Electrical impedance tomography using equipotential line method (A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)

June-Yub Lee (Ewha),

Ohin Kwon (Konkuk),

Jeong-Rock Yoon (RPI)

Physical Background (Forward) Conductivity Problem

Ω∂= ∂ ∂=

Ω Ω∈

ong n uORfu

tsinxufind xxgivenFor ..)(

,),(σ

0=∇⋅∇ uσ

Inverse Conductivity Problem

Ω∂= ∂ ∂= ong n uANDfu

givenfromxFind )(σ

Ω∂ ),( gf

Impedance ~Voltage/Current

Tomography

Electrical Impedance Electrical Impedance Electrical Impedance Electrical Impedance TomographyTomographyTomographyTomography (EIT)(EIT)(EIT)(EIT)

생체조직의저항률 (20-100KHz)

10

100

1000

10000

100000

체 액

혈 장

혈 액

근 육 (횡

방 향 ) 간 팔 근 육

신 경 조 직

심 장 근 육

근 육 (종

방 향 ) 폐

지 방 뼈

최소값

최대값

[Ω-cm]

경혈 및 경락의 전기적 특성

가 설

경혈과 경락 전기저항이 낮은 지점이다.

즉, 조직의 저항률이 낮다.

A model problem with heart and lungA model problem with heart and lungA model problem with heart and lungA model problem with heart and lung

Methods of EIT/ICP

g n uuxuf kkMin =∂

∂=∇⋅∇−∑ ,0 where)( Objective 2 σσ σ

σ

voltagefi =currentgi =

)(xσ

• Single Measurement ⇒ minimum information

• Many Measurement ⇒ full information(?) on σ

• Infinite Measurement ⇒ mathematical theory

Numerical Obstacles for EIT/ICP

σσ

σ

σ σ

σσ σ

~better Find ,0 Solve Guss

,0 where)()( Objective )(2

→= ∂ ∂=∇⋅∇→

= ∂ ∂=∇⋅∇−∑ Ω

g n uuσ(x)

g n uuxuxf

LMin

• Strongly nonlinear:

• Ill-posed problem:

)()( 22 Ω∂∈→Ω∈ LuL σσ

σσσσ ~~ uu ≈→≈/

fug n uu −→= ∂ ∂=∇⋅∇ σσσ σσ using ~better Find ,0 Solve

σσ on on restrictiu Regularity 0condition Stop →≅− fu

Ill-posed Nonlinear

• 50 iterations

EIT and MREIT (Magenetic Resonace Electrical Impedance Tomography)

0.95 mA

Current Source Voltmeter

변수형 고속 영상복원 알고리즘

진단 변수 추출

심폐기능 소화기능 배뇨기능 뇌기능

비정상 조직 검출 기타 변수

실시간실시간 영상영상 모니터링모니터링

MREIT 고해상도 영상

적응형 요소융합 및 세분화

변수형 고속 알고리즘

초기해초기해초기해초기해

(1)문제정의

(2)자속밀도의 영상화

MRCDI 및 MREIT의 원리

1E 2EΩ

∂Ω

( ), , ,Vρ J BI I

n3 \ Ω

L− L +

a

1 ( ) 0 ( )

V ρ

  ∇ ∇ = 

  r

r 1 on V J

nρ ∂ = ∂Ω ∂

1( ) ( ) ( )

V ρ

= − ∇J r r r

0 3 '( ) ( ') '

4 ' J dv

µ π Ω

−= × −∫

r rB r J r r r

3 0

3\

0 3

'( ) ( ') ( ') ' 4 '

'( ') ( ') ' 4 '

I

L

dv

I dl

µ π

µ π

±

±

Ω

−= × −

−= × −

∫

∫

r rB r J r a r r r

r rr a r r r

0

1( ) ( )B µ

= ∇×J r B r

1E

2E

x

y z

x y z

1E

2E

1E

2E

x

y z

Transversal Imaging Slice

Three Sagittal Imaging Slices

Three Coronal Imaging Slices

전류밀도 영상 실험용 팬텀

자속밀도 및 전류밀도 영상

y

z

]Tesla[

y

z

]Tesla[

x

z

]Tesla[

x

z

]Tesla[

x

y

]Tesla[

x

y

]Tesla[

Bx By

Bz

]mA/mm[ 2

x

y

]mA/mm[ 2 ]mA/mm[ 2

x

y

|J| 0

1 µ

= ∇×J B

자속밀도 및 전류밀도 영상

J-Substitution Algorithm

Original Conductivity

Reconstructed Conductivity

Current Density

Schematic Diagram for Equi-potential Line Method

Numerical Algorithm

Equipotential Line Method

Smooth conductivity distribution

Conductivity Result with noisy J

Phantom with 1%Add+10%Mul Noise

Artificially generated Human Body

1%Add+10%Mul Noise

Conclusions and further works • Fast, stable and efficient • Well-posed (error~noise) • Better interpolation method to handle discontinuous cases

• Noise handling for high conductivity ratio case where |J|~0

• 3D extension is trivial But 3D data for J is expensive Usage of Bz instead of J is better