Modification of a three-compartment muscle fatigue model to predict peak torque decline during intermittent tasks

This study aimed to test whether adding a rest recovery parameter, r, to the analytical three-compartment controller (3CC) fatigue model (Xia and Frey Law, 2008) will improve fatigue estimates during intermittent contractions. The 3CC muscle fatigue model uses differential equations to predict the flow of muscle between three muscle states: Resting (M_R), Active (M_A), and Fatigued (M_F). This model uses a feedback controller to match the active state to target loads and two joint-specific parameters: F, fatigue rate controlling flow from active to fatigued compartments) and R, the recovery rate controlling flow from the fatigued to the resting compartments. This model does well to predict intensity-endurance time curves for sustained isometric tasks. However, previous studies find when rest intervals are present that the model over predicts fatigue. Intermittent rest periods would allow for the occurrence of subsequent reactive vasodilation and post-contraction hyperemia. We hypothesize a modified 3CC-r fatigue model will improve predictions of force decay during intermittent contractions with the addition of a rest recovery parameter, r, to augment recovery during rest intervals, representing muscle re-perfusion. A meta-analysis compiling intermittent fatigue data from 63 publications reporting decline in peak torque (% torque decline) were used for comparison. The original model over-predicted fatigue development from 19 to 29% torque decline; the addition of a rest multiplier significantly improved fatigue estimates to 6–10% torque decline. We conclude the addition of a rest multiplier to the three-compartment controller fatigue model provides a physiologically consistent modification for tasks involving rest intervals, resulting in improved estimates of muscle fatigue.     

Evidence of long term muscle fatigue following prolonged intermittent contractions based on mechano- and electromyograms.

The focus of the present study is the long term element of muscle fatigue provoked by prolonged intermittent contractions at submaximal force levels and analysed by force, surface electromyography (EMG) and mechanomyogram (MMG). It was hypothesized that fatigue related changes in mechanical performance of the biceps muscle are more strongly reflected in low than in high force test contractions, more prominent in the MMG than in the EMG signal and less pronounced following contractions controlled by visual compared to proprioceptive feedback. Further, it was investigated if fatigue induced by 30 min intermittent contractions at 30% as well as 10% of maximal voluntary contraction (MVC) lasted more than 30 min recovery. In six male subjects the EMG and MMG were recorded from the biceps brachii muscle during three sessions with fatiguing exercise at 10% with visual feedback and at 30% MVC with visual and proprioceptive feedback. EMG, MMG, and force were evaluated during isometric test contractions at 5% and 80% MVC before prolonged contraction and after 10 and 30 min of recovery. MVC decreased significantly after the fatiguing exercise in all three sessions and was still decreased even after 30 min of recovery. In the time domain significant increases after the fatiguing exercise were found only in the 5% MVC tests and most pronounced for the MMG. No consistent changes were found for neither EMG nor MMG in the frequency domain and feedback mode did not modify the results. It is concluded that long term fatigue after intermittent contractions at low force levels can be detected even after 30 min of recovery in a low force test contraction. Since the response was most pronounced in the MMG this may be a valuable variable for detection of impairments in the excitation-contraction coupling.     

Electromyography and fatigue during prolonged, low-level static contractions

Findings from five separate studies of EMG changes and muscle fatigue during prolonged low-level static contractions are summarized, and the possible mechanisms behind the changes are briefly discussed. Sustained static contractions (10%, 7% and 5% MVC) of up to 1 h duration were performed by finger flexors, elbow flexors and extensors, and knee extensors. In one experiment, intermittent static arm pulling (triceps) (10 s contraction and 5 s rest, average work load 14% and 10% MVC) was performed for 7 h. The endurance time for the sustained contractions was around one hour for 10% MVC, and it was shown--all in all--that the concept of "indefinite" endurance times at contractions below 15-20% MVC cannot be maintained. After 5% MVC sustained contractions for one hour a 12% reduction in MVC was seen, and significant increases in EMG amplitude and decreases in the mean spectral frequency of the EMG-power spectrum were found. Marked differences were also seen in the EMG changes in the elbow flexors and extensors, and transcutaneous electrical stimulation of the knee extensors showed that low frequency fatigue was present after the contraction. With intermittent contractions similar changes in the EMG parameters were seen after 2-3 h of contractions at 14% MVC. On average, during contractions of 10% MVC no EMG changes were detected. Increased extracellular potassium concentration in the contracting muscles is suggested as a possible explanation of these findings.     

Muscle blood flow during isometric activity and its relation to muscle fatigue

The effect of isometric exercise on blood flow, blood pressure, intramuscular pressure as well as lactate and potassium efflux from exercising muscle was examined. The contractions performed were continuous or intermittent (5 s on, 5 s off) and varied between 5% and 50% maximal voluntary contraction (MVC). A knee-extensor and a hand-grip protocol were used. Evidence is presented that blood flow through the muscle is sufficient during low-level sustained contractions (less than 10% MVC). Despite this muscle fatigue occurs during prolonged contractions. One mechanism for this fatigue may be the disturbance of the potassium homeostasis. Such changes may also play a role in the development of fatigue during intermittent isometric contractions and even more so in the recovery from such exercise. In addition the role of impaired transport of substances within the muscle, due to long-lasting daily oedema formation, is discussed in relation to fatigue in highly repetitive, monotonous jobs.     

Parameter Sensitivity Analysis in Electrophysiological Models Using Multivariable Regression

Computational models of electrical activity and calcium signaling in cardiac myocytes are important tools for understanding physiology. The sensitivity of these models to changes in parameters is often not well-understood, however, because parameter evaluation can be a time-consuming, tedious process. I demonstrate here what I believe is a novel method for rapidly determining how changes in parameters affect outputs. In three models of the ventricular action potential, parameters were randomized, repeated simulations were run, important outputs were calculated, and multivariable regression was performed on the collected results. Random parameters included both maximal rates of ion transport and gating variable characteristics. The procedure generated simplified, empirical models that predicted outputs resulting from new sets of input parameters. The linear regression models were quite accurate, despite nonlinearities in the mechanistic models. Moreover, the regression coefficients, which represent parameter sensitivities, were robust, even when parameters were varied over a wide range. Most importantly, a side-by-side comparison of two similar models identified fundamental differences in model behavior, and revealed model predictions that were both consistent with, and inconsistent with, experimental data. This new method therefore shows promise as a tool for the characterization and assessment of computational models. The general strategy may also suggest methods for integrating traditional quantitative models with large-scale data sets obtained using high-throughput technologies.     

The influence of circadian rhythm during a sustained submaximal exercise and on recovery process.

The purpose of this study was to examine the time-of-day effects on muscle fatigue and recovery process following an isometric fatiguing contraction. Sixteen male subjects were tested at two times (06:00h and 18:00h) and were requested to perform a sustained submaximal contraction of the elbow flexors, consisting in maintaining 40% of their absolute strength as long as they could. Isometric maximal voluntary contractions (MVC) were performed before (Pre), immediately after (Post), and up to 10min after the endurance task. Endurance time, peak torque (PT) and electromyographic (EMG) activities of the biceps brachii and triceps brachii were recorded and analysed. Results showed that under Pre-test conditions, PT developed at 18:00h was higher than at 06:00h. No time-of-day effect appears for the endurance time and EMG activities during the test. No time-of-day effect was observed on either MVC or EMG recovery. From the results of this study, it seems that both muscle fatigue and recovery process are not time-of-day dependent. We conclude that circadian rhythm of the force do not influence the evaluation of muscle capacities during a submaximal exercise corresponding at 40% of MVC.     

Quantifying the history dependency of muscle recovery from a fatiguing intermittent task

Muscle fatigue and recovery are complex processes influencing muscle force generation capacity. While fatigue reduces this capacity, recovery acts to restore the unfatigued muscle state. Many factors can potentially affect muscle recovery, and among these may be a task dependency of recovery following an exercise. However, little has been reported regarding the history dependency of recovery after fatiguing contractions. We examined the dependency of muscle recovery subsequent to four different histories of fatiguing muscle contractions, imposed using two cycle times (30 and 60 s) during low to moderate levels (15% and 25% of maximum voluntary contraction (MVC)) of intermittent static exertions involving index finger abduction. MVC and low-frequency electrical stimulation (LFES) measures (i.e., magnitude, rise and relaxation rates) of muscle capacity were used, all of which indicated a dependency of muscle recovery on the muscle capacity state existing immediately after fatiguing exercise. This dependency did not appear to be modified by either the cycle time or exertion level leading to that state. These results imply that the post-exercise rate of recovery is primarily influenced by the immediate post-exercise muscle contractile status (estimated by MVC and LFES measures). Such results may help improve existing models of muscle recovery, facilitating more accurate predictions of localized muscle fatigue development and thereby helping to enhance muscle performance and reduce the risk of injury.     

Effect of low frequency fatigue on human muscle strength and fatigability during subsequent stimulated activity

Fatiguing contractions of the adductor pollicis muscle were produced by intermittent supramaximal stimulation of the ulnar nerve in a set frequency pattern, in six normal subjects. At the end of an initial fatiguing contraction series, low frequency fatigue (LFF) had been induced and persisted at 15 min of recovery. Stimulated fatiguing activity was then repeated in an identical fashion to the initial series. At high frequencies, declines in force were similar for both series. At low frequencies, declines in force were greater during the second series despite similar changes in compound muscle action potential amplitude. This confirmation that LFF persists during subsequent stimulated activity, and reduces low but not high frequency fatigue resistance, suggests that the impaired endurance of fatigued muscle during voluntary activity primarily results from peripheral changes at low frequency. These findings also have implications for therapeutic electrical stimulation of muscle.     

Development of Standard Fuel Models in Boreal Forests of Northeast China through Calibration and Validation

Understanding the fire prediction capabilities of fuel models is vital to forest fire management. Various fuel models have been developed in the Great Xing''an Mountains in Northeast China. However, the performances of these fuel models have not been tested for historical occurrences of wildfires. Consequently, the applicability of these models requires further investigation. Thus, this paper aims to develop standard fuel models. Seven vegetation types were combined into three fuel models according to potential fire behaviors which were clustered using Euclidean distance algorithms. Fuel model parameter sensitivity was analyzed by the Morris screening method. Results showed that the fuel model parameters 1-hour time-lag loading, dead heat content, live heat content, 1-hour time-lag SAV(Surface Area-to-Volume), live shrub SAV, and fuel bed depth have high sensitivity. Two main sensitive fuel parameters: 1-hour time-lag loading and fuel bed depth, were determined as adjustment parameters because of their high spatio-temporal variability. The FARSITE model was then used to test the fire prediction capabilities of the combined fuel models (uncalibrated fuel models). FARSITE was shown to yield an unrealistic prediction of the historical fire. However, the calibrated fuel models significantly improved the capabilities of the fuel models to predict the actual fire with an accuracy of 89%. Validation results also showed that the model can estimate the actual fires with an accuracy exceeding 56% by using the calibrated fuel models. Therefore, these fuel models can be efficiently used to calculate fire behaviors, which can be helpful in forest fire management.     

Therapeutic Implications from Sensitivity Analysis of Tumor Angiogenesis Models

Anti-angiogenic cancer treatments induce tumor starvation and regression by targeting the tumor vasculature that delivers oxygen and nutrients. Mathematical models prove valuable tools to study the proof-of-concept, efficacy and underlying mechanisms of such treatment approaches. The effects of parameter value uncertainties for two models of tumor development under angiogenic signaling and anti-angiogenic treatment are studied. Data fitting is performed to compare predictions of both models and to obtain nominal parameter values for sensitivity analysis. Sensitivity analysis reveals that the success of different cancer treatments depends on tumor size and tumor intrinsic parameters. In particular, we show that tumors with ample vascular support can be successfully targeted with conventional cytotoxic treatments. On the other hand, tumors with curtailed vascular support are not limited by their growth rate and therefore interruption of neovascularization emerges as the most promising treatment target.     

Are Subject-Specific Musculoskeletal Models Robust to the Uncertainties in Parameter Identification?

Subject-specific musculoskeletal modeling can be applied to study musculoskeletal disorders, allowing inclusion of personalized anatomy and properties. Independent of the tools used for model creation, there are unavoidable uncertainties associated with parameter identification, whose effect on model predictions is still not fully understood. The aim of the present study was to analyze the sensitivity of subject-specific model predictions (i.e., joint angles, joint moments, muscle and joint contact forces) during walking to the uncertainties in the identification of body landmark positions, maximum muscle tension and musculotendon geometry. To this aim, we created an MRI-based musculoskeletal model of the lower limbs, defined as a 7-segment, 10-degree-of-freedom articulated linkage, actuated by 84 musculotendon units. We then performed a Monte-Carlo probabilistic analysis perturbing model parameters according to their uncertainty, and solving a typical inverse dynamics and static optimization problem using 500 models that included the different sets of perturbed variable values. Model creation and gait simulations were performed by using freely available software that we developed to standardize the process of model creation, integrate with OpenSim and create probabilistic simulations of movement. The uncertainties in input variables had a moderate effect on model predictions, as muscle and joint contact forces showed maximum standard deviation of 0.3 times body-weight and maximum range of 2.1 times body-weight. In addition, the output variables significantly correlated with few input variables (up to 7 out of 312) across the gait cycle, including the geometry definition of larger muscles and the maximum muscle tension in limited gait portions. Although we found subject-specific models not markedly sensitive to parameter identification, researchers should be aware of the model precision in relation to the intended application. In fact, force predictions could be affected by an uncertainty in the same order of magnitude of its value, although this condition has low probability to occur.     

Caffeine effect on respiratory muscle endurance and sense of effort during loaded breathing

The present study examined respiratory muscle endurance and the magnitude of the sense of effort during inspiratory threshold loading following a dose of caffeine (600 mg) previously observed to increase diaphragm strength. Experiments were performed on 12 normal subjects. Respiratory muscle endurance at a given level of load was assessed from the time of exhaustion and from the time course of the change in the power spectrum (centroid frequency) of the diaphragm electromyogram (EMG). The intensity of the sense of effort during loaded breathing was evaluated using a category (Borg) scale. Increasingly severe loads were associated with more rapid onset of fatigue. At a given load, caffeine prolonged the time to exhaustion and decreased the rate of fall of the centroid frequency of the diaphragm EMG. Caffeine also decreased the sense of effort during loaded breathing in 9 of 11 subjects. Changes in respiratory muscle endurance after caffeine administration were not explained by changes in the pressure-time index of the respiratory muscles or the pattern of thoracoabdominal movement. We conclude that caffeine enhances inspiratory muscle endurance, while concomitantly reducing the sense of effort associated with fatiguing inspiratory muscle contractions.     

Electro-mechanical failures and lactate production during fatigue

The electrical and mechanical failures observed during sustained and intermittent electrically triggered (30 Hz) contractions of human flexor carpi ulnaris were compared with the blood lactate concentration. The changes recorded during contractions sustained for 60 s were compared with those observed during a series of sixty 1 s contractions separated by 1 s intervals, and also with the changes during the first 30 min of recovery. No significant (P less than 0.05) difference in force reduction or maximal venous lactate concentration was observed in either fatigue test, although electrical failure differed significantly (P less than 0.05). The recovery of electrical failure was poorly correlated with the reduction in lactate concentration following both sustained (r = -0.70) and intermittent contractions (r = 0.72). In contrast, the recovery in tetanic tension, rate of tension development and time to half relaxation correlated closely with the reduction in venous lactate concentration (r = -0.95, -0.93 and 0.96 respectively). It is suggested that, of the peripheral processes which appear to play a dominant role in peripheral fatigue, lactate production controls mechanical failure directly.     

Asymptotic Distribution of Density-Dependent Stage-Grouped Population Dynamics Models

Matrix models are widely used in biology to predict the temporal evolution of stage-structured populations. One issue related to matrix models that is often disregarded is the sampling variability. As the sample used to estimate the vital rates of the models are of finite size, a sampling error is attached to parameter estimation, which has in turn repercussions on all the predictions of the model. In this study, we address the question of building confidence bounds around the predictions of matrix models due to sampling variability. We focus on a density-dependent Usher model, the maximum likelihood estimator of parameters, and the predicted stationary stage vector. The asymptotic distribution of the stationary stage vector is specified, assuming that the parameters of the model remain in a set of the parameter space where the model admits one unique equilibrium point. Tests for density-dependence are also incidentally provided. The model is applied to a tropical rain forest in French Guiana.     

A submaximal test to assess back muscle capacity: Evaluation of construct validity

A functional endurance test more specific to common occupational tasks is proposed for assessing back muscle capacity. The test involves static intermittent contractions (8-s work-rest cycles) using a predefined absolute load (90 Nm) across subjects. Since the test involved an absolute endurance task, it was hypothesized that performance would be influenced by both the strength and relative endurance of the subjects, thus demonstrating the construct validity of this new test. Fifteen males and 17 females were assessed on three different days to allow familiarization and to measure their Strength as well as their absolute and relative endurance. Absolute and relative endurance were defined as the time to reach exhaustion (Tend_abs and Tend_rel, respectively) during a fatigue protocol including both an absolute (90 Nm) and a relative (40% of individual strength) load (extension moment at the L5/S1 joint). Strength and Tend_rel each explained an almost equivalent portion of Tend_abs (total variance explained: 61.5%), thus confirming the construct validity of the functional endurance test. This new test should better identify the back muscle impairments (weakness, fatigability) often observed in chronic low-back-pain patients.     

Relationship of contraction capacity to metabolic changes during recovery from a fatiguing contraction

The relationship between changes in muscle metabolites and the contraction capacity was investigated in humans. Subjects (n = 13) contracted (knee extension) at a target force of 66% of the maximal voluntary contraction force (MVC) to fatigue, and the recovery in MVC and endurance (time to fatigue) were measured. Force recovered rapidly [half-time (t 1/2) less than 15 s] and after 2 min of recovery was not significantly different (P greater than 0.05) from the precontraction value. Endurance recovered more slowly (t 1/2 approximately 1.2 min) and was still significantly depressed after 2 and 4 min of recovery (P less than 0.05). In separate experiments (n = 10) muscle biopsy specimens were taken from the quadriceps femoris muscle before and after two successive contractions to fatigue at 66% of MVC with a recovery period of 2 or 4 min in between. The muscle content of high-energy phosphates and lactate was similar at fatigue after both contractions, whereas glucose 6-phosphate was lower after the second contraction (P less than 0.05). During recovery, muscle lactate decreased and was 74 and 43% of the value at fatigue after an elapsed period of 2 and 4 min, respectively. The decline in H+ due to lactate disappearance is balanced, however, by a release of H+ due to resynthesis of phosphocreatine, and after 2 min of recovery calculated muscle pH was found to remain at a low level similar to that at fatigue.(ABSTRACT TRUNCATED AT 250 WORDS)     

Dealing with Uncertainty in Spatially Explicit Population Models

It has been argued that spatially explicit population models (SEPMs) cannot provide reliable guidance for conservation biology because of the difficulty of obtaining direct estimates for their demographic and dispersal parameters and because of error propagation. We argue that appropriate model calibration procedures can access additional sources of information, compensating the lack of direct parameter estimates. Our objective is to show how model calibration using population-level data can facilitate the construction of SEPMs that produce reliable predictions for conservation even when direct parameter estimates are inadequate. We constructed a spatially explicit and individual-based population model for the dynamics of brown bears (Ursus arctos) after a reintroduction program in Austria. To calibrate the model we developed a procedure that compared the simulated population dynamics with distinct features of the known population dynamics (=patterns). This procedure detected model parameterizations that did not reproduce the known dynamics. Global sensitivity analysis of the uncalibrated model revealed high uncertainty in most model predictions due to large parameter uncertainties (coefficients of variation CV 0.8). However, the calibrated model yielded predictions with considerably reduced uncertainty (CV 0.2). A pattern or a combination of various patterns that embed information on the entire model dynamics can reduce the uncertainty in model predictions, and the application of different patterns with high information content yields the same model predictions. In contrast, a pattern that does not embed information on the entire population dynamics (e.g., bear observations taken from sub-areas of the study area) does not reduce uncertainty in model predictions. Because population-level data for defining (multiple) patterns are often available, our approach could be applied widely.     

Nonignorable Models for Intermittently Missing Categorical Longitudinal Responses

Summary A class of nonignorable models is presented for handling nonmonotone missingness in categorical longitudinal responses. This class of models includes the traditional selection models and shared parameter models. This allows us to perform a broader than usual sensitivity analysis. In particular, instead of considering variations to a chosen nonignorable model, we study sensitivity between different missing data frameworks. An appealing feature of the developed class is that parameters with a marginal interpretation are obtained, while algebraically simple models are considered. Specifically, marginalized mixed‐effects models ( Heagerty, 1999 , Biometrics 55, 688–698) are used for the longitudinal process that model separately the marginal mean and the correlation structure. For the correlation structure, random effects are introduced and their distribution is modeled either parametrically or non‐parametrically to avoid potential misspecifications.     

Fatigue of intermittent submaximal voluntary contractions: central and peripheral factors

Central and peripheral factors were studied in fatigue of submaximal intermittent isometric contractions of the human quadriceps and soleus muscles. Subjects made repeated 6 s, 50% maximal voluntary contractions (MVC) followed by 4 s rest until the limit of endurance (Tlim). Periodically, a fatigue test was performed. This included a brief MVC, either a single shock or 8 pulses at 50 Hz during a rest period and a shock superimposed on a target force voluntary contraction. At Tlim, the MVC force had declined by 50%, usually in parallel with the force from stimulation at 50 Hz. The twitches superimposed on the target forces declined more rapidly, disappearing entirely at Tlim. In similar experiments on adductor pollicis, no reduction of the evoked M wave was seen. The results suggest that, during fatigue of quadriceps and adductor pollicis induced by this protocol, no central fatigue was apparent, but some was seen in soleus. Thus the reduced force-generating capacity could result mainly or entirely from failure of the muscle contractile apparatus.