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Space--time Isogeometric Analysis of cardiac electrophysiology

P. F. Antonietti, L. Dedè, G. Loli, M. Montardini, G. Sangalli, P. Tesini

TL;DR

The paper develops a stabilized space--time Isogeometric Analysis (IgA) framework for the monodomain equation coupled with the Rogers--McCulloch ionic model in cardiac tissue, employing a Spline Upwind (SU) stabilization to combat fronts and sharp layers. A low-rank approximation of the stabilization term and a specialized preconditioner based on generalized eigen-decompositions of pencils $( extbf{K}_l, extbf{M}_l)$ enable efficient, scalable solution of the resulting space--time linear systems with a tensor-product structure. The authors derive a nonlinear fixed-point solver and a discrete linear system that leverages the Kronecker product structure, and they demonstrate computationally efficient solves for 2D and 3D cardiac geometries, including a left-ventricle-like domain, with favorable comparisons to standard Galerkin and $C^0$-time stabilizations. Overall, the method delivers stable, accurate simulations of electrophysiological wave propagation with competitive runtimes and suggests further extensions to more detailed cardiac models such as the bidomain formulation.

Abstract

This work proposes a stabilized space--time method for the monodomain equation coupled with the Rogers--McCulloch ionic model, which is widely used to simulate electrophysiological wave propagation in the cardiac tissue. By extending the Spline Upwind method and exploiting low-rank matrix approximations, as well as preconditioned solvers, we achieve both significant computational efficiency and accuracy. In particular, we develop a formulation that is both simple and highly effective, designed to minimize spurious oscillations and ensuring computational efficiency. We rigorously validate the method's performance through a series of numerical experiments, showing its robustness and reliability in diverse scenarios.

Space--time Isogeometric Analysis of cardiac electrophysiology

TL;DR

The paper develops a stabilized space--time Isogeometric Analysis (IgA) framework for the monodomain equation coupled with the Rogers--McCulloch ionic model in cardiac tissue, employing a Spline Upwind (SU) stabilization to combat fronts and sharp layers. A low-rank approximation of the stabilization term and a specialized preconditioner based on generalized eigen-decompositions of pencils enable efficient, scalable solution of the resulting space--time linear systems with a tensor-product structure. The authors derive a nonlinear fixed-point solver and a discrete linear system that leverages the Kronecker product structure, and they demonstrate computationally efficient solves for 2D and 3D cardiac geometries, including a left-ventricle-like domain, with favorable comparisons to standard Galerkin and -time stabilizations. Overall, the method delivers stable, accurate simulations of electrophysiological wave propagation with competitive runtimes and suggests further extensions to more detailed cardiac models such as the bidomain formulation.

Abstract

This work proposes a stabilized space--time method for the monodomain equation coupled with the Rogers--McCulloch ionic model, which is widely used to simulate electrophysiological wave propagation in the cardiac tissue. By extending the Spline Upwind method and exploiting low-rank matrix approximations, as well as preconditioned solvers, we achieve both significant computational efficiency and accuracy. In particular, we develop a formulation that is both simple and highly effective, designed to minimize spurious oscillations and ensuring computational efficiency. We rigorously validate the method's performance through a series of numerical experiments, showing its robustness and reliability in diverse scenarios.
Paper Structure (13 sections, 67 equations, 13 figures, 2 tables)

This paper contains 13 sections, 67 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary notions of IgA
  3. Spline Upwind for cardiac electrophysiology
  4. Non-linear solver
  5. Discrete linear system
  6. Preconditioner
  7. Numerical Results
  8. Test with smooth solution
  9. 2D spatial domain
  10. SU stabilization with uniform and local refinement in time: comparison with Galerkin method
  11. Comparison with low regularity $C^0$ splines in time
  12. 3D spatial domain
  13. Conclusions

Figures (13)

  • Figure 1: SU relative error plots in $L^2$-norm.
  • Figure 2: 2D spatial domain with section line $s$ (A–B), test in Section \ref{['sec:McCulloch2D']}.
  • Figure 3: Two views of standard Galerkin transmembrane potential $u_h$ along section line $s$ on the 2D spatial domain for the test in Section \ref{['sec:McCulloch2D']} (the extreme values of the source interval in time $[90,100]$ are highlighted in red), with a uniform mesh in time represented with black crosses. The number of degrees of freedom is $n_1=73, n_2=11$ and $N_t=34$.
  • Figure 4: Two views of SU transmembrane potential $u_h$ along section line $s$ on the 2D spatial domain for the test in Section \ref{['sec:McCulloch2D']} (the extreme values of the source interval in time $[90,100]$ are highlighted in red), with a uniform mesh in time represented with black crosses. The number of degrees of freedom is $n_1 = 73, n_2=11$ and $N_t=34$.
  • Figure 5: The function $\theta$ evaluated at the solution $(u_h,w_h)$ along section line $s$ on 2D spatial domain, test in Section \ref{['sec:McCulloch2D']}.
  • ...and 8 more figures