Vascular segmentation plays an important role in the assessment of peripheral arterial disease. The segmentation is very challenging especially for arteries with severe stenosis or complete occlusion. We present a cascading algorithm for vascular centerline tree detection specializing in detecting centerlines in diseased peripheral arteries. It takes a three-dimensional computed tomography angiography (CTA) volume and returns a vascular centerline tree, which can be used for accelerating and facilitating the vascular segmentation. The algorithm consists of four levels, two of which detect healthy arteries of varying sizes and two that specialize in different types of vascular pathology: severe calcification and occlusion. We perform four main steps at each level: appropriate parameters for each level are selected automatically, a set of centrally located voxels is detected, these voxels are connected together based on the connection criteria, and the resulting centerline tree is corrected from spurious branches. The proposed method was tested on 25 CTA scans of the lower limbs, achieving an average overlap rate of 89% and an average detection rate of 82%. The average execution time using four CPU cores was 70 s, and the technique was successful also in detecting very distal artery branches, e.g., in the foot.