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Data assimilation performed with robust shape registration and graph neural networks: application to aortic coarctation

Francesco Romor, Felipe Galarce, Jan Brüning, Leonid Goubergrits, Alfonso Caiazzo

TL;DR

This work compares state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

Abstract

Image-based, patient-specific modelling of hemodynamics can improve diagnostic capabilities and provide complementary insights to better understand the hemodynamic treatment outcomes. However, computational fluid dynamics simulations remain relatively costly in a clinical context. Moreover, projection-based reduced-order models and purely data-driven surrogate models struggle due to the high variability of anatomical shapes in a population. A possible solution is shape registration: a reference template geometry is designed from a cohort of available geometries, which can then be diffeomorphically mapped onto it. This provides a natural encoding that can be exploited by machine learning architectures and, at the same time, a reference computational domain in which efficient dimension-reduction strategies can be performed. We compare state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

Data assimilation performed with robust shape registration and graph neural networks: application to aortic coarctation

TL;DR

This work compares state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

Abstract

Image-based, patient-specific modelling of hemodynamics can improve diagnostic capabilities and provide complementary insights to better understand the hemodynamic treatment outcomes. However, computational fluid dynamics simulations remain relatively costly in a clinical context. Moreover, projection-based reduced-order models and purely data-driven surrogate models struggle due to the high variability of anatomical shapes in a population. A possible solution is shape registration: a reference template geometry is designed from a cohort of available geometries, which can then be diffeomorphically mapped onto it. This provides a natural encoding that can be exploited by machine learning architectures and, at the same time, a reference computational domain in which efficient dimension-reduction strategies can be performed. We compare state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

Paper Structure

This paper contains 36 sections, 6 theorems, 90 equations, 30 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Forward computational hemodynamics
  3. Statistical shape modelling of patients with aortic coarctation
  4. Blood flow modelling
  5. Calibration of boundary conditions across the cohort of patients
  6. Synthetic dataset of aortic shapes and numerical simulations
  7. Registration with ResNet-LDDMM
  8. Large deformation diffeomorphic metric mapping
  9. Multigrid ResNet-LDDMM for aortic shape meshes
  10. Preliminaries
  11. Objective function
  12. Optimization
  13. Registration of the computational domain: pullback and pushforward operators
  14. Shape registration results
  15. Application of shape registration for solution manifold learning
  16. ...and 21 more sections

Key Result

Theorem 1

Assume that $S:G\rightarrow\mathbb{R}$ and $T:G\rightarrow\mathbb{R}$ are two bounded measurable functions, and that $S$(e.g., the source image) is continuous almost everywhere. Let us suppose that $f\in L^2(I, H^s)$, with $s>d/2 + 1$ (the Sobolev embedding implies $f\in \mathcal{C}^{1, \alpha}(I, \ has a minimizer.

Figures (30)

  • Figure 1: Overview of the registration-based data assimilation process.
  • Figure 2: Left: Sketch of the centerline encoding (points $p_i$ and radius $r_i$ of the associated inscribed sphere, for $i\in\{1,\dots,390\}$). Center: Clustering of the considered training ($n=724$) and test ($n=52$) shapes using t-SNE with the Euclidean distance on a geometrical encoding, based on the distance of each point from the centerline after shape registration (see section \ref{['subsec:sml_correlations']} for details). Right: Visualization of the furthest shapes (top, test cases $34$, $50$, and $44$) and the closest ones (bottom, test cases $3$, $21$, $15$) according to the metric in the center plot.
  • Figure 3: Example of a computational domain. A parabolic profile is imposed on the inlet boundary (ascending aorta), based on a given peak flow rate. Windkessel models are used at the outlets (thoracic aorta, brachiocephalic artery, left common carotid artery, and left subclavian artery).
  • Figure 4: Distribution of Windkessel parameters $C_{d,i}$ (distal capacity), $R_i=R_{d,i}+R_{p,i}$ (Total resistance), and $A_i$ (boundary area) for the different outlets, across the training and test datasets.
  • Figure 5: Top: Numerical results for te flow at inlet (AAo, with opposite sign) and outlets (BCA, LCCA, LSA, TA) boundaries. Bottom: Numerical results for the pressure at inlet (AAo) and outlets (BCA, LCCA, LSA, TA) boundaries. The $25$-th and $75$-th percentile across all the $724$ and $52$ training and test data are shown. The red vertical bands correspond to the time window $t\in[0.05s, 0.25s]$ which is the focus of our data assimilation studies, see remark \ref{['rmk:timewindow']}.
  • ...and 25 more figures

Theorems & Definitions (26)

  • Remark 1: Calibration based on flow split
  • Theorem 1: LDDMM registration, theorem 3.1 dupuis1998variational
  • Theorem 2: Existence of ResNet-LDDMM vector field
  • proof
  • Remark 2: Bijectivity of the transformation map
  • Remark 3: Choice of template geometry
  • Remark 4: Partitioned vs. monolithic rSVD
  • Remark 5: Time window
  • Remark 6
  • Remark 7
  • ...and 16 more