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Efficient simulation of cardiac electrical propagation using high order finite elements.

Arthurs CJ, Bishop MJ, Kay D - J Comput Phys (2012)

Bottom Line: We detail the hurdles which must be overcome in order to achieve theoretically-optimal errors in the approximations generated, including the choice of method for approximating the solution to the cardiac cell model component.We place our work on a solid theoretical foundation and show that it can greatly improve the accuracy in the approximation which can be achieved in a given amount of processor time.Our results demonstrate superior accuracy over linear finite elements at a cheaper computational cost and thus indicate the potential indispensability of our approach for large-scale cardiac simulation.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Science, University of Oxford, Oxford, United Kingdom.

ABSTRACT
We present an application of high order hierarchical finite elements for the efficient approximation of solutions to the cardiac monodomain problem. We detail the hurdles which must be overcome in order to achieve theoretically-optimal errors in the approximations generated, including the choice of method for approximating the solution to the cardiac cell model component. We place our work on a solid theoretical foundation and show that it can greatly improve the accuracy in the approximation which can be achieved in a given amount of processor time. Our results demonstrate superior accuracy over linear finite elements at a cheaper computational cost and thus indicate the potential indispensability of our approach for large-scale cardiac simulation.

No MeSH data available.


Noble-form LR91 with our method. Fig. 5(a) shows theL2(H1)norm of the error; this is theL2-in-time norm of the Sobolev ∥ · ∥1,2 norm of the error inspace. Fig. 5(b) shows the same dataas Fig. 4(b), but with theexponential convergence in p highlighted instead. Datafrom 20 ms simulations on a 2 cm 1D domain.Δt = 0.001 ms. The errors are against a quartic reference solution withh = 10−4 cm. The values ofh given are in cm.
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f0025: Noble-form LR91 with our method. Fig. 5(a) shows theL2(H1)norm of the error; this is theL2-in-time norm of the Sobolev ∥ · ∥1,2 norm of the error inspace. Fig. 5(b) shows the same dataas Fig. 4(b), but with theexponential convergence in p highlighted instead. Datafrom 20 ms simulations on a 2 cm 1D domain.Δt = 0.001 ms. The errors are against a quartic reference solution withh = 10−4 cm. The values ofh given are in cm.

Mentions: We stimulated a 2 cm 1D domain at one endwith a ramp stimulus using a time-step of Δt = 0.001 ms and simulatedthe first 20 ms of activation. The errors in two differentnorms are presented in Fig.4(b) and Fig. 5(a)and use as a reference a quartic solution generated withh = 0.0001 cm. Fig.5(a) shows the error measured using theL2-in-time norm of the norm in space, for which we expectO(hp)convergence gradients [15]. Notethe agreement of Figs. 4(b) and5(a) with the theoretical error gradients presented, andthe limited accuracy displayed in Fig.4(a) caused by the discontinuities in the standard LR91cell model. The exponential error convergence rate achievable usingp-refinement is emphasised in Fig. 5(b). Note that we use a conductivityof 0.5 S m−1 for allour convergence figures.


Efficient simulation of cardiac electrical propagation using high order finite elements.

Arthurs CJ, Bishop MJ, Kay D - J Comput Phys (2012)

Noble-form LR91 with our method. Fig. 5(a) shows theL2(H1)norm of the error; this is theL2-in-time norm of the Sobolev ∥ · ∥1,2 norm of the error inspace. Fig. 5(b) shows the same dataas Fig. 4(b), but with theexponential convergence in p highlighted instead. Datafrom 20 ms simulations on a 2 cm 1D domain.Δt = 0.001 ms. The errors are against a quartic reference solution withh = 10−4 cm. The values ofh given are in cm.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4067136&req=5

f0025: Noble-form LR91 with our method. Fig. 5(a) shows theL2(H1)norm of the error; this is theL2-in-time norm of the Sobolev ∥ · ∥1,2 norm of the error inspace. Fig. 5(b) shows the same dataas Fig. 4(b), but with theexponential convergence in p highlighted instead. Datafrom 20 ms simulations on a 2 cm 1D domain.Δt = 0.001 ms. The errors are against a quartic reference solution withh = 10−4 cm. The values ofh given are in cm.
Mentions: We stimulated a 2 cm 1D domain at one endwith a ramp stimulus using a time-step of Δt = 0.001 ms and simulatedthe first 20 ms of activation. The errors in two differentnorms are presented in Fig.4(b) and Fig. 5(a)and use as a reference a quartic solution generated withh = 0.0001 cm. Fig.5(a) shows the error measured using theL2-in-time norm of the norm in space, for which we expectO(hp)convergence gradients [15]. Notethe agreement of Figs. 4(b) and5(a) with the theoretical error gradients presented, andthe limited accuracy displayed in Fig.4(a) caused by the discontinuities in the standard LR91cell model. The exponential error convergence rate achievable usingp-refinement is emphasised in Fig. 5(b). Note that we use a conductivityof 0.5 S m−1 for allour convergence figures.

Bottom Line: We detail the hurdles which must be overcome in order to achieve theoretically-optimal errors in the approximations generated, including the choice of method for approximating the solution to the cardiac cell model component.We place our work on a solid theoretical foundation and show that it can greatly improve the accuracy in the approximation which can be achieved in a given amount of processor time.Our results demonstrate superior accuracy over linear finite elements at a cheaper computational cost and thus indicate the potential indispensability of our approach for large-scale cardiac simulation.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Science, University of Oxford, Oxford, United Kingdom.

ABSTRACT
We present an application of high order hierarchical finite elements for the efficient approximation of solutions to the cardiac monodomain problem. We detail the hurdles which must be overcome in order to achieve theoretically-optimal errors in the approximations generated, including the choice of method for approximating the solution to the cardiac cell model component. We place our work on a solid theoretical foundation and show that it can greatly improve the accuracy in the approximation which can be achieved in a given amount of processor time. Our results demonstrate superior accuracy over linear finite elements at a cheaper computational cost and thus indicate the potential indispensability of our approach for large-scale cardiac simulation.

No MeSH data available.