The accuracy of fiber tracking on the basis of diffusion tensor magnetic resonance imaging (DTI) is affected by many parameters. To increase accuracy of the tracking algorithm, we introduce DTI with a fourth-order tensor. Tensor elements comprise information obtained by high angular resolution diffusion imaging (HARDI). We further developed the flattened high rank tensor (FLAHRT) method and applied it to the measured fourth-order tensor. We then compared FLAHRT with: 1) the standard tracking algorithm using a second-order tensor; and 2) existing techniques involving the representation of conventional second-order tensor components as a weighted average of fourth-order tensor elements. Such techniques have been formulated in recent DT studies to link high-rank to low-rank Cartesian diffusion tensors (DTs). Diagonalization of the second-order tensor decomposes the tensor into three eigenvalues and three eigenvectors, which in turn are used to describe the diffusivity profile of a particular voxel. Diagonalization after application of the FLAHRT method reveals six eigenvalues and six eigentensors, resulting in a more accurate description of the anisotropy. We performed fiber tracking based on the eigenvalues and eigentensors calculated with the FLAHRT and standard methods. We could show that the FLAHRT technique gives more consistent and more accurate results even with a data set acquired in 15 directions only. The decomposition of the fourth-order tensor into six eigentensors has the potential to describe six different fiber orientations within a voxel.