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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

=

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