An Introduction to Current Density Impedance Imaging . Carlos - - PowerPoint PPT Presentation

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An Introduction to Current Density Impedance Imaging

Carlos Montalto

March 5, 2015

Department of Mathematics

Carlos Montalto Current Density Impedance Imaging

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. . Table of contents

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1 Introduction

Electrical Impedance Tomography (EIT) Applications and Limitations Magnetic Resonance Imaging . .

2 Coupled-Physics Inverse Problems

Current Density Impedance Imaging History of CDII . .

3 Uniqueness and Stability in CDII

Idea of the Proof

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. . Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is an imaging technique that uses electrical measurements on the surface of a body Ω to obtain the electrical conductivity σ at the interior of the body.

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. . Mathematical Formulation

In EIT, an electric potential u is generated inside a body Ω while maintaining a voltage f at the boundary. Assuming the electrostatic approximation of Maxwell’s equations, the potential solves the following Dirichlet problem ∇ · σ∇u = 0 in Ω, u|∂Ω = f , (1) for isotropic electrical conductivity σ. The Dirichlet to Neumann map, or voltage to current map, is given by Λσ : f → (σ∂u/∂ν)|∂Ω, where ν denotes the unit outer normal to ∂Ω. The inverse EIT problem is to recover σ from knowledge of Λσ.

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. . EIT in Geophysics

Electrical Tomography is successfully used for Geophysical exploration, for imaging sub-surface structures from electrical resistivity measurements from the surface. In such applications, the problem is known as Electrical Resistivity Tomography (ERT). Mathematically ERT and EIT are described by the same inverse problem, in ERT the interest is on recovering the interior resistivity

• f materials denoted by ρ and defined as

ρ(x) = 1 σ(x).

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. . Electrical Resistivity Tomography

Figure 1 : Surface of the earth using Electrical Resistivity Tomography.

(Pierce et al., 2012)

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. . Geophysical Applications

Figure 2 : Electrical Resistivity Tomography used for water exploration.

(Pierce et al., 2012)

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. . Medical Applications: Continuous Monitoring

Figure 3 : EIT used for regional ventilation monitoring.

(Teschner and Imhoff, 1998)

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. . Monitoring Lung Ventilation

Figure 4 : Changes of End-Expiratory Lung Volume (EELV) measured with an EIT machine.

(Teschner and Imhoff, 1998)

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. . Medical Applications

There are at least two important reasons for using EIT in medical applications: . .

1 Easy Monitoring: Can be applied at bedside as a continuous

monitoring technique and is relatively inexpensive. . .

2 Diagnosis: Provides images based on new and different

information, such as electrical tissue properties. High quality images could provide better differentiation of tissue or organs, resulting in enhanced diagnosis and treatment of numerous diseases.

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. . Conductivity Comparison

Figure 5 : Contrast of conductivity in biological tissue at frequencies ranging from 50Hz to 500KHz.

(Widlak and Scherzer 2012)

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. . Conductivity Comparison

Figure 6 : Contrast of conductivity in biological tissue at frequencies ranging from 50MHz to 500MHz.

(Widlak and Scherzer 2012)

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. . Obstacles of EIT for Medical Diagnosis

Unfortunately there are two obstacles on using EIT for medical diagnosis. Difference in conductivity : The conductivity differences in human and biological tissue are smaller compared to the material in geophysical exploration. Logarithmic Stability : The EIT has logarithmic stability that only guarantees very low resolution. This type of stability is sometimes refer as ’instability’

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. . Conductivity of Materials

Table 1 : Conductivity of different types of tissue or materials at 1 kHz

Tissue type/material Conductivity σ (S/m) Application copper 6 · 107 geophysics drinking water 5 · 10−2 geophysics granite (dry) 10−8 geophysics skin (wet) 3 · 10−3 medical blood 7 · 10−1 medical fat 2 · 10−2 medical liver 5 · 10−2 medical

(Widlak and Scherzer, 2012)

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. . EIT inverse problem

EIT model: Find the conductivity σ in ∇ · σ∇u = 0, u|∂Ω = f , from knowledge of the DN map Λσ = {(f , σ∂u/∂ν) : for all f }

Figure 7 : Illustration of EIT experiment.

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. . Reconstruction in EIT

Figure 8 : Reconstruction of a phantom of a heart and lungs using D-bar method D-bar in 2D.

(Motoya-Vallejo, 2012)

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. . Magnetic Resonance Imaging

Magnetic Resonance Imaging (MRI) uses magnetic fields to detect the radio frequency signal emitted by excited hydrogen atoms by using the fact that their protons are spin 1/2 particles. Usual MRI images can achieve images with spatial resolution of about 1 mm (New MRI, INUMAC (Imaging of Neuro disease Using high-field MR And Contrastophores) 11.75-Tesla resolves up to 0.1mm).

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. . Constrast Problem in MRI

Figure 9 : Defect of blood-brain barrier after stroke in MRI. (Wikipedia)

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. . Coupled Physics Inverse Problems

Coupled-Physics Inverse Problems are new medical imaging modalities that combine the best imaging properties of different type of ’waves’ to generate high contrast and high resolution images.

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. . Examples of Coupled Physics Inverse Problems

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. . Types of Coupling

These physical couplings can be explained by three potential interactions: Generation: the interaction of the first wave with the tissue can generate a second kind of wave (photo-acoustic effect or thermo-acoustic effect). Tagged: the first wave is tagged locally by a the second type

• f wave.

Movie: the first wave travels much faster than the second type of wave, this difference is used to produce a movie of the slow wave propagation.

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. . Current Density Impedance Imaging

Current Density Impedance Imaging (CDII) are two examples

• f coupled-physics inverse problems. In these models MRI

measurements are combined with EIT information to overcome the poor spatial resolution of EIT while taking advantage of its contrast.

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. . CDII Experiment

Figure 10 : An electric potential u is generated, inside a body Ω, using a boundary voltage f . With an MRI machine, we measure the current density J inside Ω. The problem in CDII becomes to recover σ in ∇ · σ∇u = 0, u|∂Ω = f with the additional internal information

• f the current density

J = −σ∇u

(Picture from Prof. Tamasan’s web page)

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. . CDII Experiment

Positive and negative current J is sequentially injected into the body for time TJ to produce a perturbation in the z-component of the Bio-Savart field BJ = (Bx

J , By J , Bz J ). This produces a shift in

the phase of the MRI signal given by m±(x, y, z0) = M0(x, y, z0)eiφ0±iγBz

J TJ

where M0(x, y, 0)eiφ0 is the initial spin magnetization of the hydrogen atoms and γ is the gyromagnetic ratio of hydrogen protons in water molecules. Hence Bz(x, y, z0) = 1 2γTJ Im ( ln (m+(x, y, z0) m−(x, y, z0) ))

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. . CDII Experiment

Finally the body is rotated two times to obtain the x and y components of the Biot-Savart field BJ. Then one determined the current density by Amp` ere’s law J = 1 µ0 ∇ × BJ. where µ0 ≈ 4π · 10−7

V· s A· m is the vacuum permeability.

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. . Summary of CDII

Table 2 : Coupled physics inverse problems based on EIT and MRI.

Technique Equation in Ω Boundary data Interior data CDII ∇ · σ∇u = 0 u J = −σ∇u MREIT ∇ · σ∇u = 0 u Bz = F(σ∇u) There are four fundamental question in all inverse problems: Existence. Uniqueness. Stability. Reconstruction.

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. . History of CDII

1991 Scott G. C., Joy L. G. and Nanping Z. Used MRI to measure the current density J. 1994 Woo EJ., Lee SY. and Mun CW. Minimization algorithm to obtain conductivity σ from J 2002-2003 Kwon O., Woo EJ., Yoon JR. and Seo JK. Introduced J-substitution with the Neumann boundary data. Uniqueness and reconstruction of CDII with two measurements.

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Table 3 : CDII with J-substitution.

Technique Equation in Ω Boundary data Interior data CDII ∇ · σ∇u = 0 u, σ∂νu |J| = |σ∇u| The J-substitution algorithm consist in substitute J = −σ∇u on ∇ · σ∇u = 0, u|∂Ω = f to get ∇ · ( |J| |∇u|∇u ) = 0, u|∂Ω = f . Notice that his is the 1-Laplacian operator.

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2007-2010 Nachman A., Tamasan A. and Timonov A. For n = 2, uniqueness of CDII for planar domains using

• nly one |J| and also with partial illumination.

Sufficient conditions on Dirichlet boundary data to guarantee uniqueness. Reconstruction with partial illumination and conditional stability. For n ≥ 3, uniqueness using only |J| and Dirichlet data. Nice geometric variational approach for recovering the potential.

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They show that u0 solves ∇ · ( |J| |∇u0|∇u0 ) = 0, u0|∂Ω = f , if and only if, u0 minimizes F[u] = ∫

Ω

|J(x)| · |∇u(x)| dx restricted to u|∂Ω = f .

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2008 Hasanov K., Ma A., Nachman A. and Joy M. Reconstruction algorithm using two current density and in-vitro experiment. 2012 Moradifam A., Nachman A. and Tamasan A. Proved uniqueness in the case of perfectly conducting and insulating inclusions. 2012 Tamasan A. and Veras J. Reconstruction method for planar conductivities with partial data and stability of the potential. 2012 Kuchment P. Steinhauer D. Proved that the linearization is elliptic using two internal measurements. 2012 Monard F. and Bal G. Proved Lipschitz stability of CDII using n + 1 measurements.

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. . Stability from the Magnitude of J

. Theorem (Stefanov-M) . . For σ0 ∈ C 2(Ω) and f ∈ C 2,α(∂Ω) for 0 < α < 1, let u0 such that ∇ · σ0∇u0 = 0, u0|∂Ω = f , with ∇u0 ̸= 0 in Ω. For any 0 < θ < 1, there exist s > 0 so that if ∥σ∥Hs(Ω) < L for some L > 0, there exist ϵ > 0 such that if ∥σ − σ0∥C 2(¯

Ω) < ϵ and (σ − σ0)|∂Ω = 0 then

∥σ − σ0∥L2(Ω) < C∥|J| − |J0|∥θ

L2(Ω).

Joint work with Plamen Stefanov.

• Remark. The main difference is that we do not use the

J-substitution to deal with the problem of CDII.

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. . Strategy

Use linearization for the CDII problem. Show that the linearization is stable by decomposing it as composition of simpler operators. Transfer stability of the linearization to the non-linear problem.

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. . Linearization

Comment: We use a more general functional of the from F(σ) = σ|∇u|p for 0 < p ≤ 1. the CDII case is when p = 1. The derivative of F at some fixed σ0 is given by dFσ0(ρ) = σ0|∇u0|p ( ρ + p∇u0 · ∇v(ρ) |∇u0|2 ) , ρ := δσ/σ0, (2) where v solves ∇ · σ0∇v = −σ0∇u0 · ∇ρ in Ω, v|∂Ω = 0. (3) Notice that (2) makes sense as long as ∇u0 ̸= 0 in Ω.

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. . Linearization

Solve (2) for the free ρ term and plug that into (3) to get −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) = ∇ · (dFσ0(ρ) |∇u0|p ∇u0 ) = σ0∇u0 · ∇ ( dFσ0(ρ) σ0|∇u0|p ) . in Ω, with v|∂Ω = 0.

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. . Representation of dFσ0

. Proposition . . Let u0 be σ0-harmonic, with ∇u0 ̸= 0 in Ω. Then σ0T0 dFσ0(ρ) σ0|∇u0|p = −L∆−1

σ0,Dσ0T0ρ,

where T0 = ∇u0 · ∇ is a transport operator along the gradient field

• f u0, ∆σ,D is the Dirichlet realization of ∆σ := ∇ · σ∇ and

Lv := −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) . Remark: The operator L is the only interesting object to study.

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. . Key Example

Example: Consider the particular case when σ0 = 1, f = xn. Then u0 = xn and L = −∆x′ − (1 − p)∂2

xn, where x = (x′, xn).

Notice that: For 0 ≤ p < 1, L is an elliptic operator. For p = 1, L becomes the restriction of the Laplacian over xn = 0. For p > 1, L is a hyperbolic operator. This characterization remains true in the general case.

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. . Local Characterization of L

. Proposition . . Let u0 be σ0-harmonic, with ∇u0 ̸= 0 in Ω. There exist local coordinates (y′, yn) with yn = u0, such that dx2 = c2(dyn)2 + gαβdyαdyβ, gαβ := ∑

i

∂xi ∂yα ∂xi ∂yβ (4) where c = |∇u0|−1. In this coordinates L = Q + − 1 − p √det g ∂ ∂yn c−2σ0 √ det g ∂ ∂yn , where Q is the restriction of ∆σ0 to the equipotential surfaces of u0.

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. . Idea of the Proof

Idea of proof. Recall Lv = −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) Let

• Π0ω the the orthogonal projection of the covector ω onto ∇u0.
• Π⊥ := Id − Π0.

Take a test function ϕ ∈ C ∞

0 (Ω), and compute

(Lv, ϕ) = (σ0∇v, ∇ϕ) − p(σ0Π0∇v, ∇ϕ), = (σ0Π⊥∇v, Π⊥∇ϕ) + (1 − p) (σ0Π0∇v, Π0∇ϕ) . (5) Hence, L = −(Π⊥∇)′ · σ0(Π⊥∇) − (1 − p)(Π0∇)′ · σ0(Π0∇), (6)

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. . Stability with Partial Data

. Theorem (Global Stability with Partial Data) . . Let σ, ˜ σ ∈ C 2,α(Ω) be positive functions in Ω and let u ∈ C 2(Ω) be σ-harmonic and ˜ u ∈ C 2(Ω) be ˜ σ-harmonic with u|∂Ω, ˜ u|∂Ω ∈ C 2,α(∂Ω), 0 < α < 1. Let Γ′ and Γ be open sets of ∂Ω such that Γ is the union of finitely many connected component and Γ′ is compactly contained in Γ. If σ|Γ = ˜ σ|Γ, u|Γ = ˜ u|Γ and ∇(u + ˜ u) ̸= 0 in the closure of I(Γ′, u + ˜ u), then there exist C > 0 such that ∥σ − ˜ σ∥L2(S(Γ′,u+˜

u)) ≤ C∥J − ˜

J∥

α 2+α

H1(I(Γ′,u+˜ u)).

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. . Stability Region Connected Γ

Γ Ω

∇w w =const. w =const. I(Γ, w) = S(Γ, w) I(Γ, w) = S(Γ, w)

Figure 11 : When Γ is connected the visible region and the trajectory region can be the same. Here w = u + ˜ u.

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. . Stability Non Connected Region Γ

Γ Γ

S(Γ, w) I(Γ, w)

Ω

w =const. w =const. ∇w S(Γ, w)

Figure 12 : The injectivity region I(Γ, u) is the light grey region that contains the stability region S(Γ, u) in dark grey. Here w = u + ˜ u.

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. . Non-linear decomposition

. Proposition . . Let u and ˜ u be σ-harmonic and ˜ σ-harmonic, respectively. If ∇(u + ˜ u) ̸= 0 in V , for open subset V of Ω, then 2∇ · Π∇(u+˜

u)(J(σ) − J(˜

σ)) = L(u − ˜ u) in V . (7) where L is a differential operator given by Lv := −∇ · (σ + ˜ σ)∇v + ∇ · ( (σ + ˜ σ)∇(u + ˜ u) · v |∇(u + ˜ u)|2 ∇(u + ˜ u) ) . (8)

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. . Idea of Proof

Strategy for partial data: Stability for potential: Reduce to the inversion of transport

• perator along gradient field (lower order terms)

∥σ − ˜ σ∥ ≤ C∥u − ˜ u∥ Stability for L operator: Similar to the case when the magnitude of the current density is available (elliptic on equipotential lines). ∥u − ˜ u∥ ≤ C∥L(u − ˜ u)∥ where L is a non-linear version of the previous operator L.

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. . Images with Real Data

Figure 13 : Postmortem animal imaging of a swine leg using a 3 T MRI scanner.

(Minhas et al., 2011)

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. . In Vivo Images

Figure 14 : In vivo MREIT imaging experiment of a human leg.

(Seo et al., 2011)

Carlos Montalto Current Density Impedance Imaging

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Carlos Montalto Current Density Impedance Imaging