A three-compartment muscle fatigue model accurately predicts joint-specific maximum endurance times for sustained isometric tasks

The development of localized muscle fatigue has classically been described by the nonlinear intensity-endurance time (ET) curve (Rohmert, 1960; El Ahrache et al., 2006). These empirical intensity-ET relationships have been well-documented and vary between joint regions. We previously proposed a three-compartment biophysical fatigue model, consisting of compartments (i.e. states) for active (M(A)), fatigued (M(F)), and resting (M(R)) muscles, to predict the decay and recovery of muscle force (Xia and Frey Law, 2008). The purpose of this investigation was to determine optimal model parameter values, fatigue (F) and recovery (R), which define the "flow rate" between muscle states and to evaluate the model's accuracy for estimating expected intensity-ET curves. Using a grid-search approach with modified Monte Carlo simulations, over 1 million F and R permutations were used to predict the maximum ET for sustained isometric tasks at 9 intensities ranging from 10% to 90% of maximum in 10% increments (over 9 million simulations total). Optimal F and R values ranged from 0.00589 (F(ankle)) and 0.0182 (R(ankle)) to 0.00058 (F(shoulder)) and 0.00168 (R(shoulder)), reproducing the intensity-ET curves with low mean RMS errors: shoulder (2.7s), hand/grip (5.6s), knee (6.7s), trunk (9.3s), elbow (9.9s), and ankle (11.2s). Testing the model at different task intensities (15-95% maximum in 10% increments) produced slightly higher errors, but largely within the 95% prediction intervals expected for the intensity-ET curves. We conclude that this three-compartment fatigue model can be used to accurately represent joint-specific intensity-ET curves, which may be useful for ergonomic analyses and/or digital human modeling applications.     

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.     

A dynamical model of muscle activation, fatigue, and recovery

A dynamical model is presented as a framework for muscle activation, fatigue, and recovery. By describing the effects of muscle fatigue and recovery in terms of two phenomenological parameters (F, R), we develop a set of dynamical equations to describe the behavior of muscles as a group of motor units activated by voluntary effort. This model provides a macroscopic view for understanding biophysical mechanisms of voluntary drive, fatigue effect, and recovery in stimulating, limiting, and modulating the force output from muscles. The model is investigated under the condition in which brain effort is assumed to be constant. Experimental validation of the model is performed by fitting force data measured from healthy human subjects during a 3-min sustained maximal voluntary handgrip contraction. The experimental results confirm a theoretical inference from the model regarding the possibility of maximal muscle force production, and suggest that only 97% of the true maximal force can be reached under maximal voluntary effort, assuming that all motor units can be recruited voluntarily. The effects of different motor unit types, time-dependent brain effort, sources of artifacts, and other factors that could affect the model are discussed. The applications of the model are also discussed.     

Mathematical Models of Localized Muscle Fatigue: Sensitivity Analysis and Assessment of Two Occupationally-Relevant Models

Muscle fatigue models (MFM) have broad potential application if they can accurately predict muscle capacity and/or endurance time during the execution of diverse tasks. As an initial step toward facilitating improved MFMs, we assessed the sensitivity of selected existing models to their inherent parameters, specifically that model the fatigue and recovery processes, and the accuracy of model predictions. These evaluations were completed for both prolonged and intermittent isometric contractions, and were based on model predictions of endurance times. Based on a recent review of the literature, four MFMs were initially chosen, from which a preliminary assessment led to two of these being considered for more comprehensive evaluation. Both models had a higher sensitivity to their fatigue parameter. Predictions of both models were also more sensitive to the alteration of their parameters in conditions involving lower to moderate levels of effort, though such conditions may be of most practical, contemporary interest or relevance. Although both models yielded accurate predictions of endurance times during prolonged contractions, their predictive ability was inferior for more complex (intermittent) conditions. When optimizing model parameters for different loading conditions, the recovery parameter showed considerably larger variability, which might be related to the inability of these MFMs in simulating the recovery process under different loading conditions. It is argued that such models may benefit in future work from improving their representation of recovery process, particularly how this process differs across loading conditions.     

Fiber type distribution in the shoulder muscles of the tree shrew, the cotton-top tamarin, and the squirrel monkey related to shoulder movements and forelimb loading

Muscle fiber type composition of intrinsic shoulder muscles was examined in tree shrews, cotton-top tamarins, and squirrel monkeys with respect to their shoulder kinematics and forelimb loading during locomotion. Enzyme- and immunohistochemical techniques were applied to differentiate muscle fiber types on serial cross-sections of the shoulder. In the majority of the shoulder muscles, the proportions of fatigue resistant slow-twitch fibers (SO) and fatigable fast-twitch fibers (FG) were inversely related to each other, whereas the percentage of intermediate FOG-fibers varied independently. A segregation of fatigue resistant SO-fibers into deep muscle regions is indicative of differential activation of histochemically distinct muscle regions in which deep regions stabilize the joint against gravitational loading. In all three species, this antigravity function was demonstrated for both the supraspinatus and the cranial subscapularis muscle, which prevent passive joint flexion during the support phase of the limb. The infraspinatus muscle showed a high content of SO-fibers in the primate species but not in the tree shrew, which demonstrates the "new" role of the infraspinatus muscle in joint stabilization related to the higher degree of humeral protraction in primates. In the tree shrew and the cotton-top tamarin, a greater proportion of the body weight is carried on the forelimb, but the squirrel monkey exhibits a weight shift to the hind limbs. The lower amount of forelimb loading is reflected by an overall lower proportion of fatigue resistant muscle fibers in the shoulder muscles of the squirrel monkey. Several muscles such as the deltoid no longer function as joint stabilizers and allow the humerus to move beyond the scapular plane. These differences among species demonstrate the high plasticity of the internal muscle architecture and physiology which is suggested to be the underlying reason for different muscle activity patterns in homologous muscles. Implications for the evolution of new locomotor modes in primates are discussed.     

On a phenomenological model for fatigue effects in skeletal muscles

This work deals with the development and implementation of a new fatigue model for simulating fatigue effects in skeletal muscles. Basic idea of this modelling strategy is an approach that divides the fibres of a muscle into three groups: fibres in the active state, those that are already fatigued and fibres in the resting state. All fibres are able to switch between the different groups by defining adequate rates. In this way a continuous transfer of fibres between those three states has been described. Rooted on an incompressible, hyperelastic constitutive law with transversely isotropic characteristics the fatigue model has been implemented in the framework of the finite element method. Numerical examples are given in order to illustrate the ability of this model. Further, we validate the model by fatigue experiments of the rat soleus muscle. In doing so, it proves that the model is able to predict physiological observations and mechanical test results.     

Folding kinetics of the S100A11 protein dimer studied by time-resolved electrospray mass spectrometry and pulsed hydrogen-deuterium exchange

This study reports the application of electrospray ionization (ESI) mass spectrometry (MS) with on-line rapid mixing for millisecond time-resolved studies of the refolding and assembly of a dimeric protein complex. Acid denaturation of S100A11 disrupts the native homodimeric protein structure. Circular dichroism and HSQC nuclear magnetic resonance measurements reveal that the monomeric subunits unfold to a moderate degree but retain a significant helicity and some tertiary structural elements. Following a rapid change in solution conditions to a slightly basic pH, the native protein reassembles with an effective rate constant of 6 s(-)(1). The ESI charge state distributions measured during the reaction suggest the presence of three kinetic species, namely, a relatively unfolded monomer (M(U)), a more tightly folded monomeric reaction intermediate (M(F)), and dimeric S100A11. These three forms exhibit distinct calcium binding properties, with very low metal loading levels for M(U), up to two calcium ions for M(F), and up to four for the dimer. Surprisingly, on-line pulsed hydrogen-deuterium exchange (HDX) reveals that each of the monomeric forms of the protein comprises two subspecies that can be distinguished on the basis of their isotope exchange levels. As the reaction proceeds, the more extensively labeled species are depleted. The exponential nature of the measured intensity-time profiles implies that the rate-determining step of the overall process is a unimolecular event. The kinetics are consistent with a sequential folding and assembly mechanism involving two increasingly nativelike monomeric intermediates en route to the native S100A11 dimer.     

Structural and functional recovery of electropermeabilized skeletal muscle in-vivo after treatment with surfactant poloxamer 188

A critical requirement for cell survival after trauma is sealing of breaks in the cell membrane [M. Bier, S.M. Hammer, D.J. Canaday, R.C Lee, Kinetics of sealing for transient electropores in isolated mammalian skeletal muscle cells, Bioelectromagnetics 20 (1999) 194-201; R.C. Lee, D.C. Gaylor, D. Bhatt, D.A. Israel, Role of cell membrane rupture in the pathogenesis of electrical trauma, J. Surg. Res. 44 (1988) 709-719; R.C. Lee, J.F. Burke, E.G. Cravalho (Eds.), Electrical Trauma: The Pathophysiology, Manifestations, and Clinical Management, Cambridge University Press, 1992; B.I. Tropea, R.C. Lee, Thermal injury kinetics in electrical trauma, J. Biomech. Engr. 114 (1992) 241-250; F. Despa, D.P. Orgill, J. Newalder, R.C Lee, The relative thermal stability of tissue macromolecules and cellular structure in burn injury, Burns 31 (2005) 568-577; T.A. Block, J.N. Aarsvold, K.L. Matthews II, R.A. Mintzer, L.P. River, M. Capelli-Schellpfeffer, R.L. Wollman, S. Tripathi, C.T. Chen, R.C. Lee, The 1995 Lindberg Award. Nonthermally mediated muscle injury and necrosis in electrical trauma, J. Burn Care and Rehabil. 16 (1995) 581-588; K. Miyake, P.L. McNeil, Mechanical injury and repair of cells, Crit. Care Med. 31 (2003) S496-S501; R.C. Lee, L.P. River, F.S. Pan, R.L. Wollmann, Surfactant-induced sealing of electropermeabilized skeletal muscle membranes in vivo, Proc. Natl. Acad. Sci. 89 (1992) 4524-4528; J.D. Marks, C.Y. Pan, T. Bushell, W. Cromie, R.C. Lee, Amphiphilic, tri-block copolymers provide potent membrane-targeted neuroprotection, FASEB J. 15 (2001) 1107-1109; B. Greenebaum, K. Blossfield, J. Hannig, C.S. Carrillo, M.A. Beckett, R.R. Weichselbaum, R.C. Lee, Poloxamer 188 prevents acute necrosis of adult skeletal muscle cells following high-dose irradiation, Burns 30 (2004) 539-547; G. Serbest, J. Horwitz, K. Barbee, The effect of poloxamer-188 on neuronal cell recovery from mechanical injury, J. Neurotrauma 22 (2005) 119-132]. The triblock copolymer surfactant Poloxamer 188 (P188) is known to increase the cell survival after membrane electroporation [R.C. Lee, L.P. River, F.S. Pan, R.L. Wollmann, Surfactant-induced sealing of electropermeabilized skeletal muscle membranes in vivo, Proc. Natl. Acad. Sci. 89 (1992) 4524-4528; Z. Ababneh, H. Beloeil, C.B. Berde, G. Gambarota, S.E. Maier, R.V. Mulkern, Biexponential parametrization of T2 and diffusion decay curves in a rat muscle edema model: Decay curve components and water compartments, Magn. Reson. Med. 54 (2005) 524-531]. Here, we use a rat hind-limb model of electroporation injury to determine if the intravenous administration of P188 improves the recovery of the muscle function. Rat hind-limbs received a sequence of either 0, 3, 6, 9, or 12 electrical current pulses (2 A, 4 ms duration, 10 s duty cycle). Magnetic resonance imaging (MRI) analysis, muscle water content and compound muscle action potential (CMAP) amplitudes were compared. Electroporation injury manifested edema formation and depression of the CMAP amplitudes. P188 (one bolus of 1 mg/ml of blood) was administrated 30 or 60 min after injury. Animals receiving P188 exhibited reduced tissue edema (p<0.05) and increased CMAP amplitudes (p<0.03). By comparison, treatment with 10 kDa neutral dextran, which produces similar serum osmotic effects as P188, had no effect on post-electroporation recovery. Noteworthy, the present results suggest that a single intravenous dose of P188 is effective to restore the structural integrity of damaged tissues with intact circulation.     

Different modes of activating phosphofructokinase,a key regulatory enzyme of glycolysis,in working vertebrate muscle

Glycolytic flux in white muscle can be increased several-hundredfold by exercise. Phosphofructokinase (PFK; EC 2.7.1.11) is a key regulatory enzyme of glycolysis, but how its activity in muscle is controlled is not fully understood. In order not to neglect integrative aspects of metabolic regulation, we have studied in frogs (Rana temporaria) a physiological form of muscle work (swimming) that can be triggered like a reflex. We analysed swimming to fatigue in well rested frogs, recovery from exercise, and repeated exercise after 2 h of recovery. At various times, gastrocnemius muscles were tested for glycolytic intermediates and effectors of PFK. All metabolites responded similarly to the two periods of exercise, with the notable exception of fructose 2,6-bisphosphate (F2,6P(2)), which we proved to be a most potent activator of frog muscle PFK. The first bout of exercise triggered a more than 10-fold increase in F2,6P(2); PFK activity and the content of F2,6P(2) in muscle were well correlated. F2,6P(2) decreased to pre-exercise levels in fatigued frogs and it virtually disappeared during recovery. Varying by a factor of 70, F2,6P(2) was the most dynamic of all metabolites in muscle. Even more surprisingly, F2,6P(2) did not respond at all to a second bout of exercise. Other activators of PFK, such as Pi, AMP and ADP, are increased as a consequence of increased ATP turnover in contracting muscle cells. This does not apply to F2,6P(2) which is likely to respond to extracellular signals and could be involved in mechanisms by which muscle metabolism is integrated into the metabolism of the whole body. Whether this phenomenon exists in vertebrates other than the frog, and maybe even in humans, and how the content of F2,6P(2)in muscle is controlled are intriguing open questions.     

Skeletal muscle energy metabolism during prolonged, fatiguing exercise.

A depletion of phosphocreatine (PCr), fall in the total adenine nucleotide pool (TAN = ATP + ADP + AMP), and increase in TAN degradation products inosine 5'-monophosphate (IMP) and hypoxanthine are observed at fatigue during prolonged exercise at 70% maximal O(2) uptake in untrained subjects [J. Baldwin, R. J. Snow, M. F. Carey, and M. A. Febbraio. Am. J. Physiol. 277 (Regulatory Integrative Comp. Physiol. 46): R295-R300, 1999]. The present study aimed to examine whether these metabolic changes are also prevalent when exercise is performed below the blood lactate threshold (LT). Six healthy, untrained humans exercised on a cycle ergometer to voluntary exhaustion at an intensity equivalent to 93 +/- 3% of LT ( approximately 65% peak O(2) uptake). Muscle biopsy samples were obtained at rest, at 10 min of exercise, approximately 40 min before fatigue (F-40 =143 +/- 13 min), and at fatigue (F = 186 +/- 31 min). Glycogen concentration progressively declined (P < 0.01) to very low levels at fatigue (28 +/- 6 mmol glucosyl U/kg dry wt). Despite this, PCr content was not different when F-40 was compared with F and was only reduced by 40% when F was compared with rest (52. 8 +/- 3.7 vs. 87.8 +/- 2.0 mmol/kg dry wt; P < 0.01). In addition, TAN concentration was not reduced, IMP did not increase significantly throughout exercise, and hypoxanthine was not detected in any muscle samples. A significant correlation (r = 0.95; P < 0. 05) was observed between exercise time and glycogen use, indicating that glycogen availability is a limiting factor during prolonged exercise below LT. However, because TAN was not reduced, PCr was not depleted, and no correlation was observed between glycogen content and IMP when glycogen stores were compromised, fatigue may be related to processes other than those involved in muscle high-energy phosphagen metabolism.     

Load transfer across the scapula during humeral abduction

Stress analysis in the individual parts of the scapula under normal physiological conditions is necessary to understand the load transfer mechanism, its relation with morphology of bone and to analyse the deviations in stress patterns due to implantation of the glenoid. The purpose of this study was to obtain stress distribution in the scapula during abduction of the arm and to obtain a qualitative estimate of the function of coracoacromial ligament. An accurate three-dimensional (3D) finite element (FE) model of the natural scapula has been developed for this purpose, using computed tomography data and shell-solid modelling approach. The model was experimentally validated. A musculoskeletal shoulder model of forces that calculates all muscle, ligament and joint reaction forces, in six load cases (30-180 degrees) during unloaded humeral abduction was used as applied loading conditions for the 3D FE model. High tensile and compressive stresses (15-60 MPa) were generated in the thick bony ridges of the scapula, like the scapular spine, lateral border, glenoid and acromion. High compressive stresses (45-58 MPa) were evoked in the glenoid and at the connection of glenoid-scapular spine-infraspinous fossa. The stresses in the infraspinous fossa and supraspinous fossa were low (0.05-15 MPa). These results indicated that the transfer of major muscle and joint reaction take place predominantly through the thick bony ridges, whereas the fossa area act more as attachment sites of large muscles. During humeral abduction, coracoacromial ligament was stretched, and presumably will be under tension.     

Posture and hand load alter muscular response to sudden elbow perturbations

Joint stiffness and stability are reliant on coordinated muscle activity which may differ depending on initial posture and loading during sudden perturbations. This study investigated the effects of arm posture and hand load on muscle activity during perturbations of the arm. Fifteen male participants experienced perturbations to the wrist causing elbow extension using a combination of three body postures (standing, supine, sitting) and three hand load conditions (no, solid, and fluid loads), with known and unknown timing. Surface EMG was collected from eight muscles of the right upper extremity. The response to sudden loading was examined using muscle activities pre (baseline) and post (reflex) perturbation. During the baseline period, known perturbation timing resulted in greater muscular activity than for unknown timing, while the opposite was found for the reflex period. During the reflex period with fluid load, biceps brachii and brachioradialis demonstrated increased activity of 2.4% and 4.0% of maximum respectively, from supine to standing. During the reflex period, the fluid load resulted in forearm co-contraction 23% and 47% greater than the solid and no load conditions. Body orientation and hand loading influenced muscular response to elbow perturbations. Muscle co-contraction at the elbow during known timing suggests a contribution to elbow joint stability that may reduce injury risk caused by sudden elbow loading.     

Investigation of the Peculiarities of Two-joint Muscles Using a 3 DOF Model of the Human Upper Limb in the Sagittal Plane: an Optimization Approach

Abstract

The purpose of this paper is an investigation of the peculiarities of biarticular muscles by means of modelling and analytical solution of the indeterminate problem. The basic model includes 10 muscle elements performing flexio/extensio in the shoulder, elbow and wrist. Four of them are biarticular muscles. Two modifications of the model with only monoarticular muscles are developed. The indeterminate problem is solved analytically using the objective criterion σc_iF_i ² where F( is the module of the i-th muscle force and Cj is a weight coefficient. The predicted muscle forces, joint reactions and moments are compared in-between the basic model and its two modifications for different joint angles, external loading and weight coefficients. The main conclusions are: it is impossible to formulate strict advantages of the biarticular muscles under quasistatical conditions, their peculiarities depend on limb position, external loading and neural control; in general, monoarticular muscles are more powerful than biarticular ones; the biarticular muscles fine tune muscle coordination, their control is more precise and graceful; the presence of biarticular muscles leads to an increase of the joint reactions and moments, thus stabilizing the limb.     

Structural and functional recovery of electropermeabilized skeletal muscle in-vivo after treatment with surfactant poloxamer 188

A critical requirement for cell survival after trauma is sealing of breaks in the cell membrane [M. Bier, S.M. Hammer, D.J. Canaday, R.C Lee, Kinetics of sealing for transient electropores in isolated mammalian skeletal muscle cells, Bioelectromagnetics 20 (1999) 194-201; R.C. Lee, D.C. Gaylor, D. Bhatt, D.A. Israel, Role of cell membrane rupture in the pathogenesis of electrical trauma, J. Surg. Res. 44 (1988) 709-719; R.C. Lee, J.F. Burke, E.G. Cravalho (Eds.), Electrical Trauma: The Pathophysiology, Manifestations, and Clinical Management, Cambridge University Press, 1992; B.I. Tropea, R.C. Lee, Thermal injury kinetics in electrical trauma, J. Biomech. Engr. 114 (1992) 241-250; F. Despa, D.P. Orgill, J. Newalder, R.C Lee, The relative thermal stability of tissue macromolecules and cellular structure in burn injury, Burns 31 (2005) 568-577; T.A. Block, J.N. Aarsvold, K.L. Matthews II, R.A. Mintzer, L.P. River, M. Capelli-Schellpfeffer, R.L. Wollman, S. Tripathi, C.T. Chen, R.C. Lee, The 1995 Lindberg Award. Nonthermally mediated muscle injury and necrosis in electrical trauma, J. Burn Care and Rehabil. 16 (1995) 581-588; K. Miyake, P.L. McNeil, Mechanical injury and repair of cells, Crit. Care Med. 31 (2003) S496-S501; R.C. Lee, L.P. River, F.S. Pan, R.L. Wollmann, Surfactant-induced sealing of electropermeabilized skeletal muscle membranes in vivo, Proc. Natl. Acad. Sci. 89 (1992) 4524-4528; J.D. Marks, C.Y. Pan, T. Bushell, W. Cromie, R.C. Lee, Amphiphilic, tri-block copolymers provide potent membrane-targeted neuroprotection, FASEB J. 15 (2001) 1107-1109; B. Greenebaum, K. Blossfield, J. Hannig, C.S. Carrillo, M.A. Beckett, R.R. Weichselbaum, R.C. Lee, Poloxamer 188 prevents acute necrosis of adult skeletal muscle cells following high-dose irradiation, Burns 30 (2004) 539-547; G. Serbest, J. Horwitz, K. Barbee, The effect of poloxamer-188 on neuronal cell recovery from mechanical injury, J. Neurotrauma 22 (2005) 119-132]. The triblock copolymer surfactant Poloxamer 188 (P188) is known to increase the cell survival after membrane electroporation [R.C. Lee, L.P. River, F.S. Pan, R.L. Wollmann, Surfactant-induced sealing of electropermeabilized skeletal muscle membranes in vivo, Proc. Natl. Acad. Sci. 89 (1992) 4524-4528; Z. Ababneh, H. Beloeil, C.B. Berde, G. Gambarota, S.E. Maier, R.V. Mulkern, Biexponential parametrization of T2 and diffusion decay curves in a rat muscle edema model: Decay curve components and water compartments, Magn. Reson. Med. 54 (2005) 524-531]. Here, we use a rat hind-limb model of electroporation injury to determine if the intravenous administration of P188 improves the recovery of the muscle function. Rat hind-limbs received a sequence of either 0, 3, 6, 9, or 12 electrical current pulses (2 A, 4 ms duration, 10 s duty cycle). Magnetic resonance imaging (MRI) analysis, muscle water content and compound muscle action potential (CMAP) amplitudes were compared. Electroporation injury manifested edema formation and depression of the CMAP amplitudes. P188 (one bolus of 1 mg/ml of blood) was administrated 30 or 60 min after injury. Animals receiving P188 exhibited reduced tissue edema (p < 0.05) and increased CMAP amplitudes (p < 0.03). By comparison, treatment with 10 kDa neutral dextran, which produces similar serum osmotic effects as P188, had no effect on post-electroporation recovery. Noteworthy, the present results suggest that a single intravenous dose of P188 is effective to restore the structural integrity of damaged tissues with intact circulation.     

A task-specific validation of homogeneous non-linear optimisation approaches

In biomechanics, musculoskeletal models are typically redundant. This situation is referred to as the distribution problem. Often, static, non-linear optimisation methods of the form “min: φ(f) subject to mechanical and muscular constraints” have been used to extract a unique set of muscle forces. Here, we present a method for validating this class of non-linear optimisation approaches where the homogeneous cost function, φ(f), is used to solve the distribution problem. We show that the predicted muscle forces for different loading conditions are scaled versions of each other if the joint loading conditions are just scaled versions. Therefore, we can calculate the theoretical muscle forces for different experimental conditions based on the measured muscle forces and joint loadings taken from one experimental condition and assuming that all input into the optimisation (e.g., moment arms, muscle attachment sites, size, fibre type distribution) and the optimisation approach are perfectly correct. Thus predictions of muscle force for other experimental conditions are accurate if the optimisation approach is appropriate, independent of the musculoskeletal geometry and other input required for the optimisation procedure. By comparing the muscle forces predicted in this way to the actual muscle forces obtained experimentally, we conclude that convex homogeneous non-linear optimisation approaches cannot predict individual muscle forces properly, as force-sharing among synergistic muscles obtained experimentally are not just scaled versions of joint loading, not even in a first approximation.     

Static optimization underestimates antagonist muscle activity at the glenohumeral joint: A musculoskeletal modeling study

Static optimization is commonly employed in musculoskeletal modeling to estimate muscle and joint loading; however, the ability of this approach to predict antagonist muscle activity at the shoulder is poorly understood. Antagonist muscles, which contribute negatively to a net joint moment, are known to be important for maintaining glenohumeral joint stability. This study aimed to compare muscle and joint force predictions from a subject-specific neuromusculoskeletal model of the shoulder driven entirely by measured muscle electromyography (EMG) data with those from a musculoskeletal model employing static optimization. Four healthy adults performed six sub-maximal upper-limb contractions including shoulder abduction, adduction, flexion, extension, internal rotation and external rotation. EMG data were simultaneously measured from 16 shoulder muscles using surface and intramuscular electrodes, and joint motion evaluated using video motion analysis. Muscle and joint forces were calculated using both a calibrated EMG-driven neuromusculoskeletal modeling framework, and musculoskeletal model simulations that employed static optimization. The EMG-driven model predicted antagonistic muscle function for pectoralis major, latissimus dorsi and teres major during abduction and flexion; supraspinatus during adduction; middle deltoid during extension; and subscapularis, pectoralis major and latissimus dorsi during external rotation. In contrast, static optimization neural solutions showed little or no recruitment of these muscles, and preferentially activated agonistic prime movers with large moment arms. As a consequence, glenohumeral joint force calculations varied substantially between models. The findings suggest that static optimization may under-estimate the activity of muscle antagonists, and therefore, their contribution to glenohumeral joint stability.     

Surface mechanomyogram reflects the changes in the mechanical properties of muscle at fatigue

The contractile properties of muscle are usually investigated by analysing the force signal recorded during electrically elicited contractions. The electrically stimulated muscle shows surface oscillations that can be detected by an accelerometer; the acceleration signal is termed the surface mechanomyogram (MMG). In the study described here we compared, in the human tibialis anterior muscle, changes in the MMG and force signal characteristics before, and immediately after fatigue, as well as during 6 min of recovery, when changes in the contractile properties of muscle occur. Fatigue was induced by sustained electrical stimulation. The final aim was to evaluate the reliability of the MMG as a tool to follow the changes in the mechanical properties of muscle caused by fatigue. Because of fatigue, the parameters of the force peak, the peak rate of force production and the peak of the acceleration of force production (d2F/dt2) decreased, while the contraction time and the half-relaxation time (1/2-RT) increased. The MMG peak-to-peak (p-p) also decreased. The attenuation rate of the force oscillation amplitude and MMG p-p at increasing stimulation frequency was greater after fatigue. With the exception of 1/2-RT, all of the force and MMG parameters were restored within 2 min of recovery. A high correlation was found between MMG and d2F/dt2 in un-fatigued muscle and during recovery. In conclusion, the MMG reflects specific aspects of muscle mechanics and can be used to follow the changes in the contractile properties of muscle caused by localised muscle fatigue.     

Induction of endogenous uncoupling protein 3 suppresses mitochondrial oxidant emission during fatty acid-supported respiration

Uncoupling protein 3 (UCP3) expression increases dramatically in skeletal muscle under metabolic states associated with elevated lipid metabolism, yet the function of UCP3 in a physiological context remains controversial. Here, in situ mitochondrial H(2)O(2) emission and respiration were measured in permeabilized fiber bundles prepared from both rat and mouse (wild-type) gastrocnemius muscle after a single bout of exercise plus 18 h of recovery (Ex/R) that induced a approximately 2-4-fold increase in UCP3 protein. Elevated uncoupling activity (i.e. GDP inhibitable) was evident in Ex/R fibers only upon the addition of palmitate (known activator of UCP3) or under substrate conditions eliciting substantial rates of H(2)O(2) production (i.e. respiration supported by succinate or palmitoyl-L-carnitine/malate but not pyruvate/malate), indicative of UCP3 activation by endogenous reactive oxygen species. In mice completely lacking UCP3 (ucp3(-/-)), Ex/R failed to induce uncoupling activity. Surprisingly, when UCP3 activity was inhibited by GDP (rats) or in the absence of UCP3 (ucp3(-/-)), H(2)O(2) emission was significantly (p < 0.05) higher in Ex/R versus non-exercised control fibers. Collectively, these findings demonstrate that the oxidant emitting potential of mitochondria is increased in skeletal muscle during recovery from exercise, possibly as a consequence of prolonged reliance on lipid metabolism and/or altered mitochondrial biochemistry/morphology and that induction of UCP3 in vivo mediates an increase in uncoupling activity that restores mitochondrial H(2)O(2) emission to non-exercised, control levels.     

Strain-dependent modulation of phosphate transients in rabbit skeletal muscle fibers.

When inorganic phosphate (Pi) is photogenerated from caged Pi during isometric contractions of glycerinated rabbit psoas muscle fibers, the released Pi binds to cross-bridges and reverses the working stroke of cross-bridges. The consequent force decline, the Pi-transient, is exponential and probes the kinetics of the power-stroke and Pi release. During muscle shortening, the fraction of attached cross-bridges and the average strain on them decreases (Ford, L. E., A.F. Huxley, and R.M. Simmons, 1977. Tension responses to sudden length change in stimulated frog muscle fibers near slack length. J. Physiol. (Lond.). 269:441-515; Ford, L. E., A. F. Huxley, and R.M. Simmons, 1985. Tension transients during steady state shortening of frog muscle fibers. J. Physiol. (Lond.). 361:131-150. To learn to what extent the Pi transient is strain dependent, muscle fibers were activated and shortened or lengthened at a fixed velocity during the photogeneration of Pi. The Pi transients observed during changes in muscle length showed three primary characteristics: 1) during shortening the Pi transient rate, Kpi, increased and its amplitude decreased with shortening velocity; Kpi increased linearly with velocity to > 110 s-1 at 0.3 muscle lengths per second (ML/s). 2) At a specific shortening velocity, increases in [Pi] produce increases in Kpi that are nonlinear with [Pi] and approach an asymptote. 3) During forced lengthening Kpi and the amplitude of the Pi transient are little different from the isometric contractions. These data can be approximated by a strain-dependent three-state cross-bridge model. The results show that the power stroke's rate is strain-dependent, and are consistent with biochemical studies indicating that the rate-limiting step at low strains is a transition from a weakly to a strongly bound cross-bridge state.