Basic investigation on identification of tissue composition based on propagation speeds of longitudinal and shear waves

Table I shows the compositions of the agar-glycerol phantoms. 21 cylindrical homogeneous phantoms (diameter: 80 mm, height: 80 mm) were prepared by combining agar (24440-1250, Junsei Chemical Co., Ltd., Japan) and glycerol (17029-70, Kanto Chemical Co., Inc., Japan). The weight concentration of agar was determined primarily to change the stiffness or SWS, while the weight concentration of glycerol was determined primarily to change the LWS. A 0.4-wt% polyethylene powder (10239, Avocado Research Chemicals Ltd., UK) was uniformly mixed as a scatterer in all phantoms.

Figure 1 shows the setup of the phantom experiment to ultrasonically measure the LWS and SWS. A homogeneous agar-glycerol phantom was placed on a sound absorber with a thickness of 10 mm, and then a linear array probe (L7-4, Philips, USA; 5.2 MHz, 128 channels, element pitch: 0.298 mm) for the LWS and SWS measurements was placed on the upper surface of the phantom. Using an ultrasound research platform (Vantage 64LE, Verasonics, USA), push pulses for the shear wave generation were transmitted and channel data for the LWS and SWS measurements were collected. The temperature inside the phantom at the time of measurement was 26.3 ± 0.1 °C. The measurement procedure used in the experiment was as follows.

• (1)  
  SWS measurement by push pulse transmission (6 times).
• (2)  
  LWS measurement by horizontally inserting a wire with a diameter of 0.65 mm at a depth of approximately 20 mm (6 times).
• (3)  
  Cut out a sample (50 mm × 50 mm × 15 mm).
• (4)  
  LWS measurement by a pulse transmission method (reference value).
• (5)  
  Density measurement using a specific gravity meter (reference value).
• (6)  
  Young's modulus measurement by a compression test (reference value).

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As shown in Fig. 2(a), a single push pulse with a frequency of 5.2 MHz and duration of 192 μs (= 1000 cycles) was transmitted toward a geometric focal depth of 15 mm by exciting 32 central elements of the L7-4 probe at 39.5 V. The focal depth of 15 mm corresponds to the conversion distance when a value of 1540 m s⁻¹ was assumed. Subsequently, the propagation of shear waves was visualized by transmitting plane waves and tracking particle displacements within a region of interest (ROI) with a width of 24 mm and height of 15 mm. The pulse repetition interval (PRI) between tracking frames was set to 100 μs. ³⁰⁾ As shown in Fig. 2(b), the position and time of the peak of the shear wave were tracked along the direction perpendicular to the propagation direction of the push pulse. Finally, the SWS was measured using the time of flight (TOF) method, ^31,32) which calculates the proportionality coefficient between the travel time and travel distance, as shown in Fig. 2(c).

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The LWS was measured using the Focusing method used in previous studies. ³³⁾ However, as there was a measurement error due to the horizontal misalignment between positions of the central element and wire target, ³³⁾ it was not possible to measure the LWS with a uniform accuracy for a wide range of LWS. Therefore, in this study, a minor improvement in correcting this misalignment was added. Figure 3 shows the geometrical positional relationship between the channel element and wire target. When the wire is not directly under the central element, the LWS measurement is performed by adding the variable ${x}_{0}$ to determine the position of the virtual central element. The delay time with this added ${x}_{0}$ is

$$\begin{eqnarray}&&{\tau }_{i}\left({x}_{0},{c}_{\mathrm{TEST}}\right)=\frac{{t}_{0}}{2}\left\{\sqrt{1+4{\left(\frac{i{\rm{\Delta }}x-{x}_{0}}{{c}_{\mathrm{TEST}}{t}_{0}}\right)}^{2}}-1\right\},\end{eqnarray} \tag{ 1 }$$

where ${t}_{0}$ is the appearance time of the wire echo received in the central element, $i$ is the channel number, ${\rm{\Delta }}x$ is the element pitch, ${c}_{{\rm{T}}{\rm{E}}{\rm{S}}{\rm{T}}}$ is the test value of the LWS (test LWS), and ${\tau }_{i}$ is the delay time given to the $i$th channel. As shown in Eq. (2), the envelope of the aperture synthesis wave of the channel RF data ${s}_{i}$ corrected for the time delay with ${\tau }_{i}$ is then calculated. In the time range of ${T}_{1}\leqslant t\leqslant {T}_{2}$ including the appearance time of the wire target echo, ($t,$ ${x}_{0},$ ${c}_{\mathrm{TEST}}$), which results in the maximum amplitude of the envelope of the aperture synthesis wave, is searched. Finally, at this time ${c}_{\mathrm{TEST}}$ is determined as the estimated value ${c}_{\mathrm{EST}}$ of the LWS.

$$\begin{eqnarray}&&{c}_{\mathrm{EST}}\leftarrow \,\max \left\{\mathrm{env}\left(\displaystyle \sum _{i=1}^{L}{s}_{i}\left(t,{\tau }_{i}\left({x}_{0},{c}_{\mathrm{TEST}}\right)\right)\right)\right\},\end{eqnarray} \tag{ 2 }$$

where $L$ is the number of all channels and $\mathrm{env}\left(\cdot \right)$ is the operation to calculate the envelope. The LWS was measured by varying the test LWS from 1300 to 1800 m s⁻¹ in increments of 1 m s⁻¹.

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For comparison, the LWS, density, and Young's modulus of the cut-out samples were measured. The LWS of the sample was measured using a method based on the pulse transmission method. ³⁴⁾ The density was measured using a specific gravity meter (MD-300S, Alpha Mirage, Japan) based on Archimedes' principle. The density of water was assumed to be 1 g cm⁻³. Young's modulus was measured by a compression test using a universal testing machine (AG-500NX, Shimadzu, Japan). Using a circular compressor with a diameter of 50 mm and load cell with a load capacity of 500 N and resolution of 1/1000, the slope of the force-displacement line was calculated when the sample was compressed at a crosshead speed of 1 mm min⁻¹. Young's modulus was then calculated based on the dimensions of the cut-out sample.

Figure 4 shows the SWS measurement process for each combination of agar and glycerol weight concentrations. Figures 4(a) and 4(b) show snapshots of the shear wave propagation inside the ROI shown in Fig. 2(a) at 0.5 ms after the push pulse was transmitted. Figure 4(a) compares the propagation distances of the shear wave with respect to the weight concentration of agar, while Fig. 4(b) compares the propagation distances of the shear wave with respect to the weight concentration of glycerol. The red circle indicates the position of the peak amplitude of the shear wave propagating from the focal point on the horizontal blue line. The contrast of these shear wave images differs between phantoms because the peak amplitude of the shear wave decreases owing to the effects of the distance attenuation corresponding to the weight concentration of agar and absorption corresponding to the weight concentration of glycerol. As the weight concentration of agar increases, the propagation distance at 0.5 ms increased and the SWS was expected to increase. However, there was no remarkable difference in the propagation distance with respect to the weight concentration of glycerol. Figures 4(c) and 4(d) show the relationship between the travel time and travel distance of the shear wave. In Fig. 4(c), the slope increased as the agar weight concentration increased under a constant glycerol weight concentration, whereas, in Fig. 4(d), the slope seemed to be independent on the glycerol weight concentration under a constant agar weight concentration.

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Figure 5 summarizes the relationship between the measured SWS and each weight concentration of agar and glycerol. The SWS measurement was repeated six times. The mean and standard deviation were calculated and are presented by error bars. As shown in Fig. 5(a), the SWS increased according to the weight concentration of agar, while the SWS was almost constant with respect to the weight concentration of glycerol, as shown in Fig. 5(b). Figure 5(c) compares the SWS converted from the Young's modulus $E$ using Eq. (3) and measured SWS. The Poisson's ratio was assumed to be 0.5. The results of Fig. 10, described later, were used as the density $\rho .$ Both were in good agreement, which confirms that the SWS was reasonably measured.

$$\begin{eqnarray}&&{\rm{SWS}}=\sqrt{\displaystyle \frac{E}{3\rho }}.\end{eqnarray} \tag{ 3 }$$

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Figure 6 shows the LWS measurement process for each combination of agar and glycerol weight concentrations. Figures 6(a) and 6(b) show the Focusing maps obtained on the right side of Eq. (2). The horizontal axis shows the test LWS ${c}_{{\rm{T}}{\rm{E}}{\rm{S}}{\rm{T}}},$ while the vertical axis shows the range of $\left[{T}_{1},{T}_{2}\right]$ at time $t.$ Figure 6(a) shows the Focusing map with respect to the weight concentration of agar, while Fig. 6(b) shows the Focusing map with respect to the weight concentration of glycerol. The redder part in the Focusing map corresponds to a larger amplitude of the aperture synthesis wave, while ${c}_{{\rm{T}}{\rm{E}}{\rm{S}}{\rm{T}}},$ which exhibits the maximum amplitude, corresponds to ${c}_{{\rm{E}}{\rm{S}}{\rm{T}}}.$ Figure 6(c) shows the maximum aperture synthesis wave with respect to the change in the agar weight concentration under a constant glycerol weight concentration, while Fig. 6(d) shows that with respect to the change in the glycerol weight concentration under a constant agar weight concentration. Contrary to the case of SWS, the LWS hardly changed when the agar weight concentration changed, while the LWS tended to change significantly when the glycerol weight concentration changed.

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Figure 7 summarizes the relationship between the measured LWS and each weight concentration of agar and glycerol. The LWS measurement was repeated six times. The mean and standard deviation were calculated and are presented by error bars. As shown in Fig. 7(a), the LWS was almost constant with respect to the weight concentration of agar, while the LWS increased in proportion to the glycerol weight concentration, as shown in Fig. 7(b). Figure 7(c) compares the values to the LWS reference value measured using the pulse transmission method. Both were in good agreement, which confirms that the LWS was reasonably measured.

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Figure 8 compares the measured SWS and LWS when (agar, glycerol) = (1.3 wt%, 0 wt%), (2.1 wt%, 0 wt%), and (1.3 wt%, 20 wt%). The error bar represents the standard deviation, i.e. the measurement error, for six measurements. This comparison was carried out because the effects of measurement errors must be considered in assessing the detectability of difference in composition by each index (SWS and LWS). Figure 8(a) shows the detectability of differences in agar and glycerol weight concentrations by the SWS. There was a significant difference in agar weight concentration, but not in glycerol weight concentration. Figure 8(b) shows the detectability of the difference in agar weight concentration and glycerin weight concentration by the LWS. Conversely, there was no significant difference in agar weight concentration, but there was a significant difference in glycerol weight concentration.

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The Poisson's ratio can be calculated using both LWS and SWS. The feasibility of identifying the compositions of the phantom by calculating the Poisson's ratio was evaluated. The Poisson's ratio ν was calculated by

$$\begin{eqnarray}&&\nu =\displaystyle \frac{1}{2}\left\{1-\displaystyle \frac{1}{{\left(\displaystyle \frac{{\rm{LWS}}}{{\rm{SWS}}}\right)}^{2}-1}\right\}.\end{eqnarray} \tag{ 4 }$$

Figure 9 shows the results of the calculated Poisson's ratio. Figure 9(a) shows the results of the difference in the agar weight concentration, while Fig. 9(b) shows the results for different glycerol weight concentrations. The Poisson's ratio tended to decrease as the agar weight concentration increased and increased as the glycerol weight concentration increased. However, Poisson's ratio seemed to be more sensitive to the agar weight concentration.

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