Bayesian Windkessel calibration using optimized 0D surrogate models

TL;DR

The study tackles the computational bottleneck of Bayesian Windkessel boundary condition calibration in 3D cardiovascular CFD by introducing an optimized 0D surrogate trained from a single 3D evaluation. The method performs two SMC runs, first on a geometric 0D model to obtain a MAP, then optimizes 0D parameters hat{α} to closely match the 3D response, followed by a second SMC on the optimized 0D model to infer the Windkessel posterior. Across 72 publicly available vascular models, the optimized 0D surrogates drastically reduce 0D-3D approximation errors (nearly an order of magnitude) and generalize over a wide range of Windkessel parameters, yielding posteriors that closely approximate the full 3D posteriors under varying noise levels. The approach is open-source, enabling broader access to robust, uncertainty-aware Windkessel calibration in subject-specific cardiovascular simulations.

Abstract

Boundary condition (BC) calibration to assimilate clinical measurements is an essential step in any subject-specific simulation of cardiovascular fluid dynamics. Bayesian calibration approaches have successfully quantified the uncertainties inherent in identified parameters. Yet, routinely estimating the posterior distribution for all BC parameters in 3D simulations has been unattainable due to the infeasible computational demand. We propose an efficient method to identify Windkessel parameter posteriors using results from a single high-fidelity three-dimensional (3D) model evaluation. We only evaluate the 3D model once for an initial choice of BCs and use the result to create a highly accurate zero-dimensional (0D) surrogate. We then perform Sequential Monte Carlo (SMC) using the optimized 0D model to derive the high-dimensional Windkessel BC posterior distribution. We validate this approach in a publicly available dataset of N=72 subject-specific vascular models. We found that optimizing 0D models to match 3D data a priori lowered their median approximation error by nearly one order of magnitude. In a subset of models, we confirm that the optimized 0D models still generalize to a wide range of BCs. Finally, we present the high-dimensional Windkessel parameter posterior for different measured signal-to-noise ratios in a vascular model using SMC. We further validate that the 0D-derived posterior is a good approximation of the 3D posterior. The minimal computational demand of our method using a single 3D simulation, combined with the open-source nature of all software and data used in this work, will increase access and efficiency of Bayesian Windkessel calibration in cardiovascular fluid dynamics simulations.

Figures (13)

• Figure 1: Illustration of different blood flow models used in this work. Inflow (Inflow) and Windkessel boundary conditions (WK) with parameters $\boldsymbol{\theta}$ are identical in all models. The geometry model is separated into branch (light) and junction (dark) regions. The geometric 0D model (middle) consists of BloodVessel elements (BV) for branches with parameters $\boldsymbol{\alpha}_\text{b}$ derived purely from the 3D geometry. The optimized 0D model (right) includes additional BloodVesselJunction elements (BVJ) for junctions and all parameters $\hat{\boldsymbol{\alpha}}$ are optimized from 3D simulation results.
• Figure 2: Framework proposed in this work for Bayesian Windkessel BC calibration, using 3D models $\mathfrak{M}_\text{3D}$ (expensive, red) and 0D models (cheap, green). The 0D models use geometric parameters (gray, dotted) or optimized parameters (blue, dashed).
• Figure 3: Changes in maximum pressure and flow error between geometric and optimized 0D models. Blue is an improvement; red is a deterioration. Darker values indicate larger changes. Statistical results are provided on the right. See Figure \ref{['fig_collage']} for an overview of the geometries.
• Figure 4: Cross-validation of optimized 0D model $\mathfrak{M}_\text{0D}(\boldsymbol{\theta}_\text{VMR},\hat{\boldsymbol{\alpha}})$ and comparison to geometric 0D model $\mathfrak{M}_\text{0D}(\boldsymbol{\theta}_\text{VMR},\boldsymbol{\alpha}_\text{b})$. In each validation, the model was calibrated to the test parameters, and the result was compared to the 49 validation parameters.
• Figure 5: 2D marginals of the 5D posterior for all parameters $\theta^{(i)}$ and signal-to-noise ratios $\text{SNR}$ based on model 0104_0001. The diagonal plots show the 1D kernel density estimate. The upper right plots the particles colored by their weight, and the lower left plots the 2D density contours. The parameter values corresponding to the ground truths are marked with crossing gray lines.