Numerical Modeling of Ultrasound Propagation in Weakly Heterogeneous Media Using a Mixed-Domain Method

Model-based navigation of transcranial focused ultrasound neuromodulation in humans: Application to targeting the amygdala and thalamus

BACKGROUND

Transcranial focused ultrasound (tFUS) neuromodulation has shown promise in animals but is challenging to translate to humans because of the thicker skull that heavily scatters ultrasound waves.

OBJECTIVE

We develop and disseminate a model-based navigation (MBN) tool for acoustic dose delivery in the presence of skull aberrations that is easy to use by non-specialists.

METHODS

We pre-compute acoustic beams for thousands of virtual transducer locations on the scalp of the subject under study. We use the hybrid angular spectrum solver mSOUND, which runs in ∼4 s per solve per CPU yielding pre-computation times under 1 h for scalp meshes with up to 4000 faces and a parallelization factor of 5. We combine this pre-computed set of beam solutions with optical tracking, thus allowing real-time display of the tFUS beam as the operator freely navigates the transducer around the subject' scalp. We assess the impact of MBN versus line-of-sight targeting (LOST) positioning in simulations of 13 subjects.

RESULTS

Our navigation tool has a display refresh rate of ∼10 Hz. In our simulations, MBN increased the acoustic dose in the thalamus and amygdala by 8-67 % compared to LOST and avoided complete target misses that affected 10-20 % of LOST cases. MBN also yielded a lower variability of the deposited dose across subjects than LOST.

CONCLUSIONS

MBN may yield greater and more consistent (less variable) ultrasound dose deposition than transducer placement with line-of-sight targeting, and thus could become a helpful tool to improve the efficacy of tFUS neuromodulation.

High-resolution acoustic mapping of tunable gelatin-based phantoms for ultrasound tissue characterization

Background: The choice of gelatin as the phantom material is underpinned by several key advantages it offers over other materials in the context of ultrasonic applications. Gelatin exhibits spatial and temporal uniformity, which is essential in creating reliable tissue-mimicking phantoms. Its stability ensures that the phantom's properties remain consistent over time, while its flexibility allows for customization to match the acoustic characteristics of specific tissues, in addition to its low levels of ultrasound scattering. These attributes collectively make gelatin a preferred choice for fabricating phantoms in ultrasound-related research. Methods: We developed gelatin-based phantoms with adjustable parameters and conducted high-resolution measurements of ultrasound wave attenuation when interacting with the gelatin phantoms. We utilized a motorized acoustic system designed for 3D acoustic mapping. Mechanical evaluation of phantom elasticity was performed using unconfined compression tests. We particularly examined how varying gelatin concentration influenced ultrasound maximal intensity and subsequent acoustic attenuation across the acoustic profile. To validate our findings, we conducted computational simulations to compare our data with predicted acoustic outcomes. Results: Our results demonstrated high-resolution mapping of ultrasound waves in both gelatin-based phantoms and plain fluid environments. Following an increase in the gelatin concentration, the maximum intensity dropped by 30% and 48% with the 5 MHz and 1 MHz frequencies respectively, while the attenuation coefficient increased, with 67% more attenuation at the 1 MHz frequency recorded at the highest concentration. The size of the focal areas increased systematically as a function of increasing applied voltage and duty cycle yet decreased as a function of increased ultrasonic frequency. Simulation results verified the experimental results with less than 10% deviation. Conclusion: We developed gelatin-based ultrasound phantoms as a reliable and reproducible tool for examining the acoustic and mechanical attenuations taking place as a function of increased tissue elasticity and stiffness. Our experimental measurements and simulations gave insight into the potential use of such phantoms for mimicking soft tissue properties.

A rapid element pressure field simulation method for transcranial phase correction in focused ultrasound therapy

Transcranial focused ultrasound ablation has emerged as a promising technique for treating neurological disorders. The clinical system exclusively employed the ray tracing method to compute phase aberrations induced by the human skull, taking into account computational time constraints. However, this method compromises slightly on accuracy compared to simulation-based methods. This study evaluates a fast simulation method that simulates the time-harmonic pressure field within the region of interest for effective phase correction. Experimental validation was carried out using a 512-element, 670 kHz hemispherical transducer for fourex vivoskulls. The ray tracing method achieved a restoration ratio of 64.81% ± 4.33% of acoustic intensity normalized to hydrophone measurements. In comparison, the rapid simulation method demonstrated improved results with a restoration ratio of 73.10% ± 7.46%, albeit slightly lower than the full-wave simulation which achieved a restoration ratio of 75.87% ± 5.40%. The rapid simulation methods exhibited computational times that were less than five minutes for parallel computation with 8 threads. The incident angle was calculated, and a maximum difference of 6.8 degrees was found when the fixed position of the skull was changed. Meanwhile, the restoration ratio of acoustic intensity was validated to be above 70% for different target positions away from the geometrical focus of the transducer. The favorable balance between time consumption and correction accuracy makes this method valuable for clinical treatment applications.

Nonlinear Acoustic Holography With Adaptive Sampling

Accurate and efficient numerical simulation of highly nonlinear ultrasound propagation is essential for a wide range of therapeutic and physical ultrasound applications. However, due to large domain sizes and the generation of higher harmonics, such simulations are computationally challenging, particularly in 3-D problems with shock waves. Current numerical methods are based on computationally inefficient uniform meshes that resolve the highest harmonics across the entire spatial domain. To address this challenge, we present an adaptive numerical algorithm for computationally efficient nonlinear acoustic holography. At each propagation step, the algorithm monitors the harmonic content of the acoustic signal and adjusts its discretization parameters accordingly. This enables efficient local resolution of higher harmonics in areas of high nonlinearity while avoiding unnecessary resolution elsewhere. Furthermore, the algorithm actively adapts to the signal's nonlinearity level, eliminating the need for prior reference simulations or information about the spatial distribution of the harmonic content of the acoustic field. The proposed algorithm incorporates an upsampling process in the frequency domain to accommodate the generation of higher harmonics in forward propagation and a downsampling process when higher harmonics are decimated in backward propagation. The efficiency of the algorithm was evaluated for highly nonlinear 3-D problems, demonstrating a significant reduction in computational cost with a nearly 50-fold speedup over a uniform mesh implementation. Our findings enable a more rapid and efficient approach to modeling nonlinear high-intensity focused ultrasound (HIFU) wave propagation.

A Generalized Split-Step Angular Spectrum Method for Efficient Simulation of Wave Propagation in Heterogeneous Media

Angular spectrum (AS) methods enable efficient calculation of wave propagation from one plane to another inside homogeneous media. For wave propagation in heterogeneous media such as biological tissues, AS methods cannot be applied directly. Split-stepping techniques decompose the heterogeneous domain into homogeneous and perturbation parts, and provide a solution for forward wave propagation by propagating the incident wave in both frequency-space and frequency-wavenumber domains. Recently, a split-step hybrid angular spectrum (HAS) method was proposed for plane wave propagation of focused ultrasound beams. In this study, we extend these methods to enable simulation of acoustic pressure field for an arbitrary source distribution, by decomposing the source and reflection spectra into orthogonal propagation direction components, propagating each component separately, and summing all components to get the total field. We show that our method can efficiently simulate the pressure field of arbitrary sources in heterogeneous media. The accuracy of the method was analyzed comparing the resultant pressure field with pseudospectral time domain (PSTD) solution for breast tomography and hemispherical transcranial-focused ultrasound simulation models. Eighty times acceleration was achieved for a 3-D breast simulation model compared to PSTD solution with 0.005 normalized root mean-squared difference (NRMSD) between two solutions. For the hemispherical phased array, aberrations due to skull were accurately calculated in a single simulation run as evidenced by the resultant-focused ultrasound beam simulations, which had 0.001 NRMSD with 40 times acceleration factor compared to the PSTD method.

Time-Resolved Passive Cavitation Mapping Using the Transient Angular Spectrum Approach

Passive cavitation mapping (PCM), which generates images using bubble acoustic emission signals, has been increasingly used for monitoring and guiding focused ultrasound surgery (FUS). PCM can be used as an adjunct to magnetic resonance imaging to provide crucial information on the safety and efficacy of FUS. The most widely used algorithm for PCM is delay-and-sum (DAS). One of the major limitations of DAS is its suboptimal computational efficiency. Although frequency-domain DAS can partially resolve this issue, such an algorithm is not suitable for imaging the evolution of bubble activity in real time and for cases in which cavitation events occur asynchronously. This study investigates a transient angular spectrum (AS) approach for PCM. The working principle of this approach is to backpropagate the received signal to the domain of interest and reconstruct the spatial-temporal wavefield encoded with the bubble location and collapse time. The transient AS approach is validated using an in silico model and water bath experiments. It is found that the transient AS approach yields similar results to DAS, but it is one order of magnitude faster. The results obtained by this study suggest that the transient AS approach is promising for fast and accurate PCM.

mSOUND: An Open Source Toolbox for Modeling Acoustic Wave Propagation in Heterogeneous Media

mSOUND is an open-source toolbox written in MATLAB. This toolbox is intended for modeling linear/ nonlinear acoustic wave propagation in media (primarily biological tissues) with arbitrary heterogeneities, in which, the speed of sound, density, attenuation coefficient, power-law exponent, and nonlinear coefficient are all spatially varying functions. The computational model is an iterative one-way model based on a mixed domain method. In this article, a general guideline is given along with three representative examples to illustrate how to set up simulations using mSOUND. The first example uses the transient mixed-domain method (TMDM) forward projection to compute the transient acoustic field for a given source defined on a plane. The second example uses the frequency-specific mixed-domain method (FSMDM) forward projection to rapidly obtain the pressure distribution directly at the frequencies of interest, assuming linear or weakly nonlinear wave propagation. The third example demonstrates how to use TMDM backward projection to reconstruct the initial acoustic pressure field to facilitate photoacoustic tomography (PAT). mSOUND (https://m-sound.github.io/mSOUND/home) is designed to be complementary to existing ultrasound modeling toolboxes and is expected to be useful for a wide range of applications in medical ultrasound including treatment planning, PAT, transducer design, and characterization.

A fast full-wave solver for calculating ultrasound propagation in the body

Therapeutic ultrasound is a promising non-invasive method for inducing various beneficial biological effects in the human body. In cancer treatment applications, high-power ultrasound is focused at a target tissue volume to ablate the malignant tumour. The success of the procedure depends on the ability to accurately focus ultrasound and destroy the target tissue volume through coagulative necrosis whilst preserving the surrounding healthy tissue. Patient-specific treatment planning strategies are therefore being developed to increase the efficacy of such therapies, while reducing any damage to healthy tissue. These strategies require to use high-performance computing methods to solve ultrasound wave propagation in the body quickly and accurately. For realistic clinical scenarios, all numerical methods which employ volumetric meshes require several hours or days to solve the full-wave propagation on a computer cluster. The boundary element method (BEM) is an efficient approach for modelling the wave field because only the boundaries of the hard and soft tissue regions require discretisation. This paper presents a multiple-domain BEM formulation with a novel preconditioner for solving the Helmholtz transmission problem (HTP). This new formulation is efficient at high-frequencies and where high-contrast materials are present. Numerical experiments are performed to solve the HTP in multiple domains comprising: (i) human ribs, an idealised abdominal fat layer and liver tissue, (ii) a human kidney with a perinephric fat layer, exposed to the acoustic field generated by a high-intensity focused ultrasound (HIFU) array transducer. The time required to solve the equations associated with these problems on a single workstation is of the order of minutes. These results demonstrate the great potential of this new BEM formulation for accurately and quickly solving ultrasound wave propagation problems in large anatomical domains which is essential for developing treatment planning strategies.

Spherical Array System for High-Precision Transcranial Ultrasound Stimulation and Optoacoustic Imaging in Rodents

Ultrasound can be delivered transcranially to ablate brain tissue, open the blood-brain barrier, or affect neural activity. Transcranial focused ultrasound in small rodents is typically done with low-frequency single-element transducers, which results in unspecific targeting and impedes the concurrent use of fast neuroimaging methods. In this article, we devised a wide-angle spherical array bidirectional interface for high-resolution parallelized optoacoustic imaging and transcranial ultrasound (POTUS) delivery in the same target regions. The system operates between 3 and 9 MHz, allowing to generate and steer focal spots with widths down to [Formula: see text] across a field of view covering the entire mouse brain, while the same array is used to capture high-resolution 3-D optoacoustic data in real time. We showcase the system's versatile beam-forming capacities as well as volumetric optoacoustic imaging capabilities and discuss its potential to noninvasively monitor brain activity and various effects of ultrasound emission.

A modified mixed domain method for modeling acoustic wave propagation in strongly heterogeneous media

In this paper, phase correction and amplitude compensation are introduced to a previously developed mixed domain method (MDM), which is only accurate for modeling wave propagation in weakly heterogeneous media. Multiple reflections are also incorporated with the one-way model to improve the accuracy. The resulting model is denoted as the modified mixed domain method (MMDM) and is numerically evaluated for its accuracy and efficiency using four distinct cases. It is found that the MMDM is significantly more accurate than the MDM for strongly heterogeneous media, especially when the phase aberrating layer is approximately perpendicular to the acoustic beam. Additionally, a convergence study suggests that the second-order reflection could be sufficient for cases involving high contrast inhomogeneities and large loss values (e.g., skulls). The method developed in this work could facilitate therapeutic ultrasound for treating brain-related diseases and disorders.

Heterogeneous Angular Spectrum Method for Trans-Skull Imaging and Focusing

Ultrasound, alone or in concert with circulating microbubble contrast agents, has emerged as a promising modality for therapy and imaging of brain diseases. While this has become possible due to advancements in aberration correction methods, a range of applications, including adaptive focusing and tracking of the microbubble dynamics through the human skull, may benefit from even more computationally efficient methods to account for skull aberrations. Here, we derive a general method for the angular spectrum approach (ASA) in a heterogeneous medium, based on a numerical marching scheme to approximate the full implicit solution. We then demonstrate its functionality with simulations for (human) skull-related aberration correction and trans-skull passive acoustic mapping. Our simulations show that the general solution provides accurate trans-skull focusing as compared to the uncorrected case (error in focal point location of 1.0 ± 0.4 mm vs 2.2 ± 0.7 mm) for clinically relevant frequencies (0.25-1.5MHz), apertures (50-100 mm), and targets, with peak focal pressures approximately 30 ± 17% of the free field case, with the effects of skull attenuation and amplitude shading included. In the case of source localization, our method leads to an average of 75% error reduction (from 2.9 ± 1.8 mm to 0.7 ± 0.5 mm) and 40-60% increase in peak intensity, evaluated over the range of frequencies (0.4-1.2 MHz), apertures (50-100 mm), and point source locations (40 mm by 50 mm grid) as compared to the homogeneous medium ASA. Overall, total computation times for both focusing and point source localization of the order milliseconds (166 ± 37 ms, compared with 44 ± 4 ms for the homogeneous ASA formulation) can be attained with this approach. Collectively our findings indicate that the proposed phase correction method based on the ASA could provide a computationally efficient and accurate method for trans-skull transmit focusing and imaging of point scatterers, potentially opening new possibilities for treatment and diagnosis of brain diseases.

Simulation of the Second-Harmonic Ultrasound Field in Heterogeneous Soft Tissue Using a Mixed-Domain Method

A mixed-domain method (MDM) dubbed frequency-specific MDM (FSMDM) is introduced for the simulation of the second-harmonic ultrasound field in weakly heterogeneous media. The governing equation for the second harmonics is derived based on the quasi-linear theory. The speed of sound, nonlinear coefficient, and attenuation coefficient are all spatially varying functions in the equation. The fundamental frequency pressure field is first solved by the FSMDM and it is subsequently used as the source term for the second-harmonics equation. This equation can be again solved by the FSMDM to rapidly obtain the second-harmonic pressure field. Five 2-D cases, including one with a realistic human tissue map, are studied to systematically verify the proposed method. Results from the previously developed transient MDM are used as the benchmark solutions. Comparisons show that the two methods give similar results for all cases. More importantly, the FSMDM has a crucial advantage over the transient MDM in that it can be two orders of magnitude faster.