Blood flow measurements using 4D Phase Contrast blood flow imaging (PCI) provide an excellent fully non-invasive technique to assess the hemodynamics clinically in-vivo. Iterative reconstruction techniques combined with parallel MRI have been proposed to reduce the data acquisition time, which is the biggest drawback of 4D PCI. The novel LoSDeCoS technique combines these ideas with the separation into a low-rank and a sparse component. The high-dimensionality of the PC data renders it ideally suited for this approach. The proposed method is not limited to a single body region, but can be applied to any 4D flow measurement. The benefits of the new method are twofold: It allows to significantly accelerate the acquisition; and generates additional images highlighting temporal and directional flow changes. Reduction in acquisition time improves patient comfort and can be used to achieve better temporal or spatial resolution, which in turn allows more precise calculations of clinically important quantitative numbers such as flow rates or the wall shear stress. With LoSDeCoS, acceleration factors of 6-8 were achieved for 16 in-vivo datasets of both the carotid artery (6 datasets) and the aorta (10 datasets), while decreasing the Normalized Root Mean Square Error by over 10 % compared to a standard iterative reconstruction and by achieving similarity values of over 0.93. Inflow-Outflow phantom experiments showed good parabolic profiles and an excellent mass conservation.