow motor neuron stimulation. It is found that despite large differences in the force response to slow and fast motor neuron stimulation, which could be accounted for by the differing physiology of the fibres they innervate, the model is able to describe the response to both fast and slow motor neuron stimulation. Thus, the presented model provides a potentially generally applicable, robust, simple model to describe the isometric force response of a range of muscles.

A physiologically and biomechanically approximate model for surface electromyography amplitude estimation

Surface electromygraphy (sEMG) provides information of the neural drive to the muscle, so muscle force estimation by sEMG is of high relevance in biomechanical studies and in bionic applications. Even though mean absolute value (MAV) has been widely used for sEMG amplitude estimation due to the probabilistic nature of sEMG, but it has been used without any comprehensive physiological justification. A physiologically and biomechanically approximate model for the force estimation would enable a clear understanding of the relationships between sEMG and the force, and it can be used as sEMG amplitude estimation method. We proposed a new sEMG amplitude estimation method comprising two procedures: MUAP (motor unit action potential) event detection and muscle force indication using a biomechanical muscle model. The estimation performances were evaluated with nine subjects and compared with MAV. The performance (R(2)) of the proposed method (0.94 ± 0.03) outperformed it of MAV (0.90 ± 0.02). The method we proposed should be widely applicable to quantitatively analysis muscle activities by sEMG.

A predictive model of the isometric force response of the locust extensor muscle

A predictive model that can be used to estimate the isometric force response of the locust hind leg extensor muscle is presented. The model consists of two first order coupled differential equations. The first of these equations is linear and relates an input pulse train to the calcium concentration in muscle filaments. The second is non-linear and relates the calcium concentration to muscle force. Experimental data was collected by stimulating the extensor muscle and measuring the force generated at the tibia. Model parameters were estimated by minimising the error between the modelled and actual force response in a set of training data. These parameters were then used to predict the isometric response when the neural activity recorded during a kick was used as an input to the model. The model was found to accurately predict the isometric force response of the locust hind leg extensor muscle.

Identification of a Modified Wiener-Hammerstein System and Its Application in Electrically Stimulated Paralyzed Skeletal Muscle Modeling

Electrical muscle stimulation demonstrates potential for restoring functional movement and preventing muscle atrophy after spinal cord injury (SCI). Control systems used to optimize delivery of electrical stimulation protocols depend upon mathematical models of paralyzed muscle force outputs. While accurate, the Hill-Huxley-type model is very complex, making it difficult to implement for real-time control. As an alternative, we propose a modified Wiener-Hammerstein system to model the paralyzed skeletal muscle dynamics under electrical stimulus conditions. Experimental data from the soleus muscles of individuals with SCI was used to quantify the model performance. It is shown that the proposed Wiener-Hammerstein system is at least comparable to the Hill-Huxley-type model. On the other hand, the proposed system involves a much smaller number of unknown coefficients. This has substantial advantages in identification algorithm analysis and implementation including computational complexity, convergence and also in real time model implementation for control purposes.

Development of a mathematical model for predicting electrically elicited quadriceps femoris muscle forces during isovelocity knee joint motion

Background

Direct electrical activation of skeletal muscles of patients with upper motor neuron lesions can restore functional movements, such as standing or walking. Because responses to electrical stimulation are highly nonlinear and time varying, accurate control of muscles to produce functional movements is very difficult. Accurate and predictive mathematical models can facilitate the design of stimulation patterns and control strategies that will produce the desired force and motion. In the present study, we build upon our previous isometric model to capture the effects of constant angular velocity on the forces produced during electrically elicited concentric contractions of healthy human quadriceps femoris muscle. Modelling the isovelocity condition is important because it will enable us to understand how our model behaves under the relatively simple condition of constant velocity and will enable us to better understand the interactions of muscle length, limb velocity, and stimulation pattern on the force produced by the muscle.

Methods

An additional term was introduced into our previous isometric model to predict the force responses during constant velocity limb motion. Ten healthy subjects were recruited for the study. Using a KinCom dynamometer, isometric and isovelocity force data were collected from the human quadriceps femoris muscle in response to a wide range of stimulation frequencies and patterns. % error, linear regression trend lines, and paired t-tests were used to test how well the model predicted the experimental forces. In addition, sensitivity analysis was performed using Fourier Amplitude Sensitivity Test to obtain a measure of the sensitivity of our model's output to changes in model parameters.

Results

Percentage RMS errors between modelled and experimental forces determined for each subject at each stimulation pattern and velocity showed that the errors were in general less than 20%. The coefficients of determination between the measured and predicted forces show that the model accounted for ~86% and ~85% of the variances in the measured force-time integrals and peak forces, respectively.

Conclusion

The range of predictive abilities of the isovelocity model in response to changes in muscle length, velocity, and stimulation frequency for each individual make it ideal for dynamic applications like FES cycling.

Predicting muscle forces of individuals with hemiparesis following stroke

Background

Functional electrical stimulation (FES) has been used to improve function in individuals with hemiparesis following stroke. An ideal functional electrical stimulation (FES) system needs an accurate mathematical model capable of designing subject and task-specific stimulation patterns. Such a model was previously developed in our laboratory and shown to predict the isometric forces produced by the quadriceps femoris muscles of able-bodied individuals and individuals with spinal cord injury in response to a wide range of clinically relevant stimulation frequencies and patterns. The aim of this study was to test our isometric muscle force model on the quadriceps femoris, ankle dorsiflexor, and ankle plantar-flexor muscles of individuals with post-stroke hemiparesis.

Methods

Subjects were seated on a force dynamometer and isometric forces were measured in response to a range of stimulation frequencies (10 to 80-Hz) and 3 different patterns. Subject-specific model parameter values were obtained by fitting the measured force responses from 2 stimulation trains. The model parameters thus obtained were then used to obtain predicted forces for a range of frequencies and patterns. Predicted and measured forces were compared using intra-class correlation coefficients, r² values, and model error relative to the physiological error (variability of measured forces).

Results

Results showed excellent agreement between measured and predicted force-time responses (r² >0.80), peak forces (ICCs>0.84), and force-time integrals (ICCs>0.82) for the quadriceps, dorsiflexor, and plantar-fexor muscles. The model error was within or below the +95% confidence interval of the physiological error for >88% comparisons between measured and predicted forces.

Conclusion

Our results show that the model has potential to be incorporated as a feed-forward controller for predicting subject-specific stimulation patterns during FES.

Mathematical models of human paralyzed muscle after long-term training

Spinal cord injury (SCI) results in major musculoskeletal adaptations, including muscle atrophy, faster contractile properties, increased fatigability, and bone loss. The use of functional electrical stimulation (FES) provides a method to prevent paralyzed muscle adaptations in order to sustain force-generating capacity. Mathematical muscle models may be able to predict optimal activation strategies during FES, however muscle properties further adapt with long-term training. The purpose of this study was to compare the accuracy of three muscle models, one linear and two nonlinear, for predicting paralyzed soleus muscle force after exposure to long-term FES training. Further, we contrasted the findings between the trained and untrained limbs. The three models' parameters were best fit to a single force train in the trained soleus muscle (N=4). Nine additional force trains (test trains) were predicted for each subject using the developed models. Model errors between predicted and experimental force trains were determined, including specific muscle force properties. The mean overall error was greatest for the linear model (15.8%) and least for the nonlinear Hill Huxley type model (7.8%). No significant error differences were observed between the trained versus untrained limbs, although model parameter values were significantly altered with training. This study confirmed that nonlinear models most accurately predict both trained and untrained paralyzed muscle force properties. Moreover, the optimized model parameter values were responsive to the relative physiological state of the paralyzed muscle (trained versus untrained). These findings are relevant for the design and control of neuro-prosthetic devices for those with SCI.

Partition between volitional and induced forces in electrically augmented dynamic isometric muscle contractions

Augmentation of force in partially deficient muscles can be achieved by combining electrical stimulation (ES) with their volitional activation (hybrid activation). However, while the overall torque results from the combination of the volitional and the electrically-induced torque components, the exact share between these components is not known. In a previous work, we described a method to resolve the share between the torque components under isometric static contractions. In this work, we extend our analysis to the case of isometric dynamic contractions. Five healthy subjects were instructed to contract their Tibialis Anterior (TA) muscles according to a typical gait-like dynamic torque pattern, that was visually displayed to them, while monitoring their actual ankle torque and TA electromyography (EMG). These experiments were done with and without augmented activation by means of ES. A computational algorithm was developed to dissociate the volitional from the overall torque, based on EMG signal processing and on precalibration of the dynamic system of the volitional torque versus EMG. The results indicated the quantitative relations between decrease in the volitional torque and the required increase in ES enhancement. The developed method also demonstrated what ES intensity profile is necessary to produce a desired overall torque output. This provides the means for designing an adaptive rehabilitation device for the hybrid activation of deficient muscles.

Predicting human chronically paralyzed muscle force: a comparison of three mathematical models

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.