Extracting centerlines of coronary arteries is a challenging but important task in clinical applications of cardiac CTA. In this paper, we propose a model-guided approach, the directional minimal path, for the centerline extraction. The proposed method is based on the minimal path algorithm and a prior coronary model is used. The model is first registered to the unseen image. Then, the start point and end point for the minimal path algorithm are provided by the model to automate the centerline extraction process. Also, the direction information of the coronary model is used to guide the path tracking of the minimal path procedure. This directional tracking improves the robustness and accuracy of the centerline extraction. Finally, the proposed method can automatically recognize the branches of the extracted coronary artery using the prior information in the model. We validated the proposed method by extracting the three main coronary branches. The mean accuracy of the 56 cases was 1.32+/-0.81 mm and the detection ratio was 88.7%.