Skip to main content
SpringerLink
Log in

Computational Mechanics of the Heart

  • Published:
Journal of elasticity and the physical science of solids Aims and scope Submit manuscript

Abstract

Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.J. Atkin and N. Fox, An Introduction to the Theory of Elasticity. Longman, London (1980).

    MATH  Google Scholar 

  2. J.B. Caulfield and T.K. Borg, The collagen network of the heart. Lab. Invest. 40(3) (1979) 364–372.

    Google Scholar 

  3. K.D. Costa, P.J. Hunter, J.M. Rogers, J.M. Guccione, L.K. Waldman and A.D. McCulloch, A three-dimensional finite element method for large elastic deformations of ventricular myocardium: Part I-Cylindrical and spherical polar coordinates. ASME J. Biomech. Engrg. 118(4) (1996) 452–463.

    Google Scholar 

  4. K.D. Costa, P.J. Hunter, J.M. Rogers, J.M. Guccione, L.K. Waldman and A.D. McCulloch, A three-dimensional finite element method for large elastic deformations of ventricular myocardium: Part II-Prolate spherical coordinates. ASME J. Biomech. Engrg. 118(4) (1996) 464–472.

    Google Scholar 

  5. K.D. Costa, K. May-Newman, D. Farr, W.G. O'Dell, A.D. McCulloch and J.H. Omens, Three-dimensional residual strain in midanterior canine left ventricle. Amer. J. Physiol. 273 (1997) H1968-H1976.

    Google Scholar 

  6. L.L. Demer and F.C.P. Yin, Passive biaxial mechanical properties of isolated canine myocardium. J. Physiol. London 339 (1983) 615–630.

    Google Scholar 

  7. S. Dokos, B.H. Smaill, A.A. Young and I.J. Le Grice, Anisotropic shear properties of passive ventricular myocardium. Amer. J. Physiol. (2001) (submitted).

  8. S. Dokos, A.A. Young, B.H. Smaill and I.J. Le Grice, A triaxial-measurement shear-test device for soft biological tissues. ASME J. Biomech. Engrg. (2001) (in press).

  9. P. Gould, D. Ghista, L. Brombolich and I. Mirsky, In vivo stresses in the human left ventricular wall: Analysis accounting for the irregular 3-dimensional geometry and comparison with idealised geometry analyses. J. Biomech. 5 (1972) 521–539.

    Article  Google Scholar 

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

    Google Scholar 

  11. P.J. Hunter, A.D. McCulloch and H.E.D.J. ter Keurs, Modelling the mechanical properties of cardiac muscle. Prog. Biophys. Molec. Biol. 69 (1998) 289–331.

    Article  Google Scholar 

  12. P.J. Hunter, M.P. Nash and G.B. Sands, Computational electromechanics of the heart. In: A.V. Panfilov and A.V. Holden (eds), Computational Biology of the Heart. Wiley, West Sussex, England (1997) Chapter 12, pp. 345–407.

    Google Scholar 

  13. J.M. Huyghe, T. Arts, D.H. van Campen and R.S. Reneman, Porous medium finite element model of the beating left ventricle. Amer. J. Physiol. 262 (Heart Circ. Physiol. 31) (1992) H1256-H1267.

    Google Scholar 

  14. J.M. Huyghe, D.H. van Campen, T. Arts and R.M. Heethaar, A two-phase finite element model of the diastolic left ventricle. J. Biomech. 24(7) (1991) 527–538.

    Article  Google Scholar 

  15. R.F. Janz and A.F. Grimm, Finite-element model for the mechanical behaviour of the left ventricle: Prediction of deformation in the potassium-arrested rat heart. Circ. Res. 30 (1972) 244–252.

    Google Scholar 

  16. R.F. Janz and A.F. Grimm, Deformation of the diastolic left ventricle. I. Nonlinear elastic effects. Biophys. J. 13 (1973) 689–704.

    Google Scholar 

  17. A.M. Katz, Physiology of the Heart, 2nd ed. Raven Press, New York (1992).

    Google Scholar 

  18. P. Kohl, P. Hunter and D. Noble, Stretch-induced changes in heart rate and rhythm: Clinical observations, experiments and mathematical models. Prog. Biophys. Molec. Biol. 71(1) (1999) 91–138.

    Article  Google Scholar 

  19. I.J. Le Grice, B.H. Smaill, L.Z. Chai, S.G. Edgar, J.B. Gavin and P.J. Hunter, Laminar structure of the heart: Ventricular myocyte arrangement and connective tissue architecture in the dog. Amer. J. Physiol. 269(38) (1995) H571-H582.

    Google Scholar 

  20. I.J. Le Grice, Y. Takayama and J.W. Covell, Transverse shear along myocardial cleavage planes provides a mechanism for normal systolic wall thickening. Circ. Res. 77 (1995) 182–193.

    Google Scholar 

  21. D.A. MacKenna, J.H. Omens and J.W. Covell, Left ventricular perimysial collagen fibers uncoil rather than stretch during diastolic filling. Basic Res. Cardiol. 91(2) (1996) 111–122.

    Article  Google Scholar 

  22. D.A. MacKenna, J.H. Omens, A.D. McCulloch and J.W. Covell, Contributions of collagen matrix to passive left ventricular mechanics in isolated rat hearts. Amer. J. Physiol. 266 (1994) H1007-H1018.

    Google Scholar 

  23. L.E. Malvern, Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ (1969).

    Google Scholar 

  24. A.D. McCulloch, Cardiac biomechanics. In: J.D. Bronzino (ed.), The Biomedical Engineering Handbook. CRC Press, Boca Raton, FL (1995) Chapter 31, pp. 418–439.

    Google Scholar 

  25. A.D. McCulloch and J.H. Omens, Non-homogeneous analysis of three-dimensional transmural finite deformation in canine ventricular myocardium. J. Biomech. 24(7) (1991) 539–548.

    Article  Google Scholar 

  26. I. Mirsky, Ventricular and arterial wall stresses based on large deformation analyses. Biophys. J. 13(11) (1973) 1141–1159.

    Article  MathSciNet  Google Scholar 

  27. M.P. Nash, Mechanics and material properties of the heart using an anatomically accurate mathematical model. PhD Thesis, The University of Auckland, New Zealand (1998).

    Google Scholar 

  28. P.M.F. Nielsen, I.J. Le Grice, B.H. Smaill and P.J. Hunter, Mathematical model of geometry and fibrous structure of the heart. Amer. J. Physiol. 260 (Heart Circ. Physiol. 29) (1991) H1365-H1378.

    Google Scholar 

  29. J.T. Oden, Finite Elements of Nonlinear Continua. McGraw-Hill, New York (1972).

    MATH  Google Scholar 

  30. J.H. Omens, K.D. May and A.D. McCulloch, Transmural distribution of three-dimensional strain in the isolated arrested canine left ventricle. Am. J. Physiol. 261 (Heart Circ. Physiol. 30) (1991) H918-H928.

    Google Scholar 

  31. T.F. Robinson, M.A. Geraci, E.H. Sonnenblick and S.M. Factor, Coiled perimysial fibers of papillary muscle in rat heart: Morphology, distribution and changes in configuration. Circ. Res. 63 (1988) 577–592.

    Google Scholar 

  32. E.K. Rodriguez, A. Hoger and A.D. McCulloch, Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4) (1994) 455–467.

    Article  Google Scholar 

  33. B.H. Smaill and P.J. Hunter, Structure and function of the diastolic heart: Material properties of passive myocardium. In: L. Glass, P.J. Hunter and A.D. McCulloch (eds), Theory of Heart: Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function. Springer, New York (1991) pp. 1–29.

    Google Scholar 

  34. D.D. Streeter, Jr., H.M. Spotnitz, D.P. Patel, J. Ross, Jr. and E.H. Sonnenblick, Fibre orientation in the canine left ventricle during diastole and systole. Circ. Res. 24 (1969) 339–347.

    Google Scholar 

  35. Y. Takayama, I.J. Le Grice, J.W. Holmes and J.W. Covell, Effects of chordal uncoupling on deformation in the papillary muscle. FASEB J. 8(5) (1994) A591.

    Google Scholar 

  36. J.V. Tyberg and E.R. Smith, Ventricular diastole and the role of the pericardium. Herz 15(6) (1990) 354–361.

    Google Scholar 

  37. G.S. Tyson, Jr., G.W. Maier, C.O. Olsen, J.W. Davis and J.S. Rankin, Pericardial influences on ventricular filling in the conscious dog: An analysis based on pericardial pressure. Circ. Res. 54 (1984) 173–184.

    Google Scholar 

  38. 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) 143–164.

    Article  MATH  Google Scholar 

  39. F.J. Villarreal, W.Y.W. Lew, L.K. Waldman and J.W. Covell, Transmural myocardial deformation in the ischemic canine left ventricle. Circ. Res. 68(2) (1991) 368–381.

    Google Scholar 

  40. W.M. Vogel, C.S. Apstein, L.L. Briggs, W.H. Gaasch and J. Ahn, Acute alterations in left ventricular diastolic chamber stiffness: Role of the erectile effect of coronary arterial pressure and flow in normal and damaged hearts. Circ. Res. 51(4) (1982) 465–478.

    Google Scholar 

  41. L.K. Waldman, Y.C. Fung and J.W. Covell, Transmural myocardial deformation in the canine left ventricle: Normal in vivo three-dimensional finite strains. Circ. Res. 57(1) (1985) 152–163.

    Google Scholar 

  42. L.K. Waldman, D. Nossan, F. Villareal and J.W. Covell, Relation between transmural deformation and local myofiber direction in the canine left ventricle. Circ. Res. 63 (1988) 550–562.

    Google Scholar 

  43. R.H. Woods, A few applications of a physical theorem to membranes in the human body in a state of tension. J. Anat. Physiol. 26 (1892) 362–370.

    Google Scholar 

  44. M. Yang and L.A. Taber, The possible role of poroelasticity in the apparent viscoelastic behaviour of passive cardiac muscle. J. Biomech. 24(7) (1991) 587–597.

    Article  Google Scholar 

  45. F.C.P. Yin, Ventricular wall stress. Circ. Res. 49(4) (1981) 829–842.

    Google Scholar 

  46. 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.

    Article  Google Scholar 

  47. A.A. Young, I.J. Le Grice, M.A. Young and B.H. Smaill, Extended confocal microscopy of myocardial laminae and collagen network. J. Microsc. 192(2) (1998) 139–150.

    Article  Google Scholar 

  48. O.C. Zienkiewicz and R.L. Taylor, The Finite ElementMethod. I. Basic Formulation and Linear Problems, 4th ed. McGraw-Hill, Berkshire, UK (1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Laboratory of Physiology, University of Oxford, Parks Road, Oxford, OX1 3PT, U.K.

    M. P. Nash

  2. Department of Engineering Science, The University of Auckland, Auckland, New Zealand

    P. J. Hunter

Authors
  1. M. P. Nash
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. J. Hunter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nash, M.P., Hunter, P.J. Computational Mechanics of the Heart. Journal of Elasticity 61, 113–141 (2000). https://doi.org/10.1023/A:1011084330767

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011084330767

Search

Navigation