[Paper Review] Cardiac Electrophysiology Meshfree Modeling through the Mixed Collocation Method

Konstantinos A. Mountris, Esther Pueyo|arXiv (Cornell University)|Oct 13, 2021

Low-power high-performance VLSI design4 citations

TL;DR

This paper proposes the meshfree Mixed Collocation Method (MCM) with radial point interpolation (RPI) and moving Kriging interpolation (MKI) to solve the monodomain model for cardiac electrophysiology simulation. MCM achieves accuracy comparable to FEM in 2D and 3D tissue sheets, slabs, and biventricular anatomies, with improved efficiency and robustness in handling complex geometries via an immersed grid approach, despite higher computational cost than FEM.

ABSTRACT

We present the meshfree Mixed Collocation Method (MCM) to solve the monodomain model for numerical simulation of cardiac electrophysiology. We apply MCM to simulate cardiac electrical propagation in 2D tissue sheets and 3D tissue slabs as well as in realistic large-scale biventricular anatomies. Capitalizing on the meshfree property of MCM, we introduce an immersed grid approach for automated generation of nodes in the modeled cardiac domains. We demonstrate that MCM solutions are in agreement with FEM solutions, thus confirming their suitability for simulation of cardiac electrophysiology both in healthy and disease conditions, such as left-bundle-branch block (LBBB) and myocardial infarction. Despite the fact that the computational time for MCM calculations is longer than for FEM, its efficiency in dealing with domains presenting irregularity, nonlinearity and discontinuity make MCM a promising alternative for heart's electrical investigations.

Motivation & Objective

• To develop a meshfree alternative to finite element methods (FEM) for simulating cardiac electrical propagation.
• To overcome FEM’s mesh generation challenges in complex, irregular, or deforming cardiac geometries.
• To evaluate the accuracy and efficiency of MCM using RPI and MKI interpolants against FEM in 2D and 3D cardiac models.
• To introduce and validate an immersed grid approach for automated node generation in patient-specific anatomies.
• To assess MCM performance in realistic biventricular models under healthy and pathological conditions (LBBB, myocardial infarction).

Proposed method

• MCM is applied to the monodomain model using the weak form derived via the Meshfree Local Petrov-Galerkin (MLPG) method with the Dirac delta as the test function.
• The method employs interpolating trial functions—RPI and MKI—enabling direct enforcement of essential boundary conditions due to their Kronecker delta property.
• RPI uses radial basis functions with polynomial enrichment; MKI uses a moving Kriging framework without polynomial terms, reducing computational cost.
• An immersed grid model is generated by distributing equidistant nodes within the bounding box of the biventricular surface mesh and filtering out nodes outside the domain using a point-in-polygon test.
• The monodomain equation is solved using collocation at nodal points, with local support domains defined by the nearest neighbors (up to 150 for optimal balance).
• Solutions are validated against FEM using local activation time (LAT) maps and convergence analysis in 3D domains.

Experimental results

Research questions

• RQ1 Can the Mixed Collocation Method (MCM) with RPI and MKI achieve FEM-level accuracy in simulating cardiac electrical activation in 2D and 3D tissue geometries?
• RQ2 How does the choice of interpolant (RPI vs. MKI) affect the computational efficiency and accuracy of MCM in cardiac electrophysiology simulations?
• RQ3 To what extent does the immersed grid approach improve the automation and robustness of node generation in complex biventricular anatomies?
• RQ4 How do MCM solutions compare to FEM in terms of local activation time (LAT) maps under healthy and pathological conditions such as LBBB and myocardial infarction?
• RQ5 What is the optimal support domain size (number of nearest neighbors) for balancing accuracy and memory usage in large-scale MCM simulations?

Key findings

• MCM-RPI and MCM-MKI solutions showed excellent agreement with FEM in 2D and 3D tissue sheets and slabs, with mean LAT differences below 1 ms in most regions.
• For the biventricular model with scar tissue, mean LAT in the endocardium was 29.8 ms (MCM-RPI/MKI) vs. 21.6 ms (FEM) on the tetrahedral mesh, and 25.2 ms (MCM) vs. 22.8 ms (FEM) on the immersed grid.
• MCM execution times were 16.1 mins (RPI) and 15.3 mins (MKI) on the tetrahedral mesh, compared to 7.8 mins for FEM, with additional time required for trial function computation.
• On the immersed grid, MCM execution time was reduced to 13.8 mins (RPI) and 13.5 mins (MKI), with 12.6 and 9.8 mins respectively for trial function and gradient evaluation.
• MKI was more efficient than RPI due to the absence of polynomial enrichment, reducing computational overhead.
• Convergence analysis showed improved accuracy with larger support domains, with 150 nearest neighbors identified as optimal for balancing accuracy and memory footprint.