Post on 27-Jul-2020
Computational Modeling of theCardiovascular System
Modeling of Electrical Conductionin Cardiac Tissue
CVRTI Frank B. Sachse, University of Utah
Computational Modeling of the Cardiovascular System - Page 2CVRTI
Overview
• Experimental Studies
• Reaction Diffusion Systems• Cable Model• Monodomain Model• Bidomain Model
• Cellular AutomataGroup work & pause
Group work
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Experimental Studies of Cardiac Electrical Conduction
Measurement methods• Electrode arrays: Extracellular voltages (similar ECG measurements on body surface) Sampling rate up to several kHz Channels up to 2000• Optical: Transmembrane voltages CCD-camera Photodiode array
Preparations• Cell strands - Purkinje fibers• Small muscles - papillary muscle, trabeculae• Sections - wedge preparations from ventricles• Atria/ventricle • Whole heart
in vivo/in vitro
Color-coded visualization ofextracellular voltages measuredon surface of canine ventricles
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Experimental Studies in Papillary Muscle
Rabbit papillary muscle TendonEG measurement
Stimulus positionFix
Oxygenated HEPES solution, 37 °C
Species: Adult New Zealand White rabbits (1.5-3.0 kg)1. Anti-coagulated with heparin and anesthetized with pentobarbital2. Hearts are rapidly excised and moved to dissection tray3. Retrograde perfusion via aorta with modified Tyrode solution4. Opening of right ventricle5. Selection and excision of papillary muscle including onset of chordae tendinae
Criteria: Small diameter, large length, unramified 6. Transfer to horizontal flow-through chamber7. Fixation of muscle8. Measurement
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Measurement Results: Electrograms
Stimulus artifact
Distance tostimulus site
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Electrical Mapping of Canine Ventricles
http://www.cvrti.utah.edu/research
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Optical Mapping of Canine Ventricular Area
http://www.cvrti.utah.edu/research
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In-/Outflow of Currents during Excitation
10 ms 20 ms 30 ms
40 ms 50 ms 60 ms
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In-/Outflow of Currents during Repolarization
110 ms 130 ms 150 ms
170 ms 190 ms 210 ms
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Dipole Approximation and Surface ECG
RA (-)
LF(+)
II
P
Q
R
S
T
RA (-)
LF(+)
II
P
Q
R
S
RA (-)
LF(+)
P
RA (-)
LF(+)
P
RA (-)
LF(+)
P
R
IIII II
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Isotropic/Anisotropic Excitation Propagation (2D)
x
y
Simplifications• Homogeneous tissue• Long axis of myocytes parallel to y-axis• Stimulus at point (0,0)
t=2 3 4
Anisotropic x/y - 1/3Velocity vx: 1 / s, vy: 3 / s
x
yt=2
34
Isotropic x/y - 1/1Velocity v: 1 / s
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Models of Electrical Conduction
• Macroscopic • Rule based / cellular automata (Moe 62, Eifler-Plonsey 75, Killmann-Wach 91, Wei-Okazaki 95, Werner-Sachse-Dössel 97, Siregar 98, Simelius 00 etc.)
• Reaction diffusion systems• Simplified (FitzHugh-Nagumo 61, Rogers-McCulloch 94 etc.)• Combined with models of cellular electrophysiology
• monodomain
(Rudy 89, Virag-Vesin-Kappenberger 98 etc.)• bidomain (Henriquez-Plonsey 89, Sepulveda-Wiksow 93, Sachse-Seemann-Riedel-Werner-Dössel 00 etc.)
• Microscopic (Spach 81)
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Reaction Diffusion System: Cable Model
Reduction to 1DDiscretization
1D Bidomain
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Cable Model: Steady State Response to Non-Excitatory Current
Length constant λ describes spatial distance between two points:1. Location of electrode for injection of
current leading to ΔVm.2. Location at which the voltage ΔVm/e
is interpolated from measurements.
Length constant λ is determined by intra-,extracellular and membrane resistances,ri, ro, and rm:
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Monodomain Modeling of Electrical Conduction in 2D
Resistor of gap junctions(average conductance betweencell pairs 250-1000 nS)
Myocyteintracellular space surroundedby sarcolemma
Membrane Voltage Source
Ground
!m
!i
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Monodomain Model for Electrical Conduction in 2/3D
Transmembrane voltage
Intracellular conductivity tensor(includes conductivity of gapjunctions)
!i
!m
Transmembrane currentIm
Is i
External intracellular current
! Surface-volume ratio of cell
!
"(# i"$i ) = %Im & Isi
"(#e"$
e) = &%I
m& I
se
m
!
"mx,t( ) is unknown
Ii=# $
i#"
m( ) + Isi
%"m
% t=
1
Cm
Ii
&' I
ion
(
) *
+
, -
!
!
Iion
Current through ion channels
Coupling with cell modelNumerical Procedure
Computational Modeling of the Cardiovascular System - Page 17CVRTI
2D-Simulation
Array of myocytes
Area: 6.42 mm2 Elements• number: 642
• size: 0.12 mm2 with fiber orientation
Electrophysiology
Noble et al. 98Monodomain model
Stimulus
1. left, middle
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2D-Simulation of Arrhythmia
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Current Flow in 3D-Model of Electrical Conduction
AnisotropicMonodomain Model
64 x 64 x 128 elementswith electrophysiology of
ventricular myocytes(Noble-Varghese-Kohl-Noble)
Stimulus at center ofplane (Z=0) at time t=0 ms
Fiber orientation parallel to Z-axis
Duration of simulation: 500ms
Colour-coded voltages and streamlines at time t=10 msin plane (Z=0). Colour indicates transmembrane voltage.
Computational Modeling of the Cardiovascular System - Page 20CVRTI
Bidomain Modeling of Electrical Conduction in 2D
Resistor of gap junctions(average conductance betweencell pairs 250-1000 nS)
Myocyteintracellular space surroundedby sarcolemma
Membrane Voltage Source
Resistor of extracellular space
Extracellular potential
Transmembrane Voltage!m
!e
Intracellular potential!i
!m
!i
!e
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Bidomain Model: Motivation
Intra- and extracellular voltage distribution relevant for:• cardiac excitation propagation• body surface potential maps (BSPM)• electrocardiogram (ECG)
Problem: Realistic cell-based modeling of tissue• complex geometry of cells• large number of cells
Idea „Bidomain Model“• division of space in two domains• separated calculation
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Bidomain Model: Basics
Continuum 1: Extracellular space (Interstitial space)
Continuum 2: Intracellular space
Tissue
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Group Work
Find and describe other applications for (non-electrical) multidomain models in • physics• biology• …
What might be the domains of a tridomain model of cardiac electrophysiology?
Computational Modeling of the Cardiovascular System - Page 24CVRTI
Bidomain Model: Basics
!i
!e
Ji
Je
!
"m ="i #"e
"m : Transmembrane voltage V[ ]
"i / e : Intra - /extracellular potential V[ ]
J = Ji + Je
J : Summary current density A/m2[ ]Ji / e : Intra - /extracellular current density A/m2[ ]
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Bidomain Model: Intracellular Space
!Im
Is i
!
"# $ i J i( ) =# $ i#%i( ) =&'m " Isi
$ i : Intracellular conductivity S
m
(
) * +
, -
Ji : Intracellular current density A
m2
(
) * +
, -
%i : Intracellular potential V[ ]
Isi : Intracellular current source density A
m3
(
) * +
, -
Im : Membrane source density A
m2
(
) * +
, -
& : Ratio of membrane surface to volume m-1[ ]
!
"i#i
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Bidomain Model: Extracellular Space
!Im
Ise
!
"# $e Je( ) =# $e#%e( ) = "&'m " Ise
$e : Intracellular conductivity S
m
(
) * +
, -
Je : Intracellular current density A
m2
(
) * +
, -
%e : Intracellular potential V[ ]
Ise : Intracellular current source density A
m3
(
) * +
, -
Im : Membrane source density A
m2
(
) * +
, -
& : Ratio of membrane surface to volume m-1[ ]
!
"e
#e
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Bidomain Model: Relationships
Generalized Poisson‘s Equation
!
J = Ji+ J
e= "#
i$%
i"#
e$%
e
with %m
= %i" %
e:
J = "#i$%
m"#
i$%
e"#
e$%
e
with #H
= #i+ #
e:
J = "#i$%
m"#
H$%
e
with $J = 0 :
#i$%
m= "#
H$%
e
Computational Modeling of the Cardiovascular System - Page 28CVRTI
Bidomain Model: Numerical Solution
Elliptical partial
differential equations
(Poisson‘s equation)
Ordinary differentialequation(Cell model)
Problem: Spatio-temporal discretization!
unkn
own
unkn
own
!
" #i"$
m( ) = %" #H"$
e( )
Istim
=" #i"$
m( ) +" #i"$
e( )
&$m
& t=1
Cm
Istim
'% I
ion
(
) *
+
, -
!
"mx,t( ) and "
ex,t( ) are unknown
Computational Modeling of the Cardiovascular System - Page 29CVRTI
Simulation of Electrophysiology in Myocardial Area
Myocyte clusterin left ventricular
free wall
128 x 128 x 128 elementswith electrophysiology of
ventricular myocytes(Noble-Varghese-Kohl-Noble)
Inclusion of wall depth dependent
• myocyte orientation• current Ito
Element couplingvia bidomain model
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Transmembrane Voltage in Static Myocardial Area
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Calcium Concentration in Static Myocardial Area
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Rotor in Static Myocardial Area
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• Wiener, Rosenblueth 2D sheets • Moe, Rheinboldt, Abildskov Atria (2D sheet) -
• Eiffler, Plonsey Ventricular myocardium (2D sheet)• Adam Human ventricles (ellipsoids)
• Killmann, Wach, Dienstl Human heart (from drawings)• Saxberg, Cohen Myocardium• Wei, Okoazaki, Harumi, Harasawa, Human ventricles (Anisotropic) Hosaka• Siregar, Sinteff, Chadine, Le Beux Human heart (2D)• Werner, Sachse, Dössel Human heart (Anisotropic)• Siregar, Sinteff, Julen, Le Beux Human heart (CAD)…
1946
today
Cellular Automatons of Cardiac Excitation Propagation
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Cellular Automaton: Basics
CellularAutomaton
AnatomicalModel
PhysiologicalParameters
• Autorhythmicity• Transmembrane voltage• Conduction velocity• Refractory period
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Cellular Automaton: Modeling of Propagation
Initial excitation
Excitation
Passive element
Active element
1 2
3 4
5 6
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Anatomical Model of Heart: Requirements
• left atrial myocardium
• right atrial myocardium
• left ventricular myocardium
• right ventricular myocardium
• Sinus node
• AV node
• His bundle
• Tawara bundle branches
• Purkinje fibers
• Fiber orientation
Image segmentation Manual/rule-based definition
Necessary: Anatomical model of all excitation triggering and conductive components
Example: Components in model of Werner et al.:
Computational Modeling of the Cardiovascular System - Page 37CVRTI
Parameters for Simulation: Lookup Tables
• Stimulus
• Time since activation tS• Stimulus frequency f
• Tissue type
•Transmembrane voltage (tS)
• Refractory period (tS)
• Autorhythmicity (tS)
• Conduction velocity (f)
• Excitable neighborhood
(constant)
Known per volume element dV
and for time t
dV
Computational Modeling of the Cardiovascular System - Page 38CVRTI
Cellular Automaton: Parameter - Transmembrane Voltage
t [ms]Vm [mV]
0
-50
t [ms]Vm [mV]
0
-50
t [ms]Vm [mV]
0
-80
Sinus node AV node Atrial myocardium
Course of transmembrane voltage is dependent on tissue type andstimulus frequency.Activation is only possible outside of absolute refractory time.
Most cellular electrophysiological properties, e.g. ion and transmitter concentrations, nervous influences, extracellular potentials etc. areneglected!
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Cellular Automaton: Parameterization
Ventricle: Noble-Varghese-Kohl-Noble 98 Atrium: Earm-Hilgemann-Noble 90
Course of transmembrane voltageLongitudinal/transversal propagation velocity
Measurements Numerical experiments} {
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Unidirectional Block/Rotation Around Obstacles (2D)
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Unidirectional Block/Rotation Around Obstacles (2D)
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Unidirectional Block in Homogeneous Slice (2D)
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Results of Whole Heart Simulations
Transmembrane voltage color-coded at heart surface for physiological excitationpropagation
8 time steps• atrial activation starting at sinus node• ...• atrial repolarisation• ventricular activation starting at subendocardium• ...• ventricular repolarisation
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Simulation of 3rd Degree AV Block
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Simulation of Infarction
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Cellular Automaton: Application in ECG/BSPM Simulation
T / msUm / mV
0
-80
AnatomieElectrophysiology
NumericalField Calculation• Volume and surface
voltages• Current densities
EKG
Cellular Automaton• Transmembrane voltages• Membrane current densities
BSPM
BodySurfacePotentialMap
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Visualization
Images/Movies
Simulation System: Overview
Electrophysiological Model Macro-/microscopic, rule-based/analytical
Electrical Model Body/Thorax
Source modelBidomain approximation
Lead Modele.g. standard leads of
Einthoven and Goldberger
Trans-membrane
voltage
Current sourcedensity
Electrical Voltages
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Example: ECG Simulation
ECG
BSPMVm
Trans-membranevoltage
fe
Currentsourcedensities
Cellular automatonof excitation propagation
Bidomainmodel
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Group Work
Compare cellular automata with mono-/bidomain models of cardiacconduction! Apply ~5 criteria for comparison.