Abstract. The cardiac valvular apparatus, composed of the aortic, mitral, pulmonary and tricuspid valve, is an essential part of the anatomical, functional and hemodynamic characteristics of the heart and the cardiovascular system as a whole. Valvular heart diseases often involve multiple dysfunctions and require joint assessment and therapy of the valves. In this paper, we propose a complete and modular patient-specific model of the cardiac valvular apparatus estimated from 4D cardiac CT data. A new constrained Multi-linear Shape Model (cMSM), conditioned by anatomical measurements, is introduced to represent the complex spatiotemporal variation of the heart valves. The cMSM is exploited within a learning-based framework to efficiently estimate the patient-specific valve parameters from cine images. Experiments on 64 4D cardiac CT studies demonstrate the performance and clinical potential of the proposed method. To the best of our knowledge, it is the first time cardiologists and c...