Introduction:: Strokes are one of the leading causes of death in the United States. The recommended time to treat an occlusive stroke patient is 2 to 3 hours from the onset of symptoms. The treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. Along with the arterial obstruction, the catheter may further reduce blood flow to the affected area of the brain potentially causing further damage to the brain tissue. A computer simulation to systematically plan such patient-specific treatments needs a network of about 107 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time.
Materials and Methods:: This work introduces a novel reduced-order mathematical formulation of the insertion of a rigid catheter in an elastic blood vessel. The governing equations are first order hyperbolic partial differential equations very similar to the 1D cylindrical fluid flow equations without a catheter. However, due to the modified flow area, a hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We solve the hyperbolic system using the Discontinuous Galerkin method to obtain the required hemodynamic variables (i.e., flow rate and pressure along the vessels). We compared this reduced-order model against a validated 3-dimensional CFD solver on a variety of idealized vessels as well as on a realistic but truncated arterial network tractable by 3D solvers.
Results, Conclusions, and Discussions:: In the context of cerebral blood flow, the results showed a clinically unimportant difference in the steady flow cases and captured the expected wave reflection phenomenon at the discontinuities in the domain for unsteady cases. At the catheter start location, the difference in the 1D and 3D steady pressure drop was ~2mmHg for an ideal cylindrical vessel and was ~6mmHg for a realistic arterial tree model when the mean intra-arterial pressure in both the models was ~100mmHg.
The major advantage of using this 1D solver as compared to 3D solvers is the ease of obtaining a discretized geometry for complex vasculatures including many arterial branches. The overall differences between the 1D and 3D models were below 7% (Fig. 1). These differences arise from 3D effects not included in the 1D model as well as the assumption of rigid walls in the 3D model. More analysis and incorporation of 3D effects including vessel curvatures, bifurcation, catheter end transition regions, etc. that are not currently included in the 1D model could improve its accuracy even further.
The accuracy and ease of use of this 1D computational model enables the use of CFD as a tool for investigating the effect of different treatment options and for planning endovascular interventions in strokes, etc.
