Micro-structurally Based Kinematic Approaches to Electromechanics of the Heart

Abstract

This contribution is concerned with a new kinematic approach to the computational cardiac electromechanics. To this end, the deformation gradient is multiplicatively decomposed into the active part and the passive part. The former is considered to be dependent on the transmembrane potential through a micro-mechanically motivated evolution equation. Moreover, the proposed kinematic framework incorporates the inherently anisotropic, active architecture of cardiac tissue. This kinematic setting is then embedded in the recently proposed, fully implicit, entirely finite-element-based coupled framework. The implicit numerical integration of the transient terms along with the internal variable formulation, and the monolithic solution of the resultant coupled set of algebraic equations result in an unconditionally stable, modular, and geometrically flexible structure. The capabilities of the proposed approach are demonstrated by the fully coupled electromechanical analysis of a generic heart model.

Keywords

Deformation Gradient Kinematic Setting Kirchhoff Stress Tensor Kinematic Approach Passive Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The research by SG leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no: PCIG09-GA-2011-294161. Work of EK has received financial support from the National Science Foundation CAREER award CMMI-0952021 and from the National Institutes of Health Grant U54 GM072970.

References

  1. Aliev RR, Panfilov AV (1996) A simple two-variable model of cardiac excitation. Chaos Solitons Fractals 7:293–301 CrossRefGoogle Scholar
  2. Ambrosi D, Arioli G, Nobile F, Quarteroni A (2011) Electromechanical coupling in cardiac dynamics: the active strain approach. SIAM J Appl Math 71:605–621 MathSciNetMATHCrossRefGoogle Scholar
  3. Ask A, Menzel A, Ristinmaa M (2012a) Electrostriction in electro-viscoelastic polymers. Mech Mater 50:9–21 CrossRefGoogle Scholar
  4. Ask A, Menzel A, Ristinmaa M (2012b) Phenomenological modeling of viscous electrostrictive polymers. Int J Non-Linear Mech 47:156–165 CrossRefGoogle Scholar
  5. Cherubini C, Filippi S, Nardinocchi P, Teresi L (2008) An electromechanical model of cardiac tissue: constitutive issues and electrophysiological effects. Prog Biophys Mol Biol 97:562–573 CrossRefGoogle Scholar
  6. Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve induction. Biophys J 1:455–466 CrossRefGoogle Scholar
  7. Göktepe S, Acharya SNS, Wong J, Kuhl E (2011) Computational modeling of passive myocardium. Int J Numer Methods Eng 27:1–12 MATHCrossRefGoogle Scholar
  8. Göktepe S, Kuhl E (2009) Computational modeling of cardiac electrophysiology: a novel finite element approach. Int J Numer Methods Eng 79:156–178 MATHCrossRefGoogle Scholar
  9. Göktepe S, Kuhl E (2010) Electromechanics of the heart: a unified approach to the strongly coupled excitation-contraction problem. Comput Mech 45:227–243 MathSciNetMATHCrossRefGoogle Scholar
  10. Göktepe S, Wong J, Kuhl E (2010) Atrial and ventricular fibrillation-computational simulation of spiral waves in cardiac tissue. Archive. Appl Mech 80:569–580 CrossRefGoogle Scholar
  11. Holzapfel GA, Ogden RW (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos T Roy Soc A 367:3445–3475 MathSciNetMATHCrossRefGoogle Scholar
  12. Keener JP, Sneyd J (1998) Mathematical physiology. Springer, New York MATHGoogle Scholar
  13. Keldermann RH, Nash MP, Panfilov AV (2007) Pacemakers in a reaction-diffusion mechanics system. J Stat Phys 128:375–392 MathSciNetMATHCrossRefGoogle Scholar
  14. Kohl P, Hunter P, Noble D (1999) Stretch-induced changes in heart rate and rhythm: clinical observations, experiments and mathematical models. Prog Biophys Mol Biol 71:91–138 CrossRefGoogle Scholar
  15. Kröner E (1960) Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch Ration Mech Anal 4:273–334 MATHCrossRefGoogle Scholar
  16. Lee EH (1969) Elastic-plastic deformation at finite strain. J Appl Mech 36:1–6 MATHCrossRefGoogle Scholar
  17. Nash MP, Panfilov AV (2004) Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog Biophys Mol Biol 85:501–522 CrossRefGoogle Scholar
  18. Niederer SA, Smith NP (2008) An improved numerical method for strong coupling of excitation and contraction models in the heart. Prog Biophys Mol Biol 96:90–111 CrossRefGoogle Scholar
  19. Nielsen PMF, LeGrice IJ, Smaill BH, Hunter PJ (1991) Mathematical model of geometry and fibrous structure of the heart. Am J Physiol, Cell Physiol 260:H1365–H1378 Google Scholar
  20. Panfilov AV, Keldermann RH, Nash MP (2005) Self-organized pacemakers in a coupled reaction-diffusion-mechanics system. Phys Rev Lett 95:258104 CrossRefGoogle Scholar
  21. Pelce P, Sun J, Langeveld C (1995) A simple model for excitation-contraction coupling in the heart. Chaos Solitons Fractals 5:383–391 MATHCrossRefGoogle Scholar
  22. Rohmer D, Sitek A, Gullberg GT (2007) Reconstruction and visualization of fiber and laminar structure in the normal human heart from ex vivo diffusion tensor magnetic resonance imaging (DTMRI) data. Invest Radiol 42:777–789 CrossRefGoogle Scholar
  23. Stålhand J, Klarbring A, Holzapfel GA (2011) A mechanochemical 3D continuum model for smooth muscle contraction under finite strains. J Theor Biol 268:120–130 CrossRefGoogle Scholar
  24. Wong J, Göktepe S, Kuhl E (2011) Computational modeling of electrochemical coupling: a novel finite element approach towards ionic models for cardiac electrophysiology. Comput Methods Appl Mech Eng 200:3139–3158 MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Civil EngineeringMiddle East Technical UniversityAnkaraTurkey
  2. 2.Institute of MechanicsTU DortmundDortmundGermany
  3. 3.Division of Solid MechanicsLund UniversityLundSweden
  4. 4.Departments of Mechanical Engineering, Bioengineering, and Cardiothoracic SurgeryStanford UniversityStanfordUSA

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