Application of Topological Sensitivity toward Tissue Elasticity Imaging Using Magnetic Resonance Data

This work investigates the application of topological sensitivity (TS) toward reconstructing and identifying tissue anomalies from magnetic resonance elastography (MRE) measurements. The basic idea of MRE imaging is to apply time–harmonic vibration to a tissue inside the magnetic resonance (MR) scanner and to capture the induced three-dimensional motion response (i.e., wave patterns) via suitable sequencing of the phase-coordinated, freeze-frame MR scans. To account for the facts that (1) the displacement of a material point captured by the MRE signifies the volume average over a reference voxel size, and (2) any given measurement voxel may contain, at least partially, the lesion of interest, the concept of TS is extended to allow for (1) averaged volumetric measurements and (2) overlapping interrogation and measurement domains. To circumvent the difficulties involved in the numerical modeling of an entire organ, the latter is subdivided into an array of cubic subdomains, each consisting of $N^3$ voxels and playing the role of the reference body for TS imaging. To accomplish the sought uncoupling, the triaxial displacements along the boundary of each subdomain (captured by the MRE) are deployed as the Dirichlet data, thus specifying a separate boundary value problem, whereas the internal measurements within the subdomain are taken as the observations for lesion reconstruction. The performance of the proposed methodology for lesion reconstruction and material characterization is examined via both numerical examples and a preliminary application to in vivo MRE data. The sensitivity of the technique to measurement errors is investigated by introducing random noise to synthetic observations. The results highlight the potential of the TS method toward elevating the resolution of tissue (visco)elasticity reconstruction by MR imaging.