The aim of this study was to investigate the influence of fiber orientation in the left ventricular (LV) wall on the ejection fraction, efficiency, and heterogeneity of the distributions of developed fiber stress, strain and ATP consumption. A finite element model of LV mechanics was used with active properties of the cardiac muscle described by the Huxley-type cross-bridge model. The computed variances of sarcomere length (SL(var)), developed stress (DS(var)), and ATP consumption (ATP(var)) have several minima at different transmural courses of helix fiber angle. We identified only one region in the used design space with high ejection fraction, high efficiency of the LV and relatively small SL(var), DS(var), and ATP(var). This region corresponds to the physiological distribution of the helix fiber angle in the LV wall. Transmural fiber angle can be predicted by minimizing SL(var) and DS(var), but not ATP(var). If ATP(var) was minimized, then the transverse fiber angle was considerably underestimated. The results suggest that ATP consumption distribution is not regulating the fiber orientation in the heart.

The dispersive effects due to the presence of microstructure in solids are studied. The basic mathematical model is derived following Mindlin's theory. In the onedimensional case the governing equations of a linear system are presented. An approximation using the slaving principle indicates a hierarchy of waves. The corresponding dispersion relations are compared with each other. The choice between the models can be made on the basis of physical effects described by dispersion relations.

Recent studies have revealed the structural and functional interactions between mitochondria, myofibrils and sarcoplasmic reticulum in cardiac cells. Direct channeling of adenosine phosphates between organelles identified in the experiments indicates that diffusion of adenosine phosphates is limited in cardiac cells due to very specific intracellular structural organization. However, the mode of diffusion restrictions and nature of the intracellular structures in creating the diffusion barriers is still unclear, and, therefore, a subject of active research. The aim of this work is to analyze the possible role of two principally different modes of restriction distribution for adenosine phosphates (a) the uniform diffusion restriction and (b) the localized diffusion limitation in the vicinity of mitochondria, by fitting the experimental data with the mathematical model. The reaction-diffusion model of compartmentalized energy transfer was used to analyze the data obtained from the experiments with the skinned muscle fibers, which described the following processes: mitochondrial respiration rate dependency on exogenous ADP and ATP concentrations; inhibition of endogenous ADP-stimulated respiration by pyruvate kinase (PK) and phosphoenolpyruvate (PEP) system; kinetics of oxygen consumption stabilization after addition of 2 mM MgATP or MgADP; ATPase activity with inhibited mitochondrial respiration; and buildup of MgADP concentration in the medium after addition of MgATP. The analysis revealed that only the second mechanism considered--localization of diffusion restrictions--is able to account for the experimental data. In the case of uniform diffusion restrictions, the model solution was in agreement only with two measurements: the respiration rate as a function of ADP or ATP concentrations and inhibition of respiration by PK + PEP. It was concluded that intracellular diffusion restrictions for adenosine phosphates are not distributed uniformly, but rather are localized in certain compartments of the cardiac cells.

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The formal structure of generalized continuum theories is recovered by means of the extension of canonical thermomechanics with dual weakly non-local internal variables. The canonical thermomechanics provides the best framework for such generalization. The Cosserat, micromorphic, and second gradient elasticity theory are considered as examples of the obtained formalization.Keywords Generalized continua · Canonical thermomechanics · Internal variables · Microstructure · Cosserat medium · Micromorphic medium MotivationGeneralized continuum theories extend conventional continuum mechanics by incorporating intrinsic microstructural effects in the mechanical behavior of materials [1][2][3][4][5]. Internal variable approach was always an alternative framework for the continuum modeling of such effects in materials [6][7][8][9][10][11][12]. However, the wellestablished theory of internal variables of state [13,14] cannot completely describe a generalized medium because an internal variable of state has no inertia, but it dissipates. If inertia is introduced, the internal variable must be treated as an actual degree of freedom [15]. Accordingly, the variable is not "internal" any more but can be controlled for instance at the boundary of a body.A more general thermodynamic framework of the internal variable theory presented recently [16] is based on a duality between internal variables, which make possible to derive evolution equations both for internal variables of state and internal degrees of freedom. A natural question relates to the ability of this duality concept to comprise inertial effects. To answer this question, we show how the dual internal variables can be introduced into continuum mechanics and how certain generalized continuum theories can be interpreted in terms of the dual internal variables.The most suitable framework for the generalization of continuum theory by weakly non-local dual internal variables enriched by an extra entropy flux is the material formulation of continuum thermomechanics [17,18]. Therefore, basic definitions of the canonical thermomechanics [17] are recalled in Sect. 2 of the paper. Then dual variables are introduced in Sect. 3 and evolution equations for both dissipative and non-dissipative processes are derived in Sect. 3.2. Linear Cosserat, micromorphic, and second gradient elasticity theories are

The propagation of action potentials in nerve fibres is usually described by models based on the ionic hypotheses. However, this hypothesis does not provide explanation of other experimentally verified phenomena like the swelling of fibres and heat production during the nerve pulse propagation. Heimburg and Jackson (Proc Natl Acad Sci USA 102(28):9790-9795, 2005, Biophys Rev Lett 2:57-78, 2007) have proposed a model describing the swelling of fibres like a mechanical wave related to changes of longitudinal compressibility of the cylindrical membrane. In this paper, the possible dispersive effects in such microstructured cylinders are analysed from the viewpoint of solid mechanics, particularly using the information from the analysis of the well-known rod models. A more general governing equation is proposed which satisfies the conditions imposed by the physics of wave processes. The numerical simulations demonstrate the influence of nonlinearities, the role of various dispersion terms and the formation and propagation of solitary waves along the wall together with the corresponding transverse displacement. It is conjectured that due to the coupling effects between longitudinal and transverse displacements of a cylinder, the transverse displacement (i.e. swelling) is related to the derivative of the longitudinal displacement. In this way, the correspondence between theoretical and experimental (Tasaki in Physiol Chem Phys Med NMR 20:251-268, 1988) results can be described.

The propagation of an action potential (AP) in a nerve fibre is accompanied by mechanical and thermal effects. In this paper, an attempt is made to build up a mathematical model which couples the AP with a possible pressure wave (PW) in the axoplasm and waves in the nerve fibre wall (longitudinal-LW and transverse-TW) made of a lipid bilayer (biomembrane). A system of differential equations includes the governing equations of single waves with coupling forces between them. The single equations are kept as simple as possible in order to carry out the proof of concept. An assumption based on earlier studies is made that the coupling forces depend on changes (the gradient, time derivative) of the voltage. In addition, it is assumed that the transverse displacement of the biomembrane can be calculated from the gradient of the LW in the biomembrane. The computational simulation is focused to determining the influence of possible coupling forces on the emergence of mechanical waves from the AP. As a result, an ensemble of waves (AP, PW, LW, TW) emerges. The further experiments should verify assumptions about coupling forces.

Wave propagation in microstructured materials is directly affected by the existence of internal space scale(s) in the compound matter. In this case the classical continuum theory cannot be used. In this paper based on the Mindlin model, the balance laws for macro-and microstructure are formulated separately. Using the slaving principles relating macro-and microdisplacements, the governing equations are derived for a single-and two-scale (scale within scale) cases. These equations exhibit hierarchical properties assigning the wave operators to internal scales. In terms of macrodisplacements, higher-order dispersive terms appear having a clear physical background (microinertia, wave speed in microstructure) related to the scale of the microstructure. Full, approximated (corresponding to hierarchical models), and simplified dispersion relations are derived and analysed to demonstrate the validity of the hierarchical governing equations. Linear theory is based on the quadratic free energy function, in nonlinear theory the cubic terms should also be taken into account. The corresponding governing equation includes nonlinearities in both macro-and microscale. Such consistent modelling opens up new possibilities to Nondestructive Testing (NDT) of material properties.