Taming cardiac disorders: with some help from applied mathematics

(1)

Taming cardiac disorders:

…with some help from applied mathematics…

Alain Pumir

Laboratoire de Physique, ENS Lyon, France

(2)

Cardiac arrhythmia :

meet the monster…

Life-threatening cardiac arrhythmias result from

spiral wave breakup leading to

electrical turbulence

in the

heart, and to lack of proper functioning of the blood pump.

(3)

Pacing and control of chaos

For lack of a better strategy, chaos control in the heart rests on electric fields.

- Defibrillation shock : one shock, with a very large currents delivered through the heart.

* High success rate but numerous undesirable side effects -- including pain.

- Anti-Tachycardia-Pacing is used to control an incipient arrhythmia

(tachycardia), by sending through one electrode several small stimuli, at frequency larger than the main frequency of the ECG.

* Success rate : ok (70-90%).

(4)

How to terminate turbulent activity in

the heart ?

An old problem for applied mathematics !

(5)

Pacing of the heart : role of

heterogeneities.

Wiener and Rosenblueth (1946)

Topological conservation :

(Wiener and Rosenblueth, 1946)

# of waves rotating counterclockwise, clockwise around the obstacle.

In the figure,

Impossible to remove the rotating wave by pacing in the case of the

cellular automata model of Wiener and Rosenblueth !



n

_



n

_



cst.



n

_

,

n

_





n

_



n

_



1

(6)

Control of vortices by pacing.

Pumir, Sinha et al. (2010)

The topological

conservation law of

Wiener and Rosenblueth

does not apply to small

obstacles in the case of

realistic models !

Possibility of pacing a wave from a small obstacle

nb : generic phenomenon in

excitable media; experimental possibilities...



(7)

Taming a large turbulent system??

• Suggestion : use many pacing sites –

(8)

Ingredients for better pacing.

Pacing from multiple sites

has been noticed many times to

be more efficient than from just one site (e.g., Allessie et al,

1991; Pande et al., 2001).

Heterogeneities

in the heart can act as

wave sources

when

an external electric field is applied (e.g., R. Plonsey, 1982,

Pumir and Krinsky, 1999).

Combine the two ideas =>

Far Field Pacing

~ use heterogeneities as ‘pacing electrodes’.

n.b. : there are

many heterogeneities

in a heart !!

(9)

(10)

Action potential propagation

In the tissue, the

electric potential

propagates due to

the resistive

coupling between

points

 C_m 



e



t  Istim (INa  IK  IL)  1





2_e



x2

(11)

Shaw, R. M. et al. Circ Res 1997;81:727-741

Dynamical model of the heart : the dynamic

Luo-Rudy ventricular cell model (LRd)

(12)

The minimal model : FitzHugh

• The simplest model that describes the

class of excitable system is the

FitzHugh system :

 dw dt (ew)  de dt  Ae(1e)(eu)w  de dt 0  dw dt 0

The horizontal motion is fast compared to the vertical motion :

Two time scale problem.



(13)

Interaction of an external

(14)

Obstacle/electric field interaction

Model problem

:

one hollow circular obstacle in a 2-dimensional piece of cardiac tissue

.

Use the monodomain model (simplest generalization of cable theory).

V

_m = membrane potential

e=(V

_m

-V

_m,rest

)

=

deviation from the resting membrane potential.

In the linear regime (weak deviations from resting potential) :

with the b.c. :

when

and at the obstacle’s boundary ( ).





2

e



e



2



0



e



0



r





ˆ

n

.



(

e



E

.

r

)



0



r



R

(15)

Obstacle/electric field interaction

The solution is obtained in 2d as a standard Bessel function :

( = angle between E and r)







e

(

r

)

 

E



K

1

(

r

/



)

K

'

₁

(

R

/



)



cos



=> Part of the tissue is

depolarized (e > 0).

If

depolarization

exceeds a

threshold

,

an Action Potential

(16)

Maximum depolarization

:



e

_max

 



E

K

1

(

R

/



)

K

'

₁

(

R

/



)

The stimulation threshold is reached when is

larger than a threshold: .

The field needed to reach the stimulation threshold

decreases with the size

R

of the obstacle:



e

_max



e

_max



(



_t

 

_rest

)



E



E

 

(



t

 

rest

)



K

₁

'(

R

/



)

K

₁

(

R

/



)

(17)

Dependence of the electric field needed to start a

wave as a function of the size of the obstacle:

Consequence

:

When an electric field of

strength

E=e,

is applied,

all obstacles with a size

emit waves.

n.b.:

when

R

_min

is small.



R



R

_min

(

e

)



E

 

(



t

 

rest

)



K

₁

'(

R

/



)

K

₁

(

R

/



)



E

~ 1/

R

_min

(

E

)

(18)

Consequence

: depolarization reaches the stimulation

threshold provided the obstacle is large enough, or the field

is strong enough.

By increasing the external field, one may increase the

number of (virtual) pacing electrodes.

(19)

Numerical illustration:

Integrate numerically the Luo-Rudy 1 model, with several

circular obstacles

of various sizes :

As the externally applied

electric field increases

,

obstacles of

decreasing sizes

trigger AP propagation

see: Pumir et al, Phys. Rev. Lett. 2007.

(20)

Far field pacing :

(21)

Far field pacing : numerical example

Heterogeneities:

collagenous septa

randomly

distributed

throughout the

medium.

From Fenton et al, Circulation 2009.

E=0.8 V/cm

E=1.14 V/cm

(22)

Far field pacing : numerical example

Apply 8

low-intensity shocks:

S3: T=970ms

S8: T=1905ms

From Fenton et al, Circulation 2009.

E=0.8 V/cm

E=1.14 V/cm

(23)

Far field pacing: numerical example

The success of the method is based on the presence of

obstacles close to the core of the spiral.

=> one needs precise information on the

distribution of heterogeneities !!

(24)

Activation time and distribution

of heterogeneities

(25)

Response time in the presence of many obstacles



V



4



3



N



 

v



3 define density as  = N/V







(3/(4



))

1/ 3

/

v

assumptions: • 3d • homogenoeus tissue • propagation velocity v

(26)



V



4



3



N



 

v



3 define density as  = N/V







(3/(4



))

1/ 3

/

v

assumptions: • 3d • homogenoeus tissue • propagation velocity v

(27)







(3/(4



))

1/ 3

/

v

tissue: distribution function of sizes p(R)





 

E



p R

 

R_min

 

E R_max



d

R

E  1/R MRI

(28)

(29)

Obstacle Size Distributions



p



1/

R

2.520.07



p



1/

R







 

E



p R

 

R_min E R_max



d

R

(30)

n

Obstacle Size Distributions



p



1/

R

2.520.07



p



1/

R







 

E



p R

 

R_min E R_max



d

R

E  1/R







E

1







(3/(4



))

1/ 3

/

v

with







E

(1) 3 prediction

(31)

(32)

(33)

Optical Mapping

• simultaneous epi/endo mapping • EMCDD (128 x 128, 16 bit) • 500 fps

•Use voltage-sensitive dyes (di-4-anepps).

(34)

(35)

(36)

Relating the density of obstacle size and the response time: Ventricular tissue (3d) MRI result: p(R) ~ R-2.75 Response time: τ ~ E-0.58 Comparison: Response time vs. MRI measurment.

(37)

(38)

Protocol:

5 pulses, 1.4 V/cm

[5 ms duration, 90 ms period]

(39)

(40)

Summary and Conclusion

• virtual electrodes exist; they reflect the fine structure of the tissue, which can be described in terms of power laws.

• Virtual electrode mechanism is consistent with observation.

• Virtual electrodes can be used to defibrillate (== synchronize) in an optimal way.

(41)

Thanks to:

• V. Krinsky (Nice and Göttingen)

• S. Sinha, S. Sridhar (Chennai)

• E. Bodenschatz, S. Luther, C. Richter, A.

Squires, D. Hornung, P. Bithin (Göttingen)

• R.Gilmour, F.Fenton, E.Cherry and N.Otani

(Cornell)

(42)

Thank You !

Ref :

A. Pumir et al., Phys. Rev. Lett. 99, 208101 (2007)

F. Fenton et al., Circulation, 120, 467 (2009)

(43)

Experimental results

The main issues addressed here :

In a real tissue :

• Possibility of terminating VT/VF by applying a small intensity

electric pulse ?

• Does increasing the electric field strength lead to more

heterogenities acting as wave sources ?

Experimental preparation (Fenton et al, 2009):

o

in-vitro

experiments with Mongrel dog hearts.

o

optical mapping of the atrium surface

o

Deliver shocks of duration 5-10ms, and of amplitude up to E

= 4.6 V/cm.

(44)

Recruitment of wave sources

A: pace with a single electrode B: apply a shock E=0.32 V/cm C: E=0.46 V/cm D: E=0.93 V/cm E: E=1.4 V/cm

(45)

Activation time

The stronger the applied Electric field, the shorter it takes to excite

completely the tissue.

Indirect evidence that when the electric field is increased,

the number of wave sources increases !

(46)

Successful termination of AF

Use 5 shocks of 5ms; Period : 45ms; E=1.4 V/cm. « Electrical turbulence» Apply 5 shocks Restoration of the normal rhythm.

(47)

Unsuccessful termination of AF

Use 5 shocks of 5ms; Period : 45ms; E=0.9 V/cm. « Electrical turbulence» Apply 5 shocks Arrhythmia continues

(48)

Success of multishock therapy

Ambrosi et al, 2011:

Proof of success of ‘multiple shock

(49)

Conclusions

Heterogeneities are known to potentially arrhythmogenic.

They can also help in getting rid of arrhythmias !

Main observations in this work :

- Heterogeneities can be used as Wave Emitting Sites, the number of WES increasing with the electric field.

- Properly placed virtual electrodes can pace spiral waves away.

- Experimental evidence : suggests that it does work… still work to be done to fully understand the mechanisms...

(50)

Obstacle/electric field interaction

3-dimensional case.

For a simple spherical obstacle of radius R, with a value  of the electrotonic length :  E_max  E   1  R 1 2 R  22 R2           E   R   R2 2 22R   R2 2            E_max ER/2 in 3d ER in 2d

(51)

In cardiac tissue, heterogeneities may have a noncircular

structure (e.g., clefts experiments on tissue culture; cf Fast et

al, 1998)

(52)

Effect of the angle :

Integrate numerically the Luo-Rudy 1 model, with several

elliptical obstacles

of various sizes, either horinzontal or

vertical :

By

flipping the direction

of the electric field, waves

start from an entirely

different set of virtual

(53)

A simple model problem

: field generated by an ellipse :

on the obstacle.

at infinity

E  a b





2

e



e



2



0



ˆ

n

.



(

e



E

.

r

)



0



e



0

(54)

Solution

: In the case where solve Laplace

equation : by using conformal transformation :

with

Solution for the membrane potential :





2

 





2

e



₀



z









2









1

2

a

2



b

2



e

(

z

)



Re

(

a



b

)

2

e

i



(

a

2



b

2

)

e

i

4















(55)

Maximum depolarization

:



e

Max

= b E



e

Max

= a E

[Hard to emit

waves]

[Easy to emit

waves]



e

Max



E

(

a



b

)

2



(

b

2



a

2

)cos(2



)

2













1/ 2



(



2

 

)

(56)

A more realistic bidomain study (Takagi et al, 2004)

Reentry “unpinning” by a low-energy

shock: Mathematical 2D bidomain model

(57)

Far field pacing : a simple

numerical example

It can happen that pacing from one single site is enough to get rid of free vortices.

… ATP Consider again the tissue with a stripe of tissue with a long refractory time, where ATP was illustrated to fail.

Assume that there exists relatively big obstacle (virtual electrode) not too far away from the cores of the vortices.

Turn on and off the external electric field :

• field on for ~5ms, then off for ~100ms, several times; • typical values of the field : ~0.5-1 V/cm)

(58)

Application:

Unpinning of vortices attached

to obstacles.

(59)

Theoretical idea: What can be done with a single pulse, using the interaction with the obstacle ?

The depolarization induced by a (relatively) moderate electric field, if properly timed, can detach a wave.

Proof of concept obtained by studying over-simplified models (Pumir and Krinsky, JTB 1999).

How to get rid of spiral waves “attached”

to obstacles (scar tissue, etc) ?

(60)

Condition of successful “unpinning”:

low intensity, but precise timing !

Successful “unpining” of waves attached to an obstacle in a rabbit heart preparation (Ripplinger et al, 2006).

(61)

Controlling electric turbulence

in the heart with low energy far

field stimulation.

Alain Pumir

(ENS Lyon, France)

.

Collaborators:

V.Krinsky

(Nice and Göttingen),

E.Bodenschatz and S.Luther (et al.)

(MPI Göttingen)

,

R.Gilmour, F.Fenton, E.Cherry and N.Otani (et al.)

(Cornell)

,

K.Yoshikawa, A.Isomura, K.Agladze & M.Hörning

(Kyoto),

(62)

Low field defibrillation of cardiac muscle via wave

emission from heterogeneities

Alain Pumir

Laboratoire de Physique, ENS Lyon, France

S. Luther, E. Bodenschatz, V. Krinsky, G. Luther, C. Richter, A. Squires, D. Hornung, P. Bithin

Max Planck Institute for Dynamics and Self-Organization &

Inst. for Nonlinear Dynamics, Georg August Universität, Göttingen, Germany

F. Fenton, R. Gilmour, E. Cherry, N. Otani.

(63)

Outline

-

Electric waves in the heart

-

Interaction of an external electric field with an obstacle.

-

Far Field Pacing : numerical examples.

-

Activation time and distribution of heterogeneities.

(64)

Pacing vortices: how many wave

sources does one need ?

“One” can be good enough…

Anti-Tachycardia Pacing : involves pacing waves from an electrode at high enough a frequency (higher than the frequency of the rotating waves in the heart).

Gottwald, Pumir and Krinsky, 2001

Recent observation of the phenomenon in cardiac cell cultures : Agladze et al., Am. J. Physiology, 2007

(65)

Far field pacing : a simple

numerical example

Pumir et al., Phys. Rev. Lett. 2007. Stripe of longer Refractory period Obstacles (virtual electrode) Final time : The pacing waves have swept away the vortices Initial time : Two vortices

(66)

Failure or success of pacing ?

key to success: Wave sources should be close enough to the spiral cores. A remote wave source can be shielded from the vortex, e.g. by a region with a longer refractory period (=> unsusccesful pacing.)

P

: Pacing site

Stripe of tissue with a longer refractory period

(67)

Far field pacing : a simple

numerical example

Lessons:

•

The study of very simple models suggests that success

can be achieved with a relatively weak external field

amplitude, provided the density of potential wave sources

is large enough.

• The success or failure of Far Field Pacing does

NOT

depend

much on the precise phase of the pacing w.r.t. the phase(s) of

the spiral(s) provided the wave sources are properly located.

(68)

Effect of Increasing Field Strength

Multisite excitation

Single Wave propagation (n = 1)

Cardioversion (n  )

(69)

Relating the density of obstacle size and the response time: Atrial tissue (2d) MRI result: p(R) ~ R-2.54 Response time: τ ~ E-0.87 Comparison: Response time vs. MRI measurment.

(70)

Obstacle/electric field interaction

The simplest problem:

one fiber with an externally applied electric field (Weidmann 1953).

Mathematical description: Use cable theory. V_m : membrane potential

e = (V_m – V_m,rest) : deviation from the resting potential.

In the linear regime ( ~ weak deviations from resting potential) :

λ = electrotonic length (~1mm)

with: far away from the boundary, and de/dx = E at the boundary.

Solution: e(x) = λ E exp(-x/λ) for x > 0;

Depolarization: + or - λ E 

d2_e_/_dx2 __e_/



2 _₀

