Electrical Impedance Tomography
General Information
Ultrasound, MRI and X-ray all have the purpose to gather
information about an unknown object. Each of these modalities makes use of
different physical properties. A very distinctive physical property is the electrical
conductivity. For example marine sand is much more conductive than granite or
cancer tissue is more conductive than healthy tissue. This raises the question
of whether it is possible to determine the conductivity of an unknown object
through measurements on the boundary.
This question is the so-called conductivity problem or ‘Calderon’
problem since he was the first who mentioned and investigated it in his article
'On an inverse boundary value problem', [1]. Calderon was inspired by
geophysical questions yet the possibilities for applications are much broader
and include, but are not restricted to, medical imaging, environmental science,
and nondestructive testing of materials. In applications, especially in medical
imaging, the conductivity problem is referred to as Electrical Impedance
Tomography (EIT).
The mathematical equation that describes the problem is the generalized
My work within EIT
I implemented the method that Calderon presented in his pioneering article
which is arguably the most cited in the EIT literature and applied it to numerical and experimental data.
Calderon's reconstruction method is meant for conductivities with small
perturbations from a constant background. David and Eli Isaacson considered the
case of homogeneous concentric conductivities and showed in [2] that Calderon's
method yields precise information about the spatial variation of the
conductivity, even when the perturbation is large. Similar conclusions could be
drawn from the reconstruction of experimental data. The following pictures show
a phantom chest (with heart and lungs) and a reconstruction image using Calderon’s
method.
A movie that shows the reconstruction of a cross-section of
a chest during breath holding can be downloaded here.
The other reconstruction algorithm that I am about to implement is a 3D direct algorithm outlined by Nachman, [3]. In my master's thesis I already worked with 3D EIT reconstructions by using a linearization method and boundary data taken from a planar electrode array. The new algorithm differs from my master thesis in that it is a direct method which solves the full nonlinear problem without iterations. Moreover the basic geometry will be a sphere to accommodate applications such as breast cancer detection and detection of centers of brain activities. A D-bar method for 2 dimensions was introduced by Nachman, [4] and implemented in [5], [6]. Knudsen et al showed a close connection between Calderon's method and the D-bar method, [7].
References:
[1] A. P. Calderon,
[2] Isaacson, David and Isaacson, Eli L., Comment on A.-P. Calderon's paper: “On an inverse boundary value problem'', Mathematics of Computation, 52, 1989
[3] A.
[4] A.
[5] S. Siltanen and J. L. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16, p.681-699, 2000
[6] D. Isaacson and J. L. Mueller and J. C. Newell S. Siltanen, Imaging Cardiac Activity by the D-bar Method for Electrical Impedance Tomography, Physiol. Meas., 27, p.S43-S50, 2006
[7] K. Knudsen and M. Lassas and J. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Applied Math., to appear
Publication:
Jutta Bikowski and Jennifer Mueller, 2D EIT Reconstructions using Calderon's method, Inverse Problems and Imaging, submitted
Jutta Bikowski
and Jennifer Mueller, Electrical
Impedance Tomography and Fast Multipole Method, procedings of annual SPIE
conference in