A review on B/A measurement methods with a clinical perspective

In the last several decades, ultrasound propagation has been treated as a nonlinear phenomenon (Duck, 2002). The importance of nonlinear propagation effects has been recognised in medical ultrasound for predicting/modelling heat deposition in tissue, relevant for safety regulations, as well as defining optimal settings for high-intensity focused ultrasound therapy (Carstensen et al., 1980; Cartersen, 1998; Duck and Starritt, 1983; Filonenko and Khokhlova, 2001; Ginter et al., 2002; Goss and Fry, 1981; Jackson et al., 2013; Muir and Carstensen, 1980) and lithotripsy (Cleveland and McAteer, 2007). In addition, tissue harmonic imaging has found wide application, improving the resolution of images with respect to those obtained in fundamental mode (Burns et al., 2000; Ma et al., 2005; Shen and Li, 2001; Zhang and Gong, 2006). Furthermore, stemming from a different underlying mechanism, contrast agents have become a useful tool in the clinic, generating a strong nonlinear signal when isonified and allowing for the extraction of information about vascular perfusion and dispersion (Panfilova et al., 2019; van Sloun et al., 2017).

Apart from the established medical applications of nonlinearity, new ultrasound modalities for the quantification of the parameter of nonlinearity (Madigosky et al., 1981) B/A have been continuously developed for the last few decades in an effort to bring it to the clinic. This parameter characterizes the degree of nonlinearity of a medium. Studies of aqueous solutions have concluded that B/A is influenced by the chemical composition and molecular structure of the solutes (Gong et al., 1993; Sarvazyan et al., 1990; Sehgal et al., 1986a), and were found to be useful to assess the structure of silicone oil used in eye surgery (Zhe et al., 2014). It has also been shown to be useful for tissue characterization, demonstrating distinct values for fatty (Law et al., 1985), malignant, healthy, and cirrotic tissue in the liver (Errabolu et al., 1988; Gong et al., 1993; Sehgal et al., 1984; Sehgal et al., 1986a). Since these different tissue conditions are also associated with different compositions, several papers have developed models defining B/A depending on the constituents of the studied substance (Apfel, 1983; Everbach et al., 1991), leading to works estimating tissue content from its B/A value combined with additional parameters (e.g., speed of sound, compressibility) (Apfel, 1986; Errabolu et al., 1987; Gong et al., 1993; Sehgal et al., 1986a).

In studies with the same chemical composition, it was shown that B/A increases with structural hierarchy of tissue (e.g., intact liver vs homogenized liver) (Law et al., 1981, 1983; Zhang et al., 1991) and that B/A was sensitive to structural changes in tissue caused by disease (Gong et al., 1993; Zhang and Gong, 1999). Moreover, there have been indications that B/A reflects the quasilattice structure of water, i.e., the ratio of bound to unbound water molecules (Sehgal et al., 1986b; Yoshizumi et al., 1987), altered in malignant tissues (Chung et al., 2008; Nikolini et al., 1987) and skin disease (Takenouchi et al., 1986).

B/A has also been used to quantify the nonlinear scattering properties of ultrasound contrast agents (Verboven, 2017; Wu and Tong, 1998; Xia, 2019). From a different perspective, B/A is sensitive to temperature, increasing for most liquids as the temperature increases (Khelladi et al., 2009; Lu et al., 2001) and increasing as tissue is heated (Choi et al., 2011; Jackson et al., 2014; Liu et al., 2008; Lu et al., 2004; Sehgal et al., 1986a). Even though some works show a small B/A increment when tissue is coagulated (Jackson et al., 2014; Saito and Kim, 2011), others state the contrary (Choi et al., 2011; Lu et al., 2004). Several works demonstrated that the B/A profile in tissue follows the temperature profile, generating images of the temperature distribution through B/A (Ichida et al., 1983; Liu et al., 2008; Lu et al., 2004) and suggesting B/A as a tool for high intensity focused ultrasound (HIFU) treatment monitoring (Dongen and Verweij, 2008; Varray et al., 2011b). Besides this, there is evidence that some tumors exhibit an increased temperature compared to surrounding parenchyma (Fear et al., 2002), and that glucose administration is able to raise their temperature by 7 °C (Jain et al., 1984), suggesting that the temperature distribution may help to identify tumor location. This body of evidence fortifies the motivation to develop B/A measurement methods, whether it is for assessment of biological liquids or nonlinear imaging aimed at tissue diagnosis or temperature monitoring.

It was demonstrated that B/A is proportional to the change of sound speed accompanied by an adiabatic change in pressure [Eq. (4)]. This dependency forms the basis of the thermodynamic method for measuring B/A (Beyer, 1960; Everbach and Apfel, 1995; Lu et al., 1998; Sehgal et al., 1984; Zhang and Dunn, 1991; Zhu et al., 1983). In this framework, one records speed of sound changes in the substance when it is subjected to different static pressures. This method is considered to be accurate (Everbach and Apfel, 1995; Hamilton and Blackstock, 1998; Zhang and Dunn, 1991); however, it is also the most cumbersome, requiring a temperature controlled environment and a special cell, where the pressure can be varied. This technique can only yield a single, averaged B/A value for the whole bulk of the studied medium.

The method for aqueous solutions observes the subtle B/A changes for varying solute concentrations of biological compounds (Chalikian et al., 1992; Sarvazyan et al., 1990). Modifying the setup for the thermodynamic method, the inventors of the aqueous solution method enabled accurate velocity measurement for a small sample volume of 1 mL and derived an equation with differential parameters between the solution and the solvent, rather than absolute values. Keeping the main disadvantages of the thermodynamic method, this technique is the most accurate of all B/A measurement methods.

The finite amplitude method (FAM) exploits the dependency of the speed of sound on B/A and the excess medium density (or particle velocity, or excess pressure). Due to this dependency [Eq. (5)], the peaks of the sinusoidal US wave travel faster than the valleys. This leads to nonlinear distortion of the wave, equivalent to the generation of higher harmonics in the frequency spectrum. Consequently, nonlinear attenuation of the wave increases due to energy transfer from the fundamental component to the higher harmonics, which experience stronger attenuation than the fundamental. This way, the energy of a nonlinear wave is attenuated to a higher extent than that of a small-signal linear wave. FAMs exploit all these alterations, quantifying distortion through direct observation of the wave profile (Hunter et al., 2016; Mikhailov and Shutilov, 1959; Takahashi, 1995), through harmonic content (Adler and Hiedemann, 1962; Beyer, 1960; Fujii et al., 2004; Gong et al., 1989; Law et al., 1985; Liu et al., 2008; Shklovskaya-Kordi, 1963; Varray et al., 2011b; Wallace et al., 2007; Zhang and Gong, 1999; Zhang and Dunn, 1987) or by observing nonlinearly induced attenuation (Byra et al., 2017; Hikata et al., 1980; Kashkooli et al., 1987; Nikoonahad and Liu, 1989). This family of methods counts the largest number of publications of all. Even though FAMs are less accurate than the thermodynamic method, they require a much simpler measurement setup and have high potential for a clinical application, enabling B/A tomography for in-transmit measurements and a few echo-mode imaging strategies.

The parametric array method requires transmission of two, typically collinear, beams that generate secondary waves at the sum and difference frequencies. The amplitude of these waves is proportional to the medium B/A (Barrière and Royer, 2001; Bereza et al., 2008; Nakagawa et al., 1984; Zhang et al., 2001a). The secondary beams are narrow, less prone to diffraction than those observed with FAM, and do not have side lobes. Parametric array tomography allows for higher resolution, compared to finite amplitude tomography (Gong et al., 2004; Wang et al., 2003). No echo-mode imaging has been performed with this method.

The pump wave method registers the speed of sound change of a low-amplitude high-frequency wave when another high-amplitude low-frequency wave modulates the pressure in the medium (Ichida et al., 1983; Kato and Watanabe, 1994; Sato et al., 1985). Uniquely, this method allows for a reconstruction of the B/A profile along the path of the low-amplitude high-frequency wave from a through-transmission measurement. A particular case of this method, the second order ultrasound field technique (SURF), has been utilized to acquire echo-mode images representing the B/A distribution (Fukukita et al., 1996; Kvam et al., 2019b).

The method of phase conjugated beams (Krutyansky et al., 2007; Preobrazhensky and Pernod, 2003) utilizes a wave phase conjugater to reverse the beam insonating it and reradiate its amplified version back to the source. The amplitude of the harmonics of the reradiated beam reflects the B/A of the propagation medium. Phase conjugation provides the unique capability to compensate for phase distortion of the wave and achieves high-quality retrofocusing. Only C-scan images of isoechogenic phantoms have been acquired with this method. The possibility of echo-mode imaging is excluded.

The vast body of literature devoted to nonlinear ultrasound has already provided material for several review papers. Beyer (1973), Duck (2002), Hamilton and Blackstock (1998), Muir and Carstensen (1980) discuss the origins of nonlinearity and the way it manifests itself with further consequences in practical applications. Some of these (Beyer, 1973; Hamilton and Blackstock, 1998) introduce separate equations for different states of matter: gas, liquid, and solids. Hamilton and Blackstock (1998) and Naugolnykh (2009) wrote historical reviews on the evolution of nonlinearity in ultrasound. Zheng et al. (1999) wrote a review on material characterization with the help of nonlinear acoustics, devoting a significant portion of it to solids. (Bjørnø, 1986, 2005, 2010; Hamilton and Blackstock, 1998; Zhang and Gong, 2006) are review papers that summarize the progress in B/A measurement methods either over a short time span, devote their attention mainly to a specific measurement strategy, or provide a brief general overview of the main concepts of the existing methods. A concise review of most techniques for B/A measurement has been given by Sato and Yamakoshi more than 30 years ago (Sato and Yamakoshi, 1986). Varray et al. (2011a) presented a review of FAMs that have the potential to be extended to echo-mode regarding the parametric array and pump wave method as one of the above.

This review aims at presenting an extensive and comprehensive up to date overview of B/A measurement and B/A imaging methods of liquids and liquid-like media (e.g., tissue). Importantly, it gives more focus to methods that are relevant for a medical application and discusses the most common pitfalls in this context. By identifying blind spots and limitations we aim at suggesting directions of research that may bring B/A to the clinic. This review paper is further separated into the following sections: Sec. II, theoretical background; Sec. III, Thermodynamic method; Sec. IV, method for aqueous solutions; Sec. V, FAM; Sec. VI, parametric array; Sec. VII, pumping waves; Sec. VIII, phase conjugate beam, and Sec. IX, conclusion.

Section II explains the origin of B/A, presents the most utilized wave equations of nonlinear acoustics, and gives a short overview of the main B/A measurement groups of methods. The sections devoted to various methods (Secs. III–VIII) start with a short introduction of the governing equations and the first published works, followed by the resulting developments, subdivided depending on the adopted measurement strategies. In cases where this has been accomplished, the sections are concluded with studies that presented B/A images. Since the review introduces the basic theory required to understand the physics behind the presented measurement strategies, it is also suited for a reader who is new to nonlinear ultrasound.

An ultrasound wave consists of a series of compressions and rarefactions. Linear acoustics views density as linearly dependent on pressure. However, this is an approximation, and ultrasound propagation, in general, is a nonlinear process. The adiabatic equation of state expresses the pressure-density relation with the Taylor expansion series (Beyer, 1960; Coppens et al., 1965; Hamilton and Blackstock, 1998),

Here, P and P₀ are instantaneous and hydrostatic pressures, ρ and ρ₀ are instantaneous and equilibrium densities of the medium under investigation and the partial derivatives are taken about the equilibrium state (indicated by the subscript 0) and constant entropy (indicated by subscript s). One can define

where c₀ is the small-signal speed of sound, and

making

The relative importance of second-order nonlinear effects to linear effects can be expressed with the nonlinear parameter B/A or the alternative nonlinear coefficient (Varray et al., 2011a), expressed as $β=1+(B/2A)$ for liquid and liquid-like media (e.g., tissue).

As stated previously (Beyer, 1973; Hamilton and Blackstock, 1998), one may differentiate Eq. (1) by ρ and, by substituting the speed of sound $c2=(∂P/∂ρ)s$⁠, obtain

for a plane progressive wave, illustrating that the local speed of sound c is dependent on B/A and u, the particle velocity. The former is an oscillating disturbance, induced by ultrasound propagation. This explains the origin of accumulating wave distortion, leading to saw-tooth waves: the compressional part of the wave (the high density region, with positive particle velocity and positive excess pressure) travels faster than the rarefractional part, contributing to wave distortion proportionally to B/A.

Tables I and II summarize the measured B/A values for liquids and animal tissues. As one can see, at atmospheric pressure and room temperature B/A is in the range of 5–11 for most liquids and liquid-like media.

To obtain the equations governing the propagation of ultrasound waves in fluid homogeneous media one may refer to the equation of motion, the continuity equation, the heat transfer equation, and the equation of state [Eq. (1)] (Naugolnykh and Ostrovskii, 1998). For an ultrasound wave, together these equations describe the relationship between the spatially varying quantities of pressure, particle velocity, density, as well as heat transfer, related to loss. When these equations model ideal fluid (lossless fluid) and only linear terms are kept, one may derive the well-known wave equation Crocker (1997),

in one-dimensional (1D) space, where we chose to describe the pressure variation P, dependent on the coordinate z and time t. When second order nonlinear effects are considered [e.g., equation of state given in the form of Eq. (1)] and propagation in three-dimensional (3D) space is addressed, one can obtain the Westervelt equation (Westervelt, 1963),

where $∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2$ is the Laplacian. The term on the right-hand of Eq. (7) models cumulative nonlinear effects, while local nonlinear effects are neglected here (Hamilton and Blackstock, 1998; Jeong et al., 2016), implying that the Westervelt equation is valid at propagation distances further than a few wavelengths from the source. Equation (7) models sound propagation of plane waves or quasi-plane waves, like directional beams (Devaney, 1980; Hamilton and Blackstock, 1998) in homogeneous lossless media. It was further expanded to include loss in a weakly thermoviscous fluid (Hamilton and Blackstock, 1998; Naugolnykh and Ostrovskii, 1998; Sapozhnikov, 2015; Szabo, 1994b; Tjotta and Tjotta, 1981),

where $δ=2c03α/ω2$ is sound diffusivity, proportional to α, the attenuation coefficient. Importantly, loss in a weakly thermoviscous fluid assumes attenuation to be proportional to the squared frequency of the wave $f2$ (Hamilton and Blackstock, 1998; Naugolnykh and Ostrovskii, 1998; Szabo, 1994b; Tjotta and Tjotta, 1981). This is valid for some liquids (e.g., water, certain oils; Chanamai and McClements, 1998; Chavrier et al., 2006; Naugolnykh and Ostrovskii, 1998; Szabo, 1994a); however, most biological media exhibit a nearly linear attenuation-frequency dependence (Cai et al., 1992; Duck, 1990; Goss et al., 1979; Purrington and Norton, 2012). Alternative time-domain equations exist, incorporating arbitrary attenuation (Cai et al., 1992; Szabo, 1994b), more appropriate for, e.g., tissues. No exact analytical solution to the Westervelt equation exists. However, expressions in the form of integrals have been obtained in Jeong et al. (2016) for the case of weak nonlinearity with the help of the Green's function and when approximating the source pressure as a sum of Gaussian beams (Wen and Breazeale, 1988).

A somewhat simpler description of nonlinearity and attenuation was provided by Zabolotskaya and Khokhlov (1969) and Kuznetsov (1970). Derived from the same original equations as the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [Eq. (9)] makes the additional assumption of the parabolic approximation that holds for “acoustic sources which are many wavelengths across and for field points that are not too close to the source or too far off axis” (Jeong et al., 2016),

Here, $∇tr2=∂2/∂x2+∂2/∂y2$ is the transverse Laplacian and τ is the retarded time $τ=t−(z/c0)$⁠. The terms on the right side of the equation from left to right represent wave attenuation, nonlinearity, and diffraction effects (Zhao and McGough, 2014). The diffraction effect describes the deviation of the field from a plane wave, due to finite source geometry and is somewhat elaborated on in Sec. V B 1. An explicit solution to the KZK equation for the case of weak nonlinearity has been derived in Froysa (1994), Jeong et al. (2015), and Ji et al. (2011).

The simplest equation describing combined effects of nonlinearity and thermoviscous loss is the Burgers equation (Burgers, 1948),

It considers only plane progressive waves, and therefore, does not account for the diffraction effects. Just as for the Westervelt equation, another form of Eq. (10) can be adopted for biological tissues and liquids, accounting for an arbitrary frequency dependence of attenuation (Blackstock, 1985; Chavrier et al., 2006; Chen and Holm, 2004). The lossless Burger's equation is derived by setting the attenuation coefficient α = 0 (and therefore δ = 0) to zero,

The exact solutions to the lossy and lossless Burger's equations are known, derived by Fubini-Ghiron (1935) and Keck and Beyer (1960), respectively. These solutions are presented further in this review (Sec. V B). Unlike the Burgers equation, the Westervelt and KZK equations are mainly solved numerically, providing much more accurate predictions of the pressure field (Demi et al., 2011; Doinikov et al., 2014; Purrington and Norton, 2012; Zhao et al., 2014). At the same time, solutions of the Burgers equation provide simple formulas for the fundamental and higher harmonic pressures. Diffraction affects can be accounted for in the solutions in a post hoc manner.

It has been shown in Eq. (4) that B/A is proportional to the change in sound velocity occurring with an isentropic (adiabatic) change of hydrostatic pressure. The term isentropic refers here to a process where no heat or matter is abstracted or conveyed to the system from outside. Once the derivative in Eq. (4) is expanded, as previously demonstrated (Beyer, 1960; Hamilton and Blackstock, 1998; Rudnick, 1958),

B/A can be expressed through the change in sound velocity caused by isothermal pressure and isobaric temperature (T) changes. The parameter $q=(1/V)(∂V/∂T)P$ is the isobaric volume coefficient of thermal expansion and C_P is the specific heat at constant pressure. B/A is sometimes referred to in the literature as the adiabatic nonlinear parameter, while the first term on the right-hand side of Eq. (12) is referred to as the isothermal nonlinear parameter (Errabolu et al., 1988; Varray, 2011) $(B/A)′$⁠,

The remaining term in Eq. (12) is referred to as the isobaric nonlinear parameter,

In general, the thermodynamic methods can be classified into two groups, where B/A is determined either from Eq. (12), as it was initially done, or from Eq. (4). Within these groups, the strategies to measure the speed of sound c differ. All studies carried out with the thermodynamic method require a velocimeter: a vessel of known length L, comprising the test liquid and the transmitter-receiver equipment, inserted in a liquid-filled pressure vessel, e.g., water (Law et al., 1983, 1985) or oil (Greenspan and Tschiegg, 1957, 1959; Wilson, 1959) that is in turn submerged in a bath with controlled temperature (Fig. 1).

The traditional thermodynamic technique determines B/A via Eq. (14). The required speed of sound measurement can be performed with different techniques, allowing to infer the travel time (time of flight) t_tr of the wave through the velocimeter of known length L (Fig. 1). All the identified works utilizing the traditional thermodynamic technique are summarized in Table III, stating the utilized technique and the parameter directly measured, as well as the specified measurement uncertainty and the investigated media.

The first paper reporting the determination of B/A by the thermodynamic method (Beyer, 1960) used thermodynamic data previously acquired by other scientists (Greenspan and Tschiegg, 1957, 1959; Wilson, 1959) for several liquids. In Greenspan and Tschiegg (1957, 1959) and Wilson (1959), t_tr was inferred with the help of a sing-around circuit (Ficken and Hiedemann, 1956; Zorebski et al., 2005). This circuit allows for triggering of the generator to send a pulse once the preceding pulse is received and, therefore, to infer t_tr through the pulse repetition frequency (PRF). In this work, to improve the accuracy of the measurement Greenspan and Tschiegg (1957, 1959) and Wilson (1959) adjusted the PRF of the generator so that a new pulse was transmitted when the echoes of the previously transmitted pulse were superimposed on the receiver. This way, the device allowed determining the speed of sound from the pulse transit time t_tr, inferred from the PRF and the distance travelled by the pulse (L) equal to twice the length of the vessel,

Hagelberg et al. (1967) and Holton et al. (1968) performed the speed of sound c measurement in water with the pulse-echo method: using one transducer as the source and receiver, where t_tr was inferred from the interval between echoes reflected from an acoustic mirror positioned at the other end of the vessel.

Law et al. (1983) measured the coefficient of nonlinearity of biological solutions and soft tissues. Since for the studied substances, values of q and C_p in Eq. (12) were not known, the authors used the well-known values for water: the values for tissues measured in former studies showed a difference with water up to 30%, and as previously discovered, the term $(B/A)″$ contributed only 3% to the B/A value. The speed of sound was inferred from a direct measurement of the time of flight based on the display of the oscilloscope, showing the driving and received signals. Law et al. (1985) observed no dependence of B/A on solute molecular weight in dextran solutions and a linear dependency on solute concentration. The authors postulated that nonlinearity is a result of solute-solvent interactions. The authors also identified that homogenization of tissue reduced B/A and, in general, B/A showed an increasing trend with the specimen's structural hierarchy. It is worth mentioning that this paper presents a comprehensible diagram and description of the apparatus used.

Zorebski and Zorebski (2009) utilized the pulse-echo-overlap method (Greenspan and Tschiegg, 1957; Zorebski et al., 2005) to determine the speed of sound in lower alkenediols by extracting the PRF at conditions of overlapping echoes coming back from a reflector. The authors acquired B/A at pressures up to 100 MPa and temperatures from 21 to 46 °C. Zorebski et al. (2016) extended the range of studied temperatures, measuring B/A from 16 to 46 °C.

The accuracy of the traditional thermodynamic techniques, taking into account further measurement uncertainties (e.g., temperature and pressure), resulted in a global uncertainty of the B/A estimation within 3% for liquid and 5% for tissue (Table III). The higher uncertainty for tissue samples accounts for the inhomogeneous speed of sound (Law et al., 1985). The studies conducted with the traditional thermodynamic method revealed that $B/A″$ is much smaller compared to $B/A′$ (constituting less than 5% for fluorocarbon fluids (Madigosky et al., 1981), 9% of B/A for the liquids studied in Zorebski and Zorebski (2009), and 12% of the pressure-dependent term for methanol-water mixtures (Coppens et al., 1965). Besides this, $B/A′$ has shown to be always positive, while $B/A″$ can exhibit positive and negative values depending on the studied material. Hagelberg et al. (1967) and Holton et al. (1968) illustrated that for a range of temperatures up to 80 °C and a very wide range of pressures B/A of water has small variability (from 4.1 to 6.8), and increases monotonically with temperature for a pressure value of 1 atm. Coppens et al. (1965) demonstrated that B/A of alcohol mixtures show low variability with temperature. Madigosky et al. (1981) reported fluorocarbon fluids to have the highest nonlinearity reported so far (B/A = 13). Law et al. (1983, 1985) demonstrated the dependence of B/A on the chemical composition of biological solutions as well as the structural hierarchy of biological material.

The isentropic thermodynamic technique, also referred to as the improved thermodynamic method (Gong et al., 1989; Lu et al., 1998; Plantier et al., 2002a), makes use of Eq. (4) rather than Eq. (12). It requires a rapid (1–3 s) change of pressure to eliminate significant heat transfer with the test vessel. This way, the pressure change can be regarded as an isentropic process. As noted by Zhu et al. (1983), this technique is simpler than its predecessor, since it eliminates the need for measurements at different temperature points. Moreover, as Sehgal et al. (1984) indicated, the traditional thermodynamic method requires knowledge of q and C_p, which are “not known with great precision for most soft tissues.” The uncertainty of the early isentropic phase method is estimated to be 4% (Everbach and Apfel, 1995). However, the development of this method is connected to the improvement in the techniques measuring the speed of sound, permitting to reduce the uncertainty to <1% (Table IV). It is velocity measurement techniques that account for the variability of the isentropic phase methods. Therefore, the following section is divided into subsections according to the measured parameter through which the speed of sound is inferred, that being phase $ϕ$⁠, voltage U_p, frequency f, or the time of flight t_tr of a pulse. A summary of the identified works, utilizing the isentropic thermodynamic technique, is presented in Table IV.

The most frequently used are phase measurement techniques, providing more accurate speed of sound estimations compared to earlier methods used in the framework of the traditional thermodynamic technique (Table III). A few others, deriving speed of sound from frequency information, or estimating the time of flight of a pulse, are also used. A detailed explanation follows below.

Among the first publications using the isentropic thermodynamic technique are Emery et al. (1979) and Zhu et al. (1983), measuring B/A of liquids. The authors considered the measurement isentropic, as the applied pressure changes were small (varying from 1 to 2 atm) and too fast (⁠$≈$2.0 s) for a significant heat exchange. The speed of sound change is connected to the wave's transit time t_tr and the distance between transducers L via Eq. (16) (Zhu et al., 1983),

Equation (16) allows us to determine B/A based on Eq. (4) as

To detect the change in transit time t_tr, Zhu et al. (1983) chose to conduct phase measurements: they compared the phase of the received tone burst to that of a reference signal with a phase mixer and acquired $Δttr$ with the help of a delay line. Since $Δφ=ωΔttr$ and $ttr=L/c0$⁠, B/A was determined according to

where L is the length of the ultrasound path through the liquid. Equation (16) forms the basis of all phase measurement techniques, which in general produced more accurate results compared to earlier strategies. A follow-up paper (Gong et al., 1989) of one of the authors of Zhu et al. (1983) used the same version of the method to measure B/A of biological solutions and soft tissues. Gong et al. (1989) confirmed the results obtained by Law et al. (1985) for homogenized liver versus whole liver, as well as biological solutions, observing that B/A increases with the structural hierarchy of the specimen.

Sehgal et al. (1984) presented the pressure jump method. During the pressure increase, the phase of the received signal was monitored. The speed of sound was inferred from Eq. (18). After the pressure change, the system was allowed to equilibrate to the ambient temperature and the authors could also measure the isothermal nonlinear parameter $(B/A)′$⁠. Sehgal et al. (1984) and Sehgal et al. (1986a) used the method described above, transmitting ultrasound in continuous-wave mode. This configuration was “most suitable for attenuating media” (Sehgal et al., 1984) as tissue since high attenuation avoided the formation of standing waves. Sehgal et al. (1984) measured B/A of several human tissues. They demonstrated that fatty breast tissue had a substantially higher value than parenchymal liver tissue, and that multiple myeloma had a substantially lower B/A compared to normal parenchymal liver tissue. Sehgal et al. (1986a) showed that B/A of fatty liver is higher compared to normal, while cirrotic and tumorous B/A is lower than for normal liver. Sehgal et al. (1984) and (1986a) are the only papers, to the best of our knowledge, reporting B/A for malignant tissues.

In Errabolu et al. (1987), Errabolu et al. (1988), and Sehgal et al. (1986b), the adiabatic nonlinear parameter and the isothermal nonlinear parameter were measured with the pressure-jump method, detecting the phase change [Eq. (18)] of a shock-excited pulse. Sehgal et al. (1986b) reported B/A of alcohol-water mixtures. Since alcohols have smaller speed of sound and larger B/A, their addition to water was expected to increase B/A. This was not the case for low concentrations of alcohol. Such an effect on B/A was attributed to the effects of solvent-solute interactions, and alteration in the water-molecule structure due to the addition of alcohol. The authors speculated that, since tissue is composed of 60%–80% of water, changes in the state of tissue due to the change in unbound-bound water ratio must have been reflected in its B/A, creating another opportunity for determining tissue properties based on B/A. Errabolu et al. (1988) demonstrated that human and animal fat tissues are highly nonlinear, with B/A values ranging between 10 and 12. Moreover, in the range of temperatures from 20 to 37 °C, B/A vs temperature exhibits a positive or flat trend. Errabolu et al. (1987) proposed a two-component model (fat and nonfat), able to predict the fat percentage based on the measured B/A and speed of sound. The model was tested for livers, fats, oil, and egg mixtures.

Lu et al. (1998) utilized a highly sensitive phase comparison technique by transmitting a tone burst signal and producing small pressure changes below 2 atm to liquids. This technique (like Zhu et al., 1983) also utilizes a phase mixer. However, different from other methods, the phase change is inferred from the amplitude of the phase detector rather than with a delay line. The measurements are done for small phase changes (below $π/20$⁠), where the output voltage of the phase detector (mixer) U_p is linearly dependent on signal phase $ϕ$⁠. Therefore, the phase change of the detector signal, $ΔUp$⁠, can be described through the phase change $Δϕ$ as

where A₁ and A₂ are the amplitudes of the received and the reference signal that are mixed in the phase detector, and k is a constant characterizing the phase detector. This way, $ΔUp$ can be utilized to estimate B/A as

The maximum pressure change was adjusted to maintain a linear relation between voltage and phase. Lu et al. (1998) report the technique to have an uncertainty of 2.2%.

This strategy was used to measure B/A of several liquids (Khelladi et al., 2009; Lu et al., 1998; Plantier et al., 2002a,b; Zhe et al., 2014). A few of these studies (Khelladi et al., 2009; Plantier et al., 2002a,b, 2003) enabled B/A measurement at pressures up to 100 MPa and temperatures up to 100 °C.

Zhang and Dunn (1991) developed an isentropic thermodynamic method capable of measuring B/A of 4-mL sample volumes. This system is important for situations when samples are products of biochemical reactions with small yields or pathological tissue areas which may be of limited locus/size. The distance between the source and receiver in the velocimeter was only 1 cm. The speed of sound was determined with Eq. (17) by calculating the time delay from cross correlation of the transmitted and received pulses. Since the receiver was in the near field, an error was introduced in the velocity measurements; however, the authors stated that their system was nevertheless able to measure within an error of 0.7%, confirmed by measurement of three mixtures.

Zhang et al. (1991) used the above described setup (Zhang and Dunn, 1991) to determine the influence of structural parameters on B/A. Performing measurements of cat and rat liver tissue as well as suspensions acquired from these livers, the authors altered their structure physically and biochemically, and reached the conclusion that structural dependence of B/A “exists at all three levels of biological structure, viz., the tissue level, the cellular level and the molecular level. The relative contributions due to structural features is 26% at the tissue level, 20% at the cellular level, and 15% at the macromolecular level.”

Everbach and Apfel (1995) automated the measurement of speed of sound, allowing for the performance of thousands of acquisitions on a sample in a reasonable time. The utilized interferometer consisted of a receiving and source transducer. A phase-locked loop circuit was used to correct for the frequency of the transmitted pulse by $Δf$ so that a constant phase relationship was kept at the receiver transducer as the speed of sound changed in the medium. Since $Δf$ required to keep the phase constant can be defined by

$B/A$ can be expressed as

where f₀ and ρ₀ are the initial frequency and density. A pressure of 180 kPa (1.85 atm) was generated in the measurement cell and then released. During this release (3 s) the source transducer transmitted 20-cycle tone bursts at 11 frequencies. Contrary to Sehgal et al. (1984) and Sehgal et al. (1986a), attenuation was an undesired effect for an interferometer, as the measurements were performed for a range of organic and aqueous solutions. The presented method is reported to have an accuracy of about 1%.

Davies et al. (2000) also measured the change in frequency associated with the pressure change [Eq. (22)]. Contrary to Everbach and Apfel (1995), they performed continuous wave phase locking since the continuous wave approach avoids the uncertainty of pulse onset identification. As the system was developed for small-volume samples, the authors encountered near-field problems (Zhang and Dunn, 1991) in an early setup when using phase locking in double-disk interferometers. To overcome this problem, they utilized a cylindrical piezoelectric cavity resonator which, coupled with the developed electronic system, “provided a real-time measurement of the change in speed of sound as function of frequency.” The authors applied a 2-s pressure sweep from 0 to 200 kPa during which 100 data frequencies $f=f0+Δf$ were acquired. This system was reported to produce uncertainty of less than 1% for B/A.

Sarvazyan et al. (1990) noted that the B/A errors were too large to study solute-solvent mixtures with a small amount of solvent. Most biological compounds cannot be diluted in high concentrations in aqueous solutions due to the low solubility. The B/A change in the possible range of concentration is estimated to be approximately 1% (Sarvazyan et al., 1990), within the error span of the most accurate thermodynamic techniques (see Tables III and IV). For this reason, Sarvazyan et al. (1990) developed a differential method that rather than measuring absolute values, estimated the differences between the solute and the solution, which is a common approach in chemical relaxation kinetics (Eggers and Funck, 1973). The accuracy of the relative measurements of the nonlinearity parameter achieved by this method was 0.3%.

The theory was derived by differentiating Eq. (12) for the traditional thermodynamic method, resulting in the following expression:

where capital C is the solute concentration, values attributed to the solvent are denoted by subscript 0, and Δ refers to the difference between the solution and the solvent for the corresponding expressions. $[c], [ρ], [γ], [Cp]$ are relative specific increments of speed of sound, solution density, thermal expansion coefficient, and heat capacity at constant pressure, respectively,

Parameters $[ρ], [γ], [Cp]$ are known from literature, leaving $[c], Δ(∂c/∂P)T$⁠, and $Δ(∂c/∂T)P$ as the values to be measured, given that the solvent parameters with subscript 0 are known.

The setup used in Sarvazyan (1982) is typical for the thermodynamic method (Sec. III). However, several important modifications were made. The measurement cell was represented by a four-channel resonator cell, each with a volume as small as 0.2 mL, all filled with the test liquids. Simultaneous velocity measurements in these chambers were made by the resonator method with an acoustic interferometer. Standing waves form in the cells at resonance frequencies at which the distance between transducers is equal to a whole number of half-wavelengths. This results in amplitude peaks at these frequencies (amplitude frequency characteristic), as well as a particular phase dependence of the received signal on the frequency (phase-frequency characteristic). The authors inferred $Δc$ from the phase frequency characteristic, identifying the shift in resonance frequency $Δf$ to keep the phase constant at its inflection point [Eq. (21)]. The resonator method is the only technique that can be applied for such small-volume samples (Sarvazyan, 1991). Different from previous thermodynamic studies (Sec. III), the authors introduced a reference cell with the solvent, placed in the same thermostated volume as the solution. This lowered the requirement for the temperature stability of the system.

The above work was employed to assess the acoustic properties of solutions of amino acids and proteins, giving insight into the molecular origins of B/A. For instance, the authors discovered that an increase in the number of charged groups that favor bonds with water molecules augments B/A, while CH₂ groups decrease it due to decreased accessibility of water to such molecules. The authors also noted a strong sensitivity of B/A to a replacement of a single atomic group within a molecule, compared to ultrasound velocity and density. These observations demonstrated that B/A may be a useful indicator of molecular structure and hydration of biomolecules in solutions. This method enables measurement of the smallest amounts of sample reported and provides the highest measurement accuracy (0.3%) reported until now (Sec. IX, Table VIII). A follow-up paper studied temperature dependencies of B/A of aqueous amino acid solutions (Chalikian et al., 1992).

As demonstrated in Sec. II, the speed of an ultrasound wave at a point in space and time is dependent on B/A (or β) and the excess density (or particle velocity) at that point and time. Due to this, as the wave propagates it distorts, which was shown to be equivalent to the generation of higher harmonics (integers of the transmitted frequency) in the frequency domain (Fubini-Ghiron, 1935; Keck and Beyer, 1960; Krasilnikov et al., 1957). As higher harmonics grow, the fundamental component is depleted due to the energy transfer from the fundamental harmonic to the higher harmonics.

All FAMs register the wave after a certain propagation distance in a medium (e.g., Fig. 2) and derive B/A from cumulative nonlinear effects observed in the registered signal. The FAMs can be classified into three main groups: deriving B/A directly from the wave's shape, from the second harmonic component, and from the fundamental component. A detailed description of each family of methods follows below. The theory presented in the introductory sections (Secs. V C 1 and V B 1) of the FAM groups concerns only weak nonlinearity (the shock parameter $σ=[2πfP1(0)zβ/ρ0c03] ⩽ 1$⁠), since the pressure amplitudes that demonstrate strong nonlinearity are above the safety regulations for assessment of biological media. Table V, presented below, summarized all the identified FAM works and the utilized strategies to measure B/A.

The earliest works observing the wave shape to infer B/A were performed with optical methods (Fig. 3). When an optical wave propagates in a direction perpendicular to the ultrasonic beam, the initially flat wave front of the optical wave is modulated in phase according to the velocity profile of the US wave, possibly distorted due to nonlinear propagation.

This way, measurement of the diffraction of light allows for the reconstruction of the US wave's velocity profile and quantification of distortion by extracting w₀ or w₁ from its shape (Fig. 4), and thereby derivation of B/A.

This method was implemented by Mikhailov and Shutilov (1960) and Shutilov (1959) and yielded B/A for water and several other optically transparent liquids. More recently, Takahashi (1995) also utilized the waveshape to quantify B/A through the assessment of w₀ or w₁. In this case, the signal was received with a hydrophone, while the source pressure was determined from the diffraction pattern of light emitted by a laser. This way, the setup was a hybrid of those presented in Figs. 2 and 3. Other works are also known (Nomoto and Negishi, 1965), using the diffraction of light to capture the nonlinear distortion of ultrasound waves and extract B/A.

Chavrier et al. (2006), Hunter et al. (2016), Jackson et al. (2014), and Jeong et al. (2016) utilized the setup presented in Fig. 2, where the signal passed through a medium of length L and was received by a hydrophone or a transducer. Chavrier et al. (2006), Hunter et al. (2016), and Jackson et al. (2014) fit nonlinear waveforms of the received pulses with the Burger's equation, where B/A (and α) was the fit parameter. These works use large source transducers (e.g., 10 cm in diameter; see Jackson et al., 2014) to avoid edge diffraction effects, eliminating the need for diffraction correction and justifying the use of the Burger's equation which cannot account for diffraction. Conversely, Jeong et al. (2016) developed multi-Gaussian beam models based on a quasilinear approximation of the Westervelt equation and the KZK equation (Jeong et al., 2015), including both diffraction and attenuation effects. This allowed describing the pressure fields of the fundamental and 2nd harmonic with no restrictions for the source size or the distance range. From a single measurement set at distances from 2 to 20 cm, the authors extracted the attenuation coefficients at the fundamental α₁ and 2nd harmonic frequency α₂, as well as the β value by fitting a model to the observed pressure profiles. A calibration procedure for $P1(0)$ was required. Works inferring B/A through the wave profile require broadband receivers, able to capture the wave shape accurately.

The amplitudes of the higher harmonics in the preshock region (⁠$σ ⩽ 1$⁠) of a plane wave in a lossless medium are given by Fox and Rock (1941), Fubini-Ghiron (1935), and Hamilton and Blackstock (1998),

where n is a positive integer indicating the number of the harmonic: the fundamental $P1(0)$ at the source and higher harmonics $P2,3,…$⁠; and σ is the shock parameter $σ=2πfP1(0)zβ/ρ0c03$⁠. Equation (25) is the Fubini solution (Fubini-Ghiron, 1935) of the lossless Burgers equation [Eq. (11)]. By expanding the Bessel function as a power series and neglecting the high order terms, the amplitude of the 2nd harmonic can be expressed as

This equation illustrates that the amplitude of the 2nd harmonic increases proportionally to B/A, to the distance z from the source, and the frequency of the transmitted signal f. It also shows a quadratic dependence on the transmitted pressure amplitude $P1(0)$ at the source.

Later, this theory was further developed to include losses in two alternative ways. The first one was based on the assumption that the attenuation of the fundamental and higher harmonics are independent of each other. Furthermore, the change of the 2nd harmonic amplitude was ascribed to its harmonic generation due to nonzero B/A value and its small-signal absorption (neglecting energy transfer to higher harmonic components) (Thuras et al., 1935). The following expression is the solution to the equation describing the change of the 2nd harmonic:

where α₁ and α₂ are the attenuation coefficients of the fundamental and its harmonic. Here, no assumption about the frequency dependence of attenuation has been made, therefore, the formula is valid for liquids and tissues. This expression can be further simplified (Dunn et al., 1982) when assuming $(α2−2α1)z$ to be small,

The previous simplification leads to an error of 1% when the value $(α2−2α1)z$ is $<1/2$ (Dunn et al., 1982). Equation (28) is particularly applicable to tissue since tissue attenuation exhibits a nearly linear frequency dependence (Duck, 1990; Goss et al., 1979).

An alternative way to take attenuation into account was proposed (Blackstock, 1966; Keck and Beyer, 1960) when solving the Burger's equation [Eq. (10)] for the weakly nonlinear case characterized by a Goldberg number $G=2πfP0β/ρ0c03α<1$ (Hamilton and Blackstock, 1998),

where α is the attenuation coefficient of the fundamental wave. The Golberg number reflects the balance between the nonlinear processes and the absorption processes (Duck, 2002). Equation (29) applies to weakly thermoviscous fluids (Sec. II B).

The theory above, as already mentioned, was developed for plane waves. Due to this, most studies use a plane piston source as a signal transmitter and perform the measurements in the near field at distances closer than the Rayleigh distance F_d (⁠$Fd=πr2/λ$⁠, where r is the source radius and λ is the signal wavelength; Kuntz et al., 1983). In reality, however, even the near field of a plane piston source differs from an ideal plane wave, due to interference of signals originated from different locations of the source (Huygens principle). These effects are diffraction effects (Duck, 2002) and result in a complicated diffraction pattern of pressure amplitude oscillations, demonstrated in Fig. 5. Only a very small triangular region of the near field remains truly a plane wave: in the proximity of the source the plane wave and the edge wave do not yet interfere (Kramer et al., 1988). It is also worth mentioning that the second harmonic beam is somewhat narrower than the fundamental beam. This can be fully appreciated at greater distances in Duck (2002), where the pressure distributions are presented up to the far field.

The deviation from plane wave theory can be accounted for by the diffraction correction term. For example, Dunn et al. (1982) considered both attenuation and diffraction effects, leading to the following expression for the 2nd harmonic:

where F(z) is the diffraction correction factor, described in detail in Ingenito and Williams (1971) and Williams (1951). Alternatively, the diffraction corrections developed using Multi-Gaussian beams have also been utilized (Haumesser and Meulen, 2019; Jeong et al., 2017; Jeong et al., 2015; Jeong et al., 2016).

The theory above serves the basis of the FAMs. Numerous works have estimated B/A through 2nd harmonic measurement [Eqs. (26)–(30)]. In most of these methods, the 2nd harmonic is measured in the near field of a plane piston source, enabling the plane wave approximation. Therefore, these conditions are implied throughout this section (Sec. V B) unless otherwise stated. These methods are presented in two sections: measurements of homogeneous media (Sec. V B 2) and heterogeneous media (Sec. V B 3).

The studies discussed in this section assume invariant B/A along the beam path in the medium, yielding a single B/A value. All the strategies suitable for bulk estimation of B/A through 2nd harmonic measurement are presented below. All the presented works, unless otherwise mentioned, utilized the setup as in Fig. 2. Unlike the wave shape method, in this case, the receiver has to be responsive at the 2nd harmonic of the fundamental signal, transmitted by the source.

Equation (29) formed the basis for one of the early B/A measurement approaches (Beyer, 1960; Keck and Beyer, 1960). One can find the distance to the source where the 2nd harmonic reaches its maximum value by differentiating Eq. (29) with respect to z and finding the zero crossing. Once the coordinate of the 2nd harmonic maximum is determined experimentally, B/A can thus be calculated.

Other strategies utilized the lossless formulation in Eq. (26). It has been experimentally verified (Krasilnikov et al., 1957) that at distances close to the source, the dependence of the 2nd harmonic on the distance z and the pressure at the source P(0) follows the lossless model described by Eq. (26) also in a dissipative medium. Therefore, one can derive B/A by extrapolating the 2nd harmonic amplitude at zero distance from the source,

For this purpose, Adler and Hiedemann (1962) utilized the optical setup (Fig. 3) to measure the transmitted fundamental signal $P1(0)$ and the 2nd harmonic at different distances from the source transducer. Extrapolation of $[P2(z)/zP1(0)2]$ to z = 0, as in Fig. 6, yielded B/A for water and m-Xylene.

Law et al. (1981) and Dunn et al. (1982) utilized a much simpler setup (Fig. 2), measuring the 2nd harmonic amplitude P₂ and $P1(0)$ with an additional planar transducer, also employed in all the following studies unless otherwise mentioned. For this purpose, a calibration procedure of the receiver is required, allowing us to convert measured voltage to pressure values and to determine $P1(0)$⁠. Extrapolating $[P2(z)/zP1(0)2]$ to zero distance from the source, they obtained B/A for several biological solutions as well as whole liver. When comparing their values to glycerol and glycerin, previously obtained by the thermodynamic method (Coppens et al., 1965), they concluded that this technique has an uncertainty of 10%.

Dunn et al. (1982) introduced attenuation in the expression for the 2nd harmonic [Eq. (28)], showing that $[P2(z)/zP1(0)2]$ decreases exponentially with distance in the proximity of the transducer. Moreover, they were the first ones to consider diffraction effects [Eq. (30)] that manifested themselves in the calibration procedure (Williams, 1951) of the receiver as well as in the quantification of the 2nd harmonic (Ingenito and Williams, 1971). Law et al. (1985) followed the same procedure as Dunn et al. (1982) to measure B/A of tissues, requiring the tissue sample to be sliced in order to obtain measurements at several distances.

The extrapolation scheme has an important advantage: it avoids measurements of the attenuation coefficient of the medium. As illustrated in Fig. 6, even if the attenuation value of the medium under study changes, the curves intersect at the same point, provided that the other acoustic parameters are equivalent. This method is sometimes referred to as the pullback method (Wallace et al., 2007).

Equations (27)–(30) also enable measuring B/A from a single measurement once the attenuation coefficients at the fundamental α₁ and harmonic α₂ frequencies are known. After first estimating the attenuation coefficients, Cobb (1983) estimated B/A, taking diffraction of the 2nd harmonic (Ingenito and Williams, 1971) into account [for the case of transducers of equal size, Eq. (27)]. The authors noted that the approximation given by Eq. (28), justifying the extrapolation scheme, may lead to errors of 7% in samples such as water and glycerin (Dunn et al., 1982). Zhang and Dunn (1987) utilized Eq. (30) to estimate B/A from a single acquisition, finding that there is no significant difference between B/A of in vivo and ex vivo cat livers. In Haumesser and Meulen (2019), the second harmonic was measured after its reflection from an aluminium plate that replaced the receiver in a typical through-transmission setup (Fig. 2). The reflection mode method increased the propagation path of the signal in the studied medium. This way an accurate B/A measurement could be acquired for a reduced amount of the investigated liquid. Li et al. (2017) presented a through-transmission method for B/A measurement of fluids, utilizing focused transducers to transmit and receive acoustic signals. Diffraction corrections were derived for focused receiver and source and a simple calibration procedure of the receiver was proposed. This enabled estimating B/A of water within an 8% error for multiple distances.

Panfilova et al. (2018) and Pantea et al. (2013) determined B/A of water based on Eqs. (27) and (28) by fitting a line to the dependence of the 2nd harmonic amplitude on the fundamental pressure $P1(z)$⁠. They require no attenuation measurement since they account for attenuation by using the fundamental and 2nd harmonic values at the measurement point. Pantea et al. (2013) reported more extensive work, performing the fit for a larger range of distances and making use of the extrapolation scheme or pullback method (Sec. V B 2 b).

Meulen and Haumesser (2008) implemented this method in echo-mode by the employment of a reflector, concluding that the reflector must have an impedance higher than that of the studied fluid. This ensures that the 2nd harmonic components generated during forth and backpropagation interfere constructively, and B/A quantification is possible. Otherwise, these components add out of phase.

Similarly, Chitnalah et al. (2007) recorded the 2nd harmonic in reflection mode at different source pressures but reflected from the interface of two liquids. They took into account attenuation and diffraction correction, decomposing the source function into a series of Gaussian beams.

Transmission mode comparative method. The first record of the comparative method was found in Gong et al. (1963). Its aim is to avoid absolute pressure measurements and therefore a receiver calibration procedure. The idea of the method is to compare the 2nd harmonic signal relative to that generated in a medium with a known B/A when the measurements are performed at the same source pressure and distance L from the source (Fig. 2). This way, when describing the 2nd harmonic component with Eq. (26) for both media, and taking their ratio, one can derive that

where the subscripts m and 0 denote the medium under study and the reference medium, respectively. In this first study, the reference liquid was acetone, with a density and speed of sound close to that of the studied nitrogen. However, most studies use water as the reference medium, as its B/A is well known and attenuation can be neglected. Gong et al. (1984) and Zhang et al. (1991) utilized the same technique, with the latter work also accounting for diffraction effects.

Wallace et al. (2007) used a hybrid of the extrapolation and comparative methods. The authors compared the ratio $[V2(z)/zV1(0)2]zV1(0)=0$ of isopropanol to that of water, where V₁ and V₂ are the voltages associated with the fundamental and the 2nd harmonic. Differently from previous works, the authors took into account the fact that the generated source pressure actually depends on the medium where the source is immersed (also explained in Jackson et al., 2014), defined by the transmission coefficient between the transducer material and the medium [see also Eq. (34)]. Besides this, they introduced a steel delay line in front of the transmitting transducer. Since steel has a speed of sound that is approximately four times higher than in water, the resulting diffraction pattern is compressed compared to that in water. this allowed positioning the natural focus inside or right after the path in steel. As a result, Wallace et al. (2007) satisfy the plane wave approximation by measuring in the far field, requiring no further diffraction corrections.

Yu et al. (2014) performed an evaluation of the comparative technique, including simulations and a phantom experiment for homogeneous tissue. The simulation study utilized a linear probe as the source and measured β for 3 tissue types at different distances. The experiment showed that the estimated β did not depend significantly on the measurement distance, demonstrating the method's robustness. The authors took into account tissue attenuation [Eq. (28)]. In this work, the formula for B/A was derived from the special case of the comparative method: the finite amplitude insert substitution (FAIS) technique explained below in Sec. V B 2 f). This resulted in an erroneous formula. However, since the transmission coefficients of the studied tissues were close to unity, the introduced error was very small. The authors assessed their technique to be within 3% of uncertainty.

Reflection mode comparative method. Kourtiche et al. (2001) used the same transducer to transmit the fundamental and receive the 2nd harmonic reflected from a reflector plate. In this case, the 2nd harmonic generation occurs both ways. The authors also performed an analysis of the electromechanical behavior of the transducer at different transmit frequencies. A transducer generates a “clean” and strong fundamental around its resonance frequency. At the same time, it must be sensitive enough to detect the 2nd harmonic of the transmitted signal. Therefore, a favorable trade-off frequency region must be determined for B/A estimation. Besides this, a transducer usually shows different sensitivities in transmission and in receiving. These are influenced by the impedance of the transducer, which varies with signal frequency and the impedance of the medium. The authors showed that neglecting these effects can lead to errors in β estimation and that there is a frequency range where B/A can be estimated most accurately. This work is relevant for all echo-mode developments as well as transmission mode measurements utilizing identical transducers as source and receiver.

Transmission mode FAIS. The FAIS method was motivated by the idea of avoiding absolute pressure measurements, similar to the comparative method. Some authors classify the comparative method (Sec. V B 2 e) as a particular case of FAIS (Yu et al., 2014), which in reality was developed later. In light of this, it should be noted that the equation utilized for the comparative method should not be derived as a general case of the FAIS. With FAIS, the medium under study is inserted in a water path and does not have the same length as the reference medium (water), in most cases resulting in 2 reflection interfaces (Fig. 7). Initially, Shklovskaya-Kordi (1963) developed this method based on the Fubini solution [Eq. (26)] for measurement of internal pressure through B/A. When the source pressure and distance between the source and receiver (L) are fixed (in this work kept within the near field distance to adhere to the plane wave approximation), the amplitude of the 2nd harmonic in a water path P₂₀, whose B/A is known, is compared to that when a sample of thickness d is inserted in this path $P2m$⁠. Through the ratio $P2m/P20$⁠, we can calculate B/A as (Gong et al., 1984),

where $D′$ and $D″$ are the sound transmission coefficients from water to the sample and from the sample to water, respectively,

Equation (33) is only valid when the investigated medium is nearly lossless. Moreover, the formula was derived merging the 2nd harmonic contribution from water before the sample and after it, which is justified if the acoustic impedance of the medium is close to water. Alternatively, the medium can be placed very close to the receiver, making the 2nd harmonic generation in the water path after the sample negligible. Because of this, the sample is conventionally positioned close to the receiver in all FAIS modifications. As for the comparative method, FAIS does not require transducer calibration for recovery of the absolute pressure values since the pressure ratio term is equivalent to the received voltage ratio. Gong et al. (1989) built on this method, introducing an attenuation correction in Eq. (28) aimed at measuring B/A of tissue, making the assumption that $(α2−2α1)z$ is small, valid for most tissues as they have a nearly linear frequency dependence (Duck, 1990; Goss et al., 1979). The resulting expression is

Here, one can see that a measurement of the attenuation coefficient at the fundamental α₁ and the harmonic α₂ is required for B/A determination. Besides this, Gong et al. (1989) also introduced a diffraction correction which for most biological tissues (speeds of sound: 1400–1600 m/s) they quantified to be within 2%. However, it must be noted that, in general, for liquids with a speed of sound further away from that of water, the diffraction correction can introduce significant errors (e.g., 5% error for ethanol).

Wu and Tong (1998) measured B/A of contrast agents. Since B/A of contrast agents can be on the order of hundreds and thousands, the harmonic component generated in water can be neglected for the configuration in Fig. 7. This resulted in a simpler expression. No diffraction correction was used, possibly due to the assumption of a similar speed of sound in contrast agents to that of water (not stated).

Dong et al. (1999) derived a formula analogous to Eq. (35), however, they utilized the general expression for attenuation in Eq. (27). Dong et al. (1999) positioned the sample and the receiver in the extreme far field of the source, another region of the piston field where the plane wave approximation may be considered valid. The authors removed all harmonic content generated in the near field by placing an acoustic absorber before the sample. Yet, due to the absorber, the insonifying amplitude is expected to have been low, and the resulting distortion weak. Cortela et al. (2020) and King et al. (2011) utilized the same configuration as Dong et al. (1999) to measure B/A of gellan gum-based tissue-mimicking phantoms. Choi et al. (2011) utilized the same formula as Dong et al. (1999) in a setup where no acoustic absorber was used. They presented the temperature dependence of B/A of porcine liver.

Harris et al. (2007) proposed another setup that allowed avoiding diffraction correction. Their solution was a large source transducer (8 cm in diameter), which provided a broad plane wave region, without diffraction effects. Figure 5 demonstrates such a triangular region in the proximity of the transducer, where the field is stable.

Kujawska et al. (2003) introduced a modified FAIS method. A model of the ratio $P2m/P20$ depending on sample thickness d was fit to experimental data when using a small receiver (0.4 mm in diameter). The best fit of the data provided an estimate B/A for liquids and homogenized tissue. The current approach required no plane wave assumption.

Zeqiri et al. (2015) conducted a detailed analysis of the influence of several factors on the accuracy of B/A measurement with the FAIS, e.g., sample positioning with respect to the receiver, source pressure amplitude, and sample thickness.

Reflection mode FAIS. Lu et al. (2004) utilized FAIS in reflection mode, transmitting and receiving with the same compound transducer whose inner ring served as the source, and outer ring received the signal from a reflector plate. In this case, the authors needed to take into account 2nd harmonic generation in the forward and backward path, making the calculations more complex. It is of notice that for FAIS, the transducer sensitivity is not altered by the studied medium. The authors showed a strong dependence of B/A in porcine liver and muscle on temperature, suggesting the possibility of temperature monitoring through B/A during HIFU.

Nonlinear acoustic microscopy. Acoustic microscopy allows studies of small-volume samples (e.g., 0.1 mL; Saito, 2010). It utilizes high frequency sources (14–19 MHz) in conjunction with acoustic lenses that provide a short focal distance (e.g., 2.3 mm; Germain et al., 1989). Altogether, this produces appreciable nonlinear effects in the focal spot already at such short distances. All the studies in this field have adapted the comparative and FAIS techniques using water as a reference medium and either completely filling the space by the sample, or positioning it only in the focal region, surrounded by water.

Banchet et al. (2000), Banchet and Cheeke (2000), and Germain et al. (1989) developed an acoustic microscope for the measurement of B/A. The 2nd harmonic was detected by a planar transducer. Its generation was assumed to be confined to the focal region and was described with plane wave theory utilizing Eq. (28). This system also allowed measuring sound velocities; however, parameters like density and attenuation needed to be measured beforehand to enable calculations with Eq. (28).

Additional work in this direction was performed in Saito (1993a,b, 2010), Saito and Kim (2011), and Saito et al. (2005). The final setup of the developed system utilized the source as a receiver, detecting the signal from a reflector. The authors made use of Gaussian beam theory that models the field as a series of beams whose spatial pressure distribution is described by the Gaussian function (Kim et al., 2006). This system allowed measurement of B/A as well as linear acoustic parameters, including sample density. At the latest stage of development, Saito and Kim (2011) generated two-dimensional (2D) images of B/A and linear acoustic parameters by mechanically translating the samples of biological tissues (e.g., fat vs nonfat; coagulated vs normal), showing that tissue B/A was variable on a microscale and exhibited different variation patterns than the linear parameters (e.g., attenuation, density). In Saito (2010) and Saito and Kim (2011) the authors observed good reproducibility of their measurements (within 1%) and stated the measurement error to be within 10%, typical for FAMs.

Dong et al. (2006) and Kushibiki et al. (1997) introduced a novel method utilizing frequencies as high as 100–200 MHz. The specimen was positioned between two SiO₂ buffer rods with transducers at their outer ends. SiO₂ has a negative β: its 2nd harmonic shows an opposite phase compared to that generated in liquid. This way, when acquiring P₂ for different sample lengths, at a certain point the 2nd harmonic was cancelled out completely. The authors utilized plane wave theory [Eq. (27)], incorporating diffraction and dispersion in Kushibiki et al. (1997), showing that this gives more stable B/A estimates. The major advantage of this method is that no pressure measurement is needed. However, the observed β for water was 20% higher than the value reported in literature. This was ascribed by the authors to uncertainties in the properties of SiO₂.

The studies discussed in this section consider B/A estimation in heterogeneous media. First, the section performing through-transmission measurements is presented. In this case, the image is reconstructed with computer tomography (CT) once the through-transmission measurement is repeated for several sample rotation and translation configurations (Fig. 8). The reconstruction is obtained by use of the Radon transform with the resulting image resolution being determined by the number of employed rotation angles. The reflection-mode measurement follows the same scheme, however, using the source as the receiver. After transmission through the tissue, the signal is reflected from a reflective plate on the opposite side of the sample. Echo-mode imaging is the last family of methods presented in this section. In this case, the signal is received by the source transducer as it is reflected along its propagation path in tissue, due to tissue inhomogeneities. This distinction between reflection-mode and echo-mode imaging will be further used in this review.

Zhang and Gong (1994, 1999) and Zhang et al. (1996) implemented B/A tomography based on the FAIS method. When considering a heterogeneous medium, the received 2nd harmonic amplitude at a distance L from the source is the result of its propagation path where different values of $β,α1,α2$ correspond to every z,

where the indice m refers to the medium under investigation. In order to avoid conversion to absolute pressure values, the authors also measure the amplitude of the 2nd harmonic $P2(L)$ in a homogeneous reference medium with a known β. The reference medium was water, making it possible to apply the lossless Fubini solution [Eq. (26)],

This way, when the sample is placed in the water path between the source and the receiver, the ratio of the received 2nd harmonics is defined by

The implemented CT system allowed for rotation of the sample and translation of the source and receiver (hydrophone) along the sample length (Fig. 8). The receiver was positioned in the near field of the transmitting transducer. The obtained projection images of the ratio P [Eq. (38)] were transferred to the β domain by the filtered convolution method and then corrected by multiplying with the attenuation matrix describing the sample's attenuation in space (estimated by attenuation tomography with the same setup). This work showed promising results; however, the authors concluded that attenuation and velocity estimates required further improvement. Yu et al. (2014) also simulated B/A tomography based on Eq. (38), modified for the case of an attenuating reference medium and utilizing a filtered back projection algorithm.

Gong et al. (2004) extended their previous work (Zhang et al., 1996) to reflection tomography. The tissue sample was positioned in water between the source and a reflective plate, where the reflective plate replaced the receiver in Fig. 8. Equation (38) was extended, now containing nonlinear generation and attenuation terms for the forward and backward path. Figure 9 presents an example of an image acquired in this work. In this image, we can see a two-layered tissue structure with porcine liver surrounded by porcine fat, submerged in water.

Reflection mode imaging poses additional challenges, compared to transmission tomography. In the latter case, the recorded signal has travelled through the whole bulk of the tissue, and therefore, the effect of varying scatterer density within the medium is averaged out. However, when utilizing echo-mode β imaging, the strength of the reflected echoes is, to a large extent, defined by the scatterer density at each reflection point (Waag, 1984), masking information about other acoustic properties. To cancel out the scatterer effect, scientists normalized the recorded signal to a signal that is assumed to have an analogous scattering pattern. Three different reference signals have been identified in literature. One utilizes an additional signal, transmitted at the 2nd harmonic frequency. Another utilizes the second harmonic amplitude reflected from a scattering homogeneous reference medium. The last reference signal is the reflected fundamental component of the received pulse.

Akiyama (2000) and Fujii et al. (2004) found a way to mitigate the influence of scatterers by assuming that the scatterer distribution affects a signal of a certain frequency in the same way, whether it is the generated 2nd harmonic at $2f0$ of the fundamental at f₀, or whether it is the transmitted fundamental at $2f0$⁠. The authors assumed that the 2nd harmonic is generated only in transmission, and not on the way back when reflected from scatterers due to low amplitudes of the reflected signal. Attenuation was taken into account in both directions of propagation, leading to

where ρ and c are medium properties, assumed constant and $γ(2f0,z)$ is the backscattering characteristic term. The received fundamental when transmitting a 2nd pulse at $2f0$ is also proportional to $γ(2f0,z)$⁠, and $∫α(2f0,z)dz$⁠. Therefore, the ratio of the generated 2nd harmonic to the fundamental at the frequency of the 2nd harmonic cancels out the backscattering and the attenuation at $2f0$ terms. The authors extracted h(z), a parameter defined by B/A, speed of sound and density of the medium with the following equation: