Solution of the Inverse Elastography Problem in Three Dimensions for a Parametric Class with A Posteriori Error Estimation

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The paper presents the solution of a special three-dimensional inverse elastography problem: given a quasistatic model of a linear-elastic isotropic body subject to surface forces, to find the Young’s modulus distribution in the biological tissues under study using the known values of vertical displacements of these tissues. This study is aimed at detecting local inclusions in the tissue interpreted as tumors with values of the Young’s modulus that are significantly different from the known background. In addition, it is assumed that Young’s modulus is a constant function inside the unknown inclusions of a parametrically given geometry. This inverse problem leads to the solution of a nonlinear operator equation, which is reduced by a variational method to the extremum problem of finding the number of inclusions, parameters defining their shape, and the Young’s modulus for each inclusion. The problem is solved algorithmically by using a modification of the method of extending compacts by V. K. Ivanov and I. N. Dombrovskaya. To illustrate how the algorithm works, we give examples of solving model inverse problems with inclusions in the form of balls. A posteriori error estimation of the obtained distribution of Young’s modulus is carried out for the found solution to one of the model problems.