Assessing the role of Ca in skeletal muscle fatigue using a multi-scale continuum model
Introduction
When skeletal muscles are activated repeatedly with intensity, it is well known that the force output declines. But the force is not the only feature affected by fatigue in this tissue: shortening velocity and relaxation behavior are also affected (Allen, Kabbara, Westerblad, 2002, Jones, 2010). Although the central nervous system, motor nerves and the neuromuscular junctions can contribute to this phenomenon, the main mechanisms are located in the muscle itself (Allen et al., 2002). A variety of intracellular processes appear to be responsible for fatigue, and these have been assessed and reported extensively in the literature (Allen, Westerblad, 2001, Allen, Kabbara, Westerblad, 2002, Allen, Lamb, Westerblad, 2008, Westerblad, Allen, 1991, Westerblad, Allen, 1993). Although all of these experiments provide invaluable insights into these mechanisms, according to Röhrle et al. (2012) computer simulations in conjunction with experimental findings can be a powerful tool for evaluating complex hypotheses and conclusions.
Modeling the behavior of skeletal muscle has typically been focused on sub-cellular processes of a half-sarcomere or on simplified phenomenological relationships to simulate the whole muscle (Röhrle et al., 2012). Regarding the first type of models, the interaction between actin and myosin filaments was initially simulated using a two state model by Huxley (1957). Although this model predicted good results under the mechanical point of view, it was later suggested that the attachment of the cross-bridge occurred at different stages to fulfill heat release outcomes (Huxley, 1973) and transient dynamic responses (Huxley and Simmons, 1971). The complexity of these models was also reduced to achieve more accessible formulation to simulate macroscopic muscle dynamics (Wu, Herzog, Cole, 1997, Zahalak, 1981). Furthermore, new formulations inspired by contractile processes appeared in order to consider dynamic contractions in a half-sarcomere (Razumova, Bukatina, Campbell, 1999, Razumova, Bukatina, Campbell, 2000).
Continuous reduction of intracellular Ca together with force deficit due to fatigue condition have been proved by experimental investigations on skeletal muscles (Allen, Westerblad, 2001, Westerblad, Allen, 1991). Moreover, the shortening velocity of the tissue is also affected by fatigue, which means a loss of power production and poor performance. These changes in the force-velocity relationship could be caused by Ca deficit (Jones, 2010, Ruiter, Didden, Jones, Haan, 2000). Since Ca is the initiator agent of active behavior in sarcomere, experimental tests suggest a direct relationship between the decay of Ca and the functional effects of fatigue (Allen, Westerblad, 2001, Jones, 2010).
In this paper, a multi-scale continuum model that fulfills thermodynamic and mechanical requirements is presented to simulate skeletal muscle contraction under fatigue conditions. First, the chemical phase involved in activation at the sarcomere level is described and formulated in Section 2. The thermodynamic basis that allows the derivation of constitutive laws is presented in Section 3 and is particularized in Section 4. Finally, the assessment and validation of the model using experimental tests is presented in Section 5. The ability of the model to predict the fatigue response of the tissue is discussed in Section 6 together with the formulation proposed and the numerical results obtained.
Section snippets
Cross-bridge kinetics
Cross-bridge cycling is responsible for the movement and force production in skeletal muscle cells. In conditions of relaxation, the tropomyosin-troponin complex on the actin filament blocks the actin binding sites and avoids the formation of the cross-bridge with the myosin head. When the Ca concentration increases above a certain level or resting threshold, Ca binds to troponin and tropomyosin exposes the myosin binding sites to cross-bridge formation. Since cross-bridges within a
Thermodynamic model
The deformation associated with muscle activity can be modeled as two fictitious steps (Hernández-Gascón, Grasa, Calvo, Rodríguez, 2013, Stålhand, Klarbring, Holzapfel, 2008). The first corresponds to the relative motion of the myosin with respect to actin, while the second relates to the elastic deformation of cross-bridges. Mathematically, it can be expressed as a multiplicative decomposition of the tissue stretch as: where λa defines the deformation associated with the contractile
Model specialization
As mentioned in the previous section, different functions have to be proposed to obtain the response of the tissue under different levels of activation induced by the calcium concentration. First, the following form of the strain energy is considered: where the total energy has been decomposed into several terms related to the energy associated with the passive behavior of the muscle Ψe(λ), the elastic energy in the cross-bridge attachment Ψa(λe, si), the
Model assessment and results
To study the ability of the model to predict muscle fatigue, different assumptions for the parameters that define its behavior were established due to the lack of experimental results for identical animal species. Initially, as shown in the following section, the unfatigued behavior of the muscle was analyzed to reproduce a single tetanic contraction and the force-velocity relationship. Then, the effect of the Ca concentration was assessed in the force/stress response. Finally, the model was
Discussion
The multi-scale model developed in this work, using the [Ca] level as an input governing parameter of the skeletal muscle contraction, is able to reproduce the most important characteristic features of fatigue such as force deficit, reduction in shortening velocity and alterations in the contraction-relaxation cycle. In previous studies related to muscle fatigue, it has been accepted that this force decreasing phenomenon is a consequence of a lower number of connections in post- and pre-power
Conclusions
A multi-scale chemo-mechanical model for predicting the active behavior of skeletal muscle under both unfatigued and fatigued conditions has been presented. The constitutive framework of the model adheres to the thermodynamic laws so that it is suitable to be applied in 3D simulations. A comparison of the computational results with experimental data demonstrates the ability of the model to simulate isometric and concentric contractions and the force-velocity relationship in skeletal muscles.
Acknowledgments
The authors gratefully acknowledge research support from the Spanish Ministry of Economy and Competitiveness (Grants DPI2014-54981-R and DPI2017-84047-R) and the Gobierno de Aragón for the support of group T24_17R. The first author is grateful for research support from the Iranian Ministry of Science, Research and Technology.
The authors also acknowledge the support of the Tissue Characterization Platform of CIBER-BBN, an initiative funded by the VI National RDi Plan 2008–2011, Iniciativa
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