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Department of Electrical and Software Engineering, Schulich School of Engineering, University of Calgary, Alberta, CanadaDepartment of Radiology, Cumming School of Medicine, University of Calgary, Alberta, Canada
The opening of the blood-brain barrier (BBB) to allow therapeutic drug passage can be achieved by inducing microbubble cavitation using focused ultrasound (FUS). This approach can be monitored through analysis of the received signal to distinguish between stable cavitation associated with safe BBB opening and inertial cavitation associated with blood vessel damage. In this study, FUS phantom and animal studies were used to evaluate the experimental conditions that generate several cross-consistent metrics having the potential to be combined for the reliable, automatic control of cavitation levels.
Methods
Typical metrics for cavitation monitoring involve observing changes in the spectrum generated by applying the discrete Fourier transform (DFT) to the time domain signal detected using a hydrophone during FUS. A confocal hydrophone was used to capture emissions during a 10 ms FUS burst, sampled at 32 ns intervals, to produce 321,500 points and a high-resolution spectrum when transformed. The FUS spectra were analyzed to show the impact that equipment-transients and well-known DFT-related distortions had on the metrics used for cavitation control. A new approach, physical sparsification (PH-SP), was introduced to sharpen FUS spectral peaks and minimize the effect of these distortions.
Discussion
It was demonstrated that the general spectral signal-to-noise ratio (SNR) could be improved by removing the initial noisy phantom hydrophone signal transient. Minor changes in the transient length digitally removed from the sampled values significantly changed the spectral bandwidths of all the harmonically related FUS signals. We evaluated signal processing techniques to minimize the impact these DFT-related distortions on area-under-the-curve (AUC) metric calculations, and we identified the advantages of using PH-SP and proposed new metrics when characterizing FUS spectral properties. The results show many second, third and sub-harmonic metrics provide cross-consistent evidence of changes between stable and inertial cavitation levels. Removing the first harmonic signal component with a hardware low-pass filter allowed the hydrophone gain to be boosted without introducing distortion, leading to an improved analysis of the sub-harmonic signal orders of magnitude smaller in intensity. Metrics that optimized the energy in the real component of the complex-valued PH-SP spectra provided a 32% increase in the sub-harmonic sensitivity compared to standard metrics.
Conclusion
A preliminary investigation of existing and proposed metrics showed that system noise could be large enough to mask the transition between stable and inertial cavitation. Strong narrowing of sub-harmonic peak shapes on applying physical sparsification (PH-SP) were seen in both phantom and animal studies. However, validating equivalent trends of the metrics with pressure were limited by the increased system noise level in the animal study combined with the natural variability between subjects studied. The combined use of hardware low-pass filters and physical sparsification to selectively removing distortions in the spectrum allowed the optimization of metrics for cavitation monitoring by improving the sub-harmonic sensitivity.
]. An increase in BBB permeability can be achieved by acoustic cavitation induced by low-intensity, burst-type ultrasonic waves combined with microbubbles, resulting in focused ultrasound (FUS) as a potential strategy for non-invasive, reversible BBB disruption [
]. A key issue is how to ensure that a safe, adequate and controllable FUS pressure level is generated in medical equipment designed to assist the passage of therapeutic drugs through the BBB [
]. This requires identifying when the pressure level generated by the ultrasound transducer causes stable cavitation with sufficient acoustic energy to safely open the BBB without inducing inertial cavitation leading to vessel damage [
Current cavitation level metrics involve monitoring the changes in the area under the curve (AUC) of a narrow frequency bandwidth around one of the sub- or ultraharmonics of the transmitter's excitation frequency (see, e.g., Lapin et al. [
]). The hydrophone signal can contain second or higher harmonics of the FUS excitation frequency. One limitation of analyzing these higher-frequency signals is their attenuation through the skull in in vivo studies, leading to our suggestion for a single metric that combines multiple metrics with complementary signal characterization. In addition, these higher harmonics may arise through the coupling media or tissue [
]. Therefore, it must be determined whether these harmonics contain valid information on cavitation levels that can be combined with other metrics.
A recent study revealed that removing the initial hydrophone startup transient removed components that lowered the overall signal-to-noise ratio (SNR) of the FUS spectrum. However, an order-of-magnitude widening/narrowing of the bandwidth of the harmonically related FUS signals could be seen with minor, less than ±0.015%, changes in the length of the transient signal digitally removed from the time signal [
]. The source of this spectral sensitivity to minor experimental conditions is directly related to the well-known distortions that occur when the discrete Fourier transform (DFT) is used to generate spectra from time domain signals gnerated in many research fields. These distortions commonly appear as Gibbs’ ringing amplitude artefacts together with peak shape changes and intensity loss [
]. However, the effect has potentially a greater impact when higher-resolution spectra are generated by transforming the orders-of-magnitude longer-length (>300,000 points) FUS time signal.
Theoretically, no distortions, that is, infinitely sharp spectra, would be produced if the experimental time domain signal could be described in terms of a minimal number of DFT basis functions. Components in a typical time domain signal do not meet this criterion and must be described as a sum of multiple DFT basis functions, leading to a wide distorted spectrum in an effect described as “spectral leakage” [
] reported that such idealized spectra could be generated when it was possible to apply physical sparsification (PH-SP)—an extension of an idea applied to optimize mechanical vibration spectral studies [
]. They determined that the harmonic structure of FUS spectra met the criterion for allowing the deliberate PH-SP manipulation of the captured hydrophone data to best match the DFT basis function criterion. This will lead to the minimization of spectral leakage, leaving sharp FUS spectra with no distortions.
After an overview of experimental FUS procedures and detailing of a range of spectral distortions found during a phantom study, we review fundamental DFT properties and how they might be manipulated through digital PH-SP to improve specific FUS spectral characteristics. We propose techniques that optimize the real component of the FUS frequency data to reject half the noise power incorporated into any AUC calculation from the complex-valued spectra. We compare existing FUS AUC metrics with those making use of PH-SP and evaluate their ability to characterize changes in the cavitation environment of an FUS phantom model and a preliminary animal study for different transducer power levels. A comparison is made between the results from a phantom study and a preliminary animal study.
Methods
Experimental phantom study equipment
A sample of Definity (Lantheus, Billerica, MA, USA) contrast agent was activated through 45 s of Vialmix agitation (Lantheus), then diluted in saline solution 1:3 and placed in a pipette bulb at the focus of the confocal FUS transducer hydrophone assembly (RK50, FUS Instruments, Toronto, ON, Canada). A series of 10-ms pulses at the first harmonic (1,420 kHz) frequency are generated every 1 s for 10 s and sent to a transducer [
]. As illustrated schematically in Figure 1, this produces ultrasound pulses that excite the phantom containing lipid microsphere ultrasound contrast agent. The bubble response is captured by a hydrophone and sampled at 32-ns intervals to produce a time domain signal of 312,500 points. Various signal processing steps are applied to this signal using MATLAB [
] before it is transformed into the frequency domain via the DFT implemented using the fast Fourier transform (FFT) algorithm. Full details of the procedure used in our experimental phantom FUS studies can be found in other reports [
Microbubbles within the phantom non-linearly expand and contract in response to the ultrasound excitation. In our study, the hydrophone spectrum was found to contain subharmonic, second and third harmonic peaks respectively at half, twice and thrice the fundamental frequency of the ultrasound emitted by the transducer. These signals are several orders of magnitude smaller in intensity than the first harmonic signal. No ultraharmonics at odd multiples of the subharmonic frequency were seen. Abrupt microbubble collapse occurred at higher power levels, and the associated shock waves appeared as broadband, noise-like, emissions to the spectrum, which can be monitored by changes in the AUC measurements ofthe peaks and the frequencies between the peaks [
The animal study data were generated on the RK-50 pre-clinical system at the University of Texas Southwestern Medical Center, Dallas. All animal experiments were performed by Dr. I Youssef, Dr. B. Shah and Dr. R. Chopra of the University of Texas Southwestern Medical Center, Dallas, with the approval of the Institutional Animal Care and Use Committee of the University of Texas Southwestern Medical Center. All methods and protocols were carried out in accordance with the guidelines set forth in the Guide for the Care and Use of Laboratory Animals. Female Swiss Webster mice (Charles River, Wilmington, MA, USA) aged 3 to 4 mo were randomly selected and divided into three groups: group I at 0.28 MPa, group II at 0.35 MPa and group III at 0.42 MPa. Each animal was anesthetized initially through inhalation using a mixture of 1.5%–3% isoflurane and 1 L/min Medical Air. Hair over the cranial surface of the skull was removed using an animal clipper and depilatory cream (VEET sensitive formula, Reckitt Benckiser, Parsippany, NJ, USA). By use of a sterile technique, a 1- to 2-cm incision was made on the skull to visualize bregma and lambda skull suture landmarks. Ultrasound gel was added to the animal's skull, and a tank filled with de-ionized, de-gassed water was lowered onto the ultrasound gel between the transducer and the skull. Definity microbubbles were activated, diluted in saline and administered through the tail veil (1.5 μL of Definity added to 50 mL of preservative-free saline delivered at a rate between 4 and 10 mL/min.
Spectral distortion under low-pressure excitation
It is important that other spectral features not mask the subtle changes in the spectrum as power levels change from inducing stable cavitation to inertial cavitation capable of damaging the BBB. Noise related to possible errors introduced by analog-to digital (A/D) sampling is a signal processing accuracy concern [
]. The important subharmonic used to monitor cavitation levels has an intensity several orders of magnitude smaller than that of the first harmonic, meaning that it is measured with a greater level of quantization error for a given B-bit A/D [
]. This effect is illustrated by the filtered signal's amplitude (Fig. 2b) being 35 times smaller than that of the unfiltered signal (Fig. 2a). The true length of the hydrophone starting transient impacting the lower intensity subharmonic becomes obvious after removing the first harmonic.
Figure 2Both the (a) unfiltered and (b) filtered hydrophone (intensity scaled by 35) responses at 0.28 MPa exhibit evidence of a startup transient. The length of the hydrophone transient, at least 0.8 ms, becomes more obvious in (b) after the intensity of the first harmonic response filtered out.
Figure 3a illustrates that the lower frequency range of the spectrum already contains broadband noise at the low 0.14 MPa pressure level, at which no broadband inertial noise is expected to have been produced. Smith et al. [
] found that digitally removing the strong initial hydrophone transient samples seen in both unfiltered and filtered time signals (Fig. 2) led to a higher SNR spectrum (Fig. 3b). Arrows in Figure 3a and 3b indicate unexpected repeated peaks, at 59.70 ± 0.05 kHz intervals, with a potential for overlapping and distorting the key FUS harmonics. Simulation studies [
] revealed that a series of peaks separated by a harmonic of 10 kHz can be generated through A/D quantization errors associated with repeated synchronous sampling of signals with multiple strong harmonic components. Another potential cause of these peaks is detailed during the discussion of the animal study results.
Figure 3(a) There is the appearance of a wide range of broadband noise close to the 710 kHz sub-harmonic even at the low 0.14 MPa excitation pressure. (b) This noise is significantly reduced when the initial 1.0 ms of the time signal is deleted. The arrows indicate the presence of harmonically related peaks approximately 60 kHz apart, potentially overlapping the focused ultrasound harmonics.
DFT and correcting spectral distortions introduced after transient removal
This section outlines the theoretical underpinnings of why the shape of the FUS harmonic signals, and hence its AUC determination, can significantly change following minor different experimental decisions on the length of the transient to be removed, the sampling frequency or the excitation frequency. Approaches to remove/minimize these distortions are discussed.
Figure 4a illustrates the first harmonic spectrum for a 0.28 MPa phantom study with a wide distorted spectral peak when a transient length of 1.000352 ms is removed. The smooth envelope of the absolute frequency values used in AUC calculations hides the considerable DFT distortions present in the real (red) and imaginary (blue) spectral components. The alternative experimental decision to remove a transient of length 1.000704 ms, a 0.035% change, gives an ideal, very narrow PH-SP peak (Fig. 4b), for all absolute, real and imaginary frequency interpretations.
Figure 4(a) Experimentally assuming that the signal length must be reduced by 1.000352 ms to remove the transient leads to a broad bandwidth for the first harmonic spectrum after taking the absolute value of the discrete Fourier transform distortions in the real (red) and imaginary (blue) components. (b) Removing a 1.000704 ms transient length, a 0.35% change, introduces a physical sparsification (PH-SP) condition producing a narrow peak exactly at 1420 kHz.
To explain this behavior, let FUSSUB[nΔT], 0 ≤ n < n, represent the N points of an FUS subharmonic sampled at a time interval ΔT satisfying the Nyquist criterion of there being no experimental signal or noise components with frequencies greater than FMAX = 1/2ΔT. Applying the DFT
(1)
generates the discrete FUS spectrum, FUSSUB[kΔF], –N/2 ≤ k < N/2, evaluated at frequency locations separated by steps of ΔF = 2FMAX/N. The term W[nΔT], 0 ≤ n < N, corresponds to a window that can be applied to the time domain signal. Windowing data can be used to advantage to achieve compromises on how DFT distortions affect a spectrum [
]. For example, an unmodified time signal, inherently a rectangular window W[nΔT] = 1, produces Gibbs’ ringing. This is a sinc function envelope with a central peak and strong side lobes spread over a wide frequency range seen in Figure 4a. In contrast, using Hann (Hanning), Hamming or Blackman–Harris window shapes produces broader central peaks but a lower bandwidth for their weakened side-lobe amplitude distortions.
Physical sparsification
Rather than introducing the known side effects of data windowing, Smith et al. [
] took advantage of a fundamental property of the DFT to reduce artefacts. By definition, the N DFT basis functions, DFTk(i) = exp(–j2πki/N), are orthogonal to each other. This orthogonality property guarantees a sharp delta function–like peak in the frequency domain for each component of a time signal that is identical to a basis function.
It is unlikely that any given signal from an arbitrary research study will meet this strict orthogonality requirement. However, an extension to a DFT artefact removal concept detailed by Smith [
] can be applied to the harmonically related FUS components to deliberately customize them as DFT basic functions. This modified approach, PH-SP, is achieved by digitally reducing the number of sampled points from N to N – S prior to generating the DFT spectrum. This changes the spectrum's frequency spacing to ΔFS = 2FMAX/(N – S). A custom choice of S to make the ratio FSUB/ΔFS an integer causes the shortened version of FSUB[nΔT] to become identical to a DFT basis function in the new (N – S) DFT space. This specific choice of S to sparsify the subharmonic spectrum also ensures that the related first, second and third harmonic ratios, F1/ΔFS, FULTRA/ΔFS, F2/ΔFS, and ultraharmonics if present, become integers. This means that all FUS components become sparsified, sharp single peaks undistorted by DFT artefacts as illustrated by the customized decision to remove one of many specific PH-SP transient lengths, 1.000704 ms, in Figure 4b.
Theoretical and practical considerations for metrics for cavitation level determination
Functional ultrasound frequency values will have a complex format R(f) + jI(f) after DFT transformation. AUC metrics can be evaluated from the sum of the absolute magnitude values
(2)
over a given bandwidth (BW) around a centre frequency point fc. Removing the transient modifies the DFT frequency resolution from ΔFS = 2FMAX/N to ΔFS = 2FMAX/(N – S). As an example, the original 100-Hz resolution changes to 111.1 and 104.2 Hz, respectively, for our phantom and animal studies following removal of a 1-ms and a 0.4-ms transient from a 10-ms burst. To compensate, the standard 300 Hz bandwidth needs to be increased to ensure that at least three digital values are summed during the AUC calculation. Other bandwidths considered were 75 Hz to evaluate an AUC containing only the maximum frequency amplitude near the harmonic together with larger AUC bandwidths when considering broadened bandwidths present during windowing or anti-PHSP processing.
Making use of magnetic resonance imaging (MRI) concepts, Smith et al. [
] indicated that both components of the frequency values are independently noisy so that the actual absolute magnitude value is determined by ABS(AMPLITUDE) = . This implies that an AUC value could be obtained with an decrease in the noise level by using the absolute value of the real component of the frequency value ABS(REAL) = .
As with FUS spectra, MR images are also generated by taking the magnitude of the complex valued transform of the captured k-space frequency values [
]. In that field, it is known that taking the magnitude of noisy values introduces a hidden bias as the experimental Gaussian broadband noise signal with zero mean changes to a noisy Rician distribution with a non-zero mean [
]. Translating this into an FUS context, the non-zero noise mean will have the potential to inherently increase AUC values. We suggest that a possible AUC metric be calculated directly from the real FUS component without applying the absolute operation. This has the advantage of generating a summation where the positive and negative noise components can be averaged out/reduced without introducing a non-zero mean.
] proposed using FUS phantom studies over a wide pressure range to optimize FUS metrics in a more controllable approach that avoided basic testing of metrics with animals. They hypothesized that the frequency resolution obtained when transforming for 312,500 points captured at 32 ns was too low to obtain good AUC values around a harmonic, given only 3 digitized points are present in a 300 Hz bandwidth. This problem was overcome by combining windowing with Fourier interpolation, zero padding, to increase the displayed frequency resolution [
]. A hidden advantage of windowing the time domain signal is that it automatically removes the hydrophone's starting transient and the distortion that causes throughout the spectrum.
] suggested that signal processing characteristics could be improved by inserting a hardware low-pass filter between the hydrophone and the A/D to remove the strong first harmonic (Fig. 1). This would allow any automated gain control to avoid A/D signal overload while becoming more sensitive to changes in the subharmonic signal's intensity.
In this article, a number of new metrics and experiments are discussed to extend that analysis.
] found that the subharmonic amplitude, and hence its AUC, changed with each cycle within a 10-cycle burst at a given pressure. They reported a mean and standard deviation calculated from the AUC values of the individual bursts. We generate the mean AUC value for each proposed metric in a similar manner. However, we estimated the error using a leave-one-out approach. This requires that the measurements be considered as coming from 10 virtual experiments, each consisting of a different selection of 9 of the available 10 cycle AUC values, effectively studying the impact on the average if one AUC measurement of 10 was experimentally corrupted . The variability of the averages calculated from each of the 10 virtual experiments, that is their standard deviation, was considered as a better estimate of the true experimental error than the standard deviation of the points from the original mean.
•
A number of studies, for example, O'Reilly et al. [
], suggest avoiding first and second harmonic AUC metrics because of potential contamination through a number of experimental sources. We hypothesize an improvement is available in the reliability of these metrics by making use of a proposed harmonic generation efficiency metric (AUC/MPa) [
] combined with the two new real component AUC determinations. Any direct transmitter/hydrophone leakage would now be identifiable as a flat linear portion in rate-of-change AUC/MPa-versus-pressure plots.
•
FUS metrics are explored in both phantom and animal studies with and without the hardware filter used to remove the strong first harmonic.
New approaches using physical sparsification metrics
The frequency components associated with the hydrophone's transient were shown to affect a wide frequency range of the phantom study data. Smith et al. [
] determined that PH-SP could be used to generate FUS spectrum peaks with narrower bandwidths by removing the transient in such a way that the components in the shortened time domain signal could individually be described as DFT basis functions. A number of metrics are explored that make use of this PH-SP concept, for example, optimizing a metric requiring all the harmonic energy to be in one major real component of the spectrum rather than spread across a wide bandwidth. However, given that FUS harmonics are noisy, we also evaluated whether a contrary approach of deliberately widening the signal employing anti-PH-SP procedures would provide greater sensitivity to cavitation level changes.
The following is an abbreviated and modified version of the steps proposed by Smith et al. [
] to correctly select the number of points that needed to be removed to generate a PH-SP spectrum for any given time signal with harmonic components. They suggested that a root-two SNR improvement could be achieved by applying PH-SP and then measuring AUC from the absolute-real component to reject the noise present in the imaginary frequency components. However, to optimize the energy in the real component requires the additional steps 5 and 6 to take into account that different cycles will have unrelated starting phases of each FUS harmonic leading to changes in their relative real and imaginary component intensities.
1. Use an existing procedure to capture N sample points with the A/D gain adjusted to maximize the FUS amplitude while ensuring neither signal nor initial transient is distorted before sampling.
2. Determine the length of time, R, before a particular harmonic intensity value is measured a second time, that is, occurring after Q samples in a time R = QΔT, with Q an integer.
3. The custom physical sparsification data length is TCUSTOM = PQΔT, with P the largest integer satisfying PQΔT < T, and T the length of the available FUS time N-point signal without initial starting transients.
4. Generate a PQ-point DFTPQ using the custom physical sparsification data length of the FUS data.
5. To optimize the real component's energy for a harmonic at frequency f, multiply the complex frequency components by the phase correction term exp(–2π tan–1 (Im(DFTPQ(f))/ Re(DFTPQ(f)))) where DFTPQ(f) is the result of step 4 at frequency f. The whole spectrum was multiplied by –1 for ease of comparison with positive absolute metrics if the maximum real component was negative after correction.
6. Reliable control of the appearance of the real and imaginary components is achieved with PQ even (not necessarily a power of 2) and application of the fftshift() operator in Matlab [
The following describes one example of meeting these criteria applied for our current experimental procedure which acquired N = 312,500 time points sampled at 32-ns intervals (RK-50, FUS Instruments). A 10-ms burst contains 7100 periods of a 710-kHz subharmonic. Thus, the impact of removing the transient could be investigated in PH-SP steps of 0.2 ms, corresponding to dual even and integer values of 142 periods containing 6,250 samples. Selection of a maximum anti-PH-SP condition requires that the time signal sans transient contain precisely an extra half-cycle in addition to an integer number of harmonic cycles. In principle, future experiments might obtain this condition exactly via a minor adjustment, less than 0.03%, in the A/D sampling frequency. An equivalent tuning of the excitation frequency to better match the hydrophone frequency response would offer advantages when background “system noise” is an experimental concern.
Results
Phantom AUC metrics based on broadband noise
Figure 5a and 5b illustrate the spectrum, before and after transient removal, respectively, for a phantom study at 1.120 MPa when a large component of broadband noise from inertial cavitation will be present. Applying a hardware filter (Fig. 5c), with a 1-MHz cutoff (blue line) reduces the first harmonic's intensity at 1,420 kHz from 6.7 to 0.05, a factor of 134. However, the reduction in the noise level below the filter's high-frequency cutoff of 1 MHz is theoretically unexpected. This result is a high-pressure validation of the low-pressure observation by Smith et al. [
] that allowing the gain of the system to be controlled by the smaller subharmonic intensity is important in bringing the true hydrophone FUS response up above the impact of any background “system noise.”
Figure 5Signal and noise level (a) before and (b) after transient removal. (c)The further reduction in noise level after application of the 1-MHz low-pass filter indicates the presence of significant background “system noise.” (d) Both unfiltered (solid blue line) and filtered (solid red line) show a significant increase in noise level after 0.60 MPa. (e) The flat response in the AUC/MPa efficiency measure versus pressure indicates that the noise level is increasing linearly with pressure prior to 0.60 MPa, a response not obvious in the plot of AUC versus pressure in (d). ABS, absolute; AUC, area under the curve.
]. There is no theoretical SNR advantage to modify the excitation of the transmitter to generate a subharmonic closer to the hydrophone's 850 kHz peak. This is because signal and noise intensities at a specific frequency are equally changed by the Q-factor, frequency sensitivity, of the hydrophone. An exception would be when modifying the subharmonic frequency to better match the Q-factor peak would increase the hydrophone's response above the background system noise.
Figure 5d compares the noise-only AUC values across a 200 kHz bandwidth for unfiltered (solid blue line) and filtered (solid red line) responses. The larger error bars correspond to the standard deviations of the AUC values across all 10 FUS bursts at a given pressure level. Both responses show a strong increase in AUC values around the 0.6 MPa level. Using the absolute real value (dashed line) reduces the noise-only AUC value by a factor of nearly root-2 as predicted theoretically, while using the real value leads to a near-zero AUC measurement (dot-dashed lines) with the noise being summed. The flat response of the proposed rate-of-change AUC/MPa efficiency measure in Figure 5e indicates that the noise level is increasing linearly with pressure prior to 0.6 MPa. We suggest that all AUC/MPa results at 0.017 MPa be considered as unreliable as they are potentially associated with a divide-by-near-zero-MPa calculation.
First, second and third harmonic phantom AUC metrics
Prior to the insertion of the low-pass filter, there are strong first, second and third harmonics present in the phantom data. Figure 6 illustrates that applying the ABS(REAL) metric to data containing both a FUS harmonic signal and noise is more complex than when applied to noise-only data (Fig. 5). Figure 6a and 6b illustrate that there are different relative intensities for the real and imaginary components for cycles 1 and 7 in the same 10-burst study of the second harmonic at 2,840 kHz. This is an indirect result of there being no deliberate synchronization between the starting phase of individual 1,420 kHz FUS transmitter excitations and when the second-harmonic hydrophone responses are sampled by the A/D. The signal energy can be concentrated in the real component of an individual cycle (Fig. 6c, 6d), by using the corrective phase factor detailed in steps 5 and 6 `of the PH-SP procedure outlined previously.
Figure 6(a) Cycle 7 and (b) cycle 1 of this 0.42-MPa phantom's second harmonic study have different relative intensities of their real (red) and imaginary (black) components because of differences in the starting phases of their excitation burst. Applying an appropriate phase correction to each cycle's frequency spectrum assists in concentrating the energy into the real component of the focused ultrasound signals for both (c) cycle 7 and (d) cycle 1. IMAG, imaginary.
Rephasing is particularly effective when the FUS burst length is precisely 10 ms as this satisfies PH-SP conditions to sharpen a harmonic component for an excitation frequency of precisely 1,420 kHz. This implies that even a minor decrease in data length to remove a short transient or a small drift in excitation frequency could significantly change spectral characteristics as in Figure 4. The accuracy of the rephrasing correction will be limited by the amount of noise at a given harmonic frequency. We consider there is currently insufficient evidence to conclude that the early negative peaks seen in the phase-corrected real components of both second harmonic spectra (Fig. 6c, 6d) are related to the subharmonic FUS spectral side lobes reported in Khan et al. [
Figure 7 illustrates the application of the AUC versus MPa metric to these harmonics. The error bars were calculated using the leave-one-out procedure discussed earlier. Without phase correction, the ABS(REAL) (dashed red) and REAL (dashed blue) versions of these metrics are inconsistent with the other metrics shown. We interpret the initial linear increase in the AUC metric with changes in pressure in Figure 7a as corresponding to a leakage signal [
], implying there is no useful cavitation information characterized within the intensity changes in the first harmonic.
Figure 7Results for AUC versus MPa metrics for the (a) first, (b) second and (c) third harmonics from a phantom FUS study. Without phase correction, the ABS (REAL) (dashed blue) and REAL (dashed red) metric variants are inconsistent with the standard ABS metric (solid black line). The first harmonic AUC increasing linearly with pressure until 0.6 MPa is attributed to direct cross-coupling between the transmitter and hydrophone and is not a true response from the phantom. The metrics for the second and third harmonics reveal a non-linear response with pressure, providing evidence that they are true responses from the microbubbles in the excited phantom. ABS, absolute; AUC, area under the curve; FUS, focused ultrasound.
In contrast, the upward curved slope of the AUC-versus-pressure (MPa) graph (Fig. 7b) indicates a faster than linear increase in the second harmonic signal strength with pressure before 0.60 MPa. The fact that the ABS(AMPLITUDE) measure (solid black) remains higher than the ABS(REPHASED REAL) metric (solid red) indicates that the second harmonic's contribution to the AUC is falling faster than the broadband noise intensity is increasing. All metrics in Figure 7c exhibit an initial slow increase in third harmonic's intensity until 0.3 MPa, with a more rapid increase until 0.6 MPa. After that their response to broadband noise levels cause the ABS(REAL), ABS(REPHASED REAL) and REPHASED REAL variants of this metric to diverge.
The harmonic efficiency rate-of-change AUC/pressure versus pressure (MPa) metrics in Figure 8 provide an alternative rate-of-change interpretation of the characteristics of the first, second and third harmonics with pressure. All metrics have a gradient slightly less than 1 for the linear section of Figure 8a. This indicates that the increases in first harmonic intensity are initially smaller than the pressure increases. The overlap of the AUC metrics at all bandwidths indicates that the first harmonic intensity initially dominates over any broadband noise associated with inertial cavitation. The strong slope of the initial second-harmonic rate-of-change AUC/MPa metrics and their similarity at all bandwidths (Fig. 8b) correspond to a quadratic increase with pressure up until around 0.42 MPa. Differences between the ABS(REAL), ABS(REPHASED REAL) and REPHASED REAL metrics after that pressure are indicative of the increased role broadband noise is beginning to play. The wider initial spread of the third-harmonic metrics indicates that the third-harmonic signal is weak, allowing broadband noise effects to be more significant. We offer no physical interpretation for the observation that although the rate of change in the second harmonic increases before and decreases after 0.42 MPa (Fig. 8c), the rate of change in the third harmonic (Fig. 4) decreases then increases. Such differences in characteristics provide the potential for a useful combination of these metrics to be used for automatic control of cavitation levels.
Figure 8Harmonic efficiency AUC/MPa versus MPa metrics for the (a) first, (b) second and (c) third harmonics for bandwidths of 75 Hz (dashed line), 330 Hz (solid line) and 660 Hz (dotted line). The flat response of this pressure normalized metric below 0.60 MPa again indicates the first harmonic increasing linearly with pressure until 0.6 MPa, attributed to direct cross-coupling between the transmitter and hydrophone. This metric makes the rates of change of the second and third harmonic intensities more obvious than the standard AUC measure (cf.Fig. 7). ABS, absolute; AUC, area under the curve; BW, bandwidth.
] determined a metric's reliability in terms of the standard deviation of that metric applied to individual burst cycles at a given pressure. The smaller error bars on the representative, overlaid black metrics are calculated using our leave-one-out approach detailed earlier.
Figure 9(a) The presence of a small subharmonic signal means that the unfiltered subharmonic AUC metrics based on a 300-Hz bandwidth exhibit a slightly faster response to increased pressure than the noise-only metrics determined using a 200 kHz bandwidth (as in Fig. 5d). In contrast, there is a distinct characteristic change at 0.28 MPa in the filtered subharmonic metrics as proposed by Khan et al.
. (b) The differences in the rate of change of signal characteristics between unfiltered and filtered data are clearly demonstrated in the efficiency AUC/MPa metrics. The overlaid black lines are examples of using a leave-one-out approach when calculating metric inaccuracies rather than the standard deviation across multiple cycles. (c) The AUC/MPa metric appears larger for the ABS(AMPLITUDE) (black line) for a 300 Hz bandwidth measurement and no transient removal. However, after transient removal of 1.007504 ms and 1.000000 ms, respectively, the (d) PH-SP and (e) anti-PH-SP metrics based on the REPHASED REAL component exhibit 32% and 23% increases, respectively, in their sensitivity to spectral changes prior to the key 0.42 MPa pressure. ABS, absolute; AUC, area under the curve; BW, bandwidth; PH-SP, physical sparsification.
Figure 9a illustrates that the unfiltered AUC metrics (dashed line) for a small bandwidth around the subharmonic increase at a slightly faster rate with pressure than the noise-only metrics in Figure 5d. We attribute this difference to the presence of a small sub-harmonic peak barely above the noise level (see figures in Smith et al. [
]). In contrast, there is a rapid change in the AUC metric characteristics between 0.14 and 0.42 MPa (solid line) after filtering the hydrophone signal to make the subharmonic a greater component of the A/D gain control. The efficiency rate-of-change AUC/MPa metric (Fig. 9b) again empathizes these metric changes which occur in a location similar to those seen with the second and third harmonics in Figures 7 and 8.
Theoretically, applying a window to the time signal before transforming will reduce the broadband spectral leakage around high-resolution peaks [
]. There is an additional advantage in a FUS context in that windowing inherently reduces/removes any effect that a starting transient might have on the frequency spectrum. The disadvantage of windowing is a broadened central peak width, hence the need to measure over a broader 1,200-Hz bandwidth detailed in Khan et al. [
]. This metric again shows the same strong peak at 0.28 MPa as the non-windowed data. Interpolating the frequency spectrum by zero-padding the time data before transforming suggested in references [
Phantom subharmonic metrics after transient removal
Figures 3 and 5 illustrate that reductions in the distortions in the frequency spectrum are achieved with the removal of the hydrophone starting transient before applying the DFT. The transient removal conditions used for the subharmonic study were 1.007504 ms and 1.003000 ms for the PH-SP and anti-PH-SP conditions, respectively. Other transient removal lengths have been reported to provide AUC results between these extremes [
]. Three AUC bandwidths were selected—120, 360 and 1,006 Hz—experimentally determined to contain 1, 3 and 9 frequency points, respectively, for both PH-SP and anti-PH-SP analyses.
Figure 9c illustrates that the use of the standard metric, ABS(AMPLITUDE) (black line), with a 300-Hz bandwidth and no transient removal [
] generates the greatest maximum AUC value at the key 0.28 MPa pressure. However, the harmonic relationship between the main FUS components has placed the following hidden complications into comparing AUC metrics after deleting the transient distorted portion of the time signal to improve spectral SNR:
•
Different decisions on how to best remove the transient can change the shape of a harmonic peak (see Fig. 4).
•
The PH-SP condition for optimally removing the transient to determine metrics using the narrowest first harmonic peak at 1,420 kHz (Fig. 4b) is the anti-PH-SP condition generating the broadest subharmonic peak at 710 kHz.
•
In addition, the anti-PH-SP condition for the broadest first harmonic peak (Fig. 4a) is one of the possible PH-SP conditions for the narrowest second harmonic peak at 2,840 kHz.
•
The standard approach of measuring AUC values from a bandwidth around a harmonic becomes problematic as only a PH-SP condition generates a value at the harmonic frequency, all other data lengths cause the maximum value to move away from the excitation frequency (cf.Fig. 4a and 4b).
•
Modifying the length of signal transformed changes both the resolution and position of the digitized frequency values. This results in an AUC calculated from a bandwidth of 330 Hz located at the first harmonic frequency to be derived from three PH-SP frequency values but four anti-PH-SP points in Figure 4, making AUC larger. This will require an adjustment in how the AUC is calculated from eqn (2) to take into account the different number of points.
•
A further AUC adjustment will be required as each of the four anti-PH-SP point contributes more to the AUC value because of the increased weighting of their digitized point spacing, ΔF, associated with a shorter transformed data length.
To overcome these issues, we propose a new metric, Norm-AUC/MPa, normalized to 1 at a specific experimental pressure. For the phantom studies we have chosen to identify the differences in metric sensitivity as determined by their relative changes before the key experimental pressure, 0.42 MPa, identified by its minimum AUC level for all metrics in Figure 9a and 9b for the filtered data. The proposed normalized rate-of-change AUC/MPa measurements for the PH-SP (Fig. 9d) and anti-PHSP (Fig. 9e) analyses indicate that the REAL measure (brown line) using a bandwidth of 120 Hz to include the subharmonic's maximum value and 1 ms of transient removed has the greatest sensitivity, 32% and 23% respectively, to spectral changes prior to the key 0.42 MPa compared with standard approaches. In addition to its greater sensitivity, we also recommend the selection of PH-SP conditions to sharpen/reduce a given harmonic's bandwidth to mitigate the possible impact on the AUC metrics of the many nearby distortion peaks from background “system noise” seen in the phantom (Fig. 3) and animal (Fig. 10b) studies.
Figure 10(a) The unfiltered time signal from the 0.35 MPa animal study reveals many distortions associated with hydrophone gain issues. Various time domain transients generate a wide band of sharp spikes (arrows) in (b) the log10() frequency spectrum, with (c) spikes larger than, and close to, the subharmonic peak. No evidence of second or higher harmonics was present. (d) The filtered response exhibits less evidence of gain control issues over a series of burst cycles with (e) spikes produced by time domain distortions less evident above the 1 MHz filter cutoff. (f) While the subharmonic is enhanced after filtering, the reduction in background “system noise” is smaller than seen with the phantom (cf.Fig. 5).
Time, frequency and broadband noise responses from an animal study
We discuss the different experimental issues found between a limited set of unfiltered and filtered animal FUS studies provided by the laboratory at the Southwestern Medical Center, Dallas, and our laboratory's phantom environment. Both laboratories used a confocal RK-50 transmitter/hydrophone. A comparison was made of the relative sensitivity of the standard and proposed metrics to changes in the animal frequency spectrum with FUS pressure levels.
Conditions are more difficult to control in animal studies than with a phantom. Some of the new experimental issues are seen in the time signal (Fig. 10a) from a 0.35-MPa animal study. Each of the first 3 cycles in the 10-cycle burst has a different appearance, with details more clearly seen in the black lines representing the same burst cycle zoomed × 150 in time. The first cycle burst response is fully amplitude clipped. The second cycle has only the extreme values of a repeated transient signal clipped with the third cycle signal having minimal signal distortions. Other cycles showed a variation between clipped and unclipped transients.
The transient signals are repeated at approximately 41.08-µs intervals (23.908 kHz) and generate a broad spectrum of spikes (Fig. 10b) surrounding a strong first harmonic (1,556 kHz). There is no clear evidence of any significant subharmonic (778 kHz) or second harmonic (3,112 kHz) in the log() display of unfiltered data. However, a subharmonic of intensity 0.013 can be seen in Figure 10c. This is a magnitude smaller compared with the first harmonic intensity of 4.08 than the 0.25-to-6.7 ratio seen with the phantom study (Fig. 5).
] proposed that using a low-pass filter (0 to 1 MHz passband) to remove the strong first harmonic would improve the signal processing characteristics of the weaker subharmonic in a phantom study. Figure 10d illustrates the time signal amplitude following filtering in the animal study. Unlike the unfiltered time signals, the time signals of all cycles of the 0.35 MPa filtered animal data appear undistorted by amplitude clipping or transients.
When transformed, the first harmonic is difficult to observe in the log() display (Fig. 10e), as at 1,556 kHz it is further past the 1 MHz filter cutoff than the phantom's first harmonic at 1,420 kHz (cf.Fig. 5c). Although the filtered time signal exhibits little indication of the transients seen in the unfiltered data, their presence as spikes is still observable in the spectrum. The fact that they are reduced above the 1 MHz filter cutoff relative to Figure 10b suggests that they are part of the true signal from the hydrophone response in this study rather than just a part of this system's background “system noise.”
In the phantom study (Fig. 2b), the gain was controlled by the initial transient, which was several times larger than the filtered signal. The animal signal transient is shorter than the phantom's transient. However, the gain is not as well adjusted to provide the optimum SNR for the subharmonic as the transient has a higher intensity relative to the filtered animal signal. This means that while filtering reduces the first-harmonic amplitude by three orders of magnitude, 106.5 to 103.4, the background noise amplitude changes by a factor of only 5, 103.5 to 102.7. This indicates that system broadband noise is still dominating this animal study in comparison with the phantom study. This conclusion is also supported by comparing the unfiltered and filtered spectra in Figure 10c and 10f. Figure 10f illustrates a small × 3 enhancement of the subharmonic peak at 778 kHz. There is a spectral noise peak of similar intensity at 781 kHz whose own DFT distortions could have an impact on the animal study's AUC metric accuracy.
Discussion
Although performed in a different laboratory environment, Figure 11a illustrates the same variations of subharmonic intensity with cycle number at a fixed pressure, 0.35 MPa, as reported for the phantom study [
]. An anti-PH-SP spectrum results when a 0.397824 ms transient is removed from the hydrophone's 10 ms received signal. The 778 kHz peaks of all 10 cycles have a broad base from spectral leakage, affecting the AUC calculations. Figure 11b illustrates that digitally removing a 0.401024 ms transient from the same data set, a 0.8% difference, generates a PH-SP spectrum. The subharmonic peaks across the 10 cycles now have a higher intensity relative to the noise as all the harmonic's spectral energy is concentrated into a narrow bandwidth, echoing the phantom first harmonic PH-SP results illustrated in Figure 4.
Figure 11(a) Removing 0.397824 ms of the transient generates an anti-PH-SP spectrum for all 10 cycles of the 0.35 MPa animal study which exhibit broadened subharmonic peaks. (b) Digitally removing a longer 0.401024 ms transient length from the same data generates a PH-SP spectrum with high intensity, narrower peaks echoing the changes seen in the phantom first harmonic study. (c) There is not a clear trend with the un-normalized AUC with pressure. (d) There is an upward growth of the unfiltered AUC/MPa animal (dashed line) reflecting that seen with the unfiltered phantom data. As with the filtered phantom data, there is a peak with the filtered animal data (solid line) with the proposed metrics not seen with the standard metrics. (e) The peak becomes more evident when the change in AUC per cycle is modeled with a linear fit (dashed line)
The trends are unclear when plotting the un-normalized AUC value versus pressure in Figure 11c. Figure 11d compares the AUC/MPa efficiency metric for the unfiltered (dotted lines) and filtered (solid lines) time data for (i) the standard ABS(AMPLITUDE) calculation using a 360 Hz bandwidth and no transient removal (coloured brown) and the proposed (ii) ABS(REPHASED REAL) (coloured red) and (iii) REPHASED REAL metrics (colored black), using a 120 Hz bandwidth and transient removal using a PH-SP specific transient length. The upward slope of the unfiltered AUC metric and the decrease at higher pressures of the filtered AUC metric are similar to the trends for the phantom metrics (cf.Fig. 9). In Figure 11d the decrease at higher pressures is stronger when the variation in AUC metrics within a 10-burst experiment is modelled as a linear change (dashed line), as suggested in [
], rather than the standard average (solid line). Generating the normalized AUC/MPa metrics was considered inappropriate given the limited size of data.
Strong peak narrowing was evident on the peaks of all cycles after applying PH-SP pre-processing (Fig. 11b). However, we are concerned with how the trends of the AUC metrics with pressure have been affected by the increased “system noise” level compared with the phantom study and the natural variability between the data from individual animals. The use of a separate amplifier to boost the hydrophone signal before filtering and sampling might provide an approach to increase confidence in future limited experiments by ensuring that major “system noise” components are not overpowering harmonics present on the hydrophone signal. In addition, we suggest that the following key difference between the broadband noise characteristics of the animal and phantom systems could provide an approach to use to identify hydrophones with an appropriate FUS harmonic response. The phantom subharmonic (Fig. 4a–c) is surrounded by a strong broadband noise response illustrating that both signal and noise are being equally enhanced by the hydrophone's frequency response. In contrast, Figure 10f illustrates that there is no opportunity for the second study's subharmonic to rise above the background “system noise” as its 778-kHz frequency is not situated within any of the several frequency bands enhanced by that system's hydrophone resonances.
Practical considerations
The phantom studies (Fig. 3, arrows) at Laboratory 1 revealed a series of background noise spikes separated in frequency by approximately 60 kHz. A more complex series of spikes with frequency separations of 60 and 120 kHz (Fig. 10, arrows) are evident in the spectra of the animal study at Laboratory 2 on a different RK50 pre-clinical system. The amplitude of the spikes is largest (Fig. 10b, solid arrows) when the time signal (Fig. 10a, blue) is strongly distorted, but becomes smaller (Fig. 10b, dash-dot arrows) when the A/D overloading is reduced (Fig. 10a, orange). Only the spikes with a separation of 60 kHz are dominant after filtering the animal data (Fig. 10e) and occur over a much wider frequency range than in the phantom study (Fig. 3). We have illustrated through simulation [
] that false peaks can be generated through A/D quantization errors. However, a more detailed investigation has recently revealed that the spikes in the phantom spectrum were reduced, but not removed, by switching off the motor positioning drivers on our RK50 system [
]. These spikes will not affect spectral analysis provided signal post-processing techniques, such as physical sparsification, optimize the sharpness of peaks of interest so that they do not broaden and overlap, for example, the true animal subharmonic peak at 778 kHz and the false peak at 781 kHz (Fig. 10e).
One reviewer asked about “the feasibility of the AUC/MPa metric in pre-clinical and clinical settings when the in situ pressure is unknown due to distortion and attenuation as ultrasound passes through the skull bone.” We expect that protocols and metrics related to pressure will already be in place for a given study to handle the fact that the lack of knowledge of the exact in situ pressure is a problem in any FUS study. An example protocol might be a metric using changes in a proxy pressure estimate based on the use of a table relating the voltage applied to a given experimental transducer and the water pressure changes generated during calibration. We are suggesting that a more sensitive metric would be obtained if (i) the existing protocol was modified to look at the rate of change of a metric with the proxy pressure estimate with (ii) multiple complementary metrics combined.
Conclusions
Focused ultrasound (FUS) can be used to excite the expansion and contraction of microbubbles to induce the BBB to open and allow passage of therapeutic drugs. Changes in the DFT spectrum of the FUS time signal have been investigated with respect to their use to distinguish between power levels that create safe stable or damaging inertial cavitation.
The 321,500 points sampled in a 10 ms FUS excitation can generate high-resolution spectra. It has been found that the characteristics of the well-known DFT-induced ringing distortions can be significantly changed by making minor modification (0.03%) in standard experimental procedures, such as how much of a starting transient is digitally deleted from the FUS time signal samples before transforming. The associated change in the width of spectral peaks affects typical AUC metrics. Using a controllable phantom study, we report under what conditions a new technique, PH-SP, can deliberately use this effect to enhance the sensitivity of FUS metrics.
Given FUS spectra are complex valued, standard AUC metrics are based on summing the absolute values of a band of frequencies near a FUS harmonic. Advantages are found by combining PH-SP concepts with optimization of the real spectral component to achieve a root-two or more reduction in the noise power incorporated withing an AUC metric, improving its sensitivity to cavitation level changes. New AUC metrics are introduced to characterize the efficiency with which certain power levels excite harmonics. Strong PH-SP effects, sharpening, narrowing and FUS peak values were seen with both phantom and animal studies. Similar trends were also seen with the AUC versus pressure metrics. However, additional animal studies are needed given the natural variability between individuals and the observed increased “system noise” level.
New subharmonic, second harmonic and third harmonic metrics were found to be self-consistent in their identification of cavitation level changes. The new real-component, PH-SP subharmonic metric provides a 32% increase in sensitivity to cavitation level changes compared with the standard AUC metric. Future work will involve identifying how the characteristics of these different metrics can combined into a single approach to control the power level within an automated medical FUS device to safely open the BBB. In addition, an investigation is underway on how to adapt the application of PH-SP improvements to the spectral analysis of nanobubble contrast agent's response to a shorter transient excitation.
Data availability
The MPS-to-MAT file conversion and AUC analysis tools used in this study, together with sample data, are available by contacting the corresponding author.
Conflict of interest
The authors declare no competing interests.
Acknowledgments
The authors thank Dr. I Youssef, Dr. B. Shah and Dr. R. Chopra of the University of Texas Southwestern Medical Center, Dallas, for providing the animal study data; Dr. C. Pellow, University of Calgary, for helping to clarify concept explanations; and Mitacs International intern undergraduate D. Nandi, Jadavpur University, West Bengal, India, for assistance in decoding the new MPS file formats. The authors thank the reviewers for their suggestions on improving the clarity of text and figures.
This work was supported in part by a Natural Sciences and Engineering Research Council of Canada Discovery Grant (L.C.), by an Analog Devices University Ambassadorship for teaching and research (M.R.S.) and by the University of Calgary.
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