Continual mechanochemical model of muscular tissue: PMM vol. 37, n≗3, 1973, pp. 448–458

Abstract

The behavior of active muscular tissue is described with the help of a closed system of equations of motion of a two-phase, multicomponent, anisotropic continuous medium, with the mechanochemical processes occurring within it taken into account. The fundamental hypotheses are based on the information of general character concerning the structure and performance of the muscular tissue. It is assumed that the phase in which the mechanochemical reactions take place is viscoelastic, while the other phase is assumed elastic. The medium is assumed to have single velocity, although a passage of components between the phases is allowed. The laws of conservation are given and the rheological equations are written in accordance with the general principles of the mechanics of continuous medium and thermodynamics of irreversible processes [1–4]. It is shown that the model constructed describes, e.g., such characteristic properties of the muscle tissue as the existence of stresses in the absence of strains, zero-load deformations, and dissipation of energy in the state of mechanical equilibrium. The activity of the muscular tissue is governed by chemical processes taking place in the tissue, within the specific ordered structures called myofibrillae and, in the final count, by the mechanochemical reactions which affect the form or the relative distribution of the protein molecules [5–8]. Outside the myofibrillae we have various auxilliary systems, the connecting tissue and other structures, including capillary blood vessels which serve as the source of initial chemical compounds. The onset of active muscular contraction is connected with the arrival of specific reagents at the myofibrillae. The study of various physiological phenomena (such as the working of heart, propagation of an excitation in the tissues, organization of movements, regulation of the blood circulation and breathing) requires, on the one hand. a rheological equation of the active muscle to be available, on the other hand it requires the knowledge of the relation connecting the basic mechanical parameters of the muscle (such as length, load, etc.) with the parameters defined by the chemical processes, such as the energy needed by the muscle during contraction. The absence of a sufficiently general model of the muscle makes necessary the use, at the present time, of much simpler models [9–11]. The following problems are usually considered in the literature on their own: the propagation of excitation throughout the muscle tissue, mechanics of the muscle as a whole in terms of the load — length, the energetics and biochemistry of the muscular activity. All existing models of the muscular tissue are, with the exception of one given in [12], one-dimensional and are given, as a rule. in the form of relations connecting the load and extension and their derivatives with respect to time, with certain additional parameters which have the dimensions of the load or extension. These additional parameters have no explicit connection with the chemical or other internal processes taking place within the muscle, and their variation with respect to time is defined a priori, differently for the muscle in the active and in the passive state [12–18]. Thus the author of [12] proposes a rheological equation of the form P ij d = k ijln ( ε ln → N ln ') (0.1) where P ij d is the deviator of the stress tensor, ε ij is the strain tensor and N ij' is a tensor parameter (“biofactor”) characterizing the activity of the tissue; N ij' = 0 and N ij' ≠ 0 for the passive and the active state, respectively. The so-called three-element models (see e. g. [19]) contain practically the same additional parameters. Another group of one-dimensional models explains the meaning of the additional parameters contained in the rheological equation using the terms and quantities chatacterizing the microstructure of the muscle and its hypothetic “internal” mechanics (see e. g. [16–18, 20, 21]). We have already noted that these models are also based on the assumption that active deformations (contractions), i. e. zero-stress deformations, are possible. The purpose of this paper is to construct a model of a continuous medium with properties typical of a muscular tissue, without assumptions of existence of active deformations and without using the discussion concerning the micro-structure of the myofibrillae and the processes taking place within them.