Computer model for harmonic ultrasound imaging

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 47, no. 5, september 2000 1259

Regular Issue Papers

Computer Model for HarmonicUltrasound Imaging

Yadong Li and James A. Zagzebski, Associate Member, IEEE

Abstract—Harmonic ultrasound imaging has receivedgreat attention from ultrasound scanner manufacturers andresearchers. In this paper, we present a computer modelthat can generate realistic harmonic images. In this model,the incident ultrasound is modeled after the “KZK” equa-tion, and the echo signal is modeled using linear propaga-tion theory because the echo signal is much weaker than theincident pulse. Both time domain and frequency domain nu-merical solutions to the “KZK” equation were studied. Re-alistic harmonic images of spherical lesion phantoms weregenerated for scans by a circular transducer. This modelcan be a very useful tool for studying the harmonic buildupand dissipation processes in a nonlinear medium, and it canbe used to investigate a wide variety of topics related to B-mode harmonic imaging.

I. Introduction

“Tissue harmonic” imaging is being pursued ac-tively in clinical ultrasound. The modality appears

to provide better resolution and less noisy B-mode imagesthan standard, fundamental frequency imaging [1]. Similarto conventional medical ultrasound, in harmonic imagingpulsed sound waves are directed into the body, and echosignals from reflectors and scatterers are detected and usedto construct an image. With harmonic imaging, however,second harmonic echo signals, rather than fundamentalfrequency signals, are acquired and displayed. A demon-strated advantage of this modality appears to be reductionin acoustic noise resulting from reverberations and phaseaberrations when the incident sound pulse penetrates thepatient’s body wall and tissue structures. Harmonic imag-ing results in “cleaner” images with greater contrast, par-ticularly for large, overweight patients [1].Second harmonic echoes result from the gradual distor-

tion of high amplitude incident ultrasound pulses as theypropagate into the body [2]. The second harmonic compo-nent of the incident beam increases with increasing am-plitude and depth, usually peaking at a depth that is wellbeyond the patient’s body wall [2]. It is believed that thisgradual buildup of the harmonic component of the beamresults in less dependence of the displayed images on rever-

Manuscript received May 26, 1999; accepted February 25, 2000.This work was supported in part by National Institutes of Healthgrants R01-CA39224 and R42GM54377.

The authors are with the Department of Medical Physics,University of Wisconsin–Madison, Madison WI 53706 (e-mail:[email protected]).

berations, phase aberrations and other types of distortionsintroduced by the body wall of the patient.Computational models are being developed by re-

searchers to study the information content of second har-monic images and the relationship between acoustic prop-erties of tissues, scanner parameters, and harmonic imagecontent [3], [4]. Models also may help us to understandwhether it is possible to produce images of the nonlinearsound propagation characteristics of tissues (such as B/Aimages), a topic that is currently of high interest.The “KZK equation” [5], [6] is the most accurate model

equation to describe the combined effects of diffraction,nonlinearity, and absorption on ultrasound wave propa-gation. Currently there are two popular algorithms forsolving KZK equations: the spectral method introducedby Aanonsen and co-workers [7], [8] and the time domainmethod developed by Lee and Hamilton [9]. The spec-tral method subsequently was expanded to the transducerfar field [10], to pulsed conditions [3], and to rectangu-lar sources [11]. A Fourier series expansion of the soundpressure is substituted into the KZK equation, and theresulting set of coupled equations are solved using finitedifference methods. Cahill and Baker [12] have calculatedthe nonlinear field of a phrased-array using the frequencydomain method and have compared the calculated re-sults with experiment measurements. For the time domainsolution, Lee and Hamilton [9] use an operator-splittingscheme to solve for diffraction, nonlinear propagation, andabsorption separately. The time domain method is moreefficient when the propagation of short pulses is of con-cern because short pulses require a very large bandwidthto represent them in the frequency domain.Christopher [13] and Christopher and Parker [4] devel-

oped a method to calculate the nonlinear field that is notbased on the “KZK equation.” They use a spatial Fouriertransform method to calculate diffraction while applyingthe Burgers equation to model nonlinear effects. Theirmethod has been shown to be able to calculate the nonlin-ear field even in regions that do not satisfy the parabolicapproximation [13], which is incorporated into the KZKequations.Thus far, nonlinear beam models have been applied

mainly to predict distortions and harmonic generation fortransmitted ultrasound fields in water. The purpose of thework described in this paper is to develop a model for

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1260 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 47, no. 5, september 2000

producing harmonic images in medical ultrasound. Inci-dent pulsed beams are modeled using established nonlinearpropagation equations. Tissue-like attenuation coefficientsand realistic B/A values are included in the transmittedfield calculations. By introducing a model for a randomlyscattering medium, quantitative predictions are providedfor the strength of harmonic echo signals vs. depth; simu-lated harmonic images are generated for study of informa-tion content. Several applications of the model are givenas examples.

II. Method

B-mode imaging is a pulse-echo process. Because non-linear phenomena are significant only when the ultrasoundpressure is high, we treat the incident pulse and the echosignal in different ways. The incident pulse is modeledusing the full KZK equation. However, linear propaga-tion is assumed for echoes resulting from scattering in themedium.

A. KZK Equation and Boundary Conditions

We computed the beam for a single element focusedtransducer whose radius is a, emitting pulsed waves into amedium in which the infinitesimal sound speed is c0. Let αbe the linear coefficient of diffusion, {ρ, θ, z} be cylindricalspatial coordinates, and t be the time. The KZK equationfor an axisymetric source geometry is [9]:

∂2P

∂σ∂τ= αr0

∂3P

∂τ3 +r0

2ld∂2P 2

∂τ2 +14∇2⊥P (1)

where r0 = ω0a2/2c0 is the Rayleigh distance for the char-

acteristic frequency ω0; σ = z/r0 is the dimensionless axialcoordinate; τ = ω0(t− z/c0) is the dimensionless retardedtime; ∇2

⊥ =1ξ∂∂ξ

(ξ ∂∂ξ

)is the transverse Laplace operator,

where ξ = ρ/a is the dimensionless transverse coordinate;P = p/p0 is the pressure normalized to the source pressurep0; ld = p0c

30/βω0p0 is the shock wave formation distance

for a plane wave of frequency ω0 and amplitude p0; andβ = 1 + 1

2B/A is the coefficient of nonlinearity.The source condition for a weakly focused, axisymetric

beam can be written within the parabolic approximationas:

p = p0f

(t+

ρ2

2c0d

)D(ρ) at z = 0 (2)

where D(ρ) is the amplitude distribution over the source;f(t) is the driving pulse; and d is the focal distance. Interms of the dimensionless coordinates, the source condi-tion becomes:

P = f(τ +

r0

dξ2)D(ξ) at σ = 0. (3)

An untransformed coordinate system is used because us-ing a transformed coordinate system provides only a slightreduction in computation time for moderate focus gains(G = ω0a

2/2c0d < 20).

B. Frequency Domain Solution

The frequency domain solution follows the approach ofAanonsen et al. [7] and Berntsen and Vefring [14]. P canbe expanded by a Fourier series:

P (ξ, σ, τ) = RE

[ ∞∑n=1

Bn(ξ, σ)e−inτ]

(4)

= RE∞∑n=1

[hn(ξ, σ) + ign(ξ, σ)] e−inτ

(5)

where Bn, hn and gn are Fourier series coefficients, and REdenotes the real part of the complex quantity. Substituting(4) into (1) yields:

∂gn∂σ

= −n2αr0gn +14n

(∂2hn∂ξ2 +

1ξ

∂hn∂ξ

)+

nr0

2ld[12

n−1∑k=1

(gkgn−k − hkhn−k)−∞∑

l=n+1

(gl−ngl + hl−nhl)

]

∂hn∂σ

= −n2αr0hn −14n

(∂2gn∂ξ2 +

1ξ

∂gn∂ξ

)+

nr0

2ld[12

n−1∑k=1

(hkgn−k + gkhn−k) +∞∑

l=n+1

(hl−ngl − gl−nhl)

].(6)

Arbitrary frequency dependent attenuation can bemodeled by replacing the n2α factor by the actual attenua-tion coefficient α(f). The coupled equations (6) are solvedby finite difference schemes [8], [14].The boundary conditions gn and hn at σ = 0 are ob-

tained by applying a Fourier transform according to (3)and (4). In our implementation, 200 harmonic frequencieswere used to represent the pulse, where the base frequencyis 100 KHz. This implies an unrealistically high pulse rep-etition frequency of 100 kHz.

C. Time Domain Solution

The time domain solution to the KZK equation was firstdeveloped by Lee and Hamilton [9]. The KZK equation in(1) can be rewritten as:

∂P

∂σ= αr0

∂2P

∂τ2 +r0

2ld∂2P 2

∂τ2 +14

∫ τ

−∞∇2⊥P dτ ′ (7)

where the terms were defined previously.The three terms on the right side represent absorption,

nonlinear propagation, and diffraction respectively. Usingthe operator-splitting technique, these three terms can besolved separately. We followed Lee’s approach [9] for thenonlinear and diffraction terms, including a similar finitedifference step size. To model the f dependence of attenu-ation found in tissue, the absorption term was replaced bya convolution between the incident pulse and an attenua-tion filter function. Unlike the frequency domain solutions,extra computational overhead is required for modeling thef dependence of the attenuation in the time domain be-cause the convolution method is more costly than the fi-nite difference scheme for the simple attenuation term. To

li and zagzebski: model for harmonic ultrasound imaging 1261

reduce this overhead, the convolution can be applied incoarser spatial steps than the spatial step size of the dif-ference scheme for the nonlinearity and diffraction terms.In our case, the convolution was done for every 10 finitedifference spatial grid steps, corresponding to about 5 con-volution steps per millimeter. This is more than adequateto accurately describe attenuation effects.

D. Modeling the Echo Pulse

The echo signal is modeled using linear propagation the-ory, applying a frequency domain solution. Linear propa-gation also can be modeled in the time domain using theimpulse response function. However, in our case, the fre-quency domain solution was convenient, even when thetime domain KZK numerical solution was used for the in-cident pulse because the convolution used for modelingattenuation was implemented using a Fourier transforma-tion.After solving the KZK equation for the incident beam,

the frequency component Bn = hn + ign can be obtainedfor all grid locations of the finite difference scheme. Assum-ing linear propagation for the echo signal, the pulse-echoresponse detected by the same transducer, for a scattererlocated at {ρ, z}, can be written as [15]:

R(ρ, z, t) =14π

∞∑n=1

e−inω0tBn(ρ, z)An(ρ, z)Φ(nω0)(8)

where Φ(nω0) is a frequency-dependent factor related tothe backscatter coefficient; An(ρ, z) is the Rayleigh inte-gral:

An(ρ, z) =∫ a

0

∫ 2π

0

eikn| r− r′|

|#r − #r′|e−i

knρ′2

2d ρ′dθdρ′ (9)

where #r(ρ, θ, z) are the coordinates of the scatterer;#r′(ρ′, θ′, 0) is a point on the transducer surface; kn is the

wave number for the n-th harmonic frequency. e−iknρ′2

2d

is a parabolic phase factor accounting for focusing; d isthe focal distance. For convenience, the modeling of linearecho propagation is done using actual coordinates insteadof the dimensionless transformed coordinates.Within the parabolic approximation:

|#r − #r′ =[r2 + ρ′2 − 2ρρ′ cos(θ − θ′)

] 12

≈ r +ρ′2

2r− ρρ′ cos(θ − θ′)

r(10)

where r = |#r| =(ρ2 + z2

) 12 . With the help of (10), (9)

reduces to a 1-D integral:

An(ρ, z) = 2πeikr

r

∫ a

0J0

(kρ′ρ

r

)eik(

12r− 1

2d )ρ′2ρ′ dρ′

(11)

where J0 is the zero degree Bessel function, (11) canbe evaluated using numerical methods [16]. Because theparabolic approximation is assumed in the derivation ofthe KZK equation, the approximation made to reach (11)does not compromise the overall accuracy of the model.

E. Forming the Simulated Harmonic Image

Images are produced for a simulated phantom formedfrom spatially randomly distributed point scatterers [15].Echo signals from multiple scatterers are formed by addingup the pulse-echo responses due to all contributing scat-terers. Then B-mode images are formed by standard sig-nal processing techniques, including timed gain control(TGC), envelope detection, log compression, scan con-version, and gray scale mapping. The TGC compensatesfor attenuation by increasing amplification as echoes ar-rive from a deeper region. Linear translation of the trans-ducer is assumed to occur between pulse-echo sequences,in which the step size simulated is 0.2 mm. A Hamming-shaped bandpass filter is applied to extract the fundamen-tal and harmonic components from the simulated non-linear echo signal. The filter has a 6 dB bandwidth of1.5 MHz. Its center frequency is 2 MHz for fundamen-tal images and 4 MHz for second harmonic images. The4 MHz fundamental frequency images that also were sim-ulated applied a 1.5 MHz bandwidth bandpass filter, cen-tered at 4 MHz. This filter simulates the combined effectsof the transducer frequency response and any additionalfiltering that might take place in the scanner’s preprocess-ing stage. Various objects (such as cysts, spherical lesions,and line targets) can be constructed in the digital phantomby positioning and grouping the point-like scatterers.Artifacts may occur in the very near field because of

the limited step sizes and limited time duration retainedin the numerical method. Fig. 1 is an example of suchan artifact when the frequency domain method is usedin the computation. In the simulation of Fig. 1, attenua-tion and nonlinearity are ignored so that the waveformsfrom numerical solutions can be compared with the an-alytical solution on the axis. The current step size andtime duration we used in the frequency domain solution(∆σ = 2.5×10−4, ∆ρ = .025, ∆f = 0.1 MHz, T = 10µs)would be inadequate to accurately depict the waveform ata 2 cm depth. However, beyond this distance, such finitesteps and time durations yield accurate depiction. For ex-ample, Fig. 1 (bottom) compares the analytical and nu-merical solutions at 4 cm from the transducer. If the nearfield is of interest, this artifact can be alleviated by usinga smaller finite step size and longer time duration in thenumerical solution. However, care must be exercised be-cause the KZK equation itself is less accurate in the nearfield due to its parabolic nature [3]. In this paper, simu-lated images start at a depth of 2 cm and assume a 2 cmdiameter source, effectively avoiding most of the numeri-cal artifacts. The choice of the start depth depends on thesize of the transducer, the finite difference steps used, andpulse duration retained in the simulation.

III. Results

Typical results obtained using this model are presented.The beam and images for a circular transducer with a di-ameter of 2 cm and a focal distance of 7 cm were simulated.


Fig. 1. Artifacts caused by limited finite difference steps and limited time duration retained in the numerical solution. The waveform is thetransmitted pulse at a point on the axis at depths of 2 cm from a 20 mm diameter transducer. Nonlinearity and attenuation are neglected inthe numerical solution for the purpose of comparing with a linear (analytical) solution. The numerical solution uses the frequency domainapproach with 200 harmonic terms, ∆f = 100 kHz, ∆σ = 2.5 × 10−4 and ∆ρ = .025. (bottom) Same as top but for an axial distance of4 cm.

The driving pulse in harmonic imaging was assumed to beGaussian shaped, with a center frequency of 2 MHz anda 6 dB bandwidth of 75%. Driving pulses producing peakamplitudes of 200 kPa and 1 MPa at the transducer sur-face were assumed. The speed of sound was assumed tobe 1540 m/s, the attenuation coefficient was taken to be0.5 dB/cm/MHz, and the B/A value was 8. These valuesare close to those of fatty liver.

A. Beam Patterns and Profiles

Fig. 2 presents a vivid depiction of the incident beampatterns for both the fundamental and harmonic compo-nents for both pulse amplitudes. Each panel representsthe pulse throughout a plane 6 cm wide and 14 cm deep.This plane includes the axis of symmetry of the source andbeam. The grayscale represents the log-compressed ampli-tude of the pulse. Each image is normalized to its own max-imum amplitude so that the brightness reflects the beampatterns themselves. However, the relative brightness doesnot reflect the relative strength between the fundamentaland harmonic components. The dynamic range depictedis 30 dB.

The two lower panels are for fundamental frequencybeams of 2 MHz [Fig. 2(C)] and 4 MHz (Fig. 2D). Theupper panels are the filtered 4 MHz harmonic componentof the 2 MHz fundamental, obtained by applying a band-pass filter centered at 4 MHz. As can be seen in Fig. 2, theharmonic component [Fig. 2(A) and (B)] builds in strengthover the first 2 cm in this case. Build-up is more rapid witha 1 MPa incident pulse [Fig. 2(B)] than the 200 kPa pulse[Fig. 2(A)], as expected. The harmonic component reachesits peak around the focal zone of the transducer, then grad-ually dissipates over the far field. The fundamental 4 MHzbeam [Fig. 2(D)] is narrowest over the focal zone, but ithas less penetration due to its higher attenuation.

B. Pulse Distortion

Fig. 3 shows the gradual distortion of the 2 MHz 1 MPainitial pulse during nonlinear propagation. Waveforms andspectral contents are shown for points on the axis of thetransducer, at distances ranging from 0 cm, 6 cm, and14 cm. The second harmonic component accumulates andreaches its maximum around 6 cm depth, then graduallydissipates over distance due to attenuation in the tissue-like medium.


Fig. 2. Computed beam patterns in tissue-like material for a 20 mm diameter 7 cm focal depth transducer. Grayscale pixel values representthe log-compressed pressure amplitude, in which the displayed dynamic range is 30 dB. (A) 4 MHz second harmonic component for a2 MHz center frequency, 200 kPa transmit pulse; (B) 4 MHz second harmonic component for a 2 MHz, 1 MPa transmit pulse; (C) 2 MHzfundamental; (D) 4 MHz fundamental.


Fig. 3. Calculated on-axis waveforms and their spectra for a 2 MHz, 1 MPa transmit pulse at 0 cm (initial pulse), 6 cm, and 14 cm depths.The initial pulse has a 6 dB bandwidth of 75%. The circular transducer modeled has a 2 cm diameter and a 7 cm focal distance.

Fig. 4 demonstrates similar waveforms when a techniquecalled “pulse inversion harmonics” is used. The “pulse in-version technique” applies a transmit pulse, then its in-verse into the medium [17]. Resultant echo signals areadded to form a superposed response. Waveforms in Fig. 4were obtained simply by summing the original transmitpulse with the inverted transmit pulse at each depth. Theexamples show that the pulse-inversion technique is veryeffective in suppressing the fundamental and odd harmoniccomponents of the echo signal, while retaining the sec-ond and even harmonic components. With the fundamen-tal component suppressed, there is little overlap betweenthe second harmonic and the subharmonic, or even higher

harmonic components. This makes it much easier to ex-tract the second harmonic component. The superposedwaveforms look irregular because of the reinforced subhar-monic, and the fourth harmonic component. (We assumeda flat receiving frequency response in this calculation toemphasize this acoustical phenomenon.)

C. Simulated Images

Figs. 5–7 are simulated images of a “spherical lesionphantom” [18]. The lesion phantom is simulated using ran-domly positioned scatterers in the background medium.However, scatterers inside lesions are assumed to have


Fig. 4. Calculated on-axis “pulse inversion” waveforms and their spectra for a 2 MHz, 1 MPa transmit pulse at 0 cm, 67 cm, and 14 cmdepths. Waveforms are produced by adding waveforms when the transmit pulse and its negative are numerically propagated.

negligible backscattering strength to simulate “sphericalvoids.” The frequency dependence of the backscatteringin the background material is assumed to be ∝ f , which issimilar to liver. The spherical lesions are 4 mm in diameter.Fig. 5 presents fundamental images for a 2 MHz and a

4 MHz transmit pulse, assuming linear propagation. Theexpected improved spatial detail at 4 MHz is evident, eventhough only a 50% transmitted pulse bandwidth was ap-plied in this case. Fig. 6 presents simulated second har-monic images of the phantom, obtained by computing thenonlinear incident pulse and applying a bandpass filter toextract the 4 MHz second harmonic component of the echo

signals. In Fig. 7 are harmonic images simulated using the“pulse inversion” technique. Again, a 4 MHz bandpass fil-ter was applied to extract the second harmonic compo-nent. All images are constructed with 200 acoustic lines;the width of each image is 4 cm and the vertical rangeextends from 2 cm to 13 cm.When comparing the images in Figs. 5–7, it is evident

that the 4 MHz harmonic images (Figs. 6 and 7) have bet-ter resolution and contrast than the 2 MHz fundamentalimages. This fact also is seen when imaging spherical lesionphantoms with clinical scanners [18]. The results suggestthat the “pulse inversion” technique slightly improves im-


Fig. 5. Simulated fundamental frequency images of phantom containing 4 mm diameter low-scatter spheres. The computation assumes ahorizontal translation of the beam axis between pulse-echo sequences. Image dimensions are 4 cm × 10.5 cm. (A) 2 MHz fundamental; (B)4 MHz fundamental.

age quality over the simple filtered version of the harmonicimage, particularly for the most shallow lesions seen. Thesecond harmonic image (Figs. 6 and 7) has resolution andcontrast comparable to that of the 4 MHz fundamentalimage [Fig. 5(B)]. However, the image speckle pattern isdifferent because the 50% bandwidth 4 MHz pulse usedin Fig. 5(B) is shorter in time duration than that of the2 MHz fundamental pulse and the 4 MHz harmonic signal.To show the versatility of this model, we also included

simulated harmonic images using a 1 MPa 35% bandwidth,2 MHz pulse. Fig. 8(A) is the filtered harmonic image;Fig. 8(B) is the pulse inversion harmonic image. Comparedwith the 75% bandwidth counterparts (Figs. 6 and 7), theshorter bandwidth results in a “blobbier” speckle pattern.Fig. 9 plots the relative strength of the echo amplitude

averaged over all beam lines of the 2 MHz fundamental sig-nal and its second harmonic signal. Fig. 9 clearly shows theharmonic signal gradually building up during propagationand peaking at a depth deeper than that of the fundamen-

tal. After the peak, the harmonic signal strength decreasesat a higher rate because of a higher attenuation coefficient.The relative amplitude of the harmonic signal componentto the fundamental may be of interest in equipment designand evaluation.

D. Reverberation Effects

It is generally believed that harmonic imaging offersbetter image quality because it is less susceptible to rever-berations and phase aberrations than fundamental modeimaging [1]. Reverberations are caused in part by reflectionbetween tissue boundaries and the surface of the trans-ducer. To illustrate that harmonic imaging is less suscep-tible to artifacts caused by interfaces, a simulation studywas performed using the computer model.Reverberations degrade image quality for several rea-

sons. First, reverberation echoes from overlying tissues willbe superimposed with echoes from deeper tissues, reducing


Fig. 6. Simulated second harmonic images resulting from 2 MHz transmit pulses. Initial pulse amplitudes are 200 kPa (A) and 1 MPa (B).

contrast. Second, if an echo from an overlying structure isof sufficient amplitude, its reflection off the transducer pro-duces a lagged pulse, trailing the primary incident pulse.If the lagged pulse is of sufficient magnitude, echoes fromit can cause degradation of image quality throughout thefield.

We simulated harmonic imaging when the second effectwas present. We assumed that a strongly reflecting inter-face produces a lagged pulse with an amplitude that is20% that of the main pulse traveling into the tissue. Thisassumption is on the strong side, considering the reflectioncoefficient for a muscle/fat interface is about 0.1. However,in the simulation, only one interface is assumed; in real-ity, there can be multiple interfaces causing this “acousticnoise.” In addition, in the simulation, we only consideredthe effects of the incident pulse, in reality, additional acous-tic noise would be produced by echoes. Moreover, if therewere bones or hard skin layers in the field, they would bemuch more reflective than a muscle/fat layer. Therefore,we deem this a reasonable assumption to illustrate the ef-

fects of image quality degradation caused by acoustic noiseand the benefits of harmonic imaging.To add the effects of a reverberation pulse, it was as-

sumed that the highly reflective interface was 1 cm fromthe transducer surface and that the reflection coefficientfor the tissue-to-transducer interface was 1. This delayed,new pulse was then modeled by applying the KZK equa-tion once again. The incident beam at interfaces and scat-terers in the model consisted of both the primary beamand the lagged pulse.Fig. 10 is the calculated incident signal along with the

lagged pulse, at a depth of 6 cm. Also shown are its funda-mental and harmonic components. The noise signal for theharmonic has a much lower amplitude relative to that ofthe fundamental. The reason is that harmonic generationis roughly proportional to the square of the fundamentalamplitude; when the reverberation pulse amplitude is 20%of the main pulse, its harmonic signal is only on the orderof 4% of the harmonic signal generated by the main pulse.Fig. 11 is a simulated 4 MHz fundamental image and


Fig. 7. Second harmonic images of the same phantom as in Figs. 5 and 6 using the pulse inversion method. A bandpass filter centered at4 MHz is applied to extract the second harmonic component of the echo signals. (A) 200 kPa transmit pulse; (B) 1 MPa transmit pulse.

a harmonic image of the spherical lesion phantom, withthe lagged pulse effects included. It is clear that, in thepresence of acoustic noise, the harmonic image [Fig. 11(B)]has better contrast at depth than the fundamental image[Fig. 11(A)].

IV. Discussion

Mathematical models can be useful in ultrasound imag-ing to help understand the relationship between image fea-tures and acoustic properties of tissues. This paper buildson previous researchers work on nonlinear wave propa-gation, computing pulsed waveforms for media exhibitingtissue-like acoustic properties. Harmonic images of tissue-like media are simulated by applying nonlinear theory tocompute the incident pulse and linear theory to computesubsequent echoes arising from randomly distributed scat-terers. Simulated images exhibit improved resolution overfundamental images, as seen experimentally when compar-ing harmonic and fundamental images generated by clin-ical scanners when imaging “spherical lesion” phantoms.The fact that harmonic images can exhibit improved con-

trast in the presence of acoustic noise also is realized withthe model by introducing acoustic noise such as would orig-inate from a strongly reflecting interface.The typical time to solve the nonlinear field for a sin-

gle element transducer is about 2 hours on a DEC alpha500 MHz 21164 workstation when the time domain solu-tion is used. Most of the computation time is spent solvingthe KZK equation numerically. Once nonlinear propaga-tion for a certain combination of transducer and mediais solved and stored to file, images of different phantomshaving the same attenuation can be obtained very quickly.Each additional image takes about a half hour to compute.Both the frequency domain and the time domain meth-

ods are viable solutions to the KZK equation. The timedomain method has an advantage when dealing with highamplitude, short pulses because such pulses generate verystrong harmonic components, and this condition requiresmore harmonic terms in the numerical scheme when car-ried out in the frequency domain. Solving the nonlinearpart of the KZK equation in the frequency domain hasa complexity of order O(M2), where M is the number ofharmonics retained. Therefore, the computation time us-


Fig. 8. Simulated harmonic images using a 2 MHz transmit pulse having a 35% bandwidth. (A) 200 kPa transmit pulse; (B) 1 MPa transmitpulse.

Fig. 9. Fundamental (2 MHz) and second harmonic echo amplitude vs. depth for a 200 kPa transmit pulse and a 1 MPa transmit pulse.The solid line is the fundamental frequency mean echo strength, and the dashed line the harmonic mean echo strength.


Fig. 10. (Top) Calculated transmit pulse at a 6 cm depth, with added acoustic noise such as caused by reverberation. (Middle) Fundamental,2 MHz component. (Bottom) Harmonic component.


Fig. 11. Simulated fundamental and harmonic images with acoustic noise as in Fig. 10 included. (A) 2 MHz fundamental image; (B) 2ndharmonic image.

ing the frequency domain method grows very rapidly withM . The time domain numerical method described by Leeand Hamilton [9] has a complexity of O(M) for diffrac-tion, nonlinearity, and absorption. However, the originalversion of that method only deals with a f2 dependenceof the absorption on frequency. The convolution methodused to model the f dependence of the attenuation has acomplexity ofM log(M) if the convolution is implementedusing a Fourier transform. In our implementation, convolu-tion was applied only for every 10 grid steps for which thenonlinearity and diffraction steps are applied; this helpsreduce the computation cost to implement tissue-like at-tenuation.

V. Conclusions

The computer model presented here is capable of gen-erating realistic harmonic images. Simulated harmonic im-ages demonstrate the enhanced resolution and better im-age contrast over fundamental images, in agreement withclinical observations. This model is reasonable in terms ofcomputational cost. It offers flexibility in dealing with dif-

ferent levels of frequency-dependent attenuation, backscat-tering, and dispersion.The model can be useful in many areas. For example, it

can be applied to study the impact of fundamental factorsaffecting harmonic imaging, such as the B/A value, atten-uation, backscattering, driving pressure, and transducergeometry. It also can be used to study how to improve theresolution, contrast, and penetration of harmonic images.Further work in developing harmonic imaging models forarray transducers is underway.

References

[1] Yadong Li and James A. Zagzebski, Associate Member, IEEE,James A. Zagzebski, Associate Member, IEEE, F. Tranquast, N.Grenier, V. Eder, and L. Pourcelot, “Clinical use of ultrasoundtissue harmonic imaging,” Ultrason. Med. Biol., vol. 25, pp. 889–894, 1999.

[2] G. Muir and E. L. Carstensen, “Prediction of nonlinear acous-tic effects at biomedical frequencies and intensities,” UltrasoundMed. Biol., vol. 6, pp. 341–344, 1980.

[3] A. C. Baker, K. Anastasiadis, and V. F. Humphrey, “The non-linear pressure field of a plane circular piston: Theory and ex-


periment,” J. Acoust. Soc. Amer., vol. 84, pp. 1483–1487, 1988.[4] T. Christopher and K. J. Parker, “New approaches to nonlinear

diffractive field propagation,” J. Acoust. Soc. Amer., vol. 90, pp.488–499, 1991.

[5] E. A. Zabloskaya and R. V. Khoklov, “Quasi-plane waves in thenonlinear acoustics of confined beams,” Sov. Phys. Acoust., vol.15, pp. 35–40, 1969.

[6] V. P. Kuznetsov, “Equations of nonlinear acoustics,” Sov. Phys.Acoust., vol. 16, pp. 467–470, 1970.

[7] S. I. Aanonsen, T. Barkve, and J. N. Tjøtta, “Distortion andharmonic generation in the nearfield of a finite amplitude soundbeam,” J. Acoust. Soc. Amer., vol. 75, pp. 749–768, 1984.

[8] S. I. Aanonsen, “Numerical computation of the nearfield of afinite amplitude sound beam,” Tech. Rep. 73, Department ofMathematics, University of Bergen. Bergen, Norway, 1983.

[9] Y.-S. Lee and M. F. Hamilton, “Time-domain modeling of pulsedfinite-amplitude sound beams,” J. Acoust. Soc. Amer., vol. 97,pp. 906–917, 1994.

[10] M. F. Hamilton, J. N. Tjøtta, and S. Tjøtta, “Nonlinear effects inthe farfield of a directive sound source,” J. Acoust. Soc. Amer.,vol. 78, pp. 202–216, 1985.

[11] T. Kamakura, M. Tani, Y. Kumamato, “Harmonic generationin finite amplitude sound beams from a rectangular aperturesource,” J. Acoust. Soc. Amer., vol. 91, pp. 3144–3151, 1992.

[12] M. D. Cahill and A. C. Baker, “Numerical simulation of theacoustic field of a phased-array medical ultrasound scanner,” J.Acoust. Soc. Amer., vol. 104, pp. 1274–1282, 1998.

[13] T. Christopher, “Experimental investigation of finite amplitudedistortion-based, second harmonic pulse echo ultrasonic imag-ing,” IEEE Trans. Ultrason., Ferroelectric., Freq. Contr., vol.45, pp. 158–162, 1998.

[14] J. Berntsen and E. Vefring, “Numerical computation of a finiteamplitude sound beam,” Tech. Rep. 81, Department of Mathe-matics, University of Bergen. Bergen, Norway, 1986.

[15] Y. Li and J. A. Zagzebski, “A frequency domain model for gen-erating b-mode images with array transducers,” IEEE Trans.Ultrason., Ferroelect., Freq. Contr., vol. 46, pp. 690–699, 1999.

[16] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numer-ical Recipes in C. Second ed. Cambridge, MA, Cambridge Univ.Press, pp. 236–252, 1992.

[17] D. H. Simpson, C. T. Chin, and P. N. Burns, “Pulse inversiondoppler: A new method for detecting nonlinear echoes from mi-crobubble contrast agents,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr., vol. 46, pp. 372–382, 1999.

[18] J. Rownd, E. Madsen, J. Zagzebski, G. Frank, and F. Dong,“Phantoms and automated system for testing resolution of ul-trasound scanners,” Ultrasound Med. Biol., vol. 23, pp. 245–260,1997.

Yadong Li received his B.S. in electrical en-gineering from Beijing University, China, in1995. Currently, he is pursuing the Ph.D. de-gree in physics and a M.S. degree in com-puter science at the University of Wisconsin-Madison.

His research interests include ultrasoundimaging, B-mode image texture analysis, andcomputer modeling in ultrasound.

James A. Zagzebski (A’89) was born inStevens Point, WI, in 1944. He received theB.S. degree in physics from St. Mary’s Col-lege, Winona, MN, and the M.S. degree inphysics and the Ph.D. degree in radiologi-cal sciences from the University of Wisconsin,Madison. He is Professor of medical physicsand of radiology and human oncology at theUniversity of Wisconsin.

His research interests include ultrasoundimaging and tissue characterization, flow de-tection and visualization using ultrasound,

and technological assessment of imaging devices.Dr. Zagzebski’s professional affiliations include the IEEE, the

American College of Radiology, the American Institute of Ultrasoundin Medicine, and the American Association of Physicists in Medicine.

Documents

Computer model for harmonic ultrasound imaging