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A model of macroscale deformation and microvibrationin skeletal muscle tissue

Published online by Cambridge University Press:  08 July 2009

Bernd Simeon ,
Radu Serban  and
Linda R. Petzold
Bernd Simeon
Affiliation:
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85748 Garching, Germany. [email protected]
Radu Serban
Affiliation:
Xulu entertainment 890 Hillview Court, Milpitas, CA 95032, USA.
Linda R. Petzold
Affiliation:
Dept. of Mechanical Engineering, University of California Santa Barbara, CA 93106, USA.
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Abstract

This paper deals with modeling the passivebehavior of skeletal muscle tissue includingcertain 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 appliedvibration 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 possessesan index up to three, depending on certain physicalparameters. In this context, the consequences forthe time integration as well as possible remedies are discussed.

Keywords

Skeletal muscle tissuemicrovibrationscoherencePDAEindextime integration.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 805 - 823
Copyright
© EDP Sciences, SMAI, 2009

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References

U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, USA (1998).
K.E. Brenan, S.L. Campbell and L.R. Petzold, The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations. SIAM, Philadelphia, USA (1996).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer (1991).
Brown, P., Hindmarsh, A. and Petzold, L.R., Using Krylow methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comp. 15 (1994) 1467–1488. CrossRef
COMSOL Multiphysics User Manual, Version 3.4 (2007).
Cross, M., Rogers, J., Lifshitz, R. and Zumdieck, A., Synchronization by reactive coupling and nonlinear frequency pulling. Phys. Rev. E  73 (2006) 036205. CrossRef
F. Dietrich, Ein Zweiskalenansatz zur Modellierung der Skelettmuskulatur. Diploma Thesis, TU München, Germany (2007).
E. Gallasch and T. Kenner, Characterisation of arm microvibration recorded on an accelometer. Eur. J. Appl. Physiol. 75 (1997) 226–232.
Gallasch, E. and Moser, M., Effects of an eight-day space flight on microvibration and physiological tremor. Am. J. Physiol. 273 (1997) R86–R92.
Gear, C., Gupta, G. and Leimkuhler, B., Automatic integration of the Euler-Lagrange equations with constraints. J. Comp. Appl. Math. 12 (1985) 77–90. CrossRef
Gielen, A., Oomens, C., Bovendeerd, P. and Arts, T., A finite element approach for skeletal muscle using a distributed moment model of contraction. Comp. Meth. Biomech. Biomed. Engng. 3 (2000) 231–244. CrossRef
A. Goldbeter, Biochemical Oscillations and Cellular Rhythms. Cambridge University Press (1996).
G. Golub and C. van Loan, Matrix Computations. Third Edition, John Hopkins University Press, Baltimore (1996).
Hill, A.V., The heat of shortening and the dynamic constants of muscle. P. Roy. Soc. Lond. B Bio. 126 (1938) 136–195. CrossRef
T.J. Hughes, The Finite Element Method. Prentice Hall, Englewood Cliffs (1987).
Huxley, A.F., Muscle structure and theories of contraction. Prog. Biophys. Biophys. Chem. 7 (1957) 255–318.
Kuhl, E., Garikipati, K., Arruda, E.M. and Grosh, K., Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network. J. Mech. Physics Solids 53 (2005) 1552–1573. CrossRef
Line, G.T., Sundnes, J. and Tveito, A., An operator splitting method for solving the Bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194 (2005) 233–248.
Ch. Lubich, Integration of stiff mechanical systems by Runge-Kutta methods. ZAMP 44 (1993) 1022–1053.
J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity. Dover Publications (1994).
W. Maurel, N.Y. Wu and D. Thalmann, Biomechanical models for soft tissue simulation. Springer (1998).
Matthews, P., Mirollo, R. and Strogatz, St., Dynamics of a large system of coupled nonlinear oscillators. Physica D 52 (1991) 293–331. CrossRef
U. Randoll, 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.
Simeon, B., Lagrange, On multipliers in flexible multibody dynamics. Comput. Methods Appl. Mech. Eng. 195 (2006) 6993–7005. CrossRef
www-m2.ma.tum.de/twiki/bin/view/Allgemeines/ProfessorSimeon/movie12.avi.
S. Thiemann, Modellierung und numerische Simulation der Skelettmuskulatur. Diploma Thesis, TU München, Germany (2006).
Zahalak, G.I. and Motabarzadeh, I., A re-examination of calcium activation in the Huxley cross-bridge model. J. Biomech. Engng. 119 (1997) 20–29. CrossRef