3D Source location in optical mapping

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3D Source location in optical mapping

Gwladys Ravon, Yves Coudière, Angelo Iollo, Olivier Bernus

To cite this version:

Gwladys Ravon, Yves Coudière, Angelo Iollo, Olivier Bernus. 3D Source location in optical mapping.

Workshop Liryc, Oct 2013, Pessac, France. 2013. �hal-00925812�

3D SOURCE LOCATION IN OPTICAL MAPPING

G ^WLADYS R ^{AVON 1} , Y ^VES C ^{OUDIÈRE 1 AND} A ^NGELO I ^{OLLO 2}

gwladys.ravon@inria.fr, yves.coudiere@inria.fr, angelo.iollo@math.u-bordeaux1.fr

1

: Université Bordeaux 1, Inria Bordeaux Sud-Ouest ;

²

: Université Bordeaux 1

This work is sponsored by the grant number ANR-10-IAHU-04 from the french government.

P ROBLEM STATEMENT AND OBJECTIVES

•

Optical mapping enables to display optical potentials on the boundary of a slab of tissue.

•

At a given time, we have 4 images: 2 from epi-illumination and 2 from endo-illumination.

•

Exploit these images to reconstruct an optimal 3D depolarization wave front.

E ^QUATIONS Domain:

Incident light

z

Surface z=0 (epicardium)

Surface z=L (endocardium) Ω (inside of the

tissue)

Incident light

finite elements discretization

Incident light:



 

 



 

 



D

_e

∆ φ

_e

− µ

_e

φ

_e

= 0 in Ω φ

_e

= I

_e

δ

_e

D

_e

on illuminated surface φ

_e

= d

_e

∂φ

_e

∂ν elsewhere

(1)

•

λ : wavelength

•

I

_e

( λ ): light intensity

•

D

_e

( λ ) , δ

_e

( λ ) , . . . : material properties

Fluorescence:

source: w = β ( V

_m

− V

₀

) φ

_e







D ∆ φ − µφ + w = 0 in Ω φ = d ∂φ

∂ν on ∂ Ω (2)

•

V

_m

: transmembrane potential

•

V

₀

: rest potential

W ^{AVE FRONT} : RESTRICTION

S ( t ) −→ V

_m

−→ W −→

^B

Φ −→

^L

Φ

^S

We look for V

_m

=

( V

₀

in Ω

_rest

V

_peak

in Ω

_peak

where Ω

_rest

∩ Ω

_peak

= S ( t )

M ATRIX RELATIONS

•

discretization of (2) gives:

A Φ = M W L Φ = Φ

^S

Φ

^S

: projection on surface, observations

•

reformulation: B = A

⁻¹

M

W ( V

_m

) −→

^B

Φ −→

^L

Φ

^S

•

under-determined problem because 19954 points in the whole domain including 1871 on the epicardium.

W ^{AVE FRONT} : RESTRICTION

2D representation:

Ωrest, V

m= V

0

Ω 1

peak, V

m= V

peak

S(t)

choice: S ( t ) = {| X − X

₀

| − c ( t − t

₀

) = 0}, expanding sphere

parameters to identify: X

₀

, t

₀

and sometimes c .

R ^ESULTS

In-silico example:

Direct simulation of a single source at X

^⋆

= (10 , 10 , 8) and t

^⋆

= 0.

Simulated observations:

Reconstructed X

₀

: X

₀

= X

^⋆

Reconstructed wavefront:

I NVERSE PROBLEM

Minimize

e ( X

₀

, t

₀

, ... ) = kΦ

^S

− Φ

^⋆

k

²_L2(S)

Φ

^⋆

: observation

Method:

•

BFGS method

•

1

^st

case functional:

e ( X

₀

, t

₀

, ... ) = kΦ

^S

( t

_k

) − Φ

^⋆_k

k

²_L2(S)

Φ

^⋆_k

: observation at time t

_k

•

2

^nd

case functional:

e ( X

₀

, t

₀

, ... ) = P

k

kΦ

^S

( t

_k

) − Φ

^⋆_k

k

²_L2(S)

1/2

R ^EFERENCES

[1] Khait et al. “Method for 3-dimensional localization [...]” In:

JBO (2006).

[2] Walton et al. “Experimental validation of alternating [...]”

In: IEEE (2011).

R ^ESULTS

Our approach:

several situations tested:

•

reflexion in epi-illumination

•

reflexion and transillumination in epi- illumination

•

reflexion in epi- en endo-illumination

•

unknown speed

•

unknown excitation time

A NY SOURCE COULD BE RECONSTRUCTED

Comparison with Khait

In [1], Khait calculates Z

^endo

and Z

^epi

and then defines the depth ( Z

^Khait

) as the mean.

The last curves were obtained with the following domain:

D ^ISCUSSION - C ^ONCLUSION

Analysis:

•

sometimes with not enough depo- larised tissue, our method does not con- verge

•

very good accuracy when it converges

•

convergence even after breakthrough

•

convergence for sources close to boundaries

•

independence of the domain

Perspectives:

•

validation with optical phantoms: find size and location of spherical fluores- cent sources [2]

•

generalize the wave front S ( t ) :

◦

Radial Basis functions

◦

Eikonal or level sets equations