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• 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_at_umit.at

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Agenda

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

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Introduction

• Atrial Fibrillation (AF) most common
  supraventricular arrhythmia, prevalence of0.5
  people gt 50 years10 people gt 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)
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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(Nernsts
  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
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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
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Ionic current models
Example Modeling a single current (INa)
3 independent gating variables m(t), h(t), j(t)
? 0, 1 INa GNa m³ h j (V
ENa) gating variables governed by aX
opening ratesßX closing rates
total conductivity
driving potential
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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/
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Ionic current models
Summary system of ordinary differential
equations of first order
C membrane capacity per area
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Ionic current models
Numerics
S, T stiffness-, and mass-matrix according to
spatial discretization scheme
Spatial discretization FEM, FD, FV (?x 0.2
mm, model sizes cm) Time discretizationexplici
t, implicit schemes(?t 20 µs, simulations 10
s for fibrillation)
Computationally demanding task ? seek for
efficient methods
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Ionic current models
Example wave propagation in 1D
40 mV
membrane potential
-80 mV
0 cm
10 cm
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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 ß(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)

?t
small time step
time ms
?t/K, K elem N
0
100
200
300
400
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Results
adaptive time stepperformance in a single cell
Luo Rudy I
duration of action potential (AP) compared to
reference time step ?t 10 µs 120
µsadaptive K 6
adaptive time step
normal time step
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Results
adaptive time stepperformance in a 1D cable
Ramirez et al
conduction velocity (CV), compared to reference
time step ?t 10 µs 55 µsadaptive K 3
adaptive time step
normal time step
1.1
600
relative CV
CPU time
1
0
0
60
0
60
?t µs
?t µs
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Results
numerical scheme
Ramirez model, FEM formulation with lumped mass
approximation time step (?t)
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, ?t/4
ODE, ?t or ?t/K
PE, ?t/4
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Atrial Geometry
Model acquisition
coarse model
segmentation from MRI
fine model
mesh generator
Additional structures Bachmanns
bundlecoronary sinusorifices (valves and
veins)fossa ovaliscrista terminalis
atrial wall represented as curved surface in
space(323.000 triangles, 163.000 nodes)
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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 activation67 min (PE)
  146 min (ODE) 213 min(Pentium, 2.8 GHz,
  single processor)

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Results
Simulating sinus rhythm (physiological pathway)
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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)

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Results
Simulating fibrillation
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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)

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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

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