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Uncovering near-wall blood flow from sparse data with physics-informed neural networks

Amirhossein Arzani, Jian-Xun Wang, Roshan M. D'Souza

arXiv:2104.08249v2physics.flu-dynphysics.comp-ph

TL;DR

High-fidelity near-wall blood flow and WSS are difficult to measure in diseased arteries because data are sparse, noisy, and uncertain, while inlet and outlet conditions may be unknown. The paper uses PINNs to combine sparse velocity measurements with partial Navier-Stokes and boundary-condition information. Across idealized transport, stenosis, and aneurysm cases, WSS was obtained with very high accuracy, though practical sensor placement and patient-specific pulsatile geometries remain limitations.

  • Problem

    Near-wall WSS is difficult to quantify accurately because diseased-flow measurements and patient-specific boundary conditions can be sparse, noisy, low-resolution, or uncertain.

  • Method

    PINN combines sparse velocity measurements away from the wall with Navier-Stokes physics and partially specified boundary conditions, without requiring training data.

  • Results

    WSS estimation was very accurate: 2D predictions were indistinguishable from ground truth, while 3D errors were very small.

  • Takeaways & Limitations

    Partial cardiovascular-flow physics and sparse localized measurements can be combined to obtain near-wall hemodynamic parameters such as WSS.

  • Takeaways & Limitations

    The study selected sparse measurement locations arbitrarily, leaving optimal sensor placement and data collection for practical applications unresolved.

Abstract

from arXiv · show

Near-wall blood flow and wall shear stress (WSS) regulate major forms of cardiovascular disease, yet they are challenging to quantify with high fidelity. Patient-specific computational and experimental measurement of WSS suffers from uncertainty, low resolution, and noise issues. Physics-informed neural networks (PINN) provide a flexible deep learning framework to integrate mathematical equations governing blood flow with measurement data. By leveraging knowledge about the governing equations (herein, Navier-Stokes), PINN overcomes the large data requirement in deep learning. In this study, it was shown how PINN could be used to improve WSS quantification in diseased arterial flows. Specifically, blood flow problems where the inlet and outlet boundary conditions were not known were solved by assimilating very few measurement points. Uncertainty in boundary conditions is a common feature in patient-specific computational fluid dynamics models. It was shown that PINN could use sparse velocity measurements away from the wall to quantify WSS with very high accuracy even without full knowledge of the boundary conditions. Examples in idealized stenosis and aneurysm models were considered demonstrating how partial knowledge about the flow physics could be combined with partial measurements to obtain accurate near-wall blood flow data. The proposed hybrid data-driven and physics-based deep learning framework has high potential in transforming high-fidelity near-wall hemodynamics modeling in cardiovascular disease.

1 Introduction

Near-wall blood flow and WSS are clinically relevant but difficult to quantify because diseased arterial flows and measurements are complex, uncertain, sparse, and noisy. The study develops PINN-based estimation of near-wall hemodynamics from sparse measurements despite unknown boundary conditions.

  • Near-wall hemodynamics and WSS matter because vessel-wall fluid dynamics regulate and correlate with major cardiovascular diseases.
  • Complex, spatially and temporally varying WSS vector fields make near-wall flow characterization sensitive to modeling assumptions.WSS depends strongly on vessel-diameter variations, illustrated by τ ∼1/D3.
  • Traditional data assimilation can improve near-wall predictions but often requires estimated modeling uncertainty, repeated expensive CFD evaluations, or intrusive adjoint solvers.
  • Scientific machine learning targets sparse, incomplete, low-fidelity, and noisy hemodynamics data, including low-resolution 4D flow MRI and near-wall PIV measurements.
  • PINN integrates governing equations with neural-network approximation and can solve fluid-flow problems with little or no training data.
  • Prior PINN studies addressed cardiovascular surrogate modeling, inverse flow reconstruction, pressure estimation, and superresolution, usually with measurements distributed across the region of interest.
  • This work estimates accurate WSS from sparse velocity measurements away from the wall without training data, while treating inlet and outlet boundary conditions as unknown.

2.1 Problem statement

The problem is to reconstruct velocity and near-wall WSS from sparse interior measurements when physical parameters or inlet and outlet boundary conditions are partially known or unknown. PINN seeks a solution satisfying both the available measurements and the governing Navier-Stokes equations.

  • Steady, incompressible, Newtonian Navier-Stokes equations describe the blood-flow problem, with velocity, pressure, spatial position, viscosity, and density as key quantities.
  • Patient-specific cardiovascular models often have partially known or uncertain parameters and boundary conditions, complicating traditional CFD solution.
  • When inlet or outlet conditions are missing, the problem becomes non-well-posed and lacks a unique solution for traditional numerical methods.
  • Sparse velocity measurements at discrete interior locations are used to identify a flow field satisfying the data, partial boundary conditions, and Navier-Stokes equations.The target is near-wall flow and WSS inferred from measurements taken away from the wall.

2.2 Physics-informed neural networks (PINN)

The PINN represents velocity and pressure with neural networks and trains them using losses for the governing physics, available boundary conditions, and sparse measurements. This combines physical constraints with data when boundary conditions are incomplete.

  • Fully connected neural networks approximate the velocity vector as a function of spatial position, using layered affine transformations and nonlinear activations.
  • The study uses ADAM optimization and Swish activation functions for the network weights and biases.Swish is used with β=1 across layers and cases.
  • PINN uses the Navier-Stokes residual as a physics loss and separately enforces boundary conditions through a boundary-condition loss.
  • When boundary conditions are incomplete, the boundary-condition loss is enforced only on the known boundary portions.Sparse measurements are added because partial boundary conditions do not generally define a well-posed traditional PDE problem.
  • The data loss compares predicted and measured velocities at sparse locations, while the total loss combines boundary and measurement contributions with weighting hyperparameters.Mean squared error, equivalently the squared L2 norm, computes the losses.

2.3 Blood flow problems

The study tests PINN-based flow reconstruction across one-dimensional transport and idealized two- and three-dimensional blood-flow geometries using sparse measurements and incomplete boundary information.

  • Computational setup: CFD simulations provide ground-truth data, while PINN seeks solutions satisfying Navier–Stokes, no-slip conditions, and sparse measurements with inlet and outlet conditions unknown.The CFD mesh nodes are reused as PINN sampling points.
  • 1D advection-diffusion transport: The one-dimensional test uses three concentration measurements, including one inside a sharp boundary layer, to infer unknown boundary conditions and the concentration field.The setup uses 100 uniformly spaced sampling points and measurement-loss weight λd = 10 without boundary-condition enforcement.
  • 2D blood-flow test cases: The two-dimensional blood-flow cases comprise an idealized stenosis and saccular aneurysm, each using five sensors and geometry-specific Reynolds numbers.The stenosis uses Re = 150, whereas the aneurysm uses Re = 320.
  • 3D aneurysm test case: The three-dimensional aneurysm extends the 2D geometry and places sensors on the outer boundary of a near-wall region located 0.13R from the wall.The computational mesh contains 1M quadratic tetrahedral elements, corresponding to approximately 8M linear elements.
  • Problem formulation: The 3D problem remains challenging because outer-wall velocity conditions are incomplete and inlet and outlet boundary conditions are unknown.These omissions distinguish the problem from a mathematically well-posed traditional formulation.
  • Inverse parameter identification: The framework can also treat viscosity as an unknown parameter in an inverse stenosis problem.The motivation includes viscosity variations in hemorheological disorders and possible non-Newtonian effects.

3 Results

Across 1D, 2D, and 3D test cases, PINN predictions closely matched analytical or CFD solutions for velocity and WSS despite unspecified boundary conditions. The method also identified viscosity and remained accurate with very sparse sensing, although reduced sensors lowered aneurysm accuracy.

  • 3.1 Test case 1: 1D advection-diffusion transport: The PINN solution matched the analytical 1D advection-diffusion solution very well, including boundary values with unknown boundary conditions and a thin boundary layer.Three measurement points were used, including one inside the sharp boundary layer.
  • 3.2 Test case 2: 2D blood flow in a stenosis: In the 2D stenosis, PINN velocity matched CFD near the sensors and produced an excellent WSS match distal to the stenosis despite unspecified inlet and outlet conditions.PINN did not identify the correct inlet boundary condition, but reconstructed the stenosed region and distal WSS accurately.
  • 3.3 Test case 3: 2D blood flow in an aneurysm: In the 2D aneurysm, PINN accurately reconstructed flow in the sensor-covered aneurysm sac and showed excellent agreement with CFD for aneurysm-wall WSS.The PINN domain was cropped to a region of interest because the inlet boundary condition was unknown.
  • 3.4 Test case 4: 3D blood flow in an aneurysm: The 3D aneurysm PINN prediction showed very good agreement with CFD for velocity and WSS, with small errors at some locations.The comparison included velocity and WSS streamlines plus WSS along a line through the aneurysm center.
  • 3.6 Test case 5: Parameter identification (viscosity): PINN converged to the ground-truth viscosity (µ = 0.001) after sufficient iterations in the stenosis model.The loss analysis showed consistently lower boundary-condition and data losses than equation loss across the 2D and 3D models.
  • 3.7 The effect of sensor numbers: With three sensors, predicted stenosis WSS remained very close to CFD, whereas reducing aneurysm sensors from four to three reduced accuracy.The four-sensor stenosis result was similarly close and was not shown.

4 Discussion

The discussion argues that PINN can estimate near-wall blood flow and WSS from sparse, localized measurements despite unknown boundary conditions, while identifying practical extensions and limitations.

  • PINN estimated WSS accurately from limited measurements, with 2D predictions indistinguishable from ground truth and 3D errors very small.The framework also used a 1D transport problem to demonstrate feasibility when two coefficients must be inferred without specified boundary conditions.
  • In 2D problems, PINN did not recover the inlet boundary condition but accurately reproduced flow in the sensor region and downstream behavior.Sparse measurements define a regression problem, and one point inside the sharp boundary layer is needed to avoid ill-conditioning.
  • The approach succeeds because WSS topology and dynamics are related to coherent flow structures away from the wall, where velocity measurements can be obtained.Taylor-series relations also connect away-from-wall velocity vectors with wall quantities such as WSS and pressure.
  • Physics-informed learning addresses concerns that large hemodynamics databases may not generalize across diverse geometries, boundary conditions, and vectorial WSS fields.The study presents sparse-data physics-informed modeling as an example of leveraging “smart data” rather than relying solely on large datasets.
  • Future cardiovascular digital twins may use physics-informed models to reduce dependence on continuous, large real-time datasets, but sensor placement remains unresolved.The study selected sparse measurements arbitrarily, so optimal placement and data collection require further investigation.
  • Potential extensions include convolutional architectures, adaptive PINN hyperparameters, uncertainty-aware handling of noisy or multi-fidelity data, and more efficient complex-flow models.The current work selected λb and λd by trial and error and did not address patient-specific geometries, pulsatile flow, or real-world experimental data.

5 Conclusion

The conclusion reports that PINN recovered near-wall hemodynamics and WSS from sparse velocity data without inlet or outlet boundary conditions. Across one-dimensional transport and idealized two- and three-dimensional blood-flow cases, WSS was obtained with very high accuracy.

  • PINN obtained near-wall hemodynamics and WSS from sparse localized velocity measurements without known inlet or outlet boundary conditions.The framework was evaluated in 1D advection-diffusion transport, 2D stenosis and aneurysm flows, and a 3D idealized aneurysm.
  • Very high WSS accuracy was achieved by combining partial cardiovascular-flow physics with sparse measurements in a localized region.
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