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Electrical Impedance Tomography is an imaging method which attempts to reveal the conductivity distribution of a domain based on the electrical boundary condition. For time difference EIT, the voltage difference at two time steps is employed for reconstruction. This is an ill-posed inverse problem, especially, it is non-linear. The currently available EIT devices are all based on linearized reconstruction algorithms. The linearized reconstruction employs a reconstruction matrix which is essentially a regularized pseudo inverse of the Jacobian matrix. This reconstruction matrix multiplying the voltage differences will provide a distribution of conductivity changes. However, the linearized reconstruction contains modelling error. In this paper, we study the modelling error caused by linearization based on the shunt model through simulations. Specifying a current injection pattern in simulation, at each time step a simulated voltage measurement can be calculated from Maxwell’s equations. The voltage difference between two time steps can be obtained. On the other hand, according to the assumption of linearized reconstruction, the voltage difference is assumed to be the Jacobian matrix multiplying the conductivity distribution changes. The discrepancy between these two voltage differences will be studied.

Electrical Impedance Tomography (EIT) is a radiation-free imaging method. It attempts to reveal the conductivity distribution changes inside the human body of two time instants through electrical data obtained via the electrodes attached to the boundary. In lung EIT, commonly 16 electrodes are placed equidistantly on the boundary of a horizontal chest plane.

We denote the conductivity of the domain changes between two time steps by

where

Jacobian matrix is commonly calculated by studying the first order perturbation of conductivity on each element [

However, such linearized forward and inverse model contains modeling error. EIT reconstruction is a non-linear inverse problem. There are many modelling error affects the imaging quality of EIT. For example, the deformations of the boundary shape the inhomogeneous background conductivity as well as the shift of contact impedance on the electrodes. In this study we focus only on the modelling error caused by linearization. To this end, we simplify the simulation models by specifying the domain shape to be unit disk and the known background conductivity.

On the currently available EIT devices, all electrical boundary conditions are collected through the electrode-band located around a plane of the human body. The electrical current injection and voltage measurement are always performed with a period scheme. The most commonly employed of the current injection and voltage measurement pattern is called adjacent pattern. At each time step, alternative currents with fixed amplitude are sequentially injected through two neighboring electrodes (

In this subsection we outline the forward model calculation based on Maxwell’s equation. In order to simplify the formulations, we employ the shunt model boundary value problem. This model neglects the effect of contact impedance. In the absence of interior current sources, there is Ohm’s law:

where

With a given conductivity distribution, the Jacobian matrix

Linearization error: =

The discrepancy between the voltage differences calculated from Maxwell’s equation and the linearized model will be studied through simulation. A 2D disk FEM model was constructed with the help of MATLAB toolbox EIDORS. This model has 1215 triangular elements. Around the boundary of the disk phantom, 16 electrodes are attached equidistantly. The contact impedance between each electrode and the disk domain is fixed to be 0.01 Ω∙m. The simulated current with the amplitude of 10 mA was injected through these electrodes with the adjacent pattern. Different conductivity contrasts are embedded into this disk phantom. In this first study, we increase the constant conductivity of the contrast region from 1.01 Sm^{−1} to 2 Sm^{−1}. These simulated phantoms reflex the different aspects of the contrasts. Such as different location, size as well as asymmetry.

In this section, we present the first simulation results. We employ the 2D FEM model with disk shape for simulation as displayed in section 2.4. The linearization modeling errors, pronounced by the voltage discrepancy descript in the last section will be studied. In below the modeling errors induced by linearization are studied with respect real to the voltage changes. The relation between modeling errors and voltage differences will be investigated according to each of the four given phantoms. For the phantom A with contrast conductivity

respect to the voltage differences are further studied by linear regression. This regression result has been displayed in the right plot of

It can be observed from left plot of

In this study, the modeling errors induced by linearization are investigated according to simulation. The first simulation results indicated that such modeling

errors may have linear correspondence with the voltage differences. An exceptional case appears when the contrast’s shape has large asymmetry. For this situation, the linear correspondence is not obvious.

This work is partially supported by the Federal Ministry of Education and Research (BMBF) under grant no. 03FH038I3 (MOSES).

Gong, B., Schullcke, B., Krueger-Ziolek, S. and Moeller, K. (2017) An Investigation of Modeling Error of EIT Reconstruction. J. Biomedical Science and Engineering, 10, 59-65. https://doi.org/10.4236/jbise.2017.105B007