Assessment of 4D flow MRI's quality by verifying its Navier-Stokes compatibility

4D Flow Magnetic Resonance Imaging (MRI) is the state-of-the-art technique to comprehensively measure the complex spatio-temporal and multidirectional patterns of blood flow. However, it is subject to artifacts such as noise and aliasing, which due to the 3D and dynamic structure is difficult to detect in clinical practice. In this work, a new mathematical and computational model to determine the quality of 4D Flow MRI is presented. The model is derived by assuming the true velocity satisfies the incompressible Navier-Stokes equations and that can be decomposed by the measurements u -> meas$$ {\overrightarrow{u}}_{meas} $$ plus an extra field w ->$$ \overrightarrow{w} $$. Therefore, a non-linear problem with w ->$$ \overrightarrow{w} $$ as unknown arises, which serves as a measure of data quality. A stabilized finite element formulation tailored to this problem is proposed and analyzed. Then, extensive numerical examples-using synthetic 4D Flow MRI data as well as real measurements on experimental phantom and subjects-illustrate the ability to use w ->$$ \overrightarrow{w} $$ for assessing the quality of 4D Flow MRI measurements over space and time.