A Meshless Solver for Blood Flow Simulations in Elastic Vessels Using Physics-Informed Neural Network

TL;DR

This work introduces a mesh-free, physics-informed neural network approach to simulate blood flow in elastic vessels by solving the incompressible Navier–Stokes equations in an Arbitrary Lagrangian-Eulerian framework coupled with a linear elastic vessel wall. The method uses three dedicated neural networks for displacement, velocity, and pressure, with a tailored training scheme that alternates between fluid and solid subproblems and employs adaptive loss weighting to ensure stable convergence. Key contributions include a novel multi-network architecture with separate velocity and pressure nets, an alternating activation scheme for improved training, and demonstrated GPU-accelerated performance on cylinder-like vessels and plaque-affected geometries, achieving accuracy comparable to high-resolution FEM with substantially reduced computation time. The results illustrate the method’s robustness to complex geometries and its potential for patient-specific vascular simulations and rapid scenario testing in clinical contexts.

Abstract

Investigating blood flow in the cardiovascular system is crucial for assessing cardiovascular health. Computational approaches offer some non-invasive alternatives to measure blood flow dynamics. Numerical simulations based on traditional methods such as finite-element and other numerical discretizations have been extensively studied and have yielded excellent results. However, adapting these methods to real-life simulations remains a complex task. In this paper, we propose a method that offers flexibility and can efficiently handle real-life simulations. We suggest utilizing the physics-informed neural network (PINN) to solve the Navier-Stokes equation in a deformable domain, specifically addressing the simulation of blood flow in elastic vessels. Our approach models blood flow using an incompressible, viscous Navier-Stokes equation in an Arbitrary Lagrangian-Eulerian form. The mechanical model for the vessel wall structure is formulated by an equation of Newton's second law of momentum and linear elasticity to the force exerted by the fluid flow. Our method is a mesh-free approach that eliminates the need for discretization and meshing of the computational domain. This makes it highly efficient in solving simulations involving complex geometries. Additionally, with the availability of well-developed open-source machine learning framework packages and parallel modules, our method can easily be accelerated through GPU computing and parallel computing. To evaluate our approach, we conducted experiments on regular cylinder vessels as well as vessels with plaque on their walls. We compared our results to a solution calculated by Finite Element Methods using a dense grid and small time steps, which we considered as the ground truth solution. We report the relative error and the time consumed to solve the problem, highlighting the advantages of our method.

Figures (11)

• Figure 1: The illustration for the Arbitrary Lagrangian Euler form. Left: reference configuration; Right: current configuration.
• Figure 1: Illustration of Network Architecture: Three networks are employed. The displacement network $N_d$ solves the displacement vector $\boldsymbol{d}$ for both fluid and solid domains. The velocity network $N_u$ computes fluid velocity $\boldsymbol{u}$, while the pressure network $N_p$ handles fluid pressure $p$. Green lines indicate data flow requiring gradient calculation, while black lines denote no gradient delivery. Dotted black lines indicate computed results for loss function calculation without enabling back-propagation, as explained in Section \ref{['sec:scheme']}.
• Figure 1: Illustration of velocity magnitude (first row), pressure (second row), displacement magnitude (third row) and displacement magnitude for wall (forth row) variations at time points 0.01, 0.21, 0.41, 0.61, and 0.81. The first and the second don't contain the vessel wall. Deformations of the shape are enlarged by a factor of 10 for better visibility.
• Figure 1: Illustration of velocity magnitude (first row), pressure (second row), displacement magnitude (third row) and displacement magnitude for wall (forth row) variations at time points 0.04, 0.07, 0.10, and 0.14. The first and the second don't contain the vessel wall. Deformations of the shape are enlarged by a factor of 10 for better visibility.
• Figure 2: Temporal profiles of velocity magnitude and pressure at different points. From left to right, the upper three plots illustrate velocity magnitude at the centerline's inlet, middle, and outlet, while the lower three represent pressure magnitude for the corresponding points. Legend notations: H1, H2, H3, and H4 indicate grids with minimum lengths of $\frac{1}{2}\times 10^{-3}$, $\frac{1}{4}\times 10^{-3}$, $\frac{1}{8}\times 10^{-3}$, and $\frac{1}{16}\times 10^{-3}$ meter, respectively. T1, T2, and T3 denote time step lengths of $1\times 10^{-1}$, $1\times 10^{-2}$, and $1\times 10^{-3}$ second.