- Email: [email protected]
Adapting parametric acoustic arrays for tomographic imaging
ULTRASONIC
IMAGING
10,
196-203
ADAPTING PARAMETRIC ACOUSTIC ARRAYS FOR TOMOGRAPHIC IMAGING Anni Cai, Jing Ao Sun and Glen Wade Department of Electrical and Computer Engineering University of California Santa Barbara, CA 93106
In this paper, we present a theoretical model of a parametric array for tomographic applications. The array in the model is assumed to extend right up to the receiver and its cross section is assumed not to be negligible. These two assumptions have not been previously made in applications concerned with tomography. We invoke conditions that permit the derivation of a simple relation for tomographic reconstruction of variations in the acoustic nonlinear parameter throughout the cross section of an object. 0 1988Academic Press, Inc. Key words: Nonlinear parameter tomography; parametric acoustic array. INTRODUCTION The acoustic nonlinear properties of biological materials have received increasing attention recently [l-3]. Nonlinear parameter tomography is expected to provide information not available from other kinds of tomography. For instance, the temperature in a carcinoma, usually higher than in the surrounding tissue, may be sensitively revealed by a measurement of its acoustic nonlinearity. In addition, the acoustic nonlinear parameter is closely related to the molecular structure of tissue. The possibility of obtaining structure information makes nonlinear parameter tomography attractive from this point of view also. A tomographic system to image acoustic nonlinearities was recently proposed [4]. This system involves a relatively high-power pumping pulse and a continuous low-intensity high-frequency probing wave propagating in a perpendicular direction with respect to the pulse. An alternative approach was later proposed which utilizes a parametric acoustic array [5]. In this approach, information carrying ultrasound at a difference frequency is generated as a result of nonlinear interaction between two high-frequency collinear beams that insonify the object. The parametric acoustic array was originally suggested by P. J. Westervelt for underwater transmitting and receiving [6]. Many theoretical models for predicting the difference-frequency sound field generated by the parametric array have been proposed since then. The accuracy of each of these models depends upon how closely the physical conditions agree with the assumptions made concerning the shape of the primary acoustical fields, the effects of the nonlinear attenuation experienced by these fields, and the position of the observation point relative to the array [7-lo]. The purpose of this paper is to present a proper model for tomographic application. BASIC EQUATIONS Figure 1 shows the system being considered. Two collinear sound beams at frequencies q and o2 are transmitted by the same transduux. Due to nonlinear parametric interaction between the two primary beams, a secondary wave at the difference frequency 0161-7346/88 $3.00 CoDright 0 I988 by Academic Press. Inc. All rights of reproduction in any form reserved.
196
PARAMETRIC ARRAY FOR TOMOGRAPHY
Fig. 1 A tomographic setup which uses a parametric acoustic array.
z
o, is generated. The pressure components of the primary waves and the secondary wave are detected simultaneously by a wide-band receiver which is coaxially located with respect to the transmitter. By scanning the transmitter-receiver pair along the dashed line, one projection of the object is obtained. Since the detected secondary sound pressure is related to the nonlinearity of the medium, a tomogram of the nonlinear parameter of the object mayber econ&ucted from a set of such projections, each taken from a different angle. In order to develop a reconstruction algorithm for the above system, a line-integral formula, which closely models the underlying physical process, is needed. In his classical paper, Westervelt presented a model for parametric arrays [6]. He assumed: (1) the primary fields consist of collimated plane waves; (2) nonlinear attenuation can be ignored; (3) the extent of the interaction region is limited by viscous absorption, and (4) the observation point is far removed from the interaction region. He showed that the secondary wave field can be thought of as being generated by a virtual line source q located along the axis of the primary waves. The strength of the virtual source is a function involving several acoustical properties of the medium along the propagation path, including the nonlinear parameter p which is the focus of our interest. At any position zO which is along the axis but far from the interaction region (see figure 2 (a)), the secondary fields generated by each segment dz of this line source arrive in phase. Therefore, the detected secondary wave pressure pS(zO)corresponds to a line integral of the strength of the source function. Unfortunately, this model does not directly apply to the system shown in figure 1, because in that system the object must be placed within the interaction region. To apply the concept of the parametric acoustic array to a tomographic system, we now consider the case where assumptions (1) and (2) above remain, but the receiver is located within the interaction region. Under these circumstances, the thickness of the interaction region must be taken into account. Instead of being a line source along the axis of the primary waves, the parametric sources are distributed in the finite (i.e. moderately nonxero) volume defined by the collimated primary beams. Figure 2 (b) depicts the geometry. It is r easonable to aSSume that the localized values of the acoustical parameters (nonlinear parameter, sound speed and the attenuation coefficients of the primary waves) of the medium in the volume 2ux 2ux a’x are uniform and isotropic. The attenuation of the secondary wave is assumed to be negligible compared to that of primary waves. We will ignore refraction and diffraction effects caused by the variation in refractive index of the medium. We also assume that the collimated primary waves have the following form over their cross section:
197
CA1 ET AL
y:‘-
z
y
(4
(b)
Fig. 2 Geometry corresponding to the line and volume integrals in Westervelt’s model and in Eq. (2). (a) Th e case corresponding to Westervelt’s model. (b) The general case corresponding to Eq. (2). The primary waves are assumed to be confined to the parallelepiped.
where k1 and kz are the wave numbers, and o1 and ct2 are the attenuation coeffkients, of the two primary waves respectively. Under the above assumptions, one may imagine that the source volume consists of an infinite number of contiguous source planes oriented at right angles to the z axis and positioned continuously along its length. These sources all have the same cross section but are of different strength. The secondary wave pressure ps generated at the point f0 can then be expressed as (see figure 2 (b)), dxdydz
where ks is the wave number of the secondary wave and k,=(q-o&co, brium density, c,, is the sound speed, and q(z) is the source function [6]:
(2) p. is the equili-
(3)
Eqs. (2) and (3) show that the secondary wave signal is a complicated function involving the nonlinear parameter, primary-wave attenuation, sound speed, and spreading of the secondary waves generated by virtual, spatially-limited planar sources within the interaction volume. In order to extract information about l3 from the measurements of the secondary wave, it is necessary to determine the importance of the various factors listed above. DISCUSSION Let us first examine the spreading effects, We will assume that the sound speed is constant in the medium. We rewrite Eq. (2) as follows P,(?,> = &J;,
q(z)R(fo,z)dz
198
(4)
PARAMETRIC ARRAY FOR TOMOGRAPHY
where R(FO,z) equals the double-integral with respect to x and y in Eq. (2) and represents the diffracted secondary field at ?” from the virtual, planar source located at z. If we introduce an aperture function +,y) into the integrand, the limits of integration for x and y can be changed to - 00 to + 03. For t(x,y), we have IXl%z,
lylla (5)
otherwise
Thus the spreading factor R(FO,z) becomes, R(~o,z)=
j-w--oof -cc t(x,y)G
dxdy 0
(6)
By using Parseval’s relationship, we obtain
where A(K,,K,,) and the quantity in the square brackets are the two-dimensional Fourier transforms of the a rture function and the Green’s function at zO--z plane respectively, and K;+K;+Kf= r s. In the limit
as A, + 0 (i.e., no secondary wave diffraction), Eq. (4) can then be simplified to:
we have a/A, - 03 and
A(K,,K,)=~T~~~(K,)~(K~).
For this limiting case, the wavefronts of the secondary-wave components generated by the various planar sources are also planar. These components arrive at the receive~Jexactly in phase because of the inherent nature of the parametric array. The phase I( te-qd ofthe parametric source function q compensates for the propagation delay d “O associated with the distance from the source plane at z to the receiving plane at z,. The detected secondary signal corresponds to a line integral involving only p and the primary-wave attenuation coefficients, a1 and a2. In tomographic systems, limits on how small the wavelengths can be are established by the frequency range useful for medical imaging. Therefore, secondary-wave diffraction is not always negligible and Eq. (8) may not be applicable. A large aperture (i.e. a large beam cross section) will minimiz.e diffraction effects. However, an aperture size of the order of the secondary wavelength is preferred in a tomographic system because increasing the aperture size will worsen the resolution of the tomogram. We performed a numerical evaluation for a/x, ranging from 1 to 8 to set if Eq. (8) can be used for such a range of secondary wavelengths to predict the secondary field generated by the parametric array. For the numerical evaluation, we assumed that the receiver aperture has the same cross-sectional shape as the transmitter, is in the same lateral position relative to the z axis and is oriented normal to that axis. Because of diffraction, the received signal will vary over the receiver aperture in both amplitude and phase. To calculate the received signal,
199
CA1
ET
AL
we must average these quantities over the receiver aperture. Thus,
(9)
where F(z,-z)
is a normalized spreading factor and is equal to
is the average value of R(ib,z) and R,(z,-z)
-=NJ,,z)> Mzo-4 ’ is the value of R(?,,z) in the limit as X, - 0.
Calculated absolute values and phases of F(z,-z) for different aperture sires and (z,-z) positions are given in figures 3 and 4. The decrease of IF(z,-z)l with (z,--z) in figure 3 is due to diffraction loss. The secondary waves generated from source elements furthest from the receiver have the smallest weighting factors in the integral of Eq. (92; Diffraction causes imperfect compensation of the propagation delay by the phase term k in q. Phase variations of F(z,-z) with respect to (z,--z) can be seen in figure 4. Since F(zO-z) is a complex function of (z,- z), it is difficult to relate the measured secondary field to the acoustical properties of the medium in a simple way. However, from figures 3 and 4, we can see that F(z,-z) is approximately constant within the range of z values plotted provided a/k, 28. Even when a/A, is only 5, the variation in F(z,-z) may also be tolerable. Under such circumstances, it can still be said that the detected amplitude of the average secondary wave corresponds to a line integral of a real function involving only the
0
50
25
75
(zo-z)
Fig. 3 Amplitude
of F(z,-z)
200
as a function of (z,-z).
100 x,
PARAMETRIC ARRAY FOR TOMOGRAPHY
\
:::_:::
\ a -5 r-
-1200
-1400
-1600
-1800 25
0
50
75
100 A,
(q-z)
Fig. 4 Phase of F&-z)
variables p, 01~and
as a function of (z,-z).
a2:
w-3 The secondary field predicted by Eq. (10) differs from that predicted by Eq. (8) only by the constant F. Eq. (10) can be used as the basis for tomographic reconstruction. The spatial resolution of the tomogram will be relatively coarse. This coarseness can be regarded as a trade-off to minimize diffraction effects and therefore to permit the modelling of the secondary-wave generation as a line integral. We will now examine the effects of variations in sound speed. As mentioned above, for the case of X,-O all secondary-wave components arrive at the receiver in phase because the propagation delay is compensated by the phase term of the source function q. Actually, the term JkJ in q originates from the phase difference between the two primary plane waves propagating from 0 to z [6]. If we assume that the variation in refractive index along the path is n(z), the phase term of the source located at z becomes j&-kz)fd e
= &+h’
phase delay ejq’~
. The secondary wave generated by this source will undergo a in propagating from z to zO. Therefore, the total phase of each
secondary-wave component at the receiver has the same value, ejk,&‘*ndz’ . All components arrive at z, in phase as they do when the variation in n is neglected. Taking n as a function
201
CA1
ET
AL
of position, Eq. (8) becomes
(11) Since in biological soft tissue II does not change very much compared to changes in 8, we can conclude that the measured pressure amplitude of the secondary wave depends primarily on p, o1 and 01~. The same arguments can be made for the cases where the secondary wavefronts are only approximately planar, i.e., where Eq. (10) can be applied. To obtain an image of 8 based on Eq. (lo), a correction for attenuation should be included in the reconstruction algorithm. A correction method similar to the one used in single-photon emission tomography has been investigated [ 111. CONCLUSIONS We have applied the parametric array concept to the question of reconstructing tome grams of the acoustic nonlinear parameter of biological specimens. We have extended Westervelt’s model to include the case of observation points within the interaction region of the array. We have shown that a trade-off exists between worsening the spatial resolution and reducing the effects of diffraction of the secondary waves generated by the parametric sources. We argue that variations in the sound speed can be neglected. A tomogram of the nonlinear parameter can be reconstructed from measured projections at the difference frequency by including a compensation for attenuation in the reconstruction process. ACKNOWLEDGEMENT This work was performed with support from the RICOH COMPANY, LTD., Japan. The authors wish to thank anonymous reviewers of an early version of this paper for their important comments and suggestions. The authors acknowledge the help of Nicholas Arnold who read the draft of this paper and made several suggestions for clarifying the explanations. REFERENCES Law, W. K., Frizzell, L. A. and Dunn, F., Determination of non-linear parameter B/A of biological media, Ultrasound Med. Biol. II, 307-308 (1985). PI Sehgal, C. M., Porter, B. and Greenleaf, J. F., Measurements of the acoustic nonlinearity parameter BJA in human tissues by a thermodynamic method, J. Acoust. Sot. Amer. 76, 1023-1029 (1984). 131 Sehgal, C. M., Brown, G. M., Bahn, R. C. and Greenleaf, J. F:, Measurement and use of acoustic nonlinearity and sound to estimate composrtron of excised livers, Ultrasound Med. Biol. 12, 865-874 (198 “g”d ). I&da, N., Sato, T. and Linzer, M., Ima ’ the nonlinear ultrasonic parameter of a PI medium, Ultrasonic Imaging 4, 295-299 (1PI83 . Nakagawa, Y., Naka awa, M., Yoneyama, M. and Kikuchi, M., Nonlinear parame PI parametric acoustic array, in IEEE 1984 Ultruter imaging compu tet sonic Symposium Proce~~~$!y67 “r -676 (IEEE Cat. No. 84CH2112-1). PI Westervelt, P. J., Parametric acoustic array, J. Acowt. Sot. Amer. 35, 535-537 (1963).
PI
202
PARAMETRIC ARRAY FOR TOMOGRAPHY
[7] Berktay, H. 0. and Leahy, D. J., Farfield performance of parametric transmitters, J. Acoust. Sot. Amer. 5.5, 539-546 (1974). [8] Fenlon, F. H., On the performance of a dual frequency parametric source via matched asymptotic solution of Burgers’ equation, J. Acoust. Sot. Amer. 5.5, 35-46 (1974). [9] RolIeigh, R. L., Difference frequency pressure within the interaction region of a parametric array, J. Acoust. Sot. Amer. 58, 964-971 (1975). [lo] Moffett, M. B. and Mellen, R. H., Neat-field characteristics of parametric acoustic sources, J. Acoust. Sot. Amer. 69, 404-409 (1981). [l l] Nakagawa, Y., Hou, W., Cai, A., Arnold, N., Wade, G., Yoneyama, M. and Nakagawa, M., Nonlinear parameter imaging with finite-am litude sound waves, in IEEE 1986 UltrasonicsSymposiumProceedings,pp. 901-904 (I If EE Cat. No. 86(X2375-4).
203
IMAGING
10,
196-203
ADAPTING PARAMETRIC ACOUSTIC ARRAYS FOR TOMOGRAPHIC IMAGING Anni Cai, Jing Ao Sun and Glen Wade Department of Electrical and Computer Engineering University of California Santa Barbara, CA 93106
In this paper, we present a theoretical model of a parametric array for tomographic applications. The array in the model is assumed to extend right up to the receiver and its cross section is assumed not to be negligible. These two assumptions have not been previously made in applications concerned with tomography. We invoke conditions that permit the derivation of a simple relation for tomographic reconstruction of variations in the acoustic nonlinear parameter throughout the cross section of an object. 0 1988Academic Press, Inc. Key words: Nonlinear parameter tomography; parametric acoustic array. INTRODUCTION The acoustic nonlinear properties of biological materials have received increasing attention recently [l-3]. Nonlinear parameter tomography is expected to provide information not available from other kinds of tomography. For instance, the temperature in a carcinoma, usually higher than in the surrounding tissue, may be sensitively revealed by a measurement of its acoustic nonlinearity. In addition, the acoustic nonlinear parameter is closely related to the molecular structure of tissue. The possibility of obtaining structure information makes nonlinear parameter tomography attractive from this point of view also. A tomographic system to image acoustic nonlinearities was recently proposed [4]. This system involves a relatively high-power pumping pulse and a continuous low-intensity high-frequency probing wave propagating in a perpendicular direction with respect to the pulse. An alternative approach was later proposed which utilizes a parametric acoustic array [5]. In this approach, information carrying ultrasound at a difference frequency is generated as a result of nonlinear interaction between two high-frequency collinear beams that insonify the object. The parametric acoustic array was originally suggested by P. J. Westervelt for underwater transmitting and receiving [6]. Many theoretical models for predicting the difference-frequency sound field generated by the parametric array have been proposed since then. The accuracy of each of these models depends upon how closely the physical conditions agree with the assumptions made concerning the shape of the primary acoustical fields, the effects of the nonlinear attenuation experienced by these fields, and the position of the observation point relative to the array [7-lo]. The purpose of this paper is to present a proper model for tomographic application. BASIC EQUATIONS Figure 1 shows the system being considered. Two collinear sound beams at frequencies q and o2 are transmitted by the same transduux. Due to nonlinear parametric interaction between the two primary beams, a secondary wave at the difference frequency 0161-7346/88 $3.00 CoDright 0 I988 by Academic Press. Inc. All rights of reproduction in any form reserved.
196
PARAMETRIC ARRAY FOR TOMOGRAPHY
Fig. 1 A tomographic setup which uses a parametric acoustic array.
z
o, is generated. The pressure components of the primary waves and the secondary wave are detected simultaneously by a wide-band receiver which is coaxially located with respect to the transmitter. By scanning the transmitter-receiver pair along the dashed line, one projection of the object is obtained. Since the detected secondary sound pressure is related to the nonlinearity of the medium, a tomogram of the nonlinear parameter of the object mayber econ&ucted from a set of such projections, each taken from a different angle. In order to develop a reconstruction algorithm for the above system, a line-integral formula, which closely models the underlying physical process, is needed. In his classical paper, Westervelt presented a model for parametric arrays [6]. He assumed: (1) the primary fields consist of collimated plane waves; (2) nonlinear attenuation can be ignored; (3) the extent of the interaction region is limited by viscous absorption, and (4) the observation point is far removed from the interaction region. He showed that the secondary wave field can be thought of as being generated by a virtual line source q located along the axis of the primary waves. The strength of the virtual source is a function involving several acoustical properties of the medium along the propagation path, including the nonlinear parameter p which is the focus of our interest. At any position zO which is along the axis but far from the interaction region (see figure 2 (a)), the secondary fields generated by each segment dz of this line source arrive in phase. Therefore, the detected secondary wave pressure pS(zO)corresponds to a line integral of the strength of the source function. Unfortunately, this model does not directly apply to the system shown in figure 1, because in that system the object must be placed within the interaction region. To apply the concept of the parametric acoustic array to a tomographic system, we now consider the case where assumptions (1) and (2) above remain, but the receiver is located within the interaction region. Under these circumstances, the thickness of the interaction region must be taken into account. Instead of being a line source along the axis of the primary waves, the parametric sources are distributed in the finite (i.e. moderately nonxero) volume defined by the collimated primary beams. Figure 2 (b) depicts the geometry. It is r easonable to aSSume that the localized values of the acoustical parameters (nonlinear parameter, sound speed and the attenuation coefficients of the primary waves) of the medium in the volume 2ux 2ux a’x are uniform and isotropic. The attenuation of the secondary wave is assumed to be negligible compared to that of primary waves. We will ignore refraction and diffraction effects caused by the variation in refractive index of the medium. We also assume that the collimated primary waves have the following form over their cross section:
197
CA1 ET AL
y:‘-
z
y
(4
(b)
Fig. 2 Geometry corresponding to the line and volume integrals in Westervelt’s model and in Eq. (2). (a) Th e case corresponding to Westervelt’s model. (b) The general case corresponding to Eq. (2). The primary waves are assumed to be confined to the parallelepiped.
where k1 and kz are the wave numbers, and o1 and ct2 are the attenuation coeffkients, of the two primary waves respectively. Under the above assumptions, one may imagine that the source volume consists of an infinite number of contiguous source planes oriented at right angles to the z axis and positioned continuously along its length. These sources all have the same cross section but are of different strength. The secondary wave pressure ps generated at the point f0 can then be expressed as (see figure 2 (b)), dxdydz
where ks is the wave number of the secondary wave and k,=(q-o&co, brium density, c,, is the sound speed, and q(z) is the source function [6]:
(2) p. is the equili-
(3)
Eqs. (2) and (3) show that the secondary wave signal is a complicated function involving the nonlinear parameter, primary-wave attenuation, sound speed, and spreading of the secondary waves generated by virtual, spatially-limited planar sources within the interaction volume. In order to extract information about l3 from the measurements of the secondary wave, it is necessary to determine the importance of the various factors listed above. DISCUSSION Let us first examine the spreading effects, We will assume that the sound speed is constant in the medium. We rewrite Eq. (2) as follows P,(?,> = &J;,
q(z)R(fo,z)dz
198
(4)
PARAMETRIC ARRAY FOR TOMOGRAPHY
where R(FO,z) equals the double-integral with respect to x and y in Eq. (2) and represents the diffracted secondary field at ?” from the virtual, planar source located at z. If we introduce an aperture function +,y) into the integrand, the limits of integration for x and y can be changed to - 00 to + 03. For t(x,y), we have IXl%z,
lylla (5)
otherwise
Thus the spreading factor R(FO,z) becomes, R(~o,z)=
j-w--oof -cc t(x,y)G
dxdy 0
(6)
By using Parseval’s relationship, we obtain
where A(K,,K,,) and the quantity in the square brackets are the two-dimensional Fourier transforms of the a rture function and the Green’s function at zO--z plane respectively, and K;+K;+Kf= r s. In the limit
as A, + 0 (i.e., no secondary wave diffraction), Eq. (4) can then be simplified to:
we have a/A, - 03 and
A(K,,K,)=~T~~~(K,)~(K~).
For this limiting case, the wavefronts of the secondary-wave components generated by the various planar sources are also planar. These components arrive at the receive~Jexactly in phase because of the inherent nature of the parametric array. The phase I( te-qd ofthe parametric source function q compensates for the propagation delay d “O associated with the distance from the source plane at z to the receiving plane at z,. The detected secondary signal corresponds to a line integral involving only p and the primary-wave attenuation coefficients, a1 and a2. In tomographic systems, limits on how small the wavelengths can be are established by the frequency range useful for medical imaging. Therefore, secondary-wave diffraction is not always negligible and Eq. (8) may not be applicable. A large aperture (i.e. a large beam cross section) will minimiz.e diffraction effects. However, an aperture size of the order of the secondary wavelength is preferred in a tomographic system because increasing the aperture size will worsen the resolution of the tomogram. We performed a numerical evaluation for a/x, ranging from 1 to 8 to set if Eq. (8) can be used for such a range of secondary wavelengths to predict the secondary field generated by the parametric array. For the numerical evaluation, we assumed that the receiver aperture has the same cross-sectional shape as the transmitter, is in the same lateral position relative to the z axis and is oriented normal to that axis. Because of diffraction, the received signal will vary over the receiver aperture in both amplitude and phase. To calculate the received signal,
199
CA1
ET
AL
we must average these quantities over the receiver aperture. Thus,
(9)
where F(z,-z)
is a normalized spreading factor and is equal to
is the average value of R(ib,z) and R,(z,-z)
-=NJ,,z)> Mzo-4 ’ is the value of R(?,,z) in the limit as X, - 0.
Calculated absolute values and phases of F(z,-z) for different aperture sires and (z,-z) positions are given in figures 3 and 4. The decrease of IF(z,-z)l with (z,--z) in figure 3 is due to diffraction loss. The secondary waves generated from source elements furthest from the receiver have the smallest weighting factors in the integral of Eq. (92; Diffraction causes imperfect compensation of the propagation delay by the phase term k in q. Phase variations of F(z,-z) with respect to (z,--z) can be seen in figure 4. Since F(zO-z) is a complex function of (z,- z), it is difficult to relate the measured secondary field to the acoustical properties of the medium in a simple way. However, from figures 3 and 4, we can see that F(z,-z) is approximately constant within the range of z values plotted provided a/k, 28. Even when a/A, is only 5, the variation in F(z,-z) may also be tolerable. Under such circumstances, it can still be said that the detected amplitude of the average secondary wave corresponds to a line integral of a real function involving only the
0
50
25
75
(zo-z)
Fig. 3 Amplitude
of F(z,-z)
200
as a function of (z,-z).
100 x,
PARAMETRIC ARRAY FOR TOMOGRAPHY
\
:::_:::
\ a -5 r-
-1200
-1400
-1600
-1800 25
0
50
75
100 A,
(q-z)
Fig. 4 Phase of F&-z)
variables p, 01~and
as a function of (z,-z).
a2:
w-3 The secondary field predicted by Eq. (10) differs from that predicted by Eq. (8) only by the constant F. Eq. (10) can be used as the basis for tomographic reconstruction. The spatial resolution of the tomogram will be relatively coarse. This coarseness can be regarded as a trade-off to minimize diffraction effects and therefore to permit the modelling of the secondary-wave generation as a line integral. We will now examine the effects of variations in sound speed. As mentioned above, for the case of X,-O all secondary-wave components arrive at the receiver in phase because the propagation delay is compensated by the phase term of the source function q. Actually, the term JkJ in q originates from the phase difference between the two primary plane waves propagating from 0 to z [6]. If we assume that the variation in refractive index along the path is n(z), the phase term of the source located at z becomes j&-kz)fd e
= &+h’
phase delay ejq’~
. The secondary wave generated by this source will undergo a in propagating from z to zO. Therefore, the total phase of each
secondary-wave component at the receiver has the same value, ejk,&‘*ndz’ . All components arrive at z, in phase as they do when the variation in n is neglected. Taking n as a function
201
CA1
ET
AL
of position, Eq. (8) becomes
(11) Since in biological soft tissue II does not change very much compared to changes in 8, we can conclude that the measured pressure amplitude of the secondary wave depends primarily on p, o1 and 01~. The same arguments can be made for the cases where the secondary wavefronts are only approximately planar, i.e., where Eq. (10) can be applied. To obtain an image of 8 based on Eq. (lo), a correction for attenuation should be included in the reconstruction algorithm. A correction method similar to the one used in single-photon emission tomography has been investigated [ 111. CONCLUSIONS We have applied the parametric array concept to the question of reconstructing tome grams of the acoustic nonlinear parameter of biological specimens. We have extended Westervelt’s model to include the case of observation points within the interaction region of the array. We have shown that a trade-off exists between worsening the spatial resolution and reducing the effects of diffraction of the secondary waves generated by the parametric sources. We argue that variations in the sound speed can be neglected. A tomogram of the nonlinear parameter can be reconstructed from measured projections at the difference frequency by including a compensation for attenuation in the reconstruction process. ACKNOWLEDGEMENT This work was performed with support from the RICOH COMPANY, LTD., Japan. The authors wish to thank anonymous reviewers of an early version of this paper for their important comments and suggestions. The authors acknowledge the help of Nicholas Arnold who read the draft of this paper and made several suggestions for clarifying the explanations. REFERENCES Law, W. K., Frizzell, L. A. and Dunn, F., Determination of non-linear parameter B/A of biological media, Ultrasound Med. Biol. II, 307-308 (1985). PI Sehgal, C. M., Porter, B. and Greenleaf, J. F., Measurements of the acoustic nonlinearity parameter BJA in human tissues by a thermodynamic method, J. Acoust. Sot. Amer. 76, 1023-1029 (1984). 131 Sehgal, C. M., Brown, G. M., Bahn, R. C. and Greenleaf, J. F:, Measurement and use of acoustic nonlinearity and sound to estimate composrtron of excised livers, Ultrasound Med. Biol. 12, 865-874 (198 “g”d ). I&da, N., Sato, T. and Linzer, M., Ima ’ the nonlinear ultrasonic parameter of a PI medium, Ultrasonic Imaging 4, 295-299 (1PI83 . Nakagawa, Y., Naka awa, M., Yoneyama, M. and Kikuchi, M., Nonlinear parame PI parametric acoustic array, in IEEE 1984 Ultruter imaging compu tet sonic Symposium Proce~~~$!y67 “r -676 (IEEE Cat. No. 84CH2112-1). PI Westervelt, P. J., Parametric acoustic array, J. Acowt. Sot. Amer. 35, 535-537 (1963).
PI
202
PARAMETRIC ARRAY FOR TOMOGRAPHY
[7] Berktay, H. 0. and Leahy, D. J., Farfield performance of parametric transmitters, J. Acoust. Sot. Amer. 5.5, 539-546 (1974). [8] Fenlon, F. H., On the performance of a dual frequency parametric source via matched asymptotic solution of Burgers’ equation, J. Acoust. Sot. Amer. 5.5, 35-46 (1974). [9] RolIeigh, R. L., Difference frequency pressure within the interaction region of a parametric array, J. Acoust. Sot. Amer. 58, 964-971 (1975). [lo] Moffett, M. B. and Mellen, R. H., Neat-field characteristics of parametric acoustic sources, J. Acoust. Sot. Amer. 69, 404-409 (1981). [l l] Nakagawa, Y., Hou, W., Cai, A., Arnold, N., Wade, G., Yoneyama, M. and Nakagawa, M., Nonlinear parameter imaging with finite-am litude sound waves, in IEEE 1986 UltrasonicsSymposiumProceedings,pp. 901-904 (I If EE Cat. No. 86(X2375-4).
203