Complete and accurate segmentation of the vessel from 3D (three dimensional) CT images is challenging due to lowcontrast, combined with noise, and high variation of vessel size. We describe a novel centerline-based method to produce the accurate vessel segmentation. It starts with locating vessel centerline which will be used as guidance, followed by graph cuts, with edge-weights depending on the intensity of the centerline. The main advantage of our framework is that it detects vessel boundary in problematic regions that contain small vessels and noise. A comparison has been made with two state-of-the-art vessel segmentation methods. Quantitative results on synthetic data indicate that our method is more accurate than these methods. Furthermore, experimental results on clinical data have shown that our method is capable of detecting more detailed information of vessel. It is more accurate and robust that these state-of-the-art methods and is, therefore, more suited for automatic vessel extraction.