Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

Abstract

We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. Intravascular pressure and velocity are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

Keywords: Finite element methods; Finite volume methods; Immersed methods; Vascularized tissues.

Figure 1

Sketch of a three-dimensional thin vessel, identified via its cross-sectional radius and its centerline.

Figure 2

Left: The three-dimensional domain considered for the bifurcation model. The corresponding one-dimensional description is obtained considering vessel centerlines. Right: Comparison between the displacement obtained in the fully resolved 3D model (left part) and the result of the hypersingular model (right part). The right part of the figure shows also the one-dimensional vessel.

Figure 3

The flow rate imposed for solving the one-dimensional hemodynamics in the bifurcation setting.

Figure 4

Comparison of the average forces measured on the lateral faces (0 and 1) (left) and on the front and back faces (2 and 3) (right), between the matching grid and the singular method.

Figure 5

Left: Randomly distributed vasculature, irrorating two thousands randomly distributed points in the sample. The root point is situated in the lower left corner. The average radius of the vessels is about 0.012 mm. Right: Flow rate imposed at the bottom-left corner for simulating the one-dimensional hemodynamics.

Figure 6

Left. Average pressure and standard deviation within the fluid vasculature over time in the considered period. The dashed lines indicated the time steps corresponding to the plots in Fig. 7. Right. Resulting absolute value of total forces on opposite faces, using the same numbering as in Fig. 2.

Figure 7

Contour plots of the 3D displacement field and plot of the 1D pressure on the vessels at different times. The considered time steps are also indicated with dashed lines in Fig. 6.

Figure 8

Visualization of the unstructured tetrahedral mesh obtained choosing $h_{1D}^{3D}=0.01\text{mm}$ and constraining the discrete hypersingular points to be vertices of tetrahedra. The red segments show the one-dimensional vasculature, while the blue lines depicts the intersection of the volume elements with the cutting planes.