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High-order finite element methods for cardiac monodomain simulations.

Vincent KP, Gonzales MJ, Gillette AK, Villongco CT, Pezzuto S, Omens JH, Holst MJ, McCulloch AD - Front Physiol (2015)

Bottom Line: The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements.Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone.Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

View Article: PubMed Central - PubMed

Affiliation: Department of Bioengineering, University of California San Diego La Jolla, CA, USA.

ABSTRACT
Computational modeling of tissue-scale cardiac electrophysiology requires numerically converged solutions to avoid spurious artifacts. The steep gradients inherent to cardiac action potential propagation necessitate fine spatial scales and therefore a substantial computational burden. The use of high-order interpolation methods has previously been proposed for these simulations due to their theoretical convergence advantage. In this study, we compare the convergence behavior of linear Lagrange, cubic Hermite, and the newly proposed cubic Hermite-style serendipity interpolation methods for finite element simulations of the cardiac monodomain equation. The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements. Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone. Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

No MeSH data available.


Related in: MedlinePlus

(A) Activation times from three unconverged electrophysiology simulations using cubic Hermite elements (black, ϕc = 2.0; blue, ϕc = 4.0; and red, ϕc = 8.0) are plotted against activation times at the same location on the mesh of a converged (ϕc = 1.0) reference solution. Regression lines are shown for each unconverged solution as a broken line. The activation times in the unconverged location were scaled so that regression line had a slope of 1.0. (B) These scaled activation times are compared with the converged activation times using a Bland-Altman plot where solid lines represent the mean difference between the two solutions and the broken lines are ± two standard deviations in the residuals from the regression line. (C) The root mean squared (RMS) error between the unconverged and fully converged activation patterns decreases with the cell Thiele modulus of the simulation. (D) The high-order methods have a smaller RMS error compared to the linear method on a degree of freedom basis particularly as the size of the problem increases.
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Figure 5: (A) Activation times from three unconverged electrophysiology simulations using cubic Hermite elements (black, ϕc = 2.0; blue, ϕc = 4.0; and red, ϕc = 8.0) are plotted against activation times at the same location on the mesh of a converged (ϕc = 1.0) reference solution. Regression lines are shown for each unconverged solution as a broken line. The activation times in the unconverged location were scaled so that regression line had a slope of 1.0. (B) These scaled activation times are compared with the converged activation times using a Bland-Altman plot where solid lines represent the mean difference between the two solutions and the broken lines are ± two standard deviations in the residuals from the regression line. (C) The root mean squared (RMS) error between the unconverged and fully converged activation patterns decreases with the cell Thiele modulus of the simulation. (D) The high-order methods have a smaller RMS error compared to the linear method on a degree of freedom basis particularly as the size of the problem increases.

Mentions: Finally, we sought to determine convergence criteria for electrophysiology solutions when only activation patterns rather than absolute conduction times are needed. This need arises commonly in patient-specific modeling, where the total activation time is known (e.g., from a measured QRS duration or electrocardiological mapping), but the conductivity is unknown. Electrical propagation in a patient-derived human biventricular mesh was simulated at four levels of spatial refinement resulting in simulations with an average cell Thiele modulus in the primary direction of propagation of 1.0, 2.0, 4.0, and 8.0 for the four mesh refinements. For this exercise, the simulation using cubic Hermite elements with ϕc = 1.0 was considered fully converged. Activation times in the three less-converged simulations were compared to the converged activation times at node locations from the coarsest mesh (Figure 5A). Since propagation of the high cell Thiele modulus simulations was too fast, the total activation time for those simulations was scaled to give a regression line with a slope of 1.0. The Bland-Altman plot in Figure 5B comparing the scaled activation times and the fully converged activation times identifies a pattern of outliers (>2 standard deviations) that were too fast compared with the fully converged solution. These outliers were located in the basal right ventricular free wall. The root-mean-squared (RMS) error between fully converged activation pattern and the scaled unconverged activation patterns decreases as the cell Thiele modulus decreases toward 1.0 for all three interpolation methods (Figure 5C). The RMS error was less than 5 ms for all unconverged simulations and twice the RMS error was less than 5 ms for the ϕc = 2.0 and ϕc = 4.0 simulations with cubic Hermite elements, the ϕc = 2.0 and ϕc = 1.0 simulations with serendipity Hermite elements, and the ϕc = 1.0 simulation with the linear Lagrange elements. This error is comparable to the lowest uncertainly in clinically measured activation times (Gold et al., 2011; Villongco et al., 2014). Figure 5D demonstrates that the high-order methods also have smaller RMS error than the linear elements on a degree of freedom basis.


High-order finite element methods for cardiac monodomain simulations.

Vincent KP, Gonzales MJ, Gillette AK, Villongco CT, Pezzuto S, Omens JH, Holst MJ, McCulloch AD - Front Physiol (2015)

(A) Activation times from three unconverged electrophysiology simulations using cubic Hermite elements (black, ϕc = 2.0; blue, ϕc = 4.0; and red, ϕc = 8.0) are plotted against activation times at the same location on the mesh of a converged (ϕc = 1.0) reference solution. Regression lines are shown for each unconverged solution as a broken line. The activation times in the unconverged location were scaled so that regression line had a slope of 1.0. (B) These scaled activation times are compared with the converged activation times using a Bland-Altman plot where solid lines represent the mean difference between the two solutions and the broken lines are ± two standard deviations in the residuals from the regression line. (C) The root mean squared (RMS) error between the unconverged and fully converged activation patterns decreases with the cell Thiele modulus of the simulation. (D) The high-order methods have a smaller RMS error compared to the linear method on a degree of freedom basis particularly as the size of the problem increases.
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4525671&req=5

Figure 5: (A) Activation times from three unconverged electrophysiology simulations using cubic Hermite elements (black, ϕc = 2.0; blue, ϕc = 4.0; and red, ϕc = 8.0) are plotted against activation times at the same location on the mesh of a converged (ϕc = 1.0) reference solution. Regression lines are shown for each unconverged solution as a broken line. The activation times in the unconverged location were scaled so that regression line had a slope of 1.0. (B) These scaled activation times are compared with the converged activation times using a Bland-Altman plot where solid lines represent the mean difference between the two solutions and the broken lines are ± two standard deviations in the residuals from the regression line. (C) The root mean squared (RMS) error between the unconverged and fully converged activation patterns decreases with the cell Thiele modulus of the simulation. (D) The high-order methods have a smaller RMS error compared to the linear method on a degree of freedom basis particularly as the size of the problem increases.
Mentions: Finally, we sought to determine convergence criteria for electrophysiology solutions when only activation patterns rather than absolute conduction times are needed. This need arises commonly in patient-specific modeling, where the total activation time is known (e.g., from a measured QRS duration or electrocardiological mapping), but the conductivity is unknown. Electrical propagation in a patient-derived human biventricular mesh was simulated at four levels of spatial refinement resulting in simulations with an average cell Thiele modulus in the primary direction of propagation of 1.0, 2.0, 4.0, and 8.0 for the four mesh refinements. For this exercise, the simulation using cubic Hermite elements with ϕc = 1.0 was considered fully converged. Activation times in the three less-converged simulations were compared to the converged activation times at node locations from the coarsest mesh (Figure 5A). Since propagation of the high cell Thiele modulus simulations was too fast, the total activation time for those simulations was scaled to give a regression line with a slope of 1.0. The Bland-Altman plot in Figure 5B comparing the scaled activation times and the fully converged activation times identifies a pattern of outliers (>2 standard deviations) that were too fast compared with the fully converged solution. These outliers were located in the basal right ventricular free wall. The root-mean-squared (RMS) error between fully converged activation pattern and the scaled unconverged activation patterns decreases as the cell Thiele modulus decreases toward 1.0 for all three interpolation methods (Figure 5C). The RMS error was less than 5 ms for all unconverged simulations and twice the RMS error was less than 5 ms for the ϕc = 2.0 and ϕc = 4.0 simulations with cubic Hermite elements, the ϕc = 2.0 and ϕc = 1.0 simulations with serendipity Hermite elements, and the ϕc = 1.0 simulation with the linear Lagrange elements. This error is comparable to the lowest uncertainly in clinically measured activation times (Gold et al., 2011; Villongco et al., 2014). Figure 5D demonstrates that the high-order methods also have smaller RMS error than the linear elements on a degree of freedom basis.

Bottom Line: The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements.Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone.Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

View Article: PubMed Central - PubMed

Affiliation: Department of Bioengineering, University of California San Diego La Jolla, CA, USA.

ABSTRACT
Computational modeling of tissue-scale cardiac electrophysiology requires numerically converged solutions to avoid spurious artifacts. The steep gradients inherent to cardiac action potential propagation necessitate fine spatial scales and therefore a substantial computational burden. The use of high-order interpolation methods has previously been proposed for these simulations due to their theoretical convergence advantage. In this study, we compare the convergence behavior of linear Lagrange, cubic Hermite, and the newly proposed cubic Hermite-style serendipity interpolation methods for finite element simulations of the cardiac monodomain equation. The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements. Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone. Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

No MeSH data available.


Related in: MedlinePlus