Patient-specific computational flow analysis of the coronary artery can provide valuable information that the doctor can use in making treatment decisions. Reliable analysis requires addressing a number of computational challenges. The challenges include the dynamic nature of the analysis due to the arterial motion, accurately representing the flow near the arterial wall, and nonlinearity of the arterial-wall material. Our core technology in addressing these challenges is the Space-Time Variational Multiscale method, which, because of its moving-mesh nature, maintains the high-resolution representation of the boundary layers near the wall as the artery moves. The isogeometric discretization increases the fluid mechanics accuracy even more by providing smoother representation of the wall geometry and more accuracy in the flow solution. The image-based geometries do not come from a zero-stress state (ZSS) of the arterial wall, which we need in nonlinear structural mechanics, and we handle that with a special method for computing the element-based ZSS. In our comparative studies, we compute the unsteady flow field i) for a given, fixed shape of the artery, ii) for wall deformation driven by the physiological blood pressure, iii) for wall motion fully prescribed based on the time-dependent medical images, and iv) for wall deformation driven by the physiological blood pressure in combination with the motion of a thin strip along the arterial surface coming from the medical images.