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Internal pressure dependence on viscoelasticity ofarterial wall by ultrasonic measurement
To cite this article: Saki Suzuki et al 2023 Jpn. J. Appl. Phys. 62 SJ1041

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Japanese Journal of Applied Physics 62, SJ1041 (2023)
https://doi.org/10.35848/1347-4065/acbf5d

Internal pressure dependence on viscoelasticity of arterial wall by ultrasonicmeasurement
Saki Suzuki1, Shohei Mori2, Masumi Iwai-Takano1,3,4, Mototaka Arakawa1,2*, and Hiroshi Kanai1,2
1Graduate School of Biomedical Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan3
Department of Epidemiology, Fukushima Medical University, Fukushima 960-1295, Japan
4
Division of Cardiovascular Surgery, Fukushima Medical University, Fukushima 960-1295, Japan2*

E-mail: arakawa@tohoku.ac.jp

Received November 15, 2022; revised February 21, 2023; accepted February 26, 2023; published online March 27, 2023
Previously, we proposed an ultrasonic measurement method of arterial wall elasticity for the early detection of arteriosclerosis. Since vascular wall
elasticity depends on blood pressure, in this study, the elasticity and viscosity were estimated using the hysteresis loop determined between theincremental strain in the wall and internal pressure by altering the internal pressure in the phantom and in vivo experiments. Consequently, both theestimated elasticity and viscosity increased with the internal pressure. Moreover, the slope of each hysteresis loop was larger than that of theapproximated curve for the lowest blood pressures of the hysteresis loops with different internal pressures, as the blood pressure changed in theorder of several hertz within a single heartbeat. Hence, we conclude that measuring both the blood pressure and the elastic and viscous moduli
would be beneficial in comprehensively investigating more vessel wall properties that change with the progression of arteriosclerosis.
© 2023 The Japan Society of Applied Physics

1. Introduction

Cardiovascular and cerebrovascular diseases are the leading
causes of death worldwide.1) Early diagnosis of arteriosclerosis, the main cause of these diseases, is necessary to
prevent them. In the early stages of arteriosclerosis, thecholesterol content in the aorta severely affects the elastic
properties of the arterial wall,2) and the early detection ofarteriosclerosis can be expected by evaluating the elasticity of
the arterial wall.
The pulse wave velocity (PWV) measurement,3–5) ﬂowmediated dilatation (FMD) test6,7) and arterial-wall thicknesschange measurement8,9) can be adopted to noninvasively andrepeatedly evaluate the elastic properties of the arterial wall.
The PWV measurement can solely evaluate the average
elastic properties of the entire measurement area. Accurate
evaluation with the FMD test is difﬁcult because the distanceresolution via ultrasonic measurement is insufﬁcient. In
contrast, ultrasound can be used to evaluate tissue properties
in vivo.10,11) The method used to measure the wall thickness
change for each ultrasonic beam can estimate the local
elasticity of the arterial wall.12–16)
Kanai et al. developed the phased-tracking method andmeasured minute displacement in heart and blood vessels in
the order of tens of microns, which were not measured with thespatial resolution of conventional ultrasound systems.17,18) The
elastic modulus of the vessel wall was estimated by measuring
the local thickness change during a heartbeat and the pulsepressure using a sphygmomanometer. Since blood vessels are
viscoelastic media, the relationship between the blood pressure
and strain draws a hysteresis loop. Assuming a viscoelastic
model, it is possible to qualitatively determine not only theelasticity, but also the viscosity of the vessel wall from thehysteresis loop between the inner diameter change and bloodpressure. Furthermore, by ﬁtting the hysteresis loop to a
viscoelastic model, such as the Voigt model, the elastic andviscous moduli can be quantitatively determined. To obtain ahysteresis loop, the blood pressure waveform and strain waveform should be measured simultaneously. In the previous study,

since the blood pressure and thickness change could not bemeasured simultaneously at the same position using the normal
ultrasound probe, only the elastic modulus was estimated,
although the blood vessel is a viscoelastic medium. Therefore,
Arakawa et al. have developed an ultrasound probe with a center
frequency of 7.5 MHz that can measure blood pressure waveform with a central element.19–21) Since the radial artery is close
to the skin and just above the radius, the pressure waveform canbe obtained by pressing the probe, including the piezoelectricelement, against the radial artery through the skin. The thickness
change in the radial artery cannot be measured because of theinsufﬁcient distance resolution. However, using this probe,changes in the inner diameter and blood pressure were
simultaneously measured at the same position.
Hypertension is one of the factors of arteriosclerosis.22,23) In
hypertension, vascular endothelial cells are damaged and thebasis for arteriosclerosis is formed.24) The elasticity of thevascular wall is known to depend on blood pressure.25–27) The
vascular wall comprises elastin, collagen, and smooth muscle.
The elastic properties are dominated by soft elastin and stiff
collagen at low and high blood pressures, respectively.28) The
relationship between the internal pressure p(t) and innerdiameter di(t) is expressed using the reference pressure Arefand inner diameter diref at the reference pressure Aref, asp (t ) = Aref · exp (b · (d i (t )/d iref - 1)), where β is the stiffness parameter.29) The stiffness parameter β is an index ofelastic properties independent of blood pressure.25,30) However,
it is a dimensionless parameter that reﬂects the apparent elastic
property of the arterial wall. If the blood pressure dependence
on the viscoelasticity of blood vessels could be measured
in vivo, the steady-state load exerted by blood pressure on
blood vessels can be determined, and this may facilitate theestablishment of diagnostic guidelines for arteriosclerosis bymeasuring the viscoelastic properties. Moreover, because thevascular wall is a viscoelastic medium, the estimated viscoelastic properties may differ under static and dynamic conditions,such as in vivo, where the vascular wall is beating.
In the present study, by comparing the blood pressure
dependence of the viscoelastic properties of the vascular wall

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under dynamic conditions, we demonstrated the importanceof measuring them in vivo. As a preliminary study, wemeasured the hysteresis loops between the internal pressureand vascular inner diameter with varying minimal internalpressure and then investigated the internal pressure dependence of the vascular wall’s viscoelasticity. First, weconstructed a phantom experimental system with an adjustable internal pressure. The internal pressure of the phantom
was varied while the pulse pressure was kept consistent, andthe hysteresis loops between the inner diameter change andinternal pressure were investigated.31) Moreover, the elasticity and viscosity were estimated. In in vivo experiments, thediastolic blood pressure was varied in the radial artery byaltering the height of the radial artery from the heart, thehysteresis loops between the blood pressure and innerdiameter were obtained, and the viscoelasticity was estimatedusing the developed probes.19–21)
2. Experimental method
2.1. Measurement of viscoelasticity using change in
the inner diameter

The elasticity and viscosity of the wall were estimated bysimultaneously measuring the inner diameter change Dd i (t )of the phantom or blood vessel and internal pressure (bloodpressure) p (t ). The inner diameter change Dd i (t ) occurringwith pulsation was calculated by integrating the minutevelocity waveforms at the inner sides of the anterior andposterior walls.32) The minute velocity waveform was
obtained using the phased-tracking method.17) It wasE=h=s (t )¶ Dd i (t ) 2¶t d i0

assumed that the phantom or vessel wall is isotropic,
elastically incompressible, uniform in wall thickness, andexpansion and contraction do not occur in the longitudinal
direction.33–35) The relationship between the strain
Dd i (t )/d i0 due to the inner diameter change Dd i (t ) andincremental internal pressure Dp (t ) can be expressed by
Eq. (1), assuming the Voigt model, and considering theviscous term:E

Dd i (t )d i0¶ Dd i (t )d i0

+ h ¶t3 (h 0 + d i0 )(2h 0 + d i0 )Dp (t ) = Ds (t ) ,h 0 d i0=8

where d i0 and h 0 denote the inner diameter and thickness,
respectively, of the phantom or vessel at the minimum
internal pressure; E and η are the elasticity and viscosity,
respectively, of the phantom or arterial wall. The right-handside of Eq. (1) shows the incremental stress Ds (t ). In thederivation of Eq. (1), it is assumed that the inner diameterchange Dd i (t ) and outer diameter change Dd o (t ) are
sufﬁciently smaller than the inner diameter d i0 and outer
diameter d o0 at the minimum internal pressure, respectively(∣ Dd i (t )∣  d i0, ∣ Dd o (t )∣  d o0 ). The inner diameter d i0
and thickness h 0 of the phantom at the minimum internalpressure were obtained by the time-of-ﬂight method using
ultrasound.36) The elasticity E and viscosity η were estimatedto minimize the squared error between the measured incre (t ), as exmental internal pressure Dp (t ) and model Dp
pressed in Eqs. (2) and (3), respectively:( ) · ( ) - s (t) ( ) · ( )(( ) · ( ) - {( )( )}Dd i (t )d i0

Dd i (t ) 2
d i0(¶ Dd i (t )¶t d i0

Dd i (t )d i0¶ Dd i (t ) 2¶t d i0

Dd i (t )d i0¶ Dd i (t )¶t d i0

Dd i (t ) 2
d i0

Dd i (t )d i0

Dd i (t )d i0

Dd i (t ) 2
d i0

Dd i (t )d i0¶ Dd i (t )¶t d i0¶ Dd i (t ) 2¶t d i0

),

(2 )

),

(3 )

¶ Dd i (t )¶t d i02) ⋅ ( ) - s (t) ( ) ⋅ ( )(( ) ⋅ ( ) - ⎧⎨⎩( )( )⎫⎬⎭¶ Dd i (t )d i0s ( t ) ¶t

(1 )¶ Dd i (t )¶t d i02where ¯· represents the average operation during one beat.

Fig. 1. (Color online) Schematic diagram of the phantom experimental system.
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S. Suzuki et al.

2.2. Phantom experiments(a)Figure 1 shows a schematic diagram of the experimental
phantom system. A pulsatile pump (EC-8, Fuyo, Japan) was
utilized to circulate water in a closed circulation path, notpassing the tank, to simulate in vivo blood circulation (valve
I: open, valve II: closed). Moreover, the water column
applied hydrostatic pressure in the pathway (valve III:open) to achieve hydrostatic pressure as high as the in vivo
condition, regardless of the ﬂow rate from the pump.
In the ultrasound measurement, a linear probe with a center
frequency of 7.5 MHz was connected to an ultrasonicdiagnostic device (SSD-6500, Aloka, Japan). The sampling
frequency, beam spacing and frame rate were 40 MHz,150 μm and 286 Hz, respectively. A silicone rubber tube
with an Asker C hardness of 10°, outer diameter of 9 mm,
inner diameter of 7 mm and wall thickness of 1 mm (allnominal values) with graphite added as a scatterer was
adopted as a phantom to mimic the carotid artery. The
pulsation rate was set at 70 times min−1.
In the phantom experiment, the probe developed by ourgroup was not used because the internal pressure wasmeasured directly using pressure sensors. The internal pressures p1 (t ) and p2 (t ) were measured by pressure sensors (I)and (II) (PS-1KC, Kyowa, Japan) positioned upstream anddownstream of the phantom, respectively. The delay time t
between the two pressure waveforms p1 (t ) and p2 (t ) was
determined using the cross-correlation function. The innerdiameter of the phantom, d i (t ), was measured using ultrasoundat equidistant positions from the two pressure sensors. The
pressure waveforms p1 (t ) and p2 (t ) were shifted by t /2 tocompensate for the delay time due to the different measurement positions between the inner diameter waveform d i (t ) andinternal pressure waveforms p1 (t ) or p2 (t ). To obtain thehysteresis loops between the internal diameter d i (t ) andinternal pressure p (t ) and estimate elasticity E and viscosity η,
p1 (t - t /2) measured upstream the phantom was adopted.
To simulate in vivo blood pressure conditions, the steadystate water column height DH was set at 130 cm so that the
maximum and minimum water pressures were 120 mmHg
(160 cmH2 O) and 75 mmHg (100 cmH2 O), respectively. For
comparison, the height DH of the water column was set at 30(the maximum and minimum water pressures were 45 and
0 mmHg, respectively), 60 (70 and 20 mmHg) and 100 cm(90 and 45 mmHg). The hysteresis loops were measured byvarying the minimum internal pressure from 0 to 75 mmHg.
Furthermore, the changes in the slope and viscoelasticity are
discussed. At each water column height DH, the hysteresis
loops between the inner diameter waveform d i (t ) and internalpressure waveform p (t ) were measured for four beats toestimate elasticity E and viscosity η. The instantaneous ﬂow
rate q (t ), measured with a ﬂowmeter (FD-XS20, Keyence,
Japan), was used to estimate the ﬂow rate per pulsation Q,
and the ﬂow volume from the pump was set so that the waterﬂow rate Q was approximately 600 ml min−1, corresponding
to the blood ﬂow rate in the carotid artery.
2.3. In vivo experiments
Figure 2(a) shows a schematic diagram of the in vivo
experimental system. The subject was a healthy female inher twenties, and her inner diameter d i (t ) and blood pressure
waveform p (t ) in the radial artery of her left hand were
measured. In in vivo experiments, pressure sensors cannot be(b)Fig. 2. (Color online) Schematic diagram of the in vivo experimental
system. (a) Block diagram and (b) arm position during the experiment.

used to measure blood pressure waveforms. Thus, the bloodpressure waveform can be measured using the linear probe
developed by our research group.19–21) The linear probe wasconnected to an ultrasonic diagnostic device (Prosound F75;
Aloka, Japan). The center frequency, sampling frequency andframe rate were set at 7.5 MHz, 40 MHz and 252 Hz,
respectively.
While viewing the B-mode short-axis image of the radial
artery, the measurement position was set so that the center ofthe ultrasound probe was just above the radial artery, and theprobe was pressed against the radial artery with an appropriate force to measure the blood pressure waveform. The
output from the piezoelectric element at the center of theultrasound probe passed through an ampliﬁer with anampliﬁcation factor of 40 dB and a low-pass ﬁlter with acut-off frequency of 30 Hz, and the blood pressure waveform
was obtained by integrating the output waveform.
To acquire the internal pressure p (t ) and inner diameterd i (t ) in the radial artery under different blood pressure
conditions, the values were measured in three different
situations, as shown in Fig. 2(b): (A) when the radial arterywas 20 cm lower than the heart (DH = −20 cm), (B) at thesame height as the heart (DH = 0 cm), and (C) 15 cm higher
than the heart (DH = 15 cm). At each arm position, thehysteresis loops between the inner diameter waveform d i (t )and blood pressure waveform p (t ) were measured for twoheartbeats to estimate elasticity E and viscosity η. The radialartery wall thickness could not be measured by the probewith a center frequency of 7.5 MHz due to the lack of
distance resolution. Hence, a probe with a center frequency of
40 MHz was connected to an ultrasonic diagnostic device(UD-8000, Tomey, Japan) to measure the wall thickness h 0of the subject’s radial artery beforehand.
As the blood pressure waveform measured with our probe
was output as a voltage, calibration was required to obtain the

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(b)

(c)

(d)

Fig. 3. (Color online) B-mode images of the phantom. Red line depicts the analyzed beam. (a) 130 cmH2O, (b) 100 cmH2O, (c) 60 cmH2O, (d) 30 cmH2O
and (e) 0 cmH2O.

(a)absolute value of the blood pressure. Diastolic and systolicblood pressures were measured at three different heights atthe same positions as the ultrasound measurement shown in
Fig. 2(b), using a wrist-measurable sphygmomanometer(WS-20J, Japan Precision Instruments, Japan) after theultrasound measurement, and the blood pressure waveforms
were calibrated.19,20) These measurements were approved by
the Ethics Committee of the Graduate School of Engineering
Tohoku University, and all the participants provided consent
before participating in the present study.

(b)

3. Results
3.1. Phantom experiments

By setting the output of the pulsating pump, the ﬂow rate Q ofwater per pulsation was 611 ± 10 ml min−1, thereby
simulating the blood ﬂow condition in the carotid artery.
Figure 3 shows B-mode images at minimum pressures (a)when the height of the water column DH was 130 cm, andin vivo blood pressure conditions were simulated (minimumpressure at a beat: 70 mmHg), (b) 100 cm (45 mmHg), (c)60 cm (20 mmHg) and (d) 30 cm (0 mmHg). The innerdiameter d i (t ) was obtained by analyzing the beam depictedby the red line. The minimum inner diameters d i0 at theminimum internal pressure were (a) 9.76 mm, (b) 8.98 mm, (c)7.51 mm and (d) 6.81 mm. The minimum posterior wall
thicknesses h 0 at the minimum internal pressure were (a)0.67 mm, (b) 0.72 mm, (c) 0.84 mm and (d) 0.90 mm. The
higher the water column and internal pressure p (t ), the larger
the inner diameter d i (t ) and the thinner the wall thickness h (t ).
Figure 4 shows (a) the internal pressure p (t ) for four
circulation cycles when the height DH of the water column
was 130 cm (minimum pressure at a beat: 70 mmHg) (in vivo
condition), 100 cm (45 mmHg), 60 cm (20 mmHg) and 30 cm(0 mmHg), (b) the incremental strain Dd i (t )/d i0 of the innerdiameter change Dd i (t ), and (c) the hysteresis loops between
the internal pressure p (t ) and inner diameter d i (t ).
Figure 4(a) demonstrates that the maximum and minimumpressures increased as the height DH of the water column
was raised. Figure 4(b) demonstrates that the maximum strain
increased from the height of the water column DH = 30 cm(blue) to 60 cm (green) and decreased from 60 cm (green) to
130 cm (red). Figure 4(c) demonstrates that the slope of thehysteresis loop increased as the height DH of the water(c)Fig. 4. (Color online) Results of the phantom obtained for four circulation
cycles at each internal pressure: (a) internal pressure p (t ), (b) incremental
strain of the thickness change, Dh (t )/h 0 , and (c) hysteresis loops between
the internal pressure p (t ) and inner diameter d i (t ).

column was increased. The hysteresis loops at the height of
the water column DH = 130 cm (red) and 100 cm (yellow)
have a larger slope at higher pressures, thus indicating anonlinear relationship between the pressure p (t ) and innerdiameter d i (t ). The relationships between the internal pressure p (t ) and inner diameter d i (t ) measured at different
heights DH of the water column were not plotted on a single
curve.

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Fig. 5. (Color online) Results of the hysteresis loops for the phantom
between the natural logarithms of p (t )/Aref , ln ( p (t )/Aref ), and incremental
strain of the inner diameter, Dd i (t )/d iref , for four circulation cycles at eachinternal pressure. Dashed line indicates the approximated straight line
obtained for the pressures at the lowest pressure in the hysteresis loops when
the heights of the water column DH were 130 (minimum pressure at a beat:
70 mmHg), 100 (45 mmHg) and 60 cm (20 mmHg).

The natural logarithm of the internal pressure p (t ) is
calculated because the relationship between the internalpressure p (t ) and inner diameter d i (t ) is usually expressed asp (t ) = Aref · exp (b · (d i (t )/d iref - 1)), considering thenonlinearity of elasticity.29) Here, Aref , d iref and β are thereference pressure, inner diameter at the reference pressure
Aref and stiffness parameter, respectively.
Figure 5 shows the hysteresis loops determined between
the natural logarithms of p (t )/Aref , ln ( p (t )/Aref ), and thestrain in the inner diameter change Dd i (t )/d iref in thephantom experiment. Aref is the minimum internal pressure
when the height DH of the water column is 60 cm. The
dashed line in Fig. 5 indicates the approximated straight line
obtained for the pressures at the lowest pressure in thehysteresis loops when the height of the water column DH is
130 (minimum pressure at beat: 70 mmHg), 100 (45 mmHg)
and 60 cm (20 mmHg). Considering the hysteresis loop at
each internal pressure, the mean values of the four beats withthe lowest internal pressure and inner diameter at each time
point were calculated. Using these values, an approximatedstraight line was obtained using the least squares method. The
slope of the relationship between the natural logarithm ofp (t )/Aref , ln ( p (t )/Aref ), and the strain in the inner diameterchange, Dd i (t )/d i0, is a straight line in the internal pressure
range of 60–160 mmHg.25) The internal pressures at thelowest pressures in the hysteresis loops were approximately
on a straight line in this range.
Next, we focused on the estimated elasticity E andviscosity η for four consecutive circulation cycles using
Eqs. (2) and (3); they are approximately the same when theheight DH of the water column is 30 (blue) or 60 cm (green).
However, these factors increase as the internal pressure
increases for DH = 100 (yellow) and 130 cm (red), as
shown in Fig. 6. Hence, when the height DH of the water

Fig. 6. (Color online) Elastic and viscous modulus values of the phantom
estimated for four beats at each internal pressure.

column was greater than DH = 60 cm (20 mmHg), both
elasticity E and viscosity η of the phantom depended on theinternal pressure p (t ).
3.2. In vivo experiments
The wall thickness of the radial artery was measured as
h 0 = 0.20 mm beforehand. Figure 7 shows the B-mode
images of the radial artery at the minimum blood pressure
when the radial artery was (a) 20 cm lower than the heart(DH = -20 cm ), (b) at the same height as the heart(DH = 0 cm ), and (c) 15 cm higher than the heart(DH = 15 cm ). The inner diameters (short diameters) d i0
were (a) 1.68 mm, (b) 1.60 mm and (c) 1.55 mm.
Figure 8 shows the hysteresis loops between the internalpressure p (t ) and inner diameter d i (t ) for two heartbeats
when the radial artery was (A) 20 cm lower than the heart(DH = -20 cm ), (B) at the same height as the heart(DH = 0 cm ), and (C) 15 cm higher than the heart(DH = 15 cm ). Similar to the phantom experimental results,
the slope of the hysteresis loop increased with increasing
blood pressure, and the hysteresis loops at the three different
blood pressures were not plotted on a single curve. The
dashed line in Fig. 8 indicates the exponential function
representing the relationship between the minimum bloodpressure and inner diameter at that time for each hysteresis
loop. The mean values of the two heartbeats with the lowest
internal pressure and inner diameter at each time point were
calculated in the hysteresis loop at each internal pressure.
Using these values, an approximate curve was obtained using
the least squares method.
Figure 9 shows the hysteresis loops between the natural
logarithm of p (t )/Aref , ln ( p (t )/Aref ), and the strain in the
inner diameter change, Dd i (t )/d iref , from the in vivo experiment and the approximated straight line for the lowest
pressure of the hysteresis loops (dashed line). Aref is theminimum internal pressure when the radial artery was (C)15 cm above the heart (DH = 15 cm ). Similar to thephantom experimental results, the blood pressure at the

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Jpn. J. Appl. Phys. 62, SJ1041 (2023)(a)S. Suzuki et al.

(b)

(c)

Fig. 7. (Color online) B-mode images of the radial artery: (a) arm position (A), (b) arm position (B), and (c) arm position (C) in Fig. 2.

Fig. 8. (Color online) Results of the hysteresis loops for the radial artery
between the internal pressure p (t ) and inner diameter d i (t ) for two
circulation cycles at each arm position. Dashed line indicates the exponentialfunction representing the relationship between the lowest blood pressure andinner diameter at that time for each hysteresis loop.

Fig. 10. (Color online) Elastic and viscous modulus values of the radial
artery estimated for two heartbeats at each arm position.

radial artery was lower than the heart (high blood pressure)(red), elasticity E and viscosity η were estimated to be higher
than when the radial artery was at the same height as the heart(blue), and when the radial artery was higher than the heart(low blood pressure) (black), they were estimated to belower. Similar to the phantom experimental results, elasticity
E and viscosity η depended on the internal pressure, i.e. theblood pressure.
4. Discussion

Fig. 9. (Color online) Results of the hysteresis loops for the radial artery
between the natural logarithms of p (t )/Aref , ln ( p (t )/Aref ), and incremental
strain of the inner diameter, Dd i (t )/d iref , for two circulation cycles at eacharm position. Dashed line indicates the approximated straight line for the
blood pressures at the lowest pressure of the hysteresis loops. Gray line oneach hysteresis loop shows the slope of each hysteresis loop.

minimum blood pressure in each hysteresis loop was
approximated as a straight line.
Elasticity E and viscosity η estimated for the two heartbeats using Eqs. (2) and (3), are shown in Fig. 10. When the

In our previous study, we conﬁrmed that the strain of theouter diameter change and elasticity E could be measured
ultrasonically with the same accuracy as those obtained via
laser.36) In the present study, elasticity E was estimated fromthe change in the inner diameter. The resulting E agreed withthat estimated from the change in the outer diameter within
4%. Therefore, the appropriate elasticity E can be estimatedfrom the change in the inner diameter. However, the validityof viscosity η needs to be veriﬁed in future studies.
In both the phantom and in vivo experiments, as theinternal pressure p (t ) increased, the estimated elasticity E
increased, as shown in Figs. 6 and 10. As the internalpressure increased, the strain Dd i (t )/d i0 became smaller, as
shown in Fig. 4(b), and the phantom and blood vessels
elongate signiﬁcantly and are less deformed.
In the phantom experiment, as shown in Fig. 4(c), when
the height DH of the water column is 60 cm, the slope of thehysteresis loop is smaller than that when DH is 30 cm;

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S. Suzuki et al.

however, both estimated elasticities E are approximately thesame. Speciﬁcally, the slope in Fig. 4(c) does not represent
elasticity E. Elasticity E is related to the slope of theincremental stress Ds (t ) concerning strain Dd i (t )/d i0, which
was nearly the same at 0 and 30 cm height DH of the watercolumn.
Next, we discuss viscosity η. In the phantom experiment,
as the internal pressure increases, the area of the hysteresis
loop decreases, as shown in Fig. 4(c), whereas viscosity η
increases, as shown in Fig. 6. This is because the area of thehysteresis loop is determined not by viscosity but by the ratioof viscosity to elasticity, η/E. When the height DH of thewater column was 130 cm, elasticity E and viscosity η wereapproximately three and two times higher, respectively, thanthose at 30 cm. This indicates that the area of the hysteresis
loop with DH = 130 cm was smaller than that with DH= 30 cm. In contrast, the areas of the hysteresis loops wereapproximately the same when the internal pressure wasvaried, and elasticity E and viscosity η changed at a constantrate in the in vivo experiment.
In the phantom and in vivo experiments, the variance inviscosity η was larger than that in elasticity E, as shown in
Figs. 6 and 10. Since the differentiation of strain results in alarger error, the difference in viscosity η is larger than that inelasticity E. Furthermore, in the phantom experiments, thevariance in viscosity η increases as the internal pressure p (t )increases. The variance in viscosity η increases as theamplitude of the differential waveform ¶ (Dd i (t )/d i0 )/¶t of
the strain decreases. The greater the internal pressure p (t ),the greater the elasticity E and the smaller the strain
Dd i (t )/d i0. Therefore, the derivative ¶ (Dd i (t )/d i0 )/¶t ofstrain is also smaller, resulting in greater viscosity variance.
Next, we compared the viscoelasticity of the phantom in
Fig. 6 and in vivo experimental results in Fig. 10. The
estimated E and η were of the same order in the phantom andin vivo experiments. This conﬁrms that the E and η of bloodvessels depend on the blood pressure in the normal in vivo
blood pressure range.
The hysteresis loops in Fig. 9 and those formed when theheights DH of the water column were 130 cm (diastolicpressure at beat: 70 mmHg), 100 cm (45 mmHg) and 60 cm(20 mmHg) in Fig. 5 were not represented in a single straightline. The slope of each hysteresis loop was larger than that of
the approximated straight line for the lowest blood pressures.
When the relationship between the internal pressure p (t ) andinner diameter d i (t ) is expressed by an exponentialfunction,25) each hysteresis loop can be expressed on astraight line. In the present study, the frequency dependence
was not considered in the viscoelasticity estimation.
However, in the Voigt model, the complex elastic modulus
E* is represented as E + j wh, and the imaginary part of thecomplex elastic modulus E* increases as the frequency
increases. The reason for the larger slope of the hysteresis
loops is that the absolute value of the complex elastic
modulus E* increased due to blood pressure changes ofseveral hertz within a single heartbeat. As shown in Fig. 9,
the slope of the hysteresis loop increased as the internalpressure increased. Since the conventional stiffness parameterβ is also measured under dynamic conditions, it is suggested
that β may also depend on the internal pressure, although it

has been reported that it does not depend on the internalpressure.25)
5. Conclusion

In the present study, to investigate the blood pressure
dependence of the viscoelastic properties of blood vessels,
the hysteresis loops between the internal pressure p (t ) andinner diameter d i (t ) were measured, and elasticity E andviscosity η were estimated by altering the internal pressurep (t ) in the phantom and in vivo experiments. The slopes of thehysteresis loops increased, and elasticity E and viscosity ηwere estimated to increase, as the internal pressure p (t )increased. The internal pressure p (t ) is expressed as an
exponential function of the inner diameter Dd i (t ). In thein vivo experiments, the relationship between the minimum
blood pressure and the inner diameter at the time of eachhysteresis loop was expressed as an exponential function.
However, the slope of each hysteresis loop was larger than thatof the approximate curve (approximate straight line for thenatural logarithm of blood pressure). This could be due to theeffect of the blood pressure waveform frequency. Therefore,
the frequency dependence of the viscoelastic properties of thevessel wall should be investigated in future. Simultaneous
measurements of the internal pressure p (t ) and the incremental
strain of the inner diameter Dd i (t )/d i0 by altering the internalpressure enabled us to measure the changes in elasticity E andviscosity η due to the blood pressure dependency. Hence,measuring the blood pressure, elasticity E and viscosity η
would be beneﬁcial in comprehensively investigating moredetailed properties of the vessel wall that change with theprogression of arteriosclerosis. In the future, we will apply thismethod to measure the blood pressure, elasticity E andviscosity η in patients with arteriosclerosis to demonstrate
the usefulness of the early diagnosis of arteriosclerosis.
Acknowledgments

This work was supported in part by JSPS KAKENHI
21H03835.

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