Full-field surrogate modeling of cardiac function encoding geometric variability

TL;DR

This work proposes a novel computational pipeline to embed cardiac anatomies into full-field surrogate models using Branched Latent Neural Maps (BLNMs) as an effective scientific machine learning method to encode activation maps extracted from physics-based numerical simulations into a neural network.

Abstract

Combining physics-based modeling with data-driven methods is critical to enabling the translation of computational methods to clinical use in cardiology. The use of rigorous differential equations combined with machine learning tools allows for model personalization with uncertainty quantification in time frames compatible with clinical practice. However, accurate and efficient surrogate models of cardiac function, built from physics-based numerical simulation, are still mostly geometry-specific and require retraining for different patients and pathological conditions. We propose a novel computational pipeline to embed cardiac anatomies into full-field surrogate models. We generate a dataset of electrophysiology simulations using a complex multi-scale mathematical model coupling partial and ordinary differential equations. We adopt Branched Latent Neural Maps (BLNMs) as an effective scientific machine learning method to encode activation maps extracted from physics-based numerical simulations into a neural network. Leveraging large deformation diffeomorphic metric mappings, we build a biventricular anatomical atlas and parametrize the anatomical variability of a small and challenging cohort of 13 pediatric patients affected by Tetralogy of Fallot. We propose a novel statistical shape modeling based z-score sampling approach to generate a new synthetic cohort of 52 biventricular geometries that are compatible with the original geometrical variability. This synthetic cohort acts as the training set for BLNMs. Our surrogate model demonstrates robustness and great generalization across the complex original patient cohort, achieving an average adimensional mean squared error of 0.0034. The Python implementation of our BLNM model is publicly available under MIT License at https://github.com/StanfordCBCL/BLNM.

Figures (15)

• Figure 1: Sketch of the computational pipeline illustrated for synthetic patient 45. We use SSM to generate a synthetic patient by sampling 12 modes. For this patient, we conduct heat simulations to define some universal ventricular coordinates Bayer2012. Next, we create the corresponding fiber fields: fiber sheets (top left), longitudinal fibers (top right), normal fibers (bottom left), and Purkinje fibers (bottom right). We use these fiber orientations to run an electrophysiology simulation and generate the activation map. BLNMs predict activation maps by using two branches for modes and spatial coordinates, respectively
• Figure 2: Comparison of myocardial volumes, ventricular surface areas, and structural metrics (wall thickness and apicobasal heights) across our original patient cohort.
• Figure 3: Overview of the synthetic cohort generation pipeline. We apply large deformation diffeomorphic metric mapping to map original anatomies onto a common template, followed by principal component analysis to encode the anatomical variability into z-scores. Using a novel sampling strategy, We generate synthetic anatomies based on their probabilistic proximity to the original samples in the multivariate space defined by these geometrical encodings. Specifically, we selected the 4 closest samples (dotted lines) to each original subject (colored lines).
• Figure 4: Selection of the initial points for the Purkinje networks. On the left, the white contour points represent universal ventricular coordinates, $\phi_{ab} = 0.97$ and $\phi_{ab} = 0.99$, derived from the apex-to-base simulation of the Laplace heat equation Bayer2012. The centroids of the $\phi_{ab}$ points for the LV and RV are marked by blue spheres. The first and second points in each ventricle are represented by white spheres, where the second point is chosen to determine the direction of the first segment. On the right, the corresponding RV and LV Purkinje networks are visualized in light blue and orange, respectively
• Figure 5: Sketch of the optimal BLNM architecture of 16 layers with 42 neurons per layer and 5 states as determined using Ray Tune with the OptunaSearch algorithm. The network starts with separate branches: the first branch processes spatial coordinates (x, y, z), and the second processes the geometry-specific modes $\theta_1$ through $\theta_{12}$. These branches operate independently up to a specified disentanglement level of 2 before merging.