Solution of the three-dimensional inverse elastography problem for parametric classes of inclusions

We study the three-dimensional inverse problem of elastography, that is finding the Young's modulus of a biological tissue from known values of its vertical displacements. In this way, one can find inclusions with Young's modulus several times higher than its known background value. Such inclusions are interpreted as tumours. A quasistatic statement of the problem is used in which the fragment of the tissue is considered as a linearly elastic body. It is assumed that the geometry of inclusions is specified parametrically, and the Young's modulus inside and outside the inclusions is constant. The task is reduced to finding the number of inclusions, parameters defining their shape and the Young's modulus inside the inclusions. To solve the problem, a special algorithm is proposed and justified. The results of numerical experiments on solving a three-dimensional model problems are presented. A comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected cross-sections of this domain. It is found that 2D inverse problems do not always allow one to find a true 3D solution. For one of solved inverse problems, a-posteriori error estimates for Young modulus and geometric parameters of inclusions are obtained.