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

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Дата последнего поиска статьи во внешних источниках: 14 октября 2020 г.

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[1] Leonov A. S., Sharov A. N., Yagola A. G. Solution of the three-dimensional inverse elastography problem for parametric classes of inclusions. published online // Inverse Problems in Science and Engineering. — 2020. — no. published online. — P. 1–14. 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 during external compression. Solving this inverse problem, one can find inclusions with Young’s modulus several times higher than its known background value. Such inclusions are interpreted as tumors. A quasistatic statement of the problem is used in which the studied fragment of the tissue is considered as a linearly elastic body undergoing small surface compressions. It is also assumed that the geometry of inclusions can be specified parametrically, and the Young’s modulus inside and outside the inclusions is constant. Then the task is reduced to finding the number of inclusions, parameters defining their shape and the values of 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 with inclusions in the form of a ball and ellipsoids. In addition, a comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected coordinate cross-sections of this domain. The data for two-dimensional problems are corresponding coordinate sections of three-dimensional data. It is established that two-dimensional inverse problems do not always allow one to find a true three-dimensional solution. For one of solved inverse problems, a-posteriori error estimates for found Young modulus and geometric parameters of inclusions are obtained. [ DOI ]

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