Source-linked AI summary
Non-invasive Inference of Thrombus Material Properties with Physics-informed Neural Networks
Minglang Yin, Xiaoning Zheng, Jay D. Humphrey, George Em Karniadakis
TL;DR
The paper addresses the difficulty of inferring patient-specific thrombus permeability and visco-elastic modulus from limited measurements. It uses PINNs with coupled Cahn-Hilliard and Navier-Stokes physics, auxiliary-network treatment of the fourth-order derivative, and joint field-parameter training. The model accurately infers material properties from noisy synthetic data and supports partial measurements and spatially varying permeability.
Problem
Patient-specific thrombus permeability and visco-elasticity are difficult to quantify, despite their importance for modeling thrombus deformation.
Method
PINNs encode coupled Cahn-Hilliard and Navier-Stokes residuals in a loss function while jointly learning material parameters and PDE fields from partial measurements.
Results
PINNs accurately infer permeability and visco-elastic modulus from noisy synthetic data, matching spectral/hp element simulations and handling spatially varying permeability.
Takeaways & Limitations
The approach provides a potential route for non-invasive parameter inference from experimental multi-modality and multi-fidelity data.
Takeaways & Limitations
The model requires the biomedical system to have explicit governing equations.
Abstract
from arXiv · showhide
We employ physics-informed neural networks (PINNs) to infer properties of biological materials using synthetic data. In particular, we successfully apply PINNs on inferring the thrombus permeability and visco-elastic modulus from thrombus deformation data, which can be described by the fourth-order Cahn-Hilliard and Navier-Stokes Equations. In PINNs, the partial differential equations are encoded into the loss function, where partial derivatives can be obtained through automatic differentiation (AD). In addition, to tackling the challenge of calculating the fourth-order derivative in the Cahn-Hilliard equation with AD, we introduce an auxiliary network along with the main neural network to approximate the second-derivative of the energy potential term. Our model can predict simultaneously unknown parameters and velocity, pressure, and deformation gradient fields by merely training with partial information among all data, i.e., phase-field and pressure measurements, and is also highly flexible in sampling within the spatio-temporal domain for data acquisition. We validate our model by numerical solutions from the spectral/\textit{hp} element method (SEM) and demonstrate its robustness by training it with noisy measurements. Our results show that PINNs can accurately infer the material properties with noisy synthetic data, and thus they have great potential for inferring these properties from experimental multi-modality and multi-fidelity data.
1. Introduction
Thrombus material properties are clinically important but patient-specific and difficult to quantify. The paper proposes PINNs as a physics-constrained, mesh-less approach for inferring these properties from limited measurements.
- Thrombus failure and emboli shedding can cause life-threatening complications in several thrombotic diseases.
- Permeability and visco-elasticity influence thrombus mechanics and are important for assessing occlusion and thromboembolism risk.
- These patient-specific properties are difficult to quantify experimentally or with traditional finite-element and finite-volume simulations.
- PINNs encode governing-equation residuals, initial conditions, and boundary conditions into a neural-network loss function.
- Automatic differentiation enables PDE residuals to be evaluated at random spatio-temporal points within this mesh-less framework.
- This work applies PINNs to infer permeability and visco-elastic modulus in coupled Cahn-Hilliard and Navier-Stokes equations, including noisy-data and training-point studies.
2. Methods
The method models thrombus–flow interaction with coupled phase-field and fluid equations, then trains neural surrogates whose physics, boundary, initial, and measurement residuals jointly identify unknown material parameters.
- The thrombus–blood-flow interaction is modeled as a fully Eulerian fluid-structure problem using coupled Cahn-Hilliard and Navier-Stokes equations derived from free-energy minimization.
- The Navier-Stokes equation includes viscous, elastic, and cohesive stresses together with a permeability-related drag term.
- The model represents velocity, pressure, stress, phase field, and deformation information, with ψ providing gradients of the deformation gradient tensor.
- The fourth-order Cahn-Hilliard equation is decoupled into two second-order equations to formulate the weak form.
- The unknown quantities are visco-elastic modulus λe and permeability κ(φ), while the other PDE parameters are assumed known.
- The formulation supports inlet Dirichlet and wall no-slip conditions, Neumann conditions on phase-related variables, and a two-dimensional proof-of-concept domain.
- Physics-Informed Neural Networks (PINNs): Two neural networks approximate PDE solutions, and automatic differentiation computes their input derivatives for the physics-informed loss.
- Physics-Informed Neural Networks (PINNs): The total loss combines PDE, initial-condition, boundary-condition, and sensor-data residuals, while optimization updates both network and PDE parameters.
3. Results
Across thrombus and biofilm cases, PINNs inferred permeability and visco-elastic modulus while reconstructing phase, velocity, and pressure fields from synthetic and partial measurements. Accuracy remained strong with reduced or noisy data, although spatially varying permeability produced localized shell-related errors.
- Method: The PINN architecture uses two networks and automatic differentiation to infer PDE fields and parameters for high-order Cahn-Hilliard–Navier-Stokes systems.An auxiliary network approximates the second derivative of the energy potential term, addressing the fourth-order derivative challenge.
- Uniform permeability: 30,000 phase-field training points produced excellent agreement with reference phase and velocity fields across permeability values from 10^-3 to 10^2.Maximum phase-field absolute error remained below 10%, while the largest local velocity error reached 10% in the bottleneck region.
- Uniform permeability: The inferred permeability agreed with reference values, with maximum mean relative L2 errors below 0.6% for velocity and 0.02% for phase.Parameter retrieval converged toward the tested reference values during training.
- Training-data sensitivity: Using more than 7.5% of 200,000 points, or 15,000 training points, generally yielded permeability convergence and phase-field errors below 1%.Velocity errors were smaller for κ = 1 and 100 than for smaller permeability values.
- Noisy measurements: At noise levels up to 20%, inferred permeability remained unaffected, while velocity-field error increased from below 1% to 25%.The results support parameter robustness to noisy phase measurements but indicate greater sensitivity in reconstructed velocity fields.
- Space-dependent permeability: For phase-dependent permeability, predicted values matched the fluid and core references closely but underestimated shell permeability, producing shell-confined velocity errors.The inferred values were κ(φ = 1) = 0.0015, κ(φ = −1) = 0.99, and κ(φ = 0) = 0.74 against the stated reference configuration.
- Visco-elastic thrombus: After reweighting losses at epoch 600,000, the visco-elastic modulus converged closer to the reference value 0.25 and reconstructed phase, velocity, and pressure accurately.Minor discrepancies remained near the interface and top periodic layer, while errors increased as the system developed over the full time window.
4. Discussion
PINNs infer thrombus permeability and visco-elastic modulus from limited, partial measurements while regressing full fields. The approach also handles spatially varying permeability and supports imaging- and multimodality-based inference.
- 4. Discussion: PINNs infer permeability and visco-elastic modulus from relatively limited data while regressing the associated physical fields.The model uses partial measurements to infer unknown parameters and full fields.
- 4. Discussion: The model infers thrombus permeability from phase-field distribution alone, suggesting a route toward estimating material properties from imaging data.Visco-elastic modulus inference additionally uses phase-field data and boundary pressure measurements.
- 4. Discussion: PINNs also address thrombi with space-dependent permeability, distinguishing different permeabilities in core and shell layers.
- 4. Discussion: Partial measurements can support field regression and parameter inference using imaging techniques and multimodality data.
Appendix
The appendix examines how visco-elastic modulus changes affect biofilm fields and how different partial-data combinations affect inference. Similar fields for widely separated modulus values make inverse inference difficult, while velocity-and-pressure training still converges with larger parameter error.
- Appendix: λe = 0.1 and 15 produce relatively similar deformation, velocity, and pressure fields, complicating inference of the unknown modulus.Differences are concentrated mainly at the biofilm interface and within the biofilm velocity field.
- Appendix: The pressure distributions remain broadly similar across λe values, with only minor differences near the center.
- Appendix: Training with velocity and pressure fields alone yields converged PINN results, but its inferred parameter has the largest error among the tested configurations.The phase-field error ends at the order of 1.