Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-18T13:24:06.494Z Has data issue: false hasContentIssue false
  • English
  • Français

Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

Published online by Cambridge University Press:  28 November 2014

Alessio Gizzi ,
Christian Cherubini ,
Simonetta Filippi  and
Anna Pandolfi
Alessio Gizzi
Affiliation:
Nonlinear Physics and Mathematical Modeling Laboratory
Christian Cherubini*
Affiliation:
Nonlinear Physics and Mathematical Modeling Laboratory International Center for Relativistic Astrophysics Network, I.C.R.A.Net, University Campus Bio-Medico of Rome, Via A. del Portillo 21, 00128 Rome, Italy
Simonetta Filippi
Affiliation:
Nonlinear Physics and Mathematical Modeling Laboratory International Center for Relativistic Astrophysics Network, I.C.R.A.Net, University Campus Bio-Medico of Rome, Via A. del Portillo 21, 00128 Rome, Italy
Anna Pandolfi
Affiliation:
Civil and Environmental Engineering Department Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
*
*Email addresses:[email protected](A. Gizzi), [email protected](C. Cherubini), [email protected](S. Filippi), [email protected](A. Pandolfi)
Get access

Abstract

We present a general theoretical framework for the formulation of the nonlinear electromechanics of polymeric and biological active media. The approach developed here is based on the additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on the multiplicative decomposition of the deformation gradient in passive and active parts. We describe a thermodynamically sound scenario that accounts for geometric and material nonlinearities. In view of numerical applications, we specialize the general approach to a particular material model accounting for the behavior of fiber reinforced tissues. Specifically, we use the model to solve via finite elements a uniaxial electromechanical problem dynamically activated by an electrophysiological stimulus. Implications for nonlinear solid mechanics and computational electrophysiology are finally discussed.

Keywords

46.25.Hf05.70.Ce62.20.F-81.40.Jj87.10.PqActive electromechanical mediaHelmholtz free energymultiplicative decomposition of the deformation gradientactive deformationactive stress
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 1 , January 2015 , pp. 93 - 126
Copyright
Copyright © Global Science Press Limited 2015 

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

[1]Aliev, R. R. and Panfilov, A. V.. A simple two-variable model of cardiac excitation. Chaos Solitons Fractals, 7(3):293–301, 1996.CrossRefGoogle Scholar
[2]Allen, D. G. and Kentish, J. C.. The cellular basis of the length-tension relation in cardiac muscle. Journal of Molecular and Cellular Cardiology, 17:821–840,1985.Google Scholar
[3]Ambrosi, D., Arioli, G., Nobile, F., and Quarteroni, A.. Electromechanical coupling in cardiac dynamics: The active strain approach. SIAM Journal of Applied Mathematics, 71(2):605–621, 2011.CrossRefGoogle Scholar
[4]Ask, A., Menzel, A., and Ristinmaa, M.. Electrostriction in electro-viscoelastic polymers. Mechanics of Materials, 50:9–21,2012.CrossRefGoogle Scholar
[5]Ask, A., Menzel, A., and Ristinmaa, M.. Phenomenological modeling of viscous electrostrictive polymers. International Journal of Non-Linear Mechanics, 47:156–165,2012.Google Scholar
[6]Beeler, G. W. and Reuter, H.. Reconstruction of the action potential of ventricular myocardial fibres. Journal of Physiology, 268:177–210,1977.Google Scholar
[7]Bini, D., Cherubini, C., and Filippi, S.. Viscoelastic FitzHugh-Nagumo models. Physical Review E, 72:041929, 2005.Google Scholar
[8]Cherubini, C., Filippi, S., and Gizzi, A.. Electroelastic unpinning of rotating vortices in biological excitable media. Physical Review E, 85:031915, 2012.Google Scholar
[9]Cherubini, C., Filippi, S., Nardinocchi, P., and Teresi, L.. An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects. Progress in Biophysics & Molecular Biology, 97:562–573, 2008.Google Scholar
[10]Cherubini, C., Filippi, S., Nardinocchi, P., and Teresi, L.. Electromechanical modelling of cardiac tissue. In Kamin, A. and Kiseleva, I., editors, Mechanosensitivity of the Heart. Vol. 3, pages 421–449. Springer, Berlin, 2010.Google Scholar
[11]Coleman, B. D. and Noll, W.. The thermodynamics of elastic materials with heat conduction and viscosity. Archives of Rational Mechanics and Analysis, 13(1):167–178,1963.Google Scholar
[12]Dokos, S., Smaill, B. H., Young, A. A., and LeGrice, I. J.. Shear properties of passive ventricular myocardium. American Journal of Physiological and Heart Circulatory Physiology, 283:H2650–2659, 2002.CrossRefGoogle ScholarPubMed
[13]Dorfmann, A. and Ogden, R. W.. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005.Google Scholar
[14]Eckart, C.. The thermodynamics of irreversible processes, IV. The theory of elasticity and anelasticity. Physical Reviews, 73:373–380,1948.Google Scholar
[15]Eisner, D. A., Choi, H. S., Diaz, M. E., O'Neill, S. C., and Trafford, A. W.. Integrative analysis of calcium cycling in cardiac muscle. Circulation Research, 87:1087–1094,2000.CrossRefGoogle ScholarPubMed
[16]Eringen, A. C. and Maugin, G. A.. Elastodynamics of Continua. Foundations and Solid Media. Spinger-Verlag, New York, 1990.Google Scholar
[17]Fenton, F. H. and Cherry, E. M.. Models of cardiac cell. Scholarpedia, 3:1868,2008.Google Scholar
[18]Fenton, F. H., Gizzi, A., Cherubini, C., and Filippi, S.. Role of temperature on nonlinear cardiac dynamics. Physical Review E, 87:042717,2013.Google Scholar
[19]Fenton, F. H. and Karma, A.. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos, 8:1054,1998.Google Scholar
[20]Filippi, S., Gizzi, A., Cherubini, C., Luther, S., and Fenton, F. H.. Mechanistic insights into hy-pothermic ventricular fibrillation: the role of temperature and tissue size. Europace, 16:424–434, 2014.Google Scholar
[21]Gizzi, A., Cherry, E. M., Gilmour, R. F. Jr, Luther, S., Filippi, S., and Fenton, F. H.. Effects of pacing site and stimulation history on alternans dynamics and the development of complex spatiotemporal patterns in cardiac tissue. Frontiers in Physiology, 4:71, 2013.Google Scholar
[22]Gizzi, A., Cherubini, C., Migliori, S., Alloni, R., Portuesi, R., and Filippi, S.. On the electrical intestine turbulence induced by temperature changes. Physical Byology, 7:016011,2010.Google Scholar
[23]Gizzi, A., Vasta, M., and Pandolfi, A.. Modeling collagen recruitment in hyperelastic bio-material models with statistical distribution of the fiber orientation. International Journal of Engineering Science, 78:48–60, 2014.CrossRefGoogle Scholar
[24]Goktepe, S., Acharya, S. N. S., Wong, J., and Kuhl, E.. Computational modeling of passive myocardium. International Journal for Numerical Methods in Biomedical Engineering, 27(1):1–12, 2011.Google Scholar
[25]Göktepe, S., Menzel, A., and Kuhl, E.. Micro-structurally based kinematic approaches to electromechanics of the heart. In Computer Models in Biomechanics, pages 175-187. Springer Netherlands, 2013.Google Scholar
[26]Granzier, H. L. and Irving, T. C.. Passive tension in cardiac muscle: contribution of collagen, titin, microtubules, and intermediate filaments. Biophysical Journal, 68:1027–1044,1995.CrossRefGoogle ScholarPubMed
[27]Guccione, J. M., McCulloch, A. D., and Waldman, L. K.. Passive material properties of intact ventricular miocardium determined from a cylindrical model. Journal of Biomechanical Engineering, 113(1):42–55, 1991.Google Scholar
[28]Guo, Z., Caner, F., Peng, X., and Moran, B.. On constitutive modelling of porous neo-Hookean composites. Journal of the Mechanics and Physics of Solids, 56:2338–2357, 2008.Google Scholar
[29]Guo, Z. Y., Peng, X. Q., and Moran, B.. A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. Journal of the Mechanics and Physics of Solids, 68:1027–1044, 2006.Google Scholar
[30]Hill, A. V.. The heat of shortening and dynamics constants of muscles. Proceedings of the Royal Society of London, B, 367(843):136–195,1938.Google Scholar
[31]Holzapfel, G. A. and Ogden, R. W.. Constitutive modelling of passive myocardium: A structurally based framework for material characterization. Phylosophycal Transactions of the Royal Society, A, 367:3445–3475, 2009.Google Scholar
[32]Hunter, P. J., Nash, M. P., and Sands, G. B.. Computational Electromechanics of the Heart. Wiley, London, 1997.Google Scholar
[33]Hutter, K., Ven, A.F.A., and Ursescu, A.. Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Lecture Notes in Physics. Springer, 2006.Google Scholar
[34]Itskov, M., Ehret, A. E., and Mavrilas, D.. A polyconvex anisotropic strain-energy function for soft collagenous tissues. Biomechanics and Modeling in Mechanobiology, 5(1):17–26,2006.CrossRefGoogle ScholarPubMed
[35]Keldermann, R. H., Nash, M. P., and Panfilov, A. V.. Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems. Physica D, 238:1000–1007,2009.CrossRefGoogle Scholar
[36]Kittel, C.. Introduction to Solid State Physics. John Wiley & Sons, New York, 2004.Google Scholar
[37]Landau, L. D., Lifshitz, E. M., and Pitaevskii, L. P.. Electrodynamics of Continuous Media, volume 8. Butterworth-Heinemann, Oxford, 1984.Google Scholar
[38]Lee, E. H.. Elastic-plastic deformation at finite strains. ASME, Journal of Applied Mechanics, 36:1–6,1969.CrossRefGoogle Scholar
[39]LeGrice, I. J., Hunter, P. J., and Smaill, B. H.. Laminar structure of the heart: A mathematical model. American Journal of Physiology - Heart Circulation Physiology, 272:H2466–H2476, 1997.Google Scholar
[40]Lubarda, V. A.. Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics. Applied Mechanics Reviews, 57(2):95–108, 2004.Google Scholar
[41]Lubliner, J.. On the thermodynamic foundations of nonlinear solid mechanics. International Journal of Non-Linear Mechanics, 7:237–254,1972.CrossRefGoogle Scholar
[42]Luther, S., Fenton, F. H., Kornreich, B. G., Squires, A., Bitthn, P., Hornung, D., Zabel, M., Flanders, J., Gladuli, A., Campoy, L., Cherry, E. M., Luther, G., Hasenfuss, G., Krinsky, V. I., Pumir, A., Gilmour, R. F. Jr, and Bodenschatz, . Low-energy control of electrical turbulence in the heart. Nature Letter, 475(235-239):1000–1007, 2011.Google Scholar
[43]Maugin, G. A.. Continuum Mechanics of Electromagnetic Solids. North-Holland Series in Applied Mathematics and Mechanics, volume 33. Elsevier Science, Amsterdam, 1988.Google Scholar
[44]McMeeking, R. M. and Landis, C. M.. Electrostatic forces and stored energy for deformable dielectric materials. Journal of Applied Mechanics, 72:581–590,2005.Google Scholar
[45]McMeeking, R. M., Landis, C. M., and Jimenez, M. A.. A principle of virtual work for combined electrostatic and mechanical loading of materials. International Journal of Non-Linear Mechanics, 42:831–838,2007.Google Scholar
[46]Miehe, C., Kiefer, B., and Rosato, D.. An incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level. International Journal of Solids and Structures, 48:1846–1866, 2011.CrossRefGoogle Scholar
[47]Montanaro, A.. On the constitutive relations for second sound in thermo-electroelasticity. Archives of Mechanics, 63(3):225–254,2011.Google Scholar
[48]Nardinocchi, P. and Teresi, L.. On the active response of soft living tissues. Journal of Elasticity, 88:27–39,2007.CrossRefGoogle Scholar
[49]Nash, M. P. and Panfilov, A. V.. Electromechanical model of excitable tissue to study reentrant cardiac arrythmias. Progress in Biophysics & Molecular Biology, 85:501–522,2004.CrossRefGoogle Scholar
[50]Niederer, A. A., ter Keurs, H. E. D. J., and Smith, N. P.. Modelling and measuring electromechanical coupling in the rat heart. Experimental Physiology, 94(5):529–540,2009.CrossRefGoogle ScholarPubMed
[51]Nolasco, J. B. and Dahlen, R. W.. A graphic method for the study of alternation in cardiac action potentials. Journal of Applied Physiology, 2:191–196,1968.Google Scholar
[52]Odegard, G. M., Donahue, T. L. Haut, Morrow, D. A., and Kaufman, K. R.. Constitutive modelling of skeletal muscle tissue with an explicit strain-energy function. Journal of Biome-chanical Engineering, 130:061017,2008.Google Scholar
[53]Ogden, R. W. and Steigman, D.. Mechanics and Electrodynamics of Magneto- and Electro-elastic Materials. CISM International Centre for Mechanical Sciences, Springer, Udine, 2011.Google Scholar
[54]Ortiz, M. and Pandolfi, A.. A variational cam-clay theory of plasticity. Computer Methods in Applied Mechanics and Engineering, 193:2645–2666,2004.Google Scholar
[55]Ortiz, M. and Stainier, L.. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171:419–444,1999.CrossRefGoogle Scholar
[56]Pandolfi, A., Conti, S., and Ortiz, M.. A recursive-faulting model of distributed damage in confined brittle materials. Journal of the Mechanics and Physics of Solids, 54:1972–2003, 2006.CrossRefGoogle Scholar
[57]Panfilov, A. V., Keldermann, R. H., and Nash, M. P.. Self-organized pacemakers in a coupled reaction-diffusion-mechanics system. Physical Review Letters, 95:258104,2005.Google Scholar
[58]Pao, Y.-H. and Hutter, K.. A dynamic theory for magnetizable elastic solids with thermal and electrical conduction. Journal of Elasticity, 2:89–114,1974.Google Scholar
[59]Paulev, P. E. and Zubieta, G. C.. Textbook in Medical Physiology and Pathophysiology Essentials and Clinical Problems. Copenhagen, Denmark, 2000.Google Scholar
[60]Podio-Guidugli, P.. A virtual power format for thermomechanics. Continuum Mechanics and Thermodynamics, 20:479–487,2009.Google Scholar
[61]Pumir, A., Sinha, S., Sridhar, S., Argentina, M., Hörning, M., Filippi, S., Cherubini, C., Luther, S., and Krinsky, V.I.. Wave-train-induced termination of weakly anchored vortices in excitable media. Physical Review E, 81:010901,2010.CrossRefGoogle ScholarPubMed
[62]Richards, A. W. and Odegard, G. M.. Constitutive modelling of electrostrictive polymers using a hyperelasticity-based approach. Journal of Applied Mechanics, 77:014502,2010.Google Scholar
[63]Rodrigez, E. K., Hoger, A., and McCulloch, A. D.. Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics, 27:455–467, 1994.Google Scholar
[64]Rogers, J. M. and McCulloch, A. D.. A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Transactions of Biomedical Engineering, 41:743–757,1994.Google Scholar
[65]Ruiz-Baier, R., Gizzi, A., Rossi, S., Cherubini, C., Laadhari, A., Filippi, S., and Quarteroni, A.. Mathematical modeling of active contraction in isolated cardiomyocytes. Mathematical Methods in Biology, doi: 10.1093/imammb/dqt009,2013.Google Scholar
[66]Schmid, H., Wang, Y. K., Ashton, J., Ehret, A. E., Krittian, S. B. S., Nash, M. P., and Hunter, P. J.. Myocardial material parameter estimation: A comparison of invariant based orthotropic constitutive equations. Computer Methods in Biomechanics and Biomedical Engineering, 12:283–295,2009.CrossRefGoogle ScholarPubMed
[67]Simo, J. C.. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Computer Methods in Applied Mechanics and Engineering, 66:199–219,1988.Google Scholar
[68]Spencer, A. J. M.. Deformations of fibre-reinforced materials. Oxford Science Research Papers, Oxford, 1972.Google Scholar
[69]Stojanovic, R., Djuric, S., and Vujosevic, L.. On finite thermal deformations. Archives of Mechanics Stosow, 16:103–108,1964.Google Scholar
[70]Suo, Z.. Theory of dielectric elastomers. Acta Mechanica Solida Sinica, 23(6):549–578, 2010.CrossRefGoogle Scholar
[71]Suo, Z., Zhao, X., and Greene, W. H.. A nonlinear field theory of deformable dielectrics. Journal of the Mechanics and Physics of Solids, 56:467–486, 2008.Google Scholar
[72]Vasta, M., Gizzi, A., and Pandolfi, A.. On three- and two-dimensional fibers distributed models of biological tissues. Probabilistic Engineering Mechanics, In Press. DOI:10.1016/j.probengmech.2014.05.003Google Scholar
[73]Vasta, M., Pandolfi, A., and Gizzi, A.. A fiber distributed model of biological tissues. Procedia IUTAM, 6:79–86,2013.Google Scholar
[74]Vogel, F., Bustamante, R., and Steinmann, P.. On some mixed variational principles in electro-elastostatics. International Journal of Non-Linear Mechanics, 47:341–354,2012.Google Scholar
[75]Vu, D. K. and Steinmann, P.. On 3-D coupled BEM-FEM simulation of nonlinear electro-elastostatics. Computer Methods in Applied Mechanics and Engineering, 201-204:82–90, 2012.CrossRefGoogle Scholar
[76]Vu, D. K. and Steinmann, P.. On the spatial and material motion problems in nonlinear electro-elastostatics with consideration of free space. Mathematics and Mechanics of Solids, 10.1177/1081286511430161,2012.Google Scholar
[77]Yang, Q., Stainier, L., and Ortiz, M.. A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. Journal of the Mechanics and Physics of Solids, 54:401–424,2006.Google Scholar
[78]Zulliger, M. A., Rachev, A., and Stergiopulos, N.. A constitutive formulation of arterial mechanics including vascular smooth muscle tone. American Journal of Physiology - Heart Circulation Physiology, 287:H1335–H1343,2004.Google Scholar