Transmission conditions for clear depiction of thoracic spine based on difference between reflection and scattering characteristics of medical ultrasound

Figure 1 shows a wave scattered from a point scatterer when a focused wave is transmitted with a linear probe, where $d$ and ${x}_{m}$ are the depth of the point scatterer and the lateral position of the $m$th received element, respectively, ${x}_{m}=0$ shows the center of the probe aperture, and $c$ is the speed of sound in the medium. The time ranging from transmission at the center element to the receipt of the wave scattered from the point scatterer at the $m$th received element, ${{\rm{\tau }}}_{m,d},$ is given by

$$\begin{eqnarray}&&{{\rm{\tau }}}_{m,d}=\frac{d+\sqrt{{d}^{2}+{x}_{m}^{2}}}{c}.\end{eqnarray} \tag{ 1 }$$

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By defining the ideal delay time by ${\{{\rm{\tau }}}_{m,d}\}$ and the received signal at $m$th element by ${y}_{m}(t),$ the angular amplitude characteristic $R\left(m;d\right),$ which shows the head amplitude of the ${y}_{m}(t)$ just at the ideal delay time ${{\rm{\tau }}}_{m,d},$ is provided by the enveloped amplitude $y{{\prime} }_{m}\left(t\right)$ of ${y}_{m}\left({{\rm{\tau }}}_{m,d}\right)$ as

$$\begin{eqnarray}&&\begin{array}{c}R\left(m;d\right)=y{{\prime} }_{m}\left({{\rm{\tau }}}_{m,d}\right).\end{array}\end{eqnarray} \tag{ 2 }$$

When an ultrasound is irradiated to a scatterer such as muscle tissue, the ultrasound is scattered along with all the directions from the object, which follows Rayleigh scattering when the scatterer is sufficiently smaller than the wavelength of the ultrasound and the angular amplitude characteristic ${R}_{{\rm{S}}}\left(m;d\right)$ for the scattering becomes a gentle parabola. ³⁰⁾ On the other hand, when ultrasound is irradiated to a reflector with a regional plate, such as a bone, the ultrasound is specularly reflected, and ${R}_{{\rm{R}}}\left(m;d\right)$ for the reflection becomes high for the direction corresponding to the specular reflection and low for other directions. ³⁰⁾

The difference between ${R}_{{\rm{R}}}\left(m;d\right)$ and ${R}_{{\rm{S}}}\left(m;d\right)$ was quantitatively evaluated using following the root mean square error (RMSE):

$$\begin{eqnarray}&&\begin{array}{c}RMSE\,[ \% ]=\sqrt{\displaystyle \frac{{\sum }_{m}{\left\{{R}_{{\rm{R}}}\left(m;d\right)-\hat{A}{R}_{{\rm{S}}}\left(m;d\right)\right\}}^{2}}{{\sum }_{m}{R}_{{\rm{R}}}{\left(m;d\right)}^{2}}}\times 100,\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\begin{array}{c}\hat{A}=\displaystyle \frac{{\sum }_{m}{R}_{{\rm{R}}}\left(m;d\right){R}_{{\rm{S}}}\left(m;d\right)}{{\sum }_{m}{\left\{{R}_{{\rm{S}}}\left(m;d\right)\right\}}^{2}},\end{array}\end{eqnarray} \tag{ 4 }$$

where $\hat{A}$ is the parameter that minimizes the RMSE. From $\partial {RMSE}/\partial A=0,$ we can determine $\hat{A}.$ In the present paper, a suitable transmission aperture was determined so that the RMSE becomes maximum for various transmission aperture widths.

Figure 2(a) shows the simulation done in this paper. The number of received elements was 96, and the element pitch was 0.2 mm, assuming the typical ultrasound diagnostic equipment with the linear probe. When the short axis of a molybdenum wire with a diameter of 30 μm was set in a water tank, the transmitted and received waveform using one element above the wire was measured using ultrasound diagnostic equipment at a center frequency of 7.5 MHz. The waveform sampled at 40 MHz was interpolated 50 times using the sinc function, and the resultant ${y}_{0}(t)$ is shown in Fig. 2(b).

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The received signals from a reflector or scatterer were generated when a focused wave with a focal depth of 30 mm was transmitted. In Fig. 2(a), ${y}_{k{s}_{i}m}(t),$ which is transmitted from the $k$th transmitted element, scattered by $i$th point scatterer ${{s}}_{i},$ and received by $m$th received element, is given by

$$\begin{eqnarray}&&\begin{array}{c}{y}_{k{{s}}_{i}m}\left(t\right)=S\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)\,{y}_{0}\left(t-{\tau }_{k{{s}}_{i}m}\right),\end{array}\end{eqnarray} \tag{ 5 }$$

where $S\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)$ shows theoretical scattering characteristics,

$$\begin{eqnarray}&&\begin{array}{l}S\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)\\ =\,A\,\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)s\left(k,{x}_{{s}_{i}},{z}_{{s}_{i}},m\right)\\ D\left({x}_{{s}_{i}},{z}_{{s}_{i}},m\right)E\left({x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),\end{array}\end{eqnarray} \tag{ 6 }$$

and ${{\rm{\tau }}}_{k{{s}}_{i}m}$ shows the delay time of propagation from the $k$th transmitted element, $i$th point scatterer ${{s}}_{i},$ and $m$th received element as

$$\begin{eqnarray}&&\begin{array}{c}{{\rm{\tau }}}_{k{{s}}_{i}m}=\displaystyle \frac{{l}_{k{{s}}_{i}}}{{c}_{{\rm{w}}}}+\displaystyle \frac{{l}_{{{s}}_{i}m}}{{c}_{{\rm{w}}}}+{{\rm{\tau }}}_{k},\end{array}\end{eqnarray} \tag{ 7 }$$

where ${c}_{{\rm{w}}}$ is the speed of sound in the medium, ${l}_{k{{s}}_{i}}$ and ${l}_{{{s}}_{i}m}$ are the path lengths from the kth transmitting element to ${{s}}_{i}$ and from ${{s}}_{i}$ to mth receiving element, respectively, and ${{\rm{\tau }}}_{k}$ is the delay time of the kth element to transmit the focused wave. $S\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)$ of Eq. (6) comprises the propagation attenuation $A\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),$ the angular characteristic of Rayleigh scattering $s\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),$ the reception directivity of piezoelectric elements $D\left({x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),$ and the effect of interference in the width of the piezoelectric element at the reception of ultrasound, $E\left({x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)$ ³²⁾, $A\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),$ $s\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right),$ and $D\left({x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)$ are shown in Eqs. (8)–(10), respectively.

$$\begin{eqnarray}&&\begin{array}{c}A\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)={10}^{-\alpha \,\left({l}_{k{{s}}_{i}}+{l}_{{{s}}_{i}m}\right){f}_{0}^{2}/20},\end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\begin{array}{c}s\left(k,{x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)\propto \displaystyle \frac{1+{\cos }^{2}{{\theta }}_{k{s}_{i}m}}{2},\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\begin{array}{c}D\left({x}_{{s}_{i}},\,{z}_{{s}_{i}},m\right)={\cos }^{2}{{\theta }}_{m{\rm{ML}}},\end{array}\end{eqnarray} \tag{ 10 }$$

where $\alpha$ is the attenuation coefficient of water $(\alpha =1.94\times {10}^{-4}\,{\rm{dB}}\,{{\rm{MHz}}}^{-2}\,{{\rm{mm}}}^{-1})$ ³³⁾; ${f}_{0}$ = 7.5 MHz, which is the center frequency of the probe used in the water tank experiments; ${\theta }_{k{s}_{i}m}\,$is the scattering angle and is defined as

$$\begin{eqnarray}&&\begin{array}{c}{\theta }_{k{s}_{i}m}={\tan }^{-1}\displaystyle \frac{{x}_{{s}_{i}}-{x}_{k}}{{z}_{{s}_{i}}}+{\tan }^{-1}\displaystyle \frac{{x}_{m}-{x}_{{s}_{i}}}{{z}_{{s}_{i}}},\end{array}\end{eqnarray} \tag{ 11 }$$

and ${\theta }_{m{\rm{ML}}}$ is the incidence angle from the acoustic matching layer to the piezoelectric element given by

$$\begin{eqnarray}&&\begin{array}{c}{\theta }_{m{\rm{ML}}}={\sin }^{-1}\left(\displaystyle \frac{{c}_{{\rm{ML}}}}{{c}_{{\rm{w}}}}\cdot \displaystyle \frac{{x}_{m}-{x}_{{s}_{i}}}{{l}_{{{s}}_{i}m}}\right),\end{array}\end{eqnarray} \tag{ 12 }$$

where ${c}_{{\rm{ML}}}$ is the speed of sound in the acoustic matching layer. By considering scattering, ${y}_{km}\left(t\right)$ transmitted from the $k$th element and received by the $m$th element is given by

$$\begin{eqnarray}&&\begin{array}{c}{y}_{km}\left(t\right)=\displaystyle \sum _{{s}_{i}}{y}_{k{s}_{i}m}\left(t\right).\end{array}\end{eqnarray} \tag{ 13 }$$

By adding $\{{y}_{km}\left(t\right)\}$ to the transmitted elements $\{k\},$ the received signal at the $m$th received element ${y}_{m}\left(t\right)$ is obtained as follows:

$$\begin{eqnarray}&&\begin{array}{c}{y}_{m}\left(t\right)=\displaystyle \sum _{k}{y}_{km}\left(t\right).\end{array}\end{eqnarray} \tag{ 14 }$$

In the actual measurement, ${y}_{m}\left(t\right)$ is observed at the $m$th element and each component of $\{{y}_{km}\left(t\right)\}$ cannot be detected. However, by the simulation experiments in the present paper, ${y}_{m}\left(t\right)$ is decomposed into the element signals $\{{y}_{km}\left(t\right)\}$ of Eq. (13).

Alternatively in the present paper, we investigated $R\left(m;d\right)$ of Eq. (2) by clarifying the difference of $\{{y}_{km}\left(t\right)\}$ between the reflector and scatterer through the simulation experiments for each of the transmission aperture's widths ${{\ell }}_{{\rm{w}}}$ (the number ${N}_{{\rm{w}}}$ of transmitting elements) of 19.2 (96 elements), 14.4 (72 elements), 9.6 (48 elements), 4.8 (24 elements), and 0.2 mm (1 element). The signal from a plate reflector was generated by adding all the signals received from point scatterers aligned densely at 20 μm intervals with a width of 50 mm. By aligning point scatterers at 20 μm intervals on a tilted line, the surface of the tilted reflector was simulated.

The ultrasound diagnostic equipment was ProSound SSD-α10 (Hitachi Aloka) with a sampling frequency of 40 MHz. A linear probe UST-5412 (center frequency: 7.5 MHz, number of elements: 192, and element pitch: 0.2 mm) was attached to the ultrasound diagnostic equipment. For transmission, each element was given a delay time so that the focal point was 30 mm, which simulated the depth from the skin surface to the thoracic spine. No apodization was applied for transmission and reception.

Figure 3 shows the water tank experimental system. The short axis of molybdenum wire with a diameter of 30 μm and the surface of an acrylic block were used as a scatterer and a reflector, respectively. The depths of both objects were 30 mm. The angular amplitude characteristics $\{R\left(m;d\right)\}$ acquired with changing the transmission aperture's widths $\{{{\ell }}_{{\rm{w}}}\}$ were compared with the simulation results.

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First, we discuss the simulation results of angular amplitude characteristics $\{{R}_{{\rm{S}}}\left(m;d\right)\}\,\,$of scattering for a transmission aperture's width (${{\ell }}_{{\rm{w}}}$) of 19.2 mm (96 transmitted elements). The received element number of $m=47$ at the center position was set to $x=0$ mm. Figures 4(a1) and 4(b1) show the received signals ${y}_{km}\left(t\right){| }_{m=47}$ and ${y}_{km}\left(t\right){| }_{m=23}$ at the received positions $x=0\,\,$and $-4.8\,\,$mm, respectively, for each $k$th transmitted element. The dashed lines in these figures show the ideal delay time $\{{{\rm{\tau }}}_{m,d}\}$ calculated using Eq. (1). By integrating $\{{y}_{km}\left(t\right){| }_{m=47}\}$ and $\left\{{y}_{km}\left(t\right){| }_{m=23}\right\}$ along the direction of the transmitted elements $\{k\},$ Figs. 4(a2) and 4(b2) show the resultant waveforms of ${y}_{m}\left(t\right){| }_{m=47}$ and ${y}_{m}\left(t\right){| }_{m=23}.$

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As shown in Figs. 4(a1) and 4(b1), $\left\{{y}_{km}\left(t\right)\right\}$ were received at the same time for each received element when the object was a scatterer and placed at the focal point. Thus, the shape of ${y}_{m}\left(t\right)$ was the same as that of the transmitted and received waveform ${y}_{0}(t)$ based on one element. When the received position $m$ moved away from above the point scatterer, the signal amplitude in Fig. 4(b2) decreased owing to the receiving directivity of elements compared with that in Fig. 4(a2).

Next, we discuss the simulation results of angular amplitude characteristics $\{{R}_{{\rm{R}}}\left(m;d\right)\}$ of reflection. Figures 5(a1), 5(b1), and 5(c1) show the received signals $\{{y}_{km}\left(t\right){| }_{m=47}\},$ $\{{y}_{km}\left(t\right){| }_{m=33}\},$ and $\left\{{y}_{km}\left(t\right){| }_{m=23}\right\}$ at $x=0,\,-2.8,$ and $-4.8$ mm, respectively. Figures 5(a2), 5(b2), and 5(c2) show ${y}_{m}\left(t\right){| }_{m=47},$ ${y}_{m}\left(t\right){| }_{m=33},$ and ${y}_{m}\left(t\right){| }_{m=23}$ at $x=0,\,-2.8,$ and $-4.8$ mm, respectively. Figures 5(a3), 5(b3), and 5(c3) show the amplitudes of the received signals ${y}_{km}\left({{\rm{\tau }}}_{m,d}\right){| }_{m=47},$ ${y}_{km}\left({{\rm{\tau }}}_{m,d}\right){| }_{m=33},$ and ${y}_{km}\left({{\rm{\tau }}}_{m,d}\right){| }_{m=23}$ at the ideal delay time ${{\rm{\tau }}}_{m,d}.$

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In Figs. 5(a1), 5(b1), and 5(c1), $\{{y}_{km}\left(t\right)\}$ was received earlier than the $\,$ideal delay time ${{\rm{\tau }}}_{m,d}$ when the target was the reflector. This is because the propagation paths are shorter than those when the ultrasound is scattered at the focal point. Therefore, as shown in Figs. 5(a3), 5(b3), and 5(c3), {${y}_{km}\left({{\rm{\tau }}}_{m,d}\right)$} are not constant for the transmitted elements $k.$

The following two things can be found in Fig. 5(a3). The first is that the angular amplitude characteristic ${R}_{{\rm{R}}}\left(m=47;d\right)$ at the central received element $m\,=$ 47 has positively large amplitudes for 24 transmitted elements with $k\,=\,$36–59. The second is that ${R}_{{\rm{R}}}\left(m=47;d\right)$ is negative for the transmitted elements around $k=30$ and $65.$ By comparing Figs. 5(a3), 5(b3), and 5(c3), the profile of $\left\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\right\}$moved toward the transmitted elements on the right side. As a result of this movement, the negative components around $k=65$ in Fig. 5(a3) were not received at $m=23,$ as shown in Fig. 5(c3).

Figure 6 shows angular amplitude characteristics of reflection and scattering when the transmission aperture's width ${{\ell }}_{{\rm{w}}}$ (the number ${N}_{{\rm{w}}}$ of transmitting elements) was 19.2 (${N}_{{\rm{w}}}$ = 96), 14.4 (${N}_{{\rm{w}}}\,$= 72), 9.6 (${N}_{{\rm{w}}}$ = 48), 4.8 (${N}_{{\rm{w}}}$ = 24), and 0.2 mm (${N}_{{\rm{w}}}$ = 1). The shape of ${R}_{{\rm{R}}}\left(m\right)$ significantly changed as ${{\ell }}_{{\rm{w}}}$ decreased, but the maximum value did not significantly change, except when ${{\ell }}_{{\rm{w}}}$ = 0.2 mm. On the other hand, the shape of ${R}_{{\rm{S}}}\left(m\right)$ did not change, although its amplitude decreased in proportion to the decrease in ${{\ell }}_{{\rm{w}}}.$

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This difference is caused by the fact that $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ for the scattering were almost the same among the transmitted elements [Figs. 4(a1) and 4(b1)], whereas $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ for the reflection significantly varied among the transmitted elements [Figs. 5(a3), 5(b3), and 5(c3)]. The decrease in ${{\ell }}_{{\rm{w}}}$ corresponds to the exclusion of signals at both ends shown in Figs. 5(a3), 5(b3), and 5(c3).

For reflection, as the received element position ($m$) departed from the center, the shape of $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ changed [Figs. 5(a3), 5(b3), and 5(c3)], and the transmitted elements $\left\{k\right\}$ with large amplitudes of $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ shifted toward the end. Therefore, as ${{\ell }}_{{\rm{w}}}$ decreased, ${R}_{{\rm{R}}}\left(m\right)$ became small.

The RMSE between $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ and $\{{R}_{{\rm{S}}}\left(m;d\right)\}$ for each transmission aperture's width (${{\ell }}_{{\rm{w}}}$) is denoted with red dots in Fig. 7. The cases of ${{\ell }}_{{\rm{w}}}$ = 2.4 mm (${N}_{{\rm{w}}}$ = 12) and ${{\ell }}_{{\rm{w}}}$ = 7.2 mm (${N}_{{\rm{w}}}$ = 36) were also calculated and added. The case of ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}$ = 24) can emphasize the difference.

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Figures 8(a1) and 8(b1) show the received signals $\{{y}_{km}\left(t\right){| }_{m=47}\}$ and $\left\{{y}_{km}\left(t\right){| }_{m=23}\right\}$ at $x=0$ and $4.8$ mm for each $k$th transmitted element, respectively, for ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}\,$ = 24). Figures 8(a2) and 8(b2) show the received signal ${y}_{m}(t)$ at $x=0$ and $-4.8$ mm, respectively. Figures 8(a3) and 8(b3) show the amplitudes of the received signals $\left\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right){| }_{m=47}\right\}$ and $\left\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right){| }_{m=23}\right\}$ at the ideal delay times ${{\rm{\tau }}}_{m,d},$ respectively.

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The reason for the RMSE being maximized for ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}$ = 24) can be explained in two points: first, as shown in Fig. 8(a3), when ${{\ell }}_{{\rm{w}}}$ was 4.8 mm (24 elements) or more, ${R}_{{\rm{R}}}\left(47;d\right)$ was almost the same for $k\,$ = 36–59, which accounted for a large fraction of ${R}_{{\rm{R}}}\left(47;d\right)$ at the center with regard to the received element; second, as shown in Figs. 5(a3), 5(b3), and 5(c3), the shape of $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ changed as the received element position shifted. When ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}$ = 24), as shown in Fig. 8(b3), for the received element position $(m)\,\,$apart from the center, the received signal decreased with the change in $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}.$ When ${{\ell }}_{{\rm{w}}}$ is larger than 4.8 mm, the number of transmitted elements with a large amplitude of $\{{y}_{km}\left({{\rm{\tau }}}_{m,d}\right)\}$ does not decrease even if the shape of ${y}_{km}\left({{\rm{\tau }}}_{m,d}\right)$ changes slightly; therefore, the width of the peak of ${R}_{{\rm{R}}}\left(m;d\right)$ increases, and the RMSE with the angular amplitude characteristic of scattering ${R}_{{\rm{S}}}\left(m;d\right)$ decreases. When ${{\ell }}_{{\rm{w}}}$ is narrower than 4.8 mm, the peak of ${R}_{{\rm{R}}}\left(m;d\right)$ decreases because all the signals from the transmitted elements of $k$ = 36–59 have not been received. Therefore, the RMSE between ${R}_{{\rm{R}}}\left(m;d\right)$ and ${R}_{{\rm{S}}}\left(m;d\right)$ is maximum when ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}$ = 24).

Next, we investigated the case in which the reflector was inclined to the probe surface. Figure 9 shows the angular amplitude characteristics of reflection and scattering when the reflector was tilted by 5° or 10° concerning the probe surface. The signals with large amplitude were detected with the end of the received elements $\left\{m\right\}$ as the tilt angle increased. When the reflector was tilted by 10°, the amplitudes decreased with decreasing ${{\ell }}_{{\rm{w}}},$ because most of the reflected signals could not be received at the receiving elements; this was because of the tilt of the reflector.

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The RMSEs between $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ and $\{{R}_{{\rm{S}}}\left(m;d\right)\}$ for each transmission aperture's width (${{\ell }}_{{\rm{w}}}$) when the reflector was tilted by 5° or 10° to the probe surface are denoted with red and blue dots, respectively in Fig. 10. When the reflector inclination was 5°, the RMSE was maximum for ${{\ell }}_{{\rm{w}}}$ = 4.8 mm (${N}_{{\rm{w}}}$ = 24). The tendency was the same as the result for the parallel reflector in Fig. 7. When the reflector was tilted by 10°, the RMSE also increased with decreasing ${{\ell }}_{{\rm{w}}},$ although the RMSE for ${{\ell }}_{{\rm{w}}}$ = 4.8 mm was almost the same as that for ${{\ell }}_{{\rm{w}}}$ = 19.2 mm (${N}_{{\rm{w}}}$ = 96). This is because most of the reflected signals could not be received at the receiving elements for ${{\ell }}_{{\rm{w}}}$ = 4.8 mm. Therefore, it suggests that the bone surface should be aligned as parallel as possible to the probe surface in clinical applications.

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In the simulation, the reflector was assumed to be flat; therefore, if the reflector was tilted to the probe surface in the elevational direction of the probe, the shape of $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ did not change. However, the angular amplitude characteristics from the curved surfaces such as the thoracic vertebrae would change due to phase aberration. We will investigate this effect further in our future work.

Figure 11 shows $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ of reflection and $\{{R}_{{\rm{S}}}\left(m;d\right)\}$ of scattering for each transmission aperture's width (${{\ell }}_{{\rm{w}}}$); these were obtained from the surface of the acrylic block and the short axis of the wire, respectively, in the water tank experiment. The received signal was too weak while measuring the scatterer when ${{\ell }}_{{\rm{w}}}$ = 0.2 mm. The results were almost similar to Fig. 6. In Fig. 7, the blue dots show the RMSE between $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ and $\{{R}_{{\rm{S}}}\left(m;d\right)\}$ for each value of ${{\ell }}_{{\rm{w}}}.$ Reflectors and scatterers can be distinguished in detail when ${{\ell }}_{{\rm{w}}}$ = 4.8 mm.

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The reduction in ${{\ell }}_{{\rm{w}}}$ affects the lateral resolution of the focused wave. Figure 12(a) shows the acoustic field distribution along the lateral direction $x$ at a focal depth of 30 mm for each ${{\ell }}_{{\rm{w}}}.$ The horizontal axis represents the lateral position centered just above the beam. The vertical axis was normalized to the maximum amplitude for each value of ${{\ell }}_{{\rm{w}}}.$ The full-width-at-half-maximum (FWHM) increased as ${{\ell }}_{{\rm{w}}}$ decreased.

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Figure 12(b) shows the relationship between the FWHM of the transmitted beam at the focal depth and RMSE of $\left\{{R}_{{\rm{R}}}\left(m;d\right)\right\}$ and $\{{R}_{{\rm{S}}}\left(m;d\right)\}.$ There is a trade-off between the lateral resolution and RMSE. In thoracic spine measurements in the clinic, it is necessary to determine the appropriate transmission aperture's width ${{\ell }}_{{\rm{w}}}$ for imaging by considering this trade-off relationship.