Modeling of pulsed finite-amplitude focused sound beams in time

Modeling of pulsed finite-amplitude focused sound beams in time domain Jahangir Tavakkoli,a) Dominique Cathignol, and Re´mi Souchon INSERM—Unite´ 281, 151 Cours Albert Thomas, 69424 Lyon Cedex 03, France

Oleg A. Sapozhnikov Department of Acoustics, Physics Faculty, Moscow State University, Moscow 119899, Russia

~Received 10 June 1997; revised 11 June 1998; accepted 23 June 1998! A time-domain numerical model is presented for simulating the finite-amplitude focused acoustic pulse propagation in a dissipative and nonlinear medium with a symmetrical source geometry. In this method, the main effects responsible in finite-amplitude wave propagation, i.e., diffraction, nonlinearity, and absorption, are taken into account. These effects are treated independently using the method of fractional steps with a second-order operator-splitting algorithm. In this method, the acoustic beam propagates, plane-by-plane, from the surface of a highly focused radiator up to its focus. The results of calculations in an ideal ~linear and nondissipative! medium show the validity of the model for simulating the effect of diffraction in highly focused pulse propagation. For real media, very good agreement was obtained in the shape of the theoretical and experimental pressure-time waveforms. A discrepancy in the amplitudes was observed with a maximum of around 20%, which can be explained by existing sources of error in our measurements and on the assumptions inherent in our theoretical model. The model has certain advantages over other time-domain methods previously reported in that it: ~1! allows for arbitrary absorption and dispersion, and ~2! makes use of full diffraction formulation. The latter point is particularly important for studying intense sources with high focusing gains. © 1998 Acoustical Society of America. @S0001-4966~98!02810-0# PACS numbers: 43.25.Jh @MAB#

INTRODUCTION

Advances in the development of acoustic microscopes,1 high intensity focused ultrasound surgery,2 lithotripsy, and cavitation-induced tissue destruction3 have generated renewed concern about finite-amplitude effects in focused sound beams. Each system is capable of transmitting focused sound that is strongly affected by the combined influence of nonlinearity, absorption, and diffraction. Existing analytical solutions fail to adequately describe these sound fields, and only recently have numerical solutions been developed that model the radiation of focused finite-amplitude sound from practical sources. A series of theoretical models for studying the focusing of intensive acoustic waves has been developed during the last two decades. A model which seems to be best suited to the study of moderately focused acoustic beams and which accounts for diffraction, nonlinearity, and absorption is based on the Khokhlov–Zabolotskaya–Kuznetsov ~KZK! parabolic equation.4,5 In the original KZK equation, the effect of absorption is modeled by incorporating viscous and thermal conductivity losses, in which the absorption coefficient is assumed to be proportional to the square of frequency. However, such an assumption is not appropriate to most biological tissues which exhibit a nearly linear frequency dependence. Concerning the diffraction effect, the KZK model uses a so-called parabolic or quasi-optical approximation, a!

Current address: Institute of Biomedical Eng., University of Toronto, 4 Taddle Creek Road, Toronto, Ontario M5S 3G9, Canada.

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J. Acoust. Soc. Am. 104 (4), October 1998

which means that the angular spectrum is supposed to be narrow or, in other words, the wave is very close to a plane wave. This is not the case for strongly focused beams or for beams with strong irregularities in the transverse structure, e.g., near the edges or at the focal point of a focused source. This limits the validity of the model to the cases in which diffraction effects are relatively weak and focusing gains are relatively low. In general, the KZK equation is an accurate model of the sound field produced by directional sound sources (ka@1, where k is the wave number and a the radius of the source! at distances beyond a few source radii and in the paraxial region. A complete discussion of the domain of validity of the KZK equation for plane and focused sources is provided in Refs. 6 and 7. A number of computer algorithms has been proposed to solve the KZK equation numerically. One of the most popular algorithms for solving the KZK equation is a frequencydomain technique, called the spectral method, introduced by Aanonsen8 and co-workers.9 Most studies, based on the spectral method, focused initially on monochromatic waves or on tone bursts. However, in many biomedical applications the acoustic wave consists of a small number of cycles or even has the form of a single cycle. If, additionally, shock fronts are developed because of nonlinearity, the numerical analysis requires large amounts of computer time. The situation becomes even more complex if the absorption is frequency dependent and the beam is strongly focused. To partially overcome these problems, Lee and Hamilton10 developed a time-domain algorithm for solving the KZK equation. A marching scheme, based on the

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© 1998 Acoustical Society of America

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operator-splitting method,11,12 was used as the basis for the algorithm. In this technique, as the code marches along, at each step it takes separate account of nonlinear distortion, absorption, dispersion, diffraction, and any other effects. Cleveland et al.13 used this algorithm to solve an augmented KZK equation that accounts for nonlinearity, thermoviscous absorption, relaxation, and diffraction in directive sound beams. Moreover, Averkiou and Hamilton14 have recently presented new results of this time-domain model for the case of short pulses radiated by plane and focused circular pistons. Within the limits of the KZK model, they obtained very good agreement between their simulations and experiments. In a related work, again based on the operator-splitting method, Cleveland et al.15 presented a comparison of several time and frequency-domain codes for the propagation of sonic booms through idealized atmosphere. The main effects considered in these codes were nonlinearity, absorption, and dispersion. A model of nonlinear diffractive field propagation, not based on the KZK equation, was developed by Christopher and Parker.16 In this model, using an operator-splitting method, they solved the equations of diffraction and absorption in the frequency domain and the equation of nonlinearity in the time domain. This model has advantages over the KZK model in that it accounts for full diffraction and arbitrary absorption effects. Using a modified version of this model, Christopher17 presented the modeling of an electrohydraulic, extracorporeal shock wave lithotripter as an example of an intense highly focused sound source. On the other hand, the effect of acoustic nonlinearity on the focused beam can be predicted on the basis of simpler theoretical approaches. We have already proposed a simple model that makes it possible to study the focusing of an intense pulse on the basis of the spherical wave theory.18 In this model, the focusing is associated with wave propagation along a rigid-wall tube. The dependence of the tube cross section on distance is chosen in such a way that the peak pulse pressure in the tube coincides with the peak pressure calculated by the Rayleigh diffraction integral in the linear regime. The nonlinear wave propagation along the tube is then described on the basis of a Burgers-type equation. Of course, such a simple model suffers from the same limitations as nonlinear spherical wave theory and is not accurate enough, especially in the focal region of a high-amplitude focused source. For this reason, we present another, more complete, model. As with the approach used by Christopher, our model accounts for full diffraction and arbitrary absorption effects. A second-order operator-splitting algorithm is used to solve a set of equations that account for the effects of diffraction, absorption, and nonlinearity. To avoid numerical errors associated with the transformations between the time and frequency domains, a pure time-domain approach has been adopted.15 The basic theoretical approach and the results obtained are presented below. The model was used for simulating the pressure field of a highly focused source developed for tissue destruction studies.19 2062

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I. BASIC EQUATIONS

In general wave theory, differential equations of the evolution type are widely used to describe the propagation process:

] v~ t ,z ! ˆ [L • v , ]z

~1!

where v is the particle velocity, z is the coordinate in the direction of wave propagation, t 5t2z/c 0 is retarded time, c 0 is the wave speed, and the operator Lˆ accounts for the effects changing the waveform. If Lˆ • v 50, we have v 5F( t ), i.e., the waveform does not change during propagation, which is the case of a plane wave traveling in an ideal linear medium. In reality, the waveform is distorted due to different effects. Consider, for instance, the equation for a linear plane acoustic wave propagating in a thermoviscous medium:20 b ] 2v ]v 5 , ] z 2 r 0 c 30 ] t 2

~2!

where b is the dissipative coefficient of the medium and r 0 is the ambient density. In many practical situations, e.g., in biological tissues, the attenuation operator differs from that of Eq. ~2!. It can be written in the general form:21

]v 5 ]z

E

t

2`

K ~ z, t 2 t 8 ! • v~ z, t 8 ! d t 8 [Lˆ A • v ,

~3!

where the kernel K can be evaluated from the frequency dependence of the attenuation. The upper limit of the integral accounts for causality. Another example is the evolution equation for a plane wave propagating in a lossless quadratically nonlinear medium:

]v b ]v ˆ 5 v [L N • v , ] z c 20 ] t

~4!

where b is the coefficient of nonlinearity of the medium. The third example is the so-called parabolic equation, describing diffraction of acoustical beams in linear lossless medium:

]v c0 2 5 ¹' ]z 2

E

t

2`

v d t 8,

~5!

where z is the distance along the beam axis, ¹'2 is a twodimensional transverse Laplacian, and v is the axial component of particle velocity. Equation ~5! describes the effect of diffraction properly only if the wave angular spectrum is narrow ~quasi-plane wave!. A more exact, wider-angle parabolic evolution equation was proposed by Hill.22 On the other hand, the diffraction effect can be described by the Rayleigh integral over the initial source surface.23 It is known that the Rayleigh integral presents an exact solution of the diffraction problem for the case of a plane surface. When the surface is curved, the integral gives an approximate, but fairly accurate solution.24,25 Let us denote the corresponding diffraction operator as Lˆ D and write a general form of the evolution equation for diffraction: Tavakkoli et al.: Modeling of pulsed finite-amplitude sound beams

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1`

]v ˆ 5L D • v . ]z

~6!

In all of the above examples, each evolution equation is associated with only one effect. In the presence of multiple effects, if each is fairly weak, the evolution equation can be derived simply by summing the corresponding operators from the ‘‘one-effect equations’’: N

]v 5 Lˆ • v . ] z i51 i

(

~7!

For example, the combination of Eqs. ~2! and ~4! gives the Burgers equation and the combination of Eqs. ~2!, ~4!, and ~5! results in the KZK equation. A more exact result than the KZK equation, accounting for the absorption and diffraction effects, is possible on the basis of combining Eqs. ~3!, ~4!, and ~6!. We write this combined equation in a general form:

]v ˆ 5L • v [Lˆ A • v 1Lˆ N • v 1Lˆ D • v , ]z

~8!

which will be used for the development of our model for finite-amplitude acoustic pulse focusing. Note that Lˆ is a fairly complex integro-differential operator. One of the possible ways of solving it numerically is the use of the method of fractional steps with an operator-splitting procedure.11,12 According to this method, the solution of the Eq. ~8! at each step Dz is obtained on the basis of separate solutions of Eqs. ~3!, ~4!, and ~6!. This is why we will first consider each of these equations separately.

v n ~ z1Dz ! 5

(

g n ~ Dz ! 5

1 2p

E

G a ~ Dz, v ! 5exp$ 2 a ~ v ! •Dz1i v @ c

21

~ v ! 2c 21 0 # •Dz % ,

v~ z1Dz, t ! 5

E

1`

2`

v~ z, t 8 ! •g ~ Dz, t 2 t 8 ! d t 8 ,

~10!

where the impulse response g is an inverse Fourier transform of the factor G a : 1 g ~ Dz, t ! 5 2p

E

1`

2`

G a ~ Dz, v ! e i v t d v .

~11!

The discrete analogy of the convolution integral ~10! has the form of a convolution sum: 2063

J. Acoust. Soc. Am., Vol. 104, No. 4, October 1998

~13!

G ~ e iV ! e iVn dV,

where 1 G~ e !5 T iV

1`

( Ga k52`

S

Dz,

V12 p k T

D

is a periodic function with a period of 2p, associated with the frequency response.26 Supposing that G a 50 for v .2 p /T, we can therefore use within the interval 2 p