Goldberg’s Number Influence on the Validity Domain of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves

Nonlinear propagation occurs widely in many acoustic systems, especially in the field of medical ultrasound. Despite the widespread use of ultrasound in diagnosis and therapy, the propagation of ultrasound through biological media was modeled as a linear process for many years. The invalidity of infinitesimal acoustic assumption, at biomedical frequencies and intensities, was demonstrated by Muir and Carstensen (Muir & Carstensen, 1980). It was realized that nonlinear effects are not negligible and must therefore be taken into account in theoretical developments of ultrasound in biomedical research. Indeed, increasing the acoustic frequency or intensity in order to enhance resolution or penetration depth may alter the beam shape in a way not predicted by linear theory.

attenuates the harmonics resulting from nonlinear effects. The shock length s l (Enflo & Hedberg, 2002;Naugolnykh & Ostrovosky, 1998) quantifies the influence of the nonlinear phenomena, and it is necessary to define another parameter, denoted Goldberg's number  (Goldberg, 1957), when dissipation is added.  represents the ratio of the absorption length a l (the inverse of the absorption coefficient  and corresponds to the beginning of the old age region) to the shock length s l at which the waveform would shock if absorption phenomena were absent: where k , M ,  and  are, respectively, the wave number, the acoustic Mach number, the acoustic nonlinearity parameter and the absorption coefficient.
It should be noted that higher harmonics may turn the wave into shock state. On the other hand, dissipation attenuates higher harmonics much more than lower harmonics, thus making it more difficult for the waves to go into shock.
The dimensionless parameter  measures the relative importance of the nonlinear and dissipative phenomena, which are in perpetual competition. Thus, the Goldberg's number is a reliable indicator for any analysis including these two phenomena. An analysis based on the Goldberg's number is important since it is an essential step for solving general problems involving ultrasound waves of finite amplitude.
Nowadays, Tissue Harmonic Imaging (THI) or second harmonic imaging offers several advantages over conventional pulse-echo imaging. Both harmonic contrast and lateral resolution are improved in harmonic mode. Tissue Harmonic Imaging also provides a better signal to noise ratio which leads to better image quality in many applications. The major benefit of Tissue Harmonic Imaging is artifact reduction resulting in less noisy images, making cysts appear clearer and improving visualization of pathologic conditions and normal structures. Indeed, Tissue Harmonic Imaging is widely used for detecting subtle lesions (e.g., thyroid and breast) and visualizing technically-challenging patients with high body mass index.
In order to create images exclusively from the second harmonic, a theoretical review with some mathematical approximations is elaborated, in this chapter, to derive an analytical expression of the second harmonic. The performance of the simplified model of the second harmonic is interesting, as it can provide a simple, useful model for understanding phenomena in diagnostic imaging.
Despite the significant advantages offered by Tissue Harmonic Imaging, theory has been partially explained. A number of works were elaborated over recent decades. Among these, are Trivett and Van Buren (Trivett & Van Buren, 1981) work which have presented an analysis of the generated harmonics based on the generalized Burgers' equation. Significant differences in the calculated harmonic content were found by Trivett and Van Buren when compared with those obtained by Woodsum (Woodsum, 1981). No explanation was given by Trivett and Van Buren to justify their results. In an author's reply, Woodsum seemed to attribute these differences to the high number of terms retained by Trivett  Similarly, Haran and Cook (Haran & Cook, 1983) have used the Burgers' equation to elaborate an algorithm for calculating harmonics generation by a finite amplitude plane wave of ultrasound propagating in a lossy and nondispersive medium. Their algorithm accounts for an absorption coefficient of any desired frequency dependence. The variation effect of the absorption coefficient on the second harmonic was demonstrated in a medium similar to carbon tetrachloride. Calculations for several types of tissue and biological fluids were presented. It was shown that for some biological media having a low absorption coefficient, a significant distortion of the plane wave can be observed for large propagation ranges.
Recently, D'hooge et al. (D'hooge et al., 1999) have analyzed the nonlinear propagation effects of pulses on broadband attenuation measurements and their implications in ultrasonic tissue characterization by using a simple mathematical model based on the numerical solution, in the time domain, of the Burgers' equation. The developed model has been validated by measuring the absorption coefficient of both a tissue-mimicking phantom in vitro and a liver in vivo at several pressure amplitudes using transmission and reflection measurements, respectively.
In the present chapter, the intensity effects on the behavior of the fundamental and the generated second harmonic, by using both the numerical solution of the Burgers' equation and the analytical expressions established with the quasi-linear approximation are examined. An analysis on the validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation is elaborated. The deviations resulting from the analytical expressions established with the quasi-linear approximation and the numerical solution of the Burgers' equation are estimated. This investigation is based on Krassilnikov et al. (Krassilnikov et al., 1957) experimental results. These experimental data concern water and glycerol that correspond, respectively, to a weakly dissipative liquid approaching the characteristics of urine or amniotic fluid (Bouakaz et al., 2004) and a strongly dissipative liquid with some similarities to soft tissues.
It should be noted that in this study all derivations are developed entirely in the frequency domain, thus avoiding both the steep waveform problems and the use of FFT, which alternates between time and frequency domains. The utility of the method resides in the ease with which it can be implemented on a digital computer.

Theoretical formulation
The description of acoustic waves in a liquid is founded on the theory of motion of a liquid, which is considered to be continuous. In the present investigation, the viscosity and the heat conduction coefficients, although in general are functions of the state variables, are assumed to be constant. The theoretical formulation of the propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is elaborated in section 2.1, and the theoretical model is based on the derivation of a nonlinear partial differential equation in which the longitudinal particle velocity is a function of time and space. In section 2.2, the dimensionless Burgers' equation is presented, which is considered to be among the most exhaustively studied equations in the theory of nonlinear waves.

Basic equations
The propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is governed by the Burgers' equation. Here, it is assumed that the ultrasonic wave propagates in the positive z direction, and the differential change of the longitudinal particle velocity with respect to z is given by (Enflo & Hedberg, 2002;Naugolnykh & Ostrovosky, 1998

  
is function of the nonlinearity parameter of the medium / BA , which represents the ratio of quadratic to linear terms in the isentropic pressure-density relation (Hamilton & Blackstock, 1988;Khelladi et al., 2007Khelladi et al., , 2009). 0 tz c   is the retarded time, 0 c is the infinitesimal sound speed and 0  is the undisturbed density of the liquid.
The term on the left hand side of equation (2) is the linear wave propagation. The first term on the right hand side of equation (2) is the nonlinear term that accounts for quadratic nonlinearity producing cumulative effects in progressive plane wave propagation, while the second term represents the loss due to viscosity and heat conduction or any other agencies of dissipation.
Nonlinear propagation in a dissipative liquid is considered using Fourier series expansion. By assuming that the solution of equation (2)  When complex notation is used, equation (3) changes to (Haran & Cook, 1983;Ngoc & Mayer, 1987 For the easiest derivation, equation (4) is substituted into equation (2) (Haran & Cook, 1983;Ngoc & Mayer, 1987): (5) describes the amplitude variation of the n th harmonic in the propagation direction z . The summation over m expresses nonlinear interactions among various spectral components caused by the energy transfer process, while the other term accounts for loss due to dissipation relative to the n th harmonic.
Equation (5) is rewritten in another form (Haran & Cook, 1983;Ngoc & Mayer, 1987 , equation (6) yields two coupled partial differential equations governing the behavior of the components n v and n u as a function of the spatial coordinate z (Aanonsen et al., 1984;Hamilton et al., 1985): For a sinusoidal source condition, Aanonsen et al., 1984;Hamilton et al., 1985;Hedberg, 1999;Menounou & Blackstock, 2004), equation (3) becomes: Equation (8) is then written more simply as: The incremental change of the particle velocity can be approximated by the first order truncated power series (Haran & Cook, 1983;Ngoc et al., 1987): By combining equations (10) and (11), an iterative description of finite amplitude plane wave propagation in a homogeneous and dissipative liquid, is obtained: The first summation term on the right hand side of equation (12) represents the contribution of lower order harmonics to the n th harmonic, while the second one is associated with the contribution of higher order harmonics. According to the sign of each contribution the n th harmonic energy can be enhanced or decreased. The last term in this equation represents losses undergone by the n th harmonic.
Generally, the absorption coefficient  depends on the propagation medium characteristics and the insonation frequency. For the considered viscous fluids, this frequency dependence is quadratic with frequency and can be represented by (Smith & Beyer, 1948;Willard, 1941): where 0  depends upon the nature of the liquid, and Therefore the Goldberg's number  , increases with the amplitude of excitation and decreases with frequency.
Equation (12) (14) allows the determination of the n th harmonic amplitude at the location zz  in terms of all harmonics at the preceding spatial coordinate z . This derivation requires an appropriate truncation of the finite series on the right hand side of equation (14) to ensure a negligibly small error in the highest harmonic of interest and to maintain some acceptable accuracy.
In the hypothesis of the quasi-linear approximation, all the harmonics of higher order than two can be neglected in the numerical solution of the Burgers' equation, so equation (14) changes to: www.intechopen.com  In many situations, the experimental studies are based on pressure measurements. Knowing that the ratio of the n th harmonic pressure to the associated particle velocity is given by Germain et al., 1989); equation (15) is rewritten as: The acoustic pressure of the fundamental can be written as (Gong et al., 1989;Thuras et al., 1935): where 0 P is the characteristic pressure amplitude (the value of the fundamental pressure at 0 z  ).

Dimensionless equations
For theoretical analysis as well as for numerical implementation, it is more convenient to define dimensionless variables, by using the characteristic particle velocity where U ,  and  are, respectively, the dimensionless longitudinal particle velocity, the dimensionless time and the dimensionless propagation path.
Insertion of equation (21) into the Burgers' equation (equation (2)), gives the dimensionless equation (BjØrnØ, 2002;Fenlon, 1971;Hedberg, 1994): The dimensionless amplitude of the n th harmonic at the dimensionless location    in terms of all harmonics at the preceding dimensionless location  can be written as: With this dimensionless notation, the acoustic pressure of the fundamental and the second harmonic can be expressed as:  Krassilnikov et al. (Krassilnikov et al., 1957) experimental data for water and for glycerol are used in order to simulate the amplitude of the first two harmonics, by using both the numerical solution of the Burgers' equation (equation (23)) and the analytical expressions established with the quasi-linear approximation (equations (24), (25) and (26)). Table 1 lists material properties.

Numerical experiments and discussions
According to Krassilnikov et al. (Krassilnikov et al., 1957) experimental work, the absorption coefficient is a quadratic function of frequency. The absorption coefficient is that obtained from an infinitesimal acoustic excitation, even though the acoustic intensity increases. In the www.intechopen.com  (Krassilnikov et al., 1957).
Nonlinear effects occur more strongly when ultrasound propagates through slightly dissipative liquids, so a special attention is given to a propagation medium characterized by a Goldberg number greater than unity. In this case, when the waveform approaches the shock length, nonlinear effects dominate dissipation phenomena. The amplitude of the generated harmonics increases at the expense of the fundamental component. After the shock length, absorption limits the generation of harmonics by decreasing theirs amplitudes gradually with the propagation path. For this reason, all the simulations of the first two harmonics are plotted as a function of the dimensionless location  up to unity and for several values of the acoustic intensity. Moreover, all the shock lengths for several intensities are greater than 19.8 cm ( Table 2). As in biomedical diagnosis the region of interest (ROI) is about 20 cm , it is absolutely useless to explore beyond 1   and the selected range 0 1    is amply appropriate for this kind of investigation.
It should be pointed out that the shock length s l depends on the medium characteristics 0  , 0 c ,  and on the external parameters such as the insonation frequency and the amplitude of excitation. In this study, the insonation frequency is fixed at 2 MHz , thus the shock length for a given medium will depend only upon the amplitude of excitation.
Among all the configurations presented in this study, including various acoustic intensities and two analyzed mediums, only one case is sensitive in biomedical diagnostic and must be analyzed with extreme caution. Indeed, a more favorable situation where nonlinear effects have sufficient time to be entirely established corresponds to the case of water, for which the acoustic intensity is equal to 2 4.7 W/cm and as a consequence a shock length equal to 19.8 cm . As the generation of harmonics occurs while moving away from the source and approaching the shock length, the greatest signal distortion may occur in the range of interest. Moreover, the irradiation of living tissue with shock waves in diagnostic processes appears risky since the damage and exposure criteria for these radiations have not been delineated.
It should be noted that all the simulations are made with intensities of 2 0.2 -4.7 W/cm (Table 2), which correspond to breast lesion diagnosis (Nightingale et al., 1999). It will be stated by the derivation of the Goldberg number that water surpasses any tissue in its ability to produce extremely distorted waveforms even at relatively low intensity. So, a special attention is given to this liquid where the possibility of distortion occurring has several implications. Indeed, water can generate extreme waveform distortion compared to glycerol, as indicated by the Goldberg's number for water, which is 200 times larger than that of glycerol for an acoustic intensity of about 2 0.2 W/cm ( Initially, the ultrasonic wave is taken to be purely sinusoidal with a frequency of 2 MHz in the two considered media. Only the fundamental wave exists at the starting location 0   , and the other harmonic modes are generated as the wave propagates from the source. Through an iterative method, the value of the Goldberg number is inserted into the Burgers' equation in order to determine its numerical solution (Table 2). 40 harmonics are retained to simulate the numerical solution of the Burgers' equation which is considered, in the deviation calculus, as an exact solution.
For a better readability and interpretation of the obtained numerical data, a symbol with a defined shape and type is inserted on the graphic layout of the analyzed functions. All the following simulations exploit equations (23), (24), (25) and (26)  The amplitude 10 p/ P ( figure 1a, figure 1b) and the amplitude 20 p/ P (figure 2a, figure 2b) increase with  (Table 2). So, the effect of the increased acoustic intensity is to enhance the amplitude of the fundamental and also that of the second harmonic.
In the hypothesis of linear acoustics, increasing the absorption coefficient leads systematically to a decrease of the wave amplitude. The finite amplitude waves do not obey to the same principle because nonlinear effects and dissipation are two phenomena in perpetual contest. The interplay between these two phenomena developed along the propagation path is not simply an additive effect as normally assumed in linear acoustics. Therefore, a measure of whether nonlinear effects or absorption will prevail is the Goldberg's number  . The larger  is, the more nonlinear effects dominate. Whereas for values of 1  , absorption is so strong that no significant nonlinear effects occur. Thus the calculation of the Goldberg's number is required to quantify the amplitude of the generated harmonics.
By taking water as an example, the most significant amplitude of the generated harmonic, for various values of intensity, corresponds to the highest Goldberg's number (figure 2a). This is in perfect agreement with physical phenomena that take place in the analyzed medium. Indeed, a high Goldberg number corresponds to a predominance of the nonlinearity phenomenon as compared to dissipation, which represents the main factor of amplitude decrease. This situation is also apparent for glycerol (figure 2b).
For a slightly dissipative liquid, it can be seen that the second harmonic component grows cumulatively with increasing the normalized length  at the expense of the fundamental ( figure 1a, figure 2a). Its growth begins to taper off at the location of the initial shock formation, beyond this location the curves decay as expected. So, the nonlinearity mechanism is a bridge that facilitates the energy exchange among different harmonic modes. An increase of the Goldberg's number enhances the transfer of energy from the fundamental to higher harmonics and between harmonics themselves. Thus, the generated harmonics can only follow the evolution of the fundamental which gives them birth.
However, for a strongly dissipative medium, the absorption is so strong that significant nonlinear effects do not occur. Indeed, the old age region begins at a range smaller than the shock length and once nonlinear effects take place, absorption dominates the behavior of the fundamental and the generated harmonic ( figure 1b, figure 2b). In absorbing media, the exchange of energy is more complicated, because absorption diminishes amplitude with increasing the propagation path and acts as a low pass filter that reduces the energy of higher harmonics (figure 2b).
The evaluation of the relative deviation, for each analytical expression in relation to the numerical solution of the Burgers' equation, is carried out in the following way: The relative deviation, on the selected range, of the analytical expression of the fundamental component (equation (24)) in relation to the numerical solution of the Burgers' equation is less than 4% for glycerol (figure 3b). Thus, for a strongly dissipative liquid, equation (24) can be considered as a good approximation of equation (23). In fact, in this case the Goldberg's number is lower than unity (Table 2); then dissipation becomes important and dominates nonlinear effects.

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As for water, the relative deviation of the analytical expression of the fundamental component (equation (24) For glycerol, the relative deviation of the analytical expression of the second harmonic (equation (25)) in relation to the numerical solution of the Burgers' equation is much weaker than that resulting from equation (26) (figure 4b).
As an example, for 0.1   the deviation obtained from equation (25) is lower than 1% , and that produced by equation (26)  So, for a strongly dissipative liquid, equation (25) is a good approximation of the numerical solution of the Burgers' equation (figure 4b). But, the equivalence of equations (25) and (26) is not checked (figure 4b). Indeed, equation (26) is a good approximation of equation (25) only if 21 (2 ) s l    is weak comparatively to unity.
In the case of water, the relative deviation of the analytical expression of the second harmonic (equation (25) (2 ) s l    is weak comparatively to unity and equations (25) and (26) are equivalent.
Consequently, the preceding comments are also applicable for the analytical expression of the approximated second harmonic (equation (26)) (figure 4a).
According to this study, all these obtained solutions are valid, since the measurement is made near the source; otherwise some assumptions must be taken into account in the analysis of the propagation of finite amplitude acoustic waves in liquids. In addition, the analytical expressions precision depends essentially on the Goldberg's number value.
Moreover, for a strongly dissipative medium, the analytical expressions of the fundamental and second harmonic (equations (24) and (25) ). This hypothesis is not always checked (equation (16)).
As mentioned at the beginning of this chapter, the performance of the simplified model (equation (26)) is interesting, as it can provide a simple, useful model for understanding phenomena in diagnostic imaging. In fact, tissue harmonic imaging offers several unique advantages over conventional imaging. The greater clarity, contrast and details of the harmonic images are evident and have been quantitatively verified, like the ability to identify suspected cysts... Despite the significant advantages offered by harmonic imaging, theory has been only partially explained. According to the theoretical development established in this chapter, equation (26)  On the other hand, as the finite amplitude method is based on pressure measurements of the finite amplitude wave distortion during its propagation, the analytical expressions of the fundamental (equation (24)), the second harmonic (equation (25)) and the approximated second harmonic (equation (26)) lead also to the measurement of the acoustic nonlinearity parameter . However, this method necessitates an accurate model taking into account diffraction effects (Labat et al., 2000;Gong et al., 1989;. The omission of this phenomenon can explain the discrepancies observed of the nonlinearity parameter values measured by the finite amplitude method compared to those achieved by the thermodynamic method (Law et al. 1983;Plantier et al., 2002;Sehgal et al., 1984;. The latter is potentially very accurate. The major advantage of the thermodynamic method is that it does not depend on the characteristics of the acoustic field (Khelladi et al., 2007(Khelladi et al., , 2009).

Conclusion
The validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation can be preset only on the derivation of the Goldberg's number, which can be considered as a reliable indicator for any analysis incorporating nonlinear effects and dissipation.
The obtained numerical results illustrate that the analytical expressions of the fundamental and the second harmonic established with the quasi-linear approximation provide a good approximation of the numerical solution of the Burgers' equation for a propagation medium characterized by a Goldberg number that is small compared to unity.
In the other hand, for a propagation medium characterized by a Goldberg number greater than unity, the analytical expressions of the fundamental and the second harmonic already established with the quasi-linear approximation are not checked and must be redefined.
For that purpose, future studies will concentrate on a new mathematical formulation of the fundamental and second harmonic for a propagation medium characterized by a Goldberg number that is large compared to unity. www.intechopen.com