This article presents the solution of a special three-dimensional inverse elastography problem: within the quasi-static model of a linear-elastic isotropic body under small surface compressions nds Young's modulus distribution in the investigated biological tissues using known vertical values of the displacements of tissues in it. A goal of this investigation is to detect local inclusions in tissue, interpreted as tumors and having Young's modulus values that are signicantly dierent from the known background value. Additionally, it is assumed that the Young's modulus is a constant function inside and outside of the required inclusions, the geometry of which is given parametrically. This inverse problem leads to the solution of a nonlinear operator equation, which by a variational method reduces to an extremal problem of nding the number of inclusions, the parameters dening their shape and the Young modulus for each inclusion. Algorithmically, the problem is solved using a modication of the method of extending compacts by V. K. Ivanov and I.N. Dombrovskaya. As an illustration of the algorithm, examples of solving model inverse problems with inclusions in the form of balls are given. For the found solution of one of the model problems a posteriori error estimate of the Young's modulus distribution is carried out.

02.60.-x Numerical approximation and analysis

62.20.de Elastic moduli

87.10.Pq Elasticity theory

$^1$National Research Nuclear University MEPhI

$^2$Moscow State University, Physics Faculty, chair of Mathematics.