Mathematical model that predicts the force-intensity and force-frequency relationships after spinal cord injuries

Electrically Elicited Force Response Characteristics of Forearm Extensor Muscles for Electrical Muscle Stimulation-Based Haptic Rendering

A haptic interface based on electrical muscle stimulation (EMS) has huge potential in terms of usability and applicability compared with conventional haptic interfaces. This study analyzed the force response characteristics of forearm extensor muscles for EMS-based haptic rendering. We introduced a simplified mathematical model of the force response, which has been developed in the field of rehabilitation, and experimentally validated its feasibility for haptic applications. Two important features of the force response, namely the peak force and response time, with respect to the frequency and amplitude of the electrical stimulation were identified by investigating the experimental force response of the forearm extensor muscles. An exponential function was proposed to estimate the peak force with respect to the frequency and amplitude, and it was verified by comparing with the measured peak force. The response time characteristics were also examined with respect to the frequency and amplitude. A frequency-dependent tendency, i.e., an increase in response time with increasing frequency, was observed, whereas there was no correlation with the amplitude. The analysis of the force response characteristics with the application of the proposed force response model may help enhance the fidelity of EMS-based haptic rendering.

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.

Implantable power generation system utilizing muscle contractions excited by electrical stimulation

An implantable power generation system driven by muscle contractions for supplying power to active implantable medical devices, such as pacemakers and neurostimulators, is proposed. In this system, a muscle is intentionally contracted by an electrical stimulation in accordance with the demands of the active implantable medical device for electrical power. The proposed system, which comprises a small electromagnetic induction generator, electrodes with an electrical circuit for stimulation and a transmission device to convert the linear motion of the muscle contractions into rotational motion for the magneto rotor, generates electrical energy. In an ex vivo demonstration using the gastrocnemius muscle of a toad, which was 28 mm in length and weighed 1.3 g, the electrical energy generated by the prototype exceeded the energy consumed for electrical stimulation, with the net power being 111 µW. It was demonstrated that the proposed implantable power generation system has the potential to replace implantable batteries for active implantable medical devices.

Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration

Background

Our previous model of the non-isometric muscle fatigue that occurs during repetitive functional electrical stimulation included models of force, motion, and fatigue and accounted for applied load but not stimulation pulse duration. Our objectives were to: 1) further develop, 2) validate, and 3) present outcome measures for a non-isometric fatigue model that can predict the effect of a range of pulse durations on muscle fatigue.

Methods

A computer-controlled stimulator sent electrical pulses to electrodes on the thighs of 25 able-bodied human subjects. Isometric and non-isometric non-fatiguing and fatiguing knee torques and/or angles were measured. Pulse duration (170–600 μs) was the independent variable. Measurements were divided into parameter identification and model validation subsets.

Results

The fatigue model was simplified by removing two of three non-isometric parameters. The third remained a function of other model parameters. Between 66% and 77% of the variability in the angle measurements was explained by the new model.

Conclusion

Muscle fatigue in response to different stimulation pulse durations can be predicted during non-isometric repetitive contractions.

Multiscale modeling of skeletal muscle properties and experimental validations in isometric conditions

In this article, we describe an approach to model the electromechanical behavior of the skeletal muscle based on the Huxley formulation. We propose a model that complies with a well established macroscopic behavior of striated muscles where force-length, force-velocity, and Mirsky-Parmley properties are taken into account. These properties are introduced at the microscopic scale and related to a tentative explanation of the phenomena. The method used integrates behavior ranging from the microscopic to the macroscopic scale, and allows the computation of the dynamics of the output force and stiffness controlled by EMG or stimulation parameters. The model can thus be used to simulate and carry out research to develop control strategies using electrical stimulation in the context of rehabilitation. Finally, through animal experiments, we estimated model parameters using a Sigma Point Kalman Filtering technique and dedicated experimental protocols in isometric conditions and demonstrated that the model can accurately simulate individual variations and thus take into account subject dependent behavior.

Predicting fatigue during electrically stimulated non-isometric contractions

Mathematical prediction of power loss during electrically stimulated contractions is of value to those trying to minimize fatigue and to those trying to decipher the relative contributions of force and velocity. Our objectives were to: (1) develop a model of non-isometric fatigue for electrical stimulation-induced, open-chain, repeated extensions of the leg at the knee; and (2) experimentally validate the model. A computer-controlled stimulator sent electrical pulses to surface electrodes on the thighs of 17 able-bodied subjects. Isometric and non-isometric non-fatiguing and fatiguing leg extension torque and/or angle at the knee were measured. Two existing mathematical models, one of non-isometric force and the other of isometric fatigue, were combined to develop the non-isometric force-fatigue model. Angular velocity and 3 new parameters were added to the isometric fatigue model. The new parameters are functions of parameters within the force model, and therefore additional measurements from the subject are not needed. More than 60% of the variability in the measurements was explained by the new force-fatigue model. This model can help scientists investigate the etiology of non-isometric fatigue and help engineers to improve the task performance of functional electrical stimulation systems.

A predictive mathematical model of muscle forces for children with cerebral palsy

AIM

The purpose of this study was to determine if our previously developed muscle model could be used to predict forces of the quadriceps femoris and triceps surae muscles of children with spastic diplegic cerebral palsy (CP).

METHOD

Twenty-two children with CP (12 males, 10 females; mean age 10y, SD 2y, range 7-13y; Gross Motor Function Classification System levels II and III) participated. A physiologically based mathematical model with four free parameters is presented.

RESULTS

For individuals with CP, the model predicted well the force profile throughout each contraction and both peak force and force-time integral responses to a wide range of stimulation frequencies (5-100Hz) and different stimulation patterns (constant-, variable-, and doublet-frequency trains) both for nonfatigued and fatigued muscles.

INTERPRETATION

The significance of this work is the insight the model can provide into the physiology of muscle in CP. Additionally, the model can potentially be applied clinically to design optimal electrical stimulation patterns for interventions to address impairments in strength and function in individuals with CP, such as functional electrical stimulation-assisted cycling.

In vivo demonstration of a self-sustaining, implantable, stimulated-muscle-powered piezoelectric generator prototype

An implantable, stimulated-muscle-powered piezoelectric active energy harvesting generator was previously designed to exploit the fact that the mechanical output power of muscle is substantially greater than the electrical power necessary to stimulate the muscle's motor nerve. We reduced to practice the concept by building a prototype generator and stimulator. We demonstrated its feasibility in vivo, using rabbit quadriceps to drive the generator. The generated power was sufficient for self-sustaining operation of the stimulator and additional harnessed power was dissipated through a load resistor. The prototype generator was developed and the power generating capabilities were tested with a mechanical muscle analog. In vivo generated power matched the mechanical muscle analog, verifying its usefulness as a test-bed for generator development. Generator output power was dependent on the muscle stimulation parameters. Simulations and in vivo testing demonstrated that for a fixed number of stimuli/minute, two stimuli applied at a high frequency generated greater power than single stimuli or tetanic contractions. Larger muscles and circuitry improvements are expected to increase available power. An implanted, self-replenishing power source has the potential to augment implanted battery or transcutaneously powered electronic medical devices.

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.