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Image formation in vibro-acoustography with depth-of-field effects夽
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Glauber T. Silva a,∗ , Alejandro C. Frery a , Mostafa Fatemi b
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a Instituto de Computa¸ ca˜ o, Universidade Federal de Alagoas, Macei´o, AL 57.072-970, Brasil Department Physiology and Biomedical Engineering, Mayo Clinic College of Medicine, Rochester 55905, USA
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Abstract
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Keywords: Vibro-acoustography; Point-spread function; Image formation
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We study the image formation of vibro-acoustography systems based on a concave sector array transducer taking into account depth-of-field effects. The system point-spread function (PSF) is defined in terms of the acoustic emission of a point-target in response to the dynamic radiation stress of ultrasound. The PSF on the focal plane and the axis of the transducer are presented. To extend the obtained PSF to the 3D-space, we assume it is a separable function in the axial direction and the focal plane of the transducer. In this model, an image is formed through the 3D convolution of the PSF with an object function. Experimental vibro-acoustography images of a breast phantom with lesion-like inclusions were compared with simulated images. Results show that the experimental images are in good agreement with the proposed model. © 2006 Published by Elsevier Ltd.
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Imaging elastic properties of tissues has become a new field in image sciences known as elastography. Variation in elasticity of soft tissues is commonly related to pathologies. Thereby, non-invasive techniques to image elasticity parameters of in vivo tissues are promising tools in medical diagnosis. The underlying principle of elastography is to measure or track the induced motion of tissue due to an external applied force [1]. One of the most effective ways to produce such force is to use radiation stress of ultrasound. Vibro-acoustography is an elastography technique that images the response of an object (or tissue) to the low-frequency dynamic radiation stress produced by a localized ultrasound beam [2]. The dynamic radiation is caused by the wave momentum transferring to an object or propagating medium [3]. In vibro-acoustography, this stress is yielded by two co-focused monochromatic ultrasound beams of slightly different frequencies (typically in the kilohertz range). The tissue within the sys-
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1. Introduction
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夽 Partially presented at the XVII Brazilian Symposium on Computer Graphics
and Image Processing (SIBGRAPI’04). ∗ Corresponding author. E-mail address:
[email protected] (G.T. Silva). 1 2
tem focal zone is subjected to deformation/vibration at the beat frequency of the ultrasound beams. It, thus, irradiates a field (acoustic emission) which can be probed by an acoustic detector. The image is formed by pixels whose brightness is determined by the acoustic emission of each point in the tissue. The acoustic emission carries information of the tissue region at low and ultrasound frequencies. The information at the ultrasound frequency is related to the medium scattering properties, as variation of density and compressibility. The low-frequency information may reveal elastic properties of tissue. Conventional ultrasound imaging does not provide this kind of information. Potential clinical applications of vibro-acoustography include imaging of lesions in soft tissues such as calcification [4,5], liver tumors [6], and following brachytherapy seeds [7]. The development of vibro-acoustography for medical imaging applications demands a detailed assessment of the image formation in this method. Under very sensible conditions, vibroacoustography image formation can be described by the system point-spread function (PSF), which is related to the radiation stress field exerted on a point-target [8]. Thus, it is necessary to study the stress field forming by acoustic transducers. Stress field forming in vibro-acoustography has been study theoretically and experimentally for a two-element confocal and x-focal transducers [9], and an eight-element sector array transducer
0895-6111/$ – see front matter © 2006 Published by Elsevier Ltd. doi:10.1016/j.compmedimag.2006.08.001
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
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2. Image formation theory
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2.1. Dynamic radiation stress
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In vibro-acoustography, the dynamic radiation stress “taps” an object embedded in the propagating medium; which, in response, emits an acoustic field (acoustic emission). The acoustic emission depends on the object’s shape and mechanical properties and is proportional to the dynamic radiation stress. The information carried by the acoustic emission is used to synthesize an image of the object. A vibro-acoustography imaging, as a linear system, can be characterized by its PSF. The vibroacoustography PSF depends on the acoustic emission of a pointtarget. This target is considered here as a small sphere of radius a. The dynamic radiation stress is produced by a transducer with two ultrasound sources. Each source is driven by a sinusoidal signal. The driving angular frequencies of the sources are ω1 = ω0 + ω/2 and ω2 = ω0 − ω/2, where ω0 and ω are the center and difference frequencies, and ω ω0 . The two ultrasound beams are focused at the same point in space producing a dynamic radiation stress at the beat frequency ω. We consider an small sphere immersed in an infinitely extended homogeneous lossless fluid with density ρ0 and speed of sound c0 . In a lossless medium, ultrasound waves can be fully described in terms of the velocity potential φ(r, t), where r is the position vector and t is the time. The two interacting beams are supposed to behave like a bichromatic plane wave in the vicinity of the system focal zone. Having ω/ω0 1, the complex amplitude of the radiation stress at ω exerted on the small sphere is, according to ref. [3], given by
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σˆ = k02 yˆ φˆ 1 φˆ 2∗ ,
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2.2. Point-spread function
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We are now interested in obtaining the acoustic emission by a point-target modelled as an small rigid sphere. The sphere is supposed to oscillate in linear regime with small amplitude around the coordinate system origin along the z axis. The sphere vibration velocity in the steady-state is vˆ ejωt , in which vˆ is the complex velocity amplitude. The sphere radius is much smaller than the wavelength of the incident ultrasound waves. Hence, ka 1, where k = ω/c0 is the wavenumber of the acoustic emission. The amplitude of the emitted pressure by the oscillating sphere is, according to ref. [12], a dipole radiation given, in spherical coordinates (r, θ, ϕ) by
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pˆ = ρ0 c0 vˆ k3 a3 cos θ
. (2) 2kr The velocity amplitude vˆ of sphere can described by the sphere mechanical impedance zˆ at the frequency ω as follows vˆ = ˆ z. The acoustic outflow (the volume of the medium which πa2 σ/ˆ is displaced per unit time due to an object vibration per unit force) by the sphere is related to the mechanical impedance by qˆ (ω) = 2πa2 /ˆz(ω). Thus, Eq. (2) may be rewritten as
(1)
where k0 = ω0 /c0 and yˆ is the complex radiation force function of the sphere which depends on the scattering properties of the sphere mostly at the ultrasound frequency ω0 . The functions φˆ 1 and φˆ 2 are the complex amplitude of the velocity potentials of the ultrasound beams.
ˆ 0 )ˆq(ω)ˆg(r, θ), pˆ = ρ0 c0 σ(ω
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e−jkr kr is the a transfer function which depends on the medium and the object boundary-conditions. By using Eqs. (1) and (3), we may write the acoustic emission amplitude of an sphere located at the position r and measured at r as follows p(r ˆ |r) =
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gˆ (r, θ) = k3 a3 cos θ
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[10]. In these studies, the spatial resolution and sidelobe levels were obtained and validated experimentally. A study based on computational simulations for linear arrays can be found in ref. [11]. Despite those investigation, no much attention has been given to the effects of the PSF depth-of-field in the system image formation. In this paper, we investigate the image formation in vibroacoustography including depth-of-field effects. To achieve this, we propose a model in which the 3D PSF can be represented as a separable function in the axial direction and the focal plane of the transducer. Images can be simulated by performing the spatial convolution of the 3D PSF and an object described by means of a function. An experimental vibro-acoustography system acquired images of a breast phantom with lesion-like inclusions. Both experimental and simulated images are compared. The obtained results are in good agreement with the proposed image formation model.
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ρ0 c0 k02 yˆ (ω0 )ˆq(ω)φˆ 1 (r)φˆ 2 (r)∗ gˆ (r |r),
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where gˆ (r |r) = k3 a3 cos θ
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e−jk|r −r|
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To define the image of an object we need to represent it through a function. As shown in Eq. (4), the acoustic emission has information of the sphere in both high and low ultrasound frequencies. This information is present in the radiation force function yˆ (ω0 ) and the acoustic outflow qˆ (ω), respectively. We assume that the acoustic emission of an object (or region) is a linear combination of the emission of every infinitesimal volume in the object. The acoustic emission at the frequency difference ω is given by Eq. (4). The functions yˆ and qˆ may vary linearly within the object. Hence, we define the object function as oˆ (r; ω0 , ω) = yˆ (r, ω0 )ˆq(r, ω).
(5)
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The vibro-acoustography image of the object is, thus, given by
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ˆ ˆı(r; ω0 , ω) = oˆ (r; ω0 , ω) h(r),
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(6)
ˆ where h(r) is the PSF of the system and the symbol denotes spatial convolution. It follows from Eq. (6) that the image of an small object brings up its high- and low-frequency characteristics.
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
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The normalized PSF of a vibro-acoustography system can be described in terms of the acoustic emission of a point-target (the small rigid sphere). We may thus define the PSF as ) p(r|r ˆ ˆ h(r) ≡ lim , ˆ 0 |r ) a→0+ p(r
where r0 is the transducer focal point at which the acoustic emission should get its maximum value. When a approaches to zero the relevant part of the acoustic emission which remains ˆ g(r |r)/σ(r ˆ 0 )ˆg(r |r0 ). We may neglect the for the PSF is σ(r)ˆ contribution of the transfer function gˆ to the system PSF if r − r0 r . In practical means this condition is always verified. Referring to Eq. (1), we see that the PSF is proportional to the radiation stress given in Eq. (1). Accordingly, the normalized PSF is ˆ σ(r) ˆ h(r) = lim = Aφˆ 1 (r)φˆ 2∗ (r), ˆ 0) a→0+ σ(r
(7)
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Vibro-acoustography applications require transducers driven by sinusoidal signals. Consider a sector array transducer with N elements and aperture radius b. The elements are grouped in symmetric pairs (see Fig. 1). Each pair is driven by the angular frequencies ω1 and ω2 alternately. The respective wave numbers of these frequencies are k1 and k2 . The transducer focal distance is z0 . The vibro-acoustography PSF of a sector array transducer in both focal plane and axial direction was derived in ref. [10] through the classical radiation theory in the Fresnel approximation (b2 z20 ). For the sake of simplicity, we omit the technical details on this derivation and only present the results. Before presenting the system PSF, let us introduce the following dimensionless variables r˜ = r/z0 , z˜ = z/z0 , k˜ = kz0 , r˜i = ki b˜r , and k˜ i = ki b2 /(4z0 ), i = 1, 2. The PSF produced by the sector array on the focal plane (plane-xy) is
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hˆ xy = A1 e−jψ
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where A = [φˆ 1 (r0 )φˆ 2∗ (r0 )]−1 is the normalization constant. The vibro-acoustography PSF is a three-dimensional complex function. The spatial resolution of an imaging system is proportional to the width of the mainlobe of the PSF [13]. Hence, we define the spatial resolution cell of the vibro-acoustography system as the volume enclosed by the mainlobe of the PSF at −6 dB.
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˜ + r˜ 2 /2), and where A1 is a constant, ψ = k(1 +∞ 1 (i) 2m,n (ϕ)J2m+1 (˜ri ) φn = r˜i m=0 +∞ 4(m + 1) J2m+2 +4 (˜ri ) , i = 1, 2. + r˜i =0
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Fig. 1. Geometric description of the sector array transducer. (a) The focal plane and a pair of elements (in gray). (b) Lateral view. Focal plane at z = z0 .
The Jm is the mth-order Bessel function of first-kind. The angular function mn is mπ 2n + 1 (−1)m mn (ϕ) = cos m ϕ − π sin , m N N m = 1, 2, . . .
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and 0n = 2π/N. The PSF along the axial direction is given by 1 ˜ ˆhz = A2 sinc k˜ 1 1 − 1 sinc k2 − 1 e−jξ , (11) z˜ 2 z˜ z˜ ˜ z−1 + (k˜ 2 − k˜ 1 )(˜z−1 − 1). where A2 is a constant and ξ = k˜ We may now assume that the 3D PSF can be expressed as the product of hˆ xy and hˆ z in the vicinity of the focal point. Accordingly, we have hˆ = hˆ xy hˆ z .
(12)
This hypothesis may fail when the observation point is far from the transducer focal point because the changes in the phase of the radiation stress may not follow this assumption. 3. Experimental setup (9)
A block diagram of a vibro-acoustography imaging system is shown in Fig. 2(a). Two function generators produce the
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
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stress produced by the incident beam on the latex sheet can be neglected. The displacement of the sphere in the experiments is very small (in the nanometer range). Hence, nonlinear vibration of the sphere and membrane is not considered. Based on these assumption the sphere and the latex sheet can be considered as a linear spring-mass system driven by the radiation stress. Therefore, the amplitude of the radiation stress field upon the sphere is σˆ = zˆ vˆ , where Z and vˆ are, respectively, the mechanical impedance and the velocity amplitude of the sphere. We assume that the impedance zˆ does not depend on the position of the sphere. It depends only on the vibration frequency of the sphere. Thus, by measuring the sphere vibrating velocity we assess the radiation stress field, i.e., σˆ ∝ vˆ . Referring to Eq. (7) the vibroacoustography PSF is also proportional to the vibration velocity of the sphere. A laser vibrometer (Polytec GmbH, Waldbronn, Germany) is used to measure the vibration velocity of the sphere by means of the Doppler effect on the reflected laser by the sphere. The use of laser vibrometer in measuring the vibro-acoustography PSF has some advantages compared to acoustic measurements. An acoustic detector such as a microphone or hydrophone detects the acoustic emission by the sphere and the latex sheet. The emission by the latex sheet may reduce the information about the radiation stress sought in the acoustic emission by the sphere. Furthermore, possible standing waves present in the water tank due to multiple reflections of the acoustic emission can be also detected. Therefore, the laser vibrometer should yield a more accurate measurement of the vibro-acoustography PSF than acoustic detectors. If the sphere has size comparable to the −3 dB-beamwidth, it may suffer a torque that misaligns the laser vibrometer producing incorrect results. Here the beamwidth is about twice as the sphere diameter, and the radiation stress measurement with the laser vibrometer is therefore valid.
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Fig. 2. Experimental apparatus used in vibro-acoustography. (a) Pictorial view of a vibro-acoustography system with a two-element confocal transducer.
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sinusoidal driving signals at frequencies ω1 and ω2 . A sector array transducer is used to produce the dynamic radiation force on an object. The object is scanned by moving the vibroacoustography transducer mechanically with step motors. The vibration of the object produces the acoustic emission which is detected by a hydrophone. A lock-in amplifier (7265-DSP, Perkin Elmer Instruments, Wellesley, MA, USA) is used as an extremely narrow passband filter with the center point selected by a reference signal at ω. The filtered signal is digitized by a 12 bits/sample digitizer (HP-E1429A, Hewlett-Packard, Palo Alto, CA, USA) at the rate higher than the Nyquist rate required for the particular ω used. This information is stored in a computer and used to form the object image. In Fig. 2(b), we depict the experimental setup to measure the vibro-acous-tography PSF. An eight-element sector array transducer of 47 mm aperture was used to produce the dynamic radiation force on the sphere. The transducer is driven by two sinusoidal signals at 2 MHz with difference frequency of 1 kHz. A sphere of 600 m diameter is used as the point-target. The sphere is attached to a 5 m thick latex sheet stretched over a rigid frame. The sheet and the frame are used to hold the sphere in space, with minimum interference to its motion. The dynamic radiation stress vibrates the sphere at the difference frequency in the water tank. The impedance of the membrane is close to that of water due to its small thickness. We, thus, assume the radiation
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4. Results In the following, we evaluate numerically the PSF given in Eq. (12) along with Eqs. (8)and (11). The transducer has the following characteristics: eight elements, aperture size a = 47 mm, focal distance z0 = 75 mm, center frequency f0 = 2 MHz, and difference frequency f = 1 kHz. The speed of sound in the water is c0 = 1500 m/s. Our analysis is done in the magniˆ In the dB-scale, we have tude of PSF expressed as h = |h|. hdB = 20 log h. 4.1. Point-spread function Fig. 3(a) and (b) show, respectively, the theoretical and the measured PSF for the sector array transducer. The measured and the theoretical PSF have the same characteristics, with sidelobes localized in eight spots separated by π/4 rad around the mainlobe with levels below −15 dB. The spatial resolution on the focal plane of the experimental vibro-acoustography system is about 1.2 mm × 1.2 mm. Fig. 4(a) shows a plot of the theoretical and the measured PSF in the lateral direction. The spatial resolution and sidelobe
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
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Fig. 4. The theoretical and experimental PSFs at f = 1 kHz. (a) PSF magnitude in the lateral direction.
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Fig. 3. Theoretical and experimental PSFs of the sector array with surface levels underneath. (a) Theoretical PSF magnitude. (b) Experimental PSF magnitude.
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4.2. Vibro-acoustography images
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Now, we present vibro-acoustography magnitude images of a breast phantom (Model 18-228, Nuclear Associates, USA) made of a flexible resin material. Fig. 5(a) shows a digital picture of the breast phantom with a selected area. The phantom contains six solid masses, 2–8 mm, and six cystic masses, 3– 10 mm. The phantom images are compared qualitatively. Our phantom was not designed or calibrated for contrast. Therefore, even if we calculate the contrast, we would not have a reference for comparison. We thus present our results for visual evaluation. To obtain the images we used the experimental apparatus described in Fig. 2(a). The transducer is driven by 2 MHz with difference frequency of 9.7 kHz. Fig. 5(b) exhibits the image by the sector array transducer. The image clearly shows the inclusions with good contrast but halo artifacts are noticeable.
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levels (under −15 dB) are in good agreement as can be seen. In Fig. 4(b), we have the PSF magnitude plotted in depth. The depth-of-field at −6 dB is 13 mm.
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Fig. 5. Breast phantom and images of the region enclosed by the dashed box. (a) Breast phantom with inclusions. (b) Vibro-acoustography image. The marked area was chosen to a digital phantom with difference frequency at 9.7 kHz.
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
CMIG 704 1–7 G.T. Silva et al. / Computerized Medical Imaging and Graphics xxx (2006) xxx–xxx
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Fig. 6. Surface levels of the theoretical vibro-acoustography PSF (in magnitude) at f = 1 kHz. (a) Resolution cell (−6 dB). (b) Surface of PSF at −20 dB.
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In Fig. 6, we see the surface levels of the PSF magnitude. The resolution cell resembles a prolate spheroid whose volume is approximately 20.35 mm3 . The level surface at −20 dB reveals surfaces corresponding to the eight sidelobes surrounding the mainlobe. Again, these surface look like prolate spheroids. The volumes of the mainlobe and each sidelobes are, approximately, 104 and 19.6 mm3 , respectively. A digital phantom of 25 mm × 25 mm × 50 mm (see Fig. 7(a)) was designed to mimic the major features exhibited by the breast phantom shown in Fig. 5(a). It consists of three opaque spheres of radii 4, 3.5 and 3 mm. Two spheres are placed at the focal plane of the transducer and other one is located behind this plane. They have the same acoustic properties for the imaging system. The discrete version of the phantom has 256 × 256 × 512 pixels. The image of the digital phantom is obtained by convolving it with the system PSF as described in Eq. (6). This convolution was computed in the Fourier domain. The magnitude image in the focal plane is shown in Fig. 7(b). It is noteworthy that, despite the differences between the real setup and the numerical experiment, the salient features of both are alike. Compare the region outlined in Fig. 5(a) with the one shown in Fig. 7(b). The two bigger objects in both images are
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Fig. 7. The digital phantom with spherical inclusions. (a) Digital phantom in 3D. (b) Obtained image of the phantom at the transducer focal plane.
in the focal plane; they provide most of the energy received by the sensor and show a white halo within the dark spot. In the experimental image, the bigger object in the focal plane exhibits a lighter center, and the same can be seen in the simulated image. In both images, objects far from the focal plane are barely distinguishable since the energy that they yield to the acoustic detector is small. 5. Conclusion We have presented image formation of vibro-acoustography systems based on a sector array transducer. A method to obtain
Please cite this article as: Glauber T. Silva et al., Image formation in vibro-acoustography with depth-of-field effects, Computerized Medical CMIG 704 1–7 Imaging and Graphics (2006), doi:10.1016/j.compmedimag.2006.08.001.
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This work was partially supported by grants DCR2003.013 (FAPEAL/CNPq, Brazil) and 473976/2004-8 (CNPq, Brazil). We kindly thank Baldoino Fonseca and Guilherme Maia for the artwork.
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[1] Parker KJ, Taylor LS, Gracewski S, Rubens DJ. A unified view of imaging the elastic properties of tissue. J Acoust Soc Am 2005;117(5):2705– 12. [2] Fatemi M, Greenleaf JF. Ultrasound-stimulated vibro-acoustic spectrography. Science 1998;280:82–5. [3] Silva GT, Chen S, Greenleaf JF, Fatemi M. Dynamic ultrasound radiation force in fluids. Phys Rev E 2005;71:056617. [4] Fatemi M, Wold LE, Alizad A, Greenleaf JF. Vibro-acoustic tissue mammography. IEEE Trans Med Imag 2001;21(1):1–8. [5] Alizad A, Fatemi M, Nishimura RA, Kinnick RR, Rambod E, Greenleaf JF. Detection of calcium deposits on heart valve leaflets by vibroacoustography: an in vitro study. J Am Soc Echocardiogr 2002;15(11): 1391–5. [6] Alizad A, Wold LE, Greenleaf JF, Fatemi M. Imaging mass lesions by vibro-acoustography: modeling and experiments. IEEE Trans Med Imag 2004;23(9):1087–93. [7] Mitri FG, Trompette P, Chapelon J. Improving the use of vibroacoustography for brachytherapy metal seed imaging: a feasibility study. IEEE Trans Med Imag 2004;23(1):1–6. [8] Fatemi M, Greenleaf JF. Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission. Proc Natl Acad Sci USA 1999;96:6603–8. [9] Chen S, Fatemi M, Kinnick R, Greenleaf JF. Comparison of stress field forming methods for vibro-acoustography. IEEE Trans Ultrason Ferr Freq Cntrl 2004;51(3):313–21. [10] Silva GT, Chen S, Frery A, Greenleaf JF, Fatemi M. Stress field forming of sector array transducers for vibro-acoustography. IEEE Trans Ultrason Ferr Freq Cntrl 2005;52(11):1943–51. [11] Silva GT, Greenleaf JF, Fatemi M. Linear arrays for vibro-acoustography: a numerical simulation study. Ultrasonic Imaging 2004;26:1–17. [12] Wallace PR. Mathematical Analysis of Physical Problems. New York: Dover; 1984. p. 239. [13] Wang G, Li Y. Axiomatic approach for quantification of image resolution. IEEE Signal Process Lett 1999;6(10):257–8.
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a 3D PSF was proposed stemming from results of the classical theory of acoustic radiation. Such method is computationally affordable once we have an analytic expression for the PSF in the vicinity of the focal point. Other techniques like the spatial impulse method or finite-difference schemes would require much heavier computational efforts. The major drawback in the presented method is the phase information of the PSF. The underlying assumption of the method is that near the transducer focal point the incident wave is a bichromatic plane wave. As we get far from the system focus this assumption is no longer valid at all due to diffraction. Only a full 3D model for the incident ultrasound beam may reveal the contents of the PSF phase. The experimental validation of the theoretical PSF in the focal plane and axial direction for a sector array transducer was presented. This transducer achieved a resolution of about 1.2 mm × 1.2 mm. Results showed that the PSF sidelobes are radially distributed in eight regions around the mainlobe. Sidelobe are as low as −15 dB. Finally, we believe the 3D PSF model opens up new possibilities of investigation in vibro-acoustography. Through this model we investigated how the system images an inhomogeneous medium with small scatterers and lesion-like inclusions. Such situation may be found in applications of a vibro-acoustography system imaging breast tissue with lesions.
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