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In magnetic particle imaging the concentration of superparamagnetic iron oxide nanoparticles is determined by measuring the particle’s nonlinear response to an applied magnetic field. The particles are highly sensitive to the dynamic magnetic field which allows a rapid data acquisition. As a result magnetic particle imaging benefits from a high temporal resolution and can reach high spatial resolutions. But model-based reconstruction techniques are still not able to reach the quality of data-based approaches. In the latter case the linear system function is determined by a time-consuming measurement process which also has negative implications for the spatial resolution of the reconstructions. Common model approaches are overly simplified leading to reconstructions of minor quality. We aim for the formulation of a nonlinear parameter identification problem which is able to deal with model errors while reconstructing a sparse concentration. For this purpose we use a total least squares approach to simultaneously reconstruct the tracer concentration and deviations in the system matrix. The starting point is a commonly used model which is investigated with respect to the simplifying assumptions to derive a formal definition of the problem. Sparsity constraints are introduced for the concentration function and reconstructions are obtained from publicly available data by minimizing a Tikhonov-type functional. Data-based as well as model-based reconstructions are computed and improved by using the total least squares approach.