Vibro-elastography: direct FEM inversion of the shear wave equation without the local homogeneity assumption

To produce ultrasound images of tissue elasticity, the vibro-elastography technique involves applying a steady-state multi-frequency vibration to tissue, estimating displacements from ultrasound echo data, and using the estimated displacements in an inverse elasticity problem with the shear modulus spatial distribution as the unknown. The governing equation used requires all three displacement components to fully solve the inverse problem. However, using ultrasound, only the axial component of the displacement can be measured accurately. Therefore, simplifying assumptions must be used. Usually, the equations of motion are transformed into a Helmholtz equation by assuming tissue incompressibility and local homogeneity. In this paper, we remove the local homogeneity assumption which causes significant imaging artifacts in areas of varying elasticity. We introduce a new finite element based direct inversion technique in which only the coupling terms in the equation of motion are ignored, so it can be used with only one component of the displacement. The use of multi-frequency excitation also allows us to obtain multiple measurements and reduce artifacts in areas where the displacement of one frequency is close to zero. The proposed method was tested in simulations and experiments against a conventional approach in which the local homogeneity is used. The results show significant improvements in elasticity imaging with the new method.