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Mathematical modeling in spectroscopic and hybrid tissue property imaging Habib Ammari Department of Mathematics and Applications Ecole Normale Sup erieure, Paris Ttissue property imaging Habib Ammari Tissue property imaging Wave


  1. Mathematical modeling in spectroscopic and hybrid tissue property imaging Habib Ammari Department of Mathematics and Applications Ecole Normale Sup´ erieure, Paris Ttissue property imaging Habib Ammari

  2. Tissue property imaging • Wave imaging techniques in medicine • Visualize contrast information on the electrical, acoustic, optical, mechanical properties of tissues. • Contrasts depend on molecular building blocks and on the microscopic and macroscopic structural organization of these blocks. • Enhance resolution, robustness, and specificity. • Perform biopsy in the operating room. • Help surgeons to make sure they removed everything unwanted around the margin of the cancer tumor. Ttissue property imaging Habib Ammari

  3. Tissue property imaging • Key concepts: • Resolution: smallest detail that can be resolved. • Robustness: stability of the imaging functionals with respect to model uncertainty, medium and measurement noises. • Specificity: physical nature (benign or malignant for cancer tumors). • Terminology: • Differential imaging: imaging small changes with respect to known (or even unkown) situations. • Super-resolution: resolve the microstructure at cellular level from macroscopic measurements at tissue level. Ttissue property imaging Habib Ammari

  4. Tissue property imaging • Spectroscopic tissue property imaging: specific dependence with respect to the frequency of the contrast. • Detect the characteristic signature of tumors; determine which are malignant and which are benign: specificity enhancement. • Classify micro-structure organization using spectroscopic tissue property imaging: resolution enhancement. • Hybrid imaging: one single imaging system based on the combined use of two kinds of waves. • Single wave imaging: sensitivity to only one contrast. • Spatial resolution: determined by the wave propagation phenomena and the sensor technology. • Hybrid imaging: Wave 1 gives its contrast and Wave 2 its spatial resolution. • 2 kinds of interactions between waves: Wave 1 can be tagged locally by Wave 2; Interaction of Wave 2 with tissues generates Wave 1. Ttissue property imaging Habib Ammari

  5. Spectroscopic electrical tissue property imaging 1 1 With J. Garnier, L. Giovangigli, W. Jing, and J.K. Seo, 2014. Ttissue property imaging Habib Ammari

  6. Spectroscopic electrical tissue property imaging • Differentiate between normal, pre-cancerous and cancerous tissues from electrical measurements at tissue level. Ttissue property imaging Habib Ammari

  7. Spectroscopic electrical tissue property imaging • Admittivities of biological tissues vary with the frequency ω ≤ 10 MHz of the applied sinusoidal current. • Admittivities of biological tissues may be anisotropic at low frequencies, but they become isotropic as the frequency increases. • Cell: homogeneous core covered by a thin membrane of contrasting electric conductivities and permittivities. • Intra and extra-cellular media: k 0 := σ 0 + i ωε 0 (conducting effect; transport of charges); • Membrane: k m := σ m + i ωε m with σ m /σ 0 ≪ 1 (capacitance effect; storage or charges or rotating molecular dipoles); • Thickness of the membrane ≪ typical size of the cell. Ttissue property imaging Habib Ammari

  8. Spectroscopic electrical tissue property imaging • Tissue model: • δ : cell period; • Ω + δ : extra-cellular medium; • Ω − δ : intra-cellular medium; • Γ δ : cell membranes. • Y : unit cell; Y ± : extra-cellular and intra-cellular (rescaled) media. Ttissue property imaging Habib Ammari

  9. Spectroscopic electrical tissue property imaging  −∇ · k 0 ∇ u + in Ω + δ ∪ Ω − δ = 0 δ ,        k 0 ∂ u + ∂ n = k 0 ∂ u −   δ δ  on Γ δ ,   ∂ n   δ − δ ξ ∂ u +  u + δ − u − δ ∂ n = 0 on Γ δ ,         ∂ u +   δ ∂ n = g on ∂ Ω .    • u δ = u ± δ in Ω ± δ ; • ξ = thickness × k m / k 0 : effective thickness; � • g : electric field applied at ∂ Ω of frequency ω ( ∂ Ω gd σ = 0). Ttissue property imaging Habib Ammari

  10. Spectroscopic electrical tissue property imaging • Homogenized problem:  −∇ · K ∗ ∇ u 0 ( x ) = 0 in Ω ,   ∂ u 0   ∂ n = g on ∂ Ω , • Effective admittivity: � � � K ∗ i , j = k 0 δ ij + ∇ w i · e j , Y Ttissue property imaging Habib Ammari

  11. Spectroscopic electrical tissue property imaging • Cell problems ( i = 1 , . . . , d ; d : space dimension):  −∇ · k 0 ∇ ( w + in Y + , i ( y ) + y i ) = 0        −∇ · k 0 ∇ ( w − in Y − ,  i ( y ) + y i ) = 0       k 0 ∂ i ( y ) + y i ) = k 0 ∂ ∂ n ( w + ∂ n ( w − i ( y ) + y i ) on Γ ,      − ξ ∂  ∂ n ( w +  w + i − w − i ( y ) + y i ) = 0 on Γ ,   i      y �− → w i ( y ) Y -periodic . • u δ two-scale converges to u 0 . • ∇ u δ two-scale converges to ∇ u 0 + χ + ∇ y u + 1 + χ − ∇ y u − 1 . • χ ± : characteristic function of Y ± . • Corrector: � 2 ∂ u 0 ∀ ( x , y ) ∈ Ω × Y , u 1 ( x , y ) = ∂ x i ( x ) w i ( y ) . i =1 Ttissue property imaging Habib Ammari

  12. Spectroscopic electrical tissue property imaging • Spectroscopic imaging: ω �→ K ∗ ( ω ); � � � • K ∗ i , j ( ω ) = k 0 δ ij + ∇ w i ( ω ) · e j ; Y •  −∇ · k 0 ( ω ) ∇ ( w + in Y + , i ( y ) + y i ) = 0        −∇ · k 0 ( ω ) ∇ ( w − in Y − ,  i ( y ) + y i ) = 0       k 0 ∂ i ( y ) + y i ) = k 0 ∂ ∂ n ( w + ∂ n ( w − i ( y ) + y i ) on Γ ,      − ξ ( ω ) ∂  ∂ n ( w +  w + i − w − i ( y ) + y i ) = 0 on Γ ,   i      y �− → w i ( y ) Y -periodic . Ttissue property imaging Habib Ammari

  13. Spectroscopic electrical tissue property imaging · 10 − 3 4 0 . 1 2 0 0 − 2 − 0 . 1 − 4 Figure : Real and imaginary parts of the cell problem solution w 1 . · 10 − 3 4 0 . 1 2 0 0 − 2 − 0 . 1 − 4 Figure : Real and imaginary parts of the cell problem solution w 2 . Ttissue property imaging Habib Ammari

  14. Spectroscopic electrical tissue property imaging Frequency dependence of the λ i for the 3 different shapes of cell: circle, an ellipse and a very elongated ellipse with the same volume 0 . 12 0 . 24 0 . 11 0 . 22 0 . 1 0 . 2 0 . 09 0 . 18 0 . 08 0 . 16 0 . 07 0 . 14 λ 2 ( C ) λ 1 ( C ) 0 . 06 0 . 12 0 . 05 0 . 1 0 . 04 0 . 08 0 . 03 0 . 06 0 . 02 0 . 04 0 . 01 0 . 02 0 0 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 ω ω Ttissue property imaging Habib Ammari

  15. Spectroscopic electrical tissue property imaging The effective admittivity of a periodic dilute suspension: � � − 1 � � I − f K ∗ = k 0 + o ( f 2 ) , I + f M 2 M • f = | Y − | = ρ 2 : volume fraction; • M : membrane polarization tensor � � � n j ψ ∗ M = m ij = β k 0 i ( y ) ds ( y ) , ρ − 1 Γ ( i , j ) ∈ [ | 1 , 2 | ] 2 � � − 1 [ n i ] . • ψ ∗ i = − I + β k 0 L ρ − 1 Γ � ∂ 2 ln | x − y | • L Γ [ ϕ ]( x ) = 1 2 π p . v . ∂ n ( x ) ∂ n ( y ) ϕ ( y ) ds ( y ) , x ∈ Γ . Γ Ttissue property imaging Habib Ammari

  16. Spectroscopic electrical tissue property imaging Maxwell-Wagner-Fricke Formula: • Case of concentric circular-shaped cells. • For ( i , j ) ∈ [ | 1 , 2 | ] 2 : m i , j = − δ ij β k 0 π r 0 . 1 + β k 0 2 r 0 • ℑ M attains one maximum with respect to ω at 1 /τ : π r 0 δω ( ε m σ 0 − ε 0 σ m ) ℑ m i , j = δ ij 2 r 0 ) 2 . 2 r 0 ) 2 + ω 2 ( ε m + ηε 0 ( σ m + ησ 0 • η : membrane thickness. • τ : relaxation time ( β -dispersion). Ttissue property imaging Habib Ammari

  17. Spectroscopic electrical tissue property imaging Frequency dependence of ℑ M for a circle: 5 · 10 − 2 4 . 5 4 3 . 5 3 λ ( C ) 2 . 5 2 1 . 5 1 0 . 5 0 10 4 10 5 10 6 10 7 10 8 10 9 ω Ttissue property imaging Habib Ammari

  18. Spectroscopic electrical tissue property imaging • Properties of the membrane polarization tensor: • M is symmetric; • M is invariant by translation; • M ( sC , ξ ) = s 2 M ( C , ξ s ) for any scaling parameter s > 0. • M ( R C , ξ ) = R M ( C , ξ ) R t for any rotation R . • ℑ M is positive and its eigenvalues have one maximum with respect to ω (membrane thickness η small enough). • Relaxation times for the arbitrary-shaped cells: 1 τ i := arg max λ i ( ω ) , ω λ 1 ≥ λ 2 : eigenvalues of ℑ M . • ( τ i ) i =1 , 2 : invariant by translation, rotation and scaling. Ttissue property imaging Habib Ammari

  19. Spectroscopic electrical tissue property imaging Properties of the relaxation times: ellipse translated, rotated and scaled · 10 − 2 5 0 . 1 4 . 5 0 . 09 4 0 . 08 3 . 5 0 . 07 3 0 . 06 λ 1 ( C ) λ 1 ( C ) 2 . 5 0 . 05 2 0 . 04 1 . 5 0 . 03 1 0 . 02 0 . 5 0 . 01 0 0 10 4 10 5 10 6 10 7 10 8 10 9 10 4 10 5 10 6 10 7 10 8 10 9 ω ω Ttissue property imaging Habib Ammari

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