The speed, reflection and intensity of waves propagating in flexible tubes with aneurysm and stenosis: Experimental investigation

Abstract

A localized stenosis or aneurysm is a discontinuity that presents the pulse wave produced by the contracting heart with a reflection site. However, neither wave speed (c) in these discontinuities nor the size of reflection in relation to the size of the discontinuity has been adequately studied before. Therefore, the aim of this work is to study the propagation of waves traversing flexible tubes in the presence of aneurysm and stenosis in vitro. We manufactured different sized four stenosis and four aneurysm silicone sections, connected one at a time to a flexible 'mother' tube, at the inlet of which a single semi-sinusoidal wave was generated. Pressure and velocity were measured simultaneously 25 cm downstream the inlet of the respective mother tube. The wave speed was measured using the PU-loop method in the mother tube and within each discontinuity using the foot-to-foot technique. The stenosis and aneurysm dimensions and c were used to determine the reflection coefficient (R) at each discontinuity. Wave intensity analysis was used to determine the size of the reflected wave. The reflection coefficient increased with the increase and decrease in the size of the aneurysm and stenosis, respectively. c increased and decreased within stenosis and aneurysms, respectively, compared to that of the mother tube. Stenosis and aneurysm induced backward compression and expansion waves, respectively; the size of which was related to the size of the reflection coefficient at each discontinuity, increases with smaller stenosis and larger aneurysms. Wave speed is inversely proportional to the size of the discontinuity, exponentially increases with smaller stenosis and aneurysms and always higher in the stenosis. The size of the compression and expansion reflected wave depends on the size of R, increases with larger aneurysms and smaller stenosis.

Keywords: Aneurysm; PU-loop; reflection wave; stenosis; wave intensity analysis.

Figure 1.

A schematic diagram of the experimental setup including all parts of this construction. Mother and daughter tubes connected in line with the pump and cylindrical reservoir. All elements of the experiment are on the same horizontal platform. The raw indicates the positive flow direction.

Figure 2.

Top panel: drawings for a respective stenosis (left) and the mould configuration (right) for a manufactured stenosis of 10 cm length and 0.5 cm minimum diameter. Bottom panel: technical drawings for a respective aneurysm (left) and the mould configuration (right) for a manufactured aneurysm of 9 cm length and 4.4 cm maximum diameter. All dimensions are in mm.

Figure 3.

Effect of the stenosis and aneurysm size on wave speed. ID is the maximum internal diameter of each aneurysm and minimum diameter of each stenosis, which were taken at the centre of each discontinuity. Highlighted in light grey are results of the stenosis and in dark grey are those of the aneurysm discontinuities. Wave speed is higher within the stenosis than the aneurysm. Data are presented as mean of four measurements, and the error bars indicate standard deviation. The dashed curve represents the exponential regression of data, described by the equation and correlation coefficient, R² = 0.95.

Figure 4.

Measured pressure (P), velocity (U) waveforms and calculated wave intensity (dI), and their separation into forward (+) and backward (–) are shown in (a), (b) and (c), respectively. Measurements are taken at (25 cm) downstream of the mother tube inlet without discontinuity. The dashed line indicates the arrival time of reflected wave, T_R = 0.168 s, in agreement with the onset of the backward wave intensity (c), and the time of the separated forward and measured pressure (a), forward and measured velocity (b). FEW: forward expansion wave.

Figure 5.

Measured pressure (P), velocity (U) and calculated wave intensity (dI) and their separation into forward (+) and backward (–) directions at 25 cm downstream inlet of the mother tube with the context of 4.4 cm aneurysm (left column) and with the context of 0.5 cm stenosis (right column). R_t is the theoretical reflection coefficient calculated from equation (10). FEW: forward expansion wave.

Figure 6.

Maximum amplitude of the reflected pressure waveform is plotted against the respective sizes of the discontinuities. Top is the impact of stenosis and lower is the impact of aneurysm. Data are presented as mean of four measurements, and the error bars indicate standard deviation. The dashed lines represent the linear regression of data, described by the equation and correlation coefficient, R² = 0.9802.

Figure 7.

Maximum amplitude of the reflected wave intensity is plotted against the respective sizes of the discontinuities. Top is the impact of stenosis and lower is the impact of aneurysm. Data are presented as mean of four measurements, and the error bars indicate standard deviation. The dashed lines represent the linear regression of data, described by the equations and correlation coefficients, R² = 0.8594 for stenosis and R² = 0.9652 for aneurysm.