This paper deals with modeling the passive behavior of skeletal muscle tissue including certain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators. The model verifies that an externally applied vibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells. From the numerical analysis point of view, one faces here a partial differential-algebraic equation (PDAE) that after semi-discretization in space by finite elements possesses an index up to three, depending on certain physical parameters. In this context, the consequences for the time integration as well as possible remedies are discussed.
Mots clés : skeletal muscle tissue, microvibrations, coherence, PDAE, index, time integration
@article{M2AN_2009__43_4_805_0,
author = {Simeon, Bernd and Serban, Radu and Petzold, Linda R.},
title = {A model of macroscale deformation and microvibration in skeletal muscle tissue},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {805--823},
publisher = {EDP-Sciences},
volume = {43},
number = {4},
year = {2009},
doi = {10.1051/m2an/2009030},
mrnumber = {2542878},
zbl = {1168.92008},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2009030/}
}
TY - JOUR AU - Simeon, Bernd AU - Serban, Radu AU - Petzold, Linda R. TI - A model of macroscale deformation and microvibration in skeletal muscle tissue JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 805 EP - 823 VL - 43 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2009030/ DO - 10.1051/m2an/2009030 LA - en ID - M2AN_2009__43_4_805_0 ER -
%0 Journal Article %A Simeon, Bernd %A Serban, Radu %A Petzold, Linda R. %T A model of macroscale deformation and microvibration in skeletal muscle tissue %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 805-823 %V 43 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2009030/ %R 10.1051/m2an/2009030 %G en %F M2AN_2009__43_4_805_0
Simeon, Bernd; Serban, Radu; Petzold, Linda R. A model of macroscale deformation and microvibration in skeletal muscle tissue. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 4, pp. 805-823. doi : 10.1051/m2an/2009030. http://archive.numdam.org/articles/10.1051/m2an/2009030/
[1] and , Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, USA (1998). | MR | Zbl
[2] , and , The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations. SIAM, Philadelphia, USA (1996). | MR
[3] and , Mixed and Hybrid Finite Element Methods. Springer (1991). | MR | Zbl
[4] , and , Using Krylow methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comp. 15 (1994) 1467-1488. | MR | Zbl
[5] COMSOL Multiphysics User Manual, Version 3.4 (2007).
[6] , , and , Synchronization by reactive coupling and nonlinear frequency pulling. Phys. Rev. E 73 (2006) 036205. | MR
[7] , Ein Zweiskalenansatz zur Modellierung der Skelettmuskulatur. Diploma Thesis, TU München, Germany (2007).
[8] and , Characterisation of arm microvibration recorded on an accelometer. Eur. J. Appl. Physiol. 75 (1997) 226-232.
[9] and , Effects of an eight-day space flight on microvibration and physiological tremor. Am. J. Physiol. 273 (1997) R86-R92.
[10] , and , Automatic integration of the Euler-Lagrange equations with constraints. J. Comp. Appl. Math. 12 (1985) 77-90. | MR | Zbl
[11] , , and , A finite element approach for skeletal muscle using a distributed moment model of contraction. Comp. Meth. Biomech. Biomed. Engng. 3 (2000) 231-244.
[12] , Biochemical Oscillations and Cellular Rhythms. Cambridge University Press (1996). | Zbl
[13] and , Matrix Computations. Third Edition, John Hopkins University Press, Baltimore (1996). | MR | Zbl
[14] , The heat of shortening and the dynamic constants of muscle. P. Roy. Soc. Lond. B Bio. 126 (1938) 136-195.
[15] , The Finite Element Method. Prentice Hall, Englewood Cliffs (1987). | MR | Zbl
[16] , Muscle structure and theories of contraction. Prog. Biophys. Biophys. Chem. 7 (1957) 255-318.
[17] , , and , Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network. J. Mech. Physics Solids 53 (2005) 1552-1573. | MR | Zbl
[18] , and , An operator splitting method for solving the Bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194 (2005) 233-248. | MR | Zbl
[19] , Integration of stiff mechanical systems by Runge-Kutta methods. ZAMP 44 (1993) 1022-1053. | MR | Zbl
[20] and , Mathematical Foundations of Elasticity. Dover Publications (1994). | MR
[21] , and , Biomechanical models for soft tissue simulation. Springer (1998).
[22] , and , Dynamics of a large system of coupled nonlinear oscillators. Physica D 52 (1991) 293-331. | MR | Zbl
[23] , Matrix-Rhythm-Therapy of Dynamic Illnesses, in Extracellular Matrix and Groundregulation System in Health and Disease, H. Heine, M. Rimpler, G. Fischer Eds., Stuttgart-Jena-New York (1997) 57-70.
[24] , On Lagrange multipliers in flexible multibody dynamics. Comput. Methods Appl. Mech. Eng. 195 (2006) 6993-7005. | MR | Zbl
[25] www-m2.ma.tum.de/twiki/bin/view/Allgemeines/ProfessorSimeon/movie12.avi.
[26] , Modellierung und numerische Simulation der Skelettmuskulatur. Diploma Thesis, TU München, Germany (2006).
[27] and , A re-examination of calcium activation in the Huxley cross-bridge model. J. Biomech. Engng. 119 (1997) 20-29.
Cité par Sources :
