acoustic scattering in three-dimensional fluid media

Acoustic scattering in three-dimensional fluid media Kavitha Chandra, a) Sylvia K. Isler, and Charles Thompson Center for Advanced Computation and Telecommunications, University of Massachusetts Lowell, Lowell, Massachusetts O1854

(Received 3 January 1995; revised 16 May 1995; accepted 29 May 1995)

The scattering of acoustic waves from three-dimensional compressible fluid scatterers is considered. Particular attention is paid to cases where the scatterers have moderate magnitude in compressibility contrast and nondimensional wave number. The perturbation method based on Pad• approximants developed by Chandra and Thompson [J. Acoust. Soc. Am. 92, 1047-1055 (1992)] is extended to allow the solution of problems in three dimensions. It is shown that the functional form afforded by the Pad• approximant model allows one to represent and evaluate the characteristic resonances and mode shapes of the scattered pressure field. These modes are a function of the compressibility contrast and the frequency of the incident pressure wave. Numerical results are shown to compare favorably with analytical solutions for scattering from a sphere. The Pad• approximant method is shown to be feasible for calculating the pressure scattered from an inhomogeneous distribution of scatterers as well as for determining internal resonance. ̧ 1995 Acoustical Society of America.

PACS numbers' 43.20.Bi, 43.20.Fn, 43.30.Ft

INTRODUCTION

When the constituents of the medium exhibit a high contrast in acoustic properties, scattering and refraction are dominant phenomena. The magnitude of the acoustic pressure waves scattered from a medium is dictated by the degree of spatial variability in the compressibility and density. For a discrete set of scatterers, spatial orientation is also important. Historically, ultrasonic interrogation and evaluation of biological media has been based on absorption • and backscat- tered pulse-echo measurements. 2'3 More recently, scattering properties of soft tissues have been analyzed 4-9 in terms of the angular variation in the scattered pressure field. These results identify the potential for using nonspecular reflections to characterize the structure and composition of tissues. When the scattered pressure amplitude is small, experimental results have verified that weak scattering theory based on the Born approximation yields satisfactory results. The theoretical and computational analysis in cases where strong scattering is important remains an area of active research.

The scattered pressure from discrete, arbitrarily shaped three-dimensional scatterers can be evaluated by either boundary integral methods •ø or wave-field expansion techniques. •'•2 Boundary integral solutions of the Helmholtz integral formula have been shown to be numerically tractable when the scatterer boundaries can be appropriately modeled as either acoustically rigid or soft. However, nonunique solutions arise at characteristic frequencies. The application of boundary integral techniques to biological media is problem- atic. This is due to the complexity of the interfacial regions and the absence of appropriate boundary conditions. Promi- nent among the wave-field expansion techniques for the solution of the Helmholtz integral equation are extensions of

a) ß Currently at AT&T Bell Laboratories, 101 Crawfords Corner Rd., Holm- del, NJ 07733.

Waterman's T-matrix method. 12 In this method, the pressure is expanded in terms of a set of a basis functions to approximate the scattered acoustic pressure field. In particular, cy- lindrical or spherical wave functions have been used to model scattering from single fluid or elastic scatterers of arbitrary shape. 13 For multiple scatterers having arbitrary shape, integral equation formulations for the unknown pressure have superior convergence and stability properties when the medium contrast is small. TM

Recently, Chandra and Thompson 15 proposed a perturbation scheme for solving two-dimensional scattering problems using the Helmholtz integral equation. Representing the pressure field in a rational fraction form, using the magnitude of the compressibility contrast as the gauge function, it was found that the region of convergence of the classical Neu- mann series solution could be extended. In this work, we

consider the application of the Pad• approximant method to three-dimensional scatterers. The paper is organized in three sections. In Sec. I, the problem geometry is presented, the governing equations are described, and the Pad• approximant technique is briefly reviewed. A single spherical scatterer is considered to validate the method. In Sec. II, the

relationship between the pole singularities in the Pad• approximant solution and frequency resonances is presented. In Sec. III, the applicability of the numerical technique for inhomogeneous media is demonstrated. In particular, the problem of acoustic scattering from an ellipsoid embedded in a sphere is considered. The impact of the inhomogeneity in contrast on the mode shapes for the pressure is examined.

I. PROBLEM GEOMETRY AND GOVERNING

EQUATIONS

The geometry for the scattering problem to be analyzed is shown in Fig. 1. The three-dimensional scattering volume, shown shaded in the figure, is denoted as region V. The

3462 J. Acoust. Soc. Am. 98 (6), December 1995 0001-4966/95/98(6)/3462/7/$6.00 ¸ 1995 Acoustical Society of America 3462

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v0

FIG. 1. The geometry of the three-dimensional scattering problem. The host medium V0 is characterized by compressibility K 0. The scatterer V is characterized by the spatially varying compressibility, Kv(X)+ K0.

compressibility in the host medium represented by the region V0 has a constant value K 0. The region V is characterized by a spatially varying compressibility Kv(X)+ K 0. The compressibility contrast is given by the expression

%•(x,k) = Kv(X,k)/K o , (la)

where k is the magnitude of the wave-number vector. Taking e to be the typical magnitude of the compressibility variations, the function -•,,(x,k) is the compressibility contrast normalized by e,

T,•(x,k) = e-•,•(x,k). (lb)

The scale factor e is considered to be the perturbation param- eter in the analysis.

The incident field is taken to be that of a time harmonic

plane wave propagating in the k direction:

pi(x,k)=e ik'x. (2)

The acoustic pressure p at the coordinate position x in the region VUV0 is given by the solution of the Helmholtz integral equation, 16

p(x,k,e)=pi(x,k)

4- • f v k2.•,•(xO,k)G(xOIx, k)p(x o ,k, e)dx 0 . (3)

The free-space Green's function G(xlx0,k), being equal to

G(xl x0 ,k) - eikR/4 erR, (4)

where R-Ix-x01, is the distance between the observation point x and the source point x 0. The method of solution of Eq. (3) is outlined next.

II. METHOD OF SOLUTION: PADI• APPROXIMANTS

A classical method for solving the Helmholtz integral equation given in Eq. (3) uses the method of successive approximations 16'•7 to determine the unknown pressure amplitude. In the limit as e tends to zero the pressure can be evaluated in terms of a perturbation series in the gauge functions e n. This perturbation expansion is called the Neumann series:

p(x,e,k)= • enpn(x,k ). (5) n=0

TABLE I. Location of zeros of •m (E, ka) in complex ß plane, le•<l.0.

ka m=0 m=l

2.0 0.002-i0.881

2.5 -0.234-i0.650 0.557-i0.829

3.0 - 0.374- i0.4 87 0.184-i0.692

Each coefficient Pn of the series can be evaluated by substi- tuting Eqs. (5) and (2) into Eq. (3) and equating like terms in e. The coefficient terms are obtained successively and the zeroth-order term is equal to the incident pressure. For n >0,

Pn(X, e,k) = f v k2 '•t<(x0, •)G(xolx, k)p n_ l(X0, •)dx 0 . (6)

The Neumann series converges if is less than the distance to the nearest singularity in the complex e plane. The location of the pressure singularities in the complex e plane can be evaluated in the cases where the exact pressure solution is known. When approximate methods of solution are used the location of these poles cannot be easily determined. The Pad6 approximant method allows these poles to be directly recovered from the resulting rational fraction expression for the pressure.

To highlight this fact, scattering from a single spherical scatterer of radius a will be considered. The pressure for this problem geometry will be evaluated numerically. An outline of the derivation of the analytical solution is given in the Appendix. The analytical solution for pressure scattered from a sphere is represented as/•(x,e,k). From Eq. (A1) given in the Appendix, it can be seen that the singularities of the pressure/•(x,e,k) are governed by the zeros of the denomi- nator of the coefficients A m and B m . The zeros are determined by the function

J•m(e,ka)= x/( 1 + e)jn(kva)hm(ka)-jm(kva)hn(ka), (7)

where the Jm and h m are mth-order spherical Bessel and Hankel functions, and j•n (ka) and h•n (ka) are their respec- tive derivatives. The prime is taken to represent differentia- tion with respect to the argument of the function. The wave number interior to the sphere is kv = k x/1 + •.

We will determine the pole locations in the e plane for the cases ka =2.0, 2.5, and 3.0. The zeros of b m with respect to e and fixed ka can be examined individually for each integral value of the index m. We will focus the search on the region I•< 1.0. The zero crossings of D m were first numerically identified on a coarse grid in the region of interest of the e plane. Using these points as the initial starting values,

, 18 Muller s method was used to identify the complex zeros of the function D m . The locations of the zeros of D m in the complex e plane that have magnitude less than one are given in Table I. The pole for m = 1 and ka =2.0 is outside of the disk, Il-

It can be seen that when ka=3.0 and m=0 the pole locations closest to the origin occur along the radius of

l eSl I = 0.61. The poles typically move closer to the origin as

3463 J. Acoust. Soc. Am., Vol. 98, No. 6, December 1995 Chandra et aL: Scattering in three dimensions 3463


4-

2-

• 1•1=o.8

........ •' ........ I•1 = 0.65

.... • .... I•1 = 0.45

ß " *. ""• "'...... •... e•• '•' ....o "& .... •' -?•2"'"" '/ '."52•.•.....•. ...'.'.• ......... o' • ..... . •..... "•"•'"" "x ..-"

0

Number of terms summed, N

FIG. 2. The partial sum of the Neumann series for a location at the center of the sphere for ka =3.0, and increasing magnitude of e.

the wave number is increased. Since the Neumann series

solution is geometric in e, one can expect series divergence

to occur for values of e greater than lest. The Neumann series coefficients in Eq. (5) were com-

puted by discretizing the sphere on a rectangular grid of Nx by Ny by N z equal to 100 points. The three-dimensional integral in Eq. (6) was evaluated using a trapezoidal quadrature scheme. The incident field is taken to propagate along the z axis. The discretized sphere is comprised of 33 401 nodes. The distance of 8x=0.01 between the center of successive

nodes was used. The magnitude of the partial sum of the Neumann series coefficients versus the number of terms used

is plotted in Fig. 2 for the case of ka=3.0. The pressure amplitude corresponds to a fixed spatial location, selected to be the center of the sphere and the curves correspond to values of e equal to 0.45, 0.65, and 0.8. As e is increased, the series coefficients decay at a slower rate. As a result, the partial sum of the Neumann series increases with the number

of terms used. It can be seen that for e >1 ers•l, the Neumann series diverges. To extend the Neumann series beyond les•l, one must consider a rational fraction form for the pressure amplitude in the contrast magnitude e. Such a representation was considered by Chandra and Thompson •5 for two- dimensional scatterers and Pad• approximants were used to represent the pressure. In this work, the result is extended to three dimensions. The pressure at each coordinate position x is represented as

• nN_-oA n( x,k ) E n P•(x,e,k)= i q- 5•Nm=lBm(x,k)E m' (8)

where N represents the order of the numerator and denomi- nator polynomials in e. One evaluates the coefficients A n and B m by equating the two representations for the pressure given in Eqs. (5) and (8),

2N

'• pj(x)erJ- P•(x, er, k). (9) j=0

TABLE II. Pole-zero distribution from Pad• approximants.

ka Poles Zeros

2.0 -0.009-i0.8 83 0.230-i0.705

-0.243-i0.645 0.386-i0.7 81

2.5 0.552-i0.855

3.0 -0.378-i0.474 -0.008-i0.522

0.170-i0.705

In doing so, the Pad• approximant P• and the Neumann series are asymptotically equivalent in the limit as e tends to zero. Having computed the Neumann coefficients Pn, the Pad• approximant coefficients can be obtained by equating like orders in E on both sides of Eq. (9). Further details on the calculation of the Pad• approximant coefficients can be found in Refs. 15 and 19. Typically, an increase in the magnitude of the wave number is manifest in an increase in the

number of pole singularities in the e plane. This point is supported by Sommerfeld, 2ø who showed that the number of terms required for the convergence of the series expansion for the scattered pressure from a sphere is proportional to ka. In this case, the pole locations are given by the zeros of the function b m . Since k v is equal to k x/1 + e, the number of terms, and accordingly the number of poles, increases with increasing e or ka. For example, with normalized wave- number parameters ka=2.0, 2.5, and 3.0, the Pad• approximants were found to converge at order N=7, 8, and 9, respectively. The residual poles after pole-zero cancellation are shown in Table II.

The scattered pressure field obtained from the Pad•- based numerical technique is compared with that of the analytical solution for the case ka=3.0 in Fig. 3. The figure depicts the scattered pressure field variations inside the sphere at a radius of r-0.75a in the x-z plane and y=0.0.

i

10-

5-

0-

-5-

-10

-15

Observation Angle

FIG. 3. Comparison of the analytical result and the Pad• approximant method. The scattered pressure is evaluated at r=0.75a for ka=3.0 and e= 1.0.--, analytical; •, Pad• approximant.

3464 J. Acoust. Soc. Am., Vol. 98, No. 6, December 1995 Chandra et aL' Scattering in three dimensions 3464


TABLE III. Pad6 approximant poles and analytically derived poles.

ka Pad6 Analytical Relative error

2.0 -0.009-i0.883 0.002-i0.881 0.008

-0.243-i0.645 -0.234-i0.650 0.014 2.5

0.552-i0.85 5 0.557-i0.829 0.026

3.0 -0.378-i0.474 -0.374-i0.48 7 0.022

0.170-i0.705 0.184-i0.692 0.027

The pressure field is shown as a function of 0 where 0•<0•<2rr. The results from the Pad• approximant method are in good agreement with the analytical result. The maxi- mum error was found to be 1.17 dB which occurred at 0= rr

rad. The slight separation between the curves is the result of the coarseness of the discretization interval used in the inte-

gration of the scattering volume. A solution to the accuracy of 10 -6 requires that the number of integration nodes be doubled.

An examination of the roots of the numerator and de-

nominator polynomials of P•(x) is in order at this juncture. These roots represent the zeros and the pole singularities of the pressure solution, respectively, in the complex e plane. For a fixed spatial location x=0, the pole and zero distribution with magnitude less than one is tabulated in Table II for the wave numbers considered. Using the Pad• method a number of pole-zero cancellations ensue. The residual poles and zeros that do not cancel characterize the pressure amplitude. These poles and zeros are given in the table.

Examination of the spatial distribution of pole singularities over the scatterer volume as well as in the far field re-

vealed that the residual poles identified in Table II remain fixed with respect to changing spatial location. Therefore these poles are common to all locations in the spatial pressure distribution. The residual pole singularities obtained from the Pad• approximant method are tabulated in Table III. The analytical results given in Table I and the magnitude of relative error between the analytical and Pad• approximant solutions normalized by the analytical result are tabulated for comparison.

The Pad• approximant poles approach those of the analytical solution as the number of quadrature points is increased. In particular, for ka = 3.0, the relative error suffered in using 50 by 50 by 50 quadrature points was 0.055 and 0.069 for the two poles that lie closest to the origin of the sphere. The corresponding relative error resulting from the use of 100 by 100 by 100 quadrature points for the same two poles and value of ka are given in Table III. It can be seen that the error decreases as the reciprocal of the number of quadrature points.

Once the characteristic singularities of the pressure field are determined, the characteristic mode shapes for the scattered pressure can be evaluated. We consider here the mode shapes as a function of the angular coordinate 0 in the x-z plane. For this purpose, the Pad• approximant form function is placed in a proper rational fraction by using its partial fraction expansion. For a fixed far-field radial position

ß -

.

• ' 0 , • 08 ß

.

N ONd

FIG. 4. The mode shapes corresponding to the first two singularities in TableI. C• ß ß ß , C2-'-.

r- 100a from the center of the sphere, the pressure can be expressed as

c(o) Cn(O) P•(0)-C0(0) + +'" q- •, (10)

E-- ES l E-- Es n

where 65i , i= 1,...,n, represents the n singularities in the pressure solution. Since the location of these singularities can be found using the aforementioned numerical analysis, the corresponding modes C•'"Cn can be evaluated. For the case ka- 3.0, the mode shapes corresponding to the first two singularities identified in Table III are plotted in Fig. 4. The angular variation of C•(O) corresponds to the zeroth-order Legendre polynomial, whereas C2(0 ) varies as the first-order Legendre polynomial multiplied by cos 0. This result is in agreement with the spherical modes given in the Appendix.

The results in this section have demonstrated the utility of the Pad6 approximant model in evaluating the scattered pressure in situations where the unmodified Neumann series is not directly applicable. In addition, the singularities of the model have been shown to accurately approximate the poles present in the pressure solution. This feature allows one to extract the characteristic mode shapes for the scattered pressure field from the Pad6 approximant representation. The results for the scattered pressure considered here served to validate the numerical approach. The objective in using the method, however, is to examine similar features for media that cannot be analyzed exactly. To this end, the next section presents results for acoustic scattering from an ellipsoid embedded within a sphere. The effect of additive noise in the compressibility distribution is also examined.

III. SCATTERING FROM INHOMOGENEOUS MEDIA

Biological media are often characterized by an inhomogeneous spatial distribution in the medium compressibility. For example, in the particular case of breast carcinomas, studies have shown 2•'22 that over 50% of breast carcinomas

appear with globular clusters of calcium. In addition, breast cancer tumors tend to be less compressible than the healthy breast tissue that encompasses them. In order to adequately describe the scattering properties of biological media and



FIG. 5. A planar section of the compressibility distribution for the inhomogeneous medium. The compressibility contrast of the sphere with respect to ¾0 is e2- The compressibility contrast of the ellipsoid with respect to the host medium is el-

detect diseased conditions, it is necessary to describe both angle- and frequency-dependent scattering. Frequently, normal and diseased tissue are characterized based on the degree of angular- or frequency-dependent scattering that occurs. Davros et al. 23 characterized normal breast tissue by study- ing frequency-dependent angular scattering properties of ex- cised tissue samples. In this section, the utility of the Pad• approximant method for computing scattered fields from inhomogeneous media is demonstrated.

A study of the effect of the heterogeneous structure of the medium on the acoustically scattered field will be under- taken using a medium comprised of an ellipsoid enclosed within a sphere. By choosing an ellipsoid, the influence of the angle of incidence of the ensonifying pressure on the nonspecular reflection of the pressure wave can be directly related to the principal axes of the ellipsoid. It has been shown that information on the structure of the medium can

be gleaned from the scattered field. 24 A planar section of the compressibility distribution in the x-z plane at y =0 is shown in Fig. 5. The compressibility contrast of the sphere with respect to the host medium is e 1 . The ellipsoid enclosed by the sphere has a compressibility contrast e 2. The ratio of minor to major axis of the ellipsoid is fixed at 1:1.5. We will assume that e2>e • . The scattered pressure field will be examined for varying compressibility contrast and angle of incidence of the ensonifying pressure.

The variation of the pole location as a function of the incident angle will be examined first. We consider a contrast distribution e2: q of 2.0. This ratio represents the approximate compressibility differential between a cyst of compressibility contrast e 2 and a calcified region within the cyst of contrast e 1. The magnitude of e 1 is approximately 0.5. Ratios of e2:e • > 1 are typical in cases where a scatterer, such as a tumor, contains a hardened interior. Values of ka > 1 are

typical for ultrasonic interrogation of biological media. In our case, ka is fixed at 3.0 where a is the radius of the

sphere. Angles of incidence considered are 0i=0.0 ø, 90 ø, and 45 ø and correspond to directions along and perpendicular to the principal axis and intermediate to the major and minor axis of the ellipsoid, respectively. The characteristic singularities obtained for the three angles of incidence are given in Table IV. All poles reside within a radius of 14-2.25.

The mode shapes corresponding to each of the singularities tabulated are depicted in Fig. 6. The mode shapes are

TABLE IV. Location of poles in complex e plane, le1•<2.25.

0 i rn =0 rn = 1 rn =2

0.0 -0.371- i 1.019 0.613-i 1.377

90.0 -0.371- i 1.019 0.613- i 1.377

45.0 -0.371 -i 1.019 0.613- i 1.377 0.876-i2.040

referenced as C•(O), C2(0), and C3(0 ) in order of increasing magnitude of the distance of the poles from the origin. It can be seen that C• and C2 correspond to modes for a spherical geometry, i.e., Legendre polynomials of order 0 and 1, respectively. These are the only modes excited for propagation along the principal axis of the ellipsoid. When the incident field is propagated in a direction off the principal axis as in the case 0i=45 ø, additional modes due to the ellipsoidal inclusion are excited. This corresponds to mode C3(0). It is seen that although the location of the individual singularities in the complex e plane are invariant to the change in the angle of incidence, the angle of incidence determines the angular mode selected. It was found that the full spectrum of modes can be excited by propagating the incident field in a direction that is intermediate to the major and minor axis of the ellipsoid.

Next, we consider the effect of nonuniformity of the contrast variation on the pole location and mode shapes. For this purpose random noise is added to uniformly distributed compressibility contrast of the sphere and ellipsoid. Gaussian distributed noise having a mean value of 1.0 and a standard deviation cr was used. As cr is varied the pole location was tracked. Table V depicts the characteristic poles of the pressure field for the values of standard deviation or=0.1 and 0.3, respectively. The incident angle is maintained at 45 deg. The presence of noise serves to drive the singularities further from the origin. This can be verified by comparing Table V

0.(

0.01

FIG. 6. The mode shapes corresponding to Table IV, 0i=45 ø. C 1 ß - ß , C2-- -, C3.

3466 d. Acoust. Soc. Am., Vol. 98, No. 6, December 1995 Chandra et aL: Scattering in three dimensions 3466


TABLE V. Effect of additive noise.

rr m=O m=l

0.1 -0.670-i0.976 0.413-il.505

0.3 - 1.226-il.752 0.731-i2.726

with the corresponding results in Table IV in the absence of noise. Therefore, randomization of the acoustic properties can render the Neumann series convergent over a wider compressibility contrast range.

The singularities can be driven further from the origin by increasing the variance of the noise component, as can be seen for the case where o-=0.3. The noise component re- duces the coherence of the scattered field, thereby smearing out the modal features of the embedded scatterer.

IV. CONCLUSIONS

The scattering of a time harmonic acoustic wave from three-dimensional fluid scatterers has been examined. It has

been shown that Pad• approximants can be used to extend the validity of the Neumann series expansion. In addition, the poles of the resulting rational fraction expression for the pressure allow one to the perform a modal decomposition of the scattered field. The introduction of noise into the com-

pressibility contrast moves the location of the poles in the ß plane farther from the origin. As a result the Neumann series can converge over a wider range of the compressibility pa- rameter.

ACKNOWLEDGMENT

The authors wish to acknowledge the support of the De- partment of Energy and the AT&T Cooperative Research Fellowship Program.

APPENDIX

The outline of the analytical solution for scattering from a sphere is given here. For a spherical inhomogeneity of radius a, the pressure at an observation location (r,O, ck) is given by the following equations:

• im(2m+ 1)Pm(0)cos(mck)•mJm(kvr), r•<a, m=0

t•(r,O, ck,ß,k)= •

• im(2m+ 1)Pm(0)cos(mqb)[jm(kr)+t•mhm(kr)], m=0

r>a,

(A1)

where Jm and h m represent mth-order spherical Bessel and Hankel functions, respectively, and Pm represents the mth- order Legendre function. The wave number in the sphere k v is equal to k x/1 + ß.

In the aforementioned equations the coefficients •m and /•m are obtained by applying the boundary conditions of con- tinuity of pressure and normal velocity at r = a. This results in

•m-- aJ•(kva)jm(ka)-jm(kva)j•(ka) (A2) h•n(ka)jm(kva)- aj•n(kva)hm(ka) '

where a = x/( 1 + ß) and

•m_Jm(ka)+Bmhm(ka) jm(kva ) . (A3)

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acoustic scattering in three-dimensional fluid media

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