A mathematical model for skeletal muscle activated by N-let pulse trains

MODELING AND SIMULATION OF SKELETAL MUSCLE BASED ON METABOLISM PHYSIOLOGY

Skeletal muscle energy metabolism plays a very important role in controlling movement of the whole body and has important theoretical guidance for making exercise training plans and losing weight. In this paper, we developed a mathematical model of skeletal muscle excitation–contraction pathway based on the energy metabolism that links excitation to contraction to explore the effects of different metabolic energy systems on calcium ion changes and the force during skeletal muscle contraction. In this paper, a membrane potential model, a calcium cycle model, a cross-bridge dynamics model and an energy metabolism model were established. Finally, the physiological phenomenon of calcium ion transport and calcium ion concentration change of the sarcoplasm was simulated. The results show that the phosphagen system has the fastest metabolic rate and the phosphagen system has the largest impact on the explosive power of skeletal muscle exercise. The specific characteristics of the three metabolic energy systems supporting skeletal muscle movement in vivo were also analyzed in detail.

Exhaustion of Skeletal Muscle Fibers Within Seconds: Incorporating Phosphate Kinetics Into a Hill-Type Model

Initiated by neural impulses and subsequent calcium release, skeletal muscle fibers contract (actively generate force) as a result of repetitive power strokes of acto-myosin cross-bridges. The energy required for performing these cross-bridge cycles is provided by the hydrolysis of adenosine triphosphate (ATP). The reaction products, adenosine diphosphate (ADP) and inorganic phosphate (P i ), are then used-among other reactants, such as creatine phosphate-to refuel the ATP energy storage. However, similar to yeasts that perish at the hands of their own waste, the hydrolysis reaction products diminish the chemical potential of ATP and thus inhibit the muscle's force generation as their concentration rises. We suggest to use the term "exhaustion" for force reduction (fatigue) that is caused by combined P i and ADP accumulation along with a possible reduction in ATP concentration. On the basis of bio-chemical kinetics, we present a model of muscle fiber exhaustion based on hydrolytic ATP-ADP-P i dynamics, which are assumed to be length- and calcium activity-dependent. Written in terms of differential-algebraic equations, the new sub-model allows to enhance existing Hill-type excitation-contraction models in a straightforward way. Measured time courses of force decay during isometric contractions of rabbit M. gastrocnemius and M. plantaris were employed for model verification, with the finding that our suggested model enhancement proved eminently promising. We discuss implications of our model approach for enhancing muscle models in general, as well as a few aspects regarding the significance of phosphate kinetics as one contributor to muscle fatigue.

Gaussian Process-Integrated State Space Model for Continuous Joint Angle Prediction from EMG and Interactive Force in a Human-Exoskeleton System

As one of the most direct indicators of the transparency between a human and an exoskeleton, interactive force has rarely been fused with electromyography (EMG) in the control of human-exoskeleton systems, the performances of which are largely determined by the accuracy of the continuous joint angle prediction. To achieve intuitive and naturalistic human intent learning, a state space model (SSM) for continuous angle prediction of knee joint is developed. When the influence of the interactive force is often ignored in the existing models of human-exoskeleton systems, interactive force is applied as the measurement model output of the proposed SSM, and the EMG signal is used as the state model input signal to indicate muscle activation. The forward dynamics of joint motion and the human-machine interaction mechanism, i.e., the biomechanical interpretations of the interactive force generation mechanism, are derived as the bases for the state model and measurement model based on Hill’s muscle model and semiphenomenological (SP) muscular model, respectively. Gaussian process (GP)-based nonlinear autoregressive with the exogenous inputs (NARX) model and back-propagation neural network (BPNN) are applied to provide better adaptivity for the SSM in practical applications. Corresponding experimental results demonstrate the validity and superiority of the method.

A highly efficient semiphenomenological model of a half-sarcomere for real-time prediction of mechanical behavior

With existent biomechanical models of skeletal muscle, challenges still exist in implementing real-time predictions for contraction statuses that are particularly significant to biomechanical and biomedical engineering. Because of this difficulty, this paper proposed a decoupled scheme of the links involved in the working process of a sarcomere and established a semiphenomenological model integrating both linear and nonlinear frames of no higher than a second-order system. In order to facilitate engineering application and cybernetics, the proposed model contains a reduced number of parameters and no partial differential equation, making it highly concise and computationally efficient. Through the simulations of various contraction modes, including isometric, isotonic, successive stretch and release, and cyclic contractions, the correctness and efficiency of the model, are validated. Although this study targets half-sarcomeres, the proposed model can be easily extended to describe the larger-scale mechanical behavior of a muscle fiber or a whole muscle.

ON THE COMPLEXITY OF BIOMECHANICAL MODELS USED FOR NEUROPROSTHESES DEVELOPMENT

The use of mathematical models has the potential to enhance the development of lower extremity neuroprostheses (NP) based on Functional Electrical Simulation (FES). The choice of model complexity is not trivial when building a model for FES control design. On the one hand, a comprehensive model might be useful to account for the many different biomechanical and neurophysiological effects that can be observed during FES-induced movements. On the other hand, too complex models are difficult to be utilized in and identified for NP applications. In this paper we discuss the disadvantages of too complex models, and propose potential simplifications on the basis of existing models that are commonly used to describe muscle activation, muscle contraction and body-segmental motion. The obtained model approach is simple enough to be identified, and sufficiently comprehensive to describe most of the relevant effects that occur during FES-induced locomotion.

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.

Assessment of the mechanical properties of the musculoskeletal system using 2-D and 3-D very high frame rate ultrasound

One of the great challenges for understanding muscular diseases is to assess noninvasively the active and passive mechanical properties of the musculoskeletal system. In this paper we report the use of ultrafast ultrasound imaging to explore with a submillimeter resolution the behavior of the contracting tissues in vivo (biceps brachii). To image the contraction, which is a very brief phenomenon (100 ms), a recently designed ultrasound scanner prototype able to take up to 6000 frames/s was used. A very high frame rate from 1000 to 2500 frames/s was used to image the cross section plane of the muscle (transverse to fibers) enabling us to catch in real time the muscle contraction during a transient electrostimulation. Tissue velocities were obtained from radiofrequency based speckle tracking techniques and their profiles are discussed with respect to electrostimulation intensities and pulse repetition frequencies for different volunteers. Three-dimensional (3-D) very high frame rate movies were also acquired by repeating the experiment for different acquisition planes while triggering the imaging system with the electrostimulation device. The reconstructed 3-D velocity field allows the full localization of the contracting fibers bundle. This ultrasound technique, referred to as echo mechanomyography, offers new perspectives for in vivo and in situ noninvasive muscle diagnosis of an active contractile tissue.

A mathematical model of fatigue in skeletal muscle force contraction

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation-contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation-contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

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

We have previously developed and tested a muscle model that predicts the effect of stimulation frequency on muscle force responses. The aim of this study was to enhance our isometric mathematical model to predict muscle forces in response to stimulation trains with a wide range of frequencies and intensities for the quadriceps femoris muscles of individuals with spinal cord injuries. Isometric forces were obtained experimentally from 10 individuals with spinal cord injuries (time after injury, 1.5-8 years) and then compared to forces predicted by the model. Our model predicted accurately the force-time integrals (FTI) and peak forces (PF) for stimulation trains of a wide range of frequencies (12.5-80 HZ) and intensities (150-600-mus pulse duration), and two different stimulation patterns (constant-frequency trains and doublet-frequency trains). The accurate predictions of our model indicate that our model, which now incorporates the effects of stimulation frequency, intensity, and pattern on muscle forces, can be used to design optimal customized stimulation strategies for spinal cord-injured patients.

Response linearity determined by recruitment strategy in detailed model of nictitating membrane control

Many models of eyeblink conditioning assume that there is a simple linear relationship between the firing patterns of neurons in the interpositus nucleus and the time course of the conditioned response (CR). However, the complexities of muscle behaviour and plant dynamics call this assumption into question. We investigated the issue by implementing the most detailed model available of the rabbit nictitating membrane response (Bartha and Thompson in Biol Cybern 68:135-143, 1992a and in Biol Cybern 68:145-154, 1992b), in which each motor unit of the retractor bulbi muscle is represented by a Hill-type model, driven by a non-linear activation mechanism designed to reproduce the isometric force measurements of Lennerstrand (J Physiol 236:43-55, 1974). Globe retraction and NM extension are modelled as linked second order systems. We derived versions of the model that used a consistent set of SI units, were based on a physically realisable version of calcium kinetics, and used simulated muscle cross-bridges to produce force. All versions showed similar non-linear responses to two basic control strategies. (1) Rate-coding with no recruitment gave a sigmoidal relation between control signal and amplitude of CR, reflecting the measured relation between isometric muscle force and stimulation frequency. (2) Recruitment of similar strength motor units with no rate coding gave a sublinear relation between control signal and amplitude of CR, reflecting the increase in muscle stiffness produced by recruitment. However, the system response could be linearised by either a suitable combination of rate-coding and recruitment, or by simple recruitment of motor units in order of (exponentially) increasing strength. These plausible control strategies, either alone or in combination, would in effect present the cerebellum with the simplified virtual plant that is assumed in many models of eyeblink conditioning. Future work is therefore needed to determine the extent to which motor neuron firing is in fact linearly related to the nictitating membrane response.

Mathematical model that predicts lower leg motion in response to electrical stimulation

Electrical stimulation of skeletal muscles of patients with upper motor neuron lesions can be used to restore functional movements such as standing or walking. Mathematical muscle models can assist in designing stimulation patterns that will enable patients to perform particular tasks more efficiently. In this study we extend our previous model to allow us to predict changes in knee joint angle in response to electrical stimulation of the human quadriceps femoris muscle. The model was tested both with and without inertial loads placed around the ankle joints of healthy subjects. Results showed that the model predicted the knee extensions with a RMS angle error that was generally <or=8 degrees. The coefficients of determination between the measured and predicted data showed the model accounted for approximately 71%, approximately 94%, approximately 73%, and approximately 89% of the variances in the experimental maximum excursion, time to maximum excursion, maximum shortening velocity, and time to maximum shortening velocity, respectively. This study showed that our general non-isometric model predicted the lower limb motion in response to a range of stimulation frequencies and patterns, and external loads. This model can be implemented in an algorithm for controlling the position of the lower leg during the swing phase of gait during functional electrical stimulation.

A mathematical model that predicts the force-frequency relationship of human skeletal muscle

In previous work we developed and validated a mathematical model that predicted force output from skeletal muscles subjected to six-pulse stimulation trains under isometric condition. The current study investigated the model's ability to predict force responses to longer stimulation trains under both nonfatigued and fatigued conditions. Using the six-pulse train model to predict the force produced by longer stimulation trains showed that the model was successful, but a modified parameter identification scheme was required. For most of the trains tested the model accounted for 95% of the variance in the experimental forces produced by stimulation trains, with mean frequencies from 12.5 to 100 HZ, train durations from 485 to 1000 ms, and number of pulses from 14 to 50 for both nonfatigued and fatigued muscles. The success of our mathematical model in predicting forces produced by stimulations with a wide range of frequencies, durations, and number of pulses implies great potential of the model for the identification of optimal activation patterns that should be used during functional electrical stimulation.

Modeling the length dependence of isometric force in human quadriceps muscles

Functional electrical stimulation is used to restore movement and function of paralyzed muscles by activating skeletal muscle artificially. An accurate and predictive mathematical model can facilitate the design of stimulation patterns that produce the desired force. The present study is a first step in developing a mathematical model for non-isometric muscle contractions. The goals of this study were to: (1) identify how our isometric force model's parameters vary with changes in knee joint angle, (2) identify the best knee flexion angle to parameterize this model, and (3) validate the model by comparing experimental data to predictions in response to a wide range of stimulation frequencies and muscle lengths. Results showed that by parabolically varying one of the free parameters with knee joint angle and fixing the other parameters at the values identified at 40 degrees of knee flexion, the model could predict the force responses to a wide range of stimulation frequencies and patterns at different muscle lengths. This work showed that the current isometric force model is capable of predicting the changes in skeletal muscle force at different muscle lengths.

A model for human skin impedance during surface functional neuromuscular stimulation

A new mathematical model for the bulk electrical impedance of human skin is presented. In particular this model describes the impedance of skin during surface functional neuromuscular stimulation (FNS) with square stimulation pulses. Experimental data are presented that illustrate the nonlinear dynamic properties of human skin during current and voltage controlled stimulation. Model predictions are compared to experimental data, measured under both constant voltage and constant current transcutaneous stimulation. It is found that this model captures a variety of nonlinear time-varying effects observed in the skin impedance when stimulating with either protocol. This model may be used as part of large neuromusculoskeletal models or in the more accurate modeling of transcutaneous FNS, which is currently the most common clinical implementation of FNS.

Model-based development of neuroprosthesis for paraplegic patients

In paraplegic patients with upper motor neuron lesions the signal path from the central nervous system to the muscles is interrupted. Functional electrical stimulation applied to the lower motor neurons can replace the lacking signals. A so-called neuroprosthesis may be used to restore motor function in paraplegic patients on the basis of functional electrical stimulation. However, the control of multiple joints is difficult due to the complexity, nonlinearity, and time-variance of the system involved. Furthermore, effects such as muscle fatigue, spasticity, and limited force in the stimulated muscle further complicate the control task. Mathematical models of the human musculoskeletal system can support the development of neuroprosthesis. In this article a detailed overview of the existing work in the literature is given and two examples developed by the author are presented that give an insight into model-based development of neuroprosthesis for paraplegic patients. It is shown that modelling the musculoskeletal system can provide better understanding of muscular force production and movement coordination principles. Models can also be used to design and test stimulation patterns and feedback control strategies. Additionally, model components can be implemented in a controller to improve control performance. Eventually, the use of musculoskeletal models for neuroprosthesis design may help to avoid internal disturbances such as fatigue and optimize muscular force output. Furthermore, better controller quality can be obtained than in previous empirical approaches. In addition, the number of experimental tests to be performed with human subjects can be reduced. It is concluded that mathematical models play an increasing role in the development of reliable closed-loop controlled, lower extremity neuroprostheses.