Journal of Medical Ultrasonics
https://doi.org/10.1007/s10396-023-01383-y

ORIGINAL ARTICLE–PHYSICS & ENGINEERING

Estimation error in speed of sound caused by rotation of measured
cross‑section from short‑axis plane of blood vessels: a preliminary
study
Shohei Mori1· Keiji Onoda2 · Mototaka Arakawa1,2 · Hiroshi Kanai1,2

Received: 12 April 2023 / Accepted: 19 September 2023
© The Author(s) 2023

Abstract
Purpose Estimating the speed of sound (SoS) in ultrasound propagation media is important for improving the quality of
B-mode images and for quantitative tissue characterization. We have been studying a method for estimating the SoS by
measuring the reception time distribution of waves scattered from a scatterer at the elements in a probe. Previously, themeasurement cross section was assumed to be perpendicular to the long axis of the blood vessel. In this study, we experimentally investigated the relationship between rotation angle 𝜃 of the probe relative to the short-axis plane of the bloodc.
vessel and the estimated SoS, ̂c and element signals
Methods Water tank and phantom experiments were conducted to investigate the characteristics of ̂
when the probe was rotated.
Results The received signal powers at the elements around both edges greatly decreased as 𝜃 increased. We introduced aparameter representing the decrease in power, Pdec, in the received signal at the elements at both edges relative to the centerc was estimated to be larger as 𝜃 increased, especially for 𝜃 ≥ 30◦. Pdec also increased as 𝜃 increased. An approxielement. ̂c and Pdec.
mately proportional relationship existed between the errors in ̂Conclusion Based on these results, we can distinguish between the presence and the absence of SoS misestimations using
the difference in power among the elements in the received signal. In the absence of misestimation, we can obtain the true
SoS, even if the target has a non-negligible size, by applying our previously proposed methods.
Keywords Ultrasound imaging · Speed of sound · Liver · Blood vessels · Element data · Delay time

Introduction
Medical ultrasound imaging is useful for observing various organs because of its noninvasiveness and real-time
performance. The formation of a B-mode image from thereceived element signals requires information on the speedof sound (SoS) in the propagation medium. The image quality degrades when an image is formed at an assumed SoS
that differs from the actual SoS. Therefore, several studies* Shohei Mori
mori@tohoku.ac.jp
1

Graduate School of Engineering, Tohoku University, 6‑6‑05
Aoba, Aramaki, Aoba, Sendai, Miyagi 980‑8579, Japan

2

Graduate School of Biomedical Engineering, Tohoku
University, 6‑6‑05 Aoba, Aramaki, Aoba, Sendai,
Miyagi 980‑8579, Japan

have been conducted to estimate the SoS in propagation
media [1–9]. Determination of the SoS in tissues can also
be used to identify diseases. For example, the SoS in theliver decreases with an increase in fat concentration [10].
Therefore, methods for estimating the SoS in the liver havebeen studied to quantitatively diagnose fatty liver [11–13].
We proposed a method for estimating the SoS in propagation media by measuring the reception time distributionof waves scattered from a scatterer at elements in a linear
array probe [4]. To accurately estimate the SoS using thismethod, the reception time distribution must be measured
accurately. For this purpose, the use of signals with a highsignal-to-noise (SNR) ratio is preferable. The liver contains
many blood vessels, and the signals received from these vessels have a higher intensity than the scattered waves from theliver parenchyma. We have studied a method to estimate theaverage SoS of an ultrasound propagation medium between

13Vol.:(0123456789)

Journal of Medical Ultrasonics

the probe and blood vessels in the liver using signals fromblood vessels with high signal intensity.
Our group developed a method to estimate the SoS using
the reception time of a scattered wave from a point scatterer
at each element [4]. However, the assumption of a point scatterer is not applicable to scatterers larger than the ultrasoundwavelength, such as blood vessels. Therefore, we verified
the effect of the size of the scatterers in the SoS estimationcmethod assuming a point scatterer. We observed that SoS ̂
was overestimated (positive bias error) when the scatterers
had non-negligible diameters [14]. Therefore, we proposedc using the appara method for correcting the overestimated ̂
ent size and depth of the scatterer in the B-mode image [15].
This method can estimate the average SoS in the ultrasoundpropagation region using signals from scatterers of a finite
size, such as blood vessels. To apply our previous studies[14, 15] to SoS estimation using signals from a blood vessel,
the measurement cross section must be perpendicular to thelong axis of the blood vessel. However, this assumption does
not always hold because of the complex directions of theblood vessels in the liver [16, 17]. In this study, the crosssection perpendicular to the long axis of the blood vessel
was defined as the short-axis plane.
Figure 1 shows a schematic of the blood vessel measurement using an ultrasound probe. Figure 1(1) shows theprobe measuring the short-axis plane of the blood vessel,and Fig. 1(2) shows the probe rotating at an angle 𝜃 aroundthe z-axis relative to the short-axis plane of the blood vessel.
In this paper, we experimentally investigated the relationshipFig. 1  Schematic of measurement of a blood vessel using
an ultrasound probe. (1-a, b)
Ultrasound probe measuring theshort-axis plane of a blood vessel and (2-a, b) when the probeis 𝜃-rotated around the z-axis
relative to the short-axis planeof a blood vessel(1-a)

between probe rotation angle 𝜃 relative to the short-axis
c , and observed
plane of the blood vessel and the estimated ̂c was overestimated by the SoS estimation methodthat ̂
assuming a point scatterer when 𝜃 was large. Furthermore,since 𝜃 is unknown in actual in vivo measurements, we propose a method to determine whether the method describedin Ref. [4] could be applied, based on the characteristics ofthe received signal measured at each element and without
using the information of 𝜃.

Methods
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 thecenter element of the linear array probe. The position of thecentral element is defined as xk = 0. A wave scattered fromthe point scatterer is received at each element in the probe.
Let c be the average SoS in the ultrasound propagation
medium, xk be the element position of element number k ,and T0 (xk ) be the theoretical value of the return propagation

blood vessel

blood vessel

linear probe

(1-b)

(2-b)
linear probe

blood vessel

13

(2-a)

linear probe

blood vessel

Journal of Medical Ultrasonics

time of the scattered wave from the point scatterer at element
k . T0 (xk ) can be expressed by Eq. (1).
√xk2 + d2
( )(1)
T0 xk =cTaking the square of Eq. (1), the right-hand side represents a linear expression for xk2.

( ( ))2
2
T0 xk= c12 xk2 + dc2 = axk2 + b.
(2){( ( )) }2
T0 xk
of theoretical values
Squared values{T 0 (xk )} for the return path propagation times are( )matchedwith the squared values of measured values {t xk } 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 thecoefficients of the second and third equations in Eq. (2),average c from the probe to the scatterer and depth of thescatterer, d , can be simultaneously estimated using Eqs. (3)and (4), respectively.
√
1
(3)
,
̂
c=
a

when the probe is 𝜃-rotated relative to the short-axis plane ofthe vessel, the irradiated ultrasonic wave is scattered on theuppermost surface of the vessel cylinder across the imaging
plane of the probe, and the reception time of the scatteredwaves received at each element becomes earlier, except forthe center element, than that when 𝜃 = 0◦, causing a positivec in Eq. (3).
bias error in the estimated ̂This change in the reception time of the scattered waves
may be altered by the transmitted beam condition and depthof the target cylinder. However, if this relationship does notdepend on the target tissue characteristics, such as SoS andattenuation of the propagation medium and SoS of the target
blood vessel, we can preliminarily measure and understandthe relationship between rotation angle 𝜃 and estimated SoS
̂c for each machine setting and target depth. Thus, in thispreliminary study, we examined a single condition of thetransmitted beam and target depth. Meanwhile, the dependc on the target tisence of the relationship between 𝜃 and ̂
sue characteristics must be confirmed. Therefore, in thisstudy, 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
̂
d=

√

b
.
a

(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 afinite size, the irradiated ultrasonic wave is scattered at eachposition on the surface of the scatterer, and the arrival timeof 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 ̂c using Eq. (3) is overestimated[14]. When the propagation path of each ultrasound wave isonly on the short-axis plane of the blood vessel, the positive
bias error in the estimated SoS caused by the size of thescatterer 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 ateach element [15].
Whereas previous studies [14, 15] assumed that the shortaxis plane of a cylindrical target was measured as shown in
Fig. 1(1), this study investigated the condition in which theprobe is 𝜃-rotated around the z-axis relative to the shortaxis 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,

First, we conducted a simple experiment in which a silicone rubber tube was placed in a water tank. The probe wasrotated by 𝜃 relative to the short-axis plane of the silicone
c andtube [Fig. 1(2)], and the characteristics of estimated ̂element signals changed by 𝜃 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 𝜃from 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 wasplaced 30 mm from the probe surface. The true SoS in thewater was obtained by measuring the water temperature [18].

Phantom experiment
A phantom experiment was conducted to simulate the livercondition 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 useda 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 𝜃 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

13Journal of Medical Ultrasonics

distances was measured, as shown in Fig. 2. The wires weresurrounded by a hydrogel with a nominal SoS of 1540 ± 10m/s.

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 beamand that of the target cylinder is 0.10 mm (half the distance
between adjacent elements). The probe was rotated by 𝜃around the z-axis, as shown in Fig. 2, and data were acquiredat θ =0°, 10°, 20°, 30°, and 60◦ to examine a broader rangeof rotation angle 𝜃.

Acoustic properties of ultrasound propagation
media and target cylinders in water tankand 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 watertank and phantom experiments differed.

Ultrasonic measurement conditions

Results

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

Water tank experimentFig. 2  Schematic of the phantom experiment

The measurement results for the silicone tube in water at θ
= 0°, 30°, and 60◦ are shown in Fig. 3. Figure 3(1) shows

3.6 cmdepth [mm]linear probe (measurement region) lateral position

0.5 mm0.25 mm

1 mm 2 mmA

depth: 30 mm
A C
B D

2 mm

3 mm
1 mm

B

4 mm

C
D

0.5 mm0.25 mm

3 mm

wires (diameter : 80 μm)

4 mm

solid elastic hydrogelwire

lateral position

phantom

Table 1  Acoustic properties in water tank and phantom experiments
Propagation medium

Water tank experiment
Phantom experiment

13

Speed of sound [m/s]Attenuation coefficient [dB/cm/MHz]

1486–1491 [18] (water)
1540 ± 10 (nominal value)(hydrogel)0.0021 (at 20 ◦ C) [19] (water)
0.7 (nominal value) (hydrogel)

Speed of sound in target
cylinder [m/s]
960 [20] (silicone rubber)
2530 [21] (nylon)

Journal of Medical Ultrasonics[mm]element position[mm]element position[mm]

amplitude

reception time [ s](2-c)

reception time [ s]amplitude

reception time [ s]

(2-b)

[mm]

amplitude [dB]

[mm]
depth
lateral position[mm]lateral position

[mm]

(2-a)

element position(1-c)

amplitude

lateral position(1-b)
amplitude [dB]

depth

[mm]

amplitude [dB]depth [mm](1-a)

element position[mm]

power [dB](3)

Fig. 3  Measurement results when the probe was rotated 𝜃 = 0◦, 30◦,
and 60◦ relative to the silicone tube arranged in a water tank. (1)B-mode images of the silicone tube for (1-a) 𝜃 = 0◦, (1-b) 𝜃 = 30◦,

and (1-c) 𝜃 = 60◦. (2) Received element signals from the silicone tube
for (2-a) 𝜃 = 0◦, (2-b) 𝜃 = 30◦, and (2-c) 𝜃 = 60◦. (3) Average powerof the received signal at each element in the window (red lines) in (2)the B-mode images. The observed shapes of the silicone
tube on the B-mode images extended along the lateral direction as 𝜃 increased. The ultrasonic beam had a width in theelevational direction, and the silicone tube had a cylindrical shape. Therefore, the observed shapes on the B-mode
images were considered to extend along the lateral direction
because the signal from the silicone tube was received on
the beams near the center of the tube, similar to that on thecenter beam, for 𝜃 = 30◦ and 60◦.
Figure 3(2) shows the received element signals before
the formation of the ultrasonic beam. The signals received
from the silicone tube had smaller amplitudes at the elements around both edges as probe rotation angle 𝜃 increased.
Figure 3(3) shows the average power of the received element signals in a 0.6-µs wide window between the two redlines shown in Fig. 3(2), calculated for each element. The
window for calculating the power of the received signals was
set along the approximated time distribution obtained using
Eqs. (1)–(4). As shown in Fig. 3(3), the signal powers at theelements around both edges greatly decreased as 𝜃 increased.

Phantom experiment
The measurement results for the phantom at 𝜃 = 0◦, 30◦, and60◦ are shown in Fig. 4. Figure 4(1) shows B-mode images.
Figure 4(2) shows the received element signals before theformation of the ultrasonic beam near wire A at each angle
𝜃, where the ranges of the signals are indicated by the arrows
in Fig. 4(1). Similar to the results of the water tank experiment, the observed shapes of the nylon wire on the B-mode
images extended along the lateral direction and the element
signals showed smaller amplitudes at the elements aroundboth edges as 𝜃 increased.
Figure 4(3) shows the average power of the received
signals in a 0.6-µs wide window between the two red lines
shown in Fig. 4(2), calculated for each element. As thedetection of signals from the nylon wire at the edge-side
elements was difficult owing to the decrease in the signal
amplitude, the parabolic approximation for SoS estimation
using Eq. (2) was applied only to the elements whose peaksignals could be detected, and the window for calculating the13

Journal of Medical Ultrasonics

Fig. 4  Measurement results when the probe was rotated 𝜃 = 0◦, 30◦,
and 60◦ relative to the phantom. (1) B-mode images of the phantom
for (1-a) 𝜃 = 0◦, (1-b) 𝜃 = 30◦, and (1-c) 𝜃 = 60◦. (2) Received ele-

power of the received signals was set along the approximatedtime distribution obtained using Eqs. (1)–(4).
The power of the received signals from the nylon wire
was lower than that from the silicone tube in water (Fig. 3)
because of the difference in attenuation between the hydrogel and water (approximately − 30 dB difference when the
7.5 MHz frequency of the ultrasound wave propagated a
total round-trip distance of 60 mm). However, similar to
the water tank experiment, the decrease in power aroundthe edge of elements due to the increase in 𝜃 was observed
for the phantom experiment, as shown in Fig. 4(2) and (3).

Relationship between the decrease in power
and the SoS estimation error
To quantify the decrease in the received power at the elements at both edges (k = ±47) relative to the center element(k = 0), we introduced a parameter representing the decreasein power, Pdec, defined by the following equation:

13

ment signals from scatterer A for (2-a) 𝜃 = 0◦, (2-b) 𝜃 = 30◦, and(2-c) 𝜃 = 60◦. (3) Average power of the received signal at each element in the window (red lines) in (2)Pdec =

Pxk=−47 +Pxk=47
2− Pxk=0 , [dB](5)

where Pxk denotes the average power of the signals in thewindow, calculated for the element at position xk.
c values were estimated by applying the SoS estiThe ̂
mation method in [4] to the signals received from the silicone tube in water and nylon wires A, B, C, and D in thephantom. The relationships between the Pdec values cal(culated)using Eq. (5) and the SoS estimation error ratios,̂c − ctr ∕ctr , where ctr is the true SoS, are shown in Fig. 5.
The results of both the water tank and phantom experiments are plotted in Fig. 5 (the results of the water tankexperiment are plotted by cross markers).
As Pdec decreased, the SoS estimation error increased
in both water tank and phantom experiments. This result
showed that even if 𝜃 is unknown, the thresholding process
for Pdec can estimate and eliminate the conditions underwhich 𝜃 is large and the SoS is misestimated.

Journal of Medical Ultrasonics

Discussion

Fig. 5  Relationship between the decrease in power, Pdec, at eachangle 𝜃 and the speed-of-sound (SoS) estimation error ratio,̂(̂c − ctr )∕ctr , where ctr is the true SoS. ĉ
A , ĉB, ĉC, and c
D : SoS in
phantom (hydrogel) estimated using the signals received from nylon
wires A, B, C, and D, respectively. ĉ
w : SoS in water estimated using
signals from silicone tube

For 𝜃 ≤ 30◦, the water tank and the phantom experiments
showed a similar relationship between Pdec and SoS estimation error ratio. This result indicates that the relationshipbetween the Pdec and the SoS estimation error ratio is independent of or less dependent on the acoustic properties ofthe propagation medium and target cylinders.
For 𝜃 = 60◦, however, Pdec in the phantom experiment
was higher than that in the water tank experiment, whereas
the SoS estimation error ratio showed similar values. Thisdifference in Pdec might be caused by the lower SNR in thephantom experiment. In the phantom, the nylon wires weresurrounded by hydrogel; therefore, the scattered signals fromthe nylon wires were affected by the interference of the weak
scattered signals from the hydrogel. The averaged power ofthe scattered signals from the hydrogel was 7 dB in both
the regions at depths of 25.0–25.5 mm and 35.0–35.5 mm,
averaged for all 95 elements. Because the transmitted focus
depth was 30 mm, the powers of the scattered signals fromthe hydrogel at depths of nylon wires were considered to behigher than 7 dB. Therefore, the powers of the scattered signals from the hydrogel were on the same order as those fromthe target wires (approximately 10 dB) at both edge elementsfor 𝜃 = 60◦, as shown in Fig. 4(3). Thus, the decrease in the
power of the wire signals around the edge elements wasconsidered to be suppressed compared with the noiselesscondition in the water tank experiment.
In Fig. 5, the results of the phantom experiment deviateslightly for wires A, B, C, and D. This may be attributed tothe interference of signals from the surrounding scattering
sources (hydrogel) and the low SNR due to attenuation in
the propagation medium (hydrogel). However, a tendency
between Pdec and SoS estimation error ratios was observed
even when considering the variation in results among wires.

c was overesThe conducted experiments revealed that ̂
timated when the probe was rotated by 𝜃 relative to thevessel short-axis plane [Fig. 1(2)], and that Pdec decreased.
Furthermore, a similar relationship between the Pdec and
the SoS estimation error ratio was observed for the watertank and the phantom experiments, although the acoustic properties of the propagation media and target cylinders were different between the experiments, as listed inTable 1.
Several studies have reported the measurement resultsof SoS in human liver [22] and rat liver [13, 23]. Ino et al.
[23] also reported the measured value of attenuation in rat
liver. The range of reported values differed among studies;
however, the differences among reported values (SoS inliver: approximately 1510–1580 m/s [13, 22, 23], attenuation coefficient in liver: approximately 0.5 dB/cm/MHz
or higher measured by 5 MHz transducer [23]) and acoustic properties (SoS and attenuation) of the phantom were
smaller than the differences between the water tank andphantom experiments shown in Table 1. These results suggest that the relationship between the decrease in power,
Pdec , and the SoS estimation error ratio is independent ofor less dependent on the acoustic properties of the propagation medium and target blood vessels, which are generally unknown in vivo.
As described in the Methods section, the relationshipbetween the Pdec and the SoS estimation error ratio may
depend on the ultrasonically transmitted beam conditions
and depth of the target cylinder. However, this relationship can be preliminarily evaluated for each beam condition (machine setting) and target depth using experiments
similar to those conducted in this study. In in vivo measurements, the actual running directions ( 𝜃 ) of the bloodvessels in the liver cannot be known. However, by preliminarily determining the threshold of Pdec for each beam condition (machine setting) and target depth, it is possible todetermine whether the scattered waves from a blood vesselin the liver can be used to correctly estimate the SoS by
evaluating Pdec . For example, when the same beam conditions as in this study are used and blood vessels positioned
at a depth of 30 mm are used for the SoS estimation target,
we can determine that a 2% or larger SoS estimation error
may occur if Pdec < −10 dB.
Thus, the SoS in the liver can be stably estimated byapplying the SoS estimation method in our previous studies [4, 15] only to vessels in which Pdec satisfies the desired
SoS estimation accuracy. If this condition is satisfied, the
SoS can be estimated using the method described in Ref.
[4]. Subsequently, if the target can be considered as a point
scatterer, the estimated SoS should be used as it is; if it

13

Journal of Medical Ultrasonics

has a non-negligible size, it should be corrected using themethod described in Ref. [15]. In this study, a target cylinder with a small diameter (200 μ m for the water tankexperiment and 80 μ m for the phantom) was examined to
distinguish the effect of probe rotation angle 𝜃 from thatof the scatterer size described in previous studies [14, 15].
The examination of more complicated conditions, where
the probe rotates to blood vessels with a non-negligible
diameter, will be the subject of our future study. The conditions in which the target cylinder does not have a straightshape should also be examined in future studies.
As discussed in the Results section, Pdec for 𝜃 = 60◦ wasaffected by weak scattered signals from the surrounding
scattering sources in the phantom experiment. This effectmay also be caused in in vivo measurements of the liver.
However, this effect becomes a problem only for the low
Pdec condition (that is, large 𝜃 condition causing a decreasein power around the edge of elements). The low Pdec condition (that is, large 𝜃 condition) causes large SoS estimation
error, as shown in Fig. 5, and this is the condition that should
be removed for accurate SoS estimation. Thus, the effect of
weak scattered signals from surrounding tissues does not
become a problem for this study.
In this study, we experimentally confirmed that thedecrease in power, Pdec, could be used to judge the target signal condition (probe rotation angle 𝜃 ) causing the SoS estimation error. However, the reason for the decrease in powerowing to rotation angle 𝜃 must be theoretically elucidated infuture studies. When the target has a cylindrical structure,
the transmitted ultrasound waves are scattered at multiple
positions on the cylinder surface, and the interfered waves
are received by the elements in the probe. This interference
may cause a decrease in power around the edge of the elements. In addition, this interference changes as a result of
the geometric relationship between the target and the element array in the probe; therefore, the amount of decreasein power may be systematically changed by rotation angle 𝜃.
The liver contains many blood vessels [16, 17]; therefore,
it is possible that signals from the neighboring blood vessels
interfered with each other. In our experiments, the results fora single target (the water experiment) and those for several
targets (the phantom experiment) showed a similar tendency;that is, the effect of this interference among the target wires
was not observed in the phantom experiment. However, theeffects of this interference should be examined in greater
detail in future studies.
In this study, a linear array probe was used; however,application of the proposed method to a convex probe will
expand its applicability in liver measurements. For this purpose, it is necessary to introduce the curvature factor of theconvex probe into the proposed method, which should be
examined in detail in future studies. Another choice is touse a linear array probe for SoS measurement in the liver,13

although applicable depth and field of view are restricted.
Shear wave elastography has been widely studied to evaluate diffuse liver diseases although measurement position
and range of the region of interest are restricted for accurate
measurements [24]. Similarly, SoS measurement using alinear array probe may be applicable to diagnosis of diffuse
liver diseases.
Several SoS estimation methods assume that the scattering source is a point scatterer. We demonstrated that the SoS
is overestimated if the target size is non-negligible [14, 15].
Based on the results of this study, the running directionsof vessels in addition to the target size must be carefully
considered for applications utilizing absolute SoS values.

Conclusion
In this study, the effect of probe rotation angle 𝜃 relative
to the short-axis plane of the blood vessel on SoS estimation was examined through water tank and phantom experiments. When the probe was rotated by 𝜃 relative to the shortaxis plane of the vessel, the SoS was misestimated, and adecrease in power at the edge elements relative to the center
element was observed. Therefore, we proposed a method todistinguish the presence or the absence of SoS misestimation caused by the probe rotation without information on 𝜃
by thresholding the decrease in the power of the received
signals at the edge elements relative to the center element.
The proposed method can be used for in vivo SoS estimation by combining the methods described in Refs. [4] and[15]. In future, this method will be used to estimate the SoS
in the liver.
Acknowledgements This study was partially supported by the Japan
Society for the Promotion of Science, KAKENHI (Grant 21K14166).
Data availability The data are available from the corresponding author
upon reasonable request.

Declarations
Conflict of interest Shohei Mori, Keiji Onoda, Mototaka Arakawa, andHiroshi Kanai declare no conflicts of interest.
Ethical approval This article does not contain any studies involving
human or animal subjects performed by any of the authors.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as longas you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article areincluded in the article’s Creative Commons licence, unless indicatedotherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is notJournal of Medical Ultrasonics
permitted by statutory regulation or exceeds the permitted use, you willneed to obtain permission directly from the copyright holder. To view acopy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References
1. Anderson ME, Trahey GE. The direct estimation of soundspeed using pulse-echo ultrasound. J Acoust Soc Am.
1998;104:3099–106.
2. Yoon C, Seo H, Lee Y, et al. Optimal sound speed estimation
using modified nonlinear anisotropic diffusion to improve spatial
resolution in ultrasound imaging. IEEE Trans Ultrason Ferroelectr
Freq Control. 2012;59:905–14.
3. Jakovljevic M, Hsieh S, Ali R, et al. Local speed of sound estimation in tissue using pulse-echo ultrasound: model-based approach.
J Acoust Soc Am. 2018;144:254–66.
4. Abe K, Arakawa M, Kanai H. Estimation method for sound velocity distribution for high-resolution ultrasonic tomographic imaging. J Med Ultrason. 2019;46:27–33.
5. Hasegawa H, Nagaoka R. Initial phantom study on estimation of
speed of sound in medium using coherence among received echo
signals. J Med Ultrason. 2019;46:297–307.
6. Nagaoka R, Yoshizawa S, Umemura S, et al. Effects from correction of speed of sound in transmit and receive beamforming using
focus beam. Jpn J Appl Phys. 2021;60:SDDE19. https://​doi.​org/​
10.​35848/​1347-​4065/​abf55b.
7. Nitta N, Washio T. Relation between statistical properties of soundspeed distribution and average sound speed estimation. Jpn J Appl
Phys. 2021;60:SDDE18. https://​doi.​org/​10.​35848/​1347-​4065/​
abf4a6.
8. Nitta N, Washio T, Numano T. Basic investigation on identification of tissue composition based on propagation speeds of longitudinal and shear waves. Jpn J Appl Phys. 2022;61:SG1023. https://​doi.​org/​10.​35848/​1347-​4065/​ac45d5.
9. Nitta N, Washio T. Ex vivo evaluation of the sensitivity of modified average speed of sound estimation method. Jpn J Appl Phys.
2023;62:SJ1055. https://​doi.​org/​10.​35848/​1347-​4065/​accead.
10. Hachiya H, Ohtsuki S, Tanaka M. Relationship between speed ofsound in and density of normal and diseased rat livers. Jpn J Appl
Phys. 1994;33:3130–3.
11. Imbault M, Faccinetto A, Osmanski BF, et al. Robust sound speed
estimation for ultrasound-based hepatic steatosis assessment. Phys
Med Biol. 2017;62:3582–98.

12. Ali R, Telichko AV, Wang H, et al. Local sound speed estimation
for pulse-echo ultrasound in layered media. IEEE Trans Ultrason
Ferroelectr Freq Control. 2022;69:500–11.
13. Telichko AV, Ali R, Brevett T, et al. Noninvasive estimation of
local speed of sound by pulse-echo ultrasound in a rat model ofnonalcoholic fatty liver. Phys Med Biol. 2022;67:015007. https://​doi.​org/​10.​1088/​1361-​6560/​ac4562.
14. Nakayama A, Mori S, Arakawa M, et al. Estimation error in soundvelocity depending on size of target scatterer. Jpn J Appl Phys.
2021;60:SDDE17. https://​doi.​org/​10.​35848/​1347-​4065/​abf39e.
15. Mori S, Kanai H, Arakawa M. Speed-of-sound estimation in ultrasound propagation medium by considering size of target scatterer.
J Med Ultrason. 2023;50:151–65.
16. Zein NN, Hanouneh IA, Bishop PD, et al. Three-dimensional printof a liver for preoperative planning in living donor liver transplantation. Liver Transpl. 2013;19:1304–10.
17. Ishii T, Fukumitsu K, Ogawa E, et al. Living donor liver transplantation in situs inversus totalis with a patient-specific three-dimensional printed liver model. Pediatr Transplant. 2020;24:e13675.
https://​doi.​org/​10.​1111/​petr.​13675.
18. Kroebel W, Mahrt KH. Recent results of absolute sound velocity
measurements in pure water and sea water at atmospheric pressure. Acta Acustica United Acustica. 1976;35:154–64.
19. Hashimoto Y, Akashi N, Kushibiki J. Measurements of ultrasonicattenuation coefficients of water in VHF/UHF range. IEICE Tech
Rep. 1997;97:37–42 [in Japanese].
20. Eggleton RC, Whitcomb JA. Tissue simulators for diagnostic
ultrasound. Ultrasonic Tissue Characterization II, Linzer M ed,
NBS Spec Publ 525, US Govt Printing Office. 1979;327–36.
21. Marsh SP. Last shock hugoniot data. University of California
Press; 1980.
22. Boozari B, Potthoff A, Mederacke I, et al. Evaluation of soundspeed for detection of liver fibrosis: prospective comparison withtransient dynamic elastography and histology. J Ultrasound Med.
2010;29:1581–8.
23. Ino M, Yoshida K, Hirata S, et al. Assessment of the frequency
dependence of acoustic properties on material, composition, andscatterer size of the medium. J Med Ultrason. 2022;49:569–78.
24. Kudo M, Shiina T, Moriyasu F, et al. JSUM ultrasound elastography practice guidelines: liver. J Med Ultrason. 2013;40:325–57.
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