A visco-hyperelastic model for skeletal muscle tissue under high strain rates

In this paper, a visco-hyperelastic skeletal muscle model is developed. The constitutive relation is based on the definition of a Helmholtz free energy function. It is assumed that the Helmholtz energy can be decomposed into volumetric and isochoric parts; furthermore, the isochoric energy can be decoupled into hyperelastic and viscous parts. The model developed involves 14 material parameters and its performance is evaluated by comparing the finite element simulation results with the published experimental studies on the New Zealand white rabbit tibialis anterior muscle. Results show that this model is able to describe the visco-hyperelastic behaviour of both passive and active skeletal muscle tissues under high strain rates (10/s and 25/s).     

A computational model of oxygen transport in skeletal muscle for sprouting and splitting modes of angiogenesis

Oxygen transport from capillary networks in muscle at a high oxygen consumption rate was simulated using a computational model to assess the relative efficacies of sprouting and splitting modes of angiogenesis. Efficacy was characterized by the volumetric fraction of hypoxic tissue and overall heterogeneity of oxygen distribution at steady state. Oxygen transport was simulated for a three-dimensional vascular network using parameters for rat extensor digitorum longus (EDL) muscle when oxygen consumption by tissue reached 6, 12, and 18 times basal consumption. First, a control network was generated by using straight non-anastomosed capillaries to establish baseline capillarity. Two networks were then constructed simulating either abluminal lateral sprouting or intraluminal splitting angiogenesis such that capillary surface area was equal in both networks. The sprouting network was constructed by placing anastomosed capillaries between straight capillaries of the control network with a higher probability of placement near hypoxic tissue. The splitting network was constructed by splitting capillaries from the control network into two branches at randomly chosen branching points. Under conditions of moderate oxygen consumption (6 times basal), only minor differences in oxygen delivery resulted between the sprouting and splitting networks. At higher consumption levels (12 and 18 times basal), the splitting network had the lowest volume of hypoxic tissue of the three networks. However, when total blood flow in all three networks was made equal, the sprouting network had the lowest volume of hypoxic tissue. This study also shows that under the steady-state conditions the effect of myoglobin (Mb) on oxygen transport was small.     

Mathematical modeling of oxygen transport in skeletal muscle

Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.     

Modulation of glucose transport in skeletal muscle by reactive oxygen species.

Glucose transport is an essential physiological process that is characteristic of all eukaryotic cells, including skeletal muscle. In skeletal muscle, glucose transport is mediated by the GLUT-4 protein under conditions of increased carbohydrate utilization. The three major physiological stimuli of glucose transport in muscle are insulin, exercise/contraction, and hypoxia. Here, the role of reactive oxygen species (ROS) in modulating glucose transport in skeletal muscle is reviewed. Convincing evidence for ROS involvement in insulin- and hypoxia-mediated transport in muscle is lacking. Recent experiments, based on pharmacological and genetic approaches, support a role for ROS in contraction-mediated glucose transport. During contraction, endogenously produced ROS appear to mediate their effects on glucose transport via AMP-activated protein kinase.     

Estimating oxygen consumption rates of arteriolar walls under physiological conditions in rat skeletal muscle

To examine the effects of vascular tone reduction on O2 consumption of the vascular wall, we determined the O2 consumption rates of arteriolar walls under normal conditions and during vasodilation induced by topical application of papaverine. A phosphorescence quenching technique was used to quantify intra- and perivascular PO2 in rat cremaster arterioles with different branching orders. Then, the measured radial PO2 gradients and a theoretical model were used to estimate the O2 consumption rates of the arteriolar walls. The vascular O2 consumption rates of functional arterioles were >100 times greater than those observed in in vitro experiments. The vascular O2 consumption rate was highest in first-order (1A) arterioles, which are located upstream, and sequentially decreased downstream in 2A and 3A arterioles under normal conditions. During papaverine-induced vasodilation, on the other hand, the O2 consumption rates of the vascular walls decreased to similar levels, suggesting that the high O2 consumption rates of 1A arterioles under normal conditions depend in part on the workload of the vascular smooth muscle. These results strongly support the hypothesis that arteriolar walls consume a significant amount of O2 compared with the surrounding tissue. Furthermore, the reduction of vascular tone of arteriolar walls may facilitate an efficient supply of O2 to the surrounding tissue.     

Effect of temperature on the myoglobin-facilitated transport of oxygen in skeletal muscle.

An analysis of thermal effects on the facilitative transport of oxygen in skeletal muscle fibers is presented. Steady-state mass and energy transport balances are written and solved analytically or numerically using a finite-difference procedure. It is shown that no significant spatial thermal gradients exist due to internal reactions or bulk conduction effects across a muscle fiber. At typical muscle conditions, it is predicted that increased global temperature reduces the fraction of oxygenated myoglobin, increases local oxygen concentrations, and increases the percentage of oxygen flux attributed to oxy-myoglobin. The maximum supportable oxygen consumption rate, mO2max, is defined as the highest consumption rate sustainable without developing anoxic regions at the center of the fiber. By considering only temperature sensitive effects within fibers, mO2max is found to increase slightly with temperature at low temperatures. This increase is due to thermal effects on the diffusion coefficients as opposed to effects associated with the kinetics of the myoglobin-oxygen reaction. If the simulations include the temperature effect associated with oxygen solubility in blood plasma, mO2max decreases with temperature. A sensitivity analysis was performed by varying the values of relevant parameters. The maximum consumption rate was least affected by parameters associated with the kinetic and equilibrium constants and most affected by the diffusion coefficients and the concentration of myoglobin.     

A model study of intracellular oxygen gradients in a myoglobin-containing skeletal muscle fiber.

A theoretical two-dimensional model is used to investigate oxygen gradients in a red skeletal muscle fiber. The model describes the steady state, free and myoglobin-facilitated diffusion of oxygen into a respiring cylindrical muscle fiber cross section. The oxygen tension at the sarcolemma is assumed to vary along the sarcolemma as an approximation to the discrete capillary oxygen supply around the fiber. Maximal oxygen gradients are studied by considering parameters relevant to a maximally-respiring red muscle fiber. The model predicts that angular variations in the oxygen tension imposed at the sarcolemma due to the discrete capillary sources do not penetrate deeply into the fiber over a range of physiological values for myoglobin concentration, diffusion coefficients, number of surrounding capillaries, and oxygen tension level at the sarcolemma. Also, the oxygen tension in the core of the fiber is determined by the average oxygen tension at the sarcolemma. The drop in oxygen tension from fiber periphery to core, however, does depend significantly on the myoglobin concentration, the oxygen tension level at the sarcolemma, and the oxygen and myoglobin diffusivities. This dependence is summarized by calculating the minimum average sarcolemmal oxygen tension for maximal respiration without the development of an intracellular anoxic region. For a myoglobin-rich muscle fiber (0.5 mM myoglobin), the model predicts that maximal oxygen consumption can proceed with a relatively flat (less than 5 mm Hg) oxygen tension drop from fiber periphery to core over a large range for diffusion coefficients.     

Regulation of glucose transport in skeletal muscle.

The entry of glucose into muscle cells is achieved primarily via a carrier-mediated system consisting of protein transport molecules. GLUT-1 transporter isoform is normally found in the sarcolemmal (SL) membrane and is thought to be involved in glucose transport under basal conditions. With insulin stimulation, glucose transport is accelerated by translocating GLUT-4 transporters from an intracellular pool out to the T-tubule and SL membranes. Activation of transporters to increase the turnover number may also be involved, but the evidence is far from conclusive. When insulin binds to its receptor, it autophosphorylates tyrosine and serine residues on the beta-subunit of the receptor. The tyrosine residues are thought to activate tyrosine kinases, which in turn phosphorylate/activate as yet unknown second messengers. Insulin receptor antibodies, however, have been reported to increase glucose transport without increasing kinase activity. Insulin resistance in skeletal muscle is a major characteristic of obesity and diabetes mellitus, especially NIDDM. A decrease in the number of insulin receptors and the ability of insulin to activate receptor tyrosine kinase has been documented in muscle from NIDDM patients. Most studies report no change in the intracellular pool of GLUT-4 transporters available for translocation to the SL. Both the quality and quantity of food consumed can regulate insulin sensitivity. A high-fat, refined sugar diet, similar to the typical U.S. diet, causes insulin resistance when compared with a low-fat, complex-carbohydrate diet. On the other hand, exercise increases insulin sensitivity. After an acute bout of exercise, glucose transport in muscle increases to the same level as with maximum insulin stimulation. Although the number of GLUT-4 transporters in the sarcolemma increases with exercise, neither insulin or its receptor is involved. After an initial acute phase, which may involve calcium as the activator, a secondary phase of increased insulin sensitivity can last for up to a day after exercise. The mechanism responsible for the increased insulin sensitivity with exercise is unknown. Regular exercise training also increases insulin sensitivity, which can be documented several days after the final bout of exercise, and again the mechanism is unknown. An increase in the muscle content of GLUT-4 transporters with training has recently been reported. Even though significant progress has been made in the past few years in understanding glucose transport in skeletal muscle, the mechanisms involved in regulating transport are far from being understood.     

Thermal compensation of peripheral oxygen transport in skeletal muscle of seasonally acclimatized trout

Seasonal changes in ultrastructure of locomotory muscle were quantified after acclimatization to natural temperature and photoperiod. Only modest changes were seen in the volume density (V(v)) of mitochondria in slow fibers ranging from 0.21 +/- 0.01 (summer) to 0.24 +/- 0.01 (winter), despite an increase in fiber size from 945 +/- 19 to 1,594 +/- 46 microm(2), respectively, resulting in a significantly greater total mitochondrial volume at low temperatures. In contrast, intracellular lipid stores showed a marked change with season, from a maximum V(v) of lipid droplets of 0.16 +/- 0.01 in winter, progressively declining through spring and summer to a minimum of 0.07 +/- 0.01 in autumn. For both organelles, the surface density reflected changes in V(v), indicating little modification of structure. Seasonal effects may dominate those of environmental temperature on mitochondrial separation, which in winter and spring fish at 4(o)C averaged 0.64 +/- 0.06 and 1.20 +/- 0.07 microm, respectively. The extracellular transport of oxygen also varies with season, the peak capillary density in autumn (2,851 +/- 88 mm(-2)) resulting in a minimum tissue supply (domain) area of 529 +/- 9 microm(2) per capillary. As a consequence, the predicted intracellular PO(2) ( approximately 2.5 kPa) is similar throughout the year.     

A mathematical model of oxygen transport in intact muscle with imposed surface oscillations

A one-dimensional (1D) reaction-diffusion equation is presented to model oxygen delivery by the microcirculation and oxygen diffusion and consumption in intact muscle. This model is motivated by in vivo experiments in which oscillatory boundary conditions are used to study the mechanisms of local blood flow regulation in response to changes in the tissue oxygen environment. An exact periodic solution is presented for the 1D 'in vivo' model and shown to agree with experimental data for the case where the blood flow regulation system is not activated. Approximate low- and high-frequency solutions are presented, and the latter is shown to agree with the pure diffusion solution in the absence of sources or sinks. For the low frequencies considered experimentally, the 1D in vivo model shows that as depth increases: (i) the mean of tissue O(2) oscillations changes exponentially, (ii) the amplitude of oscillations decreases very rapidly, and (iii) the phase of oscillations remains nearly the same as that of the imposed surface oscillations. The 1D in vivo model also shows that the dependence on depth of the mean, amplitude, and phase of tissue O(2) oscillations is nearly the same for all stimulation periods >30s, implying that experimentally varying the forcing period in this range will not change the spatial distribution of the O(2) stimulation.     

Sex differences in human skeletal muscle fatigue are eliminated under ischemic conditions.

Several studies have suggested that women may be more resistant to muscle fatigue than men (Fulco CS, Rock PB, Muza SA, Lammi E, Cymerman A, Butterfield G, Moore, LG, Braun B, and Lewis SF. Acta Physiol Scand 167: 233-239, 1999) possibly because of differences in muscle oxidative metabolism. We evaluated muscle fatigue produced by intermittent, maximal volitional isometric contractions of the dorsiflexor muscles of healthy young (21-34 yr) men (n = 8) and women (n = 8) under two conditions: free-flow (FF) circulation and ischemia. Measures of voluntary and stimulated (10- and 50-Hz) force, central activation ratio (CAR), and compound muscle action potential (CMAP) were collected in each session. The ischemic protocol induced greater fatigue than the FF protocol, in both sexes, and was associated with greater reductions in CAR, CMAP, stimulated force, and the ratio of 10- to 50-Hz force compared with the FF condition. Women fatigued less than men in FF but not during ischemia, and this difference was roughly paralleled by a difference in CAR. No sex effects on the CMAP, tetanic force, and measures of excitation-contraction coupling function were found in the FF condition, suggesting that the primary mechanism behind the difference in fatigue was a relatively greater impairment of central activation in men. The observation that ischemia eliminated the sex differences in fatigue is consistent with a number of studies (Kent-Braun JA, Ng AV, Doyle JW, and Towse TF. J Appl Physiol 93: 1813-1823, 2002) relating fatigue to muscle metabolism and might be the result of sex-based differences in metabolic pathway utilization during muscle contraction.     

Osmolality influences bistability of membrane potential under hypokalemic conditions in mouse skeletal muscle: an experimental and theoretical study

The membrane potential in mouse skeletal muscle depends on both extracellular osmolality and potassium concentration. These dependencies have been related to two membrane transporters, Na+/K+/2Cl- co-transporter and the inward potassium rectifier channel. To investigate the relation of the Na+/K+/2Cl- co-transporter and the inward potassium rectifier channel in a qualitative way, a combined electrophysiological and modelling approach was used. The experimental results show that the bistability of the membrane potential, which is related to the conductive state of the inward potassium rectifier channel, is shifted to higher extracellular potassium values when medium osmolality is increased. These results are confirmed by the computer simulation calculations for increased co-transporter flux. The combined results indicate that the co-transporter is capable of modulating the conductive state of the inward potassium rectifier channel.     

A model for the active site of skeletal muscle myosin.

A model for a main element of the active site of skeletal muscle myosin is presented that relates directly to the 92 amino acid fragment (p₁₀) of myosin recently described by Elzinga &; Collins (1977). In this model, the substrate, an eight-membered cyclic complex of MgATP, fits tightly into a 16 amino acid segment of p₁₀ and interacts with seven of its amino acids. A main feature of the model is the important role played by the one molecule of N^τ-methylhistidine² that is present in each myosin heavy chain. At the site, it is postulated that this rare amino acid functions as a donor ligand to Mg²⁺. Once N^τ-methylhistidine is put in place next to the metal, the other amino acids that appear to form a pocket come easily into position around the MgATP. These amino acids with their postulated functions are: tyrosine 72, which through a Mg-bound water, or perhaps directly, is attached to the Mg; histidine 76, which donates a proton to the P_γ of ATP; lysine 78, which binds electrostatically to P_β of ATP; phenylalanines 80 and 81, which flank the purine ring of ATP; and aspartate 66, which forms a hydrogen bond to the 6-amino group of adenine. The Mg-coordination role ascribed to N^τ-methylhistidine 69 in skeletal muscle myosin could be taken by histidine 69 in cardiac myosin and in other muscle myosins that do not contain the methylated amino acid.The choice of p₁₀ to contain a main element of the active site is based on: (a) the presence in p₁₀ of the essential sulfhydryl groups, SH1 and SH2, whose modification affects the ATPase activity of myosin; (b) the presence in ρ₁₀ of N^τ-methylhistidine, an unusual amino acid whose methylation in skeletal muscle we take as an indicator for a special function at the active site; (c) the position of p₁₀ in the primary structure near the junction between subfragment 1 and subfragment 2 (the hinge region) where, we postulate, enzymatic events at the active site are coupled to movements of the hinge that occur during contraction; (d) indications that the DTNB light chain, probably involved in regulation, is also near the hinge; (e) the effects of MgATP at the active site on the chemical reactivity of three SH groups (SH1, SH2 and SH3) located near the hinge; and (f) the effect of hinge cleavage on the oxygen exchange reaction catalyzed at the active site. The correlation of all these observations forms the basis for our placement of part of the active site on p₁₀ near the subfragment 1-subfragment 2 hinge.     

A quantitative model of actin-myosin interaction in skeletal muscle.

Biochemical schemes for the actomyosin ATPase cycle as well as the cooperative regulation of ATPase activity are incorporated into a model of the contractile process of intact muscle. This model is shown to describe accurately the tension developed by skinned muscle fibers in the absence of Ca. This work adds to the evidence that the extrapolation of results from purified protein systems to intact muscle may be valid. Extensions to the case of Ca-activated tensions are discussed.     

Effect of sepsis on skeletal muscle oxygen consumption and tissue oxygenation: interpreting capillary oxygen transport data using a mathematical model

Inherent in the inflammatory response to sepsis is abnormal microvascular perfusion. Maldistribution of capillary red blood cell (RBC) flow in rat skeletal muscle has been characterized by increased 1) stopped-flow capillaries, 2) capillary oxygen extraction, and 3) ratio of fast-flow to normal-flow capillaries. On the basis of experimental data for functional capillary density (FCD), RBC velocity, and hemoglobin O2 saturation during sepsis, a mathematical model was used to calculate tissue O2 consumption (Vo2), tissue Po2 (Pt) profiles, and O2 delivery by fast-flow capillaries, which could not be measured experimentally. The model describes coupled capillary and tissue O2 transport using realistic blood and tissue biophysics and three-dimensional arrays of heterogeneously spaced capillaries and was solved numerically using a previously validated scheme. While total blood flow was maintained, capillary flow distribution was varied from 60/30/10% (normal/fast/stopped) in control to 33/33/33% (normal/fast/stopped) in average sepsis (AS) and 25/25/50% (normal/fast/stopped) in extreme sepsis (ES). Simulations found approximately two- and fourfold increases in tissue Vo2 in AS and ES, respectively. Average (minimum) Pt decreased from 43 (40) mmHg in control to 34 (27) and 26 (15) mmHg in AS and ES, respectively, and clustering fast-flow capillaries (increased flow heterogeneity) reduced minimum Pt to 14.5 mmHg. Thus, although fast capillaries prevented tissue dysoxia, they did not prevent increased hypoxia as the degree of microvascular injury increased. The model predicts that decreased FCD, increased fast flow, and increased Vo2 in sepsis expose skeletal muscle to significant regions of hypoxia, which could affect local cellular and organ function.     

Alterations in phospholipid content of chick skeletal muscle under stress conditions

Alterations in the phospholipid levels of three gastrocnemii (G. externus, G. medius and G. internus) of chick have been studied during 56 days postembryonic growth of the three muscles. The effects of denervation and work-overload stress on their phospholipid content during the same period have been discussed in the light of denervation-induced membrane breakdown and exercise-induced fibre hypertrophy.     

The pentose phosphate pathway in skeletal muscle under patho-physiological conditions. A combined histochemical and biochemical study

Over the last 30 years, research into the neuromuscular apparatus, has expanded greatly. Multidisciplinary investigations have rapidly advanced our understanding both of diseases and of the basic neuromuscular mechanisms. The mode of pathological reaction of the neuromuscular apparatus is now quite well understood. The most notable aspect of the reaction of the injured neuromuscular apparatus is the remarkably stereotyped character of the resulting pathological changes as demonstrated by a wide variety of harmful causes, producing surprisingly similar effects. The findings of our combined histochemical and biochemical investigations presented in this monograph, are in complete harmony with the stereotyped character of the pathological changes. For example, it is particularly striking that many affected muscle fibres of patients with muscular dystrophies, congenital myopathies, inflammatory myopathies, metabolic myopathies, endocrine myopathies, or with diseases of the lower motor neuron, display an enhanced activity of both oxidative enzymes of the pentose phosphate pathway. Likewise, we found that experimental animals with disordered skeletal muscles, provoked by different types of agents or treatments, reveal the same marked rise in activity of GPDH and PGDH in the muscle fibres, with a positive correlation between the activity of both enzymes. Other findings of our investigations point to a positive correlation between the activity of GPDH and PGDH on the one hand and that of the non-oxidative enzymes of the pentose phosphate pathway, the enzymes TA, TK, RPI and RPE on the other hand. The rise in activity of PGDH and, in particular, of GPDH is regulated by two different mechanisms. The first represents a rapid control mechanism based on the stimulation of both oxidative enzymes of the pentose phosphate pathway by NADP+ and on their inhibition by NADPH. The other mechanism represents a long-term effect directed at the synthesis of the enzymes. It is this type of mechanism which is responsible for the rise in activity of GPDH and PGDH we observed. The findings obtained with the applied enzyme histochemical techniques clearly demonstrated that the rise in activity of both enzymes is not homogeneously distributed in the disordered skeletal muscles of man and experimental animals. For that reason, in order to obtain reliable quantitative information about enzyme activities in the muscle fibres themselves, the application of biochemical assays on a micro-scale was indispensable. The biochemical assay of enzyme activities was performed on histologically and histochemically selected dissected muscle specimens.(ABSTRACT TRUNCATED AT 400 WORDS)     

Molecular adaptations in human skeletal muscle to endurance training under simulated hypoxic conditions.

This study was performed to explore changes in gene expression as a consequence of exercise training at two levels of intensity under normoxic and normobaric hypoxic conditions (corresponding to an altitude of 3,850 m). Four groups of human subjects trained five times a week for a total of 6 wk on a bicycle ergometer. Muscle biopsies were taken, and performance tests were carried out before and after the training period. Similar increases in maximal O(2) uptake (8.3-13.1%) and maximal power output (11.4-20.8%) were found in all groups. RT-PCR revealed elevated mRNA concentrations of the alpha-subunit of hypoxia-inducible factor 1 (HIF-1) after both high- (+82.4%) and low (+78.4%)-intensity training under hypoxic conditions. The mRNA of HIF-1alpha(736), a splice variant of HIF-1alpha newly detected in human skeletal muscle, was shown to be changed in a similar pattern as HIF-1alpha. Increased mRNA contents of myoglobin (+72.2%) and vascular endothelial growth factor (+52.4%) were evoked only after high-intensity training in hypoxia. Augmented mRNA levels of oxidative enzymes, phosphofructokinase, and heat shock protein 70 were found after high-intensity training under both hypoxic and normoxic conditions. Our findings suggest that HIF-1 is specifically involved in the regulation of muscle adaptations after hypoxia training. Fine-tuning of the training response is recognized at the molecular level, and with less sensitivity also at the structural level, but not at global functional responses like maximal O(2) uptake or maximal power output.