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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. The objective of this paper is to develop and test algorithms whose ultimate product is images of the shear wave speed of tissue mimicking phantoms. The data used in the algorithms are the front of the propagating shear wave. Here, we first develop techniques to find the arrival time surface given the displacement data from a transient elastography experiment. The arrival time surface satisfies the Eikonal equation. We then propose a family of methods, called distance methods, to solve the inverse Eikonal equation: given the arrival times of a propagating wave, find the wave speed. Lastly, we explain why simple inversion schemes for the inverse Eikonal equation lead to large outliers in the wave speed and numerically demonstrate that the new scheme presented here does not have any large outliers. We exhibit two recoveries using these methods: one is with synthetic data; the other is with laboratory data obtained by Mathias Fink's group (the

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.

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In this paper analytical methods for recovering coefficients from spectral data are discussed. Properties of these methods are given and mathematical models for which these methods might apply are presented.

In this paper we show that the coefficients in a second-order differential equation can be determined from the positions of the nodes for the eigenfunctions. We prove uniqueness results, derive approximate solutions, give error bounds and present numerical experiments.

The results indicate the fundamental role of altered viscoelastic properties of brain tissue during disease progression and tissue repair in NPH. Clinical improvement in NPH is associated with an increasing complexity of the mechanical network whose inherent strength, however, remains degraded.

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 secondorder 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 Lamé 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.