Daniel Renzi

Using level set based inversion of arrival times to recover shear wave speed in transient elastography and supersonic imaging

Transient elastography and supersonic imaging are promising new techniques for characterizing the elasticity of soft tissues. Using this method, an ‘ultrafast imaging’ system (up to 10 000 frames s −1 ) follows in real time the propagation of a low-frequency shear wave. The displacement of the propagating shear wave is measured as a function of time and space. Here we develop a fast level set based algorithm for finding the shear wave speed from the interior positions of the propagating front. We compare the performance of level curve methods developed here and our previously developed (McLaughlin J and Renzi D 2006 Shear wave speed recovery in transient elastography and supersonic imaging using propagating fronts Inverse Problems 22 681–706) distance methods. We give reconstruction examples from synthetic data and from data obtained from a phantom experiment accomplished by Mathias Fink’s group (the Laboratoire Ondes et Acoustique, ESPCI, Universit´ e Paris VII).

Interior elastodynamics inverse problems: shear wave speed reconstruction in transient elastography

We review and present new results on the transient elastography problem, where the goal is to reconstruct shear stiness properties using interior time and space dependent displacement measurements. We present unique identifiability of two parameters for this inverse problem, establish that a Lipschitz continuous arrival time satisfies the eikonal equation, and present two numerical algorithms, simulation results, and a reconstruction example using a phantom experiment accomplished by Mathias Finks group (the Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII). One numerical algorithm uses a geometrical optics expansion and the other utilizes the arrival time surface.

Shear wave speed recovery using moving interference patterns obtained in sonoelastography experiments

Two new experiments were created to characterize the elasticity of soft tissue using sonoelastography. In both experiments the spectral variance image displayed on a GE LOGIC 700 ultrasound machine shows a moving interference pattern that travels at a very small fraction of the shear wave speed. The goal of this paper is to devise and test algorithms to calculate the speed of the moving interference pattern using the arrival times of these same patterns. A geometric optics expansion is used to obtain Eikonal equations relating the moving interference pattern arrival times to the moving interference pattern speed and then to the shear wave speed. A cross-correlation procedure is employed to find the arrival times; and an inverse Eikonal solver called the level curve method computes the speed of the interference pattern. The algorithm is tested on data from a phantom experiment performed at the University of Rochester Center for Biomedical Ultrasound.

Some Improvements for the Fast Sweeping Method

In this paper, we outline two improvements to the fast sweeping method to improve the speed of the method in general and more specifically in cases where the speed is changing rapidly. The conventional wisdom is that fast sweeping works best when the speed changes slowly, and fast marching is the algorithm of choice when the speed changes rapidly. The goal here is to achieve run times for the fast sweeping method that are at least as fast, or faster, than competitive methods, e.g. fast marching, in the case where the speed is changing rapidly. The first improvement, which we call the locking method, dynamically keeps track of grid points that have either already had the solution successfully calculated at that grid point or for which the solution cannot be successfully calculated during the current iteration. These locked points can quickly be skipped over during the fast sweeping iterations, avoiding many time-consuming calculations. The second improvement, which we call the two queue method, keeps all of the unlocked points in a data structure so that the locked points no longer need to be visited at all. Unfortunately, it is not possible to insert new points into the data structure while maintaining the fast sweeping ordering without at least occasionally sorting. Instead, we segregate the grid points into those with small predicted solutions and those with large predicted solutions using two queues. We give two ways of performing this segregation. This method is a label correcting (iterative) method like the fast sweeping method, but it tends to operate near the front like the fast marching method. It is reminiscent of the threshold method for finding the shortest path on a network, [F. Glover, D. Klingman, and N. Phillips, Oper. Res., 33 (1985), pp. 65-73]. We demonstrate the numerical efficiency of the improved methods on a number of examples.

Variance controlled shear stiffness images for MRE data

In the magnetic resonance elastography experiment we consider a harmonically oscillating mechanical force applied to the boundary surface of a phantom and synchronized with the motion encoding gradient. The phantom is symmetric in the direction of the applied mechanical force and the vector component in that direction decouples from the other components and satisfies a Helmholtz equation. We present a local inversion method to determine the shear wave speed that: (1) treats the phase and amplitude of the data differently; (2) computes derivatives of the data by using statistically justified filtering; and (3) varies filters according to SNR. We test our methods on data from Mayo Clinic and recover the position and stiffness of a 3 mm diameter inclusion

A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions

In this paper, we develop a third order accurate fast marching method for the solution of the eikonal equation in two dimensions. There have been two obstacles to extending the fast marching method to higher orders of accuracy. The first obstacle is that using one-sided difference schemes is unstable for orders of accuracy higher than two. The second obstacle is that the points in the difference stencil are not available when the gradient is closely aligned with the grid. We overcome these obstacles by using a two-dimensional (2D) finite difference approximation to improve stability, and by locally rotating the grid 45 degrees (i.e., using derivatives along the diagonals) to ensure all the points needed in the difference stencil are available. We show that in smooth regions the full difference stencil is used for a suitably small enough grid size and that the difference scheme satisfies the von Neumann stability condition for the linearized eikonal equation. Our method reverts to first order accuracy near caustics without developing oscillations by using a simple switching scheme. The efficiency and high order of the method are demonstrated on a number of 2D test problems.

Shear wave speed recovery in sonoelastography using crawling wave data

The crawling wave experiment, in which two harmonic sources oscillate at different but nearby frequencies, is a development in sonoelastography that allows real-time imaging of propagating shear wave interference patterns. Previously the crawling wave speed was recovered and used as an indicator of shear stiffness; however, it is shown in this paper that the crawling wave speed image can have artifacts that do not represent a change in stiffness. In this paper, the locations and shapes of some of the artifacts are exhibited. In addition, a differential equation is established that enables imaging of the shear wave speed, which is a quantity strongly correlated with shear stiffness change. The full algorithm is as follows: (1) extract the crawling wave phase from the spectral variance data; (2) calculate the crawling wave phase wave speed; (3) solve a first-order PDE for the phase of the wave emanating from one of the sources; and (4) compute and image the shear wave speed on a grid in the image plane.

Anisotropy Reconstruction from Wave Fronts in Transversely Isotropic Acoustic Media

This paper considers an inverse problem for a transversely isotropic three-dimensional acoustic medium, where there is one preferred direction called the fiber direction along which the wave propagates fastest and there is no preferred wave propagation direction in the isotropic plane, which is the plane orthogonal to the fiber direction. In this medium the parameters to be recovered are (1) the wave speed for a wave propagating in the direction along the fiber; (2) the wave speed for a wave propagating in any direction which is orthogonal to the fiber direction; and (3) the unit fiber direction itself. So four scalar functions are to be recovered. The data are the positions of four distinct wave fronts as the corresponding waves propagate through the medium. The mathematical relation, which is the Eikonal equation, between the wave front locations and the four unknown functions, is nonlinear. Here it is established, perhaps surprisingly, that corresponding to the given data set, there can be up to four p...