TRANSCUTANEOUS VISCOELASTICITY ESTIMATION OF
HEART WALL USING ULTRASOUNDHIROSHI KANAI
Department of Electronic Engineering, Tohoku University, Aramaki-aza-Aoba 6-6-05,
Sendai 980-8579, Japan
E-Mail: kanai@ecei.tohoku.ac.jp
This paper describes a novel method to noninvasively measure the myocardial
viscoelasticity in vivo to evaluate the heart diastolic properties. By the ultrasonicmeasurement of the myocardial motion, we have already found that some pulsive waves
are spontaneously excited by aortic-valve closure (AVC) at end-systole (T0). In thisstudy, a sparse sector scan at a sufficiently high frame rate clearly reveals wave
propagation along the heart wall. The propagation time of the wave along the heart wallis very small, namely, several milliseconds, and cannot be measured by conventional
equipment. From the measured phase velocity, we estimate the myocardial viscoelasticity
in vivo. In in vivo experiments applied to 6 healthy subjects, the propagation of thepulsive wave was clearly visible in all subjects. For the frequency component up to 90
Hz, the typical propagation speed is about several m/s and rapidly decreased around thetime of AVC. For the healthy subject, the typical value of elasticity was about 24-30 kPa
and did not change around the time of AVC. The typical transient values of viscosity
decreased rapidly from 400 Pa⋅s to 70 Pa⋅s around the time of AVC. The measured shear
elasticity and viscosity in this study are comparable to those obtained for the humantissues using audio frequency in in vitro experiments reported in the literature. Thismethod offers potential for in vivo imaging of the spatial distribution of the passive
mechanical properties of the myocardium, which cannot be obtained by conventional
echocardiography, CT, or MRI.
Keywords: viscoelasticity, myocardium, echocardiography, pulsive wave, Voigt model,
Lamb wave.

1. Introduction
Conventional ultrasonography, CT, and MRI enable clinical visualization ofcross-sectional images of the human heart, but their imaging is restricted to large
motion (> 1 mm) and low frequency components (< 30 Hz). The tissue Doppler
imaging (TDI) technique 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 [1]), that is, thesampling period is 5 ms, which is too long to detect the propagation time of the
wave.

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To measure the original vibrations of the heart sounds, we have alreadydeveloped a method to transcutaneously measure the heart-wall vibrations as a
waveform at one point or multiple points preset along an ultrasonic beam in theheart wall [2]. We found that there is a steep dip in the pulse which occurs
exactly at the time of AVC (T0) [3] as shown in Fig. 1. This notch has been alsomeasured by the TDI approach to determine the time of AVC (T0).

Fig. 1. upper: A cross-sectional image measured by conventional echocardiography in a healthy
young male (subject A). The upper-right illustration shows the scanning range of the ultrasonicbeams 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. lower: In vivo measurement results for thehealthy man at two points set along the 13th ultrasonic beam. Each waveform for six consecutive
cardiac cycles is overlaid.

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As shown in the upper figure of Fig. 1, use of the sparse sector scan [3] hasallowed us to simultaneously measure heart-wall motion at 160 points at a
sufficiently high frame rate to measure the propagation of the notch pulse along
the interventricular septum (IVS). From consecutively obtained spatial
distributions of the phase value of the vibration wave, our study [4] has revealed
for the first time that the steep dip of the notch pulse propagates along the IVS
from the base to the apex, and its phase velocity is determined. By analyzing
various frequency components up to 90 Hz, the propagation speed shows thefrequency dispersion.
This study shows that this dispersion characteristic agrees with thetheoretical one of the Lamb wave which 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, the myocardial viscoelastic properties are determined noninvasively
for the first time.

Fig. 2. Spatial distributions of color-coded phase values for 60-Hz component of the notch pulse in
Fig. 1 for 6 healthy young subjects (A-F). The analyzed time corresponds to AVC (T0).

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2. Propagation of Pulsive Wave Along Heart Wall
Using a sparse sector scan in 16 directions, multiple points were preset at 770μm intervals in the heart wall along each of 16 ultrasonic beams, and thevibrations at about 160 points were simultaneously measured as waveforms witha sampling frequency of 450 Hz by the phased tracking method [3].
Since the wavelength of the detected pulsive wave is about 100 mm for a30-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 the pulsive wave. Therefore,
using the method in [3], 2-D spatial distribution of the instantaneous phasevalues of the measured wave is shown in Fig. 2. For this imaging, the short time
Fourier transform is applied to the pulsive wave at each point in Fig. 1 after thepulsive wave is multiplied by the Hanning window with a short length of 35 ms.
The phase value is detected for each frequency component from 10 Hz to 90 Hz,
and then color-coded based on the upper-right circular figure. Figure 2 shows
the phase distributions for a 60-Hz component along the IVS at a time of AVC(T0) for 6 healthy young subjects.
From the consecutively obtained cross-sectional 2-D images, a motion
picture is presented. It can be seen that a few pulsive waves are radiated fromthe root of aortic valve and propagate along the IVS. The delay due to the
propagation of the pulse from the root to the apex is several milliseconds, which
has not been recognized at all by any other clinical technique.
3. Estimation of Viscoelasticity Using Lamb Wave and Voigt Models
To determine the instantaneous phase velocity of the pulsive wave, if the straightline with a constant gradient k [rad/m] is spatially fit to the instantaneous spatial
distribution of the measured phase in each cross-sectional image in Fig. 2, theinstantaneous phase velocity for the frequency component f0 is determined by
2πf0/k. However, since there is initially some spatial distribution of the phasewhich is independent of the propagation, the wavelength and thus the wave
number k do not correspond to the actual phase velocity.
Therefore, in this study, based on the definition of the phase velocity, thedistance Δx between two consecutively obtained phase distributions isdetermined. For this, the phase distribution θ(x;t) at a time t is compared withthe shifted phase distribution θ(x+Δx;t-ΔT) obtained at a time t-ΔT.
Regarding the phenomenon of the wave propagation along the viscoelastic
plate with thickness 2h, there are three kinds of plate wave. In the parasternallongitudinal-axis view, the direction (-y) of each ultrasonic beam is almost

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perpendicular to the IVS, and the detected motion is along each beam (ydirection in Fig. 3). The pulsive wave propagates along the IVS (x-direction).
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 and40-65 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. The separate invivo experiments show that the longitudinal component with x-displacement also
propagates along the IVS (x-direction). 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 longitudinalcomponent. Given these facts, the detected vibration signal can be modeled by aLamb wave with asymmetric mode in Fig. 3.

Fig. 3. Lamb wave with asymmetric mode of plate waves in the viscoelastic plate with thickness 2h.
The SV-wave component (y-displacement) and longitudinal component (x-displacement) arecoupled, and the Lamb wave then propagates along the x-direction.

For the IVS, the blood in RV and LV should be considered. Thus, themodel of the Lamb wave propagating along the viscoelastic plate immersed inblood, is employed in the present study. Let us define the wave number of theLamb wave by kL. By referring to [5], the function, termed f(kL,kp,ks), whichshould be zero, is given byf (k L , k p , k s ) = 4k L2 ηβ cosh(ηh) sinh(βh) − 2(k L2 − k s2 ) 2 sinh(ηh) cosh(β h)−ρ b ηk s4
cosh(ηh) cosh(βh)ρ m ηb(1)=0where kp and ks are the wave numbers of the primary wave and the secondarywave, respectively. Using the wave number kb in blood, η=(kL2-kp2)0.5, β=(kL2ks2)0.5, and ηb=(kL2-kb2)0.5. ρm and ρb are the myocardium density and the blood

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density, respectively, and ρm can be approximated by ρb=1.1×103 kg/m3. The
thickness 2h of the IVS is determined from the B-mode image.
By introducing a single Voigt dash-pot model in the frequency range up to
90 Hz, the Lamé elastic constants λ and μ become complex values as λ=λ1+jωλ2
and μ=μ1+jωμ2, respectively [6]. Since λ is about 103 times larger than μ forsoft tissue due to non-compressibility [6], kp is approximated by kb. Since kL is
close to ks, η≈kL and ηb≈kL. Therefore, f(kL, kp, ks) of Eq. (1) is approximated byf ' (c L , μ, ω) = 4k L3β cosh( k L h) sinh(βh) − 2(k L2 − k s2 ) 2 sinh( k L h) cosh(β h)− k s4 cosh(k L h) cosh(βh)

(2)

=0

where kL=ω/cL(μ,ω) and ks=ω(ρm/μ)0.5. Thus, f '(cL,μ,ω) of Eq. (2) on both thephase velocity cL(μ,ω) of the Lamb wave and the Lamé elastic constant
μ=μ1+jωμ2, that is, the elasticity μ1 and the viscosity μ2 are estimated so that thetheoretical value of the phase velocity cL(μ,ω) of the Lamb wave is to be close
to the measured the dispersion of the phase velocity cphase(ω).

Fig. 4. Measured instantaneous phase velocity values cphase and fitted theoretical curves cL(μ,f) for
the frequency components from 10 Hz to 90 Hz during the period from T0-8 ms to T0+10 ms in left
upper figure.

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4. In Vivo Experimental Results
For subject A, by analyzing the period from T0-8 ms to T0+10 ms for eachfrequency component from 10 Hz to 90 Hz, the instantaneous phase velocity
cphase(ω) was obtained as in Fig. 4. There is large dispersion among the
instantaneous phase velocities cphase(ω), and they rapidly decrease for all
frequency components in this short period around the time of AVC (T0).
The estimated elasticity μ1 is about 24-30 kPa and does not change aroundthe time of AVC. The instantaneous viscosity μ2 of the myocardium rapidly
decreases from about 400 Pa⋅s to 70 Pa⋅s. This would be due to the rapid
decrease in the LV inner pressure from about 120 mmHg to several mmHg,
which is caused by relaxation of the myocardium.

Fig. 5. Transient of the viscosity parameters μ2 measured during the period from T0-8 ms to T0+10
ms for five healthy young male subjects.

For the other four subjects, B-E, the same measurement and analysis wereapplied, and similar transient characteristics of the viscosity were obtained asshown in Fig. 5.
Using the estimated Lamé constant μ, the phase velocity cL(μ, f) is obtained.
The results are shown by the solid lines in Fig. 4 for each time t. The estimatescL(μ, f) well fit the measured dispersion characteristics cphase(f).

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Fig. 6. The elastic values {μ1} and viscosity values {μ2} reported for the myocardium and soft
tissues in the literature using audio or lower frequency components up to 10 kHz.

5. Discussion
The elastic values and viscosity values obtained in the present study arecompared with those reported for the myocardium and soft tissues in theliterature using audio or lower frequency components up to 10 kHz as shown in
Fig. 6. Except for those of the present study, all data were measured in in vitro

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experiments. Roughly speaking, the elasticity is large for very low frequency
less than 10 Hz and for frequency higher than 1 MHz. In [12], 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 toη2=626.7×f-0.722 [Pa⋅s], which shows that the viscosity decreases with theincrease in frequency f. By considering the frequency dependency and the
freshness of the specimen in Fig. 6, the elasticity and viscosity measured in thepresent study are close to those measured for the same frequency range in theliterature.
6. Conclusion
We measured rapid and minute vibrations simultaneously at multiple points in
the IVS. Clear propagation of the pulsive wave along the IVS was recognized.
From the dispersion of the phase velocities, the myocardial viscoelasticity wasdetermined noninvasively for the first time. This method offers potential for invivo imaging of the spatial distribution of the passive mechanical properties ofthe myocardium and its rapid change during the IR period, which would enabledirect assessment of diastolic properties based on myocardial relaxation in heart
failure.
Acknowledgments
This work is partially supported by grants-in-aid for scientific research from the
Ministry, Education, Culture, Sports, Science, and Technology of Japan (2005–
2007, No. 17206043.).
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