We derive a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discrete lattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discrete homogenization technique. The macroscopic constitutive law is obtained through the resolution of a nonlinear self-equilibrum system of the discrete lattice reference cell.
Mots-clés : myocardium, constitutive law, homogenization, large deformations
@article{M2AN_2003__37_4_681_0,
author = {Caillerie, Denis and Mourad, Ayman and Raoult, Annie},
title = {Cell-to-muscle homogenization. {Application} to a constitutive law for the myocardium},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {681--698},
publisher = {EDP-Sciences},
volume = {37},
number = {4},
year = {2003},
doi = {10.1051/m2an:2003054},
mrnumber = {2018437},
zbl = {1070.74030},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2003054/}
}
TY - JOUR AU - Caillerie, Denis AU - Mourad, Ayman AU - Raoult, Annie TI - Cell-to-muscle homogenization. Application to a constitutive law for the myocardium JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 681 EP - 698 VL - 37 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2003054/ DO - 10.1051/m2an:2003054 LA - en ID - M2AN_2003__37_4_681_0 ER -
%0 Journal Article %A Caillerie, Denis %A Mourad, Ayman %A Raoult, Annie %T Cell-to-muscle homogenization. Application to a constitutive law for the myocardium %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 681-698 %V 37 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2003054/ %R 10.1051/m2an:2003054 %G en %F M2AN_2003__37_4_681_0
Caillerie, Denis; Mourad, Ayman; Raoult, Annie. Cell-to-muscle homogenization. Application to a constitutive law for the myocardium. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Biological and Biomedical Applications, Tome 37 (2003) no. 4, pp. 681-698. doi : 10.1051/m2an:2003054. http://archive.numdam.org/articles/10.1051/m2an:2003054/
[1] , and, A model of the mechanics of the left ventricle. Ann. Biomed. Engrg. 7 (1979) 299-318.
[2] , and, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). | MR | Zbl
[3] and, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | MR | Zbl
[4] , Three models of non periodic fibrous materials obtained by homogenization. ESAIM: M2AN 27 (1993) 759-775. | Numdam | Zbl
[5] , Loi de comportement en grandes déformations du muscle à fibres actives. Application à la mécanique du cœur humain et à sa croissance. Thèse de l'Université de Savoie (1998).
[6] and, Les techniques de changement d'échelles dans les milieux granulaires, in Micromécanique des milieux granulaires. Hermès Sciences, Paris (2001).
[7] , Mechanics of the left ventricle. Biophys. J. 112 (1982) 333-339.
[8] ,,,, and, A Physiologically-Based Model for the Active Cardiac Muscle Contraction, in Functional Imaging and Modeling of the Heart, Katila, Magnin, Clarysse, Montagnat and Nenonen Eds., LNCS 2230. Springer (2001) 128-133. | Zbl
[9] , Mathematical Elasticity. Vol. 1: Three-Dimensional Elasticity. North-Holland, Amsterdam (1987). | MR | Zbl
[10] and, Homogenization of Reticulated Structures, Applied Mathematical Science 136. Springer-Verlag, New York (1999). | MR | Zbl
[11] , Biomechanics: Mechanical Properties of Living Tissues. 2nd ed., Springer-Verlag, New York (1993). | Zbl
[12] , An Introduction to Continuum Mechanics. Academic Press, San Diego (1981). | MR | Zbl
[13] ,, and, Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart. Anat. Embryol. 202 (2000) 103-118.
[14] , and, Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Engrg. 112 (1990) 333-339.
[15] , and, Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J. Biomech. Engrg. 112 (1990) 340-346.
[16] and, A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus. J. Biomech. Engrg. 120 (1998) 504-517.
[17] and, Continuum modeling of lattice structures in large displacement. Applications to buckling analysis. Comput. & Structures 68 (1998) 181-189. | Zbl
[18] ,,,,, and, Geometrical modelling of the fibre organization in the human left ventricle, in Functional Imaging and Modeling of the Heart, Katila, Magnin, Clarysse, Montagnat, Nenonen Eds., LNCS 2230. Springer (2001) 32-38. | Zbl
[19] and, Computational mechanics of the heart. J. Elasticity 61 (2000) 113-141. | Zbl
[20] , Fiber architecture of the left ventricular wall: An asymptotic analysis. Comm. Pure Appl. Math. XLII (1989) 79-113. | Zbl
[21] , Homogénéisation des milieux continus et discrets périodiques orientés. Thèse de l'École Nationale des Ponts et Chaussées (1998).
[22] , Non Homogeneous Media and Vibration Theory, Monographs in Physics 127. Springer-Verlag, Berlin (1980). | MR | Zbl
[23] , Gross morphology and fiber geometry of the heart, in Handbook of Physiology. The cardiovascular system, R.M. Berne, N. Sperelakis and S.R. Geiger Eds., Am. Phys. Soc. Williams & Wilkins, Baltimore (1979).
[24] and, Modeling heart development. J. Elasticity 61 (2000) 165-197. | Zbl
[25] and, Continuous modeling of lattice structures by homogenization. Adv. Engrg. Software 29 (1998) 699-705.
[26] , A First Course in Rational Continuum Mechanics. Academic Press, New York (1977). | MR | Zbl
[27] , and, Effect of laminar orthotropic myofiber architecture on regional stress and strain in the canine left ventricle. J. Elasticity 61 (2000) 143-165. | Zbl
[28] , Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press (1975). | MR | Zbl
[29] ,, and, Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J. Biomech. 20 (1987) 577-589.
[30] ,,,,,, IV abd V. Gharpuray, Gel stretch method: a new method to measure constitutive properties of cardiac muscle cells. Am. J. Physiol. 274 (1998) H2188-2202.
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