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Submitted on 10 Jan 2014
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Inverse problem in electrocardiography via factorization method of boundary problems : how reconstruct epicardial potential maps from measurements of the
torso ?
Julien Bouyssier, Nejib Zemzemi, Jacques Henry
To cite this version:
Julien Bouyssier, Nejib Zemzemi, Jacques Henry. Inverse problem in electrocardiography via factor- ization method of boundary problems : how reconstruct epicardial potential maps from measurements of the torso ?. 2013. �hal-00926344�
Inverse Problem in Electrocardiography via Factorization Method of Boundary Value Problems :
How reconstruct epicardial potential maps from measurements of the torso ?
Julien Bouyssier, Nejib Zemzemi, Jacques Henry
julien.bouyssier@inria.fr, nejib.zemzemi@inria.fr, jacques.henry@inria.fr
Motivation and goal
Motivation : Solve the inverse problem in electrocardiography from measurements of the torso.
Goal : Use factorization method to compute epicardial potential maps.
This work is a simplified presentation of the method by considering a cylinder as geometry of our problem.
Initial problem : electrical potential u in the domain Ω is governed by
(P
0)
∆ u = 0 in Ω , Ω : cylinder
u = 0 on Σ , Σ : lateral surface
u = T on Γ
T, T : potential on the torso surface Γ
T∇ u · n = Φ on Γ
T, Φ : normal derivative of the potential
With T and Φ known, find potential t and his normal derivative φ on the heart surface Γ
Hto complete this ill-posed Cauchy problem.
Optimal control problem
(P
0) is decomposed into two sub-problems, ∀( η, τ ) :
(1)
∆ u
1= 0 in Ω u
1= 0 on Σ u
1= T on Γ
T∇ u
1· n = η on Γ
Hand (2)
∆ u
2= 0 in Ω u
2= 0 on Σ u
2= τ on Γ
H∇ u
2· n = Φ on Γ
TSolve (P
0) :
= ⇒ Define the cost function E ( η, τ ) = R
Ω
(∇ u
1( η ) − ∇ u
2( τ ))
2= ⇒ Find ( φ, t ) : minimize E ( η, τ )
New approach : the factorization method by invariant embedding
Principle of invariant embbeding
Principle : transport potential data from torso surface to heart surface
= ⇒ Boundary value problems (1) and (2) are embedded into a family of similar problems on subdomains Ω
S= ⇒ Ω
Sare bounded by a moving boundary Γ
Sdefined at x = s for x = 0 −→ x = a
= ⇒ At each position x = s , we impose a Neumann boundary condition
∂u∂xS1 |ΓS
= α for (1) and a Dirichlet boundary condition ( u
S2)
|ΓS= β for (2) :
(P
S1)
∆ u
1S= 0 in Ω
Su
1S= 0 on Σ
Su
1S= T on Γ
T∇ u
1S· n = α on Γ
Sand (P
S2)
∆ u
2S= 0 in Ω
Su
2S= 0 on Σ
Su
2S= β on Γ
S∇ u
2S· n = Φ on Γ
T000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
Γ
Tx=s x=a
x=0
Evolution for the moving boundary Ω
SΓ
Svolume Growing
Γ
HΣ
SΩ boundary Moving Σ
Figure 1: Illustration of the moving boundary. The domain Ω is a cylinder of a lenght a with a section O and a lateral surface Σ. The section O = Γ
Tat x = 0 represents the torso surface and the section O = Γ
Hat x = a represents the heart surface. The moving boundary O = Γ
Sat x = s is defined between these two surfaces and moving along the axis of revolution for x ∈ [0; a ]. For each position s , a cylinder with a section O, a lateral surface Σ
Sand a length s is defined and forms a new subdomain Ω
S.
Resolution method
At each position x = s of Γ
S, we define two linear operators :
Neumann-to-Dirichlet application Q ( s ) : α 7−→ u
1S|ΓS
, associated to (P
S1) Dirichlet-to-Neumann application P ( s ) : β 7−→
∂u∂x2S|ΓS
, associated to (P
S2)
= ⇒ P and Q depend on s : variable that describes the axis of evolution
= ⇒ P and Q act on functions defined on section O
Let w
1and w
2: residual functions, defined on O. ∀ x ∈ [0; a ], we have : (3)
( u
1S( x ) = Q ( x ) α + w
1( x )
with Q (0) = 0 and w
1(0) = T and (4)
(
∂u2S
∂x
( x ) = P ( x ) β + w
2( x )
with P (0) = 0 and w
2(0) = −Φ Let ∆
y: laplacian operator, defined on the section O. ∀ x ∈ [0; a ], we have :
(5)
dP
dx
+ P
2= −∆
y, P (0) = 0
dw2
dx
+ P w
2= 0 , w
2(0) = −Φ and (6)
dQ
dx
− Q ∆
yQ = I, Q (0) = 0
dw1
dx
− Q ∆
yw
1= 0 , w
1(0) = T
= ⇒ First solve Riccati equations for P and Q for x = 0 −→ x = a
= ⇒ Then solve equations for w
1and w
2for x = 0 −→ x = a
= ⇒ Compute operators and residuals at x = a .
Rename P ( a ) = P , Q ( a ) = Q , w
1( a ) = w
1, w
2( a ) = w
2Define the matrix A as : A =
Q − QP
− P Q P
We can rewrite E ( η, τ ) as : E ( η, τ ) = C + [ η, τ ] A [ η, τ ]
′− 2 R
ΓH
( w
1P τ + Qw
2η ) Finally :
( φ, t ) = arg min E ( η, τ ) ⇐⇒ A [ φ, t ]
′= [ Qw
2, P w
1]
′= ⇒ Find ( φ, t ) : regularize previous system and inverse A
Conclusions and perspectives
Conclusions :
Direct optimal estimation of t and φ before using any discretisation :
= ⇒ Analyse ill-posedness and propose a better regularization and discretization Equations for P and Q depend only of the geometry :
= ⇒ Not necessary to repeat resolution at every time step of cardiac cycle Perspectives :
Apply the method to 3 D case where the moving boundary Γ
Swill be a deformed surface :
= ⇒ First : model of spheres
= ⇒ Then : realistic geometries : how compute 3 D surfaces ? + numerical cost ?
Figure 2: Illustration of Γ
Sfor the case of spheres, during the depolarization phase (left) and the repolarization phase (right). The smaller sphere represents the heart surface and the bigger one the torso surface. The heart was stimulated and direct problem was solved. We use computed potential data on the torso surface to apply factorization method and recover potential at heart surface. Spheres between torso and heart represent succesive positions of Γ
Sduring invariant embbeding. We also represent the potential recovered by the method.
References
[1] Jacques Henry and Angel Manuel Ramos , La m´ethode de factorisation des probl`emes aux limites, book (in preparation), (2013)
[2] Fadhel Jday, Compl´etion de donn´ees fronti`eres : la m´ethode de plongement invariant, PHD thesis, (2012)
