Passive Cavitation Mapping by Cavitation Source Localization from Aperture-Domain Signals – Part 1: Theory and Validation Through Simulations

Abstract

Passive cavitation mapping (PCM) algorithms for diagnostic ultrasound arrays based on time exposure acoustics (TEA) exhibit poor axial resolution, which is in part due to the diffraction-limited point spread function of the imaging system and poor rejection by the delay-and-sum beamformer. Here, we adapt a method for speed of sound estimation to be utilized as a cavitation source localization (CSL) approach. This method utilizes a hyperbolic fit to the arrival times of the cavitation signals in the aperture domain, and the coefficients of the fit are related to the position of the cavitation source. Wavefronts exhibiting poor fit to the hyperbolic function are corrected to yield improved source localization. We demonstrate through simulations that this method is capable of accurate estimation of the origin of coherent spherical waves radiating from cavitation/point sources. The average localization error from simulated microbubble sources was 0.12±0.12 mm (0.15±0.14λ₀ for a 1.78MHz transmit frequency). In simulations of two simultaneous cavitation sources, the proposed technique had an average localization error of 0.2 mm (0.23λ₀), while conventional time exposure acoustics (TEA) had an average localization error of 0.81 mm (0.97λ₀). The reconstructed PCM-CSL image showed a significant improvement in resolution compared to the PCM-TEA approach.

I. Introduction

CAVITATING microbubbles can produce localized mechanical effects, making them useful for various ultrasound-based applications ranging from sonoporation-induced drug delivery [1]–[3] and blood-brain barrier opening [4]–[6] to surface cleaning [7], [8]. Outcomes produced by these applications are determined by the cavitation activities associated with stable oscillations or inertial growth/collapse of microbubbles, often referred to as stable and inertial cavitation, respectively. For therapeutic ultrasound applications, monitoring cavitation is necessary to achieve a designated threshold of therapy for effective treatment or to avoid overtreatment. Conventional monitoring methods such as B-mode and contrast-enhanced ultrasound imaging can localize microbubbles under stable oscillation. These imaging methods are known as active cavitation imaging methods, where the pulses are designed to have a low mechanical index (MI) in order to avoid destruction of the microbubbles and allow echoes from the microbubbles to be detected. However, these active imaging methods are incapable of monitoring short-lived transient microbubbles that undergo inertial cavitation. Although ultrafast B-mode imaging has been shown to overcome this challenge by using frame rates higher than typical B-mode imaging [9], these methods, while providing information about the existence and location of microbubbles, do not map inertial cavitation signals and typically record echo signals when the therapeutic transducer is not transmitting, because the imaging pulses interfere with the acoustic emissions from the bubbles. Unlike pulse-echo imaging methods that localize targets based on time-of-flight between transmit pulse and receiving echo, passive cavitation imaging methods are decoupled from the transmission pulse and attempt to localize acoustic emissions from collapsing microbubbles using only receive beamforming.

Several studies have shown a significant correlation between cavitation signal levels and therapeutic outcomes [10]–[13]. For example, increased broadband emissions correspond to an increased ultrasound ablation volume [10], [11]. A study on ultrasound/microbubble-mediated drug delivery suggested that drug delivery outcomes were strongly correlated to inertial cavitation dose [12]. It was found in this study that increased ultrasound pressure lead to enhanced drug delivery in mice. However, these passive cavitation detection methods typically use a single element transducer that does not provide localized spatial information of cavitation, but rather cavitation signals spatially averaged within the focal volume.

Recently, passive cavitation mapping (PCM) using an array of transducers co-aligned with a therapeutic ultrasound transducer was proposed to obtain a spatial map of inertial cavitation activity. PCM has shown promise in microbubble-mediated therapy monitoring, such as drug delivery [14], [15], cancer therapy [16], and tissue ablation [17].

PCM algorithms implemented on transducer arrays are primarily based on time exposure acoustics (PCM-TEA) [18], which utilize fixed-focus delay-and-sum beamforming over a grid of points, followed by the time-averaging of the squared beamformed channel signals. Here, the time average is capable of capturing cavitation signals that occur long after reflected pulses. However, PCM-TEA algorithms utilizing clinical diagnostic arrays (e.g. linear, curved linear, and phased arrays) suffer from poor axial resolution. Notably, PCM-TEA exhibits a tail artifact, the appearance of an elongated region of cavitation energy beyond the focal region of the applied acoustic pulse, even though most of the cavitation should occur within the focal region. Based on the high-resolution achieved in PCM with hemispherical arrays [19], [20], this tail artifact is likely due to a combination of the diffraction-limited point spread function of the limited aperture of clinical transducer arrays [17], [21] and the poor rejection of off-focus targets with delay-and-sum beamforming.

Inspired by recent advances in adaptive beamforming using constraints, the robust Capon beamformer, also called the minimum variance distortionless response (MVDR) beamformer, was proposed to significantly reduce interference artifacts for PCM applications [16]. While the robust Capon beamformer was originally designed to address the problems in the MVDR beamformer posed by inhomogeneous tissue by minimizing the output power of the array for non-ideal signals, it it does not fully suppress off-focus signals in the TEA process and is unable to compensate for localization errors due to bubble lifetime.

Frequency analysis approaches have also been proposed to address tail artifacts [22], [23]. In these methods, ultraharmonic and inharmonic signals are separated to distinguish stable cavitation from for the broadband inertial cavitation emissions, and integrated using TEA.

Here, we adapt a speed of sound estimation technique to perform passive cavitation mapping based on cavitation source localization (PCM-CSL). Cavitation sources are localized by fitting an expected wavefront profile to the detected arrival time of the cavitation signals. A similar concept was previously proposed by Gyöngy et al. [24], [25], where the localization was based on the signal time delays obtained from the cross-correlation of the receive signal, but the approach is limited to individual sources in near-field applications and still suffers from tail artifacts appearing after the time integration process. Our proposed PCM-CSL approach is not limited by the axially-elongated point spread function of the imaging transducer, and therefore the axial resolution of PCM-CSL is significantly improved by eliminating factors contributing to the tail artifact.

The PCM-CSL method is described and analyzed in two parts. In Part I, we introduce the proposed cavitation source localization technique and compare it with the Anderson-Trahey [26] method on which the method is derived. A series of simulation experiments is carried out to demonstrate the fundamental properties and limitations of the proposed technique and we evaluate the method under specific controlled conditions. A wavefront correction technique is also introduced to correct discontinuities in the recorded wavefront.

In Part II [27], we apply the proposed technique to microbubble flow phantoms, wire targets, and in vivo experiments. A filtering technique based on the root mean square error between the estimated arrival-time profile and the expected hyperbolic shape of the signal is introduced to eliminate spurious noise from the mapping process. A detailed analysis of the wavefront correction performance is carried out in Part II.

II. Theory

A. The Anderson-Trahey Method for Sound Speed Estimation

Upon insonification with an imaging pulse, an ideal stationary point target reflects a spherical wave. The arrival times of the reflected waves as they return to a linear array of transducer elements are based on the path length differences between the reflection point and the transducer elements:

$$t\left(x_i\right)=\frac{\sqrt{{\left(x_i-x_t\right)}^2+y_t^2+z_t^2}}{c},$$ (1)

where [x_t, y_t, z_t] is the target location of the reflection point, c is the assumed sound speed of the media, and x_i is the location of transducer element i. Anderson and Trahey [26] showed that the square of the arrival times of the spherical wave impinging on a linear array, t(x_i), can be described by a parabola

$$t^2\left(x_i\right)=p_1{x}_i^2+p_2{x}_i+p_3,$$ (2)

where the coefficients of the parabola depend on the location of the target and the speed of sound of the propagation medium and are defined as [26]:

$$\begin{array}{c}{p}_1=\frac{1}{c^2},\\ p_2=\frac{-2x_t}{c^2},\\ p_3=\frac{x_t^2+y_t^2+z_t^2}{c^2}.\end{array}$$ (3)

For pulse-echo ultrasound applications, Anderson and Trahey [26] utilized a parabolic fit to the square of the arrival-time profile of the reflected waves to estimate the speed of sound of a medium and infer the position of the sonicated target. Accurate usage of the Anderson-Trahey approach requires that the round trip transit time of the pulse-echo from the transducer to the target and back to be well-characterized. In this approach, the transmit (τ_TX) and receive (τ_RX) times are computed as half the minimum round-trip transit time (τ_m) of the pulse and echo (i.e. ${\tau }_{TX}={\tau }_{RX}={\tau }_m/2$), as illustrated in Fig. 1(a). This limits the method to scattering events on the axis of the ultrasound beam, and is ill suited to scatterers off the ultrasound beam axis, where the transmit time is not equal to the receive time, as illustrated in Fig. 1(b) [28].

Fig. 1:

(a) Pulse and echo for a scatterer on the beam axis. The receive time is equal to the transmit time, i.e., τ_TX = τ_RX. (b) Pulse and echo for a scatterer off the beam axis. The receive time cannot be easily determined because it is not equal to the transmit time, i.e., τ_TX ≠ τ_RX. (c). Acoustic emission from cavitation. There exists a time delay between the transmit and the receive event, corresponding to the microbubble lifetime. The minimum arrival time of the acoustic emission at the transducer surface includes the microbubble lifetime τ_l, i.e. τ_m = τ_TX + τ_l + τ_RX.

B. Adaptation of the Anderson-Trahey Method for Cavitation Source Localization

To adapt Anderson-Trahey’s method to cavitation source localization, the method must account for sources that are located off the beam axis (x_t ≠ 0), as shown in Fig. 1(a). A coordinate transformation is applied to the arrival-time profile t(x) of the wave to account for the source’s unknown location:

$$x^{\prime }=x-x_0.$$ (4)

Here, x₀ is the lateral location of the minimum arrival time of the acoustic emission at the transducer surface (see Fig. 2(b)).

Fig. 2:

(a) A spherical wavefront, emitted by the cavitation source, is detected by a linear array of elements. Here x_t, z_t is the cavitation source’s location, x₀ is the coordinate of the transducer element that first detected the wavefront from the cavitation source, and x_i is the lateral coordinate of the i-th transducer element. (b) The RF channel data and expected wavefront’s shape (solid blue line) detected by a linear transducer array.

C. Modified Hyperbolic Fit

The actual transit time from acoustic pulse transmission to reception of emissions from cavitating microbubbles is unknown because the microbubbles can exhibit uncertainty in the time in which they may collapse in response to a therapeutic ultrasound pulse. This unknown time-delay between the therapeutic transmit event and the received cavitation event is known as the microbubble lifetime (τ_l). Thus, the total time between the transmitted therapy pulse and the received cavitation event is given by ${\tau }_m={\tau }_{TX}+{\tau }_l+{\tau }_{RX}$, as illustrated in Fig. 1(c). The presence of such a microbubble lifetime will result in “delayed” arrival-times and overestimation of the axial position of the microbubbles from the Anderson-Trahey method, Fig. 1(c).

To account for microbubble lifetime, an approach similar to Anderson-Trahey [26] is utilized, except a modified hyperbolic function is fit to the recorded cavitation wavefront. The approach is based on the geometrical characteristics of a spherical wave emission from the acoustic cavitation and eliminates the fit’s dependence on time. For a linear array transducer, the wavefront t(x_i) for a spherical wave emitted from a cavitation event at location [x_t, z_t] and received by element i at lateral location x_i is described by (1). To eliminate the time dependence due to the unknown microbubble lifetime and location, the arrival times are shifted so that the transducer element at x₀ is coincident with t = 0:

$$\tilde{t}=t_{side,max}-t.$$ (5)

Here $t_{side,max}=\max \left(t_1,t_M\right)$, and t₁ and t_M are the arrival-times of the wavefront at the first and M-th elements, respectively, for a transducer with a total of M elements. Assuming that y_t ≪ z_t and c is known, (1) is thus modified to be:

$$c\tilde{t}\left(x_i\right)=\sqrt{{\left(x_t-x_i\right)}^2+z_t^2}-z_t.$$ (6)

Here $\tilde{t}\left(x_i\right)$ is the shifted arrival time of the received wavefront for the i-th transducer element relative to the transducer element located at x₀. By fitting (6) to the shifted (and scaled by a factor of c) arrival times, the cavitation location can be estimated from the best-fit parameters x_t and z_t. The hyperbolic fit in this approach is preferred over the parabolic fit in the original Anderson-Trahey method because errors in the arrival-time estimation process are squared in the Anderson-Trahey method, which lead to larger errors in the source localization.

D. Proposed Passive Cavitation Mapping Based on Cavitation Source Localization

While the original Anderson-Trahey method was proposed to estimate the average speed of sound between the transducer and a scattering point, the method is adapted here to localize multiple cavitation sources, which may include multiple cavitation events occurring simultaneously at nearby locations. For example, an illustration of cavitation occurring at location [x_t, z_t] with an unknown microbubble lifetime τ_l is shown in Fig. 2(a). The received wavefronts sampled by the transducer array over time is composed of a hyperbolic-shaped pattern corresponding to the cavitation event (Fig. 2(b)). The arrival-time profile from an acoustic emission or scatterer can be estimated by cross-correlation of the received radio-frequency (RF) channel data in the aperture domain.

1). Extraction of Arrival-Time Profiles:

In this approach, multiple arrival-time profiles from cavitation acoustic emissions are extracted using a time-gated window of the RF channel signals. As shown in Fig. 3, an arrival-time profile is estimated from the received RF channel signals by forward-propagation of nearest-neighbor cross-correlation of element signals from the first receive element to the last receive element in the array (Fig. 3(a-c)), identical to that described in the Anderson-Trahey approach [26]. The cross-correlation procedure is then backward propagated by starting from the last receive element in the array (M-th element) and continuing to the first element to minimize any potential discontinuities (Fig. 3(d)). A detailed description of this estimation procedure is described in Appendix A.

Fig. 3:

Illustration of passive cavitation mapping based on the proposed hyperbolic fit for cavitation source localization (PCM-CSL). (a) Captured RF channel signals with emission from a cavitation. (b) A wavefront of interest. (c) The detected wavefront by forward propagation of cross-correlation. The blue region represents the time-gated window (kernel) for the i-th channel, while the red region represents the padded window for the i+1 channel used in the cross-correlation (search region) and arrival-time estimation. (d) The final wavefront, detected by backward-propagation of cross-correlation. (e) The selected wavefront using time-gating for energy calculation. (f) Extracted arrival times from (d) with respect to x₀ (solid blue line) and the hyperbolic fit (dashed black) using (6). (g) The Gaussian-shaped uncertainty region based on the 68% confidence interval of the hyperbolic fit in (f). (h) The final passive cavitation map based on cavitation source localization. The total cavitation energy from (e) (units of J) is distributed over the uncertainty region, with a white box showing the region boundaries from (g). The final PCM-CSL image is the summation of all PCMs for all cavitation sources.

The wavefronts, such as shown in Fig. 3(e), can selectively be extracted for passive cavitation mapping. Once successfully extracted, the wavefront (Fig. 3)(e)) is excluded from further analysis in order to minimize replicas of the same cavitation event.

2). Cavitation Mapping:

The cavitation energy is calculated by time-delaying and integrating the cavitation energy for the detected wavefront, as illustrated in Fig. 3(e) and described in Appendix B. The process of cavitation mapping is illustrated in Fig. 3(f-h). The cavitation source location is determined from the coefficients of the hyperbolic fit ((6), Fig. 3(f)). To construct a cavitation map showing both the cavitation source location and its magnitude, each cavitation event is given a finite area in the shape of a 2D Gaussian distribution function according to the uncertainty of the source localization, which is determined by the 68% confidence intervals (i.e. ±1σ) of the estimates for x_t and z_t from the hyperbolic fit. These 68% confidence intervals are used to define the standard deviations, in mm, of the 2D Gaussian distribution function for the detected cavitation (Fig. 3(g)). The energy of the cavitation source, E_n, is then distributed over the uncertainty region and is shown on the PCM image in Fig. 3(h). The final PCM-CSL image is obtained by summing all of the cavitation maps from the individual detected wavefronts.

3). Wavefront Correction:

In phantom or in vivo experiments, the received RF channel signals typically have high noise and/or interference from multiple cavitation events, resulting in errors and discontinuities in the arrival-time profile. Applying the proposed cross-correlation procedure to the RF channel signals may result in an arrival-time profile with discontinuities and a poor hyperbolic fit, leading to errors in source localization.

To improve source localization, a wavefront correction algorithm is introduced. First, the derivative of the arrival-time profile (6) with respect to x is computed as

$$c{t}^{\prime }\left(x_i\right)=\frac{\alpha x_i}{\sqrt{x_i^2+\beta }}+\eta ,$$ (7)

where α, β, and η are constants. Equation 7 is then fit to the measured derivative, excluding outliers. The outliers could be defined by 1) fitting (7) using all points, 2) computing the squared error, and 3) finding all points outside the M ±5×Σ, where M is mean and Σ is the standard deviation. However, as the arrival-time profile’s derivative can be very noisy, either the fit quality will be low or the fit will fail to converge. To overcome such difficulties in step 1), a linear approximation is fit to Equation 7 in the form of $c{t}^{\prime }\left(x_i\right)=a\times x_i+b$, and steps 2) and 3) are carried out as described before. At each point of discontinuity (i.e. outliers of the time derivative, as described above), the arrival times are realigned by shifting part of it in time to eliminate the discontinuity point. After realignment, (6) can be fit to the arrival times. If the fit quality of the derivative, excluding the outliers, is poor (i.e. R² < 0.9), no wavefront correction is applied.

III. Methods

A. Implementation of the Proposed Technique

In all experiments, a time gated window with parameters N = 3λ₀, n_p = 0.5λ₀, and n_o = 0.5λ₀ (2.51mm, 0.42mm, and 0.42mm, respectively, all converted and rounded to nearest integer value in samples) was used unless otherwise specified. Here N is size of the window in samples, n_p is the padding for the search region, and n_o is the overlap of the next axial window with the previous one. Wavefronts are extracted every N–n₀ samples. A detailed description of the correlation and windowing process and their corresponding parameters are described in Appendix A. The fitting of the hyperbolic model in (6) was performed in MatLab (Mathworks, Natick, MA) using the fit() function with x_t and z_t as parameters. Only hyperbolic fits with R² > 0.9 were used for localization analysis and PCM image formation.

B. Validation of the Hyperbolic Fit: Field II Simulations

1). Comparison with Anderson-Trahey Method:

RF channel signals from a grid of point scatterers were simulated with Field II [29], [30]. A 128-element linear transducer array was used and had an excitation pulse consisting of 2 periods of a f₀ = 1.78 MHz sinusoidal pulse (wavelength of λ₀ = 837μm) weighted by a Hanning window. The transducer’s impulse response consisted of a two-cycle Hanning-weighted sinusoid for both the transmit and receive apertures. The receiving transducer was defined in a similar manner, but had a center frequency of 7.8 MHz. The speed of sound of the medium was set to 1495 m/s. The simulated transducer parameters are summarized in Table I. Point scatterers were separated axially from 30 to 50 mm depth by 6λ₀ (5.02 mm) and laterally from −12 to 12 mm by 3λ₀ (2.51 mm). Each point scatterer was simulated individually to avoid interference of the signal with other scatterers. For these simulations, the transducers were both placed in the z = 0 plane at the same location. Transmit focusing was applied directly at the location of each point scatterer, and the unfocused RF channel signals were recorded with sampling frequency F_s = 31.3MHz and upsampled by a factor of 8 using cubic spline interpolation. In this simulation, the point scatterers mimicked conventional scatterers from pulse-echo ultrasound. To mimic inertial cavitation sources (i.e. broadband acoustic emissions caused by microbubble collapse several microseconds after insonification), a delay of several microseconds (values of 0, 1, 3, and 10μs and a random delay selected from a uniform distribution over 0 and 10μs) [31] to the arrival times was applied over the entire receive aperture to simulate the microbubble lifetime. While these scatterers simulate the wavefronts and timing of cavitation sources, Field II is not capable of simulating the non-linear echoes from stable cavitation sources and the complete broadband response from inertial cavitation sources. In addition, although non-overlapping transducer bandwidths are utilized in the companion paper [27] to separate stable and inertial cavitation signals, in these simulations, the echoes from the weak out-of-band reflections in a noise-free environment are used as a surrogate for cavitation signals.

TABLE I:

Simulation Parameters

	Transmit	Receive
Transducer	Linear Array	Linear Array
Frequency (f0), MHz	1.78	7.8
No. Elements	128	128
Pitch, μm	260	260
Sample Rate (samples/s)	n/a	4f0
Cycles per Pulse	2	n/a
Focusing	Focused	Unfocused
Bandwidth, MHz	1.2 – 2.2	5.7 – 9.7

For source localization, three methods were compared: the original Anderson-Trahey method (where half of the minimum transit time, t_m/2, was subtracted from the estimated arrival times); the Anderson-Trahey method with an x-coordinate transformation (where the x origin was shifted to the x₀ location of the leading edge of the wavefront, as in (4)); and the proposed source localization method using the hyperbolic fit (6). The first two methods were based on the estimated transit time, while the proposed hyperbolic fit method did not estimate transit time but required an assumed speed of sound in the medium. The estimation errors of the localization for the two types of the acoustic sources (point scatterers and cavitation sources) were calculated.

2). Off Axis Source Detection:

To estimate the detection of off-axis sources, Field II simulations were performed using the same parameters as described above with minor modifications. Two separate focused transmit excitations were applied at x, z coordinates of [0, 35] and [10, 35] mm. A grid of point scatterers were placed around the focal point with λ₀ lateral and axial spacing. Simulations were performed individually for each scatterer.

3). Multiple Sources Detection:

While the above simulations evaluate the performance of the method to single cavitation sources, the presence of multiple cavitation sources is more likely in vivo due to the size of the transmit beam. Therefore, two simulations experiments were performed involving multiple sources per transmit to evaluate the accuracy of the proposed method. In the first simulation, 10 scatterers with random scattering strength varying between 0.5 and 1 were placed randomly within the focal region of the transmit beam. In the second simulation, two scatterers were placed at similar axial locations in proximity to the focal point, but were symmetrically offset laterally by λ₀, i.e. scatterer 1 was placed at [-λ₀, 35] mm, and scatterer 2 was placed at [λ₀, 35] mm. These simulations were repeated with different combinations of axial and lateral offsets from the focal point (lateral offsets: ±2λ₀ and ±4λ₀; axial offsets: 0, ±λ₀/2, and ±λ₀). Source localization results were compared using time-gating windows of two different sizes: N = 3λ₀, n_p = 0.5λ₀, n_o = λ₀ and N = λ₀, n_p = 0.25λ₀, n_o = 0.25λ₀ (0.84 mm, 0.21 mm, and 0.21 mm, respectively). The cavitation source localization results with and without applying wavefront correction were compared.

4). Microbubble Cloud Detection:

Because cavitation signals are more likely to occur from microbubble clouds than from individual microbubbles, the resolution of the proposed passive cavitation mapping technique may be limited by the localization error with respect to these microbubble clouds. Therefore, localization error was estimated by Field II simulations of 30 and 3000 randomly-positioned scatterers within a spherical region placed near the focal point of the therapeutic transducer. The sphere of scatterers were placed within a diameter d_cloud of either 0.21 mm (1/4λ₀) or 1.14 mm (1.36λ₀, which is equal to the transmitted beam’s full-width at half-maximum in lateral dimension) with the center of the sphere [x_sca, z_sca] placed at a random position in the therapy beam. The localization error, Δr, was determined as the difference between the estimated and exact positions of the center of the diffuse scatterers:

$$\text{Δ}r=\sqrt{{\left(x_{sca}-x_t\right)}^2+{\left(z_{sca}-z_t\right)}^2}.$$ (8)

These simulations were performed with and without applying a microbubble lifetime of 10 μs. 1,000 simulations were performed for each case.

C. Passive Cavitation Mapping

Passive cavitation maps for simulations in Section III-B3 with two sources were reconstructed. The simulations with the sources placed at 1) [−2λ₀, 35+0.5λ₀] mm and [2λ₀, 35 – 0.5λ₀] mm, and at 2) [−4λ₀, 35+λ₀] mm and [4λ₀, 35 – λ₀] mm were used and the cavitation maps were reconstructed using PCM-TEA with delay-and-sum (PCM-DAS-TEA) and minimum-variance distortionless-response (PCM-MVDR-TEA) beamformers and compared to PCM-CSL. The localization errors for PCM-CSL were computed as the difference between the predicted cavitation source location from the hyperbolic fit and the known source location. The PCM-TEA localization errors were computed as the difference between the location of the peak signal strength in the PCM-TEA maps and the known source location.

The PCM-MVDR-TEA implementation here is similar to the broadband MVDR proposed in Holfort et al. [32] and is the basic method on which the robust MVDR beamformer implemented in Coviello et al. [16]. The MVDR implementation here lacks the constrained optimization steps on the steering vector, which is the key modification of the robust MVDR method. However, because there is no noise in the simulations, the MVDR beamformer implemented here is equivalent to the best-case scenario of the robust MVDR beamformer.

IV. Results

A. Cavitation Source Localization: Field II Simulations

1). Comparison of the Proposed Hyperbolic Fit with Anderson-Trahey Method:

Fig. 4 shows source localization performance using three different methods: the Anderson-Trahey method (3), the Anderson-Trahey method with the x-coordinate transformation ((3) and (4)), and the proposed hyperbolic fit (6). The estimated source locations (blue) are plotted against the actual source locations (red), with cavitation sources (i.e. “microbubbles”) simulated assuming τ_l = 3 μs. The mean error and standard deviation across all source locations (Δr; (8)) were computed for all three source localization techniques for the pulse-echo and microbubble-mimicking sources assuming different microbubble lifetimes are shown in Table II.

Fig. 4:

Source localization using the Anderson-Trahey method, with and without x-coordinate transformation, and the proposed hyperbolic fit method for simulated sources. Two types of sources are used: pulse-echo scatterers (a-c), and cavitation-mimicking sources with an assumed microbubble lifetime (3 μs) (d-f). Regardless of the source type and location, the hyperbolic fit shows good localization, while estimation by the original Anderson-Trahey method produces significant errors for scatterers or sources off the transducer’s normal axis and closer to the transducer. The errors are larger for the cavitation mimicking sources due to the delay associated with the microbubble lifetime. The transmit beams’ focal points coincide with each source location; a single scatterer was used per simulation.

TABLE II:

CSL Errors for Scatterers (τ_l = 0) and Microbubbles with Varying Lifetime (τ_l ≠ 0) from Field II Simulations

Method	Units	Source Localization Error, Δr
		τl = 0μs	τl = 1μs	τl = 3μs	τl = 10μs	τl = 0 to 10μs
Anderson-Trahey	mm	1.77 ± 1.12	2.00 ± 1.22	2.81 ± 1.35	6.00 ± 2.20	4.24 ± 1.73
Anderson-Trahey, x-transformation	mm	0.41 ± 0.18	0.29 ± 0.16	0.90 ± 0.29	3.30 ± 0.96	1.93 ± 0.58
Hyperbolic Fit	mm	0.13 ± 0.14	0.12 ± 0.12	0.12 ± 0.11	0.11 ± 0.10	0.12 ± 0.11
Anderson-Trahey	λ0	2.11 ± 1.34	2.39 ± 1.46	3.43 ± 1.62	7.17 ± 2.63	5.07 ± 2.07
Anderson-Trahey, x-transformation	λ0	0.49 ± 0.22	0.35 ± 0.19	1.08 ± 0.34	3.94 ± 1.14	2.31 ± 0.69
Hyperbolic Fit	λ0	0.16 ± 0.17	0.14 ± 0.14	0.14 ± 0.14	0.13 ± 0.12	0.14 ± 0.13

2). Off Axis Source Detection:

Detected locations of a single source when placed with an axial and/or lateral offset from the beam’s focal point are shown in Fig. 5. The mean localization error for therapy beams focused at [0, 35] and [10, 35] mm was Δr = 0.11 ± 0.12 mm (0.13 ± 0.14λ₀) and Δr = 0.09 ± 0.09 mm (0.10 ± 0.11λ₀), respectively.

Fig. 5:

The proposed source localization technique with two different beams: (a) focus at [0, 35] mm and (b) focus at [10, 35] mm. A single scatterer was used per simulation. The mean localization errors were 0.11 and 0.09 mm for (a) and (b), respectively.

3). Multiple Sources Detection:

Fig. 6 shows detected scatterers for the simulation with 10 scatterers of varying amplitude randomly placed within the focal region. Most of the points are well localized, while the sources located near the therapy beam’s focal position, where there are multiple sources in close proximity, are poorly localized. As we cannot, with 100% certainty, match each estimated wavefront to its ground truth source location due to signal interference, it is not possible to compute the localization error (Δr) for these sources.

Fig. 6:

Cavitation source localization from a simulation with 10 cavitation sources having random amplitudes and randomly placed within the focal region of the transmitted beam. All sources were simulated simultaneously.

The arrival-time profiles (R² > 0.9) and estimated locations of two simultaneous sources with varying positions are shown in Fig. 7. The larger time-gating window (N = 3λ₀, n_p = 0.5λ₀, n_o = λ₀) was used to detect the wavefronts for the plots shown in the top two rows (Fig. 7(a)-(h)), while the smaller time-gating window (N = λ₀, n_p = 0.25λ₀, n_o = 0.25λ₀) was used for the two bottoms rows (Fig. 7(i)-(p)). No arrival-time profiles are shown in Fig. 7(a,i) because the method failed to detect any wavefronts with high fit to the arrival-time profile (i.e. R² > 0.9). Arrival-time profiles in Fig. 7(b,c,j), resulted in an estimated source location. However, the estimated locations are outside of the displayed axis limits and have large localization errors, Fig. 7(f,g,n).

Fig. 7:

Arrival-time profiles and estimated source localization from the Field II simulations with two scatterers. Results for the top 2 rows (a-h) were obtained with a window of size 3λ₀. The bottom rows (i-p) used a shorter window (size of λ₀). The proposed method fails to detect any wavefronts for two scatterers placed too closely together (a & e; i & m). A small time-gating window is, in general, more accurate in source localization than the larger time-gating window size in the presence of interfering sources. Vertical and horizontal dotted lines represent z and x spacing in wavelengths. No wavefronts with good fit quality were detected in (a,i). The estimated locations of the sources are outside of the axis limits in (f,g,n).

4). Wavefront Correction:

Fig. 8(a) shows a simulation example where localization of an off-axis source at [−1, 33.5] mm is corrupted by two other sources. The corresponding RF channel signals are shown in Fig. 8(b). Due to the interfering sources, the estimated arrival-time profile contains a discontinuity at the center of the array (Fig. 8(c)) and the source is incorrectly localized at [0.27, 33.7] mm. The discontinuity is easily visible as an outlier in the derivative of the arrival-time profile (shown in Figs. 8(d) and (e)). The arrival-time profile in Fig. 8(c) is corrected by shifting the left half of the arrival-time profile (shown in blue) in the time domain to be “aligned” with the right half, resulting in the corrected arrival-time profile in Fig. 8(f) (shown in red). The corrected arrival-time profile results in a higher fit quality and more accurately localizes the source to [−1.0, 33.6] mm.

Fig. 8:

Wavefront correction procedure for noisy wavefronts. (a) Sources’ locations. The source of interest is in the top-left and highlighted with a circle. (b) The RF channel signals of interest showing the detected wavefront from the scatterer at [−1, 33.5] mm in blue; the search region is shown in red. (c) The estimated arrival-time profile (blue) and its corresponding hyperbolic fit (dashed black). (d) Derivative of the arrival-time profile (blue) and it’s fit to (6) (black). (e) The time derivative’s squared error (red). All arrival times with derivative error ≥ 5Σ (dashed black) require correction. (f) The arrival-time profile (red) and hyperbolic fit (dashed black) after correction.

Similarly, applying the wavefront correction technique to the simulations in Fig. 7(d,h) improves the source localization. For example, for the source located at [−3.35, 35.84] mm in Fig. 7(d,h), the localization error was reduced from Δr = 1.02 mm (1.22λ₀) to Δr = 0.33 mm (0.39λ₀). The wavefront correction was unable to be applied to the other wavefronts in Fig. 7 because of the poor fit quality (R² < 0.9) of the derivatives.

However, defining the outlier threshold as M ± 3 × Σ (instead of M ± 5 × Σ) allows the wavefront correction to be applied to the scatterers in Fig. 7(b,f; c,g; j,n) and reduces the localization error of these sources from Δr = 10.85±8.20 mm (12.96 ± 9.80λ₀) to Δr = 4.79 ± 2.52 mm (5.73 ± 3.01λ₀).

Most of the improvement is due to correction of the axial location from Δr_z = 10.20 ± 8.40 mm (12.19 ± 10.04λ₀) to Δr_z = 3.51 ± 3.05 mm (4.19 ± 3.64λ₀).

5). Microbubble Cloud Detection:

From the 1,000 simulations of microbubble clouds with d_cloud = 0.25λ₀ the source localization error was Δr = 0.08 ± 0.07 mm (Δr = 0.10 ± 0.08λ₀) with a median of 0.07 mm (0.08λ₀) and an interquartile range of 0.08 mm (0.09λ₀). The average number of sources detected per simulation was 5.1±1.6. The average uncertainty of the localization (i.e. σ_x and σ_z for the Gaussian shape uncertainty function) was 0.0016±0.0004 mm (0.0019±0.0005λ₀) and 0.0136±0.0042 mm (0.0162±0.0050λ₀) for x_t and z_t, respectively. For simulations with d_cloud = 1.36λ₀, the source localization error was Δr = 0.61 ± 0.41 mm (Δr = 0.72 ± 0.49λ₀) with a median of 0.50 mm (0.60λ₀) and an interquartile range of 0.50 mm (0.60λ₀). The average number of sources detected per simulation was 4.5 ± 2.4.

Applying a microbubble lifetime of 10 μs had no effect on the source localization error. For example, the source localization error for d_cloud = 1.36λ₀ was Δr = 0.57 ± 0.41 mm (Δr = 0.68 ± 0.49λ₀) with a median of 0.46 mm (0.55λ₀) and an interquartile range of 0.46 mm (0.55λ₀). The average number of sources detected per simulation was 4.5 ± 2.5.

In addition, increasing the number of sources in the microbubble cloud to 3,000 resulted in a slightly decreased source localization error. For example, in the case with d_cloud = 1.36λ₀, the source localization error was Δr = 0.43 ± 0.31 mm (Δr = 0.52 ± 0.37λ₀) with a median of 0.37 mm (0.44λ₀) and an interquartile range of 0.41 mm (0.49λ₀). The average number of sources detected per simulation was 3.2±2.5. As expected, an increase in the “cavitation energy” emitted by the cloud was observed when the number of sources increased.

B. Passive Cavitation Mapping

Several example PCM images of two simulated sources (Fig. 7(o,p)) are shown in Fig. 9 on a decibel scale. PCM-CSL maps (Fig. 9(c,d)) are shown with a 5 and 10 fold increase, for illustration purposes, in the x and z uncertainty, respectively. PCM-DAS-TEA and PCM-MVDR-TEA (Fig. 9(a,b,d,e)) can detect both scatterers but have poor tail artifacts resulting in high errors of axial position estimation. The errors in the lateral dimension are smaller compared to those in the axial dimension. Localization errors were 0.69 mm (0.82 λ₀), 0.88 mm (1.05 λ₀) and 0.16 mm (0.18 λ₀) for PCM-DAS-TEA, PCM-MVDR-TEA, and PCM-CSL images in Fig. 9(a,b,c)), respectively. Localization errors were 0.71 mm (0.84 λ₀), 0.97 mm (1.16 λ₀), and 0.23 mm (0.27 λ₀) for PCM-DAS-TEA, PCM-MVDR-TEA, and PCM-CSL images in Fig. 9(d,e,f)), respectively. In addition, there is high uncertainty in the estimated source location in PCM-DAS-TEA and PCM-MVDR-TEA as shown by the long tail artifacts. PCM-CSL localizes both sources significantly better in both axial and lateral dimensions.

Fig. 9:

Passive cavitation maps from the simulations with two sources placed at different locations. PCM-DAS-TEA (a,d), PCM-MVDR-TEA (b,e), and PCM-CSL (c,f) images are shown on a 40 dB dynamic range. The actual source location is marked with a circle. The PCM-CSL maps are shown with a 5-fold and 10-fold increase in the standard deviation in the x and z dimensions, respectively, for visualization purposes.

V. DISCUSSION

A. Cavitation Source Localization: Hyperbolic Fit

The proposed hyperbolic fit in (6) shows better localization of a single source compared to the Anderson-Trahey methods (original and with the x-transformation), as seen from Fig. 4 and Table II. The proposed hyperbolic fit estimates the scatterer’s position with high precision if the wavefront has no interference. The average error and standard deviation of scatterer location is less than a quarter of a wavelength (Δ_r = 0.16±0.17λ₀), which is significantly smaller than the Anderson-Trahey methods. When localizing the microbubble mimicking sources, the Anderson-Trahey method is highly susceptible to the errors from the microbubble lifetime τ_l. Increasing the microbubble lifetime results in increased errors for the Anderson-Trahey method, but does not introduce any additional errors to the proposed hyperbolic fit method. The hyperbolic fit also showed excellent localization of individual sources when the source was offset from the focal point of the therapy beam (Fig. 5). The localization error (Δr = 0.10 ± 0.12 mm, or 0.13 ± 0.14λ₀) was not substantially different than when the sources were located at the focal point of the therapy beam.

In addition to errors from the microbubble lifetime, the Anderson-Trahey method is also more susceptible to errors in the arrival-time profile than the proposed hyperbolic fit. This is because the Anderson-Trahey method fits a parabola to the wavefront, which requires a squaring of the estimated arrival-time profile, as seen from (2). The proposed hyperbolic fit utilizes a direct fit to the arrival-time profile, as shown in (6). If any errors are present in the estimated arrival-time profile, the squaring in the Anderson-Trahey method will amplify these errors and result in an increased localization error. This can be seen from the τ_l = 0 μs column in Table II, where the hyperbolic fit shows improved localization error compared to the Anderson-Trahey method with x-transformation.

Note that in Figs. 4 and 5, multiple sources are sometimes identified for each scatterer/source due to the longer pulse length with respect to the window size and the use of pulse-echo signals to mimic cavitation sources in the Field II simulations.

B. Multiple Sources Detection

As seen from Fig. 6, several sources located at approximately 29, 30 and 32 mm depth are well localized, while the group of sources at 34 mm is poorly localized. The proposed technique is able to estimate the lateral location of sources when there are no interfering wavefronts from other sources. However, the individual sources in the cluster that are in proximity to the focal point are not able to be properly localized because all of the sources reside within the lateral diffraction limit of the transmitting transducer (1.36λ₀). Therefore, we estimate that the lateral resolving power of the proposed technique is equal to the resolution of the transmitting system, but the interference of nearby sources must be sufficiently weaker in order for the sources to be localized accurately.

The axial locations were accurately estimated for sources separated axially by a distance greater than half the pulse length, which is relatively short compared to therapeutic beams. This is due to the simulation utilizing the pulse-echo response from scatterers in place of a true cavitation source. In physical experiments, the signal from the cavitation source would be a short “burst” in the time domain, resulting in a signal with a wide frequency spectrum, in which case the axial resolving power of the proposed technique will be defined by the receiving transducer bandwidth. We do not anticipate any significant differences in performance or axial resolution with the increased pulse lengths utilized in physical experiments because the “burst length” of the cavitation signal is independent of the transmitted pulse length.

A non-optimized window size in the proposed hyperbolic fit can result in multiple extracted locations (usually located close to each other) from a single source. For example, as shown in Figs. 4 and 5, multiple detected “sources” are shown for each scatterer location. Optimization of the window size is necessary to allow multiple cavitation sources from the same location (but perhaps with different times) while minimizing multiple detections of the same source. Even though a large window size may decrease the number of detections of the same source, it performs worse than a shorter window size (e.g. λ₀, Fig. 7(k,o; l,p)) at estimating the source location. A shorter window obtains a more accurate estimation of the source location, but could result in repeated detection of the same source if the receiving transducer bandwidth is sufficiently narrow such that it modifies the cavitation burst length to be longer than the window. Furthermore, in low signal-to-noise ratio (SNR) environments or in the presence of interfering waves, a smaller window may result in a greater number of wavefront discontinuities, although we have demonstrated the capability to correct a subset of wavefronts with discontinuities using the waterfront correction technique to improve source localization. Here, the cavitation sources are mimicked by backscatter from a point target with high SNR, resulting in a multiple-cycle reflection at the transmit frequency. Because inertial cavitations from microbubbles are expected to be short bursts with wide bandwidth, careful selection of window size is necessary.

C. Microbubble Cloud Detection

When performing in vivo experiments, we anticipate that most cavitation will occur from microbubble clouds or multiple microbubbles located in proximity to each other rather than from individual microbubbles. The detected cavitation signals will likely be an interference of multiple cavitations in a cloud, resulting in partially coherent cavitation wavefronts, much like those resulting from diffuse scattering in B-mode imaging that gives rise to speckle. However, we have shown here that the hyperbolic fit method still retains excellent localization of microbubble clouds with a mean localization error of 0.11λ₀ for the cloud with a diameter of 1/4λ₀. This error is close to the localization error of a single source. However, increasing the microbubble cloud size to the diffraction limit of the transmitting beam resulted in a larger localization error. In this case, the localization error is approximately half the diffraction limit of the transmitting beam. Note that the localization error is calculated as a radial error; thus the total uncertainty of the center of the bubble cloud is approximately the diffraction limit of the transmitted beam. This lends more confidence that the lateral resolution of the proposed cavitation map is the diffraction limit of the transmitting beam.

Because of the diffraction-limited resolution of the clinical transducers, the proposed technique is not able to distinguish individual bubbles in microbubble clouds. The average number of detected “sources” for a microbubble cloud varies between 3 and 5 and is independent of number of sources inside the cloud. Thus, the technique is not expected to map the individual microbubbles, but rather provide spatio-temporal map of the total cavitation energy distribution.

Of importance is that no significant effect on localization error of the microbubble clouds was observed by differences in microbubble lifetime or microbubble concentration. Because the proposed technique is designed to be independent of the microbubble lifetime, source localization errors are greatly reduced compared to other time-dependent techniques (see Table II).

D. Wavefront Correction

The proposed wavefront correction is necessary to improve localization in the presence of noise or interfering sources. For corrupted arrival-time profiles (Fig. 7(b,c,d,j)), applying wavefront correction resulted in 60% more accurate estimation of the axial location of the source. While the source positions were not estimated with high accuracy in these simulations, we show in a companion paper [27] that the wavefront correction is more useful in phantoms and in vivo experiments due to the expected weaker off-axis cavitation sources than was used in these simulations. The correction technique, however, is dependent on the definition of the outliers (i.e. discontinuity points in the delay profile) and will thus affect the performance of the wavefront correction. A detailed analysis of the wavefront correction technique is provided in Part II [27].

In the present study, we have analyzed the hyperbolic fit and wavefront correction technique separately. Therefore, our process is not optimized for speed or efficient use of computational resources. An alternative approach to minimize the number of steps and combine the hyperbolic fitting and wavefront correction is to perform a single fit of (7) to the arrival-time’s derivative, and all potential correction steps would be carried out by evaluating the fit quality of the time derivative.

E. Passive Cavitation Mapping

The reconstructed PCM-CSL maps show a significant improvement compared to the PCM-DAS-TEA and PCM-MVDR-TEA approaches (Fig. 9). The PCM-DAS-TEA approach results in an acceptable source location estimation (Fig. 9(a)) based on the peak cavitation value. However, the long tail artifacts reduce the confidence of the estimation. The source localization error does not improve when sources are placed further apart (Fig. 9(d)). The PCM-MVDR-TEA beamformer (Fig. 9(b,e)) improves the lateral resolution of the passive cavitation map and suppresses some of the tail artifact but, as expected, does not fully reject off-focus signals. The difference between the PCM-MVDR-TEA tail artifacts shown here and the images in Coviello et al. [16] is likely due to the logarithmic compression and larger dynamic range used in Fig. 9. As expected, the PCM-MVDR-TEA showed no improvement in the localization error compared to the PCM-DAS-TEA because the method is designed to suppress non-ideal signals in the wavefield. The PCM-CSL shows good resolution (even with the artificially-enlarged uncertainty region) with the localization error significantly lower for all the cases (Fig. 9(c,f)). Because there is no noise present in these simulations and idealized point sources are utilized, there is a high hyperbolic fit quality (when using a proper window size) with small errors in x and z, leading to a localization with high precision. Such small errors are not anticipated for phantom or in vivo experiments where noise is present and sources have finite sizes or are distributed over larger regions, as shown in the companion paper [27].

VI. Conclusion

We have developed a passive cavitation mapping (PCM) algorithm based on cavitation source localization using a modification to the Anderson-Trahey speed of sound estimation technique. This approach can realize high-resolution PCM and was shown to achieve localization of cavitation signals with greater resolution than conventional time exposure acoustics techniques. Briefly, the proposed method detects the wavefronts by applying cross-correlation of the receive channel data. The resulting arrival times are then fit to a hyperbola, and the coefficients of this fit are used to estimate the source’s axial and lateral location. A wavefront correction technique is introduced to correct delay profiles obtained in noisy environments or in the case of signal interference from multiple sources. Application of the technique to Field II simulations showed a mean localization error of 0.61 mm for microbubble clouds and yielded higher resolution and improved source localization compared to PCM based on time exposure acoustics.

APPENDIX A. Time Gating Approach to Extraction of Multiple Arrival-Time Profiles

A time-gated window is formed by first selecting a kernel of N samples from the RF signal of the first element, S₁ (the samples within the blue region of element 1 in Fig. 3 (c)). Here, the N-sample time-gated window is selected to minimally overlap with the previous window. For example, the n-th window starts at a depth of r_n samples, where r_n = r_n–1 + N − n_o and n_o is the number of overlapping samples, while the first window begins at r₁ = 1 sample. The N-sample windowed signal is then zero-padded on both sides to be N′ = N +2n_p samples and cross-correlated with an N′-sample window at the same depth from the second element signal, S₂ (the samples within the first red line on the left in Fig. 3 (c)). Cross-correlation is computed as a function of axial sample lag l, using

$$\begin{array}{c}{R}_{1,2}\left(r_n,l\right)=\\ =\frac{{\sum }_{k=1}^{N^{\prime }}\left\{\left[S_1\left(r_n+k\right)\right]\left[S_2\left(r_n+k+l\right)\right]\right\}}{\sqrt{{\sum }_{j=1}^{N^{\prime }}{\left[S_1\left(r_n+j\right)\right]}^2{\sum }_{k=1}^{N^{\prime }}{\left[S_2\left(r_n+k+l\right)\right]}^2}}\end{array},$$ (9)

where $l=-N-n_p,\dots ,+N+n_p$. The time-delay estimate between the first and second elements is determined from the location of the maximum of R_1,2:

$${\text{Δ}}_{1,2}\left(r_n\right)=argma{x}_{l\in L}{R}_{1,2}\left(r_n,l\right),$$ (10)

where L is the search region from −N − n_p to N + n_p (i.e. the red shaded region in Fig. 3 (c)) and Δ_1,2(r_n) is the time-delay estimate in samples between elements 1 and 2 at depth r_n. For the time-delay estimate between S₂ and S₃, the time-gated windows for S₂ and S₃ are shifted by Δ_1,2(r_n) samples corresponding to the previous time-delay estimate in addition to r_n samples, i.e. the exact window position is r_n+Δ_1,2(r_n). For further elements (i > 3), the additional time-delay is accumulated as

$$r_n=\sum _{j=1}^{i-1}{\text{Δ}}_{j,j+1}\left(r_n\right)$$ (11)

to allow the time-gated window to follow the wavefront across the element signals, as shown in Fig. 3 (c). Using the accumulated time-delay shift, the cross correlation of an arbitrary pair of element signals is determined by

$$\begin{array}{c}{R}_{i,i+1}\left(r_n,l\right)=\\ =\frac{{\sum }_{k=1}^{N^{\prime }}\left\{\left[S_i\left(r_i^n+k\right)\right]\left[S_{i+1}\left(r_i^n+k+l\right)\right]\right\}}{\sqrt{{\sum }_{m=1}^{N^{\prime }}{\left[S_i\left(r_i^n+m\right)\right]}^2{\sum }_{j=1}^{N^{\prime }}{\left[S_{i+1}\left(r_i^n+j+l\right)\right]}^2}}.\end{array}$$ (12)

Here i is the element number, $r_i^n=r_n+{\overline{\text{Δ}}}_{i,n}$, and ${\overline{\text{Δ}}}_{i,n}$ is the accumulated time-delay across all previous channels:

$${\overline{\text{Δ}}}_{i,n}=\sum _{k=1}^{i-1}{\text{Δ}}_{i,n},{\text{with}\overline{\text{Δ}}}_{1,n}=0\text{for}i=1.$$ (13)

After computing the time-delay for the last (M-th) element, the accumulated time-delay, ${\overline{\text{Δ}}}_{i,n}$, represents the wavefront’s relative time-delay between the first and last elements, obtained by forward-propagation of cross-correlations across the array (Fig. 3 (c)). The cross-correlation procedure is then backward propagated by starting from the last receive element in the array (M-th element) and continuing to the first element to account for any potential discontinuities (Fig. 3(d)). The wavefront’s relative arrival-time profile for the cavitation signal is the estimated time-delays

$$t_n=\frac{{\overline{\text{Δ}}}_{m,n}}{F_s},$$ (14)

where F_s is the sampling frequency. The modified hyperbolic fit (1) is then applied to the arrival-time profile obtained by (14) to localize the cavitation source.

APPENDIX B. Cavitation Energy

To calculate energy for each detected cavitation source, the time-gated detected wavefront signals from the corresponding cavitation source (Fig. 3 (e)) are time-delayed (1), summed across the channels, squared, and then integrated in the time domain over the time-gated window. The measured electrical energy detected by the ultrasound scanner (in units of J) is then expressed by

$$\begin{array}{c}{E}_n=\frac{1}{Z}{\int }_{\left(r_n+{\overline{\text{Δ}}}_{i,n}\right)/F_s}^{\left(r_n+{\overline{\text{Δ}}}_{i,n}+N\right)/F_s}{\left|\sum _{i=1}^M{S}_i\left(t-t_i\right)\right|}^{2}dt=\\ =\frac{1}{Z}\sum _{k=1}^N{\left|\sum _{i=1}^M{S}_i\left(r_n+{\overline{\text{Δ}}}_{i,n}+k\right)\right|}^2,\end{array}$$

where Z the impedance of the ultrasound system in Ohms and S_i has units of Volts. Estimation of cavitation energy is transducer and system dependent. In order to scale the approach to different transducers and systems, 1) the recorded signals should be converted to voltage, 2) the transducer’s receive voltage-pressure response should be characterized, and 3) spherical spreading of the pressure wave from cavitating sources and the transducer’s aperture size should be taken into account [33]–[36].