EIT Lung Monitoring using the Factorization Method Bastian von - PowerPoint PPT Presentation

EIT Lung Monitoring using the Factorization Method Bastian von Harrach harrach@math.uni-mainz.de Institute for Mathematics, Joh. Gutenberg-University of Mainz, Germany Conference on Applied Inverse Problems 2009, Vienna, Austria, July 20-24 B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Mathematical model EIT model using point electrodes and ”adjacent-adjacent” current patterns ξ j ∇ · ( σ ( x ) ∇ u j ( x )) = 0 in Ω ξ k +1 I U jk ξ j +1 ξ k σ ( x ) ∂ ν u j ( x ) | ∂ Ω = Iδ ( x − ξ j ) − Iδ ( x − ξ j +1 ) Ω Measured voltage: U jk := u j ( ξ k +1 ) − u j ( ξ k ) I : applied current between electrodes ξ j und ξ j +1 (here: fix I = 1 ), σ ( x ) : conductivity, u j ( x ) : resulting electrical potential. B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Measurements U jk : Voltage between k and k + 1 -th electrode needed to U = maintain current of I = 1 mA between j and j + 1 -th electrode.   U 1 , 1 U 1 , 2 · · · U 1 ,N U 2 , 1 U 2 , 2 · · · U 2 ,N     . . . ...   . . . . . .     U N, 1 U N, 2 · · · U N,N No measurements at current carrying electrodes � U j,j − 1 , U j,j , U j,j +1 missing ( j = 1 , . . . , N ). Reciprocity principle: U j,k = U k,j � only 16 · (16 − 3) / 2 = 104 non-redundant entries. B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

tdEIT In practice: Inevitable modelling errors (body shape, electrode position,. . . ) U (1) − U (0) at different times, (e.g. U (0) : exhaled state). � Reduction of model dependance by using a reference measurements with the same systematic errors � Time-difference EIT: Reconstruct σ 1 − σ 0 from measurements U (1) − U (0) . Factorization method (Kirsch 1998 for inverse scattering) : reconstructs supp( σ 1 − σ 0 ) from Neumann-Dirichlet-maps, Λ 1 − Λ 0 , i.e. from infinite-dimensional analogons of FM for EIT (1999–2009): uhl, Hakula, Hanke, H., Hyv¨ onen, Kirsch, Lechleiter, Nachman, P¨ aiv¨ Br¨ arinta, Pursiainen, Schappel, Schmitt, Seo, Teiril¨ a B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

FM for real data U (1) , U (0) approximate NtDs for large number of electrodes FM relies on infinite-dimensional NtDs Some approximation results for the FM available. (Theory: Lechleiter, Hyv¨ onen, Hakula) However, Practitioners keep electrode number small due to ill-posedness (”regularization by discretization”). Practitioners do not like infinite-dimensional arguments. No convergence theory for the FM („threshold choosing problem”)! In this talk: Physical justificiation of the FM in a realistic (discrete) setting First results of the FM for human lung data (myself) B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Two-phase lung model 1 + σ , ( σ > − 1) , For this talk: constant lung conductivity: constant background conductivity: 1 . U (1) : Analogous results for inhomogenous (but known!) background or σ = σ ( x, t ) . (As long as lung is always less conductive than background.) Measurements at current state Ω Conductivity σ 1 = 1 + σχ D 1 , U (0) : Current pattern g ∈ R 16 generates voltage u (1) g . D 1 Measurements at exhaled state Ω Conductivity σ 0 = 1 + σχ D 0 , D 0 ⊂ D 1 , Current pattern g ∈ R 16 generates voltage u (0) g . D 0 B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Bilinear form g · ( U (0) − U (1) ) h = Useful identity: � · ∇ u (0) σ ∇ u (1) h d x g D 1 \ D 0 g · ( U (0) − U (1) ) h ≈ Linearisation: u (1) ≈ u (0) ≈ u (hom) g g g � · ∇ u (hom) σ ∇ u (hom) d x g h D 1 \ D 0 (Virtual) background measurements: Ω Conductivity σ hom = 1 , Current pattern g ∈ R 16 generates voltage u (hom) . g B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Dipole Function Dipole measurements � ∆ ϕ z,d = d · ∇ δ z , Φ z,d = ( ϕ z,d ( ξ k +1 ) − ϕ z,d ( ξ k )) 16 k =1 , where ∂ ν ϕ z,d | ∂ Ω = 0 . Dipole preimage h z,d = ( U (0) − U (1) ) − 1 Φ z,d : Scalar products ( z ) = g · Φ z,d = g · ( U (0) − U (1) ) h z,d g · Φ z,d = d · ∇ u (hom) ( z ) . g d · ∇ u (hom) g � ( x ) · ∇ u (hom) σ ∇ u (hom) ≈ h z,d ( x ) d x g D 1 \ D 0 must hold for all applied current patterns g ∈ R 16 . B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Dipole preimage h z,d = ( U (0) − U (1) ) − 1 Φ z,d : Localization � ( x ) · ∇ u (hom) d · ∇ u (hom) σ ∇ u (hom) ( z ) ≈ h z,d ( x ) d x ∀ g g g D 1 \ D 0 z �∈ D 2 ”well-separated” from D 1 : z (large current in z , little current through D 1 ) � �∇ u (hom) h z,d ( x ) � L 2 ( D 1 \ D 0 ) very large. For z ∈ D 1 one can show (in R 2 ) D 1 �∇ u (hom) 1 h z,d ( x ) � L 2 ( D 1 \ D 0 ) � dist( z,∂D 1 ) Plotting z �→ �∇ u (hom) h z,d ( x ) � L 2 ( D 1 \ D 0 ) shows D 1 . B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Factorization Method Plotting z �→ �∇ u (hom) h z,d ( x ) � L 2 ( D 1 \ D 0 ) shows D 1 . Up to multiplicative constants, h z,d ( x ) | 2 d x ≈ h z,d · ( U (0) − U (1) ) h z,d �∇ u (hom) h z,d ( x ) � 2 � ( U (0) − U (1) ) − 1 / 2 Φ z,d L 2 ( D 1 \ D 0 ) � σ |∇ u (hom) � D 1 \ D 0 � ( U (0) − U (1) ) − 1 / 2 Φ z,d 2 � � = � � � � 2 shows D 1 . (Factorization Method) � � Plotting Up to multiplicative constants everything holds without linearisation! B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

� ( U (0) − U (1) ) − 1 / 2 Φ z,d Physical justification � 2 . � � FM indicator: z �→ Physical justification of the FM (H., Seo, Woo) : Plot of FM indicator distinguishes object from well-separated points. Well-separated points are those in which the current can be made large without making it large in the object. Justifies FM for realistic, discrete settings Consistent with continuous setting, where current can be concentrated everywhere in object’s connected complement (H. ’08) . B. Harrach: ’EIT Lung Monitoring with the Factorization Method’

Real data Reconstructions for real data measured on human lung (Gisa, H.) : B. Harrach: ’EIT Lung Monitoring with the Factorization Method’