The importance of simulations in assessing cardiovascular diseases has been growing over the last decades. Recent improvements in Magnetic Resonance Imaging (MRI) have improved the information on blood hemodynamics, vasculature response, and different heart phenomena increasing the understanding of the mechanism behind. Different inverse problems applied to imaging, data-driven modeling, and hemodynamics simulations have been proposed in the last years as a helpful way to extract relevant clinical information from it. This is done with the ultimate goal of personalizing the diagnosis of the whole spectrum of cardiovascular diseases. However, these problems often suffer from lengthy and expensive calculations, strong sensitivity to the images’ artifacts, and complex post-processing steps, making these methods not applicable to clinical decision-making. In my doctoral research, I tackled these problems using physics-based modeling and reduced-order approaches. Using physical conservative principles, I defined a novel strategy for quantifying the level of physics-discrepancy of a recent imagining methodology called 4D Flow MRI. For that purpose, I validate the method using numerical data, real MRI images from a realistic model (phantom), and clinical data applied to volunteers. The second focus of the research was using a fast sequential data-assimilation algorithm called “Reduced-Order Unscented Kalman Filter” (ROUKF) for cardiovascular applications. I found that this problem is very prone to fail when clinical MRI data is used, mainly because this data is often contaminated with artifacts. For that reason, I developed a new way of setting this problem using highly aliased velocity measurements to estimate unknown boundary conditions.