Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks

Georgios Kissas; Yibo Yang; Eileen Hwuang; Walter R. Witschey; John A. Detre; Paris Perdikaris

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Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks

Georgios Kissas, Yibo Yang, Eileen Hwuang, Walter R. Witschey, John A. Detre, Paris Perdikaris

arXiv:1905.04817v2May 13, 2019cs.LGstat.ML

TL;DR

The paper targets the computational burden and measurement limitations that hinder clinically useful cardiovascular flow modeling. It combines noisy, scattered 4D flow MRI with one-dimensional physics-informed neural networks to predict physically consistent flow, pressure, and wall displacement. The framework reconstructs arterial pressure waves and offers low-cost Windkessel calibration, while remaining bounded by simplified one-dimensional assumptions and substantial setup and training times.

Problem
Detailed cardiovascular simulations require costly preprocessing and calibration, while pressure is difficult to measure directly using non-invasive in-vivo techniques.
Method
Deep neural networks represent arterial velocity, wall displacement, and pressure while fitting 4D flow MRI measurements and satisfying one-dimensional conservation laws.
Results
The networks reconstruct spatio-temporal absolute pressure waves from noisy, scattered 4D flow MRI data and support Windkessel parameter estimation as post-processing.
Takeaways & Limitations
The framework provides a non-invasive pressure-diagnostics approach and a low-cost procedure for calibrating conventional blood-flow simulations.
Takeaways & Limitations
The one-dimensional model neglects vessel tortuosity, curvature, and other three-dimensional geometry effects, which can cause inaccuracies when these effects are important.

Abstract

from arXiv · show

Advances in computational science offer a principled pipeline for predictive modeling of cardiovascular flows and aspire to provide a valuable tool for monitoring, diagnostics and surgical planning. Such models can be nowadays deployed on large patient-specific topologies of systemic arterial networks and return detailed predictions on flow patterns, wall shear stresses, and pulse wave propagation. However, their success heavily relies on tedious pre-processing and calibration procedures that typically induce a significant computational cost, thus hampering their clinical applicability. In this work we put forth a machine learning framework that enables the seamless synthesis of non-invasive in-vivo measurement techniques and computational flow dynamics models derived from first physical principles. We illustrate this new paradigm by showing how one-dimensional models of pulsatile flow can be used to constrain the output of deep neural networks such that their predictions satisfy the conservation of mass and momentum principles. Once trained on noisy and scattered clinical data of flow and wall displacement, these networks can return physically consistent predictions for velocity, pressure and wall displacement pulse wave propagation, all without the need to employ conventional simulators. A simple post-processing of these outputs can also provide a cheap and effective way for estimating Windkessel model parameters that are required for the calibration of traditional computational models. The effectiveness of the proposed techniques is demonstrated through a series of prototype benchmarks, as well as a realistic clinical case involving in-vivo measurements near the aorta/carotid bifurcation of a healthy human subject.

1. Introduction

The paper addresses the computational and clinical barriers to predicting arterial pressure from non-invasive measurements by combining 4D flow MRI with physics-informed neural networks. The resulting framework enforces one-dimensional blood-flow conservation laws while reconstructing pressure and flow wave propagation and supporting Windkessel calibration.

Motivation: Non-invasive measurements can avoid the cost and practical limitations of invasive techniques, but pressure remains difficult to measure directly in vivo.These limitations motivate computational models and non-invasive imaging as complementary approaches.
Motivation: Traditional cardiovascular flow simulations provide detailed predictions but require costly preprocessing and calibration, limiting their clinical deployment.Patient-specific models commonly depend on truncated arterial topologies and boundary-condition modeling for neglected downstream dynamics.
Approach: The proposed framework represents velocity, wall displacement, and pressure with deep neural networks trained to fit sparse clinical data and satisfy one-dimensional conservation laws.The networks also enforce conservation and information propagation across interfaces in arterial networks.
Contributions: The work applies physics-informed neural networks to an inference problem involving real noisy clinical data and to conservation laws on graph topologies.The graph setting consists of disjoint arterial domains connected through appropriate interface boundary conditions.
Results: The method reconstructs the spatio-temporal absolute pressure wave from scattered noisy 4D flow MRI measurements of blood velocity and wall displacement.It is demonstrated in prototype benchmarks and a clinical case near the aorta/carotid bifurcation of a healthy subject.
Implications: The framework supports non-invasive pressure diagnostics and computationally efficient Windkessel-parameter calibration through post-processing.The surrogate can predict quantities throughout the network without mesh generation or precise boundary prescription.

2.1. A simplified model of pulsatile flow in arterial networks

The paper uses a one-dimensional reduced model to represent pulsatile flow in arterial networks, under simplifying assumptions about geometry, fluid properties, and vessel mechanics.

The reduced model describes arterial geometry with a Cartesian coordinate along vessel center-lines, assuming sufficiently small local curvature.Three-dimensional geometries are recovered from MRI data, and center-lines are extracted using VMTK.
The model assumes incompressible, Newtonian blood with constant density and dynamic viscosity in large arteries.
A(x, t), u(x, t), and p(x, t) denote cross-sectional area, velocity, and pressure, respectively.The coordinates x and t represent spatial and temporal position within each vessel.
The model parameters include equilibrium area, wall thickness, Young’s modulus, external pressure, Poisson ratio, blood density, friction, and momentum-flux correction.For blood flow, the paper typically chooses KR = −22µπ and α = 1.1; β is computed using an empirical relation.
The one-dimensional formulation neglects vessel tortuosity, curvature, and other three-dimensional geometry effects, which can cause inaccuracies when those effects are substantial.The paper identifies cerebral or coronary flows as examples where the present model may be prone to inaccuracies.

2.2. Physics-informed neural networks

The framework represents each vessel’s flow solution with a neural network and constrains it using one-dimensional flow equations, measurements, and network conditions.

Each vessel is represented by a neural network whose output approximates the solution of the one-dimensional arterial-flow equations.
Physics-informed training uses equation residuals to encourage neural-network predictions that satisfy the governing physical constraints.
The networks fit scattered measurements of cross-sectional area and velocity while predicting pressure without direct pressure measurements.The examples use measurements of A(x, t) and u(x, t), but not p(x, t), because pressure is often difficult to measure non-invasively.
The formulation links absolute pressure to area and velocity through the one-dimensional model, enabling absolute-pressure identification beyond momentum conservation alone.Momentum conservation alone exposes only the pressure gradient, whereas the wall constitutive relation supplies an area-pressure connection.
The realistic case sets external pressure and equilibrium area using nominal diastolic pressure and area values from the literature.

2.3. Loss function: Measurements

The measurement component of the loss function trains the networks to reproduce available clinical observations of area and velocity.

The measurement loss compares neural-network outputs for area and velocity with clinical measurements collected at selected vessel locations.In Figure 2, black and red crosses denote area and velocity measurements, respectively.
The measurements can come from procedures such as segmenting 2D cine images and Doppler ultrasound.
Minimizing the measurement term encourages predicted area and velocity to match the available observations in each measured vessel.

2.4. Loss function: Collocation points

The collocation component enforces the governing equations at sampled points inside the arterial domains, using automatic differentiation to evaluate residuals.

Collocation points are randomly sampled inside arterial domains using a Latin-hypercube strategy.These points correspond to the blue dots in Figure 2.
At each collocation point, the loss encourages the area, velocity, and pressure residuals to equal zero.
Automatic differentiation computes the partial derivatives appearing in the residual expressions.
The collocation objective selects network parameters whose predictions are consistent with the underlying differential equations.

2.5. Loss function: Interfaces

The interface loss enforces conservation laws at bifurcations in arterial graph topologies, allowing information from measured domains to propagate into neighboring domains.

Interface conditions: One-dimensional arterial models handle bifurcations by imposing interface boundary conditions that conserve flow and momentum.The father and daughter vessels are coupled through continuity and pressure-related conditions at each splitting point.
Interface notation: The indices identify the father and daughter vessels, while DI denotes the total number of bifurcation points and Nk_b the interface collocation count.These indices organize the loss across all bifurcations and interface boundary points.
Interface loss: The interface loss evaluates conservation-law mismatches at bifurcation points using neural-network predictions from the connected vessels.The loss compares flow rates and pressure-plus-kinetic terms for the father and daughter vessels.
Role in the network: Minimizing the interface loss encourages conservation-law satisfaction and propagates measurements from one arterial domain throughout neighboring domains.The method experiments with characteristic-variable constraints but reports no substantial accuracy improvement from adding them.

2.6. Loss function: Accounting for all constraints

The complete PINN loss combines measurement, interface, and physics-residual terms so predictions reconstruct available data while respecting conservation constraints across arterial networks.

Complete loss: The PINN objective combines measurement and residual losses, with interface losses included to enforce continuity across connected arterial domains.The full construction is assembled from terms representing measurements, physical equations, and interface conditions.
Single artery: For a single artery, the loss contains measurement and residual terms because no bifurcation interface is present.The splitting contribution vanishes when the geometry has no bifurcations.
Multiple vessels: For three vessels and one bifurcation, separate sums represent measured domains, the bifurcation continuity condition, and physics constraints in all vessels.The interface summation has one component because the example contains one bifurcation.
Complex network: For a network with 7 vessels and 3 bifurcations, the loss sums measurement terms over seven vessels, interface terms over three bifurcations, and residual terms over seven arteries.Minimization reconstructs measurements at their acquisition points while enforcing the model's physical constraints.

2.7. Non-dimensionalization and normalization

Non-dimensionalization and input normalization bring the variables and coordinates to comparable scales, addressing severe magnitude differences that otherwise hinder PINN training.

Scaling problem: Pressure, cross-sectional area, and velocity have markedly different physical scales, approximately P ∼ 10^6 Pa, A ∼ 10^-5 m^2, and u ∼ 10 m/s.These scale differences create a systematic training difficulty by producing strongly unequal loss magnitudes.
Non-dimensionalization: Characteristic length, velocity, area, and related variables are introduced so the dependent variables take O(1) magnitudes.The characteristic length is based on the mean equilibrium vessel area, while normalized wave speed is set to c = 10.
Input normalization: The spatial and temporal inputs are subsequently normalized to zero mean and unit variance for more robust back-propagation training.The procedure follows standard deep-learning normalization practices intended to mitigate vanishing-gradient pathologies.
Optimization and prediction: The resulting equations enforce conservation conditions at interfaces and define residual objectives at collocation, bifurcation, and measurement points.At prediction time, inputs are scaled and normalized before inference, and outputs are converted back to their original physical units afterward.

2.8. Windkessel model parameter identification as post processing

Windkessel parameters are identified after PINN training by fitting the predicted outlet flow and pressure through an adaptive search over resistance and compliance values.

Windkessel model: Windkessel boundary conditions model downstream arterial dynamics that are omitted when computational simulations use truncated arterial topologies.The three-element model uses outlet-specific resistance R_j and compliance C_j, with characteristic impedance Z_j and constant downstream pressure p_inf.
Calibration motivation: Traditional calibration repeatedly evaluates conventional solvers across R_j and C_j combinations, becoming costly for networks with many outlets.The proposed procedure instead uses predicted outlet data from the trained PINN.
Post-processing: The post-processing method fits the outlet flow with a 50-mode Fourier series, computes its time derivative, and solves the Windkessel ODE for predicted pressure.The resulting pressure is compared with the model-predicted pressure over randomly selected times.
Parameter search: Adaptive grid refinement searches for R_j and C_j values minimizing the pressure mismatch loss, using five consecutive refinement steps in the reported implementation.The identified parameters are obtained by refining the search near the lowest-loss grid region.
Computational cost: The reported Windkessel-parameter search requires about 10 minutes per outlet on a ThinkPad P52s laptop.The hardware includes an Intel i7-8650U CPU, 32 GiB RAM, and an NVIDIA Quadro P500 2 GiB GPU.

2.9. Computational cost

The workflow is dominated by one-time neural-network training, while trained-model prediction and Windkessel calibration are comparatively inexpensive.

Approximately 7 hours of training on a single NVIDIA Tesla P100 GPU was required for the largest geometry.This cost assumes random network initialization and is treated as a one-time offline expense for each arterial topology.
Predictions from a trained neural network take a fraction of a second on a GPU because they require only a few matrix multiplications.
Windkessel parameter calibration costs 10 minutes per outlet using the trained neural network as a surrogate.
Transfer learning, improved GPU hardware, and parallel multi-GPU implementations could reduce costs for subsequent simulations.

3. Results

The results evaluate the physics-informed neural-network framework on synthetic bifurcation and pelvic-network cases, then on measurements near an aorta/carotid bifurcation. The model generally matches reference or clinical data, preserves conservation laws, and supports Windkessel calibration, while showing limitations for fine-scale waveform features and some clinical discrepancies.

3.1.4. Comparison of model predictions and reference solution: Non-dimensionalization and normalization improve training and produce accurate predictions despite the different scales of the governing variables.The unscaled formulation struggles with quantities spanning scales such as β ∼10^8 Pa/m, ρ ∼10^3 kg/m^3, A ∼10^-5 m^2, and u ∼10 m/s.
3.1.4. Comparison of model predictions and reference solution: The predicted velocity and pressure waves show good agreement with the Discontinuous Galerkin reference solution across the prototype bifurcation domains.The comparison covers the middle points of each domain at randomly selected times.
3.3.4. Discontinuous Galerkin simulation using the identified Windkessel model parameters: The model does not fully capture secondary waveform peaks, and discrepancies remain between predictions and clinical measurements at Aorta #4 and carotid outlets.The secondary-peak limitation is attributed to insufficient representation power for small-scale waveform details.
3.1.5. Systematic studies: Mass and momentum conservation are preserved at the bifurcation, with parent-vessel mass equaling the combined daughter-vessel mass.
3.1.5. Systematic studies: Single precision is selected because it provides adequate accuracy while reducing analysis time from approximately 2 hours 30 minutes to 1 hour 20 minutes.
3.1.6. Windkessel parameter identification by adaptive grid search: An adaptive grid search identifies Windkessel resistance and compliance parameters by minimizing pressure-waveform relative error against the predicted waveform.
3.3.3. Numerical results: For the clinical aorta/carotid case, predictions agree well with measured area and velocity at Aorta #3, while predicted pressure lies within 80–120 mmHg.The framework also calibrates three-element Windkessel parameters at the carotid and Aorta #4 outlets.

4. Discussion

The discussion presents physics-informed neural networks as a flexible bridge between scattered clinical measurements and cardiovascular flow models, while identifying clinical deployment and uncertainty quantification as open challenges.

The framework integrates one-dimensional blood-flow models with clinical data to predict non-invasively inaccessible quantities such as blood pressure.It uses conservation laws together with scattered medical-imaging measurements.
The proposed methodology produced favorable results on synthetic and realistic examples, with good agreement with Discontinuous Galerkin numerical solvers.
Low-cost post-processing of neural-network outputs can calibrate Windkessel model parameters for traditional simulations.
Unlike conventional physics-based models, the approach bypasses mesh generation and does not require precise initial or boundary conditions or constitutive laws.The authors frame it as PDE-constrained filtering of noisy, scattered data rather than a classical numerical PDE solver.
Clinical deployment requires reducing geometry-extraction and training times from several hours to a few minutes, while improving prediction accuracy, robustness, and uncertainty estimates.Transfer learning is suggested as one possible route to shorter adaptation times.
Coarse, noisy measurements can conflict with conservation-law enforcement, especially in smaller arterial networks, and mean-square error may inadequately model non-Gaussian, heteroscedastic noise.