Permanent metallic implants, such as dental fillings and cardiac devices, generate streaks-like artefacts in computed tomography (CT) images. In this article, we propose a strategy to perform metal artefact reduction (MAR) that relies on the total variation-H-1 inpainting, a variational approach based on a fourth-order total variation (TV) flow. This approach has never been used to perform MAR, although it has been profitably employed in other branches of image processing. A systematic evaluation of the performance is carried out. Comparisons are made with the results obtained using classical linear interpolation and two other partial differential equation-based approaches relying, respectively, on the Fourier's heat equation and on a second order TV flow. Visual inspection of both synthetic and real CT images, as well as computation of similarity indexes, suggests that our strategy for MAR outperforms the others considered here, as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field.