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

Abstract

The author 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 pressure-volume (P-V) and pressure-wall stress (P-S) behavior using the same geometric and stree-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 stress-strain 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.