Wave propagation in a model artery

Abstract

Fluid filled pipes are ubiquitous in both man-made constructions and living organisms. In the latter, biological pipes, such as arteries, have unique properties as their walls are made of soft, incompressible, highly deformable materials. In this article, we experimentally investigate wave propagation in a model artery: an elastomer strip coupled to a rigid water channel. We measure out-of-plane waves using synthetic Schlieren imaging, and evidence a single dispersive mode which resembles the pulse wave excited by the heartbeat. By imposing an hydrostatic pressure difference, we reveal the strong influence of pre-stress on the dispersion of this wave. Using a model based on the acoustoelastic theory accounting for the material rheology and for the large static deformation of the strip, we demonstrate that the imposed pressure affects wave propagation through an interplay between stretching, orthogonal to the propagation direction, and curvature-induced rigidity. We finally highlight the relevance of our results in the biological setting, by discussing the determination of the arterial wall's material properties from pulse wave velocity measurements in the presence of pre-stress.

Figures (8)

• Figure 1: (a) Dispersion relation of the axial waves propagating along a water filled nearly incompressible elastic pipe ($\rho_s = \qty{1070}{\kilogram\per\cubic\meter}$, $\mu_0 = \rho_s V_T^2 = \qty{23}{\kilo\pascal}$) with radius $R = \qty{4}{\mm}$ and thickness $h = \qty{1}{\mm}$ surrounded by an infinite water medium. We represent only the six lowest order fluid (orange), compression (green), torsion (purple), breathing (blue), and flexion (red) modes. (b) Zoom of (a) in the small wavenumber limit, $k_x R \ll 1$ (greyed region), where one dimensional models successfully capture the dispersion of axial waves. (c) Displacement fields at 100 Hz for the solid zero order modes identified in (a), where the color code corresponds to the main component of the normalized displacement field for each mode.
• Figure 2: (a) A soft compartment consisting of an elastomer strip ($h_0 = \qty{1}{\milli\metre}$, $w_0 = \qty{1}{\centi\metre}$) clamped to a rectangular water channel ($\qty{58}{\centi\meter} \times \qty{1}{\centi\meter} \times \qty{1}{\centi\meter}$) is subjected to a pressure difference $\Delta P = \rho_f g \Delta H$, that induces a large deflection of the soft wall $\delta(z)$. We generate waves in the strip using a shaker, and measure the out-of-plane displacement field, $u_y(x,z,t)$, associated to wave propagation using Synthetic Schlieren imaging. (b) Out-of-plane displacement field averaged along the z direction for two frequencies: $f = \qty{50}{\hertz}$ and $f = \qty{150}{\hertz}$; and two imposed pressure differences: $\Delta P = \qty{0}{\pascal}$ and $\Delta P = \qty{1500}{\pascal}$. (c) Dispersion relations of the zero order mode for $\Delta P = \qty{0}{\pascal}$ and $\Delta P = \qty{1500}{\pascal}$. The mode dispersion closely resembles that of the breathing mode in a fluid-filled tube and is highly sensitive to $\Delta P$.
• Figure 3: (a) A soft strip, clamped along its edges, is subjected to an elongation in the $\boldsymbol{e}_z$ direction. Elastic waves are generated using a shaker for three imposed values of the stretch ratio $\lambda_z$, and we extract the strip out-of-plane displacement $u_y(x,t)$ by recording the in-plane motion of a laser sheet in oblique incidence. (b) Dispersion curves of the lowest order flexion mode propagating in the strip for $\lambda_z = 1.0$, $1.09$, and $1.26$. The lines are predictions for two distinct hyperelastic models, the Mooney-Rivlin model (solid lines) and the neo-Hookean model (dotted lines), whose transparency renders the ratio $\mathrm{Im}(k_x)/|k_x|$.
• Figure 4: Dispersion curves of the lowest order flexural mode propagating in clamped curved strips with normalized radius of curvature $R/w_0 = \infty$, $1.0$, and $0.7$. The solid lines are predictions obtained without fitting parameters whose transparency renders the ratio $\mathrm{Im}(k_x)/|k_x|$.
• Figure 5: (a) Dispersion relations of the lowest order mode propagating in the model soft pipe for four imposed air pressures $\Delta P = \qty{0}{\pascal}$, $ \qty{200}{\pascal}$, $ \qty{490}{\pascal}$ and $ \qty{780}{\pascal}$. The solid lines are theoretical predictions, whose transparency encodes the ratio $\mathrm{Im}(k_x)/|k_x|$, for which the input stretch ratio $\lambda_\theta$ and curvature radius $\bar{R}$ are obtained from the static problem. (b) Normalized strip deflection $\delta(z)/w_0$ and curvature $|\kappa(z)| w_0$ profiles computed for the same imposed air pressures as in (a).