Transcatheter aortic valve implantation (TAVI) has become a key treatment for severe aortic stenosis, especially for patients unsuitable for surgery. Since its introduction in 2002, TAVI has advanced significantly due to improvements in imaging, operator skills, and device engineering. Despite these innovations, challenges in device sizing and positioning remain, complicating outcome predictions. Computational modelling is a powerful tool to aid TAVI device design and to understand its interactive behaviour with the aortic root during the deployment. Previous studies often simplified tissue properties, neglected patient-specific geometries or omitted crucial elements such as leaflets and fabric. This paper presents a numerical framework capable of simulating the whole crimping and deployment process of a full TAVI device in a patient-specific aortic root including the native leaflets and calcifications. We conduct a comprehensive investigation into the mechanical behaviour of the TAVI and its interactions with patient-specific aortic root through dynamic finite element analysis during the deployment process, with validation against experimental results. Additionally, we examined the influence of applied pressure during balloon inflation on the interactive dynamics of the entire model. The study concludes that selecting optimal balloon pressures is crucial for enhancing TAVI device performance and reducing complications. Numerical simulations demonstrate that appropriate balloon pressure ensures sufficient flow area and effective contact pressure between the TAVI and the aortic root, while minimising deformation and the risk of paravalvular leak.