Solution of the two-dimensional inverse problem of quasistatic elastography with the help of the small parameter method

The direct and inverse problems of two-dimensional quasi-static elastography are considered within a tissue deformation model in which the tissue is treated as an elastic body exposed to surface compression. In the approximation of plane linear elastic deformations, the arising displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by Young’s modulus and the constant Poisson ratio of the tissue. This problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. The corresponding solution procedure is studied, and, under certain assumptions, simple formulas for solving both direct and inverse problems of two-dimensional quasi-static elastography are derived. Direct and inverse test problems are solved numerically with the help of the proposed formulas. The results agree rather well with the model solutions. The computations based on the formulas require fractions of microsecond on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.