A One-dimensional Finite Element Method for Simulation-based Medical Planning for Cardiovascular Disease

Abstract

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…Alternatively, a high-order finite volume scheme is presented in [129] and a space-time finite element method is proposed in [204]. Recently, a series of benchmark test cases with an increasing degree of complexity is presented in [35] to compare different numerical schemes.…”

Patient-Specific Operative Planning for Aorto-Femoral Reconstruction Procedures

“…Finally, the 1-D simulation methods described in the next section require medial axis paths and circular cross section segments. Porcine in-vivo studies and other experiment work (see [6]) seem to indicate that that the global impact of approximating three-dimensional junctions with the lofting techniques described above are a second order effect compared to inaccuracies in boundary conditions. See section 4 for further discussion on possible improvements to the image segmentation.…”

“…With the assumptions that (1) blood flow velocity along the vessel axis is much greater than the flow perpendicular to the vessel axis, (2) blood can be approximated as a Newtonian fluid, and (3) the velocity profile along the axis is a scaled version of a Poiseuille cross-sectional velocity profile function, a non-linear one-dimensional equation for pulse wave propagation in elastic blood vessels has been derived [6]. The space-time finite element formulation for solving the one-dimensional problem discussed in [6] has been integrated into the present software system (see Fig.…”

“…The space-time finite element formulation for solving the one-dimensional problem discussed in [6] has been integrated into the present software system (see Fig. 2).…”

“…inadequate flow to lower extremity tissue). As discussed in [6], one-dimensional finite element methods can be used to predict volumetric distributions and pressure losses. In addition to simulating resting conditions, volumetric flow distribution during exercise can be estimated by dilation and constriction of the distal vascular beds as described below.…”