Finite Element Modelling of Contracting Skeletal Muscle
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
This CellML model cannot be run in either OpenCell or COR due to the presence of the MathML operator "sum" which is not yet supported by the CellML API.
Model Structure
ABSTRACT: To describe the mechanical behaviour of biological tissues and transport processes in biological tissues, conservation laws such as conservation of mass, momentum and energy play a central role. Mathematically these are cast into the form of partial differential equations. Because of nonlinear material behaviour, inhomogeneous properties and usually a complex geometry, it is impossible to find closed-form analytical solutions for these sets of equations. The objective of the finite element method is to find approximate solutions for these problems. The concepts of the finite element method are explained on a finite element continuum model of skeletal muscle. In this case, the momentum equations have to be solved with an extra constraint, because the material behaves as nearly incompressible. The material behaviour consists of a highly nonlinear passive part and an active part. The latter is described with a two-state Huxley model. This means that an extra nonlinear partial differential equation has to be solved. The problems and solutions involved with this procedure are explained. The model is used to describe the mechanical behaviour of a tibialis anterior of a rat. The results have been compared with experimentally determined strains at the surface of the muscle. Qualitatively there is good agreement between measured and calculated strains, but the measured strains were higher.
The original paper reference is cited below:
Finite element modelling of contracting skeletal muscle, C.W.J. Oomens, M. Maenhout, C.H. van Oijen, M.R. Drost, and F.P. Baaijens, 2003, Phil. Trans. R. Soc. Lond. B , 358, 1453-1460. PubMed ID: 14561336
Muscle Activation and Contraction: Constitutive Relations Based Directly on Cross-Bridge Kinetics, George I. Zahalak and Shi-Ping Ma, 1990, Journal of Biomedical Engineering, 122, 52-62. PubMed ID: 2308304
$r=\frac{c^{2.0}}{c^{2.0}+\mathrm{mu}c+\mathrm{mu}^{2.0}}$
$\frac{d \mathrm{Ct}}{d \mathrm{time}}=\mathrm{rho}(1.0-\frac{c}{\mathrm{c\_}})\mathrm{chi}-\mathrm{tau0}^{-1.0}\frac{c}{c+\mathrm{km}}\mathrm{Ct\_initial}=c+2.0b\mathrm{Q0}+r(2.0+\frac{\mathrm{mu}}{c})(1.0-b\mathrm{Q0})$
$\frac{d \mathrm{Ca\_in}}{d \mathrm{time}}=\mathrm{rho}(2.0-\frac{c}{\mathrm{c\_}})\mathrm{chi}$
$\mathrm{chi}=\sum \mathrm{chi\_}(\mathrm{time}-\mathrm{time\_i})\mathrm{chi\_}=(e^{\frac{-\mathrm{time}}{\mathrm{tau1}}}-e^{\frac{-\mathrm{time}}{\mathrm{tau2}}})(\mathrm{tau1}-\mathrm{tau2})$
myofilament mechanics
skeletal muscle
myosin
calcium
finite element
muscle contraction
mechanical constitutive laws
2003-09-29
The University of Auckland, Auckland Bioengineering Institute
C
Oomens
J
W
Finite element modelling of contracting skeletal muscle
358
1453
1460
2004-09-08
F
Baaijens
P
Oomens et al.'s 2003 finite element model of contracting skeletal muscle.
M
Maenhout
total calcium in the myofibrillar space
Ct
Phil. Trans. R. Soc. Lond. B
C
van Oijen
H
fraction of activated actin
r
Catherine Lloyd
Catherine
Lloyd
May
c.lloyd@auckland.ac.nz
This is the CellML description of Oomens et al.'s 2003 finite element model of contracting skeletal muscle.
Sum of the normalised impulse functions
chi
keyword
rate of injection of calcium from the SR
Ca_in
M
Drost
R
The University of Auckland
Auckland Bioengineering Institute
14561336