Electrical Tissue Property Imaging

Jin Keun Seo

Medical Imaging Lab.Computational Science & Engineering

Yonsei University

KAISTApril 12, 2012

Electrical Tissue Property Imaging: conductivity σ andpermittivity ε

Develop a robust system

S X = b subject to ...

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

About X: Conductivity σ and Permittivity ε

J = σ E, E = −∇u

I = V/R = V σS/L

(V = |∇u|L, I = |J|S)

J = iωε EI = iωC V

(C = Q/V = εS/L)

In free space, ε0 ≈ 8.85× 10−12 and σ = 0.

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

J = (σ + iωε)E = −τ∇u (τ = σ + iωε)

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

Extra-CellularFluid

-Cl+Na

-Cl+Na

+Na -Cl

+ + + + + + +

+ + + + + + +

_ _ _ _ _ _ _

_ _ _ _ _ _ _~

+

v(t)

−

Cell Membrane

Intra-CellularFluid

sin) ( )(i t I tω=

, /V S LIτ τ==J E is scalarτ11 12 13

21 22 23

31 32 33

τ τ ττ τ ττ 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

· zcoil

I sinωt

Ω

τ, J, E, H

x

y

· z

∇ · J = ∇ · (τE) = 0n · J|∂Ω ≈

I|E±|

(χE+ − χE−

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

∣∣∂Ω

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

∗ As : secondary magnetic potential & Ap : primary magnetic potential

∗ In R3 \ Ω, σ = 0, ε0 = 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 − ωε∇ui)− i(ωε∇ur + σ∇ui)︸ ︷︷ ︸Jtotal = −τ∇u

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

−ω2ε∇ur: sum of D by ac-E anddisplacement eddy current by ac-Hinduced by ac-D

ωε∇ui: sum of D by ac-E induced byac-H

−σ∇ui: the sum of conductioncurrent by ac-E induced by ac-H andthe conduction eddy current by ac-Hinduced 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

· zcoil

I sinωt

Ω

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 MRIMIT (≤ 10MHz): External magnetic field using coilsMREPT (128MHz at 3T MRI): Internal H+ = 1

2(Hx + iHy) using MRI

Electrical ImpedanceTomography

S X = b

EIT system: S (X) = b.

Reciprocity Principle

uP(Q+, ω, t)− uP(Q−, ω, t) =1I

∫Ωτ∇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)

Ω

i(t) = I sin(ωt)︸ ︷︷ ︸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 τfrom the NtD datauP|∂Ω : P± ∈ ∂Ω

> Note that t and t are different. The time t is usedfor the time change of the impedance, while t isrelated to ω.

Structure of S X = b.

X = δσ : dynamic conductivity imagingApply Neumann data:∇ ·(σ(r)∇uPj(r)

)= 0 in Ω

σ(r) ∂∂n uPj(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 · ∇u0

k dr.: Data

Structure of S X = b.

Discretizing Ω into np elements as Ω = ∪npn=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. Leftfigures 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 <-> DataCheck 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 linearcombination of the column vectorsof the sensitivity matrix whichproduces the data such that | |S1

j · · · 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

1λt〈Dataj,ut〉vt

Eigenvectors of S∗S and image X.

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

=

DataPixel position

≫ ≫

λ≈ 0

=

About X = (S∗S)−1S∗b.

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

(1− δm)‖X∥∥2

2 ≤ ‖ SX ‖22 ≤ (1 + δm)‖X‖2

2, ∀‖X‖0 ≤ m

Mutual coherence depends mainly on pixel size (or resolution).] of electrodes may not help much for increasing mutualincoherence. Theoretical results of Calderon problem may not beapplied 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 <-> DataCheck RIP condition: columns

Sensitivity matrix structure.

Data =∫

Ω τ ∇uPj · ∇uPk dr

=

∫near∂Ω

τ ∇uPj · ∇uPk︸ ︷︷ ︸high

sensitivity

dr +

∫innerregion

τ ∇uPj · ∇uPk︸ ︷︷ ︸very low

sensitivity

dr

InteriorRegion

InteriorRegion

ii

v

v

Static EIT data depends mainly on conductivity distribution

near boundary.

Examples - Boundary influence in EIT-data

In the case when δτ 6= 0 near ∂Ω, it is difficult to probe internalchanges because the data depends mainly on δτ near ∂Ω.

σ=1

σ=2

σ=1

σ=2

Model Reconstructed image

δτ=0

δτ=0

δτ=1

δτ=1

Achievable spatial resolution: N-channel EIT

Imagine that σ is the true conductivity. Define

Aσj := φ ∈ C1(R) : φ′ > 0, φ(uPjσ (Pk)) = uPj

σ (Pk), k = 1, 2, · · · ,N

where uPjσ is the solution of

∇ · (σ∇uPj

σ ) = 0 in Ω

σ ∂∂n uPj

σ |∂Ω = δ(· − P+j )− δ(· − P−j )

.

Observation. Every element of∩N

j=1σ

φ′(uPjσ )

: φ ∈ Aσj are solutions.

(Achievable resolution of N-channel EIT is somehowrelated to this figure and sensitivity.)

It appears that we can increase EIT-resolution byincreasing the number of electrodes. However,when ∂Ω is covered mostly by electrodes, EITcannot probe internal conductivity. (Why?) How can we improve EIT-image quality? Needcareful arrangement of electrodes.

Difference EIT for eliminating boundary influences

To extract internal impedance information from EIT data, we need to get rid of theboundary influences. The use of difference data enable us to cancel out the boundaryinfluences.

Differencedata

=

∫near∂Ω

δτ∇ua · ∇ub dr

︸ ︷︷ ︸Hope δτ=0

+

∫interior

region

δτ∇ua · ∇ub dr

(ua =

∑ajuPj , ub =

∑bjuPj

)Time difference EIT (tdEIT) : ∂

∂t Data −→ ∂∂tτ(r, ω, t)

Frequency difference EIT (fdEIT) : ∂∂ωData −→ ∂

∂ω τ(r, ω, t)

Recovering δτ from the difference data requires to control thesensitivity part ∇ua · ∇ub.

What is visible for static electrical conducting bodyusing currently available EIT system?

Images provided by frequency difference EIT.

Sample

500 Hz 5 kHz 50 kHz 500 kHz

membranes

500 Hz

5 kHz

50 Hz

Functional Analysis Approach for SX = b.

In Factorization Method, we measure an unchangability index ateach pixel in the imaging domain by estimating a difficulty ofproducing n-th column of sensitivity matrix by a linear combination ofdata-matrix such that

Snj =

| |Data1 · · · DatanE

| |

hn

1...

hnnE

.

=Sensitivity matrix Data vector

Conductivity to be imaged

Sensitivity matrix based Factorization (S-FM).

Snj ⇔

| |Data1 · · · DatanE

| |

Square matrix

Rationale of S-FM

For characterizing Sj, we introduce φnj satisfying

∇ · ∇φnj = ∇ · (χqn∇u(0)

j ) in Ω

∇φnj · n = 0 on ∂Ω

Theorem: Denote Φnj := (Φn

j (1) · · · Φnj (nE))T where

Φnj (k) := φn

j (Pk)− φnj (Pk+1) for k = 1, · · · , nE.

Then Φnj = n-th column of Sj.

Proof. From the Neumann boundary condition of u(0)k ,∫

Ω∇φn

j · ∇u(0)k dr = φn

j (Pk)− φnj (Pk+1)

On the other hand, it follows from the property of φnj that∫

Ω∇φn

j · ∇u(0)k dr =

∫qn

∇u(0)j · ∇u(0)

k dr.

Main Observation

Let δσ = χD where D ⊂⊂ Ω. Replace n2E-dimensional data vector δV

by the nE × nE-matrix

δV := (δV1 · · · δVnE ).

For all pixel qn and j-th injection current

Φnj · (δV)−1Φn

j ≥ maxh∈RnE

∣∣∣∫qn∇u(0)

j · ∇u(0)h dr

∣∣∣2∫D |∇u(0)

h |2dr

where u(0)h (r) :=

∑nEk=1 hk u(0)

k (r).If the pixel qn lies inside the inclusion D then for all injectioncurrents ∣∣Φn

j · (δV)−1Φnj

∣∣ ≤∫

qn

∣∣∣∇u(0)j

∣∣∣2 dr.

Projective EIT system for higher resolution image ofthe conductivity distribution near the surface Γ ⊂ ∂Ω.

Two pairs of drivingelectrodes E±1 and E±2 toinject currents.

Many sensing electrodes tomeasure detailed voltageson ∂Ω

Experiments showed thatEIT-standard reconstructionalgorithms fail to providegood images.

Sensing Electrodes

: EJ Lee, Muugii, Woo, Seo (2011)

Image Υt: a projected tdEIT image of γt/γ

∇2SΥt(r) = ∇S ·

∫D∇2

SΦ(r, r′)[J1,2(r)]†J1,2(r′)∇(γt(r′)γ(r′)

)dr′︸ ︷︷ ︸

measurable quantity

, for r ∈ Γ

> ∇S is surface gradient. J1,2 =

(J1

x J1y J1

z

J2x J2

y J2z

), [J1,2]† is the pseudo-inverse.

Simulation

(a) Front view (b) Side view (c) Back view30 cm

27 cm

20 cm

13x11 cm

2x2 cm

1 cm

Front view

Side view

4 cm

backgroundγ = 1

γ = 8at t = 0

γ = 7at t = 0.1

γ = 1.3at t = 0

γ = 1.1at t = 0.1

Reconstructed image with thresholding0.05

-0.07-0.07

0.05

1x10-3 1x10-3

Breast EIT-Frequency difference projected imageProjection image[

∂φ log(γ∗)1

sinφ∂θ log(γ∗)

]= −

[∂φu1 1

sinφ∂θu1

∂φu2 1sinφ∂θu2

]−1 [∇2

s u1

∇2s u2

]> where γ∗ is function defined on Ωs satisfying γ∗|∂Ωs = γ0

Model & Simulation

Sensing Electrodes

Front Side Back

Electrical tissue property imaging using MRI:τ = σ + iωε

J = (σ + iωε)E & −∇2H =∇γγ× (∇×H)− iωµ0γH

EIT

MREIT

MREPT

MREIT-way of feeling admittivity (τ ≈ σ since ωε ≈ 0)

Biot-Savart Law: (z = (0, 0, 1)T )

Hz(r)︸ ︷︷ ︸Data

=1

4π

∫Ω

< r− r′,−

target︷ ︸︸ ︷σ(r′)∇u(r′)× z >|r− r′|3

dr′ + Harmonic term

Experiment Setting data target σAttaching ElectrodezH

Ω

∇ · (σ∇u) = 0 in Ω, n · (σ∇u)|∂Ω ≈ IE±

(χE+ − χE− )

How to measure Bz. (1989, Joy et al, Toronto Group)

Using the MRI scanner, we obtain the following complex k-space datainvolving Bz information in the slice Ωz0 :

SI+(kx, ky) =

∫ ∫Ωz0

M(x, y, z0)ei(γBz(x,y,z0)Tc+δ(x,y,z0))ei(xkx+yky)dxdy

Application of Fourier transform to the k-space MR signal SI+ yields thefollowing complex MR image,MI+(x, y, z0):

MI+(x, y, z0) := M(x, y, z0) eiγBz(x,y,z0)Tc eiδ(x,y,z0).

RecessedCopper Electrode

Saline

Agar

RecessedCopper Electrode

(a)M(x, y, z0) (b)wrapped Bz (c) Bz

MREIT using full components of H = (Hx,Hy,Hz): It

requires subject rotation inside MRI scanner. (Impractical)

Least square method(1994)

minσ

12‖∇ ×H + σ∇u ‖2

Finite element model True

1/σ

1/σ 1/σReconstructed

Poor spatial Resolution

[Zhang 1992], [Woo, Lee, Moon 1994], [Idel and Birgul

1995]

J−substitution method(2002)

∇ ·(|∇ ×H||∇u|

∇u)

= 0

True Reconstructed 1/σ 1/σ

High spatial resolution

[Kwon, Seo, Yoon, Woo 2002], [Hasanov, Ma, Nachman, Joy

2008]

Harmonic Bz Algorithm: Key Observations

Hz, z−component of the magnetic field H = (Hx,Hy,Hz) can trace achange of σ in the tangential direction to the equipotential surface:

∇2Hz = ∇lnσ · (σ∇u× z)

where Hz(r) = 14π

∫Ω

〈r−r′,−σ(r′)∇u(r′)×z〉|r−r′|3 dr′ + harmonic term

Hz is blind to the transversal change of σ in the direction ∇xyu.

To uniquely reconstruct σ from Bz, at least two currents is needed.

[Seo, Kwon, Yoon, Woo 2003 IEEE TBE], [Seo, Woo SIAM REVIEW 2011]

Harmonic Bz Algorithm

For any (x, y, z0) ∈ Ωz0 := (x, y, z) ∈ Ω : z = z0, the harmonic Bzalgorithm is based on

∇2xy lnσ(x, y, z0) = ∇xy ·

[ σ ∂u1[σ]∂y −σ ∂u1[σ]

∂x

σ ∂u2[σ]∂y −σ ∂u2[σ]

∂x

]−1 [∇2Hz,1∇2Hz,2

]

0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

Canine Pelvis Experiment Human Knee Experiment

[Seo et al 2003 IEEE Trans. Boimed. Eng., Oh et al 2003 Phys. Med. Biol.]

MREIT: Animal imaging experiment

Harmonic Bz-algorithm requires to solve the forward model in which weneed to know Ω and Neumann data (semi-)automatically.

Domain Ω: We use level set method or snake to extract ∂Ω fromMR-image automatically.Neumann boundary data: Very difficult to get automaticalgorithm to get Neumann condition. Use semi-automatic methodwith Level set.

Electrode

Lead Wire RF Coil

MRI Bore

B0x

y

z

CoReHa: Software for MREIT

MREIT Matlab Toolkit(2006: C. J. Park, S. H. Lee, Kwon, Woo, Seo),CoReHA(Conductivity Reconstructor using Harmonic Algorithm 2008: G.W. Jun, C.O. Lee, Woo, Seo)

MREIT: CoReHa

Segmentation Electrode Modeling Meshing(Level set method) (Neuman data) (FEM)

Denoising/HarmonicInpainting

[G.W. Jun, C. O. Lee, Woo, Seo 2008]

MREIT: Animal imaging

Kidney

Spinal Cord

Liver

Spleen

Stomach

IntestinesCarbon-hydrogel

Electrode

MREIT: Human imaging

FDA safety limit: 1mA. Reducing the amount of current⇒ LowSNR data⇒ Denoising process⇒ produce artifacts

Coefficient of Elliptic PDE in terms of Neumann data:macroscopic-view

The irresponsible (or precise) expression would be

∇ · ((1 + (10−8 − 1)χC)∇u) = 0 in Ω C := (thin film with holes)

M R I I m a g e

Saline

V e r y T hi n I ns ul a t i ng F i l m ha v i ng s m a l l ho l e s

Apparent conductivity & Homogenization

Considering the overall structure of J = −σ∇u due to Neumann datadeeply, a responsible expression is

∇ · (σ∇u) = 0 in Ω σ =

σ1 inside OHP filmσ2 outside OHP film

because the coefficient of elliptic PDE can be viewed as a bestobservable quantity σ which minimizes

Φ(σ) = supg∈A‖Jg + σ∇ug‖L2(Ω) + λ‖∇σ‖BV A ⊂ H−1/2(∂Ω)

Coefficient of elliptic PDE observed by MREIT

The apparent conductivity by MREIT changes with the size of holesize:

∇ · (σ∇u) = 0 in Ω σ =

σ1 inside OHP filmσ2 outside OHP film

Bz data & Reconstructed Conudctivity distribution using MREIT

Ø of hole Without hole Ø 1mm Ø 2mm Ø 3mm Ø 4mm Ø 5mm Ø 6mm Ø 10mm

Conductivity change(%) 0 0.94 1.39 1.86 2.36 2.92 3.14 4.35

MREPT-way of feeling Admittivity (τ = σ + iωε, ω2π ≥ 100 MHz)

H+ = 12(Hx + Hy) provide information of τ = σ + iωε through

−∇2 H+(r)︸ ︷︷ ︸Data

=12

((x + iy)× (∇×H(r))) · ∇ ln τ︸︷︷︸target

− iωµ0 τ︸︷︷︸target

H+(r).

Experiment Setting ( )H +ℜ ( )H +ℑ Target σ

[Haacke1991]

Measurable quantity for MREPT

According to Faraday’s law and reciprocity principle [Hoult2000], the inducedRF-signal at the coil C is

ξ︸︷︷︸RFsignal

:=

∮C

EM · d` = −iωµ0

I

∫Ω

H(r) ·M(r) dr

RF excitation with the main field B0 = −B0z generates time-harmonic field H(called B1-field) that is influenced by τ = σ + iωε.

During RF excitation, M(r, t) precess according to Bloch equation∂∂t M(r, t) = M(r, t)× γ(B0 + B1(r, t)).

If we turn off RF field H, M (M(r, t) = <eiω0tM(r)) creates HM and EM that aredictated by ∇× EM = −iω0µ0(HM + M), ∇×HM = τEM

Measurable quantity for MREPT-continue 2

According to Faraday’s law and reciprocity principle, the inducedRF-signal at the coil C in the the presence of the magnetization M is

ξ︸︷︷︸RFsignal

:=

∮C

EM · d` = −iωµ0

I

∫Ω

H(r) ·M(r) dr

This reciprocity is based on

ξ =1I

∫R3

EM · Jcdr =

∫R3

EM · (∇×H− τE)dr

= −iω0µ0

∫R3

M ·Hdr +

∫R3∇ · (EM ×H− E×HM)dr

= −iω0µ0

∫R3

M ·Hdr + 0

where Jc is the filamentary current flow along the coil.[Scott 2003] [Hoult2000]

Measurable quantity for MREPT-continue 3

NMR RF signal ξ can be expressed as

ξ = 2C1

∫Ω

M0(r)H−(r)

(sin(C2α|H+(r)|

) H+(r)

|H+(r)|

)dr

The transverse field Hxy = Hxx + Hyy can be decomposed intoHxy =

Hx−iHy2 a+ +

Hx−iHy2 a− = H+a+ + H−a− where a+ = x− iy and a− = x + iy.

The transversal component Mxy = Mxx + Myy interact with H+ components and itcan be approximated by Mxy(r) ≈ C1M0(r)

(sin(C2α|H+(r)|

) H+(r)|H+(r)|

)a+

Since a− · a− = 0 and a− · a+ = 2, the above identity follows from

ξ︸︷︷︸RFsignal

:=

∮C

EM · d` = −iωµ0

I

∫Ω

H(r) ·M(r) dr

Measurable quantity for MREPT-continue 4

For each flip angle α > 0, we can measure the following signal:

ξα(r) := 2C1M0(r)H−(r)

(sin(C2α|H+(r)|

) H+(r)

|H+(r)|

)

One simple method to extract |H+(r)| would be to use ξα(r) and ξ2α(r) to get

ξ2α(r)

ξα(r)=

sin(2C2α|H+(r)|

)sin (C2α|H+(r)|) = 2 cos

(C2α|H+(r)|

)Assuming H+(r)

|H+(r)| ≈H−(r)|H−(r)| , we can extract H+(r)

|H+(r)| by

12

Argξα(r) = Arg

H+(r)

|H+(r)|

MREPT Reconstruction Method

−∇2H =∇γγ× (∇×H)︸ ︷︷ ︸

refraction term

−iωµ0γH

Wen (2003) uses the assumption of local homogeneity of τ toget

τ(r) =i

ωµ0

∇2H+(r)

H+(r)

Katscher et al (2009) performed initial experiments on a standardclinical MRI system: For any disk Dδ(r0) ⊂ Ω where ∇τ ≈ 0,

τ =

∮∂Dr∇×H · d`

iµ0ω∫

DrH · dS

.

Nachman et al.(2008) develop τ = ∇2H·(∇×H)iωµ0H·(∇×H) .

MREPT for locally homogeneous subject (∇τ ≈ 0)

For any vector a,

−∇2H = ∇ ln τ ×∇×H− iµ0ωτH ⇒ τ =1

iωµ0

a · ∇2Ha ·H

This direct formula can not be reliable at frequencies below 10MHz.Why? [Wave number] ×|subject size| 1 ⇒ ∇2H ≈ 0 in Ω.

Here, diam(Ω) = 1m, σ = 0.1, εr = 100.

MREPT Reconstruction

Conductivity Permittivity

[Katscher et al (2009)]

MREPT - Error analysis using the direct reconstructionformula τ ∗ = 1

iωµ0

∇2H+

H+ (Note: τ = τ∗ when ∇τ = 0)

Error = τ − τ∗ =

(1

iωµ0

∇2H+

H+− ∇

2H−

iωµ0H−

)[1−

H+ ∂∂z H−

H− ∂∂z H+

]−1

where H = (H+ + H−,−iH+ + iH−,Hz).

MREPT

MREIT

Phantom Model Reconstruction Horizontal axis Vertical axis

[Seo, D H Kim, E J Woo, ... IEEE TMI 2012]

Dual Frequency: MREIT vs EPT

MREIT provides σ at frequencies below a few kHz whereas MREPTproduces σ + iωε at the Larmor frequency (about 128 MHz at 3 T)using H+ = 1

2(Hx + iHy) and

−∇2H =∇ττ× (∇×H)− iωµ0τH

−3

−2

−1

0

1

2

3

C = 0.129 ± 0.242B = 3.019 ± 0.529A = 0.901 ± 0.480

C = 0.122 ± 0.200B = 3.210 ± 0.520A = 1.146 ± 0.505

−3

−2

−1

0

1

2

3

0.8

0.9

1.0

1.1

1.2

[S/m] [S/m]0.6

0.8

1.0

1.2

1.4