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Quantification of limitations in statistical analysis ofultrasound echo envelope amplitudes
To cite this article: Shohei Mori et al 2023 Jpn. J. Appl. Phys. 62 SJ1045- Imaging the elastic properties of tissue:
the 20 year perspective
K J Parker, M M Doyley and D J Rubens- Performance criteria for quantitative
ultrasonology and image parameterisationC R Hill, J C Bamber and D O Cosgrove- Power laws prevail in medical ultrasound
K J Parker

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REGULAR PAPER

Japanese Journal of Applied Physics 62, SJ1045 (2023)
https://doi.org/10.35848/1347-4065/acc33e

Quantification of limitations in statistical analysis of ultrasound echo envelope
amplitudes
Shohei Mori1*, Mototaka Arakawa1,2, Hiroshi Kanai1,2, and Hiroyuki Hachiya3
1

Graduate School of Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan
Graduate School of Biomedical Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan3
School of Engineering, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan2*

E-mail: mori@tohoku.ac.jp

Received November 16, 2022; revised February 26, 2023; accepted March 10, 2023; published online April 12, 2023
Ultrasound echo envelope statistics have been widely studied for quantitative tissue characterization. In ultrasound measurements, the size of theregion-of-interest (ROI) is limited by several factors, such as the locality of the tissue characteristics and the depth dependence of the acoustic fieldof the ultrasound beam. In this case, the evaluated echo envelope statistics vary even when the envelope amplitudes follow the same population
without any noise. In this study, the statistical variance of the moments caused by this finite number of samples was quantified based on the central
limit theorem and the law of error propagation. The proposed principles were validated by random number simulation and used to quantify thestatistical variance of Nakagami parameter estimation. Finally, the effective number of independent samples in an ultrasonic measurement wasquantified based on the relationship between the ROI size and the ultrasound spatial resolution. © 2023 The Japan Society of Applied Physics

1. Introduction

Medical ultrasound is widely used for the diagnosis ofvarious diseases owing to its noninvasiveness and real-time
performance. Ultrasound RF echo signals obtained from
biological tissues reﬂect information on scatterer distribution
in the tissue. This scatterer distribution changes according totissue characteristics; therefore, an estimation of the scatterer
distribution characteristics from ultrasound RF echo signals
contributes to the diagnosis of various diseases.
Echo envelope statistics have been widely studied toevaluate the characteristics of scatterer distribution.1,2) In
the most common approaches for statistics-based tissue
characterization, the probability density function of theecho envelope amplitudes is ﬁtted by a model function, andthe scatterer distribution characteristics are quantiﬁed bymodel parameters. Various model functions have been used
to quantify scatterer distribution characteristics, such as
Rayleigh,3–5) Nakagami,6–8) K,9,10) homodyned-K,11–14)
multi-Rayleigh,15–18) and double-Nakagami19–22) models.
Based on these model functions, various tissue characterization methods have been proposed, such as quantitative tissue
characterization methods for the liver,15–24) skin,25,26) andlymph nodes,27,28) and evaluation methods for temperaturechanges in biological tissues.29–31)
In these model functions, the model parameters are related
to the scatterer distribution characteristics. Therefore, the tissue
characteristics can be quantiﬁed using the model parameters.
In general, the relationship between model parameters andscatterer distribution characteristics is theoretically quantiﬁed
under the condition that inﬁnite-length data can be obtained.
However, in actual ultrasonic measurements, the numberof samples, i.e. the size of the region of interest (ROI), islimited owing to several factors, such as the locality of thetissue characteristics and the depth dependence of theacoustic ﬁeld of the ultrasound beam. Under the conditionof a limited number of samples, the statistical properties ofthe obtained data vary, as indicated by the central limit
theorem. This variation is caused even under conditions inwhich there is no noise, the depth dependence of the acoustic
ﬁeld can be completely compensated, and the obtained echo
envelope signals follow the exact same population. This is alimitation in the statistical analysis of echo envelope amplitudes; therefore, it is essential to understand this limitation for
accurate tissue characterization.
In this study, we theoretically quantify this statisticallimitation based on the central limit theorem and the law oferror propagation and evaluate the relationship between theultrasonic analysis conditions and statistical limitations. In
this study, we focus on moments, which are basic statisticalproperties such as the average, variance, skewness, andkurtosis. Moments have also been used for parameter
estimation for model functions6,9–11,32–34); therefore, a quantiﬁcation of the statistical limitations in moment analysis isimportant.
In our previous study, we examined the statistical varianceof non-normalized moments.35) However, in our previous
studies,17,18,32–34) the signals were normalized by the root of
their power because the power depends on the ultrasonictransmitted/received conditions, which complicates the evaluation of the scatterer distribution characteristics. Thus, inthis study, we quantify the statistical variance for normalized
moments as well as for non-normalized moments.
Furthermore, we quantify the statistical covariance between
different orders of moments because they are obtained fromthe same samples and are correlated with each other. Then,we quantify the statistical variance of the Nakagami parameter estimated by the moment method to demonstrate theuse of the proposed principles in an analysis of the statisticallimitations of tissue characterization. Finally, we conducted
an ultrasonic simulation to quantify the effective number of
independent samples within the ROI.
2. Principles
2.1. Statistical moment

Statistical moment is a basic statistical property. When
independent random variables {xq} with inﬁnite length follow
the probability density function q (x ), the k th order of the

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S. Mori et al.

non-normalized moment around the origin, MT (k; q ), isdeﬁned as
MT (k ; q) = E [xqk ] =ò-¥ x kq (x) dx,(1 )where E[·] denotes the expectation. As shown by Eq. (1), theﬁrst order of moment MT (1; q ) is the expectation of variables{xq} following q (x ). The second, third, and fourth orders ofmoments are related to information on the power, skewness,
and kurtosis of the variables {xq} following q (x ), respectively.
2.2. Quantification of statistical variance andcovariance of non-normalized moments
Under ultrasonic measurement conditions, the number ofsamples in the ROI is ﬁnite. In this case, the statisticalproperties of the random variables, such as the statistical
moment, vary around the theoretical value even if the randomvariables follow the exact same population and are independent of each other.
Figure 1 shows examples of random number simulations inwhich random numbers following the Rayleigh distribution[Fig. 1(a)] were generated and the ﬁrst and third orders of the
non-normalized moments of the random numbers were
calculated. The results of the 100 trials are plotted in
Figs. 1(b) and 1(c). Figures 1(b) and 1(c) show the moments
calculated using 500 and 2000 independent random numbers,
respectively. As shown in Fig. 1, the moments obtained froma ﬁnite number of samples vary around the theoretical values(red), and this variation can be suppressed by increasing thenumber of independent samples. This statistical variancecaused by the ﬁnite number of samples is quantiﬁed asfollows:
Let {xq, l }l = 1,2,  , L be L independent random numbers that
follow the model function q (x ). In ultrasonic measurements,
these random numbers are related to echo envelope amplitudes. The ﬁnite number of independent samples, L , isdetermined by the relationship between the area of the ROI
and the area of the point spread function (PSF) (i.e.
ultrasound spatial resolution),5) as discussed in the Methods
and Results sections. The k th order of the non-normalized
moment ML (k; q ) for {xq, l }l = 1,2,  , L is deﬁned as follows:
ML (k ; q) =

(a)

¥

1 L k
å xq,l.
L l=1

(2 )

(b)

(c)

Fig. 1. (a) Probability density function of Rayleigh distribution. (b), (c)First and third orders of non-normalized moments obtained from (b) 500 and(c) 2000 independent random numbers generated from Rayleigh distribution.
Results of 100 trials are plotted in (b) and (c).

sq ⎞⎛ML (k ; q) Î N ⎜m qk , k ⎟ ,L⎝⎠2

Let us describe yqk, l as follows:
yqk, l = xqk, l.

(3 )As the sets of {xq, l } follow a certain model function q (x ),the sets of {yqk, l } also follow a certain model function qk ( y),where qk ( y) differs from q ( y). Subsequently, ML (k; q ) in
Eq. (2) can be expressed as
ML (k ; q) =1 Lå yq ,l ,L l=1 kwhere mqk and s 2qk are the expectation and variance of the
independent random variables {yqk = xqk } with an inﬁnite
length following qk ( y), respectively. Here, mqk and s 2qk are
given bym qk = E [ yqk ] = E [xqk ] = MT (k ; q) ,(4 )which shows that the k th order of the moment ML (k; q ) for{xq, l }l = 1,2,  , L is obtained by the average of {yqk, l }l = 1,2,  , L
following a certain model function qk ( y). As a result of thecentral limit theorem, the moment ML (k; q ) obtained from L
independent random numbers follows a normal distribution
N (mqk , s 2qk /L ).

(5 )

(6 )

s 2qk = E [( yqk - m qk )2] = E [(xqk - E [xqk ])2 ]= MT (2k ; q) - MT (k ; q)2 .

(7 )Thus, the statistical variance s 2M (L, k; q ) of the k th orderof the non-normalized moment ML (k; q ) is determined fromthe number of independent samples, L , and the theoretical k th

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S. Mori et al.

and (2k )th orders of non-normalized moments.

1

M (2k ; q) - MT (k ; q)2
s 2M (L, k ; q) = T.
L

COVM(L, k1, k 2; q) = 2 (s 2SUM (L, k1, k 2; q)(8 )Because each order of moments is obtained from the same
samples, different orders of moments are correlated with eachother. This correlation is also an important factor in evaluating the statistical properties of the echo envelope amplitudes. For example, the covariance matrix for several ordersof moments can be used to estimate the number of tissue
components in a ﬁbrotic liver.18)
The statistical covariance between the k1th and k 2 th ordersof non-normalized moments can be formulated as follows:First, let us deﬁne the summation of the k1th and k 2 th ordersof moments as- s 2M (L, k1; q) - s 2M (L, k 2; q))1= L (MT (k1 + k 2; q) - MT (k1; q) MT (k 2; q)) ,(14)

SUMT (k1, k 2; q) = MT (k1; q) + MT (k 2; q) ,(9 )SUML (k1, k 2; q) = ML (k1; q) + ML (k 2; q).

(10)which shows that the covariance of the k1th and k 2 th orders of
the non-normalized moments can be determined from thenumber of independent samples, L , and the theoretical k1th,
k 2 th, and (k1 + k 2 )th orders of the non-normalized moments
given by Eq. (1).
2.3. Quantification of statistical variance andcovariance of normalized moments
Let {xN, q, l }l = 1,2,  , L be L independent random numbers
normalized by the root of their power, i.e. the root of thesecond-order of the non-normalized moment, as follows:
xq, l
xq, l
.
(15)
xN, q, l ==1 L 2
ML (2; q)xå q, l
L l=1

Then, from the law of error propagation, the following
relationship holds:

The k th order of the normalized moment MN, L (k; q )obtained from {xN, q, l }l = 1,2,  , L is given by:

s 2SUM (L, k1, k 2; q) =
+
+2

(

(

xq, l
1 L k
1 L ⎛
⎞
MN, L (k ; q) = å xN,⎜⎟
åq, l =
L l=1
L l = 1 ⎝ ML (2; q) ⎠¶SUMT (k1, k 2; q ) 2 2
s M (L, k1; q)¶M (k ; q )T)

1¶SUMT (k1, k 2; q ) 2 2
s M (L , k 2 ; q )¶M (k ; q )T2)= s 2M (L, k1; q) + s 2M (L, k 2; q) + 2COVM(L, k1, k 2; q) ,(11)
2
(L, k1, k 2; q ) is the statistical variance of
where s SUM
SUML (k1, k 2; q ) and COVM (L, k1, k 2; q ) is the covariance of
the k1th and k 2 th orders of the non-normalized moments. We
2(L, k1, k 2; q ) in Eq. (11). By substituting
then formulate s SUM
Eqs. (4) into (10),=å1yqk , l + L
1

s 2MN (L, k ; q) =
+
T

å

å

¶MN,T (k; q ) 2 2
s M (L , k ; q )¶MT (k; q ))¶MN,T (k; q ) 2 2
s M (L, 2; q)¶MT (2; q ))(k; q ) ¶MN,T (k; q )COVM(L, k , 2; q)¶MT (2; q )
1

=L
1

+L

(12)

which is the average of {yqk , l + yqk , l }l = 1,2,  , L following a12
certain model function qk1, k 2 ( y). Similar to Eqs. (4)–(8), from2
the central limit theorem, s SUM (L, k1, k 2; q ) is given by

1

{

{

{

MT (2k; q ) - MT (k; q )2
MT (2; q ) k}

k 2·MT (k; q )2
M (4; q ) - M (2; q )2
´ T M (2; q) kT+ 2
4
T- L kMT (k ; q) ´}MT (k + 2; q ) - MT (k; q ) MT (2; q )MT (2; q ) k + 1

},

(17)

where

1

s 2SUM (L, k1, k 2; q) = L s 2qk , k

MN,T (k ; q) =1 2

MT (k ; q)
k

.

(18)

MT (2; q) 2

1
= L E [( yq k + yq k - E [ yq k + yq k ])2]
1
2
1
2
1
k1
k2
k1
k2 2
= L E [(xq + xq - E [xq + xq ]) ]= L {E [xq2k1] + E [xq2k 2] + 2E [xqk1+ k 2] - (E [xqk1] + E [xqk 2 ])2 }1

1= L {MT (2k1; q) + MT (2k 2; q)+ 2MT (k1 + k 2; q) - (MT (k1; q) + MT (k 2; q))2 }.

((+ 2 ¶MN,T(k; q)Lyqk , l
2
l=1
l=1
L
1
( yqk , l + yqk , l ) ,
L
1
2
l=1

(16)

.

Thus, the k th order of the normalized moment MN, L (k; q ) is
determined from the second and k th orders of the nonnormalized moments; therefore, from the law of error
propagation, the statistical variance s 2MN (L, k; q ) of
MN, L (k; q ) is given by

¶M

L

k

ML (2; q) 2

¶SUMT (k1, k 2; q ) ¶SUMT (k1, k 2; q )COVM(L, k1, k 2; q)¶MT (k1; q )¶MT (k 2; q )1
SUML (k1, k 2; q) = L

ML (k ; q)

=

k

(13)

Thus, by substituting Eqs. (8) and (13) into Eq. (11), thecovariance COVM (L, k1, k 2; q ) is formulated as

The last equation in Eq. (17) can be obtained bysubstituting Eqs. (8), (14), and (18) into the ﬁrst equation in
Eq. (17). Thus, the statistical variance of the k th order of thenormalized moment, s 2MN (L, k; q ), is determined by the
number of independent samples, L , and the theoreticalsecond, fourth, k th, (2k )th, and (k + 2)th orders of the nonnormalized moments given by Eq. (1).
Finally, similar to the derivation of the covariance of the
non-normalized moment in Eqs. (9)–(14), the covariance

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S. Mori et al.

COVMN (L, k1, k 2; q ) for the normalized moment MN, L (k; q )can be derived from the law of error propagation as follows:

MT (k ; qMN (x {ai , mi , Wi }i = 1,2,  , N ))=ò

COVMN (L, k1, k 2; q)
= L⎧
⎨
⎩
1

k1+ k 2MT (2; q ) 2

k k M (k ; q ) MT (k 2; q )´⎬⎭MT (4; q ) - MT (2; q )2 ⎫k1+ k 2 + 4
MT (2; q ) 2=åMT (k 2 + 2; q ) - MT (k 2; q ) MT (2; q ) ⎫- L ⎧ 2 T2 2⎨⎩´

MT (k1 + 2; q ) - MT (k1; q ) MT (2; q ) ⎫1

k M (k ; q )k1+ k 2 + 2
MT (2; q ) 2

MT (2; q )

i=1

⎬
⎭

k1+ k 2 + 2
2

⎬
⎭

.

(19)

A detailed derivation of Eq. (19) is described in the
appendix. Thus, the covariance COVMN (L, k1, k 2; q ) for
normalized moments is determined from the number of
independent samples, L , and the theoretical second, fourth,k1th, k 2 th, (k1 + 2)th, (k 2 + 2)th, and (k1 + k 2 )th orders of the
non-normalized moments given by Eq. (1).
3. Methods
3.1. Mixture model of Nakagami distributions

To validate the proposed principles, we evaluated the mixture
model of Nakagami distributions because this model can
express several model functions used for ultrasound tissue
characterization, such as Rayleigh,3–5) Nakagami,6–8)
multi-Rayleigh,15–18) and double-Nakagami19–22) models.
The N -component mixture model of the Nakagami distributions, qMN (x {ai , mi, Wi }i = 1,2,  , N ), is deﬁned as follows:
qMN (x {ai , mi , Wi }i = 1,2,  , N )NN= å {ai qN (x mi , Wi )}, ⎛⎜å ai = 1⎞⎟ ,i=1⎝i = 1⎠(20)⎜

G (m ) W m

MT (k ; qN (x m , W )) = ò

¥

0

mT =

W
m

(23)

⎭

(E [x q2N ])2
E [(x q2N - E [x q2N ])2 ]=(E [x q2N ])2
E [x q4N ] - (E [x q2N ])2

MT (2; qN )2
1=
MT (4; qN )MT (4; qN) - MT (2; qN )2-1
MT (2; qN )2
1,=
MN,T (4; qN) - 1

⎟

k
2

⎫

=

x kqN (x m , W ) dxkG ⎛m + ⎞2⎠⎝= G(m)

Wimi

To validate the proposed principles, we evaluated thestatistical variance and covariance of moments for random
numbers generated using MATLAB R2020b (MathWorks
Inc., Natick, Massachusetts, USA). We generated L independent random numbers {x qMN, l }l = 1, 2,  , L following the
mixturemodelofNakagami
distributions,qMN (x {ai , mi, Wi }i = 1,2,  , N ) given by Eq. (20). Thereafter,
the non-normalized moments ML (k; qMN ) and the normalized
moments MN, L (k; qMN ) for these L independent random
numbers were obtained using Eqs. (2) and (16), respectively.
This simulation was iterated 10 000 times, and the varianceand covariance for 10 000 sets of moments were comparedwith the theoretical statistical variance and covariancederived using the proposed principles.
3.3. Demonstration of use of proposed principles:
quantification of statistical variance of Nakagami
parameter estimated by moment method
The theoretical Nakagami parameter, m = mT in Eq. (21), isrelated to the expectation and variance of the square ofvariables {x qN} following the Nakagami distribution.6) This
relationship can be expressed using the moments as follows:

mx 2

⎞.
(21)
exp ⎛ ⎝ W ⎠Here, ai denotes the mixture rate of a single-Nakagami
distribution qN (x mi, Wi ). mi and Wi are the Nakagami
parameter related to the scatterer distribution condition andthe scale factor related to the power of the echo envelope
amplitudes, respectively. G(·) denotes gamma function.
qMN (x {ai , mi, Wi }i = 1,2,  ,N ) contains a single-Rayleigh(N = 1, m1 = 1), single-Nakagami (N = 1), a multiN -components
Rayleighwith(mi = 1foralli = 1, 2, , N ), and double-Nakagami (N = 2) models.
The theoretical k th order of the non-normalized moment
for a single-Nakagami distribution qN (x m, W ) is given by8)q N (x m , W ) =⎩k2( ) ⎬.

3.2. Validation of proposed principles by random
number simulationwhere
2 m mx 2 m - 1

kG ⎛m i + ⎞2⎠⎝
ai G(m )
i
⎨

N ⎧

⎬
⎭

´

k M (k ; q )¥= å ⎧ai ò x kqN (x mi , Wi ) dx ⎫
⎬
i=1 ⎨
⎩ 0
⎭

- L ⎧ 1 T2 1⎨⎩
1

x kqMN (x {ai , mi , Wi }i = 1,2,  , N ) dx
N

MT (k1 + k 2; q ) - MT (k1; q ) MT (k 2; q ) ⎫+ L⎧ 1 2 T 14
⎨
⎩
1

¥

0

where the last equation is obtained using Eq. (18). Based on
Eq. (24), the moment method estimates the Nakagami
parameter m as follows:
mˆ =

1
1
.
=
ML (4; qN )MN, L (4; qN) - 1
1
ML (2; qN )2(25)

Based on the law of error propagation, the statisticalvariance s 2mˆ (L; mT ) of the Nakagami parameter m̂ estimatedusing Eq. (25) is determined bys 2mˆ (L; m T) =

(22)

( ).

(

¶m T
¶MN,T (4; q N )s 2MN (L , 4; q N )From Eqs. (1), (20), and (22), the theoretical k th order of the
non-normalized moment for qMN (x {ai , mi, Wi }i = 1,2,  , N ) is
given by

(24)

= (M

N,T (4; q N ) - 1)

42

) s (L, 4; q )2
MNN= m T4 s 2MN (L, 4; qN).

(26)

By substituting Eq. (17) into Eq. (26), s 2mˆ (L; mT ) is obtained
as follows:

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m 4 MT (8; qN) - MT (4; qN )2s 2mˆ (L; m T) = T ⎧L ⎨MT (2; qN )4⎩4MT (4; qN )3
4MT (4; qN ) MT (6; qN ) ⎫+,6⎬
MT (2; qN )MT (2; qN )5⎭(27)

where the theoretical moments MT (k; qN ) can be obtainedusing Eq. (22). Thus, the statistical variance of the Nakagami
parameter m̂ estimated by the moment method is determinedby the true Nakagami parameter mT , the number of independent samples, L , and the theoretical non-normalized momentswith second, fourth, sixth, and eighth orders.
To validate Eq. (27), a random number simulation was
conducted under the same conditions as those described in
Sect. 3.2. The fourth order of the normalized moment
MN, L (4; qN ) was obtained from random numbers generated
from the Nakagami distribution, and the Nakagami parameterm was estimated from MN, L (4; qN ) using the moment methodof Eq. (25).
3.4. Determination of effective number of
independent samples in ultrasonic measurements
As shown by the proposed principles, the statistical varianceand covariance of moments can be determined using thenumber of independent samples, L , and the theoretical
moments MT (k; q ). In ultrasonic measurements, the numberof independent samples, L , is determined by the ratio of thearea of analysis ROI, AROI , and the area of the ultrasound
PSF, APSF , because the samples within the PSF area arecorrelated with each other. However, the ultrasound PSF
becomes a continuous function; therefore, it is necessary to
determine the effective PSF area that determines the numberof independent samples, L . In this study, we deﬁne the PSF
area APSF (r ) as the area of the PSF that is equal to or larger
than the threshold value r (<0) [dB] under the condition thatthe amplitudes of the PSF are normalized by their maximum.
Using APSF (r ), we deﬁned the effective number of independent samples, LUS (AROI ; APSF (r )), for ultrasonicmeasurements as follows:
LUS (AROI ; APSF (r )) = A

AROIPSF ( r )

.

(28)

Here, the ROI area AROI is deﬁned under the condition thatthe width and height of the ROI are set larger than those of
the PSF (r ) region. This condition is generally satisﬁed in astatistics-based analysis of echo envelope amplitudes.
Additionally, it is assumed that the depth and lateral samplingintervals are smaller than the height and width of theultrasound spatial resolution, respectively, and this conditionis generally satisﬁed. Equation (28) shows the effective
number of PSF (r ) that can be arranged in the ROI area
without overlap. To determine the optimal r̂ , we conducted
an ultrasonic simulation using Field II developed by Jensen etal.36,37)
Several point scatterers were randomly and densely ( 10scatterers/ APSF (-3 dB)) distributed in the ultrasonic measurement region to simulate the conditions of Rayleighdistribution qRA. Ultrasound RF signals scattered from thepoint scatterers and received by the transducer were simulated using Field II. Two types of transmitted frequencies, 3.5
and 7.5 MHz, were used to simulate the different sizes of the

PSF. The sampling frequency was set to 40 MHz. The widthof the element was set to the wavelength at a sound speed of
1540 m s−1 for each transmitted frequency. The gap between
adjacent elements was set to 0.1 mm for both transmitted
frequencies. Thus, the element pitch was 0.540 mm for
3.5 MHz and 0.305 mm for 7.5 MHz. The 64 elements
were activated with an apodization of the Hanning window
for transmitting and receiving a single beam. The impulse
response of the element was set to two cycles of a sinusoidal
wave with a Hanning window weight, and the excited signal
was set to two cycles of a non-weighted sinusoidal wave. A
transducer with 64 elements was moved in the lateral
direction at intervals of the element pitch to simulate theelectronic scan. In the simulation for calculating the PSF
area, however, the transducer was moved at intervals of
0.010 mm to accurately calculate the PSF area with sufﬁciently high lateral sampling.
To determine the optimal threshold value r̂ [dB], it isnecessary to simulate the ultrasound echo envelope amplitudes that completely follow the Rayleigh distribution;
therefore, the depth dependences of the PSF and acousticpressure must be removed in this analysis. Thus, we used the
dynamic focus technique in the depth direction, and thetransmitted focused depths were dynamically changed in 1
mm intervals. All 64 elements were used for transmitting theultrasonic beam, and all 64 element signals were used for thereceived beamforming, regardless of the focus depth. To
compensate for the depth-dependence of the acoustic pressure, the obtained RF signals were normalized by the
preliminary simulated acoustic pressure property in the depth
direction.
The ROI was set for the simulated echo envelope data, andthe moment was calculated for the echo envelope amplitudes
in the ROI. The height of the ROI was set to 5 mm for
3.5 MHz and to 3 mm for 7.5 MHz by considering the depth
dependence of the PSF. The width of the ROI was changed
from 2.16 to 88.56 mm in 2.7 mm intervals for 3.5 MHz, andfrom 1.2 to 47.7 mm in 1.5 mm intervals for 7.5 MHz. To
determine the optimal threshold value r̂ , a broad range ofwidths was examined, including the wide width condition,
which is not used in the actual measurement. Subsequently,
the ﬁrst and third orders of non-normalized moments,
MUS, k = 1 (n ; AROI ) and MUS, k = 3 (n ; AROI ), were calculated
independently for each ROI area AROI . The 500 kinds(n = 1, 2,,500 ) of echo envelope data were simulated byrandomly changing the positions of the scatterers. Thereafter,
the variance for 500 sets of moments, VUS, k (AROI ), wascalculated for each ROI area AROI as follows:
VUS, k (AROI ) =1 500
å (MUS,k (n ; AROI ) - MUS,k (n ; AROI ))2 ,500 n = 1(29)

where MUS, k (n ; AROI ) is the average of the 500 kinds ofmoments {MUS, k (n ; AROI )}n = 1,2,  ,500 .
The PSF area APSF (r ) was measured for each thresholdvalue r [dB]. The effective number of independent samples,
LUS (AROI ; APSF (r)), was deﬁned by Eq. (28). Thereafter,
the theoretical variance s 2M (LUS (AROI ; APSF (r )), k; qRA )was estimated for each ROI area AROI by substituting
LUS (AROI ; APSF (r )) for L in Eq. (8). For each r , the root-

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mean-squared error (RMSE) of the measured variances of the
moments from the theoretical variances was evaluated, andthe optimal threshold value r̂ was determined by minimizing
the RMSE, as follows:rˆ = argmin {RMSEk (r )},

(30)

r

RMSEk (r )= E AROI [{VUS, k (AROI ) - s 2M (LUS (AROI ; APSF (r )), k; qRA )}2 ] .

(31)
4. Results
4.1. Validation of principles by random number
simulation

Figure 2 shows the results for the Rayleigh distribution
qRA (x ) = qMN (x {a1 = 1, m1 = 1, W1 = 1}N = 1 ). Figure 2(a)shows the probability density function of the theoretical
Rayleigh distribution. Figures 2(b-i) and 2(b-ii) show theaverages and standard deviations of the (b-i) ﬁrst and (b-ii)
third orders of the non-normalized moments ML . The resultsof the random number simulations for 10 000 trials areplotted in black, using an error bar. The theoretical nonnormalized moment MT (k; qRA ) obtained using Eq. (1) isplotted as a red dashed line, and the theoretical standard

deviation sM (L, k; qRA ) obtained using Eq. (8) is plotted as
by
red
solid
lines.
MT (k; qRA )  sM (L, k; qRA )Figures 2(c-i) and 2(c-ii) show the averages and standard
deviations of the (c-i) ﬁrst and (c-ii) third orders of thenormalized moments MN, L , plotted using the same line types
and colors as in Figs. 2(b-i) and 2(b-ii). Figures 2(b-iii) and2(c-iii) show the covariance between the ﬁrst and third ordersof the non-normalized moments and that of the normalized
moments, respectively. The simulation results for 10 000
trials are plotted using black dots and the theoreticalcovariances COVM (L, 1, 3; qRA ) and COVMN (L, 1, 3; qRA )are plotted using solid red lines.
As shown in Fig. 2(b), the average, standard deviation,and covariance for the non-normalized moments
ML (k; qRA ) corresponded well with the theoretical values
MT (k; qRA ), sM (L, k; qRA ), and COVM (L, k1, k 2; qRA ), respectively. For the normalized moments MN, L (k; qRA )shown in Fig. 2(c), the results for random numbers
corresponded well with the theoretical values
MN,T (k; qRA ), sMN (L, k; qRA ), and COVMN (L, k1, k 2; qRA ),respectively, for a large number of independent samples.
However, bias errors are observed for a small number of
independent samples. This may be because the number of
independent samples was insufﬁcient to satisfy the central
limit theorem. By comparing Figs. 2(b) and 2(c), it was
conﬁrmed that the statistical variance could be suppressedby normalizing the moments.

Fig. 2. (a) Probability density function of Rayleigh distribution. (i), (ii) Averages and standard deviations of (i) ﬁrst and (ii) third orders of moments. (iii)
Covariances of ﬁrst and third orders of moments. (b) Results for non-normalized moments ML . (c) Results for normalized moments MN, L . Black: simulation
results using random numbers. Red: theoretical values determined by proposed principles.

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Fig. 3. (a), (b) Results for Nakagami model with model parameters (m1, W1) of (a) (0.7, 1) and (b) (1.3, 1). (i) Probability density of model function. (ii, iii)
Averages and standard deviations of (ii) ﬁrst and (iii) third orders of normalized moments. (iv) Covariances of ﬁrst and third orders of normalized moments.
Black: simulation results using random numbers. Red: theoretical values determined by proposed principles.

Similar relationships between the non-normalized andnormalized moments were obtained for other model functionsof the single-Nakagami, multi-Rayleigh, and doubleNakagami models. Therefore, we only show the results forthe normalized moments MN, L (k; q ) for these model functions. Figure 3 shows the results for the single Nakagami
model [N = 1 in Eq. (20)]. The model parameters (m1, W1)
are (0.7, 1) in Fig. 3(a) (condition pre-Rayleigh: low scattereddensity) and (1.3, 1) in Fig. 3(b) (condition post-Rayleigh:existence of periodically located or specular scatterers).
Figures 4(a) and 4(b) show the results for the multi-Rayleigh(a1 = 0.7, a2 = 0.3, m1 = m 2 = 1, W2 /W1 = 4, N = 2) anddouble-Nakagami (a1 = 0.7, a2 = 0.3, m1 = 0.7, m 2 = 1.2,
W2 /W1 = 4, N = 2) models. Figures 3(i) and 4(i) show thetheoretical probability densities of the model functions.
Figures 3(ii) and 4(ii) show the results for the averages andstandard deviations of the ﬁrst order of the normalized
moment, Figs. 3(iii) and 4(iii) show the results for the thirdorder, and Figs. 3(iv) and 4(iv) show the results for thecovariance of the ﬁrst and third orders of the normalized
moments. The results were plotted using the same line types
and colors as those in Fig. 2.
As shown in Figs. 3 and 4, the averages, standard
deviations, and covariances of random numbers corresponded
well with the theoretical values for a large number of
independent samples, and the bias errors were caused by asmall number of independent samples, which was similar to
the results for the Rayleigh distribution shown in Fig. 2. Thus,
the statistical variance and covariance for each model function
can be correctly quantiﬁed by the proposed principles under
the condition that the number of independent samples, L , issufﬁciently large to satisfy the central limit theorem.
4.2. Demonstration of use of proposed principles:
statistical variance of Nakagami parameter estimatedby moment method
Figures 5(a)–5(c) show the results of the Nakagami parameter m̂ estimated by the moment method in Eq. (25). Notethat the exact same value was obtained for the use of the nonnormalized moment ML (k; qN ) by the ﬁrst equation and forthe use of the normalized moment MN, L (k; qN ) by the lastequation in Eq. (25). The average and standard deviation of
the estimated results for 10 000 trials are shown in blackusing an error bar. Figures 5(a)–5(c) show the results fordifferent numbers of independent samples, L , of 100, 500,
and 1000, respectively. The theoretical mT  smˆ (L; mT ) of
ˆ determined using
the estimated Nakagami parameter m,
Eq. (27), are plotted as solid red lines in Figs. 5(a)–5(c).
Although random numbers were generated from the
Nakagami distribution without any noise, there were bias
errors in the average of the estimated m̂ when the number of
independent samples, L , was 100, as shown in Fig. 5(a).
When the number of independent samples, L , was 500 or

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Fig. 4. (a) Results for multi-Rayleigh model with model parameters (a1 = 0.7, a2 = 0.3, m1 = m 2 = 1, W2 /W1 = 4, N = 2). (b) Results for doubleNakagami model with model parameters (a1 = 0.7, a2 = 0.3, m1 = 0.7, m 2 = 1.2, W2 /W1 = 4, N = 2). (i) Probability density of model function. (ii), (iii)
Averages and standard deviations of (ii) ﬁrst and (iii) third orders of normalized moments. (iv) Covariances of ﬁrst and third orders of normalized moments.
Black: simulation results using random numbers. Red: theoretical values determined by proposed principles.

1000, the standard deviation of the estimated m̂ corresponded
well with the theoretical standard deviation smˆ (L; mT )obtained from Eq. (27). Thus, it was conﬁrmed that the
proposed principle can correctly quantify the standard deviation smˆ (L; mT ) of the Nakagami parameter m̂ estimated by
the moment method when the number of independent
samples, L , is sufﬁciently large to satisfy the central limit
theorem, as in the moment evaluation in Sect. 4.1.
Figure 5(d) shows the relationship between the number of
independent samples, L , and the theoretical standard deviation smˆ (L; mT ) of the estimated Nakagami parameter m̂ , foreach true parameter mT . As shown in Fig. 5(d), the standard
deviation becomes larger for a large Nakagami parameter m.
4.3. Determination of effective number of independent samples in an ultrasonic measurement
Figures 6(a) and 6(b) show the simulated B-mode images of
PSFs for transmitted frequencies of 3.5 and 7.5 MHz,
respectively. Figure 6(c) shows the relationship between thethreshold value r [dB] and the PSF area APSF (r ) [mm2].
Figures 7(a) and 7(b) show examples of simulated B-mode
images following the Rayleigh distribution for transmitted
frequencies of 3.5 and 7.5 MHz, respectively, for an ROI
width of 20 mm.

Figures 8(a) and 8(b) show the relationships between the
ROI area AROI [mm2] and the obtained variances Vk (AROI ) of
the ﬁrst- and third-order non-normalized moments, respectively. As shown in Fig. 8, the variance of the moments
decreased with the broadening of the ROI area AROI [mm2].
Furthermore, for the same ROI area AROI [mm2], the varianceof the moments decreased when using a higher transmitted
frequency. This is because the PSF area APSF (r ) [mm2] issmaller for higher transmitted frequencies, as shown in Fig. 6.
Thus, the statistical variance of the moments depends on both
the ROI area AROI [mm2] and PSF area APSF (r ) [mm2].
Figure 9 shows the relationship between the thresholdvalue r [dB] for determining the PSF area APSF (r ) and
RMSEk (r ) evaluated using Eq. (31). Because a small numberof independent samples caused a bias error for normalized
moments, as conﬁrmed by Fig. 2(c), the effective number of
independent samples, LUS, was determined from the nonnormalized moments ML (k; q ). Figures 9(a) and 9(b) show
the results for the ﬁrst- and third-order non-normalized
moments, respectively. As shown in Fig. 9, RMSEk (r ) was
minimized at approximately r = -6 dB. Thus, we determined the effective number of independent samples for anr = -6 dB)).
ROI area of AROI by LUS (AROI ; APSF (^SJ1045-8© 2023 The Japan Society of Applied Physics

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Fig. 5. Results of standard deviation of Nakagami parameter m̂ estimated by moment method. (a), (b), (c) Relationships between Nakagami parameters m̂
estimated from random numbers (black) and theoretically determined values (red). Number of independent samples, L , was set to (a) 100, (b) 500, and (c)1000. (d) Relationships between true mT and theoretical standard deviation of estimated m̂ by moment method. Results for each number of independent
samples, L , are plotted in different colors.

Figure 10 shows examples of an RMSEk evaluation using
Eqs. (30) and (31) for (a) r = -10 dB, (b) r = -6 dB, and(c) r = -2 dB. Figures 10(i) and 10(ii) show the shape of thePSF (r ) when the transmitted frequencies were 3.5 and7.5 MHz, respectively. Figures 10(iii) and 10(iv) show the
relationship between the variance of the ultrasonicallysimulated moments, VUS, k (AROI ), and the estimated variances 2M (LUS (AROI ; APSF (r )), k; qRA ) obtained using Eq. (8)under the condition that the number of independent samples,
L , is LUS (AROI ; APSF (r )).
When the threshold value r is lower than the optimalrˆ = -6dB, as shown in Fig. 10(a), the PSF area
APSF (-10dB) becomes large, as shown by comparing
Figs. 10(a-i) and 10(b-i) or Figs. 10(a-ii) and 10(b-ii);
therefore, the effective number of PSF in the ROI becomes
relatively small. In this case, the determined effective numberof independent samples, LUS (AROI ; APSF (-10dB)), becomes smaller than the actual number of independent
samples in the ROI, and the estimated variances 2M (LUS (AROI ; APSF (-10 dB)), k; qRA ) using Eq. (8) was
overestimated, as shown in Figs. 10(a-iii) and 10(a-iv).
By contrast, when the threshold value r is higher than the
optimal rˆ = -6 dB, as shown in Fig. 10(c), the PSF area
APSF (-2 dB) becomes small, and the effective number of
PSF in the ROI becomes relatively large. In this case,
LUS (AROI ; APSF (-2 dB)) becomes larger than the actual
number of independent samples in the ROI, and the estimatedvariance s 2M (LUS (AROI ; APSF (-2 dB)), k; qRA )using

Eq. (8) was underestimated, as shown in Figs. 10(c-iii) and10(c-iv).
Thus, in both cases shown in Figs. 10(a) and 10(c),RMSEk in Eq. (31), which shows the deviation of thevariance of the ultrasonically simulated moments,
VUS, k (AROI ) (dots in Figs. 10(iii) and 10(iv)), from thetheoretical variance s 2M (LUS (AROI ; APSF (r )), k; qRA ) (redlines in Figs. 10(iii) and 10(iv)), becomes larger than that forthe optimal threshold value rˆ = -6 dB (Fig. 10(b)), as
shown in Fig. 9.
Using the optimized threshold value rˆ = -6 dB, thevariance of the ultrasonically simulated moments,
VUS, k (AROI ), corresponded well with the theoretical variances 2M (LUS (AROI ; APSF (rˆ = -6 dB)), k; qRA ), as shown in
Figs. 10(b-iii) and 10(b-iv). Thus, the number of independent
samples, L , in the ROI area AROI can be quantiﬁed by
LUS (AROI ; APSF (-6 dB)) obtained using Eq. (28).
5. Discussion

In this study, we quantiﬁed the statistical variance andcovariance of moments caused by a ﬁnite number of
independent samples, L . The statistical variance and covariance can be theoretically determined from the number of
independent samples, L , and the theoretical moments given
by Eq. (1). Furthermore, we evaluated the relationshipbetween the ultrasonic measurement conditions and theeffective number of independent samples via ultrasonicsimulation. As a result, the number of independent samples

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(a)

(b)

Fig. 8. Relationship between ROI area AROI [mm2] and variance of nonnormalized moments with (a) ﬁrst and (b) third orders. Variance for 500 setsof non-normalized moments is plotted. Transmitted frequencies: (blue) 3.5
and (green) 7.5 MHz.

(c)

(a)

Fig. 6. (a), (b) Simulated ultrasound B-mode images of point spreadfunction (PSF) for transmitted frequencies of (a) 3.5 and (b) 7.5 MHz. (c)Relationship between threshold value r [dB] and PSF area APSF (r ) [mm2].
Transmitted frequencies: (blue) 3.5 and (red) 7.5 MHz.

(b)
(a)

(b)
Fig. 7. Examples of simulated ultrasound B-mode images following
Rayleigh distribution. Transmitted frequencies: (a) 3.5 and (b) 7.5 MHz.

could be estimated by the ratio of the ROI area to the PSF(-6 dB) area, LUS (AROI ; APSF (-6 dB)) given by Eq. (28).
Based on these results, we can determine the statisticallimitation of the moment analysis for given ROI and PSF
sizes and the model function used. Therefore, when the
resolution of the moment analysis required for the target
tissue characterization is determined, we can theoreticallyFig. 9. Relationship between threshold value r [dB] for determining PSF
area APSF (r ) and root-mean-squared error RMSEk (r ) evaluated for (a) ﬁrst
and (b) third orders of non-normalized moments by Eq. (31). Transmitted
frequencies: (blue) 3.5 and (green) 7.5 MHz.

determine the necessary conditions for the ROI area andultrasound spatial resolution (PSF area).
In the simulation results using random numbers, bias errors
were present for normalized moments when the number of
independent samples was insufﬁcient, although the statistical

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Fig. 10. (i), (ii) Shapes of PSFs that are equal to or larger than r [dB]. Transmitted frequencies were (i) 3.5 and (ii) 7.5 MHz. Threshold value r was set to (a)-10 , (b) -6 , and (c) -2dB. (iii), (iv) Relationships between theoretical variances and ultrasonically simulated variances of (iii) ﬁrst and (iv) third orders of
non-normalized moments, when LUS (AROI ; APSF (r )) is regarded as the number of independent samples, L . Theoretical variances of non-normalized moments
are plotted using red lines, and ultrasonically simulated variances are plotted using dots. Transmitted frequencies: (blue) 3.5 and (green) 7.5 MHz.

variance could be suppressed compared with the non-normalized moments. These bias errors cause a false evaluation of
the tissue characteristics; therefore, the ROI size must be
carefully determined, especially when a small ROI is used to
improve the locality of tissue characterization.
As shown in Fig. 5, we demonstrated the use of the
proposed principles in the analysis of the statistical limitations of tissue characterization. In this study, we theoreticallyquantiﬁed the statistical variance of the Nakagami parameterm estimated using the moment method. The Nakagami
parameter m reﬂects several scatterer-distribution conditions,such as the scattered density or the existence of periodicallylocated or specular scatterers. As several scatterer-

distribution conditions can be quantiﬁed using a single
parameter, the Nakagami parameter has been widely used
for quantitative tissue characterization, such as the quantiﬁcation of liver steatosis,7) quantiﬁcation of lymph nodes,27)
and monitoring of temperature changes in biologicaltissues.29–31)
As shown in Fig. 5(a), there were bias errors for the
average of the estimated m̂ by the moment method when thenumber of independent samples was 100. By considering theeffective number of independent samples, LUS, in the ROI,
determined by the method proposed in Sect. 3.4, it isconsidered that the Nakagami parameter is statistically overestimated by the moment method when the ROI area AROI is

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only 100 times the PSF area APSF (-6 dB);
AROI = 100 ´ APSF (-6 dB). Thus, the ROI size should
be carefully determined to prevent overestimation of the
Nakagami parameter m .
Even when the ROI area is sufﬁciently large and/or the
PSF area is sufﬁciently small to avoid an overestimation of
the Nakagami parameter, the statistical variance of theestimated Nakagami parameter m̂ depends on the numberof independent samples, L , i.e. the ratio of the ROI area to the
PSF area, AROI /APSF (-6 dB), as shown in Fig. 5(d). Whenthe resolution of the Nakagami parameter estimation required
for target tissue characterization is determined, the necessary
conditions for the ROI and PSF areas can be quantitatively
determined by the principles proposed in this paper, which ispart of a future study.
In this study, we focused on the Nakagami parameter todemonstrate the use of the proposed principles. In futurestudies, we will quantify the statistical limitations of parameter estimations for other model functions, such as themulti-Rayleigh, double-Nakagami, and homodyned-K distributions, and develop stable tissue characterization methods.

In this study, we quantiﬁed the statistical limitations of
moment analysis caused by a ﬁnite number of samples. The
statistical variance and covariance can be determined from thenumber of independent samples and theoretical moments ofthe model function. Using an ultrasonic simulation, it was
conﬁrmed that the effective number of independent samples of
the ultrasonic measurement could be estimated from therelationship between the areas of the ROI and PSF. The
proposed principles will contribute to the quantiﬁcation of thestatistical limitations of statistics-based tissue characterizationand the development of stable tissue characterization methods.
Acknowledgments

This work was partially supported by the Japan Society for
the Promotion of Science KAKENHI (Grant No. 21K14166).
Appendix

We derive the covariance for the normalized moment in
Eq. (19). Similar to Eqs. (9) and (10), we deﬁne thesummation of the k1th and k 2 th orders of the normalized
moments as follows:
SUMN,T (k1, k 2; q) = MN,T (k1; q) + MN,T (k 2; q)MT (k1; q )k1
MT (2; q ) 2+

MT (k 2; q )
k2

,

(A·1)

MT (2; q ) 2

SUMN, L (k1, k 2; q) = MN, L (k1; q) + MN, L (k 2; q)=s 2SUMN (L, k1, k 2; q)

(
+(
+(
=

ML (k1; q )k1+

ML (2; q ) 2

ML (k 2; q )

.

(A·2)

k2

ML (2; q ) 2

Similar to Eqs. (11) and (14), the covariance of the normalized moment is derived based on the law of error propagation
as follows:
1

COVMN (L, k1, k 2; q) = 2 (s 2SUMN (L, k1, k 2; q)- s 2MN (L, k1; q) - s 2MN (L, k 2; q)) ,(A·3)¶SUMN,T (k1, k 2; q ) 2 2
s M (L, k1; q)¶MT (k1; q ))) s (L , k ; q )) s (L, 2; q)¶SUMN,T (k1, k 2; q ) 2 2M¶MT (k 2; q )+2

2¶SUMN,T (k1, k 2; q ) 2 2M¶MT (2; q )¶SUMN,T (k1, k 2; q ) ¶SUMN,T (k1, k 2; q )COVM(L, k1, k 2; q)¶MT (k1; q )¶MT (k 2; q )+2+2

6. Conclusions=

2
where s SUM(L, k1, k 2; q ) is the statistical variance ofN
SUMN, L (k1, k 2; q ). Here, s 2MN (L, k1; q ) and s 2MN (L, k 2; q )are obtained using Eq. (17). Therefore, we derive
2s SUM(L, k1, k 2; q ), as follows:N
As shown by Eq. (A·2), SUMN, L (k1, k 2; q ) is composed ofthree different orders of moments, ML (k1; q ), ML (k 2; q ), andML (2; q ). Therefore, from the law of error propagation, the2statistical variance s SUM(L, k1, k 2; q ) of SUMN, L (k1, k 2; q )Nis derived as follows:¶SUMN,T (k1, k 2; q ) ¶SUMN,T (k1, k 2; q )COVM(L, k1, 2; q)¶MT (k1; q )¶MT (2; q )¶SUMN,T (k1, k 2; q ) ¶SUMN,T (k1, k 2; q )COVM(L, k 2, 2; q).
¶MT (k 2; q )¶MT (2; q )(A·4)

Here, s 2M (L, k1; q ), s 2M (L, k 2; q ), and s 2M (L, 2; q ) are thestatistical variances of the non-normalized moments deterCOVM (L, k1, k 2; q ),
mined
using
Eq.
(8),
and
COVM (L, k1, 2; q ), and COVM (L, k 2, 2; q ) are the covariances of the non-normalized moments determined using
2
Eq. (14). Thus, s SUM(L, k1, k 2; q ) can be obtained byN
substituting Eqs. (8), (14), and (A·1) into Eq. (A·4).
Finally, by substituting Eqs. (17) and (A·4) into Eq. (A·3),
thecovarianceofthenormalized
moments,
COVMN (L, k1, k 2; q ), is determined, as shown in Eq. (19).

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