Propagation of shear waves in muscle tissue

A mathematical model of the propagation of acoustic shear waves in muscle tissue is considered. Muscle is modeled as an incompressible transversely isotropic viscoelastic continuum with quasi-one-dimensional active tension. There are two types of shear waves in an infinite medium. Waves of the second type (transverse) propagate without decay even when myofibril viscosity is taken into account. A problem of standing transverse waves in a rectangular layer was investigated numerically. The values of the problem parameters are found for which one can easily estimate the active tension (or muscle tone) from the characteristics of standing waves. This value is informative for diagnostics of the 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.     

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.     

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.     

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.     

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.     

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.     

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.     

Estimation of muscle active state

A method is presented for the estimation of the complete time course of muscle active state. The method is based on the selection of a proper model for the muscle, consisting of linear and non-linear components, and on the estimation of its parameters from a simple experiment. The model's parameters are estimated, using the least square method, from measurements of a tetanized muscle's response to a change of its length. The time course of the active state is calculated from an isometric twitch tension response of the same muscle. The twitch tension response is taken as the system's output, and the active state as its input. The latter can be estimated since the system parameters have already been estimated from the tetanized muscle experiment. Experiments were performed on the gastrocnemius muscle of frogs and cats. Results are given for the whole active state time course of these muscles. The results show that the peak active state force does not reach tetanic value, and a negative force is generated during the relaxation period. Additional experiments were carried out with the purpose of verifying the existence of this force; however, no conclusive results were obtained.This research was supported by the Julius Silver Institute of Bio-Medical Engineering Sciences, Grant 050-304     

Holographic recording of the interference waves in muscle tissues

The method of interference holography is suggested for the investigation of biomechanical properties of human skeletal muscles. The method permitted the registration of a profile of deforming waves induced on the muscle surface by a special striking mechanism. Analytically and in model experiments it was demonstrated that deforming waves penetrated much deeper than skin and subcutaneous tissue layer. The experiments have confirmed that the skin has but little effect on interference pattern. The interferogram of deforming waves in the biceps muscle reflect distinctions in the muscle tissue elasticity along and transverse muscle 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 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.     

Effects of peristaltic and longitudinal wave motion of the channel wall on movement of micro-organisms: application to spermatozoa transport

A mathematical model is presented to study the motion of the spermatozoa in the cervical canal by considering the transverse waves along its tail and the transverse and longitudinal motions of the cervical wall. In an attempt to control fertility by reducing the speed of sperm, the transverse waves have been considered in the direction opposite to the motion of the spermatozoa. It has been shown that by having appropriate transverse wave motion and longitudinal velocity, the sperm may not be able to move towards the oviduct even if it could continue to have its own propelling velocity. A particular case of the motion of a thin plane sheet in a channel under peristaltic motion of its walls has also been obtained and studied.     

New insights into the behavior of muscle during active lengthening.

A muscle fiber was modeled as a series-connected string of sarcomeres, using an A. V. Hill type model for each sarcomere and allowing for some random variation in the properties of the sarcomeres. Applying stretches to this model led to the prediction that lengthening of active muscle on or beyond the plateau of the length tension curve will take place very nonuniformly, essentially by rapid, uncontrolled elongation of individual sarcomeres, one at a time, in order from the weakest toward the strongest. Such a "popped" sarcomere, at least in a single fiber, will be stretched to a length where there is no overlap between thick and thin filaments, and the tension is borne by passive components. This prediction allows modeling of many results that have previously been inexplicable, notably the permanent extra tension after stretch on the descending limb of the length tension curve, and the continued rise of tension during a continued stretch.     

Contractile properties of the shortening rat diaphragm in vitro

Diaphragmatic fatigue has been defined in terms of the failure of the muscle to continue to generate a given level of tension. Appropriate shortening of the diaphragm is, however, just as important for adequate ventilation. In this study we have examined in vitro the contractile properties of the rat diaphragm under afterloaded isotonic conditions and the effect of fatigue on the ability of the diaphragm to shorten. Shortening of the muscle strips was found to depend on size of afterload, frequency of stimulation, duration of stimulation, and initial length of the muscle. The afterloaded isotonic length-tension relationship coincided with the relationship between length and active isometric tension only for relatively small afterloads. Fatigue of the muscle strips, induced by isometric or afterloaded isotonic contractions, was associated with a decline in the extent of shortening as well as a decrease in active isometric tension. Ability to shorten and ability to develop isometric tension did not decrease to the same extent under all conditions. We conclude that active shortening, as well as active isometric tension, is decreased by muscular fatigue and that changes in these properties can be different depending on experimental conditions. The results suggest that the definition of diaphragmatic fatigue should be expanded to include the ability of the muscle to shorten by an appropriate amount. The results also suggest that measurement of isometric performance may not provide a complete estimate of the overall performance of the fatigued diaphragm.     

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.     

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.     

Low-frequency electromagnetic instabilities caused by a rotating dust flow

Low-frequency electromagnetic waves propagating obliquely to an external magnetic field in a plasma with an anisotropic dust component are considered. The cold dust is assumed to have considerable longitudinal and transverse velocity components with respect to the magnetic field. A dispersion relation demonstrating that both fast and slow waves can be unstable is derived in the framework of kinetic theory. Mechanisms and consequences of these instabilities are discussed in the context of the problem of plasma transition into a turbulent state behind the shock front of a supernova.     

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.     

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).