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 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
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
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
Spectroscopic electrical tissue property imaging 1 1 With J. Garnier, L. Giovangigli, W. Jing, and J.K. Seo, 2014. Ttissue property imaging Habib Ammari
Spectroscopic electrical tissue property imaging • Differentiate between normal, pre-cancerous and cancerous tissues from electrical measurements at tissue level. Ttissue property imaging Habib Ammari
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
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
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
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
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
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
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
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
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
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
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
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
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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