Shear Wave Propagation in Soft Tissues

The following procedure was followed for urethane rubber plates, gelatin plates and excised porcine LV free-wall myocardium samples. The samples were embedded in a gelatin mixture (80% water, 10% glycerol, 10% 300 Bloom gelatin, all by volume and 10 g/L potassium sorbate preservative, all manufactured by Sigma-Aldrich, St. Louis, MO) inside a plastic container. The container was mounted on a stand (not shown) in a water tank (Figure 3a). A window was cut out on the bottom of the container to allow for pulse-echo ultrasound measurements for motion detection. The gelatin was used as a stabilizer and was contained to the edges to minimize the affect of mechanical coupling. A mechanical shaker (V203, Ling Dynamic Systems Limited, Hertfordshire, UK) was used instead of a push transducer (Figure 3a) in order to ensure large motion and avoid ultrasound wave interference. A glass rod coupled with the shaker was glued to the hole bored through the thickness of the sample. Four cycles of sinusoidal waves were used to drive the shaker at different frequencies ranging from 40 to 500 Hz to induce cylindrical shear waves in the samples. More Lamb wave speed measurements were made below 200 Hz since the Lamb wave curve plateaus at higher frequencies and the lower frequencies provide more insight into the curvature of the Lamb wave dispersion. The motion of the shaker was parallel to the rod and the maximum amplitude of the displacement was on the order of tens of micrometers near the rod and exponentially decaying with distance (Figure 3a). At each excitation frequency, motion along the z-axis was measured at 31 points, 0.5 mm apart along the r-axis where
$r=\sqrt{x^2+y^2}$ , using a 5 MHz pulse-echo transducer with a pulse repetition rate of 4 kHz. Phase measurements at these points were used to fit a regression curve and calculate the Lamb wave speed at each frequency, as described in the SDUV methods [21 (link), 22 (link)]. The pulse-echo transducer was mounted on a robotic arm capable of micrometer size steps in three Cartesian directions. The robotic arm was used to move the transducer in four orthogonal directions in the r-plane at each excitation frequency (Figure 3a). For simplicity, we will refer to the four orthogonal directions in the r-plane at 0, π/2, π and 3π/2 radians as +x, +y, −x and −y directions. The long axis of the plate shaped samples was aligned parallel to the x-axis.