High Resolution 3D Ultrasonic Breast Imaging by Time-Domain Full Waveform Inversion

Abstract

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 introduce a novel 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 442 × 442 × 222 voxel image with a resolution of 0.5 mm using Matlab on a single GPU within 24 hours.

Coupling Deep Learning with Full Waveform Inversion

“…When the data in the inversion are collected from N s different incoming sources {h s } Ns s=1 , the forward map f (m) and the data g δ defined in (3). Let u s (1 ≤ s ≤ N s ) be solution to ( 16) with source h s , and w s be the solution to the adjoint equation ( 35 Solve (35) with the s-th component of f −1 θ * [r(m)] as the source term for w s 9: end for 10: Evaluate Φ (m) according to (38) The computational procedure is summarized in Algorithm 1. The main difference between the calculation here and the adjoint calculation for a standard FWI gradient calculation is that we need to use the network f −1 θ to backpropagate the data into the velocity field in Line 5 of Algorithm 1 to compute the residual, and then use the adjoint of the network operator (transpose of the gradient in the discrete case), f −1 θ * , to map the residual r(m) to the source of the adjoint wave equation in Line 6.…”

“…We refer interested readers to [13,25,42,61] and references therein for overviews on the recent development in the field of FWI for geophysical applications. While the term FWI was mainly coined in the seismic imaging community, FWI also has a wide range of applications in other imaging applications, such as in medical ultrasound imaging [5,8,27,30,34,38,40,62]. From the practical point of view, the main difference between geophysical and medical FWI is that the quality of the dataset collected in medical applications, both in terms of the variety of source-detector configurations can be arranged and in terms of the frequency contents of the incident sources, is much richer than that of the geophysical FWI dataset.…”

Data-Driven Joint Inversions for PDE Models

“…This wave propagation model, with slightly changes in boundary and initial conditions, serves as the forward model for inverse problems in seismic imaging [2,10,14,15,32,37,45,46], medical ultrasound imaging [4,9,22,27,31,34,36,47], among many other applications.…”

Ray-based inversion accounting for scattering for biomedical ultrasound tomography

“…Note that replacing the Hessian with the identity in the above equation will give the steepest descent search direction, which is equivalent to taking a first-order optimisation approach, in which the nonlinear objective function (35) is minimised using a search direction which uses only the information included in the gradient term in (38). First-order approaches [5] are currently in widespread use for full-wave inversion [2,[50][51][52][53], because they do not require the additional expense of computing the Hessian. Here, however, the use of a second-order minimisation is the key to incorporating the effects of scattering in the inversion.…”

“…when the numerical model accurately represents the measurement scenario, these approaches have the potential to reconstruct accurate, high resolution images. This approach, for which there is a considerable literature in the seismic community [1], has begun to be explored in earnest for medical applications [2,18,19,47,[50][51][52][53][54]56]. Full-wave approaches are discussed further in section 8, but the biggest challenge with such schemes is the computational cost.…”