Analytical solution based on spatial distortion for a time-harmonicdipole source in a transverse isotropic viscoelastic solid

Abstract

Elastography refers to mapping mechanical properties in a material by measuring wave motion in it using noninvasive optical, acoustic or magnetic resonance imaging methods. Elasticity and viscosity can depend on location and direction. A material with aligned fibers may have different elastic and viscosity values along the fibers versus across them. Reconstruction refers to converting wave measurements into a mechanical property map. To make reconstruction analytically tractable, isotropy and homogeneity are often assumed. But, isotropic homogeneity is often not the case of interest, when there are pathological conditions or hidden anomalies non-uniformly distributed in fibrous or layered structures. A strategy of spatial distortion to make an anisotropic problem become isotropic has been previously validated in two-dimensional transverse isotropic (TI) viscoelastic cases. Here, the approach is extended to the three-dimensional problem by considering a time-harmonic point force (dipole) in a TI viscoelastic material. The resulting wave field, exactly solvable using a Radon transform with numerical integration, is approximated via spatial distortion of the closed form analytical solution to the isotropic case. Different distortions are used depending on whether or not the polarization of the wave motion is orthogonal to the axis of isotropy resulting in differing levels of accuracy.