Tracking of tubular elongated structures is an important goal in a wide range of biomedical imaging applications. A Bayesian tube tracking algorithm is presented that allows to easily incorporate a priori knowledge. Because probabilistic tube tracking algorithms are computationally complex, steps towards a computational efficient implementation are suggested in this paper. The algorithm is evaluated on 2D and 3D synthetic data with different noise levels and clinical CTA data. The approach shows good performance on data with high levels of Gaussian noise.