Last updated on?October 29, 2012?by Tushar Kanti Bera
History
The first impedance imaging system, the Impedance Camera [8], was constructed by Henderson and Webster to study the pulmonary edema in 1978 [8]. Since then, EIT is being researched in different areas of science and technology due to its several advantages [9-11] over other computed tomographic techniques [12].
Advantages:
?
Figure 2: Schematic of the impedance camera
Figure 1: A circular domain with surface electrodes
Introduction
Electrical Impedance Tomography (EIT) [1-7] is an inverse problem [3] in which the electrical conductivity distribution of a closed domain (Del-Omega) in a volume conductor of interest is reconstructed from the surface potential at the boundary (Omega) developed by injecting a low frequency and low magnitude constant current signal through an array of electrodes surrounding the domain to be imaged. EIT is an ill-posed [3] inverse problem [3] in which the spatial distribution of electrical conductivity or resistivity within a closed domain is reconstructed from the surface potentials developed by injecting a constant electrical current.
Applications:
Being a non-invasive, non-radiating, non-ionizing and inexpensive methodology, EIT has been extensively studied in medical diagnosis [13]. Apart from the attempts made to develop a better medical-EIT system for industrial and process engineering [14-17], chemical engineering [18-19], civil engineering [20-22], Earth science, Geophysics and Geoscience [23-24], Microbiology and biotechnology [25-26], Defense fields [27], Thin Films, MEMS and Nanotechnology [28], Archaeology [29], Oceanography [30], Nondestructive testing (NDT) in manufacturing technology [31]. Electrical Impedance Tomography (EIT) can be used as an imaging modality for nondestructive testing in several areas like:
EIT Hardware
A modern EIT system consists of an electronic instrumentation [11, 32], a computing system, EIT sensors or electrode array and a phantom system or subject under test. EIT electronic instrumentation system is developed a constant current injector [11, 32], signal conditioner block [11, 32], electrode switching module [11, 32] and a data acquisition system [11, 32-33]. Hence a modern EIT system is developed with the following four basic parts:
1. Electronic instrumentation,
2. Computing system
3. EIT sensors or electrode array
4. Phantom or subject under test
Constant current injector
Constant current injector may be a voltage controlled current converter developed with a voltage controlled oscillator (VCO) [11, 32] or any other voltage signal generator feeding to a voltage control constant current source (VCCS) [11, 32] which injects the current signal to the object to be imaged. Constant current injector injects a low magnitude low frequency constant current signal to the domain through the surface electrodes attached to the domain boundary.
Electrode switching module
In an EIT system, the current injection and voltage measurement is performed by connecting the current injector and the voltage measuring device to the particular electrodes which are automatically switched by an electrode switching module. A modern EIT system essentially requires an automatic electrode switching module to switch the surface electrodes for current injection and voltage measurement in a particular current pattern. Electrode switching module of modern EIT system can be developed by relays or analog multiplexers controlled by PC generated signals. In EIT it is a common practice to inject known current patterns and the surface potentials are measured by a particular electrode switching protocol i.e. neighboring (Figure 2a and Figure-3) or adjacent drive protocol [1] although an effort has been made to measure the boundary currents for applied voltage patterns [34]. The differential voltage data collected by the data acquisition system is then processed by an image reconstruction algorithm in PC.
Data acquisition system
DAS may be developed by microcontroller [35] or any other controlling devices like PCI DAQ cards [11, 33] and it is used for acquiring the boundary voltage data developed due to the current conduction in the object. The data acquisition system acquires the boundary voltage data from surface electrodes acting as the EIT sensors and sends the data to the PC
Signal Conditioner
Signal conditioner circuits are required to amplify and process the acquired signal with high signal to noise ratio. The processed voltage data is sent to the PC and conductivity distribution is reconstructed from the voltage data by image reconstruction algorithm.
Computing system
The computing system of a modern EIT system consists of a personal computer (PC) with image reconstruction algorithms [36-43] like EIDORS [44-45] or else. The reconstruction algorithm is a software or computer programme which reconstructs the impedance distribution of the imaging domain from the boundary potential data.
EIT-Phantom or Subject
Before applying the EIT system to the patients it must be studied tested and calibrated to ensure the system safety, reliability and efficiency. Practical phantoms [46-49] with surface electrodes [50] are used to assess the performance of EIT systems for their validation, calibration and comparison purposes. An EIT phantom consists of two parts phantom tank and the internal medium or imaging medium contained by the tank.
References
Figure 3: A schematic of the modern EIT system and its instrumentation
Figure 4: An image reconstruction flow chart in EIT
Image Reconstruction in EIT
In EIT a low frequency and low magnitude constant current is injected through an array of 16 surface electrodes [50] surrounding the domain to be imaged using different electrode switching protocols [49] and the boundary potentials are measured although, the boundary currents can also be measured for a applied voltage signal in Applied Potential Tomography (APT) [51-54]. The voltage data collected by the data acquisition system is then processed by an image reconstruction algorithm?in PC.
Mathematical Model
To calculate the nodal potential for a known conductivity a relationship can be established between the electrical conductivity (s) and spatial potential (F). Electro-dynamics of EIT is governed by a nonlinear partial differential equation, called the Governing Equation [1-5] of Electrical Impedance Tomography, is given by,
(1)
(2)
Forward and Inverse Problem
A relation can be obtained between the voltage measurements made on the boundary?and the domain conductivity can be found [55-56] as,
Where s is elemental conductivity values,?phi is the vector of nodal potential and K is the transformation matrix constructed from the elemental conductivities and nodal coordinates.
If K and I are known, Eq.-2 can be solved numerically using finite element method (FEM) [57] to calculate the nodal potentials of the domain for the known conductivity (s). It is known as the forward problem. Using Gauss-Newton method [58] applied on EIT, update vector of s [55-56] can be expressed as:
(3)
Where, Q is a function of Jacobian matrix (J) [55-56] and regularization parameters [55-56]. Phi_{d} is the mismatch vector between calculated boundary potential (V_{c}) and measured boundary potential (V_{m}). That means if the matrix Q and the surface potentials?Phi are known then the elemental conductivity (sigma_{e}) can be mapped. This is known as the inverse problem which is discussed in the next section. Using Modified Newton Raphson (MNR) iterative technique [55-56], a suitably assumed conductivity vector (initial guess), [sigma_{o}], is modified to [sigma_{o }+ Del_sigma] for achieving a specified error limit in calculated and measured voltage ([Del_sigma] denotes the conductivity update).
Image Reconstruction: Gauss-Newton Approach
Electrical conductivity imaging is a highly nonlinear and ill-posed inverse problem. The response matrix [J^{T}J] is a singular matrix. Hence in EIT, minimization algorithm [55-56, 58-59] is used to obtain the approximate solution of the ill-posed inverse problem. In minimization algorithm, objective function formed by the difference between the experimental measurement data (V_{m}) and the computationally predicted data (V_{c}) is minimized. Generally in inverse problem a least square solution [55-56] of a minimized object function (s) [58-59] obtained from the calculated voltage data and the measured voltage data is searched by a Gauss-Newton method based numerical approximation algorithm (explained in the next sub-section) called the inverse solver [60].
If V_{m} is the measured voltage matrix and f is a function mapping an E-dimensional (E is the number of element in the FEM mesh) impedance distribution into a set of M (number of the measured data available) approximate measured voltage, then, the Gauss-Newton algorithm [55-56, 58-59] tries to find a least square solution of the minimized object function s defined as [58-63]:
(4)
Where, s_{r} is the constrained least-square error of the reconstructions, G is the regularization operator and?Lamda is a positive scalar and called as the regularization coefficient.
By Gauss Newton (GN) method,
(5)
(8)
Where the term J=F_dash?is known as the Jacobian matrix [56, 59]:
Replacing G^{T}G by I (Identity matrix) Eq.-6 reduces to:
(6)
(7)
In general, for k^{th} iteration (k is a positive integer), from the Eq.-7 the conductivity update vector can be represented as: Thus the update vector of conductivity distribution as:
(9)
Where J_{k} and (Del-V)k?are the Jacobian and voltage difference matrix respectively at the k^{th} iteration.
The EIT image reconstruction algorithm starts with the solution of forward solver (FP). The FP solution is obtained by solving the EIT governing equation [1] using domain coordinates and the domain conductivity. The calculated boundary potential data [V_{c}] is obtained from the FP for a known current matrix [C] and a known (initial guess) conductivity matrix [sigma_{o}]. The matrix [Del_V] is estimated from the matrix equation [Del_V]= [V_{m}]-[V_{c}] and then it is used to calculate the conductivity update matrix [Del_sigma] using equation 1. The [Del_sigma] matrix is then used to upgrade the [sigma_{o}] matrix to new conductivity matrix [sigma_{1} = sigma_{o} + Del_sigma]. The iteration is continued till Del_V reaches to an acceptable limit epsilon.
Hence, the MoBIIR algorithm works in the following sequence (shown in Fig.-6):