Constructing a physics-augmented digital twin of the skull is imperative for a wide range of transcranial ultrasound applications including ultrasound computed tomography and focused ultrasound therapy. The high impedance contrast as well as the acoustic-elastic coupling observed between soft tissue and bone increase the complexity of the ultrasound wavefield considerably, thus emphasizing the need for waveform-based inversion approaches. This work applies reverse time migration in conjunction with the spectral-element method to an in vitro human skull to obtain a starting model, which can be used for full-waveform inversion and adjoint-based shape optimization. Two distinct brain phantoms are considered where the cranial cavity of the in vitro human skull was filled with (1) homogeneous water and (2) gelatin with two cylindrical inclusions. A 2D slice through the posterior of the skull was collected using a ring-like aperture consisting of 1024 ultrasound transducers with a bandwidth of approximately 1MHz to 3MHz. Waveform-based reverse time migration was then used to resolve the inner and outer contours of the skull from which a conforming hexahedral finite-element mesh was constructed. The synthetically generated measurements which are obtained by solving the coupled acoustic-viscoelastic wave equation are in good agreement with the observed laboratory measurements. It is demonstrated that using this revised wave speed model for recomputing the reverse time migration reconstructions allows for improved localization of the gelatin inclusions within the cranial cavity.
We present a full-waveform inversion (FWI) of an in-vivo data set acquired with a transmission-reflection optoacoustic ultrasound imaging platform containing a cross-sectional slice through a mouse. FWI is a high-resolution reconstruction method that provides quantitative images of tissue properties such as the speed of sound. As an iterative data-fitting procedure, FWI relies on the ability to accurately predict the physics of wave propagation in heterogeneous media to account for the non-linear relationship between the ultrasonic wavefield and the tissue properties. A key component to accurately predict the ultrasonic field numerically is a precise knowledge of the source characteristics. For realistic problems, however, the source-time function is generally unknown, which necessitates an auxiliary inversion that recovers the time series for each transducer. This study presents an updated sound speed reconstruction of a cross-section through a mouse using source wavelets that are inverted individually per transducer. These source wavelets have been estimated from a set of observed data by application of a source-wavelet correction filter, which is equivalent to a water-level deconvolution. Compared to previous results, the spatial resolution of anatomical features such as the vertebral column is increased whilst artefacts are suppressed.
Using waveform-based inversion methods within transcranial ultrasound computed tomography is an attractive emerging reconstruction technique for imaging the human brain. However, such imaging approaches generally rely on possessing an accurate model of the skull in order to account for the complex interactions which occur when the ultrasound waves propagate between soft tissue and bone. In order to recover the shape of the skull within the context of full-waveform inversion, adjoint-based shape optimization is performed within this study. The gradients with respect to the acoustic properties of the tissues which are used in conventional full-waveform inversion act as a proxy for estimating the sensitivities to the shape of the skull. These shape derivatives can be utilized to update the interface between the interior brain tissue and the skull. This technique employs the spectral-element method for solving the wave equation and, thus, allows for the use of a convenient framework for representing the skull interfaces throughout the inversion. Adaptations of the Shepp-Logan phantom are used as a proof of concept to demonstrate this inversion strategy where both the shape of the skull as well as the interior brain tissue are imaged sequentially.
Full-waveform modelling serves as the basis for many emerging inversion techniques within ultrasound computed tomography. Being able to accurately depict strong material interfaces, such between soft tissue and bone, is particularly important for ensuring that these numerical methods produce physically correct results. We present a procedure for constructing digital twins of various parts of the human body through the use of conforming hexahedral meshes, which are used together with the spectral-element method to accurately model the interactions of the ultrasound wavefield at these sharp material boundaries. In silico cranial and knee phantoms are used as examples.
Full-waveform inversion (FWI) for ultrasound computed tomography is an advanced method to provide quantitative and high-resolution images of tissue properties. Two main reasons hindering the widespread adoption of FWI in clinical practice are (1) its high computational cost and (2) the requirement of a good initial model to mitigate the non-convexity of the inverse problem. The latter is commonly referred to as “cycle-skipping", which occurs for phase differences between synthetic and observed signals and usually traps the inversion in a local minimum. Source-encoding strategies, which simultaneously activate several emitters and have been proposed to reduce the simulation cost, further contribute to this issue due to the multiple arrivals of the wavefronts. We present a time-domain acoustic full-waveform inversion strategy utilizing a recently proposed misfit functional based on optimal transport. Using a graph-space formulation, the discrepancy between simulated and observed signals can be computed efficiently by solving an auxiliary linear program. This approach alleviates the common need for either a good initial model and / or low-frequency data. Furthermore, combining this misfit functional with random source-encoding and a stochastic trust-region method significantly reduces the computational cost per FWI iteration. In-silico examples using a numerical phantom for breast screening ultrasound tomography demonstrate the ability of the proposed inversion strategy to converge to the ground truth even when starting from a weak prior and cycle-skipped data.
Characterizing the spatial resolution and uncertainties related to a tomographic reconstruction are crucial to assess its quality and to assist with the decision-making process. Bayesian inference provides a general framework to compute conditional probability density functions of the model space. However, analytic expressions and closed-form solutions for the posterior probability density are limited to linear inverse problems such as straight- ray tomography under the assumption of a Gaussian prior and data noise. Resolution analysis and uncertainty quantification is significantly more complicated for non-linear inverse problems such as full-waveform inversion (FWI), and sampling-based approaches such as Markov-Chain Monte-Carlo are often impractical because of their tremendous computational cost. However, under the assumption of Gaussian priors in model and data space, we can exploit the machinery of linear resolution analysis and find a Gaussian approximation of the posterior probability density by using the Hessian of the regularized objective functional. This non-linear resolution analysis rests on (i) a quadratic approximation of the misfit functional in the vicinity of an optimal model; (ii) the idea that an approximation of the Hessian can be built efficiently by gradient information from a set of perturbed models around the optimal model. The inverse of the preconditioned Hessian serves as a proxy of the posterior covariance from which space-dependent uncertainties as well as correlations between parameters and inter-parameter trade-offs can be extracted. Moreover, the framework proposed here also allows for inter- comparison between different tomographic techniques. Specifically, we aim for a comparison between tissue models obtained from ray tomography and models obtained with FWI using ultrasound data.
Ultrasound computed tomography (USCT) is a promising imaging modality for breast cancer screening. Two challenges commonly arising in time-of-flight USCT are (1) to efficiently deal with large data sets and (2) to effectively mitigate the ill-posedness for an adequate reconstruction of the model. In this contribution, we develop an optimization strategy based on a stochastic descent method that adaptively subsamples the data, and analyze its performance in combination with different sparsity-enforcing regularization techniques. The algorithms are tested on numerical as well as real data obtained from synthetic phantom scans of the previous USCT Data Challenges.
Full-waveform inversion applied to ultrasound computed tomography is a promising technique to provide highresolution quantitative images of soft human tissues, which are otherwise difficult to illuminate by conventional ultrasound imaging. A particular challenge which arises within transcranial ultrasound is the imprint of the solid skull on the measured wavefield. We present an acoustoelastic approach to full-waveform inversion for transcranial ultrasound computed tomography that accurately accounts for the solid-fluid interactions along the skull-tissue interfaces. Using the spectral-element method on cubical meshes, we obtain a scalable and performant method to resolve such a coupled physical system. Moreover, since the volume of the skull is small compared to the entire simulation domain, solving a coupled system of the acoustoelastic wave equation increases the computational cost only by a small margin compared to the acoustic approximation. We perform an in silico forward and inverse modeling study that reveals significant coupling effects at the skull-tissue interfaces when considering the skull as an elastic medium as opposed to an acoustic medium. Applying full-waveform inversion to a set of synthetically generated acoustoelastic forward data allows for favorable reconstructions to be achieved when considering an acoustoelastic prior model of the skull.
This work presents a method to estimate average anisotropy in speed of sound of soft tissue using pulse-echo ultrasound. In particular, our setup includes a passive acoustic reflector located opposite to the ultrasound probe, with tissue in between. This enables the generation of strong reflections from which we measure their traveltimes. We use ray-based approaches to derive the forward problem that relates observed traveltimes with speed of sound anisotropy parameters. The accuracy of the forward modelling is verified using numerical wave propagation simulations. We finally show the occurrence of anisotropy in muscle tissue using in-vivo data.
We present a novel approach to obtain time-of-flight measurements between transducer pairs in an Ultrasound computed tomography (USCT) scanner by applying the interferometry principle, which has been used success- fully in seismic imaging to recover the subsurface velocity structure from ambient noise recordings. To apply this approach to a USCT aperture, random wavefields are generated by activating the emitting transducers in a random sequence. By correlating the random signals recorded by the receiving transducers, we obtain an approximation of the Green’s functions between all receiver pairs, where one is acting as a virtual source. This eliminates specific source imprints, and thus avoids the need for reference measurements and calibration. The retrieved Green’s functions between any two measurement locations can then be used as new data to invert the sound speed map. On the basis of the cross-correlation travel times a ray-based time-of-flight tomography is developed and solved with an iterative least-squares method. As a proof of concept, the algorithm is tested on numerical breast phantoms in a synthetic 2D study.
The wave equation is linear, and it scales in time and space. As a consequence, wave phenomena that occur during fractions of a millisecond in human tissue often have a close correspondence in waves travelling for hours through the interior of the Earth. The scale invariance of the wave equation is the foundation for collaboration and technology transfer between medical ultrasound and seismic imaging - the promotion of which is the main goal of this contribution.
In the first part of our presentation, we review the current state of the art in seismic imaging, with a focus on regional to global scales. Special emphasis will be on (1) high-performance modelling of seismic wave propagation through a heterogeneous, attenuating and anisotropic Earth, (2) the nature of seismic data and the resulting characteristics of the inverse problem, (3) recent images of 3D deep-Earth structure, and (4) future challenges in the field.
In the second part, we highlight efforts to translate techniques from seismic imaging to medical ultrasound. This includes optimal design to position transducers, finite-frequency traveltime tomography to image out of plane, reverse-time migration, and 3D multi-parameter full-waveform inversion.
Finally, we discuss several non-mathematical challenges that still impede technology transfer, and that remain to be addressed. These include the acceptable time to solution, and the ability of well-trained radiologists and seismic interpreters to handle entirely new types of images (and artifacts).
Waveform inversion is a promising method for ultrasound computed tomography able to produce high-resolution images of human breast tissue. However, the computational complexity of waveform inversion remains a considerable challenge, and the costs per iteration are proportional to the number of emitting transducers. We propose a twofold strategy to accelerate the time-to-solution by identifying the optimal number and location of emitters using sequential optimal experimental design (SOED). SOED is a powerful tool to iteratively add the most informative transducer or remove redundant measurements, respectively. This approach simultaneously provides optimized transducer configurations and a cost-benefit curve that quantifies the information gain versus the computational cost.
First, we propose a method to identify the emitters that provide reconstructions with minimal expected uncertainties. Using a Bayesian approach, model uncertainties and resolution can be quantified with the trace of the posterior covariance. By linearizing the wave equation, we can compute the posterior covariance using the inverse of the Gauss-Newton approximation of the Hessian. Furthermore, this posterior is independent of the breast model and the experimental data, thus enabling pre-acquisition experimental optimization. Then, for the post-acquisition inversion, we present an approach to select a subsample of sources that accurately approximates the full gradient direction in each iteration. We control the convergence of the angular differences between consecutive gradient directions by randomly adding new emitters into the subsample.
We present synthetic studies in 2D and 3D that consider a ring-shaped and a semi-ellipsoidal scanning device, respectively. Numerical results suggest that the provided methods have the potential to identify redundancies from the corresponding cost-benefit curves. Furthermore, the gradient direction rapidly converges to the direction of the full gradient, which appears to be independent of the model and the emitter locations.
Waveform inversion for ultrasound computed tomography (USCT) is a promising imaging technique for breast cancer screening. However, the improved spatial resolution and the ability to constrain multiple parameters simultaneously demand substantial computational resources for the recurring simulations of the wave equation. Hence, it is crucial to use fast and accurate methods for numerical wave propagation, on the one hand, and to keep the number of required simulations as small as possible, on the other hand. We present an efficient strategy for acoustic waveform inversion that combines (i) a spectral-element continuous Galerkin method for solving the wave equation, (ii) conforming hexahedral mesh generation to discretize the scanning device, (iii) a randomized descent method based on mini-batches to reduce the computational cost for misfit and gradient computations, and (iv) a trust-region method using a quasi-Newton approximation of the Hessian to iteratively solve the inverse problem. This approach combines ideas and state-of-the-art methods from global-scale seismology, large-scale nonlinear optimization, and machine learning. Numerical examples for a synthetic phantom demonstrate the efficiency of the discretization, the effectiveness of the mini-batch approximation and the robustness of the trust-region method to reconstruct the acoustic properties of breast tissue with partial information.
We present methods to optimize the setup of a 3D ultrasound tomography scanner for breast cancer detection. This approach provides a systematic and quantitative tool to evaluate different designs and to optimize the con- figuration with respect to predefined design parameters. We consider both, time-of-flight inversion using straight rays and time-domain waveform inversion governed by the acoustic wave equation for imaging the sound speed. In order to compare different designs, we measure their quality by extracting properties from the Hessian operator of the time-of-flight or waveform differences defined in the inverse problem, i.e., the second derivatives with respect to the sound speed. Spatial uncertainties and resolution can be related to the eigenvalues of the Hessian, which provide a good indication of the information contained in the data that is acquired with a given design. However, the complete spectrum is often prohibitively expensive to compute, thus suitable approximations have to be developed and analyzed. We use the trace of the Hessian operator as design criterion, which is equivalent to the sum of all eigenvalues and requires less computational effort. In addition, we suggest to take advantage of the spatial symmetry to extrapolate the 3D experimental design from a set of 2D configurations. In order to maximize the quality criterion, we use a genetic algorithm to explore the space of possible design configurations. Numerical results show that the proposed strategies are capable of improving an initial configuration with uniformly distributed transducers, clustering them around regions with poor illumination and improving the ray coverage of the domain of interest.
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