Title: Geometric dynamics for rotor filaments and wave fronts
1Geometric dynamics for rotor filaments and wave
fronts
- Hans Dierckx1, Olivier Bernus2,3, Henri
Verschelde1
1 Department of Mathematical Physics and
Astronomy, Ghent University, Belgium 2 Institute
of Membrane and Systems Biology, University of
Leeds, United Kingdom 3 Multidisciplinary
Cardiovascular Research Centre, University of
Leeds, United Kingdom
CPP 2009, Cambridge, UK 30 June 2009
2Outline
- Introduction
- Filaments - Filament tension - Advanced
dynamics - Anisotropy effects - Fronts - Surface tension - Anisotropy effects
- Discussion
- Conclusions
3What are scroll wave filaments?
- Filament rotation axis of a spiral wave
extended to 3D
Hans Dierckx, 2009
4Why study filament evolution?
- Number of filaments vs. arrhythmiae 0
healthy state (?) 1 monomorphic
tachycardia gt 1 polymorphic tachycardia/
torsade de pointes gtgt1 fibrillation - Sensitivity of spirals/scrolls is localized
around their centre ? filament prescribes
surrounding electrical activity - Response functions/ sensitivity functions
Biktashev Biktesheva, Phys Rev E 67,
2003 -
5How to study filament evolution?
- Start from generic reaction-diffusion
equation - Quantities for describing a scroll wave
filament - - revolution velocity w
- - arclength s
- phase angle f ? twist
- - Filament curvature
- k1/R
Hans Dierckx, 2009
6Equation of motion for filaments (isotropic)
- Equations up to order O(k², w³), derived
using Frenet-Serret (T,N,B) coordinates - Derived by Keener Tyson (SIAM rev 34(1),
1986), adapted by Biktashev et al. (Phil
Trans R Soc Lond A 347, 1994) - Minimal model for filament motion
- The motion of a filament is proven to be
governed solely by its shape, i.e. curvature
k and twist w
7Filament tension
- The coefficient g1 plays the role of filament
tension -
- Positive g1 straightens filaments
- Negative g1 can lead to filament
instability/multiplication
Biktashev, Holden Zhang, Phil Trans R Soc 347,
1994
Biktashev, Holden Zhang, Phil Trans R Soc 347,
1994
Fenton Karma, Chaos 8(1), 1998
8Further facts on filament motion
- Not yet captured in the presented equations of
motion - Twist can destabilize straight filaments(sproing
instability) - When scroll rings shrink, their radial and axial
velocity are not proportional to each other
9Geometrical theory for filaments
- Construct a full solution from lower-dimensional
counterparts - Scroll wave a stack of 2D spiral waves
- Ansatz has been used before, but this time with
a geometric perturbation scheme
(Verschelde, Dierckx Bernus Phys. Rev. Lett.
99(16), 2007) - Use Fermi-Walker frame instead of Frenet-Serret
10Gradient expansion
True solution as a perturbation to cylindrical
scroll wave
Hans Dierckx, 2009
Hans Dierckx, 2009
11Result advanced filament dynamics
- For isotropic media, we have obtained terms up
to O(k³, w³) - Observations 1. Scroll ring rotation
velocity depends on curvature 2. Coupling of
twist to motion only through filament curvature
3. Effective filament tension 4. Filament
motion in an isotropic medium is captured by
355 13 model-dependent coefficients 5.
El.phys. model via reaction term hidden in the
coefficients
12Special cases of the advanced dynamics
- Straight filament with nonzero twist
(sproing) - ? Effective filament tension gets lt0 for large
twist (if a1lt0)
13Special cases of the advanced dynamics
2. Untwisted scroll ring ? Drift
velocities need not be proportionate for large k
Keener Tyson, SIAM review 34(1), 1992
14Facts on filaments in anisotropic tissue (1/2)
- Dynamics in a medium with rotational anistropy
-
- 1. An intramural filaments drift to a layer
where the fibres run parallel or
perpendicular to the filament (Wellner
et al., Phys Rev E, 61(2), 2000) - 2. A straight transmural filament loses
stability when fibre rotation rate is
increased (Fenton et al., Chaos 8(1),
1998)
15Facts on filaments in anisotropic tissue (2/2)
- Statics look for the equilibrium position of a
filament - ? Wellners minimal principle (2003)
Wellner et al., PNAS 99(12), 2003
With fibres
Without fibres
the equilibrated filament lies along a geodesic
(curve of shortest length), when measuring
distances according to
16How to deal with anisotropy?
- Activation waves propagate faster along the
myofibres axes - Conduction velocity is related to the electric
diffusion tensor in the RDE - Effective distance connectivity
- T(a ? b) lt T(a b)
17Operational measure of distance (1/2)
B
B
Hans Dierckx, 2009
A
A
C
C
Hans Dierckx, 2009
18Operational measure of distance (2/2)
- When moving at a fixed local velocity
operational definition of distance travel time
! - Perform local rescaling according to local
velocity - The inverse diffusion tensor arises as a metric
tensor (Wellner et al., PNAS 99(12), 2003
Verschelde et al., Phys. Rev. Lett. 99(16),
2007) - Resulting space curved/non-Euclidean
19What is a metric tensor?
- A metric tensor is used in non-trivial spaces
to correlate coordinates to distances - Varying fibre orientation induces curvature of
space - Physical properties of a non-Euclidean space are
contained in second-order derivatives of the
metric tensor - - Riemann tensor Rijkl (6 components)
- Ricci tensor Rij Rkilj gkl (6
components) - Ricci scalar R Rij gij (1
component)
20Derivation of the equations of motion
- Construct a co-moving curvilinear coordinate
frame - Insert the Ansatz
- Consider the Goldstone modes of the linearized
operator - Project onto the left Goldstone-modes
(sensitivity functions) - Write the result in a coordinate invariant way
21Results filament revolution velocity
- Filament rotation velocity up to O(k³, w³, kR)
-
- Some consequences for rotational anisotropy aµz
- Rotation of a transmural filament is slower due
to rotational anisotropy (if e0gt0) - Non-constant fibre rotation, i.e. µ(z) induces
twist
22Results anisotropic filament motion (1/2)
- For translation/drift (in lowest order)
(Verschelde,Dierckx Bernus, PRL, 2007) - Equation of motion is unaltered, but now
includes anisotropy, since distances are
measured using - Steady state
-
?
Proof of the geodesic principle by Wellner et
al. (2002)
Hans Dierckx, 2009
23Results anisotropic filament motion (2/2)
- EOM for filaments up to O(k4, w4, R2), in
anisotropic medium - (in complex notation i rotation of 90 in
transverse plane) - Filament tension is altered by anisotropy
- Contains (small?) corrections to the minimal
principle -
24Filament motion in rotational anisotropy (1/2)
1. Transmural filaments can become
unstable due to filament tension
modification through R
25Filament motion in rotational anisotropy (2/2)
2. A straight, untwisted intramural
filament will drift towards a layer
with ? or fibres
Wellner, Berenfeld Pertsov Phys Rev E 61(2),
2000
26Wave fronts eikonal equation
-
- Activation waves propagate with a velocity that
depends on their curvature - Explained in terms of excited neighbouring
cells - Eikonal equation (Zykov, Keener)
27Results geometric front dynamics (1/2)
- The eikonal equation is retrieved in general
form - Anisotropy included (measure distances using
gD-1) - A model-dependent coefficient g is obtained
(can differ from 1) - g is proven to be the surface tension of the
front wavefront is stable ? g gt 0
28Results geometric front dynamics (2/2)
- Covariant eikonal equation
- Excellent correlation between theory numerical
simulation -
- Surface tension gamma depends on period of
pacing - (see poster 21/07)
29Discussion Anisotropy ? Dynamics
- Anisotropy ? intrinsic geometry/curvature of
space - When considering space as experienced by the
wavefront, motion equations are found in
simple form ( fictitious forces are
eliminated) - New terms appear (k,R) due to curvature of space
itself ( tidal forces cannot be gauged away)
30Discussion Models ? Dynamics
- Leading order dynamics involves model-dependent
coefficients - Wave fronts g(T) - Scroll wave
rotation a0, , e0 (5) - Filament drift g1,
g2, a1, a2, q1, q2 ( 222) - These constants can be assigned physical meaning
(tension, stiffness, core cross-section, )
and therefore could lead to more fundamental
understanding of wave propagation - Different electrophysiological models with
similar dynamical coefficients behave alike! - For effects included in the theory
(twistcurvatureanisotropy), faster
simulation with a simpler model could be
feasible, depending on the studys purpose
(e.g. isochrones, stability)
31Conclusions
- Geometric theory for rotor filaments -
Equation of motion for drift and rotation,
including twist, fil. curvature and tissue
anisotropy - Geometric theory for wave fronts - does not
assume steep fronts includes front tail ?
predicts surface tension g - can account for
dispersive effects - Leading order dynamical coefficients are
generated - Depend on the model used -
Can be calculated numerically - Bear
physical meaning (tension, stiffness, )
32Challenges ahead
- How can one measure the dynamical coefficients
in living tissue? - What can the geometrical theory teach us
about types of filament instability? - How to describe filament interaction
(fibrillation)? - Can we numerically simulate realistic
arrhythmias using filaments and geometry
alone? O(N³) ? O(N)
33Acknowledgements
- PhD dissertation advisors - Henri Verschelde
(Universiteit Gent) - Olivier Bernus
(University of Leeds) - Funded by Flanders Research Foundation (FWO
Flanders)