Electrical Tissue Property Imaging - Yonsei University 2014-12-30آ  Electrical Tissue Property...

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Transcript of Electrical Tissue Property Imaging - Yonsei University 2014-12-30آ  Electrical Tissue Property...

  • Electrical Tissue Property Imaging

    Jin Keun Seo

    Medical Imaging Lab. Computational Science & Engineering

    Yonsei University

    KAIST April 12, 2012

  • Electrical Tissue Property Imaging: conductivity σ and permittivity �

    Develop a robust system

    S X = b subject to ...

    X = σ + iω� to be imaged. Free to choose ω. b : measured data S is a sensitivity matrix made by Maxwell’s equations. (Key: Need to build up a robust sensitivity matrix S satisfying RIP condition.)

  • About X: Conductivity σ and Permittivity �

    J = σ E, E = −∇u I = V/R = V σS/L

    (V = |∇u|L, I = |J|S)

    J = iω� E I = iωC V

    (C = Q/V = �S/L)

    In free space, �0 ≈ 8.85× 10−12 and σ = 0.

  • The system S (X) = b should be based on Ohm’s law:

    J = (σ + iω�)E = −τ∇u (τ = σ + iω�)

    ∗ Admittivity τ of biological tissues may be anisotropic at a low frequency, but it may become an isotropic as ω increases.

    Extra- Cellular Fluid

    -Cl +Na

    -Cl+Na

    +Na -Cl

    + + + + + + +

    + + + + + + +

    _ _ _ _ _ _ _

    _ _ _ _ _ _ _ ~

    +

    v(t)

    Cell Membrane

    Intra- Cellular Fluid

    sin) ( )(i t I tω= �

    , /V S LIτ τ==J E is scalarτ 11 12 13

    21 22 23

    31 32 33

    τ τ τ τ τ τ τ τ τ

    τ ⎡ ⎤ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎣ ⎦

  • About X: Admittivity τ of biological tissue changes with the angular fre- quency ω

    [Gabriel S et al 1996b Phys. Med. Biol., Haemmerich et al 2003 Phys. Med. Biol.]

  • Making S X = b requires to produce J inside the body Ω.

    Inject current (dc / ac) Induce current (ac only) E+

    E− electrodes Ω

    τ, J, E, H

    x

    y

    · z coil

    I sinωt

    τ, J, E, H

    x

    y

    · z

    { ∇ · J = ∇ · (τE) = 0 n · J|∂Ω ≈

    I |E±|

    ( χE+ − χE−

    )  −∇

    2H = ∇ ln τ ×∇× H− iµ0ωτH ∇× A = µ0H, ∇ · A = 0, −∇u = E + iωA n · (τ(∇u + iωAs))|∂Ω ≈ −n · (iωτAp)

    ∣∣ ∂Ω

    ∗ J, E, H are time-harmonic current density, electric field and magnetic flux density.

    ∗ As : secondary magnetic potential & Ap : primary magnetic potential ∗ In R3 \ Ω, σ = 0, �0 = 8.85× 10−12 and µ0 = 4π × 10−7.

  • To set up S (X) = b

    ∇× E = −iωµH ∇×H = τE, ∇ ·H = 0

    ∃ A s.t. µH = ∇× A & ∇ · A = 0 ∃ u s.t. −∇u = E + iωA

    ∇ · (τ∇u) = iω∇τ · A & −∇2H = ∇ ln τ × (∇×H)− iωµτH

    τE(r)︸ ︷︷ ︸ J

    + iωτA︸ ︷︷ ︸ Jeddy

    = −(σ∇ur − ω�∇ui)− i(ω�∇ur + σ∇ui)︸ ︷︷ ︸ Jtotal = −τ∇u

    −σ∇ur: sum of conduction current by dc or ac-E & conduction eddy current by ac-H induced by ac-D

    −ω2�∇ur: sum of D by ac-E and displacement eddy current by ac-H induced by ac-D

    ω�∇ui: sum of D by ac-E induced by ac-H

    −σ∇ui: the sum of conduction current by ac-E induced by ac-H and the conduction eddy current by ac-H induced by ac conduction current.

  • About b. What is measurable quantity?

    Inject current (dc / ac) Induce current (ac only) E+

    E− electrodes Ω

    τ, J, E, H

    x

    y

    · z coil

    I sinωt

    τ, J, E, H

    x

    y

    · z

    ∇ · (τ∇u) = iω∇τ · A & −∇2H = ∇ ln τ × (∇×H)− iωµτH

    EIT (≤ 1MHz): Boundary voltage u|∂Ω using electrodes. MREIT (≤ 1kHz): Internal Hz using MRI MIT (≤ 10MHz): External magnetic field using coils MREPT (128MHz at 3T MRI): Internal H+ = 12(Hx + iHy) using MRI

  • Electrical Impedance Tomography

    S X = b

  • EIT system: S (X) = b. Reciprocity Principle

    uP(Q+, ω, t)− uP(Q−, ω, t) = 1 I

    ∫ Ω τ∇uP · ∇uQdr = uQ(P+, ω, t)− uQ(P−, ω, t)

    Here, ∇ · (τ∇uP) = 0 in Ω with BC n · (τ∇uP)|∂Ω = I(δ(· − P+)− δ(· − P−)).

    ε1

    ε4ε5 ε6

    ε7

    ε10

    ε11

    ε12 ε13 ε14

    ε15

    ε16

    τ(r, ω, t)

    i(̃t) = I sin(ωt̃)︸ ︷︷ ︸ Nuemann data

    P+

    P−

    Q+

    Q−

    v

    i

    stream and equipotential lines for 16 ch.phantom

    uP(Q+, ω, t)− uP(Q−, ω, t)︸ ︷︷ ︸ Dirichlet data

    Calderon’s problem: Reconstruct τ from the NtD data {uP|∂Ω : P± ∈ ∂Ω}

    > Note that t and t̃ are different. The time t is used for the time change of the impedance, while t̃ is related to ω.

  • Structure of S X = b.

    X = δσ : dynamic conductivity imaging Apply Neumann data:{ ∇ · ( σ(r)∇uPj(r)

    ) = 0 in Ω

    σ(r) ∂∂n u Pj(r) = I(δ(r− Pj)− δ(r− Pj+1)) on ∂Ω

    Measure Dirichlet Data (V1, · · · ,VnE ): k − th comp. of Vj = Vj,k[σ] = uPj(Pk)− uPj(Pk+1). Denote uj = uPj . Let σ0 be a background conductivity. Vj,k[σ]− Vj,k[σ0] = −

    ∫ Ω(σ − σ0)∇uj · ∇u

    0 k dr.: Data

  • Structure of S X = b.

    Discretizing Ω into np elements as Ω = ∪ np n=1qn,

    ∫ q1 ∇uj · ∇u(0)1 dr · · ·

    ∫ qnp ∇uj · ∇u(0)1 dr

    ... ...

    ...∫ q1 ∇uj · ∇u(0)nE dr · · ·

    ∫ qnp ∇uj · ∇u(0)nE dr

     ︸ ︷︷ ︸

    Sj

    X = Vj[σ]− Vj[σ0]︸ ︷︷ ︸ data

     S1

    · · · · · · · · · ...

    · · · · · · · · · SnE

     ︸ ︷︷ ︸

    S

     δσ1... δσnp

     ︸ ︷︷ ︸

    δσ

    =

     δV1 · · · ... · · · δVnE

     ︸ ︷︷ ︸

    δV

  • Structure of S X = b.

    = Sensitivity matrix Data vector

    Conductivity to be imaged

  • Structure of S X = b.

    Column vector⇔ pixel sensitivity of all applied currents. Row vector: sensitivity distribution for a fixed current-voltage. Left figures are eigenvectors of S∗S.

    =

    DataPixel position

    −0.1

    −0.05

    0

    0.05

    0.1

    −0.15

    −0.1

    −0.05

    0

    0.05

    0.1

    0.15

    −0.08

    −0.06

    −0.04

    −0.02

    0

    0.02

    0.04

    0.06

    0.08

    Basis of row space via SVD

    Column vector Data Check RIP condition: columns

  • Is the linearized EIT robust?

    S X = δV ⇒ X = (S∗S)−1S∗δV︸ ︷︷ ︸ + tSVD or regularization

    where

    S =

     S1... SnE

     , X =  δσ1...

    δσnp

     , δV =  δV1...

    δVnE

     . Here, Sj is the j-th block of the linearized sensitivity matrix:

    Sj =

     ∫

    q1 ∇u(0)j · ∇u

    (0) 1 dr · · ·

    ∫ qnp ∇u(0)j · ∇u

    (0) 1 dr

    ... ...

    ...∫ q1 ∇u(0)j · ∇u

    (0) nE dr · · ·

    ∫ qnp ∇u(0)j · ∇u

    (0) nE dr

     .

  • Solving S X = b.

    = Sensitivity matrix Data vector

    Conductivity to be imaged

    • In LM, we try to find a best linear combination of the column vectors of the sensitivity matrix which produces the data such that | |S1j · · · Snpj | |

     

    ... δτ ...

     = Dataj. The reconstructed image of the LM relies roughly on tSVD

    S ≈ Ut0Λt0V ∗ t0

    and the expected reconstructed image is

    X = t0∑

    t=1

    1 λt 〈Dataj,ut〉vt

  • Eigenvectors of S∗S and image X.

    ill-posedness: low sensitivity to local perturbation at inner pixel away from ∂Ω. X = (S∗S)−1S∗b : low resolution for inner region.

    =

    Data Pixel position

    ≫ ≫

    λ≈ 0

    =

  • About X = (S∗S)−1S∗b.

    Check Restricted Isometry Property (RIP) condition: S is said to have RIP of order m if ∃ δm ∈ (0, 1) s.t.

    (1− δm)‖X ∥∥2

    2 ≤ ‖ SX ‖ 2 2 ≤ (1 + δm)‖X‖22, ∀‖X‖0 ≤ m

    Mutual coherence depends mainly on pixel size (or resolution). ] of electrodes may not help much for increasing mutual incoherence. Theoretical results of Calderon problem may not be applied to practical setting (like Riemann mapping theorem).

    =

    DataPixel position

    −0.1

    −0.05

    0

    0.05

    0.1

    −0.15

    −0.1

    −0.05

    0

    0.05

    0.1

    0.15

    −0.08

    −0.06

    −0.04

    −0.02

    0

    0.02

    0.04

    0.06

    0.08

    Basis of row space via SVD

    Column vector Data Check RIP condition: columns

  • Sensitivity matrix structure.

    Data = ∫

    Ω τ ∇u Pj · ∇uPk dr

    =

    ∫ near ∂Ω

    τ ∇uPj · ∇uPk︸ ︷︷ ︸ high

    sensitivity

    dr + ∫

    inner region

    τ ∇uPj · ∇uPk︸ ︷︷ ︸ very low

    sensitivity

    dr

    Interior Region