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

. An Introduction to Current Density Impedance Imaging . Carlos Montalto March 5, 2015 Department of Mathematics . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Table of contents . . 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 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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 Λ σ : 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 Λ σ . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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 of materials denoted by ρ and defined as 1 ρ ( x ) = σ ( x ) . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Electrical Resistivity Tomography . Figure 1 : Surface of the earth using Electrical Resistivity Tomography. (Pierce et al., 2012) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Geophysical Applications . Figure 2 : Electrical Resistivity Tomography used for water exploration. (Pierce et al., 2012) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Medical Applications: Continuous Monitoring . Figure 3 : EIT used for regional ventilation monitoring. (Teschner and Imhoff, 1998) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Monitoring Lung Ventilation . Figure 4 : Changes of End-Expiratory Lung Volume (EELV) measured with an EIT machine. (Teschner and Imhoff, 1998) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Conductivity Comparison . Figure 5 : Contrast of conductivity in biological tissue at frequencies ranging from 50Hz to 500KHz. (Widlak and Scherzer 2012) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Conductivity Comparison . Figure 6 : Contrast of conductivity in biological tissue at frequencies ranging from 50MHz to 500MHz. (Widlak and Scherzer 2012) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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’ . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Conductivity of Materials . Table 1 : Conductivity of different types of tissue or materials at 1 kHz Tissue type/material Conductivity σ ( S / m ) Application 6 · 10 7 copper geophysics 5 · 10 − 2 drinking water geophysics 10 − 8 granite (dry) geophysics 3 · 10 − 3 skin (wet) medical 7 · 10 − 1 blood medical 2 · 10 − 2 fat medical 5 · 10 − 2 liver medical (Widlak and Scherzer, 2012) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. EIT inverse problem . EIT model : Find the conductivity σ in ∇ · σ ∇ u = 0 , u | ∂ Ω = f , from knowledge of the DN map Λ σ = { ( f , σ∂ u /∂ν ) : for all f } Figure 7 : Illustration of EIT experiment. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. 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). . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging

. Constrast Problem in MRI . Figure 9 : Defect of blood-brain barrier after stroke in MRI. (Wikipedia) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Carlos Montalto Current Density Impedance Imaging