Mathematical modeling in spectroscopic and hybrid tissue property - - PowerPoint PPT Presentation

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,
• ptical, 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 imaging1

1With 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: k0 := σ0 + iωε0 (conducting

effect; transport of charges);

• Membrane: km := σm + iωεm with σm/σ0 ≪ 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;

• Γδ: cell membranes.
• Y : unit cell; Y ±: extra-cellular and intra-cellular (rescaled) media.

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

                             −∇ · k0∇u+

δ = 0

in Ω+

δ ∪ Ω− δ ,

k0 ∂u+

δ

∂n = k0 ∂u−

δ

∂n

• n Γδ,

u+

δ − u− δ − δ ξ ∂u+ δ

∂n = 0

• n Γδ,

∂u+

δ

∂n = g

• n ∂Ω.
• uδ = u±

δ in Ω± δ ;

• ξ = thickness × km/k0 : effective thickness;
• g: electric field applied at ∂Ω of frequency ω (
• ∂Ω gdσ = 0).

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

• Homogenized problem:

     −∇ · K ∗ ∇u0(x) = 0 in Ω, ∂u0 ∂n = g

• n ∂Ω,
• Effective admittivity:

K ∗

i,j = k0

• δij +
• Y

∇wi · ej

• ,

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

• Cell problems (i = 1, . . . , d; d: space dimension):

                             −∇ · k0∇(w +

i (y) + yi) = 0

in Y +, −∇ · k0∇(w −

i (y) + yi) = 0

in Y −, k0 ∂ ∂n (w +

i (y) + yi) = k0 ∂

∂n (w −

i (y) + yi)

• n Γ,

w +

i − w − i

− ξ ∂ ∂n (w +

i (y) + yi) = 0

• n Γ,

y − → wi(y) Y -periodic.

• uδ two-scale converges to u0.
• ∇uδ two-scale converges to ∇u0 + χ+∇yu+

1 + χ−∇yu− 1 .

• χ±: characteristic function of Y ±.
• Corrector:

∀(x, y) ∈ Ω × Y , u1(x, y) =

2

• i=1

∂u0 ∂xi (x)wi(y).

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

• Spectroscopic imaging: ω → K ∗(ω);
• K ∗

i,j(ω) = k0

• δij +
• Y

∇wi(ω) · ej

• ;
• 

                            −∇ · k0(ω)∇(w +

i (y) + yi) = 0

in Y +, −∇ · k0(ω)∇(w −

i (y) + yi) = 0

in Y −, k0 ∂ ∂n (w +

i (y) + yi) = k0 ∂

∂n (w −

i (y) + yi)

• n Γ,

w +

i − w − i

− ξ(ω) ∂ ∂n (w +

i (y) + yi) = 0

• n Γ,

y − → wi(y) Y -periodic.

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

−0.1 0.1 −4 −2 2 4 ·10−3

Figure : Real and imaginary parts of the cell problem solution w1.

−0.1 0.1 −4 −2 2 4 ·10−3

Figure : Real and imaginary parts of the cell problem solution w2.

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

Frequency dependence of the λi for the 3 different shapes of cell: circle, an ellipse and a very elongated ellipse with the same volume

104 105 106 107 108 109 1010 1011 1012 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 ω λ2(C) 104 105 106 107 108 109 1010 1011 1012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 ω λ1(C)

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

The effective admittivity of a periodic dilute suspension: K ∗ = k0

• I + f M
• I − f

2M −1 + o(f 2),

• f = |Y −| = ρ2: volume fraction;
• M: membrane polarization tensor

M =

• mij = βk0
• ρ−1Γ

njψ∗

i (y)ds(y)

• (i,j)∈[|1,2|]2

,

• ψ∗

i = −

• I + βk0Lρ−1Γ

−1 [ni].

• LΓ[ϕ](x) = 1

2π p.v.

• Γ

∂2 ln |x − y| ∂n(x)∂n(y) ϕ(y)ds(y), x ∈ 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:

mi,j = −δij βk0πr0 1 + βk0 2r0 .

• ℑM attains one maximum with respect to ω at 1/τ:

ℑmi,j = δij πr0δω(εmσ0 − ε0σm) (σm + ησ0

2r0 )2 + ω2(εm + ηε0 2r0 )2 .

• η: membrane thickness.
• τ: relaxation time (β-dispersion).

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

Frequency dependence of ℑM for a circle:

104 105 106 107 108 109 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ·10−2 ω λ(C)

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) = s2M(C, ξ

s ) for any scaling parameter s > 0.

• M(RC, ξ) = RM(C, ξ)Rt

for any rotation R.

• ℑM 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 τi := arg max

ω

λi(ω), λ1 ≥ λ2: eigenvalues of ℑM.

• (τi)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

104 105 106 107 108 109 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ·10−2 ω λ1(C) 104 105 106 107 108 109 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 ω λ1(C)

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

Shape of the cell and relaxation times: circle, an ellipse and a very elongated ellipse

104 105 106 107 108 109 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 ω λ1(C) 104 105 106 107 108 109 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ω λ2(C)

Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

• Measure of the anisotropy:

ω → λ1(ω) λ2(ω)

• λ1 ≤ λ2: eigenvalues of ℑM(ω).
• Large ω:

λ1(ω) λ2(ω) = 1 + (l1 − l2) 2ησmρ (σ2

m + ω2ε2 m)|Γ| + O(η2),

η: membrane thickness; l1 ≤ l2: eigenvalues of

• ρ−1Γ nLρ−1Γ[n]ds.
• Anisotropic information not captured:

ω ≫ 1 εm ((l1 − l2)2ησmρ |Γ| − σ2

m)1/2. Ttissue property imaging Habib Ammari

Spectroscopic electrical tissue property imaging

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging2

2With L. Giovangigli, L. Nguyen, and J.K. Seo, 2014. Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

• The potential uω in Ω ⋐ Rd:

−∇ · (σ + iωε)∇uω = 0 in Ω, uω = ϕ

• n ∂Ω.
• ϕ = (ϕ1, . . . , ϕd): d measurements (d = space dimension).
• σ and ε: constant and known in Ω \ Ω′.
• σ0 and ε0: the constant conductivity and permittivity in Ω \ Ω′.
• Ω′ = {x ∈ Ω : dist(x, ∂Ω) > c0}; c0 > 0.
• Micro-EIT: Reconstruct σ and ε from the knowledge of uω|Ω for

ω ∈ [ω, ω].

• Measurements over a range of frequencies [ω, ω].

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

• Multifrequency measurements yield invertibility of ∇uω in Ω, ω ∈ [ω, ω].
• Proper set of boundary conditions (G. Alberti): there exists

ϕ = (ϕ1, . . . , ϕd) such that there are N > 1 open pairwise disjoint open subsets B1, B2, · · · , BN of Ω, and N frequencies ω1, · · · , ωN ∈ (ω, ω) such that

(i) Ω′ ⊂ ∪N

j=1Bj ⊂ Ω;

(ii) The matrix Aωj(x) = ∇uω is invertible for all x ∈ Bj.

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

Minimization problem:

• F : (σ, ε; ω) → uω − Uω.
• Reconstruct (σ, ε) as minimizer of the functional:

J[σ, ε] = 1 2 ω

ω

F[σ, ε; ω]2

H1(Ω)dω

• Initial guess:
• γω = log(σ + iωε):
• ∆γω = ∇ · (−(AωAT

ω)†Aω∇ · Aω)

in Ω, γω = log(σ0 + iωε0)

• n ∂Ω,

†: pseudo-inverse.

• σI and εI:

σI + i (ω) + ω 2 εI = 1 ω − ω ω

ω

eγωdω.

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

• σ∗ and ε∗: the true conductivity and permittivity of Ω, belong to the

convex subset of H2(Ω)2:

• S = {(σ, ε) := (σ0, ε0) + (η1, η2)| (q1, q2) ∈ S};

S = {(η1, η2) ∈ H2

0(Ω)2| c1 < η1 + σ0 < c2, c1 < η2 + ε0 < c2,

supp ηj ⊂ Ω′, ηjH2(Ω) ≤ c3ηjH1(Ω), ηjH1(Ω) ≤ c4 for j = 1, 2 } c1, c2, c4 and c4: positive constants.

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

• Fr´

echet-differentiability of F in (σ, ε) ∈ S: for (h, k) ∈ S, −∇ · (σ + iωε)∇DF[σ, ε; ω](h, k) = ∇ · (h + iωk)∇uω in Ω, DF[σ, ε; ω](h, k) = 0

• n ∂Ω.
• DF Lipschitz continuous in (σ, ε).
• Fr´

echet-differentiability of J in (σ, ε) ∈ S: for (h, k) ∈ S, DJ[σ, ε](h, k) = ℜe ω

ω

DF[σ, ε; ω](h, k), F[σ, ε; ω]H1, = ℜe ω

ω

(h, k), DF[σ, ε; ω]∗(F[σ, ε; ω])H2, = ℜe ω

ω

• Ω

(h + iωk)∇uω : ∇pω dω,

• pω ∈ H2(Ω) the solution to the adjoint problem:
• ∇ · (σ + iωε)∇pω = F(σ, ε; ω) − ∆F(σ, ε; ω)

in Ω, pω = 0

• n ∂Ω.

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

• Optimal control algorithm:

(σn+1, εn+1) = T[σn, εn] − µDJ[T[σn, εn]],

• µ > 0: step size; T[f ]: approximation of the Hilbert projection

from H2(Ω)2 onto S (a prior bounds).

• DJ[T[σn, εn]] = (−ℜe ∇uω : ∇pω, ωℑm ∇uω : ∇pω).
• Landweber algorithm:

(σn+1, εn+1) = T[σn, εn] − µ ω

ω

DF ∗[T[σn, εn]; ω](F[T[σn, εn]; ω]) dω.

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

Convergence result:

• For all (h, k) ∈ S:

ω

ω

DF[σ, ε; ω](h, k)H1(Ω)2dω ≥ C(h, k)H2(Ω)2.

• The sequence (σn, εn) converges to the true admittivity (σ∗, ε∗) of Ω

there is η > 0 such that if T[σI, εI] − (σ∗, ε∗)H2(Ω)2 < η, then lim

n→+∞ εn − ε∗H2(Ω) + σn − σ∗H2(Ω) = 0. Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

True conductivity σ∗:

1 2 3 4

Initial guess of the conductivity (on the left) and the permittivity (on the right):

1 2 3 3 3.5 4

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

Reconstructed conductivity after 20 (on the left) and 40 (on the right) iterations of the algorithm:

1 1.5 2 2.5 3 3.5 1 2 3 4

Ttissue property imaging Habib Ammari

Micro-impedance electrical tissue property imaging

Absolute value of the difference between the reconstructed and true conductivities after 20 (on the left) and 40 (on the right) iterations:

0.5 1 1.5 2 0.5 1 1.5 2

Reconstructed permittivity after 20 (on the left) and 40 (on the right) iterations of the algorithm:

2.8 3 3.2 3.4 3.6 3.8 3 3.2 3.4 3.6 3.8

Ttissue property imaging Habib Ammari

Acoustically modulated optical tomography3

3With E. Bossy, J. Garnier, L. Nguyen, and L. Seppecher, 2014. Ttissue property imaging Habib Ammari

Modulated imaging

• Ultrasound modulated optical imaging: combined use of light

(contrast) and ultrasound (resolution).

• µ′

s: reduced scattering coefficient; µa: absorption coefficient;

µa ≪ µ′

s.

• Diffusion:
• −∆Φ + aΦ = 0 in Ω,

l∂νΦ + Φ = g on ∂Ω, a(x) = 3µ′

sµa(x), l: extrapolation length, g: the light

illumination on the boundary.

• Measure the variations of Φ on the boundary due to the

propagation of acoustic pulses.

Ttissue property imaging Habib Ammari

Modulated imaging

• Ω: acoustically homogeneous.
• Displacement field: spherical acoustic pulse generated at y.
• P : Ω −

→ Ω: the displacement. u = P−1 − Id: small compared to |Ω|.

• Typical form of u:

uη

y,r(x) = −ηr0

r w |x − y| − r η x − y |x − y|, ∀x ∈ Rd.

• w: shape of the pulse; supp(w) ⊂ [−1, 1] and w∞ = 1. η:

thickness of the wavefront, y: source point; r: radius.

• Thin spherical shell growing at a constant speed.

Ttissue property imaging Habib Ammari

Modulated imaging

• Pulse propagation: a → au(x) = a(x + u(x)). Fluence Φu:
• −∆Φu + auΦu = 0 in Ω,

l∂nΦu + Φu = g on ∂Ω,

• au(x) = a(x + u(x)).
• Cross-correlation formula:

Mu :=

• ∂Ω

(∂νΦΦu − ∂νΦuΦ) =

• Ω

(au − a)ΦΦu

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −3 −2.5 −2 −1.5 −1 −0.5 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −2.5 −2 −1.5 −1 −0.5 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −14 −12 −10 −8 −6 −4 −2 x 10

−6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 x 10

−6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −3 −2 −1 1 2 3 4 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

Phiu−Phi −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10

−5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −4 −2 2 4 6 8 10 12 x 10

−6

Boundary measurement Measurement pulse radius

Φu − Φ (left); Mu (right).

Ttissue property imaging Habib Ammari

Modulated imaging

• u depends on the center y, the radius r and the wavefront

thickness η.

• Family of measurement functions:

Mη(y, r) = 1 η2

• Ω

(auη

y,r − a)ΦΦuη y,r

• Small η:

Mη(y, r) ≈ 1 η2

• Ω

∇a.uη

y,rΦ2.

• Extract the information in Mη (asymptotically in η).

Ttissue property imaging Habib Ammari

Modulated imaging

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −20 −15 −10 −5 5 x 10

−6

True absorbtion (left) and measurements Mu (right) for 64 pulses centered on the unit circle.

Ttissue property imaging Habib Ammari

Modulated imaging

• Asymptotic behavior:

lim

η→0 Mη(y, r) = −crd−2

• Sd−1(Φ2∇a)(y+rξ).ξdσ(ξ) =: M(y, r)

c > 0: depends on the shape of u and on d. Expansion uniform in (y, r); Error = O(η).

• M: ideal measurement function.
• Reconstruct a from M.

Ttissue property imaging Habib Ammari

Modulated imaging

• Spherical means Radon transform:

R[f ](y, r) =

• Sd−1 f (y + rξ)dσ(ξ)

y ∈ S, r > 0,

• Derivative of R:

∂r(R[f ])(y, r) =

• Sd−1 ∇f (y + rξ) · ξdσ(ξ).

Ttissue property imaging Habib Ammari

Modulated imaging

• Helmholtz decomposition of Φ2∇a:

Φ2∇a = ∇ψ + ∇ × A.

• Measurement interpretation:
• Sd−1(Φ2∇a)(y + rξ).ξdσ(ξ) =
• Sd−1 ∇ψ(y + rξ).ξdσ(ξ).
• Relate M to ∂rR[ψ] and then find ψ and Φ2∇a from the

measurements.

Ttissue property imaging Habib Ammari

Modulated imaging

• Reconstruction formula for ψ:

ψ = −1 c R−1 r M(y, ρ) ρd−2 dρ

• (up to an additive constant).

Ttissue property imaging Habib Ammari

Modulated imaging

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −20 −15 −10 −5 5 x 10

−6

10 20 30 40 50 60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.02 −0.015 −0.01 −0.005 0.005 0.01 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 −0.02 −0.01 0.01 0.02 0.03

True absorbtion a; Mu; R[ψ]; ψ.

Ttissue property imaging Habib Ammari

Modulated imaging

• Reconstruct a knowing only ψ in the Helmholtz

decomposition: Φ2∇a = ∇ψ + ∇ × A ?

• Divergence of the Helmholtz decomposition:

∇ · (Φ2∇a) = ∆ψ.

• Assume a = a0 (a known constant on Ω\Ω′) and g ≥ 0 on

∂Ω: (E2) :

• ∇ · (Φ2∇a) = ∆ψ in Ω′,

a = a0 on ∂Ω′.

• Φ: unknown in Ω.

Ttissue property imaging Habib Ammari

Modulated imaging

Coupled elliptic system: (E) :                  (E1) :

• −∆Φ + aΦ = 0 in Ω,

l∂nΦ + Φ = g on ∂Ω, (E2) :      ∇ · (Φ2∇a) = ∆ψ in Ω′, a = a0 on ∂Ω′, a = a0 in Ω\Ω′, ψ, l > 0, g, and a0 > 0: known.

Ttissue property imaging Habib Ammari

Modulated imaging

• Fixed point argument.
• Landweber scheme:
• F[a] := ∇ · (Φ2[a]∇a);
• Minimization problem: min F[a] − ∆ψ;
• Landweber sequence:

a(n+1) = P(a(n)) − µDF[P(a(n))]∗(F[P(a(n))] − ∆ψ),

• µ > 0: relaxation parameter; P: projection.
• Convergence results.
• Lipschitz stability results.
• Minimal regularity assumption on a (SBV; change of

function):

• a := a − a0 − ψ

φ2 .

• a and ψ: same set of discontinuities.

Ttissue property imaging Habib Ammari

Modulated imaging

Realistic biological light absorption map:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8

Ttissue property imaging Habib Ammari

Modulated imaging

Reconstruction of the absorption map after 10 iterations of the fixed point sequence:

0.4 0.6 0.8 1 1.2 0.5 1 1.5 (1) 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 (4) 1 1.2 1.4 1.6

Ttissue property imaging Habib Ammari

Ultrasonically-induced Lorentz force electrical impedance tomography4

4With P. Grasland-Mongrain, P. Millien, J.K. Seo, and L. Seppecher, 2014. Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

Ω Γ1 Γ2 Γ0 Γ0 y ξ τ

support of the acoustic beam I(t)

Be3 e1 e2 e3

support of x → v(x, t)

• Ω: mechanically homogeneous and conductive.
• Γ1, Γ2: perfect conductors; Γ0: perfect insulator.
• B: constant.
• Velocity field: x ∈ Ω, x = y + zξ + r with z > 0, r ∈ ξ⊥,

vy,ξ(y + zξ + r, t) = A(z, |r|)w(z − ct)ξ.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• Ω: electrolyte medium. N types of ions of charge qi and volume density

ni(x), i ∈ {1, . . . , N}. For any x ∈ Ω

• Neutrality:

i qini(x) = 0

• Conductivities as ions mobility: σ(x) = e+

i µiqini(x),

µi ∈ R, satisfying µiqi > 0; the ionic mobility; e+: the elementary charge.

• Lorentz force applied to i: Fi = qivξ × Be3
• Ion i get almost immediately an additional drift speed:

vd,i = µi qi Fi = Bµivτ, τ = ξ × e3.

• At first order in the displacement length, its total velocity:

vi = vξ + Bµivτ.

• Current as the total amount of charges displacement:

jS =

• i

niqivi =

• i

niqi

• vξ + B
• i

niµiqi

• vτ = B

e+ σvτ.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• Interaction between the velocity field v(x, t)ξ and the magnetic field Be3

creates a source of current: jS(x, t) = B e+ σ(x)v(x, t)τ.

• Indirect effect of jS on the boundary:

j = jS + σ∇u, ∇ · j = 0.

• Potential at a fixed time t:

−∇ · (σ∇u) = ∇ · jS in Ω.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

Ω Γ1 Γ2 Γ0 Γ0 y ξ τ

support of the acoustic beam I(t)

Be3 e1 e2 e3

support of jS

u :      −∇ · (σ∇u) = ∇ · jS in Ω, u = 0

• n Γ1 ∪ Γ2,

∂νu = 0

• n Γ0.

The measured intensity: I =

• Γ2

σ∂νu.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• Virtual potential:

U :=          −∇ · (σ∇U) = 0 in Ω, U = 0

• n Γ1,

U = 1

• n Γ2,

∂νU = 0

• n Γ0.
• Measured intensity:

I =

• Ω

jS · ∇U = B e+

• Ω

v(x, t)σ(x)∇U(x)dx · τ.

• Measurement function:

My,ξ(z) =

• Ω

vy,ξ

• x, z

c

• σ(x)∇U(x)dx · τξ.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• Inverse problem: Find σ : Ω → R from the knowledge of

My,ξ : z →

• Ω

vy,ξ

• x, z

c

• σ(x)∇U(x)dx · τξ

known for any y ∈ Y ⊂ Rd and ξ ∈ Θ ⊂ Sd−1.

• Y : supposed to be a bounded smooth surface of Rd.
• If Y and Θ are well chosen, then the virtual current J(x) = (σ∇U)(x)

can be recovered.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• vy,ξ
• y + z′ξ + r, z

c

• = w(z′ − z)A(z′, |r|).
• My,ξ(z) = (w ∗ Φy,ξ)(z)

Φy,ξ(z) =

• ξ⊥(σ∇U)(y + zξ + r)A(z, |r|)dr · τξ.
• Deconvolution to recover Φy,ξ.

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• Recover σ from J = σ∇U:

U = F[σ] =          −∇ · (σ∇U) = 0 in Ω, U = 0

• n Γ1,

U = 1

• n Γ2,

∂νU = 0

• n Γ0.
• Optimal control algorithm:

Kε[σ] = 1 2

• Ω

|σ∇F[σ] − J|2 + ε|σ|TV (Ω).

• Nonconvexity (numerically).

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

• D = σ∇U; F = (−D2, D1): orthogonal field computed from the data;

level sets of U.

• Orthogonal field method:
• F · ∇U = 0

in Ω, U = x2

• n ∂Ω.
• F discontinuous but uniqueness holds.
• Regularized problem by a viscosity method:
• −∇ · (εI + FF T)∇Uε = 0

in Ω, Uε = x2

• n ∂Ω.
• The sequence (Uε − U)ε>0 converges strongly to zero in H1

0(Ω). Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

Reconstruction of the conductivity σ using the transport equation method with a viscosity parameter ε = 10−3:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 2 4 6 8 10

Ttissue property imaging Habib Ammari

Lorentz force electrical impedance tomography

Reconstruction of the conductivity σ with ε = 10−3 and additive medium noise. (1) Without noise. (2) Medium noise. (3) The reconstruction with 5% of

• noise. (4) with 10% of noise. (5) with 20% of noise. (6) with 50% of noise:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 −4 −2 2 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 5 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 5 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 5 10 15

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography 5

5With E. Bretin, P. Millien, J.K. Seo, and L. Seppecher, 2014. Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

• Full-field optical coherence tomography (OCT): optical image with

sub-cellular resolution.

• Elastography: mechanical tissue properties.
• Biological tissues: quasi-incompressible.
• Apply a load on the sample.
• OCTE (C. Boccara): Use a set of optical images before and after

mechanical solicitation to reconstruct the shear modulus distribution inside the sample.

• Map of mechanical properties: added as a supplementary contrast

mechanism to enhance specificity.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

• Reconstruct the shear modulus µ from ε and εu.
• ε(x) = εu (x + u(x)) ;
• Displacement field u:

       ∇ ·

• µ(∇u + ∇uT)
• + ∇p = 0

in Ω, ∇ · u = 0 in Ω, u = f

• n ∂Ω.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

• u∗ the true displacement;

ε the measured deformed optical:

• ε = ε ◦ (I + u∗)−1 .
• Optical image: discontinuous.
• Optimal control algorithm:

I(u) = 1 2

• Ω

| ε ◦ (I + u) − ε|2 dx.

• I has a nonempty subgradient.
• ξ ∈ ∂I:

ξ : h →

• Ω

[ ε(x + u) − ε(x)]h(x) · D ε ◦ (I + u)(x) dx.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

Initial guess:

• Detect the surface of jumps of the optical image (edge detection

algorithm).

• Local recovery by linearization: data = ε − εu(≈ ∇ε · u)

Jx(u) =

• Ω

|∇ε(y) · u − data(y)|2wδ(|x − y|)dy.

• wδ =

1 δd w

·

δ

• ; w: a mollifier supported on [−1, 1].
• Least-squares solution:

uT =

• Ω

wδ(|x − y|)∇ε(y)∇εT(y)dy −1

x+δB

datawδ(|x−y|)∇ε(y)dy.

• If all vectors ∇ε in {y : wδ(|y − x|) = 0} not collinear, then
• Ω

wδ(|x − y|)∇ε(y)∇εT(y)dy invertible.

• Resolution = variation of ε.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 11 12 13 14 15 16 17 18

Figure : Optical image ε of the medium.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure : Averaging kernel wδ.

1 2.8

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 u∗ · e1 2 4 ·10−3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 u∗ · e2 2 4 ·10−3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Initial guess uδ · e1 2 4 ·10−3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Initial guess uδ · e2 2 4 ·10−3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Reconstructed u · e1 2 4 ·10−3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Reconstructed u · e2 2 4 ·10−3

Figure : Displacement field and its reconstruction.

Ttissue property imaging Habib Ammari

Full-field optical coherence elastography

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Shear modulus distribution µ 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Reconstructed shear modulus distribution µrec 0.5 1 1.5 2 2.5 3 3.5 4

Figure : Shear modulus reconstruction.

Ttissue property imaging Habib Ammari