Geometric dynamics for rotor filaments and wave fronts

1
Geometric 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
2
Outline

• Introduction
• Filaments - Filament tension - Advanced
  dynamics - Anisotropy effects
• Fronts - Surface tension - Anisotropy effects
  
• Discussion
• Conclusions

3
What are scroll wave filaments?

• Filament rotation axis of a spiral wave
  extended to 3D

Hans Dierckx, 2009
4
Why 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

5
How 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
6
Equation 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

7
Filament 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
8
Further 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

9
Geometrical 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

10
Gradient expansion
True solution as a perturbation to cylindrical
scroll wave
Hans Dierckx, 2009
Hans Dierckx, 2009
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Result 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

12
Special cases of the advanced dynamics

• Straight filament with nonzero twist
  (sproing)
• ? Effective filament tension gets lt0 for large
  twist (if a1lt0)

13
Special 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
14
Facts 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)

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Facts 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
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How 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)

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Operational measure of distance (1/2)
B
B
Hans Dierckx, 2009
A
A
C
C
Hans Dierckx, 2009
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Operational 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

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What 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)

20
Derivation 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

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Results 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

22
Results 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
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Results 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

24
Filament motion in rotational anisotropy (1/2)
1. Transmural filaments can become
unstable due to filament tension
modification through R
25
Filament 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
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Wave 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)

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Results 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

28
Results 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)

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Discussion 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)

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Discussion 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)

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Conclusions

• 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, )

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Challenges 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)

33
Acknowledgements

• PhD dissertation advisors - Henri Verschelde
  (Universiteit Gent) - Olivier Bernus
  (University of Leeds)
• Funded by Flanders Research Foundation (FWO
  Flanders)