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

Abstract

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 beam width decreases. For sufficiently high harmonic frequency, beam width 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 μm and 400 μm). Harmonic radial profiles were measured with transverse scans in the focal plane using the fiberoptic 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 beam width parameter σ n varied with harmonic number n according to a power law, σ n = σ 1 /n q where 0.5 < q < 0.6. 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% ± 1.0%) and (pcp: 28.3% ± 9.6% prp: 6.0% ± 2.4% pii: 24.3% ± 6.7%) for the 100 μm and 400 μm diameter hydrophones respectively. These errors can be suppressed by application of the inverse spatial averaging filter.

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

“…The value for a eff ( f ) chosen for each frequency and each hydrophone was the value that minimized the root mean square difference (RMSD) between the experimental directivity and model directivity (1) over angles from −30° to 30°. This angular range optimized the directivity model for beams with angular spectra mostly confined to | θ | < 30°, which can accommodate transducers with f-numbers ≥ 1 [7, 8, 37]. The relative difference between effective and nominal geometrical sensitive element radii Δaag=aeff−agag was fit to a power-law function of ka g , Δaag=C(kag)n The power law fit parameters were obtained from linear fits to log-transformed data.…”

AAPM Task Group 241: A medical physicist’s guide to MRI‐guided focused ultrasound body systems

“…Measurement accuracy depends on the hydrophone effective active element radius being comparable with or smaller than one‐quarter of the effective wavelength of the ultrasound in water to avoid spatial averaging errors, at least for the fundamental frequency 48 . However, even if this condition is not met or if the pressure wave has significant harmonic content (as is commonly true), an inverse filter method exists for correcting for spatial averaging loss across the hydrophone sensitive element for the fundamental and all harmonics 52 . (Note that the harmonic beamwidth decreases with frequency 52 ).…”

“…However, even if this condition is not met or if the pressure wave has significant harmonic content (as is commonly true), an inverse filter method exists for correcting for spatial averaging loss across the hydrophone sensitive element for the fundamental and all harmonics 52 . (Note that the harmonic beamwidth decreases with frequency 52 ). To account for the fact that the intensity is highly localized and the ultrasound may be pulsed, the IEC/TS 62556 report lists several different intensity parameters, including: I SAL : the spatial average intensity, linear conditions.…”

Review of field characterization techniques for high intensity therapeutic ultrasound

“…The spatial averaging corrections can be estimated by integrating the pressure over the effective diameter of the hydrophone [69][70][71][72], or calculated from the empirical relationship of the ratio of the measured harmonic acoustic beam width and the effective diameter of the hydrophone [73]. Assuming a quadratic [74] or Gaussian [60,75] profile distribution for the pressure, the spatial averaging model can be simplified, and the spatial averaging can be corrected by the inverse technique [74,75]. Notably, the aforementioned directional response is commonly considered only a directional modulus response.…”