Magnetic resonance imaging and X-ray computed tomography provide the two principal methods available for imaging the brain at high spatial resolution, but these methods are not easily portable and cannot be applied safely to all patients. Ultrasound imaging is portable and universally safe, but existing modalities cannot image usefully inside the adult human skull. We use in silico simulations to demonstrate that full-waveform inversion, a computational technique originally developed in geophysics, is able to generate accurate three-dimensional images of the brain with sub-millimetre resolution. This approach overcomes the familiar problems of conventional ultrasound neuroimaging by using the following: transcranial ultrasound that is not obscured by strong reflections from the skull, low frequencies that are readily transmitted with good signal-to-noise ratio, an accurate wave equation that properly accounts for the physics of wave propagation, and adaptive waveform inversion that is able to create an accurate model of the skull that then compensates properly for wavefront distortion. Laboratory ultrasound data, using ex vivo human skulls and in vivo transcranial signals, demonstrate that our computational experiments mimic the penetration and signal-to-noise ratios expected in clinical applications. This form of non-invasive neuroimaging has the potential for the rapid diagnosis of stroke and head trauma, and for the provision of routine monitoring of a wide range of neurological conditions.

Adaptive waveform inversion (AWI) reformulates the misfit function used to perform full-waveform inversion (FWI), so that it no longer contains local minima related to cycle skipping. It does this by finding a model that drives the ratio of the predicted and observed data sets to unity rather than driving the difference between these two data sets to zero as is the case for conventional FWI. We apply AWI to a 3D field data set acquired over a pervasive gas cloud in the North Sea, comparing its performance with that of conventional FWI in a variety of circumstances. When starting inversion from 3 Hz, and using a good starting model obtained from reflection tomography, FWI and AWI generate similar models although the FWI result contains edge artifacts that are not produced by AWI. However, when the starting frequency is increased to approximately 6 Hz, or when the starting model is less accurate, FWI fails to recover a good model whereas AWI continues to converge. When both of these conditions apply, FWI fails comprehensively, leading to a model that is significantly worse than the starting model, whereas the AWI result remains largely unaffected. We applied Kirchhoff depth migration to the fully-processed data using the FWI result obtained following reflection tomography, and using the AWI result obtained from a simple one-dimensional starting model. We use the resulting migrated volumes, together with measures of residual moveout throughout the volume, to show that the AWI result from a simple starting model is at least as good as the FWI result obtained following tomography. We conclude that AWI is robust in the presence of cycle skipping on this 3D field data set, and can proceed successfully from a less-accurate starting model, and from a higher starting frequency, in circumstances in which FWI fails completely.

Advanced ultrasound computed tomography techniques like full-waveform inversion are mathematically complex and orders of magnitude more computationally expensive than conventional ultrasound imaging methods. This computational and algorithmic complexity, and a lack of open-source libraries in this field, represent a barrier preventing the generalised adoption of these techniques, slowing the pace of research, and hindering reproducibility. Consequently, we have developed Stride, an open-source Python library for the solution of large-scale ultrasound tomography problems. On one hand, Stride provides high-level interfaces and tools for expressing the types of optimisation problems encountered in medical ultrasound tomography. On the other, these high-level abstractions seamlessly integrate with high-performance wave-equation solvers and with scalable parallelisation routines. The wave-equation solvers are generated automatically using Devito, a domain-specific language, and the parallelisation routines are provided through the custom actor-based library Mosaic. We demonstrate the modelling accuracy achieved by our wave-equation solvers through a comparison (1) with analytical solutions for a homogeneous medium, and (2) with state-of-the-art modelling software applied to a high-contrast, complex skull section. Additionally, we show through a series of examples how Stride can handle realistic numerical and experimental tomographic problems, in 2D and 3D, and how it can scale robustly from a local multi-processing environment to a multi-node high-performance cluster. Stride enables researchers to rapidly and intuitively develop new imaging algorithms and to explore novel physics without sacrificing performance and scalability. This will lead to faster scientific progress in this field and will significantly ease clinical translation.

Ultrasound breast imaging is a promising alternative to conventional mammography because it does not expose women to harmful ionising radiation and it can successfully image dense breast tissue. However, conventional ultrasound imaging only provides morphological information with limited diagnostic value. Ultrasound computed tomography (USCT) uses energy in both transmission and reflection when imaging the breast to provide more diagnostically relevant quantitative tissue properties, but it is often based on time-of-flight tomography or similar ray approximations of the wave equation, resulting in reconstructed images with low resolution. Full-waveform inversion (FWI) is based on a more accurate approximation of wave-propagation phenomena and can consequently produce very high resolution images using frequencies below 1 megahertz. These low frequencies, however, are not available in most USCT acquisition systems, as they use transducers with central frequencies well above those required in FWI. To circumvent this problem, we designed, trained, and implemented a two-dimensional convolutional neural network to artificially generate missing low frequencies in USCT data. Our results show that FWI reconstructions using experiment data after the application of the proposed method successfully converged, showing good agreement with X-ray CT and reflection ultrasound-tomography images.

Bayesian methods are a popular research direction for inverse problems. There are a variety of techniques available to solve Bayes’ equation, each with their own strengths and limitations. Here, we discuss stochastic variational inference (SVI), which solves Bayes’ equation using gradient-based methods. This is important for applications which are time-limited (e.g. medical tomography) or where solving the forward problem is expensive (e.g. adjoint methods). To evaluate the use of SVI in both these contexts, we apply it to ultrasound tomography of the brain using full-waveform inversion (FWI). FWI is a computationally expensive adjoint method for solving the ultrasound tomography inverse problem, and we demonstrate that SVI can be used to find a no-cost estimate of the pixel-wise variance of the sound-speed distribution using a mean-field Gaussian approximation. In other words, we show experimentally that it is possible to estimate the pixel-wise uncertainty of the sound-speed reconstruction using SVI and a common approximation which is already implicit in other types of iterative reconstruction. Uncertainty estimates have a variety of uses in adjoint methods and tomography. As an illustrative example, we focus on the use of uncertainty for image quality assessment. This application is not limiting; our variance estimator has effectively no computational cost and we expect that it will have applications in fields such as non-destructive testing or aircraft component design where uncertainties may not be routinely estimated.

Accurate wave-equation modelling is becoming increasingly important in modern imaging and therapeutic ultrasound methodologies such as ultrasound computed tomography, optoacoustic tomography or high-intensity focused ultrasound. All of them rely on the ability to accurately model the physics of wave propagation, including accurate characterisation of the ultrasound transducers, the physical devices that are responsible for generating and recording ultrasound energy. However, existing methods fail to characterise the transducer response with the accuracy required to fully exploit the capabilities of these emerging imaging and therapeutic techniques. Consequently, we have designed a new algorithm for ultrasound transducer calibration and modelling: spatial response identification. This method introduces a parametrisation of the ultrasound transducer and provides a method to calibrate the transducer model using experimental data, based on a formulation of the problem that is completely independent of the discretisation chosen for the transducer or the number of parameters used. The proposed technique models the transducer as a linear time-invariant system that is spatially heterogeneous, and identifies the model parameters that are best at explaining the experimental data while honouring the full wave equation. Spatial response identification generates a model that can accommodate the complex, heterogeneous spatial response seen experimentally for ultrasound transducers. Experimental results show that spatial response identification outperforms standard methods both in transmission and reception mode. Finally, numerical experiments using full-waveform inversion demonstrate that existing transducer-modelling approaches are insufficient to produce successful reconstructions of the human brain, whereas errors in our spatial response identification algorithm are sufficiently small to allow accurate image reconstructions.