Mathematical Modeling and Simulation of Cardiac Function

Scientific Computing Tools for Advanced Mathematical Modelling
Supervisor: Prof. Stefano Pagani
Institution: Politecnico di Milano
Academic Year: 2023–2024

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🧠 Project Overview

This is a mathematical and computational framework to reconstruct, simulate, and infer electrical propagation in the human heart from sparse electro-anatomical data.

Developed within the course Scientific Computing Tools for Advanced Mathematical Modelling, this project uses Gaussian Processes, Neural Networks, Autoencoders, and U-Net architectures to tackle challenges in cardiac activation modeling.
The codebase is structured in three main checkpoints:

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📍 Checkpoint 1: Activation Time & Velocity Field Reconstruction

🎯 Goal

Reconstruct the activation time field and conduction velocity from 20 sparse intracardiac recordings.

⚙️ Method

• Gaussian Process Regression (GPR) on 20-point input data
• Kernel tuning using cross-validated hyperparameter search
• Velocity estimated via gradient inversion + second-level GPR smoothing

Activation Time Field

Velocity Field Estimation

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📍 Checkpoint 2: Parameter Estimation with Surrogate Modeling

🎯 Goal

Infer three physiological parameters:

• Fiber angle (μ₁ ∈ [−π/10, π/10])
• Anisotropy ratio (μ₂ ∈ [1, 9])
• Activation origin y₀ (μ₃ ∈ [−1.5, 1.5])

⚙️ Method

• Train a Neural Network surrogate model for the Eikonal solver anisotropic_FMM
• Inputs: (μ₁, μ₂, μ₃, x, y); Output: activation time t
• Use grid search with progressive refinement to optimize parameters by minimizing the squared error functional:

$$ C(\mu) = \sum_{i=1}^{20} |t_i - F(x_i, \mu)|^2 $$

Loss Evolution

NN Prediction vs. Eikonal

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📍 Checkpoint 3: Speed Field Estimation (Parametric & Deep Learning)

🎯 Goal

Estimate a 2D speed field ( c(x,y) ) from only 20 activation time recordings.

1. Parametric Latent Space (Autoencoder)

• Compress 151×151 speed fields into 8 latent parameters using an autoencoder
• Optimize latent parameters to minimize mismatch between real and simulated activation time fields

Autoencoder Architecture

Original vs. Decoded Field

Latent Parameter Distributions

2. Image-to-Image U-Net Regression

An alternative approach was tested using a U-net architecture to directly learn a mapping from activation time images to speed fields.

U-net Architecture

Reconstructed Speed Field

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🧠 Code Architecture

• Checkpoint_1.ipynb – GPR for activation time & velocity field
• Checkpoint_2.ipynb + functions_CP2.py – NN surrogate + parameter optimization
• Checkpoint_3_autoencoder.ipynb – speed field compression and estimation
• Checkpoint3_U-net.ipynb – U-net speed field regression
• model_NN_new_training_2.keras – trained model used in CP2

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🧪 Numerical Highlights

• Activation time prediction accuracy: 0.96 / 1
• U-net reconstruction score: 1.51 / 2
• Autoencoder compression: effective despite minor information loss

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📚 References

• Eikonal solvers and Gaussian Processes: Rasmussen & Williams (2006)
• U-Net: Ronneberger et al. (2015)
• FMM Solver: Sethian (1996)

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📌 Educational Context

This project was completed as part of the course Scientific Computing Tools for Advanced Mathematical Modelling held by Prof. Stefano Pagani at Politecnico di Milano.