[Paper Review] Physics-informed neural networks for blood flow inverse problems

Jeremías Garay, Jocelyn Dunstan+2 more|arXiv (Cornell University)|Aug 2, 2023

Electrical and Bioimpedance TomographyEngineering3 citations

TL;DR

This paper proposes a physics-informed neural network (PINN) framework to solve inverse problems in hemodynamics by estimating Windkessel model parameters and reconstructing 3D velocity fields from 2D noisy, sparse MRI-like measurements. The method couples the 3D Navier-Stokes equations with a reduced-order Windkessel model, enforcing physical laws as loss terms, and achieves accurate parameter estimation in steady flow and moderate reconstruction accuracy in transient flow, demonstrating potential for patient-specific hemodynamic modeling.

ABSTRACT

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving inverse problems, especially in cases where no complete information about the system is known and scatter measurements are available. This is especially useful in hemodynamics since the boundary information is often difficult to model, and high-quality blood flow measurements are generally hard to obtain. In this work, we use the PINNs methodology for estimating reduced-order model parameters and the full velocity field from scatter 2D noisy measurements in the ascending aorta. The results show stable and accurate parameter estimations when using the method with simulated data, while the velocity reconstruction shows dependence on the measurement quality and the flow pattern complexity. The method allows for solving clinical-relevant inverse problems in hemodynamics and complex coupled physical systems.

Motivation & Objective

• To address the challenge of estimating hemodynamic parameters and velocity fields from incomplete, noisy, and sparse clinical measurements in patient-specific blood flow simulations.
• To overcome the limitations of traditional inverse methods that rely on complete physical models and are sensitive to modeling assumptions.
• To develop a flexible, data-driven framework using PINNs that integrates physical laws directly into the neural network training process for improved robustness and accuracy.
• To enable clinical application of personalized hemodynamic models by using only limited, realistic MRI-like data and physical constraints.

Proposed method

• A physics-informed neural network (PINN) is trained to solve the coupled 3D Navier-Stokes equations and a three-element Windkessel model for aortic hemodynamics.
• The PINN uses 2D noisy, sparse velocity measurements as input, along with a mean pressure curve and boundary conditions, to train the network.
• Physical laws—Navier-Stokes equations, non-slip wall condition, mass conservation, and Windkessel model equations—are embedded as loss functions in the training objective.
• A vector potential representation is employed to enforce mass conservation exactly in the velocity field, improving estimation accuracy at the cost of increased computational time.
• The loss function includes terms for data fidelity, physical consistency (Navier-Stokes and Windkessel), and boundary conditions, with the outlet flow condition acting as a stabilizing constraint.
• The method is evaluated in both steady and transient flow regimes, with parameter estimation and velocity reconstruction assessed against reference solutions.

Experimental results

Research questions

• RQ1 Can PINNs accurately estimate Windkessel model parameters from 2D, noisy, and sparse MRI-like velocity measurements in a patient-specific aortic flow setting?
• RQ2 How does the inclusion of physical constraints (Navier-Stokes, non-slip, mass conservation) affect the accuracy of velocity field reconstruction and parameter estimation?
• RQ3 What impact does flow regime complexity (steady vs. transient) have on the performance of the PINN framework?
• RQ4 How does the use of a vector potential representation influence the accuracy and stability of the velocity reconstruction?
• RQ5 To what extent does the outlet flow condition compensate for inaccuracies in the wall boundary condition enforcement?

Key findings

• The PINN framework achieved high accuracy in estimating Windkessel parameters in the steady flow regime, closely matching reference values.
• In the transient flow regime, parameter estimation accuracy decreased moderately due to increased parameter count and data dimensionality, but remained within acceptable clinical ranges.
• Velocity reconstruction was stable and accurate in regions with high measurement density and simple flow patterns, but degraded near vessel walls and in complex flow regions.
• The non-slip wall condition was not perfectly satisfied due to optimization constraints, leading to some velocity vectors penetrating the vessel wall, though outlet flow values remained accurate.
• The use of a vector potential representation improved overall velocity estimation by enforcing mass conservation exactly, though it increased computational cost by a factor of four.
• The outlet flow condition acted as a stabilizing mechanism, compensating for wall velocity inaccuracies by reinforcing flow continuity in the lumen.