Abstract—Diastolic dysfunction is characterized by the stiffening of the left ventricle (LV) and could cause heart failure if left untreated. We investigated the feasibility of using Shearwave Dispersion Ultrasound Vibrometry (SDUV) to quantify the stiffness of excised porcine LV myocardium by propagating Lamb and Rayleigh waves. Mathematical models for Lamb and Rayleigh wave dispersion in a viscoelastic solid submerged in a fluid were derived. The dispersion equations for the two waves converge analytically as frequency increases. The estimated material properties using Lamb and Rayleigh waves agree within one standard deviation.

I. INTRODUCTION Approximately 50% of heart failures in the United States

are caused by diastolic dysfunction, characterized by stiffening and impaired relaxation of the left ventricle (LV) [1]. A noninvasive technique capable of characterizing material properties of the myocardium would be highly beneficial in clinical settings.

Shearwave Dispersion Ultrasound Vibrometry (SDUV) is a noninvasive method for quantifying material properties of soft tissues by measuring the speed of propagation of shear waves at multiple frequencies (shear wave dispersion) [2].

We have been investigating the feasibility of using SDUV to excite anti-symmetric Lamb waves using radiation force and quantifying the material properties of the myocardium by measuring the Lamb wave dispersion. In vivo excitation of Lamb would require focusing the ultrasound beam in the middle of the LV free wall thickness. This could prove to be a challenge due to periodic motion of the heart and surrounding organs, noise and limitation in estimating the middle of the myocardium. However, the epicardial surface of the free wall is highly reflective and noticeable on a B-mode image. Focusing radiation force on the epicardial surface of the LV free wall would excite Rayleigh (surface) waves whose boundary conditions and shear wave dispersion relations are different than those of Lamb waves.

We report the results of measuring viscoelasticity of excised pig myocardium by propagating Rayleigh and Lamb waves. Mathematical models of Lamb and Rayleigh waves in an axisymmetric homogenous solid submerged in a incompressible fluid were solved in cylindrical coordinates

This work was supported in part by grant EB002167 from the National Institutes of Health.

The authors are with the Department of Physiology and Biomedical Engineering, Mayo Clinic College of Medicine, Rochester, MN 55905 USA (Contact information for Ivan Nenadic: phone: 507-266-0892; e-mail: nenadic.ivan@ mayo.edu).

for the shear wave dispersion. The models were fitted to the experimental data to estimate material properties.

II. METHODS

A. SDUV Technique Amplitude modulated (AM) ultrasound beam from the

“push” transducer is focused on the medium to generate monochromatic radiation force in the range 40 – 500 Hz, and a pulse echo transducer is used to register the motion (Figure 1). Ultrasound pulses from the pulse-echo transducer are transmitted at the location of interest at a repetition rate of few kHz. Each point in the time domain of the returning echo signal corresponds to a specific region of the tissue along the beam axis. Cross-spectral correlation and a specialized Kalman filter were used to estimate the shear wave amplitude and phase.

r

1Tissue

Push transducerPulse-echo Transducer

2

r

1Tissue

Push transducerPulse-echo Transducer

2

Fig. 1. In SDUV, a push transducer is used to excited shear waves in the medium and a pulse-echo transducer is used to record the motion.

The push transducer produces cylindrical shear waves with a phase delay that varies linearly with the distance from the excitation point [2]. By knowing the frequency of the propagating shear wave, one can estimate its velocity by measuring the change in phase over distance from the beam axis rr :

s

rc

rr. (1)

Study of Lamb and Rayleigh Wave Dispersion in Excised Myocardium Using Shearwave Dispersion Vibrometry (SDUV)

Ivan Nenadic, Student Member, IEEE, Matthew W. Urban, Member, IEEE, and James F. Greenleaf, Life Fellow, IEEE

308978-1-4244-4126-6/10/$25.00 ©2010 IEEE ISBI 2010

B. Anti-Symmetric Lamb and Rayleigh Wave Models

The system of LV free wall myocardium surrounded by blood was modeled as a solid plate submerged in a fluid. The myocardium was assumed to be a Voigt material, so the shear modulus 1 2i1 2i , where 11 and 22 are the elastic and viscous moduli.

r

z

Fluid

Solid

Fig. 2. LV free-wall myocardium surrounded by blood on both sides is modeled as a viscoelastic solid submerged in an incompressible fluid.

Applying the Stokes-Helmholtz decomposition to the Navier’s equation with zero body force, and assuming axial symmetry, the governing equations for cylindrical waves in a solid submerged in a fluid can be expressed in cylindrical coordinates as follows [3]:

2 2 2

2 2 2 222 2 2 2 2

1 1

sr r r z r c t122 2

2 1 2 22 2 2 12 2 21 2 2 2

22 2 2 2 2sr r r z r c t2 2 2 22 2 2s

2 2 2 2 2 (2)

2 2 22 2 2

22 2 2 2

1 1

pr r r z c t

22 22 1 12 2

2 22 21 12 222 2 2 2

pr r r z c t2 2 22 2p

2 2 22 2 2 (3)

2 2 21 1 1

12 2 2 2

1 1

Fr r r z c t

22 21 1 12 2

1 11 11 11 112 2 2 2

Fr r r z c t2 2 22 2F

2 2 2 1 (4)

where cp and cs are the compressional and shear wave speeds in the solid; cF is the speed of the compressional wave in the fluid.

The boundary conditions for the Rayleigh wave propagation in the solid, at 0z 0 are 2 1zz zz2zz 2 1 , 2 0zr 2zr 0

and 1 2w w2w2 , where 1 and 2 represent the liquid and solid, respectively.

The stresses and strains for the given geometry are defined as follows:

11w

z11

z (5)

2 22

( )1 rw

z r r( )12 22 2( )21

z r r (6)

( )( 2 )zz

w ruz r rzz

w ( )(( 2 ) ( )(wz r r

(7)

zr

u wz rzr

u wwu wwu wz rz r

(8)

In order to apply the boundary conditions, Eqs. 2-4 need to be solved for the potential functions. These equations are

not easily solved in the real space and time domain, so we follow the approach by Zhu et al [3] and perform a Laplace transform with respect to time and Hankel transform with respect to the radial space component and obtain the following expressions:

0 02

21 1 12

H H

z

2H0H0 02

1 1

H0

12z 1 (9)

0 02

22 2 22

H H

z

2H0H0 02

2 2

H0

22z 2 (10)

1 12

22 22

H H

z

2H1H1 12

2

H1

22z 2 (11)

Where2

2 21 2

F

pc

21

22

p , 2

2 22 2

P

pc

22

22

p and2

2 22s

pc

2 22

p .

The solutions to Eqs. 9-11 in the Laplace-Henkel space, ignoring the unstable solutions, are of the following form:

0 11 1( , )

H zp e 10

1

H0 e 1ze1( , )p) (12) 0 2

2 2 ( , )H zp e0 2

2 )H0 e) 2z

2 ( , )p) (13) 1

2 ( , )H zp e1

2

H1 e z( , )p e) (14) The stress-strain relations in the Laplace-Hankel domain

are as follows:

00 1

1

HH

wz

0H0

11

z (15)

00 12

2

HH H

wz

0H0

2 1H12

z (16)

10 0

22

2 2 22 2 2H

H H

zzs

pc z

0

2

H0

zz

H2H

2 1H1H

20H02 22p 02 202 0p 00

22 2222 22 zsscs2c

(17)

01 1

222

2 2 22 2H

H H

zrs

pz c

1

2

H1

zr

H 2H

20H0H

2 2 2 1H12 22 122 1112 222222z sssc2c (18)

Applying the stress-strain relations and satisfying the boundary conditions, the dispersion curve for Rayleigh waves on the liquid-solid interface in cylindrical coordinates can be expressed in the Henkel-Laplace domain as follows:

22 42 2 2 1

22 41 2

2 4 0s s

p pc c

4p4s1 2 c1 21 2

22 2p 01 01 p112222

4 042 2

2

pp 22 22 4 222 42 22 2 22 42 2

scs2 2c 2

(19)

By noticing the relationship between the Fourier and bilateral Laplace transform, the Rayleigh waves dispersion equation in the frequency and real space domain becomes:

2 222 2 2 2 2 2 2 41

2 22

2 4 0R pR s R R s R p s

R F

k kk k k k k k k k

k k

2k22 2 2 2 2 220s R R s R p 2

kR2 2 2 2 2 22 2 2 2442 2 2 2 2 22 2 2 2 1444 4 041 0k

44

22

2FkF 2

s2 s (20)

309

where kR is the Rayleigh wave number. This expression is identical to the Rayleigh wave dispersion in plane waves.

The potential functions, Henkel-Laplace transforms and stress strain relations for the anti-symmetric Lamb wave in an axisymmetric plate submerged in a fluid are similar as in Rayleigh waves, and without any loss of generality can be expressed as follows:

0

2 2sinh( )H

A z0

2 2 )2

H0

2i h(Asinh( (21) 1

2 cosh( )H

D z1

2 )H1 cosh(Dcosh( (22)

0 1H z

L Ne0H0

L1e 1zeNe (23)

0 1H z

U Ne0H0

U1NeN 1z (24)

Here, subscripts U and L refer to the fluid above and below the sample. Following the boundary conditions and stress-strain relations, the Lamb wave dispersion equation is of the form:

2 422 2 2 1

2 22 41 2

2 tanh( ) 4 tanh( ) 0s s

pph h

c c

4

41 2

22p 02 12 1 02 1242 tanh( ) 242

p4 tanh(222

22 0442

2

pp 22 tanh(222 )2 22 2

scs2 2c 22

(25)

As in Eq. 20, it can be shown that Eq. 25 is identical to the plane Lamb wave dispersion equation [4].

In the limit of x , tanh(x) 1 so the dispersion equations for the Lamb (Eq. 25) and Rayleigh wave (Eq. 19) converge analytically as the frequency increases. Figure 3 shows the analytical converge for four different viscoelastic parameters 11 and 22 .

Fig. 3. Lamb (blue) and Rayleigh (red) dispersion curves for the given values of elasticity and viscosity. In the limit of x , tanh(x) 1, so the Lamb and Rayleigh wave dispersion equations converge analytically (Eqs. 19 and 25).

C. Experiment The excised porcine LV free wall myocardium samples

were embedded in a gelatin (70% water, 10% glycerol, 10%

300 Bloom gelatin, 10% potassium sorbate preservative, all by volume and all manufactured by Sigma-Aldrich, St. Louis, MO) inside a plastic container. The container was mounted on a stand in a water tank. A window was cut out on the bottom of the container to allow for motion detection.

A mechanical shaker (V203, Ling Dynamic Systems Limited, Hertfordshire, UK) was used to excite monochromatic shear waves in the sample. A glass rod coupled with the shaker was glued to the hole bored through the middle of the sample to excite Lamb waves and to the surface of the sample to excite Rayleigh waves. Four cycles of sinusoidal waves were used to drive the shaker at different frequencies ranging from 40 to 500 Hz to induce cylindrical shear waves in the sample. Motion was measured at each frequency in four orthogonal directions using 7.5 MHz pulse echo transducer with a pulse repetition rate of 4 kHz. Motion was recorded at 31 points along a line, 0.5 mm apart. Phase estimates at these points were used to fit a regression curve (Eq. 1) and calculate the shear wave speed at each frequency.

III. RESULTS Continuous line in Figure 4 represents the anti-symmetric

Lamb wave model fitted to the experimental Lamb wave data for elasticity and viscosity. Estimated values of

11 and 22 in

four directions due to propagating Rayleigh waves are shown above the plots in Figure 4.

Fig. 4. Lamb wave dispersion equation in blue is fitted to the experimental Rayleigh wave dispersion data shown in red. Dispersion curves in four orthogonal directions with the estimated elasticity and viscosity coefficients are shown. .

At low excitation frequencies, the wave length of the incident Rayleigh waves are large enough to ‘feel’ the bottom surface of the sample and thus produce an anti-symmetric Lamb wave. The phase maps of the propagating Rayleigh wave at low frequencies is conserved throughout

0 100 200 300 400 5000

5

10

Frequency, Hz

c, m

/s

-X, 1 = 20 kPa, 2 = 10 Pa s

0 100 200 300 400 5000

5

10

Frequency, Hz

c, m

/s

+X, 1 = 25 kPa, 2 = 15 Pa s

0 100 200 300 400 5000

5

10

Frequency, Hz

c, m

/s

-Y, 1 = 35 kPa, 2 = 20 Pa s

0 100 200 300 400 5000

5

10

Frequency, Hz

c, m

/s

+Y, 1 = 50 kPa, 2 = 25 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

-X, 1 = 1.5 kPa,

2 = 10 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

+X, 1 = 3 kPa,

2 = 11 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

-Y, 1 = 1.5 kPa,

2 = 9 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

+Y, 1 = 2 kPa,

2 = 10 Pa s

310

the thickness, consistent with the anti-symmetric Lamb wave displacement. Displacement and phase maps at higher frequencies show that the incident wave does not propagate to the bottom surface of the sample, consistent with a Rayleigh wave. Due to the Rayleigh and Lamb wave converge at higher frequencies, the Lamb wave model was used to fit the experimental Rayleigh wave dispersion data to obtain viscoelastic parameters (Figure 4).

Figure 5 shows the estimated elasticity and viscosity of the myocardial sample by propagating Lamb waves. The experimental data was fitted by the Lamb wave model.

The results of measuring elasticity and viscosity by propagating Rayleigh and Lamb waves are summarized in Table I.

Table I. Results of measuring elastic and viscous moduli of excised LV myocardium by propagating Lamb and Rayleigh waves. The results are reported as average of the four measurements with standard deviations.

Fig. 5. Lamb wave dispersion equation in blue is fitted to the experimental Lamb wave dispersion data shown in red. Dispersion curves in four orthogonal directions with the estimated elasticity and viscosity coefficients are shown.

IV. DISCUSSION The results reported in Figures 3, 4 and 5 suggest that the

Lamb and Rayleigh waves converge analytically and experimentally in the given frequency range, 40 - 500 Hz. The derived dispersion equations fit the experimental data well. It is important to note that the Lamb and Rayleigh

dispersion equations for cylindrical waves are identical to the analogous expressions for plane waves.

The estimated values of elasticity and viscosity using two methods agree within one standard deviation. The dispersion speeds in four orthogonal directions are similar for Lamb and Rayleigh waves, suggesting that the differences in fiber orientation doe not affect shear wave speed. In other words, the tissue behaves as a bulk for the given range of excitation frequencies. This could prove helpful for in vivo experiments as one would need to estimate elasticity and viscosity along a single line on order to characterize regional material properties.

In the future, we plan to test the use of radiation force for shear wave excitation. This approach would mimic the SDUV method and would be necessary for clinical use.

V. CONCLUSIONS A modified SDUV approach was used to estimate Lamb

and Rayleigh wave dispersion velocities in excised porcine myocardium. Mathematical models were fitted to the experimental dispersion data to estimate material properties. The Lamb and Rayleigh wave excitation methods results agree within one standard deviation. The reported results are encouraging steps towards in vivo application of the SDUV method.

ACKNOWLEDGMENT The authors would like to thank to Randall R. Kinnick for

experiment support, Thomas Kinter for computer support, and Jennifer Milliken for administrative support.

REFERENCES [1] R. Gary, L. Davis, “Diastolic heart failure”. Heart Lung. 2008 Nov-

Dec; 37(6):405-16. [2] S. Chen, M. W. Urban, C. Pislaru, R. Kinnick , Y. Zheng, A. Yao, J.

F. Greenleaf. Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity. IEEE Trans Ultrason Ferroelectr Freq Control. 2009 Jan;56(1):55-62.

[3] J. Zhu, J. S. Popovics. Leaky Rayleigh and Scholte waves at the fluid–solid interface subjected to transient point loading. J. Acoust. Soc. Am. 2004. 116 (4), pp. 2101-2110. control, Vol. 54, No. 2, February 2007

[4] H. Kanai. Propagation of Spontaneously Actuated Pulsive Vibration in Human Heart Wall and In Vivo Viscoelasticity Estimation. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 52, no. 11, November 2005

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

-X, 1 = 2 kPa,

2 = 10.5 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

+X, 1 = 3 kPa,

2 = 11 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

-Y, 1 = 1.5 kPa,

2 = 9 Pa s

0 100 200 300 4000

2

4

6

Frequency, Hz

c, m

/s

+Y, 1 = 2 kPa,

2 = 10.5 Pa s

TABLE I LEFT VENTRICULAR MYOCARDIUM ELASTICITY AND VISCOSITY RESULTS

USING LAMB AND RAYLEIGH WAVE EXCITATION METHODS

Excitation Method Elasticity ( 1) (kPa) Viscosity ( 2) (Pa·s)

Lamb Wave 2.1 ± 0.6 10.3 ± 0.9 Rayleigh Wave 2.0 ± 0.7 10.0 ± 0.8

311