Hydrophone Spatial Averaging Correction for Acoustic Exposure Measurements from Arrays II: Validation for ARFI and Pulsed Doppler Waveforms

Abstract

This article reports experimental validation of a method for correcting underestimates of peak compressional pressure (p_c), peak rarefactional pressure (p_r), and pulse intensity integral (pii) due to hydrophone spatial averaging effects that occur during output measurement of clinical linear and phased arrays. Pressure parameters (p_c, p_r, and pii), which are used to compute acoustic exposure safety indexes such as mechanical index (MI) and thermal index (TI), are often not corrected for spatial averaging because a standardized method for doing so does not exist for linear and phased arrays. In a companion article (Part I), a novel analytic, inverse-filter method was derived to correct for spatial averaging for linear or nonlinear pressure waves from linear and phased arrays. In the present article (Part II), the inverse filter is validated on measurements of acoustic radiation force impulse (ARFI) and pulsed Doppler waveforms. Empirical formulas are provided to enable researchers to predict and correct hydrophone spatial averaging errors for membrane-hydrophone-based acoustic output measurements. For example, for a 400-μm membrane hydrophone, inverse filtering reduced errors (means ± standard errors for 15 linear array / hydrophone pairs) from about 34% (p_c), 22% (p_r), and 45% (pii) down to within 5% for all three parameters. Inverse filtering for spatial averaging effects significantly improves accuracy of estimates of acoustic pressure parameters for ARFI and pulsed Doppler signals.

I. INTRODUCTION

A. Hydrophone Measurements of Acoustic Exposure

The goal of this work was to experimentally validate an inverse-filter method to improve accuracy of measurements of acoustic exposure safety parameters from clinical linear and phased arrays. Mechanical Index (MI) and thermal index (TI) characterize likelihood of mechanical and thermal bioeffects respectively [1, 2] and are partly derived from pressure measurements performed with hydrophones [3–8]. However, hydrophones can underestimate pressure due to spatial averaging of the pressure field across the hydrophone sensitive element surface during output measurement.

Hydrophones integrate an incident pressure wave over the surface of a sensitive element to produce an output voltage. Choice of hydrophone geometrical sensitive element size involves trade-offs. As sensitive element size increases, sensitivity tends to increase, but spatial resolution decreases and therefore spatial averaging artifacts increase. The optimum size depends on application. Common hydrophones (and geometrical sensitive element diameters) for biomedical applications include membrane (200–1000μm), needle (40–1500 μm), capsule (85–1000 μm), and fiber-optic (10–100 μm) hydrophones.

B. Distortions in Hydrophone Measurements due to Frequency-Dependent Sensitivity and Spatial Averaging

Accurate reconstruction of an input pressure wave from a hydrophone output voltage waveform entails consideration of two effects. The first effect to consider is frequency-dependent sensitivity, which is the ratio of the output voltage spectrum to the input pressure spectrum (e.g., in V/MPa) for a normally-incident quasi-planar wave. If the hydrophone sensitivity is constant over the frequency band of the pressure wave spectrum, then it is appropriate to simply divide the voltage signal by a sensitivity scale factor (e.g., in V/MPa) to convert voltage to pressure. Otherwise, it is more accurate to deconvolve the voltage signal with the sensitivity [9–12]. Fig. 1 shows how hydrophone sensitivity magnitudes can vary over a range from 1 MHz to 60 MHz.

Fig. 1.

Sensitivity magnitudes, |M_L(f)|, for five hydrophones.

The second effect to consider is spatial averaging. If the pressure wave appears quasi-planar and perpendicularly incident across the sensitive element, then there is no need to correct for spatial averaging. Otherwise, for beams that are obliquely-incident, highly-focused or highly-nonlinear, it may be necessary to correct for spatial averaging. Nonlinear beams, such as those used in acoustic radiation force impulse (ARFI) or pulsed Doppler applications, are particularly susceptible to spatial averaging artifacts because they can contain many harmonic components with harmonic beam widths that decrease as harmonic frequency increases. Even if the fundamental beam is much broader than the hydrophone sensitive element, some harmonic components may have beam widths that are on the order of or smaller than the hydrophone sensitive element and therefore are susceptible to spatial averaging. The spatial averaging filter is a low-pass filter.

The effects of spatial averaging may be found by integrating the free field (i.e., the field in the absence of a hydrophone) over the “effective” hydrophone sensitive element surface. The effective sensitive element size depends on frequency and may be inferred from measurements of directivity [13]. Frequency-dependent effective sensitive element sizes have been reported for membrane [14–18], needle [16, 19], and fiber-optic [20] hydrophones.

C. Methods for Spatial Averaging Correction

Many methods for correction for hydrophone spatial averaging have been derived for transducers with circular symmetry. One early method, which is recognized by International Electrotechnical Commission 62127–1, involves measuring a frequency-independent scale factor that is valid for linear pressure waves [7, 21]. However, nonlinear pressure waves require a frequency-dependent scale factor because of their broad bandwidth associated with multiple harmonic components. Numerical approaches have been developed that apply to linear and nonlinear pressure waves [22–26]. An analytic, inverse-filter spatial averaging correction for linear or nonlinear pressure waves was derived [27] and validated for nonlinear signals measured with 1) needle hydrophones at diagnostic pressure levels [28], 2) membrane hydrophones at diagnostic pressure levels [29, 30], and 3) needle and fiber-optic hydrophones at therapeutic pressure levels [31]. While these methods have been demonstrated to be effective for sources with circular symmetry, most diagnostic ultrasound transducers have rectangular rather than circular symmetry. Empirical evidence suggests that hydrophone spatial averaging effects are significant for clinical ARFI [32] and pulsed Doppler [33] signals generated by linear arrays.

D. Outline of this Article

In a companion article (Part I), the first spatial averaging correction method for linear and phased arrays transmitting pressure waves with arbitrary linearity / nonlinearity was derived and tested using simulation [34]. In the present article (Part II), this spatial averaging correction method is validated with experimental data acquired using clinical arrays transmitting ARFI and pulsed Doppler waveforms. First, data acquisition and analysis methods using three clinical array transducers and five hydrophones are described. Second, plots of harmonic beam widths and effective hydrophone sensitive element diameters versus harmonic frequency are presented to illustrate how hydrophone spatial averaging effects increase with harmonic frequency. Third, theoretical and experimental spatial averaging filters for all transducer / hydrophone combinations are given. Fourth, a method described in the companion article (Part I) to produce simulated pressure waveforms based on parameters in regulatory acoustic output reporting tables is validated with measurements. (This simulation was used in Part I to assess the effects of hydrophone spatial averaging on MI and thermal index with bone at focus (TIB) for diagnostic ultrasonic imaging systems.) Fifth, spatial averaging errors in peak compressional pressure (p_c), peak rarefactional pressure (p_r), and pulse intensity integral (pii) (see 5.4.3–1 in [35]), as functions of hydrophone nominal geometrical sensitive element diameter are shown. Finally, some concluding remarks are given.

II. Methods

A. Hydrophones

Table I lists the hydrophones used. The reference hydrophone was a high-resolution capsule hydrophone (HGL-0085, Onda Corp., Sunnyvale, CA), which had a nominal geometrical sensitive element diameter of 85 μm and was expected to exhibit minimal spatial averaging effects (which will be validated in Section III.C). The frequency-dependent effective sensitive element size of the HGL-0085 was calculated from directivity measurements using methods described previously [17, 19]. Four membrane hydrophones with nominal geometrical sensitive element diameters ranging from 200 to 1000 μm (spanning the typical range for medical ultrasound) were used in order to investigate the effect of geometrical sensitive element diameter on spatial averaging artifacts.

TABLE I.

Hydrophones

Manufacturer	Model	Notes	Nominal Geometrical Sensitive Element Diameter dg (μm)
Onda (C)	HGL-0085		85
Precision Acoustics (M)	D1202	differential	200
NTR (M)	HMA-0500		500
Precision Acoustics (M)	UT1606		600
Marconi (M)	IP039	bilaminar	1000

C: capsule; M: membrane

B. Transmitted Beams

Each probe was driven with a Verasonics (Kirkland, WA) Vantage 128 system. Driving frequency, pulse duration, transmit voltage, and array-azimuthal-focusing distance were all programmed using a MATLAB (Natick, MA) script. Measurements were performed with transducer driven in two conditions: one representing ARFI mode (30 cycles) and the other representing Doppler mode (3 cycles). Although ARFI pulses can contain up to 1000 or more cycles [36], 30 cycles was considered sufficient to achieve steady state waveforms suitable for hydrophone measurement [37].

Table II lists the transducers used. The linear array transducers (with driving frequencies and transmit focal distances) included an ATL (Bothell, WA) L7–4 (5.2 MHz, 3.8 cm and 5.8 cm), a Verasonics L11–4v (7.3 MHz, 5.8 cm), and an ATL L12–5 (7.8 MHz, 4.1 cm). These transducers had frequencies and sizes that are typical for medical imaging applications. Transducers were immersed in freshly deionized water. The average water temperature was measured with a digital thermometer to be 19 ± 1° C.

TABLE II.

Transducer and Wave Properties

Name	L7-4 shallow	L7-4 deep	L11-4	L12-5
Frequency f1 (MHz)	5.2	5.2	7.3	7.8
Lx (mm)	38.4	38.4	38.4	51.2 (25.6)
Ly (mm)	7	7	7	7
Number of Elements	128	128	128	256 (128)
Elevational focus (mm)	25	25	20	20
Programmed Focal Length (mm)	38.4	58	58	41
Measured zppsi (mm)	38 ± 1	58 ± 1	58 ± 1	41 ± 1
F#	1.0	1.5	1.5	1.6
Transmit Voltage (ARFI) (V)	27.0	30.1	28.2	30.0
Transmit Voltage (pulsed Doppler) (V)	8.3	9.9	9.9	9.9
Number of cycles (ARFI)	30	30	30	30
Number of cycles (pulsed Doppler)	3	3	3	3
qx (σnx = σ1x / nqx) (ARFI)	0.72 ± 0.07	0.74 ± 0.07	0.79 ± 0.08	0.82 ± 0.08
qy (σny = σ1y / nqy) (ARFI)	0.55 ± 0.05	0.50 ± 0.07	0.28 ± 0.03	0.31 ± 0.03
qx (pulsed Doppler)	0.66 ± 0.07	0.79 ± 0.08	0.81 ± 0.08	0.76 ± 0.08
qy (pulsed Doppler)	0.58 ± 0.06	0.49 ± 0.05	0.33 ± 0.03	0.46 ± 0.06
Focal compressional pressure pc (MPa) (ARFI)	7.57 ± 0.11	7.51 ± 0.42	6.73 ± 0.05	5.97 ± 0.11
Focal rarefactional pressure pr (MPa) (ARFI)	2.65 ± 0.02	2.09 ± 0.03	1.89 ± 0.03	1.55 ± 0.08
Focal compressional pressure pc (MPa) (pulsed Doppler)	8.61 ± 0.37	9.07 ± 0.30	6.57 ± 0.08	5.62 ± 0.11
Focal rarefactional pressure pr (MPa) (pulsed Doppler)	2.59 ± 0.02	2.14 ± 0.01	1.75 ± 0.01	1.57 ± 0.01
Spectral Index SI (ARFI)	0.24 ± 0.01	0.38 ± 0.01	0.42 ± 0.01	0.42 ± 0.01
Spectral Index SI (pulsed Doppler)	0.25 ± 0.01	0.40 ± 0.01	0.39 ± 0.01	0.35 ± 0.01
σm (ARFI)	1.31 ± 0.04	2.08 ± 0.06	2.57 ± 0.08	1.79 ± 0.05
σm (pulsed Doppler)	1.42 ± 0.04	2.32 ± 0.07	2.15 ± 0.06	1.40 ± 0.04
σq (ARFI)	1.68 ± 0.05	2.66 ± 0.08	3.45 ± 0.10	2.28 ± 0.07
σq (pulsed Doppler)	1.82 ± 0.05	3.02 ± 0.09	2.92 ± 0.09	1.84 ± 0.06

The lateral dimension of the array is x, and the elevation dimension is y. Although the L12-5 has 256 elements (spanning 51.2 mm), only the central 128 elements were used (spanning 25.6 mm). Values of q are determined from curve fits σ_n= σ₁ / n^q or alternatively FWHM_n = FWHM₁ / FWHM^q for each of the x and y directions where n is harmonic number. z_ppsi is the position of the maximum measured pulse-pressure-squared integral.

C. Data Acquisition

To determine the measurement position, each hydrophone was scanned along the beam axis. Measurements were performed at the position of the maximum pulse-pressure-squared integral (z_ppsi) in water (not derated). For each transducer, all five hydrophone measurements were performed at the same transducer driving voltage. Vertical and horizontal scans between ±2 mm at increments of 50 μm were performed for each transducer / hydrophone combination.

Signals were acquired at 1 GHz with a calibrated Tektronix (Beaverton, OR) DPO-7254 digital phosphor oscilloscope. Data acquisition time was 10 μs.

D. Measurement of Spatial Averaging Filters

A reference time-domain voltage response u_p1(t) from a nonlinear tone burst pressure wave (ARFI or pulsed Doppler) was measured using the high-resolution HGL-0085 (nominal geometric sensitive element radius a_g = 42.5 μm) placed at the maximum z_ppsi of the transducer. The membrane hydrophone voltage response u_p2(t) was measured by replacing the HGL-0085 with a membrane hydrophone and repeating the measurement with the same source configuration and driving voltage. This measurement was performed for each of the four membrane hydrophones in Table I. Spectra U_p1(f) and U_p2(f) were estimated using the Fast Fourier Transform (FFT). Spatial averaging filters S_p2(f) were evaluated using a substitution method as described previously [28].

$$S_{p2}\left(f\right)=\frac{U_{p2}\left(f\right)/M_{L2}(f)}{{U_{p1}\left(f\right)/M}_{L1}\left(f\right)}{S}_{p1}(f)$$ (1)

Hydrophone sensitivities M_L1(f) and M_L2(f) in (1) were measured at the National Physical Laboratory (NPL) and are shown in Fig. 1. The quotients of functions in the numerator and denominator, U_pi(f) / M_Li(f), may be regarded as pressure spectra that have been corrected for frequency-dependent hydrophone sensitivities [7, 38–40]. The sensitivity correction bandwidth was limited by the maximum frequency for which the hydrophone sensitivities were calibrated, which was 60 MHz.

The reference hydrophone spatial averaging filter S_p1(f) in (1) was computed using (15) from Part I [34], which depends on the frequency-dependent effective sensitive element size a_eff(f). The form for a_eff(f) for the HGL-0085 was obtained from a parametric fit to a rigid piston model [41–44] for a needle hydrophone, [a_eff(f) - a_g] / a_g = Aexp(−Bka_g), where A = 1.85, B = 1.05, k = 2π/λ [28] and a_g = 42.5 μm. The validity of this model for the HGL-0085 will be experimentally tested in this article (see Section III.A).

As discussed in Part I [34], signals were low-pass filtered and rarefactional filtered [45] in order to suppress common potential errors associated with inverse filtering [34].

Harmonic beam width parameters σ_nx and σ_ny (where n = harmonic number) were obtained from Gaussian fits of the form exp[-x²/(2σ_nx²)] and exp[-y²/(2σ_ny²)] (where x is the lateral coordinate and y is the elevational coordinate) to harmonic components extracted from FFTs of transverse scans using the high-resolution HGL-0085 hydrophone. The exponents q_x and q_y, where σ_nx = σ_1x / n^qx and σ_ny = σ_1y / n^qy, were estimated by fitting harmonic full-width half maxima (FWHM) to a power law function of harmonic number n [27]. FWHM_nx = ${\sigma }_{nx}2\sqrt{2ln2}$ ≈ 2.35σ_nx. FWHM_ny ≈ 2.35σ_ny [27].

E. Validation of Simulation

A simulation was used in Part I [34] to assess the effects of hydrophone spatial averaging on MI and TIB for diagnostic ultrasound imaging systems. The simulation reconstructed radio-frequency (RF) waveforms from four parameters: driving frequency (f₁), p_r, pulse duration (PD), and pii (information available from acoustic output reporting tables prepared for regulatory purposes [34]). The simulation was experimentally tested in the present article (Part II) on the complete dataset of RF waveforms acquired for all transducer / hydrophone combinations. Waveforms were not deconvolved for sensitivity or corrected for hydrophone spatial averaging (because these operations had not been applied for the data in acoustic output reporting tables).

First, the four parameters (f₁, p_r, PD, and pii) were directly measured from the experimentally-acquired RF waveform for each transducer / hydrophone combination. Second, the simulation was used to reconstruct a waveform based on the four-parameter set for each transducer / hydrophone combination. By design, simulated waveforms matched f₁ and p_r exactly. Simulation performance was measured by computing means and standard deviations of errors in PD and pii (simulation vs. experiment). In addition, linear regressions of simulated vs. experimental waveforms were computed. Simulation performance was measured by correlation coefficients (ideally, close to one), regression intercepts (ideally, close to zero) and regression slopes (ideally, close to one).

F. Nonlinearity Indexes

Signal nonlinearity was characterized by the spectral index SI [46], the nonlinear propagation parameter σ_m [47], and the local distortion parameter σ_q [48]. SI is the fraction of the power spectrum contained in frequencies above the fundamental frequency [46, 49].

III. Results

A. Frequency-Dependent Effective Sensitive Element Diameter for HGL-0085

Fig. 2 shows measurements of frequency-dependent effective sensitive element diameter for the high-resolution HGL-0085 hydrophone, obtained from directivity measurements as described previously [17, 19]. As frequency increases, the effective sensitive element diameter asymptotically approaches the geometrical sensitive element diameter of 85 μm. Fig. 2 also shows the prediction derived from the rigid piston model directivity [43]. The rigid piston model has previously been shown to accurately predict directivity and frequency-dependent effective sensitive element size of needle [19] and reflectance-based [20] hydrophones.

Fig. 2.

Frequency-dependent effective sensitive element diameter for the HGL-0085 hydrophone. The prediction derived from the rigid piston (RP) model directivity is also shown.

B. Harmonic-Frequency-Dependent Beam Widths

Fig. 3 shows focal plane lateral beam profiles for harmonics from the L7–4 transducer, measured with the high-resolution HGL-0085 hydrophone, at two different zoom levels. Gaussian fits are also shown. Fig. 3 shows that the harmonic lateral beam profiles were approximated well by Gaussians (as assumed by the theory) at least over a range of axial distances corresponding to a nominal geometrical sensitive element diameter of 500 μm.

Fig. 3.

Focal plane lateral beam profiles for harmonics from the L7–4 transducer. The right panel is a zoomed version of the left panel. Error bars denote ± one standard deviation. Gaussian fits are shown in black. The vertical dashed red lines show the geometrical extent of a hydrophone with a sensitive element diameter of 500 μm. Harmonic numbers are denoted by n.

Fig. 4 shows lateral and elevational FWHM (measured in scans with the high-resolution HGL-0085 hydrophone) plotted vs. harmonic frequency. Fits for frequency-dependent FWHM inversely proportional to n^q (where n is harmonic number) are also shown. The fitting parameters are denoted by q_x and q_y in the lateral and elevational directions respectively and are reported in Table II. The root-mean-squared (RMS) differences between frequency-dependent FWHM and fits inversely proportional to n^q in the main spatial averaging direction (lateral direction) over the harmonic frequencies shown in Fig. 4 were 40 μm ± 17 μm (ARFI) and 38 μm ± 18 μm (pulsed Doppler). The dashed lines show frequency-dependent effective sensitive element diameters d_eff(f) for five membrane hydrophones with geometrical sensitive element diameters d_g ranging from 200 to 1000 μm based on an empirical model [17]. At high frequencies, d_eff(f) asymptotically approaches a value close to d_g but at low frequencies, d_eff(f) > d_g.

Fig. 4.

Lateral (blue asterisks) and elevational (black x’s) full-width half maxima (FWHM) plotted vs. harmonic frequency. Error bars denote ± one standard deviation. Fits for frequency-dependent FWHM inversely proportional to n^q (where n is harmonic number) are also shown. The dashed lines show frequency-dependent effective sensitive element diameters d_eff(f) for five membrane hydrophones with geometrical sensitive element diameters d_g ranging from 200 to 1000 μm.

Fig. 4 shows that spatial averaging is rarely a concern in the elevational dimension because FWHM_elev(f) is almost always much greater than d_eff(f) (except sometimes at high frequencies for hydrophones with large sensitive elements). However, spatial averaging is almost always a concern in the lateral dimension because FWHM_lat(f) is often comparable to or even smaller than d_eff(f), even for hydrophones with 200-μm-geometrical-diameter sensitive elements.

C. Spatial Averaging Filters and Estimates of p_c, p_r, and pii

Figs. 5 and 6 show spatial averaging filters (SAFs) for all combinations of transducers and hydrophones for ARFI (Fig. 5) and pulsed Doppler (Fig. 6) signals. The red curves are theoretical SAFs based on two-dimensional integrals over the frequency-dependent sensitive element surfaces, (10) in Part I [34]. The black dashed curves are analytic forms based on “equivalent square” hydrophones, (15) in Part I. The measurements are in good agreement with both theoretical forms. Averaged over all hydrophones in Table I and all transducers in Table II, the root-mean-squared errors (RMSE) between theoretical and experimental spatial averaging filters were 2.8% ± 1.3% (ARFI) and 3.2% ± 1.9% (pulsed Doppler). Figs. 5 and 6 illustrate the low-pass behavior of SAFs.

Fig. 5.

Spatial averaging filters (SAF) for ARFI measurements for all transducer / hydrophone combinations. Error bars denote ± one standard deviation. Red curves are theoretical SAFs based on two-dimensional integrals over frequency-dependent sensitive element surfaces. Black dashed curves are analytic forms based on “equivalent square” hydrophones.

Fig. 6.

Spatial averaging filters (SAF) for Doppler measurements for all transducer / hydrophone combinations. Error bars denote ± one standard deviation. Red curves are theoretical SAFs based on two-dimensional integrals over frequency-dependent sensitive element surfaces. Black dashed curves are analytic forms based on “equivalent square” hydrophones.

Figs. 7 and 8 show effects of spatial averaging on measurements of peak compressional pressure (p_c), peak rarefactional pressure (p_r), and pulse intensity integral (pii) at the focal plane (and not derated) for ARFI (Fig. 7) and pulsed Doppler (Fig. 8) signals measured with all transducer / hydrophone combinations. The plots show 1) raw data (“Raw RF”), 2) data deconvolved for hydrophone sensitivity only (“Decon Sens”), and 3) data deconvolved for hydrophone sensitivity and inverse filtered using spatial averaging filters (“Decons Sens + SAF”). Data are plotted as functions of hydrophone nominal geometrical sensitive element diameter, d_g. Exponential and linear fits to data are also shown. Figs. 7 and 8 show that inverse filtering with the spatial averaging filter reduces the dependence of measurements on hydrophone sensitive element size. Scatter in Figs. 7 and 8 may be due to uncertainties in hydrophone sensitivities and positioning.

Fig. 7.

Effects of spatial averaging on measurements of peak compressional pressure (p_c), peak rarefactional pressure (p_r), and pulse intensity integral (pii) for ARFI signals measured with all transducer / hydrophone combinations. Names of measured parameters are placed at the top of each column (instead of as y-axis labels) to support matrix format for presenting results. The plots show 1) raw data (“Raw RF”), 2) data deconvolved for hydrophone sensitivity only (“Decon Sens”), and 3) data deconvolved for hydrophone sensitivity and inverse filtered using spatial averaging filters (“Decons Sens + SAF”). Data are plotted as functions of hydrophone nominal geometrical sensitive element diameter, d_g. Error bars (which are so small they are difficult to discern) denote standard deviations. Exponential and linear fits to data are also shown.

Fig. 8.

Effects of spatial averaging on measurements of peak compressional pressure (p_c), peak rarefactional pressure (p_r), and pulse intensity integral (pii) for Doppler signals measured with all transducer / hydrophone combinations. Names of measured parameters are placed at the top of each column (instead of as y-axis labels) to support matrix format for presenting results. The plots show 1) raw data (“Raw RF”), 2) data deconvolved for hydrophone sensitivity only (“Decon Sens”), and 3) data deconvolved for hydrophone sensitivity and inverse filtered using spatial averaging filters (“Decons Sens + SAF”). Data are plotted as functions of hydrophone nominal geometrical sensitive element diameter, d_g. Error bars (which are so small they are difficult to discern) denote standard deviations. Exponential and linear fits to data are also shown.

Figs. 7 and 8 show that values for p_c, p_r, and pii measured with the high-resolution reference HGL-0085 hydrophone (d_g = 85 μm) that were not corrected for spatial averaging (leftmost red point in each panel, “Decon Sens”) were close to values that were corrected for spatial-averaging (leftmost blue point in each panel, “Decon Sens + SAF”). The RMS differences (uncorrected vs. corrected for hydrophone spatial averaging) in parameters measured with the HGL-0085 hydrophone were 6% (ARFI p_c), 3% (ARFI p_r), 6% (ARFI pii), 4% (Doppler p_c), 4% (Doppler p_r), and 5% (Doppler pii). These small values imply that the reference SAF (S_p1(f) in (1)) did not have a big effect for the HGL-0085.

The fits may be used to predict performance of the spatial averaging filter for a typical membrane hydrophone with 400-μm geometrical sensitive element diameter, which was the most common hydrophone in the FDA database for ARFI and pulsed Doppler submissions between 2015 and 2019 [34]. Table III shows that inverse filtering with the spatial averaging filter reduced spatial averaging error magnitudes from about 34% (p_c), 22% (p_r), and 45% (pii) down to within 5% for all three parameters. The latter two parameters, p_r and pii, are typically included in acoustic output reporting tables for regulatory purposes, while p_c is not.

TABLE III.

Percent Errors in Acoustic Output Before and After Inverse Filtering to Compensate for Spatial Averaging

Pressure Parameter	ARFI		Pulsed Doppler
	Before	After	Before	After
pc	−34% ± 2%	4% ± 3%	−34% ± 2%	0% ± 3%
pr	−23% ± 4%	−2% ± 7%	−21% ± 2%	−4% ± 5%
pii	−46% ± 5%	5% ± 6%	−43% ± 2%	3% ± 2%

Values assume a typical membrane hydrophone with 400 μm geometrical sensitive element diameter. Values are means ± standard errors.

D. Validation of Simulation

Fig. 9 shows measured and simulated pulsed Doppler waveforms. Simulation errors were computed as differences between parameters or waveforms measured experimentally vs. those reconstructed from only input values for f₁, p_r, PD, and pii.

Fig. 9.

Pulsed Doppler waveform (blue solid line) generated by the L7–4 transducer and measured with the NTR hydrophone. Simulated waveform (red dashed line) reconstructed from acoustic output reporting parameters: f₁, p_r, pulse duration (PD), and pii.

For simulated ARFI waveforms, the error in PD was −1.3% ± 4.4% (mean ± standard deviation) and the error in pii was 0.2% ± 3.8% for all transducer / hydrophone combinations. The correlation coefficient between simulated and experimental waveform values was 0.95 ± 0.02 (mean ± standard deviation). The intercept and slope of linear regression fits of simulated to experimental waveform values were −0.04 ± 0.02 MPa and 0.96 ± 0.02 (means ± standard deviations).

For pulsed Doppler waveforms, the error in PD was 0.1% ± 4.3% and the error in pii was 2.5% ± 8.1% for all transducer / hydrophone combinations. The correlation coefficient between simulated and experimental waveform values was 0.90 ± 0.03. The intercept and slope of linear regression fits of simulated to experimental waveforms were −0.04 ± 0.01 MPa and 0.92 ± 0.03 (means ± standard deviations).

The ARFI and pulsed Doppler results suggest that the simulation could reliably reconstruct realistic waveforms based only on information available in acoustic output reporting tables. This supports the assessment in Part I of the effects of hydrophone spatial averaging on MI and TIB for diagnostic ultrasonic imaging systems

E. Empirical Formulas for Spatial Averaging Errors

Fig. 10 shows experimental (from Part II) and simulated (from Part I) errors in pressure parameters as functions of d_g/(λ₁F/#) for ARFI (left column) and pulsed Doppler (right column) waveforms. The product λ₁F/# is an index of the fundamental focal spot width. FWHM_1x ≈ 1.2λ₁F/# [50]. Experimental and simulated data follow similar trends. This level of agreement supports some of the underlying assumptions of the simulation, including the functional form of the pulse and the dynamic aperture algorithm (i.e., the method by which the number of transmit elements, and therefore the effective array width, is reduced for shallow imaging depths).

Fig. 10.

Experimental and simulated errors in pressure parameters for ARFI (left column) and pulsed Doppler (right column) waveforms. Experimental data are fit to functions of the form (–100%)[1 – exp(−αx)] where x = d_g / (λ₁F/#) and α is a fitting parameter. Standard deviations for measurements are less than one percent. The abscissa is the ratio of the membrane hydrophone geometrical sensitive element diameter to the product of the fundamental wavelength λ₁ and the F/# (ratio of focal length to array width). The product λ₁F/# is an index of the fundamental focal spot width. FWHM_1x ≈ 1.2λ₁F/#. All experimental and simulation data were inverse filtered to deconvolve the effects of hydrophone frequency-dependent sensitivity.

Empirical formulas for relative errors in pressure parameters were generated to provide correction factors for membrane-hydrophone-based acoustic output measurements, which may be useful particularly when a_eff(f) for the specific hydrophone used (e.g., obtained from directivity measurements) is unavailable. The experimental data in Fig. 10 were fit to exponential functions of the form (−100%)[1 – exp(−αx)] where x = d_g / (λ₁F/#). Table IV gives α values that may be used to correct for spatial averaging errors in pressure parameters for a given set of experimental parameters (d_g, λ₁, and F/#). Table IV also gives RMSD values between data and fits.

TABLE IV.

Spatial Averaging Error Coefficients

Pressure Parameter	ARFI		Pulsed Doppler
	α	RMSD (%)	α	RMSD (%)
pc	0.47	7.5	0.49	8.0
pr	0.22	8.9	0.22	7.9
pii	0.63	8.2	0.62	7.9

Spatial Averaging Error = (−100%) [1 − exp(−αx)], where x = d_g / (λ₁F/#). RMSD is root-mean-squared difference between experimental data points in Fig. 10 and function fits. These coefficients are for pressure measurements performed with membrane hydrophones.

Simulations based on circular transducers suggest that errors in p_r are essentially independent of local distortion parameter σ_q (see Table IV and Figs. 12 and 13 in [30]). This is because p_r is primarily determined by low-frequency spectral content [51] while σ_q is primarily an indicator of harmonic spectral content. (See Fig. 5 in [34]).

Simulations based on circular transducers suggest that errors in pii increase with σ_q (see Table IV and Figs. 12 and 13 in [30]). However, if d_g is chosen to be small enough so that d_g/(λ₁F/#) < 1, then errors in pii show little dependence on σ_q, provided that σ_q ≤ 2, which covers a great range of signals (as can be seen in Fig. 5 of [30]). For the data shown in Table II, σ_q = 2.5 ± 0.7 (ARFI) and 2.4 ± 0.7 (pulsed Doppler (means ± standard deviations).

IV. Conclusion

An inverse-filter method for correcting for hydrophone spatial averaging artifacts derived in Part I [34] has been validated with experimental data acquired using three clinical linear array transducers and five hydrophones. Without correction for spatial averaging artifacts, errors in MI and TIB can be substantial for ARFI and pulsed Doppler waveforms using typical transducers and hydrophones. A simulation method to obtain time-domain RF waveforms from data in acoustic output reporting tables (described in Part I) has been validated with experimental data acquired using three clinical linear array transducers and five hydrophones. The experimental validation in Part II reinforces the predictions in Part I regarding quantitative estimates for spatial averaging errors in MI and TIB for clinical diagnostic ultrasound systems.