An investigation into the relationship between inhomogeneity and wave shapes in phantoms and ex vivo skeletal muscle using Magnetic Resonance Elastography and finite element analysis

Soft biological tissues such as skeletal muscle and brain white matter can be inhomogeneous and anisotropic due to the presence of fibers. Unlike biological tissue, phantoms with known microstructure and defined mechanical properties enable a quantitative assessment and systematic investigation of the influence of inhomogeneities on the nature of shear wave propagation. This study introduces a mathematical measure for the wave shape, which the authors call as the 1-Norm, to determine the conditions under which homogenization may be a valid approach. This is achieved through experimentation using the Magnetic Resonance Elastography technique on 3D printed inhomogeneous fiber phantoms as well as on ex-vivo porcine lumbus muscle. In addition, Finite Element Analysis is used as a tool to decouple the effects of directional anisotropy from those of inhomogeneity. A correlation is then established between the values of 1-Norm derived from the wave front geometry, and the spacing (d) between neighboring inhomogeneities (spherical inclusions or fibers and fiber intersections in phantoms and muscle). Smaller values of 1-Norm indicate less wave scattering at the locations of fiber intersections, which implies that the wave propagation may be approximated to that of a homogeneous medium; homogenization may not be a valid approximation when significant scattering occurs at the locations of inhomogeneities. In conclusion, the current study proposes 1-Norm as a quantitative measure of the magnitude of wave scattering in a medium, which can potentially be used as a homogeneity index of a biological tissue.