The invalidity of the Laplace law for biological vessels and of estimating elastic modulus from total stress vs. strain: a new practical method

There are strong medical motivations to measure changes in material properties of tubular organs, in vivo and in vitro. The current approach estimates hoop stress from intraluminal pressure using the Laplace law and identifies 'elastic modulus' as the slope of a curve fitted hoop stress plotted against strain data. We show that this procedure is fundamentally flawed because muscle and other soft tissue are closely incompressible, so that the total stress includes a volume-preserving material-dependent hydrostatic response that invalidates the method. Furthermore, we show that the Laplace law incorrectly estimates total stress in biological vessels. However, the great need to estimate elastic modulus leads us to develop an alternative practical method, based on shear stress-strain, i.e. insensitive to nonelastic response from incompressibility, but that uses the same measurement data as the current (incorrect) method. The individual material parameters in the underlying (unknown) constitutive relation combine into an effective shear modulus that is a true measure of elastic response, unaffected by incompressibility and without reference to the Laplace law. Furthermore, our effective shear modulus is determined directly as a function of deformation, rather than as the slope of a fitted curve. We validate our method by comparing effective shear moduli against exact shear moduli for four theoretical materials with different degrees of nonlinearity and numbers of material parameters. To further demonstrate applicability, we reanalyse an in vivo study with our new method and show that it resolves an inconsistent change in modulus with the current method.