A k-space method for moderately nonlinear wave propagation.

A k-space method for moderately nonlinear wave propagation in absorptive media is presented. The Westervelt equation is first transferred into k-space via Fourier transformation, and is solved by a modified wave-vector time-domain scheme. The present approach is not limited to forward propagation or parabolic approximation. One- and two-dimensional problems are investigated to verify the method by comparing results to analytic solutions and finite-difference time-domain (FDTD) method. It is found that to obtain accurate results in homogeneous media, the grid size can be as little as two points per wavelength, and for a moderately nonlinear problem, the Courant-Friedrichs-Lewy number can be as large as 0.4. Through comparisons with the conventional FDTD method, the k-space method for nonlinear wave propagation is shown here to be computationally more efficient and accurate. The k-space method is then employed to study three-dimensional nonlinear wave propagation through the skull, which shows that a relatively accurate focusing can be achieved in the brain at a high frequency by sending a low frequency from the transducer. Finally, implementations of the k-space method using a single graphics processing unit shows that it required about one-seventh the computation time of a single-core CPU calculation.

Time-reversal transcranial ultrasound beam focusing using a k-space method.

This paper proposes the use of a k-space method to obtain the correction for transcranial ultrasound beam focusing. Mirroring past approaches, a synthetic point source at the focal point is numerically excited, and propagated through the skull, using acoustic properties acquired from registered computed tomography of the skull being studied. The received data outside the skull contain the correction information and can be phase conjugated (time reversed) and then physically generated to achieve a tight focusing inside the skull, by assuming quasi-plane transmission where shear waves are not present or their contribution can be neglected. Compared with the conventional finite-difference time-domain method for wave propagation simulation, it will be shown that the k-space method is significantly more accurate even for a relatively coarse spatial resolution, leading to a dramatically reduced computation time. Both numerical simulations and experiments conducted on an ex vivo human skull demonstrate that precise focusing can be realized using the k-space method with a spatial resolution as low as only 2.56 grid points per wavelength, thus allowing treatment planning computation on the order of minutes.

On the use of Gegenbauer reconstructions for shock wave propagation modeling.

In therapeutic ultrasound, the presence of shock waves can be significant due to the use of high intensity beams, as well as due to shock formation during inertial cavitation. Although modeling of such strongly nonlinear waves can be carried out using spectral methods, such calculations are typically considered impractical, since accurate calculations often require hundreds or even thousands of harmonics to be considered, leading to prohibitive computational times. Instead, time-domain algorithms which generally utilize Godunov-type finite-difference schemes are commonly used. Although these time domain methods can accurately model steep shock wave fronts, unlike spectral methods they are inherently unsuitable for modeling realistic tissue dispersion relations. Motivated by the need for a more general model, the use of Gegenbauer reconstructions as a postprocess tool to resolve the band-limitations of the spectral methods are investigated. The present work focuses on eliminating the Gibbs phenomenon when representing a steep wave front using a limited number of harmonics. Both plane wave and axisymmetric 2D transducer problems will be presented to characterize the proposed method.

Verification of the Westervelt equation for focused transducers.

This study investigates the validity of the Westervelt equation for focused transducers. The angular spectrum method is employed to analyze the second-harmonic acoustic field under the weakly nonlinear approximation. Although it is well known that the Westervelt equation is accurate for the case of quasi-plane waves, the present work demonstrates accurate solution for the highly focused case of a spherically-curved ultrasound transducer having an aperture angle of 80°. It is further found that the solution error is inversely dependent on the nonlinearity coefficient.

Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation.

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backward-projection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.

Transcranial magnetic resonance imaging- guided focused ultrasound surgery of brain tumors: initial findings in 3 patients.

OBJECTIVE: This work evaluated the clinical feasibility of transcranial magnetic resonance imaging-guided focused ultrasound surgery. METHODS: Transcranial magnetic resonance imaging-guided focused ultrasound surgery offers a potential noninvasive alternative to surgical resection. The method combines a hemispherical phased-array transducer and patient-specific treatment planning based on acoustic models with feedback control based on magnetic resonance temperature imaging to overcome the effects of the cranium and allow for controlled and precise thermal ablation in the brain. In initial trials in 3 glioblastoma patients, multiple focused ultrasound exposures were applied up to the maximum acoustic power available. Offline analysis of the magnetic resonance temperature images evaluated the temperature changes at the focus and brain surface. RESULTS: We found that it was possible to focus an ultrasound beam transcranially into the brain and to visualize the heating with magnetic resonance temperature imaging. Although we were limited by the device power available at the time and thus seemed to not achieve thermal coagulation, extrapolation of the temperature measurements at the focus and on the brain surface suggests that thermal ablation will be possible with this device without overheating the brain surface, with some possible limitation on the treatment envelope. CONCLUSION: Although significant hurdles remain, these findings are a major step forward in producing a completely noninvasive alternative to surgical resection for brain disorders.

An intraoperative brain shift monitor using shear mode transcranial ultrasound: preliminary results.

OBJECTIVE: Various methods of intraoperative structural monitoring during neurosurgery are used to localize lesions after brain shift and to guide surgically introduced probes such as biopsy needles or stimulation electrodes. With its high temporal resolution, portability, and nonionizing mode of radiation, ultrasound has potential advantages over other existing imaging modalities for intraoperative monitoring, yet ultrasound is rarely used during neurosurgery largely because of the craniotomy requirement to achieve sufficiently useful signals. METHODS: Prompted by results from recent studies on transcranial ultrasound, a prototype device that aims to use the shear mode of transcranial ultrasound transmission for intraoperative monitoring was designed, constructed, and tested with 10 human participants. Magnetic resonance images were then obtained with the device spatially registered to the magnetic resonance imaging (MRI) reference coordinates. Peaks in both the ultrasound and MRI signals were identified and analyzed for both spatial localization and signal-to-noise ratio (SNR). RESULTS: The first results aimed toward validating the prototype device with MRI showed an excellent correlation (n = 38; R(2) = 0.9962) between the structural localization abilities of the two modalities. In addition, the overall SNR of the ultrasound backscatter signals (n = 38; SNR = 25.4 +/- 5.2 dB, mean +/- SD) was statistically equivalent to that of the MRI data (n = 38; SNR = 22.5 +/- 4.8 dB). CONCLUSIONS: A statistically significant correlation of localized intracranial structures between intraoperative transcranial ultrasound monitoring and MRI data was achieved with 10 human participants. We have shown and validated a prototype device incorporating transcranial shear mode ultrasound for clinical monitoring applications.

Feasibility of ultrasound phase contrast for heating localization.

Ultrasound-based methods for temperature monitoring could greatly assist focused ultrasound visualization and treatment planning based on sound speed-induced change in phase as a function of temperature. A method is presented that uses reflex transmission integration, planar projection, and tomographic reconstruction techniques to visualize phase contrast by measuring the sound field before and after heat deposition. Results from experiments and numerical simulations employing a through-transmission setup are presented to demonstrate feasibility of using phase contrast methods for identifying temperature change. A 1.088-MHz focused transducer was used to interrogate a medium with a phase contrast feature, following measurement of the baseline reference field with a hydrophone. A thermal plume in water and a tissue phantom with multiple water columns was used in separate experiments to produce a phase contrast. The reference and phase contrast field scans were numerically backprojected and the phase difference correctly identified the position and orientation of the features. The peak temperature reconstructed from the phase shift was within 0.2 degrees C of the measured temperature in the plume. Simulated results were in good agreement with experimental results. Finally, employment of reflex transmission imaging techniques for adopting a pulse-echo arrangement was simulated, and its future experimental application is discussed.

Two-dimensional localization with a single diffuse ultrasound field excitation.

Traditional ultrasound imaging methods rely on the bandwidth and center frequency of transduction to achieve axial and radial image resolution, respectively. In this study, a new modality for spatially localizing scattering targets in a two-dimensional field is presented. In this method, the bandwidth of field excitation is high, and the center frequency is lowered such that the corresponding wavelengths are substantially larger than the target profiles. Furthermore, full two-dimensional field measurements are obtained with single send-receive sequences, demonstrating a substantial simplification of the traditional scanning techniques. Field reconstruction is based on temporal-spectral cross-correlations between measured backscatter data and a library of region of interest (ROI) backscatter data measured a priori. The transducer design is based upon a wedge-shaped geometry, which was shown to yield spatially frequency-separated bandwidths of up to 156% with center frequencies of 1.38 MHz. Initial results with these send-and-receive transducer parameters and cylindrical reflection targets in a 10-mm x 10-mm ROI demonstrate two-dimensional target localization to within 0.5 mm. Spatial localization of point scatterers is demonstrated for single and multiple scattering sites.

The effects of desiccation on skull bone sound speed in porcine models.

Pre- and postdesiccation sound speeds through ex vivo porcine skull specimens were determined by time-of-flight measurements with propagated broadband pulses centered at 0.97 MHz (Os 12.7 mm, -6-dB band-width = 58%). The measured longitudinal sound speed in the 13 porcine samples (predesiccation average sound speed = 1727 +/- 57 ms(-1)) changed by a statistically significant +2.3% after deionized water reconstitution (paired t-test, alpha = 0.05, p = 0.0332).

Transcranial ultrasound focus reconstruction with phase and amplitude correction.

Therapeutic and diagnostic ultrasound procedures performed noninvasively through the skull require a reliable method for maintaining acoustic focus integrity after transmission through layered bone structures. This study used a multiple-element, phased-array transducer to reconstruct ultrasound foci through the human skull by amplitude and phase correction. It was previously demonstrated that adaptive phase correction using a multiple-element, focused transducer array yields a significant correction to an acoustic field that has been distorted by the heterogeneities of the skull bone. The introduction of amplitude correction, in a regime in which acoustic pressures from individual transducer array elements are adjusted to be normalized at the focus, has demonstrated a 6% (-0.27 dB) average decrease in acoustic sidelobe acoustic intensity relative to the focal intensity and a 2% (-0.09 dB) average decrease in the full-width-at-half-maximum (FWHM) of the acoustic intensity profile at the focus. These improvements come at the expense of significant ultrasound intensity loss--as much as 30% lower (-1.55 dB)--at the focus because the amplitude correction method requires that, at constant power, a larger proportion of energy is absorbed or reflected by regions of the skull that transmit less energy. In contrast, a second correction method that distributes pressure amplitudes such that the sections of the skull which transmit more ultrasound energy are exposed with higher ultrasound intensities has demonstrated an average sidelobe intensity decrease of 3% (-0.13 dB) with no change in the FWHM at the focus. On average, there was a 2% (0.09 dB) increase in the acoustic intensity at the focus for this inverse amplitude correction method. These results indicate that amplitude correction according to the transmission properties of various segments of the skull have a clear effect on ultrasound energy throughput into a target site within the brain parenchyma.

Spectral image reconstruction for transcranial ultrasound measurement.

An approach aimed at improved ultrasound resolution and signal strength through highly attenuating media is presented. The method delivers a series of multiple-cycle bursts in order to construct a discrete spectral (frequency domain) response in one dimension. Cross-correlation of this ultrasound A-mode response with its transmitted signal results in time-localized peaks that correspond to scattering locations. The approach is particularly relevant to the problem of transcranial ultrasound imaging, as it combines numerous smaller signals into a single signal whose net power may exceed that which could be achieved using a single burst. Tests are performed with human skull fragments and nylon-wire targets embedded in a tissue phantom. Skulls are oriented to produce both lateral and shear modes of transcranial propagation. A total of nine locations distributed over three ex vivo human skull samples are studied. Compared with pulsed and chirped signals, results indicate more localized peaks when using the multi-cycle approach, with more accurate positioning when combined with the transcranial shear mode.

A new ultrasound method for determining the acoustic phase shifts caused by the skull bone.

A potential noninvasive means for obtaining the value of ultrasound (US) phase shifts caused by the skull is examined. Knowledge of these shifts could be used in new methods that restore the focus from an US array after transcranial propagation. In the present study, a pulsed signal was emitted from a single element of a therapeutic US transducer. The reflected signal was then recorded. The data were examined over the band width of the transducer, producing amplitude data as a function of frequency. A periodic appearance of local maxima and minima was observed in the data as a function of frequency. We hypothesize that the amplitude is primarily determined by the superposition of the reflections between the interfaces at the inner and outer surfaces of the skull and between the interior interfaces of trabecular and cortical bone. A homogeneous-layer model was used to predict the forward-propagated phase using the reflection data. Good correlation was found between the numeric calculation and phases measured after propagation through single-layer plastic plates. The method was used on curved three-layer plastic phantoms and four excised human skulls. The procedure could eventually be applied toward phasing multielement arrays. Such an application could have implications in both therapeutic and diagnostic brain procedures.

A unified model for the speed of sound in cranial bone based on genetic algorithm optimization.

The density and structure of bone is highly heterogeneous, causing wide variations in the reported speed of sound for ultrasound propagation. Current research on the propagation of high intensity focused ultrasound through an intact human skull for non-invasive therapeutic action on brain tissue requires a detailed model for the acoustic velocity in cranial bone. Such models have been difficult to derive empirically due to the aforementioned heterogeneity of bone itself. We propose a single unified model for the speed of sound in cranial bone based upon the apparent density of bone by CT scan. This model is based upon the coupling of empirical measurement, theoretical acoustic simulation and genetic algorithm optimization. The phase distortion caused by the presence of skull in an acoustic path is empirically measured. The ability of a theoretical acoustic simulation coupled with a particular speed-of-sound model to predict this phase distortion is compared against the empirical data, thus providing the fitness function needed to perform genetic algorithm optimization. By performing genetic algorithm optimization over an initial population of candidate speed-of-sound models, an ultimate single unified model for the speed of sound in both the cortical and trabecular regions of cranial bone is produced. The final model produced by genetic algorithm optimization has a nonlinear dependency of speed of sound upon local bone density. This model is shown by statistical significance to be a suitable model of the speed of sound in bone. Furthermore, using a skull that was not part of the optimization process, this model is also tested against a published homogeneous speed-of-sound model and shown to return an improved prediction of transcranial ultrasound propagation.

Micro-receiver guided transcranial beam steering.

A new method for focusing ultrasound energy in brain tissue through the skull is investigated. The procedure is designed for use with a therapeutic transducer array and a small catheter-inserted hydrophone receiver placed in the brain to guide the array's focus. When performed at high-intensity, a focal intensity on the order of several hundred watts per centimeter-squared is achieved, and cells within a target volume are destroyed. The present study tests the feasibility and range of the method using an ex vivo human skull. Acoustic phase information is obtained from the stationary receiver and used to electrically shift the beam to new locations as well as correct for aberrations due to the skull. The method is applied to a 104-element 1.1 MHz array and a 120-element 0.81 MHz array. Using these array configurations, it is determined that the method can reconstruct and steer a focus over a distance of 50 mm. Application of this minimally invasive technique for ultrasound brain therapy and surgery also is investigated in vitro with a 64-element 0.664 MHz hemisphere array designed for transskull surgery. Tissue is placed inside of a skull and a catheter-inserted receiver is inserted into the tissue. A focus intense enough to coagulate the tissue is achieved at a predetermined location 10 mm from the receiver, the maximum distance that this large element array can electronically steer the focus.