  This work introduces a 4D-flow magnetic resonance imaging (MRI) pressure
reconstruction method which employs weighted least-squares (WLS) for pressure
integration. Pressure gradients are calculated from the velocity fields, and
velocity errors are estimated from the velocity divergence for incompressible
flow. Pressure gradient errors are estimated by propagating the velocity errors
through Navier-Stokes momentum equation. A weight matrix is generated based on
the pressure gradient errors, then employed for pressure reconstruction. The
pressure reconstruction method was demonstrated and analyzed using synthetic
velocity fields as well as Poiseuille flow measured using in vitro 4D-flow MRI.
Performance of the proposed WLS method was compared to the method of solving
the pressure Poisson equation which has been the primary method used in the
previous studies. Error analysis indicated that the proposed method is more
robust to velocity measurement errors. Improvement on pressure results was
found to be more significant for the cases with spatially-varying velocity
error level, with reductions in error ranging from 50% to over 200%. Finally,
the method was applied to flow in a patient-specific cerebral aneurysm.
Validation was performed with in vitro flow data collected using Particle
Tracking Velocimetry (PTV) and Shake the Box (STB) method, and in vivo flow
measurement obtained using 4D-flow MRI. Pressure calculated by WLS, as opposed
to the Poisson equation, was more consistent with the flow structures and
showed better agreement between the in vivo and in vitro data. These results
suggest the utility of WLS method to obtain reliable pressure field from
clinical flow measurement data.
