Physics-constrained coupled neural differential equations for one dimensional blood flow modeling

TL;DR

The paper tackles the challenge of delivering fast yet accurate cardiovascular flow simulations by enhancing 1D blood flow models with physics-informed, data-driven components. It introduces the physics-constrained coupled neural differential equations (PCNDE) framework, which spatially formulates the momentum equation, learns corrections for the continuity-based area evolution, and enforces temporal periodicity through a Fourier representation of the cross-sectional area. Training uses cross-sectionally averaged data from 100 3D FSI simulations, enabling the model to outperform traditional 1D FEM in unseen geometries and inlet waveforms while retaining the computational efficiency of a 1D approach. The results demonstrate sub-2% relative errors across flow rate, area, and pressure (with peak outliers in area near the stenosis) and robust behavior to input noise, supporting PCNDE as a promising tool for rapid, patient-relevant cardiovascular simulations. The work also discusses stability, loss landscapes, and future directions for extending to patient-specific geometries and broader uncertainty quantification.

Abstract

Computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid-structure interaction (FSI) simulations. 1D models offer a computationally efficient alternative, by simplifying the 3D Navier-Stokes equations through axisymmetric flow assumption and cross-sectional averaging. However, traditional 1D models based on finite element methods (FEM) often lack accuracy compared to 3D averaged solutions. This study introduces a novel physics-constrained machine learning technique that enhances the accuracy of 1D blood flow models while maintaining computational efficiency. Our approach, utilizing a physics-constrained coupled neural differential equation (PCNDE) framework, demonstrates superior performance compared to conventional FEM-based 1D models across a wide range of inlet boundary condition waveforms and stenosis blockage ratios. A key innovation lies in the spatial formulation of the momentum conservation equation, departing from the traditional temporal approach and capitalizing on the inherent temporal periodicity of blood flow. This spatial neural differential equation formulation switches space and time and overcomes issues related to coupling stability and smoothness, while simplifying boundary condition implementation. The model accurately captures flow rate, area, and pressure variations for unseen waveforms and geometries. We evaluate the model's robustness to input noise and explore the loss landscapes associated with the inclusion of different physics terms. This advanced 1D modeling technique offers promising potential for rapid cardiovascular simulations, achieving computational efficiency and accuracy. By combining the strengths of physics-based and data-driven modeling, this approach enables fast and accurate cardiovascular simulations.

Figures (10)

• Figure 1: An overview of the Physics-constrained Coupled Neural Differential Equations (PCNDE) algorithm is presented. A spatial neural PDE-based momentum equation is used for predicting the flow rate values, then an initial area approximation is formulated based on the 1D continuity equation with two correction terms. A Fourier series is fit for the final area prediction, enforcing explicit temporal periodicity. The area prediction is coupled back to the momentum equation, and the process is iterated until convergence.
• Figure 2: An overview of the training and test (extrapolation) datasets. The generated inlet waveforms are shown at the top, while the generated stenoses geometries are shown on the left. The 10 by 10 matrix represents the dataset, where gray cells are training data and red cells are test (extrapolation) data points.
• Figure 3: Temporal neural PDE results. The left panel shows the true and predicted flow rate. The middle panel shows the true and predicted spatial derivatives of the flow rate. The right panel shows the true and predicted area values. The top row shows the ground-truth 3D averaged data, the middle rows shows the model results where only the flow rate is input to the NN ($f_{\theta}(Q)$), the bottom row shows the model results where both flow rate and time are inputs to the NN ($f_{\theta}(Q,t)$).
• Figure 4: PCNDE results compared with the 1D FEM results. Panel a) shows the flow rate $Q$ results as a function of time and space. Panel b) shows the cross-sectional area $S$ results as a function of time and space. For better visibility, only one case is shown for the area. The errors are only calculated between the two black vertical lines to exclude boundary effects. Panel c) shows the normalized pressure $p/p_{ref}$ results as a function of time and space. The plots as a function of time are shown at $z=3.6$ cm, while the plots as a function of space are shown at $t=0.25$ s. Panel d) shows the PCNDE relative errors for all three variables of interest for the 100 different cases.
• Figure 5: Pressure drop predictions are shown. a) Pressure drop for the new geometry in the test dataset with all waveforms (bottom row of the data table in Fig. \ref{['fig:train-test']}). b) Pressure drop for the new waveform in the test dataset with all geometries (right column of the data table in Fig. \ref{['fig:train-test']}).