“Fast” solution of the three-dimensional inverse problem of quasi-static elastography with the help of the small parameter method

We consider direct and inverse problems of three-dimensional quasi-static elastography underlying a cancer diagnosis method. They are based on a model of a tissue exposed to surface compression with deformations obeying linear elasticity laws. The arising three-dimensional displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by a variable Young’s modulus and a constant Poisson ratio. The problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. This is the direct problem. The inverse problem is to find the Young modulus distribution from given tissue displacements. A significant increase in Young’s modulus within a certain tissue domain suggests possible malignancy. Under certain assumptions, simple formulas for solving both direct and inverse problems of three-dimensional quasi-static elastography are derived. Three-dimensional inverse test problems are solved numerically with the help of the proposed formulas. The resulting approximate solutions agree fairly well with the exact model solutions. The computations based on the formulas require only several tens of milliseconds on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.