Role of the Z band in the mechanical properties of the heart

In striated muscle the mechanism of contraction involves the cooperative movement of contractile and elastic components. This review emphasizes a structural approach that describes the cellular and extracellular components with known anatomical, biochemical, and physical properties that make them candidates for these contractile and elastic components. Classical models of contractile and elastic elements and their underlying assumptions are presented. Mechanical properties of cardiac and skeletal muscle are compared and contrasted and then related to ultrastructure. Information from these approaches leads to the conclusion that the Z band is essential for muscle contraction. Our review of Z band structure shows the Z band at the interface where extracellular components meet the cell surface. The Z band is also the interface from cell surface to myofibril, from extra-myofibrillar to myofibril, and finally from sarcomere to sarcomere. Our studies of Z band in defined physiologic states show that this lattice is an integral part of the contractile elements and can function as an elastic component. The Z band is a complex dynamic lattice uniquely suited to play several roles in muscle contraction.     

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.     

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.     

A time-dependent phenomenological model for cell mechano-sensing

Adherent cells normally apply forces as a generic means of sensing and responding to the mechanical nature of their surrounding environment. How these forces vary as a function of the extracellular rigidity is critical to understanding the regulatory functions that drive important phenomena such as wound healing or muscle contraction. In recognition of this fact, experiments have been conducted to understand cell rigidity-sensing properties under known conditions of the extracellular environment, opening new possibilities for modeling this active behavior. In this work, we provide a physics-based constitutive model taking into account the main structural components of the cell to reproduce its most significant contractile properties such as the traction forces exerted as a function of time and the extracellular stiffness. This model shows how the interplay between the time-dependent response of the acto-myosin contractile system and the elastic response of the cell components determines the mechano-sensing behavior of single cells.     

Evaluation of Muscle Function of the Extensor Digitorum Longus Muscle Ex vivo and Tibialis Anterior Muscle In situ in Mice

Body movements are mainly provided by mechanical function of skeletal muscle. Skeletal muscle is composed of numerous bundles of myofibers that are sheathed by intramuscular connective tissues. Each myofiber contains many myofibrils that run longitudinally along the length of the myofiber. Myofibrils are the contractile apparatus of muscle and they are composed of repeated contractile units known as sarcomeres. A sarcomere unit contains actin and myosin filaments that are spaced by the Z discs and titin protein. Mechanical function of skeletal muscle is defined by the contractile and passive properties of muscle. The contractile properties are used to characterize the amount of force generated during muscle contraction, time of force generation and time of muscle relaxation. Any factor that affects muscle contraction (such as interaction between actin and myosin filaments, homeostasis of calcium, ATP/ADP ratio, etc.) influences the contractile properties. The passive properties refer to the elastic and viscous properties (stiffness and viscosity) of the muscle in the absence of contraction. These properties are determined by the extracellular and the intracellular structural components (such as titin) and connective tissues (mainly collagen) ^1-2. The contractile and passive properties are two inseparable aspects of muscle function. For example, elbow flexion is accomplished by contraction of muscles in the anterior compartment of the upper arm and passive stretch of muscles in the posterior compartment of the upper arm. To truly understand muscle function, both contractile and passive properties should be studied.The contractile and/or passive mechanical properties of muscle are often compromised in muscle diseases. A good example is Duchenne muscular dystrophy (DMD), a severe muscle wasting disease caused by dystrophin deficiency ³. Dystrophin is a cytoskeletal protein that stabilizes the muscle cell membrane (sarcolemma) during muscle contraction ⁴. In the absence of dystrophin, the sarcolemma is damaged by the shearing force generated during force transmission. This membrane tearing initiates a chain reaction which leads to muscle cell death and loss of contractile machinery. As a consequence, muscle force is reduced and dead myofibers are replaced by fibrotic tissues ⁵. This later change increases muscle stiffness ⁶. Accurate measurement of these changes provides important guide to evaluate disease progression and to determine therapeutic efficacy of novel gene/cell/pharmacological interventions. Here, we present two methods to evaluate both contractile and passive mechanical properties of the extensor digitorum longus (EDL) muscle and the contractile properties of the tibialis anterior (TA) 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.     

Changes in the isoform composition of the cytoskeletal protein titn--adaptation process in hibernation

Changes in the isoform composition of the elastic protein titin from skeletal and cardiac muscles of hibernating ground squirrels were revealed for the first time. It was shown that, upon hibernation, the molecular mass of titin decreases and its functional properties change as compared with the active state of the animal. The physiological significance of the changes in titin isoform composition for the inhibition of muscle contractile activity upon hibernation is discussed in connection with similar changes during some cardiomyopathies.     

A model of multicomponent cardiac fibre

In order to simulate the contraction of a cardiac myofibre, a multicomponent fibre model has been developed. This model is composed of a series of segments which are activated in succession. Each segment is represented by the Hill's three component model of the sarcomere. The contractile element behaviour is described by the Huxley's theory and the time dependence agrees with the activation factor proposed by Julian for skeletal muscle, and modified by Wong for cardiac muscle. The two elastic elements have non-linear exponential characteristics. The isometric contraction of the multicomponent fibre has been simulated by means of a computer program. The results show the tension generated by the fibre, the propagation of the contraction along the fibre and the different contribution of each segment depending on its position inside the fibre.     

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.     

Effects of elastic loading on porcine trachealis muscle mechanics

To shorten in vivo, airway smooth muscle must overcome an elastic load provided by cartilage and lung parenchyma. We examined the effects of linear elastic loads (0.2-80 g/cm) on the active changes in porcine trachealis muscle length and tension in response to electrical field stimulation in vitro. Increasing elastic loads produced an exponential decrease in the shortening and velocity of shortening while causing an increase in tension generation of muscle strips stimulated by electrical field stimulation. Shortening was decreased by 50% at a load of 8 g/cm. At small elastic loads (less than or equal to 1 g/cm) contractile responses approximated isotonic responses (shortening approximately 60% of starting length), whereas at large loads (20 g/cm) responses approximated isometric responses with minimal shortening (20%). We conclude that elastic loading significantly alters the mechanical properties of airway smooth muscle in vitro, effects that are likely relevant to the loads against which the smooth muscle must contract in vivo.     

Cardiac sarcomere mechanics in health and disease

The sarcomere is the fundamental structural and functional unit of striated muscle and is directly responsible for most of its mechanical properties. The sarcomere generates active or contractile forces and determines the passive or elastic properties of striated muscle. In the heart, mutations in sarcomeric proteins are responsible for the majority of genetically inherited cardiomyopathies. Here, we review the major determinants of cardiac sarcomere mechanics including the key structural components that contribute to active and passive tension. We dissect the molecular and structural basis of active force generation, including sarcomere composition, structure, activation, and relaxation. We then explore the giant sarcomere-resident protein titin, the major contributor to cardiac passive tension. We discuss sarcomere dynamics exemplified by the regulation of titin-based stiffness and the titin life cycle. Finally, we provide an overview of therapeutic strategies that target the sarcomere to improve cardiac contraction and filling.     

Modeling the oscillation dynamics of activated airway smooth muscle strips

When strips of activated airway smooth muscle are stretched cyclically, they exhibit force-length loops that vary substantially in both position and shape with the amplitude and frequency of the stretch. This behavior has recently been ascribed to a dynamic interaction between the imposed stretch and the number of actin-myosin interactions in the muscle. However, it is well known that the passive rheological properties of smooth muscle have a major influence on its mechanical properties. We therefore hypothesized that these rheological properties play a significant role in the force-length dynamics of activated smooth muscle. To test the plausibility of this hypothesis, we developed a model of the smooth muscle strip consisting of a force generator in series with an elastic component. Realistic steady-state force-length loops are predicted by the model when the force generator obeys a hyperbolic force-velocity relationship, the series elastic component is highly nonlinear, and both elastic stiffness and force generation are adjusted so that peak loop force equals isometric force. We conclude that the dynamic behavior of airway smooth muscle can be ascribed in large part to an interaction between connective tissue rheology and the force-velocity behavior of contractile proteins.     

Troponin: structure, properties, and mechanism of functioning.

This review discusses the structure and properties of the isolated components of troponin, their interaction, and the mechanisms of regulation of contractile activity of skeletal and cardiac muscle. Data on the structure of troponin C in crystals and in solution are presented. The Ca2+-induced conformational changes of troponin C structure are described. The structure of troponin I is analyzed and its interaction with other components of actin filaments is discussed. Data on phosphorylation of troponin I by various protein kinases are presented. The role of troponin I phosphorylation in the regulation of contractile activity of the heart is analyzed. The structural properties of troponin T and its interaction with other components of thin filaments are described. Data on the phosphorylation of troponin T are presented and the effect of troponin T phosphorylation on contractile activity of different muscles is discussed. Modern models of the functioning of troponin are presented and analyzed.     

Modelling of non-linear elastic tissues for surgical simulation

Realistic modelling of the interaction between surgical instruments and human organs has been recognised as a key requirement in the development of high-fidelity surgical simulators. Primarily due to computational considerations, most of the past real-time surgical simulation research has assumed linear elastic behaviour for modelling tissues, even though human soft tissues generally possess non-linear properties. For a non-linear model, the well-known Poynting effect developed during shearing of the tissue results in normal forces not seen in a linear elastic model. Using constitutive equations of non-linear tissue models together with experiments, we show that the Poynting effect results in differences in force magnitude larger than the absolute human perception threshold for force discrimination in some tissues (e.g. myocardial tissues) but not in others (e.g. brain tissue simulants).     

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.     

Dynamic and steady state characteristics of the rabbit gastrocnemius muscle determined in series measurements

1. Within the range of the given conditions of measuring static and dynamic properties of the rabbit gastrocnemius muscle the following results were obtained: a) the dependence of the maxima of isotonic shortening upon the relative length of the muscle at constant load is linear; b) the parameters of the non-linear dependence of the passive elastic force of the muscle upon its relative length (measured in series) were identified using asymptotic regression; c) the time course of isotonic contractions (at an interval from 0 to 0.3 s after the beginning of stimulation) could be satisfactorily approximated by responses of a linear system to a step-function; d) the time course of isometric contractions (at an interval from 0 to 0.3 s after the beginning of stimulation) could be closely approximated by responses of a linear system to a step-function. 2. The time constants of isotonic and isometric contractions were determined as the parameters of the corresponding linear systems. 3. The maximum rates of the isometric and isotonic contractions were determined as maxima of the first derivatives of the responses of the corresponding models. 4. The experimental set-up made it possible to compare the values of the parameters concomitantly followed at various muscle lengths and at various loads.     

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.     

Measurement of the hyperelastic properties of ex vivo brain tissue slices

The elastic and hyperelastic properties of brain tissue are of interest to the medical research community as there are several applications where accurate characterization of these properties is crucial for an accurate outcome. The linear response is applicable to brain elastography, while the non-linear response is of interest for surgical simulation programs. Because of the biological differences between gray and white matter, it is reasonable to expect a difference in their mechanical properties. The goal of this work is to characterize the elastic and hyperelastic properties of the brain gray and white matter. In this method, force-displacement data of these tissues were acquired from 25 different brain samples using an indentation apparatus. These data were processed with an inverse problem algorithm using finite element method as the forward problem solver. Young's modulus and the hyperelastic parameters corresponding to the commonly used Polynomial, Yeoh, Arruda-Boyce, and Ogden models were obtained. The parameters characterizing the linear and non-linear mechanical behavior of gray and white matters were found to be significantly different. Young's modulus was 1787±186 and 1195±157Pa for white matter and gray matter, respectively. Among hyperelastic models, due to its accuracy, fewer parameters and shorter computational time requirements, Yeoh model was found to be the most suitable. Due to the significant differences between the linear and non-linear tissue response, we conclude that incorporating these differences into brain biomechanical models is necessary to increase accuracy.     

Large deformation of elastic tubes

In this paper a study is made of the large deformation of elastic tubes. Analysis is confined to the case of statical deformation. The energy of deformation is considered to be a function of the three strain invariants only. In order to determine its form for actual materials it is expanded in its Taylor's series. From the theoretical analysis of the inflation and extension of tubes, linear algebraic equations in the elastic constants are developed. Using these and appropriate measurements of diametral change and longitudinal stretch, obtained from experiments, elastic constants for the non-linear theory of elasticity are explicity determined with the aid of a computer program. The highly non-linear forms of deformation of tubes computed on the basis of the theory are compared with our experimental findings for a latex rubber tubing and for some bio-physical data taken from the literature.