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.     

Doublet stimulation protocol to minimize musculoskeletal stress during paralyzed quadriceps muscle testing.

With long-term electrical stimulation training, paralyzed muscle can serve as an effective load delivery agent for the skeletal system. Muscle adaptations to training, however, will almost certainly outstrip bone adaptations, exposing participants in training protocols to an elevated risk for fracture. Assessing the physiological properties of the chronically paralyzed quadriceps may transmit unacceptably high shear forces to the osteoporotic distal femur. We devised a two-pulse doublet strategy to measure quadriceps physiological properties while minimizing the peak muscle force. The purposes of the study were 1) to determine the repeatability of the doublet stimulation protocol, and 2) to compare this protocol among individuals with and without spinal cord injury (SCI). Eight individuals with SCI and four individuals without SCI underwent testing. The doublet force-frequency relationship shifted to the left after SCI, likely reflecting enhancements in the twitch-to-tetanus ratio known to exist in paralyzed muscle. Posttetanic potentiation occurred to a greater degree in subjects with SCI (20%) than in non-SCI subjects (7%). Potentiation of contractile rate occurred in both subject groups (14% and 23% for SCI and non-SCI, respectively). Normalized contractile speed (rate of force rise, rate of force fall) reflected well-known adaptations of paralyzed muscle toward a fast fatigable muscle. The doublet stimulation strategy provided repeatable and sensitive measurements of muscle force and speed properties that revealed meaningful differences between subjects with and without SCI. Doublet stimulation may offer a unique way to test muscle physiological parameters of the quadriceps in subjects with uncertain musculoskeletal integrity.     

Postfatigue potentiation of the paralyzed soleus muscle: evidence for adaptation with long-term electrical stimulation training.

Understanding the torque output behavior of paralyzed muscle has important implications for the use of functional neuromuscular electrical stimulation systems. Postfatigue potentiation is an augmentation of peak muscle torque during repetitive activation after a fatigue protocol. The purposes of this study were 1) to quantify postfatigue potentiation in the acutely and chronically paralyzed soleus and 2) to determine the effect of long-term soleus electrical stimulation training on the potentiation characteristics of recently paralyzed soleus muscle. Five subjects with chronic paralysis (>2 yr) demonstrated significant postfatigue potentiation during a repetitive soleus activation protocol that induced low-frequency fatigue. Ten subjects with acute paralysis (<6 mo) demonstrated no torque potentiation in response to repetitive stimulation. Seven of these acute subjects completed 2 yr of home-based isometric soleus electrical stimulation training of one limb (compliance = 83%; 8,300 contractions/wk). With the early implementation of electrically stimulated training, potentiation characteristics of trained soleus muscles were preserved as in the acute postinjury state. In contrast, untrained limbs showed marked postfatigue potentiation at 2 yr after spinal cord injury (SCI). A single acute SCI subject who was followed longitudinally developed potentiation characteristics very similar to the untrained limbs of the training subjects. The results of the present investigation support that postfatigue potentiation is a characteristic of fast-fatigable muscle and can be prevented by timely neuromuscular electrical stimulation training. Potentiation is an important consideration in the design of functional electrical stimulation control systems for people with SCI.     

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.     

Predicting optimal electrical stimulation for repetitive human muscle activation.

Functional electrical stimulation is the use of electrical currents to activate paralyzed muscles to produce functional movements. Muscle force output must meet or exceed the external load to maintain a posture or produce movements. A mathematical force-fatigue modeling system that predicts muscle force responses during repetitive electrical stimulation has been developed in our laboratory to help identify stimulation patterns that optimize force output for individual subjects. This study tests how well this model predicts the number of contractions that can be maintained above a required force level (successful contractions) during repetitive activation of a muscle. Healthy human quadriceps muscles were tested isometrically on 12 subjects. Data were first collected and used to parameterize the model. Next, the model was used to predict the number of successful contractions that were produced by trains with frequencies ranging from 5 to 100 Hz while the pulse durations and amplitudes were held constant. Finally, three clinically relevant stimulation frequencies were selected and tested to verify the model's predictions. Under these conditions, the model accurately predicted the number of successful contractions for clinically relevant stimulation frequencies. Furthermore, the model appears to have the potential to identify the stimulation frequency that maximizes muscle force output and minimizes fatigue for each subject.     

Influence of electrical stimulation on the morphological and metabolic properties of paralyzed muscle.

Selected morphological and metabolic properties of single fibers were studied in biopsy samples from the tibialis anterior of normal control and spinal cord-injured (SCI) subjects. In the SCI subjects, one muscle was electrically stimulated progressively over 24 wk, in 6-wk blocks for less than or equal to 8 h/day, while the contralateral muscle remained untreated. The percentage of fibers classified as type I [qualitative alkaline preincubation myofibrillar adenosinetriphosphatase (ATPase)] was significantly less in the unstimulated paralyzed muscles than in the muscles of normal control subjects. Electrical stimulation increased the proportion of type I fibers in the SCI subjects. For both type I and type II fibers, the cross-sectional area, activities of myofibrillar ATPase and succinate dehydrogenase, and the capillary-to-fiber ratio were also significantly less in the paralyzed muscles than in the normal control muscles. Electrical stimulation increased only the activity of succinate dehydrogenase in both fiber types of the SCI subjects. These data are discussed in relation to the electromechanical properties of the respective muscles described in an accompanying paper (J. Appl. Physiol. 72: 1393-1400, 1992). In general, the electrical stimulation protocol used in this study enhanced the oxidative capacity and endurance properties of the paralyzed muscles but had no effect on fiber size and strength.     

Skeletal muscle metabolism in individuals with spinal cord injury

With increasing survival rates in people with spinal cord injuries (SCI), detection and prevention of metabolic and cardiovascular disease have become increasingly important. Few studies have evaluated in vivo mitochondrial function in paralyzed skeletal muscle. The purpose of this study was to compare oxidative muscle metabolism using the rate of phosphocreatine (PCr) resynthesis measured by magnetic resonance spectroscopy (MRS) in people with SCI and able-bodied (AB) controls. Eight subjects with complete SCI (American Spinal Injury Association Impairment Scale A, levels T3-T12, injury duration 2-13 years) were compared with 12 AB controls. T1-weighted (1)H MR images of the thigh were taken to identify skeletal muscle. Phosphorous MRS was performed with a 13 × 13-cm(2) surface coil placed on the right vastus lateralis in a 3 Tesla clinical MRI scanner. PCr resynthesis was measured after electrical stimulation for 60 s at 4 Hz in SCI and AB and in AB subjects after 39 s of voluntary isometric contractions. Resting metabolites were not different between SCI and AB, except for an elevated phosphodiester peak. PCr recovery was slower in AB subjects using electrical stimulation compared with voluntary exercise (28.4 ± 6.1 vs. 41.5 ± 4.3 s; P < 0.05). PCr recovery rates and calculated muscle maximum oxidative capacity in SCI were both 52% of electrically stimulated AB (P < 0.001). In vivo oxidative metabolism was reduced in paralyzed muscle to a similar extent as seen in people with mitochondrial myopathies and heart failure.     

Increasing blood flow before exercise in spinal cord-injured individuals does not alter muscle fatigue.

Previous studies have shown increased fatigue in paralyzed muscle of spinal cord-injured (SCI) patients (Castro M, Apple D Jr, Hillegass E, and Dudley GA. Eur J Appl Physiol 80: 373-378, 1999; Gerrits H, Hopman MTE, Sargeant A, and de Haan A. Clin Physiol 21: 105-113, 2001). Our purpose was to determine whether the increased muscle fatigue could be due to a delayed rise in blood flow at the onset of exercise in SCI individuals. Isometric electrical stimulation was used to induce fatigue in the quadriceps femoris muscle of seven male, chronic (>1 yr postinjury), complete (American Spinal Injury Association, category A) SCI subjects. Cuff occlusion was used to elevate blood flow before electrical stimulation, and the magnitude of fatigue was compared with a control condition of electrical stimulation without prior cuff occlusion. Blood flow was measured in the femoral artery by Doppler ultrasound. Prior cuff occlusion increased blood flow in the first 30 s of stimulation compared with the No-Cuff condition (1,350 vs. 680 ml/min, respectively; P < 0.001), although blood flow at the end of stimulation was the same between conditions (1,260 +/- 140 vs. 1,160 +/- 370 ml/min, Cuff and No-Cuff condition, respectively; P = 0.511). Muscle fatigue was not significantly different between prior cuff occlusion and the control condition (32 +/- 13 vs. 35 +/- 10%; P = 0.670). In conclusion, increased muscle fatigue in SCI individuals is not associated with the prolonged time for blood flow to increase at the onset of exercise.     

Hill muscle model errors during movement are greatest within the physiologically relevant range of motor unit firing rates

This study evaluated the accuracy of Hill-type muscle models during movement. Hill-type models are ubiquitous in biomechanical simulations. They are attractive because of their computational simplicity and close relation to commonly measured experimental variables, but there have been surprisingly few experimental validations of these models during functionally relevant conditions. Our hypothesis was that model errors during movement are largest at the low motor unit firing rates most relevant to normal movement conditions. This hypothesis was evaluated in the cat soleus muscle activated either by electrical stimulation at physiological rates or via the crossed-extension reflex (CXR) thereby obtaining normal patterns of motor unit recruitment and rate modulation. These activation paradigms were applied during continuous movements approximately matched to locomotor length changes. The resulting muscle force was modeled using a common Hill model incorporating independent activation, tetanic length-tension and tetanic force-velocity properties. Errors for this model were greatest for stimulation rates between approximately 10-20Hz. Errors were especially large for muscles activated via the CXR, where most motor units appear to fire within this range. For large muscle excursions, such as those seen during normal locomotion, the errors for naturally activated muscle typically exceeded 50%, supporting our hypothesis and indicating that the Hill model is not appropriate for these conditions. Subsequent analysis suggested that model errors were due to the common Hill model's inability to account for the coupling between muscle activation and force-velocity properties that is most prevalent at the low motor unit firing rates relevant to normal activation.     

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.     

Feedback-controlled stimulation enhances human paralyzed muscle performance.

Chronically paralyzed muscle requires extensive training before it can deliver a therapeutic dose of repetitive stress to the musculoskeletal system. Neuromuscular electrical stimulation, under feedback control, may subvert the effects of fatigue, yielding more rapid and extensive adaptations to training. The purposes of this investigation were to 1) compare the effectiveness of torque feedback-controlled (FDBCK) electrical stimulation with classic open-loop constant-frequency (CONST) stimulation, and 2) ascertain which of three stimulation strategies best maintains soleus torque during repetitive stimulation. When torque declined by 10%, the FDBCK protocol modulated the base stimulation frequency in three ways: by a fixed increase, by a paired pulse (doublet) at the beginning of the stimulation train, and by a fixed decrease. The stimulation strategy that most effectively restored torque continued for successive contractions. This process repeated each time torque declined by 10%. In fresh muscle, FDBCK stimulation offered minimal advantage in maintaining peak torque or mean torque over CONST stimulation. As long-duration fatigue developed in subsequent bouts, FDBCK stimulation became most effective ( approximately 40% higher final normalized torque than CONST). The high-frequency strategy was selected approximately 90% of the time, supporting that excitation-contraction coupling compromise and not neuromuscular transmission failure contributed to fatigue of paralyzed muscle. Ideal stimulation strategies may vary according to the site of fatigue; this stimulation approach offered the advantage of online modulation of stimulation strategies in response to fatigue conditions. Based on stress-adaptation principles, FDBCK-controlled stimulation may enhance training effects in chronically paralyzed muscle.     

Blood flow and muscle fatigue in SCI individuals during electrical stimulation.

Our purpose was to measure blood flow and muscle fatigue in chronic, complete, spinal cord-injured (SCI) and able-bodied (AB) individuals during electrical stimulation. Electrical stimulation of the quadriceps muscles was used to elicit similar activated muscle mass. Blood flow was measured in the femoral artery by Doppler ultrasound. Muscle fatigue was significantly greater (three- to eightfold, P < or = 0.001) in the SCI vs. the AB individuals. The magnitude of blood flow was not significantly different between groups. A prolonged half-time to peak blood flow at the beginning of exercise (fivefold, P = 0.001) and recovery of blood flow at the end of exercise (threefold, P = 0.009) was found in the SCI vs. the AB group. In conclusion, the magnitude of the muscle blood flow to electrical stimulation was not associated with increased muscle fatigue in SCI individuals. However, the prolonged time to peak blood flow may be an explanation for increased fatigue in SCI individuals.     

Model study of time‐dependent muscle response to pulsed electrical stimulation

A systems‐level model analysis of neuromuscular response to external electrical stimulation is presented. Action potential (AP) generation, dynamics of voltage‐based calcium release at the motor endplates controlled by the arrival of APs, and muscle force production are all comprehensively included. Numerical predictions exhibit trends that are qualitatively similar to measurements of muscle response in rats from a burst of cortical stimulation and a nanosecond impulse. Modulation of neural membrane conductances (including possible electroporation) that alters the neural impulse generation frequency is hypothesized as a possible mechanism leading to observed changes in muscle force production. Other possibilities such as calcium release at nerve end endings also exist. It is also proposed that multipulsing strategies and changing the electric field direction by using multielectrode systems would be useful. Bioelectromagnetics 31:361–370, 2010. © 2010 Wiley‐Liss, Inc.     

Two-step, predictive, isometric force model tested on data from human and rat muscles

Functionalelectrical stimulation can assist paralyzed individuals to performfunctional movements, but muscle fatigue is a major limitation to itspractical use. An accurate and predictive mathematical model canfacilitate the design of stimulation patterns that optimize aspects ofthe force transient while minimizing fatigue. Solution nonuniqueness, amajor shortcoming in previous work, was overcome with a simpler model.The model was tested on data collected during isometric contractions ofrat gastrocnemius muscles and human quadriceps femoris muscles undervarious physiological conditions. For each condition tested, parametervalues were identified using the force response to one or twostimulation trains. The parameterized model was then used to predictforces in response to other stimulation patterns. The predicted forcesclosely matched the measured forces. The model was not sensitive toinitial parameter estimates, demonstrating solution uniqueness. Bypredicting the force that develops in response to an arbitrary patternof stimulation, we envision the present model helping identify optimalstimulation patterns for activation of skeletal muscle duringfunctional electrical stimulation.

     

Effects of stimulation frequency versus pulse duration modulation on muscle fatigue.

During functional electrical stimulation (FES), both the frequency and intensity can be increased to increase muscle force output and counteract the effects of muscle fatigue. Most current FES systems, however, deliver a constant frequency and only vary the stimulation intensity to control muscle force. This study compared muscle performance and fatigue produced during repetitive electrical stimulation using three different strategies: (1) constant pulse-duration and stepwise increases in frequency (frequency-modulation); (2) constant frequency and stepwise increases in pulse-duration (pulse-duration-modulation); and (3) constant frequency and pulse-duration (no-modulation). Surface electrical stimulation was delivered to the quadriceps femoris muscles of 12 healthy individuals and isometric forces were recorded. Muscle performance was assessed by measuring the percent changes in the peak forces and force-time integrals between the first and the last fatiguing trains. Muscle fatigue was assessed by measuring percent declines in peak force between the 60Hz pre- and post-fatigue testing trains. The results showed that frequency-modulation showed better performance for both peak forces and force-time integrals in response to the fatiguing trains than pulse-duration-modulation, while producing similar levels of muscle fatigue. Although frequency-modulation is not commonly used during FES, clinicians should consider this strategy to improve muscle performance.     

Predicting muscle fatigue: a response surface approximation based on proper generalized decomposition technique

A novel technique is proposed to predict force reduction in skeletal muscle due to fatigue under the influence of electrical stimulus parameters and muscle physiological characteristics. Twelve New Zealand white rabbits were divided in four groups (\(n=3\)) to obtain the active force evolution of in vitro Extensor Digitorum Longus muscles for an hour of repeated contractions under different electrical stimulation patterns. Left and right muscles were tested, and a total of 24 samples were used to construct a response surface based in the proper generalized decomposition. After the response surface development, one additional rabbit was used to check the predictive potential of the technique. This multidimensional surface takes into account not only the decay of the maximum repeated peak force, but also the shape evolution of each contraction, muscle weight, electrical input signal and stimulation protocol. This new approach of the fatigue simulation challenge allows to predict, inside the multispace surface generated, the muscle response considering other stimulation patterns, different tissue weight, etc.     

A simple means of increasing muscle size after spinal cord injury: a pilot study

This study tested that hypothesis that skeletal muscle within a year of spinal cord injury (SCI) would respond to intermittent high force loading by showing an increase in size. Three males about 46 weeks post clinically complete SCI underwent surface electrical stimulation of their left or right m. quadriceps femoris 2 days per week for 8 weeks to evoke 4 sets of ten isometric or dynamic actions each session. Conditioning increased average cross-sectional area of m. quadriceps femoris, assessed by magnetic resonance imaging, by 20+/-1% (p = 0.0103). This reversed 48 weeks of atrophy such that m. quadriceps femoris 54 weeks after SCI was the same size as when the patients were first studied 6 weeks after injury. The results suggest that skeletal muscle is remarkably responsive to intermittent, high force loading after almost one year of little if any contractile activity.     

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.     

Fatigue compensation during FES using surface EMG

Muscle fatigue limits the effectiveness of FES when applied to regain functional movements in spinal cord injured (SCI) individuals. The stimulation intensity must be manually increased to provide more force output to compensate for the decreasing muscle force due to fatigue. An artificial neural network (ANN) system was designed to compensate for muscle fatigue during functional electrical stimulation (FES) by maintaining a constant joint angle. Surface electromyography signals (EMG) from electrically stimulated muscles were used to determine when to increase the stimulation intensity when the muscle’s output started to drop.

In two separate experiments on able-bodied subjects seated in hard back chairs, electrical stimulation was continuously applied to fatigue either the biceps (during elbow flexion) or the quadriceps muscle (during leg extension) while recording the surface EMG. An ANN system was created using processed surface EMG as the input, and a discrete fatigue compensation control signal, indicating when to increase the stimulation current, as the output. In order to provide training examples and test the systems’ performance, the stimulation current amplitude was manually increased to maintain constant joint angles. Manual stimulation amplitude increases were required upon observing a significant decrease in the joint angle. The goal of the ANN system was to generate fatigue compensation control signals in an attempt to maintain a constant joint angle.

On average, the systems could correctly predict 78.5% of the instances at which a stimulation increase was required to maintain the joint angle. The performance of these ANN systems demonstrates the feasibility of using surface EMG feedback in an FES control system.