Japanese Journal of Applied Physics
Vol. 47, No. 5, 2008, pp. 4180–4187#2008 The Japan Society of Applied Physics

Threshold Setting for Likelihood Function for Elasticity-Based Tissue Classiﬁcationof Arterial Walls by Evaluating Variance in Measurement of Radial Strain
Kentaro T SUZUKI, Hideyuki H ASEGAWA, Hiroshi K ANAI, Masataka I CHIKI1 , and Fumiaki TEZUKA2
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan1
Sendai Hospital of East Railway Company, Sendai 983-8508, Japan
2
Sendai Medical Center, Sendai 980-8520, Japan(Received November 26, 2007; revised February 20, 2008; accepted March 3, 2008; published online May 23, 2008)

Pathologic changes in arterial walls signiﬁcantly inﬂuence their mechanical properties. We have developed a correlationbased method, the phased tracking method [H. Kanai et al.: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43 (1996) 791],
for measurement of the regional elasticity of the arterial wall. Using this method, elasticity distributions of lipids, blood clots,
ﬁbrous tissue, and calciﬁed tissue were measured in vitro by experiments on excised arteries (mean  SD: lipids 89  47 kPa,
blood clots 131  56 kPa, ﬁbrous tissue 1022  1040 kPa, calciﬁed tissue 2267  1228 kPa) [H. Kanai et al.: Circulation 107(2003) 3018; J. Inagaki et al.: Jpn. J. Appl. Phys. 44 (2005) 4593]. It was found that arterial tissues can be classiﬁed into soft
tissues (lipids and blood clots) and hard tissues (ﬁbrous tissue and calciﬁed tissue) on the basis of their elasticity. However,there are large overlaps between elasticity distributions of lipids and blood clots and those of ﬁbrous tissue and calciﬁed tissue.
Thus, it was diﬃcult to diﬀerentiate lipids from blood clots and ﬁbrous tissue from calciﬁed tissue by simply thresholding
elasticity value. Therefore, we previously proposed a method by classifying the elasticity distribution in each region of interest(ROI) (not a single pixel) in an elasticity image into lipids, blood clots, ﬁbrous tissue, or calciﬁed tissue based on a likelihood
function for each tissue [J. Inagaki et al.: Jpn. J. Appl. Phys. 44 (2006) 4732]. In our previous study, the optimum size of an
ROI was determined to be 1,500 mm in the arterial radial direction and 1,500 mm in the arterial longitudinal direction[K. Tsuzuki et al.: Ultrasound Med. Biol. 34 (2008) 573]. In this study, the threshold for the likelihood function used in thetissue classiﬁcation was set by evaluating the variance in the ultrasonic measurement of radial strain. The recognition rate was
improved from 50 to 54% by the proposed thresholding. [DOI: 10.1143/JJAP.47.4180]
KEYWORDS: atherosclerosis, phased tracking method, radial strain, tissue classification

1.

Introduction

Noninvasive measurement of mechanical properties of thearterial wall, such as elasticity, is useful for diagnosis ofatherosclerosis because there are signiﬁcant diﬀerences
between the elastic moduli of normal arterial walls and thoseaﬀected by atherosclerosis.1–3) In particular, mechanicalproperties of plaque are important because the rupture ofplaque may cause acute myocardial infarction and cerebral
infarction.4–6) Magnetic resonance imaging (MRI) and intravascular ultrasound (IVUS) are promising technologies for
directly imaging plaque morphology,7,8) but they are expensive or invasive. On the other hand, the dynamic changeof artery diameter due to the pulsation of the heart can bemeasured noninvasively by methods using ultrasound.9–12)
Some parameters related to artery-wall elasticity can be obtained by the measured change in diameter of the artery.13–15)
However, in the derivation of these parameters, the artery is
assumed to be a cylindrical shell with uniform wall thickness
and homogeneous elasticity and, thus, the regional elasticity
around atherosclerotic plaques cannot be evaluated.
For the measurement of the mechanical properties of thearterial wall, including walls with atherosclerotic plaques,
we have developed a method, the phased tracking method,
for measuring small vibrations in the heart wall or arterial
wall with transcutaneous ultrasound.16,17) For some years, wehave measured the displacement and small change inthickness of the arterial wall caused by the heartbeat using
this method.18–21) In our phased tracking method, a set oftwo points is assigned along an ultrasonic beam, and thechange in thickness of the layer between these two points is
estimated. Furthermore, by sliding the position of the layerE-mail address: hkanai@ecei.tohoku.ac.jp

along the ultrasonic beam by intervals of the sampled points,
the spatial distribution of changes in thickness along theultrasonic beam can be obtained.
Elasticity images of the human carotid artery have beenobtained noninvasively by the measured displacement
distribution, and the potential for transcutaneous tissue
characterization has been shown by classifying elasticity
images using elasticity reference data obtained from in-vitro
experiments.21–23)
We have already measured the elasticity distributions forlipids, blood clots, ﬁbrous tissue (mixture of the smoothmuscle and collagen), and calciﬁed tissue. In these previous
studies, it was found that arterial tissues can be classiﬁed
into soft tissues (lipids, blood clots) and hard tissues (ﬁbrous
tissue, calciﬁed tissue) on the basis of their elasticity.
However, it was diﬃcult to diﬀerentiate lipids from bloodclots and ﬁbrous tissue from calciﬁed tissue because there
are large overlaps between their elasticity distributions.
Therefore, we proposed a tissue classiﬁcation method using
the elasticity distribution in a small region.24) In this method,the elasticity distribution of each small ROI (not a single
pixel) in an elasticity image was used to classify the ROI
into lipids, blood clots, ﬁbrous tissue, or calciﬁed tissue,
which improved the precision of tissue classiﬁcation.
To optimize tissue classiﬁcation, some parameters havebeen investigated. In our previous study, the optimum size ofan ROI was determined to be 1,500 mm in the arterial radial
direction and 1,500 mm in the arterial longitudinal direction.25) In addition, in the proposed classiﬁcation method, thelikelihood function of each small ROI is calculated for eachtissue component (lipids, blood clots, ﬁbrous tissue, andcalciﬁed tissue), and the region is classiﬁed into the tissue
component that has the maximum likelihood.24) Every ROI
is always classiﬁed into one of the four tissue components

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K. T SUZUKI et al.

even when its maximum likelihood is very small. However,such a region should be deﬁned as ‘‘an unclassiﬁed region’’by setting a threshold for the maximum likelihood. In thisstudy, the threshold for the likelihood was set by evaluating
the variance in the ultrasonic measurement of radial strain.
Tissue classiﬁcation was much improved by the proposed
thresholding in comparison with that in the previous study.

frequency [%]

(a)
4

2

Methods

ðk;lÞ2Rm;nwhere pi ðEk;l Þ is the probability density that elasticity value
Ek;l in the k-th row and l-th column in the Rm;n belongs to the

0

2
4
6
elastic modulus [MPa]8(b)elasticity

low

high

. . .
. . .

1 2 3 4 5 6Ji

elasticity number
40
probability density: p (Bi )Tissue classiﬁcation based on elasticity distribution
measured by ultrasoundThe elasticity of the arterial wall is deﬁned as the tissue
strain calibrated by the average stress of the entire wall
thickness, namely, the circumferential elastic modulus Eh .22)
The strain distribution is obtained by applying the phased
tracking method to the measured demodulated signals.20,26)
In this study, each pixel in an elasticity image wasclassiﬁed into one of ﬁve categories, lipids, blood clots,
ﬁbrous tissue, calciﬁed tissue, or unclassiﬁed, using the
likelihood function Li (i ¼ 1: lipids, 2: blood clots, 3: ﬁbrous
tissue, 4: calciﬁed tissue) of the elasticity distribution in thesmall region around the pixel. To obtain the likelihood
function Li , the elasticity distribution of the i-th tissue wastranslated into the normal distribution to describe theprobability distribution using the mean and the standard
deviation.24)
For example, the elasticity distribution of ﬁbrous tissue
was translated into the normal distribution as shown in Fig. 1.
From in-vitro experiments, the elasticity distribution of eachtissue i was obtained as illustrated in Fig. 1(a). The elasticity
distribution of the i-th tissue is composed of Ji data points
with the respective elastic moduli. Using all data of Ji points(J1 : 228, J2 : 179, J3 : 19,121, J4 : 1,101) with the respective
elastic moduli, the ascending sequence was constructed fortissue i as shown in Fig. 1(b). In this sequence, the j-th data
point (j ¼ 1; 2; . . . ; Ji ) has the corresponding elastic modulus
Ej (Ej  Ejþ1 ), where j is the elasticity number. The
probability distribution of each tissue was obtained byallocating all the data of Ji points of each tissue i to boxes of
the normal distribution. The box numbers, Bi , of the normal
distribution were determined so that the number of data
points in the box at each end was only one. As shown in
Fig. 1(c), the number of data points, Di;h (h ¼ 1; 2; . . . ; Bi )included in box Bi was determined to follow the proﬁle of thenormal distribution. Thus, the ðJi =2Þ-th data point was
included in the box with the highest probability. By
allocating all the data of Ji points of each tissue to boxes of
the corresponding normal distribution, the mean elasticity ofthe data included in each box was obtained.
As shown in Fig. 2, an ROI was assigned to an elasticity
image which was obtained by ultrasonic measurement. The
likelihood function Li ðm; nÞ is deﬁned as a joint probability
that all the elasticity values in ROI Rm;n (center of ROI: n-th
sampled point along m-th beam) simultaneously belong in
the i-th category as follows:!1=N0
YLi ðm; nÞ ¼
pi ðEk;l Þ; ði ¼ 1; 2; 3; 4Þ ð2:1Þ

0

2.1

(c)

80

30

60

20

40

10

20

0
0.06

...

...
0.37
2.62
mean of elasticity [MPa]

...

1

...

...
100
box number: Bi

...

200

number of data: DBi

2.

6

0

Fig. 1. (a) Original elasticity distribution of the tissue. (b) Ascending
sequence of elastic modulus in an elasticity distribution. (c) Normal
distribution. The number of boxes in which depends on the number ofdata points in (a).

mkelasticity image

n
l

3 mm

ROI R m,m

0

0.5

1

1.5

2

3

4

elasticity [MPa]Fig. 2. (Color online) Illustration of a ROI.

i-th tissue category, and N0 denotes the number of pixels in
Rm;n . The number of pixels in the ROI near the boundary of
the elasticity image is less than that in the ROI which is
suﬃciently away from the boundary. Thus, the geometricmean 1=N0 is used to compensate for the size N0 of the ROI.
The pixel at the center of the ROI is classiﬁed into the classwhich has the maximum likelihood.
In this classiﬁcation, there may be a region which has anextremely small value for the maximum likelihood. Suchregions are classiﬁed into the unclassiﬁed region by setting
the threshold To;i ðRm;n Þ to the maximum likelihood. Thus,
the category CðRm;n Þ, to which an ROI Rm;n belongs, isexpressed as follows:

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Jpn. J. Appl. Phys., Vol. 47, No. 5 (2008)

pressure
sensor

K. T SUZUKI et al.

ultrasonicprobepressure
sensorflow pumpflow pump

phantomwater tankwater tankFig. 3. Schematic diagram of the phantom experimental system.

(
CðRm;n Þ ¼

arg max Li ðm; nÞ if max Li ðm; nÞ  To;i ðRm;n Þ,
1i4

1i4

unclassiﬁed

otherwise.

3.

Results

3.1

2.2

Phantom experiment for evaluation of variance inmeasurement of radial strain
Figure 3 shows the experimental setup for the phantom
experiment. A homogeneous cylindrical phantom (internal
radius: ri ¼ 4 mm, external radius: ro ¼ 5 mm, elastic
modulus: E ¼ 750 kPa) made from silicone rubber containing 5% carbon powder by weight was measured with a
7.5 MHz linear ultrasonic probe. The scan plane was parallelto the longitudinal direction of the phantom, and the
directions of all ultrasonic beams coincided with the radial
direction of the phantom. The change in internal pressure,which was applied using a ﬂow pump, was measured by apressure sensor. The sampling frequency of the quadrature
demodulated signal and the frame rate are 10 MHz and
200 Hz, respectively. The theoretical value of radial strain
"r ðrÞ at each radial position r is obtained by27)
3
ri2 ro2p;"r ðrÞ ¼ 
2 ðro2  ri2 Þr 2 E

rubbermembraneFig. 4. Schematic diagram of the in-vitro experimental system.

ð2:2Þ
In this study, arterial tissue was classiﬁed into one of fourclasses, lipids, blood clots, ﬁbrous tissue, and calciﬁed
tissue. A normal arterial wall is composed mainly of smoothmuscle and collagen. Atherosclerosis changes the contentsof these tissue components, and it is diﬃcult to determine
what contents of these components are abnormal. Therefore,
in this study, a tissue composed of smooth muscle andcollagen was deﬁned as a class, namely, ﬁbrous tissue.

ð2:3Þ

where p is the pressure increment. The radial strain
measured by the phased tracking method was compared withthe theoretical value "r ðrÞ.
2.3

arteryIn-vitro experiment for tissue classiﬁcation of arterial
wall
Figure 4 shows a schematic diagram of the measurement
system. The change in pressure inside the artery was realizedby circulating ﬂuid using a ﬂow pump. The ﬂuid inside theartery and that circulating in the ﬂow pump were separated
by a rubber membrane to prevent the ﬂow pump from being
contaminated, and only the change in internal pressure waspropagated to the inside of the artery. The change in internalpressure was measured by a pressure transducer (Camino
Model 110-4).
In ultrasonic measurements, excised arteries were measured with a conventional 7.5 MHz linear ultrasonic probe(Toshiba SSH-140A). The quadrature demodulated signalsof RF echoes were acquired at 10 MHz at a frame rate of
200 Hz.

Phantom experiments for evaluation of variance instrain measurement
Figures 5(a)–5(d) show the strain distribution along theradial direction of the phantom obtained by the phased
tracking method for cases of pressure increments p ¼ 40,
50, 60, and 70 mmHg, respectively. Plots and vertical bars
show means and standard deviations for 60 ultrasonicbeams. The standard deviation and the diﬀerence between
the mean and the theoretical value at each applied pressure
increment are shown in Table I. In Table I, the variancein measurement of radial strain was almost constant evenwhen the magnitude of the pressure increment p varied.
Moreover, the diﬀerence between the mean and thetheoretical value is lower than the standard deviation, andmean value follows the theoretical proﬁle.
Figure 6 shows the relationship between the mean s andthe standard deviation s normalized by the mean value. The
plots and the straight line show all the measured results in
Fig. 5 and the regression line, respectively. A positive
correlation was found between the mean s and thenormalized standard deviation s because the standard
deviation was almost constant over every pressure increment. The regression line was determined as follows:s ¼ 0:071s þ 0:40:

ð3:1Þ

3.2 Setting threshold in tissue classiﬁcation
Figure 7 shows the elasticity distribution of each tissue;that is, the frequency of the elasticity values which belong to
the range is deﬁned by the position and width of each verticalbar. The width of a vertical bar was set at every 50 kPa.
Means and the standard deviations are 89  47 (lipids),
131  56 (blood clots), 1;022  1;040 (ﬁbrous tissue), and2;267  1;228 kPa (calciﬁed tissue). Although similarities
were found in the elasticity distributions of lipids and bloodclots and in those of ﬁbrous and calciﬁed tissues, there are
diﬀerences in the elasticity distributions of these tissues.
Figure 8 shows the probability density of each tissue
obtained by the axis transformation of the elasticity
distribution using the process described in §2.1. As shownin these ﬁgures, the horizontal axis of the elastic modulus isnonlinear. Using these databases, each pixel in an elasticity
image was classiﬁed as a certain tissue component.
The threshold To;i ðRm;n Þ for likelihood function Li ðm; nÞof tissue i of each ROI Rm;n is determined using theexperimental results with the phantom. Using eq. (3.1), thenormalized standard deviation s ðRm;n Þ in the measurementof the radial strain in Rm;n was given approximately by:4182

s ðRm;n Þ ¼ 0:071 þ 0:40;

ð3:2Þ

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K. T SUZUKI et al.

0
(a)

(b)

radial strain [%]

–1
–2
–3
theoretical–4

theoretical

–5
–6
–7
0
(c)

(d)

radial strain [%]

–1
–2
–3
–4
theoretical–5

theoretical

–6
–7

0

0.25
0.5
0.75
distance from luminal surface [mm]

1.0

0

0.25
0.5
0.75
distance from luminal surface [mm]1.0

Fig. 5. Measured radial strain distribution along the radial axis of a cylindrical phantom: p = (a) 40, (b) 50, (c) 60, and(d) 70 mmHg.

Pressure p(mmHg)

Standard deviation(%)

Diﬀerence
(%)

40

0.47

0.19

50

0.49

0.35

60

0.45

0.23

70

0.54

0.29

1 X
"max ðk; lÞ;
N0 ðk;lÞ2Rm;n

 ¼

ð3:3Þ

where "max ðk; lÞ and  are the maximum radial strain during
one cardiac cycle in the k-th row and l-th column and theaverage of the maximum radial strain in Rm;n , respectively.
Figure 9 shows an illustration for determining the threshold To;i ðRm;n Þ for the likelihood function Li ðm; nÞ of tissue i.
Let us consider the elasticity distribution in an ROI.
bk;l in the
Variance 02 ðRm;n Þ of the measured elasticity E
k-th row and l-th column in Rm;n is expressed as follows:
"
2 #
2
bk;l  E ½E
bk;l 
E
0 ðRm;n Þ ¼ E
ðk;lÞ2Rm;n¼¼

Eðk;lÞ2Rm;nðk;lÞ2Rm;n2 bk;l  E
bRm;n
E


E

ðk;lÞ2Rm;n

h

0.4

0.3

Δp=40 mmHg
Δp=50 mmHg
Δp=60 mmHg
Δp=70 mmHg
approximate

0.2

0.1

0
–5

–4
–3
–2
mean of radial strain μ s [%]Fig. 6. Relationship between the mean and the normalized standard
deviation s of radial strain.

elasticity values fEk;l g are not constant in Rm;n due tothe elasticity inhomogeneity of tissue even when there isbk;l g isno measurement error, the measured elasticity fE
described by the sum of true elasticity fEk;l g and the errorfEk;l g (sum of random error fek;l g and bias error bRm;n ) as
follows:
bk;l ¼ Ek;l þ Ek;l : ðk; lÞ 2 Rm;nE
ð3:5Þðk;lÞ2Rm;n

ð3:4Þ

¼

where Eðk;lÞ2Rm;n ½ shows the averaging for the data included
bRm;n is the average of the measured
in the ROI Rm;n , and E
bk;l g in Rm;n . By assuming that the trueelasticity values fE

¼

¼

E

ðk;lÞ2Rm;n–1

bRm;n of the measured elasticity values fE
bk;l gThus, the mean E
is given bybRm;n ¼ E ½E
bk;l 
E


2 
2
bk;l
bRm;n þ EbRm;nbk;l  E
E
 2E

i 
2
2
bk;l
bRm;n ;
E
 E

standard deviation σs of radial strain
normalized by mean value

Table I. The standard deviation of radial strain and average of thediﬀerence between the mean and the theoretical value.

4183

E

½Ek;l þ Ek;l E½Ek;l  þ bRm;nðk;lÞ2Rm;nðk;lÞ2Rm;n¼ ERm;n þ bRm;n ;ð3:6Þ

Jpn. J. Appl. Phys., Vol. 47, No. 5 (2008)

K. T SUZUKI et al.

6

60

40

20

0

0

0.2

0.4
0.6
elastic modulus [MPa]

0.8

4

2

0

1

(b)

0

2

20

10

8

4
6
elastic modulus [MPa](d)4frequency [%]30
frequency [%](c)frequency [%]frequency [%]

(a)

3
2
1

0

0

0.2

0.4
0.6
elastic modulus [MPa]

0.8

0

1

0

2

4
6
elastic modulus [MPa]8Fig. 7. Elasticity distribution of each tissue. (a) Lipids (N ¼ 288). (b) Blood clots (N ¼ 178). (c) Fibrous tissue (N ¼ 19;120).
(d) Calciﬁed tissue (N ¼ 1;101).

40

40
(a)

30
25
20
15
10
5
0

1

2

24

34

4
5
6
7
8
9
box number
40 56 76 108 149 205 232
mean elasticity [kPa]

25
20
15
10
5
0

50

0.06

0.17

100
150
200
box number
0.37
1.07
2.62
mean elasticity [MPa]

250
5.98

40

35

35

(b)
probability density [%]probability density [%]

30

0

3

40

30
25
20
15
10
5
0

(c)

35
probability density [%]probability density [%]

35

1

2

39

47

3

4
5
6
box number
78
128
177
244
mean elasticity [MPa]

25
20
15
10
5
0

7
257

(d)

30

5

10
15
box number
0.92
1.62
3.57
mean elasticity [MPa]

20
6.39

Fig. 8. Probability density for each tissue. (a) Lipids. (b) Blood clots. (c) Fibrous tissue. (d) Calciﬁed tissue.

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K. T SUZUKI et al.

where ERm;n and bRm;n are the true mean elasticity in Rm;n and the bias error bRm;n ¼ Eðk;lÞ2Rm;n ½Ek;l , respectively. By
substituting eq. (3.6), eq. (3.4) is modiﬁed as follows:ðEk;l þ Ek;l Þ2  ðERm;n þ bRm;n Þ2
02 ðRm;n Þ ¼ E
ðk;lÞ2Rm;n

¼
¼
¼
¼

E

2
½Ek;l
þ2

E

2
½Ek;l
þ2

E

2
½Ek;l
þ2

E

2
½Ek;l
þ2

ðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nþ 2bRm;nE

ðk;lÞ2Rm;nE½Ek;l  Ek;l  þE½Ek;l  ek;l  þ 2bRm;nE½Ek;l  ek;l  þ 2bRm;nE½Ek;l  ek;l  þ 2bRm;nðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nE

ðk;lÞ2Rm;n2½Ek;l  ðERm;n þ bRm;n Þ2E½Ek;l  þE

½Ek;l  þE

½Ek;l  þ

ðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nE

2½Ek;l  ðERm;n þ bRm;n Þ2ðek;l þ bRm;n Þ2  ðERm;n þ bRm;n Þ2E½e2k;l Eðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;n½ek;l  þ b2Rm;n  ðERm;n þ bRm;n Þ2


2
2
½Ek;l
  ERm;n þ Eðek;l  eRm;n Þ2 þ 2 E ½Ek;l  ek;l  þ 2bRm;n E ½ek;l ðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;n
"
2 #

¼ E
Ek;l  E ½Ek;l ðek;l  eRm;n Þ2 þ 2 E ½Ek;l  ek;l  þ 2bRm;n
þ E

¼

E

ðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nðk;lÞ2Rm;nE

ðk;lÞ2Rm;n½ek;l ;

ð3:7Þwhere eRm;n is assumed to be zero because fek;l g is random error. Variance ðe ðRm;n ÞÞ2 of the measurement error fek;l g in Rm;n isbRm;n Þ2 , where s ðRm;n Þ is determined by eq. (3.2). By deﬁning the variance due to theexpressed by ðe ðRm;n ÞÞ2 ¼ ðs ðRm;n Þ  E
elastic inhomogeneity of tissue in Rm;n by ðRm;n Þ2 , eq. (3.7) is expressed as follows:2bk;l þ2 E ½Ek;l  ek;l ;02 ðRm;n Þ ¼ ððRm;n ÞÞ2 þ s ðRm;n Þ  E
ð3:8Þ
ðk;lÞ2Rm;nwhere there is no correlation between the inhomogeneity of Ek;l and fek;l g in Rm;n . Therefore, variance ððRm;n ÞÞ2 without error
bk;l g in Rm;n asbRm;n g2 subtracted from the variance 02 of the measured elasticity values fEof eq. (3.8) is given by fs ðRm;n ÞE
follows:no2bRm;n :
ððRm;n ÞÞ2 ¼ 02  s ðRm;n ÞEð3:9Þ
Based on this relation, by setting the acceptance region [from i  ai ðRm;n Þ to i þ ai ðRm;n Þ] of the probability densityNði ; i2 ðRm;n ÞÞ for each tissue i, the threshold To;i ðRm;n Þ for the likelihood function Li ðm; nÞ of tissue i in Rm;n was determined2by the probability of the distribution Nði ; 0;i
ðRm;n ÞÞ with the error at the distance from mean i as shown in Fig. 9.
Therefore, the threshold To;i ðRm;n Þ for the likelihood function Li ðm; nÞ of tissue i is given by2! 3
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
o2 2
bRm;n
6
7
a 02 ðRm;n Þ  s ðRm;n ÞE
6
7
1
4
5
;
ð3:10Þ
exp To;i ðRm;n Þ ¼ pﬃﬃﬃﬃﬃﬃ
220;i20;iwhere 0;i is the variance of measured elasticity distribution
for tissue i.
Let us deﬁne the ratio of the area of the acceptance region
i  ai ðRm;n Þ in the normal distribution Nði ; i2 ðRm;n ÞÞ to

0.4

2
N (μi, σ 0 )with error

N (μi, σ 2= σ0 -(σs(R m,n)ERm,n) )without error
2

0.3
probability

the entire area of Nði ; i2 ðRm;n ÞÞ by w. Figure 10 shows anexample of a tissue classiﬁcation image that is estimatedmanually by referring to its pathologic image. By comparing
the pathology-based classiﬁcation image shown in Fig. 10
with the tissue classiﬁcation image obtained by the proposed
method, the number Nc ðwÞ of correctly classiﬁed pixels is
obtained. The recognition rate Rr ðwÞ at w is expressed asfollows:2

lipids0.2

To,i (Rm,n)3 mm

blood clots

0.1
acceptance region0μi – 3σ0 μi – 2σ0 μi – 1σ0μiμi – a (σ0 – {σs(Rm,n)ERm,n} )22μi + 1σ0 μi + 2σ0 μi + 3σ0

iliac artery (A)μi + a (σ02 –{σs(Rm,n)ERm,n}2)

Fig. 9. Illustration for determining the threshold To;i ðRm;n Þ for the likelihood function Li ðm; nÞ.

fibroustissuecalcifiedtissueunclassified

Fig. 10. (Color online) Tissue classiﬁcation image obtained by referringto the pathologic image.

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Rr ðwÞ ¼

Nc ðwÞ
Nall ðwÞ

K. T SUZUKI et al.

100 ð%Þ;ð3:11Þ

where Nall ðwÞ is the number of all pixels classiﬁed into aclass except for an unclassiﬁed pixel. Figure 11 shows therelationship between the area ratio w of the acceptanceregion and the recognition rate Rr ðwÞ obtained using eightiliac arteries and nine femoral arteries. By setting thethreshold To;i ðSROI Þ (w  99%), the recognition rate Rr ðwÞ
was increased compared with that at w ¼ 100% (without

recognition rate Rr (w) [%]

58
57
56
55
54
53
52
51
50
100

95
90
85
area ratio w of acceptance region [%]Fig. 11. Relationship between the area ratio w of the acceptance region
and the recognition rate Rr ðwÞ.

(a)lipids(d)

3 mm

fibrous tissue

elastic modulus

4 MPa
2
1.5
1
0.5
0

blood
clots

pathologic image(b)

thresholding) and the recognition rate Rr ðwÞ slightly wasincreased by decreasing the area ratio w of the acceptanceregion at the expense of the number of classiﬁed pixels.
Figures 12(c)–12(g) show the tissue classiﬁcation images
for an excised human iliac artery (A). The regions classiﬁed
as lipids, blood clots, ﬁbrous tissue, and calciﬁed tissue arecolor-coded yellow, red, blue, and purple, respectively.
Figure 12(c) shows the classiﬁcation image obtained with an
ROI size of 5 20 pixels (1;500 1;500 mm2 ) without
thresholding. Although the pixels were roughly classiﬁed
into the correct tissues, the pixels which have low likelihood
Li ðm; nÞ were also classiﬁed. Figures 12(d)–12(g) show theclassiﬁcation images obtained with thresholds To;i ðRm;n Þdetermined at four diﬀerent area ratios of w ¼ 99, 95, 90,
and 85%, respectively. In Figs. 12(d)–12(g), a region whose
likelihood is less than the threshold To;i ðRm;n Þ for all tissue
components is colored gray. As shown in Figs. 12(d)–12(g),similarities among the classiﬁcation images seem to bequalitatively high especially for those at an area ratio w lessthan or equal to 95%. It can be expected from the result
shown in Fig. 11 that improvement in the recognition rate
Rr ðwÞ by decreasing area ratio w slightly diminish at area
ratio w less than 95%. For another specimen [iliac artery(B)], some pixels were misclassiﬁed into blood clots or
calciﬁed tissue when the threshold To;i ðRm;n Þ was not set,
as shown in Fig. 13(c). In Figs. 13(d)–13(g), by settingw  95%, the pixels misclassiﬁed in the blood clots wereclassiﬁed into unclassiﬁed region. For these results, the

3 mmlipidsblood clots

classification image(acceptance region: w =99%)

fibroustissue(e)calcifiedtissue3 mm

3 mm

unclassified

elasticity image

classification image(acceptance region: w =95%)(f)(c) 3 mmclassification image(without thresholding)

3 mmclassification image(acceptance region: w =90%)
(g)

3 mm

classification image(acceptance region: w =85%)

Fig. 12. (Color online) Iliac artery (A): (a) Pathologic image subjected to elastica-Masson staining. (b) Elasticity image. (c) Tissueclassiﬁcation image without thresholding. (d) Tissue classiﬁcation image (w ¼ 99%). (e) Tissue classiﬁcation image (w ¼ 95%).
(f) Tissue classiﬁcation image (w ¼ 90%). (g) Tissue classiﬁcation image (w ¼ 85%).

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K. T SUZUKI et al.

(a)

(d)

3 mm

3 mm
lipidsblood clots

fibrous tissue

elastic modulus

4 MPa
2
1.5
1
0.5
0

classification image(acceptance region: w =99%)

pathologic image
(b)

3 mm

(e)

3 mm

fibroustissuecalcifiedtissueunclassified

elasticity image(c)classification image(acceptance region: w =95%)

3 mm

(f)

3 mm

classification image(acceptance region: w =90%)

classification image(without thresholding)

(g)

3 mm

classification image(acceptance region: w =85%)

Fig. 13. (Color online) Iliac artery (B): (a) Pathologic image subjected to elastica-Masson staining. (b) Elasticity image. (c) Tissueclassiﬁcation image without thresholding. (d) Tissue classiﬁcation image (w ¼ 99%). (e) Tissue classiﬁcation image (w ¼ 95%).
(f) Tissue classiﬁcation image (w ¼ 90%). (g) Tissue classiﬁcation image (w ¼ 85%).

misclassiﬁcation was reduced using the threshold To;i ðRm;n Þdetermined from the area ratio of the acceptance region lessthan or equal to 95%.
4.

Conclusions

In this study, variance in measured radial strain was
evaluated to set the threshold To;i ðRm;n Þ for the likelihood
Li ðm; nÞ used in elasticity-based tissue classiﬁcation of thearterial wall. The misclassiﬁcation was reduced using thethreshold To;i ðRm;n Þ determined by setting the area ratio of
the acceptance region less than or equal to 95%.

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