Japanese Journal of Applied Physics 48 (2009) 07GJ08

REGULAR PAPER

Estimation of Scatterer Diameter Using Ultrasonic Backscattering Property
for Assessment of Red Blood Cell Aggregation
Nobutaka Saitoh1 , Hideyuki Hasegawa2;1 , and Hiroshi Kanai1;2 
12Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan

Received November 19, 2008; accepted April 5, 2009; published online July 21, 2009
Red blood cell (RBC) aggregation, a determinant of blood viscosity, plays an important role in blood flow rheology. RBC aggregation is
induced by the adhesion of RBCs when the electrostatic repulsion between RBCs weakens owing to increases in protein and saturated
fatty acid levels in blood, and excessive RBC aggregation may lead to various circulatory diseases. This study was conducted to establish anoninvasive quantitative method for the assessment of RBC aggregation. The spectrum of nonaggregating RBCs presents Rayleighbehavior, indicating that the power of a scattered wave is proportional to the fourth power of frequency. By dividing the measured power
spectrum of echoes from scatterers by that from a silicone plate reflector, the frequency responses of transmitting and receiving
transducers are removed from the former spectrum. This normalized power spectrum changes linearly with respect to logarithmic
frequency. In non-Rayleigh scattering, on the other hand, the spectral slope decreases because a larger scatterer behaves as a reflector
and echoes from a reflector do not show frequency dependence. Therefore, it is possible to assess RBC aggregation using the spectral
slope value. In this study, spherical scatterers with diameters of 5, 11, 15, and 30 mm were measured in basic experiments. The spectral
slope of the normalized power spectrum of echoes from the lumen of the vein in the dorsum manus of a 24-year-old healthy male was close
to that from microspheres with a diameter of 15 mm, and the typical RBC diameter was smaller than this value. The frequency-dependent
attenuation of ultrasound during propagation in a biomedical tissue was considered to be one reason for this. Furthermore, during
avascularization, the slope gradually decreased owing to the aggregation of RBCs. These results show the possibility of using the proposed
method for the noninvasive assessment of RBC aggregation. # 2009 The Japan Society of Applied Physics
DOI: 10.1143/JJAP.48.07GJ08

1.

Introduction

Medical ultrasound is clinically used to make a diagnosis forvarious organs, and because it is noninvasive and relatively
stress free for patients, it can be repeatedly employed to
conﬁrm time-dependent changes. Ultrasound B-mode imaging is widely used for the morphological diagnosis of thearterial wall. In addition, methods for evaluating theviscoelasticity of the arterial wall have recently beendeveloped1,2) because the mechanical properties of thearterial wall are related to the atherosclerotic change.
The condition of blood is an important factor related tovarious circulatory diseases. However, conventional ultrasonic images cannot be used to evaluate the condition of theblood in the blood vessel because red blood cells (RBCs),
which are the main components of blood, are much smaller
than the wavelength of ultrasound and the variation inacoustic impedance between blood plasma and RBCs is verysmall. However, the condition of blood is related to various
circulatory diseases, and the evaluation and diagnosis ofsuch a condition are important for the detection of a diseaseat an early stage.
As one of the determinants of blood viscosity, RBC
aggregation plays an important role in blood ﬂow rheology.
The adventitia of healthy RBCs is charged with negativeelectricity, which impedes RBC adherence by electrostatic
repulsion.3) However, owing to increases in protein andsaturated fatty acid levels in blood, such repulsion between
RBCs is gradually weakened and RBC aggregation is
induced by the overlapping of RBCs. The main function ofblood is to transport nutrients, oxygen, and essential
elements to tissues and to remove metabolic products, suchas carbon dioxide and lactic acid, produced by thosetissues.4) However, RBC aggregation signiﬁcantly degrades
this function because of the decrease in the superﬁcial areaE-mail address: kanai@ecei.tohoku.ac.jp

used to transport materials. Excessive RBC aggregation may
promote various circulatory diseases, such as atherosclerosis, hypercholesterolemia, diabetes, thrombosis, and so
on.5–8) Therefore, the assessment of RBC aggregation isessential.9) The micro channel array ﬂow analyzer (MCFAN) method is a recently developed technique for theassessment of RBC aggregation by determining whether red
blood cells pass through gaps in silicon substrates simulating
blood capillaries.10) However, this method is invasive andis not quantitative. The present study was conducted toestablish a noninvasive quantitative method for the assessment of RBC aggregation.
2.

Principles

An RBC is a very small ultrasonic scatterer whose diameteris 8 mm at most, and thus, the amplitudes of scattered
RF echoes are very small. To assess the level of RBC
aggregation, the power spectrum of the echoes from RBCs is
calculated using the fast Fourier transform (FFT), and thescattering properties of RF echoes are evaluated in thefrequency domain. In the present study, it was assumed
that the diameter of a scatterer increases depending on the
degree of RBC aggregation.
The scattering strength can be measured quantitatively interms of a backscattering coeﬃcient if the tissues can bemodeled as a random medium. Figure 1 shows the geometrical condition in the measurement of echoes from arandom medium with a circular transducer.11) The average
power spectrum Si ð f Þ of incoherent components of theecho scattered by the random medium is given by11)



2
S0 4
Si ð f Þ ¼ 4 f 0 Gð f Þ k4 R2s gð f Þ
;
ð2:1Þ
2w0
where k is the wave number of ultrasonic waves (k ¼ 2 f =c0 ; c0 is the average sound velocity), 0 is the average densityof the medium, Gð f Þ is the electric characteristic of thetransducer, Rs is the standard deviation of the amplitude

07GJ08-1# 2009 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 48 (2009) 07GJ08

N. Saitoh et al.

w

d

Z0 + Δ Z(r)
V
w
0

probe

water

blood

S

8.3 mm

Z0

N
θ

2a
T

silicone plate

r

O

y
ϕ

Fig. 2.

x

System for measurement of echoes from the silicone plate.

Transducer

2
1.5
amplitude [V]Fig. 1. Illustration of geometrical condition in measurement of
echoes from scatterers with a circular transducer.

reﬂection coeﬃcients of scatterers, S0 is the area of thetransducer aperture, and w0 is the distance between the transducer and the scatterers. Furthermore, gð f Þ is expressed as 2:5 2  b d
2 2
gð f Þ ¼
ðb2 þ T2 þ  2 Þ1 e k ;ð2:2Þ
4
where d is the thickness of the layer of the mediumpincluding
ﬃﬃﬃ
scatterers,
pﬃﬃﬃ b and T are the beam radius (b ¼ 8w0 =ka)
and a= 2 (a: aperture radius), respectively, and  is thecorrelation length of continuous inhomogeneous media,12) being related to the diameter of the scatterer.11) In this case,the correlation length is much smaller than the diameter andbeam width of the transducer. Therefore, the following
approximation holds:("  2 #)1
b 2
b
2
2
2 1
2
ðb þ T þ  Þ ¼ 
þ
þ1


’ 2:
Using eq. (2.3), eq. (2.1) is modiﬁed as 6  2:5 4 3 k2  2 !
a
 k  eSi ð f Þ ¼ ½jGð f Þj2 Z02 R2s d 2
;
w0
2

ð2:3Þ

ð2:4Þ

where Z0 is the average speciﬁc acoustic impedance of themedium. When scatterer diameters are suﬃciently smaller
2 2
than the ultrasonic wavelength [k  1, ek  ’ 1, Si ð f Þ /k4  6 ], echoes from scatterers show Rayleigh behavior, i.e.,
the power Si ð f Þ of scattered waves is proportional to thefourth power of frequency, f 4 .11,13,14) Echoes from scatterers
with larger diameters include the components of reﬂection,
which have no frequency dependence. Therefore, the spectral
slope decreases when the sizes of scatterers increase and thecomponents of reﬂection are included in the echoes.15)
The measured power spectrum Ps ð f Þ of the received
ultrasonic echo signal es ðtÞ contains both the scattering
property Si ð f Þ from a microscopic sphere and the frequency
responses Gð f Þ of transmitting and receiving transducers.
Therefore, by normalizing the power spectrum Ps ð f Þ ofechoes from scatterers by the power spectrum Pr ð f Þ from asilicone plate, the frequency responses Gð f Þ of the transmitting and receiving transducers are removed.16–18)
Figure 2 shows a system for measuring an echo er ðtÞ froma silicone plate. The distance between the probe and the

1
0.5
0
-0.5
-1
-1.5
-2

10.4

10.6

10.8
time [μs]

11

11.2

Fig. 3. (Color online) Three RF echoes from the silicone plate atdifferent points on the surface.

silicone plate was equal to that between the probe and the
middle part of the intravascular lumen in an in vivo
measurement. Figure 3 shows RF echoes reﬂected by thesurface of the silicone plate and scattered from the inside of
the silicone plate. Three lines in Fig. 3 show RF signals from
diﬀerent points of the silicone plate. The scatterer distribution in the silicone plate is assumed to be random, and only
the echo reﬂected from the surface can be obtained byaveraging the echoes measured at diﬀerent points on thesurface. Therefore, echoes from the silicone plate wereacquired at 1000 diﬀerent points by moving the transducer.
In Fig. 3, there are echoes with much smaller amplitudes ofaround 10.4 ms. Such echoes were supposed to be caused by
the interference of ultrasonic waves during focusing by the
concave transducer used.19) Such undesirable echoes were
suppressed by applying Hanning windows centered at
10.8 ms when the Fourier transform was performed. Figure 4
shows the power spectrum Pr ð f Þ from a silicone plate thatwas obtained by averaging 1000 power spectra of RF echoesfrom a number of diﬀerent points on the surface of thesilicone plate. The frequency range used for normalization
was limited because the ultrasonic pulse used in this study
had a ﬁnite frequency bandwidth. In the present study, we
estimated the slope a of the normalized power spectrum
Ps ð f Þ=Pr ð f Þ using the weighted least-mean-squared methodby considering the signal-to-noise ratio (SNR) of the echo ateach frequency.20) Let us deﬁne the mean squared diﬀerence
between the measured spectrum and a linear model as

07GJ08-2

¼

N
X

e2i

i¼0

¼

N
X

w2i ½yð fi Þ  ðaxi þ bÞ2 ;

ð2:5Þ

i¼0

# 2009 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 48 (2009) 07GJ08

N. Saitoh et al.

(a)

-20

power spectrum [dB]power spectrum [dB]

-10

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-70
0

20

40
60
frequency [MHz]

80

100

weigthing function

Fig. 4. (Color online) Means and standard deviations of power
spectra of the ultrasound echoes reflected from the silicone plate.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(4)
(3)
(2)
(1)

0

-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2
-1.3
-1.4
-1.5
-1.6
7.4
25.1

20

40
60
frequency [MHz]

80

100

(4)
(3)
(2)
(1)
7.42
26.3

7.44
27.5

7.46
28.8

7.48
30.1

7.5
31.6

7.52
33.1

7.54
34.7

log f
f [MHz]frequency
0

20

40
60
frequency [MHz]

80

100

Fig. 5. (Color online) Weighting function obtained from averaged
power spectrum of the ultrasound echoes reflected from the silicone
plate.

where yð fi Þ is the normalized logarithmic power spectrum
log10 Ps ð fi Þ=Pr ð fi Þ at an angular frequency of fi , xi is thelogarithmic frequency log fi , and wi is the weighting
function. In this study, the weighting function wi wasdeﬁned bywi ¼

Pr ð fi Þ
;
Prmax

ð2:6Þ

where Prmax is the maximum value of the power spectrum
Pr ð fi Þ from a silicone plate.
Figure 5 shows the weighting function wi . It is assumed
that the echoes due to reﬂection do not show frequency
dependence and that only the scattering property of a target
remains in the normalized power spectrum Ps ð f Þ=Pr ð f Þ.
In Rayleigh scattering, the normalized power spectrum
Ps ð f Þ=Pr ð f Þ changes linearly with respect to the logarithmic
frequency. In non-Rayleigh scattering, on the other hand, thespectral slope decreases with an increase in scatterer
diameter, such an increase being equal to that in thecorrelation length  of the Gaussian correlation function. In
this study, the diameter of the scatterer was estimated toassess RBC aggregation from the spectral slope by assumingthat the correlation length  of the Gaussian correlation
function of continuous inhomogeneous media corresponds tothe diameter of the scatterer.
3.

(b)normalized logarithmic
power spectrum

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Basic Experimental Results Obtained Using
Microspheres

The ultrasound diagnostic equipment (Tomey UD-1000)
employed is equipped with a mechanical scan probe at a
center frequency f0 of 40 MHz (wavelength is about 40 mm).

Fig. 6. (Color online) (a) Averaged power spectra of the ultrasonicechoes scattered from microspheres of various sizes and (b)normalized power spectra and the weighted least-mean-squared
regression lines of microspheres (1)–(4) in Table I.

RF echoes were acquired at a sampling frequency of 1 GHz
at a 16-bit resolution by the oscilloscope (Tektronix
DPO7054), and their power spectra fPðf Þg were obtainedby Fourier transform with a Hanning window of 1.024 ms
length. To reduce the eﬀect of random noise, 1000 power
spectra were averaged. The left-hand column of Table I
shows the diﬀerent diameters of measured microspheres.
Microspheres (1) and (3) were made of copolymer with Cl2 ,and microspheres (2) and (4) were made of copolymer
without Cl2 . Microspheres with diﬀerent diameters, whichwere mixed with water at a concentration of 3.00 g/l,
simulated nonaggregated and aggregated RBCs. This concentration is lower than that of RBCs in actual blood,
preventing the aggregation of microspheres. RF echoes fromthe spheres around the focal point were acquired fromspatially diﬀerent regions by mechanical linear scan as in themeasurement of a silicone plate to acquire many statisticallyindependent data points. Eﬀects of the diﬀerence between
acoustic impedances of microspheres and RBCs were notconsidered in the present study, although such eﬀects should
be investigated in future studies.
Figure 6(a) shows the averaged power spectra of echoesfrom the microspheres, and Fig. 6(b) shows the normalized
power spectra. The numbers correspond to the microsphere
numbers in Table I. The spectral slope and intercept values
are shown in the right-hand column of Table I. Figure 7
shows the mean and standard deviation of spectral slopes forten measurements of each microsphere. In Fig. 5, the slopeof the normalized power spectrum Ps ð f Þ=Pr ð f Þ of a larger
scatterer is smaller than that of a smaller scatterer. This
result shows that the slope of the normalized logarithmic
power spectrum log10 Ps ð f Þ=Pr ð f Þ is related to the diameterof the scatterer.

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Jpn. J. Appl. Phys. 48 (2009) 07GJ08
Table I.

N. Saitoh et al.

Microsphere diameters and experimental results.

Particle diameter
(mm)

Slope
(1= log10 f )Intercept(dimensionless)52
11  3

3.46
1.78

27:2
14:8

(1)
(2)
(3)

15  5

1.47

12:9

(4)

30  10

0.02

0:56

4

(1)

(2)

(4)

(5)

(3)

slope [1/log 10 f ]

3.5
3
2.5
2
1.5
1
0.5
0
-0.5

0

5

10
15
20
25
30
scatterer diameter [μm]3540Fig. 7. (Color online) Mean and the standard deviation of theestimated spectral slopes as a function of scatterer diameter.

4.

In vivo Experimental Results

The main components of normal blood are RBCs, which do
not aggregate in large veins because of blood ﬂow.21,22) To
measure ultrasonic echoes from aggregated RBCs in
addition to those from nonaggregated RBCs, blood ﬂow
was stopped by pressurizing a cuﬀ surrounding the upperarm at 250 mmHg. Ultrasonic echoes were acquired at restfor 2 min, during avascularization for 5 min, and afterrecirculation for 3 min. Figure 8 shows B-mode images ofa vein at the dorsum manus of a 24-year-old healthy male.
B-mode images were measured at rest (1), at the beginningof avascularization (2), at the end of avascularization (3), at
the beginning of recirculation (4), and 120 s after recirculation (5). It is possible to reduce the attenuation of highfrequency ultrasound because the vein at the dorsum manus
is a superﬁcial blood vessel. The vein measured in this study
had a large diameter and a high steady blood ﬂow rate,
which aﬀect RBC aggregation, and thus, the measurement
could be assumed to be performed with respect to thenonaggregated RBCs at rest. In addition, we measured thechange in spectral slope caused by the change in blood ﬂow
rate due to avascularization using a cuﬀ. RF echoes from the
focal point positioned in the vessel lumen were acquired
from many diﬀerent positions by mechanical linear scan toreduce the eﬀect of the increase in scatterer diametervariation due to aggregation.
Figure 9 shows the averaged power spectrum Ps ð f Þ ofechoes from the lumen of the vein at the dorsum manus of
the 24-year-old healthy male with the standard deviations.
Figure 10 shows the normalized logarithmic power spectrum
log10 Ps ð f Þ=Pr ð f Þ and the weighted least-mean-squared
regression line. Figure 11 shows the transient change inestimated spectral slope due to avascularization. The slopeof the normalized power spectrum of echoes from RBCs atrest was close to that of microsphere (3) (diameter: 15 mm) in

Fig. 8. (Color online) B-mode images and windows of the vein at
the dorsum manus: (1) at rest, (2) at the beginning of avascularization,(3) at the end of avascularization, (4) at the beginning of recirculation,and (5) 120 s after recirculation.

Table I. The RBC diameter is 8 mm at most and this diameteris between those of microspheres (1) (diameter: 5 mm) and(2) (diameter: 11 mm). The spectral slope measured for
RBCs was slightly greater than that of a sphere whose
diameter corresponds to that of an RBC. One of the reasonsfor this was considered to be the frequency dependent
attenuation of ultrasound during propagation in biologicaltissues. If the attenuation coeﬃcient and the depth to vein
are assumed to be 1 dBMHz1 cm1 and 1 mm, respectively, the diﬀerences between two logarithmic normalized
power spectra log10 Ps ð f Þ=Pr ð f Þ with and without attenuation are 5 dB at 25 MHz and 7 dB at 35 MHz. Accordingly,
the spectral slope decreases by 1.37 in the frequency bandof 25 – 35 MHz. Therefore, the measured spectral slope of
RBCs was shown to be decreased.
Furthermore, the spectral slope gradually decreased
during avascularization and returned to the levels of rest in
the in vivo measurement. Although there are still manyfactors to be investigated, such as acoustic impedances of
microspheres, methods for normalization, and calibrationof the system,22) these results showed that the slope ofthe normalized power spectrum changes with scatterer
diameter.
5.

Conclusions

From basic experiments using microspheres, it was foundthat an increase in scatterer diameter led to a decrease in theslope of the normalized power spectrum. Components ofRayleigh scattering in echoes were dominant when the
scatterer diameter was small. However, the larger the
scatterer diameter was, the more dominant the componentsof the reﬂection were. Components of reﬂection showed nofrequency dependence and, therefore, the spectral slope
decreased.
In an in vivo measurement, the estimated diameter of
RBCs was observed to be slightly larger than that of

07GJ08-4# 2009 The Japan Society of Applied Physics

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Jpn. J. Appl. Phys. 48 (2009) 07GJ08

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(5)

Fig. 9. (Color online) Means and standard deviations of power spectra of the ultrasonic echoes obtained for the lumen of the vein at thedorsum manus: (1) at rest, (2) at the beginning of avascularization, (3) at the end of avascularization, (4) at the beginning of recirculation, and (5)120 s after recirculation.

normalized logarithmic
power spectrum-0.8

Acknowledgment

(3)
(2)
(1)
(4)
(5)

-1
-1.2
-1.4

We would like to thank Tomey Corporation for its
cooperation on the use of the diagnostic equipment.

-1.6
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-2.2
-2.4
7.4
25.1

7.42
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7.44
27.5

7.46
28.8

7.48
30.1

7.5
31.6

7.52
33.1

7.54 log f34.7 f [MHz]frequency

Fig. 10. (Color online) Normalized power spectra and the leastmean-squared regression lines.

duringavascularization

rest

recirculation

4

slope [1/log 10 f ]

3
2
1
0
-1
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-100

0

100
200
time [s]

300

400

500

Fig. 11. (Color online) Transient change in the estimated spectral
slope during avascularization for 5 min.

theoretical RBCs. This may have been caused by thefrequency-dependent attenuation of ultrasound, which affected the slope. Although the estimated RBC diameter did
not exactly match the real RBC diameter, these results show
the possibility of utilizing the normalized power spectrum
for the assessment of RBC aggregation.

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