Approximate Steady State Models for Magnetic Resonance Elastography

Abstract

MRE (magnetic resonance elastography) is a new diagnostic modality for measuring the stiffness of soft tissues in a living body. MRE itself measures only the displacements of waves propagating in the tissues, but the MRE-measured data can be linked to the stiffness through a proper model for wave propagation in the tissues and by solving some inverse problem under this model. Soft tissues of a living body are usually considered to be viscoelastic and nearly incompressible. In this paper we model the soft tissues as an isotropic Kelvin–Voigt material and interpret the meaning of nearly incompressible by an asymptotic analysis. As a consequence of this analysis, we show that the so-called modified Stokes system approximates very well the isotropic nearly incompressible Kelvin–Voigt model, and this explains why we can see only transverse waves inside soft tissues even if we inject longitudinal vibrations into the tissues from their surfaces. Further, by comparing the results of numerical simulations with MRE-measured data, we show that the modified Stokes system is a proper model.