Mathematical models of human paralyzed muscle after long-term training

Modeling Ankle Torque and Stiffness Induced by Functional Electrical Stimulation

Functional electrical stimulation (FES) is commonly used for individuals with neuromuscular impairments to generate muscle contractions. Both joint torque and stiffness play important roles in maintaining stable posture and resisting external disturbance. However, most previous studies only focused on the modulation of joint torque using FES while ignoring the joint stiffness. A model that can simultaneously modulate both ankle torque and stiffness induced by FES was investigated in this study. This model was composed of four subparts including an FES-to-activation model, a musculoskeletal geometry model, a Hill-based muscle-tendon model, and a joint stiffness model. The model was calibrated by the maximum voluntary contraction test of the tibialis anterior (TA) and gastrocnemius medial (GAS) muscles. To validate the model, the estimated torque and stiffness by the model were compared with the measured torque and stiffness induced by FES, respectively. The results showed that the proposed model can estimate torque and stiffness with electrically stimulated TA or/and GAS, which was significantly correlated to the measured torque and stiffness. The proposed model can modulate both joint torque and stiffness induced by FES in the isometric condition, which can be potentially extended to modulate the joint torque and stiffness during FES-assisted walking.

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.

A New Mathematical Force Model that Predicts the Force-pulse Amplitude Relationship of Human Skeletal Muscle

Current functional electrical stimulation (FES) systems vary the stimulation intensity to control the muscle force in order to produce precise functional movements. However, mathematical model that predicts the intensity effect on the muscle force is required for model-based controller design. The most previous force model designed by Ding et al was validated only for a standardized stimulation pulse amplitude (intensity). Thus, the purpose of this study was to adapt the Ding et al model to be able to predict the force-pulse amplitude relationship. The experimental results tested on quadriceps femoris muscles of healthy subjects (N=5) show that our adapted model accurately predicts the force response for trains of a wide range of stimulation intensities (30-100 mA). The accurate predictions indicate that our adapted model could be used for designing model-based control strategies to control the muscle force through FES.

Identification of ankle plantar-flexors dynamics in response to electrical stimulation

Modeling the muscle response to functional electrical stimulation (FES) is an essential step in the design of closed-loop controlled neuroprostheses. This study was aimed at identifying the dynamic response of ankle plantar-flexors to FES during quiet standing. Thirteen healthy subjects stood in a standing frame that locked the knee and hip joints. The ankle plantar-flexors were stimulated bilaterally through surface electrodes and the generated ankle torque was measured. The pulse amplitude was sinusoidally modulated at five different frequencies. The pulse amplitude and the measured ankle torque fitted by a sine function were considered as input and output, respectively. First-order and critically-damped second-order linear models were fitted to the experimental data. Both models fitted similarly well to the experimental data. The coefficient of variation of the time constant among subjects was smaller in the case of the second-order model compared to the first-order model (18.1%vs. 79.9%, p<0.001). We concluded that the critically-damped second-order model more consistently described the relationship between isometric ankle torque and surface FES pulse amplitude, which was applied to the ankle plantar-flexors during quiet standing.

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.