Propagation of shear waves in the muscle tissue

A mathematical model of the propagation of acoustic shear waves in muscle tissue is considered. The muscle is modelled by an incompressible transversely isotropic viscoelastic continuum with quasi-one-dimensional active tension. Two types of shear waves in an infinite medium have been established. The waves of the second type (transverse) propagate without attenuation even when myofibril viscosity is taken into account. A problem of standing transverse waves in a rectangular layer has been investigated numerically. The values of the problem parameters have been found for which the active tension or muscle tonus is easily estimated from the characteristics of standing waves. This value is informative for the diagnosis of muscle state.     

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.     

Length-tension relationship of abdominal expiratory muscles: effect of emphysema

The present study examined the active and passive length-tension relationship of the abdominal expiratory muscles in vitro during electrically stimulated contractions. Studies were performed on isolated strips of transverse abdominis and external oblique muscle from nine adult hamsters with normal lung function. The effect of chronic hyperinflation on the two muscles was assessed in eight hamsters with elastase-induced emphysema. In normal animals the maximal active tension per cross-sectional area (Po) was equal in the two muscles. The absolute muscle fiber length at which Po occurred (Lo) was less for the external oblique than the transverse abdominis and the length-tension curve operated at shorter fiber lengths. However, the change in tension produced by an increase or decrease in muscle length expressed in relative terms (i.e., as %Lo) was greater for the transverse abdominis than the external oblique. Mean total lung capacity of emphysematous animals was 198% of control. Po of the transverse abdominis and external oblique were the same in emphysematous and control animals. However, Lo and the length-tension curve of the transverse abdominis occurred at shorter fiber lengths in emphysematous animals because of a reduction in the number of sarcomeres in series along the fiber. The length-tension curve and the number of sarcomeres in the external oblique was the same in emphysematous and control animals. These results in normal animals indicate that the magnitude of the change in active and passive tension produced by a change in muscle length differs in the transverse abdominis and external oblique. Moreover, chronic hyperinflation of the thorax produced by elastase injection alters the length-tension relationships of some but not all the expiratory muscles.     

Changes in shear wave propagation within skeletal muscle during active and passive force generation

Muscle force can be generated actively through changes in neural excitation, and passively through externally imposed changes in muscle length. Disease and injury can disrupt force generation, but it can be challenging to separate passive from active contributions to these changes. Ultrasound elastography is a promising tool for characterizing the mechanical properties of muscles and the forces that they generate. Most prior work using ultrasound elastography in muscle has focused on the group velocity of shear waves, which increases with increasing muscle force. Few studies have quantified the phase velocity, which depends on the viscoelastic properties of muscle. Since passive and active forces within muscle involve different structures for force transmission, we hypothesized that measures of phase velocity could detect changes in shear wave propagation during active and passive conditions that cannot be detected when considering only group velocity. We measured phase and group velocity in the human biceps brachii during active and passive force generation and quantified the differences in estimates of shear elasticity obtained from each of these measurements. We found that measures of group velocity consistently overestimate the shear elasticity of muscle. We used a Voigt model to characterize the phase velocity and found that the estimated time constant for the Voigt model provided a way to distinguish between passive and active force generation. Our results demonstrate that shear wave elastography can be used to distinguish between passive and active force generation when it is used to characterize the phase velocity of shear waves propagating in muscle.     

Lateral transmission of tension in frog myofibers: A myofibrillar network and transverse cytoskeletal connections are possible transmitters

The extensibility of the sarcolemma of single myofibers can be reduced locally by leaving a segment covered by a sleeve of surrounding tissue composed of cut myofibers, blood vessels, and connective tissue, hereafter referred to as “the splint.” Splinted fibers from frog semitendinosus muscle were used to study mechanical connections (transverse coupling) between myofibrillar components and sarcolemma. The transverse coupling is strong enough to insure a tight correlation between myofibril length and overlying sarcolemma length in both resting and activated fibers and to transmit nearly maximum isometric tension to the splint. Lateral transmission of active tension was demonstrated with a preparation which had the distal two-thirds of an intact fiber covered by a splint and the proximal third dissected clean. When the outer end of the splint was pinned down and only the distal tendon was held, tension generated in the splinted fiber was transmitted to, and recorded from, the splint. Parameters of isometric tension transmitted laterally were not significantly different from those of tension transmitted longitudinally. Myofibrils branch profusely and form a network that may act as a unitary force generator and transmitter. In splinted fibers its output is possibly picked up circumferentially and transmitted across the sarcolemma by a microfilament network. A cap of relatively inextensible sarcolemma “splints” myofiber ends. Resting tension is transmitted to and from the myofibrils by transverse coupling beyond the cap and the region of short sarcomere spacing it covers. Transverse cytoskeletal connections at Z and M regions are described. Immobilization of the sarcolemma allows study of myofibril-sarcolemma linkage in intact libers. Both active and resting tension were transmitted laterally.     

Muscle sound frequencies of the frog are modulated by skeletal muscle tension.

The purpose of this study was to determine the relationships among muscle sound frequencies, muscle tension, and stiffness. Time-frequency transformations of nonstationary acoustic signals provided measures of resonant frequency during isometric contractions of frog (Rana pipiens) semitendinosus and gastrocnemius muscles. A mathematical expression for muscle transverse resonant frequency, elastic modulus and tension, based on elastic beam theory, was formulated by the Rayleigh method adapted for muscles. For thin muscles, the elastic modulus was found to have negligible influence on transverse muscle resonant frequency. Changes in muscle tension were the major determinants of changes in transverse resonant frequency. Consequently, for thin muscles, the time course of muscle tension, but not elastic modulus, can be monitored acoustically during the early phase of contraction when muscles give rise to sounds. Muscles were found to be anisotropic with a modulus of elasticity, EL, measured via length perturbations near 0.1% muscle length peak-to-peak, that was much larger than the modulus of elasticity, Eb, that resists the lateral bending that causes sound production. The elastic and resonant behavior of a thin muscle is similar to a tensioned fibrous cable with distributed mass.     

Tension responses to increased hydrostatic pressure in glycerinated rabbit psoas muscle fibres

A method developed to study the effect of increased hydrostatic pressure on the isometric tension of a single muscle fibre is described and experiments done at room temperature (18-22 degrees C) on glycerinated rabbit psoas muscle fibres are presented. Increase of pressure (range 1-10 MPa) caused little change in tension transducer response when a muscle fibre was relaxed. However, there was a reversible depression of isometric tension with an increase of pressure when a fibre was maximally calcium-activated or in rigor; the depression was around 15% for active tension and 30% for rigor tension, for an increase of pressure of 10 MPa (ca. 100 atm).     

Investigation of optimal follower load path generated by trunk muscle coordination

It has been reported that the center of rotation of each vertebral body is located posterior to the vertebral body center. Moreover, it has been suggested that an optimized follower load (FL) acts posterior to the vertebral body center. However, the optimal position of the FL with respect to typical biomechanical characteristics regarding spinal stabilization, such as joint compressive force, shear force, joint moment, and muscle stress, has not been studied. A variation in the center of rotation of each vertebra was formulated in a three-dimensional finite element model of the lumbar spine with 117 pairs of trunk muscles. Then, the optimal translation of the FL path connecting the centers of rotations was estimated by solving the optimization problem that was to simultaneously minimize the compressive forces, the shear forces, and the joint moments or to minimize the cubic muscle stresses. An upright neutral standing position and a standing position with 200N in both hands were considered. The FL path moved posterior, regardless of the optimization criteria and loading conditions. The FL path moved 5.0 and 7.8mm posterior in upright standing and 4.1mm and 7.0mm posterior in standing with 200N in hands for each optimization scheme. In addition, it was presented that the optimal FL path may have advantages in comparison to the body center FL path. The present techniques may be important in understanding the spine stabilization function of the trunk muscles.     

The Pattern of Activation in the Sartorius Muscle of the Frog

The development of isometric twitch tension has been compared with the redevelopment of isometric tension in the fully active frog sartorius muscle following release. At 0°C the rate of rise of isometric twitch tension is the same as that for the muscle in the fully active state at the same tension but not until about 40 msec. after the stimulus and then only for a few milliseconds. The rates of rise of tension in the twitch and in the redevelopment of tension in the fully active muscle appear to be nearly the same at low tensions. Substitution of nitrate for chloride in the Ringer's solution bathing the muscle retards the development of tension during the early part of the contraction phase of the twitch and the effect reaches a maximum within 3 minutes after changing the solutions. These observations have been discussed in connection with some possible patterns of activation and the hypothesis has been advanced that the rate of activation of a sarcomere is determined mainly by the rate at which the transverse component of the link between excitation and contraction is propagated inwards from the periphery to the center of the fiber. This hypothesis has been discussed in relation to others concerning the nature of excitation-contraction coupling.     

The mechanism of low-frequency sound production in muscle.

Frog gastrocnemius muscles stimulated isometrically in a saline bath at 20 degrees C were found to produce a single ringing sound event beginning just before the tension record began to rise. The sound event was substantially over by the time the isometric tension began to fall. Results from studies correlating the spatial pattern of the sound, the amplitude and frequency of the sound as a function of the muscle length, and the response of both the passive and active muscle to a transverse pluck were found to be consistent with the conclusion that the sounds in these muscles are caused primarily by transverse resonant vibrations. As the muscle develops force, its lack of cylindrical symmetry gives rise to lateral motions, which are most likely the initiators of the bending vibrations detected as sound.     

Propagation of MHD waves in a plasma in a sheared magnetic field with straight field lines

The propagation of MHD plasma waves in a sheared magnetic field is investigated. The problem is solved using a simplified model: a cold plasma is inhomogeneous in one direction, and the magnetic field lines are straight. The waves are assumed to travel in the plane perpendicular to the radial coordinate (i.e., the coordinate along which the plasma and magnetic field are inhomogeneous). It is shown that the character of the singularity at the resonance surface is the same as that in a homogeneous magnetic field. It is found that the shear gives rise to the transverse dispersion of Alfvén waves, i.e., the dependence of the radial component of the wave vector on the wave frequency. In the presence of shear, Alfvén waves are found to propagate across magnetic surfaces. In this case, the transparent region is bounded by two turning points, at one of which, the radial component of the wave vector approaches infinity and, at the other one, it vanishes. At the turning point for magnetosonic waves, the electric and magnetic fields are finite; however, the radial component of the wave vector approaches infinity, rather than vanishes as in the case with a homogeneous field.     

Strain distribution in an elastic substrate vibrated in a bioreactor for vocal fold tissue engineering

A bioreactor previously described was used to quantify the shear strain along a bioengineered tissue scaffold driven at low audio frequencies (20–200 Hz). Standing wave patterns were calculated analytically by solving a classical boundary value problem for a vibrating string under tension and bending stiffness. Boundary conditions were non-traditional in that small pivot arms at the endpoints allowed neither the displacement nor the velocity to go to zero. The calculations were corroborated with stroboscopic measurement of the motion of the material in the bioreactor. Results indicate that shear strains up to 0.2 can be obtained at low frequencies (20 Hz), with a gradual decrease at higher frequencies due to the decaying amplitude response of the mechanical driver. The bioreactor may be useful for approximating the Young's modulus of the material in situ by probing for resonance frequencies in the standing wave pattern. A yet unsolved problem is a variable drag coefficient along the length of the material due to fluid turbulence in the culture medium.     

Comparison of erector spinae and hamstring muscle activities and lumbar motion during standing knee flexion in subjects with and without lumbar extension rotation syndrome

The aim of this study was to compare the activity of the erector spinae (ES) and hamstring muscles and the amount and onset of lumbar motion during standing knee flexion between individuals with and without lumbar extension rotation syndrome. Sixteen subjects with lumbar extension rotation syndrome (10 males, 6 females) and 14 healthy subjects (8 males, 6 females) participated in this study. During the standing knee flexion, surface electromyography (EMG) was used to measure muscle activity, and surface EMG electrodes were attached to both the ES and hamstring (medial and lateral) muscles. A three-dimensional motion analysis system was used to measure kinematic data of the lumbar spine. An independent-t test was conducted for the statistical analysis. The group suffering from lumbar extension rotation syndrome exhibited asymmetric muscle activation of the ES and decreased hamstring activity. Additionally, the group with lumbar extension rotation syndrome showed greater and earlier lumbar extension and rotation during standing knee flexion compared to the control group. These data suggest that asymmetric ES muscle activation and a greater amount of and earlier lumbar motion in the sagittal and transverse plane during standing knee flexion may be an important factor contributing to low back pain.     

Effect of the shear flow in the generation and self-organization of internal gravity wave structures in the dissipative ionosphere

A linear mechanism for the generation and amplification of internal gravity waves and their further nonlinear dynamics in the stably stratified dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) is studied. For shear flows, the operators of linear problems are non-self-conjugate and the corresponding eigenfunctions are nonorthogonal. Therefore, the canonical modal approach is poorly applicable to study such motions. In this case, the so-called nonmodal mathematical analysis is more adequate. Dynamic equations and equations for the energy transport of internal gravity perturbations in the ionosphere with shear flows are derived on the basis of the nonmodal approach. Exact analytic solutions of linear and nonlinear equations are found. The growth rate of the shear instability of internal gravity waves is determined. It is revealed that perturbations grow in time according to a power law, rather than exponentially. The frequency and wavenumber of the generated internal gravity modes depend on time; hence, a wide spectrum of wave perturbations caused by linear effects (rather than nonlinear turbulent ones) forms in the ionosphere with shear flows. The efficiency of the linear mechanism for the amplification of internal gravity waves during their interaction with the inhomogeneous zonal wind is analyzed. A criterion for the development of the shear instability of such waves in the ionospheric plasma is obtained. It is shown that, in the presence of shear instability, internal gravity waves extract the shear flow energy in the initial (linear) stage of their evolution, due to which their amplitude and, accordingly, energy increase substantially (by an order of magnitude). As the amplitude increases, the mechanism of nonlinear self-localization comes into play and the process terminates with the self-organization of strongly localized solitary nonlinear internal gravity vortex structures. As a result, a new degree of freedom of the system and a new way of the evolution of perturbations in a medium with a shear flow appear. Inductive and viscous dampings limit the lifetime of vortex internal gravity structures in the ionosphere; nevertheless, their lifetime is long enough for them to strongly affect the dynamic properties of the medium. It is revealed on the basis of the analytic solution of a set of time-independent nonlinear dynamic equations that, depending on the velocity profile of the shear flow, the nonlinear internal gravity structures can take the form of a purely monopole vortex, a dipole cyclone-anticyclone pair, a transverse vortex chain, or a longitudinal vortex path against the background of the inhomogeneous zonal wind. The accumulation of such vortices in the ionosphere can result in a strongly turbulent state.     

Is a guanine nucleotide-binding protein involved in excitation-contraction coupling in skeletal muscle?

Plasma membrane depolarization causes skeletal muscle contraction by triggering Ca2+ release from an intracellular membrane network, the sarcoplasmic reticulum. A specialized portion of the sarcoplasmic reticulum, the terminal cisternae, is junctionally associated with sarcolemmal invaginations called the transverse tubules, but the mechanism by which the action potential at the level of the transverse tubules is coupled to Ca2+ release from the terminal cisternae is still mysterious. Here we show that: (i) GTP gamma S, a non-hydrolyzable analog of GTP, elicits isometric force development in skinned muscle fibre; (ii) GTP gamma S is unable to release CA2+ from isolated sarcoplasmic reticulum fractions; (iii) the threshold for tension development is shifted to higher GTP gamma S concentrations by pre-incubation with pertussis toxin. These results suggest that a GTP-binding protein is involved in coupling the action potential of transverse tubules to Ca2+ release from the terminal cisternae.     

A mathematical model for estimating muscle tension in vivo during esophageal bolus transport

We present a model of esophageal wall muscle mechanics during bolus transport with which the active and "passive" components of circular muscle tension are separately extracted from concurrent manometric and videofluoroscopic data. Local differential equations of motion are integrated across the esophageal wall to yield global equations of equilibrium which relate total tension within the esophageal wall to intraluminal pressure and wall geometry. To quantify the "passive" (i.e. inactive) length-tension relationships, the model equations are applied to a region of the esophagus in which active muscle contraction is physiologically inhibited. Combining the global equations with space-time-resolved intraluminal pressure measured manometrically and videofluoroscopic geometry data, the passive model is used to separate active and "passive" components of esophageal muscle tension during bolus transport. The model is of general applicability to probe basic muscle mechanics including the space-time stimulation of circular muscle, the relationship between longitudinal muscle tension and longitudinal muscle shortening, and the contribution of the collagen matrix surrounding muscle fibers to passive tension during normal human esophageal bolus transport and in pathology. Example calculations of normal esophageal function are given where active tone is found to extend only over a short intrabolus segment near the bolus tail and segmental regions of active muscle squeeze are demonstrated.     

Shape and tension distribution of the passive rat diaphragm

We developed an in vitro preparation to investigate shape and stress distribution in the intact rat diaphragm. Our hypothesis was that the diaphragm is anisotropic with smaller compliance in transverse fiber direction than along fibers, and therefore shape change may be small. After the animals were killed (8 rats), the entire diaphragm was excised and fixed into a mold at the insertions. Oxygenated Krebs-Ringer solution was circulated under the diaphragm and perfused over its surface. A total of 20-23 small markers were sutured on the diaphragm surface. At transdiaphragmatic pressure (P(di)) of 3-15 cmH(2)O, curvature was smaller in transverse direction than along fibers. Using finite element analysis we computed membrane tension. At P(di) of 15 cmH(2)O, tension in central tendon was larger than muscle. In costal region maximum principal tension (sigma(1)) is essentially along the fibers and ranged from 6-10 g/cm. Minimum principal tension (sigma(2)) was 0. 3-4 g/cm. In central tendon, sigma(1) was 10-15 g/cm, compared with 4-10 g/cm for sigma(2). The diaphragm was considerably stiffer in transverse fiber direction than along the fibers.     

Is muscle co-activation a predisposing factor for low back pain development during standing? A multifactorial approach for early identification of at-risk individuals

Purpose and scopeLow back pain development has been associated with static standing postures in occupational settings. Previous work has demonstrated gluteus muscle co-activation as a predominant pattern in previously asymptomatic individuals who develop low back pain when exposed to 2-h of standing. The purpose of this work was to investigate muscle co-activation as a predisposing factor in low back pain development while including a multifactorial approach of clinical assessment tools and psychosocial assessments to identify individuals who are at risk for pain development during standing.ResultsForty percent of participants developed low back pain during the 2-h of standing. Pain developers demonstrated bilateral gluteus medius and trunk flexor–extensor muscle co-activation prior to reports of pain development. Pain developers and non-pain developers demonstrated markedly different patterns of muscle activation during the 2-h of standing. A novel screening test of active hip abduction was the only clinical assessment tool that predicted pain development.ConclusionsGluteus medius and trunk muscle co-activation appears to be a predisposing rather than adaptive factor in low back pain development during standing. A combination of a positive active hip abduction test and presence of muscle co-activation during standing may be useful for early identification of at-risk individuals.     

The fin musculature of cuttlefish and squid (Mollusca, Cephalopoda): morphology and mechanics

The lateral fins of cuttlefish and squid consist of a tightly packed three-dimensional array of musculature that lacks bony skeletal support or fluid-filled cavities for hydrostatic skeletal support. During swimming and manoeuvring, the fins are bent upward and downward in undulatory waves. The fin musculature is arranged in three mutually perpendicular planes. Transverse muscle bundles extend parallel to the fin surface from the base of the fin to the fin margin. Dorso-ventral muscle bundles extend from dorsal and ventral connective tissue fasciae to a median connective tissue fascia. A layer of longitudinal muscle bundles is situated adjacent to both the dorsal and ventral surface of the median fascia. The muscle fibres are obliquely striated and include a core of mitochondria. A zone of muscle fibres with a more extensive core of mitochondria is present in both the dorsal and the ventral transverse muscle bundles. It is hypothesized that these muscle masses include two fibre types with different aerobic capacity. A network of connective tissue fibres is present in the transverse and dorso-ventral muscle masses. These fibres, probably collagen, are oriented at 45 to the long axes of the transverse and dorsoventral muscle fibres in transverse planes.
A biomechanicayl analysis of the morphology suggests that support for fin movements is provided by simultaneous contractile activity of muscles of specific orientations in a manner similar to that proposed for other 'muscular-hydrostats'. The musculature therefore provides both the force and support for movement. Connective tissue fibres may aid in providing support and may also serve for elastic energy storage.     

On the Onset of Cracks in Arteries¹

We present a theoretical approach to study the onset of failure localization into cracks in arterial wall. The arterial wall is a soft composite comprising hydrated ground matrix of proteoglycans reinforced by spatially dispersed elastin and collagen fibers. As any material, the arterial tissue cannot accumulate and dissipate strain energy beyond a critical value. This critical value is enforced in the constitutive theory via energy limiters. The limiters automatically bound reachable stresses and allow examining the mathematical condition of strong ellipticity. Loss of the strong ellipticity physically means inability of material to propagate superimposed waves. The waves cannot propagate because material failure localizes into cracks perpendicular to a possible wave direction. Thus, not only the onset of a crack can be analyzed but also its direction. We use the recently developed constitutive theories of the arterial wall including 8 and 16 structure tensors to account for the fiber dispersion. We enhance these theories with energy limiters. We examine the loss of strong ellipticity in uniaxial tension and pure shear in circumferential and axial directions of the arterial wall. We find that the vanishing longitudinal wave speed predicts the appearance of cracks in the direction perpendicular to tension. We also find that the vanishing transverse wave speed predicts the appearance of cracks in the the direction inclined (non-perpendicular) to tension. The latter result is counter-intuitive yet it is supported by recent experimental observations.