An inverse dipole source problem in inhomogeneous media: Application to the EEG source localization in neonates

Marion Darbas; Mohamadou Malal Diallo; Abdellatif El Badia; Stephanie Lohrengel

doi:10.1515/jiip-2017-0120

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An inverse dipole source problem in inhomogeneous media: Application to the EEG source localization in neonates

Marion Darbas , Mohamadou Malal Diallo , Abdellatif El Badia und Stephanie Lohrengel

Veröffentlicht/Copyright: 15. Februar 2019

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Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 27 Heft 2

Abstract

In this paper we present some aspects of an inverse source problem in an elliptic equation with varying coefficients, using partial Dirichlet boundary measurements. A uniqueness result is established for dipolar sources including their number. Additionally, assuming the number of dipoles known, a stability result is obtained and an efficient numerical identification method is developed. Finally, numerical experiments illustrate the effectiveness of the approach and a discussion is given on electroencephalography (EEG) in neonates.

Keywords: Inverse source problem; identifiability; stability; EEG; numerical experiments

MSC 2010: 65N21; 35Q92; 49K40; 97N40

Funding statement: The present work has been realized as part of both the MIFAC and BBEEG projects receiving financial supports from the region Picardie (now Hauts-de-France) and the CNRS (Interdisciplinary mission), respectively.

A Proof of Lemma 3

Let φn=(𝒒kn,Skn)1≤k≤m∈(ℝ3×Ω1)m and ψn=(𝒑kn,Tkn)1≤k≤m∈ℝ6⁢m such that limn→∞⁡(φn,ψn)=(φ,ψ) with φ=(𝒒k,Sk)1≤k≤m and ψ=(𝒑k,Tk)1≤k≤m. Without loss of generality, we can assume that for any k, the sequence (Skn)n belongs to 𝒱k (this is at least true for sufficiently large values of n). Hence,

σ1⁢(Skn)=σ1⁢(Sk)=ck.

Then 𝒯⁢(φn,ψn)=u|Γ*1,n where u1,n is the solution of the problem

(A.1){-∇⋅(σ⁢∇⁡u1,n)=F1,nin ⁢Ω,σ⁢∂𝒏⁡u1,n=0on ⁢Γ,

with the source term

(A.2)F1,n=∑k=1m𝒑kn⋅∇⁡δSkn-𝒒kn⋅H⁢(δSkn)⁢Tkn.

In the same way, 𝒯⁢(φ,ψ)=u|Γ*1, where u1 is the solution to (4.2) with source term

F1=∑k=1m𝒑k⋅∇⁡δSk-𝒒k⋅H⁢(δSk)⁢Tk.

For better reading, let

Fk1,n=𝒑kn⋅∇⁡δSkn-𝒒kn⋅H⁢(δSkn)⁢Tkn

and

Fk1=𝒑k⋅∇⁡δSk-𝒒k⋅H⁢(δSk)⁢Tk

for 1≤k≤m, and introduce vk1,n, the solution in ℝ3 of the Poisson equation

-ck⁢Δ⁢vk1,n=Fk1,n-Fk1.

The right-hand side of the above equation tends to 0 in the distributional sense since

limn→∞⁡(𝒒kn,Skn)=(𝒒k,Sk) and limn→∞⁡(𝒑kn,Tkn)=(𝒑k,Tk).

As before, vk1,n is obtained by convolution with the fundamental solution of the Laplace equation, and it follows that vk1,n as well as its derivatives converge to 0 outside 𝒱k. More precisely, we have

vk1,n=(𝒑k,n⋅∇⁡G⁢(𝒙-Sk,n)-𝒑k⋅∇⁡G⁢(𝒙-Sk))-(𝒒k,n⋅H⁢(G⁢(𝒙-Sk,n))⁢Tk,n-𝒒k⋅H⁢(G⁢(𝒙-Sk))⁢Tk).

Now, Taylor expansion of ∇⁡G (resp. H⁢(G)) at the point 𝒙-Sk shows that ∇⁡G⁢(𝒙-Sk,n)-∇⁡G⁢(𝒙-Sk) (resp. H⁢(G⁢(𝒙-Sk,n))-H⁢(G⁢(𝒙-Sk))) is of order 𝒪⁢(∥Sk,n-Sk∥) since G(⋅-Sk) and all its derivatives are regular and bounded outside 𝒱k. Taking into account that (𝒒k,n,𝒑k,n) converges to (𝒒,𝒑), we get the following estimates:

(A.3)limn→∞⁡∥vk1,n∥H1⁢(Ω∖𝒱k)=limn→∞⁡∥∂𝒏⁡vk1,n∥L2⁢(Γ)=limn→∞⁡∥vk1,n∥L2⁢(Γ)=0.

We have to prove that

limn→∞⁡∥u1,n-u1∥H12⁢(Γ*)=0.

To this end, we decompose v1,n=u1,n-u1 into a singular and a variational part according to Section 2.1,

v1,n=∑k=1mvk1,n+w1,n,

where the auxiliary function w1,n is the unique solution in the vector space V of the problem

a⁢(w1,n,v)=∑k=1m∫Ω(ck-σ)⁢∇⁡vk1,n⋅∇⁡v⁢d⁢𝒙-∑k=1m∫Γck⁢∂n⁡vk1,n⁢v⁢d⁢s for all ⁢v∈V.

Poincaré–Friedrichs’ inequality and the continuity of the linear form yield the following estimate:

∥w1,n∥H1⁢(Ω)≲∑k=1m∥∇⁡vk1,n∥L2⁢(Ω∖𝒱k)+∥∂𝒏⁡vk1,n∥L2⁢(Γ).

Together with estimates (A.3) and the continuity of the trace application, this implies limn→∞⁡v|Γ*1,n=0 in H12⁢(Γ*) which shows the continuity of the operator 𝒯.

Acknowledgements

The authors would like to thank the anonymous referees for useful suggestions and remarks.

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Received: 2017-12-22

Revised: 2019-01-28

Accepted: 2019-01-29

Published Online: 2019-02-15

Published in Print: 2019-04-01

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Schlagwörter für diesen Artikel

Inverse source problem; identifiability; stability; EEG; numerical experiments