# Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a non-invasive real-time functional imaging modality for the continuous monitoring of physiological functions such as lung ventilation and perfusion. The image contrast represents the time change of the electrical conductivity distribution inside the human body. Using an array of surface electrodes around a chosen imaging slice, the imaging device probes the internal conductivity distribution by injecting electrical currents at tens or hundreds of kHz. The injected currents (at safe levels) produce distributions of electric potentials that are non-invasively measured from the attached surface electrodes. A portable EIT system can provide functional images with an excellent temporal resolution of tens of frames per second. EIT was introduced in the late 1970s [1]-[4], likely motivated by the success of X-ray CT. Numerous image reconstruction methods and experimental validations have demonstrated its feasibility [5]-[8] and clinical trials have begun especially in lung ventilation imaging and pulmonary function testing [9].

### Governing equations

To explain the EIT image reconstruction problem clearly and effectively, we restrict our description to the case of a 16-channel EIT system for real-time time-difference imaging applications. The sixteen electrodes ($\mathcal{E}_1,\cdots, \mathcal{E}_{16}$) are attached around a chosen imaging slice, denoted by $\Omega$. We adopt the neighboring data collection scheme, where the device injects current between a neighboring electrode pair $(\mathcal{E}_j,\mathcal{E}_{j+1})$ and simultaneously measures the induced voltages between all neighboring pairs of electrodes $(\mathcal{E}_i,\mathcal{E}_{i+1})$ for $i=1,\ldots,16$. Here, we denote $\mathcal{E}_{16+1}:=\mathcal{E}_1$. Let $\sigma$ be the electrical conductivity distribution of $\Omega$, and let $\partial\Omega$ denote the boundary surface of $\Omega$.
The electrical potential distribution corresponding to the $j$th injection current, denoted by $u_j^\sigma$, is governed by the following equations:

$\displaystyle\label{eq:govern} \left\{\begin{array}{rl} \nabla\cdot(\sigma\nabla u_j^\sigma)=0&~~\mbox{in}~~{\Omega}\\ (\sigma\nabla u_j^\sigma)\cdot\mathbf{n}=0&~~\mbox{on}~~\partial{\Omega}\setminus \cup_i^{16}\mathcal{E}_i\\ \int_{\mathcal{E}_i}\sigma \nabla u_j^\sigma\cdot\mathbf{n}=0&~~\mbox{for}~~i\in\{1,2,\ldots,16\}\setminus\{j,j+1\}\\ u_j^\sigma+z_{i}(\sigma\nabla u_j^\sigma\cdot\mathbf{n})&=U_i^j ~~\mbox{on}~~\mathcal{E}_i~~\mbox{for}~~ i=1,2,\ldots,16\\ \int_{\mathcal{E}_j}\sigma\nabla u_j^\sigma\cdot \mathbf{n}\,ds=I&=-\int_{\mathcal{E}_{j+1}}\sigma\nabla u_j^\sigma\cdot \mathbf{n}\,ds \end{array}\right.$

where $\mathbf{n}$ is the outward unit normal vector to $\partial{\Omega}$, $z_{i}$ is the electrode contact impedance of the $i$th electrode $\mathcal{E}_i$, $U^j_i$ is the potential on $\mathcal{E}_i$ subject to the $j$th injection current, and $I$ is the amplitude of the injection current between $\mathcal{E}_j$ and $\mathcal{E}_{j+1}$. Assuming that $I=1$, the measured voltage between the electrode pair $(\mathcal{E}_i,\mathcal{E}_{i+1})$ subject to the $j$th injection current at time $t$ is expressed as:

$\displaystyle\label{eq:boundary_voltage} V^{j,i}(t):=U^j_i(t)-U^j_{i+1}(t)$

### Data collection protocol

One of widely used data collection protocol is the neighboring data collection scheme, where EIT device injects current between a neighboring electrode pair and simultaneously measures the induced voltages between all neighboring pairs of electrodes that have no current injected. This forms the first projection following the term of X-ray CT, and the device repeats this for all projections by sequentially choosing the next neighboring electrode pairs for current injection. The total number of data is $13\times 16=208$ for each scan with 16 projections, which constitutes the voltage data vector $\mathbf{V}$.

### Image reconstruction

However, EIT images often suffer from measurement noise and artifacts especially in clinical environments and there still exist needs for new image reconstruction algorithms to achieve both high image quality and robustness.