. Coupled Physics Inverse Problems: EIT meets MRI Carlos Montalto - PowerPoint PPT Presentation

. . Coupled Physics Inverse Problems: EIT meets MRI Carlos Montalto Department of Mathematics cmontalto@math.purdue.edu November 14, 2014 . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. 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 Stability Result Approach . . 4 Final Remarks . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. 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 Coupled Physics Inverse Problems

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 Coupled Physics Inverse Problems

Figure 1: Electrical impedance tomography inverse problem. . . . . . . Carlos Montalto Coupled Physics Inverse Problems

There are four fundamental question in any inverse problem: . . 1 Existence : Given the DN-map is there any σ that actually yields those observations? . . 2 Uniqueness : Can we determine σ uniquely from the DN-map? (Sylvester-Uhlmann, 1986) - Good news. . . 3 Stability : How are the errors in the measuring the DN-map amplified in the reconstruction of σ ? (Alessandrini, 1988 - Logarithmic stability ) - Bad news. . . 4 Reconstruction : Is there a computational efficient formula or procedure to recover σ from the DN-map? . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Electrical Resistivity Tomography . 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 Coupled Physics Inverse Problems

. Geophysical Applications . Figure 2: Surface of the earth using Electrical Resistivity Tomography. (Pierce et al., 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. EIT in Geophysics and Imaging . Table 1: Conductivity of different types of tissue or materials at 1 kHz Tissue type/material Conductivity σ ( S / m ) 6 · 10 7 copper 5 · 10 − 2 drinking water 10 − 8 granite (dry) (Widlak and Scherzer, 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Geophysical Applications . Figure 3: Electrical Resistivity Tomography used for water exploration. (Pierce et al., 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Medical Applications . There are two important reasons for using EIT in medical applications: . . 1 Monitoring: Can be applied at bedside as a continuous monitoring technique (relatively inexpensive). . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Monitoring Applications . Figure 4: EIT used for regional ventilation monitoring. (Teschner and Imhoff, 1998) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Monitoring Lungs Ventilation . Changes in body position of healthy, spontaneously breathing individuals are associated with a major redistribution of regional ventilation. In the last few years prone positioning has been used increasingly in the treatment of acute respiratory distress syndrome patients. EIT may help monitor ventilation change due to such changes of position. It may help identify to responders to this kind of treatment . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Monitoring Examples . Figure 5: Changes of End-Expiratory Lung Volume (EELV) measured with an EIT machine, in a chest trauma patient in a rotation bed, while being turned from a 60 right lateral (1st image taken at cursor position C1, representing the reference status) to supine position. The images depict how EELV increased in the right (initially dependent) lung during the rotation (blue color), while EELV decreased by a similar magnitude in the left lung (orange color). Despite the large changes in EELV, the distribution of ventilation in this patient did not change significantly during the rotation. . (Teschner and Imhoff, 1998) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Medical Applications . There are two important reasons for using EIT in medical applications: . . 1 Monitoring: Can be applied at bedside as a continuous monitoring technique (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 Coupled Physics Inverse Problems

. Diagnosis Applications . Figure 6: Contrast of conductivity in biological tissue at frequencies ranging from 50Hz to 500KHz. (Widlak and Scherzer 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Diagnosis Applications . Figure 7: Contrast of conductivity in biological tissue at frequencies ranging from 50MHz to 500MHz. (Widlak and Scherzer 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Examples of EIT in Medical Imaging . Figure 8: Reconstruction of a phantom of a heart and lungs using D-bar method D-bar in 2D. (Motoya-Vallejo, 2012) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. 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. . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. EIT in Geophysics and Imaging . Table 2: 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 Coupled Physics Inverse Problems

. 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 Coupled Physics Inverse Problems

. 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 9: Illustration of EIT experiment. . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. Instability of EIT . For dimension n ≥ 3 Alessandrini [1988] showed that for smooth, positive and bounded conductivities the stability of the EIT problem is of logarithmic type σ ||| − µ + || Λ σ − Λ ˜ || σ − ˜ σ || L ∞ (Ω) ≤ C ( | ln || Λ σ − Λ ˜ σ || ) for 0 < µ < 1. For example, if the σ || ∼ 1 × 10 − 10 error in measurements = || Λ σ − Λ ˜ then error in reconstruction ∼ log 10 − 10 = 1 × 10 − 1 . Note: log � = ln in the example. . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. MRI . Magnetic Resonance Imaging 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 Coupled Physics Inverse Problems

. Constrast Problem in MRI . Figure 10: Defect of blood-brain barrier after stroke in MRI. (Wikipedia) . . . . . . Carlos Montalto Coupled Physics Inverse Problems

. 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. . . . . . . Carlos Montalto Coupled Physics Inverse Problems