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REGULAR PAPER

Accurate measurement of elasticity of the radial
artery wall considering changes in cross-sectional
shape of artery caused by pushing pressure
applied by ultrasound probe
To cite this article: Yuto Shoji et al 2022 Jpn. J. Appl. Phys. 61 SG1042

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REGULAR PAPER

Japanese Journal of Applied Physics 61, SG1042 (2022)
https://doi.org/10.35848/1347-4065/ac4e4a

Accurate measurement of elasticity of the radial artery wall considering changes
in cross-sectional shape of artery caused by pushing pressure applied byultrasound probe
Yuto Shoji1, Shohei Mori2 , Mototaka Arakawa1,2* , Shigeo Ohba2, Kazuto Kobayashi3, and Hiroshi Kanai2,1
1

Graduated School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduated School of Engineering, Tohoku University, Sendai 987-8579, Japan3
Honda Electronics Co. Ltd., Toyohashi 441-3193, Japan2*

E-mail: arakawa@ecei.tohoku.ac.jp

Received November 2, 2021; revised January 6, 2022; accepted January 24, 2022; published online May 27, 2022
For the early diagnosis of atherosclerosis, our group developed an ultrasound probe that can simultaneously measure blood pressure and vessel
diameter at the same position. However, because the developed probe requires the blood vessel to be deformed by pushing to measure the bloodpressure, it affects the estimation of the vessel’s elastic modulus. In the present study, we derived a series of equations to estimate the elastic
modulus of a blood vessel considering the pushing pressure applied by the ultrasound probe and the resultant deformation of the blood vessel. The
validity of the proposed method was verified by numerical calculations, and then the method was applied to in vivo measurements. The proposed
method resulted in fewer variations in the elastic modulus estimates with different pushing pressures compared with the conventional method.
© 2022 The Japan Society of Applied Physics

1. Introduction

Cardiovascular diseases, including ischemic heart disease andstroke, are the leading cause of death worldwide.1) The maincause of cardiovascular diseases is atherosclerosis. It has
been clinically diagnosed using several techniques, including
intravascular ultrasound2) and X-ray imaging.3) However,these techniques are not noninvasive. Alternatively, pulse
wave velocity,4) ankle-brachial index,5) cardio-ankle-brachialindex,6) stiffness parameter,7) and elasticity8) have been
proposed as noninvasive diagnostic methods for atherosclerosis. However, these methods can only be applied todiagnose advanced-stage atherosclerosis, which is irreversible. At the early stage of atherosclerosis, vascular endothelial dysfunction occurs, and it is possible to be completely
cured of atherosclerosis by medication and/or lifestyle
improvement.9) Therefore, early diagnosis is important.
Plethysmography10,11) and the ﬂow-mediated dilatation(FMD) test12–14) are the main clinical methods used toevaluate vascular endothelial function. In plethysmography,
vascular endothelial function is evaluated by measuring thechanges in blood ﬂow caused by injecting a vasodilator suchas acetylcholine into the blood vessel. In the FMD test,
vascular endothelial function is evaluated by ultrasonicallymeasuring the dilatation ratio %FMD of the arterial diameter
after releasing the avascularization to that at rest. As the FMD
test is simple and non-invasive, it has been widely adopted by,for example, the Framingham Heart Study, and the clinicaldata on the relationship between %FMD and various risk
factors have been accumulated worldwide.15,16) However, thedistance resolution of the ultrasound measurement is 0.15 mmwith a typical 10 MHz ultrasound probe. For healthy subjects,
the %FMD is about 6% or more, which corresponds to only
0.24–0.30 mm when the diameter of the vessel is 4–5 mm.
Accurate evaluation of the %FMD is difﬁcult.
Furthermore, a change in the diameter of an artery due toendothelial function is caused by changes in the mechanicalproperties of the arterial wall; however, these changes cannot
be used to directly evaluate endothelial function. A method
for evaluating endothelial function by analyzing the pulse

wave waveforms in the radial artery has also beenstudied,17,18) referring to a method to estimate stiffness
parameters from the pulse wave analysis.19) However, thismethod also indirectly evaluates endothelial function because
it evaluates the features of the pulse wave waveform.
In the FMD response, an increase in blood ﬂow produces
the vasodilator nitric oxide (NO), which decreases theelasticity of the arterial wall and increases the arterial
diameter.20,21) If the decrease in the vascular elasticity causedby NO can be noninvasively measured, direct evaluation ofvascular endothelial function can be achieved.
Our group has focused on ultrasonic measurement of therelationship between the changes in the strain of the vessel
wall and the blood pressure during one heartbeat to estimate
the changes in the elastic and viscous moduli during
FMD.22–25) However, accurate viscoelasticity measurements
were difﬁcult because the measurement positions of thevessel diameter for the calculation of the strain and thepressure waveform were different.
Sakai et al. proposed a method for measuring the bloodpressure waveform and the strain of the vessel wall at the same
position by a correction determined from the delay between thepressure waveforms measured by two pressure sensors, where
the ultrasound probe for the strain measurement was placed at
the center of the two pressure sensors.26,27) However, it was
difﬁcult to accurately determine the delay between the twopressure waveforms because it depends on the instantaneouspressure of the pressure waveforms.
Therefore, our group has developed an ultrasound probethat can simultaneously measure the blood pressure waveform and the change in the vascular diameter at the same
position.28,29) Furthermore, we have proposed a method formeasuring viscoelasticity over time by calibrating the bloodpressure waveform at rest in advance,30) and have measured
the changes in viscoelasticity at the radial artery during theFMD.31) However, the voltage output from the center
element of the probe depends on the pressure applied by
the ultrasound probe during the measurement.
Therefore, we introduced the pulse transit time method todetermine the blood pressure, as it is not affected by theSG1042-1© 2022 The Japan Society of Applied Physics

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Y. Shoji et al.

absolute value of the acquired voltage output, and a stable
blood pressure measurement was achieved with it during theFMD.32) However, the changes in the cross-sectional shapeof the vessel caused by the pushing pressure applied by theultrasound probe still had an effect on the viscoelasticity
estimation.
Therefore, to consider the effect of the deformation of thearterial cross-section from a circular to an elliptical shapecaused by the pushing of the ultrasound probe, we derived aseries of equations to estimate the elastic modulus33) based on
the derivation of the equations for estimating the elastic
modulus of the artery with a circular shape proposed by
Hasegawa et al.34)
In the present study, we applied the derived equations toestimate the elastic modulus using in vivo measurements.
The elastic modulus was estimated by measuring the changes
in the vessel diameter and blood pressure in the radial arteryduring one heartbeat when different pushing pressures wereapplied by the ultrasound probe. From the experiments
conducted with a healthy subject, it was conﬁrmed that theestimates of the elastic modulus obtained using the derived
equations were similar regardless of the change in the crosssectional shape of the vessel caused by the pushing pressure
applied by the ultrasound probe.

Fig. 1. (Color online) Schematic of the cylindrical coordinate system in a
blood vessel.

Dsr
Dsq
Dsz
+
,
2E
E
2E

(5 )

Dsr
Dsq
Dsz
+
.
2E
2E
E

(6 )

1
(Dsr + Dsq ).
2(7 )Deq = 0=-From Eq. (6),Dsz =By rearranging Eqs. (4) and (5) for E and substituting Eq. (7)into them, we obtainE=

3(Dsr - Dsq ) ,
4Der

(8 )

2. Experimental methods

The pushing by the probe to measure the blood pressureusing the center element of the developed ultrasound probe
causes deformation of the cross-sectional shape of the bloodvessel.28,29) By assuming that the cross-sectional shape of thevessel after deformation is elliptical, a series of equations thatcan be used to estimate the elastic modulus under the pushingby the ultrasound probe was derived. In general, the relationships among the elastic modulus E, incremental stress Ds,
and incremental strain De of viscoelastic material in acylindrical coordinate system are given by:35)
Dsr
n Dsq
n Dsz
,
Er
Eq
Ez

(1 )

Deq = -

n Dsr
Dsq
n Dsz
,
+
Er
Eq
Ez

(2 )

Dez = -

n Dsr
n Dsq
Dsz
,
+
Er
Eq
Ez

(3 )

Der =

where r, q, and z are the direction of the vessel wall, thecircumferential direction of the vessel wall, and the axial
direction of the vessel, respectively, as shown in Fig. 1, and n
is Poisson’s ratio. As the vessel is strongly constrained in theaxial direction in vivo, the incremental axial strain Dsz can beignored. In the present study, the vessel wall was assumed to
be isotropic, that is, E = Er = Eq = Ez. Moreover, assumingthat the strain caused by the heartbeat is incompressible(n = 1 /2), Eqs. (1)–(3) can be rewritten as
Dsr
Dsq
Dsz
,
Der =
E
2E
2E

3
(Dsq - Dsr ).
(9 )
4Deq
Thus, the elastic modulus E can be estimated from Der or
Deq , Dsr , and Dsq.
Let us now consider the case in which the radial artery is
deformed by a change in internal pressure p1 due to a heartbeat
under the pushing pressure p3 applied by the ultrasound probe.
The cross-sectional shape of the artery is assumed to deform froma circular to an elliptical shape with a long radius b and a shortradius a because of the uniform pressure p3 acting on the uppersurface of the artery in the y direction, as shown in Fig. 2(a).
Figure 2(b) shows a magniﬁed view of a small region of thevessel, where p2 is the atmospheric pressure, T is the tension
acting along the circumferential direction f of the wall, r is thecurvature radius of the region of the vessel, and h is the thicknessof the vessel wall. The intravascular pressure p1 is assumed to
be uniformly applied regardless of f. The thickness h of thevessel wall was assumed to be constant during the transformationof the cross-sectional shape of the vessel from a circle to an
ellipse. The tension T acting along the circumferential directionof the wall is not uniform because the cross-sectional shape of thevessel is an ellipse, and it changes even for the region.
From the balance among the forces acting on the region in
the x direction and the y direction, Eqs. (10) and (11) are
obtained as follows:(4 )SG1042-2

E=

T sin (f - df) - (T + dT ) sin (f + df)+ p1 dL1cosf - p2 dL 2 cosf = 0,(10)

- T cos (f - df) + (T + dT ) cos (f + df)+ p1 dL1sinf - p2 dL 2 sinf - p3 dL 2 sinf = 0,

(11)

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Y. Shoji et al.

(a)

{ (

dT cos f 1 +df 2
2) - df sin f}- 2df sin f {T + p1 r - ( p2 + p3 )(r + h)} = 0.

(17)

By approximating these equations to ﬁrst-order equations interms of df and dT , and rearranging them in terms of dT , weobtaindT = -

2df{T - p1 r - p2 (r + h)},
tanf

(18)

dT = 2df tanf {T + p1 r - ( p2 - p3 )(r + h)}.

(19)

Eliminating dT by substituting Eq. (19) into Eq. (18) andrearranging in terms of T gives(b)tan2 ftan2 f
T = r ⎛⎜ p1 - p2 p3 ⎞⎟ - h ⎛⎜ p2 +
p ⎞⎟.
2
2f 3
1
+
tan
f
1
+
tan
⎝
⎠
⎝
⎠
(20)

Assuming that the circumferential stress is uniform along thethrough-wall direction, the circumferential stress sq is obtained by dividing the tension T of Eq. (20) by the wall
thickness h:
sq =

2
r⎛
tan2 f⎞ - ⎛ p + tan f p ⎞.
p1 - p 2 p323h⎝
1 + tan2 f ⎠ ⎝1 + tan2 f ⎠
⎜

⎟

⎜

⎟

(21)

Fig. 2. (Color online) Cross-section of the blood vessel deformed into an
ellipse by a uniform pressure. (a) Schematic of the cross-section of a bloodvessel and (b) magniﬁed view of the small portion of the blood vessel wall.

where dL1 and dL 2 are the lengths of PP′ and QQ′,
respectively, and are given by
dL1 = 2rdf ,(12)

dL 2 = 2 (r + h) df.

(13)

By substituting Eqs. (12) and (13) into Eqs. (10) and (11), we
obtainT sin (f - df) - (T + dT ) sin (f + df)+ 2p1 rdf cos f - 2p2 (r + h) df cos f = 0,(14)

- T cos (f - df) + (T + dT ) cos (f + df)+ 2p1 rdf sinf - 2p2 (r + h) df sinf
- 2p3 (r + h) df sinf = 0.

(15)

By expanding these equations using the additive theorem oftrigonometricfunctionsandtheapproximations
sin df = df and cos df = 1 - df 2 /2, where df is assumed
to be minute, we obtain{ (- dT sin f 1 -df 2
2) - df cos f}- 2df cos f {T - p1 r - p2 (r + h)} = 0,(16)

On the other hand, because the through-wall stress sr equals-p1 at PP′ in the inner wall and it equals the sum of –p2 andthe through-wall component of -p3 at QQ′ in the outerwall, assuming the through-wall stress sr is distributed
uniformly in the direction of the through-wall, then sr isgiven by the average values of p1 and the sum of p2 andp3 cosf considering the balance of forces:1
sr = - ( p1 + p2 + p3 cosf).
2(22)

By rewriting Eqs. (21) and (22) using a , b , and q shown in
Fig. 2(a) instead of the curvature radius r, cosf, and tanf, we
obtain1

sq = abh(

a 2 + b 2 tan2q
1 + tan2q3
2) (p - p -

(

1

2

a2
p
a 2 + b 2 tan2q 3

a2

)

b

tanq

- p2 + a2 + b2 tan2q p3 ,1
sr = - ⎛⎜ p1 + p2 +2⎝

a2 tan2 q + b2)
(23)

p3 ⎞⎟.
⎠(24)

Thus, sq and sr , which are required for estimating the elastic
modulus E, are calculated from a, b, the angle q of the region
from the y axis, p1 , p2 , and the pushing pressure p3 appliedby the ultrasound probe.
By subtracting the stresses sq and sr at diastole from atsystole, the incremental stresses Dsq and Dsr can beobtained. The elastic modulus E is estimated by substituting
Dsq and Dsr , and the circumferential strain Deq measured byultrasound into Eq. (9).

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Jpn. J. Appl. Phys. 61, SG1042 (2022)

Y. Shoji et al.

The process of estimating the elastic modulus in the actual
measurements is now described. To estimate the elastic
modulus, the value of the circumferential strain Deq isrequired. However, Deq cannot be measured directly. Using
the change in the short radius b of the vessel obtained byapplying the phased-tracking method36) to the RF signal of
the beam passing through the center of the vessel, as in theconventional method,30–32) Deq can be obtained as the ratioof the change in circumference to the original circumference:
Deq =2pbsys - 2pbdias
2pbdias=bsys - bdias
bdias

,

(25)

where bsys and bdias denote the short radii at systole anddiastole, respectively, and the small region is a part of a circle
with radii bsys or bdias.
When the strain Deq is deﬁned along the beam passing
through the center of the vessel (q = 0), the stress should
also be calculated at q = 0, and Eqs. (23) and (24) are given
by
sq =

a2
( p1 - p2 - p3 ) - ( p2 + p3 ) ,
bh

(26)

1
sr = - ( p1 + p2 + p3 ).
(27)
2
Therefore, the circumferential stress sq and through-wallstress sr are determined from a , b , h , p1 , p2 , and p3 , eachof which is measured as follows.
Figure 3(a) shows the B-mode image of the radial artery ofa healthy subject. Figure 3(b) shows the normalized enveloped amplitude of the RF signals along the beam passing
through the center of the vessel, as shown by the white line in
Figs. 3(a)–3(c) shows the normalized enveloped amplitude of
the RF signals along the red line in Fig. 3(a). From theseﬁgures, a and b were determined as 2.1 mm and 1.09 mm,
respectively.
The change in b, bsys - bdias, of Eq. (25) was measured byapplying the phased-tracking method36) to the RF signal
along the white line in Fig. 3(a). p1 was measured by thecenter element of the developed ultrasound probe28,29) with acenter frequency of 7.5 MHz. The atmospheric pressure p2
was assumed to be 1013.25 hPa in the present study. p3 wasmeasured by the pressure sensors separately attached to theultrasonic probe, as described below in Fig. 4.
It is difﬁcult to accurately measure the wall thickness h of
the radial artery in vivo by ultrasound at 7.5 MHz owing to
the lack of distance resolution. Therefore, in the present
study, the wall thickness of the subject’s radial artery was
separately measured in advance using another ultrasoundprobe with a center frequency of 40 MHz, andh = 0.235 mm was used for the subject.
3. Experiments
3.1. Validation of the proposed equations by
numerical calculations

We assumed that the pushing pressure p3 was a constantduring one heartbeat because the ultrasound probe was ﬁxed
for the measurement. On the other hand, the shape of theblood vessel changes between systole and diastole, and thestress sq of Eq. (26), which includes a and b, is affected at
q = 0. In the present study, therefore, the derived equations
for the estimation of elasticity during one heartbeat wereconﬁrmed by the following numerical calculations using theactual measured values of the intravascular pressure p1 = 106mmHg, long radius asys = 1.1 mm, and short radiusbsys = 0.81 mm of the radial artery in systole, and
p1 = 55 mmHg, adias = 1.1 mm, and bdias = 0.71 mm indiastole. By assuming that the pressure p3 was greater than
p1 in diastole because the vessel was deformed by theultrasound probe, p3 = 70 mmHg was employed. These
values were substituted into Eq. (26).
First, to examine the relationship between the circumferential stress sq and the position x on the arterial cross-section,
sq at systole and diastole, and the incremental stress Dsq fromdiastole to systole were obtained for various values of q from0 to 180 using Eq. (26). Next, to examine the relationshipbetween the circumferential stress sq and the vessel shape,
Dsq from diastole to systole was obtained for various aspect

(a)

(b)

(c)

Fig. 3. (Color online) Calculation of the short and long radius of the vessel. (a) B-mode image of the radial artery, (b) normalized enveloped amplitude at thebeam through the center of the vascular lumen, and (c) normalized enveloped amplitude at the depth of the center of the vascular lumen.

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Y. Shoji et al.

(a)

(b)

(c)

Fig. 4. (Color online) Schematic of the entire experimental system. (a) Block diagram, (b) photograph around the ultrasonic probe, and (c) arrangement of
the pressure sensors.

ratios a /b of the vessel shape from 1 to 3. Finally, to examine
the relationship between sq and the pushing pressure p3 , Dsq
from diastole to systole was obtained for various pressure
values p3 from 0 to 150 mmHg using Eq. (26), with q = 0.
3.2. In vivo experiment
Figure 4(a) shows a schematic of the entire experimental
system, and Fig. 4(b) shows a photograph of the ultrasoundprobe. The long radius a of the vessel, the short radius b of
the vessel, and the blood pressure p1 were measured in theright radial artery using the developed ultrasound probe.28,29)
The output from the piezoelectric element at the center ofthe ultrasound probe was passed through an ampliﬁer with anampliﬁcation factor of 40 dB and a low-pass ﬁlter with acutoff frequency of 30 Hz, and the blood pressure waveform
was obtained by integrating the waveform. The measurement
position was set such that the center of the probe was justabove the radial artery while viewing the B-mode image. As
the measured blood pressure waveform was output as a
voltage, the absolute value of the blood pressure at systoleand diastole measured by a sphygmomanometer and theabsolute value of the waveform were calibrated by the pulse
transit time method.32) In the ultrasound measurement, thecenter frequency, sampling frequency, and frame rate were
set to 7.5 MHz, 40 MHz, and 252 Hz, respectively. The longand short radii of vessels a and b were measured at systoleand diastole, respectively. Electrocardiograms were also
obtained.
The pushing pressure p3 applied to the radial artery by theultrasound probe was measured using four pressure sensors(Kyowa, PS-1KC). A probe holder consisting of the upperand lower parts that enclosed the ultrasound probe wasfabricated, and four pressure sensors were placed between
the two parts, as shown in Fig. 4(c). The ultrasound probe
was ﬁxed to the lower part of the probe holder, and the upper
part could freely move up and down. From the force applied
to the upper part of the probe holder, the pressure p3 applied
to the ultrasound probe was measured by an oscilloscope via
strain ampliﬁers. p3 was obtained by the sum psum of theforces applied to the four pressure sensors multiplied by thecontact area Ssensors of the pressure sensors and divided by thecontact area Sprobe of the ultrasonic probe attached to the skin
as follows:
p3 =psum · Ssensors,
Sprobe(28)

where Ssensors = 36p ´ 10-6 m2 and Sprobe = 1.53 ´
10-4 m2 in the present study.
A healthy male subject in his early twenties was measured
in a sitting position at rest. The subject was asked to avoid
eating for three hours before the measurement and to rest for
30 min before the measurement to eliminate the effects ofeating and exercise on blood pressure and the elastic modulusof the blood vessel, and to approach the condition in the
FMD test. We obtained the approval of the ethics committeeof our university and informed consent from the subject
before the measurements. The blood pressure waveform andthe blood vessel diameter were measured under threedifferent conditions of various pushing pressures of theSG1042-5© 2022 The Japan Society of Applied Physics

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Y. Shoji et al.

ultrasound probe while the subject was at rest. Data wereacquired for four seconds per measurement.
The proposed method was applied to the estimation of theelastic modulus of a blood vessel after three consecutive
heartbeats using the measured vessel shape, blood pressure,and pushing pressure, and the results were compared withthose obtained with our previous conventional method.30–32)
4. Result and discussion
4.1. Results of the numerical simulations

Figure 5 shows the calculated results of the circumferential
stress sq for systole and diastole, and incremental stress Dsq
from diastole to systole in the red, blue, and green lines,
respectively. Both the circumferential stresses sq at systoleand diastole have a minimum at q = 0 and 180 and the
maximum at q = 90. The incremental stress Dsq has aminimum value at q = 90 and maximum values at q = 0and 180. The q dependence in these results is reasonablebecause the artery expands along the short-axis direction(q = 0 and 180) more than the long-axis direction(q = 90) because of the deformation from the ellipse atdiastole to the circular shape at systole.
Figure 6 shows the calculated results for the shapedependence of sq. As the aspect ratio increased and thevessel cross-section collapsed, the circumferential stress sq
became larger at systole and smaller at diastole, resulting in alarger incremental stress Dsq. The larger the cross-sectional
area of the vessel collapsed, the larger the maximum strain
during one heartbeat; therefore, this calculation result isreasonable.
Figure 7 shows the calculation results for the pushingpressure p3 dependence of sq. As the pushing pressure p3
increased, sq decreased, while its incremental stress Dsq wasalmost unchanged. The larger the pressure p3 , the greater theforce in the direction in which the vessel wall elongated in
the circumferential direction, and it caused greater stress inthe compressive direction to balance it. On the other hand,
Dsq was almost constant when p3 was constant during one
heartbeat. Therefore, we considered that this is the reason
why the elasticity has been stably estimated at rest in ourprevious study because the pushing pressure has been kept
constant at rest.30)
4.2. Results of the in vivo experiments
Pushing pressures p3 of 59.3 mmHg, 64.2 mmHg, and81.1 mmHg were applied in the three measurements. The
pushing pressure was almost constant during each measurement and was not affected by the heartbeats. Therefore, theassumption in Sect. 3.1 was conﬁrmed as reasonable. The
average of the pushing pressures measured by the four
sensors was used to estimate the elastic moduli of the radial
artery. Figure 8 shows the relationship between bloodpressure and strain during one heartbeat between the succeeding R waves of the electrocardiogram for each pushingpressure p3 . The strain increased with an increase in p3 ,which is consistent with the previous results obtained by our
research group.31) From the dashed lines in Fig. 8 obtained byconnecting the most distant points in the hysteresis loops, theelastic modulus E was estimated based on the proposed
method. For the conventional method, the elastic modulus
was estimated as the slope of the line; therefore, the elastic

Fig. 5. (Color online) q dependence of the circumferential stresses sq for
systole, diastole, and incremental stress.

Fig. 6. (Color online) Dependence of the circumferential stresses sq for
systole, diastole, and incremental stress on the aspect ratio a/b of the crosssection of the vessel.

Fig. 7. (Color online) Dependence of the circumferential stresses sq for
systole, diastole, and incremental stress on pushing pressure p3 .

modulus estimated by the conventional method greatly varied
depending on the pushing pressure p3 .
Figures 9(a–i)–9(c–ii) show the B-mode images of the radial
artery for the three different pressures p3 . Figure 9(i) correspondto the diastole, and Fig. 9(ii) correspond to the systole.
Figures 9(a)–9(c) show the results at p3 = 59.3 mmHg,
p3 = 64.2 mmHg, and p3 = 81.1 mmHg, respectively.
Table I shows the measured values of the intravascular pressure
p1 , the pushing pressure p3 , the long radius a and the shortradius b of the vessel for each B-mode image. Ellipses
determined by the obtained a and b values were superimposedon each B-mode image of Fig. 9. Each ellipse was well ﬁtted,

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Y. Shoji et al.

Fig. 8. (Color online) Relationships between the blood pressure and thestrain of the radial artery with different pushing pressures p3 of 59.3, 64.2,
and 81.1 mmHg. The dashed lines connect the most distant points in thehysteresis loops.

Fig. 10. (Color online) Relationships between the stress 3 (sq - sr ) /4 andstrain eq for the proposed method for various values of p3 .

(a)
(a-i)

(a-ii)(b-i)(b-ii)(b)(c-ii)(c-i)

Fig. 9. B-mode images of the radial artery (i) at diastole and (ii) at systole:(a) p3 = 59.3 mmHg, (b) p3 = 64.2 mmHg, and (c) p3 = 81.1 mmHg.
Ellipses drawn based on the obtained a and b were superimposed on thevessel cross-section.
Table I.

Measured results of each parameter for Figs. 9(a–i)–9(c–ii).

Datap1 [mmHg]

p3 [mmHg]

a [mm]b [mm]9(a-i)9(a-ii)9(b-i)
9(b-ii)9(c-i)9(c-ii)

71.6
119.2
72.1
120.0
72.1
119.2

59.3
59.3
64.2
64.2
81.1
81.1

2.25
2.25
2.25
2.25
2.25
2.25

1.47
1.58
1.25
1.13
0.95
1.08

and the assumption of the cross-section of an ellipse wasreasonable.
Figure 10 shows the relationships between the incremental
strain Deq and the corresponding stress 3 (Dsq - Dsr ) /4

Fig. 11. (Color online) Elastic modulus values estimated by the proposedand conventional methods. (a) Absolute values and (b) percentage deviations
from the average elasticity.

of Eq. (9) for the three p3 . The stresses 3 (Dsq - Dsr ) /4
estimated by the proposed method at the systole and diastole
were plotted. It was conﬁrmed that the slopes of the straightlines reﬂecting the elastic moduli were closer to each otherthan those by the conventional method shown in Fig. 8.
Figure 11(a) shows the relationship between the pushingpressure p3 and the averages and standard deviations of theelastic moduli E estimated by the proposed and conventional
methods. The elastic moduli estimated by the conventional
method were small and signiﬁcantly decreased as p3 increased. Figure 11(b) shows the relationship between thepushing pressure p3 and the percentage deviations from theaverages of the estimated elastic moduli obtained by the
proposed and conventional methods. The estimates of E by

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Y. Shoji et al.

the conventional method changed by –36% to +25%
depending on p3 , while those by the proposed methodchanged by –7 to +3%. Therefore, the effect of the pushingpressure on the resulting change in the shape of the vessel
was markedly reduced. In a previous study,26) the decrease inelastic modulus during FMD was approximately 40%.
Therefore, as the elastic modulus in the present study is
considered to be accurate enough, the proposed method issuitable for evaluating the decrease in the elastic modulus
during FMD.
Based on the known biomechanical attributes of the vessel
wall,37) it is expected that the measured elastic modulus
increases as the initial stresses on the vessel wall sr and sq
increase. However, in the present study, the elastic moduli
obtained by both the proposed and conventional methods
were estimated to decrease with increasing pushing pressure,as shown in Fig. 11. The increase in pushing pressure caused
the vessel to collapse, an increase in the incremental stress atthe short axis of the vessel, as shown in Fig. 5, and anincrease in the incremental stress, as shown in Fig. 6. These
increased the change in the short vessel diameter by heartbeat
and increased the strain by heartbeat. Moreover, the conventional method approximates the intravascular pressure as thestress on the vessel wall. This approximation causes a large
dependence of the elastic modulus on the pushing pressure,as shown in Fig. 11. In the proposed method, the pressure
dependence of the estimated elastic modulus was markedly
decreased by calculating the stress while considering thechanges in the vessel shape due to the pushing pressure
applied by the ultrasound probe. In the present study, theshape of the vessel wall was assumed to be part of a circle inthe calculation of the strain. However, because the actual
shape of the vessel wall was an ellipse, as shown in Fig. 9,
depending on the pushing pressure, the calculation of thestrain might have affected the estimation of the elastic
modulus. To solve this problem, it is necessary to further
consider a strain calculation based on the actual shape of thevessel or to estimate the elastic modulus using Eq. (8) tomeasure the strain in the direction of the through-wall using
an ultrasonic probe with a higher resolution.
As shown in Fig. 11, the elastic moduli E estimated by the
proposed method are larger than those of the conventional
method. The circumferential incremental stress Dsq increased
as the aspect ratio a /b increased, as shown in Fig. 6. The
increase in stress due to the change in the cross-sectional
shape of the vessel was not considered in the conventional
method. As a result, Dsq was estimated to be smaller, as
shown in Fig. 11, and the elasticity by the conventional
method was estimated to be smaller than the true elasticity
obtained with Eq. (9). Therefore, the larger estimated values
obtained using the proposed method are more reasonable.
5. Conclusions

In the present study, to accurately measure the elastic modulus
of a deformed blood vessel, a series of equations were derived
to estimate the elastic modulus by considering the pushingpressure applied by an ultrasound probe, which deforms the
blood vessel. The derived equations were conﬁrmed bychanging the stresses through numerical simulations usingparameters applied from actual measurements. Moreover, we

compared the values estimated by the proposed method withthose estimated using the conventional method in in vivo
measurements for various pushing pressure values. The results
revealed that the proposed method reduces the pressure
dependence of the estimated elastic moduli. However, theproposed method cannot completely eliminate pressure dependence. Because the calculation of the strain is still insufﬁcient,
it is necessary to introduce the shape of the vessel into a strain
calculation in future research. This method has the potential tomeasure elasticity more accurately than other methods and canbe applied to the evaluation of endothelial function during the
FMD reaction.
Acknowledgments

This work was partially supported by JSPS KAKENHI
20H02156.
ORCID iDs

Shohei Mori https://orcid.org/0000-0002-5494-1055
Mototaka Arakawa
https://orcid.org/0000-0001-9386645X
Hiroshi Kanai https://orcid.org/0000-0002-6567-1687

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