Joyce R. McLaughlin

In vivo viscoelastic properties of the brain in normal pressure hydrocephalus

Nearly half a century after the first report of normal pressure hydrocephalus (NPH), the pathophysiological cause of the disease still remains unclear. Several theories about the cause and development of NPH emphasize disease‐related alterations of the mechanical properties of the brain. MR elastography (MRE) uniquely allows the measurement of viscoelastic constants of the living brain without intervention. In this study, 20 patients (mean age, 69.1 years; nine men, 11 women) with idiopathic (n = 15) and secondary (n = 5) NPH were examined by cerebral multifrequency MRE and compared with 25 healthy volunteers (mean age, 62.1 years; 10 men, 15 women). Viscoelastic constants related to the stiffness (µ) and micromechanical connectivity (α) of brain tissue were derived from the dynamics of storage and loss moduli within the experimentally achieved frequency range of 25–62.5 Hz. In patients with NPH, both storage and loss moduli decreased, corresponding to a softening of brain tissue of about 20% compared with healthy volunteers (p < 0.001). This loss of rigidity was accompanied by a decreasing α parameter (9%, p < 0.001), indicating an alteration in the microstructural connectivity of brain tissue during NPH. This disease‐related decrease in viscoelastic constants was even more pronounced in the periventricular region of the brain. The results demonstrate distinct tissue degradation associated with NPH. Further studies are required to investigate the source of mechanical tissue damage as a potential cause of NPH‐related ventricular expansions and clinical symptoms. Copyright

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).

Unique identifiability of elastic parameters from time-dependent interior displacement measurement

We consider the question: what can be determined about the stiffness distribution in biological tissue from indirect measurements? This leads us to consider an inverse problem for the identification of coefficients in the second-order hyperbolic system that models the propagation of elastic waves. The measured data for our inverse problem are the time-dependent interior vector displacements. In the isotropic case, we establish sufficient conditions for the unique identifiability of wave speeds and the simultaneous identifiability of both density and the Lame parameters. In the anisotropic case, counterexamples are presented to exhibit the nonuniqueness and to show the structure of the set of shear tensors corresponding to the same given data.

Surveys on solution methods for inverse problems

Convergence Rates Results for Iterative Methods for Solving Nonlinear III-Posed Problems.- Iterative Regularization Techniques in Image Reconstruction.- A Survey of Regularization Methods for First-Kind Volterra Equations.- Layer Stripping.- The Linear Sampling Method in Inverse Scattering Theory.- Carleman Estimates and Inverse Problems in the Last Two Decades.- Local Tomographic Methods in Sonar.- Efficient Methods in Hyperthermia Treatment Planning.- Solving Inverse Problems with Spectral Data.- Low Frequency Electromagnetic Fields in High Contrast Media.- Inverse Scattering in Anisotropic Media.- Inverse Problems as Statistics.

Extremal eigenvalue problems for composite membranes, II

Given an open bounded connected set Ω ⊂RN and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in Ω that extremizes thekth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes toN dimensions the now classical one-dimensional work of M. G. Krein.

The Riccati method for the Helmholtz equation

The operator Riccati equation for the Dirichlet‐to‐Neumann map is derived from the exact operator factorization of the two‐dimensional variable coefficient Helmholtz equation. Numerical schemes are developed for the operator Riccati equation and its variant using a local eigenfunction expansion. This leads to a practical computational method for acoustic wave propagation over large range distances, since the boundary value problem of the Helmholtz equation is reduced to ‘‘initial’’ value problems that are solved by marching in the range. The efficiency and accuracy of the method is demonstrated by numerical experiments including the plane‐parallel range‐dependent waveguide benchmark problem proposed by the Acoustical Society of America.

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.

Inverse problems: recovery of BV coefficients from nodes

We consider the Sturm-Liouville problem on a finite interval with Dirichlet boundary conditions. Let the elastic modulus and the density be of bounded variation. Results for both the forward problem and the inverse problem are established. For the forward problem, new bounds are established for the eigenfrequencies. The bounds are sharp. For the inverse problem, it is shown that the elastic modulus is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the density is known. Similarly it is shown that the density is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the elastic modulus is known. Algorithms for finding piecewise constant approximations to the unknown elastic modulus or density are established and are shown to converge to the unknown function at every point of continuity. Results from numerical calculations are presented.

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.

Stability theorems for two inverse spectral problems

Stability results are presented for two second-order inverse spectral problems defined on a bounded interval. The asymptotic form for the spectral data is the same for both problems. It is shown that a change in the form of the differential operator produces a change in the stability result.