Excitation and propagation of surface waves on a viscoelastic half-space with application to medical...

Excitation and propagation of surface waves on a viscoelastichalf-space with application to medical diagnosis

T. J. Roystona)

University of Illinois at Chicago, Chicago, Illinois 60607

H. A. Mansy and R. H. SandlerRush Medical College, Chicago, Illinois 60612

~Received 15 June 1999; accepted for publication 10 September 1999!

An analytical solution is developed for the problem of surface wave generation on a linearviscoelastic half-space by a finite rigid circular disk located on the surface and oscillating normal toit. The solution is an incremental advancement of theoretical work reported in articles focused onseismology. Since the application of interest here is medical diagnostics, the solution is verifiedexperimentally using a viscoelastic phantom with material properties comparable to biological softtissue. Findings suggest that prior estimates in the literature of the shear viscosity in human softtissue may not be accurate in the low audible frequency range. Measurement of wave motion on theskin surface caused by internal biological functions or external stimuli has been studied by a fewresearchers for rapid, nonintrusive diagnosis of a variety of specific medical ailments. It is hopedthat the developments reported here will advance these techniques and also provide insight intorelated diagnostic methods, such as sonoelastic imaging and other methodologies that utilizedisease-related variations in tissue shear elasticity or variations in density due to gaseous inclusions.© 1999 Acoustical Society of America.@S0001-4966~99!07712-7#

PACS numbers: 43.80.Cs, 43.80.Ev, 43.80.Qf, 43.20.Bi@FD#

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INTRODUCTION

There has been limited research in the area of meddiagnosis using surface wave behavior on human soft tisSuch an approach potentially can be noninvasive, rapid,inexpensive. For example, Lee1 and Hong and Fox2,3 haveinvestigated noncontact characterization of cardiovascdynamics using optical interferometry. Diagnosis of certtypes of edema conditions and other skin diseases, incluskin cancer, via alterations in surface wave propagationbeen studied by several groups.4–11 Other research has focused on characterizing lung edema by studying surfwave propagation directly on the inflated pulmonaparenchyma12 or on the chest wall.13

Exploiting changes in subsurface shear wave propation as a means of diagnosis has also received much attein recent years. Several groups of researchers have utilow-frequency excitation of shear waves, e.g., 20 to a fhundred Hz, coupled with Doppler ultrasonic imaging.14–16Itis sometimes referred to as ‘‘sonoelastic imaging.’’ Thmethod has been proposed as a technique for locatingmors, which typically represent significant changes in stness properties in otherwise acoustically homogeneousgions. A localized stiffness, such as a tumor, will distort tshear wave vibration pattern as imaged by Doppler ulsound.

To support further development of these promisingagnostic methodologies, a better understanding of surwave behavior on soft biological tissue is needed in theaudible frequency regime, 20 to 100 Hz. Above these

a!Author to whom correspondence should be addressed. [email protected]

3678 J. Acoust. Soc. Am. 106 (6), December 1999 0001-4966/99/1

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quencies, surface waves become negligible more than acentimeters from the source due to the high viscosity of sbiological tissue. Below these frequencies~the lower-frequency limit dependent on location!, the human body ismore aptly described by a lumped parameter model17 ~or, atleast, is not reasonably approximated as a half-space!. It isnoted that surface wave behavior is relevant to subsurmeasurement schemes like sonoelastic imaging as the efof surface~Rayleigh! waves penetrate below the surface wsignificant amplitude to frequency-dependent depths comrable to their wavelength on the surface. Surface waves,shear waves, are predominantly dependent on shearcoelastic properties and density.

A critical aspect of many of these proposed diagnostechniques is an understanding of how the skin surfacesponds to external stimulation. While the general naturefree Rayleigh~surface! wave propagation on a viscoelastmedium has been investigated, the problem of surface wgeneration from a typical finite surface source has not bexplicitly addressed in the literature, to the best of thethors’ knowledge. A basic pedagogical problem is thatdetermining the surface wave response to a disk-shatransducer located on the surface of a viscoelastic half-spwith human soft-tissue properties and vibrating normal toA solution to the related elastic problem can be found inseismological literature. Seminal papers by Miller aPursey18,19 address this problem and, despite their date,frequently cited in many contemporary works. Integral reresentations of the general solution for the stated disk prlem were offered. Asymptotic expressions for the far fiewere given in explicit form.

It is the objective of the research reported in this artito first theoretically derive a more accurate surface wail:

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response function, valid for the near and far field, of an itropic viscoelastic half-space to a finite disk of radius ‘‘a’’on the surface and harmonically vibrating normal to it. Tderivation is then validated based on experimental studiea viscoelastic phantom with material properties comparato human soft tissue. Generated theoretical results areused to assess the accuracy of material parameter valuesviously reported in the literature for human soft tissue.

I. THEORY

The problem considered here is that of a circular diskfinite radius ‘a’ vibrating normally to the surface of an isotropic, viscoelastic medium. The geometry of the problemdepicted in Fig. 1. The theoretical derivations below repsent incremental advancements to the derivations reporteMiller and Pursey.18,19For the medium, the equation of wavpropagation may be expressed in the following form:18

~l12m!¹¹•u2m¹3¹3u5r]2u

]t2 . ~1!

Here, u is the displacement vector,r is the density of themedium,]/]t denotes a derivative with respect to time, al andm are the Lame constants of the medium. For a linviscoelastic material, the rate-dependent Lame ‘‘constanare expressible asl(t)5l11l2]/]t and m(t)5m1

1m2]/]t where l1 , l2 , m1 , and m2 are coefficients ofvolume compressibility, volume viscosity, shear elasticiand shear viscosity, respectively.20 The problem is solved inthe polar coordinate system depicted in Fig. 1. Thus, we hu5@uruwuz#

T and it is noted thatuw50 and derivatives withrespect tow vanish due to axisymmetry. Hence,

¹•u51

r

]

]r~rur !1

]uz

]z~2a!

and

¹3u5S ]ur

]z2

]uz

]r D j . ~2b!

Consider the case of harmonic force excitation ofmassless disk of amplitude per unit areaPin and circularfrequencyv such that it applies a uniform stress on the sface of the medium in the circular region ofr ,a. ~It is notedthat this assumption of uniform stress may limit the accurof the model to values ofa less than or comparable to surfawavelength.! Miller and Pursey18 derive the following inte-gral expression for wave propagation in ther andz directions

FIG. 1. Ideal viscoelastic half-space problem.

3679 J. Acoust. Soc. Am., Vol. 106, No. 6, December 1999 Ro

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at any location in and on the surface of the medium, i.ez>0. Note that time dependenceeivt is omitted wherei5A21.

uz

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a

m E0

` J1~zak1!Az221

F0~z!$2z2e2zk1

Az22h2

1~h222z2!e2zk1Az221%J0~zrk1!dz, ~3a!

ur

Pin5

a

m E0

` J1~zak1!z

F0~z!$2Az221Az22h2e2zk1

Az22h2

1~h222z2!e2zk1Az221%J1~zrk1!dz, ~3b!

where

F0~z!5~2z22h2!224z2Az22h2Az221, ~3c!

h5k2 /k1 , ~3d!

k15vAr/~l12m!, ~3e!

and

k25vAr

m. ~3f!

In the above equations,k1 and k2 denote the wavenumberfor compression and shear wave propagation, respectivandJ0 andJ1 refer to Bessel functions of the first kind. Thdummy variablez is used to denote integration over thwavenumber domain that has been normalized with resto k1 .

Equation~3a! can be rewritten as follows:

uz

Pin5

a

m E0

`

$x1~z!e2zk1Az22h2

1x2~z!e2zk1Az221%J0~zrk1!dz, ~4a!

where

x1~z!52z2J1~zak1!Az221/F0~z! ~4b!

and

x2~z!5J1~zak1!Az221~h222z2!/F0~z!. ~4c!

To solve this equation, we make use of the integral exprsion for the Bessel function that is valid whenuarg(rzk1)u,p/2, a reasonable assumption in this case.18 It is given by

J0~zrk1!52

p E0

`

e2j2$u~zrk1 ,j!1u~zrk1 ,j!%dj,

~5a!

where

u~zrk1 ,j!5ei ~rk1z2~1/4!p!~2zrk11 i j2!21/2, ~5b!

u~zrk1 ,j!5e2 i ~rk1z2~1/4!p!~2zrk12 i j2!21/2

52u~2zrk1 ,j!, ~5c!

and j is a dummy variable of the integration. Substitutinthese relations into Eq.~4a! and using the fact thatx1(z) andx2(z) are odd functions, we obtain

3679yston et al.: Viscoelastic surface waves and medical diagnosis

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`

$x1~z!e2zk1Az22h2

1x2~z!e2zk1Az221%E

0

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e2j2u~zrk1 ,j!dj dz. ~6!

UsingMATHEMATICA , a solution to the integral overj can befound

E0

`

e2j2u~zrk1 ,j!dj

5E0

`

e2j2ei ~zrk12p/4!~2zrk11 i j2!21/2dj

52 12~21!3/4ei ~zrk12p/4!e2 i zrk1K0~2 i zrk1!. ~7!

Here,K0 denotes the modified Bessel function of the secokind of order 0. Thus, Eq.~3a! has been transformed into thfollowing equivalent expression:

uz

Pin52~21!3/4e2 ip/4

a

pm E2`

` J1~zak1!Az221

F0~z!

3$2z2e2zk1Az22h2

1~h222z2!e2zk1Az221%

3K0~2 i zrk1!dz. ~8!

The primary advantage of this expression over that of Eq.~3!is that the Cauchy principal value theorem is now easapplied to obtain its solution. It is assumed that the princivalues of the radicals are used in the above expressthroughout the range of integration. This is consistent wwaves traveling away from the source when a time faceivt is used and is necessary for the convergence of thetegrals at infinity. To ensure continuity of the integrands,contour of integration must be indented to pass belowbranch point at21 and above the branch point at11. Underharmonic excitation conditions, we have

h5A$l112m11 iv~l212m2!%/~m11 ivm2!. ~9!

For realistic values of the coefficients in this expression,1hwill be located below the real axis in the 4th quadrant a2h will be located in the 2nd quadrant above the real ax

The poles of the integral in Eq.~8! are the zeros of theexpressionF0(z) which is a cubic equation inz2. Hence,there are six roots. It is found, however, that for realisvalues ofh the only zeros ofF0(z) which correspond toprincipal values of the radicalsAz221 and Az22h2 arethose for whichz2 is greater in magnitude thanh2. Thesepolesp will be located in like quadrants ash and will havemagnitude slightly greater thanh. Physically it follows that,in this normalized wavenumber domain, polesp associatedwith propagating surface or Rayleigh waves will have manitude greater thanh, which is associated with the propagaing shear wave in the medium. This leads to a surface wof phase speed slightly less than the phase speed of thewave. For the viscoelastic case with nonzero damping, ctour integration along the real axis with the noted indentatabout61 is needed. In the limit as damping is neglected,contour integration still follows the same route below neg


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tive poles and above positive poles, but it must be furtindented as indicated in Fig. 2. See Miller and Pursey18 forfurther discussion of this issue.

For the case thatu5p/2, i.e., along the surface, the integral foruz /Pin given in Eq.~8! is negligible except for theresidue associated with the polep. The contour integrationcan be taken counterclockwise around the 1st and 2nd qrants, thus encircling the pole at2p. For the linear elasticcase, the values ofp are independent of harmonic excitatiofrequencyv and are located on the real axis, requiring identation of the integral as noted above. For the linearcoelastic case, the poles are no longer on the real axis,are a function of harmonic excitation frequency, with timaginary part increasing with increased excitation fquency.

In Ref. 19, an asymptotic expression was used toproximate the integral of Eq.~7!. This led to results that areasymptotically valid for ‘‘larger’’ and ‘‘small a.’’ In thispaper, the exact solution of the integral is used, whichbeen obtained viaMATHEMATICA . Thus, more general expressions are given for surface waves that should be validsmall r and largea, as well. We have the following:

uz

Pin52~21!3/4e2 ip/4

a

m2i F residue of

J1~zak1!Az221

F0~z!

3$2z2e2zk1Az22h2

1~h222z2!e2zk1Az221%

3K0~2 i zrk1! at z52pG5~21!3/4e2 ip/4

2ia

m

J1~pak1!Ap221

Fo8~2p!

3$2p2e2zk1Ap22h2

1~h222p2!e2zk1Ap221%

3K0~ iprk1!, ~10a!

where

F08~2p!5]F0

]z Uz52p

. ~10b!

This expression agrees asymptotically with Ref. 19. Tasymptotic solution of the reference is given here for coparison:

FIG. 2. Location of poles and determination of contour.


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3e2 ik1pr$2p2e2k1zAp22h21~h222p2!e2k1zAp221%.

~11!

Note the approximations in Eq.~11! are with respect to theBessel functions in Eq.~10!. The functionJ1(pak1) is ap-proximated withpak1/2. This may be acceptable in manseismological applications dealing with surface wave progation on the earth caused by a highly localized excitatiBut, in the medical application, it is conceivable thupak1u.1 or'1. In other words, the radius of the transducdisk a may be of the same order of magnitude as the lenof the surface waves, given byLsurf52p/real(k1p). ~Thisfact does also call into question the assumption made abeginning of the theoretical development that the disk wimpart a uniform stress in the regionr ,a. Hence, the im-proved solution here may still have its limitations.! The ap-proximation to K0( iprk1) includes the exponentialeiprk1

and a few other terms and is only valid for larger. For theparameter values used in this article in Sec. III withupak1u.1 andr .a, this approximation is reasonable.

To obtain a solution to Eq.~3b!, it is observed thatzk1J1(zrk1)52]/]r @J0(zrk1)#. It is also noted that

d

dr@K0~2 i zrk1!#5 1

2i zk1$K21~2 i zrk1!1K1~2 i zrk1!%

5 i zk1K1~2 i zrk1!. ~12!

Here,K21 and K1 denote the modified Bessel functionsthe second kind of order,21 and 1, respectively. Thus, following the arguments given above, the following equivaleexpression to Eq.~3b! is obtained ~assuming uarg(rzk1)u,p/2):

ur

Pin5~21!3/4e2 ip/4

ia

pm E2`

` J1~zak1!z

F0~z!

3$2Az221Az22h2e2zk1Az22h2

1~h222z2!e2zk1Az221%K1~2 i zrk1!dz. ~13!

Evaluating the contour integral~finding the residue! in asimilar fashion to the case ofuz yields the following for thecase of surface waves:

ur

Pin52~21!3/4e2 ip/4

2a

m

J1~pak1!p

F08~2p!

3$2Ap221Ap22h2e2zk1Ap22h2

1~h222p2!e2zk1Ap221%K1~ iprk1!. ~14!

This expression agrees asymptotically with the far-fieldlution provided in Ref. 19. The asymptotic solution of threference is given here for comparison:


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a2e~1/4!ip

mF08~p!Apk1p3

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3$2Ap221Ap22h2e2k1zAp22h2

1~h222p2!e2k1zAp221%. ~15!

II. EXPERIMENT

To assess the validity of the theoretical derivations psented in Sec. I an experiment was constructed that is smatically depicted in Fig. 3. The viscoelastic medium cosists of a gel mixture of the following composition~per literof water!: 70 grams gelatin, 40 gramsn-propanol, and 4grams formaldhyde~37% solution!. This ‘‘recipe’’ for a soft-tissue phantom is based on prior investigations.21 By takingvelocity measurements using the laser Doppler vibrome~Polytec model CLV-800-FF 1000!, vertical velocityuz canbe measured for anyr .a at the surface,z50. An imped-ance head~PCB model 288B02! records acceleration anforce input to the Plexiglas disk atz50 and 0,r ,a. Thevibratory excitation is delivered through a stinger by an eltromagnetic shaker~Labworks model ET-132-2! that is flex-ibly suspended above the phantom. Sensor output signalrecorded with a Hewlett-Packard 35 670 Dynamic Sig~FFT! Analyzer, which also performs preliminary analysThe analyzer also provides the chirp excitation signal toamplifier that drives the shaker. Chirp~rapidly swept sinu-soidal! excitation signals were used to enable extractfrequency–amplitude–phase–time information. Thislowed detection of outgoing waves from the source indepdent from reflections due to the finite boundaries of the phtom. Beyond basic frequency-response functions availablreal time using the 35 670, more detailed analyses ofexperimental data were conducted usingMATLAB software.

FIG. 3. Schematic of experimental setup.


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III. RESULTS AND DISCUSSION

Material parameter values used to theoretically mothe gel phantom are provided in Table I. Volume compreibility and density are comparable to water. Shear viscoetic constants are based on comparison to experimentalface wave speed and attenuation measurements. Also gare typical parameter values estimated by von Gierkeet al.22

to model human soft tissue. All theoretical results presenin this section are based on the material parameter vagiven in Table I gel or tissue, unless otherwise noted.

A. Gel phantom model—theory and experiment

The experimentally measured vertical particle velocamplitudes on the surface normalized to the disk velocamplitude uuz50(r ,v)/uz50(r 50,v)u as a function of dis-tance from the sourcer and excitation frequencyv areshown in Fig. 4. The displayed results are based on calcuing the transfer function of the complete time record, durwhich the excitation frequency was swept from 0 to 100 HDue to the finite dimensions of the phantom, resonant behior is detectable at the lower frequencies that cannot bedicted using the theory developed in the previous sectThis behavior could be due both in part to reflections frothe outer radial boundary of the phantom or from its findepth. Above about 40 Hz, due to increasing attenuationdecreased wavelength, resonant behavior is less appare

In Figs. 5 and 6 experimental waveform measuremeof uz /F in at selected frequencies are directly comparedtheoretical predictions of Sec. I based on the developed ‘‘proved theory’’ and the ‘‘asymptotic theory’’ of Miller andPursey.18,19Also shown are the predictions when dampingneglected in the ‘‘improved theory.’’ Here,F in5Pinpa2 isthe force input of the disk of radiusa. The experimentalresults validate the improved theory. The limitations of tasymptotic theory are evident, particularly as frequency

TABLE I. Soft tissue and gel phantom parameter values for the theoreand experimental studies.

Parameter Soft tissuea Gel phantom

l1 (N/m2) 2.63109b 2.63109b

l2 (Ns/m2) 0c 0c

m1 (N/m2) 2.53103 4.53103d

m2 (Ns/m2) 15 4e

r ~kg/m3! 1100 1000h, p at v/2p540 Hz 6682359i , 6992375i 746282.3i , 781286.2icsurf~m/s), Lsurf~cm) 2.20, 5.51 2.06, 5.16h, p at v/2p560 Hz 5432354i , 5692370i 7312119i , 7652125icsurf~m/s), Lsurf~cm) 2.70, 4.50 2.11, 3.52h, p at v/2p580 Hz 4642335i , 4862351i 7102151i , 7442159icsurf~m/s), Lsurf~cm) 3.17, 3.96 2.17, 2.71h, p at v/2p5200 Hz 2782244i , 2912255i 5672253i , 5932265icsurf~m/s), Lsurf~cm) 5.28, 2.64 2.72, 1.36h, p whenm250~no damping!

1020, 1067 760, 796

csurf~m/s), Lsurf~cm) 1.44, 2pcsurf /v 2.03, 2pcsurf /v

aLame constants and density based on von Gierkeet al. ~Ref. 22!.bSame value as water~Ref. 20!. Primarily affects compression waves, nshear waves.

cNegligible below 20 kHz~Ref. 20!.dBased on matching surface wave speed with that observed experimeneBased on matching attenuation rate with that observed experimentally


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creases and the approximation thatJ1(pak1)'pak1/2 be-comes less valid. The limitation of the approximation fK0( iprk1) is not as evident for this set of parameter valuAlso illustrated in these figures is the influence of the viscity term, as the rate of attenuation per radial distance isnificantly increased when viscosity is taken into account. Rferring to Table I, some additional parameters that descthe surface wave behavior are provided. The phase speethe surface wave,csurf can be calculated from the followingexpression:

csurf5v/real~k1p!5real~pA~l112m11 i2vm2!/r!.~16!

If the system is viscoelastic,m2Þ0, andp is complex, then itis dispersive withcsurf dependent on frequency,v. If thesystem is elastic,m250, andp is real, then it is nondispersive, with phase speed independent of frequency. Notethe dispersive nature of the phantom medium is evident,

FIG. 4. Experimental measurements on the phantom of the vertical parvelocity amplitude on the surface normalized to the disk velocity as a fution of excitation frequency and radial position. Amplitude in d520 log10 .

FIG. 5. Comparison of gel phantom experimental measurements with tretical predictions of vertical particle velocity. Velocity is taken with respeto excitation force inputuz /F in ~m/sN) on the surface as a function of radiposition at excitation frequencyv/2p540 Hz Key: s s s experiment;——— improved theory with damping, ––– improved theory withodamping (m250); --- asymptotic theory with damping.

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though its effect on phase speed is minimal for the range40 to 80 Hz. In terms of absolute value and trend, the phspeed of surface waves on the phantom agrees with treported in the literature for human soft tissue.6,7,10,22Verysimilar differences between the improved theory aasymptotic theory were predicted for horizontal particle vlocity on the surface using the different theoretical modeUnfortunately, experimental comparison was not posswith the available instrumentation.

Having established the validity of the theoretical modit is now used to investigate wave propagation belowsurface. In Figs. 7 and 8, the predicted wave propagaprofiles are shown for vertical and horizontal motion asfunction of depth below the surface normalized by surfawavelength,Lsurf. This facilitates comparison of results

FIG. 6. Comparison of gel phantom experimental measurements with tretical predictions of vertical particle velocity. Velocity is taken with respeto excitation force inputuz /F in ~m/sN) on the surface as a function of radiposition at excitation frequencyv/2p580 Hz. Key: s s s experiment;——— improved theory with damping; ––– improved theory withodamping (m250); --- asymptotic theory with damping.

FIG. 7. Comparison of gel phantom theoretical predictions of vertical pticle velocity below the surface normalized to the vertical velocity amplituat the surface uz(z)/uz(z50) at selected excitation frequencieKey: ——— improved theory with damping atv/2p580 Hz; ––– im-proved theory with damping atv/2p540 Hz; --- improved theory withoutdamping at any frequency.


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different frequencies. It also renders predictions usingimproved theory and asymptotic theory identical, sinceproximations in the asymptotic theory are not relateddepthz but rather to radial positionr and the radius of thedisk. While for the purely elastic case the normalized qutities are independent of excitation frequency, for the vcoelastic case they have dependence on excitation frequthat is pronounced in the imaginary portion of the responNote that vertical velocity amplitude actually increasslightly below the surface. Horizontal motion reverses diretion. Both types of motion exponentially approach zerodepth increases. These predictions can be used to help athe limitations of the experimental phantom model in simlating infinite half-space conditions. They can also be usedetermine what frequency of surface waves may be alteby material anomalies located at different depths belowsurface and at different radial distances from the sourceFig. 9, calculations are shown of the amplitude of surfawave vertical-particle velocityuz as a function of normalized

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FIG. 8. Comparison of gel phantom theoretical predictions of horizonparticle velocity below the surface normalized to the vertical velocity aplitude at the surfaceur(z)/uz(z50) at selected excitation frequencieKey: ——— improved theory with damping atv/2p580 Hz; ––– im-proved theory with damping atv/2p540 Hz; --- improved theory withoutdamping at any frequency.

FIG. 9. Gel phantom theoretical prediction of the normalized amplitudevertical particle velocityuuz(r ,z)/uz(r 5a,z50)u as a function of depthzand radial positionr. Depth and radial position normalized with respectsurface wavelengthLsurf . Results shown forv/2p540 Hz.


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radial distancer and depthz at v/2p540 Hz. Due to thedispersive nature of the viscoelastic medium, results atquencies other than 40 Hz will not be identical but will bsimilar since dimensions have been normalized with respto surface wavelength,Lsurf.

B. Theory using human soft-tissue material valuesfrom literature

Human soft-tissue shear parameters estimated byGierkeet al.22 and used by other researchers17,20are differentthan those of the phantom used in the experimental stuOestreicher’s20 material values are used in the theoreticmodel to generate the results shown in Figs. 10–14, whare analogous to Figs. 5–9 for the gel phantom but withthe experimental data. The increased damping causes a

FIG. 10. Comparison of soft-tissue theoretical predictions of vertparticle velocity. Velocity is taken with respect to excitation force inpuz /F in ~m/sN) on the surface as a function of radial position at excitatfrequencyv/2p540 Hz. Key: ——— improved theory with damping; ––improved theory without damping (m250); --- asymptotic theory withdamping.

FIG. 11. Comparison of soft-tissue theoretical predictions of vertparticle velocity. Velocity is taken with respect to excitation force inpuz /F in ~m/sN) on the surface as a function of radial position at excitatfrequencyv/2p580 Hz. Key: ——— improved theory with damping; ––improved theory without damping (m250); --- asymptotic theory withdamping.


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nificantly sharper attenuation in response amplitude ofsurface wave as one moves away from the source. Whilemagnitude of surface wave particle motion below the surfrelative to the surface motion is comparable to that of thephantom, the imaginary portion of the response is increaReferring to Table I, it is noted that the increased shearcosity now has a much more significant impact on surfawave phase velocitycsurf and corresponding wavelengtLsurf. The human-tissue model is significantly more dispsive.

These results do not agree quantitatively or qualitativwith some of the experimental measurements reported inliterature on human soft tissue. Indeed, in von Gierkeet al.22

measurements taken over the thigh showed very little chain surface wave phase speed over the range of 10 to 180

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FIG. 12. Comparison of soft-tissue theoretical predictions of vertiparticle velocity below the surface normalized to the vertical velocity aplitude at the surfaceuz(z)/uz(z50) at selected excitation frequencieKey: ——— improved theory with damping atv/2p580 Hz; ––– im-proved theory with damping atv/2p540 Hz; --- improved theory withoutdamping at any frequency.

FIG. 13. Comparison of soft-tissue theoretical predictions of horizonparticle velocity below the surface normalized to the vertical velocity aplitude at the surfaceur(z)/uz(z50) at selected excitation frequencieKey: ——— improved theory with damping atv/2p580 Hz; ––– im-proved theory with damping atv/2p540 Hz; --- improved theory withoutdamping at any frequency.


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Additionally, in a stroboscopic photograph of responseexcitation at 62 Hz, three crests and troughs of a surfwave propagating away from a piston source are clearlyible. Von Gierkeet al.22 used several different approximatechniques that led to estimates ofm2 ranging from 2.5–50Ns/m2. These estimates were primarily based on impedameasurements at the surface, not surface wave behavioother studies of surface waves in soft tissue, values ofm2

52.5 to 5 Ns/m2 have been employed to obtain a good mabetween computation and experiment.4,10 Some estimatesbased on sonoelastic imaging have also been in this ran15

In other words, if one does use the soft-tissue paramereported in von Gierkeet al.22 and given in Table I of thispaper, one does not get the experimental results reportevon Gierkeet al.22 and others, at least in regard to surfawave speed and attenuation measurements.

Perhaps a more valid measure is the ratio ofm2 to m1 .Experimental data found in von Gierkeet al.22 andelsewhere15 support this. While values ofm1 and m2 mayvary by several factors for soft tissue from different partsthe anatomy or from the same location but based on diffetesting methods, the ratio ofm2 to m1 is somewhat consisten~though not exact given the variability of biological tissue!.It is this ratio that is an indicator of attenuation per cycle. Fa value ofm152500 N/m2, the observation made here thm2515 Ns/m2 is too high is consistent with the experimentfindings reported in the literature, even those of von Gieet al.22

Finally, the limited applicability of this study to the biological condition is noted. The model of a viscoelastic istropic half-space does not take into account the layevariation of viscoelastic properties one encounters wmoving from the skin surface through its layers to underlytissue. Incorporation of these complicating effects is crently under investigation.

IV. CONCLUSION

An analytical solution has been developed and expmentally verified for the problem of surface wave generaton a viscoelastic half-space by a finite rigid circular di

FIG. 14. Soft-tissue theoretical prediction of the normalized amplitudevertical particle velocityuuz(z)/uz(r 5a,z50)u as a function of depthz andradial positionr. Depth and radial position normalized with respect to sface wavelengthLsurf . Results shown forv/2p540 Hz.


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located on the surface and oscillating normal to it. The alytical solution is an incremental advancement of the thretical work reported in Miller and Pursey18,19that resulted inan asymptotic solution. Parametric studies reported heregest that, for surface wave propagation on soft biologitissue, the limitations of the asymptotic solution maymore relevant than for the seismological applicationswhich the theoretical work of the reference was originaintended.

Since the application focus here is medical diagnostthe analytical solution was verified experimentally usingviscoelastic phantom with material properties comparablebiological soft tissue. Surface vibration measurements wmade using a noncontacting laser Doppler vibrometer. Treported results furnish the reader with some physical apciation of the dimensions of surface waves on biologicalsues in the low audible frequency range, 20–100 Hz. Asfrequency increases further, these waves are attenuatedhigher and higher rate with respect to distance fromsource. Reported results have also suggested that priormates in the literature for shear viscosity in human soft tisare not accurate in the low audible frequency range.

It is hoped that the developments reported here will hto advance novel medical diagnostic techniques basedmeasuring variations in surface and shear wave behacaused by elastic and/or density anomalies that may be icators of particular illnesses or signify the presence of a geous inclusion. Research is in progress to extend the wormore realistic models that account for layered variationsmaterial properties that are more indicative of the normbiological condition. The reported results may also helpthe interpretation of surface-based measurements of vibracaused by subsurface biological sources, such as cardiocular function, pulmonary disorders, or gastrointestinal acity.

ACKNOWLEDGMENTS

The financial support of the Whitaker Foundation undthe BME grant program is acknowledged. S.-H. Kim is aknowledged for assistance with the experimental study.Zhang is acknowledged for assistance with the theoretwork.

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Excitation and propagation of surface waves on a viscoelastic half-space with application to medical...

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