Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T06:11:12.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Cell-to-Muscle homogenization. Application to a constitutive law forthe myocardium

Published online by Cambridge University Press:  15 November 2003

Denis Caillerie ,
Ayman Mourad  and
Annie Raoult
Denis Caillerie
Affiliation:
Laboratoire Sols, Solides, Structures, BP 53, 38041 Grenoble Cedex 9, France. Denis.Caillerie@inpg.fr.
Ayman Mourad
Affiliation:
Laboratoire Sols, Solides, Structures, BP 53, 38041 Grenoble Cedex 9, France. Denis.Caillerie@inpg.fr. Laboratoire de Modélisation et Calcul, BP 53, 38041 Grenoble Cedex 9, France. Ayman.Mourad@imag.fr.
Annie Raoult
Affiliation:
Laboratoire de Modélisation et Calcul, BP 53, 38041 Grenoble Cedex 9, France. Ayman.Mourad@imag.fr. Laboratoire TIMC, Domaine de la Merci, 38076 La Tronche Cedex, France. Annie.Raoult@imag.fr.
Get access

Abstract

We derive a constitutive law for the myocardium from the description of both the geometrical arrangement ofcardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discretelattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discretehomogenization technique. The macroscopic constitutive law is obtained through the resolution of anonlinear self-equilibrum system of the discrete lattice reference cell.

Keywords

Myocardiumconstitutive lawhomogenizationlarge deformations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 4: Special issue on Biological and Biomedical Applications , July 2003 , pp. 681 - 698
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arts, T., Reneman, R.S. and Veenstra, P.C., A model of the mechanics of the left ventricle. Ann. Biomed. Engrg. 7 (1979) 299-318. CrossRef
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991).
Briane, M., Three models of non periodic fibrous materials obtained by homogenization. ESAIM: M2AN 27 (1993) 759-775. CrossRef
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).
D. Caillerie and B. Cambou, Les techniques de changement d'échelles dans les milieux granulaires, in Micromécanique des milieux granulaires. Hermès Sciences, Paris (2001).
Chadwick, R.S., Mechanics of the left ventricle. Biophys. J. 112 (1982) 333-339.
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.
P.G. Ciarlet, Mathematical Elasticity. Vol. 1: Three-Dimensional Elasticity. North-Holland, Amsterdam (1987).
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Science 136. Springer-Verlag, New York (1999).
Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues. 2nd ed., Springer-Verlag, New York (1993).
M. Gurtin, An Introduction to Continuum Mechanics. Academic Press, San Diego (1981).
Jouk, P.S., Usson, Y., Michalowicz, G. and Grossi, L., Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart. Anat. Embryol. 202 (2000) 103-118. CrossRef
Humphrey, J.D., Strumpf, R.K. and Yin, F.C.P., Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Engrg. 112 (1990) 333-339. CrossRef
Humphrey, J.D., Strumpf, R.K. and Yin, F.C.P., Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J. Biomech. Engrg. 112 (1990) 340-346. CrossRef
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. Engrg. 120 (1998) 504-517.
Moreau, G. and Caillerie, D., Continuum modeling of lattice structures in large displacement. Applications to buckling analysis. Comput. & Structures 68 (1998) 181-189. CrossRef
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.
Nash, M.P. and Hunter, P.J., Computational mechanics of the heart. J. Elasticity 61 (2000) 113-141. CrossRef
C.S. Peskin, Fiber architecture of the left ventricular wall: An asymptotic analysis. Comm. Pure Appl. Math. XLII (1989) 79-113.
F. Pradel, Homogénéisation des milieux continus et discrets périodiques orientés. Thèse de l'École Nationale des Ponts et Chaussées (1998).
E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Monographs in Physics 127. Springer-Verlag, Berlin (1980).
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).
Taber, L.A. and Perucchio, R., Modeling heart development. J. Elasticity 61 (2000) 165-197. CrossRef
Tollenaere, H. and Caillerie, D., Continuous modeling of lattice structures by homogenization. Adv. Engrg. Software 29 (1998) 699-705. CrossRef
C. Truesdell, A First Course in Rational Continuum Mechanics. Academic Press, New York (1977).
Usyk, T.P., Mazhari, R. and McCulloch, A.D., Effect of laminar orthotropic myofiber architecture on regional stress and strain in the canine left ventricle. J. Elasticity 61 (2000) 143-165. CrossRef
K. Washizu, Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press (1975).
Yin, F.C.P., Strumpf, R.K., Chew, P.H. and Zeger, S.L., Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J. Biomech. 20 (1987) 577-589. CrossRef
Zile, M., Cowles, M.K., Buckley, J.M., Richardson, K., Cowles, B.A., Baicu, C.F., Cooper IV, G. abd V. Gharpuray, Gel stretch method: a new method to measure constitutive properties of cardiac muscle cells. Am. J. Physiol. 274 (1998) H2188-2202.