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Markov chain Monte Carlo for Bayesian inference of the non-conducting region in intra-atrial reentrant tachycardia

This paper presents a Bayesian framework utilizing an adapted Metropolis-Hastings algorithm with a compressed likelihood and discretization error correction to efficiently estimate the uncertain geometry of non-conducting regions in intra-atrial reentrant tachycardia from noisy electrical measurements.

Original authors: Maarten Volkaerts, Marie Cloet, Hans Dierckx, Piet Claus, Giovanni Samaey

Published 2026-04-02

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CC BY 4.0

Original authors: Maarten Volkaerts, Marie Cloet, Hans Dierckx, Piet Claus, Giovanni Samaey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your heart is a bustling city where electricity is the traffic that tells the muscles when to squeeze and pump blood. Sometimes, a roadblock appears in this city—a scar from a past surgery or injury—that stops the electricity from flowing. This blockage can cause the traffic to get stuck in a loop, circling the blockage over and over again. In medical terms, this is called Intra-Atrial Reentrant Tachycardia (IART), a type of heart rhythm disorder.

To fix this, doctors need to know exactly where the "roadblock" (the non-conducting scar) is, how big it is, and what shape it takes. But looking inside a beating heart is like trying to map a city while standing outside the city limits, listening only to the sound of traffic. It's noisy, blurry, and full of guesswork.

This paper presents a new, smarter way to map that invisible roadblock using Bayesian Inference and Markov Chain Monte Carlo (MCMC). Here is how it works, broken down into simple concepts:

1. The Problem: Guessing in the Dark

Traditionally, doctors might try to find the best single "fit" for the scar's shape. It's like trying to guess the shape of a hidden object by feeling around it once and saying, "It's probably a circle." If you're slightly off, you might think it's a square. This method ignores the fact that your measurement might be shaky or noisy.

The authors say: "Let's stop guessing one shape. Let's guess a range of shapes and figure out which ones are most likely, while also admitting how unsure we are."

2. The Solution: The "Smart Detective" (Bayesian Inference)

Instead of looking for one perfect answer, the authors use a statistical method called Bayesian Inference. Think of this as a detective who starts with a hunch (a "prior") and then gathers clues (the data) to update their theory.

The Prior: Before looking at the data, the detective thinks, "The scar could be anywhere between a small pebble and a large boulder."
The Clues: They look at electrical signals recorded outside the heart (like listening to traffic noise from a hill).
The Posterior: After analyzing the clues, the detective updates their theory. Instead of saying "It's a circle," they say, "There's a 90% chance it's an oval between 10mm and 12mm wide, and a 10% chance it's a weird shape."

This gives doctors not just a map, but a confidence score. They know exactly how reliable the map is.

3. The Challenge: The "Pixelated" Map (Discretization Error)

To simulate the heart's electricity, the computer breaks the heart tissue into a grid of tiny squares (like pixels on a screen). This is necessary, but it introduces a problem: Discretization Error.

Imagine trying to draw a smooth circle on a pixelated screen. If you move the circle just a tiny bit, the pixels might snap to a slightly different shape, making the drawing look jagged or jump suddenly. In the computer model, this causes the math to "jump" or glitch, confusing the detective.

The authors realized that if they didn't account for these "pixel jumps," their detective would become overconfident in the wrong answer. They fixed this by:

Inflating the uncertainty: Telling the detective, "Hey, the map is a bit pixelated, so let's be a little more unsure about the exact edges."
Smoothing the jumps: They developed a trick called Node Relocation. Instead of redrawing the entire pixel grid from scratch every time they test a new shape, they just "nudge" the existing pixels to fit the new shape. This keeps the map smooth and prevents the "jumps" that confuse the algorithm.

4. The Speed Trick: Compressed Data

The raw data from the heart is massive—thousands of electrical readings over time. Processing all of this is like trying to read a whole library to find one sentence.

The authors created a Compressed Likelihood. Instead of reading every single word, they summarized the story into a few key "characterizing quantities":

How long does it take for the electrical loop to go around once? (The Period)
When does the electricity hit specific sensors compared to the average? (Relative Activation Time)

By focusing on these key summaries, the computer can solve the problem much faster without losing the important details.

5. The Result: A Better Map

The authors tested their method with synthetic data (computer-generated heart signals). They found that:

It's faster: Their "Smart Detective" (the adapted algorithm) found the answer using fewer guesses than traditional methods.
It's honest: It correctly identified when the data was too noisy to give a precise answer. In some cases, it admitted, "We can't tell the exact width and height, but we can tell you the total perimeter of the scar."
It's robust: Even when the computer's "pixel grid" was a bit rough, the method didn't get confused or give a wrong answer with high confidence.

The Big Picture

This paper isn't just about math; it's about making heart treatment safer and more personalized. By using this method, doctors could eventually get a 3D map of a patient's specific heart scar along with a reliability score.

Instead of saying, "We think the scar is here," they could say, "We are 95% sure the scar is in this oval shape, and here is exactly how much we are unsure." This helps doctors decide whether to perform a procedure, how to plan it, and what risks to expect, moving from "best guess" medicine to precision medicine with a safety net.

1. Problem Statement

The paper addresses the challenge of personalizing cardiac electrophysiology (EP) models to estimate the geometry of electrically non-conducting regions (e.g., surgical scars) within heart tissue. These regions are critical drivers of Intra-Atrial Reentrant Tachycardia (IART), a type of arrhythmia where electrical waves circulate around a scar.

The Core Difficulty: Determining the shape and location of these scars from external electrical measurements (electrograms) is an inverse problem.
Limitations of Current Methods: Traditional approaches rely on deterministic optimization (finding a single "best fit"). This ignores uncertainties arising from:
- Measurement noise in clinical data.
- Discretization errors inherent in solving the underlying Partial Differential Equations (PDEs).
- Lack of identifiability (where different geometric parameters produce similar data).
Goal: To develop a Bayesian inference framework that not only estimates the geometric parameters of the non-conducting region but also quantifies the uncertainty associated with these estimates, providing a reliability measure for clinical decision-making.

2. Methodology

The authors propose a comprehensive pipeline combining mathematical modeling, data compression, and advanced sampling algorithms.

A. Forward Model (The Physics)

Governing Equations: The study uses the monodomain model coupled with the modified Mitchell-Schaeffer cell model to simulate electrical wave propagation.
Geometry: The heart tissue is modeled as a 2D slab containing an elliptical non-conducting region (where diffusion $D=0$ ). The region is defined by three geometric parameters: semi-major axis ( $a$ ), semi-minor axis ( $b$ ), and inclination angle ( $\phi$ ).
Simulation: The model simulates IART by imposing a spiral wave initial condition. The forward solver uses a splitting scheme (Rush-Larsen for reaction, Finite Element Method for diffusion) via the fenicsx-beat software.

B. Data Generation and Compression

Synthetic Data: Electrogram (EGM) data is generated synthetically using a virtual PENTARAY® mapping catheter with 20 electrodes.
Compressed Likelihood: Instead of using the full high-dimensional time-series data (which is computationally expensive and noisy), the authors extract characterizing quantities:
1. Period of the spiral: The time between reentries.
2. Relative Local Activation Times (relLAT): The arrival time of the wave at each electrode relative to the average activation time.
- Benefit: This reduces the data dimension significantly, improves noise modeling, and makes the inference invariant to temporal shifts.

C. Handling Discretization Error

A major technical contribution is addressing discretization error caused by the Finite Element Method (FEM) meshing.

The Issue: Standard meshing (Independent Meshing) creates artificial discontinuities in the likelihood function because small changes in geometry ( $\vartheta$ ) trigger a complete remeshing, causing jumps in the solution. This leads to "overconfident" but incorrect posterior distributions.
The Solution:
1. Variance Inflation: The likelihood variance is inflated to account for discretization error ( $\Sigma = \Sigma_{noise} + \Sigma_{discretization}$ ).
2. Node Relocation Strategy: Instead of generating a new mesh from scratch for every proposed sample, the algorithm attempts to relocate the nodes of the previous mesh to fit the new geometry. This ensures local continuity in the likelihood function, significantly improving the stability of the sampling algorithm.

D. Sampling Algorithm (Adapted MCMC)

To sample the posterior distribution efficiently, the authors propose an Adapted Metropolis-Hastings (MH) algorithm:

Adaptive Proposal (Adaptive Metropolis): The proposal distribution's covariance matrix is updated during the sampling process based on the history of accepted samples. This adapts to the anisotropy (correlations) of the posterior, preventing "bad mixing" where the chain gets stuck.
Robust Accept-Reject Step: Integrates the Node Relocation strategy to minimize artificial likelihood jumps caused by meshing, ensuring the acceptance probability accurately reflects the model-data discrepancy rather than numerical artifacts.

3. Key Contributions

Bayesian Framework for EP: Formulated the estimation of non-conducting regions as a Bayesian inverse problem, moving beyond single-point estimates to full probability distributions.
Compressed Likelihood: Introduced a dimensionality reduction technique using characteristic quantities (period and relLAT) to improve sampling efficiency and noise modeling.
Discretization Error Management:
- Demonstrated that ignoring discretization error leads to artificial discontinuities and overconfident predictions.
- Proposed a Node Relocation meshing strategy within the MCMC loop to maintain local continuity and robustness.
- Showed that inflating the likelihood variance is necessary to account for remaining discretization bias.
Adaptive Sampling: Developed an MH algorithm that combines Adaptive Metropolis (for correlation handling) and Node Relocation (for meshing continuity), drastically reducing the number of required forward model solves.

4. Results

The authors validated their method using synthetic IART data with varying noise levels and discretization resolutions.

Efficiency: The Adapted MH algorithm significantly outperformed the standard Random Walk Metropolis-Hastings (RWMH). The adaptive proposal reduced autocorrelation, allowing the chain to explore the parameter space faster and converge with fewer samples.
Robustness to Meshing:
- Independent Meshing (standard approach) resulted in "oscillating" likelihoods and abrupt cut-offs in the posterior, leading to inaccurate histograms when discretization error was underestimated.
- Node Relocation smoothed these discontinuities, producing stable and accurate posterior distributions even with coarse meshes.
Uncertainty Quantification:
- In low-noise scenarios, the method successfully recovered the true geometric parameters with tight uncertainty bounds.
- In high-noise/identifiability-limited scenarios, the method correctly identified that parameters $a$ and $b$ were not individually identifiable. However, it revealed that the perimeter of the non-conducting region was identifiable. This insight (that uncertainty lies along a specific manifold) is impossible to capture with deterministic fitting.

5. Significance and Outlook

Clinical Relevance: This work provides a pathway to trustworthy virtual twins of the heart. By quantifying uncertainty, clinicians can assess the reliability of model predictions before using them for treatment planning (e.g., ablation strategies).
Methodological Advancement: The paper demonstrates that discretization error is not just a numerical nuisance but a critical factor in Bayesian inference that must be explicitly modeled to avoid false confidence.
Future Work: The authors note that while the current method is efficient for low-dimensional parameters, scaling to high-dimensional, patient-specific models will require surrogate modeling (e.g., Gaussian Processes or Neural Networks) to replace the expensive PDE solves, building upon the robust inference framework established here.

In summary, the paper presents a rigorous, uncertainty-aware methodology for cardiac parameter estimation, solving critical numerical stability issues in MCMC sampling and demonstrating the superior value of stochastic inference over deterministic fitting in complex biomedical problems.