Fluid mechanics of muscle vibrations.

The pressure field produced by an isometrically contracting frog gastrocnemius muscle is described by the fluid mechanics equations for a vibrating sphere. The equations predict a pressure amplitude that is proportional to the lateral acceleration of the muscle, inversely proportional to the square of the distance from the muscle, and cosinusoidally related to the major axis of lateral movement. The predictions are confirmed by experiments that measure the pressure amplitude distribution and by photographs of muscle movement during contraction. The lateral movement of muscle has the appearance of an oscillating system response to a step function input--the oscillation may be at the resonant frequency of the muscle and therefore may provide a means to measure muscle stiffness without actually touching the muscle.     

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.     

Dynamic and static control of the human knee joint in abduction-adduction

It is unclear whether humans can voluntarily control dynamic and static properties in knee abduction-adduction, which may be important in performing functional tasks and preventing injuries, whether the main load is about the abduction axis or not. A joint-driving device was used to perturb the knee in abduction-adduction at full knee extension under both passive (muscle relaxed) and active (muscle contracted in abduction or adduction) conditions. Dynamic control properties in knee abduction-adduction were characterized by joint stiffness, viscosity, and limb inertia, and quasi-static knee torque-angle relationship was characterized by knee abduction-adduction laxity and quasi-static stiffness (at a 20Nm moment). It was found that the subjects were capable of generating net abduction and adduction moment through differential co-contraction of muscles crossing the medial and lateral sides of the knee, which helped to reduce the abduction-adduction joint laxity (p< or =0.01) and increase stiffness (p<0.027) and viscous damping. Knee abduction laxity was significantly lower than adduction laxity (p=0.043) and the quasi-static abduction stiffness was significantly higher than adduction stiffness (p<0.001). The knee joint showed significantly higher stiffness and viscosity in abduction-adduction than their counterparts in knee flexion-extension at comparable levels of joint torque (p<0.05). Similar to dynamic flexion-extension properties, the system damping ratio remained constant over different levels of contraction, indicating simplified control tasks for the central nervous system; while the natural undamped frequency increased considerably with abduction-adduction muscle contraction, presumably making the knee a quicker system during strenuous tasks involving strong muscle contraction.     

Measurements of human forearm viscoelasticity

In human subjects, stiffness of the relaxed elbow was measured by three methods, using a forearm manipulandum coupled to a.d.c. torque motor. Elbow stiffness calculated from frequency response characteristics increased as the driving amplitude decreased. Step displacements of the forearm produced restoring torques linearly related to the displacement. The stiffness was very similar to that calculated from natural frequencies at amplitudes above 0.1 rad. Thirdly, elbow stiffness was estimated from brief test pulses, 120 ms in duration, by mathematically simulating the torque-displacement functions. Stiffness values in the limited linear range (under +/- 0.1 rad) were higher than in the linear range of the first two methods. A major component of elbow stiffness appears to decay within 1 s. The coefficients of viscosity determined from the simulation were, however, very similar to those calculated from the frequency response. Test pulse simulation was then used to determine joint impedance for different, actively maintained elbow angles. Joint stiffness and viscosity increased with progressive elbow flexion.     

Postural differences in shoulder dynamics during pushing and pulling

Assessments of shoulder dynamics (e.g. the inertial, viscous, and stiffness properties of the joint) can provide important insights into the stability of the joint at rest and during volitional contraction. The purpose of this study was to investigate how arm posture influences shoulder dynamics while generating pushing or pulling torques in the horizontal plane. Sixteen healthy participants were examined in seven postures encompassing a large workspace of the shoulder. At each posture, the participant’s shoulder was rapidly perturbed while measuring the resultant change in shoulder torque about the glenohumeral axis. Participants were examined both at rest and while producing horizontal flexion and extension torques scaled to 15% of a maximum voluntary contraction. Shoulder stiffness, viscosity, and damping ratio were estimated using impedance-based matching, and changes in these outcome measures with torque level, elevation angle, and plane of elevation angle were explored with a linear mixed effects model. Shoulder stiffness was found to decrease with increasing elevation angles (p < 0.001) without subsequent changes in viscosity, leading to a greater damping ratios at higher elevation angles (p < 0.001). Shoulder stiffness, viscosity, and damping ratio (all p < 0.05) were all found to significantly increase as the plane of elevation of the arm was increased. The relationship between the viscosity, stiffness and the damping ratio of the shoulder is one that the central nervous system must regulate in order to maintain stability, protect against injury, and control the shoulder joint as the inertial and muscle contributions change across different arm postures.     

Muscle stiffness measured under conditions simulating natural sound production.

Isolated whole frog gastrocnemius muscles were electrically stimulated to peak twitch tension while held isometrically in a bath at 4 degrees C. A quartz hydrophone detected vibrations of the muscle by measuring the pressure fluctuations caused by muscle movement. A small steel collar was slipped over the belly of the muscle. Transient forces including plucks and steady sinusoidal driving were applied to the collar by causing currents to flow in a coil held near the collar. The instantaneous resonant frequencies measured by the pluck and driving techniques were the same at various times during a twitch contraction cycle. The strain produced by the plucking technique in the outermost fibers was less than 1.6 x 10(-4%), a strain three orders of magnitude less than that required to drop the tension to zero in quick-length-change experiments. Because the pressure transients recorded by the hydrophone during plucks and naturally occurring sounds were of comparable amplitude, strains in the muscle due to naturally occurring sound must also be of the order 10(-3%). A simple model assuming that the muscle is an elastic bar under tension was used to calculate the instantaneous elastic modulus E as a function of time during a twitch, given the tension and resonant frequency. The result for Emax, the peak value of E during a twitch, was typically 2.8 x 10(6) N/m2. The methods used here for measuring muscle stiffness are unusual in that the apparatus used for measuring stiffness is separate from the apparatus controlling and measuring force and length.     

Effect of cycle frequency and excursion amplitude on work done by rat diaphragm muscle

Strips of isolated rat diaphragm muscle were attached to a servomotor-transducer apparatus, and the muscle length was cycled in a sinusoidal fashion about the length at which maximum isometric twitch force was developed, Lo. The amplitude of the length displacement (excursion amplitude) and rate of cycling were varied between 3 and 13% Lo and 1-4 Hz respectively. The muscle was tetanically stimulated (100 Hz, supramaximal voltage, stimulus duration (duty cycle) 20% of the length cycle period) during the shortening stage of the imposed length cycle at the phase that yielded maximum net positive work. The force and displacement of the muscle were recorded. Work per cycle was calculated from the area of the loop formed by plotting force against length for one full stretch-shorten cycle. Work per cycle decreased, but power increased, as cycle frequency was increased from 1 to 4 Hz. Maximum work done per cycle was about 12.8 J/kg at a cycle frequency of 1 Hz. Maximum mean power developed was about 27 W/kg and occurred at a cycle frequency of 4 Hz. Work and power were maximum at an excursion amplitude of 13% of Lo (i.e., Lo +/- 6.5%). Measured work and power output are considerably less than values estimated from length-tension and force-velocity curves.     

Dissipative processes in human passive skeletal muscle

The measurement of damping of low-amplitude limb oscillations permits to evaluate the energy losses in passive human skeletal muscle during small length changes. The attenuation curve for the limb oscillations is quite different from the classical attenuation curve in the presence of viscous damping. Energy losses per oscillation cycle are practically frequency independent. Thus the damping properties of passive muscles at joint angular velocities up to 100% are due mostly to the velocity-independent resistance of "dry friction" type. The value of this "friction" is about 0.07 N per sm2 of muscle cross-section. The passive muscle also has marked thixotropy, as its resistance to small amplitude low velocity stretches strongly depends on time between stretches.     

The effects of pH on force and stiffness development in mouse muscles

Changes in force and stiffness during contractions of mouse extensor digitorum longus and soleus muscles were measured over a range of extracellular pH from 6.4 to 7.4. Muscle stiffness was measured using small amplitude (less than 0.1% of muscle length), high frequency (1.5 kHz) oscillations in length. Twitch force was not significantly affected by changes in pH, but the peak force during repetitive stimulation (2, 3, and 20 pulses) was decreased significantly as the pH was reduced. Changes in muscle stiffness with pH were in the same direction, but smaller in extent. If the number of attached cross-bridges in the muscle can be determined from the measurement of small amplitude, high frequency muscle stiffness, then these findings suggest that (a) the number of cross-bridges between thick and thin filaments declines in low pH and (b) the average force per cross-bridge also declines in low pH. The decline in force per cross-bridge could arise from a reduction in the ability of cross-bridges to generate force during their state of active force production and (or) in an increased percentage of bonds in a low force, "rigor" state.     

Dependence of elbow joint stiffness measurements on speed,angle, and muscle contraction level

Elbow joint stiffness is critical to positioning the hand. Abnormal elbow joint stiffness may affect a person's ability to participate in activities of daily living. In this work, elbow joint stiffness was measured in ten healthy young adults with a device adapted from one previously used to measure stiffness in other joints. Measurements of elbow stiffness involved applying a constant-velocity rotational movement to the elbow and measuring the resultant displacement, torque, and acceleration. Elbow stiffness was then computed using a previously-established model for joint stiffness. Measurements were made at two unique elbow joint angles, two speeds, and two forearm muscle contraction levels. The results indicate that the elbow joint stiffness is significantly affected by both rotational speed and forearm muscle contraction level.     

Electromyographic instantaneous amplitude and instantaneous mean power frequency patterns across a range of motion during a concentric isokinetic muscle action of the biceps brachii.

The purpose of this study was to examine the electromyographic (EMG) instantaneous amplitude (IA) and instantaneous mean power frequency (IMPF) patterns for the biceps brachii muscle across a range of motion during maximal and submaximal concentric isokinetic muscle actions of the forearm flexors. Ten adults (mean +/- SD age = 22.0 +/- 3.4 years) performed a maximal and a submaximal [20% peak torque (PT)] concentric isokinetic forearm flexion muscle action at a velocity of 30 degrees s(-1). The surface EMG signal was detected from the biceps brachii muscle with a bipolar electrode arrangement, and the EMG IA and IMPF versus time relationships were examined for each subject using first- and second-order polynomial regression models. The results indicated that there were no consistent patterns between subjects for EMG IA or IMPF with increases in torque across the range of motion. Some of the potential nonphysiological factors that could influence the amplitude and/or frequency contents of the surface EMG signal during a dynamic muscle action include movement of the muscle fibers and innervation zone beneath the skin surface, as well as changes in muscle fiber length and the thickness of the tissue layer between the muscle and the recording electrodes. These factors may affect the EMG IA and IMPF patterns differently for each subject, thereby increasing the difficulty of drawing any general conclusions regarding the motor control strategies that increase torque across a range of motion.     

Mechanism of thixotropic behavior at relaxed joints in the rat

When a relaxed joint is subjected to a small sinusoidal torque, the amplitude of the steady-state displacement response is increased up to severalfold by a transient larger perturbation. The original state, in which the relaxed joint is unexpectedly stiff, is restored by several seconds of inactivity. This thixotropic phenomenon has previously been observed in a variety of human joints. We have now investigated the mechanism of thixotropic behavior at relaxed joints in rats anesthetized with pentobarbital sodium, by using a series of preparations including the intact ankle joint, a blood-perfused soleus muscle preparation, an isolated soleus muscle, and ankle joint isolated by severing all muscular attachments. Thixotropic behavior was observed in all intact, isolated muscle, and isolated joint preparations. The contribution of the joint to thixotropic behavior was comparable to, and at times exceeded, the contribution of muscle. We also analyzed the short-range stiffness properties of relaxed, blood-perfused soleus muscles and found them to be similar to thixotropy with respect to range of action (0.2-0.3% of muscle length), elastic modulus (approximately 4 kg/cm2), and time course for redevelopment (time constant = 2.5 s at 34 degrees C). Thus thixotropic behavior at a relaxed joint may be attributed both to the joint structures and to short-range stiffness of muscles acting at the joint.     

Human hoppers compensate for simultaneous changes in surface compression and damping

On a range of elastic and damped surfaces, human hoppers and runners adjust leg mechanics to maintain similar spring-like mechanics of the leg and surface combination. In a previous study of adaptations to damped surfaces, we changed surface damping and stiffness simultaneously to maintain constant surface compression. The current study investigated whether hoppers maintain spring-like mechanics of the leg-surface combination when surface damping alone changes (elastic and 1000-4800 N s m(-1)). We found that hoppers adjusted leg mechanics to maintain similar spring-like mechanics of the leg-surface combination and center of mass dynamics on all surfaces. Over the range of surface damping, vertical stiffness of the leg-surface combination increased by only 12% and center of mass displacement decreased by only 6% despite up to 55% less compression of more heavily damped surfaces. In contrast, a simulation predicted a 44% decrease in vertical displacement with no adjustment to leg mechanics. To compensate for the smaller and slower compression of more heavily damped surfaces, the stance legs compressed by up to 4.1 +/- 0.2 cm further and reached peak compression sooner. To replace energy lost by damped surfaces, hoppers performed additional leg work by extending the legs during takeoff by up to 3.1 +/- 0.2 cm further than they compressed during landing. We conclude that humans simultaneously adjust leg compression magnitude and timing, as well as mechanical work output, to conserve center of mass dynamics on damped surfaces. Runners may use similar strategies on natural energy-dissipating surfaces such as sand, mud and snow.     

Strategies used to stabilize the elbow joint challenged by inverted pendulum loading

The stiffness of activated muscles may stabilize a loaded joint by preventing perturbations from causing large displacements and injuring the joint. Here the elbow muscle recruitment patterns were compared with the forearm loaded vertically (a potentially unstable inverted pendulum configuration) and with horizontal loading. Eighteen healthy subjects were studied with the forearm vertical and supinated and the elbow flexed approximately 90 degrees. In the first experiment EMG electrodes recorded activity of biceps, triceps, and brachioradialis muscles for joint torques produced (a) by voluntarily exerting a horizontal force isometrically (b) by voluntarily flexing and extending the elbow while the forearm was loaded vertically with 135N. The relationship between the EMG and the torque generated was quantified by the linear regression slope and zero-torque intercept. In a second experiment a vertical load increasing linearly with time up to 300N was applied.In experiment 1 the EMG-torque relationships for biceps and triceps had an intercept about 10% of maximum voluntary effort greater with the vertical compared to the horizontal force, the inverse was found for Brachioradialis, but the EMG-torque slopes for both agonist and antagonistic muscles were not different. In experiment 2 there were 29 trials with minimal elbow displacement and all the three muscles activated on the order of 11% of maximum activation to stabilize the elbow; 19 trials had small elbow extension and 14 trials small flexion requiring altered muscle forces for equilibrium; 7 trials ended in large unstable displacement or early termination of the test. An analysis indicate that the observed levels of muscle activation would only provide stability if the muscles' short-range stiffness was at the high end of the published range, hence the elbow was marginally stable. The stability analysis also indicated that the small elbow extension increased stability and flexion decreased stability.     

Evaluation of the use of resonant frequencies to characterize physical properties of human long bones

Human tibiae were subjected to steady state vibration over a frequency range to evaluate the use of clinical measurements of resonant frequency to characterize osteoporosis. Displacement and force transmitted to the bone were monitored and used to obtain measures of dynamic mass, stiffness, damping and resonant frequency. Initial results indicate that resonant frequency F is a less sensitive indicator of change in state than either generalized mass M or generalized stiffness K due to the functional relationship of these three parameters.     

Resonance in the human medial gastrocnemius muscle during cyclic ankle bending exercise.

We investigated the behavior of the muscle tendon unit (MTU) of the medial gastrocnemius muscle during cyclic ankle bending exercise at eight different frequencies (ranging from 1.33 to 3.67 Hz). The changes in the length of fascicle in the muscle during the exercises were determined by real-time ultrasound imaging. The coordinates of anatomical references and the ground reaction force were determined from video recording and a force plate, respectively. The length change of the MTU (the distance from the origin to insertion of the muscle) was calculated from changes in the knee and ankle joint angles. It was found that the amplitude ratio and phase difference between the fascicle and MTU lengths were both dependent on the movement frequency. At lower frequencies, the fascicle lengths varied almost in phase with the MTU length, whereas they varied out of phase at the higher frequencies. At intermediate frequency, the amplitude of the fascicle became very small compared with that of the MTU, which is considered resonance. We constructed a mechanical model of the MTU based on a notion of forced oscillation in a mass-spring system. The obtained data were well explained by the model. It was concluded that the behavior of the MTU highly depends on the movement frequency due to the viscoelasticity of the MTU.     

Fibre operating lengths of human lower limb muscles during walking

Muscles actuate movement by generating forces. The forces generated by muscles are highly dependent on their fibre lengths, yet it is difficult to measure the lengths over which muscle fibres operate during movement. We combined experimental measurements of joint angles and muscle activation patterns during walking with a musculoskeletal model that captures the relationships between muscle fibre lengths, joint angles and muscle activations for muscles of the lower limb. We used this musculoskeletal model to produce a simulation of muscle-tendon dynamics during walking and calculated fibre operating lengths (i.e. the length of muscle fibres relative to their optimal fibre length) for 17 lower limb muscles. Our results indicate that when musculotendon compliance is low, the muscle fibre operating length is determined predominantly by the joint angles and muscle moment arms. If musculotendon compliance is high, muscle fibre operating length is more dependent on activation level and force-length-velocity effects. We found that muscles operate on multiple limbs of the force-length curve (i.e. ascending, plateau and descending limbs) during the gait cycle, but are active within a smaller portion of their total operating range.     

A mathematical analysis of the force-stiffness characteristics of muscles in control of a single joint system

Feldman (1966) has proposed that a muscle endowed with its spinal reflex system behaves as a non-linear spring with an adjustable resting length. In contrast, because of the length-tension properties of muscles, many researchers have modeled them as non-linear springs with adjustable stiffness. Here we test the merits of each approach: Initially, it is proven that the adjustable stiffness model predicts that isometric muscle force and stiffness are linearly related. We show that this prediction is not supported by data on the static stiffness-force characteristics of reflexive muscles, where stiffness grows non-linearly with force. Therefore, an intact muscle-reflex system does not behave as a non-linear spring with an adjustable stiffness. However, when the same muscle is devoid of its reflexes, the data shows that stiffness grows linearly with force. We aim to understand the functional advantage of the non-linear stiffness-force relationship present in the reflexive muscle. Control of an inverted pendulum with a pair of antagonist muscles is considered. Using an active-state muscle model we describe force development in an areflexive muscle. From the data on the relationship of stiffness and force in the intact muscle we derive the length-tension properties of a reflexive muscle. It is shown that a muscle under the control of its spinal reflexes resembles a non-linear spring with an adjustable resting length. This provides independent evidence in support of the Feldman hypothesis of an adjustable resting length as the control parameter of a reflexive muscle, but it disagrees with his particular formulation. In order to maintain stability of the single joint system, we prove that a necessary condition is that muscle stiffness must grow at least linearly with force at isometric conditions. This shows that co-contraction of antagonist muscles may actually destabilize the limb if the slope of this stiffness-force relationship is less than an amount specified by the change in the moment arm of the muscle as a function of joint configuration. In a reflexive muscle where stiffness grows faster than linearly with force, co-contraction will always lead to an increase in stiffness. Furthermore, with the reflexive muscles, the same level of joint stiffness can be produced by much smaller muscle forces because of the non-linear stiffness-force relationship. This allows the joint to remain stable at a fraction of the metabolic energy cost associated with maintaining stability with areflexive muscles.This work was supported in part by grant no. 1R01 NS 24926 from the NIH (Michael Arbib, PI). R.S. was supported by an IBM Graduate Fellowship in Computer Science     

Frequency-dependent power output and skeletal muscle design

Cyclically contracting muscles provide power for a variety of processes including locomotion, pumping blood, respiration, and sound production. In the current study, we apply a computational model derived from force–velocity relationships to explore how sustained power output is systematically affected by shortening velocity, operational frequency, and strain amplitude. Our results demonstrate that patterns of frequency dependent power output are based on a precise balance between a muscle's intrinsic shortening velocity and strain amplitude. We discuss the implications of this constraint for skeletal muscle design, and then explore implications for physiological processes based on cyclical muscle contraction. One such process is animal locomotion, where musculoskeletal systems make use of resonant properties to reduce the amount of metabolic energy used for running, swimming, or flying. We propose that skeletal muscle phenotype is tuned to this operational frequency, since each muscle has a limited range of frequencies at which power can be produced efficiently. This principle also has important implications for our understanding muscle plasticity, because skeletal muscles are capable of altering their active contractile properties in response to a number of different stimuli. We discuss the possibility that muscles are dynamically tuned to match the resonant properties of the entire musculoskeletal system.     

Modulation of myocardial stiffness by β-adrenergic stimulation--its role in normal and failing heart

The acute effects of beta-adrenergic stimulation on myocardial stiffness were evaluated. New-Zealand white rabbits were treated with saline (control group) or doxorubicin to induce heart failure (HF) (DOXO-HF group). Effects of isoprenaline (10(-10)-10(-5) M), a non-selective beta-adrenergic agonist, were tested in papillary muscles from both groups. In the control group, the effects of isoprenaline were also evaluated in the presence of a damaged endocardial endothelium, atenolol (beta(1)-adrenoceptor antagonist), ICI-118551 (beta(2)-adrenoceptor antagonist), KT-5720 (PKA inhibitor), L-NNA (NO-synthase inhibitor), or indomethacin (cyclooxygenase inhibitor). Passive length-tension relations were constructed before and after adding isoprenaline (10(-5) M). In the control group, isoprenaline increased resting muscle length up to 1.017+/-0.006 L/L(max). Correction of resting muscle length to its initial value resulted in a 28.5+/-3.1 % decrease of resting tension, indicating decreased muscle stiffness, as confirmed by the isoprenaline-induced right-downward shift of the passive length-tension relation. These effects were modulated by beta(1)- and beta(2)-adrenoceptors and PKA. In DOXO-HF group, the effect on myocardial stiffness was significantly decreased. We conclude that beta-adrenergic stimulation is a relevant mechanism of acute neurohumoral modulation of the diastolic function. Furthermore, this study clarifies the mechanisms by which myocardial stiffness is decreased.