Reconstruction of nonlinear ultrasound field of an annular therapeutic array from acoustic holograms of its individual elements

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.

"HIFU Beam:" A Simulator for Predicting Axially Symmetric Nonlinear Acoustic Fields Generated by Focused Transducers in a Layered Medium

"HIFU beam" is a freely available software tool that comprises a MATLAB toolbox combined with a user-friendly interface and binary executable compiled from FORTRAN source code (HIFU beam. (2021). Available: http://limu.msu.ru/node/3555?language=en). It is designed for simulating high-intensity focused ultrasound (HIFU) fields generated by single-element transducers and annular arrays with propagation in flat-layered media that mimic biological tissues. Numerical models incorporated in the simulator include evolution-type equations, either the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation or one-way Westervelt equation, for radially symmetric ultrasound beams in homogeneous and layered media with thermoviscous or power-law acoustic absorption. The software uses shock-capturing methods that allow for simulating strongly nonlinear acoustic fields with high-amplitude shocks. In this article, a general description of the software is given along with three representative simulation cases of ultrasound transducers and focusing conditions typical for therapeutic applications. The examples illustrate major nonlinear wave effects in HIFU fields including shock formation. Two examples simulate propagation in water, involving a single-element source (1-MHz frequency, 100-mm diameter, 90-mm radius of curvature) and a 16-element annular array (3-MHz frequency, 48-mm diameter, and 35-mm radius of curvature). The third example mimics the scenario of a HIFU treatment in a "water-muscle-kidney" layered medium using a source typical for abdominal HIFU applications (1.2-MHz frequency, 120-mm diameter, and radius of curvature). Linear, quasi-linear, and shock-wave exposure protocols are considered. It is intended that "HIFU beam" can be useful in teaching nonlinear acoustics; designing and characterizing high-power transducers; and developing exposure protocols for a wide range of therapeutic applications such as shock-based HIFU, boiling histotripsy, drug delivery, immunotherapy, and others.

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.