Identifying and overcoming limitations with in situ calibration beads for quantitative ultrasound

Abstract

Ensuring the consistency of spectral-based quantitative ultrasound estimates in vivo necessitates accounting for diffraction, system effects, and propagation losses encountered in the tissue. Accounting for diffraction and system effects is typically achieved through planar reflector or reference phantom methods; however, neither of these is able to account for the tissue losses present in vivo between the ultrasound probe and the region of interest. In previous work, the feasibility of small titanium beads as in situ calibration targets (0.5–2 mm in diameter) was investigated. In this study, the importance of bead size for the calibration signal, the role of multiple echoes coming from the calibration bead, and sampling of the bead signal laterally through beam translation were examined. This work demonstrates that although the titanium beads naturally produce multiple reverberant echoes, time-windowing of the first echo provides the smoothest calibration spectrum for backscatter coefficient calculation. When translating the beam across the bead, the amplitude of the echo decreases rapidly as the beam moves across and past the bead. Therefore, to obtain consistent calibration signals from the bead, lateral interpolation is needed to approximate signals coming from the center of the bead with respect to the beam.

I. INTRODUCTION

Quantitative ultrasound (QUS) imaging can reveal subtle differences within tissue structure that might not be observable from conventional, qualitative B-mode imaging. Spectral-based QUS techniques continue to provide new sources of image contrast and valuable diagnostic information for clinical medicine (Mamou and Oelze, 2013; Oelze and Mamou, 2016). In particular, QUS enables the quantification and interpretation of tissue backscatter. The backscatter coefficient (BSC) is a fundamental property of the tissue state and contains valuable information about the structures being visualized. Several successful applications of QUS based on BSCs have been demonstrated, including improved differentiation between benign and malignant lesions (Coronado-Gutiérrez et al., 2019), quantification of steatosis in the liver (Nguyen et al., 2019), and prediction of the response to neoadjuvant chemotherapy (NAC) of breast cancer, providing an opportunity for early alterations to therapy for improved treatment (DiCenzo et al., 2020).

Accurate determination of the BSC, however, is complicated by several factors, which include tissue attenuation, system settings, and transducer diffraction. These factors will vary from setup to setup, patient to patient, and one scan to the next for a single patient. Therefore, consistency of measurements is a major inhibitor of widespread clinical use. Transducer diffraction and scanner settings can be compensated by performing reference scans using either a planar reflector approach or reference phantom, and while these scans can effectively calibrate system effects and diffraction, their ability to account for frequency-dependent attenuation and transmission losses through tissue layers is limited (Mamou and Oelze, 2013). The use of local and total attenuation techniques have been proposed to account for attenuation loss but have yet to gain clinical traction (Bigelow et al., 2005; Coila and Lavarello, 2018; Labyed and Bigelow, 2010; Pawlicki and O'Brien, 2013; Vajihi et al., 2018). Other approaches involve using full angular spatial compounding to obtain an estimate of total attenuation and have been validated ex vivo using tissue-mimicking phantoms but have not been validated in vivo (Coila et al., 2021). Furthermore, accounting for transmission losses through different layers remains a roadblock to improving BSC estimation.

Previous work by Nguyen et al. (2020) proposed that a biocompatible metal bead could be inserted in the area near the tissue of interest at a similar depth to that of the desired measurements for QUS to accurately account for the effects of total tissue attenuation and transmission losses. Due to the material and size of these beads, they may be inserted via needle and need not be removed. The studied beads were made of titanium and ranged from 0.5 to 2 mm in diameter compared to an ultrasonic wavelength between 200 and 300 μm. The results of that study demonstrated that in situ calibration beads could provide consistent estimates of BSCs when lossy layers were placed between the region of interest in a sample and the transducer. BSCs estimated with the reference phantom technique, which requires a scan of material outside of the tissue of interest, varied substantially when a lossy layer was placed on top of the sample.

Although previous work on in situ calibration targets addressed the feasibility of using titanium beads for the purpose of calibration, it studied only a single size of bead after testing multiple bead diameters to find a sufficient signal-to-noise ratio (SNR) among them. In this study, we assess the repeatability of using a titanium bead as a calibration target. Specifically, three aspects of measurement from the in situ calibration target are investigated.

The first aspect involves quantifying the effects of the number of echoes from the bead that are used to estimate its calibration spectrum. Backscatter from the bead results in multiple scattering, which manifests as a train of echoes decreasing in amplitude versus time, i.e., the so-called comet-tail artifact. The temporal distance between echoes depends on the size of the bead. Therefore, an important consideration is how many echoes from a bead are needed to provide a consistent reference spectrum.

The second aspect involves quantifying the effects of the size of the calibration bead and echo train. The separation of echoes from the calibration bead is directly related to the size of the bead. The choice of bead size informs trade-offs between echo separation, SNR, and the practicality of insertion into soft tissue.

The final aspect involves the sampling error that results from scanning across the bead. Beam formation and beam translation are used to construct different scan lines. In a linear sequential scanning mode, for example, each scan line is separated by a pitch of the array. It is possible that as the beam is translated across the bead, the beam may not be centered directly on the bead, which results in a loss of backscattered signal strength when compared to a beam directly centered on the bead. Therefore, here, we quantify the consistency of the signal from the bead when the beam is directly on and off center of the bead. To mitigate the effects of the beam position on the calibration bead, averaging and interpolation, as well as a combined approach, were evaluated.

II. BACKGROUND THEORY

Assuming plane wave insonification, Faran theory can be used to predict the scattering from the bead, which was assessed in previous work (Faran, 1951; Nguyen et al., 2020). However, it is of greater interest to experimentally determine the bead's BSC so that it can act as calibration within other media and characterize the conditions under which a suitable spectrum for calibration exists. The total response of an ultrasound system under any imaging conditions can be expressed as

$$W(\hspace{0.17em}f,\mathbf{x})=H(\hspace{0.17em}f)T(\hspace{0.17em}f,\mathbf{x})A(\hspace{0.17em}f,\mathbf{x})D(\hspace{0.17em}f,\mathbf{x})S(\hspace{0.17em}f,\mathbf{x}),$$ (1)

where the total response of the system, $W(\hspace{0.17em}f,\mathbf{x})$, is the periodogram estimate of the backscattered signal and a product of the impulse response, H( f), transmission losses, $T(\hspace{0.17em}f,\mathbf{x})$, attenuation, $A(\hspace{0.17em}f,\mathbf{x})$, diffraction and system effects, $D(\hspace{0.17em}f,\mathbf{x})$, and the scattering, $S(\hspace{0.17em}f,\mathbf{x})$ (Mamou and Oelze, 2013). Information about the medium is contained in the attenuation function and scattering function. For BSC calculation, the scattering function must be determined. To isolate $S(\hspace{0.17em}f,\mathbf{x})$ and determine the BSC, the additional factors must be obtained separately such that they can be divided out from the response, $W(\hspace{0.17em}f,\mathbf{x})$. The reference phantom method is able to capture H( f) and $D(\hspace{0.17em}f,\mathbf{x})$, and attenuation estimation methods can be used, in theory, to derive and account for $A(\hspace{0.17em}f,\mathbf{x})$. However, attenuation estimation methods will not account for attenuation as accurately as an in situ method, particularly where there are multiple thin layers of tissue as may be the case in vivo. Transmission losses, $T(\hspace{0.17em}f,\mathbf{x})$, cannot be captured by the reference phantom method.

The in situ calibration approach can be used to determine the BSC of tissue adjacent to the bead by assessing the tissue response relative to the response of the bead under approximately the same overlying conditions. This is described by

$${\sigma }_{\text{BSC},\text{tissue}}=\frac{W(\hspace{0.17em}f,\mathbf{x})}{W_{\text{bead}}(\hspace{0.17em}f,\mathbf{x})}{\sigma }_{\text{BSC},\text{bead}}(\hspace{0.17em}f,\mathbf{x}),$$ (2)

where

$$W_{\text{bead}}=H(\hspace{0.17em}f)T(\hspace{0.17em}f,\mathbf{x})A(\hspace{0.17em}f,\mathbf{x})D(\hspace{0.17em}f,\mathbf{x})S_{\text{bead}}(\hspace{0.17em}f,\mathbf{x}).$$ (3)

The overlying conditions are approximately identical, and the imaging settings are the same between the sample and the bead, which means that differences between $W(\hspace{0.17em}f,\mathbf{x})$ and $W_{\text{bead}}(\hspace{0.17em}f,\mathbf{x})$ are attributed to differences in the scattering described by $S(\hspace{0.17em}f,\mathbf{x})$. In particular, for soft tissues, S_bead is approximately independent of the imaging environment and, thus, only needs to be determined once for a bead of given material and size.

III. METHODS

In the experimental setup which follows, the aim is to find S_bead for each of the tested calibration beads and quantify the ability to consistently estimate $S_{\text{bead}}(\hspace{0.17em}f,\mathbf{x})$. The beads are suspended in an agar phantom a few millimeters from the surface with no additional scatterers or attenuators present. The phantom is scanned with a transducer probe that is coupled to the phantom through a water standoff. With this setup, attenuation is minimal and scattering can be attributed primarily to the bead.

A. Imaging system

Imaging was performed exclusively with a 10-MHz, single-element transducer (Valpey-Fisher, Hopkinton, MA). This focused transducer, $f/3$, with a diameter of 12.7 mm had a full-width, half-maximum beam width of approximately 330 μm and a −6-dB bandwidth determined to be approximately 3.8 MHz; the 10 MHz center frequency was chosen to match that of the probes used for clinical QUS imaging. This transducer was mounted on and moved via a micropositioning system (Daedal, Inc., Harrison City, PA). The phantom was submerged in a tank filled with degassed water for imaging and positioned below the transducer with the embedded calibration beads within the focal zone of the transducer beam. The transducer was operated in pulse-echo mode using a Panametrics 5900 pulser-receiver (Olympus Corporation, Tokyo, Japan). The amplified radio frequency (RF) data, sampled at a rate of 250 MHz, was processed through a workstation analog-digital converter and viewed and saved with custom LabVIEW software (National Instruments, Austin TX). The data were exported to matlab® R2020a (The MathWorks, Natick, MA) for post-processing.

We scanned spherical titanium beads with diameters of 2, 3, and 4 mm as calibration targets. For each, a total of four sets of scans were acquired. In each scan, the transducer was translated over the surface of the phantom, acquiring a grid of A-mode pulse-echo lines to produce a set of three-dimensional (3D) scans. Three of these 3D acquisitions for each bead were repeatablility trials with a step size of 200 μm between each scan line to represent the scan line separation that might occur with a clinical linear array system. The fourth acquisition for each bead was a high-resolution scan with a lateral step size of 10 μm and an elevational step size of 200 μm.

In addition to the bead scans, one additional data set was obtained: two A-mode acquisitions of a Plexiglas^TM planar reflector using the same two sets of pulser-receiver settings that were used to acquire the low- and high-resolution scans.

B. Tissue models

The phantom used for these experiments was prepared using 7.5 g of bacteriological agar (Thermo Fisher Scientific, Waltham, MA), 300 ml of water, and 4.7 g of 1.5% Germall^TM -plus (Ashland Global, Wilmington, DE) heated to a maximum temperature of 80 °C. The calibration beads consisted of titanium spheres with diameters of 2, 3, or 4 mm. To suspend the titanium beads, the liquid agar mixture was poured into a refrigerator-chilled vessel partway, allowed to sit for approximately 1 min, and then the titanium beads were added, followed by the remainder of the agar mixture. The phantom was placed into a rotisserie for 30 min, allowing it to rotate and prevent the titanium beads from settling to the bottom of the phantom. The phantom was not made with any glass beads or other scatterers/attenuators. The imaging depth to the center of the titanium beads from the surface of the phantom was approximately 18 mm in each case; the beads settled into an approximately linear formation across the phantom's diameter, and adjacent beads were spaced approximately 2 cm apart.

Additional in vivo data were collected using a rabbit tumor model. The procedure was performed according to a protocol approved by the University of Illinois at Urbana-Champaign Institutional Animal Care and Use Committee (IACUC protocol 20087). VX2 tumor fragments were injected into the mammary fat pad of a 3 month old, 2 kg, female New Zealand White rabbit (Charles River Laboratories, Wilmington, MA) anesthetized with 2% isoflurane. Tumor growth was monitored for 2–3 weeks, until the tumor reached a diameter of about 2.5 cm. At this point, the tumor area was shaved and disinfected, and a 2-mm titanium bead was embedded into the tumor with a 12-gauge stainless steel needle. Bleeding was minimal, and pressure was applied to the injection site to facilitate wound closure. Three days later, the tumor and bead were imaged using a SonixOne (BK Ultrasound, Peabody, MA) ultrasound imaging system with an L14–5/38 transducer. Imaging was performed with and without meat placed on top of the tumor. A sample of pork belly approximately 1.5–2 cm thick was purchased at a local supermarket, warmed to room-temperature, and placed over the tumor site for imaging. This additional layer provided fatty layers to better simulate human breast tissue. For the case where no pork belly was present, a water standoff was used to maintain a constant bead depth from the transducer probe.

C. Echo windowing

The RF data from the high-resolution scans were inspected to find the scan line that had the highest response, which was assumed to be the signal where the beam was directly on the bead. Multiple reverberant echoes from the beads were observed to arrive at regular time intervals. An example waveform is shown in Fig. 3. Collecting the full echo waveform would require a long time window, and knowing how long the window should be to collect all of the echoes is ambiguous. Therefore, because the multiple echoes from the beads came at regular time intervals, we assessed how the power spectrum estimate from the bead changed as the window was increased in size to collect ever-increasing numbers of echoes from the bead. For each window size, the power spectrum estimate was divided by the power spectrum obtained from the reflected signal coming from a planar reflector in water, which acted as a reference. This process was repeated across each of the three beads. The window size was determined via visual inspection to locate the beginning and end of the first and most separable echo. Subsequent echoes were determined by evaluating the length of the first echo from its midpoint to its end, which was assumed to be the midpoint of the separation between the first and second echoes; this length was doubled and added to the end of the previous echo's window to include the succeeding echo.

FIG. 3.

(Color online) (a) The time-domain signal from the 2-mm diameter bead with color-coded portions representing increasing numbers of included echoes and (b) the corresponding BSC estimates from the bead using different numbers of echoes in the range gate are shown.

The durations of the echoes from this single scan line were collected into a variable array that would be used to iterate the BSC calculation over increasing numbers of echoes. In each iteration of the loop, an increasingly larger time window was extracted from the RF data, its power spectral density (PSD) was evaluated, and this value was divided by the PSD of the planar reflector, as described in Eq. (2). The spectra were averaged together over the slice of the 3D scan data containing the center of the bead. This processing was repeated across each of the three beads.

The BSC was observed to have certain frequencies where dips were present. The processing described here allowed the BSC curves for increasing numbers of reverberant echoes to be compared in terms of the magnitude of their dips in dB.

An alternate approach to windowing was also investigated; instead of accumulating subsequent echoes in the window, each echo was collected in its own rectangular window, and its spectra were calculated and divided by those of the planar reflector; the BSC curves calculated from the first five echoes were averaged together. The window size was fixed to 70 samples (280 ns total) about the visually determined center of magnitude for the first echo and reverberant echoes thereafter occurred every 200 samples (800 ns).

D. Bead size

The calibration bead size influences the separability of echoes with larger calibration beads providing greater separation between successive echoes. However, the larger the bead size, the more invasive the placement of the bead is, and so the desired choice of calibration bead will be the smallest one that can achieve a high SNR and provide sufficient echo separability, thus, minimizing interference between echoes and BSC reference curves.

For separability to be maintained, the bandwidth of the transducer must be wide enough to produce a spatial pulse length that is smaller than the round trip distance of the echo within the bead. We further define “sufficient” separability to be related to the −6-dB bandwidth of the source, particularly because the single-element transducer used in these experiments is a more narrowband source than most diagnostic imaging probes. The −6-dB bandwidth of the transducer was calculated using the spectrum of the echo from a Plexiglas^TM planar reflector. For a 10-MHz transducer beam in titanium, ${\lambda }_{\text{bead}}=607\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$, assuming a speed of sound of 6070 m/s for titanium (Rumble, 2021). In the phantom and water standoff, the wavelength is estimated to be ${\lambda }_{\text{phantom}}=148\hspace{0.17em}\hspace{0.17em}\mu \mathrm{m}$.

E. Interpolation and averaging

When translating the beam across the calibration bead to acquire backscattered signals, the signals are sampled from the beads at finite step sizes in the lateral and elevational directions as shown in Fig. 1(a). The acquisition of the calibration bead signals may be undersampled, resulting in inconsistent calibration signals. Therefore, it is necessary to evaluate the effects of beam sampling on the acquisition of calibration signals and develop strategies for mitigating the sampling problem illustrated in Fig. 1(b). Specifically, interpolation and signal spectrum averaging approaches were applied and their ability to overcome the sampling problem was quantified. To evaluate these issues, calibration signals sampled at a 10-μm step size, i.e., high-resolution scans, were collected. This step size is much smaller than the beam width of the transducer. A B-mode image of the center of the 2-mm bead's high-resolution scan is shown in Fig. 2.

FIG. 1.

(Color online) (a) The setup for scanning a bead with a single-element transducer moving across the bead surface and (b) spatially downsampling and then recovering a maximum via interpolation are depicted.

FIG. 2.

The B-mode image of the center of bead, imaged with a 10-μm step size, is shown.

In previous work, under the best-case scenario where the center of a calibration bead was captured in a sample, a 2-mm titanium bead was found to have a SNR of 11.61 dB compared to the background phantom signal (Nguyen et al., 2020). Smaller beads with lower SNRs, e.g., a 1-mm bead, had a SNR approximately 5 dB lower than that for the 2-mm diameter bead. Poor sampling could result in a loss of the overall signal magnitude in addition to inconsistency between measurements. In the current study, the response from the center of the beads, obtained from the high-resolution scans, was used as a baseline for measuring the quality of the reconstructed signal after post-processing. Post-processing strategies, which result in no more than a 1-dB drop in the signal magnitude across all of the tested sample quality configurations, are considered desirable. The sample quality is considered highest when the scan line is centered on the bead, which results in the largest SNR.

To assess the effect of interpolation on the recovery of a consistent maximum signal from the bead, the high-resolution RF data were artificially downsampled to a step size of 200 μm laterally, a downsampling factor of 20 from the original 10-μm step size. The downsampled data were compared to the ground truth, following two-dimensional cubic interpolation in the time domain. To test the effects of interpolation, we quantified the goodness of fit to the centerline when undersampling using two configurations. In the first configuration, sampled points were acquired such that the center of the bead was halfway between two sampled points, thus, each sample nearest the true maximum was 100 μm away from it. In the second configuration, the sampled lines were selected so that the maximum would be at distances of 10 and 190 μm to the nearest interpolation lines. This approach evaluated the two extremes of interpolation.

Use of averaging to recover a consistent PSD from a bead was explored as an additional component of addressing the potential for sampling errors. Again, the high-resolution data were downsampled to a step size of 200 μm laterally, using multiple sampling configurations similar to those used in an interpolation-only approach. The time-domain RF data for each line were converted into their frequency-domain representations, and the power spectra associated with each scan line were averaged together.

Additionally, a combination of these two techniques was used: interpolation combined with averaging. The data were downsampled again to a 200-μm step size and interpolated to 10, 50, and 100-μm step sizes. After the PSDs for these arrays of lines that intersected the bead were calculated, they were averaged to form one single calibration spectrum. For each of the interpolation step sizes, four different offsets from the scan line with the maximum signal were used to place it in different regions of the interpolation field. To obtain these offsets, shifted downsampling was performed on the high-resolution data. The offsets used were −50, 0, +50, and +100 μm.

IV. RESULTS

A. Number of echoes

The first evaluations performed on the data were to quantify how windowing an increasing number of reverberant echoes from the beads affected the BSC. Artificially windowing the bead responses to only the first echo resulted in a smoothing effect on the corresponding BSC from the bead compared to capturing multiple echoes from the bead. This property was noted in every case for each bead size that was examined. Figure 3 shows the BSC versus frequency for each window on a 2-mm diameter bead. Each colored curve represents an increasing time window for the response from the center of the bead, as shown in Fig. 3(a) in terms of the axial depth in mm. These windowed echoes produce the corresponding BSC curves as shown in Fig. 3(b). Here, because the BSC is a value with arbitrary units, the dB scale was referenced to the single-echo BSC curve's value at 10 MHz for readability. Note that in Fig. 3(a), the reverberant echoes do not originate from a deeper depth than the first echo at 38 mm as the axis may imply. Spatial representation of the x axis is meant to clarify the depth of origin for the initial echo for readability; the reverberant echoes, which arrive later in time from the same origin, should not be assigned a spatial depth.

The series of dips in Fig. 3(b) appear because echoes from the bead occur at periodic time delays corresponding to the size of the bead. The magnitudes of these dips in the BSC are associated with the number of echoes captured within the time window, where longer time windows tend to result in larger dips. For example, at around 11 MHz, the curve for the two-echo window experiences a change of approximately −1.3 dB due to the dip. Over this same frequency range, the fully windowed curve experiences a dip which is approximately −8.3 dB lower than the adjacent maximum. The single-echo window does not have such apparent dips, and the difference between the adjacent local minimum and maximum in a similar frequency range for this BSC curve is −0.068 dB. When the window size was both increased and decreased by 10% of its original value for the single-echo scenario, the average differences in the BSC curves over the −6-dB bandwidth were 0.0179 dB and 0.0544 dB for the decreased and increased windows, respectively.

The same smoothing effect was observed in the repeatablility trials (n = 3) for each bead. Within the −6-dB bandwidth, the standard error of the BSC measurement was observed to increase as the frequency varied from the center frequency of the imaging probe at 10 MHz. For the case where the BSC was calculated from the 2-mm bead's first echo, as shown in Fig. 4, the error was no more than approximately 1.5 dB. At the peak of the bandwidth, the standard error varied by less than 0.1 dB.

FIG. 4.

(Color online) The BSC as a function of frequency for the 2-mm bead in the repeatablility trials is depicted.

The in vivo model was used to validate the use of only the first echo from the calibration bead to form the reference spectrum. Figure 5(a) shows the B-mode images corresponding to the scans with and without a fatty layer between the tumor/bead and transducer probe. One can observe from the B-mode images that the tumor signal at the depth of the bead has a lower SNR when the layer is present, i.e., a more attenuated signal due to the fatty layer. Figure 5(b) shows the ratio of the PSDs of the boxed regions in Fig. 5(a) to that of the first echo from the bead adjacent. The ratio of the tumor PSD over bead PSD for the data with the fatty layer above tends to remain above that of the ratio of the PSDs without the layer except for a region around 4.5 MHz.

FIG. 5.

(Color online) (a) In vivo tumor scans without and with a fatty layer and (b) the PSD ratio of boxed regions in each B-mode scan in (a) to the corresponding first echo of the bead are shown.

When the bead response was windowed to individual echoes as opposed to cumulative echoes, the dips and peaks that were present in Fig. 3 did not appear. Across the bandwidth of the transducer, the BSC varied for each individual echo. Notably, the first-echo response (labeled “echo 1”) follows closely with the dashed line representing the sum of each individual window's response. Outside of the imaging bandwidth, the two curves did not match as closely. In Fig. 6(b), the two window sizes—the wide window used for the BSC calculations in Fig. 3 and the narrower, 70-sample window used in Fig. 6(a)—are compared. In addition, a third calculation was performed using a similar narrow window size on the planar reflector as opposed to using the entire response.

FIG. 6.

(Color online) (a) The BSC of each individually windowed echo, as well as the BSC of the sum of the spectra of these responses, and (b) a comparison of the summed echo from (a) with the single-echo window from Fig. 3(b), as well as with a narrow window version which used a similar window on the planar reflector, are shown.

For each diameter of each bead, the first echo was found to be separable from successive echoes. Because the goal was a consistent and repeatable calibration spectrum, windowing only the first echo provided the smoothest spectrum and was, thus, used for subsequent analysis.

B. Bead size

The beam intensity was observed to falloff as the beam moved away from the center of the bead, as shown in Fig. 7. The −6-dB roll-off was estimated to be around 380 μm, measured as the distance between the two crossings. Imaging of a 25-μm wire target as a point source suggested an experimental beam width of approximately 330 μm. The −6-dB roll-off of 380 μm was observed for each bead regardless of diameter, although the magnitude of the response was larger for larger beads.

FIG. 7.

(Color online) A comparison of the width of each bead based on the maximum magnitude in the frequency spectrum for each lateral increment is shown.

C. Interpolation and averaging of beads

The variation of the estimate of the PSD resulting from the undersampling of the bead signal due to beam translation was assessed and techniques to mitigate the undersampling were quantified. Specifically, we first downsampled the high-resolution scan from 10 to 200 μm, effectively reducing the resolution by a factor of 20. Two cases of two-dimensional interpolation, representing interpolation in the axial and lateral directions, were explored to address the uncertainty of sampling the centerline. The interpolation points, consisting of axial data from a single scan line, were windowed to the first echo. The first interpolation was staged in such a way that the centerline was situated equidistant between two interpolation points. For the second interpolation, the data were staged so that one interpolation point was 10 μm from the centerline and the other interpolation point was 190 μm from the centerline. In both configurations, the data were interpolated cubically in matlab (The MathWorks, Natick, MA) using the interp2 function, which uses a 4 × 4 convolution matrix to perform interpolation. Figure 8 shows the spectra from a single scan line centered on the bead and from “center” scan lines interpolated from the downsampled data. For the 2-mm bead, the worst case loss of magnitude from the ground truth spectra was −0.438 dB. Interpolation provided a spectrum that closely matched the actual centerline spectrum from the bead; however, it can be a computationally and time-intensive task. If a similar or greater level of consistency could be obtained through other, more computationally efficient means, those would be preferable.

FIG. 8.

(Color online) The recovery of the maximum bead response in the frequency for the 2-mm bead via time-domain interpolation is depicted.

In a second approach, spectra from multiple scan lines across the bead's diameter were averaged as shown in Figs. 9 and 10. Only spectra from lateral lines within the −6-dB beam width of the transducer were included in the average. The number of lines satisfying this condition varied between 12 and 13, depending on which shift was performed, and was fixed at 12 for consistency within the results. When averaging was performed over the undersampled lines, i.e., separated by 200 μm as in Fig. 9, the magnitude of the response relative to the average of the spectra from lines separated by 10 μm was smaller than the interpolation-only approach; i.e., the 0-μm shift achieved only 20.47% (−13.8 dB) of the maximum magnitude of the ground truth. Therefore, the simple averaging of the spectra across the −6-dB beam width did not provide a spectrum that matched the high-resolution case.

FIG. 9.

(Color online) Spectra from averaged lines spaced at 200 μm, sampled such that the center is missed by the specified shift amount. The ground truth line was calculated by averaging all lines at the original 10-μm step size within the −6-dB beam width.

FIG. 10.

(Color online) Spectra of the combined interpolation/averaging approach for the 2 mm bead using interpolation to restrict averaging to −6 dB about the center of the data. The broader spectrum is shown in (a) and the boxed region is shown in greater detail in (b).

A hybrid method using interpolation and averaging to recover a spectrum was also tested. This approach is shown in Fig. 10. The high-resolution data were first downsampled to a 200-μm step size and the centerline was offset by −50, 0, +50, and 100 μm, where an offset of 0 μm sampled the centerline. These sampled lines were interpolated to lateral step sizes of 10, 50, and 100 μm, as shown in the legend in Fig. 10(b). The ground truth data were obtained by performing no shifting or interpolation and averaging each line within the −6-dB beam width. A difference of 0.4% in magnitude relative to the ground truth was observed between the two edgemost cases tested, whereas the two tested cases for the interpolation-only approach were separated by 1.7% of the ground truth magnitude. Because interpolation can be used to recover a point near the center scan line, these data can also be used to estimate the location of the scan lines whose spectra are less than −6 dB down from the spectrum calculated from the center scan line.

V. DISCUSSION

In this study, we considered the use of an in situ calibration bead for providing a reference spectrum for the BSC-based QUS analysis. Specifically, we quantified the ability to consistently provide a reference spectrum when scanning the bead using broadband transducers that scan samples laterally with defined step sizes. Providing consistent estimates from the in situ calibration bead is required for adoption of an in situ calibration bead approach in practice.

In the first case, we quantified the shape of the reference spectrum from the bead when multiple echoes from the bead were included for the estimate of the bead spectrum. Several observations should be noted regarding the inclusion of multiple echoes in the power spectrum estimate. First, choosing a window that includes multiple echoes from the bead resulted in a spectrum with a distinctive pattern having multiple large dips. Second, these dips were present when only two reverberant echoes were included in the range gate. Third, the dips were larger when more echoes were included. Finally, by only calculating the power spectrum from the first echo, the power spectrum estimate was smooth without the large dips.

The presence of dips makes a spectrum not ideal for use as a reference spectrum. The dips represent signal regions with very low SNRs; by dividing a sample spectrum by a reference spectrum with large dips, the low SNR in the dip regions will amplify the normalized spectrum in the region of the dips. This will result in more error in the calculated BSC in the region of the dips.

However, if the data are windowed to a single echo, for any echo, no such concerns exist because the dips are not present. Averaging the echoes together individually also mitigates the dips, suggesting that the dips arise mainly from time-shifted copies inducing sinusoidal behavior in the frequency domain. Echoes beyond the first, when averaged in individually, appear to have little impact on the BSC curve within the imaging bandwidth. The spectrum from the first echo of the bead does not have any frequency bands that have large nulls or dips, improving the QUS measurement quality. Additionally, narrower windows that attempt to exclude noise may clip the signal. Therefore, in using an in situ calibration bead for a reference, we recommend that the window size and positioning are such that the entirety of only the first echo is captured, and the size of the bead and length of the pulse are sufficient to fulfill that requirement.

The in vivo data provided evidence that the bead was able to account for different layer effects in vivo. It is impossible to image the exact same region in the tumor from one scan to the next and especially when a layer of fatty meat is placed on top between scans. Therefore, it is not anticipated that the ratio of the PSDs would perfectly match unless the tumor was homogeneous throughout, which it is not. However, these two curves are sufficiently close to suggest that the first-echo calibration bead method is indifferent to overlying conditions in an in vivo model. Therefore, the in situ bead calibration could minimize tedious total attenuation calculations or transmission losses. Furthermore, special care should be taken during in vivo RF data acquisition to prevent bead signal clipping, which may be simple to overlook.

The 2-mm bead provided the least amount of separation between the initial and subsequent echoes. The calculated spatial pulse length suggests that even though the separation between the first two echoes was less than that of the 3- and 4-mm beads, separation remained such the 2-mm bead would still be a viable calibration target with most imaging probes. A very narrowband imaging probe (bandwidth <1.5 MHz) using a similar pulse will not provide a separable first echo for a 2-mm bead.

Pulses consisting of multiple cycles or different patterns may exceed these physical limitations, affecting separability. Additionally, boundaries between echoes were determined visually by human operators for these experiments. It may be desirable to automate this process, and algorithms for isolating the first echo could face similar issues with poor separability as would a human operator. Future work is necessary to further examine echo separation and the effect it has on calibration spectra.

Post-processing in matlab^® (The MathWorks, Natick, MA) was performed solely using the first echo because all calibration beads had separable first echoes in this experimental setup. Emphasis was also placed on performing the post-processing on the 2-mm bead as it remained the most minimally invasive implant choice.

The scattered energy received by the transducer, as shown in Fig. 7, was uniform in shape across the three beads tested, and the magnitude of the response varied positively with the radius of the bead a as approximately a². This result matches with the theoretical backscatter power approaching $\pi a^2$ when ka is large (Lauchle, 1975).

Although the beads had different diameters, the falloff of the spectral magnitude laterally was approximately the same for each bead size. The larger bead did not provide more samples from which to reconstruct the bead response and, therefore, each bead was subject to the same limitations in lateral sampling.

The consistency of the averaged curves shown in Fig. 9 was highly dependent on the number of lines used for averaging and the spatial locations of those lines. Low-magnitude data from the edge of the beads provided enough of an outlier effect to cause the different spectra to group into bands based on the number of lines used in the calculation. This is problematic because it means that at step sizes on the order of 200 μm, similar to what might be produced by a scanner in a clinical setting, there are a low number of sample points across the bead. Determining the number of samples and lateral sampling required for producing estimates of the reference spectrum with low bias and variance requires standardization that can compensate for small spatial misalignments between the bead and transducer as these can dramatically affect the magnitude of the response. A standardization, such as averaging only over the −6-dB beam width relative to the maximum sample, may provide inconsistent results because the response of the bead falls off quickly relative to the step size between samples. With the data used in this work, 200-μm spacing resulted in approximately three scan lines above −6 dB to average, depending on how the samples were shifted from the centerline. Averaging over such a small number of scan lines would escalate the potential for an increased estimate variance.

Averaging the spectra generated from lines scanned across the bead did not appear to provide the consistency necessary for an accurate baseline bead BSC curve. However, the hybrid method, with fewer interpolated scan lines augmented by averaging, yielded more consistent results than interpolation alone while providing a sufficient sample density to standardize averaging and improve the signal strength as compared to the ground truth. Interpolation alone proved to be adequate per the metrics of success described in this work but because it is a more computationally demanding process with more variability depending on the sample locations, the hybrid approach is preferable.

Ultimately, these results suggest that as long as the number of lines used for spectral averaging is fixed, shifting and interpolation will yield consistent calibration curves. Based on the results from this study, interpolation of a set of scans spaced at 200 μm interpolated to 100 μm and with averaging of spectra obtained from scan lines out to a falloff of −6 dB is sufficient for a consistent signal, regardless of whether the maximum signal was captured during sampling.

VI. CONCLUSION

In this work, we demonstrate that titanium calibration beads can provide a calibration curve for BSC measurements with high precision. Time-windowing was found to have a profound effect on the quality of the calibration curve with the inclusion of multiple echoes from the bead resulting in spectra with larger dips, which leads to a higher variance in the BSC estimate. Therefore, acquiring only the first echo from a bead will provide a better calibration signal. Multiple sizes of beads were found to produce similar spectra and curves as well as have comparable directionality despite differences in size. The magnitude associated with each bead's BSC curve was the most notable distinguishing factor among them. This suggests that there are no distinct advantages that a larger 3- or 4-mm diameter bead would provide as previous work showed that 2-mm beads produced an adequate SNR underneath tissue layers (Nguyen et al., 2020). The ability to miss the maximum bead response using larger elevational step sizes commonly found in 3D ultrasound scans is addressed through combinations of averaging and cubic interpolation. Interpolation-only approaches can be computationally intensive depending on the granularity of the result and compared to an averaging-only approach, do not provide the same level of consistency across spatially shifted samples. However, the averaging-only approach encounters issues with insufficient sampling to establish a robust and easily defined region over which to average samples. A combined approach unites the shift-resistant nature of averaging with the lateral scan line sample density provided by interpolation to produce consistent calibration curves. Use of titanium beads continues to show promise as a way of calibrating BSC measurements in vivo, eliminating external calibration setups which cannot capture the full detail of tissue surrounding a region of interest.