Electrical impedance tomography (Part II) - The physical principle

The classical forward problem is to find the output response of the model due to the input impact. The model and input signal we know.

Such problems are correct in the sense of Hadamard [1], for which:

1) a solution exists for all input signals;

2) the solution is unique;

3) the solution depends continuously on the input data.

Physical problems that satisfy the conditions of existence of solution, its uniqueness and stability are correct. The solution of such problems are not sensitive to small changes (errors) of input data [2].

Traditionally it was believed that all physical problems are well posed, but today there are known numerous problems that are ill-posed.

These problems are inverse to the classical forward problems, and can be interpreted as finding the input to the model, which determines a particular output response.

Forward problem in EIT is to determine the distribution of electric potential inside the object and on its surface with given boundary conditions (see Fig. 1).

Fig. 1

The problem of finding the electrical conductivity in EIT is the inverse problem. The inverse problem in EIT is ill-posed, and the discrepancy to the third criterion of Hadamard is the biggest problem. In practice this means that there could be big changes in the distribution of electrical conductivity, which for a given accuracy of the solution does not affect the measurement of the boundary. It also means that in the absence of a priori information of the solution the inverse problem is unstable in the presence of noise.

The equation of electrical impedance tomography and boundary conditions

In the task of restoring the electrical conductivity in EIT the first step is to construct a physical model of the observations: the equation that relating the measured voltage, the injected currents and distribution of electrical conductivity. The equation relating the distribution of conductivity and electric potential within a bounded region in with low-frequency currents flow has the form [3]:

- is conductivity distribution,

- is electrical potential distribution.

The electrodes, that is superimposed on the object of investigation can serve as current or measurement electrodes. Between current electrodes current is passed. This current forms a current pattern. In that time other electrodes is served as measurement type electrodes: they used to measure the voltage drop in reference to ground electrode. The measurements aren't collected from current electrodes. Of course the role of electrodes is changed from measurement to measurement.

Models of the electrodes give a mathematical description of the physical properties of real electrodes. Mixed ("Dirichlet" and "Neumann") boundary conditions that define the uniqueness of the solution of EIT equation, formally known as the "Complete electrode model" of the electrodes [2].

For the part of the object's surface under the electrodes the current density flows has the form:

 - is the normal to the object surface,

- is the current density.


For the rest of the surface :

For each electrode, the integral of current density across the electrode surface is equal to the current flow :

 - is the electrode surface,

- is the current flow,

- is the number of electrodes in the system.

Potential value, measured at the electrode is the sum of the potential on the surface under the electrode and the voltage drop on the contact resistance of the electrode:

 - is the potential value,

- is the measuring electrode,

- is the contact resistance


Thus, the model has a unique solution, when the law of conservation of charge is satisfies:

and a selection of ground is made:

Finite element representation of the EIT equation

The solution of EIT equation with respect to the object of arbitrary shape with complex boundary conditions analytically is not possible [1]. Therefore, some numerical methods is used. These methods require spatial discretization the potential distribution, conductivity and the region in which the problem is solved.

The finite element method (FEM) is widely used numerical method for solving elliptic problems. In the finite element method, the spatial area is approximated by a finite set of elements - simplexes. In the two-dimensional space simplexes are triangles, and in three are tetrahedras. A set of simplexes is called the finite element mesh. Thus, in the two-dimensional case:

- is the number of triangles that devides the object

- is total number of triangles vertices (nodes).

 The potential is approximated by basis functions that is equal to unity in i-th node and zero at all other nodes [1]:

-   is discretized potential.

- is the basis functions.

As a result of the finite-element representation of the equation EIT direct problem given a complete model of the electrodes is as follows[1]:

- is vector that represents a discrete approximation of the potential and contains potentials at the mesh grid,

- is vector containing the potentials at the measuring electrodes,

-  is vector containing the currents flowing through the current electrodes.

 - is the approximation of the conductivity is taken constant on the simplex.

Sources:

1.     Part 1of Electrical Impedance Tomography: Methods, History and Applications / [Borsic A., Lionheart W., Polydorides N.]; editor David  Holder — Institute of Physics, Bristol Series in Medical Physics and Biomedical Engineering, 2004. — 62 p.
2.     Polydorides, N. Image Reconstruction Algorithms for Soft-Field Tomography: Ph.D. thesis / Polydorides — Manchester, United Kingdom: N. UMIST, 2002. —250 p.

3.     Vauhkonen M. Electrical impedance tomography and prior information: Ph.D. thesis / Vauhkonen M. — Kuopio University Publications, 1997. —110 p.