  Time-resolved three-dimensional flow MRI (4D flow MRI) provides a unique
non-invasive solution to visualize and quantify hemodynamics in blood vessels
such as the aortic arch. However, most current analysis methods for arterial 4D
flow MRI use static artery walls because of the difficulty in obtaining a full
cycle segmentation. To overcome this limitation, we propose a neural
fields-based method that directly estimates continuous periodic wall
deformations throughout the cardiac cycle. For a 3D + time imaging dataset, we
optimize an implicit neural representation (INR) that represents a
time-dependent velocity vector field (VVF). An ODE solver is used to integrate
the VVF into a deformation vector field (DVF), that can deform images,
segmentation masks, or meshes over time, thereby visualizing and quantifying
local wall motion patterns. To properly reflect the periodic nature of 3D +
time cardiovascular data, we impose periodicity in two ways. First, by
periodically encoding the time input to the INR, and hence VVF. Second, by
regularizing the DVF. We demonstrate the effectiveness of this approach on
synthetic data with different periodic patterns, ECG-gated CT, and 4D flow MRI
data. The obtained method could be used to improve 4D flow MRI analysis.
