The Viscoelastic Properties of Passive Eye Muscle in Primates. II: Testing the Quasi-Linear Theory

We have extensively investigated the mechanical properties of passive eye muscles, in vivo, in anesthetized and paralyzed monkeys. The complexity inherent in rheological measurements makes it desirable to present the results in terms of a mathematical model. Because Fung''s quasi-linear viscoelastic (QLV) model has been particularly successful in capturing the viscoelastic properties of passive biological tissues, here we analyze this dataset within the framework of Fung''s theory.We found that the basic properties assumed under the QLV theory (separability and superposition) are not typical of passive eye muscles. We show that some recent extensions of Fung''s model can deal successfully with the lack of separability, but fail to reproduce the deviation from superposition.While appealing for their elegance, the QLV model and its descendants are not able to capture the complex mechanical properties of passive eye muscles. In particular, our measurements suggest that in a passive extraocular muscle the force does not depend on the entire length history, but to a great extent is only a function of the last elongation to which it has been subjected. It is currently unknown whether other passive biological tissues behave similarly.     

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.     

Locomotor function shapes the passive mechanical properties and operating lengths of muscle

Locomotor muscles often perform diverse roles, functioning as motors that produce mechanical energy, struts that produce force and brakes that dissipate mechanical energy. In many vertebrate muscles, these functions are not mutually exclusive and a single muscle often performs a range of mechanically diverse tasks. This functional diversity has obscured the relationship between a muscle''s locomotor function and its mechanical properties. I use hopping in toads as a model system for comparing muscles that primarily produce mechanical energy with muscles that primarily dissipate mechanical energy. During hopping, hindlimb muscles undergo active shortening to produce mechanical energy and propel the animal into the air, whereas the forelimb muscles undergo active lengthening to dissipate mechanical energy during landing. Muscles performing distinct mechanical functions operate on different regions of the force–length curve. These findings suggest that a muscle''s operating length may be shaped by potential trade-offs between force production and sarcomere stability. In addition, the passive force–length properties of hindlimb and forelimb muscles vary, suggesting that passive stiffness functions to restrict the muscle''s operating length in vivo. These results inform our understanding of vertebrate muscle variation by providing a clear link between a muscle''s locomotor function and its mechanical properties.     

Genetic Models in Applied Physiology. Merosin deficiency leads to alterations in passive and active skeletal muscle mechanics.

The role of extracellular elements on the mechanical properties of skeletal muscles is unknown. Merosin is an essential extracellular matrix protein that forms a mechanical junction between the sarcolemma and collagen. Therefore, it is possible that merosin plays a role in force transmission between muscle fibers and collagen. We hypothesized that deficiency in merosin may alter passive muscle stiffness, viscoelastic properties, and contractile muscle force in skeletal muscles. We used the dy/dy mouse, a merosin-deficient mouse model, to examine changes in passive and active muscle mechanics. After mice were anesthetized and the diaphragm or the biceps femoris hindlimb muscle was excised, passive length-tension relationships, stress-relaxation curves, or isometric contractile properties were determined with an in vitro biaxial mechanical testing apparatus. Compared with controls, extensibility was smaller in the muscle fiber direction and the transverse fiber direction of the mutant mice. The relaxed elastic modulus was smaller in merosin-deficient diaphragms compared with controls. Interestingly, maximal muscle tetanic stress was depressed in muscles from the mutant mice during uniaxial loading but not during biaxial loading. However, presence of transverse passive stretch increases maximal contractile stress in both the mutant and normal mice. Our data suggest that merosin contributes to muscle passive stiffness, viscoelasticity, and contractility and that the mechanism by which force is transmitted between adjacent myofibers via merosin possibly in shear.     

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.     

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.     

Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Absence of desmin in skeletal muscle was found to induce an increase in passive stiffness. The present study aimed at developing rheological models of passive muscle to explain this stiffening. Models were elaborated by using experimental data depicting muscle viscoelastic behaviour. The experimental protocol included stepwise extension tests applied on control and desmin knockout soleus muscles from mice. Linear and non-linear models were composed of elastic and viscous elements. They were constructed with the aim at taking the presence or absence of desmin into account by simulating desmin as an elastic element. Furthermore, associated adaptation of connective tissues in absence of desmin was modelled as an additional elastic element. Differences in passive behaviour induced by absence of desmin were predicted by using a linear model and a non-linear one. The non-linear model was selected because: (1) it is able to predict experimental viscoelastic kinetics accounting for the increase in passive stiffness in muscles lacking desmin, (2) its design is consistent with morphological data, and (3) stiffness characteristics of its elements are in accordance with the literature. Finally, this modelling approach demonstrates that both absence of desmin and adaptation of connective tissue are required to explain the increase in passive stiffness in desmin knockout muscles.     

A simple model of bipedal walking predicts the preferred speed-step length relationship

We used a simple model of passive dynamic walking, with the addition of active powering on level ground, to study the preferred relationship between speed and step length in humans. We tested several hypothetical metabolic costs, with one component proportional to the mechanical work associated with pushing off with the stance leg at toe-off, and another component associated with several possible costs of forcing oscillations of the swing leg. For this second component, a cost based on the amount of force needed to oscillate the leg divided by the time duration of that force predicts the preferred speed-step length relationship much better than other costs, such as the amount of mechanical work done in swinging the leg. The cost of force/time models the need to recruit fast muscle fibers for large forces at short durations. The actual mechanical work performed by muscles on the swing leg appears to be of relatively less importance, although it appears to be minimized by the use of short bursts of muscle activity in near-isometric conditions. The combined minimization of toe-off mechanical work and force divided by time predicts the preferred speed-step length relationship.     

Passive skeletal muscle can function as an osmotic engine

Muscles are composite structures. The protein filaments responsible for force production are bundled within fluid-filled cells, and these cells are wrapped in ordered sleeves of fibrous collagen. Recent models suggest that the mechanical interaction between the intracellular fluid and extracellular collagen is essential to force production in passive skeletal muscle, allowing the material stiffness of extracellular collagen to contribute to passive muscle force at physiologically relevant muscle lengths. Such models lead to the prediction, tested here, that expansion of the fluid compartment within muscles should drive forceful muscle shortening, resulting in the production of mechanical work unassociated with contractile activity. We tested this prediction by experimentally increasing the fluid volumes of isolated bullfrog semimembranosus muscles via osmotically hypotonic bathing solutions. Over time, passive muscles bathed in hypotonic solution widened by 16.44 ± 3.66% (mean ± s.d.) as they took on fluid. Concurrently, muscles shortened by 2.13 ± 0.75% along their line of action, displacing a force-regulated servomotor and doing measurable mechanical work. This behaviour contradicts the expectation for an isotropic biological tissue that would lengthen when internally pressurized, suggesting a functional mechanism analogous to that of engineered pneumatic actuators and highlighting the significance of three-dimensional force transmission in skeletal muscle.     

The isometric length-force models of nine different skeletal muscles.

The length-force relations of nine different skeletal muscles in the hindlimb of the cat were determined experimentally, with electrical stimulation of the sciatic nerve as the activation mode. It was shown that the active-, passive-, and total-force patterns varied widely among the muscles. The tibialis posterior (TP), medial and lateral gastrocnemius (MG, LG) and flexor digitorum longus (FDL) had a symmetric active-force curve, whereas the tibialis anterior (TA), peroneus brevis (PB), peroneus longus (PL), extensor digitorum longus (EDL), and soleus (SOL) had an asymmetric curve which exhibits about 25% of the maximal isometric force at extreme lengths. The SOL, EDL, and LG had a low-level passive force which appeared at short muscle length, whereas all other muscles exhibited initial passive force just before the optimal length. The total force was rising quasi-linearly for the SOL, whereas the other muscles exhibited an intermediate plateau about the optimal length. The LG and FDL had a substantial but temporary intermediate dip in the total force as the muscle was elongated past the optimal length. The elongation range of the various muscles also varied, ranging from +/- 15 to +/- 30% of the optimal length. The elongation range was symmetric for the FDL, LG, MG, TP, SOL, and EDL, and asymmetric for the PL, PB, and TA, being -12 to + 17%, -12 to + 17%, and -35 to + 12%, respectively. Two different models which incorporate muscle architecture were successfully fitted to the experimental data of the muscles except for the MG and TA. The architecture of these two muscles is highly nonhomogeneous and contains compartments with two pennation patterns or two different optimal lengths. New models, which add spatially and temporally the individual characteristics of each compartment of the muscles, were constructed for these two muscles. The new models demonstrated high correlation to the experimental data obtained from the MG and TA. It was concluded that the length-force relation varies widely among various skeletal muscles and is probably dependent on the primary function of the muscle in the context of integrated movement; this is a manifestation of architectural factors such as fiber pennation pattern and angle, cross-sectional area, ratio of muscle to tendon length, distribution of the fiber length within the muscle and compartmental pennation.     

Adjustable passive length-tension curve in rabbit detrusor smooth muscle.

Until the 1990s, the passive and active length-tension (L-T) relationships of smooth muscle were believed to be static, with a single passive force value and a single maximum active force value for each muscle length. However, recent studies have demonstrated that the active L-T relationship in airway smooth muscle is dynamic and adapts to length changes over a period of time. Furthermore, our prior work showed that the passive L-T relationship in rabbit detrusor smooth muscle (DSM) is also dynamic and that in addition to viscoelastic behavior, DSM displays strain-softening behavior characterized by a loss of passive stiffness at shorter lengths following a stretch to a new longer length. This loss of passive stiffness appears to be irreversible when the muscle is not producing active force and during submaximal activation but is reversible on full muscle activation, which indicates that the stiffness component of passive force lost to strain softening is adjustable in DSM. The present study demonstrates that the passive L-T curve for DSM is not static and can shift along the length axis as a function of strain history and activation history. This study also demonstrates that adjustable passive stiffness (APS) can modulate total force (35% increase) for a given muscle length, while active force remains relatively unchanged (4% increase). This finding suggests that the structures responsible for APS act in parallel with the contractile apparatus, and the results are used to further justify the configuration of modeling elements within our previously proposed mechanical model for APS.     

A mechanical model for adjustable passive stiffness in rabbit detrusor.

Strips of rabbit detrusor smooth muscle (DSM) exhibit adjustable passive stiffness characterized by strain softening: a loss of stiffness on stretch to a new length distinct from viscoelastic behavior. At the molecular level, strain softening appears to be caused by cross-link breakage and is essentially irreversible when DSM is maintained under passive conditions (i.e., when cross bridges are not cycling to produce active force). However, on DSM activation, strain softening is reversible and likely due to cross-link reformation. Thus DSM displays adjustable passive stiffness that is dependent on the history of both muscle strain and activation. The present study provides empirical data showing that, in DSM, 1) passive isometric force relaxation includes a very slow component requiring hours to approach steady state, 2) the level of passive force maintained at steady state is less if the tissue has previously been strain softened, and 3) tissues subjected to a quick-release protocol exhibit a biphasic response consisting of passive force redevelopment followed by force relaxation. To explain these and previously identified characteristics, a mechanical model for adjustable passive stiffness is proposed based on the addition of a novel cross-linking element to a hybrid Kelvin/Voigt viscoelastic model.     

A three-dimensional muscle model: a quantified relation between form and function of skeletal muscles

A three-dimensional muscle model with complex geometry is described and tested against experimental data. Using this model, several muscles were constructed. These muscles have equal optimum length but differ in architecture. The force exerted by the constructed muscles, in relation to their actual length and velocity of shortening, is discussed. Generally speaking, the constructed muscles with considerable pennation have great fiber angles, a great physiological cross section, a narrow active and steep passive length-force relation, and a low maximal velocity of shortening. The maximal power (force times velocity) delivered by the constructed muscles is shown to be almost independent of the architecture of the muscles. The steepness of the passive length-force relation is determined mainly by the shortest fibers within the group of constructed muscles, whereas maximal velocity of shortening and the width of the active length-force relation are determined mainly by the longest fibers. The validity of the three-dimensional muscle model with respect to some morphological and functional characteristics is tested. Length-force relations of constructed muscles are compared with the actual length-force relations of mm. gastrocnemii mediales and mm. semimembranosi of male Wistar rats. Moreover, actual fiber angle, fiber length, and muscle thickness of three mm. gastrocnemii mediales are compared with values found for constructed muscles. It is concluded that the three-dimensional muscle model closely approximates the actual muscle form and function.     

Myofascial force transmission between antagonistic rat lower limb muscles: Effects of single muscle or muscle group lengthening

Effects of lengthening of the whole group of anterior crural muscles (tibialis anterior and extensor hallucis longus muscles (TA + EHL) and extensor digitorum longus (EDL)) on myofascial interaction between synergistic EDL and TA + EHL muscles, and on myofascial force transmission between anterior crural and antagonistic peroneal muscles, were investigated. All muscles were either passive or maximally active. Peroneal muscles were kept at a constant muscle tendon complex length. Either EDL or all anterior crural muscles were lengthened so that effects of lengthening of TA + EHL could be analyzed. For both lengthening conditions, a significant difference in proximally and distally measured EDL passive and active forces, indicative of epimuscular myofascial force transmission, was present. However, added lengthening of TA + EHL significantly affected the magnitude of the active and passive load exerted on EDL. For the active condition, the direction of the epimuscular load on EDL was affected; at all muscle lengths a proximally directed load was exerted on EDL, which decreased at higher muscle lengths. Lengthening of anterior crural muscles caused a 26% decrease in peroneal active force.

Extramuscular myofascial connections are thought to be the major contributor to the EDL proximo-distal active force difference. For antagonistic peroneal complex, the added distal lengthening of a synergistic muscle increases the effects of extramuscular myofascial force transmission.     

Relationships between range of motion,Lo, and passive force in five strap-like muscles of the feline hind limb

The relationships between range of motion, optimal length for force production (l_o), and passive force provide useful insights into the structure and function of muscles but are unknown for most individual muscles. We measured these values and examined their relationships in five strap-like muscles of the cat hind limb: caudofemoralis, semitendinosus, sartorius anterior, tenuissimus, and biceps femoris anterior. The range of motion relative to l_o was found to vary significantly between different muscles and even between different specimens of the same muscle. The passive force-length (FL) curve was found to be correlated with both l_o and l_max (maximal in situ muscle length) but was correlated more strongly with l_max. The mean passive force produced by these muscles at l_max was less than 7% of estimated maximal isometric force, suggesting that passive force may not be important in these muscles during normal activation patterns. The variance in passive FL curves between specimens of the same muscle was found to be significantly lower when length was scaled by l_max as opposed to l_o. These results suggest that l_max may provide a more useful scaling factor for generic models of muscle. However, the passive length-tension properties of mammalian muscle appear to reflect a complex mix of structures at both the myofilament and connective tissue levels that may differ depending on muscle-fiber architecture and perhaps on the history of trophic influences on a particular specimen. © 1996 Wiley-Liss, Inc.     

Mechanical work as predictor of force enhancement and force depression

The steady-state force following active muscle shortening or stretch differs from the maximum isometric force associated with the final length. This phenomenon proves that the isometric force production is not only dependent on current muscle length and length time derivative, but depends on the preceding contraction history. Isolated extensor digitorum longus and soleus muscles from mice (NMRI strain) were used to investigate the force produced by a muscle, and some parameters hypothetically influencing this history-dependent force modification. The muscles were pre-stimulated at a fixed length, then different stretch/shortening episodes were introduced, whereafter changes of the active force were recorded while the muscles were held isometrically to approach a steady-state force before de-stimulation. The mechanical work during active stretch and shortening was evaluated by integrating the product of force and ramp velocity over the length-varying period. The results show a negative linear correlation between the force modification and the mechanical work produced on or by the muscle, continuous between shortening and stretch. A corresponding modification of the passive force component following each stimulation was also observed. The conclusion is that the isometric force attained after stretch or shortening is well described by an asymptotic force which is determined by the mechanical work.     

Morphometric characteristics of the muscles of the lower extremities during movement in the human

Systemic investigation of the mechanisms that control locomotions in the man provides elaboration of some problems concerning properties and construction of the locomotory apparatus, especially morphometric characteristics of the muscles--length and shoulder of the thrust force. Various methods for determining length and shoulder of the thrust force are considered. The first method is based on substituting the muscles for mathematical or mechanical model. The second method makes it possible to define the degree of the muscle elongation by means of measuring the distance between the ends of the sectioned tendon; it includes regression equations, connecting a relative elongation of the muscles with the angle change in the joints. As a result of perfection of the previously known methods, mathematical models are made, owing to them it is possible to calculate morphometric characteristics of 8 muscles in the lower extremities using computer facilities. As an example, results of calculations, by means of various methods, morphometric characteristics of the gastrocnemius muscle are presented. They are obtained at registration of the kinematics of the run with a maximal speed. Satisfactive coincidence of the results is obtained.     

Serotonin modulates muscle function in the medicinal leech Hirudo verbana

The body wall muscles of sanguivorous leeches power mechanically diverse behaviours: suction feeding, crawling and swimming. These require longitudinal muscle to exert force over an extremely large length range, from 145 to 46 per cent of the mean segmental swimming length. Previous data, however, suggest that leech body wall muscle has limited capacity for force production when elongated. Serotonin (5-HT) alters the passive properties of the body wall and stimulates feeding. We hypothesized that 5-HT may also have a role in allowing force production in elongated muscle by changing the shape of the length-tension relationship (LTR). LTRs were measured from longitudinal muscle strips in vitro in physiological saline with and without the presence of 10 μM 5-HT. The LTR was much broader than previously measured for leech muscle. Rather than shifting the LTR, 5-HT reduced passive muscle tonus and increased active stress at all lengths. In addition to modulating leech behaviour and passive mechanical properties, 5-HT probably enhances muscle force and work production during locomotion and feeding.     

Mapping motor neuron activity to overt behavior in the leech

As an initial step in constructing a quantitative biomechanical model of the medicinal leech (Hirudo medicinalis), we determined the passive properties of its body wall over the physiological range of dimensions. The major results of this study were:

1. The ellipsoidal cross section of resting leeches is maintained by tonic muscle activation as well as forces inherent in the structure of the body wall (i.e., residual stress).
2. The forces required for longitudinal and circumferential stretch to maximum physiological dimensions were similar in magnitude. Cutting out pieces of body wall did not affect the passive longitudinal or circumferential properties of body wall away from the edges of the cut.
3. The strain (i.e., the percentage change in dimension) of different body segments when subject to the same force was identical, despite differences in muscle crosssections.
4. Serotonin, a known modulator of tension in leech muscles, affected passive forces at all physiological muscle lengths. This suggests that the longitudinal muscle is responsible for at least part of the passive tension of the body wall.
5. We propose a simple viscoelastic model of the body wall. This model captures the dynamics of the passive responses of the leech body wall to imposed step changes in length. Using steady-state passive tensions predicted by the viscoelastic model we estimate the forces required to maintain the leech at any given length over the physiological range.

     

Tonic finite element model of the lower limb

It is widely admitted that muscle bracing influences the result of an impact, facilitating fractures by enhancing load transmission and reducing energy dissipation. However, human numerical models used to identify injury mechanisms involved in car crashes hardly take into account this particular mechanical behavior of muscles. In this context, in this work we aim to develop a numerical model, including muscle architecture and bracing capability, focusing on lower limbs. The three-dimensional (3-D) geometry of the musculoskeletal system was extracted from MRI images, where muscular heads were separated into individual entities. Muscle mechanical behavior is based on a phenomenological approach, and depends on a reduced number of input parameters, i.e., the muscle optimal length and its corresponding maximal force. In terms of geometry, muscles are modeled with 3-D viscoelastic solids, guided in the direction of fibers with a set of contractile springs. Validation was first achieved on an isolated bundle and then by comparing emergency braking forces resulting from both numerical simulations and experimental tests on volunteers. Frontal impact simulation showed that the inclusion of muscle bracing in modeling dynamic impact situations can alter bone stresses to potentially injury-inducing levels.