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.     

The influence of track compliance on running

A model of running is proposed in which the leg is represented as a rack-and-pinion element in series with a damped spring. The rack-and-pinion element emphasizes the role of descending commands, while the damped spring represents the dynamic properties of muscles and the position and the rate sensitivity of reflexes. This model is used to predict separately the effect of track compliance on step length and ground contact time. The predictions are compared with experiments in which athletes ran over tracks of controlled spring stiffness. A sharp spike in foot force up to 5 times body weight was found on hard surfaces, but this spike disappeared as the athletes ran on soft experimental tracks. Both ground contact time and step length increased on very compliant surfaces, leading to moderately reduced running speeds, but a range of track stiffness was discovered which actually enhances speed.     

The Influence of Intrinsic Muscle Properties on Musculoskeletal System Stability: A Modelling Study

The objective of this study is to investigate how the intrinsic mechanical properties of muscles will affect the musculoskeletal system stability. A typical musculoskeletal joint driven by a pair of antagonist muscles confined only in the sigittal plane was constructed. The dynamic characteristics of the flexor and extensor muscles induced by neural inputs were represented by three dynamic processes: neural excitation, muscle activation and muscle contraction dynamics. The muscle contraction mechanics was described using a modified Hill's model with a Contractile Element (CE), a parallel elastic element and a serial elastic element. Additionally, the change of muscle Physiological Cross-Sectional Area (PCSA) and pennation angle during muscle contraction were also considered. A set of dynamic simulations were conducted by applying an external impulsive force at the distal part of the musculoskeletal system. Sensitivity analysis was conducted to investigate the effect of the CE's force-length relationship, the CE's force-velocity relationship, the force-length relationship of the serial elastic element, the parallel elastic element and the pennation angle on the system stability. The results show that the muscles with full intrinsic mechanical properties are sufficient to stabilize the system subject to an impulsive force perturbation without reflexive changes in activations. To guarantee a self-stabilizing ability, a proper CE's force-velocity relationship, the existence of a series elastic element and a sufficient muscle co-contraction level are necessary. This study would provide insight into the intrinsic design and function of the musculoskeletal system, and also give implications for the design of bionic actuators, biomimetic robotics and prosthetic devices.     

Toward an identification of mechanical parameters initiating periosteal remodeling: a combined experimental and analytic approach

The ability of bone to adapt to its mechanical environment is well recognized, although the specific mechanical parameters initiating or maintaining the adaptive responses have yet to be identified. Recently introduced mathematical models offer the potential to aid in the identification of such parameters, although these models have not been well validated experimentally or clinically. We formulated a complementary experimental/analytic approach, using an animal model with a well-controlled mechanical environment combined with finite element modeling (FEM). We selected the functionally isolated turkey ulna, since the loading could be completely characterized and the periosteal adaptive responses subsequently monitored and quantified after four and eight weeks of loading. Known loads input into a three-dimensional, linearly elastic FEM of the ulna then permitted full-field mechanical characterization of the ulna. The FEM was validated against a normal strain-gaged turkey ulna, loaded in vivo in an identical fashion to the experimental ulnae. Twenty-four candidate mechanical parameters were then compared to the quantified adaptive responses, using statistical techniques. The data supported strain energy density, longitudinal shear stress, and tensile principal stress/strain as the mechanical parameters most likely related to the initiation of the remodeling response. Model predictions can now suggest new experiments, against which the predictions can be supported or falsified.     

A viscoelastic model of mechanically induced and spontaneous contractions of vascular smooth muscle

A new viscoelastic model was developed for the mathematical characterisation of mechanically induced and intrinsic contractional responses of the vascular smooth muscle. To this end, the elastic and viscous analogue elements were supplemented with a new active element generating stress proportional to its momentary elongation. The four-element model consisting of an active element, a parallel viscous element and both series and parallel elastic elements predicted biphasic or damped oscillatory stress relaxation and creep responses which were similar to that found experimentally earlier. Above a certain exciting frequency the model exhibited dissipative and below energy producing behaviour, as indicated by the sign change of the hysteresis loop area. In the case of sinusoidal modulation of the stress generation parameter the model showed parametric resonance, which was regarded as a simulation of intrinsic oscillation of the smooth muscle.     

Mechanics of cardiac muscle, based on Huxley's model: Mathematical simulation of isometric contraction

Hill's three-component model (Maxwell model) is used to represent the mechanical property of cardiac muscle. The parallel and series elastic elements of the fibres are described according to their non-linear exponential function; and Huxley's sliding-filaments model, together with the activating role of calcium, is applied to the contractile element.

With this composite model, the following responses can be simulated mathematically: isometric twitch at various muscle lengths, tension-length relationships; isometric contraction during quick stretch; and the Bowditch Treppe and tension velocity relationships of the contractile element.     

Can a rheological muscle model predict force depression/enhancement?

A new phenomenological model of activated muscle is presented. The model is based on a combination of a contractile element, an elastic element that engages upon activation, a linear dashpot and a linear spring. Analytical solutions for a few selected experiments are provided. This model is able to reproduce the response of cat soleus muscle to ramp shortening and stretching and, unlike standard Hill-type models, computations are stable on the descending limb of the force–length relation and force enhancement (depression) following stretching (shortening) is predicted correctly. In its linear version, the model is consistent with a linear force–velocity law, which in this model is a consequence rather than a fundamental characteristic of the material. Results show that the mechanical response of activated muscle can be mimicked by a viscoelastic system. Conceptual differences between this model and standard Hill-type models are analyzed and the advantages of the present model are discussed.     

A description of arterial wall mechanics using limiting chain extensibility constitutive models

Certain aspects of the mechanical response of arterial walls can be described using nonlinear elasticity theory. Uniaxial tests on vascular walls reveal nonlinear stress-strain behavior, with higher extensibility in the low stretch range and progressively lower extensibility with increasing stretch. This phenomenon is well known in the framework of rubber-like materials where it is called a strain-hardening or strain-stiffening effect. Constitutive models of incompressible hyperelasticity that take this into account include power-law models and limiting chain extensibility models. Our purpose in this paper is to bring to the attention of the biomechanics community some essential features of one such model of the latter type due to Gent. This model is compared with isotropic versions of biomechanical constitutive models by Takamizawa-Hayashi and Fung; the latter is a limiting version of a power-law material. Two particular problems are considered for which experimental data on arterial wall deformations are available. The first concerns small oscillations superposed on a large static stretch of a vertical string of arterial tissue. It is shown that the exponential model of Fung and the Gent model match well with the experimental data. The second problem is the extension of an internally pressurized circular cylindrical tube. It is shown that an inversion phenomenon observed experimentally for the human iliac artery can be described within a membrane theory by the Gent model whereas this cannot be described using the exponential model. The foregoing considerations are carried out for isotropic elastic materials in the absence of residual stress. Extensions to include anisotropy are also indicated.     

Comparison of the exercise pressor reflex between forelimb and hindlimb muscles in cats

In thirteen cats anesthetized with alpha-chloralose, we compared the cardiovascular and ventilatory responses to both static contraction and tendon stretch of a hindlimb muscle group, the triceps surae, with those to contraction and stretch of a forelimb muscle group, the triceps brachii. Static contraction and stretch of both muscle groups increased mean arterial pressure and heart rate, and the responses were directly proportional to the developed tension. The cardiovascular increases, however, were significantly greater (P < 0.05) when the triceps brachii muscles were contracted or stretched than when the triceps surae muscles were contracted or stretched, even when the tension developed by either maneuver was corrected for muscle weight. Likewise, the ventilatory increases were greater when the triceps brachii muscles were stretched than when the triceps surae muscles were stretched. Contraction of either muscle group did not increase ventilation. Our results suggest that in the anesthetized cat the cardiovascular responses to both static contraction and tendon stretch are greater when arising from forelimb muscles than from hindlimb muscles.     

An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment

This study deals with the viscoelastic constitutive modeling and the respective computational analysis of the human passive myocardium. We start by recapitulating the locally orthotropic inner structure of the human myocardial tissue and model the mechanical response through invariants and structure tensors associated with three orthonormal basis vectors. In accordance with recent experimental findings the ventricular myocardial tissue is assumed to be incompressible, thick-walled, orthotropic and viscoelastic. In particular, one spring element coupled with Maxwell elements in parallel endows the model with viscoelastic features such that four dashpots describe the viscous response due to matrix, fiber, sheet and fiber-sheet fragments. In order to alleviate the numerical obstacles, the strictly incompressible model is altered by decomposing the free-energy function into volumetric-isochoric elastic and isochoric-viscoelastic parts along with the multiplicative split of the deformation gradient which enables the three-field mixed finite element method. The crucial aspect of the viscoelastic formulation is linked to the rate equations of the viscous overstresses resulting from a 3-D analogy of a generalized 1-D Maxwell model. We provide algorithmic updates for second Piola–Kirchhoff stress and elasticity tensors. In the sequel, we address some numerical aspects of the constitutive model by applying it to elastic, cyclic and relaxation test data obtained from biaxial extension and triaxial shear tests whereby we assess the fitting capacity of the model. With the tissue parameters identified, we conduct (elastic and viscoelastic) finite element simulations for an ellipsoidal geometry retrieved from a human specimen.     

Improved prediction of the collagen fiber architecture in the aortic heart valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.     

Effect of intercostal muscle and costovertebral joint material properties on human ribcage stiffness and kinematics

Current finite element (FE) models of the human thorax are limited by the lack of local-level validation, especially in the ribcage. This study exercised an existing FE ribcage model for a 50th percentile male under quasi-static point loading and dynamic sternal loading. Both force-displacement and kinematic responses of the ribcage were compared against experimental data. The sensitivity of the model response to changes in the material properties of the costovertebral (CV) joints and intercostal muscles was assessed. The simulations found that adjustments to the CV joints tended to change the amount of rib rotation in the sagittal plane, while changes to the elastic modulus and thickness of the intercostal muscles tended to alter both the stiffness and the direction and magnitude of rib motions. This study can lend insight into the role that the material properties of these two thoracic structures play in the dynamics of the ribcage during a frontal loading condition.     

Contribution of muscle series elasticity to maximum performance in drop jumping

The contribution of muscle in-series compliance on maximum performance of the muscle tendon complex was investigated using a forward dynamic computer simulation. The model of the human body contains 8 Hill-type muscles of the lower extremities. Muscle activation is optimized as a function of time, so that maximum drop jump height is achieved by the model. It is shown that the muscle series elastic energy stored in the downward phase provides a considerable contribution (32%) to the total muscle energy in the push-off phase. Furthermore, by the return of stored elastic energy all muscle contractile elements can reduce their shortening velocity up to 63% during push-off to develop a higher force due to their force velocity properties. The additional stretch taken up by the muscle series elastic element allows only m. rectus femoris to work closer to its optimal length, due to its force length properties. Therefore the contribution of the series elastic element to muscle performance in maximum height drop jumping is to store and return energy, and at the same time to increase the force producing ability of the contractile elements during push-off.     

A transversely isotropic constitutive model of excised guinea pig spinal cord white matter

Narrowing of the spinal canal generates an amalgamation of stresses within the spinal cord parenchyma. The tissue’s stress state cannot be quantified experimentally; it must be described using computational methods, such as finite element analysis. The objective of this research was to propose a compressible, transversely isotropic constitutive model, an augmentation of the isotropic Mooney–Rivlin hyperelastic strain energy function, to describe the guinea pig spinal cord white matter. Model parameters were derived from a combination of inverse finite element analysis on transverse compression experiments and least squared error analysis applied to quasi-static longitudinal tensile tests. A comparison of the residual errors between the predicted response and the experimental measurements indicated that the transversely isotropic constitutive law that incorporates an offset stretch reduced the error by a factor of four when compared to other commonly used models.     

Modeling of the muscle/tendon excursions and moment arms in the thumb using the commercial software anybody

A biomechanical model of a thumb would be useful for exploring the mechanical loadings in the musculoskeletal system, which cannot be measured in vivo. The purpose of the current study is to develop a practical kinematic thumb model using the commercial software Anybody (Anybody Technology, Aalborg, Denmark), which includes real CT-scans of the bony sections and realistic tendon/muscle attachments on the bones. The thumb model consists of a trapezium, a metacarpal bone, a proximal and a distal phalanx. These four bony sections are linked via three joints, i.e., IP (interphalangeal), MP (metacarpophalangeal) and CMC (carpometacarpal) joints. Nine muscles were included in the proposed model. The theoretically calculated moment arms of the tendons are compared with the corresponding experimental data by Smutz et al. [1998. Mechanical advantage of the thumb muscles. J. Biomech. 31(6), 565–570]. The predicted muscle moment arms of the majority of the muscle/tendon units agree well with the experimental data in the entire range of motion. Close to the end of the motion range, the predicted moment arms of several muscles (i.e., ADPt and ADPo (transverse and oblique heads of the adductor pollicis, respectively) muscles for CMC abduction/adduction and ADPt and FPB (flexor pollicis brevis) muscle for MP extension/flexion) deviate from the experimental data. The predicted moment potentials for all muscles are consistent with the experimental data. The findings thus suggest that, in a biomechanical model of the thumb, the mechanical functions of muscle–tendon units with small physiological cross-sectional areas (PCSAs) can be well represented using single strings, while those with large PCSAs (flat-wide attachments, e.g., ADPt and ADPo) can be represented by the averaged excursions of two strings. Our results show that the tendons with large PCSAs can be well represented biomechanically using the proposed approach in the major range of motion.     

Stretching Skeletal Muscle: Chronic Muscle Lengthening through Sarcomerogenesis

Skeletal muscle responds to passive overstretch through sarcomerogenesis, the creation and serial deposition of new sarcomere units. Sarcomerogenesis is critical to muscle function: It gradually re-positions the muscle back into its optimal operating regime. Animal models of immobilization, limb lengthening, and tendon transfer have provided significant insight into muscle adaptation in vivo. Yet, to date, there is no mathematical model that allows us to predict how skeletal muscle adapts to mechanical stretch in silico. Here we propose a novel mechanistic model for chronic longitudinal muscle growth in response to passive mechanical stretch. We characterize growth through a single scalar-valued internal variable, the serial sarcomere number. Sarcomerogenesis, the evolution of this variable, is driven by the elastic mechanical stretch. To analyze realistic three-dimensional muscle geometries, we embed our model into a nonlinear finite element framework. In a chronic limb lengthening study with a muscle stretch of 1.14, the model predicts an acute sarcomere lengthening from 3.09m to 3.51m, and a chronic gradual return to the initial sarcomere length within two weeks. Compared to the experiment, the acute model error was 0.00% by design of the model; the chronic model error was 2.13%, which lies within the rage of the experimental standard deviation. Our model explains, from a mechanistic point of view, why gradual multi-step muscle lengthening is less invasive than single-step lengthening. It also explains regional variations in sarcomere length, shorter close to and longer away from the muscle-tendon interface. Once calibrated with a richer data set, our model may help surgeons to prevent muscle overstretch and make informed decisions about optimal stretch increments, stretch timing, and stretch amplitudes. We anticipate our study to open new avenues in orthopedic and reconstructive surgery and enhance treatment for patients with ill proportioned limbs, tendon lengthening, tendon transfer, tendon tear, and chronically retracted muscles.     

A force-similarity model of the activated muscle is able to predict primary locomotor functions

Biomechanical macroscopic models of the muscle organ as whole are conceptually limited in explaining muscle function in relation to structure. The examples are Hill-type and rheological muscle models where elastic properties of the muscle's contractible element are approached by a spring arranged in series and parallel, respectively. A new scaling model of the activated muscle powering a particular function is proposed. This model is based on the physical similarity suggested between the action-production muscle force and resulting reaction elastic muscle forces. Considered at a macroscopic scale, this force similarity provides four patterns of constraints in development of muscle architecture in different-sized animals. As the result, the analytical modeling predicts the primary motor, brake, strut and spring functions of individual muscles revealed earlier in work-loop experiments and now provided in terms of the scaling exponents for muscle cross-sectional area and fiber length. The model reliability is tested via literature available from muscle allometric data. The conceptual outcome of the study is that the architecture design of skeletal muscles is likely effected by the powering contractions of last fibers known as having higher myofibril volume than slow fibers.     

The non-linear response of a muscle in transverse compression: assessment of geometry influence using a finite element model

Most recent finite element models that represent muscles are generic or subject-specific models that use complex, constitutive laws. Identification of the parameters of such complex, constitutive laws could be an important limit for subject-specific approaches. The aim of this study was to assess the possibility of modelling muscle behaviour in compression with a parametric model and a simple, constitutive law. A quasi-static compression test was performed on the muscles of dogs. A parametric finite element model was designed using a linear, elastic, constitutive law. A multi-variate analysis was performed to assess the effects of geometry on muscle response. An inverse method was used to define Young's modulus. The non-linear response of the muscles was obtained using a subject-specific geometry and a linear elastic law. Thus, a simple muscle model can be used to have a bio-faithful, biomechanical response.     

Effects of Local and Remote Muscle Pain on Stretch ReflexActivities in the Elbow Joint Flexors and Extensors of Unanesthetized Cats

In experiments on unanesthetized cats, we compared the effects of experimentally induced pain in the m. biceps brachii or in the neck muscles on EMG activity of the flexors and extensors of the elbow joint (mm. biceps et triceps brachii, respectively) evoked by a passive extension-flexion of the above joint. Muscle pain was induced by injections of 0.5 ml of a hypertonic (7%) NaCl solution into the above-mentioned muscles. In the case of pain in the biceps, i.e., in the muscle directly involved in realization of the reflex, we observed an increase in the amplitude and significant shortening of the latency of EMG responses of this muscle. The amplitude of a short-latency (supposedly monosynaptic) component of the biceps reflex (М1 response) increased by 65%, while an increment of the latter (supposedly polysynaptic) М2 component was 117%. When pain was induced in anatomically remote neck muscles, the stretch reflex in the biceps was considerably suppressed. The maximum amplitudes of the М1 and М2 components decreased by 25 and 30%, respectively, but the latencies of these components decreased significantly, similarly to what was observed in the case of induction of experimental pain in the biceps. Under both conditions of experimental pain, changes in the parameters of EMG responses of the forearm extensor (m. triceps brachii) demonstrated similarity with those of the biceps responses. The maximum effect of pain induction was observed within the first 5 min after injections of the hypertonic solution; full recovery of the stretch reflex parameters was observed on the 20th to 30th min. We conclude that the effects of pain induction on the reflex under study are not generalized. They depend on the site of such induction with respect to the muscle where the stretch reflex is elicited. Unidirectional effects of both types of pain on the antagonist muscles allow us to suppose that modulation of the reflex reactions upon pain induction is mediated by influences from the supraspinal CNS structures. Induction of pain in the biceps increased the amplitude of EMG manifestations of the stretch reflex, while such induction in the neck muscles decreased such responses; nonetheless, in both cases the latency of the reflexes decreased. This fact allows us to believe that the sensitivity of muscle spindles increased under both conditions of the pain influence.