Passive Cavitation Imaging Artifact Reduction Using Data-Adaptive Spatial Filtering

Abstract

Passive cavitation imaging (PCI) with a clinical diagnostic array results in poor axial localization of bubble activity due to the size of the point spread function. The objective of this study was to determine if data-adaptive spatial filtering improved PCI beamforming performance relative to standard frequency-domain delay, sum, and integrate (DSI) or robust Capon beamforming (RCB). The overall goal was to improve source localization and image quality without sacrificing computation time. Spatial filtering was achieved by applying a pixel-based mask to DSI- or RCB-beamformed images. The masks were derived from DSI, RCB, or phase or amplitude coherence factors using both receiver operating characteristic (ROC) and precision-recall curve analyses. Spatially filtered passive cavitation images were formed from cavitation emissions based on two simulated source densities and four source distribution patterns mimicking cavitation emissions induced by an EkoSonic^™ catheter. Beamforming performance was assessed via binary classifier metrics. The difference in sensitivity, specificity, and area under the ROC curve differed by no more than 11% across all algorithms for both source densities and all source patterns. The computational time required for each of the three spatially filtered DSIs was two orders of magnitude less than that required for time-domain RCB and thus this data-adaptive spatial filtering strategy for PCI beamforming is preferable given the similar binary classification performance.

I. Introduction

Conventional passive cavitation imaging (PCI) uses the delay, sum and integrate (DSI) algorithm [1–4] to beamform passively acquired ultrasound signals. This beamforming algorithm is based on the relative time of flight from each cavitation source location to each element of the passive array. The DSI algorithm can be implemented in the temporal or frequency domains [5]. The frequency-domain algorithm facilitates a lower computational cost than the time-domain approach if the number of frequency bins analyzed is less than the number of temporal samples acquired per channel [6] and enables independent beamforming of passively received signals from stable and inertial cavitation based on frequency characteristics [7].

The point spread function (PSF) of the DSI algorithm depends on the receiving aperture of the array [3] and the frequency and location of the source [7]. If the receiver is a clinical diagnostic linear array, the PSF lateral dimension, $W_{-6dB}$, is typically within one order of magnitude of the wavelength, $\lambda$, and the axial dimension, $L_{-6dB}$, is approximately an order of magnitude larger than $W_{-6dB}$, which results in poor axial resolution [7]. PCI axial resolution does not depend on the insonation pulse duration or shape, but only the frequency-dependent diffraction pattern of the receiving array [5]. Because the DSI algorithm is simple and easily parallelizable it has been implemented in real-time [8–11].

Robust Capon beamforming (RCB) is a data-adaptive passive beamforming algorithm that reduces the PSF size relative to DSI [12]. For each pixel location, optimal element weights are computed such that off-axis energy in the delayed signals is minimized subject to a unity gain constraint. This adaptive weighting suppresses off-axis interference, including multiple bubble scattering. Coviello et al. [12] showed that RCB passive cavitation images of bubble activity induced by a high intensity focused ultrasound transducer in a flow channel have improved axial resolution and a more accurate localization of cavitation in the focus than a time-domain DSI algorithm [13]. Although the algorithm is substantially more computationally intensive, it can be implemented in real-time with adequate computational power and technique [8].

Several decades ago, Mallart and Fink [14] introduced an adaptive focusing technique that used a single computed parameter, or factor, to evaluate and monitor the convergence of phase aberration correction techniques. Hollman et al. [15] applied this factor as the ratio of the coherent sum to the incoherent sum across the elements of an imaging array, to suppress the influence of side lobes. Li and Li [16] incorporated the metric into an efficient delay-and-sum beamformer as a scaling factor (ranging from 0 to 1), argued in favor of a generalized coherence factor, and used it to correct for both near-field and displaced-phase errors. Boulos et al. and Lu et al. applied a phase coherence factor to passive acoustic mapping to reduce the point spread function and improve the accuracy [17, 18].

Low-amplitude pixel signals in passive cavitation images have been associated with artifactual signals due to the point spread function. To remove such PCI artifacts, the objective of our study was to investigate the utility of data adaptive spatial filtering based on receiver operator characteristic (ROC) and precision-recall (PR) curve analyses. Frequency-domain amplitude and phase coherence factors, Robust Capon, and DSI beamformers were used to specify thresholds to create masks used for spatial filtering. The performance and computational cost of each spatially filtered beamforming approach were compared using simulated sources and experimentally acquired passive cavitation data. Beamformer performance was assessed via binary classifier metrics, including area under the ROC curve, sensitivity, specificity, positive predicative values, and F₁ score [19, 20].

II. Methods

A. Simulations

To assess the performance of passive cavitation beamforming algorithms, a set of simulations was designed based on the experiments performed by Lafond et al., where Definity^®, infused through the drug delivery ports of an EkoSonic^™ endovascular catheter (Boston Scientific, Natick, MA, USA), nucleated cavitation upon 2.25 MHz insonation [21]. Briefly, the catheter-mounted ultrasound source consisted of a sandwiched pair of ultrasound transducers. The transducers generated a cloverleaf-shaped acoustic pressure field composed of 4 orthogonal lobes (Fig. 1(a), adapted from Fig. 3 of [21]) in the imaging plane of a passive receiving array. For simplicity, the simulations used a uniform pressure amplitude of 3 MPa peak to peak within the lobes and zero amplitude outside the lobes, similar to the approach used by Coviello et al. [12]. The size of the lobes matched the −6 dB field pattern of the EkoSonic^™ catheter and corresponded to a 5.8 mm² area [21]. The catheter was located in the center of a 6.35 mm inner diameter tube in silico, which was adapted from the arterial flow phantom described in Lafond et al. [21]. The elevational thickness of the lobes was 1.5 mm, determined from the −6 dB beamwidth measured by Lafond et al. [21]. The passive (receiver) array simulating an L11–5v linear array (Verasonics, Kirkland, WA, USA) was positioned 23 mm from the catheter in the cross-section of the tube.

Fig. 1.

(a) Experimental schematic adapted from [21] that guided the simulations. The normalized acoustic pressure field generated by a single pair of transducers on an EkoSonic^™ MicroSonic Device^® catheter driven with 9 W electrical power (0.67 MPa peak rarefactional pressure at the surface of each transducer) is shown to the right of the schematic. The red circle indicates the location of the catheter, and the white circle is the flow tube inner lumen. (b) - (e) The four patterns used for designating the uniform random distribution of sources for the simulation. The first source pattern (b) extended throughout the entire lumen (29.2 mm²) and (c) simulates sources in all four lobes of the −6 dB EkoSonic^™ acoustic pressure field while (d) and (e) corresponded to two 5.8 mm² “clouds” of sources located either laterally or axially, respectively. The passive receiving array would be at a depth of 0 mm.

Passive emissions from inertial cavitation activity were simulated using the Vokurka statistical signal model [22], which has been previously used to investigate PCI beamforming algorithms [12], [16]. This model yields frequency spectra similar to experimental data [23]. The temporal signal received by the n^th element of the imaging array, $x_n\left(t\right)$, is given by:

$$x_n(t)={\sum }_{l=1}^L{\sum }_{k=1}^K{P}_{kl}{e}^{-\frac{\left|t-l{T}_0-{\varphi }_{kl}-{\tau }_{n,k}\right|}{{\theta }_{kl}}},$$ (1)

where $l$ indexes over $\mathrm{L}$ number of acoustic pulses, $k$ indexes over the $K$ number of insonified bubbles, $P_{kl},$ ${\varphi }_{kl}$ and ${\theta }_{kl}$ are the amplitude-, phase- and time-constants of the $l^{\text{th}}$ pulse from $k^{th}$ bubble, $T_0$ is the pulse period, and ${\tau }_{n,k}$ is the propagation time from the $k^{\text{th}}$ bubble to the $n^{\text{th}}$ element of the array. The constants, $P_{kl},$ ${\varphi }_{kl}$ and ${\theta }_{kl}$ are modeled as random variables fitted to a normal distribution, $𝒩(\mu ,\sigma )$, where $\mu$ is the mean and $\sigma$ the standard deviation. The amplitude-, phase- and time-constants were modeled as $𝒩\left(3,0.01\right)\mathrm{M}\mathrm{P}\mathrm{a},𝒩\left(0.8,0.014\right)\mathrm{\mu }\mathrm{s}$, and $𝒩\left(2,0.25\right)\mathrm{n}\mathrm{s}$, respectively, for all simulations. Note that the scattered cavitation emission amplitude was not modulated by the pressure field shown in Fig. 1(a). The simulated pulse duration was 330 μs and digitized at 31.5 MHz.

The spatial locations of the simulated sources were distributed in the tube lumen using one of four different patterns (Fig. 1). The first pattern (Fig. 1b) corresponded to emitting sources being placed in a random distribution throughout the tube lumen (excluding the location of the catheter). The second pattern (Fig. 1c) corresponded to emitting sources located randomly throughout the −6 dB acoustic output field pattern of the EkoSonic^™ catheter. The final two source patterns corresponded to emitting sources located randomly throughout the two laterally oriented (Fig. 1d) or two axially oriented (Fig. 1e) acoustic lobes of the EkoSonic^™ catheter and were used to assess the passive cavitation imaging performance in the lateral and axial directions. The source densities were either 7.5×10⁵ or 6.9×10³ sources/mL, corresponding to 1123.0 or 10.4 sources/mm² in the imaging plane and were used to simulate Definity^® particles in a flow phantom [21]. These source densities corresponded to 3.28×10⁴ or 305 total sources throughout the tube luminal area (Fig. 1(b)), or 5,465 or 51 sources per cloud (Fig. 1(c)–(e)). The high source density corresponded to the conditions of the experimental validation described in subsection II.D. The low source density corresponded to, on average, 1 microbubble per passive cavitation image pixel based on the −6 dB point spread function size at 7.8 MHz (9.56×10⁻² mm² per pixel) [24]. The average separation between microbubbles based on the Wigner-Seitz radius [25] was 68 μm or 326 μm for the two concentrations, respectively. The average separation for the two source densities was less than the wavelength of the EkoSonic^™ 2.25 MHz center frequency in 37 °C water (676 μm), but on the order of or less than the wavelength for higher harmonics and ultraharmonics. Incoherent scattering dominates when scatterers are separated by distances on the order of the wavelength or larger [26]. For each of the 4 configurations, 15 simulations with different source ensembles were used, yielding a total of 60 unique simulations.

Passive cavitation images were formed as described in section II.B. Inharmonic, 150 kHz-wide frequency bands between 3 and 9 MHz centered 558.75 kHz below each ultraharmonic (consistent with an inertial cavitation source), were selected from the Fourier transform of a 128 cycle (288 μs) duration subset of the received signal. Spatial filtering of the images is described in section II.C. Simulations and beamforming were performed in MATLAB (R2016B, MathWorks, Inc., Natick, MA, USA) on a Dell precision 5810 computer with an Intel Xeon 3.70 GHz processor (Dell Technologies, Round Rock, TX, USA). All beamforming algorithms were executed using the parallel processing functions in MATLAB on a GeForce GTX 1080 graphical processing unit (NVIDIA, Santa Clara, CA, USA).

B. Passive Cavitation Imaging

To implement the frequency-domain DSI-PCI algorithm, a Fourier transform was applied to each signal acquired by the $N$ elements of the passive array, $X_n\left(f\right)$. The signals were phase-shifted based on the computed time of flight between a pixel location and a particular receive element. The shifted signals were summed across the elements and the magnitude-squared value computed. A pixel value proportional to the energy in a frequency band of interest was calculated by summing over particular frequency bands. Mathematically the pixel energy amplitude, $B_{DSI}(\overrightarrow{r})$, is expressed as [7]:

$$B_{DSI}(\overrightarrow{r})={\int }_{f_1}^{f_2}{\left|{\sum }_{n=1}^N{X}_n(f)exp\left(i2\pi f\frac{\left|{\overrightarrow{r}}_n-\overrightarrow{r}\right|}{c}\right)\right|}^{2}df,$$ (2)

where $\overrightarrow{r}$ is the vector to a pixel location, $f_l$ and $f_2$ are the integration limits corresponding to the frequency band for the type of cavitation of interest, $n$ is the index over the elements of the passive array, ${\overrightarrow{r}}_n$ is the location of the n^th element, and $c$ is the speed of sound in the media. The integrand of (2) is proportional to the energy of ${\hat{X}}_n(\overrightarrow{r},f)$. Each passive cavitation image was normalized by the maximum pixel amplitude in the image so that the normalized images $\stackrel{-}{B_{DSI}}(\overrightarrow{r})$ had a maximum value of 1.

The RCB algorithm was developed in the time domain [12] with the pixel energy amplitude, $B_{RCB}(\overrightarrow{r})$ defined as:

$$B_{RCB}(\overrightarrow{r})=w^T{R}_{x}w,$$ (3)

where $w$ is a weighting vector, $w^T$ is the transpose of the weighting vector, and $R_x$ is the correlation matrix defined by Coviello et al. [12]. Each RCB-beamformed passive cavitation image was also normalized by the maximum pixel amplitude in the image so that the normalized image $\stackrel{-}{B_{RCB}}(\overrightarrow{r})$ had a maximum value of 1. The parameter, $\mathrm{\epsilon }$, used in the calculation of the steering vector to account for array calibration uncertainty [12, 27] was chosen to be equal to 1 for processing simulated data and 10 for experimentally acquired data, consistent with Gray et al. [28].

Coviello et al. [12] implemented time-domain filters to select specific frequency bands. To maintain consistency with the frequency-domain DSI-PCI algorithm, conventional comb filters (section II.A) were applied to the acquired signals before computing the correlation matrix, $R_s$. This step allowed the selection of particular frequency bands over which the pixel energy amplitude was computed. RCB is not fully parallelizable due to the different weights for computing each pixel amplitude, thereby increasing computing time.

Coherence images were formed based on either signal phase or amplitude. Because the pixel values from a phase coherence factor (PCF) image or an amplitude coherence factor (ACF) image do not correlate with cavitation energy (as DSI-PCI or RCB-PCI do), the PCF and ACF images were used to produce a binary mask for spatial filtering DSI-PCI, as described in sections II.C and II.D. To limit increases in computational cost relative to DSI, coherence-factor based images were computed in the frequency domain. The PCF was computed as the phase dispersion using the circular standard deviation, $S_{\theta }$, of ${\widehat{X}}_n(\overrightarrow{r},f)$, defined as [13]:

$$S_{\theta }(\overrightarrow{r},f)=\sqrt{{\sigma }^2\left(\cos \left({\theta }_{{\hat{X}}_n}(\overrightarrow{r},f)\right)\right)+{\sigma }^2\left(\sin \left({\theta }_{{\hat{X}}_n}(\overrightarrow{r},f)\right)\right)},$$ (4)

where ${\sigma }^2$ represents the variance across the $N$ elements and ${\theta }_{{\hat{X}}_n}(f)$ is the unwrapped phase at ${\hat{X}}_n(f)$. Because the variance is between 0 and 1 and the maximum variance represents the minimum coherence, the frequency-dependent PCF pixel values, $W_p(\overrightarrow{r},f)$, are defined as:

$$W_p(\overrightarrow{r},f)=1-S_{\theta }(\overrightarrow{r},f).$$ (5)

The presence of multiple bubbles reduces the maximum PCF coherence value. Thus normalized PCF pixel values, $\stackrel{-}{W_p}(\overrightarrow{r},f)$, were computed as:

$$\stackrel{-}{W_p}(\overrightarrow{r},f)=W_p(\overrightarrow{r},f)/\underset{\overrightarrow{r}}{\mathrm{m}\mathrm{a}\mathrm{x}}(W_p(\overrightarrow{r},f)),$$ (6)

where the max function was over all image space. Finally, the global phase coherence pixel energy amplitudes, $B_{PCF}(\overrightarrow{r})$, were computed similarly to (2):

$$B_{PCF}(\overrightarrow{r})={\int }_{f_1}^{f_2}\mid \stackrel{-}{W_p}(\overrightarrow{r},{\left.f)\right|}^{2}df.$$ (7)

To limit the maximum coherence image pixel value to 1, the PCF amplitudes were normalized by their maximum:

$$\stackrel{-}{B_{PCF}}(\overrightarrow{r})=B_{PCF}(\overrightarrow{r})/\underset{\overrightarrow{r}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left(B_{PCF}(\overrightarrow{r})\right).$$ (8)

The global amplitude coherence image $\stackrel{-}{B_{ACF}}(\overrightarrow{r})$ was computed in the same manner as (6) through (8), except the pixel values, $W_a(\overrightarrow{r},f)$, were defined as in [29]:

$$W_a(\overrightarrow{r},f)=\frac{1}{N}\frac{{\left|\sum _{n=1}^N{\widehat{X}}_n(f)\right|}^2}{\sum _{n=1}^N{\widehat{X}}_n\left({\left.f)\right|}^2\right.}.$$ (9)

C. Determination of Spatial Filter Thresholds

Binary masks were created to form coherence factor-based passive cavitation images that scaled with energy and to decrease the artifactual cavitation energy mapped by PCI. The binary masks were computed as:

$$M_i(\overrightarrow{r})=\left\{\begin{array}{ll}1& \text{if}{\bar{B}}_i(\overrightarrow{r})>T_i\\ 0& \text{if}{\bar{B}}_i(\overrightarrow{r})\le T_i\end{array},\right.$$ (10)

where $i$ was a place holder that referred to the beamforming strategy used to form a passive cavitation image and $T_i$ was the spatial filter threshold obtained for each beamforming strategy via ROC or PR curve analyses described below.

The spatial filter thresholds used in (10) were selected based upon an ROC and PR curve analyses of ${\bar{B}}_i(\overrightarrow{r})$ formed from simulated received cavitation emissions. Note that all passive cavitation images were normalized to have a maximum value of 1 using (6) and (8) for the ROC and PR curve analyses. Pilot studies without this normalization scheme were also performed for all the beamforming algorithms. When any of the normalizations were not included, the binary classifier metrics were worse than when they were included. Thus, all studies reported herein included the normalizations. The use of simulated received cavitation emissions provided a gold standard with known cavitation source locations used in the binary classifier analysis. A pixel was assigned a true positive value when at least one source was co-located within it and the pixel amplitude was higher than the threshold value. False positive, true negative, and false negative values were similarly assigned using standard binary classifier definitions [19]. The analysis was performed for the 120 simulated data sets (15 simulations for both source densities and all four source patterns) and the total number of true (false) positives and true (false) negatives across all data sets were used to compute the ROC curve and PR curve [19]. The thresholds $T_i$ were defined as either the point of the ROC curve closest to a true positive rate (TPR) of 1 and a false positive rate (FPR) of 0 or the point on a PR curve closest to a TPR of 1 and a positive predictive value (PPV) of 1.

The passive cavitation images obtained from 15 source ensembles were averaged and plotted on the decibel scale using a 10 dB dynamic range. Spatially filtered PCI performance was assessed via the area under the ROC curve (AUROC), sensitivity, specificity, PPV, and F₁-score to determine the effect of source density and beamforming algorithm. To determine if different source cloud patterns effected the beamforming performance, binary classifier analyses were performed for each of the four source patterns separately for the 7.5×10⁵ sources/mL density. The binary classifier metrics were computed for each source pattern using the threshold determined from the global analysis. Statistically significant differences between the binary classifier metrics calculated were determined based on non-overlapping 95% confidence intervals computed using bootstrapping [30].

D. Spatially Filtered Passive Cavitation Imaging

Coherence factor-based passive cavitation images that scaled with energy, $\stackrel{-}{B_{PDSI}}(\overrightarrow{r})$ and $\stackrel{-}{B_{ADSI}}(\overrightarrow{r})$, were computed using binary masks as:

$$\stackrel{-}{B_{PDSI}}(\overrightarrow{r})=M_{PCF}(\overrightarrow{r})\cdot \stackrel{-}{B_{DSI}}(\overrightarrow{r})$$ (11)

$$\stackrel{-}{B_{ADSI}}(\overrightarrow{r})=M_{ACF}(\overrightarrow{r})\cdot \stackrel{-}{B_{DSI}}(\overrightarrow{r}).$$ (12)

To decrease artifactual cavitation energy, pixel-wise spatially filtered passive cavitation images, $I_{DSI}(\overrightarrow{r}),I_{RCB}(\overrightarrow{r})$, $I_{PDSI}(\overrightarrow{r})$, and $I_{ADSI}(\overrightarrow{r})$, were created as:

$$I_{DSI}(\overrightarrow{r})=M_{DSI}(\overrightarrow{r})\cdot \stackrel{-}{B_{DSI}}(\overrightarrow{r})$$ (13)

$$I_{RCB}(\overrightarrow{r})=M_{RCB}(\overrightarrow{r})\cdot \stackrel{-}{B_{RCB}}(\overrightarrow{r})$$ (14)

$$I_{PDSI}(\overrightarrow{r})=M_{PDSI}(\overrightarrow{r})\cdot \stackrel{-}{B_{PDSI}}(\overrightarrow{r})$$ (15)

$$I_{ADSI}(\overrightarrow{r})=M_{ADSI}(\overrightarrow{r})\cdot \stackrel{-}{B_{ADSI}}(\overrightarrow{r}).$$ (16)

An online supplement includes a MATLAB m-file implementation of the DSI, PCF, ACF, PDSI, and ADSI algorithms and a mat-file containing a representative data set simulating the emissions from a low source density (6.9×10³ sources/mL) distributed throughout the tube lumen as shown in Fig 1b.

E. Experimental validation

Passive cavitation images were acquired of Definity infused through the drug lumens of an EkoSonic^™ catheter (product number 500–56112) in an arterial flow model with air-saturated 37°C saline [21]. Briefly, Definity^® (Lantheus, Billerica, MA, USA) at a concentration of 4.6×10⁷ microbubbles/mL was infused at a volumetric flow rate of 2 ml/min through the drug delivery ports of the catheter. Considering the 100 mL/min average flow rate in the 6.35 mm inner diameter latex tube and the particle loss during infusion [21], this resulted in an estimated Definity^® concentration of 7.5×10⁵ microbubbles/mL. The 2.25 MHz center frequency EkoSonic^™ transducers were driven by a programmable unit provided by Boston Scientific with 15 ms pulses at a 10 Hz pulse repetition frequency (PRF) and 1.5 W electrical temporal pulse average power to each transducer (9 W total electrical temporal pulse average power/catheter) corresponding to 0.67 MPa peak rarefactional pressure at the surface of the catheter, for approximately 3 min. The pressure amplitude field was reported by Lafond et al. [21]. An L11–5v linear array (Verasonics, Kirkland, WA, USA) was connected to a Vantage 256 ultrasound scanner (Verasonics) and both B-mode images and acoustic emissions were obtained [21]. B-mode images were formed using the manufacturer supplied ‘L11–5v128RyLns’ algorithm. One-hundred and fifty-five data sets were sequentially acquired. A 288 μs duration subset of each data set was beamformed using the algorithms described in sections II.B and II.C. The non-normalized passive cavitation images were summed and then normalized by the maximum value in the summed image. Duplex images were formed as described by Lafond et al. [21] but without the scalars associated with a conversion to units of energy (because all data was analyzed relative to the maximum pixel amplitude in each image). Spatial filtering was achieved using the threshold values derived from simulations.

III. Results

A. Spatial filtering of DSI-PCI

Fig. 2a shows the ROC curves obtained from (2) based on the simulations using all four source patterns (Fig. 1) for both the high and low source densities (7.5×10⁵ or 6.9×10³ sources/mL). Fig. 2b shows the PR curve of the same data sets. The thresholds based on the ROC curve (maximizing the true positive rate and minimizing the false positive rate) were 0.494 and 0.437 for the high and low source densities, respectively. The thresholds based on the PR curve (maximizing the true positive rate and the positive predictive value) were 0.643 and 0.539 for the high and low source densities, respectively.

Fig. 2.

(a) ROC and (b) precision-recall curves based on DSI-PCI without spatial filtering at the high (7.5×10⁵ sources/mL) and low (6.9×10³ sources/mL) source densities. The optimal thresholds for spatial filtering, selected by maximizing the true positive rate and minimizing the false positive rate (circles) or maximizing both the true positive rate and the positive predictive value (triangles) for the high source density (filled markers) and low source densities (open markers), are shown. The curves were obtained by analyzing all four source patterns simultaneously.

When computing ROC and PR curves based on all four source distributions and both source densities combined, the thresholds were 0.480 and 0.652, respectively. At these thresholds, the AUROC, sensitivity, specificity, and positive predictive value were 97.2±0.1, 92.9±0.3, 92.03±0.05, and 31.8±0.3 for the ROC-based spatial filtered passive cavitation images, respectively and 97.2±0.1, 69.9±0.5, 97.51±0.03, and 52.9±0.4 for the PR-based spatial filtered passive cavitation images, respectively. Fig. 3 shows DSI passive cavitation images without spatial filtering (top row), ROC-based spatial filtering (middle row), and PR-based spatial filtering (bottom row) using these global thresholds. ROC-based and PR-based spatial filtering reduced the fraction of cavitation energy mapped to locations outside the clouds from 80% to 45% and 26%, respectively, for the one cloud source pattern, 91% to 51% and 18%, respectively, for the two lateral cloud pattern, 89% to 46% and 31%, respectively, for the two axial cloud pattern, and 85% to 54% and 37%, respectively, for the four-cloud pattern.

Fig. 3.

DSI-PCI for each of the four source patterns. Without spatial filtering (top row), with spatial filtering based on ROC optimization (middle row), and with spatial filtering based on PR optimization (bottom row) for the high (7.5×10⁵ sources/mL) source density. The passive receiving array is located at the top of each image. Each passive cavitation image is the average of 15 different source ensembles for each source pattern. The position of the tube and catheter are marked by the white dotted line and the bubble cloud edges in cyan.

PR-based spatial filtering yielded a higher specificity and positive predictive value and yielded less artifactual energy mapping outside the tube lumen (false positives) at the expense of an increased number of false negative pixels. Additionally, precision-recall based analysis is documented to be more informative when there is an imbalance of classifiers [11, 20] and the passive cavitation images had more true negative pixels than the other classifiers combined. Therefore, the remainder of the results will be presented using the PR-based threshold.

B. Performance of the spatially filtered DSI, RCB, PDSI, and ADSI beamformers

Table 1 shows the binary classifier metrics combining confusion matrix data from simulations of all four geometries (Fig. 1) for each of the beamforming algorithms (DSI, RCB, PDSI, and ADSI) at both source densities. The spatial filter thresholds for the phase coherence factor and amplitude coherence factor beamforming, $T_{PCF}$ and $T_{ACF}$, were 0.401 and 0.451 for the high source density and 0.304 and 0.328 for the low source density. The difference in AUROC values for both source densities was smaller between DSI and RCB as compared to between DSI and either coherence-factor masked PCI. When PR-based optimal thresholds were used, RCB, and PDSI each were top performing depending on which metric is analyzed, though other algorithms were not statistically different. Though differences in each algorithm performance were statistically significant, the difference in the AUROC and sensitivity for either source density was less than 10%. The differences in the specificity, positive predictive value, and F₁-score were less than 5%.

TABLE I.

Beamformer performance metrics for the four algorithms (delay sum and integrate (DSI), robust Capon Beamforming (RCB), Phase Coherence Factor Masked DSI (PDSI), and Amplitude Coherence Factor Masked DSI (ADSI)) for source densities of 7.5×10⁵ or 6.9×10³ sources/mL. Spatial filter thresholds were based on precision-recall Analysis for all Source Patterns. Bolded Numbers Highlight the Maximum Values for the Beamformer Performance Binary Classifier Metrics. Italicized Numbers are Statistically Significantly Lower than the Maximum Values.

Source density (sources/mL)	Binary Classifier Metric (%)	DSI	RCB	PDSI	ADSI
7.5×105	AUROC	97.8 ± 0.1	98.5 ± 0.1	88.4 ± 0.3	88.6± 0.3
	Threshold	0.643	0.549	0.619	0.612
	Specificity	97.73 ± 0.04	97.34 ± 0.05	97.84 ± 0.04	97.78 ± 0.04
	Sensitivity	77.5 ± 0.5	82.1 ± 0.4	76.8 ± 0.5	78.0 ± 0.5
	PPV	68.3 ± 0.5	66.0 ± 0.5	69.1 ± 0.5	68.9 ± 0.5
	F1	72.6 ± 0.4	73.2 ± 0.4	72.8 ±0.4	73.2 ± 0.4
6.9×103	AUROC	95.6 ± 0.3	96.5 ± 0.3	86.8 ± 0.5	87.9 ± 0.5
	Threshold	0.539	0.432	0.462	0.482
	Specificity	94.14 ± 0.06	93.48 ± 0.07	94.29 ± 0.07	94.02 ± 0.07
	Sensitivity	79.1 ± 0.8	86.0 ± 0.7	78.4 ± 0.9	80.8 ± 0.8
	PPV	19.7 ± 0.4	19.3 ± 0.4	19.9 ± 0.4	19.7 ± 0.4
	F1	31.5 ± 0.6	31.5 ± 0.6	31.8 ± 0.6	31.6 ± 0.6

Shown in Fig. 4 are the simulated passive cavitation images, beamformed with the four algorithms (13)-(16) using the PR-based thresholds for the high source density (7.5×10⁵ sources/mL) shown in Table I. Each of the beamforming algorithms render images with qualitatively similar distributions of cavitation energy for each source pattern. For all the source patterns except the lateral 2-cloud pattern, cavitation was erroneously mapped onto the location of the catheter in the center of the tube. For the one-cloud pattern, cavitation is mapped erroneously within the catheter and outside of the tube extending 1.5 mm above and 3 mm below the tube lumen boundary. For the lateral 2-cloud pattern, cavitation energy is mapped to the location of the two clouds. For the axial 2-cloud pattern, the two clouds are not resolved. The 4-cloud source pattern is visualized based on the overall shape of the mapped cavitation, but the four clouds cannot be individually resolved. Cavitation energy is also mapped erroneously outside the tube lumen.

Fig. 4.

Spatially filtered PCI using DSI (first column), RCB (second column), PDSI (third column), ADSI (fourth column). Threshold values were based on optimization from PR curves. Each row corresponds to one of the four source patterns with a source density of 7.5×10⁵ sources/mL. The passive receiving array is located at the top of each image. Each passive cavitation image is the average of 15 different source ensembles for each source pattern. The position of the tube and catheter are marked by the white dotted line and the bubble cloud edges in cyan.

Table II lists the values of the binary classifier metrics for each beamformer and each simulated source pattern for a microbubble concentration of 7.5×10⁵ sources/mL. RCB generally outperformed the other algorithms. However, for each source pattern the algorithms perform similarly. The differences in the binary classifier metrics between spatially filtered DSI and RCB was less than 5% (except for the specificity with the 1 cloud source distribution where the difference was 7.4%). The differences in the binary classifier metrics tended to be greater when comparing either coherence factor-based beamforming algorithm to RCB. RCB yielded better AUROC and sensitivity values, worse positive predictive values, and nominally equivalent specificities and F₁-scores.

TABLE II.

Beamformer performance metrics for the four algorithms (delay sum and integrate (DSI), robust Capon Beamforming (RCB), Phase Coherence Factor Masked DSI (PDSI), and Amplitude Coherence Factor Masked DSI (ADSI)) for each Source Pattern with a Source Density of 7.5×10⁵ Sources/mL. Spatial Filter Thresholds were Based on Precision-Recall Analysis. Bolded Numbers Highlight the Maximum Values for the Beamformer Performance Binary Classifier Metrics. Italicized Numbers are Statistically Significantly Lower than the Maximum Values.

Source Pattern	Binary Classifier Metric (%)	DSI	RCB	PDSI	ADSI
1 Cloud 32,789 sources	AUROC	97.6± 0.2	98.2 ± 0.2	89.6± 0.4	89.8 ± 0.4
	Specificity	96.5 ± 0.1	96.0 ± 0.1	96.8 ± 0.1	96.8 ± 0.1
	Sensitivity	82.4 ± 0.7	89.0 ± 0.6	80.5 ± 0.7	81.4 ± 0.7
	PPV	71.7 ± 0.8	70.2 ± 0.7	72.9 ± 0.7	72.8 ± 0.7
	F1	76.7± 0.6	78.5 ± 0.6	76.5 ± 0.6	76.9 ± 0.6
4 clouds 21,860 sources	AUROC	97.0 ± 0.3	97.6 ± 0.2	88.1 ± 0.5	88.4 ± 0.5
	Specificity	96.1 ± 0.1	95.0 ± 0.1	96.3 ± 0.2	96.2 ± 0.1
	Sensitivity	78.6± 0.9	87.2 ± 0.7	77.1 ± 0.9	78.8 ± 0.9
	PPV	60.3 ± 0.9	57.0 ± 0.9	61.4 ± 1.0	61.1 ± 0.9
	F1	68.2 ± 0.8	68.9 ± 0.7	68.3 ± 0.8	68.8 ± 0.8
2 lateral clouds 10,930 sources	AUROC	97.6± 0.3	99.3 ± 0.2	87.6± 0.7	87.8 ± 0.7
	Specificity	99.33 ± 0.04	99.47 ± 0.04	99.33 ± 0.04	99.28 ± 0.05
	Sensitivity	72.7 ± 1.3	71.7 ± 1.4	73.2 ± 1.4	74.9 ± 1.3
	PPV	79.7 ± 1.3	83.0 ± 1.3	79.8 ± 1.3	79.0 ± 1.3
	F1	76.0 ± 1.0	76.9 ± 1.0	76.3 ± 1.0	76.9 ± 1.0
2 axial clouds 10,930 sources	AUROC	98.5 ± 0.3	98.7 ± 0.2	85.7± 0.7	85.1 ± 0.7
	Specificity	98.84 ± 0.06	98.70 ± 0.06	98.76± 0.06	98.79 ± 0.06
	Sensitivity	67.4 ± 1.4	64.3 ± 1.4	69.8 ± 1.4	70.5 ± 1.3
	PPV	68.1 ± 1.3	64.6 ± 1.3	67.5 ± 1.3	67.7 ± 1.3
	F1	67.7 ± 1.0	64.4 ± 1.1	68.6 ± 1.1	69.1 ± 1.1

C. Experimental validation of PCI of inharmonic and ultraharmonic cavitation energy

Fig. 5 shows summed passive cavitation images without ((2), (3), (11), and (12)) and with ((13)-(16)) spatial filtering using the respective inharmonic and ultraharmonic emissions acquired by the L11–5v linear array and Verasonics Vantage 256 ultrasound research scanner. The thresholds used in the PCI algorithms were the same for both inharmonic and ultraharmonic based PCI and were derived from the high source density thresholds in Table I. The spatially filtered passive cavitation images (Fig. 5, second and fourth rows) exhibit less artifactual cavitation activity outside the tube lumen relative to the images without spatial filtering (Fig. 5, first and third rows). In general, the two types of bubble activity, inertial and stable cavitation, appear to be colocalized. The RCB images have less cavitation activity axially mapped to the location of the catheter. RCB without spatial filtering resulted in inertial and stable cavitation mapped to the four acoustic lobe locations of the EkoSonic^™ catheter (see Fig. 1(a)). Spatially filtered RCB resulted in stable and inertial cavitation mapped to only two acoustic lobe locations. DSI without spatial filtering resulted in inertial and stable cavitation mapped to either three or four acoustic lobe locations. Spatial filtering reduced the mapping to one or two lobe locations. Spatial filtering had minimal effects relative to no thresholding for PDSI and ADSI.

Fig. 5.

B-mode and PCI duplex images from experimental data in the flow phantom, adapted from [21]. Passive cavitation images are beamformed using the acquired inharmonic (top two rows) or ultraharmonic (bottom two rows) emissions and shown before (rows 1 and 3) and after (rows 2 and 4) spatial filtering based on optimization from PR curve. The beamforming algorithms are DSI (first column) RCB (second column), PDSI (third column), and ADSI (fourth column). Plotted PCI data is the sum of 155 frames of data.

IV. Discussion

By inspecting Fig. 4 and the binary classifier metrics in Tables I and II, the overall performance of the different beamformers is similar. The AUROC, sensitivity, and specificity differ by no more than 11% for both source densities. The impact of source density on binary classifier metrics is seen most strongly in the positive predictive value and F₁-score. The masks created from DSI, phase coherence factors, and amplitude coherence factors are similar despite the algorithmic differences in computing images ((2) and (4)-(9)). The masks for DSI, PDSI, and ADSI were likely similar because the DSI algorithm was used as a step in computing each mask. The relatively small impact of spatial filtering on PDSI and ADSI might be expected given that the PDSI and ADSI algorithms already include a mask ((11) and (12)). Given the similarity of the spatial filtering results, a particular algorithm for guiding or monitoring a cavitation-mediated therapy could be based on ease of implementation rather than subtle differences in image quality.

Alternative passive cavitation beamforming algorithms have been pursued due to the poor axial point spread function of DSI with linear arrays [8, 12, 24, 31–34]. However, as shown in Fig. 3, judiciously filtering the images can significantly reduce axial artifacts. A critical feature of the frequency-domain algorithms described is that they are summed over frequencies before determining spatial filtering thresholds. The summation results in frequency compounding, which is a known speckle reduction technique [35]. Reducing the speckle pattern reduces false negatives in the passive cavitation images, which in turn impacts the determination of the threshold used for spatial filtering. When computing thresholds based on single frequency PCI, the values were much smaller (<0.1) than those reported in section III.A. A limitation of the spatial filtering approach described herein may be that low levels of cavitation activity could be artifactually lost. Fig. 5 demonstrates that spatial filtering can result in a loss of mapping cavitation activity to some of the acoustic lobes.

Despite the reduction in the axial artifact, none of the spatially filtered beamformers resolved two simulated axial microbubble clouds well. However, all beamformers resolved the two lateral microbubble clouds (Figs. 3 and 4). Note that a global threshold was used to compute the images shown in Fig. 3. These thresholds were based on either an ROC or PR analysis combining all four-source pattern for both the low and high source densities. When thresholds were computed for just the high or low source density, the threshold value changed by as much as 17%. If an expected microbubble distribution or density could be anticipated, a spatially varying threshold, $T_i(\overrightarrow{r})$, could be used to improve the localization performance. Determination of the threshold could also be optimized based on the relative value of true negatives, true positives, false negatives, and false positives in a particular application. When the threshold was selected based on an ROC curve, the two lateral source clouds resulted in artifactual energy mapping nominally to the two axial cloud locations (even though there were no sources at those locations) as seen in Fig. 3. These artifacts were removed when the threshold was selected based on a PR curve (which yielded a larger threshold value). However, the higher threshold resulted in lost mapping of cavitation activity from some of the expected acoustic lobes in the experimental data set (Fig. 5). Thus, a tradeoff exists between suppressing false positives (artifactual cavitation activity) and true positives (missing true cavitation activity). Similar concepts exist in designing image segmentation algorithms [36].

Consistent with the comparative performance of the beamformers in the simulation, spatial filtering reduced the cavitation energy spread when beamforming experimentally acquired data. RCB, PDSI, and ADSI reduced the artifactual energy spread outside the tubing, even without spatial filtering. Stable and inertial cavitation was located within the −6dB beam of the EkoSonic^™ catheter transducers.

Both RCB [12] and spatially filtered images rely on an empirically selected parameter or threshold, respectively. This selection is dependent on the particular source density and pattern relative to the dimensions and orientation of the receiving array. The choice of $\mathrm{\epsilon }$ for RCB can be determined if sufficient knowledge of the system is known [37], though more commonly in cavitation imaging it has been determined empirically by the user [12, 38]. The simulation work reported in this study assumed array elements with uniform sensitivity with a value of 1 [28]. Using PCI threshold values computed from PR analyses of all the simulations and applied to experimental data, no artifactual cavitation was evident outside the flow phantom tube lumen (Figs. 5).

Several beamforming algorithms have been developed to improve the computational complexity of frequency domain passive cavitation imaging [6, 11, 39, 40]. The times required to render spatially filtered DSI, PDSI, ADSI, and RCB PCI frames, were 289±16 ms, 290±17 ms, 297±17 ms, and 79×10³±4×10³ ms, respectively, on an Intel^® Xeon^® E5–1630 3.70 GHz CPU with an NVIDIA GeForce GTX1080 graphics processing unit (GPU). There were 246 frequencies beamformed per channel with 128 channels. The number of pixels beamformed per image was 8,316. The computational costs of ADSI and PDSI were not notably greater than DSI, whereas RCB was several orders of magnitude more computationally expensive than DSI [41]. Optimizing the code, as well as decreasing the number of pixels per frame and frequency bins could potentially reduce the computational cost further, though this approach could be applied for all algorithms. Even without such optimization, the computational times are short enough to enable real time implementation on a graphics processing unit (GPU) for DSI, PDSI, and ADSI. Real time implementation coupled with similar imaging performance to RCB makes spatial filtering an attractive approach moving forward, particularly for image-guided therapeutics for clinical applications [8–10, 42–44].

A broader range of simulated source densities or experimental Definity^® concentrations was not explored, which is a limitation of our study. As evident in the positive predictive value and F₁-score shown the Table 1, the accuracy of source localization is impacted by the total number of sources. Additionally, experimental cavitation dynamics have been reported to change over time [21]. Jones et al. reported that sampling only a subset of cavitation emissions hindered the ability of PCI to predict neural tissue damage [11]. Another limitation of this study was the use of simulated data to test the beamforming algorithms. Future studies are needed to assess the spatial correlation between the beamforming algorithms and an ultrasound-induced bioeffect distribution.

A comprehensive comparison with other data-adaptive beamforming algorithms applied to passive cavitation imaging was also not attempted. Robust beamforming by linear programming, an optimization-based method using higher-order, non-Gaussian statistics of the received signals [8], has been shown to improve source resolution over RCB. Lu et al. [33, 45] investigated an eigenspace-based RCB algorithm to improve PCI performance. The optimal weighting vector of their proposed method was found by projecting the RCB weighting vector onto the desired vector subspace constructed from the eigen decomposition of the covariance matrix, which renders this algorithm relatively computationally intensive. However, simulation and in vitro experimental histotripsy results suggested that the proposed eigenvector RCB weighting significantly outperformed RCB PCI and B-mode imaging of acoustically active versus residual bubbles.

Processing of harmonic-only components using the angular spectrum method has been shown to indicate blood brain barrier disruption while the presence of broadband acoustic emissions indicated tissue damage in the brain assessed on magnetic resonance images [6]. Any comparison of algorithms should also consider the nature of the receiving array. Large-aperture 2D arrays have a smaller diffraction-limited point spread function and thus any benefit observed for small-aperture 1D arrays may not extend to large-aperture arrays, as evidenced by the work of Jones et al. showing a 3D spatial correlation between mapped broadband acoustic emissions and MRI-measured tissue damage volumes using a time-domain DSI beamforming algorithm [11]. In the future, image guidance of bubble-mediated therapies will rely on the accurate representation of the location of these nonlinear acoustic sources. Image guidance will also play a key role in safety and efficacy studies.

V. Conclusion

These studies investigated the impact of data-adaptive spatial filtering on the performance of PCI beamformers. Spatial filters were obtained using binary masks derived from precision-recall curve analyses of simulated data sets. Although robust Capon beamforming generally outperformed the spatially filtered algorithms, the improvement was limited. For some binary classifier metrics computed from simulations using low source density patterns (7.5×10⁵ or 6.9×10³ sources/mL), robust Capon beamforming performance was slightly worse. Computing the binary masks from a delay-sum-and-integrate algorithm or coherence factor weighting did not substantially increase the computational time relative to delay-sum-and-integrate beamforming without spatial filtering. These results underscore the use of spatial filtering as a computationally efficient and accurate approach to passive cavitation imaging.

Biographies

Kevin J. Haworth (S’07–M’10–SM’20) received the B.S. degree in physics from Truman State University, Kirksville, MO, USA, in 2003, and the M.S. and Ph.D. degrees in applied physics from the University of Michigan, Ann Arbor, MI, USA, in 2006 and 2009, respectively. He was a Post-Doctoral Fellow under the tutelage of C. K. Holland with the University of Cincinnati, Cincinnati, OH, USA, where he is currently an Associate Professor of Internal Medicine, Biomedical Engineering, and Pediatrics. He is directing and conducting research in medical ultrasound including the use of bubbles for diagnostic and therapeutic applications. His current research interests include the studies of cavitation imaging and acoustic droplet vaporization for gas scavenging.

Dr. Haworth received the International Society of Therapeutic Ultrasound Fredric Lizzi Award in 2019. He is a fellow of the Acoustical Society of America and a fellow of the American Institute of Ultrasound in Medicine (AIUM). He serves on the Bioeffects and Technical Standards Committees and is the vice chair for the Basic Science and Instrumentation Community of Practice for AIUM, the Biomedical Acoustic Technical Committee, and the Public Relations Committee for the ASA.

Nuria Gonzales Salido graduated from the Universidad Politécnica de Valencia in 2010 with a B.S. in Telecommunication Engineering (Imaging and Sound) and an M.S. in Acoustics Engineering in 2012. She enrolled in the Sensor and Ultrasonic System (DSSU) Group at The Spanish National Research Council (CSIC) earning her doctorate in Electronics in 2017 under the Advanced Electronic and Intelligent Systems doctoral program at the Universidad de Alcala (UAH), Madrid, Spain. Her thesis developed Acoustic Radiation Force Impulse Images (ARFI) technique in a Full Angle Spatial Compound (FASC) automated breast imaging system to improve quality with regard to conventional images. After graduation, she joined The Image-guided Ultrasound Therapeutics Laboratories (IgUTL) as a postdoctoral fellow, under the supervision of Prof. Christy K. Holland to develop a real-time image-guided therapeutic delivery system for peripheral and coronary arteries. Dr. Salido’s research focused on passive beamforming and signal processing using frequency-domain passive cavitation imaging with circular coherence factor.

Maxime Lafond received the M.S. degree in acoustics from Université du Maine, Le Mans, France, in 2013 and the Ph.D. degree in biomedical engineering from the Laboratory of Therapeutic Applications of Ultrasound, University Claude Bernard of Lyon, France, in 2016. He held a Japan Society for the Promotion of Science Postdoctoral Fellowship at the Umemura-Yoshizawa Laboratory, Department of Biomedical Engineering, Tohoku University, Sendai, Japan, from 2017 to 2018, where he developed cavitation monitoring for sonodynamic therapy. He completed a second postdoctoral fellowship under the mentorship of Prof. Christy K. Holland at the Image-guided Ultrasound Therapeutics Laboratories, University of Cincinnati, Cincinnati, OH, USA, from 2018 to 2021 where he worked on catheter-based cavitation-mediated drug delivery and bioactive gas delivery. He joined the Laboratory of Therapeutic Applications of Ultrasound at the French National Institute of Health and Medical Research (LabTAU, INSERM U1032) as a research associate in 2021. His current research interests include cavitation monitoring and use in ophthalmology applications and cancer therapy.

Daniel Suarez Escudero received both engineering and M.S. degrees from Université de technologie de Compiègne (UTC), Compiègne, France, in 2016, and the Ph.D. degree in physics from École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI), France, in 2019. From 2019 to 2022, he worked as a research engineer for Cardiawave SA, Paris, France. He is currently a postdoctoral fellow in the Image-guided Ultrasound Therapeutics Laboratories at the University in Cincinnati, Ohio, USA. His main research interests include ultrasound therapy and cavitation monitoring with a special interest on cardiovascular applications.

Christy K. Holland (M1984–SM2021) graduated from Wellesley College (B.A.) in 1983 where she double majored in physics and music. She completed M.S. (1985), M.Phil. (1986), and Ph.D. (1988) degrees at Yale University in Engineering and Applied Science in New Haven, CT, USA. She is a tenured Professor in both the College of Medicine and the College of Engineering and Applied Sciences at the University of Cincinnati (UC), OH, USA with joint appointments in the Department of Internal Medicine, Division of Cardiovascular Health and Disease, and the Department of Biomedical Engineering. She directs the Image-guided Ultrasound Therapeutics Laboratories in the UC Cardiovascular Center.

She is a fellow of the Acoustical Society of America, the American Institute of Ultrasound in Medicine, the American Institute for Medical and Biological Engineering, and the Executive Leadership in Academic Medicine. Prof. Holland served as Editor-in-Chief of Ultrasound in Medicine and Biology, the official Journal of the World Federation for Ultrasound in Medicine and Biology, from 2006 to 2021. She served as President of the Acoustical Society of America from 2015 to 2017. Prof. Holland’s research interests include ultrasound-enhanced thrombolysis for stroke therapy, ultrasound-mediated drug and bioactive gas delivery, bioeffects of diagnostic and therapeutic ultrasound, and acoustic cavitation.