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Transcript of Electrical impedance tomography usingequipotentialline jylee/Paper/equi02.pdf · PDF file...

  • 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

    ±

    ±

    −= × −

    −= × −

    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