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

Abstract

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, ...

A relationship between ultrasonic integrated backscatter and myocardial contractile function.

“…Thus, the acoustic impedance mismatch will increase more in distal than in proximal regions by end-distention, and the cyclic variation of distal IB will be greater. These observed regional intramural differences in IB concur with expectations of greater inner wall fiber stress based on calculated transmural stress-strain relationships (23, 24,[44][45][46] (38). In contrast, series elastic elements of skeletal muscle may be stiffer than those of cardiac muscle (35).…”

Finite Element Modeling of Ventricular Mechanics

“…To keep the problem mathematically tractable, many workers have developed models of left ventricular mechanics using simple geometric approximations such as thin-walled spheres [88], thin-walled ellipsoids [16,67,86]' thick-walled ellipsoids [8,21,51,52,80,87], thick-walled spheres [1,15,53,54,69,83], thick-walled cylinders [4-6,11 ,17,24,34,57,58,76,79]' solids of revolution [35], and noncircular cylinders [39]. However, the analyses made by these models also make other simplifying assumptions about the material behavior of the heart muscle and the governing equations of motion.…”

Passive biaxial mechanical properties of isolated canine myocardium.

“…Thus, in order to obtain the fundamental data necessary for determination of the three-dimensional mechanics of myocardium, it is necessary to determine myocardial properties under multiaxial loading conditions. Indeed, several recent publications have emphasized the need for determination ofmyocardial constitutive relations under multiaxial loading conditions (Mirsky, 1976;Fung, 1973b;Panda & Natarajan, 1977;Vinson, Gibson & Yettram, 1979;Bergel & Hunter, 1979; Moriarty, 1980;Yin, 1981). If one assumes tissue incompressibility, one can generalize two-dimensional test data to obtain the full three-dimensional constitutive relations (Green & Adkins, 1960;Fung, 1973 a).…”