Viscoelastic Models Describing Stress Relaxation and Creep in Soft Tissues
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Viscoelastic Models Describing Stress Relaxation and Creep in Soft Tissues Alexa V. Kobelev1,2, Rimma M. Kobeleva2, Yuri L. Protsenko2, Irina V. Berman3 and Oleg A. Kobelev4 1 Institute of Metal Physics, Russian Academy of Sciences, Urals branch, 18 Kovalevskoj, Yekaterinburg 620219, Russian Federation 2 Institute of Immunology and Physiology, Russian Academy of Sciences, Urals branch, 91 Pervomajskaja, Yekaterinburg 620219, Russian Federation 3 San Jose State University, Department of Physics, One Washington Square, San Jose, CA, 95192-0106, U.S.A. 4 Urals State Technical University, 19 Mira, Yekaterinburg 620002, Russian Federation ABSTRACT The review of our approach to the specific rheological properties modeling of myocardium has been given. We are working on the level of ‘fascicule’ as a tissue element, which consists of several cardiomyocytes surrounded by a connective tissue shell. Actually, these properties are characteristic of large majority of living soft tissues. In order to describe essentially nonlinear static ‘force-deformation’ curves and quasi-static hysteresis loops together with non-exponential stress relaxation and creep time courses we suggest a set of 2D graphs with different topology composed of classical linear Hook’s springs and Newtonian damps. Each spring and dashpot represents a group of 3D tissue structural elements. The stress response functions of these models are found for the uniaxial step-wise, or pulse (column-like) external stretching. The inverse response functions of the longitudinal displacement for the external stress loading of the pulse shape time dependencies were also found. The values of elastic modules and viscous coefficients are estimated by comparison theoretical curves of relaxation, creep and recovery with the experimental data. The latter are obtained on rather different objects, passive muscle preparations (the stress relaxation response) and endothelium cells (the creep response). It has been stated that the proposed 2D models appear to be quite general to describe nonlinear relaxation and creep properties, which are lacking in the traditionally used uniaxial 1D models. INTRODUCTION All the major morphological components of papillary muscle are well defined [1, 2]. We suggest that they may be grouped into longitudinal, transversal and inclined ones according to their connective direction. On a fascicule level the inclined and longitudinal elements represent the amount of connective tissue fibers of the covering shell, the transversal elements represent the matrix, the intracellular protein fibers, and the intracellular fluid provides the viscosity. Until now the analysis of passive ‘stress-strain’ dependencies for heart muscles, as a rule, is based on uniaxial rheological models [1, 3]. In other words, the transversal deformations are not taken into account. For the first time, the input of transversal deformations into the origin of nonlinear ‘force-length’ dependence for isolated myocardium preparations was noted by us in [4], where 2D topological model
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