The one-shot method is an approach to solve PDE-constrained optimization problems. In this thesis we study two models describing the deformation of biological tissue due to harmonic acoustic waves. These models are the Poisson equation and the linear elasticity model. By solving PDE-constrained optimization problems using the one-shot approach, the material parameters of tissue may be reconstructed. We study how Tikhonov regularization in the optimization problem affects the solution and the robustness of the method with noise in the target data. Numerical simulations are carried out using the finite element software FEniCS. Simulations are performed with both constructed data and later on with data from magnetic resonance elastography (MRE). The numerical results obtained in simulations with constructed data yielded qualitatively good results and were promising for this method to be used in application to MRE. However, carrying out simulations with MRE data, did not yield satisfactory solutions and thus changes of the model are necessary for further work.