State of the Art Methods for Estimating Backscatter Coefficients

Abstract

Ultrasound, which is routinely used for diagnostic imaging applications, is mainly qualitative. However, novel quantitative ultrasound techniques are being developed for diagnosing disease, classifying tissue, and assessing and monitoring the application of therapy. Ultrasound is a propagating wave that interacts with the medium as a function of the spatially-dependent mechanical properties of the medium. By analyzing the backscattered wave, various properties of the propagating media can be inferred. The backscatter coefficient, which is a fundamental material property, can be estimated from backscattered ultrasound signals and can be further parameterized for quantifying tissue properties and classifying disease. In this chapter, the history of estimating backscatter coefficients will be explored and different methods and their underlying assumptions for estimating the backscatter coefficient will be compared.

1 Introduction

Ultrasonic scattering from biological tissue is relevant to noninvasive tissue characterization and medical diagnosis. Ultrasound is a propagating pressure wave at frequencies above the audible range in a medium that scatters and reflects acoustic energy based on the mechanical properties of different tissues. Conventional ultrasonic B-mode images are constructed from the envelope-detected, time-domain signals scattered from different tissue structures. In an ultrasonic B-mode image, the frequency dependent information resulting from the scattering media is not utilized. However, by transforming the scattered signals in the frequency domain using a Fourier transform method before B-mode processing, the frequency dependence of the scattered signals can be related to structural properties of the biological media. Often statistical modeling techniques are employed to extract sub-resolution microstructural information using ultrasound with wavelengths larger than the length scale of heterogeneity in the scattering media. To extract microstructural features, such as an effective scatterer size of the dominant microstructure, accurate estimation of the backscatter coefficient (BSC) versus frequency is necessary.

The BSC is defined as the time-averaged scattered intensity in the backward direction per unit solid angle per unit volume normalized by time-averaged incident intensity (cm⁻¹ Sr⁻¹). Therefore, it is a fundamental quantity of a material from which microstructural properties, such as shape, size, organization, concentration and impedance mismatch between scatterers and the surrounding media, can be estimated. BSCs can be used to estimate both microstructural and acoustical properties of the tissues.

BSC-based quantitative ultrasound (QUS) has been successfully used to characterize different aspects of tissue microstructure. Numerous researchers have estimated the BSC from different organs and tissues such as ocular, liver, prostate, pancreas, spleen, renal, myocardial tissue, cardiac tissues and lymph nodes (Nicholas 1982; O’Donnell and Miller 1981; Miller et al. 1983; Lizzi et al. 1983, 1987; Fei and Shung 1985; Insana et al. 1991; Maruvada et al. 2000; Mamou et al. 2011). Topp et al. (2001) differentiated between neoplastic and healthy tissues by studying the frequency-dependent BSC from in vivo rat mammary tumors. Kabada et al. (1980) estimated BSCs to differentiate different excised canine tissues in the frequency range of 1–10 MHz. They observed a higher frequency dependence of BSC in heart (left ventricle) muscle tissue than in liver tissues. Other researchers have used BSCs to differentiate between normal dog hearts and heart tissue from dogs subjected to ischemic injury by coronary occlusion (O’Donnell et al. 1981). They hypothesized that the ultrasonic scattering was sensitive to concentration of molecular collagen and the organizational state of the structural protein. Miller et al. (1983) estimated BSC both in vitro and in vivo and observed differences between normal and ischemic dog myocardium.

Nicholas (1982) published BSCs from various human excised tissues over a frequency range of 0.7–7 MHz. The author used a power law to fit the experimental backscatter from excised human liver, spleen and brain (white matter). The author suggested two basic scattering sites in these tissues on the order of 1 mm and 20–40 μm, respectively. D’Astous and Foster (1986) found that the BSC from tumor tissues in the female breast were virtually inseparable from those of fat at low frequencies, but began to separate at higher frequency regimes. The BSC for parenchyma of the breast was about an order of magnitude above those of fat and infiltrating ductal carcinoma.

Wear et al. (1995) published estimates of BSCs from in vivo human liver and kidney in the frequency range of 2–4 MHz. Lu et al. (1999) estimated BSC in normal human livers and human livers with diffuse diseases in vivo. Maruvada et al. (2000) estimated BSCs from bovine tissues in the frequency range of 10–30 MHz. The authors observed similar frequency dependence in BSCs between kidney and liver. Recently, Ghoshal et al. (2011) observed changes in BSCs from excised beef and rabbit livers with increasing temperature in the frequency range of 10–28 MHz. The authors observed changes in frequency dependence and amplitude of BSC with increasing temperature.

Fei and Shung (1985) used a broadband substitution technique to estimate the BSC of bovine kidney. Turnbull et al. (1989) estimated the BSC in human renal tissue and three types of renal tumors over a frequency range of 3.5–7 MHz. The authors could not differentiate between renal cell carcinoma and normal kidney tissue. Insana et al. (1991) estimated BSCs from dog kidneys using a higher frequency range of 1–15 MHz. They estimated the BSC at different angles with respect to the predominant nephron orientation and observed a frequency dependence of 1.98–2.3 depending upon the angle of incidence.

Variation in BSCs was also observed from the same in vitro tissue depending upon the histochemical fixation process. Bamber et al. (1979) demonstrated that the BSCs were different between fixed and unfixed mammalian tissue using formalin, ethyl alcohol and potassium dichromate as histochemical fixing solution. The authors suggested 4 % formalin and 5 % potassium dichromate are good for consistently preserving the ultrasonic properties within a few percent of those of the fresh tissues.

Many other researchers have utilized the BSC to classify tissues and characterize tissue states. Table 1.1 provides a compilation of some BSC results acquired by various researchers from various tissues and tissue states. Therefore, the plethora of BSC applications found in the literature suggests that the accurate and precise calculation of the BSC from ultrasonic measurements is highly important.

Table 1.1 BSCs in soft tissues reported by various researchers

When correctly implemented, the estimation of the BSC is system and user independent and the BSC is only a function of the tissue properties. To estimate system- and user-independent BSCs it is necessary to account for attenuation and system effects accurately. The typical steps involved in estimating the BSC are shown in Fig. 1.1. Radio-frequency (RF) signals are acquired from a scattering media and gated using a windowing function such as Hanning or rectangular window. Next, the power spectrum of a gated signal is calculated using the Fourier transform. To improve the estimate of the BSC, the power spectra from several gated signals can be averaged providing an average power spectrum. The average power spectrum is divided by a calibration spectrum and then compensated for attenuation and diffraction effects. The resulting frequency-dependent function is known as the BSC. The main challenge to estimating the BSC is to compensate for diffraction and attenuation effects accurately.

The ability to correctly estimate the BSC has been extensively tested across different transducers and systems. In one study, ten laboratories participated in an interlaboratory study to estimate BSCs from well-calibrated tissue-mimicking phantoms using each laboratory’s respective ultrasonic systems, operators and techniques (Madsen et al. 1999). The conclusion of the study was that considerable differences in BSC estimates from tissue mimicking phantoms with the same properties existed between laboratories. These differences were attributed to the different measurement techniques adopted by each group and different methods for calculating the BSC. A better agreement was observed in a subsequent interlaboratory study to compare BSCs from tissue mimicking phantoms (Wear et al. 2005). In this study, the estimated BSCs were compared with theoretical values. The theoretical BSCs were calculated using an exact solution for the scattering from the phantoms (micrometer-sized glass beads were the scatterers) and the properties of the phantoms (Faran 1951).

Fig. 1.1

Flowchart for estimating the BSC

From the apparent lack of agreement between laboratories, more extensive studies of BSC estimation were conducted to determine the sources of error observed between different groups when estimating the BSC. In one of the subsequent studies, BSCs from tissue-mimicking phantoms, where glass beads were used as scatterers, were conducted between two laboratories (Anderson et al. 2010). The main aim was to investigate the interlaboratory comparison of the theoretical model (Faran 1951) to predict BSCs in the frequency range 1–12 MHz from glass spheres embedded in a uniform agar-based background. Overall, the results of the study demonstrated good agreement between the two laboratories (Anderson et al. 2010). Further interlaboratory comparison of BSCs from tissue-mimicking phantoms with glass beads as scatterers were conducted using four different clinical array-based imaging systems (Nam et al. 2012a, b) and again good agreement was observed.

The first interlaboratory studies demonstrated that if attenuation and diffraction effects are not accounted for accurately, the estimate of the BSC can be biased. However, later studies demonstrated that careful measurement techniques along with careful BSC calculation could provide reasonable agreement between groups suggesting that the BSC can be system and user independent. Various researchers have developed analytical and numerical methods to accurately compensate for attenuation and diffraction effects and to estimate the BSC. In this chapter, the different techniques to estimate the BSC will be examined along with the performance associated with each technique.

2 Methods to Estimate System Independent BSCs

2.1 Estimating the BSC Using Single-Element Sources

Sigelmann and Reid (1973) developed the method of estimating backscatter power from a volume of randomly distributed scatterers using a single-element planar transducer. The authors used a substitution method where the backscattered signal from a sample and planar rigid reflector are compared to estimate the volumetric backscattering cross-section. A few years later, Bamber et al. (1979) used a cylindrical tissue sample positioned with its long axis normal to the sound propagation path and recorded the backscattered signal at different angles using a planar transducer that acted both as a source and receiver. The expression for estimating the backscatter cross-section from the cylindrical tissue sample is given by (Sigelmann and Reid 1973; Bamber et al. 1979)

$$\begin{aligned} \sigma _b (\omega ) = \frac{4\alpha \xi }{\Omega e^{-4\alpha r}[e^{\alpha c \tau } - e^{-\alpha c \tau }]}\frac{W_S}{W_R} \end{aligned}$$

(1.1)

where $\alpha$ is the frequency-dependent attenuation coefficient, $r$ is the radius of the tissue cylinder, $c$ is the speed of sound in the tissue, $\tau$ is the duration of the time gate, $\Omega$ is the solid angle subtended by the transducer face at the center of the specimen, $W_S$ and $W_R$ are the measured power scattered from the tissue and the total power returned by a plane reflector and $\xi$ is the reflection coefficient of the plane reflector.

In a later study, Nicholas et al. (1982) used the substitution method to derive the BSC given by

$$\begin{aligned} \sigma _b (\omega ) = \frac{W_r}{W_i}\frac{2\eta R^2}{\beta ^2 A}\left[ \frac{\lambda _0}{\lambda }\right] ^2 \frac{2\alpha }{e^{-2\alpha z_1}[1-e^{\alpha c \tau }]} \end{aligned}$$

(1.2)

where $W_r$ is the total power received at the transducer face due to scattering, $W_i$ is the total power received from a planar reflector, $\eta$ is the intensity reflection coefficient for the reference interface, $\beta$ is the intensity transmission coefficient for the water/tissue interface, $\tau$ is the gate duration, $c$ is the sound speed in the tissue, $R$ is the distance from the transducer face to the gated volume, and $z_1$ is the distance from the surface of the tissue to the beginning of the gated region.

D’Astous and Foster (1986) used a plane wave approximation at the focus to develop the method of computing BSCs for focused transducers. According to their calculation, the BSC may be written as

$$\begin{aligned} \sigma _b (\omega ) = \frac{R_q}{2\pi (1- \cos \theta _T)z}\frac{|S^{\prime }(\omega )|^2}{|S^{\prime }_0(\omega )|^2} \end{aligned}$$

(1.3)

where $R_q$ is the intensity reflectance of the water quartz interface, $\theta _T$ is the half angle of the transducer subtended at its focus, $S^{\prime }(\omega )$ is the energy spectrum of the gated, attenuation corrected signal, $S^{\prime }_0 (\omega )$ is the Fourier transform of the reference echo signal measured using a planar surface, and $\Delta z$ is the axial length of the range gated volume.

Ueda and Ozawa (1985), derived the reference power spectrum using the boundary integral wave equation under the first order Born approximation. An approximate closed form solution for estimating the BSC assuming a Gaussian profile for the transducer radiation pattern was proposed and developed by the authors and is given by

$$\begin{aligned} \sigma _b^{G} (\omega ) = 1.9964\frac{(G_p/2)^2}{1+(G_p/2)}\frac{\gamma ^2 R_1^2}{A_0 \Delta z} |S(\omega )|^2 \end{aligned}$$

(1.4)

where $\omega$ is the frequency, $A_0$ is the aperture area of the transducer, $\Delta z$ is the axial length of the range gated volume, $R_1$ is the on-axis distance between the transducer and the proximal surface of the gated volume, $\gamma$ is the pressure reflection coefficient of the planar reflector, $G_p = kr^2/2R_1$ is the pressure focusing gain of the transducer (Chen et al. 1993, 1994), and $a$ is the radius of the transducer. The power spectrum is defined by

$$\begin{aligned} |S(\omega )|^2 = \frac{\langle |S_m(\omega )|^2\rangle }{|S_0(\omega )|^2}e^{-4(\alpha _m-\alpha _0)(R_1 +\Delta z/2)} \end{aligned}$$

(1.5)

where $S_m (\omega )$ is the Fourier transform of the sample echo signal, $S_0 (\omega )$ is the Fourier transform of the reference echo signal measured using a planar surface, and $\langle |S_m (\omega )|^2 \rangle$ is the average of the power spectrum of several adjacent, gated scan lines $s_m (t)$. The attenuation coefficients for the sample and the reference media are denoted by $\alpha _m$ and $\alpha _0$, respectively. The authors also derived the BSC for a circular piston transducer given by

$$\begin{aligned} \sigma _b(\omega ) = \left( \frac{L(ka,kR_1)}{0.919}\right) 1.9964\frac{(G_p/2)^2}{1+(G_p/2)}\frac{\gamma ^2 R_1^2}{A_0 \Delta z} |S(\omega )|^2 \end{aligned}$$

(1.6)

where $L(ka,kR_1)$ is a frequency-dependent correction factor, which can be calculated numerically for respective transducer characteristics (Ueda and Ozawa 1985).

Insana and co-workers (Insana et al. 1990; Insana and Hall 1990) used a volumetric integral wave equation derived under the first order Born approximation to estimate the power spectrum of both weakly scattering random media and the planar reflector. The BSC can be estimated using

$$\begin{aligned} \sigma _b(\omega ) = 0.3625\frac{\gamma ^2R_1^2}{A_0 \Delta z}|S(\omega )|^2. \end{aligned}$$

(1.7)

Equation (1.7) was derived assuming a Hanning window to gate the RF data. Lavarello et al. (2011) noted an inconsistency in the derivation given by Insana et al. (1990) by a factor of 16. In particular, the source function corresponding to a planar reflector was defined as $\gamma (r_0) = \gamma ^{\prime }h(z_0-z_c)$ (Insana et al. 1990, Eq. (32)) with $h(z_0-z_c)$ a step function located at the center of the gate and $\gamma ^{\prime }$ defined to be the planar reflection coefficient by the authors. However, if $\kappa ^{\prime } = \kappa _0(1+\delta \kappa )$ and $\rho ^{\prime }=\rho ^{\prime }(1+\delta \rho )$ and under the weak scattering assumption (i.e., $\delta \kappa ,\delta \rho <<1$) used by the authors (Insana et al. 1990), then $\gamma ^{\prime } = 4\gamma$. Therefore, $\gamma ^{\prime }$ is equal to four times the pressure reflection coefficient of the planar reflector and not to $\gamma$ as stated in Insana et al (1990). As a result, the expression in Eq. (1.7) is off by a factor of 16. Therefore, Eq. (1.7) should be corrected by multiplying it by 16 (Lavarello et al. 2011).

Chen et al. determined the reference power spectrum using the mirror image method assuming a perfectly reflecting plate and derived the BSC using (Chen et al. 1994, 1997)

$$\begin{aligned} \sigma _b(\omega ) = 2.17\frac{\gamma ^2R_1^2}{A_0 \Delta z}\left\{ |e^-{jG_p}\left[ J_0(G_p)+jJ_1(G_p)\right] -1|\right\} |S(\omega )|^2 \end{aligned}$$

(1.8)

where $J_m(.)$ is the $m$th order Bessel function and $|S(\omega )|^2$ is given in Eq. (1.5).

These different formulations for calculating the BSC were constructed for single-element transducers. The variations in these different formulations can explain some of the variations observed in BSC estimates in interlaboratory comparisons of BSCs from tissue-mimicking phantoms. Additional methods for calculating BSCs from ultrasonic arrays required different normalization procedures.

2.2 Estimating the BSC Using Arrays

Insana and co-workers developed a method to estimate BSCs using array transducers by analyzing the transducer beam directivity function for linear, two-dimensional and annular-array transducers (Insana et al. 1994). The BSC is given by

$$\begin{aligned} \sigma _b(\omega ) = \frac{2}{3L}\left( \frac{\lambda R}{A}\right) ^2\frac{W(f)}{B_H(0,0)} \end{aligned}$$

(1.9)

where $L$ is the range gate length, $R$ is the focal length, $A$ is the active area of the transducer, $W$ is the normalized power density spectrum and $B_H$ is the autocorrelation function for beam directivity. The authors provided the analytical expressions for beam directivity functions and BSC for linear, two-dimensional and annular-array transducers.

Yao et al. (1990) derived the BSC from a sample by comparing the echo data acquired from the sample with the data from a well-characterized reference sample whose BSCs and attenuation coefficients were known, i.e., the reference phantom technique. The BSC using the reference phantom technique is given by

$$\begin{aligned} \sigma _b(\omega ) = \sigma _b^R(\omega ) e^{-4z(\alpha _R(\omega ) - \alpha _S(\omega ))}\frac{|S_S(\omega )|^2}{|S_R(\omega )|^2} \end{aligned}$$

(1.10)

where $z$ is the depth, $\sigma _b^S(\omega )$ and $\alpha _S(\omega )$ are the backscatter and attenuation coefficients of the sample, respectively. Similarly, $\sigma _b^R(\omega )$ and $\alpha _R(\omega )$ are the backscatter and attenuation coefficients of the reference sample, respectively. The frequency domain signal from the sample and the reference sample are denoted by $S_S(\omega )$ and $S_R(\omega )$, respectively. This technique is applicable for any transducer geometry such as single-element focused/unfocused and array systems. The method assumes that the speed of sound is approximately similar both in the sample and in the reference sample.

The various methods to estimate BSCs from backscatter measurements are listed in Table 1.2. This table provides a list of the different diffraction related corrections associated with the BSC estimation.

Table 1.2 Equations for calculating the BSC developed by different authors

2.3 Attenuation Correction

Attenuation correction is one of the major steps to estimate BSCs accurately as shown in Fig. 1.1. Typically, the frequency-dependent attenuation modifies the shape of the power spectrum which will lead to poor estimation of the BSC. Using Eq. (1.5), the normalized power spectrum is written as

$$\begin{aligned} |S(f)|^2 = \frac{|S_m(f)|^2}{|S_0(f)|^2}A(f) \end{aligned}$$

(1.11)

where $A(f)$ is the frequency-dependent attenuation correction function. Extensive research has been conducted to develop attenuation correction functions. Point attenuation correction can be used for short-gated segments and low attenuation coefficients given by Oelze and O’Brien (2002)

$$\begin{aligned} A_{PC}(f) = e^{4\alpha _T(f)z_T + 2\alpha (f)L} \end{aligned}$$

(1.12)

where $z_T$ is the distance between the source and the gated region, $L$ is the length of the gated region, and $\alpha _T(f)$, $\alpha (f)$ are the frequency-dependent attenuation coefficients in Np/cm for the intervening medium and over the gated region, respectively. Specifically, the first term accounts for the round-trip frequency-dependent attenuation losses between the gated region and the source, and the second term account for the attenuation within the gated region. The attenuation over the gated region is assumed to be constant. The point correction term also works well when using a Hanning or Hamming window to gate the RF signal.

O’Donnell and Miller (1981) used similar approximations of small gate length and low attenuation ($\alpha L>1$) to derive an attenuation correction function given by

$$\begin{aligned} A_{OM}(f) = e^{4\alpha _T(f)z_T}\left[ \frac{4\alpha (f)L}{1-e^{-4\alpha (f)L}}\right] . \end{aligned}$$

(1.13)

The above attenuation correction function is derived using a rectangular gating function. The authors chose a short gate length such that the beam function was approximately the same over the length of the gate for each frequency analyzed. The term in the square bracket accounts for frequency-dependent attenuation losses over the gated region.

Oelze and O’Brien (2002) derived another attenuation correction function by averaging the power spectra after integrating an attenuation function over the entire gate. The Oelze and O’Brien attenuation correction function is given by

$$\begin{aligned} A_{OO}(f) = e^{4\alpha _T(f)z_T}\left[ \frac{2\alpha (f)L}{1-e^{-2\alpha (f)L}}\right] ^2. \end{aligned}$$

(1.14)

The authors (Oelze and O’Brien 2002) demonstrated better performance than the O’Donnell and Miller (1981) model for $\alpha L<1$ to estimate the normalized backscatter power spectrum. The authors also derived the attenuation correction function for a Hanning window given by Oelze and O’Brien (2002)

$$\begin{aligned} A_{OO}^{Han}(f) = A_{OO}(f)\left[ 1+\left( \frac{2\alpha (f)L}{2\pi }\right) ^2\right] ^2. \end{aligned}$$

(1.15)

Bigelow and O’Brien (2004) derived attenuation correction functions by coupling the windowing function, beam pattern and attenuation coefficients. The authors used a single order Gaussian function to model the transducer beam pattern in the focal region. The attenuation correction function is given by

$$\begin{aligned} A_{OB}(f) = e^{4\alpha _T(f)z_T}\left[ \displaystyle \int \limits _{-L/2}^{L/2}dz\ g_{\text{ win }}(z)e^{-4\frac{z^2}{w^2_z}+4\alpha (f)z}\right] ^{-1} \end{aligned}$$

(1.16)

where $g_{\text{ win }}$ is the windowing function and $w_z$ is the width of the Gaussian beam in the focal region.

Attenuation correction is one of the major components in estimating BSCs accurately. The frequency-dependent attenuation correction for some of the methods explained above are shown in Fig. 1.2a, b using $L = 1$ cm, $z_T = 0.5$ cm. The comparison between different methods for low ($\alpha _T = \alpha = 0.2$ dB/cm/MHz) and high ($\alpha _T = \alpha = 1.0$ dB/cm/MHz) attenuation coefficients is shown in Fig. 1.2a, b, respectively. The results clearly show that all the methods agree with each other for low attenuation values and start deviating as the attenuation coefficient in the sample increases. Specifically, the Oelze and O’Brien attenuation correction function lies in between the point attenuation and the O’Donnell and Miller attenuation correction function. Therefore, some attenuation correction functions may over- or under-compensate while normalizing the power spectrum to estimate the BSC. A table of the different attenuation correction techniques that have been developed are provided in Table 1.3.

Fig. 1.2

Frequency dependent attenuation correction functions for (a (low attenuating and) b) high attenuating tissues

Table 1.3 Attenuation correction functions

Researchers have also developed techniques to simultaneously estimate BSC and attenuation slopes using a least squares method (Nam et al. 2011). The authors used a power law model for BSCs and developed a three-parameter model to incorporate backscatter and attenuation coefficients. A least squares technique was employed to estimate backscatter and attenuation coefficients simultaneously. By simultaneously estimating the BSC and attenuation, there was no need to compensate for attenuation effects. The authors used a glass bead phantom to compare theoretical and experimental results.

3 Accuracy of Methods to Estimate BSCs

Given the availability of several methods for BSC estimation, it becomes of high importance to evaluate the feasibility of obtaining accurate and consistent BSC estimates. Some of the analytical models developed by various researchers are explained briefly in this chapter. Over the years, authors have developed models to predict BSCs accurately based on their experimental configurations. Lavarello et al. (2011) recently provided a comparison using three different techniques (Insana et al. 1990; Chen et al. 1997; Ueda and Ozawa 1985) shown in Eqs. (1.7), (1.8) and (1.4), respectively, to estimate BSCs from two well-characterized glass bead phantoms using multiple transducers with a wide range of focal properties and frequency bandwidths.

The basic difference between the three formulations for calculating BSCs are in the terms used to correct for diffraction effects. If these diffraction effect terms are isolated, then comparisons of their relative contributions can be conducted. The frequency-dependent functions that are proportional to diffraction correction for each of the three formulations are given by

$$ \begin{array}{ll} D_1(G_p)= 1 \\ D_2(G_p)= \left| e^{-jG_p} \left[ J_0(G_p) + jJ_1(G_p)\right] -1\right| ^2 \nonumber \\ D_3^G(G_p)= 0.92\;\dfrac{(G_p/2)^2}{1+(G_p/2)^2} \nonumber \\ D_3(ka,kR_1)\;=\; \frac{1}{0.92}\;D_3^G(G_p)L(ka,kR_1) \nonumber \end{array} $$

(1.17)

where $D_1,D_2, D_3^G, D_3$ can be derived from Eqs. (1.7), (1.8), (1.4) and (1.6), respectively. The pressure focusing gain is given by $G_p = kr^2/2R_1$. The authors investigated the frequency dependence of three different calibration techniques as shown in Fig. 1.3. At low $G_p$ it can be observed that all the frequency dependent calibrations terms are similar except $D_1$ (Insana et al. 1990) which does not have the frequency dependence.

Fig. 1.3

Comparison of the frequency-dependent calibration terms to estimate system independent BSCs. [Figure taken from Lavarello et al. (2011)]

Interestingly, all the frequency dependent calibration curves asymptotically agree with $D_1$ at high $G_p$. Specifically, the curve corresponding to $D_3^G$ had good agreement with $D_2$ and $D_3$ for low $G_p$ values and with $D_1$ for large $G_p$ values as shown in Fig. 1.3. The curves corresponding to $D_2,D_3$ and $D_3^G$ agreed with $D_1$ to within 1 dB for $G_p>25$, $G_p>25$, and $G_p>5$, respectively. The authors demonstrated that different estimation methods introduced varying frequency-dependent effects to BSC curves, which could have noticeable effects when estimating other parameters from the BSC such as correlation length and scattering strength.

Estimates of the BSC from the two phantoms using: (1) the different transducers to cover a large frequency range, (2) a large range of $G_p$ values, and (3) the three methods (with the Insana et al. 1990 method corrected) are plotted in Fig. 1.4. Two agar phantoms with glass bead inclusions, labeled as "41 μm phantom," contained 47 glass spheres/mm³ ranging in diameter from 36 to 48 μm, and a second phantom labeled as "150–180 μm phantom," contained 20 g/l of glass spheres (approximately 3.2 glass spheres/mm³) ranging in diameter from 144 to 204 μm were used for the experiments. All techniques gave similar overall BSC trends; however, there were differences in terms of the magnitude and slight differences in the frequency dependence of the BSC curves using the different methods. These differences were found to provide different estimates of scatterer size, but most estimates of scatterer size based on the BSC curves were within the range of sizes present in the phantoms as long as the $ka$ range (acoustic wavenumber times the scatterer radius) was above 0.5.

Fig. 1.4

BSC estimates from a (a) 41 μm diameter and (b) a 150–180 μm diameter glass bead phantom normalized using the three methods with the theoretical BSC for the respective phantoms. The BSCs are denoted as $\eta _1-$(Insana et al. 1990), $\eta _2-$(Chen et al. 1997), $\eta _3-$(Ueda and Ozawa 1985), $\eta _3^G-$(Ueda and Ozawa 1985) assuming a Gaussian beam, and $\eta _\mathrm{th }-$Farans model. [Figure taken from (Lavarello et al. 2011)]

In summary, the authors investigated several BSC normalization methods experimentally in order to isolate and determine potential differences in BSC. From experimental results, it was found that significant BSC amplitude differences may be observed depending on the normalization method used, which has a direct effect on scattering strength estimates. The differences in BSC frequency dependence introduced by all methods may result in noticeable variations in effective scatterer size estimates, especially when considering transducers with low $G_p$ values and imaging targets with low $ka$ products. The choice of normalization method may explain much of the variations reported in earlier inter-laboratory comparisons of BSC estimation (Madsen et al. 1999; Wear et al. 2005) and the results suggest that trying to use transducers with a larger $G_p$ value when estimating BSCs will provide the most accurate and consistent results.

4 Conclusion

The ultrasonic BSC is a fundamental system-independent material quantity that can be used to characterize tissues, and monitor and assess therapies. Researchers have used BSCs to characterize different organs/tissues of the body and published results from various mammalian tissues are available for reference for both in vitro and in vivo experimental configurations. For spectral-based QUS techniques to be successful, the BSC is one of the major parameters that needs to be estimated accurately. This is due to the fact that the BSC can be used to infer microstructural parameters such as correlation length and scattering strength. Often, correlation length can be related to cell size, as various researchers have hypothesized cells may be a dominant source of scattering in specific tissues. The scattering strength can be related to the acoustical properties of the tissue.

Due to the importance of estimating system- and user-independent BSCs accurately, numerous studies have been conducted to compensate for system effects and attenuation effects. Depending on the application, particular methods to estimate the BSC may be more appropriate based on experimental configuration and tissue types. We hypothesize that coupling attenuation and diffraction effects in an integral form (Bigelow and O’Brien 2004) can be used to estimate BSCs accurately. The method developed by Chen et al. (1997) is a robust method and used widely to estimate the BSC for focused single-element transducer systems. The reference phantom technique (Yao et al. 1990) is a very powerful technique because modeling of the diffraction pattern of the beam is not required, but the sound speed in the unknown and the reference sample need to be similar for the method to be most effective. The reference phantom technique can be used for both single-element and array transducer systems. The evidence of vast experimental and theoretical investigations conducted to estimate BSCs to infer tissue properties suggests that the BSC can be used for diagnostic and therapeutic applications.