Ultrasound Coefficient of Nonlinearity Imaging - IEEE Xplore

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL ,

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Ultrasound Coefficient of Nonlinearity Imaging Ruud van Sloun, Libertario Demi, Member, IEEE, Caifeng Shan, Senior Member, IEEE, and Massimo Mischi, Senior Member, IEEE Abstract—Imaging the acoustical coefficient of nonlinearity, β, is of interest in several healthcare interventional applications. It is an important feature that can be used for discriminating tissues. In this paper, we propose a nonlinearity characterization method with the goal of locally estimating the coefficient of nonlinearity. The proposed method is based on a 1-D solution of the nonlinear lossy Westerfelt equation, thereby deriving a local relation between β and the pressure wave field. Based on several assumptions, a β imaging method is then presented that is based on the ratio between the harmonic and fundamental fields, thereby reducing the effect of spatial amplitude variations of the speckle pattern. By testing the method on simulated ultrasound pressure fields and an in vitro B-mode ultrasound acquisition, we show that the designed algorithm is able to estimate the coefficient of nonlinearity, and that the tissue types of interest are well discriminable. The proposed imaging method provides a new approach to β estimation, not requiring a special measurement setup or transducer, that seems particularly promising for in vivo imaging.

I. Introduction

I

maging the nonlinear propagation of ultrasound through tissue is of interest in several healthcare interventional applications. The acoustical coefficient of nonlinearity, β, characterizing the nonlinear propagation of ultrasound through tissue, is an important feature that may be exploited for discriminating tissues [1]. Furthermore, tissue nonlinearity is affected by temperature [2], [3]. A β-imaging modality might therefore also be considered for estimating temperature variations. A promising application area for β imaging would be cardiac ablation therapy, which is increasingly used to treat persistent atrial fibrillation (AF). Cardiac ablation relies on the formation of lesions when tissue is continuously exposed to heat. Although cardiac ablation has proven to be effective for treating AF [4], it also carries serious risk. Esophageal injury is a potential life-threatening complication after ablation of the left atrium (LA) [5]. The nonuniform thickness of the posterior LA wall and the variable fibrofatty layer between the wall and the esophagus are risk factors that must be considered during

Manuscript received January 27, 2015; accepted April 18, 2015. This work was supported by the European Research Council Starting Grant (#280209). R. van Sloun, L. Demi, and M. Mischi are with the Laboratory of Biomedical Diagnostics, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands (e-mail: [email protected]). R. van Sloun was with and C. Shan is with Philips Research Eindhoven, High Tech Campus, 5656 AE Eindhoven, The Netherlands. DOI http://dx.doi.org/10.1109/TUFFC.2015.007009

the ablation procedure [6]. Furthermore, from the anatomic viewpoint, it is not possible to predict or standardize where the safe ablation lines lie, because the esophagus is mobile in the living patient. Temperature monitoring using an esophageal temperature probe may reduce complications [7], but it does not provide any insight into the anatomical structure. The potential ability to discriminate tissue layers using β imaging with an intracardiac ultrasound (US) probe would allow quantification of the atrial wall thickness and risk assessment with respect to esophageal injury by detecting the fibrofatty layer. The coefficient of nonlinearity β is a particularly suitable feature for discriminating the tissues relevant to this application, because fatty tissues are known to have a high coefficient of nonlinearity [1]. In addition to the possibility of employing β for tissue discrimination, using it to obtain information about temperature variations might also be possible [8]. Monitoring temperature variations is not only of interest during cardiac ablation; it could also be utilized during, e.g., high-intensity focused ultrasound and hyperthermia for cancer treatment. The possible alternative application of β estimation to temperature estimation is, however, not discussed in detail in this article, because tissue characterization is the main focus. Estimation of the coefficient of nonlinearity in a clinical setting is most applicable using a single probe in pulseecho mode. In [9], a β estimation method based on a highfrequency probe and pump waves propagating in opposite directions was suggested. However, requiring an acoustic reflector plane limits its practical use. Other possibilities for coefficient of nonlinearity imaging in echo mode have been studied in [10], where extensions of the direct [11], comparative [12], and second-order ultrasound field (SURF) [13] methods are derived such that they are applicable to echo-mode imaging. The direct method is the easiest, based on the second-harmonic amplitude propagation theory proposed by Zhang and Gong [14]. The SURF method relies on multi-frequency transmission requiring a special transducer and the comparative method requires a particular setup in which propagation through an inhomogeneous medium is compared with a known homogeneous reference. The authors in [10] conclude that the comparative method, and the extension that compensates for inhomogeneous attenuation, is the most promising and an alternative application based on high-frame-rate compounding of plane wave transmissions has been presented in [15]. However, neither the SURF nor the extended comparative methods are directly feasible for in vivo US

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imaging. The fact that the comparative method requires the presence of an area where the coefficient of nonlinearity is constant throughout the entire imaging depth poses serious limitations for the method’s translation into clinical application. Also, inaccurate a priori knowledge of the attenuation coefficient in the spatial domain leads to cumulative errors in nonlinearity estimation. In this paper, we propose a tissue nonlinearity imaging method that provides a new and local approach to β estimation, reducing cumulative errors. The influence of scatterer distribution variations is also addressed by considering the ratio between the harmonic and fundamental fields. The proposed method is based on a 1-D solution of the nonlinear lossy Westervelt wave equation and does not require a special measurement setup or transducer, making it particularly promising for in vivo imaging. Its feasibility is shown by a detailed in silico analysis as well as initial in vitro results. The paper is organized as follows. First, the physical background of nonlinear wave propagation through human tissues is given (Section II-A). Then, a method for extracting the coefficient of nonlinearity β from both the complete pressure wave fields (Section II-D) and echomode images (Section II-E) is derived. Next, to assess its performance, pressure wave propagation is numerically analyzed in silico, based on several forms of the wave equations. The method is then also tested on an experimental in vitro measurement. The numerical and experimental results are given in Sections III-A and III-B, respectively. Finally, the results are discussed and conclusions derived (Section IV). II. Methods A. Physical Background of Nonlinear Wave Propagation As sound waves propagate through tissue, they distort because of medium-dependent variation of the speed of sound with respect to pressure, quantified by the coefficient of nonlinearity, β. Higher pressure results in an increased speed of sound whereas lower pressure results in a reduced speed of sound. This distortion results in the accumulation of harmonics [16]–[18]. Table I gives β for some human tissues of interest. The attenuation coefficient α also is given, describing the attenuation of a pressure wave with respect to depth as a result of medium-dependent absorption. For the sake of clarity, α is given both in units of dB/(MHzy × cm) and units of Np/((rad/s)y × m) because the latter can be directly substituted into the equations. Comparing the coefficient of nonlinearity β to the attenuation coefficient α, one can observe that the relative differences in β for these tissues are higher, making it an interesting candidate for tissue characterization. Acoustic pressure wave propagation can be compactly described in the form of wave equations, combining conservation of mass and momentum. In this section, we will first give an expression for the lossy nonlinear wave equa-

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TABLE I. Properties of Some Relevant Tissues as Described by [23, Appendix B]. Tissue

α [dB/(MHzy∙cm)]

α [Np/((rad/s)y∙m]

y

β

0.6 0.52 0.57

1.10∙10−6

1* 1* 1*

6.1 3.9 4.7

Fat Heart† Muscle

0.95∙10−6 1.04∙10−6

*Assumed value. †Heart mainly consists of muscle and blood.

tion. The modification from the lossless linear wave equation is made explicit in the form of distributed sources S. B. Lossy Nonlinear Wave Equation The general lossy 3-D Westervelt wave equation [18, ch. 3.6] may be written as

∇ 2p(x, t) −

1 2 ∂ t p(x, t) = −S at[p(x, t)] − S nl[p(x, t)], (1) c 02

where

S nl[p(x, t)] =

β(x) 2 2 ∂ t p (x, t) (2) ρ 0c 04

denotes the nonlinear source [19]. The effects of attenuation, represented by Sat[p(x,t)], can be modeled by explicitly introducing relaxation in the medium behavior. Then, the following attenuation source is obtained [20]:

S at[p(x, t)] = −

1 2 ∂ t [A(x, t) ∗t p(x, t),] (3) c 02

where ∗t denotes the time-domain convolution and A(x,t) describes the delayed response of the medium—its lowpass filtering property. Attenuation through tissue is assumed to obey a power-law dependence on frequency [2], [21]. As suggested in [19] and [22], a suitable representation of A(x,t) can be given in the Laplace domain: 2

 c 0α 0(x)s y(x)−1  ˆ(x, s) = 1 + − 1, (4) A d   cos[πy(x)/2][1 + (s/s max) ] 

where α0(x) is the attenuation coefficient in units of Np∙(rad/s)−y∙m−1, ω is the angular frequency in radians per second, y(x) is the power law exponent (which may not be an odd integer), and the factor [1 + (s/smax)d] ensures causality and finite wave speed, with smax being a parameter larger than the maximum angular frequency of interest and d being a parameter that is larger than y(x) − 1. The propagation coefficient γˆ(x, s) is then given as

ˆ x, s) = γ(

s c0

ˆ(x, s) + 1. (5) A

After setting s = jω, we observe that the function γˆ(x, ω) consists of an attenuation and phase coefficient that can be approximated by [19]

van sloun et al.: ultrasound coefficient of nonlinearity imaging

ω π   γˆ(x, ω) ≈ α 0(x)|ω|y(x) +j  + a 0(x) tan  y(x)  ω|ω|y(x)−1  ,   c 2    0 (6) where the second term of the imaginary part describes the effects of dispersion. C. Solution Using Green’s Function The solution of the lossy nonlinear wave equation, with inclusion of a primary source term Spr(x,t) [19] that models the effects of the transducer, can be formulated as an integral equation:

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we can reformulate (7): p(x, t) = p (0)(x, t) + G(x, t) ∗ x,t S nl[p(x, t)]

(13) + G(x, t) ∗ x,t S at[p f(x, t) + p h(x, t)],

and, realizing that the fundamental field can be approximated by a combination of the lossless linear field p(0)(x,t) and the attenuation source, assuming that energy loss of the fundamental component due to harmonic generation is negligible with respect to loss due to absorption, i.e., p f(x, t) ≈ p (0)(x, t) + G(x, t) ∗ x,t S at[p f(x, t)], (14)

and exploiting linearity of the attenuation contrast source, p(x, t) = G(x, t) ∗ x,t {S pr(x, t) + S nl[p(x, t)] + S at[p(x, t)]}. we obtain (7) β(x ) 2 2 p h(x, t) = G(x, t) ∗ x,t ∂ t p (x, t) Here, ∗ x,t denotes the convolution over space and time and ρ 0c 04 (15) G(x,t) is the Green’s function of the lossless linear wave 1 2 − G(x, t) ∗ x,t 2 ∂ t {A(x, t) ∗t p h(x, t)}. equation, given by c0 δ (t − c ) 0 , (8) 4π||x|| ||x||

G(x, t) =

Using (9) and exploiting the commutative property of the convolution, we obtain

where the || ∙ || operator gives the length of a vector and δ(∙) denotes the Dirac delta function. In 1-D, one can show that the Green’s function is given by

G(x, t) =

c0  |x |  H  t −  , (9) 2  c0 

c0  |x |  β(x ) 2 2 H  t −  ∗ x,t ∂ t p (x, t) 2  c0  ρ 0c 04 |x |  1 c  − 0 H  t −  ∗ x,t 2 ∂ t2{A(x, t) ∗t p h(x, t)}   2 c0 c0 (16) 1 |x |    t −  ∗ x,t β(x )∂ t p 2(x, t) = δ c0  2ρ 0c 03 

p h(x, t) =

with H(∙) being the Heaviside step function. The 1-D version of (7) can be written as

−

|x |  1  δ  t −  ∗ x,t {∂ tA(x, t) ∗t p h(x, t)}. 2c 0  c0 

p(x, t) = p (0)(x, t) + G(x, t) ∗ x,t { S nl[p(x, t)] + S at[p(x, t)]} , (10) Then, considering x ≥ 0 and working out the spatiotemporal convolution integrals leads to where x 1 p h(x, t) = β(x ′)∂ t p 2(x ′, t)dx ′ 3 0 (0) 2ρ 0c 0 p (x, t) = G(x, t) ∗ x,t S pr(x, t) (11) (17) x 1 x ′ t ) d x ′ . − ∂ A x ′ t ∗ p ( , ( , ) t h denotes the linear lossless solution. 2c 0 0 t

∫

∫

D. β Estimation Based on the Pressure Wave Field We start this analysis by assuming that the pressure wave field is generally observable. To obtain a relation between the coefficient of nonlinearity β(x) and the pressure wave field p(x,t), we will first derive an expression of the harmonic pressure field. This is done by analyzing the Green’s function based contrast source solution of the 1-D lossy nonlinear Westervelt equation (10). Rewriting the total field as a combination of the fundamental and the harmonic field, i.e.,

p(x, t) = p f(x, t) + p h(x, t), (12)

The final step is to take the derivative of (17) with respect to x, thereby using the fundamental theorem of calculus [23], such that β(x ) 1 ∂ p 2(x, t) − ∂ tA(x, t) ∗t p h(x, t). 3 t 2 c 2ρ 0c 0 0 (18)

∂ x p h(x, t) =

Rewriting (18) yields where

∂ x p h(x, t) + S a[x, t, α 0(x )] =

β(x ) ∂ t p 2(x, t), (19) 2ρ 0c 03

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S a[x, t, α 0(x )] =

1 F −1{j ωA(x, ω)F {p h(x, t)}}, (20) 2c 0

with F{} ⋅ and F −1{} ⋅ denoting the Fourier transform and its inverse, respectively. With the assumption that the pressure wave fields p(x,t) and ph(x,t) are completely observable, the sampled version of (19) can thus be solved for β(x) using least-squares optimization.

{ ppˆˆ ((xx)) }, (21) h f

where pˆf(x ) and pˆh(x ) denote the peak pressure of the fundamental and second harmonic envelopes in the time domain over depth, respectively. Such an expression can be obtained by analyzing the derivative of the ratio (ph(x,t))/ (pf(x,t)) with respect to x, allowing us to write  p f(x, t)    ∂x  p h(x, t) 

∂ p (x, t) ∂ x p f(x, t)  p h(x, t)  − . (22)  = x h p ( x , t ) p h(x, t) p f(x, t)   f

Combining (22) and (18), and considering the assumptions stated earlier, (22) can be expressed (see the Appendix) in terms of pˆf(x ) and pˆh(x ) as

 pˆf(x )    ∂x  pˆh(x ) 

β(x )ω 0 pˆf2(x )  pˆh(x )   = 2ρ 0c 03 pˆh(x )  pˆf(x )  (23) ω0  1  −  |A(x, 2ω 0)| − |A(x, ω 0)|  , c 0  2 

where

A(x, ω) =

2015

{

}

We will refer to this as the echo-mode imaging method. III. Results

For echo-mode US, only echoes of the pressure wave field p(x,t) are observable. These echoes may overlap, resulting in constructive and destructive interference. This significantly increases the difficulty of estimating β, and in this section a specific solution is presented which is derived based on several assumptions. First, we assume that the acoustic wave returning from the tissue boundary to the transducer propagates linearly, which is justified by its lower pressure amplitude. Furthermore, in the final imaging algorithm we only consider the second harmonic and we neglect dispersive effects. For backscattered US signals, β estimation is strongly dependent on spatial variations of scattering. To compensate for this phenomenon, we adopt an additional assumption, i.e., variations in the speckle pattern are similar for harmonic and fundamental components. We then consider relating β to the harmonic ratio through a function f{∙}, i.e., β(x ) = f

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 2ρ 0c 03   pˆh(x )   ∂ β(x ) ≈   ω 0pˆf(x )  x  pˆf(x )  (25) 1 ω0    pˆh(x )  +  |A(x, 2ω 0)| − |A(x, ω 0)|    . 2 c 0    pˆf(x ) 

E. β Imaging in Echo Mode

vol. 62, no. 7,

α 02(x )c 02|ω|2y(x ) 2α (x )c 0|ω|y(x ) −j 0 . (24) 2 ω ω

The resulting expression for β(x) can then be derived as

A. Numerical Results In this section, the derived methods are applied to simulated ultrasound fields considering several situations, referred to as simulations A to E. An overview of the numerical analysis is given in Table II. We start with plane waves applied to simple 1-D structures and gradually increase the complexity to eventually a finite aperture single-element transducer as well as a linear array scanning setup. The plane wave simulations (A, B, C) are analyzed by solving the sampled version of (19) for β(x) using least squares optimization (Section II-D) whereas the echo mode imaging method (Section II-E) is used for the singleelement (D) and the linear array scan (E) simulations. The resulting root mean squared errors (RMSE) of the β estimation over space for all simulations are given in Table III, were the RMSE is shown in terms of absolute β. 1) General Settings: First, Burgers’ equation is exploited as it is a recognized and simple description for 1-D progressive plane waves [24]. Dispersive effects are not modeled. Then, the iterative nonlinear contrast source (INCS) approach [22], [25] is used. INCS is based on an iterative solution of (10), and is capable of simulating 3-D pressure fields and modeling dispersive effects, as opposed to Burgers’ equation. For all simulations, the transmitted US pulse is given by

2

2

p pulse = p 0 sin(2π f 0t)e −((2t) )/(Tw), (26)

where p0 = 1 MPa is the peak pressure, f0 = 1 MHz is the fundamental frequency of the sinusoid, and Tw = 6/f0 limits the pulse duration to 6 cycles. The chosen simulation source pressure is realistic for real imaging applications; see [26], where peak negative pressures on the order of −1.5 MPa were measured in water for the ULA-OP scanner. The speed of sound c0 and mass density ρ0 are assumed constant, being 1540 m/s and 1000 kg/m3, respectively. In the algorithm, the attenuation coefficient is globally set to αˆ 0(x ) = 0.57 dB/(MHzy∙cm) and y = 1.0001 [it may not be an odd integer, see (4)], if not stated otherwise. Because of numerical and real physical constraints (transducer bandwidth), only harmonics up to third order are considered. All spatial derivatives are implemented using Gaussian derivatives [27].


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TABLE II. Overview of Simulation Settings for the Numerical Results. f0 [MHz]

#

Medium

Simulator

Dimensions

Excitation

A B C D E

1 1 2 3 3

Burgers’ INCS INCS INCS INCS

1-D 1-D 3-D 3-D 3-D

Plane wave Plane wave Plane wave Single element Linear array scan

1 1 1 1 1

The nonlinear media used throughout the simulations represent heart, fat, and muscle tissue. Their attenuations and coefficients of nonlinearity are given in Table I. 2) Plane Wave Excitation (Burgers’): For this simulation, we consider three distinct tissue layers where the coefficients of nonlinearity {β1,β2,β3} = {3.9, 6.1, 4.7} and attenuation {α1,α2,α3} = {0.52, 0.60, 0.57} in dB/(MHzy∙cm) are chosen such that they represent heart, fat, and muscle tissue, respectively. Here, we consider a simulation with {Δx1, Δx2, Δx3} = {15, 20, 15} mm based on Burgers’ equation. Fig. 1 shows the obtained β(x) plot. Because Burgers’ equation does not model dispersion, it is also ignored in the estimation algorithm. That is, the imaginary part of the propagation coefficient is set to ω/c0 for these experiments. 3) Plane Wave Excitation (1-D-INCS): Here, the method is tested by applying it to a simulation with the configuration described in Section III-1-B, now simulated using 1-D-INCS, thereby considering dispersion. Fig. 1 shows the obtained β(x) plot. Although the estimated nonlinearity is close to the ground truth, the effects of not knowing all harmonic terms and using a constant attenuation for all x can be observed. The resulting RMSEs for simulations A and B are similar (Table III). 4) Plane Wave Excitation (3-D-INCS): In this simulation we still consider plane wave excitation, but applied to a medium structure having 3-D spatial variations. Fig. 2 shows the fields as well as the obtained β for a slice of the field simulated using 3-D-INCS [20]. The region between the two dashed circles corresponds to a lossless nonlinear medium with β = 4.38, whereas the background is lossless and linear, i.e., β = 0. αˆ 0 is thus set to zero in the algorithm. In this 3-D simulation, the effects of the diffraction pattern can be seen from the harmonic pressure fields as

Attenuation

Dispersion

β estimation

Figure

Yes Yes No Yes Yes

No Yes No Yes Yes

Least squares sol. of (19) Least squares sol. of (19) Least squares sol. of (19) Echo-mode imaging method Echo-mode imaging method

Fig. Fig. Fig. Fig. Fig.

1 1 2 3 3

well as the β image. The RMSE increases mainly because of the effects of diffraction on amplitude and phase, which are not included in the 1-D model (19) is based on. Therefore, we now move to the proposed echo-mode β imaging method. This method does not consider phase changes and is hence less sensitive to the diffraction. 5) Single-Element Excitation (3-D-INCS): We start by applying the echo mode method to transmission mode data, being a slice of the pressure field simulated using 3-D-INCS, employing a rectangular single element having a width and height of 10 mm as source. The structures have contrast in attenuation and nonlinearity, and represent heart, fat, and muscle tissue (values given in Table I). The background medium has an attenuation of 0.52 dB/(MHzy∙cm) and a nonlinearity β = 1. Fig. 3 shows the test configuration, in which the color scale indicates the coefficient of nonlinearity β (top plot), as well as the second-harmonic pressure field and the estimated β within the main beam (second and third plots, respectively). The RMSE is 0.62 (Table III) in the region of interest, which excludes the estimates close to the probe. The region is annotated in Fig. 3(c). 6) Linear Array Scan (3-D-INCS): The final part of this numerical analysis concerns the evaluation of the proposed echo-mode β-imaging method when considering a 52-element linear array scan, using pressure fields simulated with 3-D-INCS. The medium configuration is identi-

TABLE III. Overview of the Numerical Analysis and Root Mean Squared Errors (RMSE) of the Estimation Over Space. #

Simulator

Excitation

A B C D E

Burgers’ 1-D-INCS 3-D-INCS 3-D-INCS 3-D-INCS

Plane wave Plane wave Plane wave Single element Linear array scan

RMSE 0.17 0.16 0.77 0.62 0.55

Note that for single-element excitation, only values within 3.5 mm of the main beam axis are considered.

Fig. 1. Medium 1, Simulators: Burgers’ equation and 1D-INCS. β estimation using the least squares solution of (19). A comparison between the results for pressure data simulated using Burgers’ equation at 1 MHz (blue) and 1-D-INCS at 1 MHz (green). The red dashed line indicates the ground-truth β.

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Fig. 2. Medium 2, Simulator: 3-D-INCS, plane wave excitation. β estimation using the least squares solution of (19) for a 3-D slice of data. Plot (a) shows the configuration where the color scale indicates the coefficient of nonlinearity β. Plots (b) and (c) show the second-harmonic maximum pressure fields and the obtained β(x), respectively.

cal to the one considered in Section III-A-5. The linear array consists of elements having a pitch of 0.77 mm. For each line, 13 elements are simultaneously excited and a β estimate is obtained within the resulting beam. The nonlinearity image is then formed by stepping along the array (step size: 2 elements) and combining the obtained β values for each scan line. As Fig. 3 shows, the original morphology is recognizable from the nonlinearity image, Fig. 3(d). Moreover, the obtained β values are close to the ground truth, yielding an RMSE of 0.55 (Table III) within the region of interest. The region is annotated in Figs. 3(d) and 3(e). The resulting β maps when varying the algorithm parameters c0 and a0 are shown in Figs. 3(f)– 3(j). Absolute β estimates are most affected by strong misestimations in speed of sound, whereas variations in attenuation play a smaller role (Table IV). B. Experimental In Vitro Results The proposed echo-mode β-imaging method was tested by acquiring 128 RF lines in vitro, using a ULA-OP ultrasound scanner [26] transmitting Gaussian windowed pulses with frequency f0 = 3 MHz and a pulse duration of 6 cycles. The number of active elements is 64 and the pitch of the Esaote LA332 probe (Esaote S.p.a., Florence, Italy) is 0.245 mm. The fundamental and secondharmonic B-mode images are obtained using eighth-order Butterworth band-pass filters around f0 = 3 MHz and f2 = 6 MHz, respectively, after which their envelopes are determined by employing the Hilbert transform. A lineby-line 5-mm moving average (boxcar) filter is applied to both the fundamental and second-harmonic images in an effort to obtain continuous estimates of the pressure, weighted by the scatterer distribution. The size has been chosen empirically, being a trade-off between the effective axial resolution of the estimator and the ability to cope with differences in summing of subwavelength scatterers for the fundamental and second-harmonic components in reception. The used phantom combines two layers of tissue-mimicking material, consisting of corn oil (4% and 50% mass respectively), water (91% and 45% mass respectively), gelatin (4%), and 10-μm aluminum oxide powder scatterers (1%). The layers with 50% and 4% corn oil are referred to as layers 1 and 2, respectively. Corn oil is known to have a high nonlinearity (β = 6.2) and was exploited to obtain contrast [28]. The ratios of water to

corn oil modify the β value in the layers. The images, as well as the normalized result of the proposed echo-mode β-imaging method are shown in Fig. 4. The echo-mode algorithm [see (25)] has been applied assuming a speed of sound c0 = 1540 m/s, mass density ρ0 = 1000 kg/m3, ω0 = 6π × 106 rad/s, attenuation α0(x) = 0.3 dB/(MHzy∙cm) and y = 1.0001. For comparison, the results of the direct method and the extended comparative method (ECM) applied to the filtered second-harmonic image are also given. In this experiment, the ground truth absolute value of β in the two media is unknown, limiting the analysis to qualitative results. However, a relative spatial distribution of β can be observed for all three methods. The ECM seems to give a stronger β contrast between the two media than the direct method. However, both methods show a strong dependency on depth. Furthermore, the region highlighted most in both maps appears to coincide with the region having the strongest harmonic speckle intensity. The proposed method shows a more homogeneous β distribution for both layers and seems less dependent on propagation depth. In layer 1, the standard deviations of the normalized β maps for the direct method, the ECM, and the proposed method are 0.29, 0.28, and 0.18, respectively. In layer 2, the standard deviations are 0.03, 0.04, and 0.03, respectively. IV. Conclusions and Discussion In this paper, we proposed a nonlinearity estimation method based on a 1-D solution of the inhomogeneous Westervelt equation. By testing the method on simulated ultrasound pressure fields, increasing the complexity of the numerical experiments with each step, we showed that the tissue types of interest are well discriminable with the designed algorithm by exploiting their contrast in nonlinearity. This was first indicated by the one-dimensional results of simulations A and B, where no diffraction was present. Simulation C showed the effects of medium diffraction by exciting a cylindrical object with a plane wave. The effects of finite aperture excitation were analyzed using simulation D. The transducer’s diffraction pattern and its effect on the β image can be observed in the near field, and the results obtained in this region are unreliable. Also, beam spreading and medium diffraction are visible, increasingly degrading the shape and accuracy of the β map over


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Fig. 3. Medium 3, Simulator: 3-D-INCS, single-element and linear array scan excitation. β estimation using the echo-mode imaging method for a 2-D slice of data. Plot (a) shows the configuration where the color scale indicates the coefficient of nonlinearity, β. Three tissue types are considered— heart, muscle, and fat. Plot (b) shows the second-harmonic beam profile for the center scan line. Plots (c) and (d) show the obtained coefficient of nonlinearity β(x) images for the single element within the beam and the linear array scan for the entire domain, respectively. The corresponding absolute error map of the latter is shown in plot (e). The RMSE measures are determined from the region on the right-hand side of the dashed line in plots (c), (d), and (e), excluding the strong near-field effects. Plots (f) to (j) show the resulting maps when varying the algorithm parameters c0 (given in m/s) and a0 [given in dB/(MHz∙cm)].

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Fig. 4. β imaging using the echo mode imaging method for experimental in vitro data acquired using a linear array. Plots (a) and (b) show the fundamental and second-harmonic images, respectively. The resulting β maps for the direct method [11], the extended comparative method (ECM) [10], and the proposed harmonic ratio method are shown in plots (c), (d), and (e), respectively. The two phantom layers are denoted 1 and 2.

depth. The final step in the numerical analysis consisted of a linear array scan (simulation E). Again, its accuracy is low in the near field and degrades over depth in the far field, because of diffraction and the unrealistic assumption of constant attenuation. However, in the region of interest, the morphology and the obtained β values were found to be close to the ground truth. Even when the assumptions on attenuation in the algorithm are crude [Figs. 3(i) and 3(j)], the β estimates are not far off. In particular, these results, being a good numerical representation of real Bmode imaging, are promising for the applicability of the proposed echo-mode β-imaging method. A limitation of the simulation study lies in the assumption of homogeneous speed of sound c0, whereas in practice it may vary considerably. Variations in speed of sound affect harmonics’ generation; if wrongly estimated, they affect the β estimate. This is also indicated in Figs. 3(i) and 3(j). A particularly troublesome case would be one in which a medium has a strongly increased speed of sound as well as coefficient of nonlinearity with respect to another medium, making it difficult to distinguish between the two. However, when inspecting the tissues considered in this article, we point out that this situation does not occur. The only relatively strong variation in speed of sound comes from fat, with c0 = 1430 m/s, as compared with heart and muscle, with c0 = 1554 m/s and c0 = 1580 m/s, respectively [29]. In fact, it actually increases the contrast

TABLE IV. Influence of Algorithm Parameters on the Root Mean Squared Errors (RMSE) of the β Estimation Over Space for the Linear Array Scan Excitation (E). Algorithm parameters c0 [m/s] 1540 1540 1540 1400 1600

α0 [dB/(MHzy∙cm)]

RMSE

0.57 0.4 0.7 0.57 0.57

0.55 0.61 0.58 0.85 0.64

of the apparent coefficient of nonlinearity estimate in this case, making identification of fat an easier task. On the other hand, distinguishing between heart and muscle becomes more challenging. When comparing this work to other β estimation methods [10], [12], [13], we point out that our method does not require a special transducer or setup, and it provides a local estimation of the coefficient of nonlinearity based on the harmonic ratio, which reduces spatial accumulation of errors resulting from unknown attenuation. We observe that not knowing the attenuation at each point is not necessarily a problem when fluctuations are small, aided by the fact that part of the attenuation is compensated by employing the harmonic ratio. This can also be understood when evaluating the second term on the right-hand side of (25). In the special case of y = 1, the amount of attenuation requiring compensation is reduced by a factor of two. Moreover, dedicated attenuation estimation algorithms [30]–[32] could be exploited to provide spatial information on α0(x). This is particularly useful when considering media with strong contrast in attenuation. The echo-mode imaging algorithm is based on several assumptions. First, we assume that the acoustic wave returning from the tissue boundary to the transducer propagates linearly, which is justified by its lower pressure amplitude. Furthermore, in the final imaging algorithm we only consider the second harmonic, and we neglect dispersive effects. The impact of scatterers is addressed and contributions to the derivative term are limited to effects related to the wavelength dependency of scattering. Although speckle intensity variations are assumed to be similar for the fundamental and harmonic frequencies, there will be errors related to the fact that scattering for fundamental and harmonic frequencies is not equal. This problem requires interpolation or averaging of both fields’ echo responses, reducing the resolution of the β map. Higher transmission frequencies or shorter pulse lengths can mitigate this problem, but lead to higher attenuation or more overlap between the harmonic and funda-


mental frequencies, respectively. Although the contribution of speckle intensity variations to the derivative term is minimized, one can note that the estimated β values are still inversely proportional to the speckle intensity in the medium because of the fundamental field factor in the denominator. This type of dependency is, however, less significant compared with the impact of speckle intensity variations on, e.g., the direct and comparative methods. For the latter, in particular, differences between spatial variation of the speckle intensity in the region of interest and the reference region are highlighted. Regarding the in vitro evaluation, the proposed method shows a more homogeneous β distribution for both layers and seems less dependent on propagation depth compared with the direct and extended comparative methods. This might be due to the fact that both reference methods suffer from significant spatial accumulation of errors resulting from unknown attenuation. The proposed method is based on the harmonic ratio and uses a different approach that avoids the integral-based attenuation compensation over depth. Although the initial in vitro results are promising, further in vitro and eventually in vivo testing is required to evaluate its use for clinical imaging. The physical arguments and mathematical formulations as well as the detailed numerical analysis described in this work are intended to provide a theoretical foundation for β imaging, encouraging future research on practical implementations. Appendix β Imaging in Echo Mode: Full Derivation

neglect any contributions ω > 2ω0, and use the following trigonometric identities

∂ x p f(x, t) ≈ −

1 ∂ A(x, t) ∗t p f(x, t), (27) 2c 0 t

thereby discarding the relatively small effect of energy transfer from the fundamental to the harmonic components when compared with energy loss due to attenuation in tissue. Then, combining this with the result of (18) and (22) yields β(x ) ∂ t p 2(x, t) p f(x, t) p (x, t) ∂x h ≈ p f(x, t) p h(x, t) 2ρ 0c 03 p h(x, t)

+ −

1 2c 0

∂ tA(x, t) ∗t p f(x, t) (28) p f(x, t)

1 2c 0

∂ tA(x, t) ∗t p h(x, t) . p h(x, t)

Next, we assume

p(x, t) = pˆf(x ) sin(ω 0t) + pˆh(x ) sin(2ω 0t), (29)

sin(x ) cos(x ) =

1 sin(2x ), (30) 2

1 [sin(3x ) − sin(x )], (31) 2 1 cos(x ) sin(2x ) = [sin(x ) + sin(3x )], (32) 2 cos(2x ) sin(x ) =

such that we obtain

∂ t p 2(x, t) = pˆf2(x )ω 0 sin(2ω 0t) − pˆf(x )pˆh(x )ω 0 sin(ω 0t) (33) ≈ pˆf2(x )ω 0 sin(2ω 0t).

With reference to the temporal convolutions used in (28), these can be solved as

∂ tA(x, t) ∗t sin(ωt) = ω|A(x, ω)| cos[ωt + φA(x, ω)]. (34)

To reduce complexity, we neglect dispersion. In this case (recall Section II-A), 2

2  α (x )c 0|ω|y(x ) ω c0   γˆ(x, ω)c 0   −1 A(x, ω) =  +  −1 =  0  jω c 0 ω   j ω  α 02(x )c 02|ω|2y(x ) 2α 0(x )c 0|ω|y(x ) −j = . ω ω2 (35)

Because α0(x)c0 ≪ 1, ϕA(x,ω) = arctan[Im(A)/Re(A)] ≈ −π/2. Using this together with (33) and (35), (28) can be rewritten as

First a description of the spatial derivative of the fundamental field is presented. Using (14), we can obtain

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 pˆf(x )    ∂x  pˆh(x ) 

 pˆh(x )    ≈  pˆf(x ) 

β(x ) 2ρ 0c 03

pˆf2(x )ω 0 sin(2ω 0t)

pˆh(x ) sin(2ω 0t)

− +

ω0 |A(x, 2ω 0)|pˆh(x ) sin(2ω 0t) c0

(36)

pˆh(x ) sin(2ω 0t)

ω0 |A(x, ω 0)|pˆf(x ) sin(ω 0t) 2c 0

pˆf(x ) sin(ω 0t)

,

leading to

 pˆf(x )    ∂x  pˆh(x ) 

β(x )ω 0 pˆf2(x )  pˆh(x )    ≈ 2ρ 0c 03 pˆh(x )  pˆf(x )  (37) ω0  1  −  |A(x, 2ω 0)| − |A(x, ω 0)|  . c 0  2 

An expression for β(x) is then given by

{

 2ρ 0c 03   pˆh(x )   ∂ β(x ) ≈   ω 0pˆf(x )  x  pˆf(x )  (38) 1 ω0    pˆh(x )  +  |A(x, 2ω 0)| − |A(x, ω 0)|    . 2 c 0    pˆf(x ) 

}

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Acknowledgments The authors wish to acknowledge M. D. Verweij for facilitating the use of INCS and thank M. P. J. Kuenen and H. J. W. Belt for the useful comments and discussion on early drafts of the manuscript. References [1] F. A. Duck, “Nonlinear acoustics in diagnostic ultrasound,” Ultrasound Med. Biol., vol. 28, no. 1, pp. 1–18, 2002. [2] F. A. Duck, Physical Properties of Tissues: A Comprehensive Reference Book. New York, NY, USA: Academic Press, 1990. [3] H. Khelladi, F. Plantier, J. L. Daridon, and H. Djelouah, “Measurement under high pressure of the nonlinearity parameter B/A in glycerol at various temperatures,” Ultrasonics, vol. 49, no. 8, pp. 668–675, 2009. [4] A. N. Ganesan, N. J. Shipp, A. G. Brooks, P. Kuklik, D. H. Lau, H. S. Lim, T. Sullivan, K. C. Roberts-Thomson, and P. Sanders, “Long-term outcomes of catheter ablation of atrial fibrillation: A systematic review and meta-analysis,” J. Am. Heart Assoc., vol. 2, no. 2, p. e004549 2013. [5] N. Doll, M. A. Borger, A. Fabricius, S. Stephan, J. Gummert, F. W. Mohr, J. Hauss, H. Kottkamp, and G. Hindricks, “Esophageal perforation during left atrial radiofrequency ablation: Is the risk too high?” J. Thorac. Cardiovasc. Surg., vol. 125, no. 4, pp. 836–842, 2003. [6] D. Sánchez-Quintana, J. A. Cabrera, V. Climent, J. Farré, M. C. Mendonça, and S. Y. Ho, “Anatomic relations between the esophagus and left atrium and relevance for ablation of atrial fibrillation,” Circulation, vol. 112, no. 10, pp. 1400–1405, 2005. [7] S. M. Singh, A. d’Avila, S. K. Doshi, W. R. Brugge, R. A. Bedford, T. Mela, J. N. Ruskin, and V. Y. Reddy, “Esophageal injury and temperature monitoring during atrial fibrillation ablation,” Circ Arrhythm Electrophysiol, vol. 1, no. 3, pp. 162–168, 2008. [8] K. V. Dongen and M. Verweij “Sensitivity study of the acoustic nonlinearity parameter for measuring temperatures during high intensity focused ultrasound treatment,” J. Acoust. Soc. Am., vol. 123, no. 5, p. 3225, 2008. [9] C. A. Cain, “Ultrasonic reflection mode imaging of the nonlinear parameter B/A: I. A theoretical basis,” J. Acoust. Soc. Am., vol. 80, no. 1, pp. 28–32, 1986. [10] F. Varray, O. Basset, P. Tortoli, and C. Cachard, “Extensions of nonlinear B/A parameter imaging methods for echo mode,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 6, pp. 1232– 1244, 2011. [11] W. K. Law, L. A. Frizzell, and F. Dunn, “Determination of the nonlinearity parameter B/A of biological media,” Ultrasound Med. Biol., vol. 11, no. 2, pp. 307–318, 1985. [12] X. Gong, F. Ruo, C. Zhu, and S. Tao, “Ultrasonic investigation of the nonlinearity parameter B/A in biological media,” J. Acoust. Soc. Am., vol. 76, no. 3, pp. 949–950, 1984. [13] S. Ueno, M. Hashimoto, H. Fukukita, and T. Yano, “Ultrasound thermometry in hyperthermia,” in Proc. IEEE Ultrasonics Symp., 1990, pp. 1645–1652. [14] D. Zhang and X.-F. Gong, “Experimental investigation of the acoustic nonlinearity parameter tomography for excised pathological biological tissues,” Ultrasound Med. Biol., vol. 25, no. 4, pp. 593–599, 1999. [15] M. Toulemonde, F. Varray, O. Basset, P. Tortoli, and C. Cachard, “High frame rate compounding for nonlinear b/a parameter ultrasound imaging in echo mode—Simulation results,” in IEEE Int. Conf. Acoustics, Speech and Signal Processing, 2014, pp. 5153–5157. [16] R. T. Beyer, “Parameter of nonlinearity in fluids,” J. Acoust. Soc. Am., vol. 31, no. 11, p. 1586, 1959. [17] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, vol. 237. San Diego, CA, USA: Academic Press, 1998. [18] L. Demi and M. Verweij, “Nonlinear acoustics,” in Comprehensive Biomedical Physics, Amsterdam, The Netherlands: Elsevier Science, 2014, pp. 387–399. [19] L. Demi, K. Van Dongen, and M. Verweij, “A contrast source method for nonlinear acoustic wave fields in media with spatially inho-

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Ruud J. G. van Sloun was born in Roermond, The Netherlands, in 1990. He received the B.Sc. degree and the M.Sc. degree (with distinction cum laude) in electrical engineering in 2012 and 2014, respectively, from the Eindhoven University of Technology, The Netherlands, where he is currently pursuing the Ph.D. degree in the Biomedical Diagnostics Laboratory. In 2013 and 2014, he was an intern at Philips Research Eindhoven, specializing in the field of medical ultrasound imaging and nonlinear acoustics. His current research interests include quantitative (contrast-enhanced) ultrasound imaging for tumor localization, compressed sensing, signal and image processing, sensor analytics, system identification, mathematics, and physics.

Libertario Demi (M’11) was born in Cecina, Italy, on November 19, 1983. In 2006, he planned and established a satellite connection in Boulsa, Burkina Faso, as a volunteer for Engineers Without Borders Pisa (Ingegneria senza Frontiere Pisa). At the end of 2007, he carried out research activities in the School of Electrical and Electronic Engineering, University of Adelaide, as part of his master’s thesis project. In 2008, he obtained his M.Sc. degree in telecommunication engineering, cum laude, from Pisa University and carried out research activities at Pisa University with a scholarship in ICT from the Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT). In 2013, he received his Ph.D. degree from the Delft University

van sloun et al.: ultrasound coefficient of nonlinearity imaging of Technology, working on modeling nonlinear medical ultrasound. Since July 2012, he has held a postdoctoral position in the biomedical diagnostics lab at the Eindhoven University of Technology, mainly working on prostate cancer detection using ultrasound contrast agents. He is author of more than 40 publications, including 13 papers in international peerreviewed scientific journals, 2 book chapters, and 26 papers in international conference proceedings. His research interests include nonlinear propagation of ultrasound, parallel beamforming, ultrasound contrast agents, and pressure wave fields modeling.

Caifeng Shan is a Senior Scientist with Philips Research, Eindhoven, The Netherlands. He received the B.Eng. degree from the University of Science and Technology of China; the M.Eng degree from the Institute of Automation, Chinese Academy of Sciences; and the Ph.D. degree from Queen Mary University of London. His research interests include computer vision, pattern recognition, (medical) image and video analysis, machine learning, and related applications. He has authored more than 60 technical papers and 30 patent applications. He was the recipient of the 2007 Chinese Government Award for Outstanding Students Abroad. He is an Associate Editor of the IEEE Transactions on Circuits and Systems for Video Technology, and an Editorial Board Member of the Journal of Visual Communication and Image Representation and the Journal on Ambient Intelligence and

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Smart Environments. He has edited three books and has been a Guest Editor of the IEEE Transactions on Multimedia, IEEE Transactions on Circuits and Systems for Video Technology, Signal Processing (Elsevier), and Machine Vision and Applications. He organized several international workshops at flagship conferences such as IEEE ICCV and ACM Multimedia. He has served as a Program Committee Member and Reviewer for numerous international conferences and journals. He is a Senior Member of IEEE.

Massimo Mischi (SM’10) received the M.Sc. degree in electrical engineering from La Sapienza University, Rome, Italy, and the Ph.D. degree from the Eindhoven University of Technology, Eindhoven, The Netherlands, in 2004. In 2007, he became assistant professor and since 2011 he has been an associate professor at the Electrical Engineering Department of the Eindhoven University of Technology, where he chairs the Biomedical Diagnostics Research Lab. His research focuses on model-based quantitative analysis of biomedical signals, with focus on electrophysiology and on ultrasound and magnetic resonance imaging. Among several prizes, in 2011 he was awarded with a Starting Grant from the European Research Council (ERC) for his research on angiogenesis imaging. He is currently vice-chairman of the IEEE EMBS Benelux Chapter and secretary of the Dutch Society of Medical Ultrasound.