A computationally efficient model capable of simulating finite-amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described. It is based on a second-order operator splitting approach [Tavakkoli et al., J. Acoust. Soc. Am. 104, 2061-2072 (1998)], with a fractional step-marching scheme, whereby the effects of diffraction, attenuation, and nonlinearity can be computed independently over incremental steps. This approach is an extension to that of Christopher and Parker [J. Acoust. Soc. Am. 90, 507-521; 90, 488-499 (1991)], wherein linear and nonlinear effects are propagated separately over incremental steps, and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources. Results of the model are compared with published data. Predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented.

Modeling of nonlinear ultrasound propagation in tissue from array transducers

Wave backpropagation is a concept that can be used to calculate the excitation signals for an array with programmable transmit waveforms to produce a specified field that has no significant evanescent wave components. This concept can also be used to find the field at a distance away from an aperture based on measurements made in the aperture. For a uniform medium, three methods exist for the calculation of wave propagation and backpropagation: the diffraction integral method, the angular spectrum method, and the shift-and-add method. The boundary conditions that are usually implicitly assumed by these methods are analyzed, and the relationship between these methods are explored. The application of the angular spectrum method to other kinds of boundary conditions is discussed, as is the relationship between wave backpropagation, phase conjugation, and the time-reversal mirror. Wave backpropagation is used, as an example, to calculate the excitation signals for a ring transducer to produce a specified pulsatile plane wave with a limited spatial extent.

Propagation and backpropagation for ultrasonic wavefront design

New algorithms are described that provide insight into linear field propagation and offer significant reductions in computational complexity. The developments presented here include the usage of a recently developed discrete Hankel transform to implement two single step, planar propagation algorithms for baffled, radially symmetric, acoustic pressure or velocity fields; an update on the single step approaches that reduce computational complexity through geometrically determined spatial frequency limitations; and algorithms for extending to multistep propagation. Two equivalent means of introducing arbitrary medium attenuation into the above schemes are presented. Finally, a planar boundary crossing algorithm that accounts for refraction and reflection (but not multiple reflections) is added to one of the multistep propagating algorithms. The resulting algorithm is then used to examine the differences between the corresponding fields of a focused piston source operating in water and in a layered fat/liver (biomedical imaging) medium. The results yield computationally efficient algorithms that can be used for linear propagation of focused or unfocused beams in attenuating, multilayer media, and also provide the basis for a novel nonlinear propagation algorithm.

New approaches to the linear propagation of acoustic fields

Ultrasonic focusing in two dimensions has been investigated by calculating the propagation of ultrasonic pulses through cross-sectional models of human abdominal wall and breast. Propagation calculations used a full-wave k-space method that accounts for spatial variations in density, sound speed, and frequency-dependent absorption and includes perfectly matched layer absorbing boundary conditions. To obtain a distorted receive wavefront, propagation from a point source through the tissue path was computed. Receive focusing used an angular spectrum method. Transmit focusing was accomplished by propagating a pressure wavefront from a virtual array through the tissue path. As well as uncompensated focusing, focusing that employed time-shift compensation and time-shift compensation after backpropagation was investigated in both transmit and receive and time reversal was investigated for transmit focusing in addition. The results indicate, consistent with measurements, that breast causes greater focus degradation than abdominal wall. The investigated compensation methods corrected the receive focus better than the transmit focus. Time-shift compensation after backpropagation improved the focus from that obtained using time-shift compensation alone but the improvement was less in transmit focusing than in receive focusing. Transmit focusing by time reversal resulted in lower sidelobes but larger mainlobes than the other investigated transmit focus compensation methods.

Simulation of ultrasonic focus aberration and correction through human tissue

Measurements of temperature elevations induced by sonications in a single intact cadaver skull filled with soft-tissue mimicking phantom material were performed using magnetic resonance thermometry. The sonications were done using a clinical transcranial ultrasound therapy device operating at 230 kHz and the measurements were compared with simulations done using a model incorporating both the longitudinal and shear wave propagation. Both the measurements and simulations showed that in some situations the temperature increase could be higher in the phantom material adjacent to the skull-base than at the focus, which could lead to undesired soft-tissue damage in treatment situations. On average the measurements of the sonicated locations, as well as the comparative simulations, showed 32 ± 64% and 49 ± 32% higher temperature elevations adjacent to the skull-base than at the focus, respectively. The simulation model was used to extend the measurements by simulating multiple sonications of brain tissue in five different skulls with and without correcting the aberrations caused by the skull on the ultrasound. Without aberration correction the closest sonications to the skulls that were treatable in any brain location without undesired tissue damage were at a distance of 19.1 ± 2.6 mm. None of the sonications beyond a distance of 41.2 ± 5.3 mm were found to cause undesired tissue damage. When using the aberration correction closest treatable, safe distances for sonications were found to be 16.0 ± 1.6  and 38.8 ± 3.8 mm, respectively. New active cooling of the skull-base through the nasal cavities was introduced and the treatment area was investigated. The closest treatable distance without aberration correction reduced to 17.4 ± 1.9 mm with the new cooling method. All sonications beyond a distance of 39.7 ± 6.6 mm were found treatable. With the aberration correction no difference in the closest treatable or the safety distance was found in comparison to sonications without nasal cavity cooling. To counteract undesired skull-base heating a new anti-focus within solid media was developed along with a new regularized phasing method. Mathematical bases for both the methods and simulations utilizing them were presented. It was found that utilizing the anti-focus in solid media and regularized phasing, the fraction of temperature increase of the brain tissue at the focus and the peak temperature increase adjacent to the skull-base can be increased from 1.00 to 1.95. This improves the efficiency of the sonication by reducing the energy transfer to the skull-base.

https://iopscience.iop.org/article/10.1088/0031-9155/56/15/003/pdf

Simulations and measurements of transcranial low-frequency ultrasound therapy: skull-base heating and effective area of treatment.

Ultrasonic fields produced by a disk transducer have been measured in two planes normal to the transducer axis and then extrapolated from one measurement plane to the other and also back to the face of the transducer. The measurements were performed with a small PVF hydrophone and a quadrature detector which yielded kthe complex values of the pressure at regularly spaced points in the scanned planes. Wave field extrapolation was carried out digitally using Fourier transformation methods which characterize the propagation of a field from one plane to another as a linear filtering operation. The filtering process was implemented as a product of the angular spectrum of the measured pressure field and a propagation matrix comprised of phase factors computer from parameters which included the wavelength of the measured field, the distance over which the field was to be extrapolated, and a maximum spatial frequency. The extrapolated field was then obtained by inverse Fourier transformation of the angular spectrum-phase matrix product. Comparisons of extrapolation results with measured data and with theoretical predictions show reasonable agreement and also demonstrate the sensitivity of the extrapolation process to the extent and shape of the spatial frequency window employed to determine the non-zero elements in the phase matrix. The results indicate fields can be extrapolated more accurately by sampling higher than the Nyquist rate and extending the size of the plane over which the measurements are made.

J. Ridder

Papers

Cross-Sectional Measurements and Extrapolations of Ultrasonic Fields