Sensitivity and specificity enhancement in medical imagingplc/ammari2012.pdfphoto-acoustic imaging...
Transcript of Sensitivity and specificity enhancement in medical imagingplc/ammari2012.pdfphoto-acoustic imaging...
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Sensitivity and specificity enhancement inmedical imaging
Habib Ammari
Department of Mathematics and ApplicationsEcole Normale Superieure, Paris
Sensitivity and specificity enhancement in medical imaging Habib Ammari
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Sensitivity and specificity in medical imaging
• Mathematical and numerical modelling in medical imaging of cancer.
• Early detect tumors and determine which are malignant and which arebenign.
• Wave imaging of cancer tumors: elastic, optical, electric contrasts;specific dependence with respect to the frequency.
• Contrasts depend on molecular building blocks and on the microscopicand macroscopic structural organization of these blocks.
• Enhance the specificity and sensitivity of cancer detection.
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Sensitivity and specificity• Single wave imaging: sensitivity to only one contrast.
• Spatial resolution: determined by the wave propagation phenomena andthe sensor technology.
• Multi-wave imaging: one single imaging system based on the combineduse of two kinds of waves.
• One wave will give its contrast and the second its spatial resolution.
• Wave 1 (high contrast + low resolution) + Wave 2 (low contrast + highresolution) = Image (high contrast + high resolution).
Image of breast cancer.
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Multi-wave medical imaging
• 3 kinds of interactions between waves:
• Interaction of Wave 1 with tissues generates Wave 2:photo-acoustic imaging (V. Jugnon), thermo-acoustic imaging;
• Wave 1 can be tagged locally by Wave 2: acousto-opticaltomography, Electrical impedance tomography with ultrasound(with E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink);
• Wave 1 (travelling much faster than Wave 2) can be used toproduce a movie of Wave 2: elastography (P. Garapon).
• Imaging systems developed at Institut Langevin.
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One single wave imaging
• One single wave imaging: bio-inspired approaches.
• Super-resolution in electro-sensing (with T. Boulier, J. Garnier, W. Jing,H. Kang, and H. Wang).
• Weakly electric fishes possess: one electro-emitter and manyelectro-receptors of 2 types. One type measures the amplitude of theelectric field and another measures its phase.
Blackghost Knifefish (weakly electric: 1mV ).
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One single wave imaging
• Source term f time periodic and separable: f (x , t) = f (x)∑
n einω0t ; ω0:
fundamental frequency.
• Target D = z + δB; z : location; δ: characteristic size of the target;k = (σ + iωε)/σ0; k, σ, and ε: the admittivity, the conductivity, and thepermittivity of the target; ω = nω0: the probing frequency.
• u : the electric potential field generated by the fish:
∆u = f , x ∈ Ω,
∇ · (1 + (k − 1)χ(D))∇u = 0, x ∈ Rd \ Ω,
∂u
∂ν
∣∣∣∣−
= 0, x ∈ ∂Ω,
[u] = ξ∂u
∂ν
∣∣∣∣+
, x ∈ ∂Ω,
|u(x)| = O(|x |−d+1), |x | → ∞.
• ξ: effective thickness.
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One single wave imaging
• The effective thickness ξ = δσ0/σs .
• σ0 ∼ 0.01S ·m−1; σb = 1S ·m−1 (highly conductive);
• Skin: very thin (δ ∼ 100µm) and highly resistive (σs ∼ 10−4S ·m−1).
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One single wave imaging
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One single wave imaging
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The 8 elements of the dictionary.The dotted lines indicate a target with different electrical parameters.
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One single wave imaging
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The real part of the electric field is plotted,for 4 (over 20) positions that the fish takes around the target (placed at the origin).
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One single wave imaging
• Dipole approximation:
• u(x)− U(x) ' p · ∇G (x − z).• G : Green’s function.• p: dipole moment
p = −M(k ,D)︸ ︷︷ ︸∇U(z)
Polarization Tensor
• Source term f real: =mu(x) ' (=mp) · ∇G (x − z).
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One single wave imaging
• Polarization tensor:
M :=
∫∂D
x(λI −K∗D)−1[ν](x) dσ(x),
• λ = (k + 1)/(2(k − 1)): k: conductivity contrast;
• K∗D (Neumann-Poincare operator): weakly singular integral operator. K∗Dcompact (in the smooth case): discrete spectrum in (−1/2, 1/2) with 0as an accumulation point.
• 0 < k 6= 1 <∞: λI −K∗D : L2(∂D)→ L2(∂D) invertible.
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Near-field imaging
• Polarization tensor: low frequency information; mixture of materialparameter and size.
• Multipolar approach (at a single frequency):
• Use the dipole + quadrupole approximation of the target.• Construct shape descriptors invariant with respect to
translation, rotation, and scaling (in two and threedimensions).
• Multipolar approximation:
u(x)− U(x) '∑α,β
(∂αG(x − z)Mαβ(k,D)∂βU(z).
• Mαβ(k,D): high-order polarization tensors.
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One single wave imaging
• Reconstruction of high-order polarization tensors from the data by a leastsquares method.
• Instability:
Mαβ(k,D) = O(|D||α|+|β|+d−2), |∂αG(x−z)| = O(|x |−|α|−d+2)(|x | → +∞)
• Resolving power= number of high-order polarization tensors reconstructedfrom the data: depends on the signal-to-noise ratio (SNR) in the data.
• ε = characteristic size of the target/ the distance to the array oftransmitters/receivers.
• SNR = ε2/standard deviation of the measurement noise (Gaussian).
• Formula for the resolving power m as function of the SNR:
(mε1−m)2 = SNR.
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Near-field imaging
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Classification from multipolar measurements with 10% (measurement) noise.
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Multi-frequency imaging
• Multi-frequency approach: ω 7→ M(k(ω),D).
• Invariance with respect to translation, rotation, and scaling.• λj(ω): singular values of M(k(ω),D); ω∞: highest probing
frequency. Plot
ω 7→ λj(ω)
λj(ω∞),
for j = 1, . . . , d .
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Multi-frequency imaging
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Classification from multi-frequency measurements with 10% noise.
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Nano-particles for imaging
• Spectral decomposition (D smooth): K∗D [ψn] = µnφn.
• Plasmonic resonances: µn.
• ψn, φn: plasmonic eigenvectors.
• Gold nano-particles (negative conductivity):
k(ω) =2µn + 1
2µn − 1+ iτ ;
• τ : Debye relaxation term (small).
• Blow-up of the polarization tensor:
M =
∫∂D
x(λ(ω)I −K∗D)−1[ν](x) dσ(x) ' (ν, φn)
λ(ω)− µn(x , ψn).
• SNR enhancement.
• Design of nano-particles: size, shape, coating (P. Millien).
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Nano-particles for imaging
• Gold nano-particles: selective accumulation in tumor cells;bio-compatibility; reduced toxicity.
• Detection: localized enhancement in radiation dose (strong scattering).
• Ablation: localized damage (high absorption).
• Functionalization: targeted drugs.
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Nano-particles for imaging
• Dilute suspension of nanoparticles: volume fraction f 1.
• Effective conductivity k∗: overall macroscopic material property of thecomposite material.
• (with H. Kang and K. Touibi)
k∗ =[I + f M(I − f
dM)−1] + o(f 2) ,
M: the polarization tensor associated with the (arbitrary shaped) scaledinclusion and the conductivity contrast k. The formula is uniform withrespect to the contrast.
• Maxwell Garnett (Clausius-Mossotti) formula (D disk or sphere); Fricke’sformula (D ellipsoid).
• k∗ blows up to infinity for some (negative) values of σ close to theeigenvalues of K∗D (in the smooth case).
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Nano-particles for imaging• Dense suspension of nano-particles: target D with negative overall
parameters.
• Solution to the conductivity problem:
u = U + SD(λI −K∗D)−1[∂U
∂ν].
• SD : single layer potential.
• Spectral decomposition (D smooth): K∗D [ψn] = µnφn.
• ψn, φn: plasmonic eigenvectors.
• Spectral decomposition of the solution:
u − U =∑n
Un
λ− µnSD [ψn]
• Far-field behavior:
u − U =∑n
Un
λ− µn(∇G
∫∂D
xψndσ(x) + ∂2G
∫∂D
x2ψndσ(x) + . . .).
• G : Green’s function.
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Membrane imaging
• Admittivities of biological tissues vary with the frequency of the appliedcurrent.
• Interface phenomena (cell membrane): super-resolution in electricalimaging of biological tissues (with L. Giovangigli).
• Cell: homogeneous core covered by a thin membrane of contrasting
electric conductivities and permittivities.
• Core: σext + iωεext (conducting effect; transport of charges);• Membrane: σint + iωεint with σint/σext 1 (capacitance
effect; storage or charges or rotating molecular dipoles);• Low frequencies: induced polarization effect due to the
membrane.• High frequencies: induced polarization effect disappears.
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Membrane imaging
• δ: thickness of the membrane.
• Effective thickness:
ξ = δ(σint + iωεint)/(σext + iωεext).
• Electrical model of the cell:
∆u = 0 inD ∪ Rd \ D,
∂u
∂ν
∣∣∣∣+
− ∂u
∂ν
∣∣∣∣−
= 0 on ∂D,
u |+ −u |−= ξ∂u
∂νon ∂D.
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Membrane imaging
• U: applied field. Integral representation of the potential u:
u = U +DD [ψ]
• DD : double-layer potential.
• Integral equation:
ξ∂DD
∂ν[ψ] + ψ = −ξ ∂U
∂ν,
ν: the outward normal to ∂D.
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Membrane imaging
• Polarization tensor of the cell membrane:
M(ω) :=ξ
(σext + iωεext)
∫∂D
ν(ξ∂DD
∂ν+ I )−1[ν].
• Far-field behavior :
u(x)− U(x) ∼ −M(ω)∇U(z) · ∇G(x − z),
• G: Green’s function.
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Membrane imaging
• Effective admittivity of a dilute suspension of cells:
σ∗(ω) = (σext + iωεext)[I + f M(ω)] + o(f ).
• Disk-shaped cells (D = |x | = r0):
DD [e inθ](x) =
1
2
(r
r0
)|n|e inθ if |x | = r < r0,
−1
2
( r0
r
)|n|e inθ if |x | = r > r0.
• Maxwell-Wagner-Fricke’s formula:
M(ω) =δ
(σint + (δ/2r0)σext) + iω(εint + (δ/2r0)εext)I2.
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Membrane imaging
• Dependence of the induced polarization on the frequency:
<eM ∝ ωτ 2
1 + ω2τ 2, =mM ∝ 1
1 + ω2τ 2,
• τ (Debye relaxation): the polarization does not occur instantaneously.
• <eM attains its maximum at ω = 1/τ .
• τ carries information on the microscopic parameters.
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Concluding remarks
• Super-resolution in one single wave imaging:
• Differential imaging;• Spectroscopic imaging: target’s admittivity changes as a
function of the frequency.
• SNR enhancement: use of high-order polarization tensors (weakly electricfish); use of Plasmonic nano-particles induced resonances.
• Spectral induced polarization effects (weakly electric fish, cellmembranes).
• Plasmonic resonance of nano-particles.
• Effective medium theory : use of plasmonic resonances (the effectiveparameters blow up and the wavelength becomes much shorter).
• Physics-based classification.
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Multi-wave imaging
• Ultrasound-modulated optical tomography (with E. Bossy, J. Garnier, L.Nguyen, and L. Seppecher).
• Thermo-acoustic tomography (with J. Garnier, W. Jing, and L. Nguyen).
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Near infrared optical tomography
• Near infrared optical tomography: wavelengths 700− 1000nm,
• Differentiate between soft tissues: different absorption at the wavelengths.
• Absorption: dominated by oxy-hemoglobin, deoxy-hemoglobin, and water.
• Non-invasive (reasonable doses repeatedly employed), inexpensive.
Absorption spectrum.
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Near infrared optical tomography
• 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 lightillumination on the boundary.
• Reconstruct a from boundary measurements of Φ.
• High contrast + low resolution.
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Low resolution of optical tomography
NIR image of a breast tumor.
• Resolution enhancement: perturb the NIR light propagationby acoustic pulses inside the body and record the variation.
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Ultrasound-modulated tomography
NIR light source
Light detectors
Focused acoustic beam
Acoustic source
Spherical acousticpulsesΩ y
6
Contrasted inclusion
• Record the variations of the light intensity on the boundarydue to the propagation of the acoustic pulses.
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Ultrasound-modulated tomography
• Ω: acoustically homogeneous.
• Displacement field: spherical acoustic pulse generated at y .
• P : Ω −→ Ω: the displacement. u = P−1 − Id : smallcompared to |Ω|.
• Typical form of u:
uηy ,r (x) = −η r0rw
(|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.
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Ultrasound-modulated tomography
• 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
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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−2
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10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4
−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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x 10−5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4
−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4
−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4
−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
Phi
u−Phi
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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x 10−5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4
−2
0
2
4
6
8
10
12x 10−6 Boundary measurement
Mea
sure
men
t
pulse radius
Φu − Φ (left); Mu (right).
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Ultrasound-modulated tomography
• u depends on the center y , the radius r and the wavefrontthickness η.
• Family of measurement functions:
Mη(y , r) =1
η2
∫Ω
(auηy,r − a)ΦΦuηy,r
• Small η:
Mη(y , r) ≈ 1
η2
∫Ω∇a.uηy ,rΦ2.
• Extract the information in Mη (asymptotically in η).
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Ultrasound-modulated tomography
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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True absorbtion (left) and measurements Mu (right)for 64 pulses centered on the unit circle.
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Ultrasound-modulated tomography
• Asymptotic behavior:
limη→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 . Expansionuniform in (y , r); Error = O(η).
• M: ideal measurement function.
• Reconstruct a from M.
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Ultrasound-modulated tomography
• 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).
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Ultrasound-modulated tomography
• Φ2∇a = ∇ψ: relate M to ∂rR[ψ] and then find ψ and Φ2∇afrom the measurements.
• 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).
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Ultrasound-modulated tomography
• Reconstruction formula for ψ:
ψ = −1
cR−1
[∫ r
0
M(y , ρ)
ρd−2dρ
](up to an additive constant).
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Ultrasound-modulated tomography
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True absorbtion a; Mu; R[ψ]; ψ.
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Ultrasound-modulated tomography
• Reconstruct a knowing only ψ in the Helmholtzdecomposition:
Φ2∇a = ∇ψ +∇× A ?
• Divergence of the Helmholtz decomposition:
∇ · (Φ2∇a) = ∆ψ.
• Assume a = a0 (a known constant on Ω\Ω′):
(E2) :
∇ · (Φ2∇a) = ∆ψ in Ω′,
a = a0 on ∂Ω′.
• Φ: unknown in Ω.
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Ultrasound-modulated tomography
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.
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Ultrasound-modulated tomography
• 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.
• Minimal regularity assumption on a.
• Lipschitz stability results.
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Ultrasound-modulated tomography
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True a, and reconstructions after 2 iterationswith 16, 32, 64 and 128 acoustic centers.
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Ultrasound-modulated tomography
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2.4
2.6
2.8
3
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
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
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
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
Reconstruction of a from noisy measurements : true a;noise level: 0%, 5%, and 10%.
Sensitivity and specificity enhancement in medical imaging Habib Ammari
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Ultrasound-modulated tomography
50 100 150 200 250 300 350 400 450 500
50
100
150
200
250
300
350
400
450
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−0.01
−0.005
0
0.005
0.01
0.015
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Reconstruction of the Shepp-Logan phantom for 128 acoustic pulses.
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Quantitative thermo-acoustic imaging
• Model: (∆ + k2 + ikq)u = 0 in Ω,
ν · ∇u − iku = g on ∂Ω.
• Reconstruct q from q|u|2 in Ω (thermal energymeasurements).
Sensitivity and specificity enhancement in medical imaging Habib Ammari
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Quantitative thermo-acoustic imaging
• The set (gj)d+1j=1 ⊂ L2(∂Ω): proper set of measurements (d :
space dimension) iff:
(i) |u1| > 0 in Ω.(ii) The matrix [uj ,∇Tuj ]1≤j≤d+1 is invertible for all x ∈ Ω.
• Ej := qu1uj : can be evaluated from the thermal energymeasurements.
• αj := Ej/E1, j = 2, . . . , d + 1.
Sensitivity and specificity enhancement in medical imaging Habib Ammari
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Quantitative thermo-acoustic imaging
• A = (∂lαj+1)j ,l=1,...,d : invertible (proper set ofmeasurements); a = A−1[(∇TAT )T ].
• Exact reconstruction formula:
q(x) =−<e(a) · =m(a) +∇ · =m(a)
2k.
• Exact formula: derivatives of the data (up to the third order).
• Noise regularization model (convolution with a smoothingkernel).
• Good initial guess.
• Resolution enhancement: optimal control approach.
Sensitivity and specificity enhancement in medical imaging Habib Ammari
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Final concluding remarks
• One single wave imaging:
• Differential imaging .• High SNR: high sensitivity.• Spectral polarization and membrane effects: high specificity.• Plasmonic resonant nano-particles (high SNR, high effective
conductivity → high sensitivity, near-field imaging → highsensitivity + high specificity.
• Physics-based classification.
• Multi-wave imaging:
• Differential imaging.• Combination of two ways in one system: High sensitivity +
high specificity.• Exact reconstruction formula: good initial guess.
Sensitivity and specificity enhancement in medical imaging Habib Ammari