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Transcript of Laboratoire de Physique, ENS Lyon, France - math.unice.frauroux/IFCAM2012/Alain_Pumir.pdf · heart,...
Taming cardiac disorders:
…with some help from applied mathematics…
Alain Pumir Laboratoire de Physique, ENS Lyon, France
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.
nb : role of heterogeneities (the heart is NOT homogeneous) ??
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%).
How to achieve better success rate without using a big shock ??
How to terminate turbulent activity in
the heart ?
An old problem for applied mathematics !
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
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...
n n cst.
Taming a large turbulent system??
• Suggestion : use many pacing sites –
theoretical suggesion, see Pande et al, 2001.
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 !!
Electric waves in the heart
Action potential propagation
In the tissue, the
electric potential
propagates due to
the resistive
coupling between
points
Cm e
t Istim (INa IK IL )
1
2e
x 2
Shaw, R. M. et al. Circ Res 1997;81:727-741
Dynamical model of the heart : the dynamic Luo-Rudy ventricular cell model (LRd)
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.
1
Interaction of an external
electric field with an obstacle.
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).
Vm = membrane potential
e=(Vm-Vm,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 ( ).
2e e
2 0
e0
r
ˆ n .(e E .r ) 0
r R
Obstacle/electric field interaction
The solution is obtained in 2d as a standard Bessel function :
( = angle between E and r)
e(r ) EK1(r /)
K '1 (R /) cos
=> Part of the tissue is
depolarized (e > 0).
If depolarization exceeds a
threshold, an Action Potential
(wave) may start.
Obstacle/electric field interaction
Maximum depolarization :
emax EK1(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:
emax
emax (t rest)
E
E (t rest)
K1'(R /)
K1(R /)
Obstacle/electric field interaction
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 Rmin is small.
R Rmin (e)
E (t rest)
K1'(R /)
K1(R /)
E ~ 1/Rmin (E)
Obstacle/electric field interaction
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.
Crucial feature of the method proposed here !
Obstacle/electric field interaction
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.
Far field pacing :
numerical examples.
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
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
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 !!
Activation time and distribution
of heterogeneities
Response time in the presence of many obstacles
V 4
3 N v
3define density as
= N/V
(3/(4))1/ 3 /v
assumptions:
• 3d
• homogenoeus tissue
• propagation velocity v
excited tissue after
Response time in the presence of many obstacles
V 4
3 N v
3define density as
= N/V
(3/(4))1/ 3 /v
assumptions:
• 3d
• homogenoeus tissue
• propagation velocity v
excited tissue after
Response time in the presence of many obstacles
(3/(4))1/ 3 /v
tissue: distribution function of sizes p(R)
E p R Rmin E
Rmax
dR
E 1/R
MRI
Obstacle Size Distributions
p1/R2.520.07
p1/R
E p R Rmin E
Rmax
dR
n
Obstacle Size Distributions
p1/R2.520.07
p1/R
E p R Rmin E
Rmax
dR
E 1/R
E1
(3/(4))1/ 3 /vwith
E(1)
3
prediction
Experimental results
Experimental Setup
Optical Mapping
• simultaneous epi/endo mapping
• EMCDD (128 x 128, 16 bit)
• 500 fps
•Use voltage-sensitive dyes
(di-4-anepps).
Response of the tissue to a field of increasing strength
Response of the tissue to a field of increasing strength
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.
Implication for defibrillation/control of the heart.
Protocol:
5 pulses, 1.4 V/cm
[5 ms duration, 90 ms period]
82 % energy reduction
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.
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)
• M. Argentina (Nice)
Thank You !
Ref :
A. Pumir et al., Phys. Rev. Lett. 99, 208101 (2007)
F. Fenton et al., Circulation, 120, 467 (2009)
S. Luther et al., Nature, 475, 235 (2011)
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.
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
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 !
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.
Unsuccessful termination of AF Use 5 shocks of 5ms;
Period : 45ms;
E=0.9 V/cm.
« Electrical turbulence»
Apply 5 shocks
Arrhythmia continues
Success of multishock therapy
Ambrosi et al, 2011: Proof of success of ‘multiple shock
therapy to stop Atrial fibrillation/Atrial flutter.
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...
… Clinical implications ?? …
Obstacle/electric field interaction 3-dimensional case.
For a simple spherical obstacle of radius R, with a value of the
electrotonic length :
Emax E 1
R
12
R
22
R2
E
R
R2
2
2 2R
R2
2
Emax ER /2 in 3d
ER in 2d
Obstacle/electric field interaction
In cardiac tissue, heterogeneities may have a noncircular
structure (e.g., clefts experiments on tissue culture; cf Fast et
al, 1998)
Obstacle/electric field interaction
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
electrodes.
Obstacle/electric field interaction
A simple model problem : field generated by an ellipse :
on the obstacle.
at infinity
E
a
b
2e e
2 0
ˆ n .(e E . r ) 0
e0
Obstacle/electric field interaction
Solution : In the case where solve Laplace
equation : by using conformal transformation :
with
Solution for the membrane potential :
2
2e 0
z 2
1
2a2 b2
e(z) Re(a b)2ei (a2 b2)ei
4
Obstacle/electric field interaction
Maximum depolarization :
eMax = b E
eMax = a E
[Hard to emit
waves]
[Easy to emit
waves]
eMax E(a b)2 (b2 a2)cos(2)
2
1/ 2
(2 )
A more realistic bidomain study (Takagi et al, 2004)
Reentry “unpinning” by a low-energy
shock: Mathematical 2D bidomain model
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)
Application:
Unpinning of vortices attached
to obstacles.
[collaboration with S. Takagi and I. Efimov]
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) ?
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).
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),
I. Efimov and V. Nikolski (Washington University).
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. Department of Biomedical Sciences, Cornell University, USA
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.
- Experimental results
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
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
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
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.
Effect of Increasing Field Strength
Multisite excitation
Single Wave propagation (n = 1)
Cardioversion
(n )
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.
Obstacle/electric field interaction
The simplest problem:
one fiber with an externally applied electric field (Weidmann 1953).
Mathematical description: Use cable theory.
Vm : membrane potential
e = (Vm – Vm,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
d2e /dx2 e /2 0
e0