Directivity and Frequency-Dependent Effective Sensitive Element Size of Membrane Hydrophones: Theory Versus Experiment

Quantitative photoacoustic tomography: modeling and inverse problems

Significance

Quantitative photoacoustic tomography (QPAT) exploits the photoacoustic effect with the aim of estimating images of clinically relevant quantities related to the tissue's optical absorption. The technique has two aspects: an acoustic part, where the initial acoustic pressure distribution is estimated from measured photoacoustic time-series, and an optical part, where the distributions of the optical parameters are estimated from the initial pressure.

Aim

Our study is focused on the optical part. In particular, computational modeling of light propagation (forward problem) and numerical solution methodologies of the image reconstruction (inverse problem) are discussed.

Approach

The commonly used mathematical models of how light and sound propagate in biological tissue are reviewed. A short overview of how the acoustic inverse problem is usually treated is given. The optical inverse problem and methods for its solution are reviewed. In addition, some limitations of real-life measurements and their effect on the inverse problems are discussed.

Results

An overview of QPAT with a focus on the optical part was given. Computational modeling and inverse problems of QPAT were addressed, and some key challenges were discussed. Furthermore, the developments for tackling these problems were reviewed. Although modeling of light transport is well-understood and there is a well-developed framework of inverse mathematics for approaching the inverse problem of QPAT, there are still challenges in taking these methodologies to practice.

Conclusions

Modeling and inverse problems of QPAT together were discussed. The scope was limited to the optical part, and the acoustic aspects were discussed only to the extent that they relate to the optical aspect.

Nominal Versus Actual Spatial Resolution: Comparison of Directivity and Frequency-Dependent Effective Sensitive Element Size for Membrane, Needle, Capsule, and Fiber-Optic Hydrophones

Frequency-dependent effective sensitive element radius [Formula: see text] is a key parameter for elucidating physical mechanisms of hydrophone operation. In addition, it is essential to know [Formula: see text] to correct for hydrophone output voltage reduction due to spatial averaging across the hydrophone sensitive element surface. At low frequencies, [Formula: see text] is greater than geometrical sensitive element radius a_g . Consequently, at low frequencies, investigators can overrate their hydrophone spatial resolution. Empirical models for [Formula: see text] for membrane, needle, and fiber-optic hydrophones have been obtained previously. In this article, an empirical model for [Formula: see text] for capsule hydrophones is presented, so that models are now available for the four most common hydrophone types used in biomedical ultrasound. The [Formula: see text] value was estimated from directivity measurements (over the range from 1 to 20 MHz) for five capsule hydrophones (three with [Formula: see text] and two with [Formula: see text]). The results suggest that capsule hydrophones behave according to a "rigid piston" model for k a _g ≥ 0.7 ( k = 2π /wavelength). Comparing the four hydrophone types, the low-frequency discrepancy between [Formula: see text] and a_g was found to be greatest for membrane hydrophones, followed by capsule hydrophones, and smallest for needle and fiber-optic hydrophones. Empirical models for [Formula: see text] are helpful for choosing an appropriate hydrophone for an experiment and for correcting for spatial averaging (over the sensitive element surface) in pressure and beamwidth measurements. When reporting hydrophone-based pressure measurements, investigators should specify [Formula: see text] at the center frequency (which may be estimated from the models presented here) in addition to a_g .

Hydrophone Measurements for Biomedical Ultrasound Applications: A Review

This article presents basic principles of hydrophone measurements, including mechanisms of action for various hydrophone designs, sensitivity and directivity calibration procedures, practical considerations for performing measurements, signal processing methods to correct for both frequency-dependent sensitivity and spatial averaging across the hydrophone sensitive element, uncertainty in hydrophone measurements, special considerations for high-intensity therapeutic ultrasound, and advice for choosing an appropriate hydrophone for a particular measurement task. Recommendations are made for information to be included in hydrophone measurement reporting.

Dissemination of the Acoustic Pascal: The Role and Experiences of a National Metrology Institute

Hydrophones are pivotal measurement devices ensuring medical ultrasound acoustic exposures comply with the relevant national and international safety criteria. These devices have enabled the spatial and temporal distribution of key safety parameters to be determined in an objective and standardized way. Generally based on piezoelectric principles of operation, to convert generated voltage waveforms to acoustic pressure, they require calibration in terms of receive sensitivity, expressed in units of [Formula: see text]Pa-1. Reliable hydrophone calibration with associated uncertainties plays a key role in underpinning a measurement framework that ensures exposure measurements are comparable and traceable to internationally agreed units, irrespective of where they are carried out globally. For well over three decades, the U.K. National Physical Laboratory (NPL) has provided calibrations to the user community covering the frequency range 0.1-60 MHz, traceable to a primary realization of the acoustic pascal through optical interferometry. Typical uncertainties for sensitivity are 6%-22% (for a coverage factor k = 2), degrading with frequency. The article specifically focuses on the dissemination of the acoustic pascal through NPL's calibration services that are based on a comparison with secondary standard hydrophones previously calibrated using the NPL primary standard. The work demonstrates the stability of the employed dissemination protocols by presenting representative calibration histories on a selection of commercially available hydrophones. Results reaffirm the guidance provided within international standards for regular calibration of a hydrophone in order to underpin measurement confidence. The process by which internationally agreed realizations of the acoustic pascal are compared and validated through key comparisons (KCs) is also described.

Spatiotemporal Deconvolution of Hydrophone Response for Linear and Nonlinear Beams-Part I: Theory, Spatial-Averaging Correction Formulas, and Criteria for Sensitive Element Size

This article reports spatiotemporal deconvolution methods and simple empirical formulas to correct pressure and beamwidth measurements for spatial averaging across a hydrophone sensitive element. Readers who are uninterested in hydrophone theory may proceed directly to Appendix A for an easy method to estimate spatial-averaging correction factors. Hydrophones were modeled as angular spectrum filters. Simulations modeled nine circular transducers (1-10 MHz; F/1.4-F/3.2) driven at six power levels and measured with eight hydrophones (432 beam/hydrophone combinations). For example, the model predicts that if a 200- [Formula: see text] membrane hydrophone measures a moderately nonlinear 5-MHz beam from an F/1 transducer, spatial-averaging correction factors are 33% (peak compressional pressure or p_c ), 18% (peak rarefactional pressure or p ), and 18% (full width half maximum or FWHM). Theoretical and experimental estimates of spatial-averaging correction factors to were in good agreement (within 5%) for linear and moderately nonlinear signals. Criteria for maximum appropriate hydrophone sensitive element size as functions of experimental parameters were derived. Unlike the oft-cited International Electrotechnical Commission (IEC) criterion, the new criteria were derived for focusing rather than planar transducers and can accommodate nonlinear signals in addition to linear signals. Responsible reporting of hydrophone-based pressure and beamwidth measurements should always acknowledge spatial-averaging considerations.

Spatiotemporal Deconvolution of Hydrophone Response for Linear and Nonlinear Beams-Part II: Experimental Validation

This article reports experimental validation for spatiotemporal deconvolution methods and simple empirical formulas to correct pressure and beamwidth measurements for spatial averaging across a hydrophone sensitive element. The method was validated using linear and nonlinear beams transmitted by seven single-element spherically focusing transducers (2-10 MHz; F /#: 1-3) and measured with five hydrophones (sensitive element diameters d_g : 85-1000 [Formula: see text]), resulting in 35 transducer/hydrophone combinations. Exponential functions, exp( -αx ), where x = d_g /( λ₁F /#) and λ₁ is the fundamental wavelength, were used to model focal pressure ratios p'/p (where p' is the measured value subjected to spatial averaging and p is the true axial value that would be obtained with a hypothetical point hydrophone). Spatiotemporal deconvolution reduced α (followed by root mean squared difference between data and fit) from 0.29-0.30 (7%) to 0.01 (8%) (linear signals) and from 0.29-0.40 (8%) to 0.04 (14%) (nonlinear signals), indicating successful spatial averaging correction. Linear functions, Cx + 1, were used to model FWHM'/FWHM, where FWHM is full-width half-maximum. Spatiotemporal deconvolution reduced C from 9% (4%) to -0.6% (1%) (linear signals) and from 30% (10%) to 6% (5%) (nonlinear signals), indicating successful spatial averaging correction. Spatiotemporal deconvolution resulted in significant improvement in accuracy even when the hydrophone geometrical sensitive element diameter exceeded the beam FWHM. Responsible reporting of hydrophone-based pressure measurements should always acknowledge spatial averaging considerations.

Localized Measurement of a Sub-Nanosecond Shockwave Pressure Rise Time

In a growing number of applications, fast and localized pressure measurement in aqueous media is desired. To perform such measurements, a custom-made single-mode fiber-optic probe hydrophone (FOPH) was designed and used to measure the pressure pulse generated by laser-induced breakdown (LIB) in water. The sensor enabled sub-nanosecond pressure rise time measurement. Both the rise time and the duration of the shockwave were found to be shorter in the direction perpendicular to the breakdown generating laser beam, compared to the shockwave observed in the parallel direction. Simultaneous high-frame-rate imaging was used to qualitatively validate the novel hydrophone data and to observe the shockwave evolution. The measurements were performed also on pressure pulses emitted during the generation of miniature ( [Formula: see text] diameter) laser-induced bubbles at very small distances (down to [Formula: see text]), further demonstrating the capabilities of the small-size sensor and the ability to measure locally. The results improve understanding of LIB shockwave characteristics dependence on laser pulse energy and duration.

Hydrophone Spatial Averaging Correction for Acoustic Exposure Measurements From Arrays-Part I: Theory and Impact on Diagnostic Safety Indexes

This article reports underestimation of mechanical index (MI) and nonscanned thermal index for bone near focus (TIB) due to hydrophone spatial averaging effects that occur during acoustic output measurements for clinical linear and phased arrays. TIB is the appropriate version of thermal index (TI) for fetal imaging after ten weeks from the last menstrual period according to the American Institute of Ultrasound in Medicine (AIUM). Spatial averaging is particularly troublesome for highly focused beams and nonlinear, nonscanned modes such as acoustic radiation force impulse (ARFI) and pulsed Doppler. MI and variants of TI (e.g., TIB), which are displayed in real-time during imaging, are often not corrected for hydrophone spatial averaging because a standardized method for doing so does not exist for linear and phased arrays. A novel analytic inverse-filter method to correct for spatial averaging for pressure waves from linear and phased arrays is derived in this article (Part I) and experimentally validated in a companion article (Part II). A simulation was developed to estimate potential spatial-averaging errors for typical clinical ultrasound imaging systems based on the theoretical inverse filter and specifications for 124 scanner/transducer combinations from the U.S. Food and Drug Administration (FDA) 510(k) database from 2015 to 2019. Specifications included center frequency, aperture size, acoustic output parameters, hydrophone geometrical sensitive element diameter, etc. Correction for hydrophone spatial averaging using the inverse filter suggests that maximally achievable values for MI, TIB, thermal dose ( t ₄₃ ), and spatial-peak-temporal-average intensity ( [Formula: see text]) for typical clinical systems are potentially higher than uncorrected values by (means ± standard deviations) 9% ± 4% (ARFI MI), 19% ± 15% (ARFI TIB), 50% ± 41% (ARFI t ₄₃ ), 43% ± 39% (ARFI [Formula: see text]), 7% ± 5% (pulsed Doppler MI), 15% ± 11% (pulsed Doppler TIB), 42% ± 31% (pulsed Doppler t ₄₃ ), and 33% ± 27% (pulsed Doppler [Formula: see text]). These values correspond to frequencies of 3.2 ± 1.3 (ARFI) and 4.1 ± 1.4 MHz (pulsed Doppler), and the model predicts that they would increase with frequency. Inverse filtering for hydrophone spatial averaging significantly improves the accuracy of estimates of MI, TIB, t ₄₃ , and [Formula: see text] for ARFI and pulsed Doppler signals.

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

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

Correction for Hydrophone Spatial Averaging Artifacts for Circular Sources

This article reports an investigation of an inverse-filter method to correct for experimental underestimation of pressure due to spatial averaging across a hydrophone sensitive element. The spatial averaging filter (SAF) depends on hydrophone type (membrane, needle, or fiber-optic), hydrophone geometrical sensitive element diameter, transducer driving frequency, and transducer F number (ratio of focal length to diameter). The absolute difference between theoretical and experimental SAFs for 25 transducer/hydrophone pairs was 7% ± 3% (mean ± standard deviation). Empirical formulas based on SAFs are provided to enable researchers to easily correct for hydrophone spatial averaging errors in peak compressional pressure ( p_c ), peak rarefactional pressure ( p_r ), and pulse intensity integral. The empirical formulas show, for example, that if a 3-MHz, F /2 transducer is driven to moderate nonlinear distortion and measured at the focal point with a 500- [Formula: see text] membrane hydrophone, then spatial averaging errors are approximately 16% ( p_c ), 12% ( p_r ), and 24% (pulse intensity integral). The formulas are based on circular transducers but also provide plausible upper bounds for spatial averaging errors for transducers with rectangular-transmit apertures, such as linear and phased arrays.

100 MHz bandwidth planar laser-generated ultrasound source for hydrophone calibration

High-frequency calibration of hydrophones is becoming increasingly important, both for clinical and scientific applications of ultrasound, and user safety. At present, the calibrations available routinely to the user community extend to 60 MHz. However, hydrophones that can measure beyond this are available, and ultrasonic fields often contain energy at higher frequencies, e.g., generated through nonlinear propagation of high-amplitude ultrasound used for therapeutic applications, and the increasing use of higher frequencies in imaging. Therefore, there is a need for calibrations up to at least 100 MHz, to allow ultrasonic fields to be accurately characterized, and the risk of harmful bioeffects to be properly assessed. Currently, sets of focused piezoelectric transducers are used to meet the pressure amplitude and bandwidth requirements of Primary Standard calibration facilities. However, when the frequency is high enough such that the size of the ultrasound focus becomes less than the hydrophone element's diameter, the uncertainty due to spatial averaging becomes significant, and can be as high as 20% at 100 MHz. As an alternate to piezoelectric transducers, a laser-generated ultrasound calibration source was designed, fabricated, and characterized. The source consists of an optically absorbing carbon-polymer nanocomposite excited by a large-diameter 1064 nm laser pulse of 2.6 ns duration. Peak pressure amplitudes of several Mega-Pascal were readily achievable, and the signal contained measurable frequency components up to 100 MHz. The variation in the pressure amplitudes was less than 2% from its mean over a three-hour test period. The ultrasound beam was sufficiently broad that the uncertainties due to spatial averaging were negligible.

Solvability for Photoacoustic Imaging With Idealized Piezoelectric Sensors

Most reconstruction algorithms for photoacoustic imaging assume that the pressure field is measured by the ultrasound sensors placed on a detection surface. However, such sensors do not measure pressure exactly due to their nonuniform directional and frequency responses, and resolution limitations. This is the case for piezoelectric sensors that are commonly employed for photoacoustic imaging. In this article, using the method of matched asymptotic expansions and the basic constitutive relations for piezoelectricity, we propose a simple mathematical model for piezoelectric transducers. The approach simultaneously models how the pressure waves induce the piezoelectric measurements and how the presence of the sensors affects the pressure waves. Using this model, we analyze whether the data gathered by the piezoelectric sensors lead to the mathematical solvability of the photoacoustic imaging problem. We conclude that this imaging problem is well posed in certain normed spaces and under a geometric assumption. We also propose an iterative reconstruction algorithm that incorporates the model for piezoelectric measurements. A numerical implementation of the reconstruction algorithm is presented.

Hydrophone Spatial Averaging Artifacts for ARFI Beams from Array Transducers

This paper reports underestimation of peak compressional pressure (p _c), peak rarefactional pressure (p _r ), and pulse intensity integral (pii) due to hydrophone spatial averaging of acoustic radiation force impulse (ARFI) beams generated by clinical linear and phased arrays. Although a method exists for correcting for hydrophone spatial averaging for circularly-symmetric beams, there is presently no analogous method for rectangularly-symmetric beams generated by linear and phased arrays. Consequently, pressure parameters (p _c, p _r , and pii) from clinical arrays are often not corrected for spatial averaging. This can lead to errors in Mechanical Index (MI) and Thermal Index (TI), which are derived from pressure measurements and are displayed in real-time during clinical ultrasound scans. ARFI beams were generated using three clinical linear array transducers. Output pressure waveforms for all three transducers were measured using five hydrophones with geometrical sensitive element sizes (d_g) ranging from 85 to 1000 μm. Spatial averaging errors were found to increase with hydrophone sensitive element size. For example, if d_g = 500 μm (typical membrane hydrophone), frequency = 2.25 MHz and F/# = 1.5, then average errors are approximately -20% (p_c), -10% (p_r), and -25% (pii). Therefore, due to hydrophone spatial averaging, typical membrane hydrophones can exhibit significant underestimation of ARFI pressure measurements, which likely compromises exposure safety indexes.

Correction for Spatial Averaging Artifacts for Circularly-Symmetric Pressure Beams Measured with Membrane Hydrophones

This paper investigates experimental underestimation of pressure measurements due to spatial averaging across a hydrophone sensitive element. Empirical relationships are measured to enable correction for hydrophone spatial averaging errors in peak compressional pressure (p _c ), peak rarefactional pressure (p _r ), and pulse intensity integral (pii). The empirical relationships show, for example, that if a 3-MHz, F/2 transducer is driven to moderate nonlinear distortion and measured at the focal point with a 500-μm membrane hydrophone, then spatial averaging errors are approximately 16% (p _c ), 12% (p _r ), and 24% (pii).

Correction for Spatial Averaging Artifacts in Hydrophone Measurements of High-Intensity Therapeutic Ultrasound: An Inverse Filter Approach

High-intensity therapeutic ultrasound (HITU) pressure is often measured using a hydrophone. HITU pressure waves typically contain multiple harmonics due to nonlinear propagation. As harmonic frequency increases, harmonic beamwidth decreases. For sufficiently high harmonic frequency, beamwidth may become comparable to the hydrophone effective sensitive element diameter, resulting in signal reduction due to spatial averaging. An analytic formula for a hydrophone spatial averaging filter for beams with Gaussian harmonic radial profiles was tested on HITU pressure signals generated by three transducers (1.45 MHz, F/1; 1.53 MHz, F/1.5; 3.91 MHz, F/1) with focal pressures up to 48 MPa. The HITU signals were measured using fiber-optic and needle hydrophones (nominal geometrical sensitive element diameters: 100 and [Formula: see text]). Harmonic radial profiles were measured with transverse scans in the focal plane using the fiber-optic hydrophone. Harmonic radial profiles were accurately approximated by Gaussian functions with root-mean-square (rms) differences between transverse scans and Gaussian fits less than 9% for frequencies up to approximately 50 MHz. The Gaussian harmonic beamwidth parameter σ_n varied with harmonic number n according to a power law, σ_n = σ₁/n^q where . RMS differences between experimental and theoretical spatial averaging filters were 11% ± 1% (1.45 MHz), 8% ± 1% (1.53 MHz), and 4% ± 1% (3.91 MHz). For the two more highly focused (F/1) transducers, the effect of spatial averaging was to underestimate peak compressional pressure (pcp), peak rarefactional pressure (prp), and pulse intensity integral (pii) by (mean ± standard deviation) (pcp: 4.9% ± 0.5%, prp: 0.4% ± 0.2%, pii: 2.9% ± 1%) and (pcp: 28.3% ± 9.6%, prp: 6% ± 2.4%, pii: 24.3% ± 6.7%) for the 100- and 400- [Formula: see text]-diameter hydrophones, respectively. These errors can be suppressed by the application of the inverse spatial averaging filter.