The Invalidity of the Laplace Law for Biological Vessels and of Estimating Elastic Modulus from Total Stress vs. Strain: a New Practical Method

Abstract

The quantification of the stiffness of tubular biological structures is often obtained, both in vivo and in vitro, as the slope of total transmural hoop stress plotted against hoop strain. Total hoop stress is typically estimated using the "Laplace law." We show that this procedure is fundamentally flawed for two reasons: Firstly, the Laplace law predicts total stress incorrectly for biological vessels. Furthermore, because muscle and other biological tissue are closely volume-preserving, quantifications of elastic modulus require the removal of the contribution to total stress from incompressibility. We show that this hydrostatic contribution to total stress has a strong material-dependent nonlinear response to deformation that is difficult to predict or measure. To address this difficulty, we propose a new practical method to estimate a mechanically viable modulus of elasticity that can be applied both in vivo and in vitro using the same measurements as current methods, with care taken to record the reference state. To be insensitive to incompressibility, our method is based on shear stress rather than hoop stress, and provides a true measure of the elastic response without application of the Laplace law. We demonstrate the accuracy of our method using a mathematical model of tube inflation with multiple constitutive models. We also re-analyze an in vivo study from the gastro-intestinal literature that applied the standard approach and concluded that a drug-induced change in elastic modulus depended on the protocol used to distend the esophageal lumen. Our new method removes this protocol-dependent inconsistency in the previous result.Comment: 34 pages, 13 figure