inverse problems in cardiac electrophysiology

Elisa SCHENONE 21st March 2013

INVERSE PROBLEMS INCARDIAC ELECTROPHYSIOLOGY

LJLL - Théorie du ContrôleElisa SCHENONE (PhD student at LJLL and INRIA-Rocq)Supervisors:Jean-Frédéric GERBEAU and Muriel BOULAKIA

1


✓ Bidomain Model

✓ Inverse Problem

✓ Reduced Methods

✓ Numerical Results

✓ Conclusions and Perspectives

Summary

2


➡ Bidomain equations (Tung 1978)

➡ Ionic Model (Mitchell-Schaeffer 2003)

➡ Torso equations

➡ Coupling conditions (weak form)

Cardiac electrophysiology modelBidomain equations

AmCm@Vm

@t+AmI(Vm, w)� div(�irVm) + div(�irVm) = AmIa

div((�i + �e)rue) + div(�irVm) = 0

@w

@t+ g(Vm, w) = 0

I(Vm

, w) = � w

⌧in

V 2

m

(1� Vm

) +1

⌧out

Vm

g(Vm

, w) =

8>>><

>>>:

w

⌧open

� 1

⌧open

(vmax

� vmin

)2if V

m

< vgate

w

⌧close

if Vm

> vgate

div(�TruT) = 0

ue = uT

3


Inverse ProblemClassical electrophysiology inverse problem‣ Thorax isolated model

➡ Inverse problem:

find g 2 H1/2(@H) s.t. R(g) = d, d 2 L2(@T )

r · (�TruT) = 0 in T(�TruT) · n = 0 on @T

uT = g on @H

➡ Minimization problem: ming2H1/2(@H)

kR(g)� dk2L2(@T )

4

@H

H@T

T

where R : H1/2(@H) ! L2(@T )R(g) = uT(g)|@T


Inverse Problem - Parameters estimationDiscrete Approach and Electrocardiogram (ECG)

5

�H = ue|@Hh

�T = uT|@TECG

➡ Discrete minimization problem:

�T = A�H


Inverse Problem - Parameters estimationDiscrete Approach and Electrocardiogram (ECG)

5

�H = ue|@Hh

�T = uT|@TECG

➡ Discrete minimization problem:

�T = A�H

➡ Parameters estimation:

‣ Several evaluations of the solution of bidomain equations in the heart

min⇡2Rn

kA�H(⇡)� dk2L2(0,T ;R9)


‣ ROM basis and approximated solution

Reduced Order ModelProper Orthogonal Decomposition (POD)

VN ⌘ span{v1, . . . , vN } ⇢ VFEM space ROM space

VN ⌘ span{'1, . . . ,'N}, N ⌧ N

�N = ['1 . . .'N ] 2 RN⇥N , eu = �TNu

‣ POD technique• Snapshots of FEM solution

• Singular Value Decomposition (SVD)

• Keep columns of as ROM basis functions

S = (u1, . . . ,up) 2 RN⇥p

S = �⌃ T , ⌃ = diag(�i)

N �

6


Parameters identification problemLong time simulations and Restitution Curve (RC)

A 75 accelerated beats simulation on the heart model results.7



QTk+1

= ⌧close

ln⇣

1�(1�hmin)e�TQk/⌧

open

hmin

⌘‣ RC

A 75 accelerated beats simulation on the heart model results.7

Simpled QT/TQ valuesAnalytical value forAnalytical value for

⌧Mcell

close

⌧ epiclose



‣ RC

7

Parameters Estimate Std. Error t-value

⌧open

298.5769 0.6389 476.3

⌧close

100.5891 0.2742 366.9

APD1 189.7231 0.0392 4839.3

APDk+1

= ⌧close

ln⇣

1�(1�hmin)e�DIk/⌧

open

hmin

⌘

➡ Single cell model

➡ extend this analysis

on an ECG-based RC

Exact values

⌧open

⌧close

APD1

300 100 189.712

Elisa SCHENONE 21st March 2013 8

Transmenbrane Potential (mV)

20.00 -5.00-30.00-55.00-80.00

Time 300.00 msec

Reference(Full Order Model)

Solution(Reduced Order Model)

‣ Method

• Cost function

• Genetic Algorithm➡ several evaluations of the cost function

• Evaluation of cost function using a ROM➡ POD Basis: snapshots from various infarcted areas

X

k2{I,...,V 6}

NTX

i=1

|Vk(ti)� Vk,ref(ti)|2

S = (u1I1, . . . ,uNT

I1,u1

I2, . . . ,uNT

Im, ) 2 RN⇥(mNT )

Parameters identification problemIdentification of an infarcted area



20.00 -5.00-30.00-55.00-80.00

Time 300.00 msec



-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

I

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

II

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

III

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVR

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVL

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVF

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V1

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V2

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V3

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V4

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V5

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V6

Exact solutionEstimated solution‣ Results




20.00 -5.00-30.00-55.00-80.00

Time 300.00 msec



-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

I

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

II

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

III

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVR

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVL

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

AVF

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V1

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V2

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V3

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V4

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V5

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

V6

Exact solutionEstimated solution

Exact solution Estimated solution

‣ Results


[Boulakia, Schenone, Gerbeau - IJNMBE2012]


Future works‣ RC approach

• improve theory features

• new parametrization of the curve

‣ Inverse problem

• Poisson problem

• new regularization technique

‣Reduced Order Basis

• Approximated Lax Pairs (ALP) decomposition (D. Lombardi, J.F. Gerbeau)


Conclusions‣ POD is a good ROM to reproduce cardiac

pathologies, e.g. infarctions and tachycardia

‣ ROM are useful in inverse problem as well

‣ RC represents a new approach to parameters estimation inverse problems

Perspectives‣ more results using RC approach

‣ a faster and efficient ROM basis (e.g. ALP)

‣ application to realistic ECGs

10

inverse problems in cardiac electrophysiology

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