Dynamic Analysis of Multiply Connected Bodies for Magnetic Resonance Elastography

This work centers around the design, manufacturing, analysis, and experimental use of hydrogel phantoms in magnetic resonance elastography. The first chapter is an introduction to some basic concepts in engineering math and medical imaging that the reader might find useful to understand the rest of the work. In the second chapter, the solution to a concentric cylindrical body undergoing harmonic oscillations is derived. Then the phantom construction, experimental procedure, parameter estimation, and the forward problem are described. To validate the model, the shear moduli estimates are used as inputs to solve the forward problem with a finite element model (FEM) which is compared with experimental results. The immediate consequences of this work as well its implications for future research are considered. In the third chapter, the scattering and diffraction of a cylindrically converging transverse shear wave in a viscoelastic isotropic medium by a spherical heterogeneity is analytically solved. The wave field is determined for a hydrogel bead suspended in a different hydrogel that fills a glass test tube. Numerical examples showing the effect on displacement fields of varying the stiffness of the inclusion are presented. The fourth chapter is very similar to the second chapter, save that the phantom has a spherical heterogeneity rather than a concentric cylinder. Further, only the results of the forward problem are compared to the experiment to validate the mathematical model. The concluding chapter re-states what has preceded it, ending with some remarks on future work.