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

Abstract

We review and present new results on the transient elastography problem, where the goal is to reconstruct shear stiffness properties using interior time and space dependent displacement measurements. We present the 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 Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Université Paris VII). One numerical algorithm uses a geometrical optics expansion and the other utilizes the arrival time surface.

Anisotropy Reconstruction from Wave Fronts in Transversely Isotropic Acoustic Media

“…Since our medium is initially at rest, the wave propagates into the medium from the boundary with a propagating front. In our next two theorems, following [10,11], we establish that the wave whose propagation is governed by the above model has: (1) finite propagation speed; and (2) an arrival time, which we assume to be Lipschitz continuous, that, under this assumption, satisfies the Eikonal equation.…”

“…Note that (2.5) is merely a necessary condition for t =T (x) to be a characteristic surface with respect to the hyperbolic equation ρu tt = ∇· (M ∇u). If we suppose that t =T (x) is a noncharacteristic surface, we can draw a contradiction as done in Theorem 2.10 in [10], which is based on Theorem 3.6 in [5] and a lemma at page 544 of [6]. See [10] …”

“…and can be determined by the methods given in [10,12,13]. In later sections, we will assume that this speed, c(x), has been determined fromT so when we solve the inverse problem: find (c 66 /ρ, c 44 /ρ, f ) from multiple arrival times, we will assume we know bothT (x) and c(x).…”

“…To give some background about what is known in the isotropic case, so as to contrast to the anisotropic case, we recall that previously we have established uniqueness results [10,11], and the arrival time algorithm [10,12,13], to reconstruct wave speed in isotropic media. There we show that the positions of one propagating front established the wave speed uniquely; that there is at most one pair, the shear stiffness µ and the density ρ, corresponding to a given single displacement data as a function of space and time, provided the medium is initially at rest.…”

Viscoelasticity in polymers: Phenomenological to molecular mathematical modeling

“…Although our models (and those of Fung) have been derived under the widely accepted assumption that biotissue has viscoelastic properties, there are other groups [30][31][32] conducting research in this field that utilize non-dissipative constitutive equations related to elastic media for their modeling efforts. This raises the question as to whether the more complicated viscoelastic models, as proposed by Fung, our group, and others, really are required.…”