Nonlinear muscles, passive viscoelasticity and body taper conspire to create neuromechanical phase lags in anguilliform swimmers

Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.     

A method to characterize the passive elasticity in contracting muscle bundles

There is concern among researchers whether the passive muscle properties, characterized by purely passive material testing procedures, are an appropriate representation of the actual passive component of the muscle. This aspect is of particular importance in the biomechanical analysis of heart muscle response where it is generally agreed that the so-called parallel elasticity cannot be ignored as is done justifiably in the analysis of skeletal muscle response. In the present article, a method of quantifying the passive elasticity in contracting muscle bundles is presented. The method consists of imposing isometric transients (such as the quick-stretch or quick release) on a muscle bundle during the contraction phase and observing the differences in decayed force levels between a normal twitch and that of a perturbed twitch. The proposed method provides a means of obtaining useful passive properties from contracting muscle bundles and circumvents the difficulty of having to characterize muscle properties from separate experiments on quiescent muscle bundles.     

Functional morphology of the M. gastrocnemius medialis of the rat during growth

Length-force relations, both active and passive, and twitch contraction characteristics were quantified for left medial gastrocnemius muscles of four young, four adult, and four old male Wistar rats. Muscle and bundle optimum length and muscle weight were also determined and subsequently used for calculation of a number of morphological characteristics of the muscles. Fiber optimum length was derived from muscle bundle optimum length. Generally, physiological characteristics remained constant during growth. There was no change either in active tension at muscle optimum length or in active working range relative to fiber optimum length, relative passive fiber stiffness, active force relative to passive force at optimum length, twitch contraction time and twitch half relaxation time at optimum length. A number of morphological changes, however, did take place in the medial gastrocnemius muscle during growth. Fiber optimum length increased but only by about 2 mm from youth to old age, whereas muscle optimum length increased by approximately 14 mm, presumably owing to extensive hypertrophy of the muscle fibers during growth. The priority for force of the medial gastrocnemius muscle (defined as the quotient of physiological cross-sectional area of a muscle and the cubed root of its volume, a measure independent of architecture and dimensions of muscles) increased during growth. This increase indicates that during growth the muscle shifts relatively more towards force generation than towards excursion generation. These findings are discussed in view of existing scaling theories.     

The shear modulus of lower-leg muscles correlates to intramuscular pressure

Shear wave elastography (SWE) is emerging as an innovative tool to evaluate muscle properties and function. It has been shown to correlate with both passive and active muscle forces, and is sensitive to physiological processes and pathological conditions. Similarly, intramuscular pressure (IMP) is an important parameter that changes with passive and active muscle contraction, body position, exercise, blood pressure, and several pathologies. Therefore, the objective of this study was to quantify the dependency of shear modulus within the lower-leg muscles on IMP in healthy individuals. Nineteen healthy individuals (age: Mean age ± SD, 23.84 ± 6.64 years) were recruited. Shear modulus was measured using ultrasound SWE on the tibialis anterior (TA) and peroneus longus (PL) muscles using pressure cuff inflation around the thigh at 40 mmHg, 80 mmHg, and 120 mmHg. Changes in IMP were verified using a catheter connected to a blood pressure monitor. It was found that IMP was correlated to TA and PL shear modulus (spearman's rank correlation = 0.99 and 0.99, respectively). Applying a gradual increase of cuff pressure from 0 to 120 mmHg increased the shear modulus of the TA and PL muscles from 15.83 (2.46) kPa to 21.88 (4.33) kPa and from 9.64 (1.97) kPa to 12.88 (5.99) kPa, respectively. These results demonstrate that changes of muscle mechanical properties are dependent on IMP. This observation is important to improve interpretation of ultrasound elastograms and to potentially use it as a biomarker for more accurate diagnosis of pathologies related to increased IMP.     

Acoustic radiation force optical coherence elastography for evaluating mechanical properties of soft condensed matters and its biological applications

Evaluating mechanical properties of biological soft tissues and viscous mucus is challenging because of complicated dynamic behaviors. Soft condensed matter models have been successfully used to explain a number of dynamical behaviors. Here, we reported that optical coherence elastography (OCE) is capable of quantifying mechanical properties of soft condensed matters, micellar fluids. A 7.5 MHz focused transducer was utilized to generate acoustic radiation force exerted on the surface of soft condensed matters in order to produce Rayleigh waves. The waves were recorded by optical coherence tomography (OCT). The Kelvin‐Voigt model was adopted to evaluate shear modulus and loss modulus of soft condensed matters. The results reported that various concentrations of micellar fluids can provide reasonable ranges of elasticity from 65.71 to 428.78 Pa and viscosity from 0.035 to 0.283 Pa·s, which are close to ranges for actual biological samples, like mucus. OCE might be a promising tool to differentiate pathologic mucus samples from healthy cases as advanced applications in the future.     

Making muscle elastic: the structural basis of myomesin stretching

Skeletal and cardiac muscles are remarkable biological machines that support and move our bodies and power the rhythmic work of our lungs and hearts. As well as producing active contractile force, muscles are also passively elastic, which is essential to their performance. The origins of both active contractile and passive elastic forces can be traced to the individual proteins that make up the highly ordered structure of muscle. In this Primer, we describe the organization of sarcomeres--the structural units that produce contraction--and the nature of the proteins that make muscle elastic. In particular, we focus on an elastic protein called myomesin, whose novel modular architecture helps explain elasticity.     

Noninvasive muscle tension measurement using the novel technique of magnetic resonance elastography (MRE)

A novel method for direct measurement of the state of skeletal muscle contraction is introduced called magnetic resonance elastography (MRE). Such a technique is useful for avoiding the indeterminacy inherent in most inverse dynamic models of the musculoskeletal system. Within a standard MRI scanner, mechanical vibration is applied to muscle via the skin, creating shear waves that penetrate the tissue and propagate along muscle fibers. A gradient echo sequence is used with cyclic motion-encoding to image the propagating shear waves using phase contrast. Individual muscles of interest are identified and the shear wavelength in each is measured. Shear wavelength increases with increasing tissue stiffness and increasing tissue tension.

Several ankle muscles were tested simultaneously in normal subjects. Applied ankle moment was isometrically resisted at several different foot positions. Shear wavelengths in relaxed muscle in neutral foot position was 2.34±0.47 cm for tibialis anterior (TA) and 3.13±0.24 cm for lateral gastrocnemius (LG). Wavelength increased in relaxed muscle when stretched (to 3.80±0.28 cm for TA in 45° plantar-flexion and to 3.95±0.43 cm for LG in 20° dorsi-flexion). Wavelength increased more significantly with contraction (to 7.71±0.97 cm in TA for 16 N m dorsi-flexion effort and to 7.90±0.34 cm in LG for 48 Nm plantar-flexion effort).

MRE has been shown to be sensitive to both passive and active tension within skeletal muscle making it a promising, noninvasive tool for biomechanical analysis. Since it is based on MRI technology, any muscle, however deep, can be interrogated using equipment commonly available in most health care facilities.     

Force enhancement above the initial isometric force on the descending limb of the force-length relationship

Edman et al. (J. General Physiol. 80 (1982) 769) observed in single fibres of frog that the steady-state forces following active fibre stretch were greater than the purely isometric force obtained at the length from which the stretch was initiated. Operating on the descending limb of the force-length relationship, such a result can only be explained within the framework of the sarcomere length non-uniformity theory, if some fibre segments shortened during the fibre stretch. However, such a result was not found, leaving Edman's observation unexplained. Force enhancement above the initial isometric force has not been investigated systematically in whole muscle, and therefore it is not known whether this property is also part of whole muscle mechanics. The purpose of this study was to test if the steady-state forces following active stretch of cat semitendinosus were greater than the corresponding purely isometric forces at the muscle length from which the stretch was started. Cat semitendinosus was stretched by various amounts on the descending limb of the force-length relationship, and the steady-state forces following these stretches were compared to the corresponding isometric forces at the initial and final muscle lengths. In 109 of 131 tests, the steady-state forces following stretching were greater than the isometric forces at the initial muscle lengths. Force enhancement increased with increasing amounts of stretching, and force enhancement above the initial isometric force was more likely to occur following stretches of great compared to small amplitude. Passive forces following active muscle stretching were often significantly greater than the passive forces at the same muscle length following an isometric contraction or a passive stretching of the muscle. This observation was made consistently at the longest muscle lengths tested. It appears, therefore, that there is a passive force that accounts for part of the force enhancement above the isometric force at the initial muscle length, and that provides increased passive force when a muscle is actively, rather than passively, stretched at long muscle lengths. We conclude that cat semitendinosus demonstrates steady-state force enhancement above the corresponding purely isometric force at the initial muscle length on the descending limb of the force-length relationship for many contractile conditions, and that a unique, and so far undetected, passive, parallel element contributes to this force enhancement, particularly at long muscle lengths where muscle is assumed to be most vulnerable to injuries associated with sarcomere length instability.     

Internal viscoelastic loading in cat papillary muscle.

The passive mechanical properties of myocardium were defined by measuring force responses to rapid length ramps applied to unstimulated cat papillary muscles. The immediate force changes following these ramps recovered partially to their initial value, suggesting a series combination of viscous element and spring. Because the stretched muscle can bear force at rest, the viscous element must be in parallel with an additional spring. The instantaneous extension-force curves measured at different lengths were nonlinear, and could be made to superimpose by a simple horizontal shift. This finding suggests that the same spring was being measured at each length, and that this spring was in series with both the viscous element and its parallel spring (Voigt configuration), so that the parallel spring is held nearly rigid by the viscous element during rapid steps. The series spring in the passive muscle could account for most of the series elastic recoil in the active muscle, suggesting that the same spring is in series with both the contractile elements and the viscous element. It is postulated that the viscous element might be coupled to the contractile elements by a compliance, so that the load imposed on the contractile elements by the passive structures is viscoelastic rather than purely viscous. Such a viscoelastic load would give the muscle a length-independent, early diastolic restoring force. The possibility is discussed that the length-independent restoring force would allow some of the energy liberated during active shortening to be stored and released during relaxation.     

Muscle material properties in passive and active stroke-impaired muscle

Stroke survivors routinely experience long-term motor and sensory impairments. In parallel with neurological changes, material properties of muscles in the impaired limbs, such as muscle stiffness, may also change progressively. However, these stiffness measures are routinely derived from individual joint stiffness, representing whole muscle groups. Here, we use shear wave (SW) ultrasound elastography to measure SW velocity, as a surrogate measure of stiffness, to quantify material properties in individual muscles. Accordingly, the purpose of this study was to compare muscle material properties of the bicep brachii in stroke survivors and in age-matched control subjects by measuring SW velocity at rest and different voluntary activation levels. Our main findings show that at rest, the SW velocity was on average 41% greater in the paretic muscle compared the contralateral non-paretic muscle. The mean passive SW velocity across all subjects were 2.34 ± 0.41 m/s for the non-paretic side, 3.30 ± 1.20 m/s for the paretic side, and 2.24 ± 0.18 for controls. SW velocity was significantly different in muscles of the paretic and non-paretic side (p < 0.001), but not between muscles of the non-paretic and controls (p = 0.47). As voluntary activation increased, SW velocity increased non-linearly, with an average power fit of r² = 0.83 ± 0.09 for the non-paretic side, r² = 0.61 ± 0.24 for the paretic side, and r² = 0.24 ± 0.15 for the healthy age-matched controls. In active muscle (10, 25, 50, 75, 100% maximum voluntary contraction), there was no significant difference in SW velocity between the non-paretic, paretic, and control muscles.These findings suggest that stroke-impaired muscles have potentially altered muscle material properties, specifically stiffness, and that passive and active stiffness may contribute differently to total muscle stiffness.     

Functional muscle synergies constrain force production during postural tasks

We recently demonstrated that a set of five functional muscle synergies were sufficient to characterize both hindlimb muscle activity and active forces during automatic postural responses in cats standing at multiple postural configurations. This characterization depended critically upon the assumption that the endpoint force vector (synergy force vector) produced by the activation of each muscle synergy rotated with the limb axis as the hindlimb posture varied in the sagittal plane. Here, we used a detailed, 3D static model of the hindlimb to confirm that this assumption is biomechanically plausible: as we varied the model posture, simulated synergy force vectors rotated monotonically with the limb axis in the parasagittal plane (r2=0.94+/-0.08). We then tested whether a neural strategy of using these five functional muscle synergies provides the same force-generating capability as controlling each of the 31 muscles individually. We compared feasible force sets (FFSs) from the model with and without a muscle synergy organization. FFS volumes were significantly reduced with the muscle synergy organization (F=1556.01, p<0.01), and as posture varied, the synergy-limited FFSs changed in shape, consistent with changes in experimentally measured active forces. In contrast, nominal FFS shapes were invariant with posture, reinforcing prior findings that postural forces cannot be predicted by hindlimb biomechanics alone. We propose that an internal model for postural force generation may coordinate functional muscle synergies that are invariant in intrinsic limb coordinates, and this reduced-dimension control scheme reduces the set of forces available for postural control.     

Control of human eye movements: I. modelling of extraocular muscle

The properties of extraocular muscle are important in consideration of the control of human eye movements. A proposed model for human extraocular muscle is based on the anatomical and physiological evidence; it considers both the static and dynamic properties of active and passive muscle. The passive parallel elasticity was determined from the length-tension curves for passive muscle, while the active series elasticity was defined utilizing quick stretch results for active muscle. The characteristics of active muscle as the tension generator were computed from length-tension data; the force-velocity relationship was used to describe the viscosity of active muscle. Simulations using the muscle model accurately depicted the quick stretch experiments of both active and passive muscle as well as the isometric development of muscle force to a state of tentanus. The model will be incorporated into an overall representation of the extraocular plant mechanism in the immediately suceeding paper.     

Control of human eye movements: III. dynamic characteristics of the eye tracking mechanism

The properties of extraocular muscle are important in consideration of the control of human eye movements. A proposed model for human extraocular muscle is based on the anatomical and physiological evidence; it considers both the static and dynamic properties of active and passive muscle. The passive parallel elasticity was determined from the length-tension curves for passive muscle, while the active series elasticity was defined utilizing quick stretch results for active muscle. The characteristics of active muscle as the tension generator were computed from length-tension data; the force-velocity relationship was used to describe the viscosity of active muscle. Simulations using the muscle model accurately depicted the quick stretch experiments of both active and passive muscle as well as the isometric development of muscle force to a state of tentanus. The model will be incorporated into an overall representation of the extraocular plant mechanism in the immediately suceeding paper.     

In vivo noninvasive measurement of spatially resolved corneal elasticity in human eyes using Lamb wave optical coherence elastography

Current elastography techniques are limited in application to accurately assess spatially resolved corneal elasticity in vivo for human eyes. The air‐puff optical coherence elastography (OCE) with an eye motion artifacts correction algorithm is developed to distinguish the in vivo cornea vibration from the eye motion and visualize the Lamb wave propagation clearly in healthy subjects. Based on the Lamb wave model, the phase velocity dispersion curve in the high‐frequency is calculated to obtain spatially resolved corneal elasticity accurately with high repeatability. It is found that the corneal elasticity has regional variations and is correlated with intraocular pressure, which suggests that the method has the potential to provide noninvasive measurement of spatially resolved corneal elasticity in clinical practice.     

Muscle kinematics for minimal work of breathing.

A mathematical model was analyzed to obtain a quantitative and testable representation of the long-standing hypothesis that the respiratory muscles drive the chest wall along the trajectory for which the work of breathing is minimal. The respiratory system was modeled as a linear elastic system that can be expanded either by pressure applied at the airway opening (passive inflation) or by active forces in respiratory muscles (active inflation). The work of active expansion was calculated, and the distribution of muscle forces that produces a given lung expansion with minimal work was computed. The calculated expression for muscle force is complicated, but the corresponding kinematics of muscle shortening is simple: active inspiratory muscles shorten more during active inflation than during passive inflation, and the ratio of active to passive shortening is the same for all active muscles. In addition, the ratio of the minimal work done by respiratory muscles during active inflation to work required for passive inflation is the same as the ratio of active to passive muscle shortening. The minimal-work hypothesis was tested by measurement of the passive and active shortening of the internal intercostal muscles in the parasternal region of two interspaces in five supine anesthetized dogs. Fractional changes in muscle length were measured by sonomicrometry during passive inflation, during quiet breathing, and during forceful inspiratory efforts against a closed airway. Active muscle shortening during quiet breathing was, on average, 70% greater than passive shortening, but it was only weakly correlated with passive shortening. Active shortening inferred from the data for more forceful inspiratory efforts was approximately 40% greater than passive shortening and was highly correlated with passive shortening. These data support the hypothesis that, during forceful inspiratory efforts, muscle activation is coordinated so as to expand the chest wall with minimal work.     

The effects of muscle stretching and shortening on isometric forces on the descending limb of the force-length relationship

The purpose of this study was to examine the effects of stretching and shortening on the isometric forces at different lengths on the descending limb of the force-length relationship. Cat soleus (N = 10) was stretched and shortened by various amounts on the descending limb of the force-length relationship, and the steady-state forces following these dynamic contractions were compared to the isometric forces at the corresponding muscle lengths. We found a shift of the force-length relationship to greater force values following muscle stretching, and to smaller force values following muscle shortening. Shifts in both directions critically depended on the magnitude of stretching/shortening and the final muscle length. We confirm recent findings that the steady-state isometric force following some stretch conditions clearly exceeded the maximal isometric forces at optimum muscle length, and that force enhancement was associated with an increase in the passive force, i.e., a passive force enhancement. When the passive force enhancement was subtracted from the total force enhancement, forces following stretch were always equal to or smaller than the isometric force at optimum muscle length. Together, these findings led to the conclusions: (a). that force enhancement is composed of an "active and a "passive" component; (b). that the "passive" component of force enhancement allows for forces greater than the maximal isometric forces at the muscle's optimum length; and (c). that force enhancement and force depression are critically affected by muscle length and stretch/shortening amplitude.     

Linear relationship between electromyography and shear wave elastography measurements persists in deep muscles of the upper extremity

Recent works have demonstrated a linear relationship between muscle activation and shear modulus in various superficial muscles. As such, it may be possible to overcome limitations of traditional electromyography (EMG) methods by assessing activation using shear wave elastography. However, the relationship has not been wholly validated in deep muscles. This study measured the association between squared shear wave velocity, which is related to shear modulus, and activation within superficial and deep muscles. This relationship was also compared between surface and intramuscular EMG electrodes. We simultaneously recorded EMG and shear wave velocity in one deep (brachialis) and one superficial (brachioradialis) muscle in ten healthy individuals during isometric elbow flexion across a wide range of contraction intensities. Muscle activation and squared shear wave velocity demonstrated good reliability (ICC > 0.75) and showed a linear relationship (P < 0.05) for all muscle/EMG electrode type combinations (study conditions) after down-sampling. Study condition was not a significant within-subject factor to the slope or intercept of the relationship (P > 0.05). This work demonstrates that activation of both superficial and deep muscles can be assessed noninvasively using ultrasound shear wave elastography and is a critical step toward demonstrating elastography’s utility as an alternative to EMG.     

Élastographie transitoire par résonance magnétique

Magnetic resonance elastography (MRE) is typically performed using harmonic shear vibrations. We are investigating a new approach in which transient shear waves are induced by the radiation force of an ultrasound beam. This technique can induce the vibration directly inside the organ under examination.     

Characterization of muscle architecture in children and adults using magnetic resonance elastography and ultrasound techniques

The purpose of this study is to characterize the muscle architecture of children and adults using magnetic resonance elastography and ultrasound techniques. Five children (8-12 yr) and seven adults (24-58 yr) underwent both tests on the vastus medialis muscle at relaxed and contracted (10% and 20% of MVC) states. Longitudinal ultrasonic images were performed in the same area as the phase image showing the shear wave's propagation. Two geometrical parameters were defined: the wave angle (α(_MRE)) corresponding to the shear wave propagation and the fascicule angle (α(_US)) tracking the path of fascicles. Moreover, shear modulus was measured at different localizations within the muscle and in the subcutaneous adipose tissue. The association of both techniques demonstrates that the shear wave propagation follows the muscle fascicles path, reflecting the internal muscle architecture. At rest, ultrasound images revealed waves propagating parallel to the children fascicle while adults showed oblique waves corresponding to already oriented (α(_US)=15.4±2.54°) muscle fascicles. In contraction, the waves' propagation were in an oblique direction for children (α(_US_10%MVC)=10.6±2.27°, α(_US_20%MVC)=10.2±2.29°) as well as adults (α(_US_10%MVC)=15.4±2.54°, α(_US_20%MVC)=17.2±2.44°). A stiffness variation (1 kPa) was found between the upper and lower parts of the adult VM muscle and a lower stiffness (1.85±0.17 kPa) was measured in the subcutaneous adipose tissue. This study demonstrates the feasibility of the MRE technique to provide geometrical insights from the children and adults muscles and to characterize different physiological media.     

Shear wave induced resonance elastography of soft heterogeneous media

In the context of ultrasound dynamic elastography imaging and characterization of venous thrombosis, we propose a method to induce mechanical resonance of confined soft heterogeneities embedded in homogenous media. Resonances are produced by the interaction of horizontally polarized shear (SH) waves with the mechanical heterogeneity. Due to such resonance phenomenon, which amplifies displacements up to 10 times compared to non-resonant condition, displacement images of the underlying structures are greatly contrasted allowing direct segmentation of the heterogeneity and a more precise measurement of displacements since the signal-to-noise ratio is enhanced. Coupled to an analytical model of wave scattering, the feasibility of shear wave induced resonance (SWIR) elastography to characterize the viscoelasticity of a mimicked venous thrombosis is demonstrated (with a maximum variability of 3% and 11% for elasticity and viscosity, respectively). More generally, the proposed method has the potential to characterize the viscoelastic properties of a variety of soft biological and industrial materials.