On the estimation of backscatter coefficients using single-element focused transducers

Abstract

The ultimate goal of quantitative ultrasound (QUS) imaging methods based on backscatter coefficient (BSC) estimates is to obtain system-independent structural information about samples. In the current study, three BSC estimation methods were compared and evaluated using the same backscattered pressure datasets in order to assess their consistency. BSC estimates were obtained from two phantoms with embedded glass spheres and compared to theoretical BSCs calculated using size distributions estimated using optical microscopy. Effective scatterer diameter and concentration estimates of the glass spheres were also obtained from the estimated BSCs. One estimation method needed to be compensated by more than an order of magnitude in amplitude in order to produce BSCs comparable to the other two methods. All calibration methods introduced different frequency-dependent effects, which could have noticeable effects on the bias of QUS estimates derived from experimental BSCs. Although in most cases the experimental QUS estimates obtained with all three methods were observed to differ by less than 10%, larger differences are expected depending on both the pressure focusing gain of the transducer (proportional to the ratio of the square of the aperture radius to the product of the wavelength and focal length) and ka range used in the estimation.

INTRODUCTION

The backscatter coefficient (BSC) is a fundamental quantity that contains information about material microstructure. Extraction of microstructural properties through analysis of BSCs has been extensively studied in the past for tissue characterization applications in biomedical ultrasound.1, 2, 3, 4 The ultimate goal of quantitative ultrasound (QUS) imaging methods based on BSC estimates is to obtain meaningful, system-independent information about the analyzed tissues. Therefore, ideally any instrumentation dependence should be removed when estimating BSCs.

Many practical implementation aspects, however, may have an impact on the estimation of BSCs from tissues. Several transducer geometries have been studied in the past including unfocused5 and focused6, 7 single-element transducers, as well as array systems.8, 9 Even for a fixed experimental system, several spectrum calibration methods have been proposed resulting in several methodologies for BSC estimation. Two of the most common ways of compensating for system-dependent effects are the substitution10, 11, 12 and reference phantom13, 14, 15 methods. Finally, several methods for compensating attenuation effects have also been proposed.16, 17

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. To that effect, inter-laboratory comparisons of ultrasonic BSC estimates have been performed in the past18, 19 but only a limited agreement among BSC estimates obtained at different laboratories was observed. The goal of the current study is to evaluate three different BSC estimation methods using the same experimental datasets. This approach allows eliminating experimental discrepancies in order to assess the consistency of these BSC estimation methods and determine under what conditions a particular method may be preferred. The three approaches for obtaining BSC estimates are representative of the methods commonly used in the literature. All measurements were obtained using focused transducers having different center frequencies and focal numbers for the analysis of two well-characterized physical phantoms. Estimated BSCs were compared to the theoretical BSCs obtained given the knowledge of the scatterer distributions in both phantoms.

METHODS

BSC estimation methods

In this section, details of BSC estimation using a planar reflector to compensate for the imaging transducer characteristics will be provided. BSCs are estimated from pulse-echo data corresponding to regions of interest (ROIs) within the sample under study. An acoustic aperture receives pulse-echo pressure waveforms s_m(t) when located at several positions along the Y axis, as shown in Fig. 1. The backscattered data is gated between depths (F − Δz/2) and (F + Δz/2) using a rectangular window, where F is the transducer focal length and Δz is the gate length. BSC estimates from a ROI require the calculation of the normalized power spectrum $|{\stackrel{-}{S}}_0(k)|{}^2$ defined as

$$|{\stackrel{-}{S}}_0(k)|{}^2=\frac{〈|S_m(k)|{}^2〉}{|S_0(k)|{}^2}H(k),$$ (1)

where 〈|S_m(k)|²〉 is the average of the power spectra of several adjacent, gated scan lines s_m(t), H(k) is a function that compensates for attenuation effects, and S₀(k) is the Fourier transform of a reference waveform. The normalized power spectrum $|{\stackrel{-}{S}}_0(k)|{}^2$ can be related to BSCs after properly compensating for the transducer geometry.

Figure 1.

(Color online) Schematic of the configuration for backscatter measurements from a phantom with randomly positioned scatterers.

Three different estimation methods have been studied in this work and are summarized below for completion. All methods assume S₀(k) is obtained by measuring the reflection of ultrasound from a planar reflector located at the center of the ROI and perpendicular to the transducer axis. Furthermore, the methods assume that circular, single-element focused transducers are used to perform the measurements with the center of the ROI located at the acoustic focus of the transducer.

Estimation method 1

The first estimation method used in this study was derived by Insana et al.11 The volumetric integral wave equation derived under the first order Born approximation was used to estimate the power spectrum of both weakly scattering random media (Ref. 11, Secs. 2A, 2B) and the planar reflector (Ref. 11, Sec. 2 C). BSC estimates can be estimated using [Ref. 11, Eqs. (34), (35), and (41)]

$${\stackrel{\wedge }{\eta }}_1^{\mathrm{Hann}}(k)=0.3625\frac{{\gamma }^2{F}^2}{A_0\Delta z}|{\stackrel{-}{S}}_0(k)|{}^2,$$ (2)

where A₀ = πR² is the aperture area of the transducer of radius R and γ is the pressure reflection coefficient of the planar reflector. The expressions in Ref. 11 were derived assuming a Hanning window to gate the radiofrequency data. Different windows can be used by updating the value of B_g(0) in Ref. 11, Eq. (40) which is equal to 0.375Δz and Δz for Hanning and rectangular windows, respectively. Therefore, BSCs can be estimated using a rectangular window as

$${\stackrel{\wedge }{\eta }}_1(k)=\frac{B_g^{\mathrm{Hann}}(0)}{B_g^{\mathrm{Rect}}(0)}{\stackrel{\wedge }{\eta }}_1^{\mathrm{Hann}}(k)=0.375{\stackrel{\wedge }{\eta }}_1^{\mathrm{Hann}}(k).$$ (3)

A notation inconsistency in the development of ${\stackrel{\wedge }{\eta }}_1(k)$ can be tracked back to Ref. 11, Sec. 2 C, where the corresponding equations for spectral calibration were derived. In particular, the source function corresponding to a planar reflector was defined as γ(r₀) = γ′h(z₀ – z_c) [Ref. 11, Eq. (32)] with h(z₀ – z_c) a step function located at the center of the gate and γ′ defined to be the planar reflection coefficient by the authors. However, if κ′ = κ₀(l + δκ) and ρ′ = ρ′(1 + δρ) and under the weak scattering assumption (i.e., δκ, δρ ≪ 1) used by the authors

$$\gamma \text{'}=\gamma \text{'}{}_{\kappa }-\gamma \text{'}{}_{\rho }\approx \delta \kappa -\delta \rho \approx 2\frac{\Delta Z}{z_0}\approx 4\frac{Z-Z_0}{Z+Z_0}=4\gamma ,$$ (4)

where Z₀ and Z = (Z₀ + ΔZ) are the characteristic acoustic impedances of the background and planar reflector, respectively. Therefore, γ′ is equal to four times the pressure reflection coefficient γ of the planar reflector and not to γ as stated in Ref. 11. As a result, the expression in Eq. 3 is off by a factor of 16. A corrected version of the BSC estimation method is given by

$$\begin{array}{c}{\eta }_1(k)=16{\widehat{\eta }}_1(k)\\ =2.17D_1(G_p)\frac{{\gamma }^2{F}^2}{A_0\Delta z}\stackrel{-}{S}(k),\end{array}$$ (5)

where G_p = (kR²/2F) is the pressure focusing gain of the transducer20, 21 and D₁(·) = 1 has been defined for convenience when comparing the studied BSC estimation methods. It must be stressed that Eq. 5 is not equal to the final expression given in Ref. 11, but rather an updated expression that corrects a magnitude inconsistency in the BSC estimation derivation.

Estimation method 2

The second estimation method was derived by Chen et al.12 The authors determined the theoretical reference power spectrum using the mirror image method assuming a perfectly reflecting plate.22 The resulting expression for |S₀(k)|² was scaled by the power reflection coefficient of the plate to account for its finite reflectivity. BSCs can be estimated as [Ref. 12, Eqs. (31), (54), and (57)]

$$\begin{array}{c}\multicolumn{1}{c}{{\eta }_2(k)=2.17D_2(G_p)\frac{{\gamma }^2{F}^2}{A_0\Delta z}|{\stackrel{-}{S}}_0(k)|{}^2,D_2(G_p)=|\mathit{exp}(-{\mathit{iG}}_p)[J_0(G_p)+{\mathit{iJ}}_1(G_p)]-1|{}^2,}\end{array}$$ (6)

where J_m(·) is the mth order Bessel function.

Estimation method 3

The third estimation method was developed by Ueda and Ozawa.10 The reference power spectrum was derived in Ref. 10, Sec. VI using the boundary integral wave equation under the first order Born approximation [Ref. 10, Eq. 4]. An approximate closed form solution for estimating BSCs assuming a Gaussian profile for the transducer radiation pattern was proposed and developed by the authors and is given by [Ref. 10, Eq. (73)]

$${\eta }_3^G(k)=2.17D_3^G(G_p)\frac{{\gamma }^2{F}^2}{A_0\Delta z}|{\stackrel{-}{S}}_0(k)|{}^2,$$ (7)

$$D_3^G(G_p)=0.92\frac{(G_p∕2){}^2}{1+(G_p∕2){}^2}.$$ (8)

The BSC for a circular focused transducer can be calculated as [Ref. 10, Eq. (74)]

$$\begin{array}{c}{\eta }_3(k)=\frac{L(kR,kF)}{0.92}{\eta }_3^G(k)\\ =2.17D_3(kR,kF)\frac{{\gamma }^2{F}^2}{A_0\Delta z}\stackrel{-}{S}(k),\end{array}$$ (9)

where L(kR, kF) is a frequency-dependent correction factor and $D_3(\mathit{kR},\mathit{kF})=[D_3^G(\mathit{Gp})\times L(\mathit{kR},\mathit{kF})∕0.92]$. The procedure for numerically calculating L(kR, kF) was outlined in Ref. 10, Sec. IV.

Experimental methods

Experimental setup

In order to evaluate the reported estimation methods, experimental BSC estimates were obtained from two agar phantoms with glass bead inclusions.23 The first phantom, labeled “41 μm phantom,” contained 47 glass spheres∕mm³ ranging in diameter from 36 to 48 μm. The second phantom, labeled “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. The scatterer size distributions of the glass spheres for both phantoms were estimated from optical microscopy images, and the estimated scatterer size probability distribution functions (PDFs) p(a), i.e., the probability that the spherical inclusion radius takes the value a, for both phantoms are reported in Fig. 2. The corresponding attenuation coefficients were estimated with through-transmission techniques and approximated with a third- and fifth-order polynomial for the 41 μm and 150–180 μm phantoms, respectively. Ultrasound was transmitted into the phantoms through a layer of thin plastic film. The transmission coefficient through the film was compensated using the method outlined in Ref. 19.

Figure 2.

(Color online) Estimated scatterer size probability distribution functions for the 41 μm (left) and 150–180 μm (right) phantoms.

It has been previously reported in the literature24 that using a reference from a planar reflector at a single location to estimate S₀(k) may affect the estimated BSCs especially when using tightly focused transducers and long gate lengths. To reduce these effects, in this work the reference spectrum S₀(k) was estimated by averaging the power spectra obtained at several depths between (F − Δz/2) and (F + Δz/2) from a planar Plexiglas^® reflector (γ ≈ 0.37).

BSC estimates were obtained using all three estimation methods described in Sec. 2A. The phantoms were raster scanned in one plane over a 4 cm by 4 cm area at 2 mm steps for a total of 441 scan locations. Attenuation compensation was performed using Ref. 17, Eq. 16. The properties of the transducers used for the experiments are listed in Table TABLE I.. For all transducers, the radio frequency data was gated using a gate length Δz of 18 wavelengths at the nominal center frequency of the transducer.

Table 1.

Properties of the transducers used for the experimental BSC measurements. In the first column, f₀ represents nominal center frequency of the transducer. G_p was calculated using the experimentally estimated focal distance F.

f0 (MHz)	f#	Diameter	Analysis bandwidth (MHz)	Gp range
2.25	2.7	0.75 in. (19.05 mm)	[1.2–3.4]	[3.6–9.9]
5	3	0.75 in. (19.05 mm)	[2.3–7.3]	[7.7–24.5]
7.5	4	0.75 in. (19.05 mm)	[3.1–11.5]	[8.0–29.4]
10	4	0.5 in. (12.7 mm)	[5.7–16.1]	[8.9–25.3]
10	2	1 in. (25.4 mm)	[4.3–14.4]	[28.2–95.0]
13	3	0.5 in. (12.7 mm)	[7.0–19.1]	[15.8–43.2]
15	2	0.5 in. (12.7 mm)	[7.1–20.1]	[23.4–66.9]
20	4	0.25 in. (6.35 mm)	[8.6–27.0]	[6.9–21.9]
20	3	0.25 in. (6.35 mm)	[8.9–28.6]	[8.9–28.6]

The theoretical BSC corresponding to the experimental phantoms were calculated assuming the analyzed media consisted of randomly located spherical scatterers embedded in an otherwise homogeneous background at a concentration of β scatterers per unit of volume. Neglecting coherent and multiple scattering effects, the theoretical BSC η(k)_th can be expressed as

$${\eta }_{\mathrm{th}}(k)=\frac{\beta }{4\pi }{\int }_0^{\infty }p(a)\sigma (k,a)\mathit{da},$$ (10)

where σ(k, a) is the backscattering cross-section of an individual scatterer. In this work, σ(k, a) was calculated using the analytic solution of the scattering of a plane wave by a solid sphere.25, 26 The values of density, longitudinal sound speed, and Poisson’s ratio of the glass spheres were assumed to be equal to 2.38 g∕ml, 5.57 mm∕μs, and 0.21, respectively.11

Performance assessment

In order to experimentally assess the performance of the different BSC estimation methods, QUS estimates were obtained from the estimated BSCs. Effective scatterer diameter (ESD) estimates were obtained by solving the optimization problem11, 27

$$\mathrm{ESD}=\underset{a}{2\mathit{arg}\min }\frac{1}{N}\sum _{i=1}^N[X(\eta (k_i),\sigma (k_i,a))-\stackrel{-}{X}]{}^2,$$ (11)

$$X(r,s)=10\mathit{log}{}_{10}(r∕s),$$ (12)

where $\stackrel{-}{X}$ is the mean value of X(k, a) within the analysis wave number bandwidth k_i ∈ [k_min, k_max]. Other methods for ESD estimation such as least mean squares (LMS)15 have also been studied in the literature. However, no significant difference was observed when using LMS to obtain ESD estimates (results not shown in this manuscript). Once an estimate of the ESD has been obtained, the effective scatterer concentration (ESC) can be estimated as

$$\mathrm{ESC}=4\pi \frac{\sum _{i=1}^N\eta (k_i)\sigma (k_i,\mathrm{ESD}∕2)}{\sum _{i=1}^N{\sigma }^2(k_i\mathrm{,}\mathrm{ESD}∕2)}.$$ (13)

The fit between the estimated η(k) and theoretical η_th(k) BSCs corresponding to both phantoms was also quantified. For consistency, the fit was quantified using the first- and second-order moments of the error function in Eq. 12, i.e.,

$${\mathrm{Err}}_{\mu }=\frac{1}{N}\sum _{i=1}^{N}X(\eta (k_i),{\eta }_{\mathit{th}}(k_i)),$$ (14)

$${\mathrm{Err}}_{\sigma }=\sqrt{\frac{1}{N}\sum _{i=1}^N[X(\eta (k_i),{\eta }_{\mathit{th}}(k_i))-{\mathrm{Err}}_{\mu }]{}^2}.$$ (15)

The quantities Err_μ and Err_σ are measures of the amplitude and frequency dependence agreements, respectively, between the estimated η(k) and theoretical η_th(k) BSCs.

RESULTS

Frequency-dependent effects of the estimation methods

For all three BSC estimation methods, it was independently demonstrated that only the amplitude, and not the frequency dependence, of 〈|S_m(k)|²〉 depended on the transducer geometry when measurements were conducted near the transducer focus.10, 11, 12 However, the frequency dependence of |S₀(k)|² varies depending on which estimation method is used. Further, it follows from Eqs. 5, 6, 7, and 9 that the studied BSC estimation methods can be related to each other using the relationships

$$\frac{{\eta }_1(k)}{D_1(G_p)}=\frac{{\eta }_2(k)}{D_2(G_p)}=\frac{{\eta }_3^G(k)}{D_3^G(G_p)}=\frac{{\eta }_3(k)}{D_3(\mathit{kR},\mathit{kF})}.$$ (16)

All BSC estimation methods were compared by plotting the diffraction correction factors D₁(G_p), D₂(G_P), D₃(kR, kF), and $D_3^G(G_p)$, which relate the estimated BSCs as given by Eq. 16. Both D₂(G_P) and $D_3^G(G_p)$ depend exclusively on the pressure focal gain G_p. In contrast, D₃(kR, kF) does not depend on G_p but rather on kR and kF separately. Therefore, nine different D₃(kR, kF) curves were computed and plotted, each one corresponding to the geometry of all nine transducers listed in Table TABLE I.. The results are shown in Fig. 3 for G_pvalues between 10⁻¹ and 10³.

Figure 3.

(Color online) Comparison of the frequency-dependent calibration terms in the denominator of Eq. 16. The curves correspond to η₁ (solid line), η₂ (dotted line), ${\eta }_3^G$ (dashed line), η₃ and (nine dash-dotted lines).

Observation of Fig. 3 indicates that all nine D₃(kR, kF) curves were almost perfectly overlapping and virtually indistinguishable from each other, which suggests L(kR, kF) is mainly dependent on G_p for moderate to large kR values. This observation is consistent with numerical evaluations presented by Ueda and Ozawa in Ref. 10, Fig. 5. Further, all D₃(G_P) curves were in excellent agreement with the curve corresponding to $D_3^G(G_p)$ for all G_p values. The curve corresponding to $D_3^G(G_p)$ had good agreement with D₂(G_P) and D₃(kR, kF) for low G_p values and with D₁(G_p) for large G_p values. The curves corresponding to D₂(G_P), D₃(kR, kF), and $D_3^G(G_p)$ agreed with G₁(G_p) to within 1 dB for G_p > 25, G_p > 25, and G_p > 5, respectively.

Figure 5.

(Color online) Fractional error (ESD/2a – 1) when estimating ESDs as a function of ka when the input η to the estimator is set proportional to η_th × D₂(G_P) (solid lines) and η_th/D₂(G_p) (dashed lines). The G_p/ka ratio was set to 50 (top), 500 (center), and 5000 (bottom).

BSC and QUS estimates

A total of 21 experimental BSC estimates per transducer per phantom were obtained using all estimation methods by using backscattered data from 21 independent scan lines per BSC estimate. The average BSCs obtained using the estimation methods from Eqs. 5, 6, and 7 together with the theoretical BSC curves expected from using Eq. 10 are shown in Fig. 4 for both experimental phantoms. The BSCs estimated using Eq. 9 were not plotted because of their high correlation with the BSCs obtained using η₂.

Figure 4.

(Color online) BSC estimates from the 41 μm (left) and 150–180 μm (right) phantoms estimated using methods 1 (solid lines), 2 (dashed lines), and 3 (dash-dotted lines) together with the theoretical BSC derived from the scatterer size PDFs reported in Fig. 2 (dotted lines).

All methods produced BSC estimates that had similar agreement with the theoretical BSCs (also plotted). The errors between the experimentally estimated and theoretical BSCs are reported in Tables TABLE II. and TABLE III. for the 41 μm and 150–180 μm phantoms, respectively. As expected from the results in Sec. 3A, the Err_μ values corresponding to different methods are close to each other, i.e., less than 2 dB error difference among all estimation methods for a fixed phantom and transducer combination. Further, the difference in Err_μ values for a fixed phantom and transducer combination is reduced when using transducers with higher G_p values.

Table 2.

Experimental BSC estimation errors when analyzing the 41 μm phantom.

Transducer	Errσ				Errμ
η1	η2	${\eta }_3^G$	η3	η1	η2	${\eta }_3^G$	η3
2.25 MHz, f∕2.7	1.34 ± 0.16	−0.51 ± 0.16	0.59 ± 0.16	−0.50 ± 0.16	1.04 ± 0.09	1.04 ± 0.09	1.04 ± 0.09	1.05 ± 0.09
5 MHz, f∕3	0.60 ± 0.16	−0.66 ± 0.16	0.15 ± 0.16	−0.65 ± 0.16	1.05 ± 0.12	1.07 ± 0.11	1.05 ± 0.12	1.07 ± 0.11
7.5 MHz, f∕4	0.84 ± 0.20	−0.35 ± 0.20	0.40 ± 0.20	−0.34 ± 0.20	0.95 ± 0.08	0.97 ± 0.07	0.95 ± 0.07	0.97 ± 0.07
10 MHz, f∕4	0.18 ± 0.17	−1.04 ± 0.17	−0.26 ± 0.17	−1.04 ± 0.17	0.97 ± 0.11	0.97 ± 0.11	0.97 ± 0.11	0.97 ± 0.11
10 MHz, f∕2	−0.06 ± 0.27	−0.71 ± 0.27	−0.43 ± 0.27	−0.71 ± 0.27	1.10 ± 0.11	1.05 ± 0.11	1.10 ± 0.11	1.05 ± 0.11
13 MHz, f∕3	1.17 ± 0.25	0.25 ± 0.25	0.79 ± 0.25	0.26 ± 0.25	0.98 ± 0.09	0.98 ± 0.09	0.97 ± 0.09	0.98 ± 0.09
15 MHz, f∕2	1.37 ± 0.34	0.60 ± 0.34	0.99 ± 0.34	0.61 ± 0.34	1.05 ± 0.14	1.01 ± 0.14	1.05 ± 0.14	1.01 ± 0.14
20 MHz, f∕4	1.53 ± 0.28	0.13 ± 0.28	1.02 ± 0.28	0.14 ± 0.28	1.30 ± 0.22	1.28 ± 0.20	1.30 ± 0.22	1.28 ± 0.20
20 MHz, f∕3	0.62 ± 0.58	−0.41 ± 0.58	0.22 ± 0.58	−0.41 ± 0.58	1.54 ± 0.22	1.47 ± 0.21	1.53 ± 0.22	1.47 ± 0.21

Table 3.

Experimental BSC estimation errors when analyzing the 150–180 μm phantom.

Transducer	Errμ				Errσ
η1	η2	${\eta }_3^G$	η3	η1	η2	${\eta }_3^G$	η3
2.25 MHz, f∕2.7	2.54 ± 0.19	0.63 ± 0.19	1.74 ± 0.19	0.63 ± 0.19	1.13 ± 0.14	1.13 ± 0.14	1.12 ± 0.14	1.13 ± 0.14
5 MHz, f∕3	2.24 ± 0.28	0.96 ± 0.28	1.79 ± 0.28	0.99 ± 0.28	1.31 ± 0.17	1.26 ± 0.16	1.30 ± 0.17	1.26 ± 0.16
7.5 MHz, f∕4	2.71 ± 0.31	1.53 ± 0.31	2.27 ± 0.31	1.53 ± 0.20	1.27 ± 0.16	1.20 ± 0.14	1.24 ± 0.15	1.20 ± 0.14

Mean and standard deviations of ESD and ESC estimates obtained from all 21 estimated BSCs per transducer are reported in Table TABLE IV. for the 41 μm phantom and Table TABLE V. for the 150–180 μm phantom. In general, a larger difference in mean QUS values corresponding to different estimation methods was observed for transducer and phantom combinations with lower ka values. The most dramatic difference was observed when analyzing the 41 μm phantom with the 2.25, f∕2.7 transducer.

Table 4.

ESD and ESC estimates derived from experimental data corresponding to the 41 μm glass bead phantom, ka and G_p were calculated using the transducer nominal center frequency, with ka corresponding to a scatterer diameter of 41 μm.

Transducer	ka	Gp	ESD (μm)				ESC (scatt./mm3)
η1	η2	${\eta }_3^G$	η3	η1	η2	${\eta }_3^G$	η3
2.25 MHz, f∕2.7	0.19	6.6	49.1 ± 19.1	15.6 ± 19.8	21.2 ± 22.81	15.4 ± 19.6	(2.4 ± 10.7) × l08	(1.3 ± 1.5) × l09	(7.2 ± 12.2) × l08	(1.3 ± 1.5) × l09
5 MHz, f∕3	0.43	16.8	54.8 ± 3.6	50.7 ± 4.0	53.8 ± 3.7	50.8 ± 4.0	16.7 ± 5.7	19.3 ± 8.1	16.7 ± 6.1	19.3 ± 8.2
7.5 MHz, f∕4	0.64	19.2	42.6 ± 1.4	40.4 ± 1.5	42.1 ± 1.5	40.4 ± 1.5	52.4 ± 7.7	51.3 ± 8.8	50.1 ± 7.6	51.3 ± 8.8
10 MHz, f∕4	0.86	15.7	41.6 ± 0.8	40.8 ± 0.8	41.4 ± 0.8	40.8 ± 0.8	48.5 ± 3.6	39.0 ± 3.0	44.5 ± 3.3	39.0 ± 3.0
10 MHz, f∕2	0.86	65.9	44.9 ± 1.0	44.4 ± 1.0	44.9 ± 1.1	44.4 ± 1.0	36.2 ± 3.9	32.5 ± 3.6	33.3 ± 3.6	32.6 ± 3.6
13 MHz, f∕3	1.12	29.4	41.7 ± 0.9	41.4 ± 1.3	41.7 ± 1.0	41.4 ± 1.3	61.9 ± 5.7	51.2 ± 5.4	56.8 ± 5.3	51.2 ± 5.4
15 MHz, f∕2	1.29	49.8	41.4 ± 0.6	41.4 ± 0.6	41.4 ± 0.6	41.4 ± 0.6	66.9 ± 6.0	56.4 ± 5.1	61.4 ± 5.5	56.4 ± 5.1
20 MHz, f∕4	1.71	16.2	39.2 ± 0.9	39.6 ± 0.9	39.3 ± 0.9	39.6 ± 0.9	79.4 ± 8.1	57.5 ± 5.9	70.9 ± 7.2	57.5 ± 5.9
20 MHz, f∕3	1.71	20.0	38.0 ± 1.0	38.3 ± 1.0	38.0 ± 1.0	38.3 ± 1.0	76.1 ± 9.8	58.8 ± 7.7	69.1 ± 8.8	58.9 ± 7.7

Table 5.

ESD and ESC estimates derived from experimental data corresponding to the 150–180 μm glass bead phantom. ka and G_p were calculated using the transducer nominal center frequency, with ka corresponding to a scatterer diameter of 165 μm.

Transducer	ka	Gp	ESD (μm)				ESC (scatt./mm3)
η1	η2	${\eta }_3^G$	η3	η1	η2	${\eta }_3^G$	η3
2.25 MHz, f∕2.7	0.78	6.6	169.4 ± 4.4	161.2 ± 4.8	163.3 ± 4.7	161.1 ± 4.8	5.9 ± 0.6	4.6 ± 0.6	5.7 ± 0.7	4.6 ± 0.6
5 MHz, f∕3	1.73	16.8	163.5 ± 5.9	165.3 ± 5.5	164.0 ± 5.9	165.3 ± 5.5	5.4 ± 0.8	4.0 ± 0.6	4.8 ± 0.7	4.0 ± 0.6
7.5 MHz, f∕4	2.59	19.2	131.8 ± 49.9	135.8 ± 48.9	131.1 ± 50.7	135.8 ± 48.9	88.7 ± 121.3	63.8 ± 94.7	84.8 ± 114.2	63.9 ± 94.9

DISCUSSION

Three representative methods for BSC estimation have been studied in this work. It should be mentioned that other approaches presented in the literature can be related to the methods evaluated here. For example, D’Astous and Foster28 presented a estimation method based on geometrical considerations that resulted in expressions similar to those provided by Insana et al. in Ref. 11. Another study in BSC estimation was presented by Madsen et al.29 Analytic expressions for BSC calculation were provided, but such expressions depended on integrals that required numerical evaluation. The formulation by Madsen et al. was shown to be equivalent to η₂ by Chen et al.12 The advantage of using the results from Ref. 12 is the availability of explicit expressions for normalizing BSC estimates.

All studied estimation methods have an effect on the magnitude and frequency dependence of the estimated BSCs. Both effects are discussed below.

BSC amplitude effects

All estimation methods studied in this work can be compared in the large G_p limit, for which it holds that

$$\begin{array}{c}\multicolumn{1}{c}{D_2(G_p\to \infty )\to 1,D_3^G(G_p\to \infty )\to 0.92,D_3(\mathit{kR},\mathit{kF},G_p\to \infty )\to 1.}\end{array}$$

The first limit can be derived by using the approximate expression D₂(k) → exp[−(2∕π)(G_p∕π)^−1∕2] valid for large G_p values given in Ref. 12, Eq. (58). The second limit follows directly from the expression in Eq. 8. The third limit was validated through numerical simulations. Therefore, the asymptotic values of the relationships in Eq. 16 are given by

$$\begin{array}{c}\multicolumn{1}{c}{{\eta }_1(G_p\to \infty )={\eta }_2(G_p\to \infty )=\frac{{\eta }_3^G(G_p\to \infty )}{0.92}={\eta }_3(G_p\to \infty ).}\end{array}$$ (17)

The expressions given in Eq. 17 indicate an excellent agreement in BSC amplitude for all studied BSC estimation methods. Methods η₁, η₂, and η₃ have a perfect agreement in the G_p → ∞ limit, and in the asymptotically large G_p limit these methods differ from ${\eta }_3^G$ by less than 10%. This analysis is consistent with the experimental values of Err_μ reported in Tables 2, TABLE III. where differences of less than 2 dB in Err_μ among different estimation methods were observed. Further, the Err_μ absolute values were less than 1.6 and 2.8 dB for the 41 μm and 150–180 μm phantoms, respectively. The small error indicates a very good correspondence between the estimated and expected BSC magnitudes. The slightly larger magnitude disagreement for the 150–180 μm phantom may be due to a less accurate assumed value for the scatterer concentration.

It should be noted that η₁ agreed with the other BSC estimation methods and experimental measurements only after the magnitude correction described in Sec. 2A1 was applied. Therefore, the use of the expressions provided in Ref. 11 is expected to result in a BSC magnitude underestimation by slightly more than an order of magnitude.

As for the validity of the η₁ magnitude correction presented in Sec. 2A1, it should be acknowledged that the approximations in Eq. 4 only hold for weakly scattering media and may appear inappropriate to characterize the reflection off a planar reflector with high impedance mismatch from water, e.g., a Plexiglas reflector. However, the scattering by a planar reflector in the original work presented in Ref. 11, Sec. 2 C was derived under the first order Born approximation for which weakly scattering media is assumed, and therefore the correction factor is self-consistent. Further, excellent agreement was found between methods η₂ and η₃ although expressions for the reflection off a planar reflector were derived by Chen et al. in Ref. 12 by assuming the plate to be a perfect acoustic reflector and Ueda and Ozawa in Ref. 10 by invoking first order Born scattering. Therefore, it appears reasonable to assume expressions for reflection off a plate derived under limiting scattering conditions (such as weak scattering conditions) can be generalized to materials of finite reflectivity by scaling the resulting expressions by the reflectivity γ at the plate surface.

Finally, other estimation methods available in the literature also suffer from discrepancies in BSC magnitude. For example, D’Astous and Foster proposed in Ref. 28, Eq. 6 to estimate BSCs as

$${\eta }_{\mathrm{DAF}}(k)=\frac{{\gamma }^2}{\Delta z}\frac{1}{2\pi (1-\mathit{cos}{\theta }_t)}{|{\stackrel{-}{S}}_0(k)|}^2,$$ (18)

where θ_t is the half angle of the transducer subtended at its focus. If a small angle approximation is performed, $(1\mathrm{-}\cos {\theta }_{\mathrm{t}})\approx {\theta }_t^2\mathrm{∕}2\approx 0\mathrm{.}5{(R∕F)}^2$ and therefore

$${\eta }_{\mathrm{DAF}}(k)\approx \frac{{\gamma }^2{F}^2}{A_0\Delta z}|{\stackrel{-}{S}}_0(k)|{}^2.$$ (19)

Therefore, this estimation method has the same frequency dependence of η₁, but its amplitude for large G_p is off by roughly a factor of 2.2 when compared to methods η₁, η₂, and η₃.

Frequency-dependent effects

The diffraction correction curves presented in Fig. 3 illustrate the different frequency dependencies of all studied estimation methods, which arise from the different assumptions involved in the derivation of the corresponding BSC estimation expressions.

When deriving η₂, and following the seminal formulation by O’Neil,30 the incident field was assumed proportional to ∫∫_Sexp(−jks)/s dS, where s is the distance between a surface element dS on the surface S of the transducer and the observation point. When deriving η₃ the incident field was assumed to be proportional to jinc(kR sin θ)exp(−jkr), where jinc(x) = 2J₁(x)/x with J₁(·) the first order Bessel function, and r is the radial coordinate of the observation point. As shown by O’Neil in Ref. 30 both expressions are in good agreement in the focal plane of the transducer. Therefore, the use of almost equivalent models for both amplitude and phase of the incident field resulted in very good agreement between η₂ and η₃.

Although Insana et al. used the same incident field model as Ueda and Osawa, the phase front was assumed to be plane instead of spherical in the evaluation of |S₀(k)|² (see Ref. 11, Sec. 2 C). The use of a different expression for the incident field phase caused frequency-dependent discrepancies between η₁ and η₂, η₃. For large G_p values, however, these three methods become essentially equivalent, i.e., frequency-dependent effects in η₂, η₃ become weak. For a focused transducer, the −6-dB angular beamwidth at the transducer focus can be approximated as Δθ ≈ arctan(λf_#/F) = arctan(π/4G_p), where f_# is the transducer focal number. Therefore, the incident field can be safely approximated as a local plane wave as G_p → ∞, which is the cause for the asymptotic agreement among η₁, η₂, and η₃.

As for ${\eta }_3^G$, although it was derived assuming a spherical wave front the use of a different amplitude function (i.e., a Gaussian function instead of a jinc function) resulted in frequency-dependent discrepancies with η₂ and η₃. Therefore, both the amplitude and phase modeling of the assumed incident field have an effect on the frequency-dependent behavior of BSC estimation methods.

The results presented in this work suggest η₁ is significantly different than η₂, η₃, and ${\eta }_3^G$ at very low G_p values, i.e., G_p < 1. This is due to the fact that the local plane wave approximation used in η₁ breaks down in the low G_p limit. In practice, however, these effects may not be observed. It has already been discussed in the literature that energy focusing cannot occur in the far field of the equivalent flat transducer, i.e, for R²/λF < 1 or equivalently G_p/π < 1.31 Therefore, transducers with G_p < π (the region where most of the significant differences in the curves presented in Fig. 3 were observed) will most likely not be used in experimental practice.

The experimental results suggest all methods are capable of producing similar frequency dependencies of BSC estimates, as suggested by the Err_σ values in Tables 2, TABLE III.. The experimental Err_σ corresponding to different estimation methods had mean absolute and relative values that differed among each other in all cases by less that 0.1 dB and 6%, respectively.

Implications for QUS estimation

Both amplitude and frequency-dependent effects introduced by BSC estimation methods will affect QUS estimates. Amplitude differences will only change estimates of ESC. It was observed in this study that η₁, η₂, η₃, and ${\eta }_3^G$ were successful in approximately reproducing the expected BSC magnitude from two well-characterized experimental phantoms. Other methods reported in the literature such as ${\stackrel{\wedge }{\eta }}_1$ in Eq. 3 and η_Daf in Eq. 19 had amplitude disagreements that would result in BSC amplitude under-determination factors of 12 and 3.4 dB, respectively. Therefore, different estimation methods reported in the literature can potentially introduce significant BSC amplitude variations.

Frequency-dependent effects are expected to affect primarily ESD estimates, although it is worth remarking that errors in ESD will ultimately propagate to ESC estimates through the use of Eq. 13. Two adimensional, frequency-dependent parameters are expected to affect the mean ESD estimate values.

• (1)G_p: Because different BSC estimation methods introduce different frequency-dependent corrections which depend on G_p, mean ESD estimates are expected in principle to change depending on the chosen BSC estimation method. However, the G_p range of the transducers should determine the extent of the spread of mean ESD estimates. This is expected because the results in Sec. 3A suggest all estimation methods become nearly equivalent when G_p → ∞ but are significantly different when G_p < π.
• (2)ka: The accuracy and precision of ESD estimates can be compromised depending on the ka range used for BSC analysis. In particular, for low ka values the dependence of BSC on the actual scatterer size becomes weak which degrades the precision of ESD estimates.32 On the other hand, for large ka values the scattering model becomes too complex which may cause convergence of the ESD estimator to inaccurate solutions due to convergence to local minima.6 It has been proposed that ka values satisfying 0.5 < ka < 1.2 provide good ESD estimates when analyzing glass bead phantoms.11

Therefore, all estimation methods are only expected to provide similar QUS estimates under certain circumstances. Simulations were conducted to explore the dependence of ESD estimates on ka and G_p values. The theoretical BSC, η_th, corresponding to a single glass sphere was calculated for 0.1 < ka < 2. Two modified BSCs were set proportional to η_th × D₂(G_P) and η_th/D₂(G_p). These two curves represent the hypothesis that η₁ is the correct estimation method but either η₂ or η₃ [because D₂(G_P) is almost equivalent to D₃(kR, kF)] are used to estimate BSCs and vice versa. Portions of these modified BSCs with different center frequencies and 100% fractional bandwidths were used as input for the estimator in Sec. 2B2. The fractional error (ESD/2a – 1) when obtaining ESD estimates using the modified BSCs are shown in Fig. 5 for the cases where G_p/ka = 50, G_p/ka = 500, and G_p/ka = 5000. It can be observed that the ESD estimation error spread was indeed dependent on both ka and G_p. The variation induced by the ka range was the most significant, but the results predict the ESD estimation error spread should be significant only when approaching Rayleigh scattering, i.e., ka < 0.5. Although the results also confirm the spread is reduced for increasing G_p, reducing the ESD estimation error spread below ±16% for ka > 0.25 required significantly large G_p values (i.e., G_p/ka ratios of around 500).

The experimental results validated the dependence of the spread of ESD mean estimates on ka range. The results from Table TABLE IV. obtained from the 41 μm phantom suggest that the differences in ESD mean values introduced by the choice of estimation method became significant when low ka values (i.e., ka < 0.5) were used. For example, the absolute spread was 33.7 μm (15.4 μm for η₃ vs 49.1 μm for η₁) when using the 2.25 MHz, f∕2.7 transducer. This corresponds to a relative spread of 82% when compared to the assumed mean particle diameter of 41 μm. Similarly, for the 5 MHz, f∕3 transducer the absolute and relative maximum spreads were 4.1 μm and 10%, respectively. In contrast, for the 20 MHz, f∕4 transducer the absolute and relative spreads of ESD mean values were 0.3 μm and 0.7%, respectively, even though the spread of Err_σ for this transducer was larger than those of the 2.25 MHz, f∕2.7 and 5 MHz, f∕3 transducers. However, in all cases the spread of ESD mean values was comparable to the standard deviation of the ESD estimates. The results presented in Table TABLE V. obtained from the 150–180 μm phantom further supports the dependence of the spread of mean ESD values on ka range. With the same 2.25 MHz, f∕2.7 transducer used when analyzing the 41 μm phantom the absolute and relative (considering a mean diameter size of 165 μm) spreads of mean ESD estimates were 8.3 μm and 5%, respectively. It can also be observed that the spread of mean ESD values increased for the 7.5 MHz transducer. For this transducer, the estimates followed a binomial distribution due to high ka convergence problems discussed above.

The dependence of the QUS estimates on G_p was also supported by the experimental results. As discussed in Sec. 3A, all estimation methods agree within less than 1 dB for G_p > 25. Two transducer pairs covered very similar frequency ranges with different G_p ranges and at least one of them satisfying G_p > 25 for most of the analysis bandwidth. The first pair was the 10 MHz, f∕4 (G_p ∈ [8.9,25.3]) and 10 MHz, f∕2 (G_p ∈ [28.2,95.0]) transducers for which the spreads of mean ESD values were 0.8 and 0.5 μm, respectively, when analyzing the 41 μm phantom. The second pair was the 13 MHz, f∕3 (G_p ∈ [15.8,43.2]) and 15 MHz, f∕2 (G_p ∈ [23.4,66.9]) transducers for which the spreads of mean ESD values were 0.3 and 0 μm, respectively, when analyzing the 41 μm phantom. Therefore, the limited experimental data in this study supports the fact that the use of sufficiently large G_p values helps reduce the dependence of mean ESD estimates on the method used for normalizing BSC estimates. However, it should be noted that the spread of mean ESD estimates for the discussed cases corresponded at most to 2% of the actual mean scatterer size present in the phantom. Therefore, and as the results from Fig. 5 suggest, the spread of mean ESD estimates was not significant when ka > 0.5 even for moderate (i.e., G_p < 100) G_p values.

Although all estimation methods provided similar experimental BSCs in this work with good fits to the theoretical BSCs as evidenced by the results in Tables 2, TABLE III., the subtle frequency-dependent effects of each method had in some cases noticeable effects on the bias of QUS estimates. These differences in mean QUS estimates can be reduced by proper choice of ka and G_p ranges. Given that G_p only depends on transducer properties whereas ka depends on properties of the imaging target, in practice it should be easier to control the G_p rather than the ka range. If G_p is made sufficiently large, the spread of mean estimation values introduced by the different estimation methods would be reduced and relatively good BSC estimation consistency would also imply relatively good QUS estimation consistency. However, in this work it was found that for the G_p values encountered in practice the frequency-dependent effects of the studied BSC estimation methods did not have a significant effect on the spread of mean QUS estimates, and any of them can be safely used for BSC estimation.

Finally, significant deviations in mean ESD values from the actual mean glass bead diameter introduced by estimation methods were noticeable for low ka values. This observation is important because it has been commonly understood that the precision and not the accuracy of ESD estimates should suffer at low ka.32 Furthermore, it has been demonstrated that methods such as spatial compounding can be used to effectively reduce the standard deviation without affecting the mean of ESD estimates.33 However, the ill-conditioning of ESD estimation in the low ka regime may cause the ESD estimates to be too sensitive to even slight experimental errors due to compensating for the transmission through the plastic film, nonlinear behavior of the amplifiers used to measure the data, attenuation compensation, measurement noise, and improper modeling of the transducer radiation pattern. Therefore, it was not possible to determine which BSC estimation method may be preferred when analyzing near Rayleigh (ka < 0.5) data. This may be yet another reason to advocate for large G_p values in order to reduce frequency-dependent effects on BSC estimates.