A three-dimensional Galerkin finite element method was developed for large deformations of ventricular myocardium and other incompressible, nonlinear elastic, anisotropic materials. Cylindrical and spherical elements were used to solve axisymmetric problems with r.m.s. errors typically less than 2 percent. Isochoric interpolation and pressure boundary constraint equations enhanced low-order curvilinear elements under special circumstances (69 percent savings in degrees of freedom, 78 percent savings in solution time for inflation of a thick-walled cylinder). Generalized tensor products of linear Lagrange and cubic Hermite polynomials permitted custom elements with improved performance, including 52 percent savings in degrees of freedom and 66 percent savings in solution time for compression of a circular disk. Such computational efficiencies become significant for large scale problems such as modeling the heart.
Issue Section:
Technical Papers
1.
Sarnoff
S. J.
Braunwald
E.
Welch
G. H.
Case
R. B.
Stainsby
W. N.
Macruz
R.
Hemodynamic Determinants of Oxygen Consumption of the Heart With Special Reference to the Tension-Time Index
,” Am J Physiol
, Vol. 192
, 1958
, pp. 148
–156
.2.
Jan
K.-M.
Distribution of Myocardial Stress and Its Influence of Coronary Blood Flow
,” J Biomech
, Vol. 18
, 1985
, pp. 815
–820
.3.
Alpert, N. R., Cardiac Hypertrophy, Academic Press, New York, 1971.
4.
Huisman
R. M.
Elzinga
G.
Westerhof
N.
Sipkema
P.
Measurement of Left Ventricular Wall Stress
,” Cardiovasc Res
, Vol. 14
, 1980
, pp. 142
–153
.5.
Horowitz
A.
Sheinman
I.
Lanir
Y.
Nonlinear Incompressible Finite Element for Simulating Loading of Cardiac Tissue—Part II: Three Dimensional Formulation for Thick Ventricular Wall Segments
,” ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol. 110
, 1988
, pp. 62
–68
.6.
Hunter, P. J., McCulloch, A. D., Nielsen, P. M. F., and Smaill, B. H., “A Finite Element Model of Passive Ventricular Mechanics,” Computational Methods in Bioengineering, ASME, Spilker, R. L., and Simon, B. R., eds., Chicago, 1988, pp. 387–397.
7.
Huyghe
J. M.
van Campen
D. H.
Arts
T.
Heethaar
R. M.
A Two-Phase Finite Element Model of the Diastolic Left Ventricle
,” J Biomech
, Vol. 24
, 1991
, pp. 527
–538
.8.
Huyghe
J. M.
Arts
T.
van Campen
D. H.
Reneman
R. S.
Porous Medium Finite Element Model of the Beating Left Ventricle
,” Am J Physiol
, Vol. 262
, 1992
, pp. H1256–H1267
H1256–H1267
.9.
Bovendeerd
P. H. M.
Arts
T.
Huyghe
J. M.
van Campen
D. H.
Reneman
R. S.
Dependence of Local Left Ventricular Wall Mechanics on Myocardial Fiber Orientation: A Model Study
,” J Biomech
, Vol. 25
, 1992
, pp. 1129
–1140
.10.
Guccione, J. M., and McCulloch, A. D., “Finite Element Modeling of Ventricular Mechanics,” Theory of Heart: Biomechanics, Biophysics and Nonlinear Dynamics of Cardiac Function, Springer-Verlag, Glass, L., Hunter, P. J., and McCulloch, A. D. eds., New York, 1991, pp. 121–144.
11.
Costa
K. D.
Hunter
P. J.
Wayne
J. S.
Waldman
L. K.
Guccione
J. M.
McCulloch
A. D.
A Three-Dimensional Finite Element Method for Large Elastic Deformations of Ventricular Myocardium: Part II—Prolate Spheroidal Coordinates
,” ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol. 118
, this issue, 1996
, pp. 462
–470
.12.
Green, A. E., and Adkins, J. E., Large Elastic Deformations, Clarendon Press, Oxford, 1970.
13.
Streeter, D. D., Jr, “Gross Morphology and Fiber Geometry of the Heart,” Handbook of Physiology, Section 2: The Cardiovascular System, Chapter 4, American Physiological Society, M. B. R., eds., Bethesda, MD, 1979, pp. 61–112.
14.
Spencer, A. J. M., Continuum Mechanics, Longman Press, London, 1980.
15.
Green, A. E., and Zema, W., Theoretical Elasticity, Oxford University Press, London, 1968.
16.
Oden, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972.
17.
Glowinski, R., and LeTallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.
18.
NAG Fortran Library, Mark 15, 1991.
19.
Zienkiewicz
O. C.
Zhu
J. Z.
A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis
,” Int J Numer Methods Eng
, Vol. 24
, 1987
, pp. 337
–357
.20.
Rivlin
R. S.
Large Elastic Deformations of Isotropic Materials VI. Further Results in the Theory of Torsion, Shear and Flexure
,” Phil Trans
, Vol. A242
, 1950
, pp. 173
–195
.21.
Guccione
J. M.
McCulloch
A. D.
Waldman
L. K.
Passive Material Properties of Intact Ventricular Myocardium Determined From a Cylindrical Model
,” ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol. 113
, 1991
, pp. 42
–55
.22.
Klingbeil
W. W.
Shield
R. T.
Large-Deformation Analyses of Bonded Elastic Mounts
,” Z angew Math Phys
, Vol. 17
, 1966
, pp. 281
–305
.23.
Ericksen
J. L.
Inversion of a Perfectly Elastic Spherical Shell
,” Z Angew Math Mech
, Vol. 35
, 1955
, pp. 382
–385
.24.
Bergel, D. A., and Hunter, P. J., “The Mechanics of the Heart,” Quantitative Cardiovascular Studies, University Park Press, Hwang, N. H. C., Gross, D. R., and Patel, D. J., eds., Baltimore, 1979, pp. 151–213.
25.
Taber
L. A.
On a Nonlinear Theory for Muscle Shells: II. Application to the Beating Left Ventricle
,” J Biomech Eng
, Vol. 113
, 1991
, pp. 63
–71
.26.
Nevo
E.
Lanir
Y.
The Effect of Residual Strain on the Diastolic Function of the Left Ventricle as Predicted by a Structural Model
,” J Biomech
, Vol. 27
, 1994
, pp. 1433
–1446
.27.
Battra
R. C.
Finite Plane Strain Deformations of Rubberlike Materials
,” Int J Num Meth Engng
, Vol. 15
, 1980
, pp. 145
–160
.28.
Glowinski
R.
LeTallec
P.
Numerical Solution of Problems in Incompressible Finite Elasticity by Augmented Lagrangian Methods: II. Three-Dimensional problems
,” SIAM J Appl Math.
Vol. 44
, 1984
, pp. 710
–733
.29.
Glowinski
R.
LeTallec
P.
Numerical Solution of Problems in Incompressible Finite Elasticity by Augmented Lagrangian Methods: I. Two-Dimensional and Axisymmetric Problems
,” SIAM J Appl Math
, Vol. 42
, 1982
, pp. 400
–429
.30.
Arts
T.
Reneman
R. S.
Veenstra
P. C.
A Model of the Mechanics of the Left Ventricle
,” Ann Biomed Eng
, Vol. 7
, 1979
, pp. 299
–318
.31.
May-Newman
K. D.
Omens
J. H.
Pavelec
R. S.
McCulloch
A. D.
Three-Dimensional Transmural Mechanical Interaction Between the Coronary Vasculature and Passive Myocardium in the Dog
,” Circ Res
, Vol. 74
, 1994
, pp. 1166
–1178
.32.
Humphrey
J. D.
Yin
F. C. P.
Constitutive Relations and Finite Deformations of Passive Cardiac Tissue II: Stress Analysis in the Left Ventricle
,” Circ Res.
Vol. 65
, 1989
, pp. 805
–817
.33.
Bogen
D. K.
Strain Energy Descriptions of Biological Swelling: I. Single Fluid Compartment Models
,” ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol. 109
, 1987
, pp. 252
–256
.34.
Fung, Y. C., Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York, 1981.
35.
Spaan
J. A. E.
Coronary Diastolic Pressure-Flow Relation and Zero Flow Pressure Explained on the Basis of Intramyocardial Compliance
,” Circ Res
, Vol. 56
, 1985
, pp. 293
–309
.36.
Vosoughi
J.
Vaishnav
R. N.
Patel
D. J.
Compressibility of the Myocardial Tissue
,” Adv Bioeng
, Vol. 1980
, pp. 45
–48
.37.
Omens
J. H.
Fung
Y. C.
Residual Strain in Rat Left Ventricle
,” Circ Res.
Vol. 66
, 1990
, pp. 37
–45
.38.
Rodriguez, E. K., “Residual Stress in the Rat Left Ventricle During Growth and Remodeling,” Doctoral Thesis, 1993, University of California San Diego, La Jolla, CA.
39.
Bovendeerd
P. H. M.
Huyghe
J. M.
Arts
T.
van Campen
D. H.
Reneman
R. S.
Influence of Endocardial-Epicardial Crossover of Muscle Fibers on Left Ventricular Wall Mechanics
,” J Biomech
, Vol. 27
, 1994
, pp. 941
–951
.
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