Reconstructing Blood Flow in Data-Poor Regimes: A Vasculature Network Kernel for Gaussian Process Regression

TL;DR

This work introduces a novel methodology to reconstruct the kernel within the vascular network based on empirical kernels constructed by data generated from physics-based simulations—enabling near-real-time reconstruction of blood flow in data-poor regimes.

Abstract

Blood flow reconstruction in the vasculature is important for many clinical applications. However, in clinical settings, the available data are often quite limited. For instance, Transcranial Doppler ultrasound (TCD) is a noninvasive clinical tool that is commonly used in the clinical settings to measure blood velocity waveform at several locations on brain's vasculature. This amount of data is grossly insufficient for training machine learning surrogate models, such as deep neural networks or Gaussian process regression. In this work, we propose a Gaussian process regression approach based on physics-informed kernels, enabling near-real-time reconstruction of blood flow in data-poor regimes. We introduce a novel methodology to reconstruct the kernel within the vascular network, which is a non-Euclidean space. The proposed kernel encodes both spatiotemporal and vessel-to-vessel correlations, thus enabling blood flow reconstruction in vessels that lack direct measurements. We demonstrate that any prediction made with the proposed kernel satisfies the conservation of mass principle. The kernel is constructed by running stochastic one-dimensional blood flow simulations, where the stochasticity captures the epistemic uncertainties, such as lack of knowledge about boundary conditions and uncertainties in vasculature geometries. We demonstrate the performance of the model on three test cases, namely, a simple Y-shaped bifurcation, abdominal aorta, and the Circle of Willis in the brain.

Figures (9)

• Figure 1: a) Y-shaped vessel schematic. b) Y-shaped vessel random inlet velocities for all samples and the randomly selected sample as the measurement for prediction and validation. c) The dominant singular values remain unchanged across different numbers of simulations.
• Figure 2: Schematic of the measurement positions. a) Case 1 with high temporal and low spatial resolution. The methodology provides predictions for all spatiotemporal locations, and we have chosen two specific prediction points for result comparison. b and c) Compare the predicted velocity at prediction points for Case 1 which has the most uncertainty due to the lack of nearby measurements. d) Case 2 with low spatiotemporal resolution. e and f) Compare the predicted velocity at prediction points for Case 2 which is improved compared with Case 1.
• Figure 3: a) Abdominal aorta schematic with the location of the velocity measurements and prediction. b) 1D map of the abdominal aorta, extracted from the vessel's centerline. c) Random inlet velocities and the equivalent inlet velocity for 3D simulation.
• Figure 4: Case 1: abdominal aorta prediction using 3D simulation as measurement. Predictions are compared with 3D simulation as the ground truth at a) point 1, b) point 2, c) point 3, and d) point 4. The location of these points and utilized measurements are indicated in Figure \ref{['Fig:Geometryex2']}. In these comparisons, despite the kernel relying on 1D simulations the GP predictions align with 3D simulations across various velocity ranges.
• Figure 5: Case 2: abdominal aorta prediction using 1D simulation as measurement. The location of these points and utilized measurements are the same as Case 1 and is indicated in Figure \ref{['Fig:Geometryex2']}. Comparisons are made using 1D simulation as the ground truth at a) point 1, b) point 2, c) point 3, and d) point 4. In these comparisons the GP predictions matches with 1D simulations.