Journal of Medical Ultrasonics
https://doi.org/10.1007/s10396-023-01282-2

ORIGINAL ARTICLE–PHYSICS & ENGINEERING

Speed‑of‑sound estimation in ultrasound propagation mediumby considering size of target scatterer
Shohei Mori1· Hiroshi Kanai1,2 · Mototaka Arakawa1,2

Received: 3 September 2022 / Accepted: 5 December 2022
© The Author(s), under exclusive licence to The Japan Society of Ultrasonics in Medicine 2023

Abstract
Purpose Accurate speed-of-sound (SoS) estimation in an ultrasound propagation medium improves imaging quality andcontributes to better diagnosis of diseases. In conventional time-delay-based SoS estimation approaches studied by several
groups, a received wave is assumed to be scattered from an ideal point scatterer. In these approaches, the SoS is overestimated when the target scatterer has a non-negligible size. In this paper, we propose the SoS estimation method that considers
target size.
Methods In the proposed method, the error ratio of the estimated SoS using the conventional time-delay-based approachis determined from measurable parameters using the geometric relationship between the received elements and target.
Subsequently, the SoS erroneously estimated using conventional estimation, assuming the ideal point scatterer as a target,
is corrected by the determined estimation error ratio. To validate the proposed method, the SoS in water was estimated forseveral wire sizes.
Results The SoS in the water was overestimated using the conventional SoS estimation method, with a maximum positive
error of 38 m/s. The proposed method corrected the SoS estimates, and the errors were suppressed to within 6 m/s, irrespective of the wire diameter.
Conclusion The present results demonstrate that the proposed method can estimate the SoS by considering the target size
without using information on the true SoS, true target depth, and true target size, which is applicable to in vivo measurements.
Keywords Ultrasound · Speed of soundIntroduction
Medical ultrasound imaging is useful for observing biological tissues owing to its noninvasiveness and real-time performance. Speed of sound (SoS) in an ultrasound propagation medium is indispensable for constructing an ultrasoundB-mode image, and using an incorrect SoS deteriorates
image quality. Moreover, SoS can be used as an indicatorfor quantifying diseases such as liver steatosis [1]; therefore, accurate SoS estimation is indispensable in quantitative
medical ultrasound.

* Shohei Mori
mori@tohoku.ac.jp
1

Graduate School of Engineering, Tohoku University, Sendai,
Miyagi 980‑8579, Japan

2

Graduate School of Biomedical Engineering, Tohoku
University, Sendai, Miyagi 980‑8579, Japan

Several studies have been conducted to estimate SoS in
propagation media [1–20]. These methods can be categorized into several groups, such as time-delay-based [4–18]
and compounding approaches [19, 20]. In time-delay-based
approaches, the SoS is estimated from the time-delay distribution of the radiofrequency (RF) waves received by elements in the ultrasound probe. This time-delay distribution
is determined by the SoS in the ultrasound propagation
medium and the geometric relationship between the target
scattering source and known positions of the received elements. One of the time-delay-based approaches directly
estimates the SoS from this relationship by detecting thetime-delay distribution [4–8]. In other approaches, the timedelays among elements are compensated using the preassigned SoS, and the best SoS is determined by optimizing
evaluation factors, such as focusing quality [9–12], spatial
coherence [11, 13, 14], and coherence factor [15–18].
In these time-delay-based methods, a time-delay distribution is formulated by assuming that the target scatterer

13

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Journal of Medical Ultrasonics

is an ideal point scatterer. This assumption may hold when
the target scatterer is sufficiently thinner than the width ofthe point spread function (PSF) of the ultrasound. However,
the amplitude of the scattered wave from a small scatterer issmall and easily affected by the interference of the scatteredwaves from the surrounding scattering sources. Thus, a large
target that exhibits high scattered intensity, such as small
vessel walls, is preferable. However, these targets may not
be sufficiently smaller than the PSF width, and the assumption of an ideal point scatterer may not hold. In this case, thetime-delay distribution must be formulated by considering
the target size.
In a previous study [7], we confirmed that the SoS was
overestimated when the SoS estimation under the target
assumption of the ideal point scatterer was applied to scattered waves from a target with a finite size. Based on thegeometric relationship between the target and the elementsin the ultrasound probe, a correction method for this overestimated SoS was examined [8]; however, this method
assumes that the target size is known. In in vivo measurements, the true target size cannot be measured because theobserved size of the target in the ultrasound B-mode image
depends on the SoS used for constructing the cross-sectional
image. In other words, the true size of the target cannot beknown under the condition of an unknown SoS.
In this paper, a method is proposed to estimate the SoS
under conditions where the true values of the SoS, target
depth, and target size are unknown. To correct the erroneously estimated SoS obtained using the previous method[6], measurable parameters related to the target observed on
the B-mode image constructed by the erroneously estimated
SoS were used. The proposed method was validated througha water tank experiment using wires with several diameters.

Materials and methods
Assumption of SoS estimation target
In this study, the reception times of the ultrasound waves
scattered from a target and received by the elements in theprobe were used to estimate the average SoS in the ultrasound propagation medium from the probe surface to thetarget. As a fundamental study, SoS was assumed to behomogeneous in an ultrasound propagation medium.
To accurately detect the propagation time of the scatteredwave from the target, a large target that exhibits high scattered intensity, such as a small blood vessel, is preferable.
Thus, SoS estimation method that can be applied to RF signals received from a target of finite size is proposed.
The surface shape of the target inside the transmitted
beam width wbeam was approximated by a circular form with13

a curvature radius r as shown in Fig. 1(a) and its expanded
version in Fig. 1(b). The shape of the target does not need
to be a circle outside the transmitted beam width wbeam . The
shallowest position of the target surface was assumed to exist
exactly below the center (0th element at x0) of the active elements used for transmitting the beam, as shown in Fig. 1(a).
The depth from the probe surface to the shallowest positionof the target surface is defined by d , and the lateral positionof the 0th element is defined as x0 = 0. Although the focal
depth of the transmitted beam was assumed to be set at thetarget surface, that is, the focal depth was equal to d , asshown in Fig. 1(a), the focused beam was approximated by aplane wave within the beam width wbeam around d . The time
for transmitting the ultrasonic pulse from the 0th element at
x0 is defined as 0 s.
In the SoS estimation considering the curvature radius
r of the target surface, the measurable{ }parameters are thelateral positions of the elements, xk (in this paper, {⋅}describes the set of variables), SoS estimate ĉ
r=0 obtainedusing the conventional method, assuming an ideal point
scatterer (r = 0) as a target, and the observed depth d̂B andcurvature radius r̂B of the target surface, that are determinedfrom the B-mode image constructed using the erroneous SoS
̂estimate ĉ
r=0 . The SoS estimate c
r=0 is not the true value forr > 0, and d̂B and r̂B do not correspond to the true values dand r , respectively.

SoS estimation considering the target size
The proposed SoS estimation method considers the curvature radius r of the target surface, where r is unmeasurable
owing to the unknown SoS c . The main concept is to determine the estimation error ratio 𝜌̂r = ĉ
r=0 ∕c , that is, the ratioof the erroneously estimated SoS ĉ
r=0 by assuming r = 0 to
the true SoS c . Here, the bias error is caused by assumingr → 0 when the target has a curvature radius r > 0. Once theerror ratio 𝜌̂r is determined in the following procedures, theerroneous SoS estimate ĉ
r=0 can be corrected as

ĉr =

ĉ
r=0
.
𝜌̂r

(1)

The error ratio 𝜌̂r is determined only from the measurable
parameters using{theprocedures: (I) detection of( following)}
reception times t xk for the{ }wavefront of the scatteredwaves received at elements xk , (II) SoS ĉ
r=0 estimation
using the conventional method assuming r = 0, (III) tentative determination of the depth d̂B and curvature radius r̂Bof the target surface observed on the B-mode image constructed by ĉ
r=0 , and (IV) determination of the estimation

Journal of Medical Ultrasonics
Fig. 1  a Schematic of measurement configuration. b Definitionof target surface within focal
region. (c, d) Propagation paths
and reception time distributionsof the wavefront of received
signals (c) from an ideal point
scatterer and (d) from a target
surface having a finite size.
e Magnitude( relationship)
betweenTb xk ;c,)d, r > 0 and(Tb xk ;c, d, r → 0 . f Enlarged
view of (d) for explaining
the()
definition of Lb lmin , xk ;d, rerror ratio 𝜌̂r from these measurable parameters to apply
Eq. (1). The details of steps (I)–(IV) are described below:(I) Time detection of the wavefront of the received signal
In the first step, the RF signal of the ultrasound wave
scattered from the target is acquired( )by each element at
position xk , and the reception time t xk of the wavefront is
detected, as shown in Fig. 2. Figure 2(a) shows an example
of a B-mode image of the short-axis view of a wire with adiameter 2r = 6 mm. The received element signals for creating the center received beam, shown by the dashed line in
Fig. 2(a), are shown in Fig. 2(b–1), and its expanded version in Fig. 2(b–2). As directly detecting the received timeof the wavefront is difficult owing to noise, the time of the
wavefront was determined from the pth peak time detectedby the peak-detection approach, as in our previous study [7].
In the present study, nylon and silicone wires were used as
the targets in the water (tank experiment.
For nylon wires, the)negative peak time tneg xk ;p( was) detected. On the contrary,
the positive peak time tpos xk ;p was detected for the silicone wires because the acoustic impedance of the silicone
was lower than that of water, and the ultrasound wave wasreflected in the opposite phase at the interface between thewater and silicone. The reception time t(xk ) of the wavefront
is determined as�� p−0.25⎧
� � ⎪ tneg xk ;p − f0 , for nylon wirest xk = ⎨ �� p−0.25⎪ tpos xk ;p − f , for silicone wires
0
⎩

(2)

13

[µs]

Journal of Medical Ultrasonicsµ( )Fig. 2  Schematic for detection of reception time t xk of the wavefront of the received signal from a target. a B-mode image of wirewith the diameter of 2r = 6 mm. (b-1) Received element signals
before beamforming at the ultrasonic beam passing through the centerof the wire (white dashed line on (a)). (b-2) Enlarged view of (b-1).
c Waveform of the element RF signal at xk = 0 in (b-2). The detected
time of the second positive peak, tpos (xk ;2), and the determined reception time of the wavefront, t(xk ), are plotted in red and blue, respectively, on (b-2) and (c)where f0 denotes the transmission frequency. The second
term represents the compensation of the p th peak time tothe time of the wavefront. In this study, the peak number
was empirically determined as p = 2 , which was a condition for stable detection{ ( )}of the peak time. The reception
time distribution t xk is determined from K elements(k = −(K − 1)∕2, ⋯ , 0, ⋯ , (K − 1)∕2), where K is assumed
to be an odd number.{An (example)} of the detected second
peak time distribution tpos xk ;2 and {the( determined
recep)}
tion time distribution of the wavefront, t xk , are shown in
red and blue, respectively, in Fig. 2(b-2). An example of a
waveform received by the center element ( xk = 0) is shownin Fig. 2(c), with the second peak time tpos (0;2) (red) and thedetermined reception time of the wavefront, t(0) (blue). The
distribution of time of the wavefront, {t(xk )}, is used for SoS
estimation using the following procedure.

In the second step, the
SoS estimation{ (conventional)}
method [6] is applied to t xk as follows. As shown inFig.
propagation time( 1(c), the forward-and-backward)
Tfb xk ;c, d, r = 0 of the ultrasound wave transmitted fromthe probe, scattered by the assumed ideal point scatterer atlateral position x = 0 and depth d , and received by the element with position x = xk , is theoretically modeled by
(
)
√
(
) 1
d + xk2 + d2 .
Tfb xk ;c, d, r = 0 =(3)c(II) Conventional SoS estimation, assuming an ideal
point scatterer as the target

Here, Tfb (0;c, d, r = 0) = 2d∕c corresponds to the measured
value t(0), which is the received time of the scattered wave
at the 0th element. From Eq. (4),13

(Subsequently,) the backward propagation time
Tb xk ;c, d, r = 0 is defined by()()
Tb xk ;c, d, r = 0 ≡ Tfb xk ;c, d, r = 0(4)
T (0;c, d, r = 0) 1 √ 2=
xk + d2 .
− fb
2cJournal of Medical Ultrasonics

(
)2
1
d2
Tb xk ;c, d, r = 0 = 2 xk2 + 2 .
c
c

(5)

In the( )actual measurement, the measured propagation
time t xk contains errors; therefore, the quadratic polynomial approximation
is applied to the measured values
{
(( )
)2 }
t xk − t(0)∕2
for K elements.
{(})( ) t(0) 2t xk −
2
(6)
(
)
{ 2}
K−1
K−1
, ⋯ , 0, ⋯ ,≈ a xk + b. k = −2
2
By minimizing{the mean squared }difference between the(( ))2and their theoreticalmeasured values t xk − t(0)∕2{ 2}model axk + b , the coefficients a and b of Eq. (6) arê
determined. Then, SoS ĉ
r=0 and the target depth dr=0 are
estimated as follows:
√
[ ]
√
√
Vk xk21 √√=√ĉ
] , [m∕s]
[ (
r=0 =
(7)
( ) t(0) )2̂a √2COVk xk , t xk − 2

√
d̂
r=0 =

√
[(
√
)]
̂
[ ]
( ) t(0) 2b √1√− Ek xk2 , [m]=Ek t xk −
2
̂
a
̂
a
(8)

where Ek [⋅], Vk [⋅], and COVk [⋅, ⋅] are the average, variance,and covariance for k , respectively. Detailed derivations ofEqs. (7) and (8) are presented in Appendix A (see the electronic supplementary material). When the target is an ideal
point scatterer (r = 0 ) and the measured propagation time{( ( ))2 }has no errors, the measured values t xk − t(0)∕2cor{ ()2 }̂
respond to Tb xk ;c, d, r = 0; therefore, ĉ
r=0 and dr=0
correspond to the true values c and d , respectively.
However, when the surface of the target has a curvatureradius r , as shown in Fig. 1(d),{ the( forward-and-backward)}
propagation time distribution Tfb xk ;c, d, r > 0 is affectedby r as follows. Based on our previous study [7], several
virtual points are set on the target surface (curvature radiusr ), as shown in Figs. 1(a, b). By approximating the focused
beam by a plane wave within the narrow ultrasonic beam andaround the focal depth, the propagation distance from theelement exactly above the m th virtual point √on the target
2 , assurface to the m th virtual point is d + r − r2 − lm
shown in Figs. 1(b, d), where lm is the lateral position of them th virtual point from the centerline ( x = 0 ) passing
through the 0th element. The propagation distance from them√ th virtual point to the k th element at xk is
√
)2
(
)2 (
2
, as shown in Figs. 1(b,xk − lm + d + r − r2 − lmd). By taking the sum of these two distances,the)forward(
and-backward propagation distance Lf b lm , xk ;d, r from theelement exactly above the m th virtual point to k th element
via m th virtual point is√)(2
Lfb lm , xk ;d, r = d + r − r2 − lm
√
)2
(
(9)
√
(
)2
2
2
+
xk − lm + d + r − r − lm .

Fig. 3  Examples of()()
Lfb lm , xk ;d, r − Lfb 0, xk ;d, robtained for d = 30 mm andr = 3 mm. The received element
positions are a xk = 0 and b
xk = 6 mm

13

Journal of Medical Ultrasonics

When the m th
( virtual )point is on the centerline (lm = 0),
the distance Lfb 0, xk ;d, r is given by√()
Lfb 0, xk ;d, r = d + xk2 + d2 ,(10)which equals the forward-and-backward propagation distance when the target is an ideal point scatterer (r → 0), as
shown in Fig. 1(c) and modeled(in Eq. (3).)()
Figure 3 shows examples of Lfb lm , xk ;d, r − Lfb 0, xk ;d, robtained for target conditions of d = 30 mm and r = 3 mm
for xk = 0 and xk = 6 mm. When the received element
is on)(
the centerline ( xk = 0),( as shown) in Fig. 3(a), Lfb lm , 0;d, r
equals
Lfb 0, xk ;d, r for lm = 0 and is longer than( its minimum)
Lfb 0, xk ;d, r for lm ≠ 0. However, when the received element
is not on the centerline ( xk ≠ 0), as shown in Fig.(3(b), there)
are propagation paths shorter than the distance Lfb 0, xk ;d, r .
Thus, the following relationship holds.
√
(
)
(
)
Lfb lmin , xk ;d, r ≤ Lfb 0, xk ;d, r = d + xk2 + d2 ,
(11)

{ (
)}
(
)
where Lfb lmin , xk ;d, r is the minimum of Lfb lm , xk ;d, r
for lm within the transmitted beam width wbeam , and is givenby()
Lfb lmin , xk ;d, r
( w
)
(
)
(12)
= min Lfb lm , xk ;d, r . ||lm || ≤ min r, beamlm
2
In our previous study [7], we confirmed that the received
time of the wavefront of the interfered waves from the virtual points on the target surface can be approximated by thereceived time of the wavefront of the “first arrival wave”
among the virtual points {m} on the target surface. Using
Eq.( (12), the) forward-and-backward propagation time
Tfb xk ;c, d, r of this “first arrival wave” is given by

)
(
) 1 (
Tfb xk ;c, d, r = Lfb lmin , xk ;d, r . (r ≥ 0)c

(13)

When the target is an ideal point scatterer (r = 0), Eq. (13)is equivalent to that in Eq. (3).
When the( target has) a size (r > 0), the backward propagation time Tb xk ;c, d, r defined by Eq. (4) is given by

(
)
(
) d
Tb xk ;c, d, r = Tfb xk ;c, d, r −
c
) d
1 (
= Lfb lmin , xk ;d, r −
(14)
c
c
)
1 (
= Lb lmin , xk ;d, r ,c()
where Lb lmin , xk ;d, r is the backward propagation distancedefined in Fig. 1(f), and is given by()()
Lb lmin , xk ;d, r ≡ Lfb lmin , xk ;d, r − d.
(15)

13

(
)
Tb xk ;c, d, r of Eq. (14) for(By substituting)T(b x)k ;c, d, r = 0 of Eq. (5) or its measured value,
t xk − t(0)∕2 of Eq. (7), the SoS estimate ĉ
r=0 of Eq. (7) byassuming that r = 0 is given by
√
[ ]
√
√
Vk xk2√ĉ[
r=0 = √
(16)(
)2 ] ,
COVk xk2 , Tb xk ;c, d, r
which is the theoretical erroneous estimate under the noisefree condition, whose bias error is caused by assumingr → 0 despite the target having a curvatureradius r > 0),(i.e., the error is caused by the(fact that Tb xk ≠ 0;c,
) d, r > 0
in Eq. (14) is shorter than Tb xk ≠ 0;c, d, r → 0 , as shownin Fig. 3(b). From Eq. (16), the theoretical estimation error
ratio 𝜌r in Eq. (1) can be expressed as follows:

𝜌r =

ĉ
r=0
c

√
√
[ ]
[ ]
√
√
√
Vk xk2
Vk xk2
1√
√
√
= √
[
[
(
)2 ] = √()2 ]c
COV x2 , T x ;c, d, r
COV x2 , c2 T x ;c, d, r
k

√
√
√
=√
√

k

b

k

[ ]
Vk xk2[()2 ] ,
COVk xk2 , Lb lmin , xk ;d, r

k

k

b

k

(17)

where the last equation is obtained from Eq. (14). Equation (17) shows that the estimation error ratio 𝜌r depends
only on the geometric variables of xk , d , and r , and does notdepend on the true SoS value c.
As shown by Eqs. (6) and (7), the SoS estimate ĉ
r=0 is deter√
mined by 1∕ a of the squared propagation time distribution{( ( ))2 }t xk − t(0)∕2. Because 2a is the curvature of{ ()2 }Tb xk ;c, d, raround xk = 0, the relationship between the{ ()2 }Tb xk ;c, d, r > 0
curvatures
of
and
{ (
)2 }
Tb xk ;c, d, r → 0, which correspond to{( ( ))2 }t xk − t(0)∕2when r > 0 and r → 0, respectively, wasevaluated to clarify the relationshipbetween
r=0 and c.
() ĉFirst, by substituting Lfb( lmin , xk ;d, )r in Eq. (11) into
Eq. (15) and the resultant Lb lmin , xk ;d, r into Eq. (14),() 1√ 2()
Tb xk ;c, d, r ≤xk + d = Tb xk ;c, d, r → 0 .
(18)cSecond,( as shown) in Fig. 1(d), the backward propagation
time Tb xk ;c, d, r is minimized at xk = 0 as

Journal of Medical Ultrasonics()
dmin Tb xk ;c, d, r = Tb (0;c, d, r) = .
xk
c

(19)

That is,()
dTb (0; c, d, r) = ≤ Tb xk ; c, d, r
c
√
(
)
1
2
≤
xk + d = Tb xk ; c, d, r → 0 .
c(20)This relationship is illustrated in Fig. 1(e). Thus, the factor
propagation timeof curvature,{a , of the squared backward()2 }is smaller than that of
distribution Tb xk ;c, d, r > 0{ ()2 }Tb xk ;c, d, r → 0 , and SoS ĉ
r=0 is always overestimatederror larger than 10 m/s is caused for target radius r ≥ 1
mm even if the transmitted beam width wbeam is narrowed
to 0.2 mm (wavelength for transmitted frequency: 7.5 MHz;SoS: 1,500 m/s). In practice, the transmitted beam width wbeam
generally becomes larger than the wavelength; therefore, SoS
correction by the following method is essential for estimating
the true SoS irrespective of the target size.
(III) Measurable parameters related to the target surface
observed in the B-mode image

which was experimentally confirmed in our previous study[7].
As( shown in)Eq. (12), the minimum propagation distance
Lfb lmin , xk ;d, r changes with the transmitted beam width
wbeam ; therefore, the estimation error ratio 𝜌r also changes
with( wbeam. By )narrowing the transmitted beam width wbeam,
Lfb lmin , xk ;d,(r approaches the) condition of an ideal point
scatterer, Lfb lmin , xk ;d, r = 0 , and the estimation error ratio𝜌r decreases. However, by theoretically evaluating the relationship between the transmitted beam width wbeam and SoS
estimation error ratio 𝜌r based on Eqs. (12–17), a positive

To correct the erroneously estimated SoS ĉ
r=0 from themeasurable parameters d̂B and r̂B on the B-mode image, therelationships between the measurable parameters (d̂B , r̂B) andthe unmeasurable true values (d, r ) were clarified as follows:
Figures 4(a, b) illustrates examples of B-mode images
constructed using the true SoS c and erroneous estimate ĉ
r=0(c�
r=0 > c), respectively. The B-mode image in Fig. 4(a) cannot be constructed owing to the unknown true SoS c, whereas
the B-mode image in Fig. 4(b) can be constructed using theerroneous(estimate) ĉ
r=0.
Let zB xbeam ;c be the depth position of the target surface in the B-mode image( constructedusing the true SoS c ,)as shown in Fig. 4(a). zB xbeam ;c corresponds to the depth
along the ultrasound-focused beam at xbeam and is describedby()√
c ⋅ t xbeam
(
)
2
2
(22)
zB xbeam ;c = d + r − r − xbeam =,2

Fig. 4  Schematic of measurable and unmeasurable parameters on the
B-mode image. a B-mode image constructed by true SoS c, whichcannot be obtained by in vivo measurement. b B-mode image con-structed by erroneously estimated SoS ĉ
r=0 , which can be obtained
without the knowledge( of )the true SoS c. r̂B shows the curvatureradius around point P 0, d̂B

for r > 0 as

𝜌r =

c�
r=0
> 1, (r > 0)
c

(21)

13

Journal of Medical Ultrasonics()
where t xbeam is the received time of the scattered wave
from the target surface with lateral position xbeam . DepthzB (0;c) at xbeam = 0 corresponds to the shallowest depth dof the target surface as follows:(23)

zB (0;c) = d.

However, when the B-mode image is constructed using
the erroneously estimated SoS ĉ
r=0 , the depth positionẑB (xbeam ;̂
cr=0 ) of the target surface observed in Fig. 4(b) canbe described by
(
)
(
) ĉ
r=0 ⋅ t xbeamẑB xbeam ;̂
cr=0 =
2 ()(24)
c ⋅ t xbeam
(
)
ĉ
r=0
⋅
= 𝜌r ⋅ zB xbeam ;c ,
=
c
2

r̂B =

⎡
⎢1+
⎢
⎣

�

�2 32
⎤
⎥
d xbeam⎥⎦����� d2 ẑB (xbeam ; ĉ
)
�
r=0
�
�
�
�
�
�
d x2
�
beam
�xbeam = 0 ���r
1= 𝜌r = 𝜌 ,
r
r
�
d ẑ
B (xbeam ; ĉ
r=0 )

�
�
�xbeam = 0

wherethe) second equation is obtained by substituting(zB xbeam ;c in( Eq. (22))into Eq. (24), and substituting theresultant ẑB xbeam ;̂
cr=0 into the first equation in Eq. (28).
The derivation of the first equation in Eq. (28) is describedin Appendix B (see the electronic supplementary(material).
)
Thus, the curvature radius r̂B around point P 0, d̂B in

where the last equation is obtained using Eqs. (21) and(22), respectively.
Thus,() the measurable depth of the target
surface, (ẑB xbeam);̂
cr=0 , is given as a multiple of the true
depth zB xbeam ;c by a factor of 𝜌r ≥ 1, as shown in Fig. 4(b).
From Eq. (24), the shallowest depth of the target surface,d̂B , observed in the B-mode image constructed using ĉ
r=0
is determined by()
d̂B = ẑB 0;̂
cr=0
(25)
= 𝜌r ⋅ zB (0;c) = 𝜌r ⋅ d,where the last equation is obtained from Eq. (23).
Next, the shape of the target surface observed in the
B-mode image was considered as follows. The shape of thetarget surface, shown in Fig. 4(a), is described by Eq. (22) as

[ (
)
]2
2
xbeam
+ zB xbeam ;c − (d + r) = r2 ,(26)

which is a circular equation with radius r . In contrast, theshape of the( target )surface in Fig. 4(b) is described by substituting zB xbeam ;c in Eq. (22) into Eq. (24),

(

xbeam
r

)2
+

( ẑ x ; ĉ −𝜌)2
B ( beam
r=0 )r (d+r)
𝜌r ⋅r

= 1.

(27)

This shows that the target of a circle with radius r in Fig. 4(a)and Eq. (26) is observed as an ellipse with minor radius r
and major radius 𝜌r ⋅ r ≥ r in Fig. 4(b).
In the proposed SoS estimation method, only the curvatureradius r around the shallowest depth d of the target surface
within the transmitted beam width wbeam was considered, as
shown in Fig. 1(a). Therefore, the relationship between)the(̂measurable curvature radius r̂B around point P 0, dB in
Fig. 4(b) and the true curvature radius r in Fig. 4(a) is formulated as follows:
From (the curvatureformula, the curvature radius r̂B)around P 0, d̂B is given by
Fig. 5  Procedure for the proposed SoS correction method

13

(28)

Journal of Medical Ultrasonics

Fig. 4(b) becomes smaller than the true radius r with a factorof (1∕𝜌r ) ≤ 1. This relationship can also be derived from theproperty of the ellipse, that is, the curvature radius( of)theellipse around the position on the major axis, P 0, d̂B in
Fig. 4(b), is given by the ratio of the squared minor radiusto the major radius, r̂B = r2 ∕(𝜌r ⋅ r) = r∕𝜌r.
Thus, we can measure d̂B = 𝜌r ⋅ d and r̂B = r∕𝜌r from the
B-mode image constructed by the erroneously estimated SoSĉ
r=0 ; however, we still cannot determine the true values ofd , r , and 𝜌r . Based on these relationships, the erroneouŝ
estimate ĉ
r=0 is corrected using dB and r̂B as follows:(IV) Determination of the estimation error ratio 𝜌r frommeasurable parameters ( d̂B , r̂B)
The actual target depth d and curvature radius r are estimated from the measurable parameters d̂B and r̂B using therelationship between the measurable parameters ( d̂B , r̂B)
and the true values ( d , r ) determined by Eqs. (25) and (28).
Figure 5 shows the proposed procedure for iterations forcorrection.
First, by substituting d̂B and r̂B for d and r in Eq. (15),the
distance distr ibution{ (backward propagation)}
̂
Lb lmin , xk ;dB , r̂B is obtained. The error ratio 𝜌r̂B is then()
o b t a i n e d by s u b s t i t u t i n g Lb lmin , xk ; d̂B , r̂B fo r()
Lb lmin , xk ;d, r in Eq. (17),√Vk [xk2 ][𝜌r̂B =()2 ] ,(29)
COVk xk2 , Lb lmin , xk ; d̂B , r̂
B

where the bias error is caused by assuming r̂B → 0 although
the target has a curvature radius r�B > 0.
The relationship between 𝜌r̂B and the theoretical estimation error ratio 𝜌r obtained using Eq. (17) under true target
conditions ( d , r ) was evaluated as follows. By dividing 𝜌r̂B
in Eq. (29) using 𝜌r in Eq. (17), and substituting d̂B in Eq.
(25) and r̂B in Eq. (28) into Eq. (29),√√ COV [x2 , L l , x ; d, r 2 ]
)
√
𝜌r̂B
k k
b ( min
k
[
=√
(
)2 ]
𝜌r

√
√
√
=√

COVk xk2 , Lb lmin , xk ; d̂B , r̂
B

(30)

[
]
2
COVk xk2 , Lb (lmin , xk ; d, r)[()2 ] .
COVk xk2 , Lb lmin , xk ; 𝜌r ⋅d, 𝜌r{ }Here, the set values ( xk , d, r ) were substituted into Eqs.
{ ()}(9), (12), and (15) to obtain Lb lmin , xk ;d, r , and the{ ()}
resultant Lb lmin , xk ;d, r were substituted into Eq. (17) to{ ()}
obtain 𝜌r . Subsequently, Lb lmin , xk ; 𝜌r ⋅ d, r∕𝜌r were{ }obtained by substituting ( xk , 𝜌r ⋅ d, r∕𝜌r ) into Eqs. (9),
[

(
)2 ]
To evaluate 𝜌r̂ ∕𝜌r of Eq. (30), COVk xk2 , Lb lmin , xk ;d, r
B

[
(
)2 ]
and COVk xk2 , Lb lmin , xk ;𝜌r ⋅ d, r∕𝜌rwere simulated inadvancefor each set of the assumed parameter values{ }( xk , d, r ), where d is changed from 10 to 60 mm at 0.1-mm
intervals,{ } r is changed from 0 to 5 mm at 0.02-mm intervals,
and xk is set from − 9.6 to 9.6 mm at 0.2-mm intervals.

)2 ]

[
(
)2 ]
is always equal to or larger
COVk xk2 , Lb lmin , xk ;𝜌r ⋅ d, r∕𝜌r

[
(
)2 ]
than COVk xk2 , Lb lmin , xk ;d, r , that is, 𝜌r is always equal

to or larger than 𝜌r̂B. Because the error ratio is equal to 1 for
r = 0 and greater than 1 for r > 0, as confirmed in Eq. (21),the following relationship holds:

1 ≤ 𝜌r̂B ≤ 𝜌r .

(31)

d(1) = d̂B ∕𝜌r̂B and curvaThus, from Eq. (31), the depth ̂(1)ture radius ̂r = 𝜌r̂B ⋅ r̂B both corrected by 𝜌r̂B approach thetrue values d and r , respectively, as follows:d̂d̂d̂B ≥ ̂d(1) = 𝜌 B ≥ 𝜌B = d,(32)r̂B ≤ ̂r(1) = 𝜌r̂B ⋅ r̂B ≤ 𝜌r ⋅ r̂B = r.

(33)

r̂
B

r

Using a similar procedure, 𝜌̂r(n) (𝜌̂r(0) ≡ 𝜌r̂B) is iteratively
obtainedfor the backward
propagation distance distribution{ ()}
Lb lmin , xk ;d̂ (n) , r̂ (n) , using Eq. (17) as√
Vk [xk2 ][𝜌̂r(n) =()2 ] .
(34)
COVk xk2 , Lb lmin , xk ; ̂d(n) , ̂r(n)Similarly with Eq. (31),𝜌̂r(n) ≤ 𝜌̂r(n+1) ≤ 𝜌r . (n ≥ 0)(35)

d(n+1) = d̂B ∕𝜌̂r(n)Then, from the relationship in Eq. (35), ̂(n+1)
and ̂r= 𝜌̂r(n) ⋅ r̂B both corrected by 𝜌̂r(n) approach theirtrue values as follows:

r

[

(

(12), and (15). Finally, COVk xk2 , Lb lmin , xk ;d, r
and
[
(
)2 ]
2
COVk xk , Lb lmin , xk ;𝜌r ⋅ d, r∕𝜌r were obtained by calcu{ }{ ()}
lating the covariance of xk and Lb lmin , xk ;d, r , or the
{ }
{ (
)}
covariance of xk and Lb lmin , xk ;𝜌r ⋅ d, r∕𝜌r , respectively. The results obtained for all the parameter values{({ })}xk , d, r are shown in Fig. 6(a). As shown in Fig. 6(a),̂d(n) ≥ ̂d(n+1) =d̂B≥ d,𝜌̂r(n)

̂r(n) ≤ ̂r(n+1) = 𝜌̂r(n) ⋅ r̂B ≤ r.

(36)
(37)

Therefore, by iterative correction, the measured target
depth d̂B and curvature radius r̂B converge to the true values
d and r , respectively.

13

Journal of Medical Ultrasonics
Fig. 6  Theoretical validation forthe proposed SoS correction
method. a Theoretical
relationship between[()2 ]COVk xk2 , Lb lmin , xk ;d, r and[()2 ]COVk xk2 , Lb lmin , xk ;𝜌r ⋅ d, r∕𝜌robtained for each set of theparameters (d, r ), which shows
the relationship of true
estimation error ratio 𝜌robtained by Eq. (17) and 𝜌r̂B
obtained by Eq. (29). b True
values of estimation error ratio𝜌r obtained by Eq. (17). c
Estimation error ratio 𝜌̂r(n=4)
determined by four iterations
using Eq. (34). d Relationshipbetween the true estimation
error ratio 𝜌r and estimated 𝜌̂r(n).
e Normalized maximum) ||(difference | 𝜌̂r(n) − 𝜌r ∕𝜌r |||Thus, based on the relationship between the measurable
parameters ( d̂B , r̂B) and true values ( d , r ) determined using
Eqs. (25) and (28), the error ratio 𝜌̂r(0) ≡ 𝜌r̂B is iteratively
corrected to the true value 𝜌r based on Eq. (35). Finally,
the erroneously estimated SoS ĉ
r=0 is corrected as follows:

ĉr =

ĉ
r=0
,
𝜌̂r(N)(38)

where N denotes the total number of iterations.
Figure 6(b) shows the theoretical estimation error ratio𝜌r for each set of parameters ( d, r ) obtained using Eq. (17).
Figure 6(c) shows 𝜌̂r(4) obtained using Eq. (34) with four
iterations. 𝜌̂r(4) in Fig. 6(c) corresponds well to the true
value 𝜌r in Fig. 6(b). Figure 6(d) shows the relationship13

between 𝜌r and 𝜌̂r(n) for each parameter set ( d , r ), andFig.
6(e) )shows the normalized maximum difference|(|| 𝜌̂r(n) − 𝜌r ∕𝜌r |. Figures 6(d, e) show that the estimated 𝜌̂r(n)||correction.
converged to the true value 𝜌r by iterative) | The|(normalized maximum differences | 𝜌̂r(n) − 𝜌r ∕𝜌r | were||
1.4% ( n = 0 ), 0.22% ( n = 1 ), 0.03% ( n = 2 ), 0.005%( n = 3), and < 0.001% ( n = 4 ). These results show that the
number of iterations, N = 4 , is sufficient for the SoS cord(n) , andrection using Eq. (38). The parameters 𝜌r̂B , 𝜌̂r(n) , ̂(n)̂r required for SoS correction are obtained by directly
calculating Eqs. (29), (34), (36), and (37), respectively,
using ( d̂B , r̂B) measured in procedure (III). Therefore, theconstruction of the B-mode image is not required in procedure (IV).

Journal of Medical Ultrasonics
Fig. 7  a Experiment configuration for estimating the SoS in
water using wires with several
diameters as the targets. b
B-mode images constructedwith erroneously estimated
SoS ĉ
r=0 of (b-i) 1493, (b-ii)
1496, (b-iii) 1496, (b-iv) 1497,(b-v) 1505, (b-vi) 1511, (b-vii)
1519, (b-viii) 1524, and (b-ix)
1530 m/s for wires with diameters of (b-i) 0.25, (b-ii) 0.5,(b-iii) 0.75, (b-iv) 0.9, (b-v) 2,(b-vi) 3, (b-vii) 4, (b-viii) 5, and(b-ix) 6 mm, respectively. The
measured target curvature radiusr̂B was plotted at the measured
depth d̂B on (b-v)–(b-ix) by theorange line. c Estimated resultsof SoS in water by: blue—conventional method using Eq. (7),ĉ
r=0 ; red—proposed method
using Eq. (38), ĉr ; green—ĉ′rwhich was corrected using thevalues of true wire diameter 2r
and true SoS c in water insteadof the measurement of r̂B. The
true SoS c was determined fromthe water temperature [21]Experimental methods
The proposed method was applied to RF signals obtained
from wires of several diameters arranged in a water tank.
The SoS in water was estimated using RF signals obtained
from the wire. The experimental configuration is shownin Fig. 7(a). The nylon wires with diameters of 0.25, 0.5,
0.75, and 0.9 mm and silicone rubber wires with diametersof 2, 3, 4, 5, and 6 mm were utilized as targets. Each wire
was placed in a water tank such that the shallowest depthof the wire surface, d , was set at approximately 30 mm
Table 1  Water temperatureand true SoS determined fromwater temperature [21] in theexperiment for each wire

Diameter of wire 2r [mm] 0.25
Water temperature [◦ C]
True SoS in water[m/s]from the probe surface. The wire depth was set by moving
the ultrasound probe using the manual stage with 0.1-mm
resolution after contacting the probe surface and wire surface by observing the water tank and the B-mode image.
The cross-sectional view of the short axis of the wire
was obtained using a linear array probe (UST-5412; Hitachi
Aloka, Japan) connected to an ultrasonic diagnostic apparatus (Prosound α10 ; Hitachi Aloka, Japan). The transmitted and sampling frequencies were set at 7.5 and 40 MHz,
respectively. The focal depth was set at 30 mm. The full
widths at half maximum (FWHM) of the transmitted and
0.50

0.75

0.90

2

3

4

5

6

23.0
23.2
23.3
23.5
23.6
23.6
23.6
23.7
23.8
1491.2 1491.8 1492.1 1492.6 1492.9 1492.9 1492.9 1493.2 1493.5

13

Journal of Medical Ultrasonics

received beams were 0.85 and 0.39 mm, respectively. The
RF signals were acquired ten times for each wire by rearranging the wire. The true SoS in water was determinedfrom the water temperature [21]. The water temperature wasmeasured immediately before the measurement of each wire.
The measured water temperature and true SoS determinedfrom the measured water temperature are listed in Table 1.
The depth of the target surface observed on the B-mode
image, d̂B , was estimated by measuring the received
time()
of the wavefront of the beam-formedRFsignal,
, at
t
x
beam
(
)
substituting
into
Eq.
(24),
and
t
x
=
0
xbeam = 0, and bybeam()
the resultant ẑB xbeam = 0;̂
cr=0 into Eq. (25).
The target radius observed in the B-mode image, r̂B, wasmeasured using a simple thresholding approach. After detecting the lateral positions of the target surface at several depths
by applying the thresholding procedure to the beam-formed RF
envelope amplitudes in the lateral direction, r̂B was estimatedusing the least squares method. The threshold value was empirically determined. For wires with small diameters 2r ≤ 0.9 mm,
we did not measure r̂B, because distinguishing whether theobserved shape on the B-mode image showed the target shapeor the shape of the PSF of ultrasound was difficult, which is discussed in the results section. Therefore, the SoS correction using
the proposed method was applied to wires with large diameters
2r ≥ 2 mm.
For wires with small diameters 2r ≤ 0.9 mm, the theoreticr=0 ∕c) was used for r̂B to discuss the feasibility
cal value r∕(̂of the proposed method if the target radius r̂B was measurable from the B-mode image.

Results
Shape of target surface observed on the B‑mode
image
Figures 7(b-i)–7(b-ix) show the B-mode images constructed using the erroneously estimated SoS ĉ
r=0 for wires
with diameters of 0.25, 0.5, 0.75, 0.9, 2, 3, 4, 5, and 6 mm,
respectively. The erroneously estimated SoS values ĉ
r=0 used
for the construction of the B-mode images were (b-i) 1493,(b-ii) 1496, (b-iii) 1496, (b-iv) 1497, (b-v) 1505, (b-vi) 1511
(b-vii) 1519, (b-viii) 1524, and (b-ix) 1530 m/s.
As shown in the B-mode images of wires with small
diameters 2r ≤ 0.9 mm in Figs. 7(b-i)–7(b-iv), distinguishing the target shape from the shape of the PSF (FWHM:
0.39 mm in the lateral direction) is difficult. Thus, forwires with small diameters 2r ≤ 0.9 mm, we did not measure the radius r̂B in the B-mode image.
For wires with large diameters 2r ≥ 2 mm, the measured
target depth d̂B and radius r̂B (orange line) well corresponded

13

to the shape of the target surface observed on the B-mode
image, as shown in Figs. 7(b-v)–7(b-ix). The differencesr̂B − r∕(̂
cr=0 ∕c) between the measured curvature radii r̂B on
cr=0 ∕c) were
the B-mode image and theoretical values r∕(̂
0.06 ± 0.06 mm (average ± standard deviation for ten measurements) for 2r = 2 mm, 0.01 ± 0.06 mm for 2r = 3 mm,− 0.06 ± 0.10 mm for 2r = 4 mm, −0.02 ± 0.11 mm for
2r = 5 mm, and 0.03 ± 0.06 mm for 2r = 6 mm.

SoS estimation
The results of the estimated SoS ĉ
r=0 yielded by Eq. (7)of the conventional method (blue) and the final corrected
SoS ĉr yielded by Eq. (38) of the proposed method (red)are shown in Fig. 7(c). The final corrected SoS values ĉr ,which were corrected without using the values of true target depth d and radius r , and true SoS c in water, are shownfor wires with diameters 2r ≥ 2 mm. For wires with diameters 2r ≤ 0.9 mm, the corrected SoS values ĉ′r , which werecorrected using the values of true wire diameter 2r and true
SoS c in water instead of the measurement of r̂B, are plotted in green. The error bars show the average and standard
deviation for ten measurements. The true SoS values c inwater, measured from the water temperatures, are plotted
as a black cross marker.
As shown in Fig. 7(c), SoS ĉ
r=0 was overestimated using
Eq. (7) of the conventional method. The positive error
increased as the wire diameter increased, which coincidedwith the results of our previous study [7].
For wires with diameters 2r ≤ 0.9 mm, the estimation|
errors yielded by the conventional method, ||ĉ
r=0 − c|, were| ̂�|
at a maximum of 6 m/s. The errors |cr − c| of the corrected||
SoS values ĉ′r (green) were within 5 m/s. Thus, theimprovement in the estimation error for the small target is
not significant even if the target diameter is measurable on
the B-mode image, while the bias error can be correctedby the SoS correction, as shown in Fig. 7(c).
For wires with diameters 2r ≥ 2 mm, the SoS estimation accuracy was significantly improved by the proposed
method, whereas the standard deviation became larger
than that of the conventional method owing to the measurement errors of the curvature radius r̂B on the B-mode
image. The absolute values of the estimation errors yielded|by the conventional method, ||ĉ
r=0 − c|, were a maximum of
38 m/s, and those yielded by the proposed method, ||ĉr − c||,were within 4 m/s. Thus, the proposed method considerably improved the SoS estimation accuracy, particularly
for large targets.

Journal of Medical Ultrasonics

Discussion
In this paper, SoS estimation method that considers thecurvature radius r of the target surface was proposed andvalidated by means of a water tank experiment using a wire
as a target. The method comprises four procedures: (I) to(IV). Procedure (II) requires only a quadratic polynomial
approximation process, that is, the calculation of Eq. (7).
Procedure (IV) requires iterative calculation of 𝜌̂r(N) using
Eq. (34). However, this calculation process can be omitted by preliminarily determining 𝜌̂r(N) for each depth d̂B andradius r̂B and preparing a look-up table. Thus, procedures (II)and (IV) are clinically applicable.
Procedure (I) requires detection of the reception time of
the wavefront of the received signal. In this study, a simple peak-detection method was used; however, the methodshould be modified for clinical applications to address thelow signal-to-noise ratio and interference of scattered waves
from the surrounding tissues in in vivo measurements.
Procedure (III) requires measuring the curvatureradius r̂B of the target in the B-mode image, as shownin Fig. 7(b). In the conducted experiment, measuring r̂Bfrom the B-mode image for small wires ( 2r ≤ 0.9 mm)
was difficult owing to the ultrasonic spatial resolution.
Therefore, the proposed method was applied only tolarge wires ( 2r ≥ 2 mm). As shown in Fig. 7(c), however, the SoS estimation error for the small targets( 2r ≤ 0.9 mm) was at a maximum of 6 m/s without correction of the target size using the proposed method.
These results show that the SoS correction using the proposed method should be applied to a large target whose
observed radius r̂B on the B-mode image is measurable,
whereas the conventional SoS estimation assuming an
ideal point scatterer is applicable to smaller targets. As
shown by the green plots in Fig. 7(c), the bias error of
the SoS estimates for small targets ( 2r ≤ 0.9 mm) can be
improved with the proposed method if the target radiusr̂B is measurable on the B-mode image. Improvement of
the ultrasonic spatial resolution and/or development of ameasurement method of r̂B for small targets will enable
the application of the proposed method to small targets
and improve the bias error of the SoS estimates.
As shown in Fig. 7(c), the standard deviation of the estimated SoS values increased more for the proposed methodthan for the conventional method, whereas the bias error
was significantly improved by the proposed method. In thisstudy, a simple threshold procedure was used to measure thetarget radius r̂B on the B-mode image. Improvement of themeasurement method for r̂B will contribute to the accurate
SoS estimation by the proposed method.

In the experiment using wires, the estimated SoS ĉryielded by the proposed method was slightly overestimatedfor the large wires, as shown by the red plots in Fig. 7(c). A
possible reason for this is the slight mismatch between theactual target depth and the focal depth. If there are systematic errors for the arranged target depth, the approximation
error of the focused beam as a plane wave within the transmitted beam width wbeam around the focal depth d may causea systematic error in the estimated SoS; therefore, detailed
examination is required in a future study.
In this study, the target was assumed to be located exactly
below the center element. In an actual case, the target may notexist exactly below the center element; however, the maximum
difference in lateral positions between the center element andtarget is half of the element pitch (0.1 mm in this study) byselecting the proper analysis beam. To validate the feasibilityof this assumption, we theoretically evaluated the SoS estimation error when the lateral position of the ideal point scatterer deviated 0.1 mm from the lateral position of the center
element. We theoretically calculated the propagation time
distribution and estimated the SoS using Eq. (7) for several
target depths d . We found that the SoS estimation errors were0.0001% for d = 30 mm, 0.003% for d = 5 mm, and 0.023%
for d = 1 mm. Thus, we confirmed that the approximation for
the target lateral position does not affect the SoS estimation.
As a fundamental study, the transmitted wave wasassumed to be focused on the target surface to approximate
the focused wave as a plane wave around the focal region.
However, focusing on all target surfaces is difficult in actual
measurements; therefore, examining the effect of the difference between the target surface depth and beam-focused
depth in the proposed method is necessary. Alternatively, thetransmission of plane waves enables the application of the
proposed method to the targets irrespective of their depths,
while it causes different concerns, such as a low signal-tonoise ratio and interference of scattered waves from the surrounding tissues.
As described in the Introduction, many SoS estimation
approaches assume that the scattering source is an ideal
point scatterer. In these methods, the SoS is overestimated
if the target used for the SoS estimation has a non-negligible
size. When the estimated SoS is directly used for diagnosis,
this overestimated SoS ĉ
r=0 results in a false diagnosis; therefore, SoS correction using the proposed method is essential.
When an overestimated SoS ĉ
r=0 is used only for the received
beamforming at the target position, it does not negatively affectthe focusing quality, because the propagation time at( each ele-)ment does not change between the true condition Tf b xk ;c, d, ran(d t h e e r r o n e o)u s l y e s t i m a t e d c o n d i t i o n
Tf b xk ;̂
cr=0 , d̂
r=0 , r = 0 , that is, the element signals can be in-13

Journal of Medical Ultrasonics()
cr=0 , d̂,r=0phase using the time delay Tfb xk ;̂
obtained
r=0
from the erroneously estimated condition. However, when overestimated SoS ĉ
r=0 is used for the received beamforming at different positions, the element( signals basically)cannot be in-phase
cr=0 , d̂using the time delay Tfb xk ;̂
r=0 , r = 0 , which is obtained
from the erroneously estimated condition, because the timedelay does not correspond to the true propagation time, which
degenerates the focusing quality. Thus, when the estimated SoS
is used for received beamforming at other positions, the effectof the target size must be corrected.
Thus, in the time-delay-based SoS estimation method, theeffect of target size must be carefully considered according
to the intended application of the estimated SoS value.

Conclusions
We herein proposed an estimation method for SoS in a propagation medium that is applicable to a signal received froma target of finite size. The proposed method can estimate the
SoS without information on the true depth and size of thetarget and the true SoS. In the experiment using wires withseveral diameters as targets, the proposed method significantly decreased the maximum SoS estimation error from
38 m/s to 6 m/s. The application of this method to in vivo
measurements will be the subject of our future work.
Supplementary Information The online version contains supplementary material available at https://d​ oi.o​ rg/1​ 0.1​ 007/s​ 10396-0​ 23-0​ 1282-2.
Acknowledgements This work was partly 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, Hiroshi Kanai, and Mototaka
Arakawa 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.

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