Finite-Element Model for the Mechanical Behavior of the Left Ventricle

Janz, R. F.; Grimm, Arthur F.

doi:10.1161/01.res.30.2.244

Cited by 82 publications

(30 citation statements)

References 2 publications

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“…The values taken for the endocardial and epicardial radii (0.25 cm and 0.50 cm, respectively) are taken from Janz and Grimm (1972) as typical values from serial sections of a potassium-arrested adult Sprague-Dawley albino male rat LV. Two spherical coordinate systems will be used throughout this paper: (R,@,$) for the undeformed (stress-free) state, and (r,#,<£) for the deformed state.…”

Section: Discussionmentioning

confidence: 99%

The law of Laplace. Its limitations as a relation for diastolic pressure, volume, or wall stress of the left ventricle.

Moriarty¹

1980

Circulation Research

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SUMMARY I made a detailed comparative examination of five mathematical models of left ventricular (LV) mechanics: the Laplace model, Lame model, Valanis-Landel model, Rivlin-Saunders model, and a nonhomogeneous version of the Valanis-Landel model. All five models are used to predict LV pressurevolume (P-V) and pressure-wall stress (P-S) behavior using the same geometric and stress-strain data (rat data is used as an example). These predictions are presented in graphical form for comparison with each other and observed LV P-V behavior. The first model, based on the Law of Laplace, uses the thin wall approximation in three distinct ways, and this approximation is not consistent with LV mechanics. The small deformation and linear stress-strain assumptions of the Lame model also are inconsistent with LV mechanics. Two more homogeneous models (Valanis-Landel and Rivlin-Saunders) avoid the errors of the first two. The discrepancy in the predictions of these two models demonstrates that uniaxial stress-strain data of papillary muscle are insufficient to characterize the multiaxial stressstrain behavior of myocardial tissue. Finally, a nonhomogeneous version of the Valanis-Landel model which explores the variation of myocardial stress-strain behavior needed to achieve constant wall stress is presented. Circ Res 46: 321-331, 1980 AN ACCURATE determination of the interrelationship of pressure, volume, and wall stress of the left ventricle (LV) is an important prerequisite for a fundamental understanding of cardiac mechanics. This interrelationship is a mathematical model which is used to predict wall stress from observed pressure and geometry (Ford, 1976;Rackley, 1976;Mirsky, 1974;Spotnitz and Sonnenblick, 1973;Streeter and Hanna, 1973a, Gould et al., 1972;Falsetti, et al., 1970Falsetti, et al., ,1971 Burns et al., 1971) and stressstrain relations for myocardial tissue from either LV pressure-volume behavior (Glantz, 1976;Rackley, 1976;Fester and Samet, 1974;Mirsky and Parmley, 1973;Lafferty et al., 1972;Diamond et al., 1971), or stress-strain relations of papillary muscle. It also is used to assess myocardial contractility (Falsetti et al., 1971;Hugenholtz et al., 1970) and predict LV volume from pressure (Burns et al., 1971) or vice versa (Sonnenblick and Strobeck, 1977). A mathematical model of LV mechanics is essential for the unified understanding of LV morphology in abnormal and diseased states. It may be based on the Law of Laplace (Woods, 1892), the work of Lame (1866), or the work of others, but whatever the source of the mathematical model, it is important that the assumptions on which it is based are satisfied for the range of deformation and stress-strain behavior exhibited by the LV; otherFrom the Engineering Science and Mechanics Department, University of Tennessee, Knoxville, Tennessee.This investigation was supported in part by U.S. Public Health Service Research Grant HL-14651 from the National Heart, Lung, and Blood Institute.Address for reprints: Thomas F. Moriarty, 317 Perkins Hall-UT, Knoxville, ...

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Section: Discussionmentioning

confidence: 99%

The law of Laplace. Its limitations as a relation for diastolic pressure, volume, or wall stress of the left ventricle.

Moriarty¹

1980

Circulation Research

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“…The effect ofthe ventricular muscle-fiber distribution has been modeled with finite elements that possess material anisotropy with respect to a continuously varying fiber axis [7,28,31] or more commonly by using a number of concentric elements, each with a constant fiber direction [12,60,63,89]. The incompressibility of the heart muscle, which is composed mostly of water, was very often accounted for incorrectly by assuming that the myocardium has a Poisson ratio close to 0.5 [12,20,27,36,37,61,63,82,89]. More correctly, in the context of finite deformations, the hydrostatic pressure-an extra dependent variable arising from the kinematic incompressibility constraint-is introduced as a Lagrange multiplier in the strain energy function 2 [7,25,28,31,81].…”

Section: The Finite Element Methodsmentioning

confidence: 99%

Finite Element Modeling of Ventricular Mechanics

Guccione

McCulloch

1991

Institute for Nonlinear Science

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Computing the distributions of stress and strain in a body with the complex geometry, boundary conditions, and material properties of the heart is a difficult yet worthwhile endeavor. The most promising method for obtaining numerical solutions to this problem is the finite element method. In this chapter, we review the use of finite element analysis for modeling ventricular mechanics. And we conclude by presenting a new axisymmetric finite element model of the passive left ventricle with a realistic geometry and fibrous architecture, physiological boundary conditions, and a three-dimensional constitutive equation. IntroductionThe distributions of stress in the ventricles of the heart are determined by (1) the three-dimensional geometry and fibrous architecture of the ventricular walls, (2) the boundary conditions imposed by the ventricular cavity and pericardial pressures and structures like the fibrous valve ring skeleton at the base of the ventricles, and (3) the three-dimensional mechanical properties of the myofibers and their collagen interconnections in the relaxed and actively contracting states. Many of these determining factors have been quantified in experimental studies, some of which are described in detail in this book.For example, Streeter and Hanna [72,73] and Nielsen and coworkers [56] have made detailed studies of the three-dimensional geometry and myocardial fiber architecture of the ventricles of the dog heart. Measurements of left and right ventricular pressures in patients are quite routine. And extensive data have been collected on the passive and active uniaxial material properties of isolated papillary muscles and trabeculae from various mammalian species [64,74]. Although fully triaxial material testing still presents significant technical difficulties, biaxial stress-strain testing of excised two-

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“…Stresses in nonaxisymmetric geometries can be estimated with finite element methods, 33 but these stresses can differ markedly from those predicted by the analytical models. 3 Regardless, since the various estimates of ventricular wall stress cannot be validated, current techniques do not allow reliable quantification of regional wall stress.…”

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Transverse stiffness: a method for estimation of myocardial wall stress.

Halperin

Chew

Weisfeldt

et al. 1987

Circulation Research

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Determination of regional ventricular wall stress would allow quantification of both regional contractile state and its interplay with global function. Current methods for quantifying regional stress include mathematical modelling and measurements with strain gauges. Both methods are difficult to validate. We hypothesized that transverse stiffness (i.e., the ratio of indentation stress to strain as the ventricular wall is indented in the direction perpendicular to the wall) would be proportional to the stresses in the plane of the wall and could be used to estimate the latter. To test this hypothesis, 6 arterial ly perfused canine ventricular septa were mounted in an apparatus that could exert biaxial load in the plane of the wall. A servo system maintained the central third of the septa isometric during active contractions while the septa were paced at 30-60 pulses/min. In the center of the isometric region, a probe of 7 mm diameter indented the septa while the transverse indentation stress and strain were measured. For values of peak systolic in-plane stress from 0.56 to 2.6 g/mm 2 , the transverse stiffness varied from 1.2 to 11.7 g/mm 2 and was linearly related to the in-plane wall stress in each septum (p<0.001, ANOVA). After cardioplegia, the transverse stiffness also correlated with passively applied wall stress for each dog (p<0.001). The slopes of the individual relations between transverse stiffness and wall stress from active contractions were similar to those from passively applied stress (mean±SEM; 1.82±0.36 versus 1.45±0.31, NS). The intercepts with the transverse stiffness axis from active contractions, however, were greater than those from passively applied stress (2.23 ± 0.57 versus -0.16±0.12 g/mm 2 , p<0.015). Moreover, at similar wall stresses, the transverse stiffness for active contractions was greater than that for passively applied stress (3.1 ± 0.7 versus 1.1 ± 0.2 g/mm 2 , p<0.005). Thus, regional transverse stiffness appears to allow quantitative estimation of regional in-plane stresses and can distinguish between actively generated and passively applied stress. This approach may allow one to accurately quantify the regional contractile state and to determine whether regional dysfunction is due to abnormal muscle that is not generating stress or to muscle capable of generating stress but which is abnormally loaded. (Circulation Research 1987;61:695-703) M ultiple factors can influence regional function in diseased hearts, including regional variations in loading, contractility, vascular supply, external restraints, nervous supply, wall thickness, and wall curvature.'"* With currently available methods, it i& difficult to separate these factors.

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Finite-Element Model for the Mechanical Behavior of the Left Ventricle

Cited by 82 publications

References 2 publications

The law of Laplace. Its limitations as a relation for diastolic pressure, volume, or wall stress of the left ventricle.

The law of Laplace. Its limitations as a relation for diastolic pressure, volume, or wall stress of the left ventricle.

Finite Element Modeling of Ventricular Mechanics

Transverse stiffness: a method for estimation of myocardial wall stress.

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