Blood flow velocity estimation from x-ray densitometric data: an efficient numerical scheme for the inverse advection problem

Abstract

In previously published studies, blood flow velocity from x-ray biplane angiography was measured by solving an inverse advection problem, relating velocity to bolus densities summed across sections. Both spatial and temporal velocity variations were recovered through a computationally expensive parameter estimation algorithm. Here we prove the existence and uniqueness of the solution on three sub-domains of the plane defined by the axial position along the vessel and the time of the angiographic sequence. A fast direct scheme was designed in conjunction with a regularization step stemming from the volume flow conservation law applied on consecutive segments. Its accuracy and immunity towards noise were tested on both simulated and real densitometric data. The relative error between the estimated and expected velocities was less than 5% for more than 90% of the points of the spatiotemporal plane with simulated densities normalized to 1.0 and a Gaussian additive noise of standard deviation 0.01. For densities reconstructed from a biplane angiographic sequence, increase in velocity is used as a functional index for the stenosis ratio and to characterize the sharing of flow at bifurcation.