ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005

1931

Propagation of Spontaneously ActuatedPulsive Vibration in Human Heart Wall and
In Vivo Viscoelasticity Estimation
Hiroshi Kanai, Member, IEEE
Abstract—Though myocardial viscoelasticity is essential
in the evaluation of heart diastolic properties, it has neverbeen noninvasively measured in vivo. By the ultrasonicmeasurement of the myocardial motion, we have alreadyfound that some pulsive waves are spontaneously excitedby aortic-valve closure (AVC) at end-systole (T0 ). Thesewaves may serve as an ideal source of the intrinsic heartsound caused by AVC. In this study, using a sparse sector
scan, in which the beam directions are restricted to about
16, the pulsive waves were measured almost simultaneously
at about 160 points set along the heart wall at a suﬃciently
high frame rate. The consecutive spatial phase distributions, obtained by the Fourier transform of the measured
waves, clearly revealed wave propagation along the heart
wall for the ﬁrst time. The propagation time of the wave
along the heart wall is very small (namely, several milliseconds) and cannot be measured by conventional equipment.
Based on this phenomenon, we developed a means to measure the myocardial viscoelasticity in vivo. In this measurement, the phase velocity of the wave is determined for eachfrequency component. By comparing the dispersion of thephase velocity with the theoretical one of the Lamb wave
(the plate ﬂexural wave), which propagates along the viscoelastic plate (heart wall) immersed in blood, the instantaneous viscoelasticity is determined noninvasively. This is
the ﬁrst report of such noninvasive determination. In in
vivo experiments applied to ﬁve healthy subjects, propagation of the pulsive wave was clearly visible in all subjects.
For the 60-Hz component, the typical propagation speed
rapidly decreased from 5 m/s just before the time of AVC
8 ms) to 3 m/s at t = T0 + 10 ms. In the ex( t = T0
periments, it was possible to determine the viscosity more
precisely than the elasticity. The typical value of elasticity was about 24–30 kPa and did not change around thetime of AVC. The typical transient values of viscosity decreased rapidly from 400 Pa s at t = T0 8 ms to 70 Pa s at
t = T0 + 10 ms. The measured shear elasticity and viscosity
in this study are comparable to those obtained for the human tissues using audio frequency in in vitro experiments
reported in the literature.

;



;



I. Introduction
or tissue characterization, shear waves are artiﬁcially
actuated in tissues or phantoms to determine theirF

Manuscript received May 15, 2004; accepted April 25, 2005. This
work was partly supported by grants-in-aid for scientiﬁc researchfrom the Ministry, Education, Culture, Sports, Science, and Technology of Japan (2003–2004, No. 15300177, 2004–2005, No. 16650120,
and 2005–2007, No. 17206043.).
The author is with the Department of Electronic Engineering,
Graduate School of Engineering, Tohoku University, Sendai 9808579, Japan (e-mail: hkanai@ecei.tohoku.ac.jp).

propagation speed and viscoelasticity [1]–[7]. However,
spontaneously actuated vibrations propagating in theheart wall, which diﬀer from electrically excited waves [8]–[10], have not been recognized at all. This is understandable given that the actual wave that propagates along theheart wall is minute (namely, only several tenths of a millimeter) and that the delay time from the base to the apexof the heart is very small, only several milliseconds.
Conventional ultrasonography, computer tomography(CT) [11], and magnetic resonant imaging (MRI) [12], [13]enable clinical visualization of cross-sectional images of thehuman heart, but their imaging is restricted to large motion (> 1 mm) and low-frequency components (< 30 Hz).
Analysis of the motion-mode (M-mode) image [14], [15] isa candidate for transthoracic measurement of the minuteand rapid heart-wall motion. However, the detectable amplitude is greater than the wavelength, which is equal to411 µm for ultrasound with a frequency of 3.75 MHz normally applied to the heart in the clinical setting. Even
when a higher frequency of 10 MHz is used, the wavelengthand the detectable amplitude are still 154 µm.
The tissue Doppler imaging (TDI) technique [16]–[21]—modiﬁed ultrasound two-dimensional (2-D) color ﬂow
mapping—enables determination of motion distribution of
the myocardium in real time. Even in current measurement, however, the sampling frequency of the motion of
the heart wall is low (at most 200 Hz [21]), that is, thesampling period is 5 ms, which is too long to detect the
propagation time of the wave. Moreover, frequency analysis such as the Fourier transform has not been applied
to the measured motion signal to detect the phase components.
To measure the original vibrations of the heart sounds(which are audible by stethoscope) using ultrasound fromthe chest wall, we have already developed a method totranscutaneously measure the heart-wall vibrations as a
waveform at one point or multiple points preset along an
ultrasonic beam in the heart wall [22], [23]. We found that
there is a steep dip in the pulse that occurs exactly at
the time of AVC (T0 ) [24]. This notch also has been measured by the TDI approach with a commercial scanner todetermine the time of AVC (T0 ) [25].
Moreover, we have developed a method to measure thevibrations simultaneously at about 160 points preset in theheart wall [24]. In [24], it was revealed that there is a shortdelay of several milliseconds in the excitation of the steep
dip of the notch pulse from the base to the apex of thec 2005 IEEE
0885–3010/$20.00 1932

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005

heart along the interventricular septum (IVS). However,no propagation phenomenon of the notch pulse along theheart wall was recognized in the previous report [24].
The use of the sparse sector scan has allowed us to simultaneously measure heart-wall motion at 160 points at
a suﬃciently high frame rate to measure the propagationof the notch pulse along the IVS. Therefore, from consecutively obtained spatial distributions of the phase value ofthe vibration wave, the present study reveals for the ﬁrst
time that the steep dip of the notch pulse, excited exactly
at the time of AVC (T0 ), propagates along the IVS from thebase to the apex, and its phase velocity is determined. By
analyzing various frequency components up to 90 Hz, thepropagation speed shows the frequency dispersion. Thisdispersion characteristic agrees with the theoretical one of
the Lamb wave that propagates in the viscoelastic plate
immersed in ﬂuid. By introducing the single Voigt model
into the equation of the Lamb wave and ﬁtting the derived
theoretical phase velocity to the measured dispersion, themyocardial viscoelastic properties were determined noninvasively for the ﬁrst time. The shear elastic and viscosity
values in the in vivo experiments are comparable to thosealready obtained for human soft tissue in in vitro experiments for a similar audible frequency range as reported
in the literature. This method oﬀers potential for in vivo
myocardial tissue characterization of diastolic properties,
which cannot be obtained by conventional echocardiography, TDI, CT, or MRI.

II. In Vivo Experimental Results for Examining
Potential Mathematical Model
A. Vibrations in and on the Heart Wall
Fig. 1(a) shows a typical cross-sectional image (thetransthoracic parasternal longitudinal-axis view) of a heartobtained by conventional echocardiography for a healthy
young subject (subject A, 21 years old). The upper-rightinset shows the scanning range of the ultrasonic beams
in this imaging. From the conventional motion picture of
Fig. 1(a), in our experience the human eye can detect motion only larger than 1 mm and slower than 1/30 second.
The recognized slow motion also includes the lateral motion from the base to the apex during systole, the directionof which is perpendicular to the ultrasonic beam. However,for a short period of about 70 ms, it can be assumed thateach point preset in the heart wall stays in the same focal
area of each ultrasonic beam. This assumption was usedin this study.
We previously developed an ultrasound-based transthoracic method (phased tracking method) to directly measure the heart wall vibrations [22], [23], [26]. Radio frequency (RF) pulses are transmitted from an ultrasonictransducer at a pulse repetition interval ∆T , and the reﬂected ultrasonic wave is received by the same transducer
and quadrature demodulated. The time-domain complex
cross-correlation technique is applied to determine thephase shift between the consecutively obtained resultant

Fig. 1. (a) A cross-sectional image measured by conventional echocardiography in a healthy young male (subject A, 21 years old). The
upper-right inset shows the scanning range of the ultrasonic beams
in this imaging. The arrows show the directions of the 16 ultrasonic beams used to measure the vibrations at about 160 points in
the heart wall. Supplemental animationshows the conventional
B-mode image in the parasternal view obtained by a conventional
ultrasonic diagnosis equipment for subject A. The cross-sectional image just before the time of AVC (T0 ) from the animation is shown
here. (b) In vivo measurement results for the healthy man in (a) at
two points set along the 13th ultrasonic beam. Each waveform forsix consecutive cardiac cycles is overlaid. The time of AVC is denoted by T0 . Exactly at this time (T0 ), a steep dip in the pulsive
wave is observed with suﬃcient reproducibility. The audible vibration signal
was detected for the point preset at the LV side of
the IVS along the 13th ultrasonic beam using the method developed
for subject A. The audible heart soundwas detected on thechest surface using the microphone for subject A. The audible vibration signal and the heart sound were simultaneously measured. LV,
left ventricle; LA, left atrium; RV, right ventricle; RA, right atrium;
US probe, ultrasonic probe; IVS, intraventricular septum; Ao, aorta;
ECG, electrocardiogram; PCG, phonocardiogram (heart sound).

kanai: medical ultrasound tissue characterization

signals. The phase shift corresponds to the displacement∆x(t) during the short-period ∆T and the average velocityv(t) = ∆x(t)/∆T during that period. By accumulating the
instantaneous displacement ∆x(t), the large motion x(t)due to the heartbeat is tracked over one heartbeat; and at
the same time, the minute vibration v(t) superimposed on
the large motion is obtained as a waveform. For this measurement, the sampling frequency (the time resolution) is
increased from the 30 Hz in conventional echocardiography
to 450 Hz to avoid aliasing. To realize this, the number of
directions in which the ultrasonic beams are transmitted
is greatly decreased from 240 to 16 as shown by the arrows
in Fig. 1(a).
By applying the above phase tracking method to subject A in Fig. 1(a), Fig. 1(b) shows a typical example of thevibrations on both sides—the right ventricular (RV) side
and left ventricular (LV) side—of the IVS. These velocity
signals in Fig. 1(b) are audible, similar to the phonocardiogram (PCG) in Fig. 1(a) detected by a microphone on thechest wall. The audible vibration on the LV side along thethe 13th ultrasonic beamand the PCG
is available
online. In Fig. 1(b), six consecutive heartbeats are overlaid.
Because it is well-known that there are some ﬂuctuationsin the duration of diastole even among consecutive cardiac
cycles, there are some diﬀerences in the six waveforms in
the right-hand side of Fig. 1(b). However, during systoleand the isovolumic relaxation (IR) period, even the rapid
components of the measured vibrations have high reproducibility. As shown by these waves, some discriminative
pulsive waves are observed, especially at end systole. Just
before the time of AVC (T0 ), a slow upward pulse continues for about 30 ms. A steep notch pulse then occurs
exactly at T0 , which coincides with the beginning of theIR period [27].
B. Spatial Distribution of the Heart Wall Vibrations
Using a sparse sector scan in 16 directions, multiple
points were preset at 770-µm intervals in the heart wallalong each of 16 ultrasonic beams, and the vibrations at
about 160 points were simultaneously measured as waveforms with a sampling frequency of 450 Hz by the phased
tracking method [24]. Though, in a strict sense, a time
lag of {(k − 1)/16} × ∆T is hidden in the sampling timeof the vibrations in the direction of the k-th scan line(k = 1, · · · , 16), this time lag can be corrected [24]. Fig. 2
shows the simultaneously obtained heart-wall vibrations
for all points, but the duration is restricted to the shortperiod from T0 − 35 ms to T0 + 35 ms. The central verticalaxis in each wavelet shows the time of AVC (T0 ). As shownby the wavelets near the root of the aortic valve, a steep
notch pulse occurs at T0 . The occurrence of this notch isdelayed gradually from the root of the aortic valve to theapex by several milliseconds. To the contrary, a steep upward pulse is delayed gradually from the root of the aortic
valve to the base.
In the present study, we roughly determined the time of
AVC (T0 ) by using the heart sound and conventional mo-

1933

Fig. 2. Spatial distribution of 160 measured wavelets in the IVS at
end systole along the 5th to 16th ultrasonic beams in Fig. 1. The
subject is the same healthy, young male (subject A) as in Fig. 1,
but the measurement was on a diﬀerent day. These wavelets were
simultaneously obtained and shown for the short period between
T0 − 35 ms and T0 + 35 ms. The time of AVC (T0 ) is determined by
the waveform at the LV side along the 9th ultrasonic beam. The two
waves A and B correspond to velocity waveforms on the RV side andLV side of the IVS along the 13th ultrasonic beam in Fig. 1(b).

tion picture, but the time resolution was not so high.
In [25], the time of AVC (T0 ) was determined by the notchon the septal radial myocardial velocity proﬁle in a TDI
study, and the correlation between the time of AVC (T0 )determined above and the one measured from the hemodynamic data was conﬁrmed. Thus, we deﬁne the exact
time of AVC (T0 ) as the time of the dip observed in thevelocity signal at a point close to the root of the aortic
valve. In Fig. 2, this point is at the LV side along the 9th
beam.
The displacement of the myocardium due to the steep
dip around the time of AVC (T0 ) is roughly estimated as−120 µm because the waveform v(t) is approximated bya downward isosceles triangle, the maximum amplitude ofthe dip is about 30 mm/s, and the duration is about 8 ms.
This displacement is far smaller than both the wavelengthand the duration of the ultrasound used. Thus, it can beassumed that the preset point stays in the same place inthe myocardium during this short period of 70 ms aroundthe time of AVC (T0 ).
Because the wavelength of the detected pulsive wave
is about 100 mm for a 30-Hz component and is comparable to the size of the whole heart, its propagation phenomenon cannot be clearly visualized by showing the spatial distribution of the instantaneous amplitude of eachresultant pulsive wave in Fig. 2. Instead, the phase value
varies from 0 to 360 degrees within one wavelength. Therefore, using the method in [24], 2-D spatial distribution of
the instantaneous phase values of the measured wave isshown in Fig. 3. For this imaging, the short-time Fouriertransform is applied to the pulsive wave at each point in
Fig. 2 after the pulsive wave is multiplied by the Hanningwindow with a short length of 35 ms. The phase value
is detected for each frequency component from 10 Hz to
90 Hz, then color coded based on the upper-right circu-

1934

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005

seen from them that there is a delay in the generated timeof the pulsive wave. However, it should be conﬁrmed thatthe steep pulsive wave does propagate from the base near
the root of the aortic valve to the apex. For this purpose,from these consecutively obtained cross-sectional 2-D imis presented. From theages in Fig. 3, a motion picture
consecutiveness observed in Fig. 3 around the time of AVC(T0 ), it can be seen that a few pulsive waves are radiatedfrom the root of the aortic valve and propagate along the
IVS. The delay due to the propagation of the pulse fromthe root to the apex is several milliseconds, which has notbeen recognized at all by any other clinical technique.
The white lines in Fig. 3 show the instantaneous phaseθ(x; t) of the pulsive wave at the LV surface of the IVS.
The phase value is spatially interpolated between each consecutive two points along the adjacent ultrasonic beams.
Because there is phase discontinuity between the IVS andthe base at the root of the aortic valve, the phase is notconnected between the 8th and 9th beams. The instantaneous phase θ(x; t) spatially varies almost linearly fromthe root to the apex.
Results of the same measurement applied to four other
healthy young male subjects (subjects B-E) can be seen inof the distribution
the motion pictures , , , andof the phase values for 60-Hz component. Propagation of
the pulsive waves is clearly observed for all these subjects.
Fig. 3. Spatial distributions of color-coded phase values for 60-Hz
component of the wavelets in Fig. 2. The spatial distributions areshown consecutively from T0 − 8 ms to T0 + 10 ms around the timeof AVC (T0 ). White lines show the instantaneous phase θ(x; t) used in(1) of the pulsive wave at the LV surface of the IVS, and blue lines
show the estimated phase θ(x; t) in (1). These phase values wereinterpolated between the adjacent ultrasonic beams and are shown
shows the distributions of thehere. The supplemental animation
phase values for 60-Hz component in the parasternal view for subject
A. Each ﬁgure around the time of AVC (T0 ) is shown here.

lar ﬁgure in both panels of Fig. 3. In Fig. 3, the phase
distributions are shown for a 60-Hz frequency component
along the IVS around the time of AVC (T0 ). For example,
at a time t = T0 + 2 ms in Fig. 3, the phase values varyfrom cyan (+180 degrees) near the root of the aortic valve,through green (+90 degrees), and to red (0 degree) at theapex. As shown in the upper-right circular ﬁgures, the cyancorresponds to the downward pulse that exists exactly at
the center position of the multiplied Hanning window. The
green corresponds to a 90-degree delay for the downward
pulse observed in the analyzed window, and the red corresponds to a 180-degree delay for the same downward pulse.
These facts show the possibility that there is a time delayin the spatial distribution and that this time delay gradually increases from 0 ms near the root of the aortic valve
to +9 ms (= +180◦/(360◦ × f0 )) near the apex.
C. Propagation Phenomenon of Pulsive Wave Along
Heart Wall
Based on the imaging method in [24], Fig. 2 and one of
the cross-sectional images in Fig. 3 are obtained. It can be

D. Propagation Phenomenon of Longitudinal Component
Along the Base-Apex Direction
Until now in the present study, the shear component
with motion along the ultrasonic beam propagating along
IVS has been shown in the parasternal view as in Figs. 2,
3, and in motion pictures
,
,
,
, and
. However, by attaching the ultrasonic probe to the apex (apicalapproach) in in vivo experiments, the longitudinal component propagating along the IVS is shown as follows. Fig. 4
shows a conventional B-mode image of the IVS from theapex just before the time of AVC (T0 ) for the same subject A as in Figs. 1, 2, and 3. The conventional motion
pictureis available online. Because it is necessary todetect a deeper area, the direction in which the ultrasonicbeams are transmitted is limited to nine as shown by thearrows in Fig. 4. By applying the same procedure as inFig. 3, Fig. 5 shows the consecutive spatial distributionsof the instantaneous phases for the 60-Hz component of
the detected vibrations at all points along the nine beams.
Their motion picture
reveals that a few pulses also are
excited around the time of AVC (T0 ) and propagate along
the IVS from the root of the aortic valve to the apex.

III. Estimation of Viscoelasticity Using Lamb
Wave and Voigt Models
Based on these results, in this section, the principle ofdetermination of the phase velocity and modeling of thepulsive wave are proposed to estimate the myocardial viscoelasticity.

kanai: medical ultrasound tissue characterization

1935

Fig. 4. Cross-sectional image in the apical view exactly at the timeof AVC (t = T0 ) measured by a conventional ultrasonic diagnosis
system for the same subject A as in Figs. 1, 2, and 3. Nine ultrasonic beams were used to measure the vibrations at points preset in
the IVS for Fig. 5. The blue lines show the directions of the nineshows the conultrasonic beams. The supplemental animation
ventional B-mode image in the apical approach obtained by a conventional ultrasonic diagnosis equipment for subject A. This ﬁgureshows the cross-sectional image just at the time of AVC (T0 ).

A. Determination of Instantaneous Phase Velocity
To determine the instantaneous phase velocity of thepulsive wave, if the straight line with a constant gradient k [rad/m] is spatially ﬁt to the instantaneous spatial
distribution of the measured phase in the region on theleft-hand side in each cross-sectional image in Fig. 3, theinstantaneous phase velocity for the frequency component
f0 is determined by 2πf0 /k. However, because there is initially some spatial distribution of the phase that is independent of the propagation, the wavelength and thus thewave number k do not correspond to the actual phase velocity.
Therefore, in this study, based on the deﬁnition of thephase velocity, the distance ∆x between two consecutively
obtained phase distributions is determined. For this, thephase distribution θ(x; t) at a time t is compared with theshifted phase distribution θ(x + ∆x; t − ∆T ) obtained ata time t − ∆T . In actual calculation, a complex exponential function is introduced so that no special unwrapping
procedure is necessary for the phase jump between −π
and +π. The root γ(∆x; ω) of the averaged squared diﬀerence between the two exponential functions of θ(x; t) andθ(x+∆x; t−∆T ) for the angular frequency ω is deﬁned by:γ(∆x; ω) =
 |exp{jθ(x; t)} − exp{jθ(x + ∆x; t − ∆T )}|2
 x

. (1)


|exp{jθ(x; t)}|2x is determined so that the
The optimum distance ∆xroot γ (∆x; ω) is minimized. The phase velocity cphaseFig. 5. For the data in Fig. 4, the spatial distributions in the apicalview of color-coded phase values for the 60-Hz component of themeasured wavelets consecutively from T0 − 8 ms to T0 + 8 ms aroundthe time of AVC (T0 ). The blue line in the left-hand side of each ﬁgureshows the instantaneous phase θ(x; t) along the 4th ultrasonic beam.
The phase jump is not unwrapped. The supplemental animationshows the distribution of the phase values for 60-Hz component in
the apical view for subject A. Each ﬁgure around the time of AVC(T0 ) is shown here.

[m/s] then is obtained for each angular frequency ω by
cphase (ω) = ∆x/∆T, where ∆T = 1.96 ms for the experiment.
B. Vibration Modeling
Regarding the phenomenon of the wave propagation
along the viscoelastic plate with thickness 2h, it is wellknown that there are three kinds of plate wave [28]–[31],that is: the shear horizontal (SH)-wave, the Lamb wave
with symmetric mode (shear vertical (SV) wave with extensional vibration), and the Lamb wave with asymmetricmode (SV-wave with ﬂexural vibration as illustrated inFig. 6), in which symmetric and asymmetric refer to thecentral plane (y = 0 in Fig. 6) of the plate. In the parasternal longitudinal-axis view of Fig. 3, the direction (−y) of
each ultrasonic beam is almost perpendicular to the IVS,
and the detected motion is along each beam (y-directionin Fig. 6). As shown in Fig. 3 and the motion picture
,

1936

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005



kL2 − kp2 [rad/m], β =kL2 − ks2 [rad/m], andηb = kL2 − kb2 [rad/m]. ρm and ρb are the myocardium
density and the blood density, respectively, and ρm can beapproximated by ρb = 1.1 × 103 kg/m3 . The thickness 2hof the IVS is determined from the conventional B-mode
image.
η =C. Introduction of Voigt Model and Approximation
Due to µ  λFig. 6. Lamb wave with asymmetric mode of plate waves in theviscoelastic plate with thickness 2h. The SV-wave component (ydisplacement) and longitudinal component (x-displacement) are coupled; the Lamb wave then propagates along the x-direction. Althougha slightly higher order mode is illustrated, the lowest mode is probably dominant in actual vibration in the IVS.

the pulsive wave propagates along the IVS (x-direction).
As shown in the spatial distribution of Fig. 2, the vibrations at the RV side and those at the LV side are almost
parallel (asymmetric). The wavelength λ is about 100 mm
for the 30-Hz component and 40–65 mm for the 90-Hz
component. The thickness 2h of the IVS is about 10 mm
in healthy adults. Thus, the thickness is suﬃciently thin.
Moreover, Fig. 5 and the motion picture
show that thelongitudinal component with x-displacement also propagates along the IVS (x-direction). By comparing Fig. 3 andthe motion picturewith Fig. 5 and the motion picture, the propagation speed of the shear component is seen
to be close to that of the longitudinal component. Thus,
there is a likelihood of coupling between the SV component and the longitudinal component. Given these facts,
the detected vibration signal can be modeled by a Lambwave with asymmetric mode in Fig. 6. Though the wavelength of 100 mm at 10 Hz corresponds to the size of thewhole heart, the duration of the notch pulse is very short(10 ms), corresponding to about 1 cm in length. Thus, thetraveling wave does not overlap the reﬂected wave in thismeasurement.
For the IVS, the blood in RV and LV should be considered. Thus, the model of the Lamb wave of Fig. 6,
propagating along the viscoelastic plate (IVS) immersedin blood, is used in the present study. Let us deﬁne thewave number of the Lamb wave by kL . By referring toequation 4.24 in [31], the function—termed f (kL , kp , ks ),which should be zero—is given by:f (kL , kp , ks ) = 4kL2 ηβ cosh(ηh) sinh(βh)− (2kL2 − ks2 )2 sinh(ηh) cosh(βh)ρb ηks4−cosh(ηh) cosh(βh) = 0,
ρm ηb

(2)

where kp and ks are the wave numbers of the primary(longitudinal) wave and the secondary (shear) wave, respectively, in the myocardium. Using kp , ks , and thewave number kb in blood, η, β, and ηb are deﬁned by

By introducing a single Voigt dash-pot model in the frequency range up to 90 Hz, the Lamé elastic constants λand µ become complex values as λ = λ1 + jωλ2 [Pa] andµ = µ1 + jωµ2 [Pa], respectively [32]. Because λ is about
103 times larger than µ for soft tissue dueto noncompressibility
[32],
k
is
approximatedbyk=ω
ρm /(λ + 2µ) ≈ppω ρm /λ = kb . Because the wave number kL of the Lambwave is close to ks , kL2 is about 103 times larger than kp2 .
Thus, η = kL2 − kp2 ≈ kL and ηb = kL2 − kb2 ≈ kL .
Therefore, f (kL , kp , ks ) of (2) is approximated by:f  (cL , µ, ω) = 4kL3 β cosh(kL h) sinh(βh)− (2kL2 − ks2 )2 sinh(kL h) cosh(βh)(3)− ks4 cosh(kL h) cosh(βh) = 0,where kL = ω/cL (µ, ω) [rad/m] and ks = ω ρm /µ[rad/m]. Thus, f  (cL , µ, ω) depends on both the phase velocity cL (µ, ω) of the Lamb wave and the Lamé elastic
constant µ = µ1 + jωµ2 .
D. Actual Procedure to Determine the Lamé Elastic
Constant µ
Based on the model described above, the theoreticalvalue of the phase velocity cL (µ, ω) of the Lamb wave
is determined to be close to the measured phase velocity cphase (ω), where cL (µ, ω) depends on µ = µ1 + jωµ2 .
For the nonlinear optimization, we follow the proceduredescribed below.
Assuming one combination of values µ1 and µ2 , µ is
given for an angular frequency ω. For the value of µ, by
setting one value of the phase velocity cL (µ, ω), the wave
number kL = ω/cL and all other parameters included inf  (cL , µ, ω) of (3) can be obtained. The phase velocitycL (µ, ω) then is determined for the assumed value µ sothat f  (cL , µ, ω) becomes zero. Actually, cL (µ, ω) is chosen
so that the absolute value f  (cL , µ, ω) is closest to zero.
By varying the angular frequency ω, the values {cL (µ, ω)}of the phase velocity are all determined for the assumed
value µ. Thus, the root of the normalized mean squared
diﬀerence, α(µ1 , µ2 ), between these theoretical phase velocities {cL (µ, ω)} and the measured ones {cphase (ω)} isdeﬁned as:2 w(ω)2 |cphase (ω) − cL (µ, ω)|ωα(µ1 , µ2 ) = [m/s],w(ω)2(4)ω

kanai: medical ultrasound tissue characterization

1937

where w(ω) is the weight for each measurement of cphase (ω)
and corresponds to its reliability. In this study, the inverse of the minimum of the squared diﬀerence γ (∆x; ω)of (1) between the consecutive phase distributions is used
as w(ω). By scanning the combination of the complex valueµ, the optimum Lamé constant µ = µ1 + jωµ2 is determined so that α(µ1 , µ2 ) is minimized:µ1 , µ2 ) = min α(µ1 , µ2 ) [m/s].
αmin (µ1 , µ2(5)

By this procedure, for each time t, from the measured dispersion, the optimum Lamé constant µ (shear elastic constant µ1 and shear viscosity constant µ2 ) is determinedusing the model of the Lamb wave that propagates along
the thin viscoelastic plate (IVS) immersed in blood.

IV. In Vivo Experimental Results for Healthy
Subjects
A. Phase Velocity of the Pulsive Wave
By analyzing the spatial distribution of the phase ofeach wavelet in Fig. 2 for each time from T0 − 8 ms to
T0 + 10 ms and for each frequency component from 10 Hz
to 90 Hz, the instantaneous phase velocity cphase (ω) was
obtained as in Fig. 7(b). There is dispersion among the instantaneous phase velocities {cphase (ω)}, and they rapidly
decrease for all frequency components in this short periodaround the time of AVC (T0 ). Exactly at the time of AVC(T0 ), the instantaneous phase velocities {cphase (ω)} of the27-, 60-, and 87-Hz components are 2.0, 4.0, and 4.5 m/s,
respectively. As described in Section V, the order of magnitude of these velocity values is similar to those measured
in in vitro experiments in the literature.
 ω of the root ofFig. 7(c) shows the minimum γ ∆x;
the averaged squared diﬀerence of (1) for the frequency off = ω/2π hertz. For higher frequencies, the errors becomelarger. However, for less than 70 Hz, the velocity estimate
cphase (ω) is fairly reliable. These errors are used as theinverse of the weight w(ω) in (4).
The phase velocity of the longitudinal component propagating from the root of the aortic valve to the apex wasdetermined as follows. The blue line in each panel of Fig. 5
shows the spatial distribution of the phase θ(x; t) for the
60-Hz component. In the depth range from 60 mm to
90 mm, the phase velocity cphase (f ) was determined basedon the procedure in Section III-A. At a time T0 − 10 ms,
cphase (f ) is 5.5 m/s for the 60-Hz component. As timeincreases, cphase (f ) gradually decreases, and at a time
T0 + 10 ms, cphase (f ) is 4.0 m/s. These velocity values areclose to those of the shear component obtained in Fig. 7(b).
B. Viscoelasticity of IVS and the Transient
For subject A, the estimated elasticity is about 24–
30 kPa and does not change around the time of AVC.
In the experiments, the viscosity can be determined more

Fig. 7. (a) The analyzed period from T0 −8 ms to T0 +10 ms in Fig. 3
is shown by using the waveforms of electrocardiogram (ECG), thoseof PCG, velocity signals at the RV side along the 9th ultrasonic beam,
velocity signals at the LV side along the 9th ultrasonic beam, velocity
signals at the RV side along the 13th ultrasonic beam, and velocity
signals at the LV side along the 13th ultrasonic beam. Each waveform
contains the signals for two consecutive cardiac cycles. (b) Measured
instantaneous phase velocity values {cphase } and ﬁtted theoreticalcurves cL (µ, f ) using the optimization procedure in Section III-D forthe frequency components from 10 Hz to 90 Hz during the period
from T0 − 8 ms to T0 + 10 ms in (a). (c) For each time from T0 − 8 ms
x; ω) of the root of theto T0 + 10 ms, the achieved minimum γ(∆
averaged squared diﬀerence between the phase components of (1),where f = ω/2π Hz.

1938

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005

Using the estimated Lamé constant µ, the phase velocµ, f ) is obtained in the procedures in Section III-D.
ity cL (The results are shown by the solid lines in Fig. 7(b) for
each time t. The estimates {cL (µ, f )} well ﬁt the measured
dispersion characteristics cphase (f ) in Fig. 7(b).

V. Discussion

Fig. 8. (a) Transient of the viscosity parameters {µ2 } measured during the period from T0 − 8 ms to T0 + 10 ms in Fig. 7 for ﬁve healthy
young male subjects, including subject A. Supplemental animationof the distribution of the phase values for 60-Hz component is available, showing the distribution of the phase values for 60-Hz component in the parasternal view for subjects B
,C
,D
, and
E

. Subject B, 25 years old; subject C, 21 years old; subject D,

25 years old; and subject E, 23 years old. (b) The minimum value
αmin (µ1 , µ2 ) [m/s] of the normalized mean squared diﬀerence of (5)for each subject.

precisely than the elasticity. In Section V, the precisenessin the estimation of the elasticity and the viscosity is discussed. The transient of the viscosity {µ2 } determined foreach time from T0 − 8 ms to T0 + 10 ms in Fig. 7 for subject A is shown in Fig. 8(a). Around the beginning of theIR period, the instantaneous viscosity of the myocardium
rapidly decreases from about 400 Pa·s to 70 Pa·s. As described in Section V, this would be due to the rapid decrease in the LV inner pressure from about 120 mmHg toseveral milliters of mercury, which is caused by relaxationof the myocardium.
For the other four subjects, B–E, the same measurement
and analysis were applied, and similar transient characteristics of the viscosity were obtained as shown in Fig. 8(a).
Fig. 8(b) shows the minimum value αmin (µ1 , µ2 ) of themean squared diﬀerence of (5) between the measured phasevelocities {cphase (ω)} and the theoretical ones {cL (µ, ω)}for each analysis. The errors are at most 0.3–0.8 m/s inthe analysis. These errors correspond to about 10–20% of
the phase velocity at the center frequency (50 Hz).

As shown in Section IV-A, the phase velocity measured
in the present study ranges from 1 to 7 m/s for the relaxation phase. As shown in Table I, the order of magnitudeof the phase velocity values is similar to those measured recently using low-frequency, shear-wave vibrations reported
in the literature for human quadriceps muscle [2], for human bicep muscle [5], for pig leg/shoulder muscles [1], andfor beef muscle [4]. It is known that the propagation speedof the electrically excited waves along the cardiac muscle
and the Purkinje ﬁber in the myocardium is about 0.3 m/s[8], [9] and 2–3 m/s [10], respectively. However, the electrical excitation mainly occurs around the beginning of theejection period. Therefore, the pulsive wave propagation
found around the AVC in the present study is a mechanical phenomenon.
For the IR period, in which the myocardial viscoelasticity is estimated, the myocardium can be characterized as
being passive. The clinical eﬀect of such viscous properties
before the onset of ﬁlling on myocardium relaxation wasinvestigated in [33]. Application of the present method tothe beginning of systole, in which the mitral valve closes
and the ﬁrst heart sound is radiated, could possibly contribute to the determination of the active properties of the
myocardium as modeled in [34], [35].
The elastic values and viscosity values obtained in thepresent study are compared with those reported for the
myocardium and soft tissues in the literature as follows.
The methods used can be divided into the following two
categories with regard to the frequency (see Table II). Except for those of the present study, all data were measured
in in vitro experiments.
In [36], by the ultrasonic measurement of transversemode acoustic impedance at 2–14 MHz, the shear stiﬀnessµ1 and viscosity µ2 were obtained as 1 MPa and 0.004–
0.03 Pa·s for canine soft tissue. In [37], using a similar
method, µ1 and µ2 were 124 kPa and 0.0159 Pa·s, respectively, for the cardiac muscle. In [38], using a 3-MHz transverse mode propagated perpendicular to the ﬁber axis of
a formalin-ﬁxed normal human myocardium, the stiﬀness(C66 ) and viscosity, calculated from the velocity and attenuation, were 8.45 MPa and 0.404 Pa·s, respectively.
On the contrary, with mechanical measurement using
audio or lower frequency components, using a surface wave[39], the elastic component (C66 ) and viscosity (µ2 ) were
measured as 5 kPa and 1.6 Pa·s, respectively, at 2 kHz
for bovine myocardium [40]. In [32], for the low frequencyof 500–5,000 Hz, the shear parameters were measured asµ = 2.5 kPa + j15 Pa·s for human muscle tissue. In [41], forthe canine left ventricle, using pressure perturbations, µ1

kanai: medical ultrasound tissue characterization

1939

TABLE I
Typical Values of Speed of the Shear Wave Measured for Soft Tissues in the Literature and Present Study.
Frequency used

Speed of sound

20 Hz
30–120 Hz
50 Hz
50 Hz
90 Hz
100–225 Hz
About 150 Hz

1–2 m/s
4–5.4 m/s
3–4 m/s
About 2 m/s
3–7 m/s
5–6.5 m/s
5.3 m/s

Measured region

Reference

Human myocardium (during IR, in vivo)
Human quadriceps (in vivo)
Human myocardium (during IR, in vivo)
Human bicep (no load)
Human myocardium (during IR, in vivo)
Pig leg and shoulder muscle
Beef muscle (2-ms pulsed excitation)

Present study (Fig. 7)[2]
Present study (Fig. 7)[5]
Present study (Fig. 7)[1][4]

TABLE II
Typical Values of the Elasticity µ1 and Viscosity µ2 Measured for Soft Tissues in the Literature and Present Study.1
Frequency used

Elasticity µ1

Viscosity µ2

About 2 Hz

290–1,460 kPa

2,800–19,000 Pa·s

10 Hz

1,000–7,000 kPa

5,000–30,000 Pa·s

20 Hz
22 Hz
10–90 Hz
50 Hz
50 Hz
150 Hz500–5,000 Hz
2,000 Hz7.5 kPa
30–400 kPa
30 kPa30–400 kPa
30 kPa
21 kPa2.5 kPa
5 kPa

3 MHz
2–14 MHz
2–14 MHz8,450 kPa
124 kPa
1,000 kPa

Measured region

Reference

75 Pa·s
200–1,400 Pa·s
70–400 Pa·s
30–200 Pa·s
30 Pa·s
23 Pa·s
15 Pa·s
1.6 Pa·s

Canine descending thoracic
aorta to femoral artery
Human carotid artery to humanfemoral arteryPig kidney
Canine left ventricle
Human myocardium (in vivo)
Cow muscle
Cow muscle (freshest)
Beef muscle
Human muscle tissue
Bovine myocardium

[43]

[44]
[41]
Present study
[45]
[45]
[4]
[32]
[40]

0.4 Pa·s
0.02 Pa·s
0.004–0.03 Pa·s

Formalin ﬁxed human myocardium
Cardiac muscle
Canine soft tissue

[38]
[37]
[36]

[42]

1 Except for the present study, all data were measured in in vitro experiments.

and µ2 were determined as 30–400 kPa and 200–1,400 Pa·s,
respectively, for 22 Hz. In [42], the artery was subjected topressure oscillations, and µ1 and µ2 were measured fromthe human carotid artery to the femoral artery. For 10 Hz,µ1 and µ2 were 1–7 MPa and 5–30 kPa·s, respectively.
Similar experiments were conducted in [43]. In [44], using
a standard strain-controlled rheometer, the shear elasticity and viscosity of fresh pig kidney were 1–7.5 kPa and
20–75 Pa·s, respectively, for 0.01 to 20 Hz with a strain of
0.2%. In [45], by applying an electromagnetic transducer
to muscle tissue taken from freshly killed cows in a frequency range from 25–2,500 Hz, in which the shear stress
was applied parallel to the muscle ﬁbers, µ1 = 30–400 kPa
and µ2 = 30–200 Pa·s for 50 Hz. Especially in the freshest specimen, the values were lower at µ1 = 30 kPa andµ2 = 30 Pa·s for 50 Hz. The elasticity slightly increased
with increased frequency, but the viscosity remarkably decreased with the increase in frequency. In [45], moreover,it was experimentally found that both µ1 and µ2 increase
with time after slaughtering. That paper [45] discussed
that this dependency on the elapsed time is related to theloss of water from the fresh tissue.
Roughly speaking, the elasticity is large for very low
frequency less than 10 Hz and for frequency higher than
1 MHz. In [44], the experimental results for pig kidney

showed that the elasticity increases with frequency in thefrequency range from 0.01 Hz to 20 Hz. The viscosity data
are well ﬁtted to µ2 = 626.7 × f −0.722 [Pa·s], which showsthat the viscosity decreases with the increase in frequency
f . Especially at megahertz frequencies (in the ﬁrst category above), the viscosity values in the literature are much
lower than those reported at audio frequencies (in the second category). By considering the frequency dependency
and the freshness of the specimens in Table II, the elasticity and viscosity measured in the present study are close
to those measured for the same frequency range in theliterature.
In [46], the anisotropy of the speed of sound and theelastic properties have been measured. In the actual heartwall, along the radial axis (the thickness direction) of theLV, the myocardial ﬁber orientation changes gradually. For
patients with hypertrophic cardiomyopathy, parallel alignment of muscle cells becomes irregular (myocardial disarray). However, in the present study, such anisotropy andchange in ﬁbrous orientation are not considered at all.
When the preciseness of elasticity and viscosity is compared, it is apparent that the elasticity is not precisely
estimated. The reason is due to the ﬁtting of the theoretically derived phase values {cL (µ, f )} to the measured
ones {cphase (f )}. Because the Voigt model µ = µ1 +jωµ2 is

1940

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005

used, the gradient of the dispersion characteristics in Fig. 7
mainly depend on the viscosity µ2 , and there is large dispersion in the frequency range from 10 to 90 Hz in thesein vivo experiments. Thus, the viscosity is relatively precisely estimated. However, the elasticity µ1 corresponds tothe bias component (the velocity of the frequency components close to 0 Hz) of the dispersion characteristics.
Because the detected notch pulse is nonstationary and theshort-time Fourier transform is applied to the pulse, it isdiﬃcult to detect its very low frequency components andthe propagation. Thus, the viscosity can be precisely determined, but the elasticity cannot.
In Section III-B, the model in which the viscoelastic
plate is immersed in blood is used. The blood density ρb ,the myocardium density ρm , the longitudinal velocity cb inblood, and the longitudinal velocity cm in the myocardium
can be approximated as being almost constant for bloodpressure ranging from 0–120 mmHg. It is known that theviscosity in muscle highly depends on the velocity of thestretch [47] or the instantaneous muscle length [34]. However, in the short period analyzed in the present study, thechange in length is negligible. According to [2] and [5], theshear wave velocity and shear elasticity of the human muscle increase as a load is applied in in vivo experiments. As
in [48] and [49], the viscosity also depends on the pressure.
Thus, there is a close relationship between the transient of
the viscosity obtained in Fig. 8(a) and the change in innerpressure of the LV. Thus, the blood pressures in RV and
LV and their transient should be further introduced into
the boundary conditions for the wave equations.
The estimation process of the viscoelasticity in thisstudy should be validated using an elastic spherical shell
as a heart phantom; this is currently being conducted. Bysuch phantom experiments, it also should be conﬁrmed
whether the transient characteristics in viscosity correspond to the change in inner pressure.
In [50], experimental results on soft tissue have been
discussed in the framework of the linear theory of viscoelasticity relating stress and strain on the basis of the
Voigt, Maxwell, and Kelvin models. The main objective of
these models is to identify the parameters that are directly
related to the mechanical behavior of the constituents ofthe tissue to produce a simple description of the gross mechanical behavior of the tissue. Such behavior is inﬂuenced
by collagen and its interactions with other componentsof the tissue [44], by water content [45], and by time after death [51]. Not only the myocardial elasticity but also
the viscosity depend on the interaction between the giantmuscle protein titin and actin ﬁlaments in the sarcomere during both diastolic stretch and systolic shortening[52]. However, the mechanical response of many soft tissues to mechanical loading is not well described by current models in [44]. Thus, the physiological origin of the
myocardial viscoelasticity should be further investigated.
At the same time, more accurate modeling, such as theMaxwell or Kelvin models and their combinations, should
be undertaken, whereas the Voigt model was used in thepresent study. Many researchers have measured the relax-

ation time in the step response of the soft tissue [34], [53],in which the relaxation time can be compared with theviscosity using appropriate modeling.

VI. Conclusions
We measured rapid and minute vibrations simultaneously at multiple points in the IVS. We found that the pulsive wave is spontaneously actuated due to the AVC at the
beginning of the IR period. Clear propagation of the pulsive wave along the IVS was recognized for the ﬁrst time.
From the dispersion of the phase velocities, the myocardial viscoelasticity was determined noninvasively for theﬁrst time. This method oﬀers potential for in vivo imaging of the spatial distribution of the passive mechanicalproperties of the myocardium and its rapid change during
the IR period, which would enable direct assessment of diastolic properties based on myocardial relaxation in heart
failure [54], [55], which cannot be obtained by conventional
echocardiography, CT, or MRI.

Acknowledgments
The author is grateful to the reviewers of the present
paper for helpful and valuable advice. The author also
is grateful to Prof. Emeritus Motonao Tanaka of Tohoku
University, Prof. Floyd Dunn of the Bioacoustics ResearchLaboratory of the University of Illinois, and Dr. Jens E.
Wilhjelm of the Technical University of Denmark for useful discussions. The author also acknowledges the experimental contributions of Dr. Hideyuki Hasegawa in ourlaboratory.

References[1] Y. Yamakoshi, J. Sato, and T. Sato, “Ultrasonic imaging of internal vibration of soft tissue under forced vibration,” IEEE Trans.
Ultrason., Ferroelect., Freq. Contr., vol. 37, no. 2, pp. 45–53,
1990.
[2] S. F. Levinson, M. Shinagawa, and T. Sato, “Sonoelastic determination of human skeletal muscle elasticity,” J. Biomechan.,vol. 28, no. 10, pp. 1145–1153, 1995.
[3] S. Catheline, J.-L. Thomas, F. Wu, and M. A. Fink, “Diﬀraction
ﬁeld of a low frequency vibrator in soft tissues using transient
elastography,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr.,
vol. 46, no. 4, pp. 1013–1019, 1999.
[4] S. Catheline, F. Wu, and M. Fink, “A solution to diﬀraction
biases in sonoelasticity: The acoustic impulse technique,” J.
Acoust. Soc. Amer., vol. 105, no. 5, pp. 2941–2950, 1999.
[5] L. Sandrin, M. Tanter, J.-L. Gennisson, S. Catheline, and M.
Fink, “Shear elasticity probe for soft tissues with 1-D transient
elastography,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr.,
vol. 49, no. 4, pp. 436–446, 2002.
[6] V. Dutt, R. R. Kinnick, and J. F. Greenleaf, “Acoustic shear
wave displacement using ultrasound,” in Proc. IEEE Ultrason.
Symp., 1996, pp. 1185–1188.
[7] S. Chen, M. Fatemi, and J. F. Greenleaf, “Complex stiﬀness
quantiﬁcation using ultrasound stimulated vibrometry,” in Proc.
IEEE Ultrason. Symp., 2003, pp. 941–944.
[8] O. Anosov, S. Berdyshev, I. Khassanov, M. Schaldach, and B.
Hensel, “Wave propagation in the atrial myocardium: Dispersion

kanai: medical ultrasound tissue characterization
properties in the normal state and before ﬁbrillation,” IEEE
Trans. Biomed. Eng., vol. 49, no. 12, pp. 1642–1645, 2002.
[9] S. P. Thomas, J. P. Kucera, L. Bircher-Lehmann, Y. Rudy, J. E.
Saﬃtz, and A. G. Kléber, “Impulse propagation in synthetic
strands of neonatal cardiac myocytes with genetically reduced
levels of connexin43,” Circ. Res., vol. 92, pp. 1209–1216, 2003.
[10] N. Sperelakis, “Origin of the cardial resting potential,” in Handbook of Physiology. The Cardiovascular System. R. M. Berne,
N. Sperelakis, and S. R. Geiger, Eds. Bethesda, MD: American
Physiological Society, 1979, p. 190.
[11] L. T. Mahoney, W. Smith, M. P. Noel, M. Florentine, D. J.
Skorton, and S. M. Collins, “Measurement of right ventricularvolume using cine computed tomography,” Invest. Radiol., vol.
22, no. 6, pp. 451–455, 1987.
[12] E. A. Zerhouni, D. M. Parish, W. J. Rogers, A. Yang, and E. P.
Shapiro, “Human heart: Tagging with MR imaging—A method
for noninvasive assessment of myocardial motion,” Radiology,
vol. 169, pp. 59–63, 1988.
[13] L. Axel, R. C. Goncalves, and D. Bloomgarden, “Regional heart
wall motion: Two-dimensional analysis and functional imaging
with MR imaging,” Radiology, vol. 183, no. 3, pp. 745–750, 1992.
[14] A. D. Fleming, X. Xia, W. N. McDicken, G. R. Sutherland, and
L. Fenn, “Myocardial velocity gradients detected by Doppler
imaging,” Br. J. Radiol., vol. 67, no. 799, pp. 679–688, 1994.
[15] R. S. Adler, J. M. Rubin, P. H. Bland, and P. L. Carson, “Quantitative tissue motion analysis of digitized M-mode images: Gestational diﬀerences in fetal lung,” Ultrasound Med. Biol., vol.
16, no. 6, pp. 561–569, 1990.
[16] G. R. Sutherland, M. J. Stewart, K. W. E. Groundstroem, C. M.
Moran, A. Fleming, F. J. Guell-Peris, R. A. Riemersma, L. N.
Fenn, K. A. A. Fox, and W. N. McDicken, “Color Doppler myocardial imaging: A new technique for the assessment of myocardial function,” J. Amer. Soc. Echocardiogr., vol. 7, pp. 441–458,
1994.
[17] A. Heimdal, A. Støylen, H. Torp, and T. Skjærpe, “Real-timestrain rate imaging of the left ventricle by ultrasound,” J. Amer.
Soc. Echocardiogr., vol. 11, pp. 1013–1019, 1998.
[18] K. Miyatake, M. Yamagushi, N. Tanaka, M. Uematsu, N. Yamazaki, Y. Mine, A. Sano, and M. Hirama, “New method forevaluating left ventricular wall motion by color-coded tissue
Doppler imaging: in vitro and in vivo studies,” J. Amer. Coll.
Cardiol., vol. 25, no. 3, pp. 717–724, 1995.
[19] P. Palka, A. Lange, A. D. Fleming, G. R. Sutherland, L. N.
Fenn, and W. N. McDicken, “Doppler tissue imaging: Myocardial wall motion velocities in normal subject,” J. Amer. Soc.
Echocardiogr., vol. 8, pp. 659–668, 1995.
[20] J. Gorcsan, III, V. K. Gulati, W. A. Mandarino, and W. E.
Katz, “Color-coded measures of myocardial velocity throughout
the cardiac cycle by tissue Doppler imaging to quantify regional
left ventricular function,” Amer. Heart J., vol. 131, no. 6, pp.
1203–1213, 1996.
[21] G. R. Sutherland, G. D. Salvo, P. Claus, J. D’hooge, and B.
Bijnens, “Strain and strain rate imaging: A new clinical approach to quantifying regional myocardial function,” J. Amer.
Soc. Echocardiogr., vol. 17, no. 7, pp. 788–802, 2004.
[22] H. Kanai, M. Sato, Y. Koiwa, and N. Chubachi, “Transcutaneous measurement and spectrum analysis of heart wall vibrations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43,
no. 5, pp. 791–810, 1996.
[23] H. Kanai, H. Hasegawa, N. Chubachi, Y. Koiwa, and M. Tanaka,“Noninvasive evaluation of local myocardial thickening and its
color-coded imaging,” IEEE Trans. Ultrason., Ferroelect., Freq.
Contr., vol. 44, no. 4, pp. 752–768, 1997.
[24] H. Kanai and Y. Koiwa, “Myocardial rapid velocity distribution,” Ultrasound Med. Biol., vol. 27, no. 4, pp. 481–498, 2001.
[25] F. Jamal, T. Kukulski, J. Strotmann, M. Szilard, J. D’hooge,
B. Bijnens, F. Rademakers, L. Hatle, I. D. Scheerder, and G. R.
Sutherland, “Quantiﬁcation of the spectrum of changes in regional myocardial function during acute ischemia in closed chest
pigs: An ultrasonic strain rate and strain study,” J. Amer. Soc.
Echocardiogr., vol. 14, no. 9, pp. 874–885, 2001.
[26] H. Kanai, Y. Koiwa, and J. Zhang, “Real-time measurements oflocal myocardium motion and arterial wall thickening,” IEEE
Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, no. 5, pp.
1229–1241, 1999.

1941[27] H. Kanai, S. Yonechi, I. Susukida, Y. Koiwa, H. Kamada, and
M. Tanaka, “Onset of pulsatile waves in the heart walls at endsystole,” Ultrasonics, vol. 38, no. 1, pp. 405–411, 2000.
[28] B. A. Auld, Acoustic Fields and Waves in Solids. 2nd ed. vol.
II, Malabar, FL: Krieger, 1990, pp. 63–220.
[29] H. F. Tiersten, Linear Piezoelectric Plate Vibrations. New York:
Plenum, 1969, pp. 95–118.
[30] I. A. Viktorov, Rayleigh and Lamb Waves. New York: Plenum,
1967, pp. 67–121.
[31] T. Kundo, Ultrasonic Nondestructive Evaluation. New York:CRC Press, 2003, pp. 223–310.
[32] H. L. Oestreicher, “Field and impedance of an oscillating spherein a viscoelastic medium with an application to biophysics,” J.
Acoust. Soc. Amer., vol. 23, no. 6, pp. 707–714, 1951.
[33] S. D. Nikolic, K. Tamura, T. Tamura, M. Dahm, R. W. M.
Frater, and E. L. Yellin, “Diastolic viscous properties of the intact canine left ventricle,” Circ. Res., vol. 67, no. 2, pp. 352–359,
1990.
[34] L. Loeﬄer, III and K. Sagawa, “A one-dimensional viscoelastic
model of cat heart muscle studied by small length perturbations
during isometric contraction,” Circ. Res., vol. 36, no. 4, pp. 498–
512, 1975.
[35] L. B. Katsnelson, L. V. Nikitina, D. Chemla, O. Solovyova, C.
Coirault, Y. Lecarpentier, and V. S. Markhasin, “Inﬂuence ofviscosity on myocardium mechanical activity: A mathematicalmodel,” J. Theoretical Biol., vol. 230, no. 3, pp. 385–405, 2004.
[36] L. A. Frizzell, E. L. Carstensen, and J. F. Dyro, “Shear properties of mammalian tissues at low megahertz frequencies,” J.
Acoust. Soc. Amer., vol. 60, no. 6, pp. 1409–1411, 1976.
[37] E. L. Madsen, H. J. Sathoﬀ, and J. A. Zagzebski, “Ultrasonicshear wave properties of soft tissues and tissue-like materials,” J.
Acoust. Soc. Amer., vol. 74, no. 5, pp. 1346–1355, 1983.
[38] B. K. Hoﬀmeister, S. E. Gehr, and J. G. Miller, “Anisotropy of
the transverse mode ultrasonic properties of ﬁxed tendon andﬁxed myocardium,” J. Acoust. Soc. Amer., vol. 99, no. 6, pp.
3826–3836, 1996.
[39] Y. Onodera and P.-K. Choi, “Surface-wave modes on soft
gels,” J. Acoust. Soc. Amer., vol. 104, no. 6, pp. 3358–3363,
1998.
[40] P.-K. Choi, H. Takahashi, and Y. Onodera, “Determination ofshear elasticity of ﬁbrous soft tissues using surface waves,” IEICE Trans., vol. J84-A, no. 12, pp. 1439–1443, 2001. (inJapanese)[41] G. H. Templeton and L. R. Nardizzi, “Elastic and viscous stiﬀness of the canine left ventricle,” J. Appl. Physiol., vol. 36, no.
1, pp. 123–127, 1974.
[42] B. M. Learoyd and M. G. Taylor, “Alterations with age in theviscoelastic properties of human arterial walls,” Circ. Res., vol.
18, no. 3, pp. 278–292, 1966.
[43] B. S. Gow and M. G. Taylor, “Measurement of viscoelastic properties of arteries in the living dog,” Circ. Res., vol. 23, no. 1, pp.
111–122, 1968.
[44] S. Nasseri, L. E. Bilston, and N. Phan-Thien, “Viscoelastic properties of pig kidney in shear, experimental results and modelling,” Rheol. Acta, vol. 41, pp. 180–192, 2002.
[45] E. R. Fitzgerald, E. Ackerman, and J. W. Fitzgerald, “Preliminary measurement of the viscoelastic properties of animal tissues
at audio-frequencies,” J. Acoust. Soc. Amer., vol. 29, no. 1, pp.
61–64, 1957.
[46] E. D. Verdonk, S. A. Wickline, and J. G. Miller, “Anisotropyof ultrasonic velocity and elastic properties in normal humanmyocardium,” J. Acoust. Soc. Amer., vol. 92, no. 6, pp. 3039–
3050, 1992.
[47] M. I. Noble, “The diastolic viscous properties of cat papillarymuscle,” Circ. Res., vol. 40, no. 3, pp. 288–292, 1977.
[48] W. Philippoﬀ, “Viscoelasticity of polymer solutions at high pressures and ultrasonic frequencies,” J. Appl. Phys., vol. 34, no. 5,
pp. 1507–1511, 1963.
[49] T. A. Litovitz and E. H. Carnevale, “Eﬀect of pressure on ultrasonic relaxation in liquids. II,” J. Acoust. Soc. Amer., vol. 30,
no. 2, pp. 134–136, 1958.
[50] T. L. Szabo, “A model for longitudinal and shear wave propagation in viscoelastic media,” J. Acoust. Soc. Amer., vol. 107,
no. 5, pp. 2437–2446, 2000.
[51] E. R. Fitzgerald, “Dynamic mechanical measurements during
the lift to death transition in animal tissues,” Biorheology, vol.
12, pp. 397–408, 1975.

1942

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 11, november 2005[52] M. Kulke, S. Fujita-Becker, E. Rostkova, C. Neagoe, D. Labeit,
D. J. Manstein, M. Gautel, and W. A. Linke, “Interaction between PEVK-titin and actin ﬁlaments, origin of a viscous force
component in cardiac myoﬁbrils,” Circ. Res., vol. 89, no. 10, pp.
874–881, 2001.
[53] J. G. Pinto and P. J. Patitucci, “Visco-elasticity of passive cardiac muscle,” Trans. ASME, J. Biomechan. Eng., vol. 102, no.
2, pp. 57–61, 1980.
[54] T. J. Fraites, Jr., A. Saeki, and D. A. Kass, “Eﬀect of altering
ﬁlling pattern on diastolic pressure-volume curve,” Circulation,
vol. 96, pp. 4408–4414, 1997.
[55] S. R. Solomon and S. A. Glantz, “Regional ischemia increases sensitivity of left ventricular relaxation to volume inpigs,” Amer. J. Physiol., vol. 276, pp. H1994–H2005, 1999.

Hiroshi Kanai (M’89) was born in Matsumoto, Japan, on November 29, 1958. He received a B.E. degree in Electrical Communications from Tohoku University, Sendai, Japan,
in 1981, and M.E. and Ph.D. degrees, also
from Tohoku University, in 1983 and in 1986,
both in electrical engineering.
From 1986 to 1988 he was with the Education Center for Information Processing, Tohoku University, as a research associate. From1990 to 1992 he was a lecturer in the Department of Electrical Engineering, Faculty of Engineering, Tohoku University. From 1992 to 2001 he was an associate
professor in the Department of Electrical Engineering, Faculty of Engineering, Tohoku University. Since 2001 he has been a professor in
the Department of Electronic Engineering, Graduate School of Engineering, Tohoku University.
His present interests are in ultrasonic measurement and digital
signal processing for diagnosis of heart diseases and arteriosclerosis.
Dr. Kanai is a member of IEEE, the Acoustical Society of Japan,
the Institute of Electronics Information and Communication Engineering of Japan, the Japan Society of Mechanical Engineers, theJapan Society of Ultrasonics in Medicine, the Japan Society of Medical Electronics and Biological Engineering, and the Japanese Circulation Society.