Agenda

1. Mathematical Models for Electrical Activation on the Atrial WallWorkshop on Computational Life Sciences, Innsbruck, October 12 – 14, 2005 1Wieser L., 1Fischer G., 1Nowak C.N., Tilg B. Institute for Biomedical Engineering 1Research Group for Biomedical ModelingUniversity for Health Sciences, Medical Informatics and Technology (UMIT), Hall i. T., Austria Email: leonhard.wieser@umit.at

2. Agenda • Introduction • Ionic current models and tissue coupling • Methods • Results • Atrial Geometry • Methods • Results • Discussion

3. Introduction • Atrial Fibrillation (AF) most common supraventricular arrhythmia, prevalence of0.5 % people > 50 years10 % people > 80 years • Underlying mechanisms for initiation and maintenance poorly understood • Use of computer models (historical background: experiments on squid axon by A.L. Hodgkin and A.F. Huxley) • mechanisms • therapies S. Nattel, Nature (2002)

4. Ionic current models Resting state of the single cardiac cell: • different ionic concentrations (e.g. Na+, Ca2+, K+) in intra- and extracellular space electric potential V ~ -80 mV • driving potential for ion X:(Nernst’s formula) • current  IX = g(t) * (V – EX) • sum of all currents(stable equilibrium) concentrations valency from: J. Malmivou & R. Plonsey: Bioelectromagnetism – Principles and Applications of Bioelectric and Biomagnetic Fields

5. Ionic current models Action potential of the single cardiac cell: ↑Ca2+ • Stimulating current (above threshold) excitation • time dependent conductivities depolarization (Na+)plateau (Ca2+)repolarization (K+) • cell only reexcitableafter complete return to rest ↑Na+ ↓K+ from: J. Malmivou & R. Plonsey: Bioelectromagnetism – Principles and Applications of Bioelectric and Biomagnetic Fields

6. Ionic current models Example: Modeling a single current (INa): 3 independent gating variables: m(t), h(t), j(t) є [0, 1] INa = GNa∙ m³ ∙ h ∙ j ∙ (V – ENa) gating variables governed by αX … opening ratesβX … closing rates total conductivity driving potential

7. Ionic current models Example … … for a classical ventricular cell model Luo-Rudy I (Luo & Rudy, Circ Res 1991): - 6 currents- 8 independent variables … for a recent atrial cell model Ramirez(Ramirez et al., Am J Physiol Heart Circ Physiol 2000) :- 13 currents- 27 independent variables http://www.cellml.org/

8. Cell coupling (Monodomain) continous domain, “membrane potential flow” by diffusion equation β … ratio surface/volume σ … conductivity tensor Ionic current part diffusion part Ionic current models Summary – system of ordinary differential equations of first order C … membrane capacity per area

9. Ionic current models Numerics S, T … stiffness-, and mass-matrix according to spatial discretization scheme Spatial discretization: FEM, FD, FV (Δx ≈ 0.2 mm, model sizes ≈ cm) Time discretizationexplicit, implicit schemes(Δt ≈ 20 µs, simulations ≈ 10 s for fibrillation) Computationally demanding task  seek for efficient methods

10. Ionic current models Example: wave propagation in 1D 40 mV membrane potential -80 mV 0 cm 10 cm

11. Ionic current models Implementation techniques • Use of lookup tablestypical expression contains exp, log, ... (computationally expensive)example: opening rate of j (Na+ channel)V takes values between -85 mV and 100 mV store β(V) in a table for discrete values of V • Use of adaptive time steps(Qu & Garfinkel, IEEE Trans Biomed Eng, 1999)small time step (20 µs) only needed for depolarization (variables change rapidly) Δt small time step time [ms] Δt/K, K elem N 0 100 200 300 400

12. 20 1.04 relative AP duration CPU time 1 0 0 100 0 100 Δt [µs] Δt [µs] Results adaptive time stepperformance in a single cell – Luo Rudy I duration of action potential (AP) compared to referencetime step: Δt = 10 µs … 120 µsadaptive: K = 6 adaptive time step normal time step

13. Results adaptive time stepperformance in a 1D cable – Ramirez et al conduction velocity (CV), compared to reference time step: Δt = 10 µs … 55 µsadaptive: K = 3 adaptive time step normal time step 600 1.1 relative CV CPU time 1 0 0 60 0 60 Δt [µs] Δt [µs]

14. Results numerical scheme Ramirez model, FEM formulation with lumped mass approximation:time step (Δt) diffusion part (PE) ≈ 10 % of total CPU time membrane kinetics (ODE) ≈ 90 % of total CPU time 1 time step, split up into 3 parts PE, Δt/4 ODE, Δt or Δt/K PE, Δt/4

15. Atrial Geometry Model acquisition coarse model segmentation from MRI fine model mesh generator Additional structures: Bachmann’s bundlecoronary sinusorifices (valves and veins)fossa ovaliscrista terminalis atrial wall represented as curved surface in space(323.000 triangles, 163.000 nodes)

16. Atrial Geometry Monolayer – finite element method (FEM) • software development • standard FEM for 2D elements, adapted (additional coordinate transformation) • capable to handle curved surfaces, including branchings • CPU time for 1 second of activation:67 min (PE) + 146 min (ODE) = 213 min(Pentium, 2.8 GHz, single processor)

17. Results Simulating sinus rhythm (physiological pathway)

18. Results Simulating fibrillation and other arrhythmias • usually longer observation periods (10s of seconds) • shorter action potentials by electrical remodeling (decreased Ca2+ current)  reentry waves • anatomical heterogenities (e.g. fibres)

19. Results Simulating fibrillation

20. Discussion • Ionic current models – additional technique to study atrial fibrillation, complementary to experiments • Approaches for efficient implementation of models • Algorithms parallelizable straightforwardly  simulations on clusters • FEM: capable to easily handle unstructured meshes (atrial geometry)

21. Outlook • Test theories for initiation (conduction block, formation of reentry) and maintenance (periodic driving mechanism, multiple wavelet)of atrial fibrillation (AF): • Use atrial geometry and smaller pieces of tissue • Study effects of tissue alteration (e.g by drugs or catheter ablation) on these mechanisms • Extract data (electrograms) from models to compare to clinical measurements

22. Acknoledgement This study was supported by the Austrian Science Fund (FWF) under the grant P16759-N04. … thank you for your attention