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Focused ultrasound spinal cord neuromodulation studies have demonstrated spinal cord neuromodulation in small animals. The safe and efficacious translation of these approaches to human scale requires an understanding of ultrasound propagation and heat deposition within the human spine. To address this, combined acoustic and thermal modelling was used to assess the pressure and heat distributions produced by a 500 kHz source focused to the C5/C6 level of the cervical spine via two approaches a) the posterior acoustic window between vertebral posterior arches, or b) the lateral intervertebral foramen from which the C6 spinal nerve exits. Pulse trains of 150 0.1 s pulses with a pulse repetition frequency of 0.33 Hz and free-field spatial peak pulse-averaged intensity of 10 W/cm^2 were simulated for the CT volumes of four subjects and for $\pm$10 mm translational and $\pm$10{\deg} rotational source positioning errors. Target pressures ranged between 20% and 70% of free-field spatial peak pressures with the posterior approach, and 20% and 100% with the lateral approach. When the source was optimally positioned with the posterior approach, peak spine heating values were below 1{\deg}C, but source mis-positioning resulted in bone heating up to 4{\deg}C. Heating with the lateral approach did not exceed 2{\deg}C within the mispositioning range. There were substantial inter-subject differences in target pressures and peak heating values. Target pressure varied three to four-fold between subjects, depending on approach, while peak heating varied approximately two-fold between subjects. This results in a near ten-fold range in the target pressure achieved per degree of peak heating between subjects. This highlights the importance of developing trans-spine ultrasound simulation software for the assurance of subject-specific safety and efficacy of focused ultrasound spinal cord therapies.

Advances in additive manufacturing have enabled the realisation of inexpensive, scalable, diffractive acoustic lenses that can be used to generate complex acoustic fields via phase and/or amplitude modulation. However, the design of these holograms relies on a thin-element approximation adapted from optics which can severely limit the fidelity of the realised acoustic field. Here, we introduce physics-based acoustic holograms with a complex internal structure. The structures are designed using a differentiable acoustic model with manufacturing constraints via optimisation of the acoustic property distribution within the hologram. The holograms can be fabricated simply and inexpensively using contemporary 3D printers. Experimental measurements demonstrate a significant improvement compared to conventional thin-element holograms.

Fast imaging methods are needed to promote widespread clinical adoption of Ultrasound Tomography (UST), and more widely available UST hardware could support the experimental validation of new measurement configurations. In this work, an open-source 256-element transducer ring array was developed (morganjroberts.github.io/open-UST) and manufactured using rapid prototyping, for only {\pounds}2k. Novel manufacturing techniques were used, resulting in a 1.17$^{\circ}$ mean beam axis skew angle, a 104 $\mu$m mean element position error, and a $\pm$13.6 $\mu$m deviation in matching layer thickness. The nominal acoustic performance was measured using hydrophone scans and watershot data, and the 61.2 dB SNR, 55.4$^{\circ}$ opening angle, 16.3 mm beamwidth and 54% transmit-receive bandwidth (-12 dB), were found to be similar to existing systems, and compatible with full waveform inversion reconstruction methods. The inter-element variation in acoustic performance was typically <10% without using normalisation, meaning that the elements can be modelled identically during image reconstruction, removing the need for individual source definitions based on hydrophone measurements. Finally, data from a phantom experiment was successfully reconstructed. These results demonstrate that the open-UST system is accessible for users, and suitable for UST imaging research., Comment: 16 pages, 12 figures

A new method for solving the wave equation is presented, called the learned Born series (LBS), which is derived from a convergent Born Series but its components are found through training. The LBS is shown to be significantly more accurate than the convergent Born series for the same number of iterations, in the presence of high contrast scatterers, while maintaining a comparable computational complexity. The LBS is able to generate a reasonable prediction of the global pressure field with a small number of iterations, and the errors decrease with the number of learned iterations., Comment: 6 pages, 1 figure

The development of methods to safely rewarm large volume cryopreserved biological samples remains a barrier to the widespread adoption of cryopreservation. Here, experiments and simulations were performed to demonstrate that ultrasound can increase rewarming rates relative to thermal conduction alone. An ultrasonic rewarming setup based on a custom 444 kHz tubular piezoelectric transducer was designed, characterized, and tested with 2 mL cryovials filled with frozen ground beef. Rewarming rates were characterized in the -20$^{\circ}$C to 5$^{\circ}$C range. Thermal conduction-based rewarming was compared to thermal conduction plus ultrasonic rewarming, demonstrating a ten-fold increase in rewarming rate when ultrasound was applied. The maximum recorded rewarming rate with ultrasound was 57$^{\circ}$C per minute, approximately 2.5 times faster than with thermal conduction alone. Coupled acoustic and thermal simulations were developed and showed good agreement with the heating rates demonstrated experimentally and were also used to demonstrate spatial heating distributions with small ($<$3$^{\circ}$C) temperature differentials throughout the sample when the sample was below 0$^{\circ}$C. The experiments and simulations performed in this work demonstrate the potential for ultrasound as a rewarming method for cryopreserved tissues, as faster rewarming rates may improve the viability of cryopreserved tissues and reduce the time needed for cells to regain normal function., Comment: 13 pages, 11 figures

In recent years high-resolution 3D printing has enabled a diverse range of new, low-cost, methods for ultrasonic wave-front shaping. Acoustic holograms, particularly, allow for the generation of arbitrary, diffraction limited, acoustic fields at MHz frequencies from single element transducers. These are phase plates that function as direct acoustic analogues to thin optical holograms. In this work it is shown that, by using multiple polymer 3D printing, acoustic analogues to 'thick' or volume optical holograms can also be generated. First, an analytic approach for designing a volume hologram that diffracts a set of input fields onto a desired set of output fields is briefly summarised. Next, a greedy optimisation approach based on random downhill binary search able to account for the constraints imposed by the chosen fabrication method is introduced. Finally, an experimental test-case designed to diffract the field generated by a 2.54 cm, planar, PZT transducer onto 8 distinct patterns dependent on the direction of the incident field is used to validate the approach and the design method. Field scans of the 8 target fields demonstrate that acoustic analogues of optical volume holograms can be generated using multi-polymer printing and that these allow the multiplexing of distinct fields onto different incident field directions., Comment: 9 pages, journal article

We present an open-source differentiable acoustic simulator, j-Wave, which can solve time-varying and time-harmonic acoustic problems. It supports automatic differentiation, which is a program transformation technique that has many applications, especially in machine learning and scientific computing. j-Wave is composed of modular components that can be easily customized and reused. At the same time, it is compatible with some of the most popular machine learning libraries, such as JAX and TensorFlow. The accuracy of the simulation results for known configurations is evaluated against the widely used k-Wave toolbox and a cohort of acoustic simulation software. j-Wave is available from https://github.com/ucl-bug/jwave.

Model-based treatment planning for transcranial ultrasound therapy typically involves mapping the acoustic properties of the skull from an x-ray computed tomography (CT) image of the head. Here, three methods for generating pseudo-CT images from magnetic resonance (MR) images were compared as an alternative to CT. A convolutional neural network (U-Net) was trained on paired MR-CT images to generate pseudo-CT images from either T1-weighted or zero-echo time (ZTE) MR images (denoted tCT and zCT, respectively). A direct mapping from ZTE to pseudo-CT was also implemented (denoted cCT). When comparing the pseudo-CT and ground truth CT images for the test set, the mean absolute error was 133, 83, and 145 Hounsfield units (HU) across the whole head, and 398, 222, and 336 HU within the skull for the tCT, zCT, and cCT images, respectively. Ultrasound simulations were also performed using the generated pseudo-CT images and compared to simulations based on CT. An annular array transducer was used targeting the visual or motor cortex. The mean differences in the simulated focal pressure, focal position, and focal volume were 9.9%, 1.5 mm, and 15.1% for simulations based on the tCT images, 5.7%, 0.6 mm, and 5.7% for the zCT, and 6.7%, 0.9 mm, and 12.1% for the cCT. The improved results for images mapped from ZTE highlight the advantage of using imaging sequences which improve contrast of the skull bone. Overall, these results demonstrate that acoustic simulations based on MR images can give comparable accuracy to those based on CT.

Differentiable simulators are an emerging concept with applications in several fields, from reinforcement learning to optimal control. Their distinguishing feature is the ability to calculate analytic gradients with respect to the input parameters. Like neural networks, which are constructed by composing several building blocks called layers, a simulation often requires computing the output of an operator that can itself be decomposed into elementary units chained together. While each layer of a neural network represents a specific discrete operation, the same operator can have multiple representations, depending on the discretization employed and the research question that needs to be addressed. Here, we propose a simple design pattern to construct a library of differentiable operators and discretizations, by representing operators as mappings between families of continuous functions, parametrized by finite vectors. We demonstrate the approach on an acoustic optimization problem, where the Helmholtz equation is discretized using Fourier spectral methods, and differentiability is demonstrated using gradient descent to optimize the speed of sound of an acoustic lens. The proposed framework is open-sourced and available at \url{https://github.com/ucl-bug/jaxdf}, Comment: 10 pages, 2 figures

Ultrasound tomography (UST) scanners allow quantitative images of the human breast's acoustic properties to be derived with potential applications in screening, diagnosis and therapy planning. Time domain full waveform inversion (TD-FWI) is a promising UST image formation technique that fits the parameter fields of a wave physics model by gradient-based optimization. For high resolution 3D UST, it holds three key challenges: Firstly, its central building block, the computation of the gradient for a single US measurement, has a restrictively large memory footprint. Secondly, this building block needs to be computed for each of the $10^3-10^4$ measurements, resulting in a massive parallel computation usually performed on large computational clusters for days. Lastly, the structure of the underlying optimization problem may result in slow progression of the solver and convergence to a local minimum. In this work, we design and evaluate a comprehensive computational strategy to overcome these challenges: Firstly, we exploit a gradient computation based on time reversal that dramatically reduces the memory footprint at the expense of one additional wave simulation per source. Secondly, we break the dependence on the number of measurements by using source encoding (SE) to compute stochastic gradient estimates. Also we describe a more accurate, TD-specific SE technique with a finer variance control and use a state-of-the-art stochastic LBFGS method. Lastly, we design an efficient TD multi-grid scheme together with preconditioning to speed up the convergence while avoiding local minima. All components are evaluated in extensive numerical proof-of-concept studies simulating a bowl-shaped 3D UST breast scanner prototype. Finally, we demonstrate that their combination allows us to obtain an accurate 442x442x222 voxel image with a resolution of 0.5mm using Matlab on a single GPU within 24h.

Transcranial ultrasound therapy is increasingly used for the non-invasive treatment of brain disorders. However, conventional numerical wave solvers are currently too computationally expensive to be used online during treatments to predict the acoustic field passing through the skull (e.g., to account for subject-specific dose and targeting variations). As a step towards real-time predictions, in the current work, a fast iterative solver for the heterogeneous Helmholtz equation in 2D is developed using a fully-learned optimizer. The lightweight network architecture is based on a modified UNet that includes a learned hidden state. The network is trained using a physics-based loss function and a set of idealized sound speed distributions with fully unsupervised training (no knowledge of the true solution is required). The learned optimizer shows excellent performance on the test set, and is capable of generalization well outside the training examples, including to much larger computational domains, and more complex source and sound speed distributions, for example, those derived from x-ray computed tomography images of the skull., Comment: 23 pages, 13 figures

Simulating the propagation of elastic waves in multi-layered media has many applications. A common approach is to use matrix methods where the elastic wave-field within each material layer is represented by a sum of partial-waves along with boundary conditions imposed at each interface. While these methods are well-known, coding the required matrix formation, inversion, and analysis for general multi-layered systems is non-trivial and time-consuming. Here, a new open-source toolbox called ElasticMatrix is described which solves the problem of acoustic and elastic wave propagation in multi-layered media for isotropic and transverse-isotropic materials where the wave propagation occurs in a material plane of symmetry. The toolbox is implemented in MATLAB using an object oriented programming framework and is designed to be easy to use and extend. Methods are provided for calculating and plotting dispersion curves, displacement and stress fields, reflection and transmission coefficients, and slowness profiles.

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation., Comment: Update to section 4 (application to PDEs with non-constant coefficients)

Moving mesh methods provide an efficient way of solving partial differential equations for which large, localised variations in the solution necessitate locally dense spatial meshes. In one-dimension, meshes are typically specified using the arclength mesh density function. This choice is well-justified for piecewise polynomial interpolants, but it is only justified for spectral methods when model solutions include localised steep gradients. In this paper, one-dimensional mesh density functions are presented which are based on a spatially localised measure of the bandwidth of the approximated model solution. In considering bandwidth, these mesh density functions are well-justified for spectral methods, but are not strictly tied to the error properties of any particular spatial interpolant, and are hence widely applicable. The bandwidth mesh density functions are demonstrated by applying periodic spectral and finite-difference moving mesh methods to a number of model problems in acoustics. These problems include a heterogeneous advection equation, the viscous Burgers' equation, and the Korteweg-de Vries equation. Simulation results demonstrate solution convergence rates that are up to an order of magnitude faster using the bandwidth mesh density functions than uniform meshes, and around three times faster than those using the arclength mesh density function.

Model-based treatment planning and exposimetry for high-intensity focused ultrasound (HIFU) requires the numerical simulation of nonlinear ultrasound propagation through heterogeneous and absorbing media. This is a computationally demanding problem due to the large distances travelled by the ultrasound waves relative to the wavelength of the highest frequency harmonic. Here, the k-space pseudospectral method is used to solve a set of coupled partial differential equations equivalent to a generalised Westervelt equation. The model is implemented in C++ and parallelised using the message passing interface (MPI) for solving large-scale problems on distributed clusters. The domain is partitioned using a 1D slab decomposition, and global communication is performed using a sparse communication pattern. Operations in the spatial frequency domain are performed in transposed space to reduce the communication burden imposed by the 3D fast Fourier transform. The performance of the model is evaluated using grid sizes up to 4096 x 2048 x 2048 grid points distributed over a cluster using up to 1024 compute cores. Given the global nature of the gradient calculation, the model shows good strong scaling behaviour, with a speed-up of 1.7x whenever the number of cores is doubled. This means large-scale simulations can be distributed across high numbers of cores on a cluster to minimise execution times with a relatively small computational overhead. The efficacy of the model is demonstrated by simulating the ultrasound beam pattern for a HIFU sonication of the kidney.