Cell-to-muscle homogenization. Application to a constitutive law for the myocardium
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, p. 681-698

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.

DOI : https://doi.org/10.1051/m2an:2003054
Classification:  74L15,  74Q05,  74Q15,  92B05
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {4},
     year = {2003},
     pages = {681-698},
     doi = {10.1051/m2an:2003054},
     zbl = {1070.74030},
     mrnumber = {2018437},
     language = {en},
     url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, pp. 681-698. doi : 10.1051/m2an:2003054. http://www.numdam.org/item/M2AN_2003__37_4_681_0/

[1] T. Arts, R.S. Reneman and P.C. Veenstra, A model of the mechanics of the left ventricle. Ann. Biomed. Engrg. 7 (1979) 299-318.

[2] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). | MR 503330 | Zbl 0404.35001

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[4] M. Briane, Three models of non periodic fibrous materials obtained by homogenization. ESAIM: M2AN 27 (1993) 759-775. | Numdam | Zbl 0795.92006

[5] H. Cai, 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] D. Caillerie and B. Cambou, Les techniques de changement d'échelles dans les milieux granulaires, in Micromécanique des milieux granulaires. Hermès Sciences, Paris (2001).

[7] R.S. Chadwick, Mechanics of the left ventricle. Biophys. J. 112 (1982) 333-339.

[8] D. Chapelle, F. Clément, F. Génot, P. Le Tallec, M. Sorine and J.M. Urquiza, 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 1052.68824

[9] P.G. Ciarlet, Mathematical Elasticity. Vol. 1: Three-Dimensional Elasticity. North-Holland, Amsterdam (1987). | MR 936420 | Zbl 0648.73014

[10] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Science 136. Springer-Verlag, New York (1999). | MR 1676922 | Zbl 0929.35002

[11] Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues. 2nd ed., Springer-Verlag, New York (1993). | Zbl 0743.92007

[12] M. Gurtin, An Introduction to Continuum Mechanics. Academic Press, San Diego (1981). | MR 636255 | Zbl 0559.73001

[13] P.S. Jouk, Y. Usson, G. Michalowicz and L. Grossi, Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart. Anat. Embryol. 202 (2000) 103-118.

[14] J.D. Humphrey, R.K. Strumpf and F.C.P. Yin, Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Engrg. 112 (1990) 333-339.

[15] J.D. Humphrey, R.K. Strumpf and F.C.P. Yin, Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J. Biomech. Engrg. 112 (1990) 340-346.

[16] D.H.S. Lin and F.C.P. Yin, A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus. J. Biomech. Engrg. 120 (1998) 504-517.

[17] G. Moreau and D. Caillerie, Continuum modeling of lattice structures in large displacement. Applications to buckling analysis. Comput. & Structures 68 (1998) 181-189. | Zbl 0940.74034

[18] A. Mourad, L. Biard, D. Caillerie, P.-S. Jouk, A. Raoult, N. Szafran and Y. Usson, 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 1052.68917

[19] M.P. Nash and P.J. Hunter, Computational mechanics of the heart. J. Elasticity 61 (2000) 113-141. | Zbl 1071.74659

[20] C.S. Peskin, Fiber architecture of the left ventricular wall: An asymptotic analysis. Comm. Pure Appl. Math. XLII (1989) 79-113. | Zbl 0664.92005

[21] F. Pradel, Homogénéisation des milieux continus et discrets périodiques orientés. Thèse de l'École Nationale des Ponts et Chaussées (1998).

[22] E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Monographs in Physics 127. Springer-Verlag, Berlin (1980). | MR 578345 | Zbl 0432.70002

[23] D.D. Streeter, 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] L.A. Taber and R. Perucchio, Modeling heart development. J. Elasticity 61 (2000) 165-197. | Zbl 0987.74049

[25] H. Tollenaere and D. Caillerie, Continuous modeling of lattice structures by homogenization. Adv. Engrg. Software 29 (1998) 699-705.

[26] C. Truesdell, A First Course in Rational Continuum Mechanics. Academic Press, New York (1977). | MR 559731 | Zbl 0866.73001

[27] T.P. Usyk, R. Mazhari and A.D. Mcculloch, Effect of laminar orthotropic myofiber architecture on regional stress and strain in the canine left ventricle. J. Elasticity 61 (2000) 143-165. | Zbl 0974.92002

[28] K. Washizu, Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press (1975). | MR 391680 | Zbl 0339.73035

[29] F.C.P. Yin, R.K. Strumpf, P.H. Chew and S.L. Zeger, Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J. Biomech. 20 (1987) 577-589.

[30] M. Zile, M.K. Cowles, J.M. Buckley, K. Richardson, B.A. Cowles, C.F. Baicu, G. Cooper 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.