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Adaptive Mesh Refinement and Adaptive Time Integration for Electrical Wave Propagation on the Purkinje System.

Ying W, Henriquez CS - Biomed Res Int (2015)

Bottom Line: The equations governing the distribution of electric potential over the system are solved in time with the method of lines.At each timestep, by an operator splitting technique, the space-dependent but linear diffusion part and the nonlinear but space-independent reactions part in the partial differential equations are integrated separately with implicit schemes, which have better stability and allow larger timesteps than explicit ones.The linear diffusion equation on each edge of the system is spatially discretized with the continuous piecewise linear finite element method.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, China.

ABSTRACT
A both space and time adaptive algorithm is presented for simulating electrical wave propagation in the Purkinje system of the heart. The equations governing the distribution of electric potential over the system are solved in time with the method of lines. At each timestep, by an operator splitting technique, the space-dependent but linear diffusion part and the nonlinear but space-independent reactions part in the partial differential equations are integrated separately with implicit schemes, which have better stability and allow larger timesteps than explicit ones. The linear diffusion equation on each edge of the system is spatially discretized with the continuous piecewise linear finite element method. The adaptive algorithm can automatically recognize when and where the electrical wave starts to leave or enter the computational domain due to external current/voltage stimulation, self-excitation, or local change of membrane properties. Numerical examples demonstrating efficiency and accuracy of the adaptive algorithm are presented.

No MeSH data available.


Traces of action potentials during the simulation period [0,400] msecs at the point marked as “7” in the two-dimensional branch shown in Figure 4. The solid curves were from the adaptive simulation. The dotted curves were from the uniform simulation. The dashed curves were from the “no-refinement” simulation. In these simulations, a voltage stimulation is applied from the right side of the thickness/radius-variable branch structure. The right plot is a close-up of the left plot.
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fig6: Traces of action potentials during the simulation period [0,400] msecs at the point marked as “7” in the two-dimensional branch shown in Figure 4. The solid curves were from the adaptive simulation. The dotted curves were from the uniform simulation. The dashed curves were from the “no-refinement” simulation. In these simulations, a voltage stimulation is applied from the right side of the thickness/radius-variable branch structure. The right plot is a close-up of the left plot.

Mentions: In each of the simulations above, the action potential is initiated at the right end of branch segment “1” and propagates across the whole computational domain. We collected action potentials at those seven marked points (see Figure 4) during the simulations. See Figures 5 and 6 for traces of action potentials at the marked points “4” and “7,” respectively. The action potentials from the adaptive and uniform mesh refinement simulations match very well, and their difference at the peak of the upstroke phase is uniformly bounded by 0.1 mV. The action potential from the “no-refinement” simulation has least accurate results. Potential oscillation is even observed at point “7” (see Figure 6) in the “no-refinement” simulation.


Adaptive Mesh Refinement and Adaptive Time Integration for Electrical Wave Propagation on the Purkinje System.

Ying W, Henriquez CS - Biomed Res Int (2015)

Traces of action potentials during the simulation period [0,400] msecs at the point marked as “7” in the two-dimensional branch shown in Figure 4. The solid curves were from the adaptive simulation. The dotted curves were from the uniform simulation. The dashed curves were from the “no-refinement” simulation. In these simulations, a voltage stimulation is applied from the right side of the thickness/radius-variable branch structure. The right plot is a close-up of the left plot.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4637156&req=5

fig6: Traces of action potentials during the simulation period [0,400] msecs at the point marked as “7” in the two-dimensional branch shown in Figure 4. The solid curves were from the adaptive simulation. The dotted curves were from the uniform simulation. The dashed curves were from the “no-refinement” simulation. In these simulations, a voltage stimulation is applied from the right side of the thickness/radius-variable branch structure. The right plot is a close-up of the left plot.
Mentions: In each of the simulations above, the action potential is initiated at the right end of branch segment “1” and propagates across the whole computational domain. We collected action potentials at those seven marked points (see Figure 4) during the simulations. See Figures 5 and 6 for traces of action potentials at the marked points “4” and “7,” respectively. The action potentials from the adaptive and uniform mesh refinement simulations match very well, and their difference at the peak of the upstroke phase is uniformly bounded by 0.1 mV. The action potential from the “no-refinement” simulation has least accurate results. Potential oscillation is even observed at point “7” (see Figure 6) in the “no-refinement” simulation.

Bottom Line: The equations governing the distribution of electric potential over the system are solved in time with the method of lines.At each timestep, by an operator splitting technique, the space-dependent but linear diffusion part and the nonlinear but space-independent reactions part in the partial differential equations are integrated separately with implicit schemes, which have better stability and allow larger timesteps than explicit ones.The linear diffusion equation on each edge of the system is spatially discretized with the continuous piecewise linear finite element method.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, China.

ABSTRACT
A both space and time adaptive algorithm is presented for simulating electrical wave propagation in the Purkinje system of the heart. The equations governing the distribution of electric potential over the system are solved in time with the method of lines. At each timestep, by an operator splitting technique, the space-dependent but linear diffusion part and the nonlinear but space-independent reactions part in the partial differential equations are integrated separately with implicit schemes, which have better stability and allow larger timesteps than explicit ones. The linear diffusion equation on each edge of the system is spatially discretized with the continuous piecewise linear finite element method. The adaptive algorithm can automatically recognize when and where the electrical wave starts to leave or enter the computational domain due to external current/voltage stimulation, self-excitation, or local change of membrane properties. Numerical examples demonstrating efficiency and accuracy of the adaptive algorithm are presented.

No MeSH data available.