3D finite element solution to the dynamic poroelasticity problem for use in MR elastography

Magnetic Resonance Elastography (MRE) has emerged as a noninvasive, quantitative physical means of examining the elastic properties of biological tissues. While it is common to assume simplified elasticity models for purposes of MRE image reconstruction, it is well-accepted that many soft tissues display complex time-dependent behavior not described by linear elasticity. Understanding how the mechanical properties of biological materials change with the frequency of the applied stresses and strains is paramount to the reconstructive imaging techniques used in steady-state MRE. Alternative continuum models, such as consolidation theory, offer the ability to model tissue and other materials comprised of two distinct phases, generally consisting of an elastic solid phase and an infiltrating fluid. For these materials, the time-dependent response under a given load is a function not only of the elastic properties of the solid matrix, but also of the rate at which fluid can flow through the matrix under a pressure gradient. To better study the behavior of the dynamic poroelasticity equations, a three-dimensional finite element model was constructed. Confined, time-harmonic excitation of simulated soil and tissue-like columns was performed to determined material deformation and pore pressure distributions, as well as to identify the influence of the key model parameters under loading conditions and frequencies relevant in steady-state MRE. The results show that the finite element implementation is able to represent the analytical behavior with errors on the order of 1% over a broad range of frequencies. Further, differences between poroelastic and elastic responses in the column can be significant over the frequency range relevant to MRE depending on the value of hydraulic conductivity assumed for the medium.