geometric dynamics for rotor filaments and wave fronts

CPP 2009, H. Dierckx, Geometric dynamics for filaments and fronts, sl.1

Geometric dynamics for rotor filaments and wave fronts

Hans Dierckx1, Olivier Bernus2,3, Henri Verschelde1

CPP 2009, Cambridge, UK 30 June 2009

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


Outline

• Introduction

• Filaments- Filament tension- Advanced dynamics- Anisotropy effects

• Fronts- Surface tension- Anisotropy effects

• Discussion

• Conclusions


What are scroll wave filaments?

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

Hans Dierckx, 2009


Why study filament evolution?

• Number of filaments vs. arrhythmiae# = 0 : healthy state (?)# = 1 : monomorphic tachycardia# > 1 : polymorphic tachycardia/ torsade de pointes#>>1 : 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


How to study filament evolution?

• Start from generic reaction-diffusion equation:

• Quantities for describing a scroll wave filament:

- revolution velocity

- arclength - phase angle twist

- Filament curvature

k=1/R

w

ijt i jD u Pu f u

Hans Dierckx, 2009


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

20 0 0

1

2

a w c w

X N k

X B k


Filament tension• The coefficient 1 plays the role of filament tension

• Positive 1 straightens filaments

• Negative 1 can lead to filament instability/multiplication

Biktashev, Holden & Zhang,

Phil Trans R Soc 347, 1994

Fenton & Karma, Chaos 8(1), 1998

1X N k

Biktashev, Holden & Zhang,

Phil Trans R Soc 347, 1994


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

20 0 0

1

2

??

??

??

a w b w

X N k

X B k


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

1 2 1 2 1 2 20( , , , ) ( , ) ( , , , ) ( ) u u u O

12

= + …filament thicknessradius of curvature


Gradient expansion

True solution as a perturbation to cylindrical scroll wave:

20 ( , , )kd wd d u u RO

20 ( , , )X kd wd d RO

Hans Dierckx, 2009Hans Dierckx, 2009


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 3+5+5 = 13 model-dependent coefficients 5. El.phys. model via reaction term hidden in the coefficients

2 20 0 0 0

2 21 1 1 1 1

2 22 2 2 2 2

a w b w c k

X N k a w b w c k f w k

X B k a w b w c k f w k


Special cases of the advanced dynamics:1. Straight filament with nonzero twist (‘sproing’)

Effective filament tension gets <0 for large twist (if a1<0)

21 1 1X N k a w b w

21 1 1 1eff a w b w


Special cases of the advanced dynamics:2. Untwisted scroll ring:

Drift velocities need not be proportionate for large k

21 1

22 2

X N k c k

X B k c k

Keener & Tyson, SIAM review 34(1), 1992


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)

z


Facts on filaments in anisotropic tissue (2/2)

• Statics: look for the equilibrium position of a filament

Wellner’s minimal principle (2003)

Without fibres

Wellner et al., PNAS 99(12),

2003

31 1

, 1

d ² ( ) d d d di j Tij

i j

s D x x

x D x

the equilibrated filament lies along a geodesic (curve of shortest length),

when measuring distances according to

With fibres


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) < T(a ||| b)

ijt i jD u Pu f u

c D


Operational measure of distance (1/2)

A

B

C

A

B

C

Hans Dierckx, 2009

Hans Dierckx, 2009


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


• 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)

What is a metric tensor?


Derivation of the equations of motion

1. Construct a co-moving curvilinear coordinate frame

2. Insert the Ansatz

3. Consider the Goldstone modes

of the linearized operator

4. Project onto the left Goldstone-modes (sensitivity functions)

5. Write the result in a coordinate invariant way

20 ( ) u u u O

2 0 0ˆ ' L P A F u

, ,x y

:Y Y


• Filament rotation velocity up to O(k³, w³, kR):

• Some consequences for rotational anisotropy =µz:

1. Rotation of a transmural filament is slower due to rotational anisotropy (if e0>0)

2. Non-constant fibre rotation, i.e. µ(z) induces twist

Results: filament revolution velocity

22 02

L T

L

D D

D

R

2 20 0 0 0 0 0a w b w c k d e R


• 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)

Results: anisotropic filament motion (1/2)

1 2X k N k B

1d ² d dTs x D x

20 0X X D

Hans Dierckx, 2009


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:

0 ( , )k X O w R R


1. Transmural filaments can become unstable due to

filament tension modification through R

Filament motion in rotational anisotropy (1/2)

1 0?eff R

z


Filament motion in rotational anisotropy (2/2)

2. A straight, untwisted intramural filament will drift towards a layer with or || fibres:

z

Wellner, Berenfeld & Pertsov Phys Rev E 61(2), 2000


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):

0c c DK


Results: geometric front dynamics (1/2)

The eikonal equation is retrieved in general form:

• Anisotropy included (measure distances using g=D-1)

• A model-dependent coefficient is obtained (can differ from 1)

• is proven to be the surface tension of the front:

wavefront is stable > 0

0ij

ijc c D K

0

0

d

d

Y

Y

Y P uP

A Y A u


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)

Model theory simulation

Barkley 1.19 1.19

Aliev-Panfilov 1.00 1.04

FitzHugh-Nagumo 1.19 1.17

Reduced Noble 0.68 0.67

0ij

ijc c D K

( ) ( )T T


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 (,R) due to curvature of space itself (~ tidal forces cannot be gauged away)


Discussion: Models Dynamics

• Leading order dynamics involves model-dependent coefficients:- Wave fronts: (T)- Scroll wave rotation: a0, …, e0 (#=5)- Filament drift: 1, 2, a1, a2, … q1, q2 (#= 2+22)

• 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 (twist+curvature+anisotropy), faster simulation with a simpler model could be feasible, depending on the study’s purpose (e.g. isochrones, stability)


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 - 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, …)


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) ]


Acknowledgements

• PhD dissertation advisors: - Henri Verschelde (Universiteit Gent)

- Olivier Bernus (University of Leeds)

• Funded by Flanders Research Foundation (FWO Flanders)

geometric dynamics for rotor filaments and wave fronts

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