Speed-of-sound estimation assuming an ideal point scatterer as a target [4]

In our previous study [4], we proposed a method for estimating the SoS distribution when the medium is not homogeneous. In this paper, we describe an SoS estimation method using a linear array probe for a homogeneous medium because the present study is a basic investigation.

The target of the ultrasound irradiation is assumed to be an ideal point scatterer at depth \(d\) immediately below the center element of the linear array probe. The position of the central element is defined as \(_=0\). A wave scattered from the point scatterer is received at each element in the probe. Let \(c\) be the average SoS in the ultrasound propagation medium, \(_\) be the element position of element number \(k\), and \(_(_)\) be the theoretical value of the return propagation time of the scattered wave from the point scatterer at element \(k\). \(_(_)\) can be expressed by Eq. (1).

$$ \left( } \right) = \frac^ + d^ } }}}$$

(1)

Taking the square of Eq. (1), the right-hand side represents a linear expression for \(_^\).

$$\begin_\left(_\right)\right)}^=\frac^}_^+\frac^}^}=a_^+b.\end$$

(2)

Squared values \(\left\_\left(_\right)\right)}^\right\}\) of theoretical values \(_(_)\}\) for the return path propagation times are matched with the squared values of measured values \(\_\right)\}\) of the return path propagation times of the scattered wave at element \(k\). Coefficients \(a\) and \(b\) in Eq. (2) are determined using the least-squares method. Furthermore, by comparing the coefficients of the second and third equations in Eq. (2), average \(c\) from the probe to the scatterer and depth of the scatterer, \(d\), can be simultaneously estimated using Eqs. (3) and (4), respectively.

$$\widehat=\sqrt} ,$$

(3)

$$\widehat=\sqrt} .$$

(4)

Effects of using signals from blood vessels in the speed-of-sound estimation method

An ideal point scatterer is assumed in the aforementioned SoS estimation method. However, when the scatterer has a finite size, the irradiated ultrasonic wave is scattered at each position on the surface of the scatterer, and the arrival time of the scattered wave to each element is earlier than that of the ideal point scatterer, except for the center element. Thus, \(a\) in Eq. (2), decreases, and \(\widehat\) using Eq. (3) is overestimated [14]. When the propagation path of each ultrasound wave is only on the short-axis plane of the blood vessel, the positive bias error in the estimated SoS caused by the size of the scatterer can be corrected based on the geometric relationship between the apparent size and depth of the scatterer measured on the B-mode image and the reception time at each element [15].

Whereas previous studies [14, 15] assumed that the short-axis plane of a cylindrical target was measured as shown in Fig. 1(1), this study investigated the condition in which the probe is \(\theta\)-rotated around the z-axis relative to the short-axis plane of the vessel [Fig. 1(2)]. The beam width along the elevational direction of the probe is several millimeters, and the blood vessels have a cylindrical shape. Therefore, when the probe is \(\theta\)-rotated relative to the short-axis plane of the vessel, the irradiated ultrasonic wave is scattered on the uppermost surface of the vessel cylinder across the imaging plane of the probe, and the reception time of the scattered waves received at each element becomes earlier, except for the center element, than that when \(\theta =0^\circ\), causing a positive bias error in the estimated \(\widehat\) in Eq. (3).

This change in the reception time of the scattered waves may be altered by the transmitted beam condition and depth of the target cylinder. However, if this relationship does not depend on the target tissue characteristics, such as SoS and attenuation of the propagation medium and SoS of the target blood vessel, we can preliminarily measure and understand the relationship between rotation angle \(\theta\) and estimated SoS \(\widehat\) for each machine setting and target depth. Thus, in this preliminary study, we examined a single condition of the transmitted beam and target depth. Meanwhile, the dependence of the relationship between \(\theta\) and \(\widehat\) on the target tissue characteristics must be confirmed. Therefore, in this study, we conducted two experiments using a silicone tube in a water tank and nylon wires in a phantom, with different acoustic properties (SoS and attenuation of propagation media and SoS of target cylinders).

Water tank experiment

First, we conducted a simple experiment in which a silicone rubber tube was placed in a water tank. The probe was rotated by \(\theta\) relative to the short-axis plane of the silicone tube [Fig. 1(2)], and the characteristics of estimated \(\widehat\) and element signals changed by \(\theta\) were investigated. The diameters of the target blood vessels in the human liver range from a few hundred micrometers to a few millimeters [16]. However, to distinguish the effect of probe rotation angle \(\theta\) from that of the scatterer size described in previous studies [14, 15], we used a thin tube with inner and external diameters of 100 and 200 μm, respectively. The silicone tube was placed 30 mm from the probe surface. The true SoS in the water was obtained by measuring the water temperature [18].

Phantom experiment

A phantom experiment was conducted to simulate the liver condition in which the target cylinder (blood vessel) was surrounded by a weak scattering source (such as liver parenchyma) and to examine the conditions of different acoustic properties from those in the water tank experiment. We used a nylon wire with a diameter of 80 μm, which is shorter than the wavelength, as the target signal for SoS estimation to distinguish the effect of probe rotation angle \(\theta\) from that of the scatterer size described in previous studies [14, 15]. A general-purpose ultrasound phantom (Model 054GS; CIRS, USA) was used. A region with 13 wires aligned at different distances was measured, as shown in Fig. 2. The wires were surrounded by a hydrogel with a nominal SoS of \(1540\pm 10\) m/s.

Fig. 2

Schematic of the phantom experiment

Acoustic properties of ultrasound propagation media and target cylinders in water tank and phantom experiments

The SoS and attenuation coefficient in the propagation media in the water tank (water [18, 19]) and phantom (hydrogel) experiments and the SoS in the target cylinders (silicone tube [20] and nylon wire [21]) are summarized in Table 1. As shown in Table 1, the acoustic properties in the water tank and phantom experiments differed.

Table 1 Acoustic properties in water tank and phantom experimentsUltrasonic measurement conditions

An ultrasound diagnostic apparatus (Prosound SSD-α10; Hitachi Aloka, Japan) with a linear array probe (UST-5412; Hitachi Aloka, Japan) was used for data acquisition. The element pitch was 0.20 mm, transmitting frequency was 7.5 MHz, and sampling frequency was 40 MHz. A total of 95 elements were used for the transmission and reception of the ultrasound beam. The focal depth was set at 30 mm. For each target cylinder, the received element signals acquired using the ultrasonic beam transmitted from the element group centered on the element above the target cylinder were analyzed to estimate the SoS. The possible maximum deviation of lateral positions between the center of the beam and that of the target cylinder is 0.10 mm (half the distance between adjacent elements). The probe was rotated by \(\theta\) around the z-axis, as shown in Fig. 2, and data were acquired at θ =0°, 10°, 20°, 30°, and \(60^\circ\) to examine a broader range of rotation angle \(\theta\).