Abstract
Of the four basic biomechanics modeling steps outlined in the Introduction, determining the constitutive equations for cardiovascular tissue is often the most difficult step, especially when the tissue properties vary with time and sarcomere length history, as is the case with contracting myocardium. Using a cylindrical model to study transmural variations in stress and strain rather than a finite element model of the entire left ventricle allows for the implementation of a time- and sarcomere length history-dependent constitutive equation. The cylindrical model simulations can then be repeated with progressively simpler constitutive equations and the resulting transmural stress and strain distributions compared to determine under what conditions the most computationally efficient constitutive equations are valid. This chapter is primarily concerned with an instructive review of the constitutive equations we have implemented in cylindrical and finite element models of the passive and beating left ventricle, including that of diseased and surgically treated hearts. The last section of this chapter is concerned with experimental measurements that we have used to validate these models.
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References
Fung YC. Biomechanics: mechanical properties of living tissues. New York: Springer-Verlag, 1993.
ter Keurs HE, Rijnsburger WH, van Heuningen R, Nagelsmit MJ. Tension development and sarcomere length in rat cardiac trabeculae. Evidence of length-dependent activation. Circ Res. 1980b;46:703–14.
Leach JK, Priola DV, Grimes LA, Skipper BJ. Shortening deactivation of cardiac muscle: physiological mechanisms and clinical implications. J Investig Med. 1999;47:369–77.
Zahalak GI, Ma SP. Muscle activation and contraction: constitutive relations based directly on cross-bridge kinetics. J Biomech Eng. 1990;112:52–62.
Guccione JM, Motabarzadeh I, Zahalak GI. A distribution-moment model of deactivation in cardiac muscle. J Biomech. 1998;31:1069–73.
Guccione JM, Waldman LK, McCulloch AD. Mechanics of active contraction in cardiac muscle: Part II – Cylindrical models of the systolic left ventricle. J Biomech Eng. 1993;115:82–90.
Guccione JM, McCulloch AD, Waldman LK. Passive material properties of intact ventricular myocardium determined from a cylindrical model. J Biomech Eng. 1991;113:42–55.
Guccione JM, McCulloch AD. Mechanics of active contraction in cardiac muscle: Part I – Constitutive relations for fiber stress that describe deactivation. J Biomech Eng. 1993;115:72–81.
Tozeren A. Constitutive equations of skeletal muscle based on cross-bridge mechanism. Biophys J. 1985a;47:225–36.
Tozeren A. Continuum rheology of muscle contraction and its application to cardiac contractility. Biophys J. 1985b;47:303–9.
Guccione JM, Costa KD, McCulloch AD. Finite element stress analysis of left ventricular mechanics in the beating dog heart. J Biomech. 1995;28:1167–77.
Lin DH, Yin FC. A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus. J Biomech Eng. 1998;120:504–17.
Usyk TP, Mazhari R, McCulloch AD. Effect of laminar orthotropic myofiber architecture on regional stress and strain in the canine left ventricle. J Elasticity. 2000;61:143–64.
Walker JC, Ratcliffe MB, Zhang P, Wallace AW, Fata B, Hsu EW, Saloner D, Guccione JM. MRI-based finite-element analysis of left ventricular aneurysm. Am J Physiol Heart Circ Physiol. 2005;289:H692–700.
Zahalak GI, de Laborderie V, Guccione JM. The effects of cross-fiber deformation on axial fiber stress in myocardium. J Biomech Eng. 1999;121:376–85.
Landesberg A, Sideman S. Mechanical regulation of cardiac muscle by coupling calcium kinetics with cross-bridge cycling: a dynamic model. Am J Physiol. 1994;267:H779–95.
Guccione JM, McCulloch AD. Finite element modeling of ventricular mechanics. In: Theory of heart: biomechanics, biophysics, and nonlinear dynamics of cardiac function, edited by Glass L, Hunter PJ, McCulloch AD, New York: Springer-Verlag, 1991, p. xvii, 611p, pp. 121–44.
Waldman LK, Fung YC, Covell JW. Transmural myocardial deformation in the canine left ventricle. Normal in vivo three-dimensional finite strains. Circ Res. 1985;57:152–63.
Waldman LK, Nosan D, Villarreal F, Covell JW. Relation between transmural deformation and local myofiber direction in canine left ventricle. Circ Res. 1988;63:550–62.
Omens JH, May KD, McCulloch AD. Transmural distribution of three-dimensional strain in the isolated arrested canine left ventricle. Am J Physiol. 1991;261:H918–28.
Guccione JM, Le Prell GS, de Tombe PP, Hunter WC (1997a) Measurements of active myocardial tension under a wide range of physiological loading conditions. J Biomech 30, 189–92.
Guccione JM, O’Dell WG, McCulloch AD, Hunter WC. Anterior and posterior left ventricular sarcomere lengths behave similarly during ejection. Am J Physiol. 1997b;272:H469–77.
ter Keurs HE, Rijnsburger WH, van Heuningen R. Restoring forces and relaxation of rat cardiac muscle. Eur Heart J. 1980a;Suppl A:67–80.
Acknowledgments
This work was supported by National Institutes of Health grant 5R01 HL077921 (Dr. Guccione). This support is gratefully acknowledged.
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Jhun, CS., Wenk, J.F., Sun, K., Guccione, J.M. (2010). Constitutive Equations and Model Validation. In: Guccione, J., Kassab, G., Ratcliffe, M. (eds) Computational Cardiovascular Mechanics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0730-1_3
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DOI: https://doi.org/10.1007/978-1-4419-0730-1_3
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