Harmonic generation with a dual frequency pulse Christina P. Keravnou and Michalakis A. Averkioua) Department of Mechanical and Manufacturing Engineering, University of Cyprus, 75 Kallipoleos, 1678 Nicosia, Cyprus
(Received 19 April 2013; revised 4 October 2013; accepted 2 April 2014) Nonlinear imaging was implemented in commercial ultrasound systems over the last 15 years offering major advantages in many clinical applications. In this work, pulsing schemes coupled with a dual frequency pulse are presented. The pulsing schemes considered were pulse inversion, power modulation, and power modulated pulse inversion. The pulse contains a fundamental frequency f and a specified amount of its second harmonic 2f. The advantages and limitations of this method were evaluated with both acoustic measurements of harmonic generation and theoretical simulations based on the KZK equation. The use of two frequencies in a pulse results in the generation of the sum and difference frequency components in addition to the other harmonic components. While with single frequency pulses, only power modulation and power modulated pulse inversion contained odd harmonic components, with the dual frequency pulse, pulse inversion now also contains C 2014 Acoustical Society of America. odd harmonic components. V [http://dx.doi.org/10.1121/1.4871356] PACS number(s): 43.25.Cb, 43.80.Qf, 43.25.Zx [ROC]
I. INTRODUCTION
Nonlinear imaging started developing in the early 1990s. It originated from the idea of establishing a specific technique for imaging nonlinear echoes from ultrasound contrast agents (Burns, 1996), as at that time it was assumed that there would not be any nonlinear echoes from tissue. Later works (Averkiou et al., 1997; Christopher, 1997; Ward et al., 1997) showed that tissue produces nonlinear echoes. While microbubbles are nonlinear scatterers, tissue nonlinearity arises from the distortion of the pulse due to nonlinear propagation of sound (Hamilton and Blackstock, 1998). This realization led to a new imaging method, Tissue Harmonic Imaging (THI), in 1997 (Averkiou et al., 1997). In THI the image is formed from the backscattered signal at twice the frequency (second harmonic, 2ƒ) of the transmitted pulse (fundamental, ƒ). THI offers advantages in many clinical applications of diagnostic ultrasound over the conventional B-mode technique (Becher et al., 1998; Tranquart et al., 1999), such as reduced reverberation noise, improved border delineation, increased contrast resolution, and reduced phase aberration artifacts (Averkiou, 2001). Various methods have been proposed to detect nonlinear signals other than the second harmonic for imaging purposes. Sub-harmonic imaging of contrast agents was suggested by Forsberg et al. (2000), where the image was formed from the received signal at f/2. This method aimed to improve lateral resolution and penetration due to less attenuation of the scattered sub-harmonic signals. Bouakaz and de Jong (2003) and Bouakaz et al. (2002) proposed combining higher order harmonics (third, fourth, and fifth) into a single component called “superharmonic” (SHI), to improve resolution and quality of the images. However this method suffers from an artifact caused by the low signal in the areas a)
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J. Acoust. Soc. Am. 135 (5), May 2014
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between the harmonics. Coded excitation and frequency modulation techniques are also found in literature (Arshadi et al., 2007; Borsboom et al., 2005; Song et al., 2010). These techniques aim to offer a better signal-to-noise ratio without the loss of resolution, since they use long narrowband pulses with more acoustic power. The use of dual frequency pulses for nonlinear imaging has also been proposed. Deng et al. (2000) used a superimposed high frequency (10 MHz) broadband ultrasound pulse with a low frequency (0.5 MHz) pulse for contrast imaging. Later, Angelsen and Hansen (2007) used dual frequency pulse to suppress signal from tissue and enhance the signal scattered from microbubbles. Their method transmits pulses with a low to high frequency ratio of 1:10 and the high frequency is placed on the peak or trough of the low frequency. Van Neer et al. (2011) proposed a method that uses two transmissions with slightly different frequencies to solve the issues of ghosting artifacts and to retain the axial resolution of SHI. Shen et al. (2011) proposed a method where a pulse with two high frequencies (f1 ¼ 9 MHz and f2 ¼ 6 MHz) is used to further enhance the contrast-to-tissue ratio of sub-harmonic imaging. In this study only the phase between the two frequencies was investigated. However, none of the above dual frequency methods addressed nonlinear imaging of tissue. Pulsing schemes such as pulse inversion (PI) (Simpson et al., 1999), power modulation (PM) (Brock-Fisher and Prater, 2006), and power modulated pulse inversion (PMPI) have been developed to detect nonlinear signals while maintaining good axial resolution (Averkiou et al., 2008). These schemes are found today on diagnostic ultrasound systems. While PI has excellent resolution by detecting the even harmonic components only, PM and PMPI benefit from good sensitivity due to the detection of the nonlinear fundamental component. Sometimes hardware limitations impose the use of only a specific pulsing scheme (some systems may be capable of inverting pulses only and some others of modulating the amplitude).
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C 2014 Acoustical Society of America V
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In the present work, a new method is presented where a pulsing scheme with a dual frequency pulse is investigated for nonlinear imaging. It may be utilized in either nonlinear imaging of contrast microbubbles or tissue; however, with the present paper we only consider the latter. The pulse contains a fundamental frequency f and its second harmonic 2f. The performance and limitations of the new method for application to nonlinear tissue imaging was evaluated with both acoustic pressure measurements and theoretical simulations. The harmonic generation of the dual frequency pulse with pulsing schemes (PI, PM, and PMPI) was compared to that of a conventional single frequency pulse. It is shown that the proposed technique efficiently extracts harmonics both at low and high ends of the frequency spectrum and thus may offer improvements in penetration and resolution, respectively. PI greatly benefits from dual frequency pulses as it now contains a large amount of nonlinear fundamental that was not present before with single frequency pulses.
The same applies for the absorption coefficient a, a ¼ a0 f y ;
(3)
where a0 is a constant (thermoviscous absorption at 1 MHz), and the different frequency components of the pulse experience different absorption. Since the two frequency components have different focusing gains and different absorptions, it is possible to design for a specific pulse to arrive at the focus by changing the coefficients b1 and b2 using Eq. (1). All the nonlinear combinations (harmonics, sum frequencies, difference frequencies, etc.) are multiples of f. For instance, the harmonics of f are 2f, 3f,…,nf; and for 2f they are 4f, 6f,…, 2nf. The sum frequency components are n(f þ 2f) and the difference frequency components are n(2f f). The end result is a nonlinear signal with f, 2f, 3f, 4f,…, where all components come from multiple level nonlinear interactions. B. Theoretical model
II. MATERIALS AND METHODS A. Description of the dual frequency (f 1 2f) pulse method
The dual frequency pulse contains two frequencies that are a mixture of a fundamental frequency f and its weighted and phase-shifted second harmonic 2f (see Fig. 1). Such pulses are generated by y ¼ ½b1 sinðxtÞ þ b2 sinð2xt þ /Þ " # xt 2m exp ; Ncyc p
(1)
A time-domain numerical solution of the Khokhlov– Zabolotskaya–Kuznetsov (KZK) parabolic wave equation (Khokhlov and Zabolotskaya, 1969; Kuznetsov, 1970) was used to model the nonlinear distortion of the dual frequency pulse after nonlinear propagation. Details regarding the numerical solution of the KZK equation can be found elsewhere (Averkiou and Hamilton, 1997; Lee and Hamilton, 1995). Briefly, Eq. (4) is a non-dimensional form of the KZK equation, ! ð @P 1 s @ 2 P 1 @P @2P @P2 0 ; ¼ ds þ þ A þ N @s @r 4G 1 @q2 q @q @s2 (4)
where x ¼ 2pf, b1, and b2 are the weighted coefficients of f and 2f, respectively, and b1 þ b2 ¼ 1. u is a phase that shifts the second harmonic relative to the fundamental. The exponential term is the envelope of the pulse, where m is proportional to the envelope slope and Ncyc is the number of cycles between the 1/e points. Since the focusing gain of a source (G) is defined as G¼k
a2 ; 2d
(2)
where k is the wave number, a is the radius, and d is the focal length, each frequency component of the pulse corresponds to a different value of G.
FIG. 1. Example of a dual frequency pulse at the source with equal weights (b1 ¼ 0.5, b2 ¼ 0.5) and phase u ¼ p/2. Spectrum (S) is normalized with respect to the maximum value of the pulse. 2546
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where P ¼ p/p0, r ¼ z/d, q ¼ r/a, and s ¼ x0(t z/c0). The first term on the right-hand side of Eq. (4) accounts for diffraction, the second term for thermoviscous absorption, and the third for nonlinear propagation. A and N are dimensionless parameters that account for absorption and nonlinearity, respectively, where A ¼ az and N ¼ z0 =z (z is the distance to the point at which a shock first forms). Equation (4) accounts only for a value y ¼ 2 in Eq. (3), i.e., quadratic frequency dependence. C. Pulsing schemes
The pulsing schemes considered in the present work are PI, PMPI, and PM, which are the most commonly found in diagnostic ultrasound scanners. Two pulses (p1, p2) are sent consecutively where the second pulse (p2) is the inverse of the first in PI, the inverse scaled at half amplitude in PMPI, and the scaled at half amplitude in PM. After nonlinear propagation, pulses are summed according to 1 1 PI ¼ p1 þ p2 ; 2 2 1 2 PMPI ¼ p1 þ p2 ; 3 3
(5) (6)
C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 2. (Color online) Schematic representation of the setup used for nonlinear propagation.
1 2 PM ¼ p1 p2 : 3 3
(7)
D. Experimental setup
All experiments were conducted with a focused single element circular transducer (a ¼ 12.7 mm, d ¼ 76.2 mm, f ¼ 2.25 MHz, G ¼ 10.07). Note that the selected transducer has a fractional bandwidth (at 6 dB) of 93%, thus allows effective transmission of both 1.4 and 2.8 MHz. Furthermore, 1.4 and 2.8 MHz experience the same drop from the center frequency (about 4 dB). A 0.4 mm membrane hydrophone (Precision Acoustics Ltd., Dorchester, United Kingdom) with a flat frequency response in the range of 1–30 MHz was used as a receiver, and it was placed at the focus of the transducer. A motorized micro-positioning system (Newport, Irvine, CA) was used to align the transducer and the hydrophone and to move the hydrophone to the desired location. Between the acoustic source and the membrane hydrophone, a 57 89 118 mm acrylic container was placed to hold the tissue mimicking fluid. The sides of the container had acoustically transparent windows made from a 0.01 mm Mylar membrane. The container was filled with castor oil which has the following properties: c0 ¼ 1521 m/s, q ¼ 950 kg/m3, and a 0 ¼ 0.33 dB/cm/MHz2. The transducer, hydrophone, and castor oil container were submerged in a tank filled with de-ionized water as shown in Fig. 2. At the back wall of the tank, a 200 150 mm piece of absorptive material (Precision Acoustics Ltd., Dorchester, United Kingdom) was placed to reduce reflections. Dual frequency pulses were designed in MATLAB (The MathWorks, Inc., Natick, MA) with Eq. (1). The pulses were transferred to a Tektronix AFG 3102 Function Generator (Tektronix, Inc., Beaverton, OR), and amplified by a 150A100B RF Amplifier (Amplifier Research, Souderton, PA). The fundamental frequency f was selected to be 1.4 MHz, which is similar to what is currently used in commercial diagnostic ultrasound systems for abdominal nonlinear imaging. Consequently, the high frequency component of the pulse was 2.8 MHz. The envelope and duration of the dual frequency pulses were Ncyc ¼ 4 and m ¼ 2. Five combinations of b1 and b2 (weights of the primary frequencies of the pulse) were examined. Three cases were chosen, as shown in Fig. 3, so that at the focus their f and 2f components had the following relative amplitudes: (1) Pulse with 6 dB more f than 2f [b1 ¼ 0.60, b2 ¼ 0.40, as shown in Figs. 3(a) and 3(b)]. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
(2) Pulse with equal f and 2f [b1 ¼ 0.43, b2 ¼ 0.57, as shown in Figs. 3(c) and 3(d)]. (3) Pulse with 6 dB more 2f than f [b1 ¼ 0.27, b2 ¼ 0.73, as shown in Figs. 3(e) and 3(f)]. For comparison purposes, two single frequency pulses (f ¼ 1.4 MHz, b1 ¼ 1, b2 ¼ 0, and f ¼ 2.8 MHz, b1 ¼ 0, b2 ¼ 1) were also propagated at the focus. For each case, three pulses were sent with amplitudes of 1, 1, and 0.5. Finally, a GPIB-USB2 interface and LabVIEW (National Instruments, Austin, Texas) computer software were used to acquire the received pulses and transfer to a personal computer. Data analysis was performed with MATLAB software. Prior to the nonlinear propagation experiments, detailed characterization of the ultrasound field produced by the transducer was carried out. Low amplitude pulses were used for the transducer characterization to avoid nonlinear distortion. Measurements of the variation of the acoustic pressure along the source axis (propagation curve) and normal to the propagation axis at the focus (focal beam pattern) were collected with the membrane hydrophone in de-ionized water.
FIG. 3. Dual frequency pulses with different coefficients (b1, b2). The left column shows time waveforms and the right column shows the frequency spectra normalized with respect to the maximum value of each case. (a) and (b): A pulse with 6 dB more signal at low frequency; (c) and (d): A pulse with equal amplitude at both frequencies; (e) and (f): A pulse with 6 dB more signal at high frequency. C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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B. Nonlinear propagation in a tissue mimicking fluid
FIG. 4. Measurements (solid lines) and theoretical predictions (dashed lines) of the acoustic field of the source. (a) Axial field and (b) transverse field at the focus.
III. RESULTS A. Ultrasound transducer characterization
Figure 4 shows measurements of the acoustic field of the ultrasound source in water (solid line) compared with the linear and lossless analytical solution of the KZK equation (dashed line). The pressure along the source axis (propagation curve) is shown in Fig. 4(a), while the pressure normal to the propagation axis (beam pattern) at the focus of the transducer is shown in Fig. 4(b). The source pressure at low amplitude was calculated from the measurements by dividing the focal pressure by the focusing gain. At higher amplitudes linear extrapolation was used according to the input voltage. We have used the theoretical predictions to make small adjustments on the source dimensions due to small errors in the dimensions of the ceramic and the influence of the custom-made housing. We have thus calculated the effective source dimensions (Averkiou, 1994) which they differ by less than 5% of the original specifications (a ¼ 13.7 mm, d ¼ 80.2 mm). These effective dimensions have been used throughout this work. Good agreement is observed between measurements and theoretical predictions, whereas small discrepancies occur only near the transducer.
Three combinations of the weights (b1, b2) of the dual frequency pulse as explained in Sec. II D have been used. The case of a single frequency pulse f ¼ 1.4 MHz, b1 ¼ 1, b2 ¼ 0 is first shown in Fig. 5. The time domain of the pulses and their spectra are presented in Figs. 5(a)–5(f). Measurements are shown as solid lines and theoretical predictions as dashed lines. The PI waveform which resulted from adding the pulses of Figs. 5(a) and 5(c) is shown in Fig. 5(g). Its spectrum, Fig. 5(h), contains only nonlinear components at even harmonics (second, fourth, etc.). Good agreement is observed between theoretical predictions and measurements. Figures 5(i) and 5(j) present the waveform and spectrum of PMPI, respectively. To form PMPI, the pulses of Figs. 5(c) and 5(e) were used. It is observed that PMPI, despite that it has a strong second harmonic component, also contains the odd harmonic components [Fig. 5(j)]. The PM technique which resulted from the mixing of the pulses of Figs. 5(a) and 5(e) is shown in Figs. 5(k) and 5(l). This method also extracts both even and odd harmonic components in a somewhat similar fashion with PMPI. In the cases of PMPI and PM, discrepancies between theoretical and experimental results are present concerning the amount of nonlinear fundamental. We will further discuss this in Sec. IV. Measurements and theoretical predictions of nonlinear propagation of a dual frequency pulse are shown next. Figures 6(a)–6(f) present the waveforms and spectra of the propagated dual frequency pulses. As seen from the spectra, a pulse with equal contributions of the two primary frequencies at the focus was selected. The PI technique isolates only the even harmonics of the nonlinear signal when single frequency pulse is used. However, PI with a dual frequency pulse [Figs. 6(g) and 6(h)], has both even and odd harmonic
FIG. 5. Measurements (solid lines) and theoretical predictions (dashed lines) of harmonic generation of a tone burst (f ¼ 1.4 MHz). Time waveforms and spectra of full pulse (a) and (b), inverse pulse (c) and (d), half pulse (e) and (f), PI (g) and (h), PMPI (i) and (j), and PM (k) and (l). Spectra are normalized with respect to the maximum value of the measured pulse. 2548
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C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 6. Measurements (solid lines) and theoretical predictions (dashed lines) of harmonic generation of a dual frequency pulse with equal contributions at f, 2f. Time waveforms and spectra of full pulse (a) and (b), inverse pulse (c) and (d), half pulse (e) and (f), PI (g) and (h), PMPI (i) and (j), and PM (k) and (l). Spectra are normalized with respect to the maximum value of the measured pulse.
components including a nonlinear fundamental frequency component. Good agreement between theoretical and experimental results is observed. PMPI [Figs. 6(i) and 6(j)] and PM [Figs. 6(k) and 6(l)] methods formed by dual frequency pulses also produce both even and odd harmonic components at different levels. In Figs. 7 and 8, the effect of the relative amplitude of the two frequency components on nonlinear signal generation for every pulsing scheme is presented based on theoretical predictions. The horizontal axis of the figures denotes the amount of the low frequency on the pulse. Consequently the
weight of the high frequency will be b2 ¼ 1 b1. When b1 ¼ 1, the dual frequency pulse is reduced to a single frequency pulse with f ¼ 1.4 MHz. When b1 ¼ 0, the dual frequency pulse is reduced to a single frequency pulse with f ¼ 2.8 MHz. The vertical axis displays the nonlinear signal in dB for each pulsing scheme. These values are calculated by normalizing the resulting pulse of the pulsing scheme in discussion with respect to the largest measured pulse, and then calculating the spectrum. Theoretical predictions only have been used in these figures because they easily allow the inclusion of a much larger range of values of b1 and b2 and
FIG. 7. Effect of the relative amplitude of the two frequency components in the dual frequency pulse on nonlinear signal generation with pulsing schemes per harmonic component (theoretical predictions at r ¼ 1).
FIG. 8. Effect of the relative amplitude of the two frequency components in the dual frequency pulse on nonlinear signal generation with pulsing schemes per harmonic component (theoretical predictions at r ¼ 1.5).
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examination of depths other than the focal plane with simple runs of the code (as opposed to repeating the experiment multiple times). In addition, KZK has been experimentally verified multiple times in the past and fair agreement with measurements was observed in the measurements collected here. An important finding is demonstrated in Fig. 7(a). PI with a dual frequency pulse produces a large amount of nonlinear fundamental (f ¼ 1.4 MHz) whereas with a single frequency pulse, PI cancels out all odd harmonics. PMPI also benefits at nonlinear fundamental (f) from the use of dual frequency pulses with a gain of about 5 dB. As it is also seen in Fig. 7(a), PI with a dual frequency pulse produces the most amount of nonlinear fundamental when compared to the other two methods (PMPI and PM). The second harmonic component (2f ¼ 2.8 MHz), shown in Fig. 7(b), is reduced for all pulsing schemes when a dual frequency pulse is used. It is observed that the maximum nonlinear signal is obtained at b1 ¼ 1, i.e., single frequency pulse. The signal loss of the second harmonic can be minimized without affecting all other nonlinear components (first, third, and fourth), by adding a higher contribution of the low frequency at the dual frequency pulse. Figures 7(c) and 7(d) show the third and fourth harmonic contributions, respectively. It can be seen that a larger amount of higher order harmonics is produced when a dual frequency pulse is utilized. It is also noticed that the PI method produces about 5 dB more signal at the third and fourth harmonic components compared to the other two techniques. The same trends as those of Fig. 7 are seen in Figs. 8(a)–8(d), which shows pulses propagated at larger depth (r ¼ 1.5) than the transducer’s focus. The nonlinear signal in dB detected by the pulsing schemes is lower at greater depth, but it is important to notice that the new proposed method still offers more signal at nonlinear fundamental, thus helping imaging for cases that lower frequency and penetration depth are required. It is also shown that PI continues to benefit from dual frequency pulses and contains a more nonlinear signal compared to the other two techniques. Figure 9 shows the spectra of all considered pulsing schemes in comparison to the full amplitude pulse, based on theoretical predictions. The single-frequency pulse [Fig. 9(a)] and a dual frequency pulse with equal weights at f and 2f [Fig. 9(b)] are shown. It is observed that for the case of a single frequency pulse [Fig. 9(a)], PI can fully extract the
FIG. 9. (Color online) Comparison of the spectra of the nonlinear signal detected with the various pulsing schemes for (a) the single and (b) the dual frequency pulse. 2550
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even harmonic components from the signal. PMPI spectrum contains both odd and even harmonics, with the even harmonics being a little lower than those of PI. Looking at the PM method, it extracts even components at much lower amplitude than the other two techniques, while it manages to preserve the same amount of odd harmonics as PMPI. However, for the case of a dual frequency pulse [Fig. 9(b)], odd and even harmonics are mixed up due to the influence of nonlinearities that arise from the sum and difference frequencies of f and 2f (2f f ¼ f, f þ 2f ¼ 3f, 2f þ 2f ¼ 4f, etc.). Sum and difference frequency components are considered to be even harmonic components, thus harmonic components for the case of dual frequency pulse, cannot be clearly separated to odd and even. We note that PI extracts the most amount of nonlinear signal at all harmonics. Interestingly we observe that PMPI and PM do not have the same amount of signal at f and 3f anymore which suggests that the signal at those frequencies is not solely an odd component. IV. DISCUSSION
This work evaluates the use of a dual frequency pulse in nonlinear imaging. The method relies on established multipulse schemes to isolate specific harmonic components. Experimental measurements and theoretical predictions were used to study in detail the differences between a dual frequency pulse and the conventional single frequency pulse, which is being used today in diagnostic ultrasound. The main objective of the present work was to examine the nonlinear generation of a high amplitude dual frequency pulse, using pulsing schemes but no in-depth consideration to imaging application was given. Comparisons between our measurements and the theoretical predictions based on the KZK equation showed very good agreement. Discrepancies were present only in the cases of PMPI and PM schemes, where the experiment exhibits a higher nonlinear fundamental than the simulation [Figs. 5(j), 5(l), 6(j), and 6(l)]. Similar differences were seen in previously published work (Mannaris, 2008). These discrepancies are mainly caused by the physical limitations of the equipment. While in theory the “half pulse” is the exact half amplitude of the first pulse, the experimental system may not be able to transmit the exact half voltage (and the half harmonic distortion) of the full amplitude pulse. Thus, linear signal cancellation is not perfect and experimental results at pulsing schemes that involve amplitude modulation (PMPI, PM) appear to have a higher nonlinear fundamental. For demonstration of the above issue, Fig. 10 shows pulsing schemes that were formed using very low amplitude (no harmonic signals present) pulses after propagation in water. While in PI [Fig. 10(a)] all linear signal is eliminated, in PMPI and PM [Figs. 10(b) and 10(c)] there is a nonlinear fundamental component present. Similar trends were observed when the electrical signal coming out of the power amplifier and before going to the transducer was recorded. Castor oil was selected as the medium since it was easy to handle, was homogeneous, and had acoustic parameters (density, speed of sound, and absorption) close to those of C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 10. Spectra of measured pulses at the focus (solid line) and of linear signal suppression (dashed line) for (a) PI, (b) PMPI, and (c) PM.
tissue. Non-homogeneous media which are usually encountered in diagnostic ultrasound will probably affect the results presented here. The dual frequency pulse combined with pulsing schemes generates more harmonics over a larger bandwidth, compared to the normal single frequency pulse. The usage of the dual frequency pulse in diagnostic imaging may therefore enhance image esthetics and resolution (due to the increased harmonics). Harmonics can be used to create multi-band frequency compounding images, since frequency compounding imaging already demonstrated improved image quality (Mesurolle et al., 2007; Varray et al., 2012). In nonlinear propagation of sound beams, the fundamental component is the linear response and the harmonics comprise the nonlinear response. There is no nonlinear fundamental component in the propagation of a single pulse. The nonlinear fundamental component in PM (and PMPI) is the result of scaling and subtracting two pulses of unequal nonlinearity. It is only specific to multi-pulse processing. The nonlinear fundamental in PI when using a dual frequency pulse is simply the difference frequency. Nevertheless, it still generates a slightly more nonlinear fundamental component than single or dual frequency PM and PMPI (as shown in Figs. 7–9). All pulsing schemes studied in this work can be found today implemented on diagnostic ultrasound systems. In commercial diagnostic ultrasound systems, PI has demonstrated to have excellent resolution, while PM and PMPI offer better sensitivity. A unique advantage of the proposed method is the generation of a nonlinear signal at f with PI since with single frequency pulses only PM techniques (PM and PMPI) can detect the nonlinear fundamental component. In that way, imaging with dual frequency pulses may improve sensitivity while maintaining excellent resolution, at ultrasound systems that are capable of implementing PI, and not PM or PMPI. The dual frequency method may also be utilized for bubble imaging, using lower acoustic amplitude to avoid bubble destruction. The proposed method offers the ability to design bubble-specific pulses with the weights b1, b2, and the phase /. For example, the addition of / ¼ p/2 at the second frequency component, would result to a substantially different response from a microbubble. Moreover, the use of pulses that contain two frequencies (f, 2f) can excite a larger population of bubbles.
new proposed method can generate harmonics over a larger bandwidth. If the dual frequency pulse is utilized in diagnostic ultrasound it could lead to improvements in penetration depth and resolution. While with single frequency pulses, only PM and PMPI contained a nonlinear fundamental signal, with the dual frequency pulse, PI now contains a nonlinear fundamental signal. APPENDIX: THERMOVISCOUS ABSORPTION MEASUREMENTS IN CASTOR OIL
A plane circular single element transducer (a ¼ 3.175 mm, f ¼ 2.25 MHz) was used to measure the thermoviscous absorption of castor oil. Pressure measurements at two locations (greater than the Rayleigh distance, R0 ¼ ka2/2) on the axis of the source were recorded. Since the Rayleigh distance marks the end of the nearfield and the beginning of the far-field (Fig. 11), the acoustic pressure is spreading spherically (Blackstock, 2000). The absorption ða0 Þ was calculated using pressure measurements (P1, P2) at two known distances (r1, r2) (which were beyond 2R0 to ensure spherical spreading) using Eq. (A1), r1 P1 ln r2 P2 a0 ¼ : (A1) r2 r1 With our far-field axial pressure measurements and Eq. (A1) we have found that for castor oil a0 ¼ 0.33 dB/cm/MHz2. We also note that for castor oil the speed of sound and density (see Table I), and hence its impedance, is very close to that of water and reflections from the water-castor oil interface are thus significantly limited. Finally, castor oil is also an excellent choice for tissue mimicking fluid as it has similar values for speed of sound, density, and absorption.
V. CONCLUSIONS
A new nonlinear imaging technique based on a dual frequency pulse and pulsing schemes has been developed and evaluated for nonlinear tissue imaging. It was shown that the J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
FIG. 11. Acoustic field along the axis of a plane circular piston (a ¼ 3.175 mm, f ¼ 2.25 MHz). C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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TABLE I. Properties of castor oil. Castor Oil c0 (m/s) a0 (dB/cm/MHz2) q (kg/m3)
1521 0.33 920
Angelsen, B. A. J., and Hansen, R. (2007). “7A-1 SURF Imaging - A new method for ultrasound contrast agent imaging,” Proc. - IEEE Ultrason. Symp. 2007, 531–541. Arshadi, R., Yu, A. C. H., and Cobbold, R. S. C. (2007). “Coded excitation methods for ultrasound harmonic imaging,” Can. Acoust. 35, 35–46. Averkiou, M. A. (1994). “Experimental investigation of propagation and reflection phenomena in finite amplitude sound beams,” Ph.D. dissertation, The University of Texas at Austin, Austin, Texas. Averkiou, M. A. (2001). “Tissue harmonic ultrasonic imaging,” C. R. Acad. Sci., Ser. IV: Phys., Astrophys. 2, 1139–1151. Averkiou, M. A., and Hamilton, M. F. (1997). “Nonlinear distortion of short pulses radiated by plane and focused circular pistons,” J. Acoust. Soc. Am. 102, 2539–2548. Averkiou, M. A., Mannaris, C., Bruce, M., and Powers, J. (2008). “Nonlinear pulsing schemes for the detection of ultrasound contrast agents,” J. Acoust. Soc. Am. 123, 3110. Averkiou, M. A., Roundhill, D. N., and Powers, J. E. (1997). “New imaging technique based on the nonlinear properties of tissues,” Proc. - IEEE Ultrason. Symp. 1997, 1561–1566. Becher, H., Tiemann, K., Schlosser, T., Pohl, C., Nanda, N. C., Averkiou, M. A., Powers, J., and L€ uderitz, B. (1998). “Improvement in endocardial border delineation using tissue harmonic imaging,” Echocardiography 15, 511–518. Blackstock, D. T. (2000). Fundamentals of Physical Acoustics (Wiley, New York), Chap. 13, pp. 440–471. Borsboom, J. M. G., Chin, C. T., Bouakaz, A., Versluis, M., and de Jong, N. (2005). “Harmonic chirp imaging method for ultrasound contrast agent,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 52, 241–249. Bouakaz, A., and de Jong, N. (2003). “Native tissue imaging at superharmonic frequencies,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 50, 496–506. Bouakaz, A., Frigstad, S., Ten Cate, F. J., and de Jong, N. (2002). “Super harmonic imaging: A new imaging technique for improved contrast detection,” Ultrasound Med. Biol. 28, 59–68. Brock-Fisher, G. A., and Prater, D. M. (2006). “Acoustic border detection using power modulation,” J. Acoust. Soc. Am. 120, 584. Burns, P. N. (1996). “Harmonic imaging with ultrasound contrast agents,” Clin. Radiol. 51 Suppl. 1, 50–55.
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Christopher, T. (1997). “Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 44, 125–139. Deng, C. X., Lizzi, F. L., Kalisz, A., Rosado, A., Silverman, R. H., and Coleman, D. J. (2000). “Study of ultrasonic contrast agents using a dualfrequency band technique,” Ultrasound Med. Biol. 26, 819–831. Forsberg, F., Shi, W. T., and Goldberg, B. B. (2000). “Subharmonic imaging of contrast agents,” Ultrasonics 38, 93–98. Hamilton, M. F., and Blackstock, D. T. (1998). Nonlinear Acoustics (Academic Press, San Diego, CA), 455 p. Khokhlov, R. V., and Zabolotskaya, E. A. (1969). “Quasi-plane waves in the non-linear acoustics of confined beams,” Sov. Phys. Acoust. 15, 35–40. Kuznetsov, V. P. (1970). “Equation of nonlinear acoustics,” Sov. Phys. Acoust. 16, 467–470. Lee, Y.-S., and Hamilton, M. F. (1995). “Time-domain modeling of pulsed finite-amplitude sound beams,” J. Acoust. Soc. Am. 97, 906–917. Mannaris, C. (2008). “Experimental investigation of nonlinear pulsing schemes used in diagnostic ultrasound,” MS thesis, University of Cyprus, Nicosia, Cyprus. Mesurolle, B., Helou, T., El-Khoury, M., Edwardes, M., Sutton, E. J., and Kao, E. (2007). “Tissue harmonic imaging, frequency compound imaging, and conventional imaging: Use and benefit in breast sonography,” J. Ultrasound Med. 26, 1041–1051. Shen, C.-C., Cheng, C.-H., and Yeh, C.-K. (2011). “Phase-dependent dualfrequency contrast imaging at sub-harmonic frequency,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 58, 379–388. Simpson, D. H., Chien, T. C., and Burns, P. N. (1999). “Pulse inversion Doppler: A new method for detecting nonlinear echoes from microbubble contrast agents,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 46, 372–382. Song, J., Kim, S., Sohn, H.-Y., Song, T.-K., and Yoo, Y. M. (2010). “Coded excitation for ultrasound tissue harmonic imaging,” Ultrasonics 50, 613–619. Tranquart, F., Grenier, N., Eder, V., and Pourcelot, L. (1999). “Clinical use of ultrasound tissue harmonic imaging,” Ultrasound Med. Biol. 25, 889–894. Van Neer, P. L. M. J., Danilouchkine, M. G., Matte, G. M., Van Der Steen, A. F. W., and De Jong, N. (2011). “Dual-pulse frequency compounded superharmonic imaging,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 58, 2316–2324. Varray, F., Cachard, C., Kybic, J., Novell, A., Bouakaz, A., and Basset, O. (2012). “A multi-frequency approach to increase the native resolution of ultrasound images,” in Proceedings of the Signal Processing Conference (EUSIPCO), pp. 2733–2737. Ward, B., Baker, A. C., and Humphrey, V. F. (1997). “Nonlinear propagation applied to the improvement of resolution in diagnostic medical ultrasound,” J. Acoust. Soc. Am. 101, 143–154.
C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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(Received 19 April 2013; revised 4 October 2013; accepted 2 April 2014) Nonlinear imaging was implemented in commercial ultrasound systems over the last 15 years offering major advantages in many clinical applications. In this work, pulsing schemes coupled with a dual frequency pulse are presented. The pulsing schemes considered were pulse inversion, power modulation, and power modulated pulse inversion. The pulse contains a fundamental frequency f and a specified amount of its second harmonic 2f. The advantages and limitations of this method were evaluated with both acoustic measurements of harmonic generation and theoretical simulations based on the KZK equation. The use of two frequencies in a pulse results in the generation of the sum and difference frequency components in addition to the other harmonic components. While with single frequency pulses, only power modulation and power modulated pulse inversion contained odd harmonic components, with the dual frequency pulse, pulse inversion now also contains C 2014 Acoustical Society of America. odd harmonic components. V [http://dx.doi.org/10.1121/1.4871356] PACS number(s): 43.25.Cb, 43.80.Qf, 43.25.Zx [ROC]
I. INTRODUCTION
Nonlinear imaging started developing in the early 1990s. It originated from the idea of establishing a specific technique for imaging nonlinear echoes from ultrasound contrast agents (Burns, 1996), as at that time it was assumed that there would not be any nonlinear echoes from tissue. Later works (Averkiou et al., 1997; Christopher, 1997; Ward et al., 1997) showed that tissue produces nonlinear echoes. While microbubbles are nonlinear scatterers, tissue nonlinearity arises from the distortion of the pulse due to nonlinear propagation of sound (Hamilton and Blackstock, 1998). This realization led to a new imaging method, Tissue Harmonic Imaging (THI), in 1997 (Averkiou et al., 1997). In THI the image is formed from the backscattered signal at twice the frequency (second harmonic, 2ƒ) of the transmitted pulse (fundamental, ƒ). THI offers advantages in many clinical applications of diagnostic ultrasound over the conventional B-mode technique (Becher et al., 1998; Tranquart et al., 1999), such as reduced reverberation noise, improved border delineation, increased contrast resolution, and reduced phase aberration artifacts (Averkiou, 2001). Various methods have been proposed to detect nonlinear signals other than the second harmonic for imaging purposes. Sub-harmonic imaging of contrast agents was suggested by Forsberg et al. (2000), where the image was formed from the received signal at f/2. This method aimed to improve lateral resolution and penetration due to less attenuation of the scattered sub-harmonic signals. Bouakaz and de Jong (2003) and Bouakaz et al. (2002) proposed combining higher order harmonics (third, fourth, and fifth) into a single component called “superharmonic” (SHI), to improve resolution and quality of the images. However this method suffers from an artifact caused by the low signal in the areas a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (5), May 2014
Pages: 2545–2552
between the harmonics. Coded excitation and frequency modulation techniques are also found in literature (Arshadi et al., 2007; Borsboom et al., 2005; Song et al., 2010). These techniques aim to offer a better signal-to-noise ratio without the loss of resolution, since they use long narrowband pulses with more acoustic power. The use of dual frequency pulses for nonlinear imaging has also been proposed. Deng et al. (2000) used a superimposed high frequency (10 MHz) broadband ultrasound pulse with a low frequency (0.5 MHz) pulse for contrast imaging. Later, Angelsen and Hansen (2007) used dual frequency pulse to suppress signal from tissue and enhance the signal scattered from microbubbles. Their method transmits pulses with a low to high frequency ratio of 1:10 and the high frequency is placed on the peak or trough of the low frequency. Van Neer et al. (2011) proposed a method that uses two transmissions with slightly different frequencies to solve the issues of ghosting artifacts and to retain the axial resolution of SHI. Shen et al. (2011) proposed a method where a pulse with two high frequencies (f1 ¼ 9 MHz and f2 ¼ 6 MHz) is used to further enhance the contrast-to-tissue ratio of sub-harmonic imaging. In this study only the phase between the two frequencies was investigated. However, none of the above dual frequency methods addressed nonlinear imaging of tissue. Pulsing schemes such as pulse inversion (PI) (Simpson et al., 1999), power modulation (PM) (Brock-Fisher and Prater, 2006), and power modulated pulse inversion (PMPI) have been developed to detect nonlinear signals while maintaining good axial resolution (Averkiou et al., 2008). These schemes are found today on diagnostic ultrasound systems. While PI has excellent resolution by detecting the even harmonic components only, PM and PMPI benefit from good sensitivity due to the detection of the nonlinear fundamental component. Sometimes hardware limitations impose the use of only a specific pulsing scheme (some systems may be capable of inverting pulses only and some others of modulating the amplitude).
0001-4966/2014/135(5)/2545/8/$30.00
C 2014 Acoustical Society of America V
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In the present work, a new method is presented where a pulsing scheme with a dual frequency pulse is investigated for nonlinear imaging. It may be utilized in either nonlinear imaging of contrast microbubbles or tissue; however, with the present paper we only consider the latter. The pulse contains a fundamental frequency f and its second harmonic 2f. The performance and limitations of the new method for application to nonlinear tissue imaging was evaluated with both acoustic pressure measurements and theoretical simulations. The harmonic generation of the dual frequency pulse with pulsing schemes (PI, PM, and PMPI) was compared to that of a conventional single frequency pulse. It is shown that the proposed technique efficiently extracts harmonics both at low and high ends of the frequency spectrum and thus may offer improvements in penetration and resolution, respectively. PI greatly benefits from dual frequency pulses as it now contains a large amount of nonlinear fundamental that was not present before with single frequency pulses.
The same applies for the absorption coefficient a, a ¼ a0 f y ;
(3)
where a0 is a constant (thermoviscous absorption at 1 MHz), and the different frequency components of the pulse experience different absorption. Since the two frequency components have different focusing gains and different absorptions, it is possible to design for a specific pulse to arrive at the focus by changing the coefficients b1 and b2 using Eq. (1). All the nonlinear combinations (harmonics, sum frequencies, difference frequencies, etc.) are multiples of f. For instance, the harmonics of f are 2f, 3f,…,nf; and for 2f they are 4f, 6f,…, 2nf. The sum frequency components are n(f þ 2f) and the difference frequency components are n(2f f). The end result is a nonlinear signal with f, 2f, 3f, 4f,…, where all components come from multiple level nonlinear interactions. B. Theoretical model
II. MATERIALS AND METHODS A. Description of the dual frequency (f 1 2f) pulse method
The dual frequency pulse contains two frequencies that are a mixture of a fundamental frequency f and its weighted and phase-shifted second harmonic 2f (see Fig. 1). Such pulses are generated by y ¼ ½b1 sinðxtÞ þ b2 sinð2xt þ /Þ " # xt 2m exp ; Ncyc p
(1)
A time-domain numerical solution of the Khokhlov– Zabolotskaya–Kuznetsov (KZK) parabolic wave equation (Khokhlov and Zabolotskaya, 1969; Kuznetsov, 1970) was used to model the nonlinear distortion of the dual frequency pulse after nonlinear propagation. Details regarding the numerical solution of the KZK equation can be found elsewhere (Averkiou and Hamilton, 1997; Lee and Hamilton, 1995). Briefly, Eq. (4) is a non-dimensional form of the KZK equation, ! ð @P 1 s @ 2 P 1 @P @2P @P2 0 ; ¼ ds þ þ A þ N @s @r 4G 1 @q2 q @q @s2 (4)
where x ¼ 2pf, b1, and b2 are the weighted coefficients of f and 2f, respectively, and b1 þ b2 ¼ 1. u is a phase that shifts the second harmonic relative to the fundamental. The exponential term is the envelope of the pulse, where m is proportional to the envelope slope and Ncyc is the number of cycles between the 1/e points. Since the focusing gain of a source (G) is defined as G¼k
a2 ; 2d
(2)
where k is the wave number, a is the radius, and d is the focal length, each frequency component of the pulse corresponds to a different value of G.
FIG. 1. Example of a dual frequency pulse at the source with equal weights (b1 ¼ 0.5, b2 ¼ 0.5) and phase u ¼ p/2. Spectrum (S) is normalized with respect to the maximum value of the pulse. 2546
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where P ¼ p/p0, r ¼ z/d, q ¼ r/a, and s ¼ x0(t z/c0). The first term on the right-hand side of Eq. (4) accounts for diffraction, the second term for thermoviscous absorption, and the third for nonlinear propagation. A and N are dimensionless parameters that account for absorption and nonlinearity, respectively, where A ¼ az and N ¼ z0 =z (z is the distance to the point at which a shock first forms). Equation (4) accounts only for a value y ¼ 2 in Eq. (3), i.e., quadratic frequency dependence. C. Pulsing schemes
The pulsing schemes considered in the present work are PI, PMPI, and PM, which are the most commonly found in diagnostic ultrasound scanners. Two pulses (p1, p2) are sent consecutively where the second pulse (p2) is the inverse of the first in PI, the inverse scaled at half amplitude in PMPI, and the scaled at half amplitude in PM. After nonlinear propagation, pulses are summed according to 1 1 PI ¼ p1 þ p2 ; 2 2 1 2 PMPI ¼ p1 þ p2 ; 3 3
(5) (6)
C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 2. (Color online) Schematic representation of the setup used for nonlinear propagation.
1 2 PM ¼ p1 p2 : 3 3
(7)
D. Experimental setup
All experiments were conducted with a focused single element circular transducer (a ¼ 12.7 mm, d ¼ 76.2 mm, f ¼ 2.25 MHz, G ¼ 10.07). Note that the selected transducer has a fractional bandwidth (at 6 dB) of 93%, thus allows effective transmission of both 1.4 and 2.8 MHz. Furthermore, 1.4 and 2.8 MHz experience the same drop from the center frequency (about 4 dB). A 0.4 mm membrane hydrophone (Precision Acoustics Ltd., Dorchester, United Kingdom) with a flat frequency response in the range of 1–30 MHz was used as a receiver, and it was placed at the focus of the transducer. A motorized micro-positioning system (Newport, Irvine, CA) was used to align the transducer and the hydrophone and to move the hydrophone to the desired location. Between the acoustic source and the membrane hydrophone, a 57 89 118 mm acrylic container was placed to hold the tissue mimicking fluid. The sides of the container had acoustically transparent windows made from a 0.01 mm Mylar membrane. The container was filled with castor oil which has the following properties: c0 ¼ 1521 m/s, q ¼ 950 kg/m3, and a 0 ¼ 0.33 dB/cm/MHz2. The transducer, hydrophone, and castor oil container were submerged in a tank filled with de-ionized water as shown in Fig. 2. At the back wall of the tank, a 200 150 mm piece of absorptive material (Precision Acoustics Ltd., Dorchester, United Kingdom) was placed to reduce reflections. Dual frequency pulses were designed in MATLAB (The MathWorks, Inc., Natick, MA) with Eq. (1). The pulses were transferred to a Tektronix AFG 3102 Function Generator (Tektronix, Inc., Beaverton, OR), and amplified by a 150A100B RF Amplifier (Amplifier Research, Souderton, PA). The fundamental frequency f was selected to be 1.4 MHz, which is similar to what is currently used in commercial diagnostic ultrasound systems for abdominal nonlinear imaging. Consequently, the high frequency component of the pulse was 2.8 MHz. The envelope and duration of the dual frequency pulses were Ncyc ¼ 4 and m ¼ 2. Five combinations of b1 and b2 (weights of the primary frequencies of the pulse) were examined. Three cases were chosen, as shown in Fig. 3, so that at the focus their f and 2f components had the following relative amplitudes: (1) Pulse with 6 dB more f than 2f [b1 ¼ 0.60, b2 ¼ 0.40, as shown in Figs. 3(a) and 3(b)]. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
(2) Pulse with equal f and 2f [b1 ¼ 0.43, b2 ¼ 0.57, as shown in Figs. 3(c) and 3(d)]. (3) Pulse with 6 dB more 2f than f [b1 ¼ 0.27, b2 ¼ 0.73, as shown in Figs. 3(e) and 3(f)]. For comparison purposes, two single frequency pulses (f ¼ 1.4 MHz, b1 ¼ 1, b2 ¼ 0, and f ¼ 2.8 MHz, b1 ¼ 0, b2 ¼ 1) were also propagated at the focus. For each case, three pulses were sent with amplitudes of 1, 1, and 0.5. Finally, a GPIB-USB2 interface and LabVIEW (National Instruments, Austin, Texas) computer software were used to acquire the received pulses and transfer to a personal computer. Data analysis was performed with MATLAB software. Prior to the nonlinear propagation experiments, detailed characterization of the ultrasound field produced by the transducer was carried out. Low amplitude pulses were used for the transducer characterization to avoid nonlinear distortion. Measurements of the variation of the acoustic pressure along the source axis (propagation curve) and normal to the propagation axis at the focus (focal beam pattern) were collected with the membrane hydrophone in de-ionized water.
FIG. 3. Dual frequency pulses with different coefficients (b1, b2). The left column shows time waveforms and the right column shows the frequency spectra normalized with respect to the maximum value of each case. (a) and (b): A pulse with 6 dB more signal at low frequency; (c) and (d): A pulse with equal amplitude at both frequencies; (e) and (f): A pulse with 6 dB more signal at high frequency. C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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B. Nonlinear propagation in a tissue mimicking fluid
FIG. 4. Measurements (solid lines) and theoretical predictions (dashed lines) of the acoustic field of the source. (a) Axial field and (b) transverse field at the focus.
III. RESULTS A. Ultrasound transducer characterization
Figure 4 shows measurements of the acoustic field of the ultrasound source in water (solid line) compared with the linear and lossless analytical solution of the KZK equation (dashed line). The pressure along the source axis (propagation curve) is shown in Fig. 4(a), while the pressure normal to the propagation axis (beam pattern) at the focus of the transducer is shown in Fig. 4(b). The source pressure at low amplitude was calculated from the measurements by dividing the focal pressure by the focusing gain. At higher amplitudes linear extrapolation was used according to the input voltage. We have used the theoretical predictions to make small adjustments on the source dimensions due to small errors in the dimensions of the ceramic and the influence of the custom-made housing. We have thus calculated the effective source dimensions (Averkiou, 1994) which they differ by less than 5% of the original specifications (a ¼ 13.7 mm, d ¼ 80.2 mm). These effective dimensions have been used throughout this work. Good agreement is observed between measurements and theoretical predictions, whereas small discrepancies occur only near the transducer.
Three combinations of the weights (b1, b2) of the dual frequency pulse as explained in Sec. II D have been used. The case of a single frequency pulse f ¼ 1.4 MHz, b1 ¼ 1, b2 ¼ 0 is first shown in Fig. 5. The time domain of the pulses and their spectra are presented in Figs. 5(a)–5(f). Measurements are shown as solid lines and theoretical predictions as dashed lines. The PI waveform which resulted from adding the pulses of Figs. 5(a) and 5(c) is shown in Fig. 5(g). Its spectrum, Fig. 5(h), contains only nonlinear components at even harmonics (second, fourth, etc.). Good agreement is observed between theoretical predictions and measurements. Figures 5(i) and 5(j) present the waveform and spectrum of PMPI, respectively. To form PMPI, the pulses of Figs. 5(c) and 5(e) were used. It is observed that PMPI, despite that it has a strong second harmonic component, also contains the odd harmonic components [Fig. 5(j)]. The PM technique which resulted from the mixing of the pulses of Figs. 5(a) and 5(e) is shown in Figs. 5(k) and 5(l). This method also extracts both even and odd harmonic components in a somewhat similar fashion with PMPI. In the cases of PMPI and PM, discrepancies between theoretical and experimental results are present concerning the amount of nonlinear fundamental. We will further discuss this in Sec. IV. Measurements and theoretical predictions of nonlinear propagation of a dual frequency pulse are shown next. Figures 6(a)–6(f) present the waveforms and spectra of the propagated dual frequency pulses. As seen from the spectra, a pulse with equal contributions of the two primary frequencies at the focus was selected. The PI technique isolates only the even harmonics of the nonlinear signal when single frequency pulse is used. However, PI with a dual frequency pulse [Figs. 6(g) and 6(h)], has both even and odd harmonic
FIG. 5. Measurements (solid lines) and theoretical predictions (dashed lines) of harmonic generation of a tone burst (f ¼ 1.4 MHz). Time waveforms and spectra of full pulse (a) and (b), inverse pulse (c) and (d), half pulse (e) and (f), PI (g) and (h), PMPI (i) and (j), and PM (k) and (l). Spectra are normalized with respect to the maximum value of the measured pulse. 2548
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C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 6. Measurements (solid lines) and theoretical predictions (dashed lines) of harmonic generation of a dual frequency pulse with equal contributions at f, 2f. Time waveforms and spectra of full pulse (a) and (b), inverse pulse (c) and (d), half pulse (e) and (f), PI (g) and (h), PMPI (i) and (j), and PM (k) and (l). Spectra are normalized with respect to the maximum value of the measured pulse.
components including a nonlinear fundamental frequency component. Good agreement between theoretical and experimental results is observed. PMPI [Figs. 6(i) and 6(j)] and PM [Figs. 6(k) and 6(l)] methods formed by dual frequency pulses also produce both even and odd harmonic components at different levels. In Figs. 7 and 8, the effect of the relative amplitude of the two frequency components on nonlinear signal generation for every pulsing scheme is presented based on theoretical predictions. The horizontal axis of the figures denotes the amount of the low frequency on the pulse. Consequently the
weight of the high frequency will be b2 ¼ 1 b1. When b1 ¼ 1, the dual frequency pulse is reduced to a single frequency pulse with f ¼ 1.4 MHz. When b1 ¼ 0, the dual frequency pulse is reduced to a single frequency pulse with f ¼ 2.8 MHz. The vertical axis displays the nonlinear signal in dB for each pulsing scheme. These values are calculated by normalizing the resulting pulse of the pulsing scheme in discussion with respect to the largest measured pulse, and then calculating the spectrum. Theoretical predictions only have been used in these figures because they easily allow the inclusion of a much larger range of values of b1 and b2 and
FIG. 7. Effect of the relative amplitude of the two frequency components in the dual frequency pulse on nonlinear signal generation with pulsing schemes per harmonic component (theoretical predictions at r ¼ 1).
FIG. 8. Effect of the relative amplitude of the two frequency components in the dual frequency pulse on nonlinear signal generation with pulsing schemes per harmonic component (theoretical predictions at r ¼ 1.5).
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C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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examination of depths other than the focal plane with simple runs of the code (as opposed to repeating the experiment multiple times). In addition, KZK has been experimentally verified multiple times in the past and fair agreement with measurements was observed in the measurements collected here. An important finding is demonstrated in Fig. 7(a). PI with a dual frequency pulse produces a large amount of nonlinear fundamental (f ¼ 1.4 MHz) whereas with a single frequency pulse, PI cancels out all odd harmonics. PMPI also benefits at nonlinear fundamental (f) from the use of dual frequency pulses with a gain of about 5 dB. As it is also seen in Fig. 7(a), PI with a dual frequency pulse produces the most amount of nonlinear fundamental when compared to the other two methods (PMPI and PM). The second harmonic component (2f ¼ 2.8 MHz), shown in Fig. 7(b), is reduced for all pulsing schemes when a dual frequency pulse is used. It is observed that the maximum nonlinear signal is obtained at b1 ¼ 1, i.e., single frequency pulse. The signal loss of the second harmonic can be minimized without affecting all other nonlinear components (first, third, and fourth), by adding a higher contribution of the low frequency at the dual frequency pulse. Figures 7(c) and 7(d) show the third and fourth harmonic contributions, respectively. It can be seen that a larger amount of higher order harmonics is produced when a dual frequency pulse is utilized. It is also noticed that the PI method produces about 5 dB more signal at the third and fourth harmonic components compared to the other two techniques. The same trends as those of Fig. 7 are seen in Figs. 8(a)–8(d), which shows pulses propagated at larger depth (r ¼ 1.5) than the transducer’s focus. The nonlinear signal in dB detected by the pulsing schemes is lower at greater depth, but it is important to notice that the new proposed method still offers more signal at nonlinear fundamental, thus helping imaging for cases that lower frequency and penetration depth are required. It is also shown that PI continues to benefit from dual frequency pulses and contains a more nonlinear signal compared to the other two techniques. Figure 9 shows the spectra of all considered pulsing schemes in comparison to the full amplitude pulse, based on theoretical predictions. The single-frequency pulse [Fig. 9(a)] and a dual frequency pulse with equal weights at f and 2f [Fig. 9(b)] are shown. It is observed that for the case of a single frequency pulse [Fig. 9(a)], PI can fully extract the
FIG. 9. (Color online) Comparison of the spectra of the nonlinear signal detected with the various pulsing schemes for (a) the single and (b) the dual frequency pulse. 2550
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even harmonic components from the signal. PMPI spectrum contains both odd and even harmonics, with the even harmonics being a little lower than those of PI. Looking at the PM method, it extracts even components at much lower amplitude than the other two techniques, while it manages to preserve the same amount of odd harmonics as PMPI. However, for the case of a dual frequency pulse [Fig. 9(b)], odd and even harmonics are mixed up due to the influence of nonlinearities that arise from the sum and difference frequencies of f and 2f (2f f ¼ f, f þ 2f ¼ 3f, 2f þ 2f ¼ 4f, etc.). Sum and difference frequency components are considered to be even harmonic components, thus harmonic components for the case of dual frequency pulse, cannot be clearly separated to odd and even. We note that PI extracts the most amount of nonlinear signal at all harmonics. Interestingly we observe that PMPI and PM do not have the same amount of signal at f and 3f anymore which suggests that the signal at those frequencies is not solely an odd component. IV. DISCUSSION
This work evaluates the use of a dual frequency pulse in nonlinear imaging. The method relies on established multipulse schemes to isolate specific harmonic components. Experimental measurements and theoretical predictions were used to study in detail the differences between a dual frequency pulse and the conventional single frequency pulse, which is being used today in diagnostic ultrasound. The main objective of the present work was to examine the nonlinear generation of a high amplitude dual frequency pulse, using pulsing schemes but no in-depth consideration to imaging application was given. Comparisons between our measurements and the theoretical predictions based on the KZK equation showed very good agreement. Discrepancies were present only in the cases of PMPI and PM schemes, where the experiment exhibits a higher nonlinear fundamental than the simulation [Figs. 5(j), 5(l), 6(j), and 6(l)]. Similar differences were seen in previously published work (Mannaris, 2008). These discrepancies are mainly caused by the physical limitations of the equipment. While in theory the “half pulse” is the exact half amplitude of the first pulse, the experimental system may not be able to transmit the exact half voltage (and the half harmonic distortion) of the full amplitude pulse. Thus, linear signal cancellation is not perfect and experimental results at pulsing schemes that involve amplitude modulation (PMPI, PM) appear to have a higher nonlinear fundamental. For demonstration of the above issue, Fig. 10 shows pulsing schemes that were formed using very low amplitude (no harmonic signals present) pulses after propagation in water. While in PI [Fig. 10(a)] all linear signal is eliminated, in PMPI and PM [Figs. 10(b) and 10(c)] there is a nonlinear fundamental component present. Similar trends were observed when the electrical signal coming out of the power amplifier and before going to the transducer was recorded. Castor oil was selected as the medium since it was easy to handle, was homogeneous, and had acoustic parameters (density, speed of sound, and absorption) close to those of C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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FIG. 10. Spectra of measured pulses at the focus (solid line) and of linear signal suppression (dashed line) for (a) PI, (b) PMPI, and (c) PM.
tissue. Non-homogeneous media which are usually encountered in diagnostic ultrasound will probably affect the results presented here. The dual frequency pulse combined with pulsing schemes generates more harmonics over a larger bandwidth, compared to the normal single frequency pulse. The usage of the dual frequency pulse in diagnostic imaging may therefore enhance image esthetics and resolution (due to the increased harmonics). Harmonics can be used to create multi-band frequency compounding images, since frequency compounding imaging already demonstrated improved image quality (Mesurolle et al., 2007; Varray et al., 2012). In nonlinear propagation of sound beams, the fundamental component is the linear response and the harmonics comprise the nonlinear response. There is no nonlinear fundamental component in the propagation of a single pulse. The nonlinear fundamental component in PM (and PMPI) is the result of scaling and subtracting two pulses of unequal nonlinearity. It is only specific to multi-pulse processing. The nonlinear fundamental in PI when using a dual frequency pulse is simply the difference frequency. Nevertheless, it still generates a slightly more nonlinear fundamental component than single or dual frequency PM and PMPI (as shown in Figs. 7–9). All pulsing schemes studied in this work can be found today implemented on diagnostic ultrasound systems. In commercial diagnostic ultrasound systems, PI has demonstrated to have excellent resolution, while PM and PMPI offer better sensitivity. A unique advantage of the proposed method is the generation of a nonlinear signal at f with PI since with single frequency pulses only PM techniques (PM and PMPI) can detect the nonlinear fundamental component. In that way, imaging with dual frequency pulses may improve sensitivity while maintaining excellent resolution, at ultrasound systems that are capable of implementing PI, and not PM or PMPI. The dual frequency method may also be utilized for bubble imaging, using lower acoustic amplitude to avoid bubble destruction. The proposed method offers the ability to design bubble-specific pulses with the weights b1, b2, and the phase /. For example, the addition of / ¼ p/2 at the second frequency component, would result to a substantially different response from a microbubble. Moreover, the use of pulses that contain two frequencies (f, 2f) can excite a larger population of bubbles.
new proposed method can generate harmonics over a larger bandwidth. If the dual frequency pulse is utilized in diagnostic ultrasound it could lead to improvements in penetration depth and resolution. While with single frequency pulses, only PM and PMPI contained a nonlinear fundamental signal, with the dual frequency pulse, PI now contains a nonlinear fundamental signal. APPENDIX: THERMOVISCOUS ABSORPTION MEASUREMENTS IN CASTOR OIL
A plane circular single element transducer (a ¼ 3.175 mm, f ¼ 2.25 MHz) was used to measure the thermoviscous absorption of castor oil. Pressure measurements at two locations (greater than the Rayleigh distance, R0 ¼ ka2/2) on the axis of the source were recorded. Since the Rayleigh distance marks the end of the nearfield and the beginning of the far-field (Fig. 11), the acoustic pressure is spreading spherically (Blackstock, 2000). The absorption ða0 Þ was calculated using pressure measurements (P1, P2) at two known distances (r1, r2) (which were beyond 2R0 to ensure spherical spreading) using Eq. (A1), r1 P1 ln r2 P2 a0 ¼ : (A1) r2 r1 With our far-field axial pressure measurements and Eq. (A1) we have found that for castor oil a0 ¼ 0.33 dB/cm/MHz2. We also note that for castor oil the speed of sound and density (see Table I), and hence its impedance, is very close to that of water and reflections from the water-castor oil interface are thus significantly limited. Finally, castor oil is also an excellent choice for tissue mimicking fluid as it has similar values for speed of sound, density, and absorption.
V. CONCLUSIONS
A new nonlinear imaging technique based on a dual frequency pulse and pulsing schemes has been developed and evaluated for nonlinear tissue imaging. It was shown that the J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
FIG. 11. Acoustic field along the axis of a plane circular piston (a ¼ 3.175 mm, f ¼ 2.25 MHz). C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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TABLE I. Properties of castor oil. Castor Oil c0 (m/s) a0 (dB/cm/MHz2) q (kg/m3)
1521 0.33 920
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C. P. Keravnou and M. A. Averkiou: Dual frequency harmonic
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