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A three dimensional finite element method for biological active soft tissue Formulation in cylindrical polar coordinates

Published online by Cambridge University Press:  15 November 2003

Christian Bourdarias ,
Stéphane Gerbi  and
Jacques Ohayon
Christian Bourdarias
Affiliation:
Laboratoire de Mathématiques, Université de Savoie, Savoie Technolac, 73376 Le Bourget du Lac, France. Christian.Bourdarias@univ-savoie.fr., Stephane.Gerbi@univ-savoie.fr.
Stéphane Gerbi
Affiliation:
Laboratoire de Mathématiques, Université de Savoie, Savoie Technolac, 73376 Le Bourget du Lac, France. Christian.Bourdarias@univ-savoie.fr., Stephane.Gerbi@univ-savoie.fr.
Jacques Ohayon
Affiliation:
Laboratoire TIMC-IMAG, Dynacell, UMR CNRS 5525, Domaine de la Merci, 38706 Grenoble, France. Jacques.Ohayon@univ-savoie.fr.
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Abstract

A hyperelastic constitutive law, for use in anatomically accurate finite element models ofliving structures, is suggested for the passive and the active mechanical properties of incompressiblebiological tissues. This law considers the passive and active states as a same hyperelastic continuummedium, and uses an activation function in order to describe the whole contraction phase.The variational and the FE formulations are also presented, and the FE code has been validatedand applied to describe the biomechanical behavior of a thick-walled anisotropic cylinder underdifferent active loading conditions.

Keywords

Constitutive lawfinite element methodbiological tissuehyperelasticitynonlinear partial differential equationsanisotropic material.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 4: Special issue on Biological and Biomedical Applications , July 2003 , pp. 725 - 739
Copyright
© EDP Sciences, SMAI, 2003

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References

Bayliss, W.M., On the local reaction of the arterial wall to changes of internal pressure. J. Physiol. London 28 (1902) 220-231. CrossRef
Berntsen, J., Espelid, T.O. and Genz, A., Algorithm 698: DCUHRE: An adaptive multidimensional integration routine for a vector of integrals. ACM Trans. Math. Softw. 17 (1991) 452-456. CrossRef
Bovendeerd, P.H.M., Arts, T., Huyghe, J.M., van Campen, D.H. and Reneman, R.S., Dependance of local left ventricular wall mechanics on myocardial fiber orientation: a model study. J. Biomech. 25 (1992) 1129-1140. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, Vol. 4 of Studies in Mathematics and its Applications. North-Holland, Amsterdam-New York (1980).
Costa, K.D., Hunter, P.J., Wayne, J.S., Waldman, L.K., Guccione, J.M. and McCulloch, A.D., A three-dimensional finite element method for large elastic deformations of ventricular myocardium: Part I. Cylindrical and spherical polar coordinates. ASME J. Biomech. Eng. 118 (1996) 452-463. CrossRef
R. Glowinski and P. LeTallec, Augmented lagrangian and operator-splitting methods in nonlinear mechanics. SIAM, Philadelphia, PA (1989).
Lin, D.H.S. and Yin, F.C.P., A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus. J. Biomech. Eng. 120 (1998) 504-517.
L.E. Malvern, Introduction to the mechanics of a continuous medium. Prentice-Hall (1969).
McCulloch, A.D., Waldman, L.K., Rogers, J. and Guccione, J., Large scale finite element analysis of the beating heart. Crit. Rev. Biomed. Eng. 20 (1992) 427-449.
J.J. Morge, B.S. Garbow and K.E. Hillstrom, User Guide for MINPACK-1. Technical Report ANL-80-74, Argonne National Laboratory (March 1980).
J.T. Oden, Finite elements of nonlinear continua. McGraw-Hill, New York (1972).
Ohayon, J. and Chadwick, R.S., Effects of collagen microstructure on the mechanics of the left ventricle. Biophys. J. 54 (1988) 1077-1088. CrossRef
M.J.D. Powell, A hybrid method for nonlinear equations, in Numerical methods for nonlinear algebraic equations, P. Rabinowitz Ed. Gordon and Breach, New York (1970) 87-114.
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Vol. 23 of Springer Series in Computational Mathematics. Springer Verlag, Berlin (1994).
G.M. Rubanyi, Mechanoreception by the vascular wall. Futura Publishing Company, Inc. (1993).
Taber, L.A., On a nonlinear theory for muscle shells: Part II. Application to the beating left ventricle. J. Biomech. Eng. 113 (1991) 63-71. CrossRef
Teppaz, P., Ohayon, J. and Herbin, R., Interaction fluide-structure active : écoulement artériel. C.R. Acad. Sci. Paris 324 (1997) 37-45.
Usyk, T.P., Omens J.H. and A.D. McCulloch, Regional septal dysfunction in a three-dimensional computational model of focal myofiber dissaray. Am. J. Physiol. Heart Circ. Physiol. 281 (2001) 506-514.
Zhang, J. and Xu, C., A class of indefinite dogleg path mehods for unconstrained minimization. SIAM J. Optim. 9 (1999) 646-667. CrossRef