Inverse Problems In Dynamic Elasticity Imaging

Since the early 1990’s, elasticity imaging techniques are developed as a powerful supplement of the medical toolbox in diagnostic analysis and computer aided surgery. By solving a so-called inverse problem, information about the spatial variation of material parameters of soft (human) tissue are derived from displacement data, which can be measured noninvasively using standard imaging devices such as ultrasound or magnetic resonance tomography.
The terms of quasi-static and dynamic elastography refer to the type of load situation, by which the tissue in question is excited. The extension of the theoretical formulation and implementation of the underlying inverse problem in quasi-static elastography to time-harmonic approaches poses several additional challenges, which are addressed in detail within the course of this study. We propose a robust strategy for the reconstruction, which takes advantage of the high sensitivity of the accuracy in harmonic elastography to the choice of the starting point.
While not being reported in the literature up to now, the quite competing claims of quasi-static and time-harmonic elastography motivate a comprehensive comparison of these two techniques. Via a spectral decomposition of the curvature information of the underlying inverse problem, a clear explanation for an improved robustness of time- harmonic elastography in the presence of inaccuracies due to noise and/or numerical approximations can be given.
Several numerical examples confirm these findings as well as the efficiency of the proposed reconstruction strategy. In particular, it is shown that for moderately low frequencies, it is sufficient to use very coarse finite element meshes, so that the only additional computational cost stems from the worse conditioning of the system matrix.