# A model of macroscale deformation and microvibration in skeletal muscle tissue

Bernd Simeon; Radu Serban; Linda R. Petzold

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 4, page 805-823
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topSimeon, Bernd, Serban, Radu, and Petzold, Linda R.. "A model of macroscale deformation and microvibration in skeletal muscle tissue." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 805-823. <http://eudml.org/doc/250598>.

@article{Simeon2009,

abstract = {
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.
},

author = {Simeon, Bernd, Serban, Radu, Petzold, Linda R.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Skeletal muscle tissue; microvibrations; coherence; PDAE; index; time integration.; skeletal muscle tissue; time integration},

language = {eng},

month = {7},

number = {4},

pages = {805-823},

publisher = {EDP Sciences},

title = {A model of macroscale deformation and microvibration in skeletal muscle tissue},

url = {http://eudml.org/doc/250598},

volume = {43},

year = {2009},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2009/7//

PB - EDP Sciences

VL - 43

IS - 4

SP - 805

EP - 823

AB -
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.

LA - eng

KW - Skeletal muscle tissue; microvibrations; coherence; PDAE; index; time integration.; skeletal muscle tissue; time integration

UR - http://eudml.org/doc/250598

ER -

## References

top- U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, USA (1998). Zbl0908.65055
- 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). Zbl0844.65058
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer (1991). Zbl0788.73002
- P. Brown, A. Hindmarsh and L.R. Petzold, Using Krylow methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comp.15 (1994) 1467–1488. Zbl0812.65060
- COMSOL Multiphysics User Manual, Version 3.4 (2007).
- M. Cross, J. Rogers, R. Lifshitz and A. Zumdieck, Synchronization by reactive coupling and nonlinear frequency pulling. Phys. Rev. E 73 (2006) 036205.
- 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.
- E. Gallasch and M. Moser, Effects of an eight-day space flight on microvibration and physiological tremor. Am. J. Physiol.273 (1997) R86–R92.
- C. Gear, G. Gupta and B. Leimkuhler, Automatic integration of the Euler-Lagrange equations with constraints. J. Comp. Appl. Math.12 (1985) 77–90. Zbl0576.65072
- A. Gielen, C. Oomens, P. Bovendeerd and T. Arts, A finite element approach for skeletal muscle using a distributed moment model of contraction. Comp. Meth. Biomech. Biomed. Engng.3 (2000) 231–244.
- A. Goldbeter, Biochemical Oscillations and Cellular Rhythms. Cambridge University Press (1996). Zbl0837.92009
- G. Golub and C. van Loan, Matrix Computations. Third Edition, John Hopkins University Press, Baltimore (1996). Zbl0865.65009
- A.V. Hill, The heat of shortening and the dynamic constants of muscle. P. Roy. Soc. Lond. B Bio.126 (1938) 136–195.
- T.J. Hughes, The Finite Element Method. Prentice Hall, Englewood Cliffs (1987). Zbl0634.73056
- A.F. Huxley, Muscle structure and theories of contraction. Prog. Biophys. Biophys. Chem.7 (1957) 255–318.
- E. Kuhl, K. Garikipati, E.M. Arruda and K. Grosh, Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network. J. Mech. Physics Solids53 (2005) 1552–1573. Zbl1120.74635
- G.T. Line, J. Sundnes and A. Tveito, An operator splitting method for solving the Bidomain equations coupled to a volume conductor model for the torso. Math. Biosci.194 (2005) 233–248. Zbl1063.92018
- Ch. Lubich, Integration of stiff mechanical systems by Runge-Kutta methods. ZAMP44 (1993) 1022–1053. Zbl0784.70002
- J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity. Dover Publications (1994). Zbl0545.73031
- W. Maurel, N.Y. Wu and D. Thalmann, Biomechanical models for soft tissue simulation. Springer (1998).
- P. Matthews, R. Mirollo and St. Strogatz, Dynamics of a large system of coupled nonlinear oscillators. Physica D52 (1991) 293–331. Zbl0742.34035
- 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.
- B. Simeon, On Lagrange multipliers in flexible multibody dynamics. Comput. Methods Appl. Mech. Eng.195 (2006) 6993–7005. Zbl1120.74517
- 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).
- G.I. Zahalak and I. Motabarzadeh, A re-examination of calcium activation in the Huxley cross-bridge model. J. Biomech. Engng.119 (1997) 20–29.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.