[Paper Review] Meshfree implementation of the cardiac monodomain model through the Fragile Points Method

Konstantinos A. Mountris, Leiting Dong+3 more|arXiv (Cornell University)|Nov 12, 2021

Numerical methods in engineering4 citations

TL;DR

This paper proposes the Fragile Points Method (FPM) as a meshfree alternative to the Finite Element Method (FEM) for simulating the cardiac monodomain model. FPM uses local polynomial trial and test functions to enable accurate, efficient integration and direct imposition of boundary conditions, achieving accuracy and convergence comparable to or better than FEM in 2D and 3D benchmarks, including a large-scale biventricular infarction model.

ABSTRACT

Meshfree methods for in silico modelling and simulation of cardiac electrophysiology are gaining more and more popularity. These methods do not require a mesh and are more suitable than the Finite Element Method (FEM) to simulate the activity of complex geometrical structures like the human heart. However, challenges such as numerical integration accuracy and time efficiency remain and limit their applicability. Recently, the Fragile Points Method (FPM) has been introduced in the meshfree methods family. It uses local, simple, polynomial, discontinuous functions to construct trial and test functions in the Galerkin weak form. This allows for accurate integration and improved efficiency while enabling the imposition of essential and natural boundary conditions as in the FEM. In this work, we consider the application of FPM for cardiac electrophysiology simulation. We derive the cardiac monodomain model using the FPM formulation and we solve several benchmark problems in 2D and 3D. We show that FPM leads to solutions of similar accuracy and efficiency with FEM while alleviating the need for a mesh. Additionally, FPM demonstrates better convergence than FEM in the considered benchmarks.

Motivation & Objective

• To develop a meshfree alternative to FEM for cardiac electrophysiology simulations that avoids mesh dependency and maintains high accuracy.
• To address key limitations of existing meshfree methods, including integration inaccuracy and inefficient boundary condition enforcement.
• To evaluate FPM’s performance in simulating action potential propagation across 2D and 3D benchmark problems and in a complex biventricular infarction model.
• To compare FPM’s accuracy, efficiency, and convergence with FEM under varying discretizations and pathological conditions.
• To assess the impact of the penalty coefficient on solution quality in coarse and fine discretizations.

Proposed method

• FPM employs local, simple, discontinuous polynomial functions as trial and test functions in the Galerkin weak form, enabling exact and efficient integration via single-point quadrature.
• Numerical flux corrections are applied to assemble a consistent, sparse, and symmetric global stiffness matrix, overcoming the inconsistency from discontinuous shape functions.
• Essential and natural boundary conditions are imposed directly, similar to FEM, due to the method’s strong enforcement capability.
• The method is applied to the cardiac monodomain model by deriving its weak form using FPM shape functions and solving the resulting system with implicit time integration.
• A dual polyhedral mesh generation algorithm is used to define FPM cells from unstructured nodal points, preserving geometric fidelity in complex anatomies.
• The O’Hara et al. human cardiac cell model is used to simulate action potential dynamics, with pacing applied at base or apex to drive activation.

Experimental results

Research questions

• RQ1 Can FPM achieve comparable accuracy and efficiency to FEM in simulating the cardiac monodomain model without requiring a mesh?
• RQ2 How does FPM’s convergence behavior compare to FEM across different spatial discretizations in 2D and 3D benchmarks?
• RQ3 What is the effect of the penalty coefficient on solution accuracy in coarse and fine discretizations?
• RQ4 Can FPM accurately simulate activation patterns in complex 3D biventricular geometries with infarcted tissue?
• RQ5 How well does FPM reproduce activation times and local activation time (LAT) histograms compared to FEM in patient-specific models?

Key findings

• FPM achieved activation times in the 3D cuboid benchmark that closely matched FEM, with mean differences of less than 2 ms across all nodes at 0.1 mm resolution.
• In the 3D biventricular infarction model, mean local activation times were 170 ms (FPM) vs. 173 ms (FEM) for basal pacing and 151 ms (FPM) vs. 148 ms (FEM) for apical pacing.
• LAT histograms from FPM and FEM simulations showed excellent agreement, indicating consistent activation sequence reproduction.
• FPM demonstrated superior convergence rates compared to FEM in all benchmark problems, with faster convergence to reference solutions as mesh was refined.
• Solutions remained accurate across all discretizations when the penalty coefficient was set in the range [1, 2], while higher values degraded accuracy in fine meshes.
• FPM maintained high accuracy and efficiency without requiring mesh connectivity, making it suitable for image-based, personalized cardiac modeling.