A new vascular structure segmentation method, which is based on a cylindrical flux-based higher order tensor (HOT), is presented. On a vessel structure, the HOT naturally models branching points, which create challenges for vessel segmentation algorithms. In a general linear HOT model embedded in 3D, one has to work with an even order tensor due to an enforced antipodal-symmetry on the unit sphere. However, in scenarios such as in a bifurcation, the antipodally-symmetric tensor embedded in 3D will not be useful. In order to overcome that limitation, we embed the tensor in 4D and obtain a structure that can model asymmetric junction scenarios. During construction of a higher order tensor (e.g. third or fourth order) in 4D, the orientation vectors lie on the unit 3-sphere, in contrast to the unit 2-sphere in 3D tensor modeling. This 4D tensor is exploited in a seed-based vessel segmentation algorithm, where the principal directions of the 4D HOT is obtained by decomposition, and used in a HOT tractography approach. We demonstrate quantitative validation of the proposed algorithm on both synthetic complex tubular structures as well as real cerebral vasculature in Magnetic Resonance Angiography (MRA) datasets and coronary arteries from Computed Tomography Angiography (CTA) volumes.