Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities
description
Transcript of Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities
![Page 1: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/1.jpg)
Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities
Jianfeng Lv
Advisor: Sima Setayeshgar
May 15, 2009
![Page 2: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/2.jpg)
Outline
Motivation
Numerical Implementation
Numerical Results
Conclusions and Future Work
![Page 3: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/3.jpg)
Motivation:
Ventricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths.
Strong experimental evidence suggests that self-sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias.
And … the heart is an interesting arena for applying the ideas of pattern formation.
Patch size: 5 cm x 5 cm Time spacing: 5 msec
[1] W.F. Witkowski, et al., Nature 392, 78 (1998)
![Page 4: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/4.jpg)
Spiral Waves and Cardiac ArrhythmiasTransition from ventricular tachycardia to fibrillation is conjectured to occur as a result of breakdown of a single spiral (scroll) into a spatiotemporally disordered state, resulting from various mechanisms of spiral (scroll) wave instability. [1]
Tachychardia Fibrillation
Courtesy of Sasha Panfilov, University of Utrecht
Goal is to use analytical and numerical tools to study the dynamics of reentrant waves in the heart on physiologically realistic domains.
[1] A. V. Panfilov, Chaos 8, 57-64 (1998)
![Page 5: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/5.jpg)
Cardiac Tissue Structure
Cells are typically30 – 100 µm long8 – 20 µm wide
Propagation Speeds = 0.5 m / s = 0.17 m / s
Guyton and Hall, “Textbook of Medical Physiology”
Nigel F. Hooke, “Efficient simulation of action potential propagation in a bidomain”, 1992
||CC
![Page 6: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/6.jpg)
Cable Equation and Monodomain Model Early studies used the 1-D cable equation to describe the electrical behavior of a cylindrical fiber.
mm m m
VC V I
t
D
����������������������������
Adapted from J. P. Keener and J. Sneyd, Mathematical Physiology
transmembrane potential: intra- (extra-) cellular potential:
capacitance per unit area of membrane:conductivity tensor:
transmembrane current (per unit length):
mC
mV
tI( )i eV V
axial currents:
resistances (per unit length):
ionic current:, i eI I
D
, i er r
mI
![Page 7: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/7.jpg)
Bidomain Model of Cardiac Tissue
From Laboratory of Living State Physics, Vanderbilt University
The bidomain model treats the complex microstructure of cardiac tissue as a two-phase conducting medium, where every point in space is composed of both intra- and extracellular spaces and both conductivity tensors are specified at each point.[1-
3]
[1] J. P. Keener and J. Sneyd, Mathematical Physiology[2] C. S. Henriquez, Critical Reviews in Biomedical Engineering 21, 1-77 (1993)[3] J. C. Neu and W. Krassowska, Critical Reviews in Biomedical Engineering 21, 137-1999 (1993)
![Page 8: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/8.jpg)
Bidomain Model
Ohmic axial currents:
Conservation of total currents: 0i i e eV V D D������������������������������������������
, i i i e e eI V I V D D����������������������������
, 0a i e aI I I I ��������������
Transmembrane current:
Transmembrane current:
t i i e eI V V D D��������������������������������������������������������
( )mt m m e e
VI C I V
t
D
����������������������������
mt m m
VI C I
t
![Page 9: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/9.jpg)
|| 0 0
0 0
0 0
i
ii
i
D
D
D
D
|| 0 0
0 0
0 0
e
ee
e
D
D
D
D
||
||
i i
e e
D D
D D
Bidomain:
Conductivity Tensors
Cardiac tissue is more accurately described as a three-dimensional anisotropic bidomain, especially under conditions of applied external current such as in defibrillation studies. [1-2]
||
||
ii
e e
DD
D D
The ratio of the intracellular and extracellular conductivity tensors;
Monodomain:
[1] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, 232-240 (1994)[2] J. P. Wikswo, et al., Biophysical Journal 69, 2195-2210 (1995)
![Page 10: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/10.jpg)
Monodomain ReductionBy setting the intra- and extra-cellular conductivity matrices proportional to each other, the bidomain model can be reduced to monodomain model.
1
i i e e m aV V I D D D
����������������������������
1 1mm m i i e e m i i e a
VC I V I
t
D D D D D D D������������������������������������������
, a i i e e m i eI V V V V V D D����������������������������
If , then we obtain the monodomain model.i eD D
mm m m
VC I V
t
D
����������������������������
Substitute (1) into ( )mm m i i
VC I V
t
D
����������������������������
(1)
1( )i i e e D D D D D
![Page 11: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/11.jpg)
Rotating AnisotropyLocal Coordinate Lab Coordinate
1lab localR RD D
cos sin
sin cos
1
R
From Streeter, et al., Circ. Res. 24, p.339 (1969)
![Page 12: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/12.jpg)
Coordinate System
![Page 13: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/13.jpg)
Governing Equations
![Page 14: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/14.jpg)
Perturbation Analysis
![Page 15: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/15.jpg)
Scroll Twist Solutions
Scroll Twist, z
Rotating anisotropy generated scroll twist, either at the boundaries or in the bulk.
Tw
istT
wi
st
![Page 16: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/16.jpg)
Significance?
In isotropic excitable media ( = 1), for twist > twistcritical, straight filament undergoes buckling (“sproing”) instability [1]
Henzi, Lugosi and Winfree, Can. J. Phys. (1990).
What happens in the presence of rotating anisotropy ( > 1)??
![Page 17: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/17.jpg)
Filament Motion
![Page 18: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/18.jpg)
Filament motion (cont’d)
![Page 19: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/19.jpg)
Filament Tension
Destabilizing or restabilizing role of rotating anisotropy!!
![Page 20: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/20.jpg)
Phase SingularityTips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively.
![Page 21: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/21.jpg)
Focus of this work
Analytical and numerical works[1-5] have been done on studying the dynamic of scroll waves in monodomain in the presence of rotating anisotropy .
[1] Biktashev, V. N. and Holden, A. V. Physica D 347, 611(1994)[2] Keener, J. P. Physica D 31, 269 (1988) [3] S. Setayeshgar and A. J. Bernoff, PRL 88, 028101 (2002) [4] A. V. Panfilov and J. P. Keener, Physica D 84, 545 (1995)[5] Fenton, F. and Karma, A. Chaos 8, 20 (1998):
The focus of this work is computational study of the role of rotating anisotropy on the dynamics of phase singularities in bidomain model of cardiac tissue as a conducting medium.
• Rotating anisotropy can induce the breakdown of scroll wave;• Rotating anisotropy leads to “twistons”, eventually destabilizing scroll filament;
![Page 22: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/22.jpg)
Numerical Implementationof the Bidomain Equations with Rotating Anisotropy
![Page 23: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/23.jpg)
Transmembrane potential propagation
: transmembrane potential: intra- (extra-) cellular potential: ionic current: conductivity tensor in intra- (extra-) cellular space
Governing equations describing the intra- and extracellular potentials:
( ) (( ) ) 0i m i e eV V D D D��������������������������������������������������������
Governing Equations
Conservation of total current
mV
mI( )i eD D
( )i eV V
( )me e m
VV I
t
D
����������������������������
![Page 24: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/24.jpg)
Ionic current models Ionic current, , described by a FitzHugh-Nagumo-like kinetics [1]
( )
( )( )
m m
m m
I f V w
dwV kV w
dt
1 1 1
2 2 1 2
3 3 2
1 2
1 2 3
1 1
( ) , ( ) , when V
( ) , ( ) , when e
( ) ( 1), ( ) , when V
where 0.0065, 0.841, 0.15, 3
20, 3, 15;
0.14; 0
m m m m
m m m m
m m m m
f V c V V e
f V c V a V V e
f V c V V e
e e a k
c c c
3.0589; 2.5
[1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)
mI
These parameters specify the fast processes such as initiation of the action potential. The refractoriness of the model is determined by the function . ( )mV
![Page 25: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/25.jpg)
Boundary conditions No-flux boundary conditions:
Normal vector to the domain boundary: Conductivity tensors in natural frame:
n
( ) 0
) 0
i m e
e e
n V V
n V
D
D
��������������
��������������
,i eD D
or , or ( )i e e e mV V V V D D D
11 12
21 22
33
0
0 0
0 0
D D V x
n D D V y
D V z
Let
(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1) and (0,0,-1)n For a rectangular,
![Page 26: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/26.jpg)
Numerical Implementation
1 ( )n
n
m mn e e m
n m
V VD V I V
t
1
1
1( ) ( )
2n n
n n n
m me e e e m
m
V VD V D V I V
t
Numerical solution of parabolic PDE (for Vm )
Forward Euler scheme:
Crank-Nicolson scheme:
( )me e m
VV I
t
D
����������������������������
The spacial operator is approximated by the finite difference matrix operator ( )e eV D����������������������������
eD
![Page 27: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/27.jpg)
Numerical solution of elliptic PDE (for Ve )
Direct solution of the resulting systems of linear algebraic equations by LU decomposition.
(( ) ) ( )i e e i mV V D D D��������������������������������������������������������
1 1 1 111 1
2 2 2 2 211 2
1 3 3 3 311 3
( )
( )
( )
e m
e m
e m
m a b V f V
c m a b V f V
d c m a V f V
Numerical Implementationcont’d
ai , bi , ci , mi are coefficients of terms after discretization of LHS.
, ,e
i j kV denotes the extracellular potential Ve on node (x=i, y=j, z=k).
( )mif V denotes the corresponding RHS after discretization.
![Page 28: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/28.jpg)
Index re-ordering to reduce size of band-diagonal system
1 1 11
2 2 2 22
3 3 33
1 1 1
2 2 2 2
3 3 3
, 1
111 211 311 11 112 212 312 1 121 221 321
x
x x x
x x x x
x x x
x x z
N
N N N
N N N N
N N N
N N jx z
N N N
m a b cd m a b c
d m b c
m
e m a
e d m a
e d m
Elements ai, bi, ci … are constants obtained in finite difference approximation to the elliptic equation.
Numerical Implementationcont’d
![Page 29: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/29.jpg)
Numerical Convergence A time sequence of a typical action potential with various time-steps.
The figures show that time step δt = 0.01 is suitable taking both efficiency and accuracy of computation into account.
![Page 30: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/30.jpg)
Filament-finding algorithm
Search for the closest tip
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 31: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/31.jpg)
Filament-finding algorithm
Make connection
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 32: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/32.jpg)
Filament-finding algorithm
Continue doing search
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 33: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/33.jpg)
Filament-finding algorithm
Continue
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 34: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/34.jpg)
Filament-finding algorithm
Continue
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 35: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/35.jpg)
Filament-finding algorithm
Continue
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 36: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/36.jpg)
Filament-finding algorithm
The closest tip is too far
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 37: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/37.jpg)
Filament-finding algorithm
Reverse the search direction
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 38: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/38.jpg)
Filament-finding algorithm
Continue
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 39: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/39.jpg)
Filament-finding algorithm
Complete the filament
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 40: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/40.jpg)
Filament-finding algorithm
Start a new filament
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 41: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/41.jpg)
Filament-finding algorithm
Repeat until all tips are consumed
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
![Page 42: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/42.jpg)
Numerical Results
![Page 43: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/43.jpg)
Numerical ResultsFilament dynamics of Bidomain
Examples of filament-finding results used to characterize breakup.
Time (s)
|| ||/ 0.06, / 0.4i i e eD D D D
Time (s)
Time (s) Time (s)
|| ||/ 0.3, / 0.4i i e eD D D D
![Page 44: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/44.jpg)
Numerical Resultsof previous work in Monodomain
Previous study has shown rotating anisotropy can induce the breakdown of scroll wave.[1]
[1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)
Iso surfaces of 3D view of scroll wave in the medium with = 0.1111||/D D
Model size : 60x60x9 for 10mm thickness
No break-up while the fiber rotation is less then 60o or total thickness is less than 3.3mm.
||/D D
![Page 45: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/45.jpg)
Results of computational experiments with different parameters of cardiac tissue.
TwistThickness
(layer)
Irregular behavior
Monodomain[1] Monodomain Bidomain
∆x=0.5 ∆x=0.2 ∆x=0.5
0.3 120o 9 No No No
0.1 120o 9 Yes No Yes
0.06 120o 9 Yes Yes Yes
0.1 60o 9 Yes No Yes
0.1 40o 9 No No Yes
0.1 60o 5 Yes No Yes
0.1 40o 3 No No No
[1] A. V. Panfilov and J. P. Keener Physica D 1995
Numerical ResultsBidomain/Monodomain Comparison
||/D D
For ∆x=0.5, the size of rectangular grid is 60x60x9 pointsFor ∆x=0.2, the size of rectangular grid is 150x150x23 points
![Page 46: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/46.jpg)
Numerical Results:Larger Domain Size Result
Time (s)
Time (s)
Contour plots of transmembrane potential selected tissue layers at t = 750 time units. Scroll wave breakup is evident in the middle layers.
Model size: 140x294x48; ∆x = ∆y = ∆z = 0.25 (space units) Time step: ∆t = 0.01 (time units) ;
![Page 47: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/47.jpg)
Conclusions so far …
We have numerically implemented electrical wave propagation in the bidomain model of cardiac tissue in the presence of rotating anisotropy using FHN-like reaction kinetics.
In the finer monodomain model and bidomain model, the boundaries of irregular behavior shift;
Numerical Limitation:
• Large space step in previous study causes mesh effect;• Model size is too small. Increasing model size in bidomain model is limited by the physical memory;
![Page 48: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/48.jpg)
Multigrid Techniques:Multigrid Hierarchy
Relax
InterpolateRestrict
Relax
Relax
Relax
RelaxDragica Vasileska, “Multi-Grid Method”
![Page 49: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/49.jpg)
Multigrid Techniques:Multigrid method
Coarse-grid correction•Compute the defect on the fine grid;•Restrict the defect;•Solve exactly on the coarse grid for the correction;•Interpolate the correction to the fine grid;•Compute the next approximation
Relaxation
Structure of multigrid cycles
S denotes smoothing; E denotes exact solution on the finest grid.Descending line \ denotes restriction, each ascending line / denotes prolongation.William L. Briggs, “A Multigrid Tutorial”
“Numerical Recipes in C”, 2nd Editoin
![Page 50: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/50.jpg)
Multigrid Techniques:Full Multigrid Algorithm
Multigrid method starts with some initial guess on the finest grid and carries out enough cycles to achieve convergence. Efficiency can be improved by using the Full Multigrid Algorithm (FMG)
FMG with the exact solution at the coarsest level. It uses V-cycles (W-cycles) as the solver on each grid level.
“Numerical Recipes in C”, 2nd Editoin
![Page 51: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/51.jpg)
Multigrid Techniques:Interpolation
Trilinear interpolation between the grids
2D interpolation
1 1 1
4 2 41 1
12 21 1 1
4 2 4
The arrows denote the coarse grid points to be used for interpolating the fine grid point. The numbers attached to the arrows denote the contribution of the specific coarse grid point.
3D interpolation
Dragica Vasileska, “Multi-Grid Method”
![Page 52: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/52.jpg)
Multigrid Techniques:Restriction
2D Restriction 3D Restriction
16
1
8
1
16
18
1
4
1
8
116
1
8
1
16
1
In 3D, A 27-point full weighting scheme is used. The number in front of each grid point denotes its weight in this operation.
Dragica Vasileska, “Multi-Grid Method”
![Page 53: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/53.jpg)
Multigrid ResultsConvergence in 2D Typical action potential with various Pre and Post Relaxation-steps.
The figures show that in 2D relaxation step 200 is suitable taking both efficiency and accuracy of computation into account.
The domain is 127x127
![Page 54: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/54.jpg)
Multigrid ResultsConvergence in 3D Typical action potential with various Pre and Post Relaxation-steps.
In the case of 3D, relaxation step 200 is also an appropriate number taken both efficiency and accuracy into account.
The domain is 127x127x7, the convergence plot and density plot are taken at Z=4.
![Page 55: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/55.jpg)
Future Work
Improve numerical efficiency, optimize the multigrid code to reduce the computation time;
Systematic exploration of the role of cell electrophysiology in rotating anisotropy-induced scroll break-up in the Bidomain model;
![Page 56: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/56.jpg)
Thank you
![Page 57: Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities](https://reader035.fdocuments.in/reader035/viewer/2022081519/56813c2a550346895da5a602/html5/thumbnails/57.jpg)
Ionic current models cont.
Ionic current described by a FitzHugh-Nagumo-like kinetics[1]
1(1 )[ ( )]m m m m
m
I V V V f w
dwV w
dt
( ) ( ) f w w b a
[1] Barkley D. (1991) "A model for fast computer simulation of waves in excitable media". Physica 49D, 61–70.